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Neutrosophic Sets and Systems, vol. 35/2020
Florentin Smarandache“Neutrosophic Sets and Systems” has been created for publications on advanced studies in neutrosophy, neutrosophic set, neutrosophic logic, neutrosophic probability, neutrosophic statistics that started in 1995 and their applications in any field, such as the neutrosophic structures developed in algebra, geometry, topology, etc.
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Neutrosophic Sets and Systems, vol. 33/2020
“Neutrosophic Sets and Systems” has been created for publications on advanced studies in neutrosophy, neutrosophic set, neutrosophic logic, neutrosophic probability, neutrosophic statistics that started in 1995 and their applications in any field, such as the neutrosophic structures developed in algebra, geometry, topology, etc.
Neutrosophic Sets and Systems, vol. 31/2019
“Neutrosophic Sets and Systems” has been created for publications on advanced studies in neutrosophy, neutrosophic set, neutrosophic logic, neutrosophic probability, neutrosophic statistics that started in 1995 and their applications in any field, such as the neutrosophic structures developed in algebra, geometry, topology, etc.
Neutrosophic Sets and Systems, vol. 21/2018
2018
“Neutrosophic Sets and Systems” has been created for publications on advanced studies in neutrosophy, neutrosophic set, neutrosophic logic, neutrosophic probability, neutrosophic statistics that started in 1995 and their applications in any field, such as the neutrosophic structures developed in algebra, geometry, topology, etc.
Neutrosophic Sets and Systems, vol. 20/2018
“Neutrosophic Sets and Systems” has been created for publications on advanced studies in neutrosophy, neutrosophic set, neutrosophic logic, neutrosophic probability, neutrosophic statistics that started in 1995 and their applications in any field, such as the neutrosophic structures developed in algebra, geometry, topology, etc.
Neutrosophic Sets and Systems, vol. 34/2020
“Neutrosophic Sets and Systems” has been created for publications on advanced studies in neutrosophy, neutrosophic set, neutrosophic logic, neutrosophic probability, neutrosophic statistics that started in 1995 and their applications in any field, such as the neutrosophic structures developed in algebra, geometry, topology, etc
Neutrosophic Sets and Systems, vol. 18/2017
“Neutrosophic Sets and Systems” has been created for publications on advanced studies in neutrosophy, neutrosophic set, neutrosophic logic, neutrosophic probability, neutrosophic statistics that started in 1995 and their applications in any field, such as the neutrosophic structures developed in algebra, geometry, topology, etc.
Neutrosophic Sets and Systems, vol. 26/2019
2019
“Neutrosophic Sets and Systems” has been created for publications on advanced studies in neutrosophy, neutrosophic set, neutrosophic logic, neutrosophic probability, neutrosophic statistics that started in 1995 and their applications in any field, such as the neutrosophic structures developed in algebra, geometry, topology, etc.
Neutrosophic Sets and Systems, vol. 15/2017
”Neutrosophic Sets and Systems” has been created for publications on advanced studies in neutrosophy, neutrosophic set, neutrosophic logic, neutrosophic probability, neutrosophic statistics that started in 1995 and their applications in any field, such as the neutrosophic structures developed in algebra, geometry, topology, etc.
Neutrosophic Sets and Systems, vol. 13, 2016
”Neutrosophic Sets and Systems” has been created for publications on advanced studies in neutrosophy, neutrosophic set, neutrosophic logic, neutrosophic probability, neutrosophic statistics that started in 1995 and their applications in any field, such as the neutrosophic structures developed in algebra, geometry, topology, etc.
Neutrosophic Sets and Systems, vol. 14, 2016
”Neutrosophic Sets and Systems” has been created for publications on advanced studies in neutrosophy, neutrosophic set, neutrosophic logic, neutrosophic probability, neutrosophic statistics that started in 1995 and their applications in any field, such as the neutrosophic structures developed in algebra, geometry, topology, etc.
Volume 35, 2020
Neutrosophic Sets and Systems
An International Journal in Information Science and Engineering
<A> <neutA> <antiA>
Florentin Smarandache . Mohamed Abdel-Basset
Editors-in-Chief
ISSN 2331-6055 (Print)
ISSN 2331-608X (Online)
Neutrosophic Science
International Association (NSIA)
ISSN 2331-6055 (print)
ISSN 2331-608X (online)
Neutrosophic
Sets
and
Systems
An International Journal in Information Science and Engineering
University of New Mexico
ISSN 2331-6055 (print)
ISSN 2331-608X (online)
University of New Mexico
Neutrosophic Sets and Systems
An International Journal in Information Science and Engineering
Copyright Notice
Copyright @ Neutrosophics Sets and Systems
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Information for Authors and Subscribers
“Neutrosophic Sets and Systems” has been created for publications on advanced studies in neutrosophy, neutrosophic
set, neutrosophic logic, neutrosophic probability, neutrosophic statistics that started in 1995 and their applications in any field,
such as the neutrosophic structures developed in algebra, geometry, topology, etc.
The submitted papers should be professional, in good English, containing a brief review of a problem and obtained results.
Neutrosophy is a new branch of philosophy that studies the origin, nature, and scope of neutralities, as well as their interactions with different ideational spectra.
This theory considers every notion or idea <A> together with its opposite or negation <antiA> and with their spectrum of
neutralities <neutA> in between them (i.e. notions or ideas supporting neither <A> nor <antiA>). The <neutA> and <antiA>
ideas together are referred to as <nonA>.
Neutrosophy is a generalization of Hegel's dialectics (the last one is based on <A> and <antiA> only).
According to this theory every idea <A> tends to be neutralized and balanced by <antiA> and <nonA> ideas - as a state of
equilibrium.
In a classical way <A>, <neutA>, <antiA> are disjoint two by two. But, since in many cases the borders between notions are
vague, imprecise, Sorites, it is possible that <A>, <neutA>, <antiA> (and <nonA> of course) have common parts two by two,
or even all three of them as well.
Neutrosophic Set and Neutrosophic Logic are generalizations of the fuzzy set and respectively fuzzy logic (especially of
intuitionistic fuzzy set and respectively intuitionistic fuzzy logic). In neutrosophic logic a proposition has a degree of truth
(T), a degree of indeterminacy (I), and a degree of falsity (F), where T, I, F are standard or non-standard subsets of ]-0, 1+[.
Neutrosophic Probability is a generalization of the classical probability and imprecise probability.
Neutrosophic Statistics is a generalization of the classical statistics.
What distinguishes the neutrosophics from other fields is the <neutA>, which means neither <A> nor <antiA>.
<neutA>, which of course depends on <A>, can be indeterminacy, neutrality, tie game, unknown, contradiction, ignorance, imprecision, etc.
All submissions should be designed in MS Word format using our template file:
http://fs.unm.edu/NSS/NSS-paper-template.doc.
A variety of scientific books in many languages can be downloaded freely from the Digital Library of Science:
http://fs.unm.edu/ScienceLibrary.htm.
To submit a paper, mail the file to the Editor-in-Chief. To order printed issues, contact the Editor-in-Chief. This journal
is non-commercial, academic edition. It is printed from private donations.
Information about the neutrosophics you get from the UNM website:
http://fs.unm.edu/neutrosophy.htm. The
home page of the journal is accessed on
http://fs.unm.edu/NSS.
Copyright © Neutrosophic Sets and Systems, 2020
ISSN 2331-6055 (print)
ISSN 2331-608X (online)
University of New Mexico
Neutrosophic Sets and Systems
An International Journal in Information Science and Engineering
** NSS has been accepted by SCOPUS. Starting with Vol. 19, 2018, the NSS articles are
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NSS ABSTRACTED/INDEXED IN
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Beijing, China),
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Publons,
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Google Dictionaries have translated the neologisms "neutrosophy" (1) and"neutrosophic"
(2), coined in 1995 for the first time, into about 100 languages.
FOLDOC Dictionary of Computing (1, 2), Webster
Dictionary (1, 2), Wordnik (1),Dictionary.com, The Free
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Recently, NSS was also approved by Clarivate Analytics for Emerging Sources Citation
Index (ESCI) available on the Web of Science platform, starting with Vol. 15, 2017.
Copyright © Neutrosophic Sets and Systems, 2020
Clarivate Analytics
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Tel (215)386-0100 (800)336-4474
Fax (215)823-6635
March 20, 2019
Prof. Florentin Smarandache
Univ New Mexico, Gallup Campus
Dear Prof. Florentin Smarandache,
I am pleased to inform you that Neutrosophic Sets and Systems has been selected for coverage in Clarivate Analytics products and
services. Beginning with V. 15 2017, this publication will be indexed and abstracted in:
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ISSN 2331-6055 (print)
Editorial Board
ISSN 2331-608X (online)
University of New Mexico
Editors-in-Chief
Prof. Dr. Florentin Smarandache, Postdoc, Department of Mathematics, University of New Mexico, Gallup,
NM 87301, USA, Email: smarand@unm.edu.
Dr. Mohamed Abdel-Basset, Faculty of Computers and Informatics, Zagazig University, Egypt,
Email: mohamed.abdelbasset@fci.zu.edu.eg.
Associate Editors
Dr. Said Broumi, Laboratory of Information Processing, Faculty of Science Ben M’Sik, University of Hassan
II, Casablanca, Morocco, Email: s.broumi@flbenmsik.ma.
Prof. Dr. W. B. Vasantha Kandasamy, School of Computer Science and Engineering, VIT, Vellore 632014,
India, Email: vasantha.wb@vit.ac.in.
Assoc. Prof. Dr. Huda E. Khalid, Head of Scientific Affairs and Cultural Relations Department, Nineveh
Province, Telafer University, Iraq,
Email: dr.huda-ismael@uotelafer.edu.iq.
Prof. Dr. Xiaohong Zhang, Department of Mathematics, Shaanxi University of Science &Technology, Xian
710021, China, Email: zhangxh@shmtu.edu.cn.
Editors
Yanhui Guo, University of Illinois at Springfield, One
University Plaza, Springfield, IL 62703, United States,
Email: yguo56@uis.edu.
Giorgio Nordo, MIFT - Department of Mathematical
and Computer Science, Physical Sciences and Earth
Sciences,
Messina
University,
Italy,
Email:
giorgio.nordo@unime.it.
Le Hoang Son, VNU Univ. of Science, Vietnam
National
Univ.
Hanoi,
Vietnam,
Email:
sonlh@vnu.edu.vn.
A. A. Salama, Faculty of Science, Port Said University,
Egypt, Email: ahmed_salama_2000@sci.psu.edu.eg.
Young Bae Jun, Gyeongsang National University, South
Korea,
Email:
skywine@gmail.com.
Yo-Ping Huang, Department of Computer Science and
Information, Engineering National Taipei University,
New Taipei City, Taiwan, Email: yphuang@ntut.edu.tw.
Vakkas Ulucay, Kilis 7 Aralık University, Turkey, Email:
vulucay27@gmail.com.
Peide Liu, Shandong University of Finance and
Economics, China, Email: peide.liu@gmail.com.
Jun Ye, Department of Electrical and Information
Engineering, Shaoxing University, 508 Huancheng West
Road,
Shaoxing
312000,
China;
Email:
yejun@usx.edu.cn.
Memet Şahin, Department of Mathematics, Gaziantep
University, Gaziantep 27310, Turkey, Email:
mesahin@gantep.edu.tr.
Muhammad Aslam & Mohammed Alshumrani, King
Abdulaziz Univ., Jeddah, Saudi Arabia, Emails
magmuhammad@kau.edu.sa,
maalshmrani@kau.edu.sa.
Mutaz Mohammad, Department of Mathematics, Zayed
University, Abu Dhabi 144534, United Arab Emirates.
Email:
Mutaz.Mohammad@zu.ac.ae.
Abdullahi
Mohamud Sharif, Department of Computer Science,
University of Somalia, Makka Al-mukarrama Road,
Mogadishu, Somalia,
Email: abdullahi.shariif@uniso.edu.so.
NoohBany Muhammad, American University of
Kuwait, Kuwait,
Email: noohmuhammad12@gmail.com.
Soheyb Milles, Laboratory of Pure and Applied
Mathematics, University of Msila, Algeria, Email:
soheyb.milles@univ-msila.dz.
Pattathal Vijayakumar Arun, College of Science and
Technology,
Phuentsholing,
Bhutan,
Email:
arunpv2601@gmail.com.
Endalkachew Teshome Ayele, Department of
Mathematics, Arbaminch University, Arbaminch,
Ethiopia, Email: endalkachewteshome83@yahoo.com.
Xindong Peng, School of Information Science and
Engineering, Shaoguan University, Shaoguan 512005,
China,
Email:
952518336@qq.com.
Xiao-Zhi Gao, School of Computing, University of
Eastern Finland, FI-70211 Kuopio, Finland, xiaozhi.gao@uef.fi.
Madad Khan, Comsats Institute of Information
Technology,
Abbottabad,
Pakistan,
Email:
madadmath@yahoo.com.
Dmitri Rabounski and Larissa Borissova, independent
researchers, Emails: rabounski@ptep-online.com,
Copyright © Neutrosophic Sets and Systems, 2020
ISSN 2331-6055 (print)
Editorial Board
ISSN 2331-608X (online)
University of New Mexico
lborissova@yahoo.com.
Selcuk Topal, Mathematics Department, Bitlis Eren
University, Turkey, Email: s.topal@beu.edu.tr.
G. Srinivasa Rao, Department of Statistics, The
University of Dodoma, Dodoma, PO. Box: 259,
Tanzania,
Email:
gaddesrao@gmail.com.
Ibrahim El-henawy, Faculty of Computers and
Informatics, Zagazig University, Egypt, Email:
henawy2000@yahoo.com.
A. A. A. Agboola, Federal University of Agriculture,
Abeokuta, Nigeria, Email: agboolaaaa@funaab.edu.ng.
Abduallah Gamal, Faculty of Computers and
Informatics, Zagazig University, Egypt, Email:
abduallahgamal@zu.edu.eg.
Luu Quoc Dat, Univ. of Economics and Business,
Vietnam National Univ., Hanoi, Vietnam, Email:
datlq@vnu.edu.vn.
Maikel Leyva-Vazquez, Universidad de Guayaquil,
Ecuador,
Email:
mleyvaz@gmail.com.
Tula Carola Sanchez Garcia, Facultad de Educacion de
la Universidad Nacional Mayor de San Marcos, Lima,
Peru,
Email:
tula.sanchez1@unmsm.edu.pe.
Tatiana Andrea Castillo Jaimes, Universidad de Chile,
Departamento de Industria, Doctorado en Sistemas de
Ingeniería, Santiago de Chile, Chile, Email:
tatiana.a.castillo@gmail.com.
Muhammad Akram, University of the Punjab, New
Campus,
Lahore,
Pakistan,
Email:
m.akram@pucit.edu.pk.
Irfan Deli, Muallim Rifat Faculty of Education, Kilis 7
Aralik University, Turkey, Email: irfandeli@kilis.edu.tr.
Ridvan Sahin, Department of Mathematics, Faculty of
Science, Ataturk University, Erzurum 25240, Turkey,
Email:
mat.ridone@gmail.com.
Ibrahim M. Hezam, Department of computer, Faculty
of Education, Ibb University, Ibb City, Yemen, Email:
ibrahizam.math@gmail.com.
Aiyared Iampan, Department of Mathematics, School of
Science, University of Phayao, Phayao 56000, Thailand,
Email:
aiyared.ia@up.ac.th.
Ameirys Betancourt-Vázquez, 1 Instituto Superior
Politécnico de Tecnologias e Ciências (ISPTEC),
Luanda, Angola, Email: ameirysbv@gmail.com.
Karina Pérez-Teruel, Universidad Abierta para Adultos
(UAPA), Santiago de los Caballeros, República
Dominicana,
Email:
karinapt@gmail.com.
Neilys González Benítez, Centro Meteorológico Pinar
del
Río,
Cuba,
Email:
neilys71@nauta.cu.
Jesus Estupinan Ricardo, Centro de Estudios para la
Calidad Educativa y la Investigation Cinetifica, Toluca,
Mexico,
Email:
jestupinan2728@gmail.com.
Victor Christianto, Malang Institute of Agriculture
(IPM),
Malang,
Indonesia,
Email:
victorchristianto@gmail.com.
Wadei Al-Omeri, Department of Mathematics, Al-Balqa
Applied University, Salt 19117, Jordan, Email:
wadeialomeri@bau.edu.jo.
Ganeshsree Selvachandran, UCSI University, Jalan
Menara Gading, Kuala Lumpur, Malaysia, Email:
Ganeshsree@ucsiuniversity.edu.my.
Ilanthenral Kandasamy, School of Computer Science
and Engineering (SCOPE), Vellore Institute of
Technology (VIT), Vellore 632014, Tamil Nadu, India,
Email:
ilanthenral.k@vit.ac.in
Kul Hur, Wonkwang University, Iksan, Jeollabukdo,
South Korea, Email: kulhur@wonkwang.ac.kr.
Kemale Veliyeva & Sadi Bayramov, Department of
Algebra and Geometry, Baku State University, 23 Z.
Khalilov Str., AZ1148, Baku, Azerbaijan, Email:
kemale2607@mail.ru, Email: baysadi@gmail.com.
Irma Makharadze & Tariel Khvedelidze, Ivane
Javakhishvili Tbilisi State University, Faculty of Exact
and
Natural
Sciences,
Tbilisi,
Georgia.
Inayatur Rehman, College of Arts and Applied Sciences,
Dhofar
University
Salalah,
Oman,
Email:
irehman@du.edu.om.
Riad K. Al-Hamido, Math Department, College of
Science, Al-Baath University, Homs, Syria, Email: riadhamido1983@hotmail.com.
Faruk Karaaslan, Çankırı Karatekin University, Çankırı,
Turkey,
Email:
fkaraaslan@karatekin.edu.tr.
Morrisson Kaunda Mutuku, School of Business,
Kenyatta
University,
Kenya
Surapati Pramanik, Department of Mathematics,
Nandalal Ghosh B T College, India, Email:
drspramanik@isns.org.in.
Suriana Alias, Universiti Teknologi MARA (UiTM)
Kelantan, Campus Machang, 18500 Machang, Kelantan,
Malaysia, Email: suria588@kelantan.uitm.edu.my.
Arsham Borumand Saeid, Dept. of Pure Mathematics,
Faculty of Mathematics and Computer, Shahid Bahonar
University of Kerman, Kerman, Iran, Email:
arsham@uk.ac.ir.
Ahmed Abdel-Monem, Department of Decision
support,
Zagazig
University,
Egypt,
Email:
aabdelmounem@zu.edu.eg.
V.V. Starovoytov, The State Scientific Institution «The
United Institute of Informatics Problems of the
National Academy of Sciences of Belarus», Minsk,
Belarus,
Email:
ValeryS@newman.bas-net.by.
E.E. Eldarova, L.N. Gumilyov Eurasian National
University, Nur-Sultan, Republic of Kazakhstan, Email:
Doctorphd_eldarova@mail.ru.
Mohammad Hamidi, Department of Mathematics,
Payame Noor University (PNU), Tehran, Iran. Email:
m.hamidi@pnu.ac.ir.
Lemnaouar Zedam, Department of Mathematics,
Faculty of Mathematics and Informatics, University
Mohamed
Boudiaf,
M’sila,
Algeria,
Email:
l.zedam@gmail.com.
Copyright © Neutrosophic Sets and Systems, 2020
ISSN 2331-6055 (print)
Editorial Board
ISSN 2331-608X (online)
University of New Mexico
M. Al Tahan, Department of Mathematics, Lebanese
International University, Bekaa, Lebanon, Email:
madeline.tahan@liu.edu.lb.
Rafif Alhabib, AL-Baath University, College of Science,
Mathematical Statistics Department, Homs, Syria,
Email:
ralhabib@albaath-univ.edu.sy.
R. A. Borzooei, Department of Mathematics, Shahid
Beheshti
University,
Tehran,
Iran,
borzooei@hatef.ac.ir.
Sudan Jha, Pokhara University, Kathmandu, Nepal,
Email:
jhasudan@hotmail.com.
Mujahid Abbas, Department of Mathematics and
Applied Mathematics, University of Pretoria Hatfield
002,
Pretoria,
South
Africa,
Email:
mujahid.abbas@up.ac.za.
Željko Stević, Faculty of Transport and Traffic
Engineering Doboj, University of East Sarajevo,
Lukavica, East Sarajevo, Bosnia and Herzegovina,
Email:
zeljkostevic88@yahoo.com.
Michael Gr. Voskoglou, Mathematical Sciences School
of Technological Applications, Graduate Technological
Educational Institute of Western Greece, Patras,
Greece,
Email:
voskoglou@teiwest.gr.
Angelo de Oliveira, Ciencia da Computacao,
Universidade Federal de Rondonia, Porto Velho Rondonia,
Brazil,
Email:
angelo@unir.br.
Valeri Kroumov, Okayama University of Science,
Okayama,
Japan,
Email:
val@ee.ous.ac.jp.
Rafael Rojas, Universidad Industrial de Santander,
Bucaramanga,
Colombia,
Email:
rafael2188797@correo.uis.edu.co.
Walid Abdelfattah, Faculty of Law, Economics and
Management,
Jendouba,
Tunisia,
Email:
abdelfattah.walid@yahoo.com.
Akbar Rezaei, Department of Mathematics, Payame
Noor University, P.O.Box 19395-3697, Tehran, Iran,
Email:
rezaei@pnu.ac.ir.
Galina Ilieva, Paisii Hilendarski, University of Plovdiv,
4000 Plovdiv, Bulgaria, Email: galili@uni-plovdiv.bg.
Paweł Pławiak, Institute of Teleinformatics, Cracow
University of Technology, Warszawska 24 st., F-5, 31155 Krakow, Poland, Email: plawiak@pk.edu.pl.
E. K. Zavadskas, Vilnius Gediminas Technical
University,
Vilnius,
Lithuania,
Email:
edmundas.zavadskas@vgtu.lt.
Darjan Karabasevic, University Business Academy,
Novi
Sad,
Serbia,
Email:
darjan.karabasevic@mef.edu.rs.
Dragisa Stanujkic, Technical Faculty in Bor, University
of
Belgrade,
Bor,
Serbia,
Email:
dstanujkic@tfbor.bg.ac.rs.
Luige Vladareanu, Romanian Academy, Bucharest,
Romania,
Email:
luigiv@arexim.ro.
Hashem Bordbar, Center for Information Technologies
and Applied Mathematics, University of Nova Gorica,
Slovenia,
Email:
Hashem.Bordbar@ung.si.
Quang-Thinh Bui, Faculty of Electrical Engineering and
Computer Science, VŠB-Technical University of
Ostrava, Ostrava-Poruba, Czech Republic, Email:
qthinhbui@gmail.com.
Mihaela Colhon & Stefan Vladutescu, University of
Craiova, Computer Science Department, Craiova,
Romania,
Emails:
colhon.mihaela@ucv.ro,
vladutescu.stefan@ucv.ro.
Philippe Schweizer, Independent Researcher, Av. de
Lonay 11, 1110 Morges, Switzerland, Email:
flippe2@gmail.com.
Madjid Tavanab, Business Information Systems
Department, Faculty of Business Administration and
Economics University of Paderborn, D-33098
Paderborn, Germany, Email: tavana@lasalle.edu.
Saeid Jafari, College of Vestsjaelland South, Slagelse,
Denmark,
Email:
jafaripersia@gmail.com.
Fernando A. F. Ferreira, ISCTE Business School, BRUIUL, University Institute of Lisbon, Avenida das Forças
Armadas, 1649-026 Lisbon, Portugal, Email:
fernando.alberto.ferreira@iscte-iul.pt.
Julio J. Valdés, National Research Council Canada, M50, 1200 Montreal Road, Ottawa, Ontario K1A 0R6,
Canada,
Email:
julio.valdes@nrc-cnrc.gc.ca.
Tieta Putri, College of Engineering Department of
Computer Science and Software Engineering, University
of Canterbury, Christchurch, New Zeeland.
Mumtaz Ali, Deakin University, Victoria 3125, Australia,
Email:
mumtaz.ali@deakin.edu.au.
Phillip Smith, School of Earth and Environmental
Sciences, University of Queensland, Brisbane, Australia,
phillip.smith@uq.edu.au.
Sergey Gorbachev, National Research Tomsk State
University,
634050
Tomsk,
Russia,
Email:
gsv@mail.tsu.ru.
Willem K. M. Brauers, Faculty of Applied Economics,
University of Antwerp, Antwerp, Belgium, Email:
willem.brauers@uantwerpen.be.
M. Ganster, Graz University of Technology, Graz,
Austria,
Email:
ganster@weyl.math.tu-graz.ac.at.
Umberto Rivieccio, Department of Philosophy,
University
of
Genoa,
Italy,
Email:
umberto.rivieccio@unige.it.
F. Gallego Lupiaňez, Universidad Complutense, Madrid,
Spain,
Email:
fg_lupianez@mat.ucm.es.
Francisco Chiclana, School of Computer Science and
Informatics, De Montfort University, The Gateway,
Leicester, LE1 9BH, United Kingdom, Email:
chiclana@dmu.ac.uk.
Jean Dezert, ONERA, Chemin de la Huniere, 91120
Palaiseau, France, Email: jean.dezert@onera.fr.
Copyright © Neutrosophic Sets and Systems, 2020
ISSN 2331-6055 (print)
ISSN 2331-608X (online)
University of New Mexico
Contents
Ibrahim Yasser, Abeer Twakol, A. A. Abd El-Khalek, Ahmed Samrah and A. A. Salama COVID-X: Novel
Health-Fog Framework Based on Neutrosophic Classifier for Confrontation Covid-19…………....1
A.A. Salama, Ahmed Sharaf Al-Din, Issam Abu Al-Qasim, Rafif Alhabib and Magdy Badran, Introduction
to Decision Making for Neutrosophic Environment “Study on the Suez Canal Port, Egypt”..……22
Remya. P. B and Francina Shalini. A, Neutrosophic Vague Binary BCK/BCI-algebra ……………..45
Masoud Ghods and Zahra Rostami, Introduction to Topological Indices in Neutrosophic Graphs...68
A. Sahaya Sudha, Luiz Flavio Autran Monteiro Gomes and K.R. Vijayalakshmi, Assessment of MCDM
problems by TODIM using aggregated weights..........................................................................................78
Prakasam Muralikrishna and Surya Manokaran, MBJ – Neutrosophic – Ideal of – Algebra………….99
Nidhi Singh, Avishek Chakraborty, Soma Bose Biswas, Malini Majumdar, A Study of Social Media linked
MCGDM Skill under Pentagonal Neutrosophic Environment in the Banking Industry…………119
Rakhal Das and Binod Chandra Tripathy, Neutrosophic Multiset Topologica Space……………...142
Nidhi Singh, Avishek Chakraborty, Soma Bose Biswas and Malini Majumdar, Impact of Social Media in
Banking Sector under Triangular Neutrosophic Arena Using MCGDM Technique…………….153
Nilabhra Paul, Deepshikha Sarma, Akash Singh and Uttam Kumar Bera, A Generalized Neutrosophic
Solid Transportation Model with Insufficient Supply……………………………………………...177
Fatimah M. Mohammed and Sarah W. Raheem, Generalized b Closed Sets and Generalized b Open
Sets in Fuzzy Neutrosophic bi-Topological Spaces………………………………………………..188
Muhammad Riaz, Florentin Smarandache, Faruk Karaaslan, Masooma Raza Hashmi and Iqra Nawaz,
Neutrosophic Soft Rough Topology and its Applications to Multi-Criteria Decision-Making…..198
Kousik Das, Sovan Samanta, Kajal De and Xavier Encarnacion, A Study on Discrete Mathematics: Sum
Distance in Neutrosophic Graphs with Application……………………………………………….220
Nivetha Martin, Florentine Smarandache, I.Pradeepa, N.Ramila Gandhi and P.Pandiammal, Exploration
of the Factors Causing Autoimmune Diseases using Fuzzy Cognitive Maps with Concentric
Neutrosophic Hypergraphic Approach……………………………………………………….……232
Aiman Muzaffar, Md Tabrez Nafis and Shahab Saquib Sohail, Neutrosophy Logic and its Classification:
An Overview…………………………………………………………………………………….……239
Abhijit Saha, Florentin Smarandache, Jhulaneswar Baidya and Debjit Dutta, MADM Using mGeneralized q-Neutrosophic Sets …………………………………………………………………..252
Abhijit Saha, Irfan Deli, and Said Broumi, HESITANT Triangular Neutrosophic Numbers and Their
Applications to MADM ……………………………………………………………………………..269
Copyright © Neutrosophic Sets and Systems, 2020
ISSN 2331-6055 (print)
ISSN 2331-608X (online)
University of New Mexico
Mohsin Khalid, Florentin Smarandache, Neha Andaleeb Khalid and Said Broumi, Translative and
Multiplicative Interpretation of Neutrosophic Cubic Set……………299
Md. Hanif PAGE and Qays Hatem Imran, Neutrosophic Generalized Homeomorphism…………340
Shilpi Pal and Avishek Chakraborty, Triangular Neutrosophic Based Production Reliability Model of
Deteriorating Item with Ramp Type Demand under Shortages and Time Discounting ……..…347
S. Satham Hussain, Saeid Jafari, Said Broumi and N. Durga, Operations on Neutrosophic Vague Graphs
……………………………………………………………………………………….………….……368
Hanieh Shambayati, Mohsen Shafiei Nikabadi, Seyed Mohammad Ali Khatami Firouzabadi and
Mohammad Rahmanimanesh, Partner Selection in Virtual Enterprises using the Interval
Neutrosophic Fuzzy Approach ……………………………………………………………………..387
Mujahid Abbas, Ghulam Murtaza, and Florentin Smarandache, Basic operations on hypersoft sets and
hypersoft point ………………………………………………………………………………………407
K. Porselvi, B. Elavarasan, F. Smarandache and Y. B. Jun, Neutrosophic ℵ −bi-ideals in semigroups.422
Chinnadurai V and Sindhu M P, A Novel Approach for Pairwise Separation Axioms on Bi-Soft
Topology Using Neutrosophic Sets and An Output Validation in Real Life Application ………435
K.Radhika and K.Arun Prakash, Ranking of Pentagonal Neutrosophic Numbers and its Applications
to solve Assignment Problem………………………………………………………………………..464
Said Broumi, Malayalan Lathamaheswari, Ruipu Tan ,Deivanayagampillai Nagarajan, Talea Mohamed,
Florentin Smarandache and Assia Bakali, A new distance measure for trapezoidal fuzzy neutrosophic
numbers based on the centroids……………………………………………………..……………....478
Nivetha Martin and Florentin Smarandache, Introduction to Combined Plithogenic Hypersoft
Sets……………………………………………………………………………………………..……..503
Qays Hatem Imran, R. Dhavaseelan, Ali Hussein Mahmood Al-Obaidi, and Md. Hanif PAGE, On
Neutrosophic Generalized Alpha Generalized Continuity ………………………………………...511
Suman Das and Surapati Pramanik, Generalized neutrosophic b-open sets in neutrosophic
topological space ……………………………………………………………………………………522
Madad Khan, Muhammad Zeeshan, Saima Anis, Abdul Sami Awan and Florentin Smarandache,
Neutrosophic Soft Fixed Points……………………………………………………………………..531
M. Sarwar Sindhu, Tabasam Rashid and Agha Kashif, Selection of Alternative under the Framework of
Single-Valued Neutrosophic Sets…………………………………………………………………...547
Rajesh Kumar Saini, Atul Sangal and Manisha, Application of Single Valued Trapezoidal Neutrosophic
Numbers in Transportation Problem…………………………………………………..…………..563
Copyright © Neutrosophic Sets and Systems, 2020
Neutrosophic Sets and Systems, Vol. 53, 0202
University of New Mexico
COVID-X: Novel Health-Fog Framework Based on
Neutrosophic Classifier for Confrontation Covid-19
Ibrahim Yasser 1,*, Abeer Twakol 2, A. A. Abd El-Khalek 3, Ahmed Samrah 4 and A. A. Salama 5
1
2
3
4
Communications and Electronics Engineering Department, Nile Higher Institute for Engineering and Technology
Mansoura, Egypt; ibrahimyasser14@gmail.com.
Department of Computer Engineering, Faculty of Engineering, Benha University, Egypt; atwakol2013@gmail.com.
Communications and Electronics Engineering Department, Nile Higher Institute for Engineering and Technology
Mansoura, Egypt; ayayakot94@gmail.com.
Electronics and Communications Engineering Department, Faculty of Engineering, Mansoura University, Egypt;
shmed@mans.edu.eg.
5
Department of Mathematics and Computer Science, Faculty of Sciences, Port Said University, Egypt;
drsalama44@gmail.com.
* Correspondence: ibrahimyasser14@gmail.com
Abstract: The newly identified Coronavirus pneumonia, subsequently termed COVID-19, is highly
transmittable and pathogenic with no clinically approved antiviral drug or vaccine available for
treatment. Technological developments like edge computing, fog computing, Internet of Things
(IoT), and Big Data have gained importance due to their robustness and ability to provide diverse
response characteristics based on target application. In this paper, we present a novel Health-Fog
framework universal system to automatically assist the early diagnosis, treatment, and preventive
of people with COVID-19 in an efficient manner. Achieving an empirical of the proposed framework
which mix between deep learning and Neutrosophic classifiers in the task of classifying COVID-19.
There are some proposed applications based on the proposed COVID-X framework such as smart
mask, smart medical suit, safe spacer, and Medical Mobile Learning (MML) will be presented.
Computer-aided diagnosis systems could assist in the early detection of COVID-19 abnormalities
and help to monitor the progression of the disease, potentially reduce mortality rates.
Keywords: Coronavirus Pneumonia; COVID-19; Intelligent Medical System; Fog Computing; Health-Fog;
Neutrosophic; Deep Learning; Computer-Aided Diagnosis.
1. Introduction
The Coronavirus disease 2019-2020 pandemic (COVID-19) poses unprecedented challenges for
governments and societies around the world. In addition to medical measures, non-pharmaceutical
measures have proven to be critical for delaying and containing the spread of the virus. This includes
(aggressive) testing and tracing, bans on large gatherings, school and university closures,
international and domestic mobility restrictions and physical isolation, up to total lockdowns of
regions and countries. However, effective and rapid decision-making during all stages of the
pandemic requires reliable and timely data not only about infections, but also about human behavior,
especially on mobility and physical co-presence of people [1]. There are growing privacy concerns
I. Yasser; A. Twakol; A. A. Abd El-Khalek; A. Samrah; A.A.Salama. COVID-X: Novel Health-Fog Framework Based on
Neutrosophic Classifier for Confrontation Covid-19
Neutrosophic Sets and Systems, Vol. 55, 0202
2
about the ways governments use data to respond to the COVID-19 crisis. As new technologies emerge
that aim to collect, disseminate and use data in order to support the fight against COVID-19, we need
to ensure they respect ethical best practices. Even in times of crisis, we need to comply with data
privacy regulations and ensure that the data is used ethically. One way to do that is to establish
independent ethical committees or data trusts. Their role will be to create data governance
mechanisms to find the balance between competing public interests, while protecting individual
privacy. Examples of such rules include setting up clear guidelines on the purpose and timeline for
the use of the data, defining clear processes for the access, processing and termination of use of
personal data at the end of the crisis. Tracking a patient from symptoms, lab results and treatments
can help a hospital understand how a disease is progressing through a community, how effective
treatments are and what isn’t working [0].
Technological developments like edge computing, fog computing, Internet of Things (IoT), and
Big Data have gained importance due to their robustness and ability to provide diverse response
characteristics based on target application. These emerging technologies provide storage,
computation, and communication to edge devices, which facilitate and enhance mobility, privacy,
security, low latency, and network bandwidth so that fog computing can perfectly match latencysensitive or real-time applications [3]. Healthcare is one of the prominent application areas that
requires accurate and real-time results, and people have introduced Fog Computing in this field
which leads to a positive progress. With Fog computing, we bring the resources closer to the users
thus decreasing the latency and thereby increasing the safety measure. Getting quicker results implies
fast actions for critical COVID- 19 patients. But faster delivery of results is not enough as with such
delicate data we cannot compromise with the accuracy of the result [4]. One way to obtain high
accuracies is by using state-of-the-art analysis software typically those that employ deep learning and
their variants trained on a large dataset. Deep learning techniques showed in the last years promising
results to accomplish radiological tasks by automatic analyzing multimodal medical images [5]. Deep
convolutional neural networks (DCNNs) are one of the powerful deep learning architectures and
have been widely applied in many practical applications such as pattern recognition and image
classification in an intuitive way [6]. DCNNs are able to handle four manners as follow [7]: 1) training
the neural network weights on very large available datasets; 2) fine-tuning the network weights of a
pre-trained DCNN based on small datasets; 3) Applying unsupervised pre-training to initialize the
network weights before putting DCNN models in an application; and 4) using pre-trained DCNN is
also called an off-the-shelf CNN being used as a feature extractor. Convolutional neural networks are
sensitive to unknown noisy condition in the test phase and so their performance degrades for the
noisy data classification task including noisy recognition. In this research, a convolutional neural
network (CNN) model with data uncertainty handling; referred as NCNN (Neutrosophic
Convolutional Neural Network); is proposed for classification task. The Neutrosophic is a new view
of Modeling , designed to effectively deal underlying doubts in the real world, as it came to replace
binary logic that recognized right and wrong by introducing a third neutral case which could be
interpreted as non-specific or uncertain. Founded by Florentin Smarandache [8], he presented it in
1999 as a generalization of fuzzy logic. As an extension of this, A. A. Salama introduced the
Neutrosophic crisp sets Theory as a generalization of crisp sets theory [9] and developed, inserted
and formulated new concepts in the fields of mathematics, statistics, and computer science and
information systems through Neutrosophic [10-12]. Neutrosophic has grown significantly in recent
years through its application in measurement, sets and graphs and in many scientific and practical
fields [13- 17].
I. Yasser; A. Twakol; A. A. Abd El-Khalek; A. Samrah; A. A. Salama. COVID-X: Novel Health-Fog Framework Based on
Neutrosophic Classifier for Confrontation Covid-19
Neutrosophic Sets and Systems, Vol. 55, 0202
3
In this work, a proposed novel COVID- X framework was developed as universal Health-Fog
system for automatic diagnosis, treatment, and preventive of people with COVID-19 in an efficient
manner using deep learning, Neutrosophic and IoT. Health-Fog provides healthcare as a fog service
and efficiently manages the data of COVID-19 patients which is coming from different IoT devices.
Health-Fog provides this service by using the proposed framework and demonstrates application
enablement and engineering simplicity for leveraging fog resources to achieve the same.
In the following, the contributions of this paper are summarized:
Building altogether a novel framework universal system to automatically assist the early
diagnosis, treatment, and preventive of people with COVID-19 in an efficient manner.
Proposed a generic system architecture for development of ensemble NCNN on fog
computing
Achieving an empirical of the proposed framework which mix between deep learning and
Neutrosophic classifiers in the task of classifying COVID-19.
The proposed COVID-X framework supports interdisciplinary researchers to continue
developing advanced artificial intelligence techniques to fight the COVID-19 outbreak.
This study demonstrated the useful applications of deep learning models to classify COVID19 based on the proposed COVID-X framework such as smart mask, smart medical suit, safe
spacer, and Medical Mobile Learning (MML). These applications are the next milestone of
this research work.
The rest of this paper is structured as follows. Section 2 presents the related works. Section 3
gives a review on the state-of-the-art deep convolutional neural network models as image classifiers.
Also, a detailed description of the COVIDX-Net framework is presented. Experimental results and
comparative performance of the proposed deep learning classifiers are investigated and discussed in
section 4. Finally, limitations and this study is concluded with the main prospects in sections 4, 5.
2. Related Work
Some studies have shown the use of imaging techniques such as X-rays or Computed
Tomography (CT-scans) for finding characteristic symptoms of the novel corona virus in these
imaging techniques. Hemdan et al. [18] developed a deep learning framework, COVIDX-Net, to
diagnose COVID- 19 in X-Ray Images. A comparative study of different deep learning architectures
including VGG19, DenseNet201, ResNetV2, InceptionV3, InceptionResNetV2, Xception and
MobileNetV2 is provided by authors. Barstugan et al. [19] proposed a machine learning approach for
COVID-19 classification from CT images. Kassani et al. [20] presented a feature extractor-based deep
learning and machine learning classifier approach for computer-aided diagnosis (CAD) of COVID19 pneumonia. Loey et al. [21] presented a detection model based on GAN network with deep transfer
learning for COVID-19 detection in limited chest X-ray images. Table 1 compares the proposed model
(HealthFog) with existing models. Recent studies suggest the use of chest radiography in the
epidemic areas for the initial screening of COVID-19 [22]. Therefore, the screening of radiography
images can be used as an alternate to the PCR method, which exhibit higher sensitivity in some cases
[23]. Nevertheless, the main bottleneck that the radiologists experience in analyzing radiography
images is the visual scanning of the subtle insights. This entails the use of intelligent approaches that
can automatically extract useful insights from the chest X-rays those are characteristics of COVID-19.
I. Yasser; A. Twakol; A. A. Abd El-Khalek; A. Samrah; A. A. Salama. COVID-X: Novel Health-Fog Framework Based on
Neutrosophic Classifier for Confrontation Covid-19
Neutrosophic Sets and Systems, Vol. 55, 0202
Work
Hemdan
et al. [18]
Barstugan
et al. [19]
Kassani et
al. [20]
Loey et al.
[21]
Proposed
work
Fog
Computing
4
Table 1: Comparison of existing models
Deep
IoT Neutrosophic
Dataset
Learning
Diagnosis
Healthcare
applications
3. Proposed COVID_X Description framework
Fog and Cloud computing paradigms have emerged as a backbone of modern economy and
utilize Internet to provide on-demand services to users [24]. Both of these domains have captured
significant attention of industries and academia. In this section will proposed a new deep learning
framework for automatically identifying the status of COVID-19 extend support to emerging
application paradigms such as IoT, Fog computing, Edge, and Big Data through service and
infrastructure. The data generated from Things layer can vary in size, for instance, the data sent from
sensors. The diversity in data packages size influence the behavior of Fog node during the processing
event, thus, data packages will require more time to process than light data packages. Therefore, in
the proposed model, there is a distinction processing tasks. In addition, the fog nodes were adopted
collaboration framework to achieve the minimal request processing time for heavy data packages. In
Figure 1 the collaboration concept was elaborated and the distinction different processing tasks
received from Things layer. In addition, in this framework the advance approach was adopted to
identify the suitable treatment process, such as, Fog reputation to process specific type of data (e.g.,
health data).
I. Yasser; A. Twakol; A. A. Abd El-Khalek; A. Samrah; A. A. Salama. COVID-X: Novel Health-Fog Framework Based on
Neutrosophic Classifier for Confrontation Covid-19
Neutrosophic Sets and Systems, Vol. 55, 0202
5
Figure 1. Proposed framework universal system for confrontation covid-19
3.1 Edge Layer:
The edge layer (perception layer), is the starting point of the IoT structure where data is been
generated. This layer contains the networked Things (i.e., wireless sensors) such as heart-rate, bloodoxygen and etc., which operate to feed the system with patient symptoms data. Each Thing
device/object in this layer is facilitated with communication protocol (such as IEEE 802.15.4, WiFi,
Bluetooth, MQTT, and etc.) in which permit the Thing node to transmit the generated data to Fog
I. Yasser; A. Twakol; A. A. Abd El-Khalek; A. Samrah; A. A. Salama. COVID-X: Novel Health-Fog Framework Based on
Neutrosophic Classifier for Confrontation Covid-19
Neutrosophic Sets and Systems, Vol. 55, 0202
6
nodes over the IoT network. In our proposed architecture, A TN denoted by Ƭ, is defined as a sixtuple: 𝑇 = 〈 𝑇𝑖𝑑 , 𝑇𝑠𝑡 , 𝜏𝑖 , ℒ, ℋ, ℐ[𝑞] 〉 where, 𝑇𝑖𝑑 is an integer representing the unique ID of the TN,
𝑇𝑠𝑡 = {0,1}, defines whether the node is in active state or not, (𝜏𝑖 ) indicates the type of event that a
node senses. (ℒ) is refer to the geo-spatial location of a TN. (ℋ) is represented the specifications of an
edge device. ℐ[𝑞] is a linear data structure, such as a 1-D array (with q elements) that stores the
instance IDs of the application instances running on the device. These tuples are essential to represent
the Thing node over the IoT network.
3.1.1 Thermal Screen
The smart helmet can also detect high body’s temperature in the crowds and send the measured
data to be displayed on a phone application. Smart Helmet system work is presented in Figure 2. As
the high body temperature of people is one of the very common symptoms, a real time monitoring
system of the screening process that automatically appearing the thermal image of temperature of
people is needed. So, the diagnosis of the screening process will be less time consuming and less
human interactions that might cause the spreading of the coronavirus faster. It can be concluded that
the remote sensing procedures, which provide an assortment of ways to identify, sense, and
monitoring of coronavirus, give an awesome promise and potential in order to fulfil the demands
from the healthcare system [25].
Figure 2. Smart Helmet system work
3.1.2 Sensing Node
Smart City and Intelligent Transportation System (ITS) as shown in Figure 3 offer a futuristic
vision of smart, secure and safe experience to the end user, and at the same time efficiently manage
the sparse resources and optimize the efficiency of city operations. However, outbreaks and
pandemics like COVID-19 have revealed limitations of the existing deployments, therefore,
architecture, applications and technology systems need to be developed for swift and timely
enforcement of guidelines, rules and government orders to contain such future outbreaks. The
proposed architecture and AI assisted applications discussed in the article can be used to effectively
and timely enforce social distancing community measures, and optimize the use of resources in
critical situations. It offers a conceptual overview and serves as a steppingstone to extensive research
and deployment of automated data driven technologies in smart city and intelligent transportation
systems [26].
I. Yasser; A. Twakol; A. A. Abd El-Khalek; A. Samrah; A. A. Salama. COVID-X: Novel Health-Fog Framework Based on
Neutrosophic Classifier for Confrontation Covid-19
Neutrosophic Sets and Systems, Vol. 55, 0202
7
Figure 3. Smart City and ITS Architecture.
3.1.3 Smart Mask
Smart mask can be developed that can record air quality among other features. The Smart Mask
is more than your average face mask, as its name suggests. Figure 4 shows the proposed Smart Mask,
can record air quality information thanks to various sensors and electronics. Additionally, it can
inform wearers of possible changes in lung capacity. While this may prove useful in areas of poor air
quality,
Figure 4. The proposed Smart Mask
Specifications; Type: Head-mounted, rated voltage: DC 5V, rated power: 0.4W, Charging time:
2 hours Standby time: 5~8 hours, Filtering effect: 95%, Protection level: KN95, Function: Dustproof,
anti-haze, anti-pollen, anti-tail gas, etc. Feature; Unique ventilation design, a plurality of holes,
excellent permeability, Exhale, the valve is opened without resistance, air breathing valve, air
resistance is smaller, smooth breathing, uses efficient and low-resistance filter material, combined
with the smart electric air supply module to provide fresh air into the mask. The edge is protected by
3D sponge for effective sealing. best protection: The allergy mask separation of 98% of the dust,
chemicals, smoke and particles, it can be used for dust, anti-vehicle exhaust, anti-pollen allergy,
PM2.5, for cycling, hiking, skiing and other outdoor activities. High-performance breathing valve
I. Yasser; A. Twakol; A. A. Abd El-Khalek; A. Samrah; A. A. Salama. COVID-X: Novel Health-Fog Framework Based on
Neutrosophic Classifier for Confrontation Covid-19
Neutrosophic Sets and Systems, Vol. 55, 0202
8
that reduces heat and moisture build-up for smoother breathing. Built-in adjustable nose clip for a
good fit and comfort with the face. Charge once for 5~8-hour endurance to ensure commuting. KN95
industrial safety protection level. Low noise.one mask can be used for 5-8 days. Can be reused and
Comfortable ear band made of soft cotton, easy to wear and remove ear loop design.
3.1.4 Smart Medical Suit
The nature of Health care workers job puts them health care at an increased risk of catching any
communicable disease, including COVID-19. Where they spend a lot of time up close with the patient
doing high risk activities, those high-risk activities include things like placing patients on ventilators
or collecting samples of sputum from their lungs. That’s why it’s so important that they achieve the
highest level of protective equipment. The proposed smart medical suits is showed in Figure 5.
Figure 5. The proposed Smart Medical Suit.
3.1.5 Mobile App.
The new MobileDetect COVID-19 test kit in Figure 6 was planned to launch in April 2020. The
currently available free MobileDetect App for Apple and Android smartphone and tablet platforms
will be updated with the additional COVID-19 testing capability upon launch. Due to the novel
design incorporating simplistic operation along with credible field-testing capability, the COVID-19
test kits can be used by federal, state, local response, medical agencies and are also planned to be
available to the general public [27].
Figure 6. MobileDetect Application.
I. Yasser; A. Twakol; A. A. Abd El-Khalek; A. Samrah; A. A. Salama. COVID-X: Novel Health-Fog Framework Based on
Neutrosophic Classifier for Confrontation Covid-19
Neutrosophic Sets and Systems, Vol. 55, 0202
9
3.1.6 X-ray and CT Images
Medical imaging is also playing a critical in monitoring the progression of the disease and
patient care. Extracting features from radiology modalities is an essential step in training machine
learning models since the model performance directly depends on the quality of extracted features.
Figure 7. Illustrates the visual features extracted by VGGNet architecture from an X-ray image of a
COVID-19 positive patient. Motivated by the success of deep learning models in computer vision,
the focus of this research is to provide an extensive comprehensive study on the classification of
COVID-19 pneumonia in chest X-ray and CT imaging using features extracted by the state-of-the-art
deep CNN architectures and trained on machine learning algorithms [20].
Figure 7. Framework of the method with VGGNet as feature extractor.
3.1.7 Community Acquired Pneumonia on Chest CT
In this study, a 3D deep learning framework was proposed for the detection of COVID-19 as
shown in Figure 8. This framework is able to extract both 2D local and 3D global representative
features. Deep learning has achieved superior performance in the field of radiology. RT-PCR is
considered as the reference standard; however, it has been reported that chest CT could be used as a
reliable and rapid approach for screening of COVID-19 [28]
Figure 8. COVID-19 detection neural network (COVNet) architecture.
I. Yasser; A. Twakol; A. A. Abd El-Khalek; A. Samrah; A. A. Salama. COVID-X: Novel Health-Fog Framework Based on
Neutrosophic Classifier for Confrontation Covid-19
Neutrosophic Sets and Systems, Vol. 55, 0202
10
3.2 Fog Layer:
The Fog layer contains number of decentralized nodes in each given location. This layer handles
the primary refining, compute, and processing of data generated in the Things layer. Fog nodes aim
to improve efficiency of IoT applications, thus, Fog has the potential to reduce the amount of data
transmitted to the Cloud layer and minimizing the requests-response time for IoT applications. This
is often required to enhance the Quality of Service (QoS), such as reducing latency and improve
network bandwidth. For example, in reference to our scenario the Fog will receive patient’s data from
their wearable, analyze the data according to predetermined artificial intelligent training, and make
outcome available to caregiver over the dashboard and notify cloud with outcome for complex
analysis.
3.2.1 Data pre-processing
Covid-19 tested data e.g. the images within the dataset were collected from multiple imaging
clinics with different equipment and image acquisition parameters; therefore, considerable variations
exist in images' intensity. The proposed method in this study avoids extensive pre-processing steps
to improve the generalization ability of the convolution neural network (CNN) architecture. This
helps to make the model more robust to noise, artifacts and variations in input images during feature
extraction phase. Hence, we only employed two standard pre-processing steps in training deep
learning models to optimize the training process [29].
3.2.2 Neutrosophic Classifier
Neutrosophic classifier: a classifier that would use Neutrosophic logic principles and
Neutrosophic sets for the classification. Neutrosophic classifier incorporates a simple, Neutrosophic
rule based approach like: IF X and Y THEN Z, for solving problem rather than attempting to model
a system mathematically similar to fuzzy classifier [30]. Designing of Neutrosophic classification
inference system using fuzzy methodology is based on the principles of Mamdani fuzzy inference
method [25]. Figure 9 gives the block diagram representation of a Neutrosophic classification system
using fuzzy logic toolbox of Matlab. Values of T, I and F Neutrosophic components are independent
of each other. So using fuzzy logic toolbox of Matlab, three FIS have been designed: one for
Neutrosophic truth component, second for Neutrosophic indeterminacy component and third for
Neutrosophic falsity component. Though the working of these components are independent of each
other but a correlation is drawn between membership functions of Neutrosophic T, I and F
components so as to capture the truthness, indeterminacy and falsity of the input as well as the
output.
I. Yasser; A. Twakol; A. A. Abd El-Khalek; A. Samrah; A. A. Salama. COVID-X: Novel Health-Fog Framework Based on
Neutrosophic Classifier for Confrontation Covid-19
Neutrosophic Sets and Systems, Vol. 55, 0202
11
Figure 9. Block diagram for a Neutrosophic components
Neutrosophic Rule-based Classification System (NRCS) which is a rule based system where
Neutrosophic logic is used as a tool for representing different forms of knowledge about the problem
at hand, as well as for modeling the interactions and relationships that exist between its variables
[23]. The generic structure of a NRCSs shown in Figure 10.
Figure 10. Basic structure of a Neutrosophic Rule-Based Classification System
Let U be a universe of discourse and W is a set in U which composed of bright pixels. A
Neutrosophic images 𝑃𝑁𝑆 is characterized by three sub sets T, I, and F. that can be defined as T is the
degree of membership, I is the degree of indeterminacy, and F is the degree of non-membership. In
the image, a pixel P in the image is described as P(T,I,F) that belongs to W by its t% is true in the
bright pixel, i% is the indeterminate and f% is false where t varies in T, i varies in I, and f varies in F.
I. Yasser; A. Twakol; A. A. Abd El-Khalek; A. Samrah; A. A. Salama. COVID-X: Novel Health-Fog Framework Based on
Neutrosophic Classifier for Confrontation Covid-19
Neutrosophic Sets and Systems, Vol. 55, 0202
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The pixelp(i,j)in the image domain, is transformed to
𝑁𝐷𝑃𝑁𝑆 (𝑖, 𝑗) = {𝑇(𝑖, 𝑗), 𝐼(𝑖, 𝑗), 𝐹(𝑖, 𝑗)}
(1)
Where belongs to white set, belongs to indeterminate set and belongs to non-white set. Which can be
defined as [31]:
𝑃𝑁𝑆 (𝑖, 𝑗) = {𝑇(𝑖, 𝑗), 𝐼(𝑖, 𝑗), 𝐹(𝑖, 𝑗)}
𝑇(𝑖, 𝑗) =
̅̅̅̅̅̅̅̅
𝑔(𝑖, 𝑗) − 𝑔̅𝑚𝑖𝑛
𝑔̅𝑚𝑎𝑥 − 𝑔̅𝑚𝑖𝑛
𝐼(𝑖, 𝑗) = 1 −
𝐻𝑜 (𝑖, 𝑗) − 𝐻𝑜
𝐻𝑜𝑚𝑎𝑥 − 𝐻𝑜𝑚𝑖𝑛
(2)
(3)
(4)
𝐹(𝑖, 𝑗) = 1 − 𝑇(𝑖, 𝑗)
(5)
𝐻𝑜 (𝑖, 𝑗) = 𝑎𝑏𝑠(𝑔(𝑖, 𝑗) − ̅̅̅̅̅̅̅̅
𝑔(𝑖, 𝑗)
(6)
̅̅̅̅̅̅̅̅
Where 𝑔(𝑖,
𝑗) represents the local mean value of the pixels of window size, and 𝐻𝑜 (𝑖, 𝑗) which
can be defined as the homogeneity value of T at (i,j), that described by the absolute value of difference
between intensity 𝑔(𝑖, 𝑗) and its local mean value ̅̅̅̅̅̅̅̅
𝑔(𝑖, 𝑗).
The Content Based Image Retrieval (CBIR) goal is to retrieve images relevant to a query images
which selected by a user. The image in CBIR is described by extracted low-level visual features, such
as color, texture and shape. Retrieval System for images embedded in the Neutrosophic domain. In
this first phase, extract a set of features to represent the content of each image in the training database.
In the second phase, a similarity measurement is used to determine the distance between the image
under consideration (query image), and each image in the training database, using their feature
vectors constructed in the first phase. Hence, the N most similar images are retrieved. Several
distance metrics were suggested for both content and texture image retrieval, respectively. In this
paper, we are using a Neutrosophic version of the Euclidean distance, which was presented in [31].
For any two Neutrosophic Sets, the Content Based Image Retrieval (CBIR) goal is to retrieve images
relevant to a query images which selected by a user. The image in CBIR is described by extracted
low-level visual features, such as color, texture and shape. Retrieval System for images embedded in
the Neutrosophic domain. In this first phase, extract a set of features to represent the content of each
image in the training database. In the second phase, a similarity measurement is used to determine
the distance between the image under consideration (query image), and each image in the training
database, using their feature vectors constructed in the first phase. Hence, the N most similar images
are retrieved. Several distance metrics were suggested for both content and texture image retrieval,
respectively. In this paper, we are using a Neutrosophic version of the Euclidean distance, which was
presented in [31]. For any two Neutrosophic Sets,
𝐴 = {𝑇𝐴 (𝑥), 𝐼𝐴 (𝑥), 𝐹𝐴 (𝑥)), 𝑥 ∈ 𝑈} 𝑎𝑛𝑑
(7)
𝐵 = {𝑇𝐵 (𝑥), 𝐼𝐵 (𝑥), 𝐹𝐵 (𝑥)), 𝑥 ∈ 𝑈} 𝑖𝑛
(8)
𝑈 = {𝑢1 , 𝑢2 , 𝑢3 , … , 𝑢𝑛 } 𝑡ℎ𝑒𝑛
(9)
I. Yasser; A. Twakol; A. A. Abd El-Khalek; A. Samrah; A. A. Salama. COVID-X: Novel Health-Fog Framework Based on
Neutrosophic Classifier for Confrontation Covid-19
Neutrosophic Sets and Systems, Vol. 55, 0202
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The Neutrosophic Euclidean distance is equal to
𝑛
𝑑(𝐴, 𝐵) = √∑
𝑖=1
((𝑇𝐴 (𝑥𝑖 ) − 𝑇𝐵 (𝑥𝑖 ))2 + ((𝐼𝐴 (𝑥𝑖 ) − 𝐼𝐵 (𝑥𝑖 ))2 + ((𝐹𝐴 (𝑥𝑖 ) − 𝐹𝐵 (𝑥𝑖 ))2
(10)
Figure 11. Neutrosophic COVID-19 image classifier Architecture
The algorithm for the proposed system is given below which presented in Figure 11:
1.
2.
3.
4.
Convert each image in the database from spatial domain to Neutrosophic domain.
Create a database containing various COVID-19.
Extract Texture feature of COVID-19 in the database.
Construct a combined feature vector for T, I, F and Stored in another database called
Featured Database.
5. Find the distance between feature vectors of query COVID-19 and that of featured
databases.
6. Sort the distance and Retrieve the N-top most similar.
The RNN structure replaces the traditional neuron by two neurons (lower neuron, upper neuron) to
represent lower and upper approximations of each attribute in the CTG data set, its structure formed
from 4 layers input, 2 hidden and output layers. The hidden layers have rough neurons, which
overlap and exchange information between each other, While the input and output layers consists of
traditional neurons as in Figure 12 [32].
I. Yasser; A. Twakol; A. A. Abd El-Khalek; A. Samrah; A. A. Salama. COVID-X: Novel Health-Fog Framework Based on
Neutrosophic Classifier for Confrontation Covid-19
Neutrosophic Sets and Systems, Vol. 55, 0202
14
Figure 12. Rough Neural Network (RNN) structure.
Input layer is composed of neuron for each data attribute. The output layer represents the three
FHR classes, the hidden layers rough neurons are determined by the Baum-Haussler rule [33].
𝑁ℎ𝑛 =
𝑁𝑡𝑠 × 𝑇𝑒
𝑁𝑖 + 𝑁𝑜
(11)
Where 𝑁ℎ𝑛 is the number of hidden neurons, 𝑁𝑡𝑠 is the number of training samples, 𝑇𝑒 is the
tolerance error, 𝑁𝑖 is the number of inputs (attributes or features), and 𝑁𝑜 is the number of the
output.During training process, the normalized input data is multiplied by its weight and computed
in sigmoid activation function.
𝑓(𝑥) =
1
1 + 𝑒 −𝜆𝑥
(12)
Step II: Training phase
1.
2.
3.
Initialize random (upper, lower) weights of network
Feed forward of attribute values and multiply in both direction (Uw, Lw)
Compute (IU, IL) of hidden layers by relations:
𝑛
𝐼𝐿𝑛 = ∑
𝑊𝐿𝑛𝑗 𝑂𝑛𝑗
(13)
𝑊𝑈𝑛𝑗 𝑂𝑛𝑗
(14)
𝐽=1
𝑛
𝐼𝑈𝑛 = ∑
4.
𝐽=1
Compute (OU, OL) of hidden layers by relations:
𝑂𝐿𝑛 = 𝑀𝑖𝑛(𝑓(𝐼𝐿𝑛 ), 𝑓(𝐼𝑈𝑛 ))
5.
(15)
Check fetus according to comparing between actual output (T) and output value (O),
where output represent by
𝑂 = 𝑂𝐿𝑛 + 𝑂𝑈𝑛
(16)
I. Yasser; A. Twakol; A. A. Abd El-Khalek; A. Samrah; A. A. Salama. COVID-X: Novel Health-Fog Framework Based on
Neutrosophic Classifier for Confrontation Covid-19
Neutrosophic Sets and Systems, Vol. 55, 0202
6.
15
If output is error, then use back propagation algorithm, and compute error.
∆= 𝑇 − 𝑂
7.
(17)
Update (upper, lower) weights of network by derivation of activation function:
New weight = old weight + (Δ * η *derivative* activation of (input))
(18)
where η is learning rate of model
8.
Repeat 5, 6, 7, 8 and 8.1 until reduction error as possible as.
Step III: Testing phase Classify new sample of objects and determine the accuracy rate of the
model by using relation Accuracy = 1–absolute error, also calculate time consumption in model
processing. The proposed model for neutrosophic algorithms and source codes based on the works
presented in [34-37] and others.
3.2.3 Classification Performance Analysis
In order to evaluate the performance for each deep learning model in the COVID-X, different
metrics have been applied in this study to measure the true and/or misclassification of diagnosed
COVID-19 in the tested X-ray images as follow. First, the cross validation estimator was used and
resulted in a confusion matrix as illustrated in Table 2. The confusion matrix has four expected
outcomes as follows. True Positive (TP) is a number of anomalies and has been identified with the
right diagnosis. True Negative (TN) is an incorrectly measured number of regular instances. False
Positive (FP) is a collection of regular instances that are classified as an anomaly diagnosis FP. False
Negative (FN) is a list of anomalies observed as an ordinary diagnosis [18].
Table 2. Confusion Matrix.
Predicted Positive
True Positive (TP)
False Positive (FP)
Actual Positive
Actual Negative
Predicted Negative
False Negative (FN)
True Negative (TN)
After calculating the values of possible outcomes in the confusion matrix, the following performance metrics
can be calculated.
A) Accuracy: Accuracy is the most important metric for the results of our deep learning classifiers, as given
in (1). It is simply the summation of true positives and true negatives divided by the total values of
confusion matrix components. The most reliable model is the best but it is important to ensure that there
are symmetrical datasets with almost equal false positive values and false adverse values. Therefore, the
above components of the confusion matrix must be calculated to assess the classification quality of our
proposed COVIDX-Net framework.
Accuracy(%) =
𝑇𝑃 + 𝑇𝑁
100%
𝑇𝑃 + 𝐹𝑃 + 𝑇𝑁 + 𝐹𝑁
(19)
B) Precision: Precision is represented in (2) to give relationship between the true positive predicted values
and full positive predicted values.
Precision =
𝑇𝑃
𝑇𝑃 + 𝐹𝑃
(20)
I. Yasser; A. Twakol; A. A. Abd El-Khalek; A. Samrah; A. A. Salama. COVID-X: Novel Health-Fog Framework Based on
Neutrosophic Classifier for Confrontation Covid-19
Neutrosophic Sets and Systems, Vol. 55, 0202
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C) Recall: In (3), recall or sensitivity is the ratio between the true positive values of prediction and the
summation of predicted true positive values and predicted false negative values.
Recall =
𝑇𝑃
𝑇𝑃 + 𝐹𝑁
(21)
D) F1-score: F1-score is an overall measure of the model’s accuracy that combines precision and recall, as
represented in (4). F1-score is the twice of the ratio between the multiplication to the summation of
precision and recall metrics.
F1 − score = 2(
𝑃𝑟𝑒𝑐𝑖𝑠𝑖𝑜𝑛 × 𝑅𝑒𝑐𝑎𝑙𝑙
)
𝑃𝑟𝑒𝑐𝑖𝑠𝑖𝑜𝑛 + 𝑅𝑒𝑐𝑎𝑙𝑙
(22)
3.3 Cloud Layer:
Cloud or data-centres layer is the top layer of the IoT architecture in which enabling
omnipresent, convenient, and proper network access to shared resources (e.g., storage, and services)
over the IoT network. Thus, Cloud perform the heavy services of healthcare data analysis and
processing that Fog cannot perform.
3.3.1 Covid-19 Tracer
Interactive tracker offers users map and graphical displays for COVID-19 disease global spread,
including total confirmed, active, recovered cases, and deaths. The live dashboard pulls data from
the proposed framework as well as the centers for disease control to show all confirmed and
suspected cases of COVID-19, along with recovered patients and deaths. The data is visualized
through a real-time graphic information system (GIS) as shows in Figure 13 [38].
Figure 13. COVID-19 Tracer
3.3.2 Safe Spacer
Limiting face-to-face contact with others is the best way to reduce the spread of coronavirus
disease 2020 (COVID-19). Safe spacer, also called “social distancing,” means keeping space between
yourself and other people outside of your home. The proposed safe spacer was showed in Figure 14.
To practice social or physical distancing using Ultra-wideband technology, Safe Spacer runs
wirelessly on a rechargeable battery and precisely senses when other devices come within 2m/6ft,
I. Yasser; A. Twakol; A. A. Abd El-Khalek; A. Samrah; A. A. Salama. COVID-X: Novel Health-Fog Framework Based on
Neutrosophic Classifier for Confrontation Covid-19
Neutrosophic Sets and Systems, Vol. 55, 0202
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alerting wearers with a choice of visual, vibrating or audio alarm. Each device features a unique ID
tag and built-in memory to optionally associate with workers' names for tracing any unintentional
contact. To maintain high privacy standards, no data except the device's ID and proximity is stored.
For advanced workplace use, an optional iOS/Android app allows human resources or safety
departments to associate IDs to specific workers, log and export daily tracing without collecting
sensitive data, configure the alarms, set custom distance/alert thresholds and more.
Figure 14. The proposed safe spacer
3.3.3 Health System Response Monitor
The COVID-19 Health System Response Monitor (HSRM) assists healthcare organizations and
governments assess patient risk profiles and connects them with virtual care capabilities. It has been
designed in response to the COVID-19 outbreak to collect and organize up-to-date information
responding to the crisis. It focuses primarily on the responses of health systems but also captures
wider public health initiatives. It can be presented the main subsystem in medical system as
following:
Medical analysis subsystem It records the results of the tests for the patients either manually
or automatically by connecting the analytical devices to the system It provides a set of statistics
such as: the number of analyzes required by a particular laboratory in a specific period and the
number of analyzes that have already been done - analyzes of a particular patient divided
according to his medical visits This system is linked to a database that includes all medical
analyzes divided by type (chemistry - hematology - microbiology - immunology - pathology)
and it is related to a set of applications that record the analyzes of each laboratory and the
standard data for these analyzes (Normal Value) according to the kit used in the lab.
Radiology subsystem. It records the data of the examination staff, showing the type of radiation
required for each of them, along with some clinical data about some of the rays, such as CTrays and records the radiology report. It contains a system Picture Archiving and
Communication System (PACS) that links the radiology devices to the system so that the xrays are sent to the x-ray. It provides a set of statistics, such as: the number of radiation
transferred to a particular x-ray department in a specific period, the number of radiation
already done, and the number of x-rays sent. This system is linked to a database that includes
all the rays divided by type (therapeutic - diagnostic) or (ultrasound - CT scan - resonance) and
it is linked to a set of applications that record the radiation of each section and the standard
report for each radiator, as well as determining the work schedule for each section rays.
Medical archive subsystem. It provides a set of statistics, such as: the numbers of patients
attending a specific clinic in a specific period classified by type or age group or geographically
distributed in the governorate, center or city. The system scans patient documents, whether
paper documents or x-ray films, with scanners with special specifications. These documents
I. Yasser; A. Twakol; A. A. Abd El-Khalek; A. Samrah; A. A. Salama. COVID-X: Novel Health-Fog Framework Based on
Neutrosophic Classifier for Confrontation Covid-19
Neutrosophic Sets and Systems, Vol. 55, 0202
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are stored as part of patient data on dedicated servers. The system contains the ability to record
the type of document (x-rays-tests-good checks-surgeries -...) and the document history and
some other data that can be used to create statistics for these documents can be added. The
system contains a special viewer to display these documents with special capabilities for
dealing with these images such as enlarging, reducing or rotating the images. The Digitizer can
be used so that x-ray films are stored in the form of dicom files which is the same format that
x-ray devices output so that they can be viewed through the PACS Viewer.
3.3.4 Medical Mobile Learning subsystem
Medical Mobile Learning (MML) is an unavoidable alternative during COVID-19. It developed
to meet the needs of the education for medical sector, managing all aspects of providing educational,
training and development programs with software that looks after administration, documentation,
tracking, reporting and delivery. MML denote learning involving the use of a mobile device. It has
several advantages and benefits. First, this teaching method can occur at anyplace, anytime, and
anywhere and the learning process is not limited to one particular place. Besides, it allows doctors to
personalize instruction and allow to self-regulate learning. Generally, mobile learning can helps
doctors to develop technological skills, conversational skills, find answers to their questions for any
update for COVID-19, develop a sense of collaboration, allow knowledge sharing, and hence leverage
their learning.
3.3.5 Robotics and Telehealth system
Health systems broadly, to encompass the full continuum between public health (populationbased services) and medical care (delivered to individual patients). When we think about digital
transformation in healthcare, we usually think about some new software doctors are using or a new
medical imaging machine. However, since doctors are now scrambling to contain the COVID-19
pandemic, they have to do so without endangering themselves as well. The proposed robotics and
telehealth system shown in Figure 15. This is where robotics comes in instead of going into the room
to see the patient, a robot goes in, and the doctors operate it via an iPad from the other side of the
door—this digital innovation in healthcare currently being used in hospitals in Washington and other
states. In fact, the robot even has a stethoscope to take the patients’ vitals [39].
Figure 15. The proposed robotics and telehealth system
I. Yasser; A. Twakol; A. A. Abd El-Khalek; A. Samrah; A. A. Salama. COVID-X: Novel Health-Fog Framework Based on
Neutrosophic Classifier for Confrontation Covid-19
Neutrosophic Sets and Systems, Vol. 55, 0202
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4. Limitations
This research is interested in aspects related to Fog computing applied to the healthcare area. In
this sense, this paper focuses on the characteristics of fog computing architectures directly related to
healthcare, disregarding models. This research is limited in availability of data makes it difficult to
process due to the limited hardware availability. Interoperability, data processing, CPU
management, memory and disk resources, and big data issues are still weaknesses in architectures
that require a large number of heterogeneous devices such as healthcare applications.
5. Conclusion and Future Works
Infectious COVID-19 disease shocked the world and is still threating the lives of billions of
people. In this study, a new CVOID-X framework has been proposed to automatically identify or
COVID-19 based on deep learning classifiers. Technological developments like edge computing, fog
computing, IoT, and Big Data have gained importance due to their robustness. In this retrospective
and multi-center study, a deep learning model, COVID-19 detection neural network using
Neautrosophic classifier, was developed to extract visual features from volumetric exams for the
detection of COVID-19. The proposed system facilitates communication between people and medical
centers so that the appropriate COVID-19 patient can be reached just on time. It also integrates the
information scattered among different medical centers and health organizations across the country
to confrontation COVID-19 Stakeholders are able to use the confrontation as an applications installed
on their smartphones or as wearable devices. So the diagnosis of the screening process will be less
time consuming and less human interactions that might cause the spreading of the coronavirus faster.
It can be concluded that the remote sensing procedures, which provide an assortment of ways to
identify, sense, and monitoring of COVID-19, give an awesome promise and potential in order to
fulfil the demands from the healthcare system. As part of the future work, the proposed framework
can be stimulated and analysis the results for every Thing device/object in Edge layer presented in
this work. Moreover, to obtain the most accurate feature which is an essential component of learning,
MobileNet, DenseNet, Xception, ResNet, InceptionV3, InceptionRes- NetV2, VGGNet, NASNet will
be applied amongst a pool of deep convolutional neural networks. Furthermore, the proposed
framework can also be extended towards other important domains of healthcare such as diabetes,
cancer and hepatitis, which can provide efficient services to corresponding patients.
Acknowledgments: We would like to thank Prof. Florentin Smarandache [Department of Mathematics,
University of New Mexico, USA] for helping us to understand Neutrosophic approach. In addition, we like to
show gratitude to Prof. Mohamed A. Mohamed [Dean of the Faculty of Engineering, Mansoura University,
Egypt] for his helping and advising during the research.
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I. Yasser; A. Twakol; A. A. Abd El-Khalek; A. Samrah; A. A. Salama. COVID-X: Novel Health-Fog Framework Based on
Neutrosophic Classifier for Confrontation Covid-19
Neutrosophic Sets and Systems, Vol. 53, 0202
University of New Mexico
Introduction to Decision Making for Neutrosophic
Environment “Study on the Suez Canal Port, Egypt”
A.A. Salama1, Ahmed Sharaf Al-Din2, Issam Abu Al-Qasim3, Rafif Alhabib4, Magdy Badran5
1Dept.
of Math and Computer Sci., Faculty of Science, Port Said Univ., Egypt
drsalama44@gmail.com
2Faculty of Computers and Information, Helwan University, Egypt
3Statistics Department, Faculty of Commerce, Helwan University, Egypt
4Department of Mathematical Statistics, Faculty of Science, Albaath University, Homs, Syria;
rafif.alhabib85@gmail.com
5Business Information Systems, Faculty of Commerce, Helwan University, Egypt.
Abstract: Paper aims to use the programming codes in calculating the values of neutrosophic
grades and their representation in proving the certainty and uncertainty associated with the data
of navigational projects development in the Suez Canal, Egypt. Added to, we reach a more
descriptive of the data in terms of certainty and uncertainty, and that is through the neutrosophic
representation of both the total revenue and the revenues of the Suez Canal from the transit
carriers and ships. Finally, we will present a study of the decision-making process regarding the
better investment in the Suez Canal. Is it investing in the oil tankers or investing in cargo ships,
as this is done based on neutrosophic data. This will be done by studying optimistic, pessimistic,
and remorse entrances to the neutrosophic data, to see which oil tankers or cargo ships offer
better returns to the Canal.
Keywords: Neutrosophic categories; neutrosophic analysis; Neutrosophic data; Suez Canal;
Neutrosophic information models; Decision Making.
1. Introduction
In real-life problems, the data associated are often imprecise, or non-deterministic. Not all real
data can be precise because of their fuzzy nature. Imprecision can be of many types: nonmatching data values, imprecise queries, inconsistent data misaligned schemas, etc. The
fundamental concepts of neutrosophic set, introduced by Smarandache in [2, 3] and Salama et al.
in [2-19]. Decision-making method developed on the accuracy of the information resulting from
the neutrosophic data processing. The data has converted from the classic situation using the
neutrosophic technique, which helps in the process of decision-making. Thus, we can rank all
alternatives and make a better choice according to the degrees of certainty, uncertainty, and
impartiality. Paper is limited to the data for the ships crossing the Suez Canal Port, Egypt, such
as the oil tankers, cargo ships, passenger ships and rescue ships from 1976 to 2019, because they
are considered the most important main types that cross the Suez Canal, due to the nature and
characteristics of each of them, and this requires special attention to that types of ships
A.A. Salama, Ahmed Sharaf Al-Din, Issam Abu Al-Qasim, Rafif Alhabib, Magdy Badran, Introduction to Decision
Making for Neutrosophic Environment “Study on the Suez Canal Port, Egypt”
Neutrosophic Sets and Systems, Vol. 53, 0202
23
1.1 Preliminaries & Related Works
We recollect some relevant basic preliminaries, and in particular, the work of Smarandache in
[2, 3] and Salama et al. [16]. The data was relied on the bulletins of the Suez Canal Authority
Egypt, in [1].
2-
Proposed frameworks
In 2014, Salama et al. [16] designed and implemented an object oriented programming [OOP] to
deal with neutrosophic data operations.
The following are neutrosophic package class, some software algorithms and codes designed
to generate neutrosophic data related to projects for the development of the navigation of the
Suez Canal, Egypt:
1) The following diagram represent the neutrosophic structure.
Truth
Certainty
Falsity
Ambiguity
Neutrosophic
structure
Ignorance
Uncertainty
Contradiction
Neutrality
Saturation
Figure 1. Neutrosophic Data Structure
2)
The following diagram represent the neutrosophic Package
Figure 2: Neutrosophic Package Class Diagram.
3) The first input parameter to the neutrosophic variable has three-neutrosophic components membership
function, indeterminacy and non-membership of data is illustrated in Figure 3.
A.A. Salama, Ahmed Sharaf Al-Din, Issam Abu Al-Qasim, Rafif Alhabib, Magdy Badran, Introduction to Decision
Making for Neutrosophic Environment“Study on the Suez Canal Port, Egypt”
Neutrosophic Sets and Systems, Vol. 53, 0202
24
Figure 3: Neutrosophic Chart .
4) Some Neutrosophic codes
using System;
using System.Collections.Generic;
using System.Linq;
using System.Text;
namespace RibbonCustomize
{
class NeutrosophicValueException:Exception
{
public NeutrosophicValueException()
: base("Neutrosophic value must be between 0 and 1")
{
}
}
class NeutrosophicSet:List<Neutrosophic>
{
public NeutrosophicSet Complement1()
{
NeutrosophicSet complementSet = new NeutrosophicSet();
foreach (Neutrosophic n in this)
{
complementSet.Add(n.Complement1());
}
return complementSet;
}
public NeutrosophicSet Complement2()
{
NeutrosophicSet complementSet = new NeutrosophicSet();
foreach (Neutrosophic n in this)
{
complementSet.Add(n.Complement2());
}
return complementSet;
}
public NeutrosophicSet Complement3()
{
A.A. Salama, Ahmed Sharaf Al-Din, Issam Abu Al-Qasim, Rafif Alhabib, Magdy Badran, Introduction to Decision
Making for Neutrosophic Environment“Study on the Suez Canal Port, Egypt”
Neutrosophic Sets and Systems, Vol. 53, 0202
25
NeutrosophicSet complementSet = new NeutrosophicSet();
foreach (Neutrosophic n in this)
{
complementSet.Add(n.Complement3());
}
return complementSet;
}
public Boolean
BelongTo1(NeutrosophicSet nSet)
{
for (int i = 0; i < this.Count; i++)
{
if (!this[i].BelongTo1(nSet[i]))
return false;
}
return true;
}
public Boolean BelongTo2(NeutrosophicSet nSet)
{
for (int i = 0; i < this.Count; i++)
{
if (!this[i].BelongTo2(nSet[i]))
return false;
}
return true;
}
}
class Neutrosophic
{
double t, i, f;
public Neutrosophic(double t,double i,double f)
{
T = t;
I = i;
F = f;
}
public double T
{
get
{
return Convert.ToDouble( Math.Round( t,4));
}
set
{
if (t < 0 || t > 1)
throw new NeutrosophicValueException();
t = value;
}
A.A. Salama, Ahmed Sharaf Al-Din, Issam Abu Al-Qasim, Rafif Alhabib, Magdy Badran, Introduction to Decision
Making for Neutrosophic Environment“Study on the Suez Canal Port, Egypt”
Neutrosophic Sets and Systems, Vol. 53, 0202
26
3- Neutrosophic Data Related to Projects for the Development of the Navigation
Channel of the Suez Canal
In this section, software algorithms present the values of the neutrosophic grades
(Membership, Indeterminacy, Non-membership) associated with the most important variables
for the waterway development projects are introduced. Which in the future helps in the process
of support and decision-making through the neutrosophic environment. The following tables
represent for neutrosophic fuzzy data related to the development of Suez Canal projects.
3-1 Neutrosophic construction for the revenue of the oil tankers
The following table shows the neutrosophic functions membership, indeterminacy
and non-membership for the revenue from oil tankers
membership
indeterminacy
non-membership
2922840507
2920003.42.
29..7380058
29222544024
29220834444
29..5707.0
29225544348
292240702.
29..8787230
29225300..
29223030.8.
29..4540507
292208045.3
29225457445
29...0.83
2922285.084
292227233
29..45702.0
2922000520.
29225404.24
29..4434724
.9808E-05
292205850.0
29..4023.50
29220243003
292257.8244
29..4204343
29225550440
29225.40803
29..44803.3
29220003480
29220537823
29..707.44.
29220407428
29220402000
29..70.2.04
29222...50
2922042.240
29...2248.
29222042440
29222..530
29..403...7
29220745444
29220482225
29.447820.0
A.A. Salama, Ahmed Sharaf Al-Din, Issam Abu Al-Qasim, Rafif Alhabib, Magdy Badran, Introduction to Decision
Making for Neutrosophic Environment“Study on the Suez Canal Port, Egypt”
Neutrosophic Sets and Systems, Vol. 53, 0202
27
29..4403300
2922254.240
29220078844
29...074..0
.970427E-05
2922240022.
29..7447433
2922283.083
29220500583
29..7823327
29220534374
292203.88.5
29..4.034.3
292224.4470
29225278523
29..4002500
894455E-05
2922077.444
29..432730
292202.453
292208.084
29...548574
29222045323
29222403408
29...2.878.
29222500334
29222.23030
29...004800
2922285405
29222745374
29...445750
2922204048
29222504044
29...040340
292225.0520
2922275485.
29..4033045
29222.45230
29220488707
29..478.300
29222.04284
29220032874
29..7..24.8
29222357578
2922022.524
29..7.22.83
292224244.4
292202..233
29..480043
29222727755
2922034703
29..4008745
29220077433
29220443007
29..407833.
29220548433
29220703880
29..4304758
2922244040.
29220845044
29..403740.
29220404035
29220480040
29..43.4233
29222784.07
29220825.83
A.A. Salama, Ahmed Sharaf Al-Din, Issam Abu Al-Qasim, Rafif Alhabib, Magdy Badran, Introduction to Decision
Making for Neutrosophic Environment“Study on the Suez Canal Port, Egypt”
Neutrosophic Sets and Systems, Vol. 53, 0202
28
29...00543.
5944043E-05
29222474580
29...720048
29222020437
292220.4454
3-2 Neutrosophic construction for the revenue of the casting cargo ships
The following table shows the neutrosophic functions membership, indeterminacy and
non-membership for the revenue for casting cargo ships.
membership
indeterminacy
non-membership
29.7.3.7534
29200000833
29202820480
29...773.4
292220.24..
2922200820
29.47480000
29200055.44
2920003477.
29.447005.0
292283.8440
2920007742.
29...2805.4
09.842.E-05
29222.37420
29..4205483
2922530245.
29225.44533
29..5503080
29220872073
29224448734
29...044.00
29222407007
29222700244
29..7042234
292200.8205
2922070..80
29..745.474
292208583.0
29220042000
29...505024
29222043403
292224444.0
29..45.2005
29222808084
2922042.477
29..4788304
0944420E-05
29220033848
29..4443803
2922040.754
29225508343
29...03..53
29222408544
29222482243
29...54070.
29222403025
29222457040
29..4004354
2922278473
29220440840
A.A. Salama, Ahmed Sharaf Al-Din, Issam Abu Al-Qasim, Rafif Alhabib, Magdy Badran, Introduction to Decision
Making for Neutrosophic Environment“Study on the Suez Canal Port, Egypt”
Neutrosophic Sets and Systems, Vol. 53, 0202
29
29...240054
292224850.4
29222.04740
29..4747504
29222057700
29220050478
29..4002800
2922253455
2922544.37.
29..4008253
292204.3485
29225743.43
29..7400348
29220774253
29220577804
29..7.70384
29220735444
29220204838
29..4023343
2922054.225
292257.8853
29..754805
29220030474
2922040347
29..757.05
2922244035
2922040277
29...203.3
29222300048
29222.4823
29..4744047
29220200324
29220055455
29...370050
2922207.387
29222807744
29...4083.
29222074434
2922254380
29..44407
2922200087.
292205045
29..4504834
2922243.88.
29220475380
29..40454.5
292220.38.5
29220404027
29...54004
29222534374
2922240470
29...484200
29222584048
29222530.7.
29..4304754
292223748.0
29220845048
29..482.83
29222.03024
292203.233
29..4.07054
494072.E-05
29220240448
A.A. Salama, Ahmed Sharaf Al-Din, Issam Abu Al-Qasim, Rafif Alhabib, Magdy Badran, Introduction to Decision
Making for Neutrosophic Environment“Study on the Suez Canal Port, Egypt”
Neutrosophic Sets and Systems, Vol. 53, 0202
29...80024.
30
098234.E-05
29222374.50
3-3 Neutrosophic construction for the total revenue
The following table shows the neutrosophic functions membership, indeterminacy and
non-membership for the total Revenue.
membership
indeterminacy
non-membership
0.99453
0.68563
0.00547
0.994298
0.258039
0.005702
0.999356
0.694458
0.000644
0.998015
0.759271
0.001985
0.998491
0.182653
0.001509
0.99755
0.842113
0.00245
0.999763
0.4457
0.000237
0.99826
0.747169
0.00174
0.999419
0.657663
0.000581
0.997668
0.176678
0.002332
0.999044
0.469494
0.000956
0.998904
0.408643
0.001096
0.999146
0.936579
0.000854
0.999439
0.72134
0.000561
0.998743
0.156889
0.001257
0.999089
0.752516
0.000911
0.999257
0.014837
0.000743
A.A. Salama, Ahmed Sharaf Al-Din, Issam Abu Al-Qasim, Rafif Alhabib, Magdy Badran, Introduction to Decision
Making for Neutrosophic Environment“Study on the Suez Canal Port, Egypt”
Neutrosophic Sets and Systems, Vol. 53, 0202
31
0.999994
0.553359
6.25E-06
0.998467
0.60711
0.001533
0.998461
0.670432
0.001539
0.998631
0.026247
0.001369
0.999688
0.585905
0.000312
0.99864
0.069339
0.00136
0.999698
0.843087
0.000302
0.99874
0.135408
0.00126
0.999383
0.682067
0.000617
0.999497
0.336895
0.000503
0.998713
0.076038
0.001287
0.99899
0.042053
0.00101
0.9991
0.299292
0.0009
0.9996
0.725316
0.0004
0.9998
0.08398
0.0002
0.9999
0.040729
1E-04
0.99994
0.497875
6E-05
0.99997
0.528715
3E-05
0.97811
0.4454
0.02189
0.9888
0.390296
0.0112
0.9999
0.715649
1E-04
A.A. Salama, Ahmed Sharaf Al-Din, Issam Abu Al-Qasim, Rafif Alhabib, Magdy Badran, Introduction to Decision
Making for Neutrosophic Environment“Study on the Suez Canal Port, Egypt”
Neutrosophic Sets and Systems, Vol. 53, 0202
32
0.99999
0.947433
1E-05
0.99999
0.243405
1E-05
0.999996
0.724688
4E-06
0.999998
0.512305
2E-06
0.999999
0.092309
1.5E-06
3-4 Neutrosophic construction for the oil tankers load sizes
The following table shows the neutrosophic functions membership, indeterminacy and
non-membership for the oil tanker load sizes.
membership
indeterminacy
non-membership
0.999999683
1.67646E-07
3.1713E-07
0.999988666
4.86643E-06
1.13338E-05
0.999986858
1.139E-05
1.31419E-05
0.99999763
1.71931E-06
2.37026E-06
0.999998331
7.01436E-08
1.66936E-06
0.999994502
9.67939E-07
5.49785E-06
0.999999965
8.26009E-09
3.45204E-08
0.999995597
3.43245E-06
4.40344E-06
0.999992593
2.66158E-06
7.40721E-06
0.99999205
3.97483E-06
7.95034E-06
0.999995382
2.24479E-06
4.61806E-06
0.999996817
2.23691E-06
3.18252E-06
0.999996942
1.23941E-06
3.05795E-06
0.999994138
5.19437E-06
5.86174E-06
A.A. Salama, Ahmed Sharaf Al-Din, Issam Abu Al-Qasim, Rafif Alhabib, Magdy Badran, Introduction to Decision
Making for Neutrosophic Environment“Study on the Suez Canal Port, Egypt”
Neutrosophic Sets and Systems, Vol. 53, 0202
33
0.999994783
3.67107E-06
5.21695E-06
0.999998407
1.42519E-06
1.59336E-06
0.999999916
1.99228E-08
8.41154E-08
0.999998438
1.13182E-06
1.5621E-06
0.999994327
4.0787E-07
5.67272E-06
0.999994318
6.75615E-07
5.68204E-06
0.999988657
1.12072E-05
1.13432E-05
0.999999955
3.06172E-08
4.47353E-08
0.999997963
3.01945E-07
2.03699E-06
0.999987344
1.14491E-05
1.26559E-05
0.999995549
2.2139E-06
4.45134E-06
0.999992198
5.62777E-08
7.80226E-06
0.999994384
4.76185E-07
5.61551E-06
0.999994926
4.56611E-06
5.07413E-06
0.999994185
2.79521E-06
5.81518E-06
0.999993616
7.78117E-07
6.3837E-06
0.999995882
1.66929E-06
4.11835E-06
0.999996945
2.47541E-06
3.05471E-06
0.999993363
2.91819E-06
6.63674E-06
0.999997621
1.61063E-06
2.3791E-06
0.999994426
5.07718E-06
5.57402E-06
A.A. Salama, Ahmed Sharaf Al-Din, Issam Abu Al-Qasim, Rafif Alhabib, Magdy Badran, Introduction to Decision
Making for Neutrosophic Environment“Study on the Suez Canal Port, Egypt”
Neutrosophic Sets and Systems, Vol. 53, 0202
34
0.999994968
3.88646E-06
5.03216E-06
0.99999355
3.73223E-06
6.44966E-06
0.999998549
6.62626E-07
1.45134E-06
0.999998329
3.01809E-07
1.67063E-06
0.999996023
2.93175E-06
3.97664E-06
0.999995523
2.20011E-07
4.47681E-06
0.999995884
3.7598E-07
4.11594E-06
0.999996923
2.20412E-06
3.07715E-06
3-5 Neutrosophic construction for the sizes of tonnage of the cargo ships casting
The following table shows the neutrosophic functions membership, indeterminacy and
non-membership of the sizes of tonnage for cargo ships casting.
membership
indeterminacy
non-membership
29....42424
0944548E-06
09.0.03E-05
29....73030
4902.35E-06
0984847E-05
29.....4..3
8904.3.E-07
09228.0E-06
29.....3707
0945.35E-06
8907044E-06
29....7.404
0984035E-05
0925400E-05
29....44347
79343.4E-07
0908555E-05
29....48007
09330.4E-05
0934707E-05
29....44220
5973528E-06
090..74E-05
29.....805
09.5805E-07
3974.48E-06
29....40880
0934040E-05
0973374E-05
A.A. Salama, Ahmed Sharaf Al-Din, Issam Abu Al-Qasim, Rafif Alhabib, Magdy Badran, Introduction to Decision
Making for Neutrosophic Environment“Study on the Suez Canal Port, Egypt”
Neutrosophic Sets and Systems, Vol. 53, 0202
35
29......4.8
794.000E-08
0923774E-07
29.....544
89.5007E-06
4950.34E-06
29....4.030
7933705E-06
092787.E-05
29.....5207
092.487E-06
49.703.E-06
29.....4053
0930.08E-06
5944845E-06
29.....8553
095424E-06
3944387E-06
29.....403
09274.0E-06
5978.70E-06
29.....4058
597040.E-06
5944450E-06
29.....74.3
4978004E-07
0902874E-06
29....44487
0983548E-06
0905355E-05
29.....5407
0948743E-06
4907550E-06
29.....5.23
09428.3E-06
492.304E-06
29.....75.4
090847.E-06
094205E-06
29.....0474
09007..E-06
4950803E-06
29.....4545
0975.88E-06
594547.E-06
29......8.
0947220E-07
3902038E-07
29......408
09832.7E-07
5974537E-07
29.....7578
59.4880E-07
094040.E-06
29.....003.
4955507E-07
7978240E-06
29.....833
3940.30E-07
3988.40E-06
29......070
39.80.E-07
7904844E-07
A.A. Salama, Ahmed Sharaf Al-Din, Issam Abu Al-Qasim, Rafif Alhabib, Magdy Badran, Introduction to Decision
Making for Neutrosophic Environment“Study on the Suez Canal Port, Egypt”
Neutrosophic Sets and Systems, Vol. 53, 0202
36
29.....0725
3970023E-06
790.754E-06
29......473
.94448.E-08
090834.E-07
29.....2040
8950548E-06
.940747E-06
29......484
49.4234E-08
5930.0E-07
29.....557.
0942444E-06
494000E-06
29.....5348
395287.E-06
498034E-06
29.....4400
0903440E-06
0954404E-06
3-6 Neutrosophic construction for the total transit ship sizes
The following table shows the neutrosophic functions membership, indeterminacy and nonmembership for the total transit ship sizes
membership
indeterminacy
non-membership
29.....23.0
292403.3485
.982407E-06
29.....5055
29470380.43
4974437E-06
29.....7.4.
29.37.43047
09252.3E-06
29.....40.4
29082440348
59720.7E-06
29.....4005
2934400.43
5947700E-06
29.....8454
294252443.7
3904085E-06
29.....3720
29407.20484
890.7.4E-06
29.....44.
29.27450487
0902.43E-06
29.....4048
2983.077343
5945425E-06
29.....3533
29024482704
894830E-06
29.....3440
294243.480.
8900443E-06
A.A. Salama, Ahmed Sharaf Al-Din, Issam Abu Al-Qasim, Rafif Alhabib, Magdy Badran, Introduction to Decision
Making for Neutrosophic Environment“Study on the Suez Canal Port, Egypt”
Neutrosophic Sets and Systems, Vol. 53, 0202
37
29.....7377
29344457203
0980587E-06
29......747
298345.4440
0955503E-07
29.....400.
297570584..
0944270E-06
29.....4724
29200480878
590.058E-06
29.....4473
29548005030
5900304E-06
29.......34
29483353058
890087.E-08
29.....3.30
2902.232243
89284.4E-06
29.....8045
292.3.2725
3975704E-06
29.....4443
29084.04440
0955840E-06
29.....43.3
290780.5880
5982324E-06
29.....470.
29..0.37..4
0904234E-06
29.....3.35
290404223.0
8928700E-06
29.....477.
294458.227.
090000E-06
29......374
2903.533457
8905.83E-07
29.....7204
29028500..4
09.485E-06
29.....303.
2900400274.
8978233E-06
29......44.
290.7440405
5902408E-07
29......540
290557275.7
4954270E-07
29......700
29272704555
0944584E-07
29.....4704
29440544555
0904544E-06
29......540
294532.4087
490.80.E-07
A.A. Salama, Ahmed Sharaf Al-Din, Issam Abu Al-Qasim, Rafif Alhabib, Magdy Badran, Introduction to Decision
Making for Neutrosophic Environment“Study on the Suez Canal Port, Egypt”
Neutrosophic Sets and Systems, Vol. 53, 0202
38
29......50
29874540344
4942050E-07
29.....4004
29072335833
597480.E-06
29.....7077
29052.24843
09700.3E-06
29.....40.4
29457434730
09420.4E-06
29.....4430
2975808844.
5958448E-06
29......448
29478374447
5903373E-07
4 - Graphic Representation for Data in the Neutrosophic Environment
4-1 Neutrosophic functions of the Suez Canal revenues
The following graph shows the neutrosophic functions membership, indeterminacy and
non-membership of the Suez Canal revenues from oil tankers
1.008
1.006
1.004
1.002
1
0.998
0.996
0.994
0.992
0.99
0.988
0.986
1
3
5
7
9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43
Fig.1, The neutrosophic functions of the Suez Canal revenue collected from oil tankers. (1976:
2019)
4 -2 Neutrosophic functions of the Suez Canal revenue collected from bulk cargo ships
The following graph shows the neutrosophic functions membership, indeterminacy and
non-membership of the Suez Canal revenues received from bulk cargo ships.
A.A. Salama, Ahmed Sharaf Al-Din, Issam Abu Al-Qasim, Rafif Alhabib, Magdy Badran, Introduction to Decision
Making for Neutrosophic Environment“Study on the Suez Canal Port, Egypt”
Neutrosophic Sets and Systems, Vol. 53, 0202
39
1.02
1.01
1
0.99
0.98
0.97
0.96
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43
Fig.2. Neutrosophic functions of the Suez Canal revenue collected from bulk cargo ships.
(1976: 2019)
4 -3 Neutrosophic functions of the Suez Canal revenue (total revenue)
The following graph shows the neutrosophic functions membership, indeterminacy and
non-membership of the Suez Canal total revenue.
1.2
1
0.8
0.6
0.4
0.2
0
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43
Fig.4. Neutrosophic functions of the Suez Canal revenues in million dollars (total revenue).
4 -4 Neutrosophic functions for volumes of shiploads
The following figure shows the neutrosophic functions membership, indeterminacy and
non-membership of tonnage of tankers crossing the channel
A.A. Salama, Ahmed Sharaf Al-Din, Issam Abu Al-Qasim, Rafif Alhabib, Magdy Badran, Introduction to Decision
Making for Neutrosophic Environment“Study on the Suez Canal Port, Egypt”
Neutrosophic Sets and Systems, Vol. 53, 0202
40
1.00003
1.00002
1.00001
1
0.99999
0.99998
0.99997
0.99996
0.99995
0.99994
0.99993
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43
Fig.5. Neutrosophic functions for the volumes of tonnage of tankers crossing the channel.
4 - 5 Neutrosophic functions of tonnage of cargo vessels casting trans-channel
The following figure shows the neutrosophic functions membership, indeterminacy and nonmembership of the tonnage of cargo ships for casting trans-shipment vessels.
1.00004
1.00003
1.00002
1.00001
1
0.99999
0.99998
0.99997
0.99996
0.99995
0.99994
0.99993
1 3 5 7 9 1113151719212325272931333537
1976: 2019
Fig.6. Neutrosophic functions of tonnage of cargo vessels casting trans-channel.
4 - 6 Neutrosophic functions of the tonnage of vessels transiting the channel
The following figure shows the neutrosophic functions membership, indeterminacy and
non-membership of the sizes of tonnage of ships crossing the channel.
A.A. Salama, Ahmed Sharaf Al-Din, Issam Abu Al-Qasim, Rafif Alhabib, Magdy Badran, Introduction to Decision
Making for Neutrosophic Environment“Study on the Suez Canal Port, Egypt”
Neutrosophic Sets and Systems, Vol. 53, 0202
41
1.2
1
0.8
0.6
0.4
0.2
0
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41
Fig.7. Neutrosophic functions of the sizes of tonnage of transit ships.
5 Decision Making for Neutrosophic Environment
Here we will present a study of the decision-making process regarding the better
investment in the Suez Canal. Is it investing in oil tankers or investing in cargo ships, as this is
done based on the previous neutrosophic data. This will be done by studying optimistic,
pessimistic, and remorse entrances to the neutrosophic data, to see which oil tankers or cargo
ships offer better returns to the Canal.
Study of entrances:
i.
The Optimistic entrance:
We know that this entrance depends on evaluating the alternatives, in preparation for choosing
the alternative that guarantees the best possible returns under optimistic natural states. Without
any consideration for the pessimistic cases of this alternative. Which we express by the term
(Max, Max). So that the first "Max" indicates the highest value, and the second "Max" denotes
the optimistic natural state:
Max Max
oil tankers
Max (0.999701164,0.000102857, 0.000298836)
= 0.999701164
cargo ships
Max(0.99977598, 0.000190899, 0.00022402)
= 0.99977598
Thus, according to the optimistic entrance, investing in the cargo ships is the best alternative
considering that it includes the highest possible return, which is (0.99977598).
ii.
The conservative (pessimistic) entrance:
We know that this entrance depends on evaluating alternatives. As a prelude to choosing the
alternative, that guarantees the best possible returns in the light of pessimistic natural states.
A.A. Salama, Ahmed Sharaf Al-Din, Issam Abu Al-Qasim, Rafif Alhabib, Magdy Badran, Introduction to Decision
Making for Neutrosophic Environment“Study on the Suez Canal Port, Egypt”
Neutrosophic Sets and Systems, Vol. 53, 0202
42
Without regard for optimistic cases of that alternative. It is called the term (Max, Min), where
"Max" means the highest value here, but it is related to the second part of the term "Min", which
means the pessimistic natural state:
Max Min
oil tankers
Max (0.988740191 ,0.00461327, 0.011259809)
= 0.988740191
cargo ships
Max(0.979597358, 0.011222455, 0.020402642)
=0.979597358
According to this entrance, investing in oil tankers is the best alternative, as it guarantees the
highest possible return is (0.988740191).
iii.
The entrance to remorse:
This entrance is not optimistic or pessimistic, but rather an intermediate entrance. It depends on
the evaluation of the alternatives as a prelude to choosing the alternative that contains the least
missed opportunities.
Choosing the most appropriate alternative in the light of this entrance requires creating a new
matrix, as follows, we replace the alternative that achieves the highest value with a value of
zero, given that there are no missed opportunities for this alternative.
Highest neutrosophic return
Lowest neutrosophic return
oil tankers
(0.999701164,0.000102857, 0.000298836)
(0.988740191 ,0.00461327, 0.011259809)
cargo ships
(0.99977598, 0.000190899, 0.00022402)
(0.979597358, 0.011222455, 0.020402642)
Highest neutrosophic return
Lowest neutrosophic return
oil tankers
(0.000074816,0.000088042, -0.000074816)
(0,0,0)
cargo ships
(0,0,0)
(0.009142833, -0.006609185, -0.009142833)
We subtract the highest value in the event of high return from the rest of the values present in
this normal state. The same applies to the case of low return, and we subtract the highest value
in the case of low return from the rest of the values found in this case.
Then now we create a short matrix that includes the highest missed opportunity values for each
alternative, as follows:
A.A. Salama, Ahmed Sharaf Al-Din, Issam Abu Al-Qasim, Rafif Alhabib, Magdy Badran, Introduction to Decision
Making for Neutrosophic Environment“Study on the Suez Canal Port, Egypt”
Neutrosophic Sets and Systems, Vol. 53, 0202
43
Missed opportunities
oil tankers
(0.000074816, 0.000088042, -0.000074816)
cargo ships
(0.009142833, -0.006609185, -0.009142833)
Consequently, according to this entrance, the appropriate alternative is oil tankers as it contains
the least missed opportunities.
From the study of the previous three entrances in the light of the neutrosophic logic, we have
different options for decision according to the entrances. This matter we can view positively as
it enriches the decision-making process and is only a reflection of the circumstances of the
decision-maker and the views that affect him.
6. Conclusion and Future Work:
Neutrosophic techniques as a generalization of crisp and fuzzy techniques that may better
model imperfect information, which is omnipresent in any conscious decision making. In
neutrosophic system, each attack is determined by membership, indeterminacy and nonmembership degrees. In this paper, we have designed a program to generate neutrosophic
grades for the most important variables of the waterway of the Suez Canal. In future studies we
will design a statistical model to support and make decisions using the neutrosophic statistics.
The future importance of the research paper is the use of neutrosophic in proposing a model for
optimal decision-making in the neutrosophic environment.
The study aims at the possibility of proposing a general framework to support decision-making
to maximize the profitability of the Suez Canal Authority by crossing ships using the
neutrosophic analysis of navigation traffic data.
This is achieved through a set of objectives, as follows:
1. Neutrosophic analysis through the generation of organic functions with three degrees,
for the navigation traffic in the Suez Canal .
2. Neutrosophic analysis of the numbers and volumes of tonnage of oil tankers transiting
the Suez Canal through neutrosophic data.
3. Studying neutrosophic triple vehicles to predict future tanker and ship volumes .
4. Using the neutrosophic method to predict the value of revenues .
References
1.
Data of the annual bulletins of the Suez Canal Authority, Egypt, for different years (1976 - 2019).
2.
Smarandache Florentin, Neutrosophy and Neutrosophic Logic , First International Conference
on Neutrosophy, Neutrosophic Logic , Set, Probability, and Statistics University of New Mexico,
Gallup, NM 87301, USA(2002).
3.
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Set, Neutrosophic Probability. American Research Press, Rehoboth, NM, 1999.
4.
ELwahsh, H., Gamala, M., Salama, A.A. & El-Henawy, I.M. Modeling Neutrosophic Data by SelfOrganizing Feature Map: MANETs Data Case Study. Procdica Computer, 2017, 121, 152-157.
5.
Salama A.A., Eisa M., ElGhawalby H., Fawzy A.E. (2019) A New Approach in Content-Based
Image Retrieval Neutrosophic Domain9 In: Kahraman C9, Otay İ9 (eds) Fuzzy Multi-criteria DecisionMaking Using Neutrosophic Sets. Studies in Fuzziness and Soft Computing, vol 369 (pp. 361-369)
Springer, Cham
6.
Salama, A. A., and Florentin Smarandache. "Neutrosophic crisp probability theory & decision
making process, Critical Review. Volume XII, 2016, pp 34-48, 2016.
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Making for Neutrosophic Environment“Study on the Suez Canal Port, Egypt”
Neutrosophic Sets and Systems, Vol. 53, 0202
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ElWahsh, H., Gamal, M., Salama, A., & El-Henawy, I. (2018). Intrusion detection system and
neutrosophic theory for MANETs:A comparative study, Neutrosophic Sets and Systems, 23, pp16-22
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Hanafy IM, Salama AA, Mahfouz K. Correlation of neutrosophic Data. International Refereed
Journal of Engineering and Science (IRJES). 2012 Oct;1(2):39-43.I.M.
9.
Hanafy IM, Salama AA, Mahfouz K.," Neutrosophic Classical Events and Its Probability"
International Journal,Computer Applications Research(IJMCAR) Vol.(3),Issue 1,Mar (2013) pp171-178.
10. Salama, A. A., "Neutrosophic Crisp Points & Neutrosophic Crisp Ideals", Neutrosophic Sets and
Systems, Vol.1, No. 1, (2013) pp 50-54.
11.
Alhabib, Rafif; Salama A. A. "The Neutrosophic Time Series-Study Its Models (Linear-
Logarithmic) and test the Coefficients Significance of Its linear model." Neutrosophic Sets and
Systems 33.1 (2020) pp105-115.
12. A. A. Salama, F.Smarandache, Neutrosophic Crisp Set Theory, Educational. Education
Publishing 1313 Chesapeake, Avenue, Columbus, Ohio 43212, (2015).
13. Salama, A. A., Abdelfattah, M., El-Ghareeb, H. A., & Manie, A. M. (2014). Design and
implementation of neutrosophic data operations using object oriented programming. International
Journal of Computer Application, 4(10).
14.
Salama, A. A.; Abdelfattah, M.; Eisa, M. Distances, Hesitancy Degree and Flexible Querying
via Neutrosophic Sets. International Journal of Computer Applications, 0975-8887.2014
15. Salama, A.A: Basic Structure of Some Classes of Neutrosophic Crisp Nearly Open Sets & Possible
Application to GIS Topology, Neutrosophic Sets and Systems, vol. 7, 2015, pp. 18-22.
16. Salama, A.A., El-Ghareeb, H. A., Manie, A. M., & Smarandache, F. Introduction to Develop Some
Software Programs for Dealing with Neutrosophic Sets. Neutrosophic Sets and Systems, (2014).3(1), 8.
17. Alhabib.R, Ranna.M, Farah.H; Salama, A. A, Neutrosophic decision-making & neutrosophic
decision tree. Albaath- University Journal, Vol (02), 2018. (Arabic version).
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Neutrosophic Sets and Systems, 22, 30-38., (2018).
Received: Apr 11, 2020. Accepted: July 2, 2020
A.A. Salama, Ahmed Sharaf Al-Din, Issam Abu Al-Qasim, Rafif Alhabib, Magdy Badran, Introduction to Decision
Making for Neutrosophic Environment“Study on the Suez Canal Port, Egypt”
Neutrosophic Sets and Systems, Vol. 35, 2020
University of New Mexico
Neutrosophic Vague Binary BCK/BCI-algebra
Remya. P. B 1, * and Francina Shalini. A 2
Ph.D Research Scholar, P.G & Research Department of Mathematics, Nirmala College for Women, Affiliated to
Bharathiar University, Red Fields, Coimbatore-18, Tamil Nadu, India ; krish3thulasi@gmail.com
Assistant Professor, P.G & Research Department of Mathematics, Nirmala College for Women, Affiliated to Bharathiar
University, Red Fields, Coimbatore-18, Tamil Nadu, India ; francshalu@g-mail.com
1
2
* Correspondence: krish3thulasi@gmail.com ; Tel.: (91-9751335441)
Abstract: Ineradicable hindrances of the existing mathematical models widespread from
probabilities to soft sets. These difficulties made up way for the opening of “neutrosophic set
model’. Set theory of ‘vague’ values is an already established branch of mathematics. Complex
situations which arose in problem solving, demanded more accurate models. As a result,
‘neutrosophic vague’ came into screen. At present, research works in this area are very few. But it
is on the way of its moves. Algebra and topology are well connected, as algebra and geometry.
So, anything related to geometric topology is equally important in algebraic topology too. Separate
growth of algebra and topology will slow down the development of each branch. And in one sense
it is imperfect! In this paper a new algebraic structure, BCK/BCI is developed for ‘neutrosophic’ and
to ‘neutrosophic vague’ concept with ‘single’ and ‘double’ universe. It’s sub-algebra, different kinds
of ideals and cuts are developed in this paper with suitable examples where necessary. Several
theorems connected to this are also got verified.
Keywords: Vague H - ideal, neutrosophic vague binary BCK/BCI - algebra, neutrosophic vague binary
– subalgebra, neutrosophic vague binary BCK/BCI - ideal, neutrosophic vague binary BCK/BCI
p- ideal, neutrosophic vague binary BCK/BCI q - ideal, neutrosophic vague binary BCK/BCI a-ideal,
neutrosophic vague binary BCK/BCI H - ideal, neutrosophic vague binary BCK/BCI - cut
BCK/BCI
Notations: NVBS : neutrosophic vague binary set, NVBSS : neutrosophic vague binary subset, NVBI :
neutrosophic vague binary ideal, N BCK/BCI - algebra : neutrosophic BCK/BCI-algebra, NV BCK/BCI algebra : neutrosophic vague BCK/BCI-algebra, NVB BCK/BCI - algebra : neutrosophic vague binary
BCK/BCI - algebra, N BCK/BCI - subalgebra : neutrosophic BCK/BCI -
subalgebra, NV BCK/BCI - subalgebra
: neutrosophic vague BCK/BCI - subalgebra, NVB BCK/BCI – subalgebra : neutrosophic vague binary
BCK/BCI
- subalgebra, N BCK/BCI - ideal : neutrosophic BCK/BCI –ideal, NV BCK/BCI - ideal : neutrosophic
vague BCK/BCI - ideal , NVB BCK/BCI- ideal : neutrosophic vague binary BCK/BCI - ideal, NVB BCK/BCI
p-ideal : neutrosophic vague binary BCK/BCI p-ideal, NVB BCK/BCI q - ideal : neutrosophic vague
binary BCK/BCI q - ideal, NVB BCK/BCI a - ideal : neutrosophic vague binary BCK/BCI a - ideal, NVB
BCK/BCI
H
- ideal : neutrosophic vague binary BCK/BCI H - ideal
1. Introduction
Before 1990’s, mathematicians and researchers made use of different mathematical models
for problem solving viz. , Probability theory, Hard set theory, Fuzzy set theory, Rough set theory,
Remya. P.B & Francina Shalini. A, Neutrosophic Vague Binary BCK/BCI-algebra
Neutrosophic Sets and Systems, Vol. 35, 2020
46
Intuitionistic Fuzzy set theory etc., for problem solving. In 1993, W. L. Gau and D. J. Buehrer [16]
introduced vague sets, with “truth and false” membership values as measurement tools. In 1995,
Florentin Smarandache [13] introduced, “Neutrosophic set theory”, in which an additional data
‘uncertainty’, is also got added besides ‘truth and false’. In 2015, Shawkat Alkhazaleh [45] introduced
‘Neutrosophic Vague’ set theory, by inserting vague values, to each neutrosophic value –‘truth,
uncertainty & false’. With its several operations, he gave a rich explanation about the concept, in his
pioneer paper itself. Neutrosophic set’s main difference with Neutrosophic Vague is, with its outlook
as an “interval” (imposed with certain conditions). An algebraic structure is a universal set with a set
of operations applicable to that set, together with a set of axioms to be satisfied. BCK/BCI-logical
algebra- is a new type of algebraic structure developed in 1966, by Yasuyuki Imai and Kiyoshi Is𝑒́ ki
[48]. It is now found to be an active research area. MV-algebras, Boolean algebras etc. are some
t-related logical algebras which extend to BCK-algebra. BCK-algebra further extends to BCI-algebra.
In 2015, Samy M. Mostafa and Reham Ghanem [42] gave cubic structures of medial ideal on
BCI- algebras. Paper introduced cubic medial – ideal, and it illustrates a relation between cubic
medial – ideal and cubic medial BCI – ideal. Homomorphism and Cartesian product of this concept
have been duly verified. In 2017, M. Kaviyarasu and K. Indira [22] gave a review on BCI/BCKalgebras and its developmental scenario. In 1999, Khalid and Ahmad [25] introduced fuzzy H- ideals
in BCI-algebras. In 2007, Kordi and Moussavi [26] gave a detailed study on fuzzy ideals of BCIalgebras. In 2012, Borumand Saeid. A, Prince Williams. D. R and Kuchaki Rafsanjani [10] gave a
preliminary note on anti-fuzzy BCK/BCI-subalgebra. Paper mainly contributed on generalized notion
of fuzzy BCK/BCI-algebra. In 2018, based on hyper fuzzy structure, Young Bae Jun, Seok-Zun Song
and Seon Jeong Kim [50] introduced length - fuzzy subalgebras (length -k-fuzzy; k = 1≤ k≤ 4) in
BCK/BCI- algebras. In 2018, Anas Al-Masarwah and Abd Ghafur Ahmad [4] discussed some
properties of bipolar fuzzy H-ideals in BCK/BCI--algebra. In 2019, Anas Al-Masarwah, Abd Ghafur
Ahmad [5] introduced m-Polar fuzzy subalgebras, m-polar fuzzy closed ideal and m-polar fuzzy
commutative ideal of m-polar fuzzy sets. They also investigated their several characterizations and
theorems. In 2019, Alcheikh. M & Anas Sabouh [3] proved several theorems connected to the already
existing notions of fuzzy ideal, anti-fuzzy ideal and anti-fuzzy p-ideal of BCK-algebra. In 1983,
Hu. Q. P and Li. X [19] defined BCH-algebra as a generalization of BCK/BCI-algebra. In 2001,
Muhammad Anwar Chaudhary and Hafiz Fakhar-ud-din [34] studied some classes of BCH- algebras.
In 1998, Jun. Y. B, Roh. E. H and Kim. H. S [21] introduced BH -algebra as a generalization of
BCH/BCI/BCK-algebra and they discussed it’s ideals and homomorphisms. In 2001, Qun Zhang,
Young Bae Jun and Eun Hwan Roh [41] studied the connection of BH- algebras with ‘BCH’ and
‘BCK/BCI’-algebras. They defined BH1 -algebra and normal BH - algebra. In 1996, Neggers. J and Kim.
H. S [38] introduce d-algebras as a generalization of BCK-algebras and proved that oriented digraphs
correspond to class of edge d-algebras. They also gave several notions of d-algebra with examples
and also defined direct product and direct sum of d-algebras. In 1996, Stanley Gudder [46] introduced
D-algebras as a generalization of D-poset (without assuming a partial order in D poset). He explained
(interval) effect algebras, based on group structure and proved several lemmas and theorems
regarding to this in a deep manner. In 2012, Muhammad Anwar Chaudhry and Faisal Ali [32]
introduced multipliers in d-algebras. He remarked with example that every BCK-algebra is a
d-algebra but the converse does not hold, in general. He defined positive implicative d - algebra and
proved related theorems. In 2005, Akram. M and Dar. K. H [2] defined Fuzzy d-algebras, Fuzzy
d-ideals, Fuzzy d-subalgebras, Fuzzy d-homomorphisms. In 2014, S. R. Barbhulya. K. Dutta.
Choudhury [43] defined (ε, εvq)- fuzzy ideals of d - algebra, it’s cartesian product, homomorphism
and also investigated a few theorems. In 2002, Neggers. J and Kim. H. S [39] introduced B – algebra
which is closely related to BCH/BCI/BCK - algebras. Using a digraph on algebras, they gave a
connection between B - algebras and groups. They also defined commutative B - algebras. In 2006,
BM - algebras are introduced by Kim. C. B and Kim. H. S [24] as a specialization of B - algebras. They
proved BM - algebra as a proper subclass of B - algebras. They showed that BM - algebra is equivalent
to a 0 - commutative B - algebra. In 2011, A. Borumand Saeid and A. Zarandi [11] applied Vague Set
Remya. P.B & Francina Shalini. A, Neutrosophic Vague Binary BCK/BCI-algebra
Neutrosophic Sets and Systems, Vol. 35, 2020
47
theory to BM - algebras and discussed its cuts, Artinian and Noetherian concepts. In 2017, Arsham
Borumand Saeid, Hee Sik Kim and Akbar Rezaei [7] introduced BI - algebras as a generalization of
(dual) implication algebra. They defined ideals and congruence relations in BI - algebras. In 2006, Hee
Sik Kim and Young Hee Kim [18] studied a generalization of BCK – algebras known as BE - algebras.
They discussed its filter and self - distributive property along with some theorems. Various structures
formed within short periods of time, along with smaller or bigger changes are B, BE, BF, BF1 , BF2 ,
BG, BI , BL, BM, BN, BO, BP, BQ, BZ, CI, Coxeter - algebra, FL, FLew (bounded integral commutative
residuated lattice), GK, HW, KU, PS, Q, QS, QP, RG, TM, TP, TU, BCC (or BIK), BCL, BBG, SB𝐿¬ ,
Smarandache BCH - algebra , SU, UP, Z etc. In 2006, group theory of vague sets is introduced by
Hakimuddin Khan, Musheer Ahmad and Ranjit Biswas [17]. In 2008, Lee. K. J, So. K. S and Bang.
K. S [27] introduced vague BCK/BCI- algebras with several theorems and propositions. It was one of
the pioneer work in the area of BCK/BCI - algebraic structure with vague sets. Notion of vague ideals
are introduced with properties. A condition for a vague set to become a vague ideal is also provided.
Several characteristics for vague ideal are investigated and established. Arsham Borumand Saeid [6]
also introduced vague BCK/BCI- algebras in 2008, but his work has been published in 2009. He
discussed on cuts, subalgebras and their related theorems of vague BCK/BCI – algebra. In 2017, Jafari.
A, Mariapresenti. L and Arockiarani. I [20] discussed on vague direct product in BCK- algebra.
In 2002, Neggers. J and Kim Hee Sik [40] introduced, β −algebra as generalization of BCK-algebras.
In 2016, B. Nageswararao, N. Ramakrishna, T. Eswarlal [37] introduced vague β − algebras,
vague β −ideals, translation operators on vague β −algebras, translation operators on vague β −
ideals, vague β −ideal extension of vague β −algebra etc. In 2013, Yun Sun Hwang and Sun Shin
Ahn [52] developed vague p-ideals and vague a-ideals in BCI-algebras. In 2006, neutrosophic
algebraic structures are introduced by Vasantha Kandasamy. W. B and Florentin Smarandache [47].
Neutrosophic group structure, neutrosophic ring structure etc., with lots of theorems and
propositions are investigated. Based on this, in 2015, A. A. A. Agboola and B. Davvaz [1] introduced
neutrosophic BCI/BCK- algebras and their elementary properties. In 2018, Young Bae Jun, Seok - Zun
Song, Florentin Smarandache and Hashem Bordbar [51] discussed neutrosophic quadruple BCK/BCIalgebras. Paper consists of the newly defined definition of neutrosophic quadruple BCK/BCI-number,
neutrosophic quadruple BCK/BCI-ideals etc., with proper verification of inter-connected notions.
In 2019, Muhiuddin. G, Smarandache. F, Young Bae Jun, [35] gave a new idea - neutrosophic
Quadruple ideals in neutrosophic Quadruple BCI- algebras. In 2018, Seon Jeong Kim, Seok-Zun Song
and Young Bae Jun [44] discussed generalizations of neutrosophic subalgebras in BCK/BCI--algebras
based on neutrosophic points. In 2018, Muhammad Akram, Hina Gulzar, Florentin Smarandache and
Saeid Broumi [34] defined single-valued neutrosophic topological K-algebras and investigated some
of the properties like 𝐶5 -connected, super-connected, compact and hausdorff. They also investigated
the image and pre-image of these algebras under homomorphism. In 2018, Young Bae Jun, Florentin
Smarandache, Mehmat AliÖztü rk [49] introduced commutative falling neutrosophic ideals in BCKalgebras. In 2017, Bijan Daavavaz, Samy M. Mostafa and Fatema F. Kareem [8] developed
Neutrosophic ideals of neutrosophic KU - algebras. In 2018, Muhiuddin. G, Bordbar. H,
Smarandache. F, Jun. Y. B, [36] gave certain results on (ε, ε) – neutrosophic subalgebras and ideals in
BCK/BCI- algebras. They defined commutative (ε, ε) neutrosophic ideal and commutative falling
neutrosophic ideal for a BCK-algebra. In 2019, Chul Hwan Park [12] developed neutrosophic ideals
of subtraction algebras. Khadaeman. S, Zahedi. M. M, Borzooei. R. A, Jun. Y. B [23] developed
neutrosophic hyper BCK-ideals in 2019. Neutrosophic sets handle uncertainty in a remarkable way.
But its generalization named as plithogenic set handles uncertainty in a more powerful level than
neutrosophic! Application works using plithogenic are very few as it is a recently introduced work
in set theory. But, in 2019, Mohamed Abdel-Basset and Rehab Mohamed [31], used a plithogenic
TOPSIS-CRISIS method to sustainability supply chain risk management in telecommunication
industry. Problem is well and systematically explained with adequate assistance of diagrams like bar
diagram, pie diagram etc. In 2019, Mohamed Abdel-Basset, Mai Mohamed, Mohamed Elhoseny,
Le Hoang Son, Francisco Chiclana, Abd El-Nasser H Zaied [28] pointed out some draw backs of
Remya. P.B & Francina Shalini. A, Neutrosophic Vague Binary BCK/BCI-algebra
Neutrosophic Sets and Systems, Vol. 35, 2020
48
Dice and Jaccard similarity measures in bipolar neutrosophic set with examples. They provided a
cosine similarity measure and weighted cosine similarity measure methods for ‘bipolar and intervalvalued bipolar’- neutrosophic set. They used the above method for diagnosing bipolar disorder
diseases. A computational algorithm for MADM (Multi Attribute Decision Making) has also given in
the paper. In 2019, using ‘neutrosophic sets’, Mohamed Abdel-Basset, Mumtaz Ali, Asma Atef [30]
framed a resource levelling problem to construction projects. To improve work efficiency and to
minimize cost were underlying principle. For calculating activity durations, trapezoidal neutrosophic
numbers were used in this model. In 2019, Mohamed Abdel-Basset, Mumtaz Ali, Asma Atef [29]
designed uncertainty assessments of linear Time-Cost Tradeoffs using neutrosophic sets. In 2020,
Florentin Smarandache [14] introduced neutro - algebra as a generalization of partial algebra with
examples and showed their differences. Points of odds between universal algebra, neutro - algebra
and anti-algebra are well explained in the paper. Neutro - functions are more useful when range or
domain is not clear. Several applications to neutro functions are given with a well explanation.
In 2020, Bordbar. H, Mohseni Takallo. M, Borzooei. R. A, Young Bae Jun [9], defined BMBJ neutrosophic subalgebra in BCI/BCK – algebras. Authors introduced BMBJ neutrosophic set as a
generalization of neutrosophic set. Its subalgebra, images, translations, S - extension and its
application to BCI/BCK – algebra are defined and explained. Neutrosophic vague binary sets are
developed by Francina Shalini. A and Remya. P. B [15] in 2019. Authors developed a neutrosophic
vague set with 2 universes and discussed its properties.
.
In this paper, BCK/BCI-algebraic structure is introduced to neutrosophic vague binary sets and
it is simply called as neutrosophic vague binary BCK/BCI - algebra. It’s ideal, neutrosophic vague
binary BCK/BCI – ideal is also developed. Moreover, different neutrosophic vague binary BCK/BCIideals like neutrosophic vague binary BCK/BCI p-ideal, neutrosophic vague binary BCK/BCI
q-ideal, neutrosophic vague binary BCK/BCI a-ideal and neutrosophic vague binary BCK/BCI
H-ideal are also developed and compared. Neutrosophic vague binary BCK/BCI - subalgebra,
neutrosophic vague binary BCK/BCI-cut and their relationships, properties and several theorems are
also investigated and illustrated with examples.
Without algebra we can’t even imagine mathematics. In one sense, geometry and algebra are
equally important in mathematics. Even a layman can understand geometry because it deals with
lines and shapes. It’s applicational use in day to day life can’t neglect. But algebra is like a silent
player. In geometry, for finding out the solutions to lot of situations like, to get co-ordinates of
centroid or to find out solution space to equations which represents lines, ellipse, hyperbolas, etc.
- common way is to adopt the method of algebra. Study of surfaces is the main concept behind
topology. Topological objects can bend, twist or stretch but are not allowed to tear, since there it loses
its continuity. As a result, topological objects will become non – topological! Automatically they
admit lack of homeomorphism in these situations. Geometrical nature of topology needs the
assistance of algebra in several circumstances. This inevitable need of a mixed strategy, produced a
new branch of mathematics called ‘algebraic topology’. So developmental moments in any branch
connected to topology from basic sets to neutrosophic sets via “fuzzy, rough, intuitionistic fuzzy,
vague, interval mathematics, soft”- will equally demand the developments of it’s counterpartalgebra. Thus both of them developed equally and produced vivid outputs like fuzzy BCK/BCI
algebra, intuitionistic fuzzy BCK/BCI-algebra, rough BCK/BCI-algebra, vague BCK/BCI- algebra, soft
BCK/BCI algebra and so on. So to stabilize neutrosophic branch, developments in various algebraic
structures like BCK/BCI, BCH, BH etc are very critical and essential. This work will be important to
neutrosophic due to its ‘easy way approach’ than [1] to reach to the same destination.
Method given in [1] is equally good but the concept of generating element is a little bit
perplexing. Since [1] is closely connected to [47], it will be helpful, to verify lot of deep ideas given in
[47]. Neutrosophic ‘group and loop’ concepts are well defined with examples and explanations in
Remya. P.B & Francina Shalini. A, Neutrosophic Vague Binary BCK/BCI-algebra
Neutrosophic Sets and Systems, Vol. 35, 2020
49
[47]. It is to be noted that, as per [47], neutrosophic group does not possess a direct group structure,
but it always contains one! Neutrosophic vague is a mixed form of neutrosophic and vague. It draws
every positives and negatives of both the aforementioned sets. Numerical calculations for
‘neutrosophic’ are more than ‘vague’ due to its additional component - uncertainty. In real life
problems, complex situations demand a more clear and easily accessible method to use with ‘neutrosophic, neutrosophic vague or neutrosophic vague binary’- set values. In group theory or ring
theory algebraic structure is formed in such a manner that it includes set itself as a first member of
the structure, then provide various algebraic operations as example shows : (𝑍, +4 ), (mZ, +), (𝑅, +, . )
etc. Vague BCK/BCI algebraic-structure is defined as (U,∗, 0) by enclosing only universal set and by
omitting the corresponding vague set A. But in this context, universal set and vague set are
simultaneously essential and available: since problem is being to be checked for a ‘vague BCK/BCIalgebra and not for mere BCK/BCI-algebra’ ! Our conclusion is that, being a core object in taking a
decision to the question ‘vague BCK/BCI-algebra or not? ’ : inclusion of vague set, ‘inside the
structure’ is important. It will avoid more confusions while doing theoretical work! Same thing is
referable to fuzzy BCK/BCI-algebra, intuitionistic BCK/BCI-algebra, neutrosophic BCK/BCI-algebra
and so on. This will be useful and applicable to all other existing structures like BCH, BH, B etc., with
uncertain sets.
It is hoped that, when comparing to [1], concept developed in this paper, will be more
useful to common people, since it uses values directly and hence easily accessible. This method
depends on vague BCK/BCI paper [6]. In this paper our primary interest is to develop BCK/BCIalgebraic concept to neutrosophic vague binary sets. For this neutrosophic BCK/BCI algebraic
concept and neutrosophic vague BCK/BCI-algebraic concept are needed as a base. Since it is not
developed yet, in this paper, those are also developed with neutrosophic vague binary!
An
alternative structure approach, to vague BCK/BCI-algebra mentioned in [6] can be given as follows:
A vague BCK/BCI-algebra is a structure 𝔅A = (A, U 𝔅A = (U,∗, 0),∗, 0)=(A, U 𝔅A ,∗, 0) , where A is the
vague set under consideration and U 𝔅A = (U,∗, 0) is the underlying BCK/BCI-algebraic structure
for A with universal set U, binary operation “∗” and with constant “0”. Similarly, when A becomes
fuzzy set, the structure got is fuzzy BCK/BCI-algebraic. For theoretical applications, new approach is
found to be more helpful and clear. Throughout this paper, new structure is used for
neutrosophic/neutrosophic vague/ neutrosophic vague binary BCK/BCI-algebra.
Primary objective of this work is to develop a BCK/BCI-algebraic structure to neutrosophic
vague binary set. Along with, care is taken, to use this novel concept, in ‘theoretical applications’.
Secondary objective is kept as the formation of various ideals to this new concept and their
verification in theory part.
Paper consists of 8 sections. 1st section, provides an introduction, in which literature review
has given. 2nd paragraph gives a general format of the work. 3rd paragraph explains why this work is
essential to neutrosophic branch. 4th paragraph, points out 2 limitations of existing approaches.
5th paragraph mentions the alternative approaches to the limitations. 6th paragraph gives 2 objectives
for the work. 7th paragraph, clearly explains how the paper is organized. 8th paragraph, summarizes
all contributions of this paper in bullets. 2nd section of the paper describes materials for the work. In
3rd, neutrosophic vague binary/ neutrosophic vague/ neutrosophic BCK/BCI -algebras are developed.
In 4th, neutrosophic vague binary BCK/BCI-subalgebra and neutrosophic vague binary BCK/BCIideal are developed. In 5th section, various neutrosophic vague binary BCK/BCI-ideals are formed
Remya. P.B & Francina Shalini. A, Neutrosophic Vague Binary BCK/BCI-algebra
Neutrosophic Sets and Systems, Vol. 35, 2020
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and compared using a table. In 6th section, neutrosophic vague binary/ neutrosophic vague/
neutrosophic BCK/BCI - cuts are defined. In 7th section, propositions and lemmas related to this novel
concept are discussed as a theoretical application. In 8th section, a conclusion to the paper is given.
Contributions in this paper are given in bullets below:
Vague H-ideal
Neutrosophic Vague Binary BCK/BCI-algebra
Neutrosophic Vague Binary BCK/BCI-subalgebra
Neutrosophic Vague Binary BCK/BCI-ideal
Neutrosophic Vague Binary BCK/BCI- p ideal
Neutrosophic Vague Binary BCK/BCI- q ideal
Neutrosophic Vague Binary BCK/BCI- a ideal
Neutrosophic Vague Binary BCK/BCI- H ideal
Neutrosophic vague binary BCK/BCI- cut
2. Preliminaries
Some preliminaries are given in this section
Definition 2.1 [45] (Neutrosophic Vague Set)
A neutrosophic vague set ANV (NVS in short) on the universe of discourse X can be written as
̂A (X) ; ÎA (X), F̂A (X)〉; 𝑥 ∈ 𝑋} whose truth-membership, indeterminacy-membership
ANV = {〈𝑥 ; T
NV
NV
NV
and falsity-membership functions are defined as
̂A (x) = [T − , T + ], ÎA (x) = [I − , I + ] and F̂A (x) = [F − , F + ]
T
NV
NV
NV
where (1) T + = (1− F − ) ; F + = (1− T − ) and
(2) − 0 ≤ T − + I − + F − ≤ 2+
− 0 ≤ T + + I + + F + ≤ 2+
Definition 2.2 [15] (Neutrosophic Vague Binary Set)
A neutrosophic vague binary set (NVBS in short) MNVB over a common universe
{U1 = {xj / 1 ≤ j ≤ n}; U2 = {yk /1 ≤ k ≤ p}} is an object of the form
MNVB = {〈
̂M
̂M
T
(x ), ÎMNVB (xj ), F
(x )
NVB j
NVB j
;
xj
∀ xj ∈ U1 〉 〈
̂M
̂M
T
(y ), ÎMNVB (yk ), F
(y )
NVB k
NVB k
;
yk
∀ yk ∈ U2 〉}
is defined as, (∀ xj ∈ U1 & ∀ yk ∈ U2 )
̂M
̂ M (xj ) = [F − (xj ), F + (xj )] and
T
(xj ) = [T − (xj ), T + (xj )], ÎMNVB (xj )= [I − (xj ), I + (xj )] and F
NVB
NVB
−
+
−
+
̂M
̂ M (yk ) = [F − (yk ), F + (yk )]
T
(yk ) = [T (yk ), T (yk )], ÎMNVB (yk )= [I (yk ), I (yk )] and F
NVB
NVB
where (1) T + (xj ) = 1− F − (xj ); F + (xj ) = 1− T − (xj ) & T + (yk ) = 1− F − (yk ); F + (yk ) = 1− T − (yk )
(2) − 0 ≤ T − (xj )+I − (xj )+F − (xj ) ≤ 2+ ; − 0 ≤ T − (yk)+I − (yk)+F − (yk ) ≤ 2+ or – 0≤ T − (xj )+I − (xj )+F − (xj )+T − (yk )+I − (yk )+F − (yk ) ≤ 4+
− 0≤
T + (xj )+I + (xj )+F + (xj ) ≤ 2+; − 0 ≤ T + (yk )+I + (yk )+F + (yk) ≤ 2+ or − 0 ≤ T + (xj )+I + (xj )+F + (xj )+T + (yk )+I + (yk )+F + (yk) ≤ 4+
−
(3) T (xj ), I − (xj ), F − (xj ) : V(U1 ) ⟶ [0, 1] and T − (yk ), I − (yk ), F − (yk ) : V(U2 ) ⟶ [0, 1]
T + (xj ), I + (xj ), F + (xj ) : V(U1 ) ⟶ [0, 1] and T + (yk ), I + (yk ), F + (yk ) : V(U2 ) ⟶ [0, 1]
Here V(U1 ), V(U2 ) denotes power set of vague sets on U1 , U2 respectively
Definition 2.3 [48] (BCI-algebra)
Let X be a non-empty set with a binary operation ∗ and a constant 0. Then (X, ∗, 0) is called a
BCI-algebra if it satisfies the following conditions:
(i) ((x∗y) ∗(x∗z)) ∗(z∗y) = 0
Remya. P.B & Francina Shalini. A, Neutrosophic Vague Binary BCK/BCI-algebra
Neutrosophic Sets and Systems, Vol. 35, 2020
51
(ii) ((x ∗ (x ∗ y)) ∗ y = 0
(iii) (x ∗ x) = 0
(iv) (x ∗ y) = 0 and (y ∗ x) = 0 imply x = y, for all x, y , z ∈ X
Remark 2.4 [48]
We can define a partial ordering ≤ by x ≤ y if and only if (x ∗ y) = 0
Remark 2.5 [48] (BCK – algebra)
If a BCI-algebra X satisfies (0 ∗ x) = 0 for all x ∈ X, then we say that X is a BCK- algebra
Remark 2.6 [48]
A BCI-algebra X has the following properties:
(i) (x ∗ 0) = x ; ∀ x ∈ X
(ii) (x ∗ y) ∗ z = (x ∗ z) ∗ y ; ∀ x, y, z ∈ X
(iii) 0 ∗ (x ∗ y) = (0 ∗ x) ∗ (0 ∗ y) ; ∀ x, y ∈ X
(iv) x ∗ (x ∗ (x ∗ y)) = (x ∗ y)
; ∀ x, y ∈ X
(v) x ≤ y ⇒ (x ∗ z) ≤ (y ∗ z) ; (z ∗ y) ≤ (z ∗ x) ; ∀ x, y, z ∈ X
(vi) (x ∗ z) ∗ (y ∗ z) ≤ (x ∗ y) ; ∀ x, y, z ∈ X
(vii) 0 ∗ (0 ∗ ((x ∗ z) ∗ (y ∗ z))) = ((0 ∗ y) ∗ (0 ∗ x)) ; ∀ x, y, z ∈ X
(viii) 0 ∗ (0 ∗ (x ∗ y)) = ((0 ∗ y) ∗ (0 ∗ x)) ; ∀ x, y ∈ X
Definition 2.7 [2, 4, 6, 25, 52] (Sub-algebra, Ideal, p-ideal, q-ideal, a-ideal, H-ideal)
Any non-empty subset I of a BCK/BCI- algebra X is called,
- subalgebra /ideal /p-ideal /q-ideal /a-ideal /H-ideal - of X, if it satisfies the axioms given table:
Condition 1
Condition 2
Subalgebra of X
Nil
(x∗ 𝐲) ∈ I ; ∀ x, y ∈ I
BCK/BCI-subalgebra of X
Nil
(x∗ 𝐲) ∈ I ; ∀ x, y ∈ I
µ be a fuzzy BCK/BCI-algebra of X
Nil
A be a vague BCK/BCI-algebra of X
Nil
Ideal of X
0 ∈ I
µ be a fuzzy BCI- ideal of X
µ (0) ≥ µ (x)
BCK/BCI - Ideal of X
0 ∈ I
p - ideal of X
0 ∈ I
q - ideal of X
0 ∈ I
a - ideal of X
0 ∈ I
H - ideal of X
0 ∈ I
µ be a fuzzy BCK/BCI-ideal of X
µ (0) ≥ µ (x)
vague BCI-ideal of X
VA (0) ≥ VA (x)
µ (𝐱 ∗ 𝐲) ≥ 𝐦𝐢𝐧{µ (𝐱) ∗ µ(𝐲)} ; ∀ 𝐱, 𝐲
{𝐢. 𝐞.,
∈ 𝐗; µ 𝐛𝐞 𝐚 𝐟𝐮𝐳𝐳𝐲 𝐬𝐞𝐭 𝐢𝐧 𝐚 𝐁𝐂𝐊/𝐁𝐂𝐈 – 𝐚𝐥𝐠𝐞𝐛𝐫𝐚 𝐗
𝐕𝐀 (𝐱 ∗ 𝐲) ≽ 𝐫 𝐦𝐢𝐧 {𝐕𝐀 (𝐱), 𝐕𝐀 (𝐲)} ; ∀ 𝐱, 𝐲 ∈ 𝐗 ;
𝐭𝐀 (𝐱 ∗ 𝐲) ≥ 𝐦𝐢𝐧 {𝐭𝐀 (𝐱), 𝐭𝐀 (𝐲)}; 𝟏 − 𝐟𝐀 (𝐱 ∗ 𝐲) ≥ 𝐦𝐢𝐧 {𝟏 − 𝐟𝐀 (𝐱), 𝟏 − 𝐟𝐀 (𝐲)}
𝐀 𝐛𝐞 𝐚 𝐯𝐚𝐠𝐮𝐞 𝐬𝐞𝐭 𝐢𝐧 𝐚 𝐁𝐂𝐊/𝐁𝐂𝐈 – 𝐚𝐥𝐠𝐞𝐛𝐫𝐚 𝐗
(x ∗ y) ∈ I & y ∈ I ⇒ x ∈ 𝐈
; ∀ 𝐱 ∈ 𝐈, ∀ 𝐲 ∈ X
µ (𝐱) ≥ 𝐦𝐢𝐧{µ (𝐱 ∗ 𝐲), µ(𝐲)} ; ∀ 𝐱, 𝐲 ∈ 𝐗 ; µ 𝐛𝐞 𝐚 𝐟𝐮𝐳𝐳𝐲 𝐬𝐞𝐭 𝐢𝐧 𝐚 𝐁𝐂𝐈 – 𝐚𝐥𝐠𝐞𝐛𝐫𝐚 𝐗
(x ∗ y) ∈ I & y ∈ I ⇒ x ∈ 𝐈
[(x ∗ z) ∗ (y ∗ z)] ∈ 𝐈 & y ∈ 𝐈 ⇒ x ∈ 𝐈
; ∀ 𝐱 ∈ 𝐈, ∀ 𝐲 ∈ X
; ∀ 𝐱, 𝐲, 𝐳 ∈ X
[x ∗ (y ∗ z)] ∈ 𝐈 & y ∈ 𝐈 ⇒ (x ∗ z) ∈ 𝐈 ; ∀ 𝐱, 𝐲, 𝐳 ∈ X
[(x ∗ z) ∗ (0 ∗ y)] ∈ 𝐈 & z ∈ 𝐈 ⇒ (y ∗ x) ∈ 𝐈 ; ∀ 𝐱, 𝐲, 𝐳 ∈ X
[x ∗ (y ∗ z)] ∈ 𝐈 & y ∈ 𝐈 ⇒ (x ∗ z) ∈ 𝐈 ; ∀ 𝐱, 𝐲, 𝐳 ∈ X
µ (x) ≥ min {µ (x ∗ y), µ(y)}; ∀ x, y ∈ X; µ be a fuzzy set in a BCK/BCI – algebra X
VA(x)≥ r min { VA(x ∗ y), VA (y) } ; ∀ x, y ∈ X ; A- vague set in X
Remark 2.8 [6] (r min & r max)
Let D[0, 1] denote the family of all closed sub-intervals of [0, 1]. Now we define the refined minimum
(briefly, r min) and an order “≤” on elements D1 = [a1, b1] and D2 = [a2, b2] of D[0, 1] as : r min (D1 , D2)
= [min {a1, a2}, min{b1, b2}]. Similarly, we can define ≥, = and r max. Then the concept of r min and r
max could be extended to define r inf and r sup of infinite number of elements of D [0, 1]. It is a
known fact that L = {D [0, 1], r inf, r sup, ≤} is a lattice with universal bounds [0, 0] and [1, 1].
Definition 2.9 [6] (Vague–cuts)
Let A be a vague set of a universe X with the true-membership function tA and false-membership
function fA. The (α, β)-cut of the vague set A is a crisp subset A (α, β) of the set X given by
Remya. P.B & Francina Shalini. A, Neutrosophic Vague Binary BCK/BCI-algebra
Neutrosophic Sets and Systems, Vol. 35, 2020
52
A(α, β) = {x ∈ X / VA(x) ≥ [α, β] } where α ≤ β. Clearly, A (0, 0) = X. The (α, β)-cuts are also called vaguecuts of the vague set A.
The α-cut of the vague set A is a crisp subset Aα of the set X given by Aα = A (α, α).
Note that 𝐀𝟎 = X and if α ≥ β then Aβ ⊆ Aα and A (β, α) = Aα.
Equivalently, we can define the α-cut as Aα = {x ∈ X / 𝐭 𝐀 (𝐱) ≥ α}
Definition 2.10 [50]
Given a non-empty set X, let BK(X) and BI(X) denote the collection of all BCK-algebras and all BCIalgebras, respectively. Also, B(X): = BK(X) ∪ BI(X).
For any (X, ∗, 0) ∈ B(X), a fuzzy structure (X, µ) over (X, ∗, 0) is called a
Fuzzy subalgebra of (X, ∗, 0) with type 1 (briefly, 1-fuzzy subalgebra of (X, ∗, 0) if
µ (x ∗ y) ≥ min {µ(x), µ(y)}; ∀ 𝐱, 𝐲 ∈ X
Fuzzy subalgebra of (X, ∗, 0) with type 2 (briefly, 2 - fuzzy subalgebra of (X, ∗, 0) if
µ (x ∗ y) ≤ min {µ(x), µ(y)}; ∀ 𝐱, 𝐲 ∈ X
Fuzzy subalgebra of (X, ∗, 0) with type 3 (briefly, 3-fuzzy subalgebra of (X, ∗, 0) if
µ (x ∗ y) ≥ max {µ(x), µ(y)}; ∀ 𝐱, 𝐲 ∈ X
Fuzzy subalgebra of (X, ∗, 0) with type 4 (briefly, 4-fuzzy subalgebra of (X, ∗, 0)
µ (x ∗ y) ≤ max {µ(x), µ(y)}; ∀ 𝐱, 𝐲 ∈ X
3. Neutrosophic vague binary BCK/BCI-algebra
In this section neutrosophic BCK/BCI-algebra is developed first, based on paper [6]. Neutrosophic
BCK/BCI-algebraic structure developed in this paper is a little bit different from the definition given
in paper [1]. Concept is extended to neutrosophic vague sets and to neutrosophic vague binary sets.
Definition 3.1 (Neutrosophic BCK/BCI-algebra)
A neutrosophic BCK/BCI-algebra is a structure 𝔅MN = (MN , U 𝔅MN = (U,∗ ,0), ∗, 0) = (MN , U 𝔅MN , ∗, 0)
where,
(1) MN is a non-empty neutrosophic set
(2) U 𝔅MN = (U, ∗, 0) is the underlying BCK/BCI- algebraic structure, to the neutrosophic set
MN with a universal set U, a binary operation “∗” & a constant “0” . It satisfies the following
axioms :
(i) ((ux ∗ uy ) ∗(ux ∗ uz )) ∗(uz ∗ uy ) = 0 (ii) ((ux ∗ (ux ∗ uy )) ∗ uy = 0 (iii) (ux ∗ ux ) = 0
(iv) (ux ∗ uy ) = 0 and (uy ∗ ux ) = 0 imply ux = uy , ∀ ux , uy , uz ∈ U (v) (0 ∗ ux ) = 0 ∀ ux ∈ U
(3) “∗” and “0” are taken as defined in (2)
which satisfies the following condition,
NMN (ux ∗ uy ) ≽ r min {NMN (ux ), NMN (uy )}
;
∀ ux , uy ∈ U. That is,
TMN (ux ∗ uy ) ≥ min {TMN (ux ), TMN (uy )} ; IMN (ux ∗ uy ) ≤ max {IMN (ux), IMN (uy )} ; FMN (ux ∗ uy ) ≤ max {FMN (ux ), FMN (uy )}
Definition 3.2. (Neutrosophic vague BCK/BCI-algebra)
A neutrosophic vague BCK/BCI - algebra is a structure,
Remya. P.B & Francina Shalini. A, Neutrosophic Vague Binary BCK/BCI-algebra
Neutrosophic Sets and Systems, Vol. 35, 2020
53
𝔅MNV = (MNV , U 𝔅MNV = (U, ∗, 0) , ∗, 0) = (MNV , U 𝔅MNV , ∗, 0), where
(1) MNV is a non-empty neutrosophic vague set
(2) U 𝔅MNV = (U, ∗, 0) is the underlying BCK/BCI- algebraic structure to the neutrosophic
vague set MNV with a universal set U, a binary operation “∗” & a constant “0” satisfies
the following axioms:
(i) ((ux ∗ uy ) ∗(ux ∗ uz )) ∗(uz ∗ uy ) = 0 (ii) ((ux ∗ (ux ∗ uy )) ∗ uy = 0 (iii) (ux ∗ ux ) = 0
(iv) (ux ∗ uy ) = 0 and (uy ∗ ux ) = 0 imply ux = uy , ∀ ux , uy , uz ∈ U
(v) (0 ∗ ux ) = 0 ∀ ux ∈ U
(3) “∗” and “0” are taken as defined in U 𝔅MNV
which satisfies the following condition,
NVMNV (ux ∗ uy ) ≽ r min {NVMNV (ux ), NVMNV (uy )}
∀ ux , uy ∈ U . That is,
;
̂M (ux ∗ uy ) ≥ min {T
̂M (ux ), T
̂M (uy )} ; ÎM (ux ∗ uy ) ≤ max {ÎM (ux ), ÎM (uy )} ; F̂M (ux ∗ uy ) ≤ max {F̂M (ux ), F̂M (uy )}
T
NV
NV
NV
NV
NV
NV
NV
NV
NV
General Outline
Let U = {0, u1p , u2p , u3p , ----, ----, ukp , ----, ---- uip } be a universal set with algebraic structure
U 𝔅MNV = (U, ∗, 0) where ∗ is the given binary operation and 0 is the constant . Let U 𝔅MNV forms a BCK/BCIalgebra. Corresponding Cayley table is given below:
∗
0
𝐮𝟏𝐩
𝐮𝟐𝐩
----
----
𝐮𝐤𝐩
----
----
𝐮𝐢𝐩
0
0
0
0
0
0
0
----
0
0
𝐮𝟏𝐩
𝐮𝟏𝐩
0
----
----
----
----
----
----
----
𝐮𝟐𝐩
𝐮𝟐𝐩
----
0
----
----
----
----
----
----
----
----
----
----
0
----
----
----
----
----
----
----
----
----
----
0
----
----
----
----
𝐮𝐤𝐩
𝐮𝐤𝐩
----
----
----
----
0
----
----
----
----
----
----
----
----
----
----
0
----
----
----
----
----
----
----
----
----
----
----
----
𝐮𝐢𝐩
𝐮𝐢𝐩
----
----
----
----
----
----
----
0
By taking U as underlying set, form a neutrosophic vague set 𝐌𝐍𝐕 with neutrosophic vague
membership grades, for any 𝐮𝐤𝐩 ∈ U,
[𝛂 , 𝛂 ] ; 𝐮𝐤𝐩 = 𝟎
̂𝐌 (𝐮𝐤𝐩 ) = { 𝟏 𝟐
𝐓
𝐍𝐕
[𝛂𝟑 , 𝛂𝟒 ] ; 𝐮𝐤𝐩 ≠ 𝟎
; 𝐈̂𝐌𝐍𝐕 (𝐮𝐤𝐩 ) = {
[𝛃𝟏 , 𝛃𝟐 ] ; 𝐮𝐤𝐩 = 𝟎
[𝛄 , 𝛄 ] ; 𝐮𝐤𝐩 = 𝟎
̂𝐌 (𝐮𝐤𝐩 ) = { 𝟏 𝟐
;
𝐅
𝐍𝐕
[𝛃𝟑 , 𝛃𝟒 ] ; 𝐮𝐤𝐩 ≠ 𝟎
[𝛄𝟑 , 𝛄𝟒 ] ; 𝐮𝐤𝐩 ≠ 𝟎
∴ Corresponding neutrosophic vague set is,
𝐌𝐍𝐕 = {〈
[𝛂𝟏 ,𝛂𝟐 ],[𝛃𝟏 ,𝛃𝟐 ],[𝛄𝟏 ,𝛄𝟐 ] [𝛂𝟑 ,𝛂𝟒 ],[𝛃𝟑 ,𝛃𝟒 ],[𝛄𝟑 ,𝛄𝟒 ] [𝛂𝟑 ,𝛂𝟒 ],[𝛃𝟑 ,𝛃𝟒 ],[𝛄𝟑 ,𝛄𝟒 ]
𝟎
,
𝐮𝐩𝟏
,
𝐮𝐩𝟐
, −−,
[𝛂𝟑 ,𝛂𝟒 ],[𝛃𝟑 ,𝛃𝟒 ],[𝛄𝟑 ,𝛄𝟒 ]
𝐮𝐩𝐤
, − −,
[𝛂𝟑 ,𝛂𝟒 ],[𝛃𝟑 ,𝛃𝟒 ],[𝛄𝟑 ,𝛄𝟒 ]
𝐮𝐩𝐢
〉}
Algebraic structure 𝕭𝐌𝐍𝐕 = (𝐌𝐍𝐕 , 𝐔 𝕭𝐌𝐍𝐕 , ∗, 0) is called a neutrosophic vague BCK/BCI-algebra if it
satisfies, 𝐍𝐕𝐌𝐍𝐕 (𝐮𝐤𝐩 ∗ 𝐮𝐤𝐪 ) ≽ 𝐫 𝐦𝐢𝐧 {𝐍𝐕𝐌𝐍𝐕 (𝐮𝐤𝐩 ), 𝐍𝐕𝐌𝐍𝐕 (𝐮𝐤𝐪 )} ; ∀ 𝐮𝐤𝐩 , 𝐮𝐤𝐪 ∈ 𝐔
Remark 3.3
Different neutrosophic vague membership grades are also applicable. It is explained in the general
outline of definition 3.4
Remya. P.B & Francina Shalini. A, Neutrosophic Vague Binary BCK/BCI-algebra
Neutrosophic Sets and Systems, Vol. 35, 2020
54
Definition 3.4 (Neutrosophic vague binary BCK/BCI- algebra)
A neutrosophic vague binary BCK/BCI- algebra is a structure,
𝔅MNVB = (MNVB , U 𝔅MNVB = (U, ∗, 0), ∗, 0) = (MNVB , U 𝔅MNVB , ∗, 0), where
(1) MNVB is a non-empty neutrosophic vague binary set
(2) U 𝔅MNVB = (U = {U1 ∪ U2 }, ∗, 0) is the underlying BCK/BCI - algebraic structure to the
neutrosophic vague binary set MNVB with a universal set U = {U1 ∪ U2 } [where U1 and U2 are
universes of MNVB & “ ∪ ” is the usual set-theoretic union], a binary operation “ ∗ ”
&
a constant “0” satisfies the following axioms:
(i) ((ux ∗ uy ) ∗(ux ∗ uz )) ∗(uz ∗ uy ) = 0 (ii) ((ux ∗ (ux ∗ uy )) ∗ uy = 0 (iii) (ux ∗ ux ) = 0
(iv) (ux ∗ uy ) = 0 and (uy ∗ ux ) = 0 imply ux = uy ∀ ux , uy , uz ∈ U (v) (0 ∗ ux ) = 0 ∀ ux ∈ U
(3) “∗” and “0” are same as defined in U 𝔅MNVB
which satisfies the following condition,
NVBMNVB (ux ∗ uy ) ≽ r min {NVBMNVB (ux ), NVBMNVB (uy )} , ∀ ux , uy ∈ U = {U1 ∪ U2 } . That is,
̂M (ux ∗ uy ) ≥ min {T
̂M (ux ), T
̂M (uy )} ; ÎM (ux ∗ uy ) ≤ max {ÎM (ux ), ÎM (uy )} ; F̂M (ux ∗ uy ) ≤ max {F̂M (ux ), F̂M (uy )}
T
NVB
NVB
NVB
NVB
NVB
NVB
NVB
NVB
NVB
Remark 3.5
(i) Every NVB BCK-algebra is NVB BCI–algebra too. Generally, converse not true! (proved: Theorem 7.3).
So distinguishing between structures of these two are important! To denote NVB BCK-algebra,
𝔅MNVB
following structures can be used: 𝔅BCK
, ∗, 0) or simply as 𝔅KMNVB = (MNVB , U𝔅MNVB , ∗, 0) .
MNVB = (MNVB , U
BCK
K
𝔅MNVB
Similarly, to denote NVB BCI – algebra, following structures can be used : 𝔅BCI
, ∗, 0) or simply
MNVB = (MNVB , U
BCI
as 𝔅IMNVB = (MNVB , U𝔅MNVB , ∗, 0).
I
(ii) For NVB BCK algebra, notation for NVB BCK/BCI – algebra, i.e., 𝔅MNVB = (MNVB , U𝔅MNVB , ∗, 0) is used in
this paper instead of using, those given in remark 3.5 (i)
(iii) Similarly structures for:
Neutrosophic :
BCK
K
𝔅MN
N BCK – algebra : 𝔅BCK
, ∗, 0) or 𝔅KMN = (MN , U𝔅MN , ∗, 0) or 𝔅MN = (MN , U𝔅MN , ∗, 0)
MN = (MN , U
BCI
I
𝔅MN
N BCI – algebra : 𝔅BCI
, ∗, 0) or simply as 𝔅IMN = (MN , U𝔅MN , ∗, 0)
MN = (MN , U
Neutrosophic vague:
BCK
K
𝔅MNV
, ∗, 0) or 𝔅KMNV = (MNV , U𝔅MNV , ∗, 0) or 𝔅MNV = (MNV , U𝔅MNV , ∗, 0)
NV BCK – algebra : 𝔅BCK
MNV = (MNV , U
BCI
I
𝔅MNV
NV BCI – algebra : 𝔅BCI
, ∗, 0) or simply as 𝔅IMNV = (MNV , U𝔅MNV , ∗, 0)
MNV = (MNV , U
General Outline
𝐣
Let 𝐔𝟏 = {0, 𝐮𝟏𝐩 , 𝐮𝟐𝐩 , 𝐮𝟑𝐩 , ---------, 𝐮𝐢𝐩 } and 𝐔𝟐 = {0, 𝐮𝟏𝐪 , 𝐮𝟐𝐪 , 𝐮𝟑𝐪 , ---------, 𝐮𝐪 } be two universes under
consideration. Let the combined universe U = {𝐔𝟏 ∪ 𝐔𝟐 } = {0, 𝐮𝟏𝐩 , 𝐮𝟐𝐩 , 𝐮𝟑𝐩 , ----, 𝐮𝐢𝐩 , 𝐮𝟏𝐪 , 𝐮𝟐𝐪 , 𝐮𝟑𝐪 , ----,
𝐣
𝐮𝐪 } = {𝟎, 𝐮𝟏𝐫 , 𝐮𝟐𝐫 , 𝐮𝟑𝐫 , ---------, 𝐮𝐤𝐫 } (obtained by recording once, the common elements) be a set with a
binary operation ∗ and constant 0 . Let 𝐔 𝕭𝐌𝐍𝐕𝐁 = (U = {𝐔𝟏 ∪ 𝐔𝟐 }, ∗, 0) forms a BCK/BCI-algebra. By
Remya. P.B & Francina Shalini. A, Neutrosophic Vague Binary BCK/BCI-algebra
Neutrosophic Sets and Systems, Vol. 35, 2020
55
taking U = {𝐔𝟏 ∪ 𝐔𝟐 } as underlying set, form a neutrosophic vague binary set 𝐌𝐍𝐕𝐁 .
Let neutrosophic vague binary membership grades are as follows:
for any 𝐮𝐤𝐩 ∈ 𝐔𝟐 :
[𝜶𝟎𝟏 , 𝜶𝟎𝟐 ] ; 𝐮𝐤𝐩 = 𝟎
[𝜶𝟏𝟏 , 𝜶𝟏𝟐 ] ; 𝐮𝐤𝐩 = 𝐮𝟏𝐩
[𝛅𝟎𝟏 , 𝛅𝟎𝟐 ] ; 𝐮𝐤𝐪 = 𝟎
[𝛅𝟏𝟏 , 𝛅𝟏𝟐 ] ; 𝐮𝐤𝐪 = 𝐮𝟏𝐪
̂𝐌 (𝐮𝐤𝐩 ) = [𝜶𝟐𝟏 , 𝜶𝟐𝟐 ] ; 𝐮𝐤𝐩 = 𝐮𝟐𝐩
𝐓
𝐍𝐕𝐁
−−−−−
−−−−−
𝒊
𝒊
𝐤
𝐢
{[𝜶𝟏 , 𝜶𝟐 ] ; 𝐮𝐩 = 𝐮𝐩
[𝛃𝟎𝟏 , 𝛃𝟎𝟐 ] ; 𝐮𝐤𝐩 = 𝟎
[𝛃𝟏𝟏 , 𝛃𝟏𝟐 ] ; 𝐮𝐤𝐩 = 𝐮𝟏𝐩
𝟐 𝟐
𝐤
𝟐
̂𝐌 (𝐮𝐤𝐪 ) = [𝛅𝟏 , 𝛅𝟐 ] ; 𝐮𝐪 = 𝐮𝐪
𝐓
𝐍𝐕𝐁
−−−−−
−−−−−
𝒋
𝒋
𝐣
𝐤
{[𝛅𝟏 , 𝛅𝟐 ] ; 𝐮𝐪 = 𝐮𝐪
𝟎
𝟎
[𝛒𝟏 , 𝛒𝟐 ] ; 𝐮𝐤𝐪 = 𝟎
[𝛒𝟏𝟏 , 𝛒𝟏𝟐 ] ; 𝐮𝐤𝐪 = 𝐮𝟏𝐪
𝟐 𝟐
𝐤
𝟐
𝐈̂𝐌𝐍𝐕𝐁 (𝐮𝐤𝐩 ) = [𝛃𝟏 , 𝛃𝟐 ] ; 𝐮𝐩 = 𝐮𝐩
−−−−−
−−−−−
𝒊
𝒊
𝐤
𝐢
{ [𝛃𝟏 , 𝛃𝟐 ] ; 𝐮𝐩 = 𝐮𝐩
𝟎 𝟎
𝐤
[𝛄𝟏 , 𝛄𝟐 ] ; 𝐮𝐩 = 𝟎
& for any 𝐮𝐤𝐪 ∈ 𝐔𝟐 :
[𝛄𝟏𝟏 , 𝛄𝟏𝟐 ] ; 𝐮𝐤𝐩 = 𝐮𝟏𝐩
𝟐 𝟐
𝐤
𝟐
𝐅̂𝐌𝐍𝐕𝐁 (𝐮𝐤𝐩 ) = [𝛄𝟏 , 𝛄𝟐 ] ; 𝐮𝐩 = 𝐮𝐩
−−−−−
−−−−−
𝒊
𝒊
𝐤
𝐢
{
{[𝛄𝟏 , 𝛄𝟐 ] ; 𝐮𝐩 = 𝐮𝐩
𝟐 𝟐
𝐤
𝟐
𝐈̂𝐌𝐍𝐕𝐁 (𝐮𝐤𝐪 ) = [𝛒𝟏 , 𝛒𝟐 ] ; 𝐮𝐪 = 𝐮𝐪
−−−−−
−−−−−
𝒋
𝒋
𝐣
𝐤
{[𝛒𝟏 , 𝛒𝟐 ] ; 𝐮𝐪 = 𝐮𝐪
𝟎
𝟎
𝐤
[𝛝𝟏 , 𝛝𝟐 ] ; 𝐮𝐪 = 𝟎
[𝛝𝟏𝟏 , 𝛝𝟏𝟐 ] ; 𝐮𝐤𝐪 = 𝐮𝟏𝐪
𝟐
𝟐
𝐤
𝟐
𝐅̂𝐌𝐍𝐕𝐁 (𝐮𝐤𝐪 ) = [𝛝𝟏 , 𝛝𝟐 ] ; 𝐮𝐪 = 𝐮𝐪
−−−−−
−−−−−
𝒋
𝒋
𝐣
𝐤
{[𝛝𝟏 , 𝛝𝟐 ] ; 𝐮𝐪 = 𝐮𝐪
{
From this neutrosophic vague binary set 𝐌𝐍𝐕𝐁 , form neutrosophic vague binary membership grade
for U = {𝐔𝟏 ∪ 𝐔𝟐 } = {𝟎, 𝐮𝟏𝐫 , 𝐮𝟐𝐫 , − − −, 𝐮𝐤𝐫 } as :
𝐍𝐕𝐁𝐌𝐍𝐕𝐁 (𝐮𝐤𝐩 = 𝟎) ∪ 𝐍𝐕𝐁𝐌𝐍𝐕𝐁 (𝐮𝐤𝐪 = 𝟎) ; 𝐮𝐤𝐫 ∈ 𝐔𝟏 ; 𝐮𝐤𝐫 ∈ 𝐔𝟐 𝐮𝐤𝐫 = 𝟎
𝐍𝐕𝐁𝐌𝐍𝐕𝐁 (𝐮𝐤𝐩) ; 𝐮𝐤𝐫 ∈ 𝐔𝟏 ; 𝐮𝐤𝐫 ∉ 𝐔𝟐 ; 𝐮𝐤𝐫 ≠ 𝟎
for any 𝐮𝐤𝐫 ∈ 𝐔 : 𝐍𝐕𝐁𝐌𝐍𝐕𝐁 (𝐮𝐤𝐫 ) =
𝐍𝐕𝐁𝐌𝐍𝐕𝐁 (𝐮𝐤𝐪 ) ; 𝐮𝐤𝐫 ∉ 𝐔𝟏 , 𝐮𝐤𝐫 ∈ 𝐔𝟐 ; 𝐮𝐤𝐫 ≠ 𝟎
{
i.e. , for any 𝐮𝐤𝐫 ∈ 𝐔 :
̂𝐌
𝐦𝐚𝐱{𝐓
̂𝐌 (𝐮𝐤𝐫 ) =
𝐓
𝐍𝐕𝐁
𝐍𝐕𝐁
𝐍𝐕𝐁𝐌𝐍𝐕𝐁 (𝐮𝐤𝐩 ) ∪ 𝐍𝐕𝐁𝐌𝐍𝐕𝐁 (𝐮𝐤𝐪 ) ; 𝐮𝐤𝐫 ∈ 𝐔𝟏 ; 𝐮𝐤𝐫 ∈ 𝐔𝟐 ; 𝐮𝐤𝐫 ≠ 𝟎
̂𝐌 (𝐮𝐤𝐪 = 𝟎)} = 𝐦𝐚 𝐱{[𝜶𝟎𝟏 , 𝜶𝟎𝟐 ], [𝛅𝟎𝟏 , 𝛅𝟎𝟐 ]} ; 𝐮𝐤𝐫 ∈ 𝐔𝟏 ; 𝐮𝐤𝐫 ∈ 𝐔𝟐 𝐮𝐤𝐫 = 𝟎
(𝐮𝐤𝐩 = 𝟎), 𝐓
𝐍𝐕𝐁
̂𝐌 (𝐮𝐤𝐩 ) ; 𝐮𝐤𝐫 ∈ 𝐔𝟏 ; 𝐮𝐤𝐫 ∉ 𝐔𝟐 ; 𝐮𝐤𝐫 ≠ 𝟎
𝐓
𝐍𝐕𝐁
̂𝐌 (𝐮𝐤𝐪 ) ; 𝐮𝐤𝐫 ∉ 𝐔𝟏 , 𝐮𝐤𝐫 ∈ 𝐔𝟐 ; 𝐮𝐤𝐫 ≠ 𝟎
𝐓
𝐍𝐕𝐁
{
𝐈̂𝐌𝐍𝐕𝐁 (𝐮𝐤𝐫 ) =
𝐅̂𝐌𝐍𝐕𝐁 (𝐮𝐤𝐫 ) =
{
̂𝐌 (𝐮𝐤𝐩 ), 𝐓
̂𝐌 (𝐮𝐤𝐪 )} ; 𝐮𝐤𝐫 ∈ 𝐔𝟏 ; 𝐮𝐤𝐫 ∈ 𝐔𝟐 ; 𝐮𝐤𝐫 ≠ 𝟎
𝐦𝐚 𝐱{𝐓
𝐍𝐕𝐁
𝐍𝐕𝐁
𝐤
𝐤
̂
̂
𝐦𝐢𝐧{𝐈𝐌𝐍𝐕𝐁 (𝐮𝐩 = 𝟎), 𝐈𝐌𝐍𝐕𝐁 (𝐮𝐪 = 𝟎)} = 𝐦𝐚𝐱{[𝛃𝟎𝟏 , 𝛃𝟎𝟐 ], [𝛒𝟎𝟏 , 𝛒𝟎𝟐 ]} ; 𝐮𝐤𝐫 ∈ 𝐔𝟏 ; 𝐮𝐤𝐫 ∈ 𝐔𝟐 𝐮𝐤𝐫 = 𝟎
𝐈̂𝐌 (𝐮𝐤𝐩) ; 𝐮𝐤𝐫 ∈ 𝐔𝟏 ; 𝐮𝐤𝐫 ∉ 𝐔𝟐 ; 𝐮𝐤𝐫 ≠ 𝟎
𝐍𝐕𝐁
𝐈̂𝐌𝐍𝐕𝐁 (𝐮𝐤𝐪 ) ; 𝐮𝐤𝐫 ∉ 𝐔𝟏 , 𝐮𝐤𝐫 ∈ 𝐔𝟐 ; 𝐮𝐤𝐫 ≠ 𝟎
𝐦𝐢 𝐧{𝐈̂𝐌𝐍𝐕𝐁 (𝐮𝐤𝐩 ), 𝐈̂𝐌𝐍𝐕𝐁 (𝐮𝐤𝐪 )} ; 𝐮𝐤𝐫 ∈ 𝐔𝟏 ; 𝐮𝐤𝐫 ∈ 𝐔𝟐 ; 𝐮𝐤𝐫 ≠ 𝟎
{
𝐤
𝐦𝐢𝐧{𝐅̂𝐌𝐍𝐕𝐁 (𝐮𝐩 = 𝟎), 𝐅̂𝐌𝐍𝐕𝐁 (𝐮𝐤𝐪 = 𝟎)} = 𝐦𝐢𝐧{[𝛃𝟎𝟏 , 𝛃𝟎𝟐 ], [𝛒𝟎𝟏 , 𝛒𝟎𝟐 ]} ; 𝐮𝐤𝐫 ∈ 𝐔𝟏 ; 𝐮𝐤𝐫 ∈ 𝐔𝟐 ; 𝐮𝐤𝐫 = 𝟎
𝐅̂𝐌 (𝐮𝐤𝐩 ) ; 𝐮𝐤𝐫 ∈ 𝐔𝟏 ; 𝐮𝐤𝐫 ∉ 𝐔𝟐 ; 𝐮𝐤𝐫 ≠ 𝟎
𝐍𝐕𝐁
{
𝐅̂𝐌𝐍𝐕𝐁 (𝐮𝐤𝐪 ) ; 𝐮𝐤𝐫 ∉ 𝐔𝟏 , 𝐮𝐤𝐫 ∈ 𝐔𝟐 ; 𝐮𝐤𝐫 ≠ 𝟎
𝐦𝐢 𝐧{𝐅̂𝐌𝐍𝐕𝐁 (𝐮𝐤𝐩 ), 𝐅̂𝐌𝐍𝐕𝐁 (𝐮𝐤𝐪 )} ; 𝐮𝐤𝐫 ∈ 𝐔𝟏 ; 𝐮𝐤𝐫 ∈ 𝐔𝟐 ; 𝐮𝐤𝐫 ≠ 𝟎
Corresponding Cayley table is given by:
∗
0
𝐮𝟏𝐫
𝐮𝟐𝐫
---
𝐮𝐤𝐫
0
0
0
0
0
0
𝐮𝟏𝐫
𝐮𝟏𝐫
0
---
---
---
𝐮𝟐𝐫
𝐮𝟐𝐫
---
0
---
---
---
---
---
---
0
---
𝐮𝐤𝐫
𝐮𝐤𝐫
---
---
---
0
Algebraic structure 𝕭𝐌𝐍𝐕𝐁 = (𝐌𝐍𝐕𝐁 , 𝐔 𝕭𝐌𝐍𝐕𝐁 , ∗, 𝟎) is called a NVB BCK/BCI-algebra, if it satisfies:
𝐍𝐕𝐁𝐌𝐍𝐕𝐁 (𝐮𝐤𝐫 ∗ 𝐮𝐤𝐬 ) ≽ 𝐫 𝐦𝐢𝐧{𝐍𝐕𝐁𝐌𝐍𝐕𝐁 (𝐮𝐤𝐫 ), 𝐍𝐕𝐁𝐌𝐍𝐕𝐁 (𝐮𝐤𝐬 )} ; ∀ 𝐮𝐤𝐫 , 𝐮𝐤𝐬 ∈ 𝐔
̂𝐌 (𝐮𝐤𝐫 ∗ 𝐮𝐤𝐬 ) ≥ 𝐦𝐢𝐧 {𝐓
̂𝐌 (𝐮𝐤𝐫 ), 𝐓
̂𝐌 (𝐮𝐤𝐬 )} ; 𝐈̂𝐌 (𝐮𝐤𝐫 ∗ 𝐮𝐤𝐬 ) ≤ 𝐦𝐚𝐱 {𝐈̂𝐌 (𝐮𝐤𝐫 ), 𝐈̂𝐌 (𝐮𝐤𝐬 )} ; 𝐅̂𝐌 (𝐮𝐤𝐫 ∗ 𝐮𝐤𝐬 ) ≤ 𝐦𝐚𝐱 {𝐅̂𝐌 (𝐮𝐤𝐫 ), 𝐅̂𝐌 (𝐮𝐤𝐬 )}
𝐓
𝐍𝐕𝐁
𝐍𝐕𝐁
𝐍𝐕𝐁
𝐍𝐕𝐁
𝐍𝐕𝐁
𝐍𝐕𝐁
𝐍𝐕𝐁
𝐍𝐕𝐁
𝐍𝐕𝐁
Remya. P.B & Francina Shalini. A, Neutrosophic Vague Binary BCK/BCI-algebra
Neutrosophic Sets and Systems, Vol. 35, 2020
56
Remark 3.6
(i) Neutrosophic vague binary membership grade of common elements of 𝐔𝟏 and 𝐔𝟐 is got by
taking their neutrosophic vague binary union.
For eg., let 𝐔𝟏 = {𝟎, 𝟏} and 𝐔𝟐 = {𝟎, 𝟏, 𝟐} be two universes; ∴ {𝐔𝟏 ∪ 𝐔𝟐 } = {𝟎, 𝟏, 𝟐}; 𝐔𝟏 ∩ 𝐔𝟐 = {𝟎, 𝟏}
𝐔
𝐔
𝐔
∴ 𝐍𝐕𝐁𝐌𝐍𝐕𝐁 (𝟎) = 𝐍𝐕𝐁𝐌𝟏𝐍𝐕𝐁 (𝟎) ∪ 𝐍𝐕𝐁𝐌𝟐𝐍𝐕𝐁 (𝟎) ; 𝐍𝐕𝐁𝐌𝟏𝐍𝐕𝐁 (𝟎) is the neutrosophic vague binary
membership grade of 0 in universe 1. Similarly, to other common elements.
(ii) It is to be noted that, neutrosophic vague binary membership grade of 0 is not same in 𝐔𝟏 , 𝐔𝟐
generally. Similarly, to other common elements!
Example 3.7
Let 𝐔𝟏 = {𝟎, 𝐚} and let 𝐔𝟐 = {𝟎, 𝟏, 𝟐} be the universes under consideration. Combined universe
𝐔 = {𝐔𝟏 ∪ 𝐔𝟐 } = {𝟎, 𝐚, 𝟏, 𝟐} with (𝐔𝟏 ∩ 𝐔𝟐 ) = {0}. Cayley table to the binary operation ∗ for U is
given as:
∗
0
a
1
2
0
0
0
0
0
a
a
0
0
1
1
1
a
0
1
2
2
2
2
0
Clearly, 𝐔 𝕭𝐌𝐍𝐕𝐁 = (𝐔 = {𝐔𝟏 ∪ 𝐔𝟐 }, ∗, 𝟎) is a BCK/BCI-algebra. Let a non-empty neutrosophic
vague binary set 𝐌𝐍𝐕𝐁 with underlying set 𝐔, is given as:
[𝟎.𝟑,𝟎.𝟖],[𝟎.𝟏,𝟎.𝟑],[𝟎.𝟐,𝟎.𝟕] [𝟎.𝟐,𝟎.𝟑],[𝟎.𝟐,𝟎.𝟓],[𝟎.𝟕,𝟎.𝟖]
𝐌𝐍𝐕𝐁 = {〈
⇒
𝟎
,
𝐚
[𝟎.𝟏,𝟎.𝟕],[𝟎.𝟕,𝟎.𝟖],[𝟎.𝟑,𝟎.𝟗] [𝟎.𝟐,𝟎.𝟔],[𝟎.𝟓,𝟎.𝟕],[𝟎.𝟒,𝟎.𝟖] [𝟎.𝟐,𝟎.𝟔],[𝟎.𝟓,𝟎.𝟕],[𝟎.𝟒,𝟎.𝟖]
〉,〈
𝟎
,
,
𝟏
𝟐
〉}
∀ 𝐮𝐤𝐩 ∈ 𝐔𝟏 and ∀ 𝐮𝐤𝐪 ∈ 𝐔𝟐 ,
̂𝐌 (𝐮𝐤𝐩 ) = {
𝐓
𝐍𝐕𝐁
[𝟎. 𝟑, 𝟎. 𝟖] 𝐢𝐟 𝐮𝐤𝐩 = 𝟎
[𝟎. 𝟐, 𝟎. 𝟑] 𝐢𝐟 𝐮𝐤𝐩 = 𝐚 𝐨𝐫 𝐮𝐤𝐩 ≠ 𝟎
̂𝐌 (𝐮𝐤𝐪 ) = {
𝐓
𝐍𝐕𝐁
[𝟎. 𝟏, 𝟎. 𝟕]; 𝐢𝐟 𝐮𝐤𝐪 = 𝟎
[𝟎. 𝟕, 𝟎. 𝟖]; 𝐢𝐟 𝐮𝐤𝐪 = 𝟎
[𝟎. 𝟑, 𝟎. 𝟗]; 𝐢𝐟 𝐮𝐤𝐪 = 𝟎
(𝐮𝐤 ) = {
(𝐮𝐤 ) = {
; 𝐈̂
; 𝐅̂
[𝟎. 𝟐, 𝟎. 𝟔]; 𝐢𝐟 𝐮𝐪𝐤 = {𝟏, 𝟐} 𝐨𝐫 𝐮𝐪𝐤 ≠ 𝟎 𝐌𝐍𝐕𝐁 𝐪
[𝟎. 𝟓, 𝟎. 𝟕]; 𝐢𝐟 𝐮𝐪𝐤 = {𝟏, 𝟐} 𝐨𝐫 𝐮𝐪𝐤 ≠ 𝟎 𝐌𝐍𝐕𝐁 𝐪
[𝟎. 𝟒, 𝟎. 𝟖]; 𝐢𝐟 𝐮𝐪𝐤 = {𝟏, 𝟐} 𝐨𝐫 𝐮𝐪𝐤 ≠ 𝟎
; 𝐈̂𝐌𝐍𝐕𝐁 (𝐮𝐤𝐩 ) = {
[𝟎. 𝟏, 𝟎. 𝟑] 𝐢𝐟 𝐮𝐤𝐩 = 𝟎
[𝟎. 𝟐, 𝟎. 𝟓] 𝐢𝐟 𝐮𝐤𝐩 = 𝐚 𝐨𝐫 𝐮𝐤𝐩 ≠ 𝟎
; 𝐅̂𝐌𝐍𝐕𝐁 (𝐮𝐤𝐩 ) = {
[𝟎. 𝟐, 𝟎. 𝟕] 𝐢𝐟 𝐮𝐤𝐩 = 𝟎
[𝟎. 𝟕, 𝟎. 𝟖] 𝐢𝐟 𝐮𝐤𝐩 = 𝐚 𝐨𝐫 𝐮𝐤𝐩 ≠ 𝟎
∴ 𝐍𝐕𝐁𝐌𝐍𝐕𝐁 (𝟎) = ([𝟎. 𝟑, 𝟎. 𝟖], [𝟎. 𝟏, 𝟎. 𝟑], [𝟎. 𝟐, 𝟎. 𝟕]) ∪ ([𝟎. 𝟏, 𝟎. 𝟕], [𝟎. 𝟕, 𝟎. 𝟖], [𝟎. 𝟑, 𝟎. 𝟗]) =
([𝟎. 𝟑, 𝟎. 𝟖], [𝟎. 𝟏, 𝟎. 𝟑], [𝟎. 𝟐, 𝟎. 𝟕])
𝐍𝐕𝐁𝐌𝐍𝐕𝐁 (𝐚) = ([𝟎. 𝟐, 𝟎. 𝟑], [𝟎. 𝟐, 𝟎. 𝟓], [𝟎. 𝟕, 𝟎. 𝟖])
[since a is not a common element]
𝐍𝐕𝐁𝐌𝐍𝐕𝐁 (𝟏) = 𝐍𝐕𝐁𝐌𝐍𝐕𝐁 (𝟐) = ([𝟎. 𝟐, 𝟎. 𝟔], [𝟎. 𝟓, 𝟎. 𝟕], [𝟎. 𝟒, 𝟎. 𝟖]); [since 1 and 2 are not a common element]
[𝟎. 𝟑, 𝟎. 𝟖], [𝟎. 𝟏, 𝟎. 𝟑], [𝟎. 𝟐, 𝟎. 𝟕] ; 𝐮𝐤𝐫 = 𝟎
; (for any 𝐮𝐤𝐫 ∈ U)
⇒
= { [𝟎. 𝟐, 𝟎. 𝟑], [𝟎. 𝟐, 𝟎. 𝟓], [𝟎. 𝟕, 𝟎. 𝟖] ; 𝐮𝐤𝐫 = {𝐚} 𝐚𝐧𝐝 𝐮𝐤𝐫 ≠ 𝟎
[𝟎. 𝟐, 𝟎. 𝟔], [𝟎. 𝟓, 𝟎. 𝟕], [𝟎. 𝟒, 𝟎. 𝟖] ; 𝐮𝐤𝐫 = {𝟏, 𝟐} 𝐚𝐧𝐝 𝐮𝐤𝐫 ≠ 𝟎
It is clear after verification that, 𝕭𝐌𝐍𝐕𝐁 = (𝐌𝐍𝐕𝐁 , 𝐔 𝕭𝐌𝐍𝐕𝐁 , ∗, 𝟎) is a NVB BCK/BCI- algebra.
Remark. 3.8
(1) If 𝐔𝟏 ⊆ 𝐔𝟐 then 𝐔 = 𝐔𝟐 (2) If 𝐔𝟐 ⊆ 𝐔𝟏 then 𝐔 = 𝐔𝟏
(2) The symbols ≽ and ⋡ does not imply our usual ≥ or ≱
(3) In a Cayley table,
(i) principal diagonal elements of a BCK/BCI-algebra U is always zero, since (𝐱 ∗ 𝐱) = 𝟎, ∀ 𝐱 ∈ 𝐔
(ii) Using the property (𝐱 ∗ 𝟎) = 𝐱 ; ∀ 𝐱 ∈ 𝐔 of BCI- algebra, it is clear that (𝟎 ∗ 𝟎) = 𝟎
Every BCK-algebra is a BCI-algebra. Hence the above is true for BCK-algebra also
(iii) Body of first column of Cayley table for a BCI- algebra will be an exact copy of column of
operands, by using the property (𝐱 ∗ 𝟎) = 𝐱 ∀ 𝐱 ∈ 𝐔. But 𝟏𝐬𝐭 row need not be!
𝐍𝐕𝐁𝐌𝐍𝐕𝐁 (𝐮𝐤𝐫 )
Remya. P.B & Francina Shalini. A, Neutrosophic Vague Binary BCK/BCI-algebra
Neutrosophic Sets and Systems, Vol. 35, 2020
57
(iv) Above is true for a BCK- algebra also, since every BCK- algebra is a BCI-algebra. In addition, for
a BCK- algebra, body of first row takes only 0, using the property (𝟎 ∗ 𝐱) = 𝟎 ; ∀ 𝐱 ∈ 𝐔
Binary Operation ∗
Row of operands (Elements of 𝐔)
Column of operands (Elements of 𝐔)
Body of Cayley table (occupy with elements got after binary operation taken via column vise row operations)
4. Neutrosophic vague binary BCK/BCI-subalgebra & Neutrosophic vague binary BCK/BCI-ideal
In this section N/NV/NVB BCK/BCI- subalgebra (neutrosophic/neutrosophic vague/neutrosophic
vague binary BCK/BCI – subalgebra) & N/NV/NVB BCK/BCI– ideal (neutrosophic/neutrosophic
vague/neutrosophic vague binary BCK/BCI – ideal) are developed. Priority is given for developing
sub-algebraic and ideal concepts to neutrosophic vague binary BCK/BCI- algebra [NVB BCK/BCIalgebra]. For neutrosophic and neutrosophic vague, things are similar.
Definition 4.1 (Neutrosophic vague binary BCK/BCI-subalgebra)
A NVBSS 𝐏𝐍𝐕𝐁 of a NVB BCK/BCI-algebra 𝕭𝐌𝐍𝐕𝐁 = (𝐌𝐍𝐕𝐁 , 𝐔 𝕭𝐌𝐍𝐕𝐁 = (𝐔, ∗, 𝟎), ∗, 𝟎) is called
NVB- BCK/BCI - subalgebra of 𝕭𝐌𝐍𝐕𝐁 if,
𝐍𝐕𝐁𝐏𝐍𝐕𝐁 (𝐮𝐱 ∗ 𝐮𝐲 ) ≽ 𝐫 𝐦𝐢𝐧 {𝐍𝐕𝐁𝐏𝐍𝐕𝐁 (𝐮𝐱 ), 𝐍𝐕𝐁𝐏𝐍𝐕𝐁 (𝐮𝐲 )}
; ∀ 𝐮𝐱 , 𝐮𝐲 ∈ 𝐔
̂𝐏 (𝐮𝐱 ∗ 𝐮𝐲 ) ≥ 𝐦𝐢𝐧{𝐓
̂𝐏 (𝐮𝐱 ), 𝐓
̂𝐏 (𝐮𝐲 )} ; 𝐈̂𝐏 (𝐮𝐱 ∗ 𝐮𝐲 ) ≤ 𝐦𝐚𝐱{𝐈̂𝐏 (𝐮𝐱 ), 𝐈̂𝐏 (𝐮𝐲 )}; 𝐅̂𝐏 (𝐮𝐱 ∗ 𝐮𝐲 ) ≤ 𝐦𝐚𝐱{𝐅̂𝐏 (𝐮𝐱 ), 𝐅̂𝐏 (𝐮𝐲 )}
𝐓
𝐍𝐕𝐁
𝐍𝐕𝐁
𝐍𝐕𝐁
𝐍𝐕𝐁
𝐍𝐕𝐁
𝐍𝐕𝐁
𝐍𝐕𝐁
𝐍𝐕𝐁
𝐍𝐕𝐁
Definition 4.2 (Neutrosophic vague binary BCK/BCI- Ideal)
A non-empty NVBSS 𝐏𝐍𝐕𝐁 of a NVB BCK/BCI-algebra, 𝕭𝐌𝐍𝐕𝐁 = (𝐌𝐍𝐕𝐁 , 𝐔 𝕭𝐌𝐍𝐕𝐁 , ∗, 𝟎) is called a
NVB BCK/BCI- ideal of 𝕭𝐌𝐍𝐕𝐁 if
(i) 𝐍𝐕𝐁𝐏𝐍𝐕𝐁 (𝟎) ≽ 𝐍𝐕𝐁𝐏𝐍𝐕𝐁 (𝐮𝐤 ) ; for any 𝐮𝐤 ∈ 𝐔
̂𝐏 (𝟎) ≥ 𝐓
̂𝐏 (𝐮𝐤); 𝐈̂𝐏 (𝟎) ≤ 𝐈̂𝐏 (𝐮𝐤 ); 𝐅̂𝐏 (𝟎) ≤ 𝐅̂𝐏 (𝐮𝐤 )
i.e. , 𝐓
𝐍𝐕𝐁
𝐍𝐕𝐁
𝐍𝐕𝐁
𝐍𝐕𝐁
𝐍𝐕𝐁
𝐍𝐕𝐁
(ii) 𝐍𝐕𝐁𝐏𝐍𝐕𝐁 (𝐮𝐚 ) ≽ 𝐫 𝐦𝐢𝐧 {𝐍𝐕𝐁𝐏𝐍𝐕𝐁 (𝐮𝐚 ∗ 𝐮𝐛 ), 𝐍𝐕𝐁𝐏𝐍𝐕𝐁 (𝐮𝐛 )} ; for any 𝐮𝐚 , 𝐮𝐛 ∈ 𝐔
̂𝐏 (𝐮𝐚 ∗ 𝐮𝐛 ), 𝐓
̂𝐏 (𝐮𝐛 )}; 𝐈̂𝐏 (𝐮𝐚 ) ≤ 𝐦𝐚𝐱 {𝐈̂𝐏 (𝐮𝐚 ∗ 𝐮𝐛 ), 𝐈̂𝐏 (𝐮𝐛 )} ; 𝐅̂𝐏 (𝐮𝐚 ) ≤ 𝐦𝐚𝐱{𝐅̂𝐏 (𝐮𝐚 ∗ 𝐮𝐛 ), 𝐅̂𝐏 (𝐮𝐛 )}
̂𝐏 (𝐮𝐚 ) ≥ 𝐦𝐢𝐧{𝐓
𝐓
𝐍𝐕𝐁
𝐍𝐕𝐁
𝐍𝐕𝐁
𝐍𝐕𝐁
𝐍𝐕𝐁
𝐍𝐕𝐁
𝐍𝐕𝐁
𝐍𝐕𝐁
𝐍𝐕𝐁
Remark 4.3
For NVB BCK – ideal underlying structure will confine to BCK -algebra and for NVB BCI – ideal
it will confine to BCK -algebra. For different ideals mentioned in definition 5.2, the same principle
follows.
Remark 4.4
Similarly, for neutrosophic and neutrosophic vague. Only difference is with sets 𝐏𝐍 , 𝐏𝐍𝐕 instead of
𝐏𝐍𝐕𝐁 in above definitions taken in order. It is trivial. Moreover, instead of 𝐔 = {𝐔𝟏 ∪ 𝐔𝟐 }, for both
of them U is applied.
5. Various neutrosophic vague binary BCK/BCI-ideals
In this section vague H-ideal is developed first. Then p-ideal, q-ideal, a-ideal and H-ideal are
developed for NVB BCK/BCI-algebra 𝕭𝐌𝐍𝐕𝐁 = (𝐌𝐍𝐕𝐁 , 𝐔 𝕭𝐌𝐍𝐕𝐁 = (𝐔 = {𝐔𝟏 ∪ 𝐔𝟐 }, ∗, 𝟎) , ∗, 𝟎)
Definition 5.1 (Vague 𝐇-ideal)
A vague set A of X is called a vague H - ideal of a BCI – algebra X if it satisfies
(i)
𝐕𝐀 (𝟎) ≽ 𝐕𝐀 (𝐱)
(∀ 𝐱 ∈ 𝐗) ;
Remya. P.B & Francina Shalini. A, Neutrosophic Vague Binary BCK/BCI-algebra
Neutrosophic Sets and Systems, Vol. 35, 2020
58
𝐭 𝐀 (𝟎) ≥ 𝐭 𝐀 (𝐱)
i.e., {
𝟏 − 𝐟𝐀 (𝟎) ≥ 𝟏 − 𝐟𝐀 (𝐱)
(ii)
{
i.e.,
𝐭 𝐀 (𝟎) ≥ 𝐭 𝐀 (𝐱)
𝐟𝐀 (𝟎) ≤ 𝐟𝐀 (𝐱)
and
𝐭 𝐀 (𝐱 ∗ 𝐳) ≥ 𝐦𝐢𝐧 {𝐭 𝐀 (𝐱 ∗ (𝐲 ∗ 𝐳)), 𝐭 𝐀 (𝐲)}
𝐕𝐀 (𝐱 ∗ 𝐳) ≽ 𝐫 𝐦𝐢𝐧{𝐕𝐀 (𝐱 ∗ (𝐲 ∗ 𝐳)), 𝐕𝐀 (𝐲)} ; (∀ 𝐱, 𝐲, 𝐳 ∈ 𝐗); i.e., {𝟏 − 𝐟 (𝐱 ∗ 𝐳) ≥ 𝐦𝐢𝐧{𝟏 − 𝐟 (𝐱 ∗ (𝐲 ∗ 𝐳)), 𝟏 − 𝐟 (𝐲)}
𝐀
𝐀
𝐀
Definition 5.2 (Comparison of different NVB BCK/BCI- ideals)
Let 𝕭𝐌𝐍𝐕𝐁 = (𝐌𝐍𝐕𝐁 , 𝐔 𝕭𝐌𝐍𝐕𝐁 = (𝐔 = {𝐔𝟏 ∪ 𝐔𝟐 }, ∗, 𝟎) , ∗, 𝟎) be a NVB BCK/BCI-algebra. Conditions
for a non-empty NVBSS 𝐏𝐍𝐕𝐁 of 𝕭𝐌𝐍𝐕𝐁 to become a neutrosophic vague binary BCK/BCI - p ideal,
neutrosophic vague binary BCK/BCI - q ideal, neutrosophic vague binary BCK/BCI - a ideal and
neutrosophic vague binary BCK/BCI - H ideal are given in the table below:
Condition (1) ; (∀ 𝐮𝐤 ∈ 𝐔)
Condition (2) ; (for any 𝐮𝐚 , 𝐮𝐛 , 𝐮𝐜 ∈ 𝐔)
𝐍𝐕𝐁𝐏𝐍𝐕𝐁 (𝟎) ≽ 𝐍𝐕𝐁𝐏𝐍𝐕𝐁 (𝐮𝐤 )
𝐍𝐕𝐁𝐏𝐍𝐕𝐁 (𝐮𝐚 ) ≽ 𝐫 𝐦𝐢𝐧{𝐍𝐕𝐁𝐏𝐍𝐕𝐁 ((𝐮𝐚 ∗ 𝐮𝐜 ) ∗ (𝐮𝐛 ∗ 𝐮𝐜 )), 𝐍𝐕𝐁𝐏𝐍𝐕𝐁 (𝐮𝐛 )}
NVB BCK/BCI 𝐪-ideal
𝐍𝐕𝐁𝐏𝐍𝐕𝐁 (𝟎) ≽ 𝐍𝐕𝐁𝐏𝐍𝐕𝐁 (𝐮𝐤 )
𝐍𝐕𝐁𝐏𝐍𝐕𝐁 (𝐮𝐚 ∗ 𝐮𝐜 ) ≽ 𝐫 𝐦𝐢𝐧 {𝐍𝐕𝐁𝐏𝐍𝐕𝐁 ((𝐮𝐚 ∗ (𝐮𝐛 ∗ 𝐮𝐜 ))) , 𝐍𝐕𝐁𝐏𝐍𝐕𝐁 (𝐮𝐛 )}
NVB BCK/BCI 𝐚-ideal
𝐍𝐕𝐁𝐏𝐍𝐕𝐁 (𝟎) ≽ 𝐍𝐕𝐁𝐏𝐍𝐕𝐁 (𝐮𝐤 )
𝐍𝐕𝐁𝐏𝐍𝐕𝐁 (𝐮𝐛 ∗ 𝐮𝐚) ≽ 𝐫 𝐦𝐢𝐧 {𝐍𝐕𝐁𝐏𝐍𝐕𝐁 (((𝐮𝐚 ∗ 𝐮𝐜 ) ∗ (𝟎 ∗ 𝐮𝐛 ))) , 𝐍𝐕𝐁𝐏𝐍𝐕𝐁 (𝐮𝐜 )}
NVB BCK/BCI 𝐇-ideal
𝐍𝐕𝐁𝐏𝐍𝐕𝐁 (𝟎) ≽ 𝐍𝐕𝐁𝐏𝐍𝐕𝐁 (𝐮𝐤 )
𝐍𝐕𝐁𝐏𝐍𝐕𝐁 (𝐮𝐚 ∗ 𝐮𝐜 ) ≽ 𝐫 𝐦𝐢𝐧 {𝐍𝐕𝐁𝐏𝐍𝐕𝐁 ((𝐮𝐚 ∗ (𝐮𝐛 ∗ 𝐮𝐜 ))) , 𝐍𝐕𝐁𝐏𝐍𝐕𝐁 (𝐮𝐛 )}
NVB BCK/BCI 𝐩-ideal
6. Neutrosophic vague binary BCK/BCI - cuts
In this section N BCK/BCI-cut, NV BCK/BCI-cut and NVB BCK/BCI-cut are developed
Definition 6.1 (Neutrosophic BCK/BCI-(𝛂, 𝛃, 𝛄) − cut or Neutrosophic BCK/BCI-cut)
Let the neutrosophic set MN is a N BCK/BCI-algebra with algebraic structure 𝔅MN = (MN , U 𝔅MN ,∗ ,0).
Truth membership function, indeterminacy membership function and false membership function of
MN are TMN , IMN , FMN respectively. A neutrosophic BCK/BCI (α, β, γ) - cut of 𝔅MN is a crisp subset
MN (α,β,γ) of the neutrosophic set MN given by:
MN (α,β,γ) = {uk ∈ U ∕ NMN (uk ) ≽ (α, β, γ) ; with α, β, γ ∈ [0, 1] }
= {uk ∈ U⁄TMN (uk ) ≥ α ; IMN (uk ) ≤ β ; FMN (uk ) ≤ γ ; with α, β, γ ∈ [0, 1] }
Definition 6.2 (Neutrosophic Vague BCK/BCI ([𝜶𝟏 , 𝜶𝟐 ], [𝜷𝟏 , 𝜷𝟐 ], [𝜸𝟏 , 𝜸𝟐 ]) - cut or Neutrosophic Vague BCK/BCI- cut)
Let the neutrosophic vague set MNV is a NV BCK/BCI-algebra with algebraic structure
𝔅MNV = (MNV , U 𝔅MNV = (U ,∗, 0), ∗, 0) . Truth membership function, indeterminacy membership
̂M , ÎM , F̂M respectively. A neutrosophic
function and false membership function of MNV are T
NV
NV
NV
],
[β
],
[γ
])
- cut of 𝔅MNV is a crisp subset MNV ([α ,α ],[β ,β ],[γ ,γ ]) of
vague BCK/BCI ([α1 , α2
1 , β2
1 , γ2
1
2
1
2
1 2
the neutrosophic vague set MNV given by :
MNV ([α ,α ],[β ,β ],[γ ,γ ])
1
2
1
2
1 2
= {u k
∈ U ∕ NVMNV (uk ) ≽ ([α1 , α2 ], [β1 , β2 ], [γ1, γ2 ]) ; where α1 ≤ α2 , β1 ≤ β2 , γ1 ≤ γ2 ; with α1, α2 , β1 , β2 , γ1 , γ2 ∈ [0, 1]}
̂
̂
̂
= { uk ∈ U⁄ T
MNV (uk ) ≥ [α1 , α2 ] ; IMNV (uk ) ≤ [β1 , β2 ]; FMNV (uk ) ≤ [γ1 , γ2 ] } ;
i.e., T − (uk ) ≥ α1 and T + (uk ) ≥ α2 ; I − (uk ) ≤ β1 and I + (uk ) ≤ β2 ; F − (uk ) ≤ γ1 and F + (uk ) ≤ γ2
Definition 6.3 (Neutrosophic Vague Binary BCK/BCI 〈([𝜶𝟏 , 𝜶𝟐 ], [𝜷𝟏 , 𝜷𝟐 ], [𝜸𝟏 , 𝜸𝟐 ]), ([𝜹𝟏 , 𝜹𝟐 ], [𝝆𝟏 , 𝝆𝟐 ], [𝝑𝟏 , 𝝑𝟐 ])〉 - cut
or Neutrosophic Vague Binary BCK/BCI- cut)
Let the NVBS MNVB is a NVB BCK/BCI-algebra with algebraic structure,
𝔅MNVB = (MNVB , U 𝔅MNVB = (U = {U1 ∪ U2 } ,∗, 0), ∗, 0). Truth membership function, indeterminacy
̂M , ÎM , F̂M respectively.
membership function and false membership function of MNVB are T
NVB
NVB
NVB
A neutrosophic vague binary BCK/BCI 〈([α1 , α2 ], [β1 , β2 ], [γ1 , γ2 ]), ([δ1 , δ2 ], [ρ1 , ρ2 ], [ϑ1 , ϑ2 ])〉 - cut of
𝔅MNVB is a crisp subset MNVB 〈([α ,α ],[β ,β ],[γ ,γ ]),([δ ,δ ],[ρ ,ρ ],[ϑ ,ϑ ])〉 of the NVBS MNVB given by :
1
2
1
2
1 2
1 2
1 2
MNVB 〈([α1,α2 ],[β1,β2],[γ1 ,γ2 ]),([δ1,δ2 ],[ρ1,ρ2],[ϑ1 ,ϑ2 ])〉 = {uk ∈ U/ NVBMNVB (uk) ≽ {
1 2
[α1 , α2 ], [β1 , β2 ], [γ1 , γ2 ] ; if uk ∈ U1
[δ1 , δ2 ], [ρ1 , ρ2 ], [ϑ1 , ϑ2 ] ; if uk ∈ U2
[χ1 , χ2 ], [ϕ1 , ϕ2 ], [π1 , π2 ] ; if uk ∈ U1 ⋂U2
max{[α1 , α2 ], [δ1 , δ2 ]} = [𝜒1 , 𝜒2 ] (say); min[β1 , β2 ], [ρ1 , ρ2 ]} = [𝜙1 , 𝜙2 ] (say); min{[γ1 , γ2 ], [ϑ1 , ϑ2 ] = [𝜋1 , 𝜋2 ] (say) ;
Remya. P.B & Francina Shalini. A, Neutrosophic Vague Binary BCK/BCI-algebra
Neutrosophic Sets and Systems, Vol. 35, 2020
59
with α1, α2 , β1 , β2 , γ1 , γ2 , δ1 , δ2 , ρ1, ρ2, ϑ1, ϑ2 , χ1 , χ2 , ϕ1, ϕ2 , π1, π2 ∈ [0, 1]
and α1 ≤ α2 ,
β1 ≤ β2 ,
γ1 ≤ γ2 ; δ1 ≤ δ2 , ρ1 ≤ ρ2 ,
̂M (uk ) ≥ [α1 , α2 ] ;
i.e., T
NVB
̂
TMNVB (uk ) ≥ [δ1 , δ2 ] ;
̂M (uk ) ≥ [χ1 , χ2 ] ;
T
NVB
ϑ1 ≤ ϑ2 , χ1 ≤ χ2 , ϕ1 ≤ ϕ2 ;
ÎMNVB (uk ) ≤ [β1 , β2 ] ;
ÎMNVB (uk ) ≤ [ρ1 , ρ2 ] ;
ÎMNVB (uk ) ≤ [ϕ1 , ϕ2 ] ;
π1 ≤ π2
F̂MNVB (uk ) ≤ [γ1 , γ2 ]
F̂MNVB (uk ) ≤ [ϑ1 , ϑ2 ]
F̂MNVB (uk ) ≤ [π1 , π2 ]
i.e., T − (uk ) ≥ α1 and T + (uk ) ≥ α2 ; I − (uk ) ≤ β1 and I + (uk ) ≤ β2 ; F − (uk ) ≤ γ1 and F + (uk ) ≤ γ2
T − (uk ) ≥ δ1 and T + (uk ) ≥ δ2 ; I − (uk ) ≤ ρ1 and I + (uk ) ≤ ρ2 ; F − (uk ) ≤ ϑ1 and F + (uk ) ≤ ϑ2
T − (uk ) ≥ χ1 and T + (uk ) ≥ χ2 ; I − (uk ) ≤ ϕ1 and I + (uk ) ≤ ϕ2 ; F − (uk ) ≤ π1 and F + (uk ) ≤ π2
Remark 6.4
(i)
(a) MNVB ([0,0], [1,1], [1,1]) = U
(b) MNVB 〈([0,0], [1,1], [1,1]), ([0,0], [1,1], [1,1])〉 = U = {U1 ∪ U2 }
(ii)
If [α1 , α2 ] and [δ1 , δ2 ] coincides; [β1 , β2 ] and [ρ1 , ρ2 ]coincides; [γ1 , γ2 ] and [ϑ1 , ϑ2 ]
coincides, then (〈[α1 , α2 ], [β1 , β2 ], [γ1 , γ2 ]〉, 〈[δ1 , δ2 ], [ρ1 , ρ2 ], [ϑ1 , ϑ2 ]〉) − cuts are called
(〈[α1 , α2 ], [β1 , β2 ], [γ1 , γ2 ]〉, 〈[α1 , α2 ], [β1 , β2 ], [γ1 , γ2 ]〉) - cuts and is denoted by
MNVB ([α ,α ],[β ,β ],[γ ,γ ],[δ ,δ ]) instead of MNVB (〈[α ,α ],[β ,β ],[γ ,γ ]〉,〈[α ,α ],[β ,β ],[γ ,γ ]〉)
1
(iii)
2
1
2
1 2
1 2
1
2
1
2
1 2
1
2
1
2
1 2
If (〈[α1∗ , α2∗ ], [β1∗ , β∗2 ], [γ1∗ , γ∗2 ]〉, 〈[δ1∗ , δ∗2 ], [ρ1∗, ρ∗2 ], [ϑ1∗ , ϑ∗2 ]〉) ≥ (〈[α1 , α2 ], [β1, β2 ], [γ1 , γ2 ]〉, 〈[δ1 , δ2 ], [ρ1, ρ2 ], [ϑ1 , ϑ2 ]〉)
then MNVB (〈[α ,α ],[β ,β ],[γ ,γ ]〉,〈[δ ,δ ],[ρ ,ρ ],[ϑ ,ϑ ]〉) ⊆ MNVB (〈[α∗ ,α∗ ],[β∗ ,β∗ ],[γ∗ ,γ∗ ]〉,〈[δ∗ ,δ∗ ],[ρ∗ ,ρ∗ ],[ϑ∗ ,ϑ∗ ]〉)
1 2
1 2
1 2
1 2
1 2
1 2
1 2
1 2
1 2
1 2
1 2
1 2
7. Application
In this section theoretical application of NVB BCK/BCI algebra is developed. Various theorems and
propositions are found good to this concept.
Lemma 7.1
𝔅BCI
MNVB satisfies:
Every NVB BCI – algebra 𝔅BCI
MNVB of a BCI -algebra U
NVBMNVB (0) ≽ NVBMNVB (uk ) ; ∀ uk ∈ U = {U1 ∪ U2 }
Proof
For a 𝔅BCI
MNVB , underlying BCI - algebraic structure satisfies, (uk ∗ uk ) = 0, ∀ uk ∈ U
[By property (iii)of definition 2.3]
⇒ ∀ uk ∈ U, NVBMNVB (0) = NVBMNVB (uk ∗ uk ) ≽ r min{NVBMNVB (uk ), NVBMNVB (uk )} = NVBMNVB (uk )
[By definition 3.4]
Lemma 7.2
Every 𝔅BCK
MNVB satisfies NVBMNVB (0) ≽ NVBMNVB (uk ) ; ∀ uk ∈ U
Proof
For a 𝔅BCK
MNVB , underlying BCK- algebraic structure satisfies, an additional condition,
[By remark 2.5]
(0∗ uk ) = 0,∀uk ∈ U besides (uk ∗ uk ) = 0, ∀ uk ∈ U ;
⇒ Additional to, NVBMNVB (0) ≽ NVBMNVB (uk ) ; ∀ uk ∈ U [by lemma 7.1], we get,
NVBMNVB (0) = NVBMNVB (0 ∗ uk ) ≽ r min{NVBMNVB (0), NVBMNVB (uk )} ; ∀ uk ∈ U
⇒ NVBMNVB (0) ≽ r min{NVBMNVB (0), NVBMNVB (uk )} ; ∀ uk ∈ U, for 𝔅BCK
MNVB
& r min{NVBMNVB (0), NVBMNVB (uk )} will depend upon the given NVBS MNVB
⇒ NVBMNVB (0) ≽ NVBMNVB (uk ) and NVBMNVB (0) ≽ r min{NVBMNVB (0), NVBMNVB (uk )}
Even if r min{NVBMNVB (0), NVBMNVB (uk )} will depend upon the given NVBS, using lemma 7.1,
NVBMNVB (0) ≽ r min{NVBMNVB (0), NVBMNVB (uk )} will become NVBMNVB (0) ≽ NVBMNVB (uk )
⇒ NVBMNVB (0) ≽ NVBMNVB (uk ) and NVBMNVB (0) ≽ NVBMNVB (uk ) ∀ uk ∈ U
So, combining both, for a 𝔅BCK
MNVB too, NVBMNVB (0) ≽ NVBMNVB (uk ) ; ∀ uk ∈ U
Remark 7.3
BCI
Every 𝔅BCK
MNVB / 𝔅MNVB satisfies: NVBMNVB (0) ≽ NVBMNVB (uk ) ; ∀ uk ∈ U
i.e., Every NVB BCK/BCI – algebra satisfies: NVBMNVB (0) ≽ NVBMNVB (uk ) ; ∀ uk ∈ U
Remya. P.B & Francina Shalini. A, Neutrosophic Vague Binary BCK/BCI-algebra
Neutrosophic Sets and Systems, Vol. 35, 2020
60
Theorem 7.4
BCI
BCI
BCK
Every 𝔅BCK
MNVB is a 𝔅MNVB . But converse not true, generally. i.e., every 𝔅MNVB is not a 𝔅MNVB generally.
Proof
For a fixed universal set U, underlying BCK – algebraic structure of 𝔅BCK
MNVB consists the underlying
BCI
BCK
BCI
BCI – structure of 𝔅MNVB ⇒ Every 𝔅MNVB is 𝔅MNVB . But converse does not hold. It is illustrated with
the case (i) of remark 7.5.
Remark 7.5
Following example illustrates both the cases:
Let U𝟏 = {0} and let U𝟐 = {0, 1} be the universes under consideration. Combined universe
U = {U1 ∪ U2 } = {0, 1} with (U1 ∩ U2 ) = {0}.
∴ Cayley table to the binary operation ∗ for U is given as:
∗
0
1
0
0
1
1
1
0
BCI-algebra [fig (i)]
∗
0
1
0
1
0
1
0
0
BCK/BCI-algebra[fig(ii)]
𝐁𝐂𝐈
Clearly, 𝐔 𝕭𝐌𝐍𝐕𝐁 = (𝐔 = {𝐔𝟏 ∪ 𝐔𝟐 }, ∗ , 𝟎) is a BCI-algebra [fig (i)].
𝐁𝐂𝐊
𝐔 𝕭𝐌𝐍𝐕𝐁 = (𝐔 = {𝐔𝟏 ∪ 𝐔𝟐 }, ∗ , 𝟎) is a BCI-algebra [fig (ii)].
BCK
Case (i) : Example for a 𝔅BCI
MNVB which is a 𝔅MNVB
Let MNVB be a non-empty NVBS with U
MNVB = {〈
[0.1,0.8],[0.1,0.5],[0.2,0.9]
0
〉,〈
𝔅BCI
M
NVB
as underlying algebraic structure:
[0.3,0.7],[0.2,0.4],[0.3,0.7] [0.1,0.4],[0.3,0.5],[0.6,0.9]
,
0
〉} ; Here, (U1 ∩ U2 ) = {0}
1
NVBMNVB (0) = ([0.1, 0.8], [0.1, 0.5], [0.2, 0.9]) ∪ ([0.3, 0.7], [0.2, 0.4], [0.3, 0.7]) = [0.3, 0.8], [0.1, 0.4], [0.2, 0.7]
BCI
BCK
After verification, clearly MNVB is a 𝔅BCI
MNVB . Next question is that, - “whether 𝔅MNVB is a 𝔅MNVB or
not “ ? ∴ Additional condition to be satisfied is that, for a BCK-algebra is, (0 ∗ 1) = 0 from Cayley
table fig (ii). Correspondingly,
NVBMNVB (0– 1) ≽ r min {NVBMNVB (0), NVBMNVB (1)} ⇒ NVBMNVB (0) ≽ r min {NVBMNVB (0), NVBMNVB (1)}
⇒ [0.3, 0.8], [0.1, 0.4], [0.2, 0.7] ≽ r min {[0.3, 0.8], [0.1, 0.4], [0.2, 0.7], [0.1, 0.4], [0.3, 0.5], [0.6, 0.9]}
⇒ [0.3, 0.8], [0.1, 0.4], [0.2, 0.7] ≽ [0.1, 0.4], [0.3, 0.5], [0.6, 0.9]
BCK
Since additional condition got satisfied, 𝔅BCI
MNVB is clearly a 𝔅MNVB .
𝐁𝐂𝐈
𝐁𝐂𝐊
Case (ii) : Example for a 𝕭𝐏𝐍𝐕𝐁 which is not a 𝕭𝐏𝐍𝐕𝐁
Take binary operation and Cayley table as taken in Case (i).
Consider another NVBS 𝐏NVB with same conditions as in case (i)
𝐏NVB = {〈
[𝟎.𝟏,𝟎.𝟓],[𝟎.𝟐,𝟎.𝟓],[𝟎.𝟓,𝟎.𝟗]
𝟎
〉,〈
[𝟎.𝟏,𝟎.𝟔],[𝟎.𝟑,𝟎.𝟑],[𝟎.𝟒,𝟎.𝟗] [𝟎.𝟏,𝟎.𝟕],[𝟎.𝟑,𝟎.𝟒],[𝟎.𝟑,𝟎.𝟗]
𝟎
,
𝟏
〉}
NVBPNVB (0) = ([0.1, 0.5], [0.2, 0.5], [0.5, 0.9]) ∪ ([0.1, 0.6], [0.3, 0.3], [0.4, 0.9]) = [0.1, 0.6], [0.2, 0.3], [0.4, 0.9]
By verification PNVB is a 𝔅BCI
PNVB . But in this case, additional condition not got satisfied:
NVBPNVB (0 ∗ 1) ⋡ r min {NVBPNVB (0), NVBPNVB (1)} [Since, NVBPNVB (0) ⋡ r min {NVBPNVB (0), NVBPNVB (1)}.
Since, [0.1, 0.6], [0.2, 0.3], [0.4, 0.9] ⋡ r min {[0.1, 0.8], [0.2, 0.3], [0.2, 0.9], [0.1, 0.7], [0.3, 0.4], [0.3, 0.9]}
Since, [0.1, 0.6], [0.2, 0.3], [0.4, 0.9] ⋡ [0.1, 0.7], [0.3, 0.4], [0.3, 0.9]]
BCK
In this case, clearly, 𝔅BCI
PNVB is not a 𝔅PNVB
Theorem 7.6
Intersection of two NVB BCK/BCI -algebra remains as a NVB BCK/BCI-algebra itself.
Proof
Let MNVB and PNVB be two NVB BCK/BCI -algebras with structures 𝔅MNVB = (MNVB , U 𝔅MNVB ,∗, 0)
Remya. P.B & Francina Shalini. A, Neutrosophic Vague Binary BCK/BCI-algebra
Neutrosophic Sets and Systems, Vol. 35, 2020
61
and 𝔅PNVB = (PNVB , U 𝔅PNVB ,∗, 0) respectively, with same universal sets U1 and U2 .
So, ∀ u1 , u2 ∈ U, NVB(MNVB ∩PNVB) (u1 ∗ u2 ) = r min{NVBMNVB (u1 ∗ u2 ), NVB𝐏NVB (u1 ∗ u2 )}
≽ r min{r min{NVBMNVB (u1 ), NVBMNVB (u2 )} , r min{NVBPNVB (u1 ), NVBPNVB (u2 )}}
= r min {NVB(MNVB ∩PNVB ) (u1 ), NVB(MNVB ∩PNVB) (u2 ) }
Therefore, NVB(MNVB∩PNVB ) (u1 ∗ u2 ) ≽ r min {NVB(MNVB ∩PNVB) (u1 ), NVB(MNVB ∩PNVB) (u2 ) }
⇒ (MNVB ∩ PNVB ) is also a NVB BCK/BCI - algebra
Proposition 7.7
Every NVB BCI - ideal PNVB of a 𝔅BCI
MNVB satisfies:
(i) ua ≤ ub ⇒ NVBPNVB (ua ) ≽ NVBPNVB (ub ) ; (∀ ua , ub ∈ U)
(ii) NVBPNVB (ua ∗ uc ) ≽ r min {NVBPNVB ((ua ∗ ub ) ∗ uc ), NVBPNVB (ub )}; ∀ ua , ub , uc ∈ U
Proof
(i) Let ua , ub ∈ U be such that ua ≤ ub .
Since PNVB is a NVB BCI - ideal of 𝔅BCI
MNVB
⇒ NVBPNVB (ua ) ≽ r min{NVBPNVB (ua ∗ ub ), NVBPNVB (ub )}, [By condition (2) of definition 4.2]
= r min {NVBPNVB (0), NVBPNVB (ub )} , take (ua ∗ ub ) = 0
= NVBPNVB ((ub )) [By lemma 7.1]
⇒ NVBPNVB (ua ) ≽ NVBPNVB (ub )
(ii) Let PNVB be a NVB BCI - ideal of 𝔅BCI
MNVB
⇒ NVBPNVB (ua ) ≽ r min{NVBPNVB (ua ∗ ub ), NVBPNVB (ub ) } ; ∀ ua , ub ∈ U
⇒ NVBPNVB (ua ∗ uc ) ≽ r min{NVBPNVB ((ua ∗ uc ) ∗ ub ), NVBPNVB (ub ) } ;
[by putting ua = (ua ∗ uc ); ∀ ua , ub , uc ∈ U]
⇒ NVBPNVB (ua ∗ uc ) ≽ r min{NVBPNVB ((ua ∗ ub ) ∗ uc ), NVBPNVB (ub )};[By property (ii)of remark 2.6]
Lemma 7.8
Let PNVB be a NVB BCI-ideal of 𝔅BCI
MNVB . Then, NVBPNVB (0 ∗ (0 ∗ uk )) ≽ NVBPNVB (uk ); ∀ uk ∈ U
Proof
NVBPNVB (ua ) ≽ r min{NVBPNVB (ua ∗ ub ), NVBPNVB (ub )} ; for any ua , ub ∈ U [ By definition 4.2]
Let ua = (0 ∗ (0 ∗ uk )) and ub = uk .
∴ For any uk ∈ U, NVBPNVB (0 ∗ (0 ∗ uk )) ≽ r min {NVBPNVB ((0 ∗ (0 ∗ uk )) ∗ uk ) , NVBPNVB (uk )}
[By property (ii)of remark 2.6]
= r min{NVBPNVB ((0 ∗ uk ) ∗ (0 ∗ uk )), NVBPNVB (uk )};
[By property (iii)of remark 2.6]
= r min{NVBPNVB (0 ∗ (uk ∗ uk )), NVBPNVB (uk )};
[By condition (iii)of definition 2.3]
= r min{NVBPNVB (0 ∗ 0), NVBPNVB (uk )};
[By property (i)of remark 2.6]
= r min{NVBPNVB (0), NVBPNVB (uk )};
[By lemma 7.1]
= NVBPNVB (uk )
∴ It is concluded that, NVBMNVB (0 ∗ (0 ∗ uk )) ≽ NVBMNVB (uk ) ; ∀ uk ∈ U
Proposition 7.9
BCI
If the NVBS R NVB of 𝔅BCI
MNVB is a NVB BCI - ideal of 𝔅MNVB , then it satisfies: for any ua , ub , uc ∈ U ;
(ua ∗ ub ) ≤ uc ⇒ NVBRNVB (ua ) ≽ r min{NVBRNVB (ub ), NVBRNVB (uc )}
Proof
Let R NVB be a NVB BCI - ideal of 𝔅BCI
MNVB with (ua ∗ ub ) ≤ uc where ua , ub , uc ∈ U
⇒ NVBRNVB (ua ∗ ub ) ≽ NVBRNVB (uc ) [By proposition 7.7]
Since R NVB be a NVB BCI - ideal of 𝔅BCI
MNVB
⇒ NVBRNVB (ua ) ≽ r min {NVBRNVB (ua ∗ ub ), NVBRNVB (ub )} for any ua , ub ∈ U
≽ r min {NVBRNVB (uc ), NVBRNVB (ub )} = r min {NVBRNVB (ub ), NVBRNVB (uc )}
⇒ NVBRNVB (ua ) ≽ r min {NVBRNVB (ub ), NVBRNVB (uc )} for any ua , ub , uc ∈ U
Proposition 7.10
BCI
If the NVBS R NVB of 𝔅BCI
MNVB is a NVB BCI - algebra 𝔅RNVB of then it satisfies for any ux , uy , uz ∈ U
(ua ∗ ub ) ≤ uc ⇒ NVBRNVB (ua ) ≽ r min {NVBRNVB (ub ), NVBRNVB (uc )}
Proof
Let R NVB be a NVBS of 𝔅BCI
MNVB with (ua ∗ ub ) ≤ uc ⇒ NVBRNVB (uc ) ≽ NVBRNVB (ua ∗ ub )
Remya. P.B & Francina Shalini. A, Neutrosophic Vague Binary BCK/BCI-algebra
Neutrosophic Sets and Systems, Vol. 35, 2020
62
R NVB is a 𝔅BCI
RNVB ⇒ NVBRNVB (ua ∗ ub ) ≽ r min{NVBRNVB (ua ), NVBRNVB (ub )}
⇒ NVBRNVB (uc ) ≽ NVBRNVB (ua ∗ ub ) ≽ r min{NVBRNVB (ua ), NVBRNVB (ub )}
⇒ NVBRNVB (uc ) ≽ r min{NVBRNVB (ua ), NVBMNVB (ub ) }
⇒ NVBRNVB (ua ) ≽ r min{NVBRNVB (uc ), NVBRNVB (ub )}; [By putting uc = ua &ua = uc ]
⇒ NVBRNVB (ua ) ≽ r min{NVBRNVB (ub ), NVBRNVB (uc )};
Theorem 7.11
BCI
Let SNVB be both a NVB BCI–algebra 𝔅BCI
SNVB and a NVB BCI-ideal of a NVB BCI - algebra 𝔅SNVB .
Then NVBSNVB (0 ∗ uk ) ≽ NVBSNVB (uk ) for all uk ∈ U
Proof
Let SNVB be a NVB BCI- algebra 𝔅BCI
SNVB
⇒ NVBSNVB (ua ∗ ub ) ≽ r min{NVBSNVB (ua ), NVBSNVB (ub )}; for all ua , ub ∈ U
⇒ NVBSNVB (0 ∗ ub ) ≽ r min{NVBSNVB (0), NVBSNVB (ub )}; [By putting ua = 0 ]
⇒ NVBSNVB (0 ∗ ub ) ≽ NVBSNVB (ub ) [By definition 4.2 (i)]
⇒ NVBSNVB (0 ∗ uk ) ≽ NVBSNVB (uk ) [By putting ub = uk ]
∴ For any uk ∈ U, NVBSNVB (0 ∗ uk ) ≽ NVBSNVB (uk )
Proposition 7.12
Let TNVB be a NVB BCI - ideal of a NVB BCI -algebra 𝔅BCI
MNVB .
If TNVB satisfies NVBTNVB (ua ∗ ub ) ≽ NVBTNVB ((ua ∗ uc ) ∗ (ub ∗ uc )) for all ua , ub , uc ∈ U,
then TNVB is a NVB BCI p - ideal of 𝔅BCI
MNVB
Proof
TNVB be a NVB BCI - ideal of a NVB BCI - algebra 𝔅BCI
MNVB .
⇒ NVBTNVB (ua ) ≽ r min{NVBTNVB (ua ∗ ub ), NVBTNVB (ub )} for all ua , ub , uc ∈ U
⇒ NVBTNVB (ua ) ≽ r min{NVBTNVB ((ua ∗ uc ) ∗ (ub ∗ uc )), NVBMNVB (ub )} for all ua , ub , uc ∈ U
[From given condition]
[By definition 5.2]
⇒ TNVB is a NVB BCI – p ideal of 𝔅BCI
MNVB
Proposition 7.13
Any NVB BCI - ideal DNVB of a NVB BCI -algebra 𝔅BCI
MNVB is a NVB BCI -p ideal
⇔ NVBDNVB (ua ) ≽ NVBDNVB (0 ∗ (0 ∗ ua )) ; for all ua ∈ U
Proof
Let DNVB be a NVB BCI - ideal of a NVB BCI -algebra 𝔅BCI
MNVB . Also let DNVB is a NVB BCI -p ideal.
∴ NVBDNVB (ua ) ≽ r min{NVBDNVB ((ua ∗ uc ) ∗ (ub ∗ uc )), NVBDNVB (ub )} for all ua , ub , uc ∈ U
[By definition 5.2 of NVB BCI − p ideal]
Put uc = ua and ub = 0 in the above,
∴ NVBDNVB (ua ) ≽ r min{NVBDNVB ((ua ∗ ua ) ∗ (0 ∗ ua )), NVBDNVB (0)} for all ua , ub , uc ∈ U
⇒ NVBDNVB (ua ) ≽ r min{NVBDNVB (0 ∗ (0 ∗ ua )), NVBDNVB (0)} for all ua , ub ∈ U
[By condition (iii)of definition 2.3]
[By lemma 7.1]
= NVBDNVB (0 ∗ (0 ∗ ua )) for all ua ∈ U
⇒ NVBDNVB (ua ) ≽ NVBDNVB (0 ∗ (0 ∗ ua )) ; for all ua ∈ U
Conversely, let a NVB BCI - ideal DNVB of a NVB BCI - algebra 𝔅BCI
MNVB satisfies the given condition,
NVBDNVB (ua ) ≽ NVBMNVB (0 ∗ (0 ∗ ua )) ; for all ua ∈ U. By lemma 7.8,
“ Let PNVB be a NVB BCI-ideal of 𝔅BCI
MNVB . Then, NVBPNVB (0 ∗ (0 ∗ uk )) ≽ NVBPNVB (uk ); ∀ uk ∈ U”
⇒ NVBDNVB ((ua ∗ uc ) ∗ (ub ∗ uc )) ≼ NVBMNVB (0 ∗ (0 ∗ ((ua ∗ uc ) ∗ (ub ∗ uc ))))
[By putting uk = (ua ∗ uc ) ∗ (ub ∗ uc ) in lemma 7.8]
= NVBDNVB ((0 ∗ ub ) ∗ (0 ∗ ua )) [By property (vii)of remark 2.6]
= NVBDNVB (0 ∗ (0 ∗ (ua ∗ ub ))) ; [By property (viii) of remark 2.6]
= NVBDNVB (0 ∗ (ua ∗ ub )) ; [By property (i) of remark 2.6]
= NVBDNVB (ua ∗ ub ) ; [By property (i) of remark 2.6]
⇒ NVBDNVB (ua ∗ ub ) ≽ NVBDNVB ((ua ∗ uc ) ∗ (ub ∗ uc ))
Remya. P.B & Francina Shalini. A, Neutrosophic Vague Binary BCK/BCI-algebra
Neutrosophic Sets and Systems, Vol. 35, 2020
63
⇒ NVBDNVB is a NVB BCI - p ideal [ By proposition 7.12]
Theorem 7.14
BCI
Every NVB BCI - p ideal of a NVB BCI - algebra 𝔅BCI
MNVB is a NVB BCI - ideal of 𝔅MNVB .
Proof
Let MNVB be a NVB BCI - p ideal of a NVB BCI - algebra 𝔅BCI
MNVB . By definition,
NVBMNVB (ux ) ≽ r min {NVBMNVB ((ux ∗ uz ) ∗ (uy ∗ uz )) , NVBMNVB (uy )} for all ux , uy , uz ∈ U
Put uz = 0 then the above becomes,
NVBMNVB (ux ) ≽ r min {NVBMNVB ((ux ∗ 0) ∗ (uy ∗ 0)) , NVBMNVB (uy )} for all ux , uy ∈ U
= r min{NVBMNVB (ux ∗ uy ), NVBMNVB (uy )} for all ux , uy ∈ U
[By property (iii)of 2.3 &By property (i)of remark 2.6]
⇒ NVBMNVB (ux ) ≽ r min{NVBMNVB (ux ∗ uy ), NVBMNVB (uy )} for all ux , uy ∈ U
Obviously, MNVB is a NVB BCI - ideal
Converse of this statement need not be true and it can be verified with an example and is trivial.
Theorem 7.15
Every NVB BCK/BCI H - ideal of a NVB BCK/BCI - algebra 𝔅MNVB acts both as
(i) NVB BCK/BCI - ideal of 𝔅MNVB (ii) NVB BCK/BCI - subalgebra 𝔅MNVB
Proof
Let INVB be a NVB BCK/BCI- H ideal of a NVB BCK/BCI – algebra 𝔅MNVB
(i) From definition of NVB BCK/BCI- H ideal,
NVBINVB (ua ∗ uc ) ≽ min{NVBINVB (ua ∗ (ub ∗ uc )), NVBINVB (ub )} for all ua , ub , uc ∈ U
Put uc = 0
NVBINVB (ua ∗ 0) ≽ min{NVBINVB (ua ∗ (ub ∗ 0)), NVBMNVB (ub )} for all ua , ub , uc ∈ U
⇒ NVBINVB (ua ) ≽ min{NVBINVB (ua ∗ ub ), NVBMNVB (ub )} for all ua , ub , uc ∈ U
[Using property (i)of remark 2.6]
Since INVB is a NVB BCK/BCI- H ideal ⇒ NVBINVB (0) ≽ NVBINVB (uk ) ; for any uk ∈ U
[By definition 4.2]
∴ INVB is a NVB BCK/BCI - ideal of 𝔅MNVB
(ii) Let INVB be a NVB BCK/BCI- H ideal of 𝔅MNVB
∴ NVBINVB (ua ∗ uc ) ≽ r min{NVBINVB (ua ∗ (ub ∗ uc )), NVBINVB (ub )}; for all ua , ub , uc ∈ U
⇒ NVBINVB (ua ∗ ub ) ≽ r min{NVBINVB (ua ∗ (ub ∗ ub )), NVBINVB (ub )}; [By putting uc = ub ]
⇒ NVBINVB (ua ∗ ub ) ≽ r min{NVBINVB (ua ∗ 0), NVBINVB (ub )}; [By condition (iii)of definition 2.3]
[By condition (i)of remark 2.6]
⇒ NVBINVB (ua ∗ ub ) ≽ r min{NVBINVB (ua ), NVBINVB (ub )};
⇒ INVB be a NVB BCK/BCI - subalgebra of 𝔅MNVB
Theorem 7.16
PNVB be a NVBS of a NVB BCK/BCI - algebra 𝔅MNVB . Then PNVB is a NVB BCK/BCI -ideal of 𝔅MNVB
⟺ it satisfies the following conditions:
(i) NVBPNVB (ua ∗ ub ) ≽ NVBPNVB (ub ) ; (∀ ua , ub ∈ U)
(ii) NVBPNVB (ua ∗ ((ua ∗ um ) ∗ un )) ≽ r min {NVBPNVB (um ), NVBPNVB (un ) } ;
(∀ ua , um , un ∈ U)
Proof
Let PNVB be NVB BCK/BCI - ideal of 𝔅MNVB . By definition,
NVBPNVB (ua ) ≽ r min{NVBPNVB (ua ∗ ub ), NVBPNVB (ub ) } ; ∀ ua , ub ∈ U
(i) Put ua = (ua ∗ ub ) and ub = ua in the above,
NVBPNVB (ua ∗ ub ) ≽ r min{NVBPNVB ((ua ∗ ub ) ∗ ua ), NVBPNVB (ua ) }
⇒ NVBPNVB (ua ∗ ub ) ≽ r min{NVBPNVB ((ua ∗ ua ) ∗ ub ), NVBPNVB (ua ) } ; [By property (ii)of remark 2.6]
[By condition (iii)of definition 2.3]
⇒ NVBPNVB (ua ∗ ub ) ≽ r min{NVBPNVB (0 ∗ ub ), NVBPNVB (ua ) } ;
[By assumption ua = ub ]
(u
)
≽
r
min{NVB
(0
∗
u
),
NVB
(u
)
};
⇒ NVBPNVB a ∗ ub
PNVB
b
PNVB
b
[By remark 2.5]
⇒ NVBPNVB (ua ∗ ub ) ≽ r min{NVBPNVB (0), NVBPNVB (ub ) } ;
[Using lemma 7.1]
⇒ NVBPNVB (ua ∗ ub ) ≽ NVBPNVB (ub )
(ii) Consider, (ua ∗ ((ua ∗ um ) ∗ un )) ∗ um
= (ua ∗ um ) ∗ ((ua ∗ um ) ∗ un ) ≤ un
Remya. P.B & Francina Shalini. A, Neutrosophic Vague Binary BCK/BCI-algebra
Neutrosophic Sets and Systems, Vol. 35, 2020
64
By condition (ii)of remark 2.3, (x ∗ (x ∗ y)) ∗ y = 0 ⇒ x ∗ (x ∗ y) ≤ y, by remark 2.4
𝐻𝑒𝑟𝑒, (𝑢𝑎 ∗ ((𝑢𝑎 ∗ 𝑢𝑚 ) ∗ 𝑢𝑛 )) ∗ 𝑢𝑚 = (𝑢𝑎 ∗ 𝑢𝑚 ) ∗ ((𝑢𝑎 ∗ 𝑢𝑚 ) ∗ 𝑢𝑛 ) = ((𝑢𝑎 ∗ 𝑢𝑚 ) ∗ (𝑢𝑎 ∗ 𝑢𝑚 )) ∗ 𝑢𝑛 = 0 ∗ 𝑢𝑛 = 0. 𝑆𝑜 𝑟𝑒𝑚𝑎𝑟𝑘 2.4 𝑖𝑠 𝑎𝑝𝑝𝑙𝑖𝑐𝑎𝑏𝑙𝑒 𝑖𝑛 𝑡ℎ𝑖𝑠 𝑐𝑎𝑠𝑒
Since (𝑢𝑎 ∗ 𝑢𝑚 ) ∗ ((𝑢𝑎 ∗ 𝑢𝑚 ) ∗ 𝑢𝑛 ) = 0 𝑤𝑒 ℎ𝑎𝑣𝑒 (𝑢𝑎 ∗ 𝑢𝑚 ) ∗ ((𝑢𝑎 ∗ 𝑢𝑚 ) ∗ 𝑢𝑛 ) ≤ 𝑢𝑛
[
]
Above can be written as, (ua ∗ ((ua ∗ um ) ∗ un )) ∗ um ≤ un
[By proposition 7.7]
⇒ NVBPNVB ((ua ∗ ((ua ∗ um ) ∗ un )) ∗ um ) ≽ NVBPNVB (un )
PNVB is a NVB BCK/BCI -ideal of 𝔅MNVB ⇒ NVBPNVB (ua ) ≽ r min{NVBPNVB (ua ∗ ub ), NVBPNVB (ub )}
Put ua = (ua ∗ ((ua ∗ um ) ∗ un )) & ub = um in above,
NVBPNVB (ua ∗ ((ua ∗ um ) ∗ un )) ≽ r min {NVBPNVB ((ua ∗ ((ua ∗ um ) ∗ un )) ∗ um ) , NVBPNVB (um )}
= r min {NVBPNVB (un ), NVBPNVB (um )} [proved above]
≽ r min {NVBPNVB (um ), NVBPNVB (un )}
⇒ NVBMNVB (ua ∗ ((ua ∗ um ) ∗ un )) ≽ r min {NVBPNVB (um ), NVBPNVB (un )} ; [∀ ua , um , un ∈ U ]
Conversely,
Let PNVB be a NVBS of a NVB BCK/BCI – algebra 𝔅MNVB satisfying, the given conditions,
(i)
NVBPNVB (ua ∗ ub ) ≽ NVBPNVB (ua ) ; [∀ ua , ub ∈ U ]
NVBPNVB (ua ∗ ((ua ∗ um ) ∗ un )) ≽ r min {NVBPNVB (um ), NVBPNVB (un )} ;
(ii)
[∀ ua , um , un ∈ U ]
To prove condition (1) of a NVB BCK/BCI - ideal, take ub = ua in (i) and (ii) respectively,
(i) ⇒ NVBPNVB (ua ∗ ua ) ≽ NVBPNVB (ua ) ⇒ NVBPNVB (0) ≽ NVBPNVB (ua );
[By property (iii)of definition 2.3]
To prove condition (2) of a NVB BCK/BCI - ideal,
take, NVBPNVB (ua ) = NVBPNVB (ua ∗ 0) [By property(i)of remark 2.6]
= NVBPNVB (ua ∗ ((ua ∗ ub ) ∗ (ua ∗ ub )))
; [By property (iii)of definition 2.3]
= NVBPNVB (ua ∗ ((ua ∗ (ua ∗ ub )) ∗ ub ))
; [By property (ii) of remark 2.6]
= NVBPNVB (ua ∗ ((ua ∗ um ) ∗ un ))
;
[By putting (ua ∗ ub ) = um and ub = un ]
≽ r min {NVBPNVB (um ), NVBPNVB (un )} ; [By condition (ii) in the assumption]
= r min {NVBPNVB (ua ∗ ub ), NVBPNVB (ub )}; [By putting (ua ∗ ub ) = um and ub = un ]
⇒ NVBPNVB (ua ) ≽ r min {NVBPNVB (ua ∗ ub ), NVBPNVB (ub )}
∴ PNVB is a NVB BCK/BCI - ideal of 𝔅MNVB
Theorem 7.17
Let MNVB be a NVB BCK/BCI - algebra 𝔅MNVB . Then any NVB BCK/BCI - cut of MNVB is a crisp
NVB BCK/BCI - subalgebra of 𝔅MNVB
Proof
Let for any α1 , α2, β1 , β2 , γ1 , γ2 , δ1 , δ2 , ρ1 , ρ2 , ϑ1 , ϑ2 ∈ [0, 1],
MNVB ([α ,α ],[β ,β ],[γ ,γ ]),([δ ,δ ],[ρ ,ρ ],[ϑ ,ϑ ]) be a NVB BCK/BCI -cut of MNVB .
1
2
1
2
1 2
1 2
Assume ux , uy ∈ MNVB ([α
1 2
1 2
1 ,α2 ],[β1 ,β2 ],[γ1 ,γ2 ]),([δ1 ,δ2 ],[ρ1 ,ρ2 ],[ϑ1 ,ϑ2 ])
⇒ NVBMNVB (ux ) ≥ [α1 , α2 ], [β1 , β2 ], [γ1 , γ2 ] & NVBMNVB (ux ) ≥ ([δ1 , δ2 ], [ρ1 , ρ2 ], [ϑ1 , ϑ2 ])
NVBMNVB (uy ) ≥ [α1 , α2 ], [β1 , β2 ], [γ1 , γ2 ] & NVBMNVB (uy ) ≥ ([δ1 , δ2 ], [ρ1 , ρ2 ], [ϑ1 , ϑ2 ])
̂M (ux ) ≥ [α1 , α2 ] ; ÎM (ux ) ≤ [β1 , β2 ] ; F̂M (ux ) ≤ [γ1 , γ2 ] &
⇒ T
NVB
NVB
NVB
̂M (ux ) ≥ [δ1 , δ2 ] ; ÎM (ux ) ≤ [ρ1 , ρ2 ] ; F̂M (ux ) ≤ [ϑ1 , ϑ2 ]
T
NVB
NVB
NVB
̂M (uy ) ≥ [α1 , α2 ] ; ÎM (uy ) ≤ [β1 , β2 ] ; F̂M (uy ) ≤ [γ1 , γ2 ]
T
NVB
NVB
NVB
̂
TMNVB (uy ) ≥ [δ1 , δ2 ] ; ÎMNVB (uy ) ≤ [ρ1 , ρ2 ] ; F̂MNVB (uy ) ≤ [ϑ1 , ϑ2 ]
MNVB is a NVB BCK/BCI -algebra 𝔅MNVB ⇒ NVBMNVB (ux ∗ uy ) ≽ r min{NVBMNVB (ux ), NVBMNVB (uy )}
̂M (ux ∗ uy ) ≥ min {T
̂M (ux ), T
̂M (uy )} ; ÎM (ux ∗ uy ) ≤ max {ÎM (ux ), ÎM (uy )} ; F̂M (ux ∗ uy ) ≤ max {F̂M (ux ), F̂M (uy )}
⇒T
NVB
NVB
NVB
NVB
NVB
NVB
NVB
NVB
NVB
⇒ (ux ∗ uy ) ∈ MNVB ([α
1 ,α2 ],[β1 ,β2 ],[γ1 ,γ2 ]),([δ1 ,δ2 ],[ρ1 ,ρ2 ],[ϑ1 ,ϑ2 ])
⇒ NVB BCK/BCI - cut MNVB ([α
1 ,α2 ],[β1 ,β2 ],[γ1 ,γ2 ]),([δ1 ,δ2 ],[ρ1 ,ρ2 ],[ϑ1 ,ϑ2 ])
of MNVB is a crisp NVB BCK/BCI -
subalgebra of 𝔅MNVB
Remya. P.B & Francina Shalini. A, Neutrosophic Vague Binary BCK/BCI-algebra
Neutrosophic Sets and Systems, Vol. 35, 2020
65
8. Conclusions
In this paper, two logical algebras viz., BCK and BCI are developed for neutrosophic vague binary
sets. It’s subalgebra, ideal and cuts are also got discussed. Different kinds of ideals like p ideal, q
ideal, a ideal, H ideal for neutrosophic vague binary BCK/BCI -algebra have been investigated.
Theorems and propositions related to this concept are verified. In this paper BCK/BCI-algebra for
neutrosophic sets are firstly developed. Then it is extended to neutrosophic vague and to
neutrosophic vague binary. Work can be further extended to higher concepts like its group, rings,
filter, near-rings etc. Behavior differences of these two algebras in different algebraic notions have to
be addressed more deeply and properly to get a correct vision. This area demands some more notice
and filtering to find out its correct drawbacks. Further investigations will make it, to balance its moves
to the correct direction. Medial BCI -algebra, commutative BCK-algebra, Associative BCI – algebra,
BCK/BCI- homomorphisms, bounded commutative BCI-algebra are a few points have to be
addressed and have to be analyzed more.
Funding: “This research received no external funding”
Acknowledgments: Authors are very grateful to the anonymous reviewers and would like to thank for their
valuable and critical suggestions to structure the paper systematically and to remove the errors.
Conflicts of Interest: “The authors declare that they have no conflict of interest.”
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Remya. P.B & Francina Shalini. A, Neutrosophic Vague Binary BCK/BCI-algebra
Neutrosophic Sets and Systems, Vol. 35, 2020
University of New Mexico
Introduction to Topological Indices in Neutrosophic Graphs
Masoud Ghods1,*, Zahra Rostami 2
1Department
2Department
of Mathematics, Semnan University, Semnan 35131-19111, Iran, mghods@semnan.ac.ir
of Mathematics, Semnan University, Semnan 35131-19111, Iran, zahrarostami.98@semnan.ac.ir
* Correspondence: mghods@semnan.ac.ir; Tel.: (09122310288)
Abstract: Neutrosophic Graphs are graphs that follow three-valued logic. They may be considered
a fuzzy graph, although in some cases, it is difficult to optimize and model them using fuzzy graphs.
In this paper, the first and second Zagreb indices, the Harmonic index, the Randic’ index and the
Connectivity index for these graphs are investigated and some of the theorems related to these
indices are discussed and proven. These indices are also calculated for some specific types of
Neutrosophic Graphs, such as regular Neutrosophic Graphs and regular complete Neutrosophic
Graphs.
Keywords: Neutrosophic Graphs; Zagreb indices; Harmonic index; Randic’ index; Connectivity
index
1. Introduction
Graph theory has many applications for modeling problems in various fields of computer
science such as systems analysis, computer networks, transportation, operations research and
economics. The vertices and edges of the graphs are used to represent objects and the relationships
between them, respectively. Many of the optimization issues are caused by inaccurate information
due to factors like lack of evidence, incomplete statistical data, and lack of sufficient information; this
creates uncertainty in various issues. Classical Graphic Theory uses the basic concept of classical set
theory, as proposed by Contour. In a classic graph, for each vertex or edge, there are two possibilities:
either in the graph or not in the graph. Therefore, classical graphs cannot model uncertain
optimization problems. Real-life issues are often unclear, making modeling by classical graphs
difficult. Zadeh introduced the degree of membership/truth (T) in 1965 and defined the fuzzy set.
Atanassov [14] introduced the degree of nonmembership/falsehood (F) in 1983 and defined the
intuitionistic fuzzy set. Smarandache [15] introduced the degree of indeterminacy/neutrality (I) as an
independent component in 1995 and defined the neutrosophic set on three components (T, I, F) [4].
Fuzzy set [1] is a generalized version of the classical set in which objects have different
membership degrees. A fuzzy set gives the degree of different members between zero and one. Much
work has already been done on fuzzy graphs, including the calculation of various topology indices,
indicators such as Zagreb index, Randic’, harmonic, and so on. However, there is another class of
graphs that is a broad case of fuzzy graphs. In this type of graph, known as neutrosophic graphs, in
addition to the degree of accuracy of each membership function, the degree of its membership is
uncertain, as well as its inaccuracy. So in many cases, it may be more logical to use this model than
graphs in real-world problems.
Masoud Ghods*, Zahra Rostami, Introduction To Topological Indices in Neutrosophic Graphs
Neutrosophic Sets and Systems, Vol. 35, 2020
69 of 10
Since that neutrosophic graphs are more efficient than fuzzy graphs for modeling real problems.
Therefore, in this paper, we try for the first time to calculate some topological indices for this type of
graph.
2. Preliminaries
This section, provides some definitions and theorems needed.
Definition 1. [13] Let 𝐺 = (𝑁, 𝑀) be a single-valued Neutrosophic graph, where 𝑁 is a
Neutrosophic set on 𝑉 and, 𝑀 is a Neutrosophic set on 𝐸, which satisfy the following
𝑇𝑀 (𝑢, 𝑣) ≤ 𝑚𝑖𝑛(𝑇𝑁 (𝑢), 𝑇𝑁 (𝑣)),
𝐼𝑀 (𝑢, 𝑣) ≥ 𝑚𝑎𝑥(𝐼𝑁 (𝑢), 𝐼𝑁 (𝑣)),
𝐹𝑀 (𝑢, 𝑣) ≥ 𝑚𝑎𝑥(𝐹𝑁 (𝑢), 𝐹𝑁 (𝑣)),
Where 𝑢 and 𝑣 are two vertices of 𝐺, and (𝑢, 𝑣) ∈ 𝐸 is an edge of 𝐺.
Definition 2. [2] Let 𝐺 = (𝑁, 𝑀) be a Single-Valued Neutrosophic Graph and 𝑃 is a path in 𝐺. 𝑃 is
a collection of different vertices, 𝑣0 , 𝑣1 , 𝑣2 , … , 𝑣𝑛 such that (𝑇𝑀 (𝑣𝑖−1 , 𝑣𝑖 ), 𝐼𝑀 (𝑣𝑖−1 , 𝑣𝑖 ), 𝐹𝑀 (𝑣𝑖−1 , 𝑣𝑖 )) > 0
for 0 ≤ 𝑖 ≤ 𝑛. 𝑃 is a Neutrosophic cycle if 𝑣0 = 𝑣𝑛 and 𝑛 ≥ 3.
Definition 3. [2] Suppose 𝐺 = (𝑁, 𝑀) a single-valued Neutrosophic graph. 𝐺 is a connected SingleValued Neutrosophic Graph if there exists no isolated vertex in 𝐺. (𝑣 ∈ 𝑉𝐺 is isolated vertex, if there
exists no incident edge to the vertex 𝑣.)
Definition 4. [2] Let 𝐺 = (𝑁, 𝑀) be a Single-Valued Neutrosophic Graph, and 𝑣 ∈ 𝑉 is vertex of 𝐺.
The degree of vertex 𝑣 is the sum of the truth membership values, the sum of the indeterminacy
membership values, and the sum of the falsity membership values of all the edges that are adjacent
to vertex 𝑣. And is denoted by 𝑑(𝑣), that
𝑑(𝑣) = (𝑑𝑇 (𝑣), 𝑑𝐼 (𝑣), 𝑑𝐹 (𝑣)) = (∑ 𝑇𝑀 (𝑣, 𝑢) , ∑ 𝐼𝑀 (𝑣, 𝑢) , ∑ 𝐹𝑀 (𝑣, 𝑢)).
𝑣∈𝑉
𝑣≠𝑢
𝑣∈𝑉
𝑣≠𝑢
𝑣∈𝑉
𝑣≠𝑢
Definition 5. [2] Let 𝐺 = (𝑁, 𝑀) be a Single-Valued Neutrosophic Graph, and the 𝑑𝑚 –degree of any
vertex 𝑣 in 𝐺 is denoted as 𝑑𝑚 (𝑣) where
𝑚 (𝑢,
𝑑𝑚 (𝑣) = ( ∑ 𝑇𝑀𝑚 (𝑢, 𝑣) , ∑ 𝐼𝑀
𝑣) , ∑ 𝐹𝑀𝑚 (𝑢, 𝑣))
𝑢≠𝑣∈𝑉
𝑢≠𝑣∈𝑉
𝑢≠𝑣∈𝑉
Here, the path 𝑣 = 𝑣0𝑣1 𝑣2 … 𝑣𝑛 = 𝑢 is the shortest path between the vertices 𝑣 and 𝑢, when the
length of this path is 𝑚.
Definition 6. [2] Let 𝐺 = (𝑁, 𝑀) be a Single-Valued Neutrosophic Graph, 𝐺 is a regular
neutrosophic graph if it satisfies the following,
∑ 𝑇𝑀 (𝑣, 𝑢) = 𝑐,
∑ 𝐼𝑀 (𝑣, 𝑢) = 𝑐,
∑ 𝐹𝑀 (𝑣, 𝑢) = 𝑐,
𝑣≠𝑢
𝑣≠𝑢
𝑣≠𝑢
Where 𝑐 is a constant value.
3. Topological Indices in Neutrosophic Graphs
In the section, we introduce Topological Indices in Neutrosophic Graphs and provide a number
of examples. We define Zagreb indices, Harmonic index, and Randic’ index, and in finally
Connectivity index on the neutrosophic graphs.
Masoud Ghods*, Zahra Rostami, Introduction To Topological Indices in Neutrosophic Graphs
Neutrosophic Sets and Systems, Vol. 35, 2020
70 of 10
3.1. Zagreb index of First and Second Kind in Neutrosophic Graphs
Definition 8. Let 𝐺 = (𝑁, 𝑀) be the Neutrosophic Graph whit non-empty vertex set. The first Zagreb
index is denoted by 𝑀(𝐺) and defined as
𝑛
𝑀(𝐺) = ∑(𝑇𝑁 (𝑢𝑖 ), 𝐼𝑁 (𝑢𝑖 ), 𝐹𝑁 (𝑢𝑖 ))𝑑2 (𝑢𝑖 ),
∀ 𝑢𝑖 ∈ 𝑉.
𝑖=1
Example 1. Consider the Neutrosophic Graph 𝐺 = (𝑁, 𝑀) as shown in figure 1, with the vertex set
(𝑇𝑁 , 𝐼𝑁 , 𝐹𝑁 )(𝑎) = (0.3, 0.6, 0.7), (𝑇𝑁 , 𝐼𝑁 , 𝐹𝑁 )(𝑏) = (0.3, 0.5, 0.6),
𝑉 = {𝑎, 𝑏, 𝑐}
such
that
and
(𝑇𝑀 , 𝐼𝑀 , 𝐹𝑀 )(𝑎, 𝑏) = (0.2, 0.6, 0.8),
(𝑇𝑁 , 𝐼𝑁 , 𝐹𝑁 )(𝑐) = (0.4, 0.5, 0.6), The
edge
set
contains
(𝑇𝑀 , 𝐼𝑀 , 𝐹𝑀 )(𝑏, 𝑐) = (0.2, 0.6, 0.7), and (𝑇𝑀 , 𝐼𝑀 , 𝐹𝑀 )(𝑎, 𝑐) = (02, 0.8, 0.9). We have,
Figure 1. A neutrosophic graph with 𝑉 = {𝑎, 𝑏, 𝑐}
The first Zagreb index is
𝑑(𝑎) = (0.2 + 0.2, 0.6 + 0.8, 0.8 + 0.9) = (0.4, 1.4, 1.7),
𝑑(𝑏) = (0.2 + 0.2, 0.6 + 0.6, 0.8 + 0.7) = (0.4, 1.2, 1.5),
𝑑(𝑐) = (0.2 + 0.2, 0.8 + 0.6, 0.9 + 0.7) = (0.4, 1.4, 1.6).
Now, we have
𝑑2 (𝑎) = (0.04 + 0.04, 0.36 + 0.64, 0.64 + 0.81) = (0.08, 1, 1.45),
𝑑2 (𝑏) = (0.04 + 0.04, 0.36 + 0.36, 0.64 + 0.49) = (0.08, 0.72, 1.13),
𝑑2 (𝑐) = (0.04 + 0.04, 0.64 + 0.36, 0.81 + 0.49) = (0.08, 1, 1.3).
4
𝑀(𝐺) = ∑(𝑇𝑁 (𝑢𝑖 ), 𝐼𝑁 (𝑢𝑖 ), 𝐹𝑁 (𝑢𝑖 ))𝑑2 (𝑢𝑖 )
𝑖=1
= (0.3, 0.6, 0.7)(0.08, 1, 1.45) + (0.3, 0.5, 0.6)(0.08, 0.72, 1.13)
+ (0.4, 0.5, 0.6)(0.08, 1. 1.3)
= (0.024 + 0.6 + 1.015) + (0.024 + 0.36 + 0.678) + (0.032 + 0.5 + 0.78) = 4.013.
Definition 9. The second Zagreb index is denoted by 𝑀∗ (𝐺) and defined as
1
𝑀 ∗ (𝐺) = ∑[(𝑇𝑁 (𝑢𝑖 ), 𝐼𝑁 (𝑢𝑖 ), 𝐹𝑁 (𝑢𝑖 ))𝑑(𝑢𝑖 )][(𝑇𝑁 (𝑣𝑗 ), 𝐼𝑁 (𝑣𝑗 ), 𝐹𝑁 (𝑣𝑗 ))𝑑(𝑣𝑗 )], ∀𝑖 ≠ 𝑗 𝑎𝑛𝑑 (𝑢𝑖 , 𝑣𝑗 ) ∈ 𝐸.
2
Example 2. If 𝐺 is the same Neutrosophic Graph as example 1, we have
Masoud Ghods*, Zahra Rostami, Introduction To Topological Indices in Neutrosophic Graphs
Neutrosophic Sets and Systems, Vol. 35, 2020
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1
𝑀 ∗ (𝐺) = [(0.3, 0.6, 0.7). (0.4, 1.4, 1.7) × (0.3, 0.5, 0.6). (0.4, 1.2, 1.5) + (0.3, 0.6, 0.7). (0.4, 1.4, 1.7)
2
× (0.4, 0.5, 0.6). (0.4, 1.4, 1.6) + (0.3, 0.5, 0.6). (0.4, 1.2, 1.5)
× (0.4, 0.5, 0.6). (0.4, 1.4, 1.6)]
1
= [(0.12 + 0.84 + 1.19) × (0.12 + 0.6 + 0.9) + (0.12 + 0.84 + 1.19)
2
× (0.16 + 0.7 + 0.96) + (0.12 + 0.6 + 0.9) × (0.16 + 0.7 + 0.96)
1
1
= [(2.15)(1.62) + (2.15)(1.82) + (1.62)(1.82)] = (10.3444) = 5.1722.
2
2
∗ (𝐺)
Note 1. As we have seen, the value of 𝑀
is less than the value of 𝑀(𝐺), and this is always the
case.
Theorem 1. Let 𝐺 is the Neutrosophic Graph and 𝐻 is the Neutrosophic sub graph of 𝐺 such that
𝐻 = 𝐺 − 𝑢 then 𝑀(𝐻) < 𝑀(𝐺) and 𝑀∗ (𝐻) < 𝑀 ∗ (𝐺).
Proof. Given that by omitting a vertex of 𝐺, a positive value, the sum is lost, so the proof is obvious.
Theorem 2. Let 𝐺 be the regular neutrosophic graph. Then, we have
𝑛
2
𝑀(𝐺) = 𝑐 × ∑(𝑇𝑁 (𝑢𝑖 ) + 𝐼𝑁 (𝑢𝑖 )+ 𝐹𝑁 (𝑢𝑖 )),
∀ 𝑢𝑖 ∈ 𝑉.
𝑖=1
Where ∑𝑣≠𝑢 𝑇𝑀 (𝑣, 𝑢) = 𝑐, ∑𝑣≠𝑢 𝐼𝑀 (𝑣, 𝑢) = 𝑐, ∑𝑣≠𝑢 𝐹𝑀 (𝑣, 𝑢) = 𝑐.
Proof. Given the degree of definition of each vertex,
𝑑(𝑣) = (𝑑𝑇 (𝑣), 𝑑𝐼 (𝑣), 𝑑𝐹 (𝑣)) = (∑ 𝑇𝑀 (𝑣, 𝑢) , ∑ 𝐼𝑀 (𝑣, 𝑢) , ∑ 𝐹𝑀 (𝑣, 𝑢)).
𝑣∈𝑉
𝑣≠𝑢
𝑣∈𝑉
𝑣≠𝑢
𝑣∈𝑉
𝑣≠𝑢
On the other hand, for regular neutrosophic graphs, we know that
∑ 𝑇𝑀 (𝑣, 𝑢) = 𝑐,
∑ 𝐼𝑀 (𝑣, 𝑢) = 𝑐,
∑ 𝐹𝑀 (𝑣, 𝑢) = 𝑐,
𝑣≠𝑢
𝑣≠𝑢
𝑣≠𝑢
Therefore
𝑑(𝑣) = (𝑑𝑇 (𝑣), 𝑑𝐼 (𝑣), 𝑑𝐹 (𝑣)) = (𝑐, 𝑐, 𝑐).
Now, by embedding the formula in the first Zagreb index, we will get the desired result. The
proof is complete.
Theorem 3. Let 𝐺 be the regular neutrosophic graph. Then, we have
1
𝑀 ∗ (𝐺) = (𝑐 2 ) ∑[𝑇𝑁 (𝑢𝑖 ) + 𝐼𝑁 (𝑢𝑖 ) + 𝐹𝑁 (𝑢𝑖 )][𝑇𝑁 (𝑣𝑗 ) + 𝐼𝑁 (𝑣𝑗 ) + 𝐹𝑁 (𝑣𝑗 )],
2
∀𝑖 ≠ 𝑗 𝑎𝑛𝑑 (𝑢𝑖 , 𝑣𝑗 ) ∈ 𝐸,
Where ∑𝑣≠𝑢 𝑇𝑀 (𝑣, 𝑢) = 𝑐, ∑𝑣≠𝑢 𝐼𝑀 (𝑣, 𝑢) = 𝑐, ∑𝑣≠𝑢 𝐹𝑀 (𝑣, 𝑢) = 𝑐.
Proof. Assume 𝐺 is a regular neutrosophic graph, using the second Zagreb index formula for 𝐺, we
have ∀ 𝑖 ≠ 𝑗 𝑎𝑛𝑑 (𝑢𝑖 , 𝑣𝑗 ) ∈ 𝐸,
Masoud Ghods*, Zahra Rostami, Introduction To Topological Indices in Neutrosophic Graphs
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1
𝑀∗ (𝐺) = ∑[(𝑇𝑁 (𝑢𝑖 ), 𝐼𝑁 (𝑢𝑖 ), 𝐹𝑁 (𝑢𝑖 ))𝑑(𝑢𝑖 )][(𝑇𝑁 (𝑣𝑗 ), 𝐼𝑁 (𝑣𝑗 ), 𝐹𝑁 (𝑣𝑗 ))𝑑(𝑣𝑗 )]
2
1
= ∑[(𝑇𝑁 (𝑢𝑖 ), 𝐼𝑁 (𝑢𝑖 ), 𝐹𝑁 (𝑢𝑖 ))𝑑(𝑑𝑇 (𝑢𝑖 ), 𝑑𝐼 (𝑢𝑖 ), 𝑑𝐹 (𝑢𝑖 ))]
2
× [(𝑇𝑁 (𝑣𝑗 ), 𝐼𝑁 (𝑣𝑗 ), 𝐹𝑁 (𝑣𝑗 ))𝑑 (𝑑𝑇 (𝑣𝑗 ), 𝑑𝐼 (𝑣𝑗 ), 𝑑𝐹 (𝑣𝑗 ))]
1
= ∑[(𝑇𝑁 (𝑢𝑖 ), 𝐼𝑁 (𝑢𝑖 ), 𝐹𝑁 (𝑢𝑖 )). (𝑐, 𝑐, 𝑐)][(𝑇𝑁 (𝑣𝑗 ), 𝐼𝑁 (𝑣𝑗 ), 𝐹𝑁 (𝑣𝑗 )). (𝑐, 𝑐, 𝑐)]
2
1
= ∑[𝑐. 𝑇𝑁 (𝑢𝑖 ) + 𝑐. 𝐼𝑁 (𝑢𝑖 ) + 𝑐. 𝐹𝑁 (𝑢𝑖 )][𝑐. 𝑇𝑁 (𝑣𝑗 ) + 𝑐. 𝐼𝑁 (𝑣𝑗 ) + 𝑐. 𝐹𝑁 (𝑣𝑗 )]
2
1
= ∑ 𝑐[𝑇𝑁 (𝑢𝑖 ) + 𝐼𝑁 (𝑢𝑖 ) + 𝐹𝑁 (𝑢𝑖 )]. c[𝑇𝑁 (𝑣𝑗 ) + 𝐼𝑁 (𝑣𝑗 ) + 𝐹𝑁 (𝑣𝑗 )]
2
1
= 𝑐 2 ∑[𝑇𝑁 (𝑢𝑖 ) + 𝐼𝑁 (𝑢𝑖 ) + 𝐹𝑁 (𝑢𝑖 )][𝑇𝑁 (𝑣𝑗 ) + 𝐼𝑁 (𝑣𝑗 ) + 𝐹𝑁 (𝑣𝑗 )] .
2
The desired result was obtained.
3.2. Harmonic index in Neutrosophic Graphs
Definition 10. The Harmonic index of Neutrosophic Graph 𝐺 is defined as
1
, ∀𝑖 ≠ 𝑗 𝑎𝑛𝑑 (𝑢𝑖 , 𝑣𝑗 ) ∈ 𝐸.
𝐻(𝐺) = ∑
(𝑇𝑁 (𝑢𝑖 ), 𝐼𝑁 (𝑢𝑖 ), 𝐹𝑁 (𝑢𝑖 ))𝑑(𝑢𝑖 ) + (𝑇𝑁 (𝑣𝑗 ), 𝐼𝑁 (𝑣𝑗 ), 𝐹𝑁 (𝑣𝑗 ))𝑑(𝑣𝑗 )
Example 3. We have the previous example,
1
𝐻(𝐺) =
(0.3, 0.6, 0.7)(0.4, 1.4, 1.7) + (0.3, 0.5, 0.6)(0.4, 1.2, 1.5)
1
+
(0.3, 0.6, 0.7)(0.4, 1.4, 1.7) + (0.4, 0.5, 0.6)(0.4, 1.4, 1.6)
1
+
(0.3, 0.5, 0.6)(0.4, 1.2, 1.5) + (0.4, 0.5, 0.6)(0.4, 1.4, 1.6)
1
1
1
1
1
1
=
+
+
=
+
+
= 0.8078.
2.15 + 1.62 2.15 + 1.82 1.62 + 1.82 3.77 3.97 3.44
3.3. Randic’ index in Neutrosophic Graphs
Definition 11. The Randic’ index of Neutrosophic Graph 𝐺 is defined as
−1
𝑅(𝐺) = ∑((𝑇𝑁 (𝑢𝑖 ), 𝐼𝑁 (𝑢𝑖 ), 𝐹𝑁 (𝑢𝑖 ))𝑑(𝑢𝑖 )(𝑇𝑁 (𝑣𝑗 ), 𝐼𝑁 (𝑣𝑗 ), 𝐹𝑁 (𝑣𝑗 ))𝑑(𝑣𝑗 )) 2 , ∀𝑖 ≠ 𝑗 𝑎𝑛𝑑 (𝑢𝑖 , 𝑣𝑗 ) ∈ 𝐸.
Example 3. For above example, by simple calculations, it is easy to see that
1
𝑅(𝐺) =
√(0.3, 0.6, 0.7). (0.4, 1.4, 1.7) × (0.3, 0.5, 0.6). (0.4, 1.2, 1.5)
1
+
√(0.3, 0.6, 0.7). (0.4, 1.4, 1.7) × (0.4, 0.5, 0.6). (0.4, 1.4, 1.6)
1
+
√(0.3, 0.5, 0.6). (0.4, 1.2, 1.5) × (0.4, 0.5, 0.6). (0.4, 1.4, 1.6)
1
1
1
=
+
+
= 1.6237.
√2.15 × 1.62 √2.15 × 1.82 √1.62 × 1.82
3.4. Connectivity index in Neutrosophic Graphs
Connectivity index is an important parameter in the graph. Using it, we can study and study
some of the features of graph models.
Definition 12. Let 𝐺 = (𝑁, 𝑀) be the Neutrosophic Graph. The connectivity index of 𝐺 is defined by
Masoud Ghods*, Zahra Rostami, Introduction To Topological Indices in Neutrosophic Graphs
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𝐶𝐼(𝐺) = ∑ (𝑇𝑁 (𝑢𝑖 ), 𝐼𝑁 (𝑢𝑖 ), 𝐹𝑁 (𝑢𝑖 ))(𝑇𝑁 (𝑣𝑗 ), 𝐼𝑁 (𝑣𝑗 ), 𝐹𝑁 (𝑣𝑗 )) × 𝐶𝑂𝑁𝑁𝐺 (𝑢𝑖 , 𝑣𝑗 ).
𝑢𝑖 ,𝑣𝑗 ∈𝑉
Where 𝐶𝑂𝑁𝑁𝐺 (𝑢𝑖 , 𝑣𝑗 ) is the strength of connectedness between 𝑢𝑖 and 𝑣𝑗 .
Definition 13. The strength of connectedness between 𝑢𝑖 and 𝑣𝑗 is defined as
𝐶𝑂𝑁𝑁𝑃 (𝑢𝑖 , 𝑣𝑗 ) = ( min 𝑇𝑀 (𝑒) , max 𝐼𝑀 (𝑒) , max 𝐹𝑀 (𝑒)),
𝑒∈𝑃𝑢 𝑣
𝑖 𝑗
𝑒∈𝑃𝑢 𝑣
𝑖 𝑗
𝑒∈𝑃𝑢 𝑣
𝑖 𝑗
Where 𝑃𝑢𝑖𝑣𝑗 is the path between 𝑢𝑖 and 𝑣𝑗 .
|𝐶𝑂𝑁𝑁𝑃 (𝑢𝑖 , 𝑣𝑗 )| = 2 ( min 𝑇𝑀 (𝑒)) − ( max 𝐼𝑀 (𝑒)) − ( max 𝐹𝑀 (𝑒)),
𝑒∈𝑃𝑢 𝑣
𝑖 𝑗
𝑒∈𝑃𝑢 𝑣
𝑖 𝑗
𝑒∈𝑃𝑢 𝑣
𝑖 𝑗
Then
𝐶𝑂𝑁𝑁𝐺 (𝑢𝑖 , 𝑣𝑗 ) = max{|𝐶𝑂𝑁𝑁𝑃 (𝑢𝑖 , 𝑣𝑗 )|}.
𝑃
Example 4. For example, in the above figure, the strength of connectedness between:
𝑎 and 𝑏 from the direct path 𝑃1 = 𝑎𝑏 is
𝐶𝑂𝑁𝑁𝑃1 (𝑎, 𝑏) = 𝑀𝑎𝑏 = (0.2, 0.6, 0.8),
From path 𝑃2 = 𝑎𝑐𝑏 is
𝐶𝑂𝑁𝑁𝑃2 (𝑎, 𝑏) = (𝑚𝑖𝑛{0.2, 0.2},
𝑚𝑎𝑥{0.8, 0.6},
𝑚𝑎𝑥{0.9, 0.7}) = (0.2, 0.8, 0.9);
𝑎 and 𝑐 from the direct path 𝑃1 = 𝑎𝑐 is
𝐶𝑂𝑁𝑁𝑃1 (𝑎, 𝑐) = 𝑀𝑎𝑐 = (0.2, 0.8, 0.9),
From path 𝑃2 = 𝑎𝑏𝑐 is
𝐶𝑂𝑁𝑁𝑃2 (𝑎, 𝑐) = (𝑚𝑖𝑛{0.2, 0.2},
𝑚𝑎𝑥{0.6, 0.6},
𝑚𝑎𝑥{0.8, 0.7}) = (0.2, 0.6, 0.8);
𝑏 and 𝑐 from the direct path 𝑃1 = 𝑏𝑐 is
𝐶𝑂𝑁𝑁𝑃1 (𝑏, 𝑐) = 𝑀𝑏𝑐 = (0.2, 0.6, 0.7),
From path 𝑃2 = 𝑏𝑎𝑐 is
𝐶𝑂𝑁𝑁𝑃2 (𝑏, 𝑐) = (𝑚𝑖𝑛{0.2, 0.2},
𝑚𝑎𝑥{0.6, 0.8},
𝑚𝑎𝑥{0.8, 0.9}) = (0.2, 0.8, 0.9).
Then, we have for 𝑎 and 𝑏
|𝐶𝑂𝑁𝑁𝑃1 (𝑎, 𝑏)| = 2 × (0.2) − 0.6 − 0.8 = −1,
|𝐶𝑂𝑁𝑁𝑃2 (𝑎, 𝑏)| = 2 × (0.2) − 0.8 − 0.9 = −1.3.
For 𝑎 and 𝑐,
|𝐶𝑂𝑁𝑁𝑃1 (𝑎, 𝑐)| = 2 × (0.2) − 0.8 − 0.9 = −1.3,
|𝐶𝑂𝑁𝑁𝑃2 (𝑎, 𝑐)| = 2 × (0.2) − 0.6 − 0.8 = −1.
For 𝑏 and 𝑐,
|𝐶𝑂𝑁𝑁𝑃1 (𝑏, 𝑐)| = 2 × (0.2) − 0.6 − 0.7 = −0.9,
|𝐶𝑂𝑁𝑁𝑃2 (𝑏, 𝑐)| = 2 × (0.2) − 0.8 − 0.9 = −1.3.
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Since we have
𝐶𝑂𝑁𝑁𝐺 (𝑎, 𝑏) = −1; 𝐶𝑂𝑁𝑁𝐺 (𝑎, 𝑐) = −1;
𝐶𝑂𝑁𝑁𝐺 (𝑏, 𝑐) = −0.9.
Then 𝐶𝐼(𝐺) is calculated as follows
𝐶𝐼(𝐺) = ∑ (𝑇𝑁 (𝑢𝑖 ), 𝐼𝑁 (𝑢𝑖 ), 𝐹𝑁 (𝑢𝑖 ))(𝑇𝑁 (𝑣𝑗 ), 𝐼𝑁 (𝑣𝑗 ), 𝐹𝑁 (𝑣𝑗 )) × 𝐶𝑂𝑁𝑁𝐺 (𝑢𝑖 , 𝑣𝑗 )
𝑢𝑖,𝑣𝑗 ∈𝑉
= (0.3, 0.6, 0.7). (0.3, 0.5, 0.6) × (−1) + (0.3, 0.6, 0.7). (0.4, 0.5, 0.6) × (−1)
+ (0.3, 0.5, 0.6). (0.4, 0.5, 0.6) × (−0.9)
= (0.09 + 0.3 + 0.42)(−1) + (0.12 + 0.3 + 0.42)(−1) + (0.12 + 0.25 + 0.36)(−0.9)
= (0.81)(−1) + (0.84)(−1) + (0.73)(−0.9) = −2.307.
The connectivity index of 𝐺 is equal -2.307, which the negative sing indicates the high level of
false and indeterminacy information in the problem.
Theorem 4. Let 𝐺 and 𝐻 be the two Neutrosophic Graphs are isomorphic, then the topological
indices values of two Neutrosophic Graphs are equal.
Proof. To prove, let 𝐺 = (𝑉𝐺 , 𝑁𝐺 , 𝑀𝐺 ) and 𝐻 = (𝑉𝐻 , 𝑁𝐻 , 𝑀𝐻 ) be isomorphic Neutrosophic Graphs.
Hence there is an identity function 𝜇𝑁 : 𝑁𝐺 (𝑢) → 𝑁𝐻 (𝑢∗ ), for all 𝑢 ∈ 𝑉𝐺 there exist 𝑢∗ ∈ 𝑉𝐻 as well as
𝜇𝑀 : 𝑀𝐺 (𝑢, 𝑣) → 𝑀𝐻 (𝑢∗ , 𝑣 ∗ ), then each vertex of 𝐺 corresponds to an vertex in 𝐻 , with the same
membership value and the same edges. Hence, the Neutrosophic graph structure may differ but
collection of vertices and edges are same gives the equal topological indices value.
Theorem 5. Let 𝐺 = (𝑉𝐺 , 𝑁𝐺 , 𝑀𝐺 ), is a neutrosophic Graph and 𝐻 is the neutrosophic sub graph of 𝐺,
Such that 𝐻 is made by removing edge 𝑢𝑣 ∈ 𝑀𝐺 from 𝐺. Then, we have, 𝐶𝐼(𝐻) < 𝐶𝐼(𝐺) iff 𝑢𝑣 is
a bridge.
Proof. To prove the first side of the theorem we consider two cases:
Case 1. Let 𝑢𝑣 be an edge with all three components having the least value, Therefore the edge 𝑢𝑣
will have no effect on the result. Then we have 𝐶𝐼(𝐻) = 𝐶𝐼(𝐺).
Case 2. Now suppose that 𝑢𝑣 is an edge that has maximum components, so they will have an effect
on 𝐶𝑂𝑁𝑁𝐺 (𝑢, 𝑣). Therefore, by removing edge 𝑢𝑣, the value of 𝐶𝑂𝑁𝑁𝐺 (𝑢, 𝑣) will decrease, then we
have 𝐶𝐼(𝐻) < 𝐶𝐼(𝐺). Since the bridge is called the edge that has its deletion reducing the
𝐶𝑂𝑁𝑁𝐺 (𝑢, 𝑣), However, 𝑢𝑣 is a bridge.
Conversely, given that 𝑢𝑣 is a bridge. According to the definition of bridge we have, for the
edge 𝑢𝑣, 𝐶𝑂𝑁𝑁𝐺 (𝑢, 𝑣) > 𝐶𝑂𝑁𝑁𝐺−𝑢𝑣 (𝑢, 𝑣), So we conclude that, 𝐶𝐼(𝐻) < 𝐶𝐼(𝐺).
4. Applications
Fuzzy set theory and intuitionistic fuzzy set theory are useful models for modelling problems in
real life. But they may not be sufficient in modelling of indeterminate and inconsistent information
encountered in real word. In cases where our information is incomplete or part of our information is
incompatible with each other, depending on the features of the neutrosophic graphs, we can use them
for modeling. However, neutrosophic graphs have many application in real life. For example, social
network model, detection of a safe root for an Airline journey and military problems are application
neutrosophic graph theory [4]. Note that to many applications that neutrosophic graphs have,
obtaining topological indices can be a way to compare the different problems that are modeled by
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neutrosophic graphs. For example, by obtaining different indicators for the two social networks
Telegram and Whatsapp, we can analyze some of the features of the network and their impact.
To see more applications of the neutrosophic graphs, you can refer to [5-12].
Here we refer to one of the applications of the connectivity index for an example of [4].
4.1. Optimal flight path for weather emergency landing
In this application, we use the concept of rough neutrosophic digraph for decision-making in
real-life problems [4]. There, provided a formula for obtaining the desired result, and after
performing the calculations, reached the desired result.
Now, using the connectivity index for different paths, it is possible to predict the optimal path
for flying in weather emergency landing.
Suppose 𝑉 = {𝐶ℎ𝑖𝑐𝑎𝑔𝑜(𝐶𝐻), 𝐵𝑒𝑖𝑗𝑖𝑛𝑔(𝐵𝐽), 𝐿𝑎ℎ𝑜𝑟𝑒(𝐿𝐻), 𝑃𝑎𝑟𝑖𝑠(𝑃𝐴), 𝐼𝑠𝑡𝑎𝑛𝑏𝑢𝑙(𝐼𝑆)} , be the set of
cities under consideration and R an equivalence relation on V, where equivalence classes represent
cities having same characteristics.
Assume that a flight Boeing 747 of Pakistan International Airways (PIA) travels to these cities.
In case of bad weather, the flight will be directed to the city with good weather condition among the
cities under consideration.
Let
𝑁 = {𝐶𝐻, 0.1, 0.2, 0.8), (𝐵𝐽, 0.9, 0.7, 0.5), (𝐿𝐻, 0.8, 0.4, 0.3), (𝑃𝐴, 0.6, 0.5, 0.4), (𝐼𝑆, 0.2, 0.4, 0.6)},
And
𝑀 = {((𝐵𝐽, 𝐶𝐻), 0.1, 0.1, 0.3), ((𝐿𝐻, 𝐶𝐻), 0.1, 0.2, 0.3), ((𝐵𝐽, 𝐿𝐻), 0.1, 0.3, 0.2),
((𝐼𝑆, 𝐵𝐽), 0.2, 0.1, 0.1), ((𝑃𝐴, 𝐵𝐽), 0.1, 0.1, 0.4), ((𝑃𝐴, 𝐿𝐻), 0.2, 0.2, 0.3)}.
Now, we obtain the connectivity index for all paths.
The direct path 𝐵𝐽 _ 𝐶𝐻
𝐶𝑂𝑁𝑁𝑃 (𝐵𝐽, 𝐶𝐻) = 2(0.1) − 0.1 − 0.3 = −0.2 ,
The direct path 𝐵𝐽 _ 𝐿𝐻
𝐶𝑂𝑁𝑁𝑃 (𝐵𝐽, 𝐿𝐻) = 2(0.1) − 0.3 − 0.2 = −0.3,
The direct path 𝐿𝐻 _ 𝐶𝐻
𝐶𝑂𝑁𝑁𝑃 (𝐿𝐻, 𝐶𝐻) = 2(0.1) − 0.2 − 0.3 = −0.3,
The direct path 𝐼𝑆 _ 𝐵𝐽
𝐶𝑂𝑁𝑁𝑃 (𝐼𝑆, 𝐵𝐽) = 2(0.2) − 0.1 − 0.1 = 0.2,
The direct path 𝑃𝐴 _ 𝐵𝐽
𝐶𝑂𝑁𝑁𝑃 (𝐵𝐽, 𝐶𝐻) = 2(0.1) − 0.1 − 0.4 = −0.3,
The direct path 𝑃𝐴 _ 𝐿𝐻
𝐶𝑂𝑁𝑁𝑃 (𝑃𝐴, 𝐿𝐻) = 2(0.2) − 0.2 − 0.3 = −0.1.
Hence, as expected from [4], the weather condition between Beijing and Istanbul is good, and
Boeing 747 can use this path in case of weather emergency. We were able to achieve the desired result
with much shorter calculations. Also, if needed, we can calculate the connectivity index for indirect
paths and finally for neutrosophic graph.
For connectivity index of 𝐺 we have,
Masoud Ghods*, Zahra Rostami, Introduction To Topological Indices in Neutrosophic Graphs
Neutrosophic Sets and Systems, Vol. 35, 2020
76 of 10
𝐶𝐼(𝐺) = ∑ (𝑇𝑁 (𝑢𝑖 ), 𝐼𝑁 (𝑢𝑖 ), 𝐹𝑁 (𝑢𝑖 ))(𝑇𝑁 (𝑣𝑗 ), 𝐼𝑁 (𝑣𝑗 ), 𝐹𝑁 (𝑣𝑗 )) × 𝐶𝑂𝑁𝑁𝐺 (𝑢𝑖 , 𝑣𝑗 )
𝑢𝑖,𝑣𝑗 ∈𝑉
= (0.63)(−0.2) + (0.76)(−0.3) + (0.4)(−0.3) + (1.09)(−0.3) + (0.8)(−0.1)
+ (0.76)(0.2) + (0.5)(−0.3) + (0.58)(−0.2) + (0.8)(−0.5) + (0.48)(−0.3)
+ (0.58)(−0.4) + (0.48)(−0.5)
= −0.126 − 0.228 − 0.12 − 0.327 − 0.08 + 0.152 − 0.15 − 0.116 − 0.4 − 0.144
− 0.232 − 0.24 = −1.783.
As you can see, the negative numerical connectivity index was obtained, which means that our
incorrect information was less than our correct information.
Conclusion
In this paper, for the first time, some topological indices for neutrosophic graphs are defined.
This topic has a lot of work to do, and it can also be used for its results on various issues related to
this category of graphs. In the rest of our research and in future articles, we will address more of these
theorems and their applications.
Funding: “This research received no external funding”
Acknowledgments:
The authors thank the reviewers for their constructive feedback.
Conflicts of Interest: “The authors declare no conflict of interest.”
References
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neutrosophic environment. The Journal of Supercomputing, 76(2), 964-988.
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Decision Making Framework for Professional Selection. Applied Sciences, 10(4), 1202.
Abdel-Basset, M., Mohamed, R., Zaied, A. E. N. H., Gamal, A., & Smarandache, F. (2020). Solving the supply
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Received: Apr 14, 2020. Accepted: July 4, 2020
Masoud Ghods*, Zahra Rostami, Introduction To Topological Indices in Neutrosophic Graphs
Neutrosophic Sets and Systems, Vol. 35, 2020
University of New Mexico
Assessment of MCDM problems by TODIM using aggregated
weights
A. Sahaya Sudha1 Luiz Flavio Autran Monteiro Gomes2 and K.R. Vijayalakshmi3
1Assistant
Professor, Dept. of Maths ,Nirmala College for Women, Coimbatore-18,India
sudha.dass@yahoo.com
2
Professor, Ibmec University Center, Rio de Janeiro, Brazil. luiz.gomes@ibmec.edu.br
3 Ph.D.
Research Scholar, Nirmala College for Women Coimbatore- 18, India krviji71@gmail.com
Correspondence: krviji71@gmail.com
Abstract: Sustainability of sheep and goat production systems is a significant task for any
organization that aims for long term goals. Housing and feeding selection for goat farming is the
most important factor that should be considered before setting out the goat farm. The decision
framework of housing selection should include environmental, social and human impact for the
long term, rather than on short-term gains. In the selection process, various parameters are
involved such as housing materials, area to prevent water stagnation, ventilation, enough space for
the pen and run system, space for feeders and water troughs. Those parameters highlight the quality
of housing in relation to aspects of traditional breeding provided by the organizations. However, the
process of housing selection is often led by hands on experience which contains vague, ambiguous
and uncertain decisions. To overcome this issue it is necessary to frame an efficient algorithm which
could remove the entire barrier in the decision making process. In this paper we propose a
neutrosophic multi-criteria decision making framework that combines the TODIM method with the
SD- HNWA operator. The resulting multi-criteria decision analytical MCDM framework is then
applied in selecting the best system in housing and feeding of goats at a mixed farming agrofarm in
India. The proposed approach allows us to establish the neutrosophic based value function that
measures the degree to which one alternative is superior to others by calculating accurate number of
information in pair wise comparison in terms of gain and loss. The outcomes of the proposed
method are compared with the use of the TOPSIS method to prove its efficiency and validate the
results.
Keywords: MCDM, Hexagonal Neutrosophic numbers, Similarity Degree, Aggregated Weights,
TODIM, TOPSIS.
1. Introduction
Live stock management is considered as one of the most important study topic as it plays a vital
role in self employment for the younger generation with higher level of educational qualifications in
a country like India, with a traditionally high rate of population growth. It is also considered as an
A. Sahaya Sudha, Luiz Flavio Autran Monteiro Gomes and K.R. Vijayalakshmi, Assessment of MCDM problems by
TODIM using aggregated weights
Neutrosophic Sets and Systems, Vol. 35, 2020
79
employment intervention strategy for the younger generation for the self employment of the youth.
Goats are among the main meat producing animal in India where it has huge domestic demand. As a
result, goat production system in India is shifting to intensive system of management.The goat
rearing using improved management practices concentrates on maximization of the returns from the
view of the entrepreneur.
However, without any systematic study it is difficult to assess the economic viability of the goat
farming, as the whole system is built upon nature. The good management practice in livestock
management is the key for the resilience, social, economical and ecological sustainability and
preservation of bio-diversity in pastoral eco-systems, especially in the rural areas where goat
production plays a relevant role in the livehood for farmers. For example, Shalander [25] has
proposed a multi-disciplinary project on transfer of technology for sustainable goat production in
which he indicates that lack of technical knowledge in housing and feeding management system per
capita income in goat rearing is not being up to the expected margin of the goat farmers. Biswas et al.
[9] shows that the growth rate of goat feeder with supplements by additional concentrate with
grazing was more when compared with the normal grazing goats.
In the real world, just like other decision making problem such as supplier selection or
candidate selection, the challenge of uncertainty in the process of housing and feeding selection in
live-stock management is inevitable owing to the fact that the consequences of events are not
precisely known. In addition human judgmental analysis also contributes to its intricacy in the
decision making analysis. To overcome this vagueness and intricacy in decision making this study
aims to propose an integrated framework under neutrosophic environment to evaluate alternative
choices in terms of management system of housing and feeding.
In this research the TODIM and TOPSIS methods will be applied in the processing of selecting
such alternatives. The TODIM method (an acronym for Interactive Multi-Criteria Decision Making
in Portuguese) is a discrete multi-criteria method founded on prospect theory which underlies a
psychological theory in it, while in practice all other discrete multi-criteria methods assume that the
decision maker always looks for the solution corresponding to the maximum of some global
measure. In this way, the method is based on a descriptive theory, proved by empirical evidence, of
how people effectively make decisions when they are under risk. The mathematical structure of
TODIM allows measuring the degree to which one alternative is superior to others and then ranking
the alternatives by computing the global value of each alternative. That structure is embedded in the
paradigm of prospect theory. Gomes and Lima [18] first applied TODIM in its classical formulation
as a tool for ranking projects based on the environmental impacts of alternative road standards in
Brazil. A number of other applications of TODIM has appeared in the literature since then as it is
commented in the section 2.2. Similarly, the TOPSIS method [23] is used to weight and compare
alternatives against a set of criteria and then select the best one. The application of both TODIM and
TOPSIS are then compared one against the other. The novelty of this framework lies in studying the
behavioral risk analysis under neutrosophic environment as pointed out in the above paragraph.
The main contribution of this article is as follows
A. Sahaya Sudha, Luiz Flavio Autran Monteiro Gomes and K.R. Vijayalakshmi, Assessment of MCDM problems by
TODIM using aggregated weights
Neutrosophic Sets and Systems, Vol. 35, 2020
80
A framework is designed that emphasizes the importance of shelter and feeding system for
sustainable and productive goat farming.
Two well established Multi-Criteria Decision Making (MCDM) methods dealing with
imprecise information are applied to a quite important problem in India and compared.
Relevant criteria and sub-criteria are defined for the alternatives to maintain accuracy and
consistency in selecting the alternatives.
2. Literature review
2.1. Commercial goat farming
Raising animals lie upon a set of activities that are dependent upon biotic and socio-economic
factors. Choudhary et al. [35] highlights that India is the rich in its repository of goat genetic resource
with 28 recognized breeds with higher proportion of non-descriptive or mixed breeds. A study was
undertaken by Patil et al. [28] to compare the grazing system and stall feeding system in goats in
Gulbarga District in Karnataka which highlighted that in stall feeding system of goat rearing, goats
are found healthier and weight gain was much faster than grazing system. Kumar [26] investigated
on commercial goat farming in India and presented that planned management and technology
based system would help in increasing the goat productivity in goat farming and bridge the
demand-supply gap. Argüello [8] has presented a review on trends in goat research which talks
about the pathology, reproduction, milk and cheese production and quality, production systems,
nutrition, hair production, drugs knowledge and meat production.
2.2. Multi Criteria Decision Making
Zadeh [42] put forward the concept of fuzzy sets in 1965. Later the theory of fuzzy sets
gradually developed in the further years. The theory of ‘intuitionistic fuzzy set’ [IFS] was proposed
by Atanassov [10] in 1986. Intuitionistic fuzzy set [IFS] was extended to ‘Interval intuitionistic fuzzy
sets’ [IIFS] by Atanassov and Gargov [11]. A number of researchers have contributed their research
to the study of MCDM and a commendable accomplishment has been obtained in fuzzy sets.
Smarandache [36] proposed neutrosophic set based on Neutrosophy in 1998. The neutrosophic
theory takes into account the dynamic features of all limitations to handle uncertain, indeterminate
situations. Abdel-Basset et al. [2] proposed uncertainty assessments of linear time-cost tradeoffs
using neutrosophic set considering the neutrosophic activity duration of time-cost tradeoffs in
project management such as the tradeoffs between the project completion time and the cost and the
uncertain conditions of environment of projects. Abdel-Basset [6] developed and applied a novel
decision making model for sustainable supply chain under uncertainty environment.
Wang et al. [38] developed ‘Single Valued Neutrosophic Set’ (SVNS) and proposed various
properties of set-theoretic operators to deal with uncertain, indeterminate and inconsistent data. Ye
[40] proposed trapezoidal neutrosophic number an extension from SVNS and trapezoidal fuzzy
number and defined its score and accuracy function with aggregating operators in [41].
Smarandache [37] introduced the plithogenic set as generalization of crisp, fuzzy, intuitionistic fuzzy
and neutrosophic sets whose elements are characterized by many attribute values which have
A. Sahaya Sudha, Luiz Flavio Autran Monteiro Gomes and K.R. Vijayalakshmi, Assessment of MCDM problems by
TODIM using aggregated weights
Neutrosophic Sets and Systems, Vol. 35, 2020
81
corresponding contradiction degree values between each attribute value and the dominant attribute
value. Abdel-Basset [1] developed an evaluation framework based on plithogenic set theory for
smart disaster response systems in uncertainty environment that deals more effectively with
disaster by the effective communication of the information provided by the sensors with the
response teams.
Decision making situations in real life are much complicated when the decision makers (DMs)
have to fit in the best alternatives with respect to the given multiple criteria. Biswas et al. [14]
established TOPSIS strategy for (MCDM) in trapezoidal neutrosophic environment using the
maximum deviation strategy and also developed an optimization model to obtain the weight of the
attributes which are incompletely known or completely unknown. Abdel-Basset [5] proposed a
decision making problem to solve a supply chain problem of inventory location using the best-worst
method based on a novel plithogenic model.
Pramanik and Mallick [30] proposed a VIKOR method for group Decision Making Problem
involving trapezoidal neutrosophic number and they adapted a problem of Investment Company
from [16] and provided a comparative analysis. Mondal and Pramanik [29] proposed MCDM
approach for teacher recruitment in higher education with unknown weights based on score and
accuracy function, hybrid score and accuracy functions under simplified neutrosophic environment.
Biswas et al. [12,13] developed a new methodology for neutrosophic MCDM with unknown weight
information and a Cosine similarity measure based MCDM with trapezoidal fuzzy neutrosophic
numbers. Abdel-Basset [3] designed resource levelling problem to minimize the cost of daily
resource fluctuation in construction projects under neutrosophic environment to overcome the
ambiguity caused by real world problems.
Based on observations of human behaviour, studies have found that human decision making is
not completely rational under practical decision situations. After undertaking a number of surveys
and experiments, Kahneman and Tversky [24] proposed Prospect theory partially the subject of the
Nobel Prize for Economics awarded in 2002, which belongs to the field of cognitive psychology and
describes how people make decision under conditions of risk.
Gomes and Lima [20] used the TODIM method in order to show how human judgements in
practical multi-criteria analysis fit in to the framework of Prospect Theory and additive difference
model. Gomes et al. [19] used the classical TODIM formulation to recommend alternatives for
destination of natural gas reserves recently discovered in Santos Basin in Brazil. Gomes et al. [22]
proposed a behavioural multi-criteria decision analysis by using the TODIM method with criteria
interactions. Gomes and Rangel [21] developed a novel approach using TODIM method on rental
evaluation of residential properties carried out together with real estate agents in the city of Volta
Redonda, Brazil which has made many successful applications in selection problems. Zindani et al.
[44] proposed a material selection approach using the TODIM method and applied it to find the best
suited materials for two products, engine flywheel and metallic gear.Duarte [7] proposed the use of
multi criteria decision analysis to valuation of six Brazilian banks by applying the fuzzy TODIM
method.
A. Sahaya Sudha, Luiz Flavio Autran Monteiro Gomes and K.R. Vijayalakshmi, Assessment of MCDM problems by
TODIM using aggregated weights
Neutrosophic Sets and Systems, Vol. 35, 2020
82
Sang and Liu [34] developed the IT2 FSs-based TODIM method to green supplier selection for
automobile manufacturers by introducing a new distance computing method. Wang et al. [39]
proposed a likehood-based TODIM approach on multi-hesitant fuzzy linguistic information
(MHFLSs) which is an extension of (HFLSs) for selection and evaluation of contractors in logistics
outsourcing. Chakraborty and Chakraborty [15] used TODIM in identifying the most attractive and
affordable under-construction housing project in the city of Kolkata in India. Rangel et al. [32] used
TODIM a multi-criteria decision aiding method in the evaluation of the various types of access to the
broadband internet available in Volta Redonda, Brazil. Candidate selection is a significant task for
any organization that aims to select the most appropriate candidates who lead the firm forward
through his strong organizational skill. To overcome this tough task Abdel-Basset [4] proposed a
bipolar neutrosophic multi criteria decision making framework for professional selection that
employs a collection of neutrosophic analytical network process and TOPSIS under bipolar
neutrosophic numbers.
Lourenzutti and Krohling [27] combined TOPSIS and TODIM methods to propose the
Hellinger distance in MCDM which serves as an illustration to both methods. Fan et al. [17]
proposed an extension of TODIM (H-TODIM) to solve the hybrid MCDM problem in which
attribute values have three forms crisp number, interval number and fuzzy number. Ren et al. [33]
proposed a Pythagorean fuzzy TODIM approach to analyse MCDM problem. Qin et al. [31]
proposed generalizing of the TODIM method under triangular intuitionistic fuzzy environment.
Zhang et al. [43] proposed an extended multiple attribute group decision making based on the
TODIM method to solve the MCDM problem in which the attribute values are expressed with
neutrosophic number.
3. Preliminaries
3.1. Hexagonal Neutrosophic Weighted Aggregated Operator (HNWA)
~
Let A
be a collection of
1 = ( a1 ,b1 ,c1 , d1 ,e1 , f1 ), (l1 , m1 , n1 , p1 , q1 , r1 ), (u1 ,v1 , w1 , x1 , y1 , z1 )
hexagonal neutrosophic numbers, then the HNWA: is defined as follows
n
~ ~
~
n
~
HNWA ( A1 , A2 ,......... . An ) j A j
j 1
A. Sahaya Sudha, Luiz Flavio Autran Monteiro Gomes and K.R. Vijayalakshmi, Assessment of MCDM problems by
TODIM using aggregated weights
Neutrosophic Sets and Systems, Vol. 35, 2020
83
n
n
n
ωj
ω
ω
, 1 (1 b j ) j , 1 (1 c j ) j ,
1 (1 a j )
j =1
j =1
j =1
n
n
n
ωj
ω
ω
1 (1 d j )
, 1 (1 e j ) j , 1 (1 f j ) j
j =1
j =1
j =1
n
n
n
n
n
n
l j ω j , m j ω j , n j ω j , p j ω j q j ω j , j =1 r j ω j
j =1
j =1
j =1
j =1
j =1
j =1
(1)
n
n
n
n
n
n
u j ω j , vj ω j , wj ω j , xj ω j , y j ω j , z j ω j
j =1
j =1
j =1
j =1
j =1
j =1
where
T
w ( w1 , w2 , w3 , w4 ......wn ) is a
weight
n
~
vector of A j and w j 1 , w j 0
j =1
3.2. Distance between two Hexagonal Neutrosophic numbers
~
Let A
1 = ( a1 ,b1 , c1 , d1 , e1 , f1 ), (l1 , m1 , n1 , p1 , q1 , r1 ), (u1 , v1 , w1 , x1 , y1 , z1 )
~
A2
= (a2 , b2 , c2 , d 2 , e2 , f 2 ), (l 2 , m2 , n2 , p 2 , q2 , r2 ), (u 2 , v2 , w2 , x2 , y 2 , z 2 ) be two
~
~
hexagonal neutrosophic numbers then the weighted distance between A1 and A2 is defined as
follows.
a1 a 2 b1 b2 c1 c 2 d1 d 2 e1 e2 f1 f 2
1
~ ~
l1 l 2 m1 m2 n1 n2 p1 p 2 q1 q 2 r1 r2 (2)
d ( A1 , A2 )
18
u u v v w w x x y y z z
2
1
2
1
2
1
2
1
2
1
2
1
3.3. Similarity Degree between two Hexagonal Neutrosophic numbers
~
Let A1 (a1 , b1, , c1, , d1, , e1, , f1, ), (l1, , m1, , n1, , p1, , q1 , r1, ), (u1 , v1 , w1 , x1 , y1 , z1 ) and
~
A2 [(a 2 , b2 , c 2 , d 2 , e 2 , f 2 ), (l 2 , m 2 , n 2 , p 2 , q 2 , r2 ) , (u 2 , v 2 , w2 , x 2 , y 2 , z 2 )]
be two hexagonal neutrosophic numbers and let
~
A2C [(u2 ,v2 , w2 , x2 , y2 , z2 ), (1 l2 ,1 m2 ,1 n2 ,1 p2 ,1 q2 ,1 r2 ), (a2 , b2 , c2 , d 2 , e2 , f 2 )] be the
~
complement of A2 then the Degree of Similarity between
~
~
~
~
A1 and A2 is defined as follows.
~ ~
d ( A1 , A2C )
~ ~
~ ~ C (3)
d ( A1 , A2 ) d ( A1 , A2 )
( A1 , A2 )
3.4. Hexagonal Neutrosophic Decision Matrix
~
rij ) m n
Let R ( ~
.If all ~
rij are hexagonal neutrosophic number then
A. Sahaya Sudha, Luiz Flavio Autran Monteiro Gomes and K.R. Vijayalakshmi, Assessment of MCDM problems by
TODIM using aggregated weights
Neutrosophic Sets and Systems, Vol. 35, 2020
84
~
~
~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
~
R~
rij a~ij , bij , c~ij , dij , e~ij , f ij , lij , m
ij , nij , pij , qij , rij , uij , vij , wij , xij , yij , zij
is a hexagonal neutrosophic decision matrix.
3.5. Aggregated Hexagonal Neutrosophic Decision Matrix
~ (k )
Let R
(~
rij( k ) ) mn (k 1,2,3,......t ) be a‘t’ neutrosophic decision matrix evaluated
by the decision makers DM d (d 1,2,3.....m) respectively, then the aggregated hexagonal
~
neutrosophic decision matrix R (~
rij ) m n is defined as
~
~
~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
~
rij a~ij , bij , c~ij , d ij , e~ij , f ij , lij , m
ij , nij , pij , qij , rij , u ij , vij , wij , xij , y ij , z ij
t
1
(i 1,2,3.....m), ( j 1,2....n) where ~
r ~
rij( k )
t k 1
3.6. Degree of Similarity
~
Let R ( k ) ( ~
rij( k ) ) m n (k 1,2,3,......t ) be a‘t’ neutrosophic decision matrix and
~
R (~
rij ) m n be their aggregated hexagonal neutrosophic decision matrix then
~
~
( R ( k ) , R )
1 m n ~ (k ) ~
(r , rij ) (4)
m n i 1 j 1 ij
~ (k )
is called the degree of similarity between R
~
and R
3.7. Determine the weight of experts using Degree of Similarity:
If the hexagonal neutrosophic decision matrix R~ ( k ) (~rij ( k ) )
are
m n (k ,1,2,.......t )
non-identical,then the weight vectors of the experts are expressed as follows.
w
(k )
~
~
( ( R ( k ) , R ))
t
~
~
( ( R ( k ) , R ))
(5)
k 1
4. A Comparative Analysis of TODIM and TOPSIS Methods.
4.1. TODIM
To solve the MCDM problem with hexagonal neutrosophic information’s we propose a
hexagonal neutrosophic aggregation TODIM method based on prospect theory under the decision
maker’s behavioral risk and arithmetic mean operator.
Let Ai ( A1 , A2 ,....Am ) be the alternatives, and C j {C1 , C 2 ,.......,C n } be the criteria.
A. Sahaya Sudha, Luiz Flavio Autran Monteiro Gomes and K.R. Vijayalakshmi, Assessment of MCDM problems by
TODIM using aggregated weights
Neutrosophic Sets and Systems, Vol. 35, 2020
85
Let w ( w1 , w2 ,....wn ) be the weights of C j ,0 w j 1, and
~
Rk ~
rij (k )
mn
n
wj
1. Let,
j 1
~
~
~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~
(Tij , I ij , Fij ) mn (a~ij ,bij , c~ij , d ij , e~ij , f ij ),(lij , m
ij , nij , pij , qij , rij ),(uij , vij , wij , xij , yij , z ij )
be a hexagonal neutrosophic decision matrix, where ~
rij (Tij , I ij , Fij ) is an attribute value
given by the experts for the alternatives Ai with the criteria C j , Tij [0,1], I ij [0,1], Fij [0,1] ,
0 Tij I ij Fij 3 (i 1,2,......,m), ( j 1,2,....,n)
The proposed method is presented as follows.
Stage 1.
Step 1. Construct a decision matrix of dimension m n by using the information provided by the
decision maker for the alternatives Ai under the criteria C j . The mth hexagonal neutrosophic decision
matrix denoted by the decision maker is defined as follows.
(a1n ,b1n ,c1n ,d1n ,e1n , f1n )
(a11 ,b11 ,c11 ,d11 ,e11 , f11 )
(l ,m ,n , p ,q ,r )
(l1n ,m1n ,n1n , p1n ,q1n ,r1n )
11 11 11 11 11 11
(u11 ,v11 , w11 , x11 , y11 , z11 )
(u1n ,v1n , w1n , x1n , y1n , z1n )
~
Rk
(a ,b ,c ,d ,e , f )
(a mn ,bmn ,cmn ,d m ,emn , f mn )
m1 m1 m1 m1 m1 m1
(l m1 ,mm1 ,nm1 , pm1 ,qm1 ,rm1 ) (l mn ,mmn ,nmn , pmn ,qmn ,rmn )
(u mn ,vmn , wmn , xmn , y mn , z mn )
(u m1 ,vm1 , wm1 , xm1 , y m1 , z m1 )
Step 2: Find the aggregated hexagonal neutrosophic decision matrix of all the three decision
~
makers..The aggregated hexagonal neutrosophic decision matrix R (~
rij) m n is defined as given
below.
~
rij aij , bij , cij , d ij , eij , f ij , lij , mij , nij , pij , qij , rij , uij , vij , wij , xij , yij , zij
(i 1,2,3.....,m), ( j 1,2....,n)
where r
1 t ~ (k )
r
t k 1 ij
~ ~
Step 3. Calculate the normalized hamming distance for each ( R , R ) using the equation (2)
Step 4. Calculate the Degree of Similarity between A1 and A2
using equation (3) and (4)
A. Sahaya Sudha, Luiz Flavio Autran Monteiro Gomes and K.R. Vijayalakshmi, Assessment of MCDM problems by
TODIM using aggregated weights
Neutrosophic Sets and Systems, Vol. 35, 2020
Step 5. Calculate the weight vector w
86
(k )
using equation (5)
Step 6. Using equation (1) calculate HNWA operator
Step 7. Calculate the score value using the equation
1
a b c d e f l m n p q r u v w x y z
S ( A) 2
(6)
3
6
6
6
Step 8. Calculate the normalized hamming distance for the aggregated decision matrix using (2)
Step 9. When the aggregated matrix is brought into expression (7), matrix ( Ai , A p ) will be
derived .The function ( Ai , Ap ) is used to represent the degree to which alternative i is better than j.
j Ai , Ap is the sum of the sub-function where j 1...., n . Sub-function j Ai , Ap indicates the
degree to which i is better than j when a particular criteria c is given
w jr d (~
rij ~
r pj )
n
w jr
j
1
j Ai , A p 0
n
rij ~
r pj )
w jr d (~
1 j 1
w jr
if
~
rij ~
r pj 0
~
rij ~
r pj 0
if
if
(7 )
~
rij ~
r pj 0
The parameter shows the dilution factor of the loss. If ~
rij ~
r pj 0 then j Ai , A p
represents the gain and if ~
rij ~
r pj 0 then
j Ai , A p represents the loss.
Step 10. On the basics of the above equation the overall dominance degree is obtained as
x
n
j ( Ai , Ap ), (i, p 1,2,.....m)
j 1
Step 11. Calculate the aggregated dominance matrix
n
( Ai , A p ) x x ( Ai , A p ), (i, p 1,2,.....m) (8)
x 1
Step 12. Calculate the overall dominance degree matrix
Step 13. Then the overall value of each
[ ( Ai , A p )] mn
Ai can be calculated using the equation
A. Sahaya Sudha, Luiz Flavio Autran Monteiro Gomes and K.R. Vijayalakshmi, Assessment of MCDM problems by
TODIM using aggregated weights
Neutrosophic Sets and Systems, Vol. 35, 2020
87
m
m
( Ai , A p ) min ( Ai , A p )
p 1
(9)
m
m
max ( Ai , A p ) min ( Ai , A p )
i p 1
i p 1
Step 14. Rank all alternatives and select the most desirable one in accordance with Ai . The
i
p 1
( Ai )
alternative with maximum value is the best one.
4.2. TOPSIS
Stage 2: Applying the information’s derived from step 1 to 6 in stage 1, move on to step 7 of stage 2
Step 7: Let B1 be the set of benefit attributes and B2 be the set of cost attributes, of the alternatives
respectively. Let B
be the hexagonal neutrosophic positive ideal solution and B be the
hexagonal neutrosophic negative ideal solution. Then B
and B are defined as follows.
B r j (1,1,1,1,1,1), (0,0,0,0,0,0), (0,0,0,0,0,0) j B1 , r j (0,0,0,0,0,0), (1,1,1,1,1,1), (1,1,1,1,1,1) j B2
B r j (0,0,0,0,0,0), (1,1,1,1,1,1), (1,1,1,1,1,1) j B1 , r j , (1,1,1,1,1,1), (0,0,0,0,0,0), (0,0,0,0,0,0) j B2
Step 8: Calculate the separation measures, Si and Si of each alternative from the hexagonal
neutrosophic positive ideal solution and the hexagonal neutrosophic negative ideal solution as
follows.
1 n
w j d (rij , r j ) (10)
n j 1
1 n
S i
w j d (rij , r j ) (11)
n j 1
S i
Step 9: Calculate the relative closeness coefficient of the hexagonal neutrosophic ideal solution. The
relative closeness coefficient of the alternative Ai is given as follows.
Ci
Si
Si Si
,0 Ci 1 (12)
Step 10: Make a decision for selecting the preference alternative by ranking the closeness
coefficient in the descending order of C i to select the best choice.
5. Case Analysis:
In this section, a case study is represented for the proposed multi-criteria group decision-making
method. This is related to assessing the best system of housing and feeding of goats in the existing
A. Sahaya Sudha, Luiz Flavio Autran Monteiro Gomes and K.R. Vijayalakshmi, Assessment of MCDM problems by
TODIM using aggregated weights
Neutrosophic Sets and Systems, Vol. 35, 2020
88
Goat farm rearing in which goats grow healthier, gain better body weight, and are safer on health
grounds. A group of three decision-makers (D1, D2 and D3) are requested to assess the four
alternatives (A1to A4) with respect to the four criteria’s, (C1 to C4) defined by this group of
decision-makers to appraise the alternatives. These criteria and their definitions are represented as
follows:
Alternatives:
A1 -
Stall feeding system with normal flooring (intensive system)
A2 - Grazing system (extensive system)
A3 - Elevated floor shed with rotational grazing system
A4 - A part of both extensive and intensive grazing system
The consideration of the criteria and sub criteria’s after a brief study on the previous literature
review and discussion with the experts are stated below.
Criteria:
C1 - Floor space requirements
(Covered area, Open area, Ventilation, Bedding, Confinement, Site location)
C 2 - Feeding (Feeder) and watering space requirement
(Feeder size, Fodder type, Quantity, Food Schedule, immunization feeder, feed storage room)
C 3 - Maintenance of health and sanitization
(Nutritional ratio, Vaccination, Climate pattern, Temperature, Supplementary feeding,
Cleanliness)
C 4 - Productivity
(Capital, Typologies of farms, Technology integration, Agro climatic characteristics, Market
value, Place of selling)
A questionnaire is prepared and handed over to the domain experts. These experts further
graded the degree of the statement as given below.
Statement Very
high
Score
0.9
High
Fair
Average
Medium
Satisfactory
Low
Very
low
Not
sure
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
Table 5.1. Rating scale used by experts
Solution.
Step 1. The judgment of the three decision makers for the alternatives Ai under the four criteria were
presented using hexagonal neutrosophic number as shown in Table 5.2 .
Criteria
DMs
Alternatives
A1
C1
C2
[(.1,.2,.3,.4,.5,.6),
[(.1,.2,.3,.4,.5,.6),
(.1,.1,.1,.1,.1,.1),
(.5,.6,.7,.8,.9,.9)]
(.1,.1,.1,.1,.1,.1),
(.5,.6,.7,.8,.9,.9)]
C3
[(.8,.8,.8,.8,.9,.9),
(.3,.4,.5,.6,.7,.8),
(.1,.1,.1,.1,.1,.1)]
C4
[(.2,.3,.4,.5,.6,.7),
(.1,.2,.2,.3,.3,.3),
(.5,.5,.6,.6,.7,.7)]
A. Sahaya Sudha, Luiz Flavio Autran Monteiro Gomes and K.R. Vijayalakshmi, Assessment of MCDM problems by
TODIM using aggregated weights
Neutrosophic Sets and Systems, Vol. 35, 2020
A2
A3
D1
A4
A1
A2
[(.3,.4,.5,.6,.7,.8),
(.2,.3,.4,.5,.5,.6),
(.6,.6,.7,.8,.8,.8)]
A3
A4
A1
A3
A4
[(.8,.8,.8,.9,.9,.9),
(.2,.3,.3,.3,.4,.5),
(.1,.1,.2,.2,.2,.2)]
[(.2,.3,.4,.5,.6,.7),
[(.1..2,.3,.5,.6,.7),
(.3,.3,.4,.5,.8,.9.)
(.5,.5,.5,.6,.6,.6)]
[(.9,.9,.9,.9,.9,.9)
(.1,.2,.2,.3,.3,.3),
(.5,.5,.6,.6,.7,.7)]
[(.1,.2,.2,.3,.3,.3)
, (.2,.2,.3,.3,.5,.5)
(.4,.5,.6,.7,.8,.9, )]
[(.3,.3,.4,.4,.4,.5),
[(.1,.2,.3,.4,.5,.6)
[(.6,.6,.6,.7,.7,.8),
(.1,.2,.2,.3,.4,.4)
(.5,.5,.5,.6,.6,.6)]
, (.3,.3,.4,.6,.6,.6)
(.3,.4,.5,.5,.6,.6)]
(.5,.5,.5,.6,.6,.6)
(.2,.2,.3,.3,.3,.3)]
[(.6,.7,.8,.9,.9,.9),
(.1,.2,.3,.3,.4,.5)
(.3,.4,.4,.4,.5,.5)]
[(.3,.4,.5,.6,.7,.8)
[(.3,.4,.4,.4,.4,.4)
[(.8,.8,.8,.8,.8,.8)
, (.2,.3,.4,.5,.6,.7)
(.4,.5,.6,.7,.8,.9)]
, (.2,.2,.2,.2,.2,.2)
(.4,.4,.5,.5,.5,.5)]
, (.2,.2,.3,.3,.4,.4)
(.2,.3,.3,.4,.4,.5)]
[(.5,.5,.5,.5,.5,.5),
[(.4,.5,.6,.7,.8,.8),
(.3,.3,.3,.4,.4,.4),
(.2,.3,.4,.5,.6,.7)]
[(.5,.6,.6,.7,.7,.8),
(.3,.4,.5,.6,.6,.7),
(.5,.5,.6,.7,.8,.8)]
[(.8,.8,.9,.9,.9,.9),
(.2,.2,.3,.3,.4,.4),
(.2,.3,.3,.4,.4,.5)]
(.3,.4,.4,.4,.4,.4),
(.4,.4,.4,.5,.5,.5)]
[(.1,.3,.3,.3,.4,.4),
[(.6,.7,.7,.8,.8,.8),
(.1,.2,.3,.4,.5,.6),
(.4,.4,.5,.5,.6,.6)]
[(.4,.4,.5,.6,.7,.7),
(.2,.2,.3,.3,.4,.4),
(.1,.2,.3,.4,.4,.5)]
[(.8,.9,.9,.9,.9,.9),
(.1,.1,.2,.2,.3,.3),
(.1,.1,.1,.1,.1,.1)]
[(.4,.4,.5,.6,.7,.8),
[(.6,.7,.8,.8,.9,.9),
(.3,.3,.3,.4,.4,.5),
(.4,.5,.6,.6,.6,.6)]
[(.2,.3,.4,.5,.6,.7),
(.6,.6,.6,.6,.6,.6),
(.2,.2,.2,.3,.4,.5)]
[(.3,.4,.4,.5,.5,.6),
(.1,.1,.1,.2,.2,.2),
(.1,.2,.3,.4,.5,.6)]
[(.1,.2,.2,.2,.3,.3),
[(.4,.5,.5,.6,.6,.7)
, (.4,.4,.5,.6,.6,.7)
(.4,.5,.6,.7,.8,.9)]
D3
(.5,.6,.7,.8,.8,.8),
(.6,.6,.7,.7,.8,.8)]
[(.1,.1.1,.1,.1,.1)
(.1,.1,.1,.2,.3,.4),
(.5,.6,.7,.7,.7,.7)]
A2
[(.2,.2,.2,.2,.2,.2),
[(.4,.5,.6,.7,.8,.9),
(.2,.3,.4,.6,.7,.8),
(.2,.2,.2,.2,.2,.2)]
(.2,.3,.4,.5,.6,.7),
(.5,.5,.5,.5,.5,.5)]
D2
89
[(.6,.7,.7,.8,.9,.9),
(.5,.6,.7,.7,.8,.8),
(.3,.4,.5,.5,.6,.6)]
[(.6,.6,.6,.7,.7,.7),
(.4,.5,.6,.6,.7,.7),
(.4,.5,.5,.5,.5,.6)]
, (.3,.4,.5,.6,.7,.8)
(.4,.4,.5,.5,.6,.6)]
[(.5,.6,.7,.8,.9,.9),
(.4,.5,.6,.6,.7,.7),
(.3,.4,.4,.5,.5,.5)]
[(.7,.7,.7,.8,.8,.8),
(.5,.6,.6,.6,.6,.6),
(.4,.4,.5,.5,.6,.6)]
[(.4,.5,.6,.7,.8,.9),
(.3,.3,.3,.3,.3,.3),
(.4,.5,.5,.6,.6,.6)]
[(.6,.7,.7,.8,.8,.8),
(.4,.5,.5,.6,.6,.7),
(.2,.3,.4,.5,.6,.7)]
(.7,.7,.7,.8,.8,.8),
(.1,.2,.3,.4,.5,.6)]
[(.8,.8,.8,.9,.9,.9),
(.5,.5,.6,.6,.7,.7),
(.2,.3,.3,.4,.4,.5)]
[(.8,.8,.8,.9,.9,.9),
(.4,.4,.4,.4,.4,.4),
(.3,.4,.4,.5,.5,.5)]
[(.3,.4,.5,.6,.7,.8),
(.5,.5,.5,.5,.5,.5),
(.4,.5,.5,.6,.7,.8)]
(.1,.2,.3,.4,.5,.6)
(.3,.4,.4,.5,.6,.7)]
[(.5,.5,.5,.6,.6,.7),
(.2,.2,.2,.3,.3,.3),
(.3,.3,.4,.5,.6,.7)]
[(.3,.3,.4,.4,.4,.5),
(.4,.5,.6,.7,.8,.9),
(.3,.4,.4,.5,.5,.6)]
[(.2,.2,.2,.3,.3,.3),
(.2,.2,.2,.3,.4,.5),
(.4,.5,.5,.6,.7,.8)]
[(.6,.7,.7,.8,.8,.9),
(.1,.1,.1,.1,.1,.1),
(.2,.2,.2,.3,.3,.3)]
[(.2,.3,.3,.3,.4,.4),
(.6,.6,.6,.7,.7,.7),
(.8,.8,.8,.9,.9,.9)]
Table 5.2 Opinion of decision makers on performance values
~k
Step 2. Normalize the hexagonal neutrosophic decision matrix R
(~
rijk ) mn given by the
~
rij ) mn
experts Dk (k 1,2,3) to get the matrix R (~
Criteria
Alternative
C1
C2
C3
C4
A. Sahaya Sudha, Luiz Flavio Autran Monteiro Gomes and K.R. Vijayalakshmi, Assessment of MCDM problems by
TODIM using aggregated weights
Neutrosophic Sets and Systems, Vol. 35, 2020
A1
[(.3,.4,.5,.6,.7,.7 ),
[(.3,.3,.4,.5,.6,.7 ),
[(.6,.6,.7,.7,.7,.7 ),
(.1,.1,.2,.2,.3,.3),
(.3,.4,.4,.5,.6,.6),
(.3,.3,.3,.4,.4,.5),
(.2,.3,.3,.4,.4,.4 ),
(.4,.5,.6,.6,.7,.7 )]
(.4,.5,.6,.7,.7,.7 )]
(.3,.3,.4,.4,.4,.4)]
(.4,.4,.4,.5,.5,.6 )]
[(.6,.7,.7,.8,.8,.8),
(.3,.4,.4,.5,.6,.7),
(.3,.3,.4,.5,.5,.6)]
[(.5,.5,.6,.6,.6,.7).
(.4,.5,.5,.6,.6,.7),
A2
A3
A4
90
(.3,.3,.4,.5,.6,.7),
(.5,.5,.6,.7,.7,.7)]
[(.4,.5,.5,.6,.6,.6),
(.4,.5,.5,.6,.6,.6),
(.4,.4,.5,.6,.6,.7)]
[(.4,.5,.6,.7,.8,.8),
[(.3,.4,.5,.5,.6,.6),
(.4,.5,.5,.6,.6,.7),
(.2,.3,.3,.3,.4,.4)]
(.4,.5,.5,.6,.6,.7),
(.3,.3,.4,.5,.5,.6)]
[(.3,.4,.5,.6,.6,.7),
(.4,.4,.4,.5,.6,.6),
(.3,.4,.4,.5,.5,.6)]
[(.3,.4,.4,.5,.5,.5),
[(.5,.5,.6,.6,.7,.7),
[(.4,.5,.6,.6,.7,.8),
(.2,.3,.3,.4,.5,.5),
(.2,.3,.3,.3,.4,.4),
(.3,.4,.4,.5,.5,.5)]
(.3,.3,.4,.4,.5,.5),
(.3,.3,.3,.4,.4,.5)]
(.3,.4,.5,.6,.6,.7)]
[(.4,.5,.5,.6,.6,.7 ),
(.2,.2,.3,.3,.4,.4),
(.2,.3,.3,.4,.5,.6)]
[(.7,.8,.8,.8,.8,.9),
(.1,.2,.2,.3,.4,.4),
(.3,.3,.4,.4,.5,.5)]
[(.4,.5,.5,.5,.6,.6),
(.4,.4,.4,.5,.5,.5),
(.4,.4,.5,.6,.6,.6)]
Table 5.3 Normalized hexagonal neutrosophic decision matrix
Step 3.
Once the decision makers provide the decision matrix we calculate the relative weight of each
criterion C j Consider the weight of each criterion as w 0.15,0.15,0.20,0.50
wr max{ w j / j 1,2,.....n}
wr max 0.15,0.15,0.20,0.50
wr 0.50
Since wr 0.50 then C 4 is the reference criterion and the reference criterion weight is 0.50. Then
calculate the relative weights of the criterion C j ( j 1,2,3,4) as
w1r
w1 0.15
0.3, w2r 0.3, w3r 0.4, w4r 1
wr 0.50
The parameter
the dilution factor of the loss is
4
w jr 0.3 0.3 0.4 1 2
j 1
Step 4. Consider the alternative A1 of DM 1 and the criteria C1
Calculate the distance between A1 and A1 , A1 and A1 C of DM 1
C1 [(.1,.2,.3,.4,.5,.6), (.1,.1,.1,.1,.1,.1) (.5,.6,.7,.8,.9,.9)],
C1 [(.3,.4,.4,.5,.6,.6), (.1,.1,.2,.2,.3,.3) (.4,.5,.6,.6,.7,.7)]
C1C [(.4,.5,.6,.6,.7,.7) (.9,.9,.8,.8,.7,.7) (.3,.4,.4,.5,.6,.6)] d (C1, C1)
1
1
(2.2), d (C1, C1C ) (6.9)
18
18
Step 5. The Degree of Similarity between A1 and A is defined as follows.
A. Sahaya Sudha, Luiz Flavio Autran Monteiro Gomes and K.R. Vijayalakshmi, Assessment of MCDM problems by
TODIM using aggregated weights
Neutrosophic Sets and Systems, Vol. 35, 2020
91
d (C1, C1C )
(C1 , C1)
C
d (C1, C1) d C1, C1
1
(6.9)
18
0.76
1
(2.2 6.9)
18
Continuing the above process for all decision makers the consolidated Degree of Similarity is
tabulated below.
Degree of
A1 of
A2 of
A3 of
A4 of
A1 of
A2 of
A3 of
A4 of
A1 of
A2 of
A3 of
A4 of
Similarity
DM1
DM1
DM1
DM1
DM2
DM2
DM2
DM2
DM3
DM3
DM3
DM3
(C1 , C1)
0.76
0.70
0.76
0.68
0.56
0.69
0.66
0.51
0.77
0.77
0.66
0.42
(C1 , C 2 )
0.68
0.53
0.53
0.57
0.74
0.56
0.50
0.81
0.45
0.57
0.59
0.72
(C3 , C3)
0.66
0.61
0.60
0.48
0.54
0.61
0.38
0.65
0.72
0.83
0.56
0.45
(C4 , C4)
0.62
0.57
0.81
0.50
0.68
0.78
0.80
0.51
0.44
0.54
0.80
0.45
Table 5.4 Degree of Similarity between the alternatives compared with the criteria
Step 6. Calculate the weight vectors of the decision makers using degree of similarity
~
~
( R ( k ) , R )
~
~
( R (1) , R )
w (1)
1 m n ~ (k ) ~
(rij , rij )
m n i 1 j 1
10.524
10..097
9.9000
~
~
~
~
.877, ( R ( 2) , R )
0.8814, ( R (3) , R )
0.825
12
12
12
~
~
( ( R (1) , R )
t
~
~
( ( R (k ) , R )
~
~
( ( R ( 2) , R )
.877
.34 w ( 2)
2.59
t
k 1
w (3)
~
~
( ( R ( 2) , R )
t
~
~
( ( R (k ) , R ))
~
~
( ( R (k ) , R ))
.8814
.33
2.59
k 1
.825
.33
2.59
k 1
Step 7. Using equation (1) HNWA the aggregated decision matrix is as follows.
Criteria
Alternative
C1
C2
C3
C4
A1
[(.4,.5,.6,.7,.8,.9),
(0,.1,.2,.3,.4,.5),
(.1,.1,.1,.1,.1,.1)]
[(0,.1,.2,.3,.4,.5)
(0,.1,.2,.3,.4,.5)
(.2,.3,.4,.5,.6,.7)]
[(.3,.4,.5,.6,.7,.8),
(0,.1,.2,.3,.4,.5),
(.1,.1,.1,.1,.1,.1)]
[(.3,.4,.5,.6,.7,.8),
(.1,.1,.1,.1,.1,.1),
(.1,.2,.3,.4,.5,.6)]
A2
[(.3,.4,.5,.6,.7,.8),
(.1,.1,.2,.2,.3,.3),
(0,.1,.1,.1,.2,.2)]
[(.1,.2,.3,.4,.5,.6),
(0,.1,.1,.2,.3,.3),
(0,.1,.1,.1,.2,.2)]
[(0,.1,.1,.2,.2,.2),
(.1,.1,.1,.1,.1,.1),
(.5,.6,.6,.7,.7,.8)]
[(.3,.4,.5,.5,.6,.6),
(0,0,.1,.1,.1,.1),
(.1,.1,.1,.1,.1,.1)]
A. Sahaya Sudha, Luiz Flavio Autran Monteiro Gomes and K.R. Vijayalakshmi, Assessment of MCDM problems by
TODIM using aggregated weights
Neutrosophic Sets and Systems, Vol. 35, 2020
92
A3
[(.1,.1,.1,.1,.2,.2),
(.2,.2,.2,.2,.3,.3),
(.4,.4,.5,.5,.6,.6)]
[(.2,.2,.2,.3,.3,.3),
(0,.1,.2,.2,.3,.3),
(.4,.5,.6,.7,.8,.9)]
[(.3,.4,.5,.6,.7,.8),
(0,.1,.2,.3,.4,.5),
(.1,.2,.2,.2,.3,.3)]
[(.1,.2,.3,.4,.5,.6),
(.2,.2,.2,.2,.2,.2),
(.4,.5,.6,.7,.8,.9)]
A4
[(.1,.2,.3,.4,.5,.6),
(0,.1,.1,.1,.2,.4),
(.1,.1,.2,.2,.4,.4)]
[(.5,.5,.5,.5,.5,.5),
(.2,.3,.3,.4,.4,.5),
(.1,.1,.1,.1,.2,.2)]
[(.4,.5,.5,.6,.7,.8),
(.1,.1,.1,.1,.1,.1),
(.1,.2,.3,.4,.4,.4)]
[(.1,.1,.1,.1,.1,.1),
(0,.1,.1,.2,.3,.3),
(.5,.5,.6,.6,.7,.8)]
Table. 5.5 Aggregated decision matrix
Step 8 . Calculate the score value using the equation (6)
C1
A1 0.59
S ( A) A2 0.49
A3 0.58
A4 0.55
C2
0.48
0.47
0.54
0.64
C4
0.62
0.66
0.73
0.54
C3
0.48
0.65
0.56
0.67
Step 9. Using the score function we check for the conditions and find
d r , r
ij pj
Here we consider j 1, i 1,2,3,4 and p 1,2,3,4 and check for the conditions in (7)
1) ~
rij ~
r pj or 2) ~
rij ~
r pj
A
1
d r , r
A2
ij pj
m n A
3
A4
or 3) ~
rij ~
r pj for i 1,2,3,4 j 1 and p 1,2,3,4
C
C
C
C1
2
3
4
0
0.1222 0.2266 0.1055
0
0.1666 0.111
0.1222
0
0.2111
0.2266 0.1666
0.1055 0.111 0.2111
0
To construct the dominance matrix we check for (~
rij is , or to ~
r pj )
Since we have
(r r ) , 1( A1, A1) 0 and as (r11 r21)
11 11
and (r r ) , ( A , A )
1 2 1
21 11
(A , A )
1
1
2
w d (r , r )
1r 11 21
0.11055
4
w
jr
j 1
4
w jr d (r21, r11)
j 1
w1r
0.4401
Using equation (7) calculate the dominance matrix 1 Ai , A p as follows.
A1
A , A
A
2
1 i p mn
A3
A4
C1
0
0.4401
0.6523
0.4108
C2
0.1105
C3
0.1632
0
0.5155
0.1290
0
0.1053 0.5810
0.4214
0.1452
0
C4
0.1027
A. Sahaya Sudha, Luiz Flavio Autran Monteiro Gomes and K.R. Vijayalakshmi, Assessment of MCDM problems by
TODIM using aggregated weights
Neutrosophic Sets and Systems, Vol. 35, 2020
93
Similarly for the values j 2, i 1,2,3,4 and p 1,2,3,4 ,
j 3, i 1,2,3,4 and p 1,2,3,4 and j 4, i 1,2,3,4 and p 1,2,3,4 the dominance
matrix are calculated.
A1
A , A
2 i p mn A2
A3
A4
C2
C3
C4
C1
0
0.1393 0.2508 0.3154
0.2229
0
0.1977 0.3040
0
0.2598
0.1624 0.1235
0.1971 0.1900 0.1624
0
A
1
A , A
3 i p mn A2
A3
A4
C2
C3
C4
C1
0
0.1581 0.1786 0.1624
0.2529
0
0.1624 0.2304
0
0.2665
0.2858 0.2598
0.2598 0.1440 0.1440
0
Step 10. On the basics of the above equation the overall dominance degree is obtained as
n
( Ai , A p ) j ( Ai , A p ), (i, p 1,2,.....m)
j 1
0
0.238 0.1331 0.1656
0.7789
0
0.7508 0.7298
0.5598 0.1927
0
0.0963
0
0.656 0.2133 0.8454
Now
4
j ( Ai , A p ), (i, p 1,2,.....m
are (0.2363,-2.2266,-0.4654,-1.000)
j 1
Step 12. Then the overall value of each Ai can be calculated using the equation (9)
( A1) 1.000,
( A2 ) 0, ( A3) 0.7150, ( A4 ) 0.4980
Step 13. Ranking the values of all alternatives
( Ai ) and selecting the most desirable alternatives in
accordance with Ai , among the four alternatives A1 is the best choice and the ranking order is
A1 A3 A4 A2
Stage 2.
A. Sahaya Sudha, Luiz Flavio Autran Monteiro Gomes and K.R. Vijayalakshmi, Assessment of MCDM problems by
TODIM using aggregated weights
Neutrosophic Sets and Systems, Vol. 35, 2020
94
Step 7. Floor space requirement C1 , Feeding (Feeder) and watering space requirement C 2 are
benefiting type criteria B1 C1 , C 2 . Maintenance of health and sanitization C3 and Productivity
C 4 are cost type B2 C3 , C 4 .The hexagonal neutrosophic positive-ideal solution
B and
hexagonal neutrosophic negative-ideal solution B are obtained as follows
(1,1,1,1,1,1)(0,0,0,0,0,0)(0,0,0,0,0,0), (1,1,1,1,1,1)(0,0,0,0,0,0)(0,0,0,0,0,0),
B
(0,0,0,0,0,0)(1,1,1,1,1,1)(1,1,1,1,1,1) , (0,0,0,0,0,0)(1,1,1,1,1,1)(1,1,1,1,1,1)
(0,0,0,0,0,0)(1,1,1,1,1,1)(1,1,1,1,1,1), (0,0,0,0,0,0)(1,1,1,1,1,1)(1,1,1,1,1,1)
B
, (1,1,1,1,1,1)(0,0,0,0,0,0)(0,0,0,0,0,0) , (1,1,1,1,1,1)(0,0,0,0,0,0)(0,0,0,0,0,0)
Step 8. The vector of the attribute weight is w (0.15,0.15,0.20,0.50) . By using equation (10)
calculate the separation measure S i of the each alternative from the hexagonal neutrosophic
positive ideal solution where d ( rij , r j ) is calculated using equation (2).
The calculated values are as follows
S1 0.1482 S 2 0.1408 S 3 0.1370 S 4 0.1164
By using equation (11) calculate the separation measure S i of the each alternative from the
hexagonal neutrosophic negative ideal solution. The calculated values are as follows
S1 0.0989 S2 0.1094 S3 0.1129 S4 0.1335
Step 9. Using equation (12) calculate the relative closeness coefficient of the hexagonal neutrosophic
ideal solution. The relative closeness coefficient values are as follows
C1 0.4002 C2 0.4372 C3 0.4517 C4 0.5342
Step 10. Rank the alternatives in the decreasing order of closeness coefficient values.
A4 A3 A2 A1
A. Sahaya Sudha, Luiz Flavio Autran Monteiro Gomes and K.R. Vijayalakshmi, Assessment of MCDM problems by
TODIM using aggregated weights
Neutrosophic Sets and Systems, Vol. 35, 2020
95
6. Graphical Representation of the Comparative study
Figure 6.1 Ranking of the four alternatives using TODIM and TOPSIS
The ranking results of TODIM show that A1 is the best alternative with maximum global
value ( A1 ) 1 and the least global value is ( A2 ) 0 The ranking of the four
alternatives using TODIM is A1 A3 A4 A2
The ranking result using TOPSIS shows that A4 is the best suited alternative as it ranking is
in first position and A1 is considered to be last as it takes fourth position in ranking .
The ranking of the four alternatives using TOPSIS is A4 A3 A2 A1 .
In both the methods A3 take the same position and A4 is in the third level in TODIM
which is nearest to the ranking of TOPSIS. Similarly, A2 is in the fourth level in TODIM
which is very close to the ranking of TOPSIS.
Both the MCDM ranking results shows that they are similar by large percentage which
provides decision maker to increase the flexibility in choosing the optimal alternative.
Conclusion
The research presented in this article is an assessment study of the sustainability of commercial
goat farming and its recent impact on self-employment for youth has been carried out in a context
characterized by two MCDM methods, TODIM and TOPSIS. Using those methods the social,
economic and ecological sustainability in housing and feeding systems of goat farming are evaluated
by three experts and the evaluation was considered as hexagonal neutrosophic numbers in order to
remove the ambiguity and increase the accuracy in the decision making process. Using the TODIM
approach which is able to distinguish between risks based alternative and definite alternative in
A. Sahaya Sudha, Luiz Flavio Autran Monteiro Gomes and K.R. Vijayalakshmi, Assessment of MCDM problems by
TODIM using aggregated weights
Neutrosophic Sets and Systems, Vol. 35, 2020
96
uncertain circumstances is analyzed .At the same time, by using the TOPSIS method the ranking is
performed based on distance of each alternatives to its positive and negative ideal solutions. The
ranking results of TODIM show a large percentage of similarity with ranking resulting from
TOPSIS.The result shows that stall feeding system with normal flooring and a part with both
intensive and extensive grazing system are best suited for sustainable commercial goat farming. This
study may be applied in several other fields like livestock management systems with technology
adaptation as well as in the economics of goat farming and other livestock sectors..
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Received: Apr 15, 2020. Accepted: July 5, 2020
A. Sahaya Sudha, Luiz Flavio Autran Monteiro Gomes and K.R. Vijayalakshmi, Assessment of MCDM problems by
TODIM using aggregated weights
Neutrosophic Sets and Systems, Vol. 35, 2020
University of New Mexico
MBJ – Neutrosophic 𝜷 – Ideal of 𝜷 – Algebra
Prakasam Muralikrishna1 and Surya Manokaran2
Assistant Professor 1 and Research Scholar 2
PG & Research Department of Mathematics,
Muthurangam Govt. Arts College (A),
Vellore-632002. TN. India
pmkrishna@rocketmail.com , suryamano95@gmail.com
Abstract: This paper extends the concept of ideal of a 𝛽 – algebra to MBJ – Neutrosophic 𝛽 – Ideal
of a 𝛽 – algebra. Further discusses about the homomorphic image, inverse image, cartesian
product and related results.
Keywords: Neutrosophic Set, MBJ – Neutrosophic Set, MBJ – Neutrosophic 𝛽 – Ideals, Cartesian Product
of MBJ – Neutrosophic 𝛽 – algebra.
1. Introduction
Zadeh [21, 22] first presented the idea of Fuzzy Set by which shown a meaningful application in
many fields and this theory is welcomed to handle the uncertainty. As a generalization of fuzzy set
Atanassov [7, 11] introduced Intuitionistic Fuzzy Set which assigns a pair with membership degree
and non – membership degree. The Interval Valued Fuzzy Set [6, 10, 12] represents the membership
degree with interval values to reflect the uncertainty in assigning membership degree. As an
extension for all elements in any set, Neutrosophic Fuzzy Set provides with truth, intermediate and
false membership function by Smarandache, F [16, 17, 18] and is further developed to
MBJ – Neutrosophic fuzzy set [19, 20] with truth membership function, intermediate interval valued
membership function and false membership function.
Neggers and Kim [18] brought a new structure of algebra called 𝛽 – algebra and Jun [17] dealt some
related topics on 𝛽 – algebra. The fusion of fuzzy with algebra and the notion was initiated by
Rosenfeld [15]. Further many researchers correlated some algebras with fuzzy sets. Ansari [5, 8]
initialized the fuzzy
𝛽 – algebra.
𝛽 – subalgebra of 𝛽 – algebra and also introduced fuzzy 𝛽 – ideal of
With these inspirations, this paper extends to MBJ – Neutrosophic 𝛽 – ideal of
𝛽 – algebra and analyzed some result.
2. Preliminaries
In this section, some definitions and examples of 𝛽 – algebra and fuzzy set are discussed.
2.1 Definition: [5, 8, 14]
i.
𝑥−0=𝑥
ii.
(0 − 𝑥) + 𝑥 = 0
A non-empty set (𝑋, +, −, 0) is called a 𝛽 – algebra if
Prakasam Muralikrishna and Surya Manokaran, MBJ – Neutrosophic 𝛽 – Ideal of 𝛽 – Algebra
Neutrosophic Sets and Systems, Vol. 35, 2020
iii.
100
(𝑥 − 𝑦) − 𝑧 = 𝑥 − (𝑧 + 𝑦) ∀ 𝑥, 𝑦, 𝑧 ∈ 𝑋.
2.2 Example: [9] The following Cayley’s table is formed using a set 𝑋 = { 0, 1, 2, 3, 4, 5 } with a
constant 0 and two binary operations + and –
+
0
1
2
3
4
5
-
0
1
2
3
4
5
0
0
1
2
3
4
5
0
0
1
2
3
5
4
1
1
0
4
5
2
3
1
1
0
4
5
3
2
2
2
5
0
4
3
1
2
2
5
0
4
1
3
3
3
4
5
0
1
2
3
3
4
5
0
2
1
4
4
3
1
2
5
0
4
4
3
1
2
0
5
5
5
2
3
1
0
4
5
5
2
3
1
4
0
∴ The set 𝑋 is a 𝛽 – algebra.
2.3 Definition: [5] A non – empty subset S of a
𝛽 – algebra ( 𝑋, +, −, 0 ) is known as
𝛽 – subalgebra if
i.
𝑥−𝑦∈𝑆
ii.
𝑥 + 𝑦 ∈ 𝑆 ∀ 𝑥, 𝑦 ∈ 𝑆
2.4 Example: Let 𝑈1 = { 0 , 2 } and 𝑈2 = { 0 , 1 } be any two subset of a 𝛽 – algebra
𝑋 = { ( 0, 1, 2, 3, 4, 5), +, −, 0 } .
Here 𝑈1 is a 𝛽 – subalgebra of 𝑋 where as 𝑈2 is not a
𝛽 – subalgebra of 𝑋.
2.5 Definition: [8] A non – empty subset 𝐼 of a 𝛽 – algebra is said to be 𝛽 – ideal of ( X, +, −, 0 ) if
it has the following conditions
i.
0∈𝐼
ii.
𝑥+𝑦∈𝐼
iii.
𝑥 − 𝑦 and 𝑦 ∈ 𝐼 then 𝑥 ∈ 𝐼 ∀ 𝑥, 𝑦 ∈ 𝑋
2.6 Exercise: [12] Consider a 𝛽 – algebra( 𝑋, +, −, 0 ) in the Cayley’s table
+
0
1
2
3
-
0
1
2
3
0
0
1
2
3
0
0
3
2
1
1
1
2
3
0
1
1
0
3
2
2
2
3
0
1
2
2
1
0
3
3
3
0
1
2
3
3
2
1
0
The subset 𝐼1 = { 0 , 3 } of 𝑋 is a 𝛽 – ideal of 𝑋.
2.7 Definition: [5] A mapping 𝑓 ∶ 𝑋 → 𝑌 is said to be a 𝛽 – homomorphism where 𝑋 and 𝑌 are
two 𝛽 – algebras with constant 0 and two binary operations + and – if
i.
𝑓(𝑥 + 𝑦) = 𝑓(𝑥) + 𝑓(𝑦)
Prakasam Muralikrishna and Surya Manokaran, MBJ – Neutrosophic 𝛽 – Ideal of 𝛽 – Algebra
Neutrosophic Sets and Systems, Vol. 35, 2020
ii.
101
𝑓(𝑥 − 𝑦) = 𝑓(𝑥) − 𝑓(𝑦) ∀ 𝑥, 𝑦 ∈ 𝑋.
2.8 Definition: [22] A Fuzzy Set in 𝑋 is a mapping, 𝜌 ∶ 𝑋 → [0,1] for each 𝑥 in 𝑋, 𝜌(𝑥) is called
the membership value of 𝑥 ∈ 𝑋.
2.9 Definition: [7] A non – empty set 𝑋 is said to be Intuitionistic Fuzzy Set and is defined by
where 𝜌𝐴 ∶ 𝑋 → [0,1] is a membership function of 𝐴 and
𝐴 = { < 𝑥, 𝜌(𝑥), 𝜂(𝑥) >/𝑥 ∈ 𝑋}
𝜂𝐴 ∶ 𝑋 → [0,1] is a non – membership function of 𝐴 with 0 ≤ 𝜌𝐴 (𝑥) + 𝜂𝐴 ≤ 1.
2.10
Definition:
[6]
𝐴 = {(𝑥, 𝜌̅𝐴 (𝑥))} 𝑥 ∈ 𝑋
subintervals of [0,1].
An
Interval
where
Valued
𝜌̅𝐴 ∶ 𝑋 → 𝐷[0,1]
Also 𝜌̅𝐴 (𝑥) = [
𝜌𝐴𝐿 (𝑥)
,
𝜌𝐴𝑈 (𝑥)]
Fuzzy
Set
on
𝑋
is
represented
as
where 𝐷[0,1] is the family of all closed
where 𝜌𝐴𝐿 and 𝜌𝐴𝑈 are two fuzzy sets in 𝑋
such that 𝜌𝐴𝐿 (𝑥) ≤ 𝜌𝐴𝑈 (𝑥) ∀ 𝑥 ∈ 𝑋.
Remark: Now let us illustrate refined minimum (𝑟𝑚𝑖𝑛) and refined maximum (𝑟𝑚𝑎𝑥) of two
elements in 𝐷[0,1]. Also characterized the symbols ≤ , ≥ , = in case of two elements in 𝐷[0,1].
Let 𝐷1 = [𝑎1 , 𝑏1 ] & 𝐷2 = [𝑎2 , 𝑏2 ] ∈ 𝐷[0,1] then
𝑟𝑚𝑖𝑛(𝐷1 , 𝐷2 ) = [min(𝑎1 , 𝑎2 ) , min(𝑏1 , 𝑏2 )]
𝑟𝑚𝑎𝑥(𝐷1 , 𝐷2 ) = [max(𝑎1 , 𝑎2 ) , max(𝑏1 , 𝑏2 )].
For 𝐷𝑖 = [𝑎𝑖 , 𝑏𝑖 ] ∈ 𝐷[0,1] for 𝑖 = 1, 2, 3, ….
𝑟𝑠𝑢𝑝𝑖 (𝐷𝑖 ) = [𝑠𝑢𝑝𝑖 (𝑏𝑖 ), 𝑠𝑢𝑝𝑖 (𝑏𝑖 )] & 𝑟𝑖𝑛𝑓𝑖 (𝐷𝑖 ) = [𝑖𝑛𝑓𝑖 (𝑏𝑖 ), 𝑖𝑛𝑓𝑖 (𝑏𝑖 )]
Now 𝐷1 ≥ 𝐷2 if and only if 𝑎1 ≥ 𝑎2 , 𝑏1 ≥ 𝑏2 . Likewise, for 𝐷1 ≤ 𝐷2 and 𝐷1 = 𝐷2 are
defined.
2.11 Definition: [6] The representation of an Interval Valued Intuitionistic Fuzzy Set 𝐴 on 𝑋 is
𝐴 = { < 𝑥, 𝜌̅𝐴 (𝑥), 𝜂̅𝐴 (𝑥) >/𝑥 ∈ 𝑋}
𝜌̅𝐴 (𝑥) = [ 𝜌𝐴𝐿 (𝑥) , 𝜌𝐴𝑈 (𝑥)] and
and 0 ≤ 𝜌𝐴𝑈 (𝑥) + 𝜂𝐴𝑈 ≤ 1.
where
𝜂̅𝐴 (𝑥) = [
𝜌̅𝐴 ∶ 𝑋 → 𝐷[0,1]
𝜂𝐴𝐿 (𝑥)
, 𝜂𝐴𝑈 (𝑥)]
and
𝜂̅𝐴 : 𝑋 → 𝐷[0,1]
where
𝜌𝐴𝐿 (𝑥)
+ 𝜂𝐴𝐿 ≤ 1
with the condition that 0 ≤
2.12 Definition: [16, 17] The definition of an Neutrosophic Fuzzy Set 𝐴 on 𝑋 is characterized by a
Truth – membership function 𝜌𝑇 , an indeterminacy membership function
𝜉𝐼 , and a
falsity – membership function 𝜂𝐹 where 𝜌𝑇 , 𝜉𝐼 , 𝜂𝐹 are subsets of [0,1] that is 𝜌𝑇 , 𝜉𝐼 , 𝜂𝐹 ∶ 𝑋 → [0,1].
Thus, the Neutrosophic Set is defined as 𝐴 = { < 𝑥, 𝜌𝑇 (𝑥), 𝜉𝐼 (𝑥), 𝜂𝐹 (𝑥) >/𝑥 ∈ 𝑋}.
𝐴 = { < 𝑥, 𝜌𝑇 (𝑥), 𝜉𝐼̅ (𝑥), 𝜂𝐹 (𝑥) >/𝑥 ∈ 𝑋} is called
MBJ – Neutrosophic Set in 𝑋 where 𝜌𝑇 , 𝜂𝐹 ∶ 𝑋 → [0,1] and 𝜉𝐼̅ ∶ 𝑋 → 𝐷[0,1] with 𝜌𝑇 (𝑥) denotes
the truth membership function , 𝜉𝐼̅ (𝑥) denotes an intermediate interval valued membership
2.13
Definition:
[19,20]
The
structure
function and 𝜂𝐹 (𝑥) denotes an false membership function.
2.14 Definition: An Fuzzy set is said to have a supremum property for any subset 𝑊 of 𝑋 there
exists 𝑥0 ∈ 𝑊 such that 𝜌𝐴 (𝑥0 ) = 𝑠𝑢𝑝𝑥 ∈𝑊 𝜌𝐴 (𝑥).
Prakasam Muralikrishna and Surya Manokaran, MBJ – Neutrosophic 𝛽 – Ideal of 𝛽 – Algebra
Neutrosophic Sets and Systems, Vol. 35, 2020
102
2.15 Definition: An Intuitionistic Fuzzy Set 𝐴 is said to have a 𝑠𝑢𝑝 − 𝑖𝑛𝑓 property for any subset
𝑊 of 𝑋, there exists 𝑥0 ∈ 𝑊 such that 𝜌𝐴 (𝑥0 ) = 𝑠𝑢𝑝𝑥 ∈𝑊 𝜌𝐴 (𝑥) and 𝜂𝐴 (𝑥0 ) = 𝑖𝑛𝑓𝑥 ∈𝑊 𝜂𝐴 (𝑥).
2.16 Definition: An Interval Valued Intuitionistic Fuzzy Set 𝐴 in any set 𝑋 is said to have
𝑟𝑠𝑢𝑝 − 𝑟𝑖𝑛𝑓
property
𝜌̅𝐴 (𝑥0 ) = 𝑟𝑠𝑢𝑝𝑥 ∈𝑊 𝜌̅𝐴 (𝑥)
if
and
for
𝑊
subset
of
𝑋
there
exists
𝑥0 ∈ 𝑊
such
that
𝜂̅𝐴 (𝑥0 ) = 𝑟𝑖𝑛𝑓𝑥 ∈𝑊 𝜂̅𝐴 (𝑥).
2.17 Definition: [19] An MBJ – Neutrosophic Fuzzy Set 𝐴 in 𝑋 has 𝑠𝑢𝑝 − 𝑟𝑠𝑢𝑝 − 𝑖𝑛𝑓 property if
for subset 𝑊 of 𝑋 there exists 𝑥0 ∈ 𝑊 such
𝜉𝐴̅ (𝑥0 ) = 𝑟𝑠𝑢𝑝𝑥 ∈𝑊 𝜉𝐴̅ (𝑥); 𝜂𝐴 (𝑥0 ) = 𝑖𝑛𝑓𝑥 ∈𝑊 𝜂𝐴 (𝑥) respectively.
that
𝜌𝐴 (𝑥0 ) = 𝑠𝑢𝑝𝑥 ∈𝑊 𝜌𝐴 (𝑥)
;
2.18 Definition: [12] An Interval Valued Fuzzy Set 𝐴 = {< 𝑥, 𝜌̅𝐴 (𝑥) >/𝑥 ∈ 𝑋} in 𝑋 is said to be
Interval Valued Fuzzy 𝛽 – ideal of 𝑋 if
i.
𝜌̅𝐴 (0) ≥ 𝜌̅𝐴 (𝑥)
ii.
𝜌̅𝐴 (𝑥 + 𝑦) ≥ 𝑟min{𝜌̅𝐴 (𝑥), 𝜌̅𝐴 (𝑦)}
iii.
𝜌̅𝐴 (𝑥) ≥ rmin{𝜌̅𝐴 (𝑥 − 𝑦), 𝜌̅𝐴 (𝑦)} ∀ 𝑥 , 𝑦 ∈ 𝑋.
2.19 Definition: An Intuitionistic Fuzzy Set 𝐴 = { < 𝑥, 𝜌(𝑥), 𝜂(𝑥) >/𝑥 ∈ 𝑋} in 𝑋 is known as
Intuitionistic Fuzzy 𝛽 - ideal of 𝑋 if
i.
𝜌𝐴 (0) ≥ 𝜌𝐴 (𝑥)
;
𝜂𝐴 (0) ≤ 𝜂𝐴 (𝑥)
ii.
𝜌𝐴 (𝑥 + 𝑦) ≥ min{𝜌𝐴 (𝑥), 𝜌𝐴 (𝑦)}
;
𝜂𝐴 (𝑥 + 𝑦) ≤ max{𝜂𝐴 (𝑥), 𝜂𝐴 (𝑦)}
iii.
𝜌𝐴 (𝑥) ≥ min{𝜌𝐴 (𝑥 − 𝑦), 𝜌𝐴 (𝑦)}
;
𝜂𝐴 (𝑥) ≤ max{𝜂𝐴 (𝑥 − 𝑦), 𝜂𝐴 (𝑦)}
Definition: [19] Let 𝑋 be a 𝛽 – algebra and an MBJ Neutrosophic
𝐴 = { 𝜌𝐴 , 𝜉𝐴̅ , 𝜂𝐴 } is called an MBJ – Neutrosophic 𝛽 – subalgebra of 𝑋 if it satisfies
2.20
ii.
𝜌𝐴 (𝑥 + 𝑦) ≥ min{𝜌𝐴 (𝑥), 𝜌𝐴 (𝑦)}
𝜉𝐴̅ (𝑥 + 𝑦) ≥ 𝑟min{𝜉𝐴̅ (𝑥), 𝜉𝐴̅ (𝑦)}
;
𝜌𝐴 (𝑥 − 𝑦) ≥ min{𝜌𝐴 (𝑥), 𝜌𝐴 (𝑦)}
𝜉𝐴̅ (𝑥 − 𝑦) ≥ 𝑟min{𝜉𝐴̅ (𝑥), 𝜉𝐴̅ (𝑦)}
iii.
𝜂𝐴 (𝑥 + 𝑦) ≤ max{𝜂𝐴 (𝑥), 𝜂𝐴 (𝑦)}
;
𝜂𝐴 (𝑥 − 𝑦) ≤ max{𝜂𝐴 (𝑥), 𝜂𝐴 (𝑦)}
i.
;
Set
3 MBJ – Neutrosophic 𝜷 – Ideal of 𝜷 – Algebra
This part frames the structure of MBJ – Neutrosophic 𝛽 – Ideal of 𝛽 – Algebra and studied the
related results.
3.1 Definition: Let (𝑋, +, −, 0) be a β – algebra. An MBJ – Neutrosophic Set 𝐾 = { 𝜌𝐾 , 𝜉𝐾̅ , 𝜂𝐾 } in
𝑋 is called an MBJ – Neutrosophic 𝛽 – Ideal of 𝑋 if it satisfies the following conditions:
i.
𝜌𝐾 (0) ≥ 𝜌𝐾 (𝑥)
𝜌𝐾 (𝑥 + 𝑦) ≥ min{ 𝜌𝐾 (𝑥) , 𝜌𝐾 (𝑦)}
𝜌𝐾 (𝑥) ≥ min{ 𝜌𝐾 (𝑥 − 𝑦) , 𝜌𝐾 (𝑦)}
Prakasam Muralikrishna and Surya Manokaran, MBJ – Neutrosophic 𝛽 – Ideal of 𝛽 – Algebra
Neutrosophic Sets and Systems, Vol. 35, 2020
ii.
103
𝜉𝐾̅ (0) ≥ 𝜉𝐾̅ (𝑥)
𝜉𝐾̅ (𝑥 + 𝑦) ≥ 𝑟min{ 𝜉𝐾̅ (𝑥) , 𝜉𝐾̅ (𝑦)}
𝜉𝐾̅ (𝑥) ≥ 𝑟min{ 𝜉𝐾̅ (𝑥 − 𝑦) , 𝜉𝐾̅ (𝑦)}
iii.
𝜂𝐾 (0) ≤ 𝜂𝐾 (𝑥)
𝜂𝐾 (𝑥 + 𝑦) ≤ max{ 𝜂𝐾 (𝑥) , 𝜂𝐾 (𝑦)}
𝜂𝐾 (𝑥) ≤ max{𝜂𝐾 (𝑥 − 𝑦) , 𝜂𝐾 (𝑦)} ∀𝑥 , 𝑦 ∈ 𝑋
3.2 Example : A β – algebra 𝑋 in example 2.6 defines a MBJ – Neutrosophic set as 𝜌𝐴 ∶ 𝑋 → [0,1] ;
𝜉𝐴̅ ∶ 𝑋 → 𝐷[0,1] and 𝜂𝐴 ∶ 𝑋 → [0,1] such that
0.4 ,
𝜌𝐴 (𝑥) = {0.2 ,
0.3 ,
𝑥=0
𝑥 = 1,3
𝑥=2
[0.3 , 0.7]
𝜉𝐾̅ 𝐴 (𝑥) = {[0.1 , 0.5]
[0.2 , 0.6]
0.1 ,
𝜂𝐴 (𝑥) = {0.4 ,
0.5 ,
𝑥=0
𝑥 = 1,3
𝑥=2
𝑥=0
𝑥 = 1,3
𝑥=2
is an MBJ – Neutrosophic 𝛽 – Ideal of 𝑋.
3.3 Theorem: The intersection of any two MBJ – Neutrosophic 𝛽 – Ideal of a 𝛽 – algebra is also an
MBJ – Neutrosophic 𝛽 – Ideal.
Proof: Let 𝐾1 & 𝐾2 be two MBJ – Neutrosophic 𝛽 – Ideal of 𝑋.
Now, (𝜌𝐾1∩𝐾2 )(0) ≥ min{ 𝜌𝐾1 (0), 𝜌𝐾2 (0) }
= min{ 𝜌𝐾1 (𝑥), 𝜌𝐾2 (𝑥) }
= (𝜌𝐾1∩𝐾2 )(𝑥)
(𝜌𝐾1∩𝐾2 )(𝑥 + 𝑦) ≥ min{ 𝜌𝐾1 (𝑥 + 𝑦), 𝜌𝐾2 (𝑥 + 𝑦) }
= min {min { 𝜌𝐾1 (𝑥) , 𝜌𝐾1 (𝑦) } , min { 𝜌𝐾2 (𝑥) , 𝜌𝐾2 (𝑦)} }
= min {min { 𝜌𝐾1 (𝑥) , 𝜌𝐾2 (𝑥) } , min { 𝜌𝐾1 (𝑦) , 𝜌𝐾2 (𝑦) } }
= min { 𝜌𝐾1∩𝐾2 (𝑥) , 𝜌𝐾1∩𝐾2 (𝑦)}
𝜌𝐾1 ∩𝐾2 (𝑥) ≥ min{ 𝜌𝐾1 (𝑥), 𝜌𝐾2 (𝑥) }
= min {min { 𝜌𝐾1 (𝑥 − 𝑦) , 𝜌𝐾1 (𝑦) } , min { 𝜌𝐾2 (𝑥 − 𝑦) , 𝜌𝐾2 (𝑦)} }
= min {min { 𝜌𝐾1 (𝑥 − 𝑦) , 𝜌𝐾2 (𝑥 − 𝑦) } , min { 𝜌𝐾1 (𝑦) , 𝜌𝐾2 (𝑦)} }
(𝜉𝐾̅ 1∩𝐾2
= min {min { 𝜌𝐾1∩𝐾2 (𝑥 − 𝑦) , 𝜌𝐾1∩𝐾2 (𝑦)} }
)(0) ≥ 𝑟min{𝜉𝐾̅ (0), 𝜉𝐾̅ (0) }
1
2
= 𝑟min{𝜉𝐾̅ 1 (𝑥), 𝜉𝐾̅ 2 (𝑥) }
= (𝜉𝐾̅ ∩𝐾 )(𝑥)
1
(𝜉𝐾̅
1 ∩𝐾2
2
)(𝑥 + 𝑦) ≥ 𝑟min{𝜉𝐾̅ 1 (𝑥 + 𝑦), 𝜉𝐾̅ 2 (𝑥 + 𝑦) }
= rmin {rmin { 𝜉𝐾̅ 1 (𝑥) , 𝜉𝐾̅ 1 (𝑦) } , rmin { 𝜉𝐾̅ 2 (𝑥) , 𝜉𝐾̅ 2 (𝑦)} }
= rmin {rmin { 𝜉𝐾̅ (𝑥) , 𝜉𝐾̅ (𝑥) } , rmin { 𝜉𝐾̅ (𝑦) , 𝜉𝐾̅ (𝑦) } }
1
2
1
2
= rmin { 𝜉𝐾̅ 1∩𝐾2 (𝑥) , 𝜉𝐾̅ 1∩𝐾2 (𝑦)}
𝜉𝐾̅ 1 ∩𝐾2 (𝑥) ≥ 𝑟min{𝜉𝐾̅ 1 (𝑥), 𝜉𝐾̅ 2 (𝑥) }
Prakasam Muralikrishna and Surya Manokaran, MBJ – Neutrosophic 𝛽 – Ideal of 𝛽 – Algebra
Neutrosophic Sets and Systems, Vol. 35, 2020
104
= rmin { rmin { 𝜉𝐾̅ 1 (𝑥 − 𝑦) , 𝜉𝐾̅ 1 (𝑦) } , rmin { 𝜉𝐾̅ 2 (𝑥 − 𝑦) , 𝜉𝐾̅ 2 (𝑦)} }
= rmin { rmin { 𝜉𝐾̅ (𝑥 − 𝑦) , 𝜉𝐾̅ (𝑥 − 𝑦) } , rmin { 𝜉𝐾̅ (𝑦) , 𝜉𝐾̅ (𝑦)} }
1
2
1
2
= rmin { rmin { 𝜉𝐾̅ 1∩𝐾2 (𝑥 − 𝑦) , 𝜉𝐾̅ 1∩𝐾2 (𝑦)} }
(𝜂𝐾1∩𝐾2 )(0) ≤ max{ 𝜂𝐾1 (0), 𝜂𝐾2 (0) }
= max{ 𝜂𝐾1 (𝑥), 𝜂𝐾2 (𝑥) }
= (𝜂𝐾1∩𝐾2 )(𝑥)
(𝜂𝐾1∩𝐾2 )(𝑥 + 𝑦) ≤ max{ 𝜂𝐾1 (𝑥 + 𝑦), 𝜂𝐾2 (𝑥 + 𝑦) }
= max {max { 𝜂𝐾1 (𝑥) , 𝜂𝐾1 (𝑦) } , max { 𝜂𝐾2 (𝑥) , 𝜂𝐾2 (𝑦)} }
= max {max { 𝜂𝐾1 (𝑥) , 𝜂𝐾2 (𝑥) } , max { 𝜂𝐾1 (𝑦) , 𝜂𝐾2 (𝑦) } }
= max { 𝜂𝐾1∩𝐾2 (𝑥) , 𝜂𝐾1∩𝐾2 (𝑦)}
𝜂𝐾1∩𝐾2 (𝑥) ≤ max{ 𝜂𝐾1 (𝑥), 𝜂𝐾2 (𝑥) }
= max {max { 𝜂𝐾1 (𝑥 − 𝑦) , 𝜂𝐾1 (𝑦) } , max { 𝜂𝐾2 (𝑥 − 𝑦) , 𝜂𝐾2 (𝑦)} }
= max {max { 𝜂𝐾1 (𝑥 − 𝑦) , 𝜂𝐾2 (𝑥 − 𝑦) } , max { 𝜂𝐾1 (𝑦) , 𝜂𝐾2 (𝑦)} }
= max {max { 𝜂𝐾1∩𝐾2 (𝑥 − 𝑦) , 𝜂𝐾1∩𝐾2 (𝑦)} }
Hence 𝐾1 ∩ 𝐾2 is an MBJ – Neutrosophic β – Ideal of 𝑋.
3.4 Theorem: The intersection of any set of MBJ – Neutrosophic β – Ideal of a β – Algebra 𝑋 is also
an MBJ – Neutrosophic β – Ideal.
3.5 Theorem: Let 𝐾 = { 𝜌𝐾 , 𝜉𝐾̅ , 𝜂𝐾 } be an MBJ – Neutrosophic β – Ideal. If
𝜌𝐾 (𝑥) ≥ 𝜌𝐾 (𝑦) ; 𝜉𝐾̅ (𝑥) ≥ 𝜉𝐾̅ (𝑦) and 𝜂𝐾 (𝑥) ≤ 𝜂𝐾 (𝑦).
𝑥 ≤ 𝑦 then
Proof: For any 𝑥 , 𝑦 ∈ 𝑋 , 𝑥 ≤ 𝑦 ⟹ 𝑥 − 𝑦 = 0 .
𝜌𝐾 (𝑥) ≥ min { 𝜌𝐾 (𝑥 − 𝑦) , 𝜌𝐾 (𝑦) }
= min { 𝜌𝐾 (0) , 𝜌𝐾 (𝑦) }
= 𝜌𝐾 (𝑦)
𝜌𝐾 (𝑥) ≥ 𝜌𝐾 (𝑦)
𝜉𝐾̅ (𝑥) ≥ rmin { 𝜉𝐾̅ (𝑥 − 𝑦) , 𝜉𝐾̅ (𝑦) }
= rmin { 𝜉𝐾̅ (0) , 𝜉𝐾̅ (𝑦) }
= 𝜉𝐾̅ (𝑦)
𝜉𝐾̅ (𝑥) ≥ 𝜉𝐾̅ (𝑦)
𝜂𝐾 (𝑥) ≤ max { 𝜂𝐾 (𝑥 − 𝑦) , 𝜂𝐾 (𝑦) }
= max { 𝜂𝐾 (0) , 𝜂𝐾 (𝑦) }
= 𝜂𝐾 (𝑦)
𝜂𝐾 (𝑥) ≤ 𝜂𝐾 (𝑦).
3.6 Theorem: Let 𝐾 be an MBJ – Neutrosophic 𝛽 – Ideal of 𝑋 whenever
𝑥 ≤ 𝑧 + 𝑦 then
̅
̅
̅
;
𝜉𝐾 (𝑥) ≥ 𝑟min{ 𝜉𝐾 (𝑧) , 𝜉𝐾 (𝑦)}
and
𝜌𝐾 (𝑥) ≥ min{ 𝜌𝐾 (𝑧) , 𝜌𝐾 (𝑦)}
𝜂𝐾 (𝑥) ≤ max{ 𝜂𝐾 (𝑧) , 𝜂𝐾 (𝑦)}
Proof: For 𝑥 , 𝑦 , 𝑧 ∈ 𝑋
𝜌𝐾 (𝑥) ≥ min { 𝜌𝐾 (𝑥 − 𝑦) , 𝜌𝐾 (𝑦) }
= min { min { 𝜌𝐾 ((𝑥 − 𝑦) − 𝑧) , 𝜌𝐾 (𝑧) } , 𝜌𝐾 (𝑦) }
Prakasam Muralikrishna and Surya Manokaran, MBJ – Neutrosophic 𝛽 – Ideal of 𝛽 – Algebra
Neutrosophic Sets and Systems, Vol. 35, 2020
105
= min { min { 𝜌𝐾 (𝑥 − (𝑧 + 𝑦)) , 𝜌𝐾 (𝑧) } , 𝜌𝐾 (𝑦) }
= min { min { 𝜌𝐾 (0) , 𝜌𝐾 (𝑧) } , 𝜌𝐾 (𝑦) }
≥ min{ 𝜌𝐾 (𝑧) , 𝜌𝐾 (𝑦)}
𝜉𝐾̅ (𝑥) ≥ rmin { 𝜉𝐾̅ (𝑥 − 𝑦) , 𝜉𝐾̅ (𝑦) }
= rmin { rmin { 𝜉𝐾̅ ((𝑥 − 𝑦) − 𝑧) , 𝜉𝐾̅ (𝑧) } , 𝜉𝐾̅ (𝑦) }
= rmin { rmin { 𝜉𝐾̅ (𝑥 − (𝑧 + 𝑦)) , 𝜉𝐾̅ (𝑧) } , 𝜉𝐾̅ (𝑦) }
= rmin { rmin { 𝜉𝐾̅ (0) , 𝜉𝐾̅ (𝑧) } , 𝜉𝐾̅ (𝑦) }
≥ 𝑟min{ 𝜉𝐾̅ (𝑧) , 𝜉𝐾̅ (𝑦)}
𝜂𝐾 (𝑥) ≤ max { 𝜂𝐾 (𝑥 − 𝑦) , 𝜂𝐾 (𝑦) }
= max { max { 𝜂𝐾 ((𝑥 − 𝑦) − 𝑧) , 𝜂𝐾 (𝑧) } , 𝜂𝐾 (𝑦) }
= max { max { 𝜂𝐾 (𝑥 − (𝑧 + 𝑦)) , 𝜂𝐾 (𝑧) } , 𝜂𝐾 (𝑦) }
= max { max { 𝜂𝐾 (0) , 𝜂𝐾 (𝑧) } , 𝜂𝐾 (𝑦) }
≤ max{ 𝜂𝐾 (𝑧) , 𝜂𝐾 (𝑦)}
3.7 Theorem: Let 𝐾 = { 𝜌𝐾 , 𝜉𝐾̅ , 𝜂𝐾 } be an MBJ – Neutrosophic β – Ideal of 𝑋 , then sets
𝑋𝜌𝐾 = { 𝑥 ∈ 𝑋 ∶ 𝜌𝐾 (𝑥) = 𝜌𝐾 (0)}
;
𝑋𝜉̅𝐾 = { 𝑥 ∈ 𝑋 ∶ 𝜉𝐾̅ (𝑥) = 𝜉𝐾̅ (0)}
and
𝑋𝜂𝐾 = { 𝑥 ∈ 𝑋 ∶ 𝜂𝐾 (𝑥) = 𝜂𝐾 (0)} are β – ideals of 𝑋.
Proof: Since 𝜌𝐾 (𝑥) = 𝜌𝐾 (0) ⟹ 0 ∈ 𝑋𝜌𝐾
If 𝑥 − 𝑦 , 𝑦 ∈ 𝑋𝜌𝐾
⟹ 𝜌𝐾 (𝑥 − 𝑦) = 𝜌𝐾 (0) ; 𝜌𝐾 (𝑦) = 𝜌𝐾 (0)
Now, 𝜌𝐾 (𝑥) ≥ min { 𝜌𝐾 (𝑥 − 𝑦) , 𝜌𝐾 (𝑦) }
= min { 𝜌𝐾 (0) , 𝜌𝐾 (0) }
= 𝜌𝐾 (0)
𝜌𝐾 (𝑥) ≥ 𝜌𝐾 (0)
But 𝜌𝐾 (𝑥) ≤ 𝜌𝐾 (0) implies 𝜌𝐾 (𝑥) = 𝜌𝐾 (0)
⟹ 𝑥 ∈ 𝑋𝜌𝐾
𝑥 − 𝑦 , 𝑦 ∈ 𝑋𝜌𝐾 ⟹ 𝑥 ∈ 𝑋𝜌𝐾
∴ 𝑋𝜌𝐾 is an β – Ideal of 𝑋
𝜉𝐾̅ (𝑥) = 𝜉𝐾̅ (0) ⟹ 0 ∈ 𝑋𝜉̅
𝐾
If 𝑥 − 𝑦 , 𝑦 ∈ 𝑋𝜉̅𝐾
⟹ 𝜉𝐾̅ (𝑥 − 𝑦) = 𝜉𝐾̅ (0) ; 𝜉𝐾̅ (𝑦) = 𝜉𝐾̅ (0)
Now, 𝜉𝐾̅ (𝑥) ≥ rmin { 𝜉𝐾̅ (𝑥 − 𝑦) , 𝜉𝐾̅ (𝑦) }
= rmin { 𝜉𝐾̅ (0) , 𝜉𝐾̅ (0) }
= 𝜉𝐾̅ (0)
𝜉𝐾̅ (𝑥) ≥ 𝜉𝐾̅ (0)
But 𝜉𝐾̅ (𝑥) ≤ 𝜉𝐾̅ (0) implies 𝜉𝐾̅ (𝑥) = 𝜉𝐾̅ (0)
⟹ 𝑥 ∈ 𝑋𝜉̅𝐾
𝑥 − 𝑦 , 𝑦 ∈ 𝑋𝜉̅𝐾 ⟹ 𝑥 ∈ 𝑋𝜉̅𝐾
∴ 𝑋𝜉̅𝐾 is an β – Ideal of 𝑋.
Similarly, 𝑋𝜂𝐾 is also an β – Ideal of 𝑋.
Prakasam Muralikrishna and Surya Manokaran, MBJ – Neutrosophic 𝛽 – Ideal of 𝛽 – Algebra
Neutrosophic Sets and Systems, Vol. 35, 2020
106
3.8 Theorem: Suppose 𝐽 is subset of 𝑋. An MBJ – Neutrosophic set 𝐾 = { 𝜌𝐾 , 𝜉𝐾̅ , 𝜂𝐾 } such that
𝑡, 𝑥 ∈ 𝐽
𝜌𝐾 = {
𝑠, 𝑥 ∉ 𝐽
;
𝑡̅, 𝑥 ∈ 𝐽
𝜉𝐾̅ = {
𝑠̅, 𝑥 ∉ 𝐽
and
𝛼, 𝑥 ∈ 𝐽
𝜂𝐾 = {
𝛽, 𝑥 ∉ 𝐽
where 𝑡 , 𝑠 , 𝛼 , 𝛽 ∈ [ 0 , 1 ]
and
𝑡̅ , 𝑠̅ ∈ 𝐷[ 0 , 1 ] with [𝑡0 , 𝑡1 ] ≥ [𝑠0 , 𝑠1 ]. Then the MBJ – Neutrosophic set 𝐾 = { 𝜌𝐾 , 𝜉𝐾̅ , 𝜂𝐾 } is an
MBJ – Neutrosophic 𝛽 – ideal of 𝑋 if and only if 𝐽 𝑖𝑠 𝑎𝑛 𝛽 – ideal of 𝑋.
Proof: Consider an MBJ – Neutrosophic set 𝐾 = { 𝜌𝐾 , 𝜉𝐾̅ , 𝜂𝐾 } is an MBJ - Neutrosophic 𝛽 – ideal of
𝑋
i)
𝑎) 𝜌𝐾 (0) ≥ 𝜌𝐾 (𝑥) ∀ 𝑥 ∈ 𝑋
𝜌𝐾 (0) = 𝑡 ⟹ 0 ∈ 𝐽
𝑏) For 𝑥 , 𝑦 ∈ 𝐽
⟹ 𝜌𝐾 (𝑥) = 𝑡 = 𝜌𝐾 (𝑦)
∴ 𝜌𝐾 (𝑥 + 𝑦) ≥ min{𝜌𝐾 (𝑥) , 𝜌𝐾 (𝑦)}
= min{ 𝑡 , 𝑡 }
𝜌𝐾 (𝑥 + 𝑦) = 𝑡
⟹𝑥+𝑦 ∈𝐽
𝑐) For 𝑥 , 𝑦 ∈ 𝐽 if 𝑥 − 𝑦 𝑎𝑛𝑑 𝑦 ∈ 𝐽
⟹ 𝜌𝐾 (𝑥 − 𝑦) = 𝑡 = 𝜌𝐾 (𝑦)
∴ 𝜌𝐾 (𝑥) ≥ min{𝜌𝐾 (𝑥 − 𝑦) , 𝜌𝐾 (𝑦)}
= min{ 𝑡 , 𝑡 } = 𝑡
𝜌𝐾 (𝑥) = 𝑡
ii)
⟹𝑥 ∈𝐽
𝑎) 𝜉𝐾̅ (0) ≥ 𝜉𝐾̅ (𝑥) ∀ 𝑥 ∈ 𝑋
𝜉𝐾̅ (0) = [𝑡0 , 𝑡1 ] ⟹ 0 ∈ 𝐽
𝑏) For 𝑥 , 𝑦 ∈ 𝐽
⟹ 𝜉𝐾̅ (𝑥) = [𝑡0 , 𝑡1 ] = 𝜉𝐾̅ (𝑦)
∴ 𝜉𝐾̅ (𝑥 + 𝑦) ≥ rmin{𝜉𝐾̅ (𝑥) , 𝜉𝐾̅ (𝑦)}
= rmin{ [𝑡0 , 𝑡1 ] , [𝑡0 , 𝑡1 ] }
𝜉𝐾̅ (𝑥 + 𝑦) = [𝑡0 , 𝑡1 ]
⟹𝑥+𝑦 ∈𝐽
𝑐) For 𝑥 , 𝑦 ∈ 𝐽 if 𝑥 − 𝑦 𝑎𝑛𝑑 𝑦 ∈ 𝐽
⟹ 𝜉𝐾̅ (𝑥 − 𝑦) = [𝑡0 , 𝑡1 ] = 𝜉𝐾̅ (𝑦)
∴ 𝜉𝐾̅ (𝑥) ≥ rmin{𝜉𝐾̅ (𝑥 − 𝑦) , 𝜉𝐾̅ (𝑦)}
= rmin{ [𝑡0 , 𝑡1 ] , [𝑡0 , 𝑡1 ] } = [𝑡0 , 𝑡1 ]
𝜉𝐾̅ (𝑥) = [𝑡0 , 𝑡1 ]
⟹𝑥 ∈𝐽
iii)
𝑎) 𝜂𝐾 (0) ≤ 𝜂𝐾 (𝑥) ∀ 𝑥 ∈ 𝑋
𝜂𝐾 (0) = 𝛼 ⟹ 0 ∈ 𝐽
𝑏) For 𝑥 , 𝑦 ∈ 𝐽
⟹ 𝜂𝐾 (𝑥) = 𝛼 = 𝜂𝐾 (𝑦)
∴ 𝜂𝐾 (𝑥 + 𝑦) ≤ max{𝜂𝐾 (𝑥) , 𝜂𝐾 (𝑦)}
= max{ 𝛼 , 𝛼 }
𝜂𝐾 (𝑥 + 𝑦) = 𝛼
Prakasam Muralikrishna and Surya Manokaran, MBJ – Neutrosophic 𝛽 – Ideal of 𝛽 – Algebra
Neutrosophic Sets and Systems, Vol. 35, 2020
107
⟹𝑥+𝑦 ∈𝐽
𝑐) For 𝑥 , 𝑦 ∈ 𝐽 if 𝑥 − 𝑦 𝑎𝑛𝑑 𝑦 ∈ 𝐽
⟹ 𝜂𝐾 (𝑥 − 𝑦) = 𝛼 = 𝜂𝐾 (𝑦)
∴ 𝜂𝐾 (𝑥) ≤ max {𝜂𝐾 (𝑥 − 𝑦) , 𝜂𝐾 (𝑦)}
= max{ 𝛼 , 𝛼 } = 𝛼
𝜂𝐾 (𝑥) = 𝛼
⟹𝑥 ∈𝐽
∴ 𝐽 is an 𝛽 – ideal of 𝑋
Conversely, assuming 𝐽 is an 𝛽 – ideal of 𝑋. Then
i)
𝑎) If 0 ∈ 𝐽
Implies 𝜌𝐾 (0) = 𝑡
Also ∀ 𝑥 ∈ 𝑋 , 𝐼𝑚 ( 𝜌𝐾 ) = [𝑡 , 𝑠 ] & 𝑡 > 𝑠
⟹ 𝜌𝐾 (0) ≥ 𝜌𝐾 (𝑥) ∀ 𝑥 ∈ 𝑋
𝑏) For any 𝑥 , 𝑦 ∈ 𝐽
⟹𝑥+𝑦 ∈𝐽
⟹ 𝜌𝐾 (𝑥) = 𝜌𝐾 (𝑥 + 𝑦) = 𝑡 = 𝜌𝐾 (𝑦)
= min{ 𝜌𝐾 (𝑥) , 𝜌𝐾 (𝑦)}
∴ 𝜌𝐾 (𝑥 + 𝑦) ≥ min{𝜌𝐾 (𝑥) , 𝜌𝐾 (𝑦)}
c)
For any 𝑥 , 𝑦 ∈ 𝐽
If 𝑥 − 𝑦 𝑎𝑛𝑑 𝑦 ∈ 𝐽 ⟹ 𝑥 ∈ 𝐽
𝜌𝐾 (𝑥) = 𝑡 = min[ 𝑡, 𝑡] = min { 𝜌𝐾 (𝑥 − 𝑦) , 𝜌𝐾 (𝑦) }
ii)
𝑎) If 0 ∈ 𝐽
Implies 𝜉𝐾̅ (0) = 𝑡̅
Also ∀ 𝑥 ∈ 𝑋 , 𝐼𝑚 (𝜉𝐾̅ ) = [𝑡̅ , 𝑠̅] & 𝑡̅ > 𝑠̅
⟹ 𝜉𝐾̅ (0) ≥ 𝜉𝐾̅ (𝑥) ∀ 𝑥 ∈ 𝑋
𝑏) For any 𝑥 , 𝑦 ∈ 𝐽
⟹𝑥+𝑦 ∈𝐽
⟹ 𝜉𝐾̅ (𝑥) = 𝜉𝐾̅ (𝑥 + 𝑦) = 𝑡̅ = 𝜉𝐾̅ (𝑦)
= rmin{ 𝜉𝐾̅ (𝑥) , 𝜉𝐾̅ (𝑦)}
∴ 𝜉𝐾̅ (𝑥 + 𝑦) ≥ rmin{𝜉𝐾̅ (𝑥) , 𝜉𝐾̅ (𝑦)}
c)
For any 𝑥 , 𝑦 ∈ 𝐽
If 𝑥 − 𝑦 𝑎𝑛𝑑 𝑦 ∈ 𝐽 ⟹ 𝑥 ∈ 𝐽
𝜉𝐾̅ (𝑥) = 𝑡̅ = rmin[𝑡̅, 𝑡̅] = rmin { 𝜉𝐾̅ (𝑥 − 𝑦) , 𝜉𝐾̅ (𝑦) }
iii)
𝑎) If 0 ∈ 𝐽
Implies 𝜂𝐾 (0) = 𝛼
Also ∀ 𝑥 ∈ 𝑋 , 𝐼𝑚 ( 𝜂𝐾 ) = [𝛼 , 𝛽 ]& 𝛼 < 𝛽
⟹ 𝜂𝐾 (0) ≤ 𝜂𝐾 (𝑥) ∀ 𝑥 ∈ 𝑋
𝑏) For any 𝑥 , 𝑦 ∈ 𝐽
⟹𝑥+𝑦 ∈𝐽
⟹ 𝜂𝐾 (𝑥) = 𝜂𝐾 (𝑥 + 𝑦) = 𝛼 = 𝜂𝐾 (𝑦)
= max { 𝜂𝐾 (𝑥) , 𝜂𝐾 (𝑦)}
∴ 𝜂𝐾 (𝑥 + 𝑦) ≤ max{𝜂𝐾 (𝑥) , 𝜂𝐾 (𝑦)}
Prakasam Muralikrishna and Surya Manokaran, MBJ – Neutrosophic 𝛽 – Ideal of 𝛽 – Algebra
Neutrosophic Sets and Systems, Vol. 35, 2020
c)
108
For any 𝑥 , 𝑦 ∈ 𝐽
If 𝑥 − 𝑦 𝑎𝑛𝑑 𝑦 ∈ 𝐽 ⟹ 𝑥 ∈ 𝐽
𝜂𝐾 (𝑥) = 𝛼 = max[ 𝛼, 𝛼] = max { 𝜂𝐾 (𝑥 − 𝑦) , 𝜂𝐾 (𝑦) }
∴ 𝐾 is an MBJ – 𝛽 – Ideal of 𝑋.
3.9 Definition: Let 𝐾 = { < 𝑥, 𝜌𝐾 (𝑥), 𝜉𝐾̅ (𝑥), 𝜂𝐾 (𝑥) >/𝑥 ∈ 𝑋 } be an MBJ- Neutrosophic Set in 𝑋
and 𝑓 ∶ 𝑋 → 𝑌
be a mapping then the image of 𝐾 under 𝑓 , 𝑓(𝐾) is defined as
𝑓𝑠𝑢𝑝 (𝜌𝐾 )(𝑦) = 𝑠𝑢𝑝𝑥 ∈ 𝑓−1(𝑦) 𝜌𝐾 (𝑥) ;
𝑓(𝐾) = { < 𝑥, 𝑓𝑠𝑢𝑝 (𝜌𝐾 ), 𝑓𝑟𝑠𝑢𝑝 (𝜉𝐾̅ ), 𝑓𝑖𝑛𝑓 𝜂𝐾 (𝑥) >/𝑥 ∈ 𝑌} where
𝑓𝑟𝑠𝑢𝑝 (𝜉𝐾̅ )(𝑦) = 𝑟𝑠𝑢𝑝𝑥 ∈ 𝑓−1(𝑦) 𝜉𝐾̅ (𝑥) and 𝑓𝑖𝑛𝑓 (𝜂𝐾 )(𝑦) = 𝑖𝑛𝑓𝑥 ∈ 𝑓−1(𝑦) 𝜂𝐾 (𝑥)
3.10 Definition: Let 𝑓 ∶ 𝑋 → 𝑌 be a function and let 𝐾 and 𝐿 be two MBJ – Neutrosophic
𝛽 –
𝑓
Ideal
−1 (𝐿)
in
= { 𝑥,
𝑓 −1
𝑋&𝑌
respectively then the preimage of 𝐿 under
(𝜌𝐾 (𝑥)), 𝑓 −1 (𝜉𝐾̅ (𝑥)), 𝑓 −1 ( 𝜂𝐾 (𝑥)) >/𝑥 ∈ 𝑋 } such that
𝑓
is
defined
by
𝑓 −1 (𝜌𝐾 (𝑥)) = 𝜌𝐾 (𝑓(𝑥)) ; 𝑓 −1 (𝜉𝐾̅ (𝑥)) = 𝜉𝐾̅ (𝑓(𝑥)) and 𝑓 −1 (𝜂𝐾 (𝑥)) = 𝜂𝐾 (𝑓(𝑥)).
3.11 Theorem: Let 𝑓 ∶ 𝑋 → 𝑌
be an onto homomorphism of
MBJ – Neutrosophic 𝛽 – Ideal of 𝑌, then the preimage of 𝑓
𝛽 - algebra.
−1 (𝐾)
is an MBJ – Neutrosophic
𝛽 – Ideal of 𝑋.
Proof: Suppose 𝐾 be an MBJ - Neutrosophic 𝛽 - ideal of 𝑌
i)
For 𝑥 ∈ 𝑋
𝑓 −1 (𝜌𝐾 (0)) = 𝜌𝐾 (𝑓(0))
= 𝜌𝐾 (0)
≥ 𝜌𝐾 (𝑥)
For some 𝑥 , 𝑦 ∈ 𝑋
𝑓 −1 (𝜌𝐾 )(𝑥 + 𝑦) = 𝜌𝐾 (𝑓(𝑥 + 𝑦))
= 𝜌𝐾 (𝑓(𝑥) + 𝑓(𝑦))
≥ 𝑚𝑖𝑛 { 𝜌𝐾 (𝑓(𝑥)) , 𝜌𝐾 (𝑓(𝑦)) }
= min{ 𝑓 −1 (𝜌𝐾 (𝑥)), 𝑓 −1 (𝜌𝐾 (𝑦))}
𝑓 −1 (𝜌𝐾 )(𝑥) = 𝜌𝐾 (𝑓(𝑥))
≥ 𝑚𝑖𝑛 { 𝜌𝐾 (𝑓(𝑥) − 𝑓(𝑦)) , 𝜌𝐾 (𝑓(𝑦)) }
= 𝑚𝑖𝑛 { 𝜌𝐾 (𝑓(𝑥 − 𝑦)) , 𝜌𝐾 (𝑓(𝑦)) }
= 𝑚𝑖𝑛 { 𝑓 −1 (𝜌𝐾 (𝑥 − 𝑦)) , 𝑓 −1 (𝜌𝐾 (𝑦)) }
ii)
Suppose 𝐾 is an
𝑓 −1 (𝜉𝐾̅ (0)) = 𝜉𝐾̅ (𝑓(0))
= 𝜉𝐾̅ (0)
≥ 𝜉𝐾̅ (𝑥)
For some 𝑥 , 𝑦 ∈ 𝑋
𝑓 −1 (𝜉𝐾̅ )(𝑥 + 𝑦) = 𝜉𝐾̅ (𝑓(𝑥 + 𝑦))
= 𝜉𝐾̅ (𝑓(𝑥) + 𝑓(𝑦))
≥ 𝑟𝑚𝑖𝑛 { 𝜉𝐾̅ (𝑓(𝑥)) , 𝜉𝐾̅ (𝑓(𝑦)) }
= rmin{ 𝑓 −1 (𝜉𝐾̅ (𝑥)) , 𝑓 −1 (𝜉𝐾̅ (𝑦))}
Prakasam Muralikrishna and Surya Manokaran, MBJ – Neutrosophic 𝛽 – Ideal of 𝛽 – Algebra
Neutrosophic Sets and Systems, Vol. 35, 2020
109
𝑓 −1 (𝜉𝐾̅ )(𝑥) = 𝜉𝐾̅ (𝑓(𝑥))
≥ 𝑟𝑚𝑖𝑛 { 𝜉𝐾̅ (𝑓(𝑥) − 𝑓(𝑦)) , 𝜉𝐾̅ (𝑓(𝑦)) }
= 𝑟𝑚𝑖𝑛 { 𝜉𝐾̅ (𝑓(𝑥 − 𝑦)) , 𝜉𝐾̅ (𝑓(𝑦)) }
= 𝑟𝑚𝑖𝑛 { 𝑓 −1 (𝜉𝐾̅ (𝑥 − 𝑦)) , 𝑓 −1 (𝜉𝐾̅ (𝑦)) }
iii)
𝑓 −1 (𝜂𝐾 (0)) = 𝜂𝐾 (𝑓(0))
= 𝜂𝐾 (0)
≤ 𝜂𝐾 (𝑥)
For some 𝑥 , 𝑦 ∈ 𝑋
𝑓 −1 (𝜂𝐾 )(𝑥 + 𝑦) = 𝜂𝐾 (𝑓(𝑥 + 𝑦))
= 𝜂𝐾 (𝑓(𝑥) + 𝑓(𝑦))
≤ 𝑚𝑎𝑥 { 𝜂𝐾 (𝑓(𝑥)) , 𝜂𝐾 (𝑓(𝑦)) }
= max{ 𝑓 −1 (𝜂𝐾 (𝑥)), 𝑓 −1 (𝜂𝐾 (𝑦))}
𝑓 −1 (𝜂𝐾 )(𝑥) = 𝜂𝐾 (𝑓(𝑥))
≤ 𝑚𝑎𝑥 { 𝜂𝐾 (𝑓(𝑥) − 𝑓(𝑦)) , 𝜂𝐾 (𝑓(𝑦)) }
= 𝑚𝑎𝑥 { 𝜂𝐾 (𝑓(𝑥 − 𝑦)) , 𝜂𝐾 (𝑓(𝑦)) }
= 𝑚𝑎𝑥 { 𝑓 −1 (𝜂𝐾 (𝑥 − 𝑦)) , 𝑓 −1 (𝜂𝐾 (𝑦)) }
Hence 𝑓 −1 (𝐾) is an MBJ – 𝛽 – Ideal of 𝑋.
3.12 Theorem: Let 𝑓 ∶ 𝑋 → 𝑋 be an endomorphism on 𝑋 . If 𝐾 is an MBJ – Neutrosophic
̅
𝜂𝑓 (𝑥) = 𝜂(𝑓(𝑥)) >/𝑥 ∈ 𝑋 } is
𝛽 – Ideal of 𝑋 then 𝑓(𝐾) = { < 𝑥, 𝜌𝑓 (𝑥) = 𝜌(𝑓(𝑥)) , 𝜉𝑓̅ (𝑥) = 𝜉 (𝑓(𝑥)),
an MBJ – Neutrosophic 𝛽 – Ideal of 𝑋.
Proof: Suppose 𝐾 be an MBJ – Neutrosophic 𝛽 - ideal of X. Then,
i)
𝜌𝑓 (0) = 𝜌(𝑓(0))
= 𝜌(0) ≥ 𝜌(𝑥) ∀ 𝑥 ∈ 𝑋
𝜌𝑓 (𝑥 + 𝑦) = 𝜌(𝑓(𝑥 + 𝑦))
= 𝜌(𝑓(𝑥) + 𝑓(𝑦))
= min { 𝜌(𝑓(𝑥)) + 𝜌(𝑓(𝑦)) }
= min { 𝜌𝑓 (𝑥) , 𝜌𝑓 (𝑦) } ∀ 𝑥 , 𝑦 ∈ 𝑋
Also, 𝜌𝑓 (𝑥) = 𝜌(𝑓(𝑥))
≥ min { 𝜌(𝑓(𝑥) − 𝑓(𝑦)) , 𝜌(𝑓(𝑦))) }
= min { 𝜌(𝑓(𝑥 − 𝑦)) , 𝜌(𝑓(𝑦)) }
= min { 𝜌𝑓 (𝑥 − 𝑦) , 𝜌𝑓 (𝑦) }
ii)
̅
𝜉𝑓̅ (0) = 𝜉 (𝑓(0))
= 𝜉 (̅ 0) ≥ 𝜉 ̅(𝑥) ∀ 𝑥 ∈ 𝑋
̅
+ 𝑦))
𝜉𝑓̅ (𝑥 + 𝑦) = 𝜉 (𝑓(𝑥
= 𝜉 ̅(𝑓(𝑥) + 𝑓(𝑦))
̅
= rmin { 𝜉 ̅ (𝑓(𝑥)) + 𝜉 (𝑓(𝑦))
}
= rmin { 𝜉𝑓̅ (𝑥) , 𝜉𝑓̅ (𝑦) } ∀ 𝑥 , 𝑦 ∈ 𝑋
Also, 𝜉𝑓̅ (𝑥) = 𝜉 ̅(𝑓(𝑥))
Prakasam Muralikrishna and Surya Manokaran, MBJ – Neutrosophic 𝛽 – Ideal of 𝛽 – Algebra
Neutrosophic Sets and Systems, Vol. 35, 2020
110
≥ rmin { 𝜉 ̅ (𝑓(𝑥) − 𝑓(𝑦)) , 𝜉 ̅(𝑓(𝑦)) }
= rmin { 𝜉 ̅ (𝑓(𝑥 − 𝑦)) , 𝜉 ̅(𝑓(𝑦)) }
= rmin { 𝜉𝑓̅ (𝑥 − 𝑦) , 𝜉𝑓̅ (𝑦) }
iii)
𝜂𝑓 (0) = 𝜂(𝑓(0))
= 𝜂(0) ≤ 𝜂(𝑥) ∀ 𝑥 ∈ 𝑋
𝜂𝑓 (𝑥 + 𝑦) = 𝜂(𝑓(𝑥 + 𝑦))
= 𝜂(𝑓(𝑥) + 𝑓(𝑦))
= max { 𝜂(𝑓(𝑥)) + 𝜂(𝑓(𝑦)) }
= max { 𝜂𝑓 (𝑥) , 𝜂𝑓 (𝑦) } ∀ 𝑥 , 𝑦 ∈ 𝑋
Also, 𝜂𝑓 (𝑥) = 𝜂(𝑓(𝑥))
≤ max { 𝜂(𝑓(𝑥) − 𝑓(𝑦)) , 𝜂(𝑓(𝑦))) }
= max { 𝜂(𝑓(𝑥 − 𝑦)) , 𝜂(𝑓(𝑦)) }
= max { 𝜂𝑓 (𝑥 − 𝑦) , 𝜂𝑓 (𝑦) }
∴ 𝑓(𝐾) is an MBJ – 𝛽 – Ideal of 𝑋.
3.13
Theorem:
Let 𝑓 ∶ 𝑋 → 𝑌
be
a homomorphism of
𝛽
–
algebra.
If
𝐾
is an
MBJ – Neutrosophic 𝛽 – Ideal of 𝑋, with sup – rsup – inf property and ker(𝑓) ⊆ 𝑋𝐾 then the image
of the set 𝐾 , 𝑓(𝐾) is an MBJ – Neutrosophic 𝛽 – ideal of 𝑌.
Proof: Suppose 𝐾 is an MBJ – Neutrosophic 𝛽 – Ideal of 𝑋, with sup – rsup – inf property and
ker(𝑓) ⊆ 𝑋𝐾 then
i)
𝑓(𝜌𝐾 )(0) = 𝑠𝑢𝑝𝑥 ∈ 𝑓−1(0) { 𝜌𝐾 (𝑥) }
= 𝜌𝐾 (0)
≥ 𝜌𝐾 (𝑥) ∀ 𝑥 ∈ 𝑋
Hence, 𝑓(𝜌𝐾 )(0) = 𝑠𝑢𝑝𝑥 ∈ 𝑓−1(0) { 𝜌𝐾 (𝑥) }
= 𝑓(𝜌𝐾 )(𝑦) ∀ 𝑦 ∈ 𝑌
Let 𝑦1 , 𝑦2 ∈ 𝑌
Then there exists 𝑥1 , 𝑥2 ∈ 𝑋 such that 𝑓(𝑥1 ) = 𝑦1 , 𝑓(𝑥2 ) = 𝑦2 .
𝑓(𝜌𝐾 )(𝑦1 + 𝑦2 ) = sup { 𝜌𝐾 (𝑥1 + 𝑥2 ) ∶ 𝑥 ∈ 𝑓 −1 (𝑦1 + 𝑦2 ) }
≥ sup { 𝜌𝐾 (𝑥1 + 𝑥2 ) ∶ 𝑥1 ∈ 𝑓 −1 (𝑦1 ) & 𝑥2 ∈ 𝑓 −1 (𝑦2 ) }
≥ sup{min{ 𝜌𝐾 (𝑥1 ) , 𝜌𝐾 (𝑥2 )} , 𝑥1 ∈ 𝑓 −1 (𝑦1 ), 𝑥2 ∈ 𝑓 −1 (𝑦2 )}
≥ min{sup{ 𝜌𝐾 (𝑥1 ) ∶ 𝑥1 ∈ 𝑓 −1 (𝑦1 )} , sup{ 𝜌𝐾 (𝑥2 ) ∶ 𝑥2 ∈ 𝑓 −1 (𝑦2 )}}
= min{ 𝑠𝑢𝑝𝑥1 ∈ 𝑓−1(𝑦1) { 𝜌𝐾 (𝑥1 ) } , 𝑠𝑢𝑝𝑥2 ∈ 𝑓−1(𝑦2) { 𝜌𝐾 (𝑥2 ) }
= min{ 𝑓(𝜌𝐾 )(𝑦1 ) , 𝑓(𝜌𝐾 )(𝑦2 )}
Suppose that for some 𝑦1 , 𝑦2 ∈ 𝑌
Then 𝑓(𝜌𝐾 )(𝑦1 ) ≤ min{ 𝑓(𝜌𝐾 )(𝑦1 − 𝑦2 ) , 𝑓(𝜌𝐾 )(𝑦2 )}
Since 𝑓 is onto ∃ 𝑥1 , 𝑥2 ∈ 𝑋 such that 𝑓(𝑥1 ) = 𝑦1 & 𝑓(𝑥2 ) = 𝑦2
𝑓(𝜌𝐾 )(𝑓(𝑥1 )) < min{ 𝑓(𝜌𝐾 )(𝑓(𝑥1 ) − 𝑓(𝑥2 )) , 𝑓(𝜌𝐾 )( 𝑓(𝑥2 )) }
= min{ 𝑓(𝜌𝐾 )(𝑓(𝑥1 − 𝑥2 )) , 𝑓(𝜌𝐾 )( 𝑓(𝑥2 )) }
< min{ 𝑓 −1 ( 𝑓(𝜌𝐾 )) (𝑥1 − 𝑥2 ) , 𝑓 −1 ( 𝑓(𝜌𝐾 ))(𝑥2 )}
Prakasam Muralikrishna and Surya Manokaran, MBJ – Neutrosophic 𝛽 – Ideal of 𝛽 – Algebra
Neutrosophic Sets and Systems, Vol. 35, 2020
ii)
𝜌𝐾 (𝑥1 ) < min{ 𝜌𝐾 (𝑥1 − 𝑥2 ) , 𝜌𝐾 (𝑥2 )}
𝑓(𝜉𝐾̅ )(0) = 𝑟𝑠𝑢𝑝𝑥 ∈ 𝑓−1(0) { 𝜉𝐾̅ (𝑥) }
= 𝜉𝐾̅ (0)
≥ 𝜉𝐾̅ (𝑥) ∀ 𝑥 ∈ 𝑋
Hence, 𝑓(𝜉𝐾̅ )(0) = 𝑟𝑠𝑢𝑝𝑥 ∈ 𝑓−1(0) { 𝜉𝐾̅ (𝑥) }
= 𝑓(𝜉𝐾̅ )(𝑦) ∀ 𝑦 ∈ 𝑌
Let 𝑓(𝑥1 ) = 𝑦1 , 𝑓(𝑥2 ) = 𝑦2 .
𝑓(𝜉𝐾̅ )(𝑦1 + 𝑦2 ) = 𝑟 sup { 𝜉𝐾̅ (𝑥1 + 𝑥2 ) ∶ 𝑥 ∈ 𝑓 −1 (𝑦1 + 𝑦2 ) }
≥ rsup { 𝜉𝐾̅ (𝑥1 + 𝑥2 ) ∶ 𝑥1 ∈ 𝑓 −1 (𝑦1 ) & 𝑥2 ∈ 𝑓 −1 (𝑦2 ) }
≥ rsup{rmin{ 𝜉𝐾̅ (𝑥1 ) , 𝜉𝐾̅ (𝑥2 )} , 𝑥1 ∈ 𝑓 −1 (𝑦1 ), 𝑥2 ∈ 𝑓 −1 (𝑦2 )}
≥ rmin{ rsup{ 𝜉𝐾̅ (𝑥1 ) ∶ 𝑥1 ∈ 𝑓 −1 (𝑦1 )} , rsup{ 𝜉𝐾̅ (𝑥2 ) ∶ 𝑥2 ∈ 𝑓 −1 (𝑦2 )}}
= rmin{ 𝑟𝑠𝑢𝑝𝑥1 ∈ 𝑓−1(𝑦1) { 𝜉𝐾̅ (𝑥1 ) } , 𝑟𝑠𝑢𝑝𝑥2 ∈ 𝑓−1(𝑦2 ) { 𝜉𝐾̅ (𝑥2 ) } }
= rmin{ 𝑓(𝜉𝐾̅ )(𝑦1 ) , 𝑓(𝜉𝐾̅ )(𝑦2 )}
For 𝑦1 , 𝑦2 ∈ 𝑌
𝑓(𝜉𝐾̅ )(𝑦1 ) ≤ rmin{ 𝑓(𝜉𝐾̅ )(𝑦1 − 𝑦2 ) , 𝑓(𝜉𝐾̅ )(𝑦2 )}
𝑓(𝜉𝐾̅ )(𝑓(𝑥1 )) < rmin{ 𝑓(𝜉𝐾̅ )(𝑓(𝑥1 ) − 𝑓(𝑥2 )) , 𝑓(𝜉𝐾̅ )( 𝑓(𝑥2 )) }
= rmin{ 𝑓(𝜉𝐾̅ )(𝑓(𝑥1 − 𝑥2 )) , 𝑓(𝜉𝐾̅ )( 𝑓(𝑥2 )) }
< rmin{ 𝑓 −1 ( 𝑓(𝜉𝐾̅ )) (𝑥1 − 𝑥2 ) , 𝑓 −1 ( 𝑓(𝜉𝐾̅ ))(𝑥2 )}
𝜉𝐾̅ (𝑥1 ) < rmin{ 𝜉𝐾̅ (𝑥1 − 𝑥2 ) , 𝜉𝐾̅ (𝑥2 ) }
iii)
𝑓(𝜂𝐾 )(0) = 𝑖𝑛𝑓𝑥 ∈ 𝑓−1(0) { 𝜂𝐾 (𝑥) }
= 𝜂𝐾 (0)
≤ 𝜂(𝑥) ∀ 𝑥 ∈ 𝑋
Hence, 𝑓(𝜂𝐾 )(0) = 𝑖𝑛𝑓𝑥 ∈ 𝑓−1(0) { 𝜂𝐾 (𝑥) }
= 𝑓(𝜂𝐾 )(𝑦) ∀ 𝑦 ∈ 𝑌
Let 𝑓(𝑥1 ) = 𝑦1 , 𝑓(𝑥2 ) = 𝑦2 .
𝑓(𝜂𝐾 )(𝑦1 + 𝑦2 ) = inf { 𝜂𝐾 (𝑥1 + 𝑥2 ) ∶ 𝑥 ∈ 𝑓 −1 (𝑦1 + 𝑦2 ) }
≤ inf { 𝜂𝐾 (𝑥1 + 𝑥2 ) ∶ 𝑥1 ∈ 𝑓 −1 (𝑦1 ) & 𝑥2 ∈ 𝑓 −1 (𝑦2 ) }
≤ inf {max{ 𝜂𝐾 (𝑥1 ) , 𝜂𝐾 (𝑥2 )} , 𝑥1 ∈ 𝑓 −1 (𝑦1 ), 𝑥2 ∈ 𝑓 −1 (𝑦2 )}
≤ max{ inf { 𝜂𝐾 (𝑥1 ) ∶ 𝑥1 ∈ 𝑓 −1 (𝑦1 )} , inf { 𝜂𝐾 (𝑥2 ) ∶ 𝑥2 ∈ 𝑓 −1 (𝑦2 )} }
= max{ 𝑖𝑛𝑓𝑥1 ∈ 𝑓−1(𝑦1 ) { 𝜂𝐾 (𝑥1 ) } , 𝑖𝑛𝑓𝑥2 ∈ 𝑓−1(𝑦2) { 𝜂𝐾 (𝑥2 ) }
= max{ 𝑓(𝜂𝐾 )(𝑦1 ) , 𝑓(𝜂𝐾 )(𝑦2 )}
For 𝑦1 , 𝑦2 ∈ 𝑌
𝑓(𝜂𝐾 )(𝑦1 ) ≤ max{ 𝑓(𝜂𝐾 )(𝑦1 − 𝑦2 ) , 𝑓(𝜂𝐾 )(𝑦2 )}
𝑓(𝜂𝐾 )(𝑓(𝑥1 )) < max{ 𝑓(𝜂𝐾 )(𝑓(𝑥1 ) − 𝑓(𝑥2 )) , 𝑓(𝜂𝐾 )( 𝑓(𝑥2 )) }
= max{ 𝑓(𝜂𝐾 )(𝑓(𝑥1 − 𝑥2 )) , 𝑓(𝜂𝐾 )( 𝑓(𝑥2 )) }
< max{ 𝑓 −1 ( 𝑓(𝜂)) (𝑥1 − 𝑥2 ) , 𝑓 −1 ( 𝑓(𝜂𝐾 ))(𝑥2 )}
𝜂𝐾 (𝑥1 ) < max{ 𝜂𝐾 (𝑥1 − 𝑥2 ) , 𝜂𝐾 (𝑥2 )}
Thus, 𝑓(𝐾) is an MBJ – 𝛽 – ideal of 𝑌.
Prakasam Muralikrishna and Surya Manokaran, MBJ – Neutrosophic 𝛽 – Ideal of 𝛽 – Algebra
111
Neutrosophic Sets and Systems, Vol. 35, 2020
112
3.14 Theorem: Let 𝑓 ∶ 𝑋 → 𝑌 be an onto homomorphism of 𝛽 – algebra.
MBJ – Neutrosophic 𝛽 – ideal of 𝑋, with ker(𝑓) ⊆ 𝑋𝐾 then 𝑓
Proof: To prove 𝑓
−1
−1
(𝑓(𝐾)) = 𝐾.
(𝑓(𝐾)) = 𝐾.
It’s necessary to prove
𝑓 −1 (𝑓(𝜌𝐾 ))(𝑥) = 𝜌𝐾 (𝑥) ; 𝑓 −1 (𝑓(𝜉𝐾̅ )) (𝑥) = 𝜉𝐾̅ (𝑥) and 𝑓 −1 (𝑓(𝜂𝐾 ))(𝑥) = 𝜂𝐾 (𝑥).
For 𝑥 ∈ 𝑋 ; 𝑓(𝑥) = 𝑦
i)
Now, 𝑓 −1 (𝑓(𝜌𝐾 ))(𝑥) = 𝑓(𝜌𝐾 )(𝑓(𝑥))
= 𝑓(𝜌𝐾 )(𝑦)
= 𝑠𝑢𝑝𝑥 ∈ 𝑓−1(𝑦) { 𝜌𝐾 (𝑥) }
For 𝑥 ′ ∈ 𝑋 , 𝑥 ′ ∈ 𝑓 −1 (𝑦) ⟹ 𝑓( 𝑥 ′ ) = 𝑦
𝑓( 𝑥 ′ ) = 𝑓(𝑥)
⟹ 𝑓( 𝑥 ′ ) − 𝑓(𝑥) = 0
𝑓( 𝑥 ′ − 𝑥) = 0
This implies 𝑥 ′ − 𝑥 ∈ 𝐾𝑒𝑟 𝑓
𝑥 ′ − 𝑥 ∈ 𝑋𝜌𝐾
𝜌𝐾 (𝑥 ′ − 𝑥) = 𝜌𝐾 (0)
𝜌𝐾 (𝑥 ′ ) ≥ min{𝜌𝐾 (𝑥 ′ − 𝑥) , 𝜌𝐾 (𝑥)}
= min{𝜌𝐾 (0) , 𝜌𝐾 (𝑥)}
= 𝜌𝐾 (𝑥)
𝜌𝐾 (𝑥 ′ ) ≥ 𝜌𝐾 (𝑥) and similarly, 𝜌𝐾 (𝑥) ≥ 𝜌𝐾 (𝑥 ′ )
Therefore, 𝜌𝐾 (𝑥 ′ ) = 𝜌𝐾 (𝑥)
𝑓 −1 (𝑓(𝜌𝐾 ))(𝑥) = 𝑓(𝜌𝐾 )(𝑓(𝑥))
= 𝑓(𝜌𝐾 )(𝑓(𝑥 ′ ))
= 𝑠𝑢𝑝𝑥 ∈ 𝑓−1(𝑦) { 𝜌𝐾 (𝑥 ′ ) }
= 𝜌𝐾 (𝑥)
𝑓
ii)
−1
(𝑓(𝜌𝐾 ))(𝑥) = 𝜌𝐾 (𝑥)
𝑓 −1 (𝑓(𝜉𝐾̅ )) (𝑥) = 𝑓(𝜉𝐾̅ )(𝑓(𝑥))
= 𝑓(𝜉𝐾̅ )(𝑦)
= 𝑟𝑠𝑢𝑝𝑥 ∈ 𝑓−1(𝑦) { 𝜉𝐾̅ (𝑥) }
′
′
For 𝑥 ∈ 𝑋 , 𝑥 ∈ 𝑓 −1 (𝑦) ⟹ 𝑓( 𝑥 ′ ) = 𝑦
𝑓( 𝑥 ′ ) = 𝑓(𝑥)
⟹ 𝑓( 𝑥 ′ ) − 𝑓(𝑥) = 0
𝑓( 𝑥 ′ − 𝑥) = 0
This implies 𝑥 ′ − 𝑥 ∈ 𝐾𝑒𝑟 𝑓
𝑥 ′ − 𝑥 ∈ 𝑋𝜉̅𝐾
𝜉𝐾̅ (𝑥 ′ − 𝑥) = 𝜉𝐾̅ (0)
𝜉𝐾̅ (𝑥 ′ ) ≥ rmin{𝜉𝐾̅ (𝑥 ′ − 𝑥) , 𝜉𝐾̅ (𝑥)}
= rmin{𝜉𝐾̅ (0) , 𝜉𝐾̅ (𝑥)}
𝜉𝐾̅
(𝑥 ′ )
= 𝜉𝐾̅ (𝑥)
≥ 𝜉𝐾̅ (𝑥) and similarly, 𝜉𝐾̅ (𝑥) ≥ 𝜉𝐾̅ (𝑥 ′ )
Prakasam Muralikrishna and Surya Manokaran, MBJ – Neutrosophic 𝛽 – Ideal of 𝛽 – Algebra
If 𝐾 is an
Neutrosophic Sets and Systems, Vol. 35, 2020
113
Therefore, 𝜉𝐾̅ (𝑥 ′ ) = 𝜉𝐾̅ (𝑥)
𝑓 −1 (𝑓(𝜉𝐾̅ )) (𝑥) = 𝑓(𝜉𝐾̅ )(𝑓(𝑥))
= 𝑓(𝜉𝐾̅ )(𝑓(𝑥 ′ ))
= 𝑟𝑠𝑢𝑝𝑥 ∈ 𝑓−1(𝑦) { 𝜉𝐾̅ (𝑥 ′ ) }
= 𝜉𝐾̅ (𝑥)
𝑓 −1 (𝑓(𝜉𝐾̅ )) (𝑥) = 𝜉𝐾̅ (𝑥)
iii)
Proceeding in the same way,
𝑓 −1 (𝑓(𝜂𝐾 ))(𝑥) = 𝑓(𝜂𝐾 )(𝑓(𝑥))
= 𝑓(𝜂𝐾 )(𝑦)
= 𝑖𝑛𝑓𝑥 ∈ 𝑓−1(𝑦) { 𝜂𝐾 (𝑥) }
′
′
For 𝑥 ∈ 𝑋 , 𝑥 ∈ 𝑓 −1 (𝑦) ⟹ 𝑓( 𝑥 ′ ) = 𝑦
𝑓( 𝑥 ′ ) = 𝑓(𝑥)
⟹ 𝑓( 𝑥 ′ ) − 𝑓(𝑥) = 0
𝑓( 𝑥 ′ − 𝑥) = 0
This implies 𝑥 ′ − 𝑥 ∈ 𝐾𝑒𝑟 𝑓
𝑥 ′ − 𝑥 ∈ 𝑋𝜂𝐾
𝜂𝐾 (𝑥 ′ − 𝑥) = 𝜂𝐾 (0)
𝜂𝐾 (𝑥 ′ ) ≤ max{𝜂𝐾 (𝑥 ′ − 𝑥) , 𝜂𝐾 (𝑥)}
= max{𝜂𝐾 (0) , 𝜂𝐾 (𝑥)}
= 𝜂𝐾 (𝑥)
𝜂𝐾
(𝑥 ′ )
≥ 𝜂𝐾 (𝑥) and similarly, 𝜂𝐾 (𝑥) ≥ 𝜂𝐾 (𝑥 ′ )
Therefore, 𝜂𝐾 (𝑥 ′ ) = 𝜂𝐾 (𝑥)
𝑓 −1 (𝑓(𝜂𝐾 ))(𝑥) = 𝑓(𝜂𝐾 )(𝑓(𝑥))
= 𝑓(𝜂𝐾 )(𝑓(𝑥 ′ ))
= 𝑖𝑛𝑓𝑥 ∈ 𝑓−1(𝑦) { 𝜂𝐾 (𝑥 ′ ) }
= 𝜂𝐾 (𝑥)
𝑓
−1
(𝑓(𝜂𝐾 ))(𝑥) = 𝜂𝐾 (𝑥)
Therefore, all these conditions are proved and hence 𝑓 −1 (𝑓(𝐾)) = 𝐾.
4 Cartesian Product of MBJ – Neutrosophic 𝛃 – Ideal
This section introduces the cartesian product of MBJ – Neutrosophic 𝛽 – ideal and discusses few
associated results.
𝐾 = { < 𝑥, 𝜌𝐾 (𝑥), 𝜉𝐾̅ (𝑥), 𝜂𝐾 (𝑥) >/𝑥 ∈ 𝑋 }
and
̅
𝐿 = { < 𝑦, 𝜌𝐾 (𝑦), 𝜉𝐾 (𝑦), 𝜂𝐾 (𝑦) >/𝑦 ∈ 𝑌 } be two MBJ – Neutrosophic sets 𝑋 & 𝑌 respectively. The
4.1.Definition:
Let
𝐾 𝑎𝑛𝑑 𝐿
is
denoted
by
𝐾 × 𝐿
is
defined
as
̅
𝐾 × 𝐿 = { < (𝑥 , 𝑦 ) , 𝜌𝐾×𝐿 (𝑥, 𝑦), 𝜉𝐾×𝐿 (𝑥, 𝑦), 𝜂𝐾×𝐿 (𝑥, 𝑦) >/(𝑥, 𝑦) ∈ 𝑋 × 𝑌}
where
̅
𝜉𝐾×𝐿 ∶ 𝑋 × 𝑌 → 𝐷[0,1]
and
𝜂𝐾×𝐿 : 𝑋 × 𝑌 → [0,1].
𝜌𝐾×𝐿 (𝑥, 𝑦) =
𝜌𝐾×𝐿 ∶ 𝑋 × 𝑌 → [0,1] ;
Cartesian
product
of
Prakasam Muralikrishna and Surya Manokaran, MBJ – Neutrosophic 𝛽 – Ideal of 𝛽 – Algebra
Neutrosophic Sets and Systems, Vol. 35, 2020
min{ 𝜌𝐾 (𝑥) , 𝜌𝐿 (𝑦)}
;
114
̅ (𝑥, 𝑦) = rmin{ 𝜉𝐾̅ (𝑥) , 𝜉𝐿̅ (𝑦)}
𝜉𝐾×𝐿
and
𝜂𝐾×𝐿 (𝑥, 𝑦) = max{ 𝜂𝐾 (𝑥) , 𝜂𝐿 (𝑦)}
4.2 Theorem: If 𝐾 and 𝐿 be two MBJ – Neutrosophic 𝛽 – Ideal of 𝑋 & 𝑌 respectively then 𝐾 × 𝐿 is
an MBJ – Neutrosophic 𝛽 – Ideal of 𝑋 × 𝑌.
Proof: Let 𝐾 = { < 𝑥, 𝜌𝐾 (𝑥), 𝜉𝐾̅ (𝑥), 𝜂𝐾 (𝑥) >/𝑥 ∈ 𝑋 } and
𝐿 = { < 𝑦, 𝜌𝐾 (𝑦), 𝜉𝐾̅ (𝑦), 𝜂𝐾 (𝑦) >/𝑦 ∈ 𝑌 } be two MBJ – Neutrosophic sets 𝑋 & 𝑌.
Take (𝑥, 𝑦) ∈ 𝑋 × 𝑌
i)
𝜌𝐾×𝐿 (0,0) = min{ 𝜌𝐾 (0,0) , 𝜌𝐿 (0,0)}
≥ min{min{𝜌𝐾 (0), 𝜌𝐾 (0)} , min{𝜌𝐿 (0), 𝜌𝐿 (0)}}
= min{min{𝜌𝐾 (𝑥), 𝜌𝐾 (𝑦)} , min{𝜌𝐿 (𝑥), 𝜌𝐿 (𝑦)}}
= min{min{𝜌𝐾 (𝑥), 𝜌𝐿 (𝑥)} , min{𝜌𝐾 (𝑦), 𝜌𝐿 (𝑦)}}
= min{ 𝜌𝐾×𝐿 (𝑥) , 𝜌𝐾×𝐿 (𝑦)}
≥ 𝜌𝐾×𝐿 (𝑥, 𝑦)
Take (𝑢, 𝑣)) ∈ 𝑋 × 𝑌 where 𝑢 = (𝑥1 , 𝑦1 ), 𝑣 = (𝑥2 , 𝑦2 )
𝜌𝐾×𝐿 (𝑢 + 𝑣) = 𝜌𝐾×𝐿 ((𝑥1 , 𝑦1 ) + (𝑥2 , 𝑦2 ))
= 𝜌𝐾×𝐿 ((𝑥1 + 𝑥2 ), (𝑦1 + 𝑦2 ))
= min{ 𝜌𝐾 (𝑥1 + 𝑥2 ), 𝜌𝐿 (𝑦1 + 𝑦2 )}
≥ min{min{ 𝜌𝐾 (𝑥1 ), 𝜌𝐾 (𝑥2 )} , 𝑚𝑖𝑛{𝜌𝐿 (𝑦1 ), 𝜌𝐿 (𝑦2 )}}
= min{min{ 𝜌𝐾 (𝑥1 ), 𝜌𝐿 (𝑦1 )} , 𝑚𝑖𝑛{(𝜌𝐾 (𝑥2 ), 𝜌𝐿 (𝑦2 )}}
= min{ 𝜌𝐾×𝐿 (𝑥1 , 𝑦1 ) , 𝜌𝐾×𝐿 ((𝑥2 , 𝑦2 ))}
≥ min{ 𝜌𝐾×𝐿 (𝑢) , 𝜌𝐾×𝐿 (𝑣)}
𝜌𝐾×𝐿 (𝑢) = 𝜌𝐾×𝐿 (𝑥1 , 𝑦1 )
= min{ 𝜌𝐾×𝐿 (𝑥1 ) , 𝜌𝐾×𝐿 (𝑦1 )}
≥ min{min{ 𝜌𝐾 (𝑥1 − 𝑥2 ), 𝜌𝐾 (𝑥2 )} , 𝑚𝑖𝑛{𝜌𝐿 (𝑦1 − 𝑦2 ), 𝜌𝐿 (𝑦2 )}}
= min{min{ 𝜌𝐾 (𝑥1 − 𝑥2 ), 𝜌𝐿 (𝑦1 − 𝑦2 ))} , 𝑚𝑖𝑛{𝜌𝐿 (𝑥2 ), 𝜌𝐿 (𝑦2 )}}
= min{ 𝜌𝐾×𝐿 ((𝑥1 , 𝑦1 ) − (𝑥2 , 𝑦2 )), 𝜌𝐾×𝐿 (𝑥2 , 𝑦2 )}
ii)
≥ min{ 𝜌𝐾×𝐿 (𝑢 − 𝑣), 𝜌𝐾×𝐿 (𝑣)}
̅ (0,0) = rmin{ 𝜉𝐾̅ (0,0) , 𝜉𝐿̅ (0,0)}
𝜉𝐾×𝐿
≥ 𝑟min{rmin{𝜉𝐾̅ (0), 𝜉𝐾̅ (0)} , rmin{𝜉𝐿̅ (0), 𝜉𝐿̅ (0)}}
= 𝑟min{rmin{𝜉𝐾̅ (𝑥), 𝜉𝐾̅ (𝑦)} , rmin{𝜉𝐿̅ (𝑥), 𝜉𝐿̅ (𝑦)}}
= 𝑟min{rmin{𝜉𝐾̅ (𝑥), 𝜉𝐿̅ (𝑥)} , rmin{𝜉𝐾̅ (𝑦), 𝜉𝐿̅ (𝑦)}}
̅ (𝑦)}
̅ (𝑥) , 𝜉𝐾×𝐿
= rmin{ 𝜉𝐾×𝐿
̅ (𝑥, 𝑦)
≥ 𝜉𝐾×𝐿
̅ ((𝑥1 , 𝑦1 ) + (𝑥2 , 𝑦2 ))
̅ (𝑢 + 𝑣) = 𝜉𝐾×𝐿
𝜉𝐾×𝐿
̅ ((𝑥1 + 𝑥2 ), (𝑦1 + 𝑦2 ))
= 𝜉𝐾×𝐿
= 𝑟min{ 𝜉𝐾̅ (𝑥1 + 𝑥2 ), 𝜉𝐿̅ (𝑦1 + 𝑦2 )}
≥ 𝑟min{rmin{ 𝜉𝐾̅ (𝑥1 ), 𝜉𝐾̅ (𝑥2 )} , 𝑟𝑚𝑖𝑛{𝜉𝐿̅ (𝑦1 ), 𝜉𝐿̅ (𝑦2 )}}
= 𝑟min{𝑟 min{ 𝜉𝐾̅ (𝑥1 ), 𝜉𝐿̅ (𝑦1 )} , 𝑟𝑚𝑖𝑛{(𝜉𝐾̅ (𝑥2 ), 𝜉𝐿̅ (𝑦2 )}}
̅ ((𝑥2 , 𝑦2 ))}
̅ (𝑥1 , 𝑦1 ) , 𝜉𝐾×𝐿
= rmin{ 𝜉𝐾×𝐿
̅ (𝑣)}
̅ (𝑢) , 𝜉𝐾×𝐿
≥ rmin{ 𝜉𝐾×𝐿
̅ (𝑢) = 𝜉𝐾×𝐿
̅ (𝑥1 , 𝑦1 )
𝜉𝐾×𝐿
Prakasam Muralikrishna and Surya Manokaran, MBJ – Neutrosophic 𝛽 – Ideal of 𝛽 – Algebra
Neutrosophic Sets and Systems, Vol. 35, 2020
115
= rmin{ 𝜉𝐾×𝐿 (𝑥1 ) , 𝜉𝐾×𝐿 (𝑦1 )}
≥ 𝑟min{rmin{ 𝜉𝐾 (𝑥1 − 𝑥2 ), 𝜉𝐾 (𝑥2 )} , 𝑟𝑚𝑖𝑛{𝜉𝐿 (𝑦1 − 𝑦2 ), 𝜉𝐿 (𝑦2 )}}
= 𝑟min{rmin{ 𝜉𝐾 (𝑥1 − 𝑥2 ), 𝜉𝐿 (𝑦1 − 𝑦2 ))} , 𝑟𝑚𝑖𝑛{𝜉𝐿 (𝑥2 ), 𝜉𝐿 (𝑦2 )}}
= rmin{ 𝜉𝐾×𝐿 ((𝑥1 , 𝑦1 ) − (𝑥2 , 𝑦2 )), 𝜉𝐾×𝐿 (𝑥2 , 𝑦2 )}
≥ rmin{ 𝜉𝐾×𝐿 (𝑢 − 𝑣), 𝜉𝐾×𝐿 (𝑣)}
i)
𝜂𝐾×𝐿 (0,0) = max{ 𝜂𝐾 (0,0) , 𝜂𝐿 (0,0)}
≤ max{max{𝜂𝐾 (0), 𝜂𝐾 (0)} , max{𝜂𝐿 (0), 𝜂𝐿 (0)}}
= max{max{𝜂𝐾 (𝑥), 𝜂𝐾 (𝑦)} , max{𝜂𝐿 (𝑥), 𝜂𝐿 (𝑦)}}
= max{max{𝜂𝐾 (𝑥), 𝜂𝐿 (𝑥)} , max{𝜂𝐾 (𝑦), 𝜂𝐿 (𝑦)}}
= max{ 𝜂𝐾×𝐿 (𝑥) , 𝜂𝐾×𝐿 (𝑦)}
≤ 𝜂𝐾×𝐿 (𝑥, 𝑦)
𝜂𝐾×𝐿 (𝑢 + 𝑣) = 𝜂𝐾×𝐿 ((𝑥1 , 𝑦1 ) + (𝑥2 , 𝑦2 ))
= 𝜂𝐾×𝐿 ((𝑥1 + 𝑥2 ), (𝑦1 + 𝑦2 ))
= max{ 𝜂𝐾 (𝑥1 + 𝑥2 ), 𝜂𝐿 (𝑦1 + 𝑦2 )}
≤ max{max{ 𝜂𝐾 (𝑥1 ), 𝜂𝐾 (𝑥2 )} , 𝑚𝑎𝑥{𝜂𝐿 (𝑦1 ), 𝜂𝐿 (𝑦2 )}}
= max{max{ 𝜂𝐾 (𝑥1 ), 𝜂𝐿 (𝑦1 )} , 𝑚𝑎𝑥{(𝜂𝐾 (𝑥2 ), 𝜂𝐿 (𝑦2 )}}
= max{ 𝜂𝐾×𝐿 (𝑥1 , 𝑦1 ) , 𝜂𝐾×𝐿 ((𝑥2 , 𝑦2 ))}
≤ max{ 𝜂𝐾×𝐿 (𝑢) , 𝜂𝐾×𝐿 (𝑣)}
𝜂𝐾×𝐿 (𝑢) = 𝜂𝐾×𝐿 (𝑥1 , 𝑦1 )
= max{ 𝜂𝐾×𝐿 (𝑥1 ) , 𝜂𝐾×𝐿 (𝑦1 )}
≤ max{max{ 𝜂𝐾 (𝑥1 − 𝑥2 ), 𝜂𝐾 (𝑥2 )} , 𝑚𝑎𝑥{𝜂𝐿 (𝑦1 − 𝑦2 ), 𝜂𝐿 (𝑦2 )}}
= max{max{ 𝜂𝐾 (𝑥1 − 𝑥2 ), 𝜂𝐿 (𝑦1 − 𝑦2 ))} , 𝑚𝑎𝑥{𝜂𝐿 (𝑥2 ), 𝜂𝐿 (𝑦2 )}}
= max{ 𝜂𝐾×𝐿 ((𝑥1 , 𝑦1 ) − (𝑥2 , 𝑦2 )), 𝜂𝐾×𝐿 (𝑥2 , 𝑦2 )}
≤ max{ 𝜂𝐾×𝐿 (𝑢 − 𝑣), 𝜂𝐾×𝐿 (𝑣)}
Hence 𝐾 × 𝐿 is an MBJ – Neutrosophic 𝛽 – Ideal of 𝑋 × 𝑌.
4.3 Theorem: If 𝐾1 , 𝐾2 , … . . 𝐾𝑛 be an MBJ – Neutrosophic 𝛽 – Ideals of 𝑋1 , 𝑋2 , … 𝑋𝑛 respectively,
then ∏𝑛𝑖=1 𝐾𝑖 is also a MBJ – Neutrosophic 𝛽 – Ideal of ∏𝑛𝑖=1 𝑋𝑖 .
Proof:
i)
By induction on Theorem 4.2,
∏𝑛𝑖=1 𝜌𝐾 𝑖 (0) ≥ ∏𝑛𝑖=1 𝜌𝐾 𝑖 (𝑥𝑖 )
∏𝑛𝑖=1 𝜌𝐾 𝑖 (𝑥𝑖 + 𝑦𝑖 ) ≥ min{ ∏𝑛𝑖=1 𝜌𝐾 𝑖 (𝑥𝑖 ) , ∏𝑛𝑖=1 𝜌𝐾 𝑖 (𝑦𝑖 ) }
ii)
∏𝑛𝑖=1 𝜌𝐾 𝑖 (𝑥𝑖 ) ≥ min{ ∏𝑛𝑖=1 𝜌𝐾 𝑖 (𝑥𝑖 − 𝑦𝑖 ) , ∏𝑛𝑖=1 𝜌𝐾 𝑖 (𝑦𝑖 ) }
∏𝑛𝑖=1 𝜉𝐾̅ (0) ≥ ∏𝑛𝑖=1 𝜉𝐾̅ (𝑥𝑖 )
𝑖
𝑖
∏𝑛𝑖=1 𝜉𝐾̅ 𝑖 (𝑥𝑖 + 𝑦𝑖 ) ≥ 𝑟min{ ∏𝑛𝑖=1 𝜉𝐾̅ 𝑖 (𝑥𝑖 ) , ∏𝑛𝑖=1 𝜉𝐾̅ 𝑖 (𝑦𝑖 ) }
∏𝑛𝑖=1 𝜉𝐾̅ (𝑥𝑖 ) ≥ 𝑟min{ ∏𝑛𝑖=1 𝜉𝐾̅ (𝑥𝑖 − 𝑦𝑖 ) , ∏𝑛𝑖=1 𝜉𝐾̅ (𝑦𝑖 ) }
iii)
𝑖
𝑖
𝑖
𝑛
𝑛
∏𝑖=1 𝜂𝐾 𝑖 (0) ≤ ∏𝑖=1 𝜂𝐾 𝑖 (𝑥𝑖 )
∏𝑛𝑖=1 𝜂𝐾 𝑖 (𝑥𝑖 + 𝑦𝑖 ) ≤ max{ ∏𝑛𝑖=1 𝜂𝐾 𝑖 (𝑥𝑖 ) , ∏𝑛𝑖=1 𝜂𝐾 𝑖 (𝑦𝑖 ) }
∏𝑛𝑖=1 𝜂𝐾 𝑖 (𝑥𝑖 ) ≤ max{ ∏𝑛𝑖=1 𝜂𝐾 𝑖 (𝑥𝑖 − 𝑦𝑖 ) , ∏𝑛𝑖=1 𝜂𝐾 𝑖 (𝑦𝑖 ) }
Hence the proof is clear.
Prakasam Muralikrishna and Surya Manokaran, MBJ – Neutrosophic 𝛽 – Ideal of 𝛽 – Algebra
Neutrosophic Sets and Systems, Vol. 35, 2020
116
4.4 Theorem: For the MBJ – Neutrosophic subsets 𝐾 𝑎𝑛𝑑 𝐿 of 𝑋 & 𝑌 , if 𝐾 × 𝐿 is an
MBJ – Neutrosophic β – ideal of 𝑋 × 𝑌 then
ii)
𝜌𝐾 (0) ≥ 𝜌𝐿 (𝑦) & 𝜌𝐿 (0) ≥ 𝜌𝐾 (𝑥)
𝜉𝐾̅ (0) ≥ 𝜉𝐿̅ (𝑦) & 𝜉𝐿̅ (0) ≥ 𝜉𝐾̅ (𝑥)
iii)
𝜂𝐾 (0) ≤ 𝜂𝐿 (𝑦) & 𝜂𝐿 (0) ≤ 𝜂𝐾 (𝑥)
i)
Proof:
Let
𝐾&𝐿
be
MBJ
–
Neutrosophic
subsets
of
𝑋&𝑌
with
𝐾×𝐿
is
an
MBJ – Neutrosophic β – ideal of 𝑋 × 𝑌.
Suppose 𝜌𝐿 (𝑦) ≥ 𝜌𝐾 (0) and 𝜌𝐾 (𝑥) ≥ 𝜌𝐿 (0) for some 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑌.
𝜌𝐾×𝐿 (𝑥, 𝑦) = min{𝜌𝐾 (𝑥) , 𝜌𝐿 (𝑦)}
≥ min{𝜌𝐿 (0) , 𝜌𝐾 (0)}
= 𝜌𝐾×𝐿 (0,0)
which is a contradiction.
Thus, 𝜌𝐾 (0) ≥ 𝜌𝐿 (𝑦) & 𝜌𝐿 (0) ≥ 𝜌𝐾 (𝑥)
Similarly, 𝜉𝐿̅ (𝑦) ≥ 𝜉𝐾̅ (0) and 𝜉𝐾̅ (𝑥) ≥ 𝜉𝐿̅ (0) for some 𝑥 ∈ 𝑋 , 𝑦 ∈ 𝑌.
̅ (𝑥, 𝑦) = 𝑟min{𝜉𝐾̅ (𝑥) , 𝜉𝐿̅ (𝑦)}
𝜉𝐾×𝐿
≥ rmin{𝜉𝐿̅ (0) , 𝜉𝐾̅ (0)}
̅ (0,0)
= 𝜉𝐾×𝐿
Now, 𝜂𝐿 (𝑦) ≤ 𝜂𝐾 (0) and 𝜂𝐾 (𝑥) ≤ 𝜂𝐿 (0)
𝜂𝐾×𝐿 (𝑥, 𝑦) = max{𝜂𝐾 (𝑥) , 𝜂𝐿 (𝑦)}
≤ max{𝜂𝐿 (0) , 𝜂𝐾 (0)}
= 𝜂𝐾×𝐿 (0,0)
Hence the condition is satisfied.
4.5 Theorem: Let 𝐾 & 𝐿 be two MBJ – Neutrosophic β – ideals of 𝑋 & 𝑌 such that 𝐾 × 𝐿 is an
MBJ – Neutrosophic β – ideals of 𝑋 × 𝑌 . Then, either 𝐾 is an
MBJ - β – ideals of 𝑋 or 𝐿 is
an MBJ – Neutrosophic β – ideals of 𝑌.
Proof: By using the above theorem
i)
We consider 𝜌𝐾 (0) ≥ 𝜌𝐿 (𝑦) then
𝜌𝐾×𝐿 (0, 𝑦) ≥ min{𝜌𝐾 (0) , 𝜌𝐿 (𝑦)}
…… (1)
Given 𝐾 × 𝐿 is an MBJ – Neutrosophic β – ideals of 𝑋 × 𝑌
𝜌𝐾×𝐿 ((𝑥1 , 𝑦1 ), (𝑥2 , 𝑦2 )) ≥ min{𝜌𝐾×𝐿 ((𝑥1 , 𝑦1 ) − (𝑥2 , 𝑦2 )) , 𝜌𝐾×𝐿 (𝑥2 , 𝑦2 )}
∵ 𝜌𝐾×𝐿 ((𝑥1 , 𝑦1 ) − (𝑥2 , 𝑦2 )) ≥ min { 𝜌𝐾×𝐿 (𝑥1 , 𝑦1 ) , 𝜌𝐾×𝐿 (𝑥2 , 𝑦2 ) }
𝜌𝐾×𝐿 (𝑥1 , 𝑦1 ) ≥ min{ 𝜌𝐾×𝐿 ((𝑥1 − 𝑥2 ), (𝑦1 − 𝑦2 )), 𝜌𝐾×𝐿 (𝑥2 , 𝑦2 ) }
……. (2)
Now,
𝜌𝐾×𝐿 ((𝑥1 − 𝑥2 ), (𝑦1 − 𝑦2 )) ≥ min{ 𝜌𝐾×𝐿 (𝑥1 , 𝑦1 ) , 𝜌𝐾×𝐿 (𝑥2 , 𝑦2 ) }
……(3)
Put 𝑥1 = 𝑥2 = 0 in Equation (2 & 3)
𝜌𝐾×𝐿 (0, 𝑦1 ) ≥ min{ 𝜌𝐾×𝐿 (0, (𝑦1 − 𝑦2 )), 𝜌𝐾×𝐿 (0, 𝑦2 ) } and
𝜌𝐾×𝐿 (0 , (𝑦1 − 𝑦2 )) ≥ min{ 𝜌𝐾×𝐿 (0, 𝑦1 ) , 𝜌𝐾×𝐿 (0, 𝑦2 )}
……..(4)
From (1) & (4)
𝜌𝐿 (𝑦1 ) ≥ min{ 𝜌𝐿 (𝑦1 − 𝑦2 ) , 𝜌𝐿 (𝑦2 ) } and
𝜌𝐿 (𝑦1 − 𝑦2 ) ≥ min{ 𝜌𝐿 (𝑦1 ) , 𝜌𝐿 (𝑦2 )}
Prakasam Muralikrishna and Surya Manokaran, MBJ – Neutrosophic 𝛽 – Ideal of 𝛽 – Algebra
Neutrosophic Sets and Systems, Vol. 35, 2020
117
Consider 𝜉𝐾̅ (0) ≥ 𝜉𝐿̅ (𝑦) . Then
̅ (0, 𝑦) ≥ 𝑟min{𝜉𝐾̅ (0) , 𝜉𝐿̅ (𝑦)}
…… (5)
𝜉𝐾×𝐿
̅ ((𝑥1 , 𝑦1 ), (𝑥2 , 𝑦2 )) ≥ 𝑟min{𝜉𝐾×𝐿
̅ ((𝑥1 , 𝑦1 ) − (𝑥2 , 𝑦2 )) , 𝜉𝐾×𝐿
̅ (𝑥2 , 𝑦2 )}
𝜉𝐾×𝐿
ii)
̅ (𝑥1 , 𝑦1 ) , 𝜉𝐾×𝐿
̅ (𝑥2 , 𝑦2 ) }
̅ ((𝑥1 , 𝑦1 ) − (𝑥2 , 𝑦2 )) ≥ rmin { 𝜉𝐾×𝐿
∵ 𝜉𝐾×𝐿
̅ ((𝑥1 − 𝑥2 ), (𝑦1 − 𝑦2 )), 𝜉𝐾×𝐿
̅ (𝑥2 , 𝑦2 ) }
̅ (𝑥1 , 𝑦1 ) ≥ rmin{ 𝜉𝐾×𝐿
𝜉𝐾×𝐿
…….(6)
Now,
̅ ((𝑥1 − 𝑥2 ), (𝑦1 − 𝑦2 )) ≥ rmin{ 𝜉𝐾×𝐿
̅ (𝑥1 , 𝑦1 ) , 𝜉𝐾×𝐿
̅ (𝑥2 , 𝑦2 ) }
𝜉𝐾×𝐿
…(7)
Put 𝑥1 = 𝑥2 = 0 in Equation (6 & 7)
̅ (0, (𝑦1 − 𝑦2 )), 𝜉𝐾×𝐿
̅ (0, 𝑦2 ) } and
̅ (0, 𝑦1 ) ≥ rmin{ 𝜉𝐾×𝐿
𝜉𝐾×𝐿
̅ (0, 𝑦1 ) , 𝜉𝐾×𝐿
̅ (0, 𝑦2 )}
̅ (0 , (𝑦1 − 𝑦2 )) ≥ rmin{ 𝜉𝐾×𝐿
𝜉𝐾×𝐿
……..(8)
From (5) & (7)
𝜉𝐿̅ (𝑦1 ) ≥ rmin{ 𝜉𝐿̅ (𝑦1 − 𝑦2 ) , 𝜉𝐿̅ (𝑦2 ) } and
𝜉𝐿̅ (𝑦1 − 𝑦2 ) ≥ rmin{ 𝜉𝐿̅ (𝑦1 ) , 𝜉𝐿̅ (𝑦2 )}
iii)
As in the same way if we proceed, we get
𝜂𝐿 (𝑦1 ) ≤ max{ 𝜂𝐿 (𝑦1 − 𝑦2 ) , 𝜂𝐿 (𝑦2 ) } and
𝜂𝐿 (𝑦1 − 𝑦2 ) ≤ max{ 𝜂𝐿 (𝑦1 ) , 𝜂𝐿 (𝑦2 )}
∴ B is an MBJ - β – ideals of 𝑌.
5. Conclusion
This paper presents the characterization of MBJ – Neutrosophic 𝛽 – Ideal of 𝛽 – algebra. In
depth, the study analysed the homomorphic image, pre – image, cartesian product and related
results. The concept can be explored to other substructures of a 𝛽 – algebra.
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Neutrosophic Sets and Systems, Vol. 35, 2020
University of New Mexico
A Study of Social Media linked MCGDM Skill under Pentagonal
Neutrosophic Environment in the Banking Industry
Nidhi Singh1,2, Avishek Chakraborty3*, Soma Bose Biswas4, Malini Majumdar5
1Registrar,
2Department
Narula Institute of Technology, Kolkata-700109, W.B, India.
of Management, Maulana Abul Kalam Azad University of Technology, West Bengal, Haringhata,
Nadia-741249, W.B, India. Email: nidhi.singh@jisgroup.org
3Department
of Basic Science & Humanities, Narula Institute of Technology, Kolkata-700109, W.B, India.
Email:tirtha.avishek93@gmail.com
4Heritage
Business School, 994, Madurdaha, Chowbaga Road, Anandapur, P.O. East Kolkata Township, Kolkata-700107,
West Bengal, India. Email: somabbiswas@gmail.com
Army Institute of Management, Judges Court Road, Alipore, Kolkata 700027. Email: malini_majumdar@hotmail.com
5
*Corresponding author email address: tirtha.avishek93@gmail.com
ABSTRACT:
Social media is a new observable fact in computer-based technology and neutrosophic theory.
Researchers are now thinking of the power of social media in banks as it is the fastest expanding
online noticeable fact and banks with poor presence in social media are facing identity crisis under
uncertainty fields. Through social media we can share ideas and information through establishing
virtual networks. Initially it was evident that people used it for personal interaction with friends and
relatives but with changing time it is established that business houses and financial institutions
including Banks are using this popular technology to reach out to the prospective customers.
Especially in the banking industry digital communication is becoming most popular and powerful
as here consumers' interface is obligatory. Online communication has become a powerful medium
between banks and consumers. The power of social media is to connect and share information with
people across globe. Social Media in Indian Bank is not only a medium of advertising but it also
helps the Banks to be a part of their customers’ life as this relation involves conversation beyond
business under neutrosophic environment. The aim of this study is to find out the best social media
as per users’ preference and explore its impact on Banks’ business in pentagonal neutrosophic
(PNN) arena by increasing customer satisfaction and augment customer relationship management
in banking industry.
Keywords: Social Media, Customer Relationship Management, Customer satisfaction, Banking
Industry, PNN.
Nidhi Singh, Avishek Chakraborty, Soma Bose Biswas, Malini Majumdar; A Study of Social Media linked MCGDM Skill under Pentagonal
Neutrosophic Environment in the Banking Industry
Neutrosophic Sets and Systems, Vol. 35, 2020
1.
120
INTRODUCTION
1.1 Social Media:
The traditional marketing media consisting of radio, print, television etc offered a shotgun approach
as they represent communication in One to Many modes which we may call as Passive Approach.
However Social media marketing is following Many to Many mode, may be called Active Approach
with the power of implementation of Word Of Mouth. They are interactive in nature and believe in
peer to peer relationship [1] (Githa Heggde, 2018)
The substantial and considerable use of social media for last few years has elucidated that it is
amongst a few powerful weapons that has shown tremendous impacts on social life of human
beings and has hastened the mingling of people with each other. Previously, it was an encumbrance
for us to keep ourselves in touch with all those who were a little distant from us. Things have
apparently changed and social networking sites can take every credit for this prodigious platform
which enabled people to create their own identity. Whether it is about uploading personal posts,
surfing across the globe, getting all the indispensable information or even if one wants to express
their cavernous feelings then social media can act as a gullible platform for everyone. At times a few
of our problematic situations, disturbing sentiments need to get some succor and support by our
loved ones. At times only a single post of ours explains everything about what we are actually
feeling. Social media and its comprehensive enhancement is undeniable reality in this modern era.
Verily speaking social networking sites has made our socialization a bit easier with the rest of the
world. Data and statistics distinctly show the massive use of social media. Social Media has grown
tremendously due to increase in penetration of Internet Connectivity and easy availability of smart
phones and mobile gadgets. The conventional use of social media has changed from mere
entertainment to opportunities for trade and commerce. An estimate confirms that nearly two third
of Internet users are active on social media as well and this number is expected to cross approx three
billion by the end of 2020.
Nidhi Singh, Avishek Chakraborty, Soma Bose Biswas, Malini Majumdar; A Study of Social Media linked MCGDM Skill under Pentagonal
Neutrosophic Environment in the Banking Industry
Neutrosophic Sets and Systems, Vol. 35, 2020
121
1.2 Social Media Platforms:
Social Media is a blur of likes, tweets, shares, posts and contents. It has spread its wings in every
corner of the world. The numbers are staggering. 70% of the total internet users are now using social
media as per the research [2] (Bullas, 2014). In a research by Pew Research, 2014 [3] (DUGGAN, 2014)
, it is established that globally people are getting addicted to social media regardless of age, gender
and profession.
There are a variety of technological driven services in social media like sharing of pictures, videos
and audios, blogging, social games, social networking, business networks, reviews and much more.
Social media consists of a variety of internet-based mediums that enable users to network, share
content, interact with each other, and create communities around common interests. Social media
is therefore the media that we use to be sociable online and it can be divided into three main
categories:
•
Messaging and communication, e,g. blogging and micro-blogging such as Twitter.
•
Communities and social groups, e.g. Facebook
•
Photo and video sharing, e.g. YouTube
Statistically speaking, number of people using social media has considerably increased. The number
of people across globe who uses social media has extended 3 billion. As per a report Face book
reported 1.871, Whatsapp a billion and Instagram 600 million active users in January 2017 due to the
intensified use of social media.
1.3 Social Media in Bank:
The bank with no social media marketing strategy is at a risk of being left behind its competitors as
social media is playing a big role in marketing field. The tremendous growth and popularity of this
Nidhi Singh, Avishek Chakraborty, Soma Bose Biswas, Malini Majumdar; A Study of Social Media linked MCGDM Skill under Pentagonal
Neutrosophic Environment in the Banking Industry
Neutrosophic Sets and Systems, Vol. 35, 2020
122
medium is forcing banks to learn different social media platforms available to them and their
customers, different strategies to be adapted for proper selection of right social media channel so as
to reach out to maximum customers and improve their business. [4] (L, 2010) Banks are opting social
media channels due to the following main points:
•
To increase engagement with customers
•
To enhance their brand image by connecting with customers
•
To find out ways to distinguish themselves with competitors
•
To reduce cost as implementation of social media channels are less expensive in comparison
to the traditional marketing methods and with higher results
•
To boost innovations as through proper market research through social media more
customized products/services can be incorporated
•
To increase revenue as satisfies customers result in more business which in turn brings
revenue
The advancement faced by the banking sector today in the field of digitalization is an amalgamation
of social media and the wise users of this powerful tool which helps innumerable people in their
everyday work. With the help of digital feed people can access different social media sites like Face
book, Twitter, YouTube, Instagram etc to expand required knowledge about different products and
services offered by banks.
Many people opined that the new generations with proper knowledge of digital technologies are
more prone to use of social media but our response rate of seniors above 50 years was good. It was
observed that this number is gradually on the rise. Customers are an integral part of Banking
Industry and social media is an easiest and fastest way to reach to existing and prospective
customers. All the leading banks worldwide are trying to create business opportunities through
enhancing their creativity and innovative capacities. Through social media Banks can inform their
customers about their product & services offered in a most unique, attractive and innovative way. It
also helps the customers to consider sensibly about their investments and eradicate all the
complexity involved with the traditional banking processes. Traditional banks focused on providing
services to customers through different strategies such as advertising, direct mail or face to face
whereas banks and other financial institutions' is focusing on establishing relations with customers
through continue digital interaction vide different social media channels. By this continuous
interaction through social media Banks can discover customers' interests, feelings and behavior.
Customers of today look ahead to personalized services and they need to be heard and answered
promptly. Banks may fulfill their expectations through different digital media platforms like face
book, twitter and you tube instead of face to face interactions between customers and managers.
Bank’s Monitoring centers may follow comments, posts and tweets on social Medias which can give
a broad standpoint of customer insight about products and services banks can achieve an accurate
perceptive of customers. Today Banks need to have an effective presence in social media due to the
customers' anticipation and their obsession for the same.
Now a day social media has become crucial tool for banks. Banks are using the platform to discover
and keep up the relation with customers, motivating sales through advertisement and sales
Nidhi Singh, Avishek Chakraborty, Soma Bose Biswas, Malini Majumdar; A Study of Social Media linked MCGDM Skill under Pentagonal
Neutrosophic Environment in the Banking Industry
Neutrosophic Sets and Systems, Vol. 35, 2020
123
endorsement, guess change in consumer behavior and follow their trends and finally providing
customized services and support. Social media also helps in building customer relationships
through its reliability programs. It is an emerging concept in marketing especially in relation to
Banking Industry. Banks have now realized the influence of this medium over traditional form of
marketing strategy as it is the fastest growing online trend. Its influence has increased to the power
that the Banks with no social media connections are at a risk of being left out from competitors.
1.4 Survey of Uncertainty & Neutrosophic Theory:
In this current epoch, vagueness theory plays a vital position in social sciences and management
fields. Initially, it was discovered by prof. L.A Zadeh [5] & further, advancement of triangular [6],
trapezoidal [7], pentagonal [8], hexagonal [9], heptagonal [10] fuzzy number are established by
distinct researchers. It was extended by Prof. Attasonov [11] incorporating the idea of intuitionistic
fuzzy & further by Prof. Smarandache [12] discovering the concept of neutrosophic set. Nowadays,
researchers from distinct area are specializing in neutrosophic idea and advanced lots of exciting
articles in this domain. Recently, categorization of triangular [13], trapezoidal [14], pentagonal
[15-18] neutrosophic numbers has been developed by Chakraborty et.al. Recently, some MCGDM
based articles [19-23] are established in this neutrosophic arena which plays an essential impact in
this research domain.
Nidhi Singh, Avishek Chakraborty, Soma Bose Biswas, Malini Majumdar; A Study of Social Media linked MCGDM Skill under Pentagonal
Neutrosophic Environment in the Banking Industry
Neutrosophic Sets and Systems, Vol. 35, 2020
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124
Literature Review and Preliminaries:
This study focuses on identifying the services provided by banks through social media and
measuring its effect on customer satisfaction. The study also tries to find out the ways through
which Bank support the customers with the help of social media and the problems faced by the
customers to approach banks through social media.
Different customer services that can be provided by banks are;
•
Sharing of financial offers and upcoming promotions
•
Posting of education information and financial guidance
•
Allow clients to post reviews, complaints and suggestions
•
Reward them for recommending them
These virtual services are giving same level of personal interaction which was normally found in
physical banking as well but the advantage is that clients need not physically visit the banks. The
bank provides different services like Corporate banking, Investment banking, Asset Management,
Treasury services, Retail Banking etc. With the growth of information technology and advent of
Internet now banks are also using online banking. Internet banking is a convenient virtual banking
activity that is available for all the customers of the banks with easy and secured access to their
accounts. [2] Justified that now a day’s social media is being regularly accessed by almost 72% of the
internet users. Social Media helps the customers in providing utmost customer satisfaction through
obtaining real time comments, suggestions, complaints and addressing them instantly.
2.1 Safety & Reliability as Social Media Attribute:
Users’ Safety & Reliability is an important tool in consideration of social media channels. Data
should be handled without breaching the users’ privacy and data protection should be enormously
scrutinized. The most grounded measure that needs to be taken is to make undaunted quality of
one’s privacy whoever has affiliated with the social media channel [24] (Senthil Kumar N*, 2016).
Many a time’s users’ share their personal data intentionally and sometimes unknowingly. Often data
are extracted from them extrinsically by offering them some payback , for e.g, Location-Based Social
Network Services (LBSNS) like Google Latitude can trace the location of a person and his/her friends
[25] (Paul Lowry, 2011).
According to the safety analytics viewpoint, many people supervise the benefits and threats
associated while unveiling their credentials. It is often observed that customers are ready to forego
some privacy for a satisfactory range of danger. But reliability may be attacked significantly if
personal information is not utilized rationally and unvaryingly [26] (Patrick Van Eecke, 2010).
Proper implementation of security settings may improvise the Safety & Reliability of users’ data as
per their will [27] (Gail-Joon Ahn, 2011). Hence the quality of services provided by the social media
platforms, in terms of Safety & Reliability becomes an important criterion for its selection.
2.2 Responsiveness & Effectiveness as Social Media Attribute:
Responsiveness and Effectiveness of a social media site is measured by the internet speed,
expediency, response time etc with which customers access and use bank’s social media sites. [28]
Nidhi Singh, Avishek Chakraborty, Soma Bose Biswas, Malini Majumdar; A Study of Social Media linked MCGDM Skill under Pentagonal
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(Frederic Marimon, 2012). Efficiency of a bank’s social media is observed by timely and convenient
completion of all required interaction [29] (Chung Tin Fah, 2012). Social media can enhance the
conventional personnel–client bonding with an
effective technological
knowledge-based
relationship [30] (Rahimi & Me, 2016) .
Prompt responses can effectively be done in social media by providing customers relevant and quick
information as & when required. It is surely required for enhancement of quick responses to
customers’ queries for the improvement of e-services and improved customer satisfaction [31]
(Chinedu-Okeke & Obi, 2016). Banks can provide unique banking experience to their clients by
giving them services combined with technology [32] (Kalia, 2013). Banks may respond to its
customers’ query effectively through its social media sites but it needs to carefully monitor its
personnel’s’ response on social media sites to assess effectiveness of its response.
2.3 Ease of Use & Customer’s Satisfaction as Social Media Attribute:
Social media platform should fulfill the customers’ requirement and should be easy to be used with
minimum response time. Customers generally choose the media which is easy to operate. By ease of
use it means the service reliability and methods to use relevant information provided on a bank’s
social media websites [33] (Emel Kursunluoglu, 2015). Customers need punctual response for
acknowledgement of their complaints. The satisfaction dimension concentrates on evaluating the
banks promptness in responding to customers’ requirements [34] (Ajimon George, 2013).
For
getting customer loyalty the banks create user generated customized content for getting the
customers’ satisfaction dimension [35] (Norman Gwangwava, 2014). Customer’s confidence on
Bank’s social media platform to the extent their requirements are satisfied is termed as fulfillment
or satisfaction. Recently, several articles are established [36-40] in this research arena which plays an
essential role in research domain.
Definition 2.4: Neutrosophic Set: A set
in the universal discourse
, symbolically denoted by
, it is called a neutrosophic set if
, where
is said to be the true membership function, which has the degree of
belongingness,
of uncertainty, and
is said to be the indeterminacy membership, having degree
is said to be the incorrect membership, which has the
degree of non-belongingness of the decision maker.
exhibits the
following relation:
.
Nidhi Singh, Avishek Chakraborty, Soma Bose Biswas, Malini Majumdar; A Study of Social Media linked MCGDM Skill under Pentagonal
Neutrosophic Environment in the Banking Industry
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Definition 2.5: Single Typed Neutrosophic Number: Single Typed Neutrosophic Number
denoted
is
as
where
, where
,
and
is
given as:
2.6 Definition: Single-Valued Pentagonal Neutrosophic Number: A Single-Valued Pentagonal
Neutrosophic Number
is defined
as
,
where
membership function
. The accuracy membership function
, the indeterminacy
and the falsity membership function
are
given as:
Nidhi Singh, Avishek Chakraborty, Soma Bose Biswas, Malini Majumdar; A Study of Social Media linked MCGDM Skill under Pentagonal
Neutrosophic Environment in the Banking Industry
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2.7 Proposed Score Function:
Score function of a PNN completely depends on the value of truth, falsity and hesitation
membership indicator degree. The necessity of score function is to draw comparison or transfer a
PNN into a crisp number. In this section we will generate a score function as follows. For any
Pentagonal Single typed Neutrosophic Number (PSNN)
We define the score function as
2.7.1 Relationship between any two pentagonal neutrosophic fuzzy numbers:
Let us consider any two pentagonal neutrosophic fuzzy number defined as follows
,
1)
2)
3)
2.8 Basic Operations:
Let
=<(
,
,
be two IPFNs and
2.8.1
+
,
);
,
,
> and
=<(
,
,
,
);
,
,
>
. Then the following operational relations hold:
= < (
+
,
+
,
+
,
+
,
+
);
+
,
,
2.8.2
= < (
,
,
,
);
,
,
2.8.3
=<
2.8.4
=<(
,
,
,
,
,
,
);1
);
)
,
Nidhi Singh, Avishek Chakraborty, Soma Bose Biswas, Malini Majumdar; A Study of Social Media linked MCGDM Skill under Pentagonal
Neutrosophic Environment in the Banking Industry
Neutrosophic Sets and Systems, Vol. 35, 2020
3.
128
OBJECTIVE OF THE STUDY:
• To understand the factors affecting acceptance of Social Media Banking Technology across
Gender.
• To understand the best suitable social media channel for Banking Industry as per
customers’ preference.
4.
RESEARCH METHODOLOGY
The data have been collected from various respondents working in different organizations
categorized mainly as education sector, service sectors as banks, hospitals, etc. engineering works
and Government and Public sector companies in the Kolkata metro area. The study consisted of 94
respondents. A five point Likert scale is used where 5 indicates strongly agree, and 1 indicates
strongly disagree. 40.43% respondents are female and 59.57% are male. Age wise respondents
below the age <25 was 29.79 %, between 25 – 45 yrs was 52.13%, and >45 yrs was 18.08%
Research Instrument: The questinnaire is mainly focussed on: Social Media platforms used by the
banks and users adaptability of the same.
TABLE 4.1 DEMOGRAPPHIC DETAILS OF RESPONDENTS
CHARACTERISTICS
GENDER
AGE
FREQUENCY
%
MALE
56
59.57
FEMALE
38
40.43
<25
28
29.79
25-45
49
52.13
>45
17
18.08
SOURCE: QUESTIONNAIRE
Nidhi Singh, Avishek Chakraborty, Soma Bose Biswas, Malini Majumdar; A Study of Social Media linked MCGDM Skill under Pentagonal
Neutrosophic Environment in the Banking Industry
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Table 4.2 Indicate acceptance of Online Banking Technology across Gender
GEND
ER
Safety & Reliability
SD
FACEBOOK
YOU TUBE
TWITTER
F
Ease of Use & Customer’s
Effectiveness
Satisfaction
ATTRIBUTES
TWITTER
M
Responsiveness &
FACEBOOK
YOU TUBE
D
N
A
5
20
28
34
5.3
21.2
29.7
36.1
2
8
9
7
3
24
26
30
3.1
25.5
27.6
9
3
5
SA
7
SD
D
N
A
SA
SD
D
N
18
39
28
6
6
24
29
29
6
3.1
19.1
41.4
29.7
6.3
6.3
25.5
30.8
30.8
6.3
9
5
9
9
8
8
3
5
5
8
11
5
19
45
20
5
4
22
39
20
9
31.9
11.7
5.3
20.2
47.8
21.2
5.3
4.2
23.4
41.4
21.2
9.5
6
1
0
2
1
7
8
2
6
0
9
8
7
18
23
39
9
7
15
42
26
4
5
22
29
31
7
5.3
19.1
24.4
41.4
7.4
15.9
44.6
27.6
4.2
5.3
23.4
30.8
32.9
7.4
2
5
7
9
5
6
8
6
6
2
0
5
8
5
9
31
26
24
9
21
44
16
4
3
26
38
21
6
9.5
32.9
27.6
25.5
9.5
22.3
46.8
17.0
4.2
3.1
27.6
40.4
22.3
6.3
7
8
6
3
7
4
1
2
6
9
6
3
4
8
8
23
22
34
5
19
34
30
6
6
21
34
25
8
8.5
24.4
23.4
36.1
5.3
20.2
36.1
31.9
6.3
6.3
22.3
36.1
26.6
8.5
1
7
0
7
2
1
7
1
8
8
4
7
0
1
7
32
20
29
4
30
31
24
5
7
27
28
25
7
7.4
34.0
21.2
30.8
4.2
31.9
32.9
25.5
5.3
7.4
28.7
29.7
26.6
7.4
5
4
8
5
6
1
8
3
2
5
2
9
0
5
ATTRIBUTES
TWITTER
<25
FACEBOOK
YOU TUBE
25-45
SA
3
7.45
9.57
4
4.26
7
7.45
6
6.38
Table 4.3 Indicate acceptance of Social Media in Online Banking Technology across Age Gap
GENDER
A
TWITTER
FACEBOOK
Ease of Use &
Safety &
Responsiveness
Reliability
& Effectiveness
16
6
6
57.14
21.43
21.43
16
7
5
57.14
25.00
17.86
17
5
6
60.71
17.86
21.43
29
11
8
59.18
22.45
16.33
29
14
6
Customer’s
Satisfaction
Nidhi Singh, Avishek Chakraborty, Soma Bose Biswas, Malini Majumdar; A Study of Social Media linked MCGDM Skill under Pentagonal
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YOU TUBE
TWITTER
>45
FACEBOOK
YOU TUBE
59.18
28.57
12.24
31
12
8
63.27
24.49
16.33
9
5
3
52.94
29.41
17.65
8
4
5
47.06
23.53
29.41
10
4
3
58.82
23.53
17.65
5. Multi-Criteria Group Decision Making Problem in Pentagonal Neutrosophic Environment
In this current decade, researchers are very much interested in doing MCGDM problem in different
fields. Its main goal of this problem is to find out the best option among finite number of different
options in presence of distinct attributes, different decision maker’s choice and hesitation in human
thinking.
5.1 Illustration of the MCGDM problem
Let
is
is
the
the
distinct
distinct
attribute
alternative
set
set
respectively.
and
Let
be the weight set associated with the decision maker
and each
0 and also satisfies the relation
weight vector of the attribute function is defined as
0 and also satisfies the relation
. Also,
where each
.
5.2 Normalisation Algorithm of MCGDM Problem:
Step 1: Framework of Decision Matrices
Here, we considered all decision matrices according to the decision maker’s choice related with
finite alternatives and finite attribute functions. It is noted that the member’s
for each matrices
are of triangular fuzzy numbers. Thus, the final matrix is defined as follows:
Nidhi Singh, Avishek Chakraborty, Soma Bose Biswas, Malini Majumdar; A Study of Social Media linked MCGDM Skill under Pentagonal
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………………...(4.1)
Step 2: Framework of Standardized Decision matrix
We consider the following skill of normalization to obtain the standardized decision matrix where
=
= ([
in which the entity
,
,
,
,
];
= ([
,
,
,
,
];
,
,
) is formulated as
) where S=
.
Hence the new matrix becomes,
………………...(5.2)
Step 3: Framework of Single Decision matrix
To formulate a single group decision matrix M we utilized these logical operations of PNN [2.8]
are the weights of the decision makers for individual decision
matrix
So, the matrix becomes as follows:
………………… (5.3)
Step 4: Framework of Final matrix
To make the final decision matrix we used the logical operation [2.8] for different weights of the
attribute values and also finally operated
and converted the total
matrix into a Colum matrix form, finally we get the decision matrix as,
Nidhi Singh, Avishek Chakraborty, Soma Bose Biswas, Malini Majumdar; A Study of Social Media linked MCGDM Skill under Pentagonal
Neutrosophic Environment in the Banking Industry
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(5.4)
Step 5: Ranking
Now, by considering the Score value (2.7) and converting the matrix (5.3) into crisp form, so that we
could evaluate the best alternative corresponding to the best attributes.
5.3 Flowchart:
Figure 5.3.1: Flowchart for the problem
5.4 Illustrative Example: Here, we constructed a social media selection problem based on the
questionnaire table from which we have three different social medias are available. The problem is
to find out the best social media platform among these after computing the decision maker’s opinion
and maintain the attribute weights properly for this problem. Generally, social media platforms are
related with the attributes like safety & reliability, Responsiveness & Effectiveness, Ease of Use &
Customer’s Satisfaction of the system. Keeping these points in mind decision maker’s (Male/Female)
gives their opinion in hesitation arena and using verbal phrase we set the problem in pentagonal
neutrosophic domain. According to their suggestions we constructed the distinct decision matrices
in PNN environment as shows below:
,
,
are the alternatives.
,
Nidhi Singh, Avishek Chakraborty, Soma Bose Biswas, Malini Majumdar; A Study of Social Media linked MCGDM Skill under Pentagonal
Neutrosophic Environment in the Banking Industry
Neutrosophic Sets and Systems, Vol. 35, 2020
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,
are the attributes.
According to our problem there are two distinct decision makers are available in our environment,
having
and
the
weight
vector
related
with
weight
the
distribution
attribute
function
5.5 List of Verbal Phrase
No.
Quantitative Attributes
1
Verbal phrase
Strongly Agree (SA), Agree(A),
Neutral(N), Disagree(D), Strongly
Disagree (SD)
2
Strongly Agree (SA), Agree(A),
Neutral(N), Disagree(D), Strongly
Disagree (SD)
3
Strongly Agree (SA), Agree(A),
Neutral(N), Disagree(D), Strongly
Disagree (SD)
Step 1
According to the decision maker’s opinion from the questionnaire table we constructed the decision
matrices are as follows:
Nidhi Singh, Avishek Chakraborty, Soma Bose Biswas, Malini Majumdar; A Study of Social Media linked MCGDM Skill under Pentagonal
Neutrosophic Environment in the Banking Industry
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Step 2: Framework of Standardized decision matrix
Step 3: Framework of weighted Single Decision matrix
Step 4: Framework of Final Single Decision matrix
Step 4: Ranking
Now, we consider the established Score function (2.7), to convert the pentagonal neutrosophic
numbers into crisp one, thus we get the final ideal decision matrix as
Thus, ranking of the social media service is as
.
5.6 Results and Sensitivity Analysis
To understand how the attribute weights of each criterion affecting the relative matrix and their
ranking a sensitivity analysis is done. The below table is the evaluation table which shows the
sensitivity results.
Nidhi Singh, Avishek Chakraborty, Soma Bose Biswas, Malini Majumdar; A Study of Social Media linked MCGDM Skill under Pentagonal
Neutrosophic Environment in the Banking Industry
Neutrosophic Sets and Systems, Vol. 35, 2020
Decision Maker’s Weight
<(
>
<(
>
<(
>
<(
>
<(
135
Final Decision Matrix
Ordering
>
Figure 5.6.1: Sensitivity analysis table on Decision Maker’s Weight.
Nidhi Singh, Avishek Chakraborty, Soma Bose Biswas, Malini Majumdar; A Study of Social Media linked MCGDM Skill under Pentagonal
Neutrosophic Environment in the Banking Industry
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Figure 5.6.2: Best Alternative Social Media Service Table
5.7 Comparison Table
We compared this proposed work with the established works proposed by the researchers to find
the best social media and it is noticed that in each cases the alternative G2 becomes the best social
media service. The comparative table given as follows:
Approach
Ranking
(Chakraborty et al.) [18]
(Biswas et al.) [41]
Our Proposed
6. Implication:
Different social media platforms are available for communication with customers and digital
marketing like face book, twitter, Google plus, linked in, you tube etc. This study was primarily
done to identify the best suited social media platform for Banking Industry especially for customers
of West Bengal. We wanted to discover the right social media platform based on different attributes
as desired by customers. The perception of neutrosophy plays a critical role in designing
Nidhi Singh, Avishek Chakraborty, Soma Bose Biswas, Malini Majumdar; A Study of Social Media linked MCGDM Skill under Pentagonal
Neutrosophic Environment in the Banking Industry
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137
mathematical calculations. In this research work, we set the MCGDM problem in PNN environment
using the realistic data set. Applying the verbal phrases we formulate the MCGDM problem and
hence applied our logical operations of PNN on it to get the best alternatives. Finally, sensitivity
analysis is also performed here to which has a crucial impact in the ranking results. This novel
thought will help the other researchers in doing MCGDM problem from realistic data in social
media platform.
There are a lot of researches already done in social media implementation in Banking Industry.
However many results are still unknown. Our work is to explore the idea in the following points:
•
Defining the attributes necessary for social media platform for Banking Industry in West
Bengal.
•
Discovering the best suitable social media site for Banking Industry in West Bengal as per
customers’ preference.
•
Finding the best social media site which satisfies customers and generate revenue by
increasing business.
•
The graphical representation of adaptation of social media platform based on its attributes.
•
Covert the problem into PNN environment using verbal phrases.
•
Apply proposed MCGDM method in PNN arena.
•
Sensitivity analysis for Ranking in different cases.
7. Discussion
The main focus of this study was to find out the best social media platform for Banks. In total 94
respondents were asked varied questions and their choices and preferences about use of social
media in banks. Three parameters focusing their requirement were fixed as Safety, Efficiency and
Ease of use. The study examined different social media platform like Messaging and
communication, e,g. Twitter, Communities and social groups, e.g. Face book and Photo and video
sharing, e.g. YouTube. Face book was found to be most preferred channels both by the male and
female considering all the three factors. However other two channels have different opinion based
on different factors. In the sample considered here men respondents are more than women; most of
the respondents are under 45 years of age and they frequently uses social media. Both men and
women are equally boasting the use of social media however the worldwide trend also applied here
as it was observed that youngsters are dominating the social media sites. Social media mainly has
not only impacted the life of youngsters but it has also become drastically momentous since last ten
years across all age groups. It was also observed that awareness about the use of social media for
banking transactions is comparatively low in this region. It is agreed that Banks must publicize the
use of social media as an important tool for banking transactions. Social media has proven to be the
fastest communication mode and banks may use it for satisfying the ever increasing customized
needs of its customers. The more satisfied customers would result in more improved business for
banks. Moreover in the long run these satisfied customers would foster the brand loyalty and
customer loyalty would further result in improved customer relationship management.
Quantification of social media quality and its effects has got very less attention in the state. It is
accepted that the overall social media quality should be measured by banks to satisfy customers.
Nidhi Singh, Avishek Chakraborty, Soma Bose Biswas, Malini Majumdar; A Study of Social Media linked MCGDM Skill under Pentagonal
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Long tern connectivity with banks will improve if the services experienced by customers are
satisfactory. Better customer satisfaction in turn will bring customer loyalty. The objective of
adaption of social media for banking sector is not merely for likes and shares but it goes beyond that.
It is more of creating brand awareness and brand advocacy. Hence Banks should design their social
media strategy focusing realistic goals.
8. Findings:
Customers basically want three things from Banks like, better and responsive services, easier way to
bank and most importantly they want to be understood. Customers do not want generic ads and
offers, they want products and services tailored to them and will exchange data in order to receive
this. All the above is possible if the banks implement social media methodically and keep a proper
follow up for the same. As of now it is the best, easier and fastest responsive way to communicate
with customers. The following findings were done:
•
Face book is most preferred social media medium in comparison to other options like you
tube and twitter etc. considering all the three attributes
•
After applying pentagonal neutrosophic numbers into crisp one, we get the final ideal
decision matrix which gives the ranking of the social media as follows,
Facebook>Twitter>YouTube.
•
In spite of changing the weight age of attributes Face book remains the most preferred
choice across gender.
•
The three different attributes like Security, Efficiency and Ease of use have a strong impact
on overall customers’ satisfaction which resulted in selection of Bank’s social media
platform
•
Banks profit margin would be boosted with the help of proper implementation of social
media strategies. This will increase customers’ base without expansion of physical branches
which will result in reduction in cost..
9. Conclusions:
It may be concluded that Social Medias can greatly influence and enhance the function which is
being carried out in banks. This research found out that almost big banks in the state are using social
media for banking operations. On the questionnaire received from respondents the main concern or
obstacle for using social media was Security and privacy issues. Almost majority preferred social
media in terms of its efficiency and ease of use. Face book was found to be most acceptable mode
compare to any other media across gender and age. Majority of the respondents showed positive
indications for use of social media for banking operations in case of higher security. Hence we can
conclude that customers are willing to accept the social media for banking operations if Banks take
complete care of their security and privacy of data.
Therefore for banks in West Bengal all
conditions are met and it is up to the Banks’ policy of achieving the highest security in order to help
the customers to adapt the transactional social media. Our forecast is that transactional social media
will become more acceptable and popular in banking industry in coming years.
Nidhi Singh, Avishek Chakraborty, Soma Bose Biswas, Malini Majumdar; A Study of Social Media linked MCGDM Skill under Pentagonal
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Our future study includes more questionnaire collection and feedback received from customers and
banks to analysis the functionality of transactional social media and to suggest the ways to improve
the same.
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[2] Bullas, J. (2014). Social Media facts and statistics you should know in 2014. Springer.
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Nidhi Singh, Avishek Chakraborty, Soma Bose Biswas, Malini Majumdar; A Study of Social Media linked MCGDM Skill under Pentagonal
Neutrosophic Environment in the Banking Industry
Neutrosophic Sets and Systems, Vol. 35, 2020
University of New Mexico
Neutrosophic Multiset Topological Space
Rakhal Das1 and Binod Chandra Tripathy2
1Department
of Mathematics, Tripura University Agartala -799022; Tripura, INDIA;
and rakhal.mathematics@tripurauniv.in
E mail: 1rakhaldas95@gmail.com
2Department
of Mathematics, Tripura University Agartala -799022; Tripura, INDIA; E mail binodtripathy@tripurauniv.in,
tripathybc@yahoo.com and tripathybc@gmail.com
Abstract: In this article we have investigated some properties of netrosophic multiset topology. The
behavior of compactness and connectedness in netrosophic multiset topology, continuous function
on netrosophic multiset topology etc have been examined. Neutrosophic multiset is a generalization
of multisets and neutrosophic sets. Several properties of neutrosophic topological space in view of
neutrosophic multiset topological space have been studied.
Keywords: Neutrosophic Multiset; Neutrosophic Minimal set; Neutrosophic Maximal set;
Neutrosophic Multiset topology; Compactness, Connectedness; Continuous Neutrosophic Multiset;
Separation axioms; Distance function.
1. Introduction
In recent years, multisets and neutrosophic sets have become a subject of great interest for
researchers. Mathematicians always like to solve a complicated problem in a simple way and to find
out the most feasible solution. Neutrosophy has been introduced and studied by Smarandache [13,
15] as a new branch of philosophy. Recently various papers published on neutrosophic topology and
many researchers doing very well, neutrosophic decision making had been studied in [15, 17].
Algebraic properties of neutrosophic set studied in [9, 13], Neutrosophic Bipolar Vague Soft Set, and
its property studied in [9]. Smarandache generalizes intuitionistic fuzzy sets (IFSs) and other kinds
of sets to neutrosophic sets (NSs). In Smarandache [12, 13], some distinctions between NSs and IFSs
are underlined. decision-making problem, algebraic property one can analysis by topological
property connectedness and compactness property that property can help to take the decision into a
more reliable way. Smarandache [13, 14, 15] also defined various notions of neutrosophic topologies
on the non-standard interval. The logic of the neutrosophic set is very clear and its utilization on
topology is very beneficial for many standard problems like diagnosis of bipolar disorder
diseases group decision making and analytical property and evaluation Hospital medical care
systems etc. [1, 9, 13]. The relation between the intuitionistic fuzzy topology (IFT) on an IFS and the
neutrosophic topology are also analyzed by Smarandache.
Multiset theory was introduced by Bilzard [3]. Later on multiset topological space was studied by
many researcher Shravan and Tripathy [17, 18, 19]. The purpose of this paper is to construct a new
Rakhal Das and Binod Chandra Tripathy, Neutrosophic Multiset Topological Space
Neutrosophic Sets and Systems, Vol. 35, 2020
143
generalization of topological space called the neutrosophic multiset topological space. The possible
application of neutrosophic multiset topological space has been studied. For the different types of
behavior of objects in nature sometimes set theory and multiset theory fails to describe some
particular situation. Sometimes it is observed that Neutrosophic Multiset can be described in an
easier way to handle such cases. Neutrosophic set and topological space have been studied by
Salama and Alblowi [10, 11]. The concept of multiset topological space has been applied for studying
different properties of spatial objects. In this article we have used multiset neutrosophic topological
space for studying various spatial topological properties, like closeness connectedness and the
completeness property and its application further in various fields.
2. Materials and Methods
We procure some existing definitions in this paper, one may refer to Smarandache ([13], [15]) and S.
Alias, et.al [2].
We define functions
,
and from X to [0, 1]. Where T is membership value, F fails membership
value and I is the indeterminacy value.
The definition of neutrosophic multiset was first define by Smarandache [12] as follows.
Definition 2.1. [12] A Neutrosophic Multiset is a neutrosophic set where one or more elements are
repeated with the same neutrosophic components, or with different neutrosophic components.
Definition 2.2. The Empty neutrosophic multiset is denoted by N and define by
N = {<
>:
xX} where x can be repeated.
Definition 2.3. The Whole neutrosophic multiset is denoted by WX and define by
WX = {<
>:
xX} where x can be repeated.
The power set of neutrosophic multiset is denoted by P(X).
The collection of all possible subsets of X is called the power set of the netrosophic multiset.
Definition 2.4. Let A = {(
): xX} be a neutrosophic multiset on X then the
compliment of A is denoted by Ac and define by
Ac = {(
): xX}.
Where x can be repeated based on its multiplicity and the corresponding T, F, I values may or may
not be equal.
Rakhal Das and Binod Chandra Tripathy, Neutrosophic Multiset Topological Space
Neutrosophic Sets and Systems, Vol. 35, 2020
144
Definition 2.5. The intersection of NM sets are defined by A ∩B = {x: xA and xB}.
Definition 2.6. The union of NM sets are defined by
A∪B= { x: xA or xB}.
Definition 2.7. In the NM sets A⊂B if xA implies that xB.
Definition 2.8. Cardinality of a NM set A is denote the number of elements in a set A which is define
by card(A).
Definition 2.9. The Cartesian product of two neutrosophic multiset is defined by A×B = {(x, y) : xA
and yB}.
Definition 2.10 The difference of two NM sets A and B is the collection of members such that all
members belong to A but not in B.
Now we introduce two new types of operation maximal union NM set and minimal intersection NM
set.
Definition 2.11. Let X be a non-empty set, and neutrosophic multiset A and B in the form A = {(
): xX} and B = {(
): xX}, then the operations of maximal union
and minimal intersection NM set relation are defined as follows:
1.
(AB)max = {(
: xX}, where
,
= max{
},
,
} and
=
}.
min{
2. (AB)min
= min{
=
{(
= min{
= max{
:
,
} and
= max{
xX},
,
where
} and
}.
Example 2.1. Let X = {x, y, z, t} and A = { x<0.7, 0.2, 0.3>, x<0.7, 0.2, 0.3>, y<0.3, 0.2, 0.7>, y<0.9, 0.3, 0.1>, z<0.0, 1, 1>, t<0.5, 0.7, 0.5>},
B = { x<0.7, 0.2, 0.3>, x<0.8, 0.5, 0.2>, y<0.3, 0.2, 0.7>, y<0.3, 0.2, 0.7>, z<0.7, 0.8, 0.3>, t<0.0, 1, 1>} be neutrosophic multisets, then
the maximal union and minimal intersection are
(A B)Max = {x<0.8, 0.2, 0.2>, y<0.9, 0.2, 0.1>, z<0.7, 0.7, 0.3>, t<0.5, 0.7, 0.5>} and
Rakhal Das and Binod Chandra Tripathy, Neutrosophic Multiset Topological Space
Neutrosophic Sets and Systems, Vol. 35, 2020
145
(A B)Min = {x<0.7, 0.5, 0.3>, y<0.3, 0.3, 0.7>, z<0.0, 1, 1>, t<0.0, 1, 1>}
We formulate the following results without proof.
Result 2.1. Union of any family of neutrosophic multisets is always a neutrosophic multiset.
Result 2.2. Intersection of any family of neutrosophic multisets is always a neutrosophic multiset.
Result 2.3. The compliment of a neutrosophic multiset is always a neutrosophic multiset.
Result 2.4. Every neutrosophic set is a neutrosophic multiset but not necessarily conversely.
Example 2.2. Let A = {8〈0.6, 0.3, 0.2〉, 8〈0.6, 0.3, 0.2〉, 8〈0.4, 0.1, 0.3〉, 7〈0.2,0.7,0.0〉}.
Here A is a neutrosophic multiset but not a neutrosophic set.
Result 2.5. Let {Aj : j} be an arbitrary family of NM set in X , then the arbitrary maximal union and
arbitrary minimal intersection is also a NM set.
Remark 2.1. A neutrosophic multiset is a natural generalization of multiset as well as Cantor set.
We introduced neutrosophic multiset topological space and study some of its properties.
Definition 2.12. Let X be neutrosophic multiset and a non-empty family
subsets of WX is said to
be neutrosophic multiset topological space if the following axioms hold:
1.
N, WX
2.
AB , for A, B .
3.
.
, for {Ai :i}
In this case the pair (WX,
) is called a neutrosophic multiset topological space (NMTS in short) and
any neutrosophic multiset in
The elements of
is known as open neuterosophic multiset (ONMS in short) in WX .
are called closed neutrosophic multisets, otherwise a neutrosophic set F is
closed if and only if its complement
is an open neutrosophic multiset.
Rakhal Das and Binod Chandra Tripathy, Neutrosophic Multiset Topological Space
Neutrosophic Sets and Systems, Vol. 35, 2020
Definition 2.13. Let (WX,
Then
that
1
1
) and (WX,
1
is said be contained in
is coarser than
2
146
) be two neutrosophic multiset topological spaces on WX.
2
that is if
1
2
i.e, A
2
for each A
. In this case, we also say
1
.
2
Definition 2.14. Let (WX,
) be a neutrosophic multiset topological space on WX. A non-empty
family of subsets of X is called neutrosophic multiset basis of the neutrosophic multiset topological
space WX if any element of
can be express as the union of the element of .
Remark 2.2. As usual, basis of a neutrosophic multiset topological space is not unique.
Definition 2.15. Let (WX,
) be a neutrosophic multiset topological space with base . The interior of
the neutrosophic multiset A is the union of basis element of
which is contained in A and it is
denoted by NMintA, i.e, NMint (A) = { i : iA and i}.
Definition 2.16. Let (WX,
) be a neutrosophic multiset topological space. The closure of the
neutrosophic multiset A is the intersection of all closed neutrosophic multiset containing the set A it
is denoted by NMCl(A), i.e, NMCl(A) = {Fi : AFi and
}.
In view of the definitions, we formulate the following result.
Proposition 2.1. Let (WX,
) be a neutrosophic multiset topological space and A, B be two
neutrosophic multiset on WX, then the following property hold:
1.
NMintAA.
2.
AB NMint (A) NMint (B).
3.
A NMCl(A).
4.
AB NMCl(A) NMCl(B)
5.
NMint (NMint (A)) = NMint(A).
6.
NMCl (NMCl(A)) = NMCl(A).
7.
NMCl(AB) = NMCl(A)NMCl(B).
8.
NMint(WX) = WX.
9.
NMCl(N) = N.
Rakhal Das and Binod Chandra Tripathy, Neutrosophic Multiset Topological Space
Neutrosophic Sets and Systems, Vol. 35, 2020
Definition 2.17. Let (WX,
147
) be a neutrosophic multiset topological space a non-empty set S is called
a subbasis if the finite intersection of the elements of S can form a basis for
Definition 2.18. Let (WX,
.
) be a neutrosophic multiset topological space a point PA WX is said
to be a limit point of A if for every basis element containing p contains one element of A other than
p, i.e, A
N.
3. Results
3.1. Compactness, Connectedness and Continuous map.
Definition 3.1.1. Let (WX,
) be a neutrosophic multiset topological space. A neutrosophic multiset
A is said to be disjoint if two neutrosophic multisubsets B, C such that BC N and A = BC.
Definition 3.1.2. Let (WX,
) be a neutrosophic multiset topological space. The space WX is said to be
connected if WX cannot be express as the union of two disjoint neutrosophic multisets.
Definition 3.1.3. Let (WX,
) be a neutrosophic multiset topological space. The space WX is said to be
compact if every open cover of WX has a finite subcover.
Proposition 3.1.1. Every finite neutrosophic multiset topological space is compact.
Definition 3.1.4. Let (WX,
function f : (WX,
2
) and (WX,
1
) (WX,
1
2
2
) be two neutrosophic multiset topological space. The NMS
) is said to be continuous if for each open neutrosophic multiset V of
the neutrosophic multiset f -1 (V) is an open submset of
.
1
Proposition 3.1.2. Let f be a continuous function from a NMS topological space (WX,
NMS topological space (WX,
where A, B
1
) to another
1
), the function f is said to be a homomorphism if f(AB) = f(A) f(B)
2
and f(A), f(B)
2
.
Rakhal Das and Binod Chandra Tripathy, Neutrosophic Multiset Topological Space
Neutrosophic Sets and Systems, Vol. 35, 2020
148
3.2. Separation axioms on neutrosophic multiset.
We have defined disjoint neutrosophic multiset, connectedness, compactness and the continuous
image of neutrosophic multiset topological space. In this section we define separation axioms on
NMS topological space.
:
In the NMS a singleton set {p} is define by {p} = {
0,
-
, when x = p otherwise
=
for all xWX}.
-
Where x can be occurs more than one times it’s depends on its multiplicity and then T, F, I value may
or may not be equal.
Definition 3.2.1. Let (WX,
) be a neutrosophic multiset topological space. If there exist only two
open neutrosophic multiset in (WX,
Definition 3.2.2. Let (WX,
) is called indiscrete NMS topological space.
) be a neutrosophic multiset topological space. If every singleton
neutrosophic multiset is an open NMS set then (WX,
Definition 3.2.3. Let (WX,
) be a neutrosophic multiset topological space. If for every two distinct
NMS singleton sets, {x1}; {x2} then there exist V, U
{x1} U. Hence, (WX,
) is called discrete NMS topological space.
such that{x1} V and {x2} V or {x2} U and
) is NMSTo-space. i.e., there exists
-open NMS which contains one of them
but not the other.
Definition 3.2.4. Let (WX,
) be a neutrosophic multiset topological space. If for every two distinct
NMS singleton sets, {x1}; {x2} then there exist V,U
{x1} U. Hence, (WX,
such that{x1} V and {x2} V and {x2} U and
) is NMST1-space.
Definition 3.2.5. Let (WX,
) be a neutrosophic multiset topological space. If for every two distinct
NMS singleton sets, {x1}; {x2} then there exist V,U
{x1} U and U V = N. Hence, (WX,
such that{x1} V and {x2} V and {x2} U and
) is NMST2-space.
Rakhal Das and Binod Chandra Tripathy, Neutrosophic Multiset Topological Space
Neutrosophic Sets and Systems, Vol. 35, 2020
149
Proposition 3.2.1. Every NMST2-space is NMST1-space but it is not necessarily conversely.
Example 3.2.1. In co-finite neutrosophic multiset topological space is not a NMST2-space but when
the space has the finite neutrosophic multiset topology then it is NMST1-space.
So when we consider the infinite neutrosophic multiset topology we can get our desire result.
Proposition 3.2.2. Every NMST1-space is NMST0-space but it is not necessarily conversely.
Example 3.2.2. Let WX = {x<0.5, 0.7, 0.5>, x<0.5, 0.7, 0.5>, y<0.3, 0.4, 0.7>} and
Here (WX,
= { WX, N, {y}}.
) is a NMST0-space but it is not a NMST1.
Proposition 3.2.3. Every NMST2-space is NMST0-space but it is not necessarily conversely.
Example 3.2.3. Since every NMST0-space is not a NMST1-space and every NMST1-space is not a
NMST2-space so every NMST0-space is not a NMST2-space.
Proposition 3.2.4. Every discrete NMS topological space is NMST2-space.
3.3. Distance function on NMS.
In this section we are going to define a distance function on Neutrosophic set. Since in Neutrosophic
set we have defined Neutrosophic elements, Neutrosophic subset so it is natural to ask, can we
measure the distance between two Neutrosophic points or two Neutrosophic sets or is there any
distance between a Neutrosophic point to a Neutrosophic set?
The distance function between multiset points is defined by Shravan and Tripathy [12], based on the
multiplicity and the elements.
The Neutrosophic point p of a Neutrosophic multiset WX is define by p = {(
when x=p, otherwise
= 0,
-
-
,
):
,
for all xWX}.
Note: The Neutrosophic point p can have multiple time it’s depends on its multiplicity.
Definition 3.3.1. Let x, y be two Neutrosophic points on a Neutrosophic set WX. The distance
between the points is denoted by
|
|, |
(x, y) and is define by
|}, where the distance function
(x, y) = sup{|x-y|, |
is define by,
Rakhal Das and Binod Chandra Tripathy, Neutrosophic Multiset Topological Space
:WX R+{0}.
|,
Neutrosophic Sets and Systems, Vol. 35, 2020
150
Definition 3.3.2. Let x be a Neutrosophic point and A be a subset on a Neutrosophic set WX. The
(x, A) and is define by
distance between the point x and set the A is denoted by
=inf-sup{|x-yi |, |
|, |
(x, B)
| : for all yiA}.
|, |
Definition 3.3.3. Let A, B be two Neutrosophic subset of a Neutrosophic set WX the distance between
the sets A and set B is denoted by
(A, B) =inf sup{|xi - yi |, |
|,
| : xiA, and yiB}.
|, |
|
(A, B) and is define by
From the definition 5.1, 5.2 and 5.3 we can define another definition of matric space on a
Neutrosophic multiset.
Definition 3.3.4. A non-empty Neutrosophic set WX is said to be a Neutrosophic metric space with
: WXXWX R+{0}, if WX satisfy following:
the distance function
1.
0, x, yWX.
2.
=0, iff x=y and
3.
=
4.
Theorem 3.3.1. If
,
,
.
,x,yWX
+
and
, x,y,zWX
be two Neutrosophic metric spaces then
= max{
,
} is also a
be two Neutrosophic metric space then
= min{
,
} is not a
Neutrosophic metric space.
Theorem 3.3.2. If
and
Neutrosophic metric space.
The proof of the above two theorem is obvious using the concept of general matric space.
4. Applications
The work done in this paper is based on the application of neutrosophic sets in multiset topological
space. These can be further applicable for the development of neutrosophic topology separation
axioms on neutrosophic multiset topology and neutrosophic multisets.
5. Conclusions
Rakhal Das and Binod Chandra Tripathy, Neutrosophic Multiset Topological Space
Neutrosophic Sets and Systems, Vol. 35, 2020
151
In this paper we have established some properties of the neutrosophic multiset topological space
such as compactness and connectedness, continuous function on netrosophic multiset topology,
separation axioms on neutrosophic multiset topology. Also we have introduced the notion of the
distance function in neutrosophic multiset and examined some properties. This paper can be useful
for further development of neutrosophic multiset theory and neutrosophic topology.
Acknowledgments: The authors express their sincere thanks to Prof. F. Smarandache, Department
of Mathematics, University of New Mexico, USA for his comments, those improved the presentation
of the article.
Conflicts of Interest: The authors declare that no conflicting interest is involved in this article.
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Received: Apr 18, 2020. Accepted: July 8, 2020
Rakhal Das and Binod Chandra Tripathy, Neutrosophic Multiset Topological Space
Neutrosophic Sets and Systems, Vol. 35, 2020
University of New Mexico
Impact of Social Media in Banking Sector under Triangular
Neutrosophic Arena Using MCGDM Technique
Nidhi Singh1,2, Avishek Chakraborty3*, Soma Bose Biswas4, Malini Majumdar5
1Registrar,
2Department
Narula Institute of Technology, Kolkata-700109, W.B, India.
of Management, Maulana Abul Kalam Azad University of Technology, West Bengal, Haringhata, Nadia-741249,
W.B, India. Email: nidhi.singh@jisgroup.org
3Department
4Heritage
of Basic Science & Humanities, Narula Institute of Technology, Kolkata-700109, W.B, India.
Business School, 994, Madurdaha, Chowbaga Road, Anandapur, P.O. East Kolkata Township, Kolkata-700107, West
Bengal, India. Email: somabbiswas@gmail.com
5Army
Institute of Management, Judges Court Road, Alipore, Kolkata 700027. Email: malini_majumdar@hotmail.com
*Corresponding author email address: tirtha.avishek93@gmail.com
Abstract: This paper aims to uncover the position of social media in customer relationship
management (CRM) in banking industry in West Bengal (W.B) under neutrosophic environment. It
also tries to identify the attributes that influence the adaptation of different social media platforms
for marketing by Banks and finally its use in CRM approaches. The scope of this research is,
however, limited to the West Bengal (India) state. In this study a qualitative in-depth questionnaire
has been used in presence of impreciseness. Three case studies were developed, which explained the
adaptation and implementation of social media in retail banks in W.B. The responses, gathered
through in-depth interviews with top bank officials and estimated data from official web sites of the
banks have been used for MCGDM and sensitivity analysis. Different attributes like Safety &
Privacy, Effectiveness & Efficiency and Fulfillment & Responsiveness have a significant impact on
the overall service quality perception for Banks using social media and its platforms. We have
performed comparative analysis with the established method to find out the best social media
platform under neutrosophic environment in WB’s banking Industry. Successful implementation of
these platforms would then ensure Customer Loyalty and effective CRM. It was also noted that
customers mainly refrain from Banking through social media due to safety and privacy concerns.
The study was done to suggest betterment of social media marketing performance for banks in WB
in presence of uncertainty. It recommended managers to continuously monitor the overall service
quality of social media platforms as they lead to customer loyalty and CRM.
Keywords: West Bengal, Social media, Customer loyalty, Service quality, Customer Relationship
Management, Neutrosophic, CRM, Retail banking.
Nidhi Singh, Avishek Chakraborty, Soma Bose Biswas, Malini Majumdar; Impact of Social Media in Banking Sector under Triangular
Neutrosophic Arena Using MCGDM Technique
Neutrosophic Sets and Systems, Vol. 35, 2020
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1. INTRODUCTION:
1.1. SOCIAL MEDIA: Social Media is a communication platform that facilitates communication via
virtual networks. It is a virtual medium which is designed to aid people to share contents, pictures,
videos, and views swiftly and in real-time through websites and applications. The ability to share
photos, opinions, events, etc instantaneously has transformed the way we communicate and, also,
the way we do business. It provides the facility of continuously communicating with a large number
of people at a time. The revolution of Social media and its increasing impact has transformed its old
conventional image of amusement to an opportunity to work and trade. This vibrant use of social
media has affected almost every business sectors either positively or negatively. It has changed the
way business was done and Marketing has taken a new shift after this. Social media offers different
ways to promote business either through organic marketing (free) or by paid marketing. Web 2.0
technologies are the stage of Internet expansion where static web pages were converted to user
generated content [1]. The business communication is enhanced to a new height via online mode
through Social media [2]. According to [3] People share a lot of information about their personal
lives, their needs and preferences on social media and it may assist the institutions to design their
marketing policies. Based on the above data it can be said that the social media set-up facilitate in
building virtual group for individuals with similar mind-set, hobbies, work culture etc [4].
Therefore, use of social networking could assist Banks build up their brand awareness and brand
loyalty which ultimately help in customer acquirement and retention [5]. Communication between
clients and Banks has improved a lot after successful implementation of Internet mainly because it
has eliminated geographical hindrances [6]. Now it has almost become mandatory for all the banks
to adapt social media for getting customer loyalty and effective CRM.
1.2. Social media statistics in India: India is the 2nd largest country in the world in terms of
Population with over 1.36 billion people.
•
India currently has a population of 1,369,566,180 - this is 17.1% of the world’s total
population
•
Median age is 27.1 years - it’s a young country
•
Life expectancy is 69 years
•
Internet penetration is low in India - yet, in December 2018, 566 million users were online in
India. Out of this - 493 million are regular users of the internet. (source: livemint).
•
At the end of 2018, the number of social media users in India stood at 326.1 million. (statista)
•
At the end of 2019, this number has been estimated to grow to 351.4 million.
•
On average, Indian users spend 2.4 hours on social media a day (slightly below the global
average of 2.5 hours a day). (Source: The Hindu)
•
290 million active social media users in India access social networks through their mobile
devices. (Source: Hootsuite)
Nidhi Singh, Avishek Chakraborty, Soma Bose Biswas, Malini Majumdar; Impact of Social Media in Banking Sector under Triangular
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India: social network penetration 2017-2023
Based on customer’s requirement and rapid market the number of social media sites is increasing
day by day to cater to the needs of different audience groups. Before choosing social media platform,
it is essential for banks to realize the available social media platforms and location of their customer
base in these Medias. Some of the social media categories are as follows:
1.2.1 Communities and social groups:
“We build technologies to give people the power to connect with friends and family,
find communities and grow businesses”- face book
These sites allow connecting people of similar interests and background. This is used to share
information and events to large number of customers and building relationship by regular
interaction. Banks may also pose their brand on social network as an expert information source. This
may also be used for educating and training customers regarding different products and services
provided by banks.
Face book Statistics in India:
•
India ranks first in terms of face book users. Currently is has 269 million active users in India
(Source: Investopedia)
•
The largest user group by age on Face book is 18-24 years, with a massive 97.2 million users.
Nidhi Singh, Avishek Chakraborty, Soma Bose Biswas, Malini Majumdar; Impact of Social Media in Banking Sector under Triangular
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Face book usage penetration in India from 2015 to 2023
Messaging and communication: (e.g. blogging and micro-blogging such as Twitter):
“Follow everything from breaking news and entertainment, to sports, politics, and everyday
interests. Then, join the conversation”- Twitter
Blogging and Micro Blogging are used for creating online communities where customers can seek
out information and answers to their questions. It is used to listen and resolve customer
queries/issues in banking world. It creates a vast online, viral, and word of mouth, which is optimal
for establishing brand loyalty and monitoring reputation.
Twitter Statistics in India:
•
India has 7.75 million users on Twitter. (Source: statista)
•
18% of social media users in India look at Twitter as a source of news. (Source: Reuters)
•
Twitter usage unlike other platforms is actually decreasing = 2.2% per quarter (Source:
Digital 2019 report from Hootsuite)
Nidhi Singh, Avishek Chakraborty, Soma Bose Biswas, Malini Majumdar; Impact of Social Media in Banking Sector under Triangular
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Content Communities: (Photo and video sharing, e.g. YouTube):
“Enjoy the videos and music you love, upload original content, and share it all with friends, family,
and the world” – YouTube. They are content specific. These could be used for brand promotion,
engaging customer through sharing pictures, videos etc.
You Tube statistics in India
•
As per Google announcement, as of August 2018, there were 245 million active You Tube
users in India.
•
This figure is predicted to double over the next two years.
•
Online video accounts for 75% of data traffic in the country – and with 4G networks
improving; this is likely to further increase.
Nidhi Singh, Avishek Chakraborty, Soma Bose Biswas, Malini Majumdar; Impact of Social Media in Banking Sector under Triangular
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The literature on the banking sector has abundant references to online and electronic services (e.g.
e-banking), but has paid relatively little attention to the adoption and use of social media [7-9].
1.3. BANKING AND SOCIAL MEDIA
Banking sector is the backbone of any emerging economy. Banks are instrumental in implementing
the economic reforms. Any revolution in the banking sector because of the acceptance of technology
is bound to have a broad impact on an economy’s growth. These days, banks are seeking
unconventional ways to provide and differentiate amongst their various services. Customers now
demand a facility to conduct their banking activities at any time and place according to their
convenience [10].
Banking sector is the backbone of any emerging economy. Banks are instrumental in implementing
the economic reforms. Any revolution in the banking sector because of the acceptance of technology
is bound to have a broad impact on an economy’s growth. These days, banks are seeking
unconventional ways to provide and differentiate amongst their various services. Customers now
demand a facility to conduct their banking activities at any time and place according to their
convenience [11].
Social media has changed the entire gamut of business and marketing and Banking Industry is no
exception to this because here the Customer Interaction is a must. Today Social media is universal
and pervasive, so banks can rely on it. Digital communication is becoming a strong communication
medium between Banks and customers. This media is proving itself indispensable in connecting to
the potential clients. By allowing transfer of money, getting credit and even simply opening a bank
account, it has improved customer services which in turn are improving the customer relationship.
Assessing people’s sentiments is a very significant and staggering job, particularly in case of service
industry. Social media has a unique ability to create and sustain associations with customers,
creating better Customer relations. Hence banks need to consider social media as an integral part of
their overall marketing strategy [12].
People use Face book, Twitter, YouTube, Instagram, LinkedIn etc to understand different
information regarding the different products and services provided by banks only after
understanding the facilities and prospects of various social media platforms. Banks are using this
network to inform their customers about their products and upgrade them according to customers'
feedback. On the other hand, there is the talk of turnover in social networks. Also, purchases can be
made through social networks.
Physical Banking opted tactics like advertising, direct mail or face to face communication for
customer interaction so far but now the approaches have changed from providing customer service
to affiliation and long term relationship with customers. For doing it, banks need to diagnose
customers’ interests, emotions and behavior and with help of social media this analysis are being
done easily. Today, customers expect that they should be heard and answered and receive the
services they need through social media.
Social Medias can greatly affect the reputation and the brand image of the banks. Banks need a
transparent understanding of the key elements in the development of social media and adopt a road
Nidhi Singh, Avishek Chakraborty, Soma Bose Biswas, Malini Majumdar; Impact of Social Media in Banking Sector under Triangular
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map and a strategy. The banks may use the following pathway in social media to listen to the
customers.
•
As Is: Banks need to understand the customers’ requirements initially by analyzing their
data in social networks.
•
Listen: The next step would be to analyze the data carefully. Then the bank should design
and provide support as per their expectation,
•
Engage: Information can be collected through customers and through feedback taken Bank’s
can fulfill the customers’ needs.
•
Optimize: In the last step bank should attract fans and increase the loyalty of existing
customers by using customers' feedback and analyzing their interactions with each other.
In a media landscape increasingly dominated by social media, Bank’s marketing strategy for
these platforms can make or break its success as a brand. Banks need to hold their social media
efforts to high standard, creating custom made strategies that build their brand, win customers,
and yield high ROI. Therefore social media techniques have become essential communication
tools for banks to communicate with people across globe. Banks are adapting social media
because they are finding it difficult to fight with traditional banking methods such as interest
rates and product differentiation to attract new clients and sustain the existing ones. In today’s
aggressive atmosphere customer loyalty can be gained through allocation of finer service quality
to ensure maximum customer satisfaction [13].The purpose of this study is thus, to explore the
implication of social media on service quality perception and client loyalty in the banking
industry of West Bengal. Social media service quality can be used to boost customers’ loyalty by
Banks in the India banking industry [14]. There are limited studies on social media service
quality and client loyalty for Indian Banking industry. This study will contribute towards
reducing the knowledge gap between impact of social media on service quality and customers’
loyalty. These attributes so discussed would be able to improve the quality of social media
performance.
The article is structured as follows: The next section will provide a discussion on the use of social
media in the Indian banking industry, followed by a discussion on the methodology that was
used for data collection, and a presentation of the results. The last section provides the study’s
findings and conclusion.
2. Literature Review: Indian Banks have started using social media in their regular operations in
various capacities a little lately and are at different stages of maturity. As of April 2013, some private
banks provide regular updates on the latest offers and allow basic customer operations through
popular social media sites. A large private bank in India hosted Face book application on its secure
servers allowing balance amount check, cheque book request, stop payment, etc. Some of the private
banks are using their social media websites to provide their customers, distinct offers, detailed
product information and consumer care services. With some banks taking the lead by setting
Nidhi Singh, Avishek Chakraborty, Soma Bose Biswas, Malini Majumdar; Impact of Social Media in Banking Sector under Triangular
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example, the others also have started following their footsteps. In a survey by the Financial Brand
newsletter in July 2013, it was established that ICICI, Axis and HDFC Banks are among the top 10
Banks with Social Media presence. Of late public sector banks have also started using this media in a
grand way. As per present scenario, Indian banks can no longer live in denial by avoiding and not
using Social Media if they do not want threatening their own business. The Indian banking industry
has envisaged some social media channels to attract tech-savvy clients and improve customer
services to bring customer loyalty [15]. The use of social media in India has gained its importance.
2.1 Social Media Safety & Privacy: Privacy refers to the extent by which the customers’ details are
protected by bank’s social media platform [16].Banks need to give their customers enough
confidence to use their social media accounts so that they may perceive that their personal
information will be secured and not to be misused by banks [17]. Banks can build new healthy
relationship with customers if the privacy is perceived positively by customers [18]. The information
get disclosed and shared through social media so easily, that it has raised doubts about its privacy
among the users [19]. Maintenance of privacy in bank’s social media channel has been a big
challenge for the banking industry. The main challenge is to monitor and control the posts in these
sites [20]. A proper privacy setting of social media site is very essential in banks because privacy
invasion may lead to theft of personal identification and may lead to criminal proceedings. In case
of low security features hackers may hack the social media sites and/or may clone the original,
befooling customers and duping them [21].
2.2 Social Media Efficiency & Effectiveness: Effectiveness refers to the ease of use, internet speed,
expediency etc with which customers may access and use bank’s social media sites [22].
Effectiveness measures the efficiency of bank’s social media and it estimates the speed of accessing
and working on the bank’s social media sites to ensure timely and convenient completion of all
required interaction [23]. Social media can augment the conventional personnel–client bonding with
an effective technological knowledge-based relationship [24].
Today’s customers need prompt responses and it can effectively be done in social media by
providing them relevant and quick information as & when required. It is surely required for
enhancement of quick responses to customers’ queries for the improvement of e-services and clients’
improved customer satisfaction [25]. Banks can provide unique banking experience to their clients
by giving them services combined with technology. Hence the primary task of the bank is to find
out and respond to customers’ queries effectively on Bank’s social media sites. By monitoring the
response of bank personnel on social media sites, Banks need to assess the service quality. As per the
above discussion we can make the following hypothesis:
2.3 Social Media Fulfillment & Responsiveness: Fulfillment concentrate on the service truthfulness
and ease of use of relevant information provided on a bank’s social media websites [26]. Customers
need prompt response and acknowledgement of their complaints or suggestions. The fulfillment
dimension concentrates on evaluating the banks promptness in responding to customers’
requirements [27]. For getting customer loyalty the banks create user generated customized content
Nidhi Singh, Avishek Chakraborty, Soma Bose Biswas, Malini Majumdar; Impact of Social Media in Banking Sector under Triangular
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for getting the Fulfillment dimension [28]. Hence Fulfillment refers to the customer’s confidence on
Bank’s social media platform to the extent their requirements are fulfilled.
2.4 Theory of Vagueness and Multi-Criteria Decision-Making Problem (MCDM): Due to the
complication of detached things and hesitation in human thinking, [29] manifested a remarkable
perception of neutrosophic set theory, which has been widely applied on disjunctive arenas of
science and engineering. Recently, researchers developed pentagonal [30], Hexagonal [31],
Heptagonal [32] fuzzy numbers in research domain. Researchers also established some useful
techniques [33-35] which linked the hesitant number and the crisp number in real life scenario. In
this era, MCDM is the paramount topic in decision scientific research. Recently, it is more essential in
such problems where a group of criteria is apprised. For such problems involving multi-criteria
group, decision-making problems (MCGDM) have come into existence. In this current epoch,
several works has been already published in this arena. [36] Introduced MCDM skill in Pythagorean
fuzzy set field, [37] focused on linguistic aggregation operators based on MCGDM problem, [38]
surveyed intuitionistic interval fuzzy information and applied it in MCGDM problem, [39] derived
MCGDM methodology using type-2 neutrosophic linguistic judgments, [40] manifested the idea of
MCGDM in human resource development arena, [41]
developed MCGDM skill in thermal
enovation of masonry buildings field,[42] introduced best-Worst-Method and ELECTRE Method
using MCGDM, [43] applied MCGDM in garage location selection based civil engineering problems,
[44] derived decision making method in intuitionistic neutrosophic environment, [45] utilized
MCDM in bipolar neutrosophic set arena, [46] wielded MCGDM in entropy based problem, [47]
used MCGDM in smart phone selection problem, [48] developed MCGDM in selection of advanced
manufacturing technology in neutrosophic set, [49] derived attribute based MCDM in linguistic
variable in intuitionistic fuzzy set.
Motivated by Smarandache’s neutrosophic theory [52], researchers established several articles
[53-62] in this domain and it is fruitfully applied in various field of mathematics. Also, a few new
techniques are manifested in neutrosophic theory which can grab and solve MCDM, MCGDM
problems in disjunctive domain. In this phenomenon, Vikor [63], TOPSIS [64], MOORA [65], GRA
[66] skills are developed to solve decision making problems using some suitable and logical
operators in neutrosophic theory. So, in case of social science related hesitant data, decision making
problem becomes one of the key topics in neutrosophic ambient.
In this research article, we consider a triangular neutrosophic based MCGDM technique to select the
best social media for online marketing in banking sector. Here, we collect all the information’s from
different banks based on their online marketing report. But, we observed that these data’s are
fluctuating and filled with lots of hesitations. Now, due to the presence of impreciseness we need to
improve our general established method. Thus, we have introduced triangular neutrosophic
number to tackle this system for better results. Additionally, we also incorporate different weights in
distinct attribute functions as well as decision maker’s choice. Finally, we performed a sensitivity
analysis and comparative study which reflects different case studies in disjunctive scenario.
Nidhi Singh, Avishek Chakraborty, Soma Bose Biswas, Malini Majumdar; Impact of Social Media in Banking Sector under Triangular
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2.5 Preliminaries:
Definition 2.5.1: Fuzzy Set: A set F̃ , generally defined as F̃ = {(α, μF̃ (α)) : α ∈ S, μF̃ (α) ∈ [0,1]},
denoted by the pair(α, μF̃ (α)), where 𝛼 belongs to the crisp set 𝐹 and μS̃ (α) belongs to the
interval[0, 1], then set S̃ is called a fuzzy set.
̃ = (s1 , s2 , s3 ) should
Definition 2.5.2: Triangular Fuzzy Number: A triangular fuzzy number A
satisfy the following condition
(1) μà (x) is a continuous function which is in the interval [0,1]
(2) μà (x) is strictly increasing and continuous function on the intervals [s1 , s2 ].
(3) μà (x) is strictly decreasing and continuous function on the intervals[s2 , s3 ].
Definition 2.5.3: Linear Triangular Fuzzy Number (TFN): A linear triangular fuzzy number can be
̃ TFN = (s1 , s2 , s3 ) whose membership function is defined as follows:
written as A
Figure 2.5.3.1: Graphical Representation of Linear Triangular Fuzzy Number
̃ in the universal discourse 𝑋, symbolically
Definition 2.5.4: Neutrosophic Set: [52] A set 𝑛𝑒𝑢𝑆
̃ = {〈𝑥; [𝑇𝑛𝑒𝑢𝑆
denoted by 𝑥, it is called a neutrosophic set if 𝑛𝑒𝑢𝑆
̃ (𝑥), 𝐼𝑛𝑒𝑢𝑆
̃ (𝑥), F𝑛𝑒𝑢𝑆
̃ (𝑥)]〉 ⋮ 𝑥 ∈ 𝑋},
where 𝑇𝑛𝑒𝑢𝑆
̃ (𝑥): 𝑋 →] − 0,1 + [ is said to be the true membership function, which has the degree of
belongingness, 𝐼𝑛𝑒𝑢𝑆
̃ (𝑥): 𝑋 →] − 0,1 + [ is said to be the indeterminacy membership, having degree
of uncertainty, and 𝐹𝑛𝑒𝑢𝑆
̃ (𝑥): 𝑋 →] − 0,1 + [ is said to be the incorrect membership, which has the
degree of non-belongingness of the decision maker. 𝑇𝑛𝑒𝑢𝑆
̃ (𝑥), 𝐼𝑛𝑒𝑢𝑆
̃ (𝑥)& 𝐹𝑛𝑒𝑢𝑆
̃ (𝑥) exhibits the
following relation:
−0 ≤ 𝑆𝑢𝑝{𝑇𝑛𝑒𝑢𝑆
̃ (𝑥)} + 𝑆𝑢𝑝{𝐼𝑛𝑒𝑢𝑆
̃ (𝑥)} + 𝑆𝑢𝑝{𝐹𝑛𝑒𝑢𝑆
̃ (𝑥)} ≤ 3 +.
2.5.5: Triangular Single Valued Neutrosophic number: [33] A Triangular Single Valued
Neutrosophic number is defined as 𝐴̃𝑁𝑒𝑢 = (𝑝1 , 𝑝2 , 𝑝3 ; 𝑞1 , 𝑞2 , 𝑞3 ; 𝑟1 , 𝑟2 , 𝑟3 ) whose truth membership,
indeterminacy and falsity membership is defined as follows:
Nidhi Singh, Avishek Chakraborty, Soma Bose Biswas, Malini Majumdar; Impact of Social Media in Banking Sector under Triangular
Neutrosophic Arena Using MCGDM Technique
Neutrosophic Sets and Systems, Vol. 35, 2020
𝑥−𝑝1
𝑝2 −𝑝1
𝑇𝐴̃𝑁𝑒𝑢 (𝑥) =
1
𝑝3 −𝑥
𝑝3 −𝑝2
163
𝑞2 −𝑥
𝑤ℎ𝑒𝑛 𝑝1 ≤ 𝑥 < 𝑝2
𝑤ℎ𝑒𝑛 𝑥 = 𝑝2
,
𝑤ℎ𝑒𝑛 𝑝2 < 𝑥 ≤ 𝑝3
{ 0
𝐼𝐴̃𝑁𝑒𝑢 (𝑥) =
0
𝑥−𝑞2
𝑞3 −𝑞2
𝑤ℎ𝑒𝑛 𝑥 = 𝑞2
𝑤ℎ𝑒𝑛 𝑞2 < 𝑥 ≤ 𝑞3
{ 1
𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
𝑟2 − 𝑥
𝑟2 − 𝑟1
0
𝐹𝐴̃𝑁𝑒𝑢 (𝑥) = 𝑥 − 𝑟
2
𝑟3 − 𝑟2
{ 1
𝑤ℎ𝑒𝑛 𝑞1 ≤ 𝑥 < 𝑞2
𝑞2 −𝑞1
𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
𝑤ℎ𝑒𝑛 𝑟1 ≤ 𝑥 < 𝑟2
𝑤ℎ𝑒𝑛 𝑥 = 𝑟2
𝑤ℎ𝑒𝑛 𝑟2 < 𝑥 ≤ 𝑟3
𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
Where, 0 ≤ 𝑇𝐴̃𝑁𝑒𝑢 (𝑥) + 𝐼𝐴̃𝑁𝑒𝑢 (𝑥) + 𝐹𝐴̃𝑁𝑒𝑢 (𝑥) ≤ 3, 𝑥 ∈ 𝐴̃𝑁𝑒𝑢
2.5.6: Score Function: [50] If 𝐴̃𝑁𝑒𝑢 = (𝑝1 , 𝑝2 , 𝑝3 ; 𝜋, 𝜌, 𝜎) be a triangular neutrosophic number then its
1
score function is defined as 𝑆𝐶 = (𝑝1 + 𝑝2 + 𝑝3 ) × (2 + 𝜋 − 𝜌 − 𝜎) and accuracy value is defined as,
8
1
𝐴𝐶 = (𝑝1 + 𝑝2 + 𝑝3 ) × (2 + 𝜋 − 𝜌 + 𝜎)
8
3. Purpose/ Objectives of the Study:
1.
To understand the factors affecting the customers’ attitude towards acceptance of Social
Media Channels,
2.
To help Banks understand the impact of Social Media Channels on customer satisfaction and
customer loyalty.
4. Research Methodology:
The data have been collected from various respondents working in different organizations
categorized mainly as education sector, service sectors as banks, hospitals, etc. engineering works
and Government and Public sector companies in the Kolkata metro area. The study consisted of 234
respondents whose income is above 15,000 per month as it is assumed that those people at least
above Rs. 15000 earning/ month will be transacting more through online mode and can afford a
smart phone. We have used a five point Likert scale where 5 indicates strongly agree, and 1 indicates
strongly disagree. 64.9% respondents are male and 35.1% are female.
Research Instrument: Demographic Profile is the independent variable in this paper. Technology
acceptance model by Ajzen & Fishbein, 1980, Davis, 1989 and Ajzen, 1991 are used for validating
questionnaire. The questionnaire is mainly focused on: Social Media platforms used by the banks
and attributes affecting the users’ adaptability of the same.
TABLE 4.1.1 DEMOGRAPPHIC DETAILS OF RESPONDENTS
CHARACTERISTICS
GENDER
TYPES
FREQUENCY
%
MALE
135
57.69
FEMALE
99
42.31
Nidhi Singh, Avishek Chakraborty, Soma Bose Biswas, Malini Majumdar; Impact of Social Media in Banking Sector under Triangular
Neutrosophic Arena Using MCGDM Technique
Neutrosophic Sets and Systems, Vol. 35, 2020
164
<25
75
32.05
25-40
154
65.81
>40
5
02.14
EMPLOYED
92
39.32
UNEMPLOYED
22
9.40
PROFESSIONAL
14
5.98
STUDENT
95
40.60
BUSINESS
10
4.27
OTHERS
1
0.43
FACEBOOK
132
56.41
TWITTER
47
20.09
YOUTUBE
55
23.50
DAILY
149
63.68
WEEKLY
13
5.55
MONTHLY
6
2.56
VERY RARE
66
28.21
AGE
OCCUPATION
SOCIAL MEDIA PLATFORM
HOURS OF SURFING THROUGH SOCIAL MEDIA
Table 4.1.2 Indicate acceptance of Social Media based on various attributes
BANK
PLATFORM
SAFETY &
PRIVACY
1
2
3
(%)
EFFICIENCY &
FULFILLMENT &
EFFECTIVENESS
RESPONSIVENESS
(%)
(%)
FACEBOOK
10
65
54
TWITTER
6
16
50
YOUTUBE
5
26
28
FACEBOOK
15
76
56
TWITTER
12
37
26
YOUTUBE
21
24
15
FACEBOOK
23
29
45
TWITTER
13
15
16
YOUTUBE
45
9
7
4.1 Multi-Criteria Group Decision Making Problem in Triangular Neutrosophic Environment
Nidhi Singh, Avishek Chakraborty, Soma Bose Biswas, Malini Majumdar; Impact of Social Media in Banking Sector under Triangular
Neutrosophic Arena Using MCGDM Technique
Neutrosophic Sets and Systems, Vol. 35, 2020
165
One of the most dependable, logistical and widely used topic in this recent era is Multi criteria
decision making problem. Its main objective is to find out the finest option among finite number of
different alternatives based on finite unlike attribute values. Its execution process was quiet tough to
estimate in triangular neutrosophic environment. To handle this MCGDM problem an algorithm
was developed using some mathematical operator and de-fuzzification technique.
4.1.1 Illustration of the MCGDM problem
We consider the problem as follows:
Let 𝑃 = { 𝑃1 , 𝑃2 , 𝑃3 … … … . . 𝑃𝑚 } is the distinct alternative set and 𝑅 = { 𝑅1 , 𝑅2 , 𝑅3 … … … . . 𝑅𝑛 } is the
distinct attribute set respectively. Let 𝜔 = { 𝜔1 , 𝜔2 , 𝜔3 … … … . . 𝜔𝑛 } be the weight set associated with
the attributes R where each 𝜔 ≥0 and also satisfies the relation∑𝑛𝑖=1 𝜔𝑖 = 1. We also consider the set
of decision maker 𝐷 = { 𝐷1 , 𝐷2 , 𝐷3 … … … . . 𝐷𝐾 } associated with alternatives whose weight vector is
defined as ∆= {∆1 , ∆2 , ∆3 … … … . . ∆𝑘 } where each ∆𝑖 ≥0 and also satisfies the relation ∑𝑘𝑖=1 ∆𝑖 = 1.
4.1.2 Normalisation Algorithm of MCGDM Problem:
Step 1: Framework of Decision Matrices
Here, we considered all decision matrices according to the decision maker’s choice related with
finite alternatives and finite attribute functions. It is noted that the member’s 𝑦𝑖𝑗 for each matrices
are of triangular neutrosophic numbers. Thus, the final matrix is defined as follows:
.
𝑃1
𝑃
2
𝑋𝐾 =
𝑃3
.
(𝑃𝑚
𝑅1
𝑘
𝑦11
𝑘
𝑦21
.
..
𝑘
𝑦𝑚1
𝑅2
𝑘
𝑦12
𝑘
𝑦22
.
.
𝑘
𝑦𝑚2
. . . 𝑅𝑛
𝑘
. . . . 𝑦1𝑛
𝑘
. . . 𝑦2𝑛 ………………...(4.1)
. . .
.
. . .
.
𝑘
. . . 𝑦𝑚𝑛
)
𝑅3
𝑘
𝑦13
𝑘
𝑦23
.
.
𝑘
𝑦𝑚3
Step 2: Framework of normalised matrix
To formulate a single group decision matrix X we utilized this logical operation 𝑦𝑖𝑗′ = {∑𝑘𝑖=1 𝜔𝑖 𝑋 𝑖 } for
individual decision matrix 𝑋 𝑖 . hence, the final matrix becomes as follows:
.
𝑃1
𝑋 = 𝑃2
𝑃3
.
(𝑃𝑚
𝑅1
′
𝑦11
′
𝑦21
.
..
′
𝑦𝑚1
𝑅2
′
𝑦12
′
𝑦22
.
.
′
𝑦𝑚2
𝑅3
′
𝑦13
′
𝑦23
.
.
′
𝑦𝑚3
.
.
.
.
.
.
. . 𝑅𝑛
′
. . . 𝑦1𝑛
′
. . 𝑦2𝑛 …………………(4.2)
. . .
. . .
′
. . 𝑦𝑚𝑛
)
Step 3: Framework of Final matrix
′
To formulate the final decision matrix we utilized the logical operation 𝑦𝑖𝑗′′ = { ∑𝑛𝑖=1 ∆𝑖 𝑦𝑐𝑖
,𝑐 =
1,2 … . 𝑚} for each individual Colum and finally, we get the decision matrix as,
Nidhi Singh, Avishek Chakraborty, Soma Bose Biswas, Malini Majumdar; Impact of Social Media in Banking Sector under Triangular
Neutrosophic Arena Using MCGDM Technique
Neutrosophic Sets and Systems, Vol. 35, 2020
166
. 𝑅1
′′
𝑃1 𝑦11
′′
𝑋 = 𝑃2 𝑦21
… … … … … … … … … …(4.3)
.
.
.
.
′′
( 𝑃𝑚 𝑦𝑚1
)
Step 4: Ranking
Now, by considering the score and accuracy value (2.5.6) and converting the matrix (4.3) into crisp
form, so that we could evaluate the best alternative corresponding to the best attributes.
4.1.3 Flowchart:
Figure 4.1.3.1: Flowchart for the problem
4.1.4 Illustrative Example:
Here, we constructed a social media selection problem in which we have considered three different
social media services. Among these different social media platforms we want to select the best social
media service in a logical way. Normally, social media services are fully dependent on the attributes
like Safety & Privacy, efficiency & effectiveness and fulfilment & responsiveness of the system.
Keeping these points in mind different banks provided some realistic information in which
vagueness was present. Thus, we considered the data in the form of triangular neutrosophic number
and according to their suggestions we constructed the distinct decision matrices in triangular
neutrosophic environment as shows below: 𝑃1 = 𝐹𝑎𝑐𝑒𝑏𝑜𝑜𝑘, 𝑃2 = 𝑇𝑤𝑖𝑡𝑡𝑒𝑟, 𝑃3 = 𝑌𝑜𝑢𝑡𝑢𝑏𝑒 are the
alternatives.𝑅1 = 𝑆𝑎𝑓𝑒𝑡𝑦 & 𝑃𝑟𝑖𝑣𝑎𝑐𝑦, 𝑅2 = 𝐸𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑐𝑦 & 𝐸𝑓𝑓𝑒𝑐𝑡𝑖𝑣𝑛𝑒𝑠𝑠 , 𝑅3 =
𝐹𝑢𝑙𝑓𝑖𝑙𝑙𝑚𝑒𝑛𝑡 & 𝑅𝑒𝑝𝑜𝑛𝑠𝑖𝑣𝑒𝑛𝑒𝑠𝑠 are the attributes.
Let us select four distinct decision makers from our environment, 𝐷1 = 𝐵𝑎𝑛𝑘 1, 𝐷2 = 𝐵𝑎𝑛𝑘 2, 𝐷3 =
𝐵𝑎𝑛𝑘 3 having weight distribution 𝐷 = { 0.35, 0.33, 0.32 } and the weight vector related with the
attribute function ∆= {0.32,0.35,0.33}.
Nidhi Singh, Avishek Chakraborty, Soma Bose Biswas, Malini Majumdar; Impact of Social Media in Banking Sector under Triangular
Neutrosophic Arena Using MCGDM Technique
Neutrosophic Sets and Systems, Vol. 35, 2020
167
Step 1
According to the decision maker’s opinion the decision matrices are shown as follows:
.
𝑃
𝐷1 = ( 1
𝑃2
𝑃3
𝑅1
< 8.5,10,11; 0.8,0.5,0.4 >
< 3,6,8; 0.6,0.4,0.5 >
< 3,5,7; 0.5,0.3,0.2 >
𝑅2
𝑅3
< 62,65,67; 0.7,0.4,0.5 > < 51,54,57; 0.6,0.5,0.5 >
)
< 13,16,18; 0.7,0.3,0.4 > < 47,50,54; 0.5,0.2,0.3 >
< 23,26,30; 0.6,0.3,0.4 > < 24,28,30; 0.4,0.6,0.7 >
𝐵𝑎𝑛𝑘 1 𝑜𝑝𝑖𝑛𝑖𝑜𝑛
.
𝑃
𝐷2 = ( 1
𝑃2
𝑃3
𝑅1
< 12,15,17; 0.6,0.4,0.3 >
< 10,12,15; 0.5,0.4,0.3 >
< 18,21,25; 0.5,0.6,0.4 >
𝑅2
< 72,76,79; 0.5,0.6,0.4 >
< 35,37,39; 0.5,0.2,0.3 >
< 21,24,27; 0.5,0.3,0.4 >
𝑅3
< 53,56,60; 0.6,0.4,0.5 >
)
< 24,26,29; 0.5,0.4,0.5 >
< 11,15,18; 0.8,0.5,0.4 >
𝐵𝑎𝑛𝑘 2 𝑜𝑝𝑖𝑛𝑖𝑜𝑛
.
𝑃
𝐷3 = ( 1
𝑃2
𝑃3
𝑅1
< 21,23,25; 0.6,0.4,0.5 >
< 10,13,17; 0.5,0.2,0.3 >
< 42,45,49; 0.6,0.4,0.5 >
𝑅2
< 26,29,31; 0.6,0.4,0.5 >
< 12,15,19; 0.7,0.5,0.5 >
< 6,9,13; 0.6,0.4,0.5 >
𝑅3
< 41,45,47; 0.7,0.3,0.2 >
)
< 14,16,18; 0.8,0.5,0.4 >
< 5,7,10; 0.4,0.2,0.3 >
𝐵𝑎𝑛𝑘 3 𝑜𝑝𝑖𝑛𝑖𝑜𝑛
Step 2: Framework of Normalised decision matrix
𝑀
.
𝑃1
=(
𝑃2
𝑃3
𝑅1
< 13.65,15.81,17.46; 0.8,0.4,0.3 >
< 7.55,10.22,13.19; 0.6,0.2,0.3 >
< 20.43,23.08,26.38; 0.6,0.3,0.2 >
𝑅2
𝑅3
< 53.78,57.11,59.44; 0.7,0.4,0.4 > < 48.46,51.78,54.79; 0.7,0.3,0.2 >
)
< 19.94,22.61,25.25; 0.7,0.2,0.3 > < 28.85,31.2,34.23; 0.8,0.2,0.3 >
< 16.9,19.9,23.57; 0.6,0.3,0.4 > < 13.63,16.99,19.64; 0.8,0.2,0.3 >
Step 3: Framework of Final matrix
< 39.18,42.13,44.47; 0.74,0.36,0.26 >
𝑀 = ( < 18.92,21.48,24.35; 0.68,0.2,0.3 > )
< 16.95,19.96,23.17; 0.7,0.25,0.32 >
Step 4: Ranking
Now, we consider the score and Accuracy function technique (2.5.6), to convert the triangular
neutrosophic numbers into crisp one, thus we get the final ideal decision matrix as
< 33.34 >
𝑀 = (< 17.65 >)
< 16.01 >
Thus, ranking of the social media service is as 𝑃1 > 𝑃2 > 𝑃3 .
Nidhi Singh, Avishek Chakraborty, Soma Bose Biswas, Malini Majumdar; Impact of Social Media in Banking Sector under Triangular
Neutrosophic Arena Using MCGDM Technique
Neutrosophic Sets and Systems, Vol. 35, 2020
168
4.1.5 Results and Sensitivity Analysis
To understand how the attribute weights of each criterion affecting the relative matrix and their
ranking a sensitivity analysis is done. The basic idea of sensitivity analysis is to exchange weights of
the attribute values keeping the rest of the terms are fixed. The below table is the evaluation table
which shows the sensitivity results.
Attribute Weight
Final Decision Matrix
Ordering
<(𝟎. 𝟒, 𝟎. 𝟑, 𝟎. 𝟑)>
< 28.26 >
(< 15.56 >)
< 14.42 >
𝑃1 > 𝑃2 > 𝑃3
<(𝟎. 𝟑, 𝟎. 𝟒, 𝟎. 𝟑)>
< 31.45 >
(< 16.42 >)
< 16.20 >
𝑃1 > 𝑃2 > 𝑃3
<(𝟎. 𝟑, 𝟎. 𝟑, 𝟎. 𝟒)>
< 30.54 >
(< 16.44 >)
< 17.30 >
𝑃1 > 𝑃3 > 𝑃2
<(𝟎. 𝟑𝟐, 𝟎. 𝟑𝟓, 𝟎. 𝟑𝟑)>
< 33.34 >
(< 17.65 >)
< 16.01 >
𝑃1 > 𝑃2 > 𝑃3
<(𝟎. 𝟑𝟕, 𝟎. 𝟑𝟐, 𝟎. 𝟑𝟏)>
< 35.62 >
(< 16.23 >)
< 15.45 >
𝑃1 > 𝑃2 > 𝑃3
Figure 4.1.5.1: Sensitivity analysis table on attribute function.
Nidhi Singh, Avishek Chakraborty, Soma Bose Biswas, Malini Majumdar; Impact of Social Media in Banking Sector under Triangular
Neutrosophic Arena Using MCGDM Technique
Neutrosophic Sets and Systems, Vol. 35, 2020
169
40
35
30
P1
25
P2
20
15
P3
10
5
0
1
2
3
4
5
Figure 4.1.5.2: Best Alternative Social Media Service Table
4.1.6 Comparison Table
We compared this proposed work with the established works proposed by the researchers to find
the best social media and it is noticed that in each cases 𝑃1 (facebook) becomes the best social media
service. The comparison table given as follows:
Approach
Ranking
(Deli, Ali, & Smarandache, 2015) [51]
𝑃1 > 𝑃2 > 𝑃3
(H.Garg, 2016) [36]
Our Proposed
𝑃1 > 𝑃3 > 𝑃2
𝑃1 > 𝑃2 > 𝑃3
5. Implication:
There are a lot of social media sites like face book, twitter, Google plus, linked in, you tube etc.
available for online marketing. This study was primarily done to identify the impact of social media
marketing especially in Banking Industry based on different social media attributes. We wanted to
discover the right social media platform best suited for Banking Industry in West Bengal. The
perception of vagueness plays a vital role in designing mathematical calculations. In this study we
wanted to check the functionality of this system to find out the impact of different social media
Nidhi Singh, Avishek Chakraborty, Soma Bose Biswas, Malini Majumdar; Impact of Social Media in Banking Sector under Triangular
Neutrosophic Arena Using MCGDM Technique
Neutrosophic Sets and Systems, Vol. 35, 2020
170
attributes on its acceptance in Online banking system in WB. Later we pioneered some more
fascinating outcome on score and exactness function.
There are a lot of researches already done in social media implementation in Banking Industry.
However many results are still unknown. Our work is to explore the idea in the following points:
•
Defining the attributes necessary for social media platform for Banking Industry in West
Bengal.
•
Discovering the best suitable social media site for Banking Industry in West Bengal.
•
The graphical representation of adaptation of social media platform based on its attributes.
•
Application of Triangular neutrosophic number based MCGDM problem for selection of
social media platforms.
Discussion
This study was done primarily to understand the perceptions of the people of West Bengal to use
social media for their banking transactions. The study examined the three different types of websites
i.e. Face book, Twitter and You Tube individually using three different attributes: Safety & Privacy,
Efficiency & Effectiveness and Fulfillment & Responsiveness.
The study yielded new viewpoints that are useful to both academicians and Banks. This study
showed that the selection of social media for Banking depends on various attributes which differs
based on customers’ perception.
All the three social media considered in this paper is different in nature. Communications & Social
groups like Face book, Messaging & Communication like Twitter, and Content & Communication
like You tube. Publicity in these three different social media sites differ both in content and context.
In the sample considered here men respondents are more than women; most of the respondents are
under 40 years of age and they frequently uses social media. Like the worldwide trend here also it
was observed that youngsters are dominating the social media sites. Social media mainly has
impacted the life of youngsters. It has become radically significant since last ten years and it has
attracted all age groups.
In West Bengal banking industry very less attention has been given to the measurement of social
media quality and its effects. It is agreed that Banks must consider the overall social media quality
measurement to satisfy customers. If the services experienced by customers are satisfactory, then it
will induce them for long tern connectivity with banks. Long term connectivity with improved
customer satisfaction in turn will bring customer loyalty.
Adaption of social media for banking industry is something beyond likes, comments and shares. The
main aim of adaption of social media is brand awareness, creation of leads and ultimately
conversions and finally brand advocacy.
Banks should design their social media strategy
considering their pragmatic goals. Once the goals are set it is important to find their KPIs (Key
Performance Indicator) before implementing social media campaigns. A KPI is a quantifiable
measurement to evaluate their campaign in relation to their defined goals. The common social
media KPIs for banks can include Leads generation (through email signups or fulfilling some contact
Nidhi Singh, Avishek Chakraborty, Soma Bose Biswas, Malini Majumdar; Impact of Social Media in Banking Sector under Triangular
Neutrosophic Arena Using MCGDM Technique
Neutrosophic Sets and Systems, Vol. 35, 2020
171
forms), Conversions (account sign ups, deposits), Referral traffic (from social media to website),
Brand Advocacy (Like, comment and share)
Figure 5.1: Example of Social Media KPI
6. Findings:
•
All the three websites; Face book, Twitter and YouTube have gained attention among the
social media users in India, but Face book is the widely used social media website.
•
Banks are mostly using all international brands of social media channels for their operations
due to lack of availability of good national social media networks. There is a great chance of
development of some social media channels locally by the Govt.
•
Bank’s Social media Privacy drastically influences the endorsement of social media platform
in the banking industry of West Bengal.
•
Social media Efficiency appreciably control the acceptance of bank’s social media platform
in the West Bengal Banking Industry.
•
Social media Fulfillment extensively influences the acceptance of social media platform in
the West Bengal banking industry.
Nidhi Singh, Avishek Chakraborty, Soma Bose Biswas, Malini Majumdar; Impact of Social Media in Banking Sector under Triangular
Neutrosophic Arena Using MCGDM Technique
Neutrosophic Sets and Systems, Vol. 35, 2020
•
172
Customers’ prefer a bank that proposes them an experience that comprises all their service
needs.
•
All the three mentioned attributes have significant impact on overall customers’ satisfaction
which resulted in selection of Bank’s social media platform
•
Social media privacy appreciably persuades overall customer decision in selecting Banks
social media sites in West Bengal banking industry. The study findings discovered that
customers worth the social media privacy highly in banking operations.
•
Face book is most preferred platform for all demography regardless of age, gender and
occupation for all the Banks services.
•
For You Tube and Twitter websites, people have different perceptions and choices
depending on different Banks.
•
Banks may augment their profit margin by increased customers’ base through
implementing proper social media strategies and reduction in cost due to lesser no of
physical branches.
7. Conclusions:
In this current era, the West Bengal Banking Industry has conventionally been a high contact service
submission. As implementation of social media reduces direct human interaction, hence there arises
the need of continuous evaluation of service quality offered by Banks’ social media sites and
monitoring client’s perception on it. It was observed that clients were satisfied with the traditional
banking; still their expectations have grown bigger after introduction of e-services including social
media.
This study concluded that the following attributes of social media like Safety & Privacy, efficiency &
effectiveness and fulfillment & responsiveness have a significant influence on the service quality of
social media in the West Bengal Banking Industry under neutrosophic environment. It was observed
that customers mainly focuses on the attributes and service quality of Bank’s social media, hence it is
suggested that West Bengal Banking sector may priorities social media factors in their marketing
mixes. Additionally, comparison analysis is done with the established methods and sensitivity
analysis is performed in MCGDM technique under triangular neutrosophic arena. Finally it was
concluded that successful implementation of social media in banking industry generates customer
satisfaction and long term association which in turn converts to customer loyalty.
Further, researchers can apply this conception of triangular neutrosophic number in various fields
like social business problem, diagnoses problem, mathematical modeling, pattern recognition
problem, industrial problem, banking problem, marketing policy problem etc.
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Accepted: July 9, 2020.
Nidhi Singh, Avishek Chakraborty, Soma Bose Biswas, Malini Majumdar; Impact of Social Media in Banking Sector under Triangular
Neutrosophic Arena Using MCGDM Technique
Neutrosophic Sets and Systems, Vol.35, 2020
University of New Mexico
A Generalized Neutrosophic Solid Transportation Model with
Insufficient Supply
Nilabhra Paul1, Deepshikha Sarma2, Akash Singh3and Uttam Kumar Bera4
1Department
of Mathematics, NIT Agartala, Jirania, Tripura, 799046, India. E-mail: nilabhrapaul@gmail.com
of Mathematics, NIT Agartala, Jirania, Tripura, 799046, India. E-mail: deepshikhasarma4@gmail.com
3Department of Mathematics, NIT Agartala, Jirania, Tripura, 799046, India. E-mail: akssngh1242@gmail.com
4Department of Mathematics, NIT Agartala, Jirania, Tripura, 799046, India. E-mail: bera_uttam@yahoo.co.in
2Department
Abstract.The classical transportation problem and the solid transportation problem are special types of linear programming problems which are very important in Operations Research. In this paper, a solid transportation model is
described, where the total supply of goods is insufficient to fulfil the total demand of goods, due to which the supplier
company tries to obtain the required remaining goods from another source. An expression is derived to determine the
import plan. The parameters of the model are considered to be uncertain and imprecise and are taken as trapezoidal
neutrosophic numbers. The paper gives a general formulation of such a model and an algorithm is proposed to solve
the model. The main objective function of the model present in the manuscript is to minimize the total cost. A formula is provided to check the degree of sufficiency of such a solution. The model is elucidated with a numerical example and its solution shows its efficiency and optimality in practical aspect. Finally, the paper provides a brief discussion about the computational time and some relative points of research.
Keywords: Solid Transportation Model, Insufficient supply, Trapezoidal Neutrosophic Number, Ranking function.
1 Introduction
Transportation is the movement of humans, animals, commodities, etc. from one location to another. Modes of transport include air, land (rail and road), water cable, pipeline and space. The field can
be divided into
infrastructure, vehicles and operations. Transportation is important because it enables trade between people, which is essential for the development of civilizations. It is a key component of growth and globalization.
The transportation problem (TP) was first forwarded by Hitchcock [1] in 1941. It is a popular type
of
problem in Operations Research where the decision maker wants to find the optimal way to
transport goods from source warehouses to destination warehouses. So, there are two types of constraints, namely source constraints and demand constraints. But, real systems may contain other type
of constraints too such as product type
constraints or transportation mode constraints. This gives a
third dimension to the transportation problem and converts the classical transportation problem into
the solid transportation problem (STP).
The STP was first stated by Schell [2] in 1955 and later, in 1962, it was formally introduced by Haley [3]. In this paper, we consider that different types of conveyances are required for shipping goods
and so, the third type of constraints here are the conveyance constraints.
The classical theories of Mathematics cannot solve problems which simulate real life situations.
The
information is imprecise and uncertain in nature. To deal with vague information, the fuzzy
set theory was
introduced by Zadeh [4] in 1965. But, fuzzy sets cannot represent imprecise information efficiently as they only consider the truth membership values of the data. Then, Atanassov [5,
6] introduced the concept of intuitionistic fuzzy sets, where the data are represented by their membership and non-membership values. But, they can only handle incomplete information, not indeterminate or inconsistent information.
Smarandache [8] proposed the concept of neutrosophic set theory by adding an independent indeNilabhra Paul, Deepshikha Sarma, Akash Singh and Uttam Kumar Bera, A Generalized Neutrosophic Solid Transportation
Model with Insufficient SupplyN. Paul, D. Sarma, A. Singh and U.K. Bera. A Generalized NSTM with Insufficient Supply
Neutrosophic Sets and Systems, Vol.35, 2020
178
terminacy membership. The neutrosophic set theory generalizes the concepts of classical set theory,
fuzzy set theory,
intuitionistic fuzzy set theory, and so on, since it considers all three aspects of decision-making, viz. “agree”, “disagree” and “not sure”.Basset et al. [24] used Neutrosophic theory to
solve transition difficulties of Internet of Things identifying some challenge affectingthe process by
non-traditional methods. In the article [26], an advance type of Neutrosophic set called type-2 Neutrosophic number are defined with TOPSIS method. A green supply chain model is developed incorporated with neutrosophic set and robust ranking technique and its performance is shown in decision
making process [25].
Various researchers like Jiménez and Verdegay [7], Yang and Liu [9], Hussain and Kumar [10],
Kundu et al. [11], Singh and Yadav [13], Das et al. [14], Giri et al. [16], Das et al. [18], Aggarwal and
Gupta [19], etc. have studied the classical and solid transportation models in different fuzzy and intuitionistic fuzzy environments. A supply chain model is formulated based on some importance matrices based on economic, environment, social aspect as well as information gathering [23]. A hybrid
pliogenic decision making approach is developed in this regard. Basset et al. [22] developed an evaluation model to show the performance and efficiency of medical care system with pliogenic set.
In this paper, a mathematical model is developed for the solid transportation model. The model is
considered in neutrosophic environment so that we can address the fact of truth, indeterminacy and
falsity arises in the data due to factors like unawareness of the scale of the problem, imperfection in
data, poor forecasting, etc. As the concept of neutrosophic set theory is relatively new, a few of article
is available dealing the transportation or solid transportation models with neutrosophic parameters in
literature. A few of them in this context are by Thamaraiselvi and Santhi [15], and Rizk-Allah et al. [21].
The mathematical model present in this paperdescribes a transportation model shipping a homogeneous product from some source warehouses to some destination warehouses by means of heterogeneous conveyances. It is assumed that the conveyances have the necessary overall capacity to
transport the whole demanded quantity of the commodity. In this research work, it is considered that
the source warehouses do not have the sufficient quantity of goods to supply at a time and they fall
short of some amount. At that time, the supplier decides to import the goods from another source.
Again, if this new source does not have the requisite amount of goods, it imports the remaining
amount from another source, and so on. This process is continued until the fulfilment of the total demand. It terminates after a certain number of sources, since the total original demand of goods is a
fixed quantity. The paper addresses the general notion of the situation and also the presence of uncertainties in the data.
The main contribution of the paper is to develop the mathematical model for solid transportation
plan to satisfy the demand of customer with insufficient supply of source point.The main objective
function of the model is to minimize the total cost. In this research work, parameters of the model are
considered in neutrosophic environment. Consideration of neutrosophic number gives an ideal approach of a decision making process dealing the uncertainty with truth, false and in determinant state
of information. In this regard, trapezoidal neutrosophic number is used in this STP model. A proposition is provided to establish the relation between the import goods and the cost which define a degree
of insufficiency. Hereby, a solution algorithm is given in his manuscript. A numerical example is also
shown to discuss the performance of the model.
In this paper, Section 2 contains some preliminary definitions and concepts regarding the model.
Section 3 describes the model and gives a general formulation of the model. Section 4 is all about the
solution approach to the problem, concerned with the model and Section 5 helps in understanding the
model with the help of a
numerical example and its solution by the given procedure. Finally, Section 6 briefly discusses the model along with the computational time of the solution process, exemplified by the numerical example. It also suggests some relative points of research and is followed by the
conclusion.
2 Preliminaries
In this section, we recall some important definitions and concepts.
2.1 Single-valued neutrosophic set [20]
Let X be a non-empty set. Then a single-valued neutrosophic (SVN) set à of X is defined as
Ã= {⟨ x, 𝑇𝐴̃ (x), I𝐴̃ (x), 𝐹𝐴̃ (x)⟩ | x ∈ X},
Nilabhra Paul, Deepshikha Sarma, Akash Singh and Uttam Kumar Bera, A Generalized Neutrosophic Solid Transportation
Model with Insufficient SupplyN. Paul, D. Sarma, A. Singh and U.K. Bera. A Generalized NSTM with Insufficient Supply
Neutrosophic Sets and Systems, Vol.35, 2020
179
where 𝑇𝐴̃ (x), I𝐴̃ (x), 𝐹𝐴̃ (x) ∈[0, 1] and 0 ≤ 𝑇𝐴̃ (x) + I𝐴̃ (x) + 𝐹𝐴̃ (x) ≤ 3, ∀ x ∈ X. 𝑇𝐴̃ (x), I𝐴̃ (x) and𝐹𝐴̃ (x) respectively represent truth membership, indeterminacy membership and falsity membership degrees of x in Ã.
2.2 Trapezoidal neutrosophic number [20]
A trapezoidal neutrosophic number (TNN) Ã is a neutrosophic set in R with the following truth,
indeterminacy and falsity membership functions:
where𝛼𝐴̃ , 𝜃𝐴̃ and 𝛽𝐴̃ represent the maximum degree of truthiness, minimum degree of indeterminacy
and minimum degree of falsity respectively, 𝛼𝐴̃ , 𝜃𝐴̃ , 𝛽𝐴̃ ∈ [0, 1]. Also, 𝑎1′′ ≤ 𝑎1 ≤ 𝑎1′ ≤ 𝑎2 ≤ 𝑎3 ≤
𝑎4′ ≤ 𝑎4 ≤ 𝑎4′′ .
The membership functions of trapezoidal neutrosophic number are shown in Fig. 1.
Figure 1: Truth, indeterminacy and falsity membership functions of trapezoidal neutrosophic number.
2.3 Ranking function [20]
A ranking function of neutrosophic numbers is a function ℜ : N(R) → R, where N(R) is a set of
neutrosophic numbers defined on the set of real numbers, which convert each neutrosophic number
into the real line.
Nilabhra Paul, Deepshikha Sarma, Akash Singh and Uttam Kumar Bera, A Generalized Neutrosophic Solid Transportation
Model with Insufficient SupplyN. Paul, D. Sarma, A. Singh and U.K. Bera. A Generalized NSTM with Insufficient Supply
Neutrosophic Sets and Systems, Vol.35, 2020
180
Let à = ⟨( 𝑎1 , 𝑎2 , 𝑎3 , 𝑎4 ); 𝛼𝐴̃ , 𝜃𝐴̃ , 𝛽𝐴̃ ⟩ and 𝐵̃ = ⟨( 𝑏1 , 𝑏2 , 𝑏3 , 𝑏4 ); 𝛼𝐵̃ , 𝜃𝐵̃ , 𝛽𝐵̃ ⟩ betwo trapezoidal
neutrosophic numbers.
̃ 𝐵̃ ,
If ℜ(𝐴̃) > ℜ(𝐵̃ ), then 𝐴̃ >
̃ 𝐵̃ ,
If ℜ(𝐴̃) < ℜ(𝐵̃ ), then 𝐴̃ <
If ℜ(𝐴̃) = ℜ(𝐵̃ ), then 𝐴̃ ≈ 𝐵̃ .
3 Description and formulation of model
Real life situations regarding transportation of commodities are complex which give rise to various
transportation models. This paper discusses one such situation where the primary “supplier” company (say, Y1) has shortage of goods to meet the adequate demand of the primary “purchaser” company
(say, Y0).
It may happen that the total required amount of goods cannot be produced due to shortage of time
or lack of raw materials or some other factors to fulfill the total demand. So, Company Y 1 decides to
import the remaining amount of goods from another company (say, Y 2) and then transport the aggregate amount to Company Y0. Again, it may happen that Company Y2 faces the same problem, where it
is unable to fulfill the total demand of Company Y1. So, Company Y2 imports the remaining amount
continues until Company YN (say) fulfills the total defrom another company (say, Y3). The chain
terminates, since the total original demand of
mand of Company YN – 1 (say). The process surely
Company Y0 is a finite quantity. Here, N is at least 2.
While stating its demand, Company Y0 may not be sure about the exact quantity of goods it needs.
This may be due to the nature of the commodities, uncertain market trend and business scope, etc.
Similarly, due to
possible production and technical issues, the supply quantity of goods may be uncertain. Also, uncertainty may arise in determining the costs of transportation and the exact capacities
of the conveyances due to road issues, weather issues, etc. So, here, all of these parameters in all the N
steps are considered as trapezoidal neutrosophic numbers.
3.1 Assumptions
The total supply (in stock) of Company Yp from its origin warehouses is insufficient to fulfill
the total demand of the destination warehouses of Company Yp – 1 (p = 1, 2, …, N – 1).
Company Yp – 1 is indifferent to the arrangement of goods by Company Yp and Company Yp + 1 is
indifferent to the use of the goods imported by Company Yp (p = 1, 2, …, N – 1).
Company Yp does not have any extra warehouse to import goods. It imports the remaining
amount of goods to its existing warehouses (p = 1, 2, …, N – 1).
The warehouses of company Yp have the capacity to hold the remaining amount, but the whole
amount cannot be stored in a single warehouse and is transported to each of the warehouses in
parts (p = 1, 2, …, N – 1).
The total conveyance capacity of Company Yp is greater than or equal to the total demand of
Company Yp – 1 (p = 1, 2, …, N).
Company YN can supply the remaining quantity of goods, demanded (required) by Company
YN – 1, from its warehouses sufficiently. So, the model saturates in the Nth step and thus it is an
N-step model.
3.2 Notations
: Per unit cost of transportation from the ith origin warehouse to the jth destination warehouse by the
k conveyance in the pth step.
th
: Amount of goods to be transported from the ith origin warehouse to the jth destination warehouse
by the kth conveyance in the pth step.
Nilabhra Paul, Deepshikha Sarma, Akash Singh and Uttam Kumar Bera, A Generalized Neutrosophic Solid Transportation
Model with Insufficient SupplyN. Paul, D. Sarma, A. Singh and U.K. Bera. A Generalized NSTM with Insufficient Supply
181
Neutrosophic Sets and Systems, Vol.35, 2020
: Amount by which the supply falls short in the pth step.
: Original amount of supply of the ith origin in the pth step.
: Total amount of supply of the ith origin in the pth step.
: Amount of demand of the jth destination in the pth step.
: Capacity of the kth conveyance in the pth step.
: Number of origin warehouses in the pth step.
: Number of destination warehouses in the pth step.
: Number of conveyances in the pth step.
: Average per unit cost of transportation from the ith origin in the pth step.
: Harmonic mean of
’s ( i = 1, 2, …,
) in the pth step.
3.3 Formulation
The model is formulated mathematically as follows:
𝑚
𝑘
𝑚
(𝑝) (𝑝)
𝑝
𝑝−1
𝑝
𝑀𝑖𝑛 𝑧 (𝑝) = ∑𝑖=1
∑𝑗=1
∑𝑘=1
𝑐𝑖𝑗𝑘 𝑥𝑖𝑗𝑘
; 𝑝 = 1,2, … . . , 𝑁
(1)
Subject to
𝑚
𝑘
(𝑝)
(𝑝)
𝑝−1
𝑝
∑𝑗=1
∑𝑘=1
𝑥𝑖𝑗𝑘
≤ 𝑎𝑖 ;
𝑚
𝑘
𝑚
𝑚
(𝑝)
(𝑝)
𝑝
𝑝
∑𝑖=1
∑𝑘=1
𝑥𝑖𝑗𝑘
≥ 𝑏𝑗 ;
𝑝 = 1,2, … . . , 𝑁;
𝑝 = 1,2, … . . , 𝑁;
(𝑝)
(𝑝)
𝑝
𝑝−1
∑𝑖=1
∑𝑗=1
𝑥𝑖𝑗𝑘 ≤ 𝑒𝑘 ;
𝑖 = 1 ,2, … … . . , 𝑚𝑝
(2)
𝑗 = 1 ,2, … … . . , 𝑚𝑝−1
(3)
𝑘 = 1 ,2, … … . . , 𝑘𝑝
(4)
𝑝 = 1,2, … . . , 𝑁;
(𝑝)
and 𝑥𝑖𝑗𝑘 ≥ 0 Ɏ p, i, j, k
(5)
where
(𝑝)
𝑎𝑖
(𝑝)
= 𝐴𝑖
(𝑁)
𝑎𝑖
(𝑝)
𝑏𝑗
(𝑝)
𝑥𝑠
𝐻 (𝑝−1)
𝑚
(𝑝)
𝑝−1
= ∑𝑗=1
𝑏𝑗
(𝑝)
𝐴𝐶𝑖
𝐻 (𝑝) =
𝑖 = 1,2, … … … … , 𝑚𝑝
(𝑁)
(𝑝−1)
𝐴𝐶𝑗
𝑚𝑝−1
𝑥𝑠
𝑝 = 1,2, … … … . , 𝑛 − 1;
= 𝐴𝑖 𝑖 = 1,2, … … … … , 𝑚𝑁
(𝑝−1)
=⌊
(𝑝+1)
+ 𝑏𝑖
𝑚
⌋;
𝑘
(7)
𝑝 = 2,3, … … … … , 𝑁;
𝑚
(𝑝)
𝑝
𝐴𝑖
− ∑𝑖=1
> 0;
(𝑝)
𝑚𝑝
𝑚𝑝
1
(𝑝)
𝐴𝐶
𝑖
;
𝑝 = 1,2, … … . . 𝑁 − 1
𝑗 = 1,2, … … 𝑚𝑝−1
𝑝 = 1,2, … … . . 𝑁 − 1
𝑝−1
𝑝
= ∑𝑗=1
∑𝑘=1
𝑐𝑖𝑗𝑘 ; 𝑝 = 1,2, … . . , 𝑁 − 1;
∑𝑖=1
(6)
(8)
(9)
𝑖 = 1 ,2, … … . . , 𝑚𝑝 (10)
(11)
As it can be seen, there are N objective functions in (1) for N steps (p = 1, 2, …, N) of the model. Here, the
value of N is always a finite natural number greater than or equal to 2. (2), (3) and (4) are the supply, demand
and conveyance constraints respectively. The non-negativity constraints (5) are must, since the quantity of goods
is always non-negative.
Nilabhra Paul, Deepshikha Sarma, Akash Singh and Uttam Kumar Bera, A Generalized Neutrosophic Solid Transportation
Model with Insufficient SupplyN. Paul, D. Sarma, A. Singh and U.K. Bera. A Generalized NSTM with Insufficient Supply
182
Neutrosophic Sets and Systems, Vol.35, 2020
(𝑝)
Here, all the parameters and the decision variables𝑥𝑖𝑗𝑘 are taken as trapezoidal neutrosophic numbers. But,
(𝑝)
𝑥𝑖𝑗𝑘 denote quantities of goods to be transported and in reality, any manager or decision maker would want to obtain the crisp optimal solution of the problem through considering vague, imprecise and inconsistent information
while defining the problem.
Equation (8) is used to calculate bjb2’s (crisp values) after all the given parameters are converted into their
corresponding crisp values by a suitable ranking function. So, (8) becomes
(12)
Proposition 3.3.1
If the import plan due to insufficient supply for each supplier Company Yp (p = 1, 2, …, N – 1) is –
“import the highest quantity of goods from Yp + 1 to that warehouse jfrom which the average per unit cost of
transportation of goods to Yp – 1 is minimum”, then the import plan (quantity of goods to be imported to each
warehouse j) is mathematically given by:
Proof:
Here, bbl ’s denote the demands of the destination warehouses in the pth step, which are also the origin
warehouses in the (p – 1)th step. In the pth step, we want to import the highest quantity of goods to that
warehouse jfrom which the average per unit cost of transportation of goods A
is minimum. So,
is inversely proportional to
, i.e.,
i.e.,
,
where κ is the proportionality constant.
(𝑝−1)
Now,
total demand = 𝑥𝑠
𝑚𝑝−1
(𝑝)
i.e.,∑𝑗=1 𝑏𝑗
𝑚
𝑝−1
𝐾
i.e.,∑𝑗=1
i.e., 𝐾 =
(𝑝−1)
= 𝑥𝑠
1
(𝑝−1)
𝐴𝐶𝑗
(𝑝−1)
= 𝑥𝑠
(𝑝−1)
𝑥𝑠
𝑚𝑝−1
1
∑𝑗=1
(𝑝−1)
𝐴𝐶𝑗
But,𝐻 (𝑝−1) =
Therefore,
And so,
𝑚𝑝−1
𝑚𝑝−1
1
∑𝑖=1
(𝑝−1)
𝐴𝐶
𝑖
=
𝑚𝑝−1
𝑚𝑝−1
1
∑𝑗=1
(𝑝−1)
𝐴𝐶
𝑗
(𝑝−1)
𝐾=
(𝑝)
𝑏𝑗
𝑥𝑠
𝐻(𝑝−1)
𝑚𝑝−1
(𝑝−1)
=
𝑥𝑠
𝐻 (𝑝−1)
(𝑝−1)
𝐴𝐶𝑗
𝑚𝑝−1
(13)
Nilabhra Paul, Deepshikha Sarma, Akash Singh and Uttam Kumar Bera, A Generalized Neutrosophic Solid Transportation
Model with Insufficient SupplyN. Paul, D. Sarma, A. Singh and U.K. Bera. A Generalized NSTM with Insufficient Supply
183
Neutrosophic Sets and Systems, Vol.35, 2020
Now, while calculating
l ’s using (13), rounding off their decimals may result in loss of significant quantity of
a a ’s. So, we use the ceiling function and hence, obtain
4 Solution approach
An approach is suggested to find the optimal solution of such problems. The step by step procedure is as
follows:
Step I.
Step II.
Collect the information for a given problem as trapezoidal neutrosophic numbers from the decision
makers with the information that we always want to maximize the truth degree and minimize the
indeterminacy and falsity degrees of the data.
Construct the neutrosophic solid transportation table of the given problem for p = 1.
Step III. Convert all the trapezoidal neutrosophic numbers into their equivalent crisp values by the use of the
ranking function, proposed by M. Abdel-Basset et al. [20], which is given by:
or mathematically,
(14)
where Tã, Iã and Fãare respectively the truth, indeterminacy and falsity degrees of the trapezoidal neutrosophic number ã = (al, au, am1, am2). Here, al, au, am1and am2are the lower bound, upper bound, first
median and second median values of ã respectively, which can be obtained from the form
ã=
(a1, a2, a3, a4) by the transformations: al = a2, au = a3, am1 = a2 – a1 and am2 = a4 – a3.
Step IV. Compute
for p = 1 using the crisp form of (9).
Step V.
Calculate
Step VI.
Construct the crisp solid transportation table for p = 2 using the ranking function (14) for the values
of supply and conveyance capacity.
( ’s for p = 2 using (12).
Step VII. Repeat Steps (IV – VI) until some
the next supplier company (YN).
Step VIII. Calculate
(say, for p = N) is found, which can be totally satisfied by
( ’s for p = N using (12).
Step IX.
Construct the crisp solid transportation table for p = N using the ranking function (14) for the values
of supply and conveyance capacity.
Step X.
Solve the crisp solid transportation table for p = N using a standard method as used for solving a
general crisp STP and obtain the optimal solution for this step.
Step XI.
Compute the new crisp values of supply for p = N – 1 using the crisp form of (6) and similarly,
solve the table for p = N – 1 as solved for p = N.
Step XII. Repeat Step XI and similarly, solve the tables for p = N – 2, N – 3, …, 1.
Step XIII. Conclude the solution with the degree of sufficiency η, which is defined as:
,
where
(15)
(16)
Nilabhra Paul, Deepshikha Sarma, Akash Singh and Uttam Kumar Bera, A Generalized Neutrosophic Solid Transportation
Model with Insufficient SupplyN. Paul, D. Sarma, A. Singh and U.K. Bera. A Generalized NSTM with Insufficient Supply
184
Neutrosophic Sets and Systems, Vol.35, 2020
The proposed solution approach in the paper is a first of its kind. The solution approach is depicted as follows:
DESTINATION
S
SOURCES
(TOTAL
AVAILABILITY = X)
CONVEYANCES
(TOTAL
REQUIREMENT =
Y)
Transportation Possible if X ≥ Y
DESTINATION
S
SOURCES
(TOTAL
AVAILABILITY = X)
CONVEYANCES
(TOTAL
REQUIREMENT =
Y)
Transportation Not Possible if X < Y
The model proposed in the paper makes the transportation possible in the second case (shown above).
5Numerical example
Suppose, Company Y1 has to transport a commodity (e.g., wheat) to Company Y 0. But, it falls short of some
amount and wants to import the required amount from Company Y 2. Similarly, Company Y2 does not have the
sufficient amount of wheat to fulfill the total demand of Company Y1, so Company Y2 imports the required
amount from Company Y3. It is assumed that Company Y3 has the right amount of wheat to fulfill the total
demand of Company Y2.
The neutrosophic data for the transportations Y1→ Y0, Y2→ Y1 and Y3 → Y2 are given in Table 1, Table 2
and Table 3 respectively. For the sake of simplicity, (Tã, Iã, Fã) is taken as (0.9, 0.1, 0.1) for all the trapezoidal
neutrosophic numbers. The costs are considered in INR and the commodity is measured in kilograms.
Conveyance
Capacity
Y1 → Y0
(3150,4500,170,185)
(4100,5200,200,180)
(2900,3850,175,190)
Supply
Demand
(50,65,7,6)
(40,60,7,7)
(90,110,8,9)
(80,100,6,8)
(40,55,5,7)
(70,80,6,9)
(80,95,6,4)
(65,75,4,6)
(30,45,7,5)
(60,80,6,5)
(45,60,4,5)
(70,85,9,6)
(60,85,6,7)
(4500, 5700,250,230)
(55,70,4,3)
(1300,1600,140,170)
(50,70,3,5)
(65,85,6,7)
(1650,2000,165,150)
(75,95,8,5)
(95,115,7,4)
(1050,1400,120,135)
(5000,6500,245,260)
Table 1:Neutrosophic data table for Y1 → Y0.
Conveyance
Capacity
Y2 → Y1
(3400,4150,150,130)
(3250,3900,155,140)
Supply
(90,110,10,8)
(50,70,4,6)
(40,55,5,7)
(70,85,9,6)
(35,55,4,5)
(75,85,8,6)
(1000,1300,120,135)
Nilabhra Paul, Deepshikha Sarma, Akash Singh and Uttam Kumar Bera, A Generalized Neutrosophic Solid Transportation
Model with Insufficient SupplyN. Paul, D. Sarma, A. Singh and U.K. Bera. A Generalized NSTM with Insufficient Supply
185
Neutrosophic Sets and Systems, Vol.35, 2020
(75,90,7,8)
(65,80,6,5)
(85,95,6,6)
(50,60,4,3)
(60,80,4,6)
(60,85,9,7)
(1250,1500,145,160)
(80,90,6,7)
(90,105,9,8)
(55,75,8,7)
(75,90,5,5)
(100,120,5,7)
(40,55,6,5)
(1000,1350,105,125)
(30,50,4,4)
(70,85,5,6)
(80,95,5,7)
(60,80,4,6)
(65,75,8,7)
(50,65,4,3)
(1200,1450,110,100)
Demand
(
)
(
)
(
)
Table 2:Neutrosophic data table for Y2 → Y1.
Conveyance
Capacity
Y3 → Y2
(1650,1950,135,140)
(1800,2200,140,150)
Supply
(85,95,7,8)
(35,50,4,5)
(60,75,8,6)
(70,80,9,7)
(50,60,3,6)
(50,70,4,6)
(80,95,7,5)
(30,50,4,6)
(70,85,6,7)
(45,60,6,5)
(50,70,4,6)
(40,60,4,5)
(80,95,8,6)
(60,70,4,6)
(50,65,4,3)
(85,95,8,7)
(800,1100,90,95)
(50,70,5,4)
(65,75,7,4)
(85,95,7,7)
(75,90,9,6)
(45,60,5,4)
(55,70,5,5)
(35,55,4,5)
(50,65,5,6)
(1400,1700,115,135)
Demand
(
)
(
)
(
)
(
(1150,1350,105,100)
)
Table 3:Neutrosophic data table for Y3 → Y2.
The crisp tables are solved with LINGO 17.0 software. Table 4, Table 5 and Table 6 show the optimal crisp
solutions for Y3 → Y2, Y2→ Y1 and Y1→ Y0 respectively.
The minimum values of the objective functions are given below:
z(1) :
z(2) :
z(3) :
₹308719.43
₹216642.65
₹90474.05
The degree of sufficiency η is found out to be 0.00062.
Conveyance
Capacity
Y3 → Y2
1388.2
1565.7
Supply
Demand
0
(68.20)
737
(29.70)
0
(47.20)
0
(51.70)
0
(42.20)
0
(45.70)
0
(70.20)
184.7
(25.70)
943.2
0
(58.70)
0
(36.70)
0
(45.70)
644
(37.20)
0
(67.20)
0
(50.70)
0
(47.70)
0
(68.20)
673.2
0
(47.20)
0
(54.20)
0
(69.70)
0
(60.70)
585
(39.70)
0
(48.20)
517.3
(32.20)
0
(41.70)
1175.7
737
644
585
702
Table 4:Optimal solution table for Y3 → Y2.
Conveyance
Capacity
Y2 → Y1
3355.7
3133.2
Supply (new)
0
(73.70)
0
(45.70)
1087.9
(30.20)
0
(55.70)
417.3
(32.20)
0
(59.70)
1505.2
0
507.3
0
1052.1
0
0
1562.2
Nilabhra Paul, Deepshikha Sarma, Akash Singh and Uttam Kumar Bera, A Generalized Neutrosophic Solid Transportation
Model with Insufficient SupplyN. Paul, D. Sarma, A. Singh and U.K. Bera. A Generalized NSTM with Insufficient Supply
186
Neutrosophic Sets and Systems, Vol.35, 2020
(60.70)
(56.70)
(72.70)
(45.20)
(55.70)
(49.20)
0
(66.20)
0
(72.70)
0
(43.20)
0
(68.20)
0
(92.70)
1415.7
(31.70)
1415.7
1712.7
(28.70)
0
(61.70)
0
(70.20)
0
(55.70)
0
(48.20)
0
(47.70)
1712.7
Demand
2220
2140
1833
Table 5:Optimal solution table for Y2 → Y1.
Conveyance
Capacity
Y1 → Y0
3293.2
4080.7
2828.2
Supply (new)
Demand
0
(38.70)
1705.7
(29.70)
0
(75.20)
0
(69.70)
1500
(30.20)
0
(52.70)
3205.7
0
(53.20)
0
(73.20)
0
(55.70)
3293.2
(20.20)
0
(48.70)
200
(56.20)
3493.2
0
(54.20)
875
(39.70)
1800
(55.70)
0
(53.70)
0
(66.20)
0
(89.20)
2676.2
4380.7
4993.2
Table 6:Optimal solution table for Y1 → Y0.
6Discussion
The model, discussed in this paper, is a very interesting solid transportation model, which can be
useful in the business sector. It can be safely concluded that the problem of insufficient supply that
arises in the model, is dealt with effectively, as the degree of sufficiency η is positive for the given example and is very close to 0 (zero).
η should always be non-negative and the closer η is to zero, the
more sufficient the solution is for the model.
The numerical example given above is a 3-step model with fewer amounts of data. The computational time for this problem is not too high, but in real systems, the data is greater in amount and so,
higher will be the
computational time. The computational time T for an N-step model may roughly be given by the expression:
T = Ca(n) + Co(n) + Sol(n),
where
Ca(n) is the total time component for calculation of s ’s,
Co(n) is the total time component for conversion of the trapezoidal neutrosophic numbers into crisp
values,
Sol(n) is the total time component for solving the crisp data,
and all these components depend on the amount of data n, each of them varying directly with n.
Clearly, the amount of data n consists of N heterogeneous components. So, Ca(n)has N – 1 subcomponents, Co(n) has N sub-components and Sol(n) also has N sub-components. For the given numerical example, Ca(n) has 2 components, while Co(n) and Sol(n) have 3 components each.
Equation (12) is a key part of the solution method and a point of research for constructing a more efficient model. The degree of sufficiency η is evidently dependent on (12). The ranking function (14)
converts the
trapezoidal neutrosophic numbers into their equivalent crisp values effectively by considering the degrees of all three aspects of decision, but efforts can be made to construct a better ranking function to get more accurate crisp models and better results. The model may also be considered
with a time constraint (along with some time
penalty) for each supplier. Also, as we know that the
notion of neutrosophic set theory is relatively new and it broadly covers all the aspects of decision makNilabhra Paul, Deepshikha Sarma, Akash Singh and Uttam Kumar Bera, A Generalized Neutrosophic Solid Transportation
Model with Insufficient SupplyN. Paul, D. Sarma, A. Singh and U.K. Bera. A Generalized NSTM with Insufficient Supply
Neutrosophic Sets and Systems, Vol.35, 2020
ing, so there is a good potential for its extensive research and
tems.
187
applications in complex logistic sys-
7 Conclusion and future scope
The solid transportation problem is a significant problem in Operations Research, where the primary goal is to transport commodities from some source warehouses to some destination warehouses
via different modes of conveyance. This paper formulates a model, where the source cannot fulfill the
total demand and brings in the required amount from another source, which in turn, if unable to supply the necessary amount, brings in the remaining amount from another source, and so on. An expression is derived to provide the distribution of
demand of the deficient quantity of goods among
the importing warehouses. The paper also considers the
impreciseness and uncertainty that may
exist in the data and takes the input as trapezoidal neutrosophic numbers. An approach is presented
to solve the model and the quality of the solution is checked with the degree of
sufficiency. Also,
the computational time is shown for the model and it is believed that the model is useful and has an
interesting scope.
In this manuscript, the mathematical model has considered the minimization of cost as an objective
function. But it is very important to complete the fulfilment of demand of customers as early as possible. Therefore, for future research, one can consider the minimization of time as an objective function.
In the business purpose, profit is essential to grow. In this regard, maximization of profit can be treated as an objective function for further study. In this manuscript, uncertainty is used in terms of trapezoidal neutrosophic number. But in the direction of future research, one can use different parameters
e.g. uncertain number, fuzzy number, type-2 fuzzy number etc. To solve the problem, genetic algorithm can be developed in future research.
References
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[18]A. Das, U.K. Bera and M. Maity, A profit maximizing solid transportation model under a rough interval approach, IEEE Transaction of Fuzzy System, vol. 25(3), 2017, pp. 485 – 498.
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[20]M. Abdel-Basset, M. Gunasekaran, M. Mohamed and F. Smarandache, A novel method for solving the fully
Nilabhra Paul, Deepshikha Sarma, Akash Singh and Uttam Kumar Bera, A Generalized Neutrosophic Solid Transportation
Model with Insufficient SupplyN. Paul, D. Sarma, A. Singh and U.K. Bera. A Generalized NSTM with Insufficient Supply
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neutrosophic linear programming problems, Neural Computing and Applications, Springer, 2018, pp. 1 – 11.
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environment, Computers and Electrical Engineering, 2018, pp. 1 – 15.
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Received: Apr 20, 2020. Accepted: July 10, 2020
Nilabhra Paul, Deepshikha Sarma, Akash Singh and Uttam Kumar Bera, A Generalized Neutrosophic Solid Transportation
Model with Insufficient SupplyN. Paul, D. Sarma, A. Singh and U.K. Bera. A Generalized NSTM with Insufficient Supply
Neutrosophic Sets and Systems, Vol. 35, 2020
University of New Mexico
Generalized b Closed Sets and Generalized b Open Sets in Fuzzy
Neutrosophic bi-Topological Spaces
Fatimah M. Mohammed 1,* and Sarah W. Raheem 2
1 Department
of Mathematics, College of Education for Pure Sciences, Tikrit University, Tikrit, IRAQ;
dr.fatimahmahmood@tu.edu.iq
2 Department
of Mathematics, College of Education for Pure Sciences, Tikrit University, Tikrit, IRAQ;
sarahwaad470@gmail.com
* Correspondence: dr.fatimahmahmood@tu.edu.iq
Abstract: In this paper, the authors study and introduce a new class of sets called generalized b-closed sets and its
complement generalized b-open sets via fuzzy neutrosophic bi-topological spaces. Also, we prove some theorem
related to this definitions. Then, we investigate the relations between the new defined sets by hand and some other
fuzzy neutrosophic sets on the other hand. Some applications and many examples are presented and discussed.in
fuzzy neutrosophic bi-topological spaces.
Keywords: fuzzy neutrosophic set; fuzzy neutrosophic bi-topology; fuzzy neutrosophic b-open set; fuzzy
neutrosophic b-closed set; fuzzy neutrosophic generalized b-closed sets.
1. Introduction
At the beginning use of the concept of fuzzy sets "FS" was submitted by L. Zadeh's conference
paper in 1965 [1] where each element had a degree of membership. Then many extension done by
several studies. Intuitionistic fuzzy set "IFS" was one of the extension proved and known by K
.Atanassov in 1983 [2- 4], when he has proved the degree of membership of an item of any set in"FS"
and added a degree of non-membership in "IFS". Then many studies are being on the generalizations
of the notion of "IFS", one of them proved was by F. Smarandache in 2005 [5,6], when he developed
something else in membership and added indeterminacy membership between the last two
membership and non-membership which were known in "IFS" and called it neutrosophic sets "NSs".
After that, A Salama et.al. in 2014 [7,8] introduced neutrosophic topological spaces "NTSs".
The term of neutrosophic sets "NSs" was defined with membership, non-membership not
specified degree. In the last three year ago, Veereswari [9] submitted his paper in fuzzy neutrosophic
topological spaces "FNTSs" to be the solution and representation of the problems different fields
where he takes all values between the closed interval 0 and 1 instead of the unitary non-standard
interval ]-0,1+[ in NSs.
In this work, as generalized of the work of R.K. Al-Hamido [10] and the last papers which
studied by F. Mohammed [11-13], we have identified a new category of fuzzy neutrosophic sets
Fatimah M. Mohammed, and Sarah W. Raheem, Generalized b Closed Sets and Generalized b Open Sets in Fuzzy
Neutrosophic bi-Topological Spaces
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189
"FNSs" called fuzzy neutrosophic generalized b-closed sets in fuzzy neutrosophic bi-topological
spaces. Finally, on the basis of our manster's we will discuss some new characteristics and apply it.
Finally, there are many application of NSs in many fields see [14-19 ], so before we ended our work
we added some applications based in our new sets via fuzzy neutrosophic bi-topological spaces.
2. Preliminaries:
In this part of our study, we will refer to some basic definitions and operations which are useful
in our work.
Definition 2.1 [9]: Let U be a non-empty fixed set. The fuzzy neutrosophic set "FNS" µN is an object
having the form µN ={˂ u, λµN(u), ɣµN(u), VµN(u) ˃ : uϵ U} where the functions λµN(u), ɣµN(u),VµN(u):
U→[0,1] denote the degree of membership function (namely λµN(u)), the degree of indeterminacy
function (namely ɣµN(u)) and the degree of non-membership function (namely VµN(u)) respectively
of each element uϵ U to the set µN and 0 ≤ λµN(u)+ ɣµN(u) +VµN(u) ≤ 3, for each u∈ U.
Remark 2.2: FNS µN = {˂ u, λµN(u), σµN(u), VµN(u) ˃: u ∈U} can be identified to an ordered triple ˂ u,
λµN, σµN, VµN ˃ in [0,1] on U.
Lemma 2.3 [9]: Let U be a non-empty set and the "FNS" µN and ɣN be in the form µN = {˂ u, λµN, σµN,
VµN ˃ } and ɣN ={˂ u, λɣN, σɣN, VɣN ˃} on U. Then,
i.
µN ⊆ ɣN iff λµN ≤ λɣN, σµN ≤ σɣN and VµN ≥ VɣN,
ii.
µN = ɣN iff µN ⊆ ɣN and ɣN ⊆ µN,
iii.
(µN)c ={˂ u, VµN, 1-σµN, λµN ˃},
iv.
µN ∪ ɣN ={˂ u, Mx( λµN, λɣN ), Mx( σµN, σɣN ), Mn( VµN, VɣN ) ˃},
v.
µN ∩ ɣN ={˂ u, Mn( λµN, λɣN ), Mn( σµN, σɣN ), Mx( VµN, VɣN ) ˃},
vi.
0N = {˂ u, 0, 0, 1˃} and 1N = { ˂ u, 1, 1, 0 ˃}.
Definition 2.4 [9]: Fuzzy neutrosophic topology ( for short, FNT) on a non-empty set U is a family TN
of fuzzy neutrosophic subset in U satisfying the following axioms:
i. 0N, 1N ∈ TN,
ii. µN1 ∩ µN2 ∈ TN ∀ µN1, µN2 ∈ TN,
iii. ∪ µNj ∈ TN, ∀ { µNj : j ∈ J} ⊆ TN.
In this case the pair (U, TN) is called fuzzy neutrosophic topological space ( for short, FNTS ).
The elements of TN are called fuzzy neutrosophic-open sets ( for short, FN-OS ). The complement of
FN-OS in the FNTS ( U, TN ) is called fuzzy neutrosophic- closed set (for short, FN-CS).
Definition 2.5 [9]: Let (U, TN ) is FNTS and
µN
= ˂ u, λµN, σµN, VµN ˃ is FNS in U. Then the fuzzy
neutrosophic-closure (for short, FN-Cl ) and the fuzzy neutrosophic-interior (for short, FN-In) of µN
are defined by:
FN-Cl (µN ) = ∩ { ɣN : ɣN is FN-CS in U and µN ⊆ ɣN },
Fatimah M. Mohammed, and Sarah W. Raheem, Generalized b Closed Sets and Generalized b Open Sets in Fuzzy
Neutrosophic bi-Topological Spaces
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FN-In ( µN ) = ∪ { ɣN : ɣN is FN-OS in U and ɣN ⊆ µN }.
Now, the FN-Cl (µN ) is FN-CS and FN-In(µN) is FN-OS in U.
Further,
i. µN is FN-CS in U iff FN-Cl(µN) = µN,
ii. µN is FN-OS in U iff FN-In(µN) = µN.
Definition 2.6: Let (UN, TN1, TN2) is FNTS and µN =˂ u, λµN, σµN, VµN ˃ is FNS in UN. Then the fuzzy
neutrosophic semi-closure ( resp. fuzzy neutrosophic Pre-closure and fuzzy neutrosophic α-closure)
of µN and denoted by FN-SCl (µN) (resp. FN-PCl( µN ) and FN-αCl ( µN ) are defined by:
FN-SCl( µN ) = ∩ { ɣN : ɣN is FN-SCS set in U and µN ⊆ ɣN } = µN ∪ FN-In(FN-Cl(µN)),
FN-PCl( µN ) = ∩ { ɣN : ɣN is FN-PCS set in U and µN ⊆ ɣN } = µN ∪ FN-Cl(FN-In(µN)),
FN- α Cl( µN ) = ∩ { ɣN : ɣN is FN- αCS set in U and µN ⊆ ɣN } = µN ∪ FN-Cl (FN-In(FN-Cl(µN))),
Definition 2.7 [11, 12]: FNS λN in FNTS (U, TN) is called:
i. Fuzzy neutrosophic-regular open set (FN-ROS) if µN =FN-In(FN-Cl(µN)),
ii. Fuzzy neutrosophic-regular closed set (FN-RCS) if µN = FN-Cl(FN-In(µN),
iii. Fuzzy neutrosophic-semi open set (FN-SOS) if µN ⊆ FN-Cl(FN-In(µN)),
iv. Fuzzy neutrosophic-semi closed set(FN-SCS) if FN-In(FN-Cl(µN)) ⊆ µN,
v. Fuzzy neutrosophic pre-open set(FN-POS) if µN ⊆ FN-In(FN-Cl(µN)),
vi. Fuzzy neutrosophic pre-closed set( FN-PCS) if FN-Cl(FN-In(µN)) ⊆ µN,
vii. Fuzzy neutrosophic-α-open set(FN-αOS) if µN ⊆ FN-In(FN-Cl(FN-In(µN))),
viii. Fuzzy neutrosophic-α-closed set( FN-αCS) if FN-Cl(FN-In(FN-Cl(µN))) ⊆ µN,
ix. Fuzzy neutrosophic generalized closed set ( FN-GCS ) if FN-Cl(K ⊆N ) whenever K ⊆ N and N is a
FN-OS,
x. Fuzzy neutrosophic generalized pre closed set ( FN-GPCS) if FN-PCl(K) ⊆ N, whenever K⊆ N and
N is a FN-OS,
xi. Fuzzy neutrosophic α generalized closed set (FN-αGCS) if FNα-Cl(K) ⊆N whenever K⊆ N and N
is a FN-OS,
xii. Fuzzy neutrosophic generalized semi closed set ( FN-GSCS) if FN-SCl(K) ⊆ N, whenever K⊆ N
and N is a FN-OS.
Definition 2.8 [13]: A fuzzy neutrosophic set K in FNTs (U, TN) is called fuzzy neutrosophic b-closed
set (FN-b-CS) set if and only if FN-In(FN-Cl (K)) ⋁ FN-Cl(FN-In (K)) ≤ K.
Definition 2.9 [13]: Let UN be a non-empty set and (U, TN1), (U, TN2) be two topological spaces then,
the triple (UN, TN1, TN2) is a fuzzy neutrosophic bi-topological space ( for short, FN-bi-TS ).
Fatimah M. Mohammed, and Sarah W. Raheem, Generalized b Closed Sets and Generalized b Open Sets in Fuzzy
Neutrosophic bi-Topological Spaces
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191
Generalized b-Open Sets and Generalized b-Closed Sets in Fuzzy Neutrosophic biTopological Spaces
In this section, we generalized our work [13] and study the concept of generalized b-closed sets
and generalized b-open sets based of fuzzy neutrosophic bi- topological spaces and introduced it
after giving the definition of fuzzy neutrosophic bi- topological spaces as follows:
Definition 3. 1: Let U be a non-empty set and TN1, TN2 be two topologies on FNTS (U, TN), then the
triple (U, TN1, TN2) is a fuzzy neutrosophic bi- topological space ( for short, FN-bi-TS).
Definition 3.2: Let U be a non-empty set and TN1, TN2 be two topologies on FNTS (U, TN). A subset A
of U is called fuzzy neutrosophic open set ( for short, FN-OS) set if A∈ TN1 ∪ TN2. A is called fuzzy
neutrosophic closed set ( for short, FN-CS) if 1N-A is FN-OS.
Note: In this work we refer to TN1∪TN2 by TN.
Example
3.3:
Let
U
=
{
k1,
k2},
TN1
={0N,
1N},
TN2
=
{0N,
1 N,
E1}
and,
TN ={ 0N, E1, 1N} be a FN-bi-TS on U,
Where, E1 = ˂ u, ( k1(0.2) , k1(0.5), k1(0.8) ), ( k2(0.3), k2(0.5), k2(0.7) ) ˃ .
Then the neutrosophic set Z = ˂ u, (k1(0.7), k1(0.5), k1(0.3)), ( k2(0.6), k2(0.5), k2(0.4 )) ˃ is a FN-b-CS in U.
Definition 3.4: Let (U, TN ) be any FN-bi-TS and µN = ˂ u, λµN, σµN, VµN ˃ be FNS in U. Then the fuzzy
neutrosophic-b-closure (for short, FN-bCl ) and the fuzzy neutrosophic-b-interior (for short, FN-bIn)
of µN are defined by:
FN-bCl (µN ) = ∩ { ɣN : ɣN is FN-bCS in U and µN ⊆ ɣN },
FN-bIn ( µN ) = ∪ { ɣN : ɣN is FN-bOS in U and ɣN ⊆ µN }.
Definition 3.5: Let (U,TN) be a FN-bi-TS, then, for each µ1, λ1 ∈ IU the fuzzy set µ1 is called fuzzy
neutrosophic- generalized b-open set (for short, FN-gb-OS ) set if µ1 ≤ FN-bIn (λ1) such that µ1 ≤ λ1
and µ1 is FN-CS.
Theorem 3.6: A fuzzy neutrosophic set Z of FN-bi-TS ( U, TN ) is a FN-gb-OS iff N ⊆ FN-bIn( Z)
whenever N is a FN-CS and N ⊆ Z.
Proof: Necessity : Suppose Z is a FN-gb-OS in FN-bi-TS (U, TN) and let E be a FN-CS and N ⊆ Z.
Then Hc = 1N-H is a FN-OS in U such that Zc =1N-Z ⊆ Nc =1N-N
⟹ 1N-Z is a FN-gb-CS and FN-bCl(1N-Z) ⊆ 1N-N ,
Hence, (1N-FN-bIn(Z)) ⊆ 1N-N ⟹ N ⊆ FN-bIn(Z).
Sufficiency: Let Z be any FNS of U and let N ⊆ FN-bIn(Z) whenever, N is a FN-CS and N ⊆ Z.
Theorem 3.7: Let (U, TN) be FN-bi-TS, then:
Fatimah M. Mohammed, and Sarah W. Raheem, Generalized b Closed Sets and Generalized b Open Sets in Fuzzy
Neutrosophic bi-Topological Spaces
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(1) Every FN-CS is a FN-gb-CS,
(2) Every FN-αCS is a FN-gb-CS,
(3) Every FN-PCS is a FN-gb-CS,
(4) Every FN-b-CS is a FN-gb-CS,
(5) Every FN-RCS is a FN-gb-CS,
(6) Every FN-GCS is a FN-gb-CS,
(7) Every FN-αGCS is a FN-gb-CS,
(8) Every FN-GPCS is a FN-gb-CS
(9) Every FN-SCS is a FN-gb-CS.
(10) Every FN-GSCS is FN-gb-CS.
Proof : (1): Let Z ⊆ N and N be a FN-CS in FN-bi-TS (U, TN) with FN-bCl(Z) ⊆ FN-Cl(Z).
But, FN-bCl(Z) = Z ⊆ N. Therefore, Z is a FN-gb-CS in FN-bi-TS (U, TN).
(2): Let Z ⊆ N and N ∈ TN,⟹ Z is a FN-αCl(Z) = Z. Therefore, FN-bCl(Z) ⊆ FN-αCl(Z) = Z ⊆ N.
Hence, Z is a FN-gb-CS in FN-bi-TS (U, TN).
(3): Let Z ⊆ N and N ∈ TN .
Since Z is a FN-PCS, and FN-Cl( FN-In(Z)) ⊆ Z.
Therefore, FNCl(FN-In(Z)) ∩ FN- In(FN-Cl(Z)) ⊆ FN-Cl(Z) ∩ FN-Cl(FN-In(Z)) ⊆ Z.
⟹ FN-bCl(Z) ⊆ N. Hence, Z is a FN-gb-CS in U.
(4): Let Z ⊆ N and N be a FN-OS in FN-bi-TS (U, TN)
⟹ Z is a FN-b-CS and FN-bCl(Z) = Z.
Therefore, FN-bCl(Z) = Z ⊆ N. Hence, Z is a FN-gb-CS in FN-bi-TS (U, TN).
(5): Let Z ⊆ N and N ∈ TN and let Z be a FN-RCS.
But, FN-Cl(FN-In(Z)) = Z ⟹ FN-Cl(Z) = FN-Cl(FN-In(Z)). Therefore, FN-Cl(Z) = Z.
Hence, Z is a FN-CS in U. By (1), we get Z is a FN- gb-CS in FN-bi-TS (U, TN).
(6): Let Z ⊆ N and N ∈ TN ⟹ Z is a FN-GCS, FN-Cl(Z) ⊆ N.
Therefore, FN-bCl(Z) ⊆ FN-Cl(Z).
But, FN-bCl(Z) ⊆ N. Hence, Z is a FN-gb-CS in FN-bi-TS (U, TN).
(7): Let Z ⊆ N and N ∈ TN ⟹ Z is a FN-αGCS.
But, FN-αCl(Z) ⊆ N. Therefore, FNbCl(Z) ⊆ FN-αCl(Z),
So, FN-bCl(Z) ⊆ N. Hence, Z is a FN-gb-CS in FN-bi-TS (U, TN).
(8): Let Z ⊆ N and N ∈ TN ⟹ Z is a FN-gp-CS and FN-PCl(Z) ⊆ N.
Therefore, FNbCl(Z) ⊆ FN-pCl(Z), so FN-bCl(Z) ⊆ N.
Fatimah M. Mohammed, and Sarah W. Raheem, Generalized b Closed Sets and Generalized b Open Sets in Fuzzy
Neutrosophic bi-Topological Spaces
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Hence, Z is a FN-gb-clos. set in FN-bi-TS (U, TN).
(9): Let Z ⊆ N and N ∈ TN ⟹ Z is a FN-SCS.
But, FN-bCl(Z) ⊆ FN-SCl(Z) ⊆ N. Therefore, Z is a FN-gb-CS in FN-bi-TS (U, TN).
(10): Obivious
Proposition 3.8: The converse of theorem 3.7 is not true in general for all cases and we can see it in
(Diagram 1.)
FN-CS
FN-αCS
FN-GSCS
FN-SCS
FN-PCS
FN-b-CS
FN-gb-CS
FN-RCS
FN-GCS
FN-GPCS
FN-αGCS
( Diagram 1)
Example 3.9: (i): Let U = { k1, k2 }, TN1 = { 0N, 1N}, TN2 = {0N, 1N, E1}.
Then, TN = { 0N, E1, 1N } is a FN-bi-TS on U ,
1-
Take E1 = ˂ u, ( k1(0.3), k1(0.5), k1(0.6) ), ( k2(0.2), k2(0.5), k2(0.7) ) ˃.
Then, the FNS "Z" = ˂ u, ( k1(0.5), k1(0.5), k1(0.4) ), ( k2(0.6), k2(0.5), k2(0.3)) is a FN-gb-CS but, not a FN-CS in U
⟹ FN-Cl("Z") = E1 ≠ "Z".
2- Let E1 = ˂ u, ( k1(0.3), k1(0.5), k1(0.6)), (k2(0.2), k2(0.5), k2(0.8)) ˃ . Then, the FNS "Z" = ˂ u, ( k1(0.5), k1(0.5),
k1(0.3)), (k2(0.6), k2(0.5), k2(0.3)) ˃ is a FN-αCS in U⟹ FN-Cl(FN-Cl(Z)) = 1N- E1⊈ "Z".
3- Let E1 = ˂ u, (k1(0.9), k1(0.5), k1(0.8)), (k2(0.3), k2(0.5), k2(0.7)) ˃.
Then, the FNS "Z" = ˂ u, (k1(0.4), k1(0.5), k1(0.6) ), ( k2(0.5), k2(0.5), k2(0.5)) ˃ is a FN-gb-CS but, not a
FN-PCS in U
⟹ FN-Cl(FN-In("Z")) = E1 ⊈ "Z".
4- Let E1 = ˂ u, (k1(0.6), k1(0.5), k1(0.4) ), (k2(0.8),k2(0.5),k2(0.2)) ˃ .
Fatimah M. Mohammed, and Sarah W. Raheem, Generalized b Closed Sets and Generalized b Open Sets in Fuzzy
Neutrosophic bi-Topological Spaces
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Then, the FNS "Z" = ˂ u, (k1(0.8), k1(0.5), k1(0.2)), (k2(0.9), k2(0.5), k2(0.1)) ˃ is a FN-gb-CS but, not a FN-b-CS in
FN-bi-TS (U, TN).⟹ FN-RCS is a FN-gb-CS but, not a FN-b-CS in FN-bi-TS (U, TN),
⟹ FN-Cl(FN-In("Z")) ∩ FN-In(FN-Cl("Z")) = 1N ⊈ "Z".
5- Let E1 = ˂ u, (k1(0.2), k1(0.5), k1(0.8)), (k2(0.4), k2(0.5), k2(0.6)) ˃ .
Then, the FNS "Z" = ˂ u, (k1(0.7), k1(0.5), k1(0.3) ), (k2(0.5), k2(0.5), k2(0.5)) ˃ is a FN-gb-CS but, not a FN-RCS in
FN-bi-TS (U, TN)
⟹ FN-Cl(FN-In("Z")) = 1N-E1 ≠ "Z".
6- Let E1 = ˂ u, (k1(0.2), k1(0.5), k1(0.8)), (k2(0.4), k2(0.5), k2(0.6)) ˃.
Then, the FNS "Z" = ˂ u, ( k1(0.1), k1(0.5), k1(0.8)), (k2(0.3), k2(0.5), k2(0.7)) ˃ is a FN-gb-CS but, not a FN-GCS in
FN-bi-TS (U, TN).
⟹ FN-Cl("Z") = E1c ⊈ E1.
7- Let E1 = ˂ u, ( k1(0.5), k1(0.5), k1(0.4) ), ( k2(0.5), k2(0.5), k2(0.5) ) ˃ .
Then, the FNS "Z" = ˂ u, ( k1(0.5), k1(0.5), k1(0.5) ), ( k2(0.3), k2(0.3), k2(0.7) ) ˃ is a FN-gb-CS but, not a FN-αGCS in
FN-bi-TS (U, TN).
8-
⟹ FN-Cl( FN-In( FN-Cl("Z"))) = 1N ⊈ E1.
Let E1 = ˂ u, ( k1(0.9), k1(0.5), k1(0.1) ), ( k2(0.7), k2(0.5), k2(0.2) ) ˃.
Then, the FNS "Z" = ˂ u, ( k1(0.7), k1(0.5), k1(0.3) ), ( k2(0.6), k2(0.5), k2(0.4) ) ˃ is a FN-gb-CS but, not a FN-SCS in
FN-bi-TS (U, TN),
9-
⟹ FN-In( FN-Cl("Z")) = 1N ⊈ "Z".
Let E1 = ˂ u, ( k1(0.8), k1(0.5), k1(0.6) ), ( k2(0.0), k2(0.5), k2(0.1) ) ˃.
Then, the FNS "Z" = ˂ u, ( k1(0.6), k1(0.5), k1(0.5) ), ( k2(0.2), k2(0.5), k2(0.3) ) ˃ is a FN-gb-CS but, not a FN-GSCS in
FN-bi-TS (U, TN),
⟹ FN-In( FN-Cl("Z")) = 1N ⊈ "Z".
10- Let U = { k1, k2 },TN1 = {0N, E1}, TN2 = {0N,1N, E1, E2} = TN be a FN-bi-TS on U.
Where, E1 = ˂ u, (k1(0.2), k1(0.5), k1(0.8) ), (k2(0.3), k2(0.5), k2(0.7) ) ˃ ,
E2 = ˂ u, ( k1(0.4), k1(0.5), k1(0.6) ), ( k2(0.5), k2(0.5), k2(0.5) ) ˃.
Then, the FNS "Z" = ˂ u, ( k1(0.4), k1(0.5), k1(0.6)), ( k2(0.5), k2(0.5), k2(0.5) ) ˃ is a FN-gb-CS but, not a FN-GPCS in U
⟹ FN-PCl( "Z) = 1N-E2 ⊈ E2.
Theorem 3.10: The union of any two FN-gb-CS need not be a FN-gb-CS in general as seen from the
following example:
Example 3.11: Let U = { k1, k2 },TN1 = { 0N, E1} and TN2 = { 0N, 1N, E1} = TN be a FNT on U, where
E1 = ˂ u, ( k1(0.6), k1(0.5), k1(0.4)), ( k2(0.8), k2(0.5), k2(0.2)) ˃.
Then, the FNS "Z" = ˂ u, ( k1(0.1), k1(0.5), k1(0.9)), ( k2(0.8), k2(0.5), k2(0.2)) ˃ ,
M = ˂ u, ( k1(0.6), k1(0.5), k1(0.4)), ( k2(0.7), k2(0.5), k2(0.3) ) ˃ is a FN-gb-CS but, Z ∩ M is not a FN-gb-CS in U
⟹ FN-bCl( "Z"∩M ) = 1N ⊈E1.
Fatimah M. Mohammed, and Sarah W. Raheem, Generalized b Closed Sets and Generalized b Open Sets in Fuzzy
Neutrosophic bi-Topological Spaces
Neutrosophic Sets and Systems, Vol. 35, 2020
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Theorem 3.12: If Z is a FN-gb-CS in FN-bi-TS (U, TN) , such that Z
⊆ M ⊆ FN-bCl( Z ) then, M is a
FN-gb-CS in (U, TN)
Proof : Let M be any FNS in a FN-bi-TS (U, TN), such that M
FN-gb-CS and FN-bCl( Z)
⊆ N and N ∈ TN ⟹ Z ⊆ N , since Z is a
⊆ N.
By hypothesis, we have FN-bCl(M)
⊆ FNbCl( FN-bCl( Z )) = FN-bCl( Z ) ⊆ N.
Hence, M is FN-gb-CS in U.
Theorem 3.13: If Z is a FN-b-OS and FN-gb-CS in FN-bi-TS (U, TN), then Z is a FN-b-CS.
Proof : Since Z is a FN-b-OS and FN-gb-CS in FN-bi-TS (U, TN) such that FN-bCl( Z )
But, Z
⊆ Z.
⊆ FN-bCl( Z ) .
Thus, FN-bCl( Z ) = Z and hence, Z is FN-b-CS in FN-bi-TS (U, TN).
Definition 3.14: A fuzzy neutrosophic set Z is said to be a fuzzy neutrosophic generalized b open set (
FN-gb-OS) in FN-bi-TS (U, TN). If the complement 1N-Z is a FN-gb-CS in U. The family of all FN-gb-OS of
FN-bi-TS (U, TN) is denoted by FN-gb-O (U).
Example 3.15: Let U = { k1, k2 },TN1 = {0N, E1}, TN2 = {0N, 1N, E1} = TN be FN-bi-TS on U, where
E1 = ˂ u, ( k1(0.3), k1(0.5), k1(0.7)), ( k2(0.4), k2(0.5), k2(0.6) ) ˃.
Then, the FNS Z = ˂ u, ( k1(0.4), k1(0.5), k1(0.6)), ( k2(0.5), k2(0.5), k2(0.5) ) ˃ is a FN-gb-OS in U.
4.
Some Applications of Generalized b-Closed Sets in Fuzzy Neutrosophic bi-Topological Spaces
In [14] they propose two models for solving Neutrosophic Goal Programming Problem (NGPP), and
in [15-19], we can see many applications of neutrosophic so, we will try in our study to give some application of
our new studies concepts.
Definition 4.1: A FN-bi-TS (U, TN) is called:
1
1
i.
a fuzzy neutrosophic b
ii.
a fuzzy neutrosophic gb space ( for short, FN-gb S) if every FN-gb-CS is a FN-CS.
iii.
a fuzzy neutrosophic gbUb space ( FN-gbbS) if every FN-gb-CS is a FN-b-CS.
2
space ( for short, FN-b S) if every FN-bCS is a FN-CS.
2
1
1
2
2
1
Theorem 4.2: Every FN-gb S is a FN-gbUb S in any FN-bi-TS (U, TN),.
2
1
Proof : Let ( U, TN ) be a FN-gb S and let Z be any FN-gb-CS in FN-bi-TS (U, TN), By hypothesis, Z is a
2
FN-CS in U.
Since every FN-CS is a FN-b-CS in U. Hence, ( U, TN) , is a FN-gbUb S.
The converse of above theorem need not be true in general as seen from the following example:
Fatimah M. Mohammed, and Sarah W. Raheem, Generalized b Closed Sets and Generalized b Open Sets in Fuzzy
Neutrosophic bi-Topological Spaces
Neutrosophic Sets and Systems, Vol. 35, 2020
196
Example 4.3: Let U = { k1, k2 }, TN1 = TN = {0N, 1N, E1} and TN2 = { 0N, 1N} be a FNT on U, where,
E1 = ˂ u, ( k1(0.9), k1(0.5), k1(0.9) ), ( k2(0.1), k2(0.5), k2(0.1) ) ˃.
1
Then, the FNS "Z" = ˂ u, ( k1(0.2), k1(0.5), k1(0.3) ), ( k2(0.8), k2(0.5), k2(0.7) ) ˃ is a FN-gbUb S but, not a FN-gb S.
2
1
Theorem 4.4: Let ( U, TN) be a FN-bi-TS and ( U, TN ). A FN-gb S. Then we have the following statement:
2
i- Any union of FN-gb-CS is a FN-gb-CS.
ii- Any intersection of any FN-gb-OS is a FN-gb-OS.
1
Proof : (i ) Let {Ni }i ∈ J be a collection of FN-gb-CS in a FN-gb S, ( U, TN ).
2
Therefore, every FN-gb-CS is a FN-CS.
But, the union of FN-CS is a FN-CS. Hence, the union of FN-gb-CS is a FN-gb-CS in U.
(ii) It can be proved by taking complement in (i).
Theorem 4.5: A FN-bi-TS (U, TN) is a FN-gbUb S if and only if FN-gb(U) = FNb-O (U)
Proof : Necessity : Let "Z" be a FN-gb-OS in a FN-bi-TS (U, TN). Then, 1N-Z is a FN-gb-CS.
By hypothesis , 1N-Z is a FN-b-CS in U. Therefore, Z is a FN-b-OS
Hence, FN-gb-O(U) = FNb-O (U).
Sufficiency : Let Z be a FN-gb-CS in any FN-bi-TS (U, TN). Then, 1N-Z is a FN-gb-OS in U.
By hypothesis , 1N-Z is a FN-b-OS in U.
Therefore, Z is a FN-b-CS in U. Hence, ( U, TN) is a FN-gbUbS.
1
Theorem 4.8: A FN-bi-TS (U, TN) is a FN-gb if and only if FN-gb-O(U) = FN-O(U).
2
Proof : Necessity : Let Z be a FN-gb-OS in a FN-bi-TS (U, TN). Then 1N-Z is a FN-gb-CS in U.
By hypothesis, 1N-Z
is a FN-CS in U. Therefore, Z is a FN-OS in U.
Hence, FN-gb-O(U) = FN-O(U)
Sufficiency : Let Z be a FN-gb-CS. Then, 1N-Z is a FN-gb-OS in U. By hypothesis, 1N-Z is a FN-OS
1
in U. Therefore, Z is a FN-CS in U. Hence, (U, TN) is a FN-gb .
2
5.
Conclusions
In this paper, the new concept of a new class of sets was studied and called fuzzy neutrosophic
generalized b-closed sets and its complement fuzzy neutrosophic generalized b-open sets. We
investigated the relations between fuzzy neutrosophic generalized b closed sets and other fuzzy
neutrosophic sets such as α closed sets, regular closed sets, semi closed sets pre closed sets,
generalized closed sets, b closed sets, α generalized closed sets and semi generalized closed sets
based of fuzzy neutrosophic bi-topological spaces and applied some new spaces to be applications
of the new defined sets.
Fatimah M. Mohammed, and Sarah W. Raheem, Generalized b Closed Sets and Generalized b Open Sets in Fuzzy
Neutrosophic bi-Topological Spaces
Neutrosophic Sets and Systems, Vol. 35, 2020
6.
197
Acknowledgements
The authors would like to thanks the reviewers for their valuable suggestions to improve the
paper and get it as in this design.
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Received: Apr 21, 2020. Accepted: July 11, 2020
Fatimah M. Mohammed, and Sarah W. Raheem, Generalized b Closed Sets and Generalized b Open Sets in Fuzzy
Neutrosophic bi-Topological Spaces
Neutrosophic Sets and Systems, Vol. 35, 2020
University of New Mexico
Neutrosophic Soft Rough Topology and its Applications to
Multi-Criteria Decision-Making
Muhammad Riaz 1, Florentin Smarandache 2, Faruk Karaaslan3, Masooma Raza Hashmi4,
Iqra Nawaz5
1 Department of Mathematics, University of the Punjab Lahore, Pakistan. E-mail: mriaz.math@pu.edu.pk
Department of Mathematics, University of New Mexico, Gallup, NM 87301, USA. E-mail: smarand@unm.edu
3 Department of Mathematics, Çankiri Karatekin University, Çankiri, Turkey. Email : fkaraaslan@karatekin.edu.tr
4 Department of Mathematics, University of the Punjab, Lahore, Pakistan. E-mail: masoomaraza25@gmail.com
5
Department of Mathematics, University of the Punjab, Lahore, Pakistan. E-mail: iqra.nawaz245@gmail.com
2
Abstract: In this manuscript, we introduce the notion of neutrosophic soft rough topology (NSRtopology) defined on neutrosophic soft rough set (NSR-set). We define certain properties of NSRtopology including NSR-interior, NSR-closure, NSR-exterior, NSR-neighborhood, NSR-limit point,
and NSR-bases. Furthermore, we aim to develop some multi-criteria decision-making (MCDM)
methods based on NSR-set and NSR-topology to deal with ambiguities in the real-world problems.
For this purpose, we establish algorithm 1 for suitable brand selection and algorithm 2 to determine
core issues to control crime rate based on NSR-lower approximations, NSR-upper approximations,
matrices, core, and NSR-topology.
Keywords: Neutrosophic soft rough (NSR) set, NSR-topology, NSR-interior, NSR-closure, NSRexterior, NSR-neighborhood, NSR-limit point, NSR-bases, Multi-criteria group decision making.
1. Introduction
The limitations of existing research are recognized in the field of management, social sciences,
operational research, medical, economics, artificial intelligence, and decision-making problems.
These limitations can be dealt with the Fuzzy set [1], rough set [2, 3], neutrosophic set [4, 5], soft set
[6], and different hybrid structures of these sets. Rough set theory was initiated by Pawlak [2], which
is an effective mathematical model to deal with vagueness and imprecise knowledge. Its boundary
region gives the concept of vagueness, which can be interpreted by using the vagueness of Frege's
idea. He invented that vagueness can be dealt with the upper and lower approximations of precise
set using any equivalence relation. In the real life, rough set theory has many applications in different
fields such as social sciences, operational research, medical, economics, and artificial intelligence, etc.
Many real-world problems have neutrosophy in their nature and cannot handle by using fuzzy or
intuitionistic fuzzy set theory. For example, when we are dealing with conductors and nonconductors there must be a possibility having insulators. For this purpose, Smarandache [4, 5]
inaugurated the neutrosophic set theory as a generalization of fuzzy and intuitionistic fuzzy set
theory. The neutrosophic set yields the value from real standard or non-standard subsets of ]− 0, 1+ [.
It is difficult to utilize these values in daily life science and technology problems. Therefore, the
concept of a single-valued neutrosophic set, which takes value from the subset of [0,1], as defined by
Wang et al. [7]. The beauty of this set is that it gives the membership grades for truth, indeterminacy
M. Riaz, F. Smarandache, F. Karaaslan M.R. Hashmi, I. Nawaz, Neutrosophic Soft Topology and its Applications to MultiCriteria Decision Making
Neutrosophic Sets and Systems, Vol. 35, 2020
199
and falsity for the corresponding attribute. All the grades are independent of each other and provide
information about the three shades of an arbitrary attribute. Smarandache [8] extended the
neutrosophic set respectively to neutrosophic Overset (when some neutrosophic components are >
1), Neutrosophic Underset (when some neutrosophic components are < 0), and to Neutrosophic
Offset (when some neutrosophic components are off the interval [0,1] , i.e. some neutrosophic
components are > 1 and other neutrosophic components < 0). In 2016, Smarandache introduced
the Neutrosophic Tripolar Set and Neutrosophic Multipolar Set, also the Neutrosophic Tripolar
Graph and Neutrosophic Multipolar Graph [8].
The soft set is a mathematical model to deal ambiguities and imprecisions in parametric manners.
This is another abstraction of the crisp set theory. In 1999, Molodtsov [9] worked on parametrizations
of the universal set and invented a parameterized family of subsets of the universal set called soft set.
In recent years, many mathematicians worked on different hybrid structures of the fuzzy and rough
sets. Ali et al. [10, 11] established some novel operations in the soft sets, rough soft sets and, fuzzy
soft set theory. Aktas and Çağman [12] introduced various results on soft sets and soft groups. Bakier
et al. [13] introduced the idea of soft rough topology. Çağman et al. [14] introduced various results
on soft topology. Chen [15] worked on parametrizations reduction of soft sets and gave its
applications in decision-making. Feng et al. [16, 17] established various results on soft set, fuzzy set,
rough set and soft rough sets with the help of illustrations. Hashmi et al. [18] introduced the notion
of m-polar neutrosophic set and m-polar neutrosophic topology and their applications to multicriteria decision-making (MCDM) in medical diagnosis and clustering analysis. Hashmi and Riaz [19]
introduced a novel approach to the census process by using Pythagorean m-polar fuzzy Dombi's
aggregation operators. Kryskiewicz [20] introduced the rough set approach to incomplete
information systems. Karaaslan and Çağman [21] introduced bipolar soft rough sets and presented
their applications in decision-making. Kumar and Garg [22] introduced the TOPSIS method based on
the connection number of set pair analyses under an interval-valued intuitionistic fuzzy set
environment. Maji et al. [23, 24, 25] worked on some results of a soft set and gave its applications in
decision-making problems. He also invented the idea of a neutrosophic soft set and gave various
results to intricate the concept with numerous applications. Naeem et al. [26] introduced the novel
concept of Pythagorean m-polar fuzzy sets and the TOPSIS method for the selection of advertisement
mode. Peng and Garg [27] introduced algorithms for interval-valued fuzzy soft sets in emergency
decision making based on WDBA and CODAS with new information measures. Peng and Yang [28]
presented some results for Pythagorean fuzzy sets. Peng et al. [29] introduced Pythagorean fuzzy
information measures and their applications. Peng et al. [30] introduced a Pythagorean fuzzy soft set
and its application. Peng and Dai [31] introduced certain approaches to single-valued neutrosophic
MADM based on MABAC, TOPSIS and, new similarity measure with score function. Marei [32]
invented some more results on neutrosophic soft rough sets and worked on its modifications. Pei and
Miao [33] worked on the information system using the idea of a soft set. Quran et al. [34] introduced
a novel approach to neutrosophic soft rough set under uncertainty. Riaz et al. [35] introduced soft
rough topology with its applications to group decision making.
M. Riaz, F. Smarandache, F. Karaaslan M.R. Hashmi, I. Nawaz, Neutrosophic Soft Topology and its Applications to MultiCriteria Decision Making
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Riaz and Hashmi [36] introduced the notion of linear Diophantine fuzzy Set (LDFS) and its
applications towards the MCDM problem. Linear Diophantine fuzzy Set (LDFS) is superior to IFS,
PFS and, q-ROFS. Riaz and Hashmi [37] introduced novel concepts of soft rough Pythagorean mPolar fuzzy sets and Pythagorean m-polar fuzzy soft rough sets with application to decision-making.
Riaz and Tehrim [38] established the idea of cubic bipolar fuzzy ordered weighted geometric
aggregation operators and, their application using internal and external cubic bipolar fuzzy data.
They presented various illustrations and decision-making applications of these concepts by using
different algorithms. Roy and Maji [39] introduced a fuzzy soft set-theoretic approach to decisionmaking problems. Salama [40] investigated some topological properties of rough sets with tools for
data mining. Shabir and Naz [41] worked on soft topological spaces and presented their applications.
Thivagar et al. [42] presented some mathematical innovations of a modern topology in medical
events. Xueling et al. [43] presented some decision-making methods based on certain hybrid soft set
models. Zhang et al. [44, 45, 46] established fuzzy soft β-covering based fuzzy rough sets, fuzzy soft
coverings based fuzzy rough sets and, covering on generalized intuitionistic fuzzy rough sets with
their applications to multi-attribute decision-making (MADM) problems. Broumi et al. [47]
established the concept of rough neutrosophic sets. Christianto et al. [48] introduced the idea about
the extension of standard deviation notion with neutrosophic interval and quadruple neutrosophic
numbers. Adeleke et al. [49, 50] invented the concepts of refined eutrosophic rings I and refined
neutrosophic rings II. Parimala et al. [51] worked on 𝛼𝜔-closed sets and its connectedness in terms
of neutrosophic topological spaces. Ibrahim et al. [52] introduced the neutrosophic subtraction
algebra and neutrosophic subraction semigroup.
The neutrosophic soft rough set and neutrosophic soft rough topology have many applications in
MCDM problems. This hybrid erection is the most efficient and flexible rather than other
constructions. It is constructed with a combination of neutrosophic, soft and, rough set theory. The
interesting point in this structure is that by using this idea, we can deal with those type of models
which have roughness, neutrosophy and, parameterizations in their nature.
The motivation of this extended and hybrid work is presented step by step in the whole manuscript.
This model is generalized form and use to collect data at a large scale and applicable in medical,
engineering, artificial intelligence, agriculture and, other daily life problems. In the future, this work
can be gone easily for other approaches and different types of hybrid structures.
The layout of this paper is systematized as follows. Section 2, implies some basic ideas including soft
set, rough set, neutrosophic set, neutrosophic soft set and, neutrosophic soft rough set. We elaborate
on these ideas with the help of illustrations. In Section 3, we establish neutrosophic soft rough
topology (NSR-topology) with some examples. We introduce some topological structures on NSRtopology named NSR-interior, NSR-closure, NSR-exterior, NSR-neighborhood, NSR-limit point and,
NSR-bases. In Section 4 and 5, we present multi-criteria decision-making problems by using two
different algorithms on NSR-set and NSR-topology. We use the idea of upper and lower
approximations for NSR-set and construct algorithms using NSR-sets and NSR-bases We discuss the
optimal results obtained from both algorithms and present a comparitive analysis of proposed
approach with some existing approaches. Finally, the conclusion of this research is summarized in
section 6.
M. Riaz, F. Smarandache, F. Karaaslan M.R. Hashmi, I. Nawaz, Neutrosophic Soft Topology and its Applications to MultiCriteria Decision Making
Neutrosophic Sets and Systems, Vol. 35, 2020
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2. Preliminaries
This section presents some basic definitions including soft set, rough set, neutrosophic soft set, and
neutrosophic soft rough set .
Definition 2.1 [18]
Let U be the universal set. Let I(U) is collection of subsets of U. A pair (Θ, 𝔄) is said to be a soft set
over the universe U, where 𝔄 ⊆ E and Θ: 𝔄 → I(U) is a set-valued function. We denote soft set as
(Θ, 𝔄) or Θ𝔄 and mathematically write it as
Θ𝔄 = {(ξ, Θ(ξ)): ξ ∈ 𝔄, Θ(ξ) ∈ I(U)}.
For any ξ ∈ 𝔄, Θ(ξ) is ξ-approximate elements of soft set Θ𝔄 .
Definition 2.2 [21]
Let U be the initial universe and Y ⊆ U. Then, lower, upper, and boundary approximations of Y are
defined as
ℜå (Y) = ⋃g∈U {ℜ(g): ℜ(g) ⊆ Y},
ℜå (Y) = ⋃g∈U {ℜ(g): ℜ(g) ∩ Y ≠ ∅},
and
Bℜ (Y) = ℜå (Y) − ℜå (Y),
respectively. Where ℜ is an indiscernibility relation ℜ ⊆ U × U which indicates our information
about elements of U. The set Y is said to be defined if ℜå (Y) = ℜå (Y). If ℜå (Y) ≠ ℜå (Y) i.e BR (Y) ≠
∅, the set Y is rough set w.r.t ℜ.
Definition 2.3 [41] Let U be the initial universe. Then, a neutrosophic set N on the universe U is
defined as
N = {< g, 𝔗N (g), ℑN (g), 𝔉N (g) >: g ∈ U}, where
−
0 ≤ 𝔗N (g) + ℑN (g) + 𝔉N (g) ≤ 3+ , where
𝔗, ℑ, 𝔉: U →]− 0, 1+ [.
Where 𝔗, ℑ and 𝔉 represent the degree of membership, degree of indeterminacy and degree of nonmembership for some g ∈ U, respectively.
Definition 2.4 [16] Let U be an initial universe and E be a set of parameters. Suppose 𝔄 ⊂ E, and
let ℐ(U) represents the set of all neutrosophic sets of U. The collection (Φ, 𝔄) is said to be the
neutrosophic soft set over U, where Φ is a mapping given by
Φ: 𝔄 → ℐ(U).
The set containing all neutrosophic soft sets over U is denoted by NSU .
Example 2.5
Consider U = {g1 , g 2 , g 3 , g 4 , g 5 } be set of objects and attribute set is given by 𝔄 =
{ξ1 , ξ2 , ξ3 , ξ4 } = E = 𝔄, where
The neutrosophic soft set represented as Φ𝔄 . Consider a mapping Φ: 𝔄 → I(U) such that
Φ(ξ1 ) = {< g1 , 0.7,0.7,0.3 >, < g 2 , 0.5,0.7,0.7 >, < g 3 , 0.7,0.5,0.2 >, < g 4 , 0.7,0.4,0.4 >, < g 5 , 0.9,0.3,0.4 >},
Φ(ξ2 ) = {< g1 , 0.9,0.5,0.4 >, < g 2 , 0.7,0.3,0.5 >, < g 3 , 0.9,0.2,0.4 >, < g 4 , 0.9,0.3,0.3 >, < g 5 , 0.9,0.4,0.3 >},
Φ(ξ3 ) = {< g1 , 0.8,0.5,0.4 >, < g 2 , 0.7,0.5,0.4 >, < g 3 , 0.8,0.3,0.6 >, < g 4 , 0.6,0.3,0.7 >, < g 5 , 0.8,0.4,0.5 >},
Φ(ξ4 ) = {< g1 , 0.9,0.7,0.5 >, < g 2 , 0.8,0.7,0.7 >, < g 3 , 0.8,0.7,0.5 >, < g 4 , 0.8,0.6,0.7 >, < g 5 , 1.0,0.6,0.7 >}.
The tabular representation of neutrosophic soft set K = (Φ, 𝔄) is given in Table 1.
M. Riaz, F. Smarandache, F. Karaaslan M.R. Hashmi, I. Nawaz, Neutrosophic Soft Topology and its Applications to MultiCriteria Decision Making
Neutrosophic Sets and Systems, Vol. 35, 2020
(Φ, 𝔄)
202
g1
g2
g3
g4
g5
(0.7,0.7,0.3)
(0.5,0.7,0.7)
(0.7,0.5,0.2)
(0.7,0.4,0.4)
(0.9,0.3,0.4)
(0.9,0.5,0.4)
(0.7,0.3,0.5)
(0.9,0.2,0.4)
(0.9,0.3,0.3)
(0.9,0.4,0.3)
(0.8,0.5,0.4)
(0.7,0.5,0.4)
(0.8,0.3,0.6)
(0.6,0.3,0.7)
(0.8,0.4,0.5)
(0.9,0.7,0.5)
(0.8,0.7,0.7)
(0.8,0.7,0.5)
(0.8,0.6,0.7)
(1.0,0.6,0.7)
ξ1
ξ2
ξ3
ξ4
Table 1: Neutrosophic soft set (Φ, 𝔄)
Definition 2.6 Let (Φ, 𝔄) be a neutrosophic soft set on a universe U. For some elements g ∈ U, a
neutrosophic right neighborhood, regarding ξ ∈ 𝔄 is interpreted as follows;
g ξ = {g i ∈ U: 𝔗ξ (g i ) ≥ 𝔗ξ (g), ℑξ (g i ) ≥ ℑξ (g), 𝔉ξ (g i ) ≤ 𝔉ξ (g)}.
Definition 2.7 Let (Φ, 𝔄) be a neutrosophic soft set over a universe U. For some elements g ∈ U, a
neutrosophic right neighborhood regarding all parameters 𝔄 is interpreted as follows;
g]𝔄 =∩ {g ξi : ξi ∈ 𝔄}.
Example 2.8 Consider Example 2.5 then we find the following neutosophic right neighborhood
regarding all parameters 𝔄 as
g1 ξ = g1 ξ = g1 ξ = g1 ξ = {g1 }, g 2 ξ = g 2 ξ = {g1 , g 2 }, g 2 ξ = {g1 , g 2 , g 4 , g 5 }, g 2 ξ = {g1 , g 2 , g 3 }, g 3 ξ
1
2
3
4
1
3
2
4
1
= g 3 ξ = {g1 , g 3 }, g 3 ξ = {g1 , g 3 , g 4 , g 5 }, g 3 ξ = {g1 , g 3 , g 5 }, g 4 ξ = {g1 , g 3 , g 4 }, g 4 ξ
4
2
1
3
2
= {g 4 , g 5 }, g 4 ξ = U, g 4 ξ = U, g 5 ξ = g 5 ξ = g 5 ξ = {g 5 }, g 5 ξ = {g1 , g 5 }.
3
4
1
2
4
3
It follows that,
g1 ]𝔄 = {g1 },
g 2 ]𝔄 = {g1 , g 2 },
g 3 ]𝔄 = {g1 , g 3 },
g 4 ]𝔄 = {g 4 },
g 5 ]𝔄 = {g 5 }.
Definition 2.9 Let (Φ, 𝔄) be a neutrosophic soft set over U. For any X ⊆ U, neutrosophic soft lower
(aprNSR ) approximation, neutrosophic soft upper (aprNSR ) approximation, and neutrosophic soft
boundary (BNSR ) approximation of X are defined as
aprNSR (X) =∪ {g]𝔄 : g ∈ U, g]𝔄 ⊆ X}
aprNSR (X) =∪ {g]𝔄 : g ∈ U, g]𝔄 ∩ X ≠ ∅}
BNSR (X) = aprNSR (X) − aprNSR (X),
respectively. If aprNSR (X) = aprNSR (X) then X is neutrosophic soft definable set.
Example 2.10 Consider Example 2.5 , If X = {g1 } ⊆ U , then aprNSR (X) = {g1 } and aprNSR (X) =
{g1 , g 2 , g 3 }. Since its clear aprNSR (X) ≠ aprNSR (X), so X is neutrosophic soft rough set on U.
3 Neutrosophic Soft Rough Topology
M. Riaz, F. Smarandache, F. Karaaslan M.R. Hashmi, I. Nawaz, Neutrosophic Soft Topology and its Applications to MultiCriteria Decision Making
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In this section, we introduce and study the idea of neutrosophic soft rough topology and its related
properties. Concepts of (NSR)-open set, (NSR)-closed set, (NSR)-closure, (NSR)-interior, (NSR)exterior, (NSR)-neighborhood, (NSR)-limit point, and (NSR)-bases are defined.
Definition 3.1 Let U be the initial space, 𝔜 ⊆ U and G = (U, K) be a neutrosophic soft
approximation space, where K = (Φ, 𝔄) is a neutrosophic soft set. The upper and lower
approximations are calculated on the basis of neutrosophic soft approximation space and
neighborhoods. Then, the collection
τNSR (𝔜) = {U, ∅, aprNSR (𝔜), aprNSR (𝔜), BNSR (𝔜)}
is called neutrosophic soft rough topology (NSR-topology) which guarantee the following postulates:
• U and ∅ belongs to τNSR (𝔜).
• Union of members of τNSR (𝔜) belongs to τNSR (𝔜).
• Finite Intersection of members of τNSR (𝔜) belongs to τNSR (𝔜).
Then (U, τNSR (𝔜), E) is said to be NSR-topological space, if τNSR (𝔜) is Neutrosophic soft
rough topology.
Note that Neutrosophic soft rough topology is based on lower and upper approximations of
neutrosophic soft rough set.
Example 3.2
From Example 2.5, if 𝔜 = {g 2 , g 4 } ⊆ U , we obtain aprNSR (𝔜) = {g 4 } , aprNSR (𝔜) =
{g1 , g 2 , g 4 } and BNSR (𝔜) = {g1 , g 2 }. Then,
τNSR (𝔜) = {U, ∅, {g 4 }, {g1 , g 2 , g 4 }, {g1 , g 2 }}
is a NSR-topology.
Definition 3.3 Let (U, τNSR (𝔜), E) be an NSR-topological space. Then, the members of τNSR (𝔜) are
called NSR-open sets. An NSR-set is said to be an NSR-closed set if its complement belongs to
τNSR (𝔜).
Proposition 3.4 Consider (U, τNSR (𝔜), E) as NSR-space over U. Then,
• U and ∅ are NSR-closed sets.
• The intersection of any number of NSR-closed sets is an NSR-closed set over U.
• The finite union of NSR-closed sets is an NSR-closed set over U.
Proof. The proof is straightforward.
Definition 3.5 Let (U, τNSR (𝔜), E) be an NSR-space over U and τNSR (𝔜) = {U, ∅}. Then, τNSR is
called NSR-indiscrete topology on U w.r.t 𝔜 and corresponding space is said to be an NSRindiscrete space over U.
Definition 3.6 Let (U, τNSR (𝔜), E) is an NSR-topological space and A ⊆ B ⊆ U. Then, the collection
τNSR A = {Bi ∩ A: Bi ∈ τNSR , i ∈ L ⊆ N} is called NSR-subspace topology on A . Then, (A, τNSR A ) is
called an NSR-topological subspace of (B, τNSR ).
Definition 3.7 Let (U, τNSR′ (𝔜), E) and (U, τNSR (𝔛), E) be two NSR-topological spaces. τNSR′ (𝔜) is
finer than τNSR (𝔛), if τNSR′ (𝔜) ⊇ τNSR (𝔛).
Definition 3.8 Let (U, τNSR (𝔜), E) be a NSR-topological space and βNSR ⊆ τNSR . If we can write
members of τNSR as the union of members of βNSR , then βNSR is called NSR-basis for the NSRtopology τNSR .
Proposition 3.9 If τNSR (𝔜) is an NSR-topology on U w.r.t 𝔜 the the collection
βNSR = {U, aprNSR (𝔜), BNSR (𝔜)}
is a base for τNSR (𝔜)
M. Riaz, F. Smarandache, F. Karaaslan M.R. Hashmi, I. Nawaz, Neutrosophic Soft Topology and its Applications to MultiCriteria Decision Making
Neutrosophic Sets and Systems, Vol. 35, 2020
204
Theorem 3.10 Let (U, τNSR (𝔜), E) and (U, τNSR′ (𝔜′), E) be two NSR-topological spaces w.r.t 𝔜 and
𝔜′ respectively. Let βNSR and βNSR′ be NSR-bases for τNSR and τNSR′ , respectively. If βNSR′ ⊆ βNSR
then τNSR is finer than τNSR′ and τNSR′ is weaker than τNSR .
Theorem 3.11 Let (𝑈, 𝜏𝑁𝑆𝑅 (𝔜), 𝐸) be an NSR-topological space. If 𝛽𝑁𝑆𝑅 is an NSR-basis for 𝜏𝑁𝑆𝑅 .
Then, the collection 𝛽𝑁𝑆𝑅𝐵 = {𝐴𝑖 ∩ 𝐵: 𝐴𝑖 ∈ 𝛽𝑁𝑆𝑅 , 𝑖 ∈ 𝐼 ⊆ ℕ} is an NSR-basis for the NSR-subspace
topology on 𝐵.
Proof. Consider 𝐴𝑖 ∈ 𝜏𝑁𝑆𝑅 𝐵 . By definition of NSR-subspace topology, 𝐶 = 𝐷 ∩ 𝐵,where 𝐷 ∈ 𝜏𝑁𝑆𝑅 .
Since 𝐷 ∈ 𝜏𝑁𝑆𝑅 , it follows that 𝐷 = ⋃𝐴𝑖 ∈𝛽𝑁𝑆𝑅 𝐴𝑖 . Therefore,
𝐶 = (⋃𝐴𝑖 ∈𝛽𝑁𝑆𝑅 𝐴𝑖 ) ∩ 𝐵 = ⋃𝐴𝑖 ∈𝛽𝑁𝑆𝑅 (𝐴𝑖 ∩ 𝐵).
3.1 Main Results
We present some results of neutrosophic soft rough topology including NSR-interior, NSR-exterior,
NSR-closure, NSR-frontier, NSR-neighbourhood and NSR-limit point. These are some topological
properties of NSR-topology and can be used to prove various results related to NSR-topological
spaces.
Definition 3.12 Let (𝑈, 𝜏𝑁S𝑅 (𝔜), 𝐸) be an NSR-topological space w.r.t 𝔜 , where 𝑇 ⊆ 𝑈 be an
arbitrary subset. The NSR-interior of 𝑇 is union of all NSR-open subsets of 𝑇 and we denote it as
𝐼𝑛𝑡𝑁𝑆𝑅 (𝑇).
We verify that 𝐼𝑛𝑡𝑁𝑆𝑅 (𝑇) is the largest NSR-open set contained by 𝑇.
Theorem 3.13 Let (𝑈, 𝜏𝑁𝑆𝑅 (𝔜), 𝐸) be a NSR-topological space over 𝑈 w.r.t 𝔜, 𝑆 and 𝑇 are NSRsets over 𝑈. Then
• 𝐼𝑛𝑡𝑁𝑆𝑅 (∅) = ∅ and 𝐼𝑛𝑡𝑁𝑆𝑅 (𝑈) = 𝑈,
• 𝐼𝑛𝑡𝑁𝑆𝑅 (𝑆) ⊆ 𝑆,
• 𝑆 is NSR-open set ⇔ 𝐼𝑛𝑡𝑁𝑆𝑅 (𝑆) = 𝑆,
• 𝐼𝑛𝑡𝑁𝑆𝑅 (𝐼𝑛𝑡𝑁𝑆𝑅 (𝑆)) = 𝐼𝑛𝑡𝑁𝑆𝑅 (𝑆),
• 𝑆 ⊆ 𝑇 implies 𝐼𝑛𝑡𝑁𝑆𝑅 (𝑆) ⊆ 𝐼𝑛𝑡𝑁𝑆𝑅 (𝑇),
• 𝐼n𝑡𝑁𝑆𝑅 (𝑆) ∪ 𝐼𝑛𝑡𝑁𝑆𝑅 (𝑇) ⊆ 𝐼𝑛𝑡𝑁𝑆𝑅 (𝑆 ∪ 𝑇),
• 𝐼𝑛𝑡𝑁𝑆𝑅 (𝑆) ∩ 𝐼𝑛𝑡𝑁𝑆𝑅 (𝑇) = 𝐼𝑛𝑡𝑁𝑆𝑅 (𝑆 ∩ 𝑇).
Proof. (i) and (ii) are obvious.
(iii) First, suppose that 𝐼𝑛𝑡𝑁𝑆𝑅 (𝑆) = 𝑆. Since 𝐼𝑛𝑡𝑁𝑆𝑅 (𝑆) is an NSR-open set, it follows that 𝑆 is NSRopen set. For the converse, if 𝑆 is a NSR-open set, then the largest NSR-open set that is contained in
𝑆 is 𝑆 itself. Thus, 𝐼𝑛𝑡𝑁𝑆𝑅 (𝑆) = 𝑆.
(iv) Since 𝐼𝑛𝑡𝑁𝑆𝑅 (𝑆) is an NSR-open set, by part (iii) we get 𝐼𝑛𝑡𝑁𝑆𝑅 (𝐼𝑛𝑡𝑁𝑆𝑅 (𝑆)) = 𝐼𝑛𝑡𝑁𝑆𝑅 (𝑆).
(v) Suppose that 𝑆 ⊆ 𝑇. By (ii) 𝐼𝑛𝑡𝑁𝑆𝑅 (𝑆) ⊆ 𝑆. Then 𝐼𝑛𝑡𝑁𝑆𝑅 (𝑆) ⊆ 𝑇. Since 𝐼𝑛𝑡𝑁𝑆𝑅 (𝑆) is NSR-open set
contained by 𝑇. So by definition of NSR-interior 𝐼𝑛𝑡𝑁𝑆𝑅 (𝑆) ⊆ 𝐼𝑛𝑡𝑁𝑆𝑅 (𝑇).
(vi) By using (ii) 𝐼𝑛𝑡𝑁𝑆𝑅 (𝑆) ⊆ 𝑆 and 𝐼𝑛𝑡𝑁𝑆𝑅 (𝑇) ⊆ 𝑇 . Then, 𝐼𝑛𝑡𝑁𝑆𝑅 (𝑆) ∪ 𝐼𝑛𝑡𝑁𝑆𝑅 (𝑇) ⊆ 𝑆 ∪ 𝑇 . Since
𝐼𝑛𝑡𝑁𝑆𝑅 (𝑆) ∪ 𝐼𝑛𝑡𝑁𝑆𝑅 (𝑇) is an NSR-open, it follows that 𝐼𝑛𝑡𝑁𝑆𝑅 (𝑆) ∪ 𝐼𝑛𝑡𝑁𝑆𝑅 (𝑇) ⊆ 𝐼𝑛𝑡𝑁𝑆𝑅 (𝑆 ∪ 𝑇).
(vii) By using (ii) 𝐼𝑛𝑡𝑁𝑆𝑅 (𝑆) ⊆ 𝑆 and 𝐼𝑛𝑡𝑁𝑆𝑅 (𝑇) ⊆ 𝑇 . Then, 𝐼𝑛𝑡𝑁𝑆𝑅 (𝑆) ∩ 𝐼𝑛𝑡𝑁𝑆𝑅 (𝑇) ⊆ 𝑆 ∩ 𝑇 . Since
𝐼𝑛𝑡𝑁𝑆𝑅 (𝑆) ∩ 𝐼𝑛𝑡𝑁𝑆𝑅 (𝑇) is NSR-open, it follows that 𝐼𝑛𝑡𝑁𝑆𝑅 (𝑆) ∩ 𝐼𝑛𝑡𝑁𝑆𝑅 (𝑇) ⊆ 𝐼𝑛𝑡𝑁𝑆𝑅 (𝑆 ∩ 𝑇). For the
converse, 𝑆 ∩ 𝑇 ⊆ 𝑆 also 𝑆 ∩ 𝑇 ⊆ 𝑇 . Then, 𝐼𝑛𝑡𝑁𝑆𝑅 (𝑆 ∩ 𝑇) ⊆ 𝐼𝑛𝑡𝑁𝑆𝑅 (𝑆) and 𝐼𝑛𝑡𝑁𝑆𝑅 (𝑆 ∩ 𝑇) ⊆
𝐼𝑛𝑡𝑁𝑆𝑅 (𝑇). Hence 𝐼𝑛𝑡𝑁𝑆𝑅 (𝑆 ∩ 𝑇) ⊆ 𝐼𝑛𝑡𝑁𝑆𝑅 (𝑆) ∩ 𝐼𝑛𝑡𝑁𝑆𝑅 (𝑇).
M. Riaz, F. Smarandache, F. Karaaslan M.R. Hashmi, I. Nawaz, Neutrosophic Soft Topology and its Applications to MultiCriteria Decision Making
Neutrosophic Sets and Systems, Vol. 35, 2020
205
Definition 3.14 Let (𝑈, 𝜏𝑁𝑆𝑅 (𝔜), 𝐸) be an NSR-topological space w.r.t 𝔜, where 𝔜 ⊆ 𝑈. Let 𝑇 ⊆ 𝑈.
Then, NSR-exterior of 𝑇 is defined as 𝐼𝑛𝑡𝑁𝑆𝑅 (𝑇 𝑐 ), where 𝑇 𝑐 is complement of 𝑇. NSR-exterior of 𝑇
is denoted by 𝐸𝑥𝑡𝑁𝑆𝑅 (𝑇).
Definition 3.15 Let (𝑈, 𝜏𝑁𝑆𝑅 (𝔜), 𝐸) be an NSR-topological space w.r.t 𝔜, where 𝔜 ⊆ 𝑈. Let 𝑇 ⊆ 𝑈.
Then, NSR-closure of 𝑇 is defined to be intersection of all NSR-closed supersets of 𝑇 and is denoted
by 𝐶𝑙𝑁𝑆𝑅 (𝑇).
Example 3.16 Consider the NSR-topology given in Example 3.2, taking 𝑇 = {𝑔1 , 𝑔2 , 𝑔3 }, so 𝑇 𝑐 =
{𝑔4 , 𝑔5 }. Then 𝐼𝑛𝑡𝑁𝑆𝑅 (𝑇) = {𝑔1 , 𝑔2 }, 𝐸𝑥𝑡𝑁𝑆𝑅(𝑇) = 𝐼𝑛𝑡𝑁𝑆𝑅 (𝑇 𝑐 ) = {𝑔4 } and 𝐶𝑙𝑁𝑆𝑅 (T) = {𝑔1 , 𝑔2 , 𝑔3 , 𝑔5 }.
Theorem 3.17 Let (𝑈, 𝜏𝑁𝑆𝑅 (𝔜), 𝐸) be a NSR-topological space over 𝑈 w.r.t 𝔜, 𝑆 and 𝑇 are NSRsets over 𝑈. Then
• 𝐶𝑙𝑁𝑆𝑅 (∅) = ∅ and 𝐶𝑙𝑁𝑆𝑅 (𝑈) = 𝑈,
• 𝑆 ⊆ 𝐶𝑙𝑁𝑆𝑅 (𝑆),
• 𝑆 is NSR-closed set ⇔ 𝑆 = 𝐶𝑙𝑁𝑆𝑅 (𝑆),
• 𝐶𝑙𝑁𝑆𝑅 (𝐶𝑙𝑁𝑆𝑅 (𝑆)) = 𝐶𝑙𝑁𝑆𝑅 (𝑆),
• 𝑆 ⊆ 𝑇 implies 𝐶𝑙𝑁𝑆𝑅 (𝑆) ⊆ 𝐶𝑙𝑁𝑆𝑅 (𝑇),
• 𝐶𝑙𝑁𝑆𝑅 (𝑆 ∪ 𝑇) = 𝐶𝑙𝑁𝑆𝑅 (𝑆) ∪ 𝐶𝑙𝑁𝑆𝑅 (𝑇),
• 𝐶𝑙𝑁𝑆𝑅 (𝑆 ∩ T) ⊆ 𝐶𝑙𝑁𝑆𝑅 (𝑆) ∩ 𝐶𝑙𝑁𝑆𝑅 (𝑇).
Proof. (i) and (ii) are straightforward.
(iii) First, consider 𝑆 = 𝐶𝑙𝑁𝑆𝑅 (𝑆). Since 𝐶𝑙𝑁𝑆𝑅 (𝑆) is an NSR-closed set, so 𝑆 is an NSR-closed set over
𝑈. For the converse, suppose that 𝑆 be an NSR-closed set over 𝑈. Then, 𝑆 is NSR-closed superset of
𝑆. So that 𝑆 = 𝐶𝑙𝑁𝑆𝑅 (𝑆).
(iv) By definition 𝐶𝑙𝑁𝑆𝑅 (𝑆) is always NSR-closed set. Therefore, by part (iii) we have
𝐶𝑙𝑁𝑆𝑅 (𝐶𝑙𝑁𝑆𝑅 (𝑆)) = 𝐶𝑙𝑁𝑆𝑅 (𝑆).
(v) Let 𝑆 ⊆ 𝑇. By (ii) 𝑇 ⊆ 𝐶𝑙𝑁𝑆𝑅 (𝑇). Then, 𝑆 ⊆ 𝐶𝑙𝑁𝑆𝑅 (𝑇). Since 𝐶𝑙𝑁𝑆𝑅 (𝑇) is a NSR-closed superset of
𝑆, it follows that 𝐶𝑙𝑁𝑆𝑅 (𝑆) ⊆ 𝐶𝑙𝑁𝑆𝑅 (𝑇).
(vi) Since 𝑆 ⊆ 𝑆 ∪ 𝑇 and 𝑇 ⊆ 𝑆 ∪ 𝑇 , by part (v), 𝐶𝑙𝑁𝑆𝑅 (𝑆) ⊆ 𝐶𝑙𝑁𝑆𝑅 (𝑆 ∪ 𝑇) and 𝐶𝑙𝑁𝑆𝑅 (𝑇) ⊆
𝐶𝑙𝑁𝑆𝑅 (𝑆 ∪ 𝑇) . Hence 𝐶𝑙𝑁𝑆𝑅 (𝑆) ∪ 𝐶𝑙𝑁𝑆𝑅 (𝑆) ⊆ 𝐶𝑙𝑁𝑆𝑅 (𝑆 ∪ 𝑇) .
For the converse, let
𝑆⊆
𝐶𝑙𝑁𝑆𝑅 (𝑆) and 𝑇 ⊆ 𝐶𝑙𝑁𝑆𝑅 (𝑇). Then, 𝑆 ∪ 𝑇 ⊆ 𝐶𝑙𝑁𝑆𝑅 (𝑆) ∪ 𝐶𝐿𝑁𝑆𝑅 (𝑇). Since 𝐶𝑙𝑁𝑆𝑅 (𝑆) ∪ 𝐶𝑙𝑁𝑆𝑅 (𝑇) is a
NSR-closed superset of 𝑆 ∪ 𝑇. Thus, 𝐶𝑙𝑁𝑆𝑅 (𝑆 ∪ 𝑇) = 𝐶𝑙𝑁𝑆𝑅 (𝑆) ∪ 𝐶𝑙𝑁𝑆𝑅 (𝑇).
(vii) Since 𝑆 ∩ 𝑇 ⊆ 𝑆 and 𝑆 ∩ 𝑇 ⊆ 𝑇 , by part(5) 𝐶𝑙𝑁𝑆𝑅 (𝑆 ∩ 𝑇) ⊆ 𝐶𝑙𝑁𝑆𝑅 (𝑆) and 𝐶𝑙𝑁𝑆𝑅 (𝑆 ∩ 𝑇) ⊆
𝐶𝑙𝑁𝑆𝑅 (𝑇). Thus, we obtain 𝐶𝑙𝑁𝑆𝑅 (𝑆 ∩ 𝑇) ⊆ 𝐶𝑙𝑁𝑆𝑅 (𝑆) ∩ 𝐶𝑙𝑁𝑆𝑅 (𝑇).
Definition 3.18 Let (𝑈, 𝜏𝑁𝑅 (𝔜), 𝐸) be a NSR-topological space w.r.t 𝔜, where 𝔜 ⊆ 𝑈. Let 𝑇 ⊆ 𝑈.
Then, NSR-frontier or NSR-boundary of 𝑇 is denoted by 𝐹𝑟 𝑁𝑆𝑅 (𝑇) or 𝑏𝑁𝑆𝑅 (𝑇) and mathematically
defined as
𝐹𝑟 𝑁𝑆𝑅 (𝑇) = 𝐶𝑙𝑁𝑆𝑅 (𝑇) ∩ 𝐶𝑙𝑁𝑆𝑅 (𝑇 𝑐 ).
Clearly NSR-frontier 𝐹𝑟 𝑁𝑆𝑅 (𝑇) is an NSR-closed set.
Example 3.19 Consider the NSR-topology given in Example 3.2, taking 𝑇 = {𝑔1 , 𝑔2 , 𝑔3 }, so 𝑇 𝑐 =
{𝑔4 , 𝑔5 }. Then, 𝐶𝑙𝑁𝑆𝑅 (𝑇) = {𝑔1 , 𝑔2 , 𝑔3 , 𝑔5 } and 𝐶𝑙𝑁𝑆𝑅 (𝑇 𝑐 ) = {𝑔3 , 𝑔4 , 𝑔5 }.
𝐹𝑟 𝑁𝑆𝑅 (𝑇) = 𝐶𝑙𝑁𝑆𝑅 (𝑇) ∩ 𝐶𝑙𝑁𝑆𝑅 (𝑇 𝑐 ) = {𝑔3 , 𝑔5 }
M. Riaz, F. Smarandache, F. Karaaslan M.R. Hashmi, I. Nawaz, Neutrosophic Soft Topology and its Applications to MultiCriteria Decision Making
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Definition 3.20 Let (𝑈, 𝜏𝑁𝑅 (𝔜), 𝐸) be an NSR-topological space. A subset 𝑋 of 𝑈 is said to be NSRneighborhood of 𝑔 ∈ 𝑈 if there exist an NSR-open set 𝑊𝑔 containing 𝑔 so that
𝑔 ∈ 𝑊𝑔 ⊆ 𝑋.
Definition 3.21 The set of all the NSR-limit points of 𝑆 is known as NSR-derived set of 𝑆 and is
𝑑
.
denoted by 𝑆𝑁𝑆𝑅
4 NSR-set in multi-criteria decision-making
In this section, we present an idea for multi-criteria decision-making method based on the
neutrosophic soft rough sets 𝑁𝑆𝑅 − 𝑠𝑒𝑡.
Let 𝑈 = {𝑔1 , 𝑔2 , 𝑔3 , . . . , 𝑔𝑚 } is the set of objects under observation, 𝐸 be the set of criteria to analyze
the objects in 𝑈 . Let 𝔄 = {𝜉1 , 𝜉2 , 𝜉3 , . . . , 𝜉𝑛 } ⊆ 𝐸 and (𝛷, 𝔄) be a neutrosophic soft set over 𝑈.
Suppose that 𝐻 = {𝑃1 , 𝑃2 , . . . , 𝑃𝑘 } be a set of experts, 𝔜1 , 𝔜2 , . . . , 𝔜𝑘 are subsets of 𝑈 which indicate
results of initial evaluations of experts 𝑃1 , 𝑃2 , . . . , 𝑃𝑘 , respectively and 𝔗1 , 𝔗2 , . . . 𝔗𝑟 ∈ 𝑁𝑆𝑈 are real
results that previously obtained for same or similar problems in different times or different places.
Definition 4.1 Let 𝑎𝑝𝑟𝑁𝑆𝑅𝔗𝑞 (𝔜𝑗 ),𝑎𝑝𝑟𝑁𝑆𝑅 (𝔜𝑗 ) be neutrosophic soft lower and upper approximations
𝔗𝑞
of 𝔜𝑗 (𝑗 = 1,2, . . . , 𝑘) related to 𝔗𝑞 (𝑞 = 1,2, . . . , 𝑟). Then,
𝑎=
(
𝑎=
𝑛11
𝑛12
⋮
𝑛1𝑟
𝑛12
𝑛22
⋮
𝑛2𝑟
𝑛1
1
𝑛2
2
𝑛2
⋮
𝑟
𝑛2
𝑛1
⋮
𝑟
(𝑛1
⋯
⋯
⋱
⋯
𝑛1𝑘
𝑛𝑘2
⋮
𝑛𝑘𝑟
(1)
)
1
⋯ 𝑛𝑘
1
2
⋯ 𝑛𝑘
⋱ ⋮
𝑟
⋯ 𝑛𝑘 )
2
(2)
are called neutrosophic soft lower and neutrosophic upper approximations matrices, respectively,
and represented by 𝑎 and 𝑎. Here
𝑞
𝑞
𝑞
𝑞
𝑞
𝑞
𝑞
𝑞
𝑛𝑗 = (𝑔1𝑗 , 𝑔2𝑗 , . . . , 𝑔𝑛𝑗 )
𝑛𝑗 = (𝑔1𝑗 , 𝑔2𝑗 , . . . , 𝑔𝑛𝑗 )
(3)
(4)
Where
𝑞
𝑔𝑖𝑗 = (
1, 𝑔𝑖 ∈ 𝑎𝑝𝑟𝑁𝑆𝑅𝔗𝑞 (𝔜𝑗 )
0, 𝑔𝑖 ∈ 𝑎𝑝𝑟𝑁𝑆𝑅𝔗𝑞 (𝔜𝑗 )
and
𝑞
𝑔𝑖𝑗 = (
1, 𝑔𝑖 ∈ 𝑎𝑝𝑟𝑁𝑆𝑅 (𝔜𝑗 )
𝔗𝑞
0, 𝑔𝑖 ∈ 𝑎𝑝𝑟𝑁𝑆𝑅 (𝔜𝑗 )
𝔗𝑞
M. Riaz, F. Smarandache, F. Karaaslan M.R. Hashmi, I. Nawaz, Neutrosophic Soft Topology and its Applications to MultiCriteria Decision Making
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207
Definition 4.2 Let 𝑛 and 𝑛 be neutrosophic soft lower and neutrosophic upper approximations
matrices based on 𝑎𝑝𝑟𝑁𝑆𝑅𝔗𝑞 (𝔜𝑗 , 𝑎𝑝𝑟𝑁𝑆𝑅 (𝔜𝑗 for 𝑞 = 1,2, . . . 𝑟 and 𝑗 = 1,2, . . . , 𝑘. Neutrosophic soft
𝔗𝑞
lower approximation vector represented by (𝑛) and neutrosophic soft upper approximation vector
represented by (𝑛) are defined by, respectively,
𝑘
𝑟
𝑞
𝑛 = ⊕ ⊕ 𝑛𝑗
𝑗=1𝑞=1
𝑘
𝑟
𝑞
𝑛 = ⊕ ⊕ 𝑛𝑗
𝑗=1𝑞=1
Here the operation ⊕
(5)
(6)
represents the vector summation.
Definition 4.3 Let 𝑛 and 𝑛 be neutrosophic soft 𝔗𝑞 − lower approximation vector and
neutrosophic soft 𝔗𝑞 − upper approximation vector, respectively. Then, vector summation 𝑛 ⊕ 𝑛 =
(𝑤1 , 𝑤2 , . . . , 𝑤𝑛 ) is called decision vector.
Definition 4.4 Let 𝑛 ⊕ 𝑛 = (𝑤1 , 𝑤2 , . . . , 𝑤𝑛 ) be the decision vector. Then, each 𝑤𝑖 is called a weighted
number of 𝑔𝑖 ∈ 𝑈 and 𝑔𝑖 is called an optimum element of 𝑈 if it weighted number is maximum of
𝑤𝑖 ∀𝑖 ∈ 𝐼𝑛 . In this case, if there are more then one optimum elements of 𝑈, select one of them.
Algorithm 1 for neutrosophic soft rough set:
Input
Step-1: Take initial evaluations 𝔜1 , 𝔜2 , . . . , 𝔜𝑘 of experts 𝑃1 , 𝑃2 , . . . , 𝑃𝑘 .
Step-2: Construct 𝔗1 , 𝔗2 , . . . 𝔗𝑟 neutrosophic soft sets using real results.
Step-3: Compute 𝑎𝑝𝑟𝑁𝑆𝑅𝔗𝑞 (𝔜𝑗 ) and 𝑎𝑝𝑟𝑁𝑆𝑅 (𝔜𝑗 ) for each 𝑞 = 1,2, . . . , 𝑟 and 𝑗 = 1,2, . . . , 𝑘.
𝔗𝑞
Step-4: Construct neutrosophic soft lower and neutrosophic soft upper approximations matrices 𝑎
and 𝑎.
Step-5: Compute 𝑛 and 𝑛,
Step-6: Compute 𝑛 ⊕ 𝑛,
Output
Step-7: Select 𝑚𝑎𝑥𝑖∈𝐼𝑛 𝑤𝑖 .
The flow chart of proposed algorithm 1 is represented in Figure.1
M. Riaz, F. Smarandache, F. Karaaslan M.R. Hashmi, I. Nawaz, Neutrosophic Soft Topology and its Applications to MultiCriteria Decision Making
Neutrosophic Sets and Systems, Vol. 35, 2020
208
Fig 1: Flow chart diagram of proposed algorithm 1 for NSR-set.
Example 4.5 In finance company three finance experts 𝑃1 , 𝑃2 , 𝑃3 want to make investment one of the
clothing brand
{𝑔1 = 𝐽𝑜𝑟, 𝑔2 = 𝐴𝑒𝑟𝑜, 𝑔3 = 𝐶ℎ𝑎𝑛, 𝑔4 = 𝐿𝑖, 𝑔5 = 𝑆𝑟𝑘}.
The set of parameters include the following parameters
𝔄 = {𝜉1 = 𝑀𝑎𝑟𝑘𝑒𝑡 𝑆ℎ𝑎𝑟𝑒, 𝜉2 = 𝐴𝑐𝑘𝑛𝑜𝑤𝑙𝑒𝑑𝑔𝑒𝑚𝑒𝑛𝑡, 𝜉3 = 𝑈𝑛𝑖𝑞𝑢𝑒𝑛𝑒𝑠𝑠, 𝜉4 =
𝐸𝑐𝑜𝑛𝑜𝑚𝑖𝑐𝑎𝑙 𝑀𝑎𝑔𝑛𝑖𝑓𝑖𝑐𝑎𝑡𝑖𝑜𝑛}
𝑆𝑡𝑒𝑝1: 𝔜1 = {𝑔1 , 𝑔2 , 𝑔4 }, 𝔜2 = {𝑔1 , 𝑔3 , 𝑔5 }, 𝔜3 = {𝑔2 , 𝑔4 , 𝑔5 } are primary evaluations of experts
𝑃1 , 𝑃2 , 𝑃3 , respectively.
𝑆𝑡𝑒𝑝2: Neutrosophic soft sets 𝔗1 , 𝔗2 , 𝔗3 are the actual results in individual three periods
and tabular representations of these neutrosophic soft sets are given in Table 2, Table 3 and Table 4,
respectively.
𝔗1
𝜉1
𝜉2
𝜉3
𝜉4
𝑔1
(0.6,0.6,0.2)
(0.8,0.4.0.3)
(0.7,0.4,0.3)
(0.8,0.6,0.4)
𝑔2
(0.4,0.6,0.6)
(0.6,0.2,0.4)
(0.6,0.4,0.3)
(0.7,0.6,0.6)
𝑔3
(0.6,0.4,0.2)
(0.8,0.1,0.3)
(0.7,0.2,0.5)
(0.7,0.6,0.4)
𝑔4
(0.6,0.3,0.3)
(0.8,0.2,0.2)
(0.5,0.2,0.6)
(0.7,0.5,0.6)
𝑔5
(0.8,0.2,0.3)
(0.8,0.3,0.2)
(0.7,0.3,0.4)
(0.9,0.5,0.7)
Table 2: Neutrosophic soft set 𝔗1
𝔗2
𝜉1
𝜉2
𝜉3
𝜉4
𝑔1
(0.6,0.4,0.2)
(0.8,0.1,0.3)
(0.7,0.2,0.5)
(0.7,0.6,0.4)
𝑔2
(0.4,0.6,0.6)
(0.6,0.2,0.4)
(0.6,0.4,0.3)
(0.7,0.6,0.6)
𝑔3
(0.8,0.2,0.3)
(0.8,0.3,0.2)
(0.7,0.3,0.4)
(0.9,0.5,0.7)
𝑔4
(0.6,0.3,0.3)
(0.8,0.2,0.2)
(0.5,0.2,0.6)
(0.7,0.5,0.6)
𝑔5
(0.6,0.6,0.2)
(0.8,0.4.0.3)
(0.7,0.4,0.3)
(0.8,0.6,0.4)
M. Riaz, F. Smarandache, F. Karaaslan M.R. Hashmi, I. Nawaz, Neutrosophic Soft Topology and its Applications to MultiCriteria Decision Making
Neutrosophic Sets and Systems, Vol. 35, 2020
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Table 3: Neutrosophic soft set 𝔗2
𝔗3
𝜉1
𝜉2
𝜉3
𝜉4
𝑔1
(0.6,0.6,0.2)
(0.8,0.4.0.3)
(0.7,0.4,0.3)
(0.8,0.6,0.4)
𝑔2
(0.6,0.3,0.3)
(0.8,0.2,0.2)
(0.5,0.2,0.6)
(0.7,0.5,0.6)
𝑔3
(0.6,0.4,0.2)
(0.8,0.1,0.3)
(0.7,0.2,0.5)
(0.7,0.6,0.4)
𝑔4
(0.4,0.6,0.6)
(0.6,0.2,0.4)
(0.6,0.4,0.3)
(0.7,0.6,0.6)
𝑔5
(0.8,0.2,0.3)
(0.8,0.3,0.2)
(0.7,0.3,0.4)
(0.9,0.5,0.7)
Table 4: Neutrosophic soft set 𝔗3
The tabular representation of the neutrosophic right neighborhoods of 𝔗1 , 𝔗2 , 𝔗3 are given in
Table5, Table 6 and Table 7 respectively.
Neighborhoods of 𝔗1
𝑔1 ]𝔄
{𝑔1 }
𝑔2 ]𝔄
{𝑔1 , 𝑔2 }
𝑔3 ]𝔄
{𝑔1 , 𝑔3 }
𝑔4 ]𝔄
{𝑔4 }
𝑔5 ]𝔄
{𝑔5 }
Table 5: Neutrosophic right neighborhoods of 𝔗1 w.r.t set 𝔄
Neighborhoods of 𝔗2
𝑔1 ]𝔄
{𝑔1 , 𝑔5 }
𝑔2 ]𝔄
{𝑔2 , 𝑔5 }
𝑔3 ]𝔄
{𝑔3 }
𝑔4 ]𝔄
{𝑔4 }
𝑔5 ]𝔄
{𝑔5 }
Table 6: Neutrosophic right neighborhoods of 𝔗2 w.r.t set 𝔄
Neighborhoods of 𝔗3
𝑔1 ]𝔄
{𝑔1 }
𝑔2 ]𝔄
{𝑔2 }
𝑔3 ]𝔄
{𝑔1 , 𝑔3 }
𝑔4 ]𝔄
{𝑔1 , 𝑔4 }
𝑔5 ]𝔄
{𝑔5 }
Table 7: Neutrosophic right neighborhoods of 𝔗3 w.r.t set 𝔄
M. Riaz, F. Smarandache, F. Karaaslan M.R. Hashmi, I. Nawaz, Neutrosophic Soft Topology and its Applications to MultiCriteria Decision Making
Neutrosophic Sets and Systems, Vol. 35, 2020
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𝑆𝑡𝑒𝑝3: Next we find 𝑎𝑝𝑟𝑁𝑆𝑅𝔗1 and 𝑎𝑝𝑟𝑁𝑆𝑅
𝔗1
for each 𝔜𝑗 , where 𝑗 = 1,2,3.
𝑎𝑝𝑟𝑁𝑆𝑅𝔗1 (𝔜1 ) = {𝑔1 , 𝑔2 , 𝑔4 },
𝑎𝑝𝑟𝑁𝑆𝑅 (𝔜1 ) = {𝑔1 , 𝑔2 , 𝑔3 , 𝑔4 },
𝔗1
𝑎𝑝𝑟𝑁𝑆𝑅𝔗1 (𝔜2 ) = {𝑔1 , 𝑔3 , 𝑔5 },
𝑎𝑝𝑟𝑁𝑆𝑅 (𝔜2 ) = {𝑔1 , 𝑔2 , 𝑔3 , 𝑔5 },
𝔗1
𝑎𝑝𝑟𝑁𝑆𝑅𝔗1 (𝔜3 ) = {𝑔4 , 𝑔5 },
𝑎𝑝𝑟𝑁S𝑅 (𝔜3 ) = {𝑔1 , 𝑔2 , 𝑔3 , 𝑔4 , 𝑔5 }
𝔗1
Similarly we find 𝑎𝑝𝑟𝑁𝑆𝑅𝔗2 ,𝑎𝑝𝑟𝑁𝑆𝑅
𝔗2
and 𝑎𝑝𝑟𝑁𝑆𝑅𝔗3 , 𝑎𝑝𝑟𝑁𝑆𝑅
𝔗3
corresponding to each 𝔜𝑗 , where 𝑗 =
1,2,3.
𝑎𝑝𝑟𝑁𝑆𝑅𝔗2 (𝔜1 ) = {𝑔4 },
𝑎𝑝𝑟𝑁𝑆𝑅 (𝔜1 ) = {𝑔1 , 𝑔2 , 𝑔4 , 𝑔5 },
𝔗2
𝑎𝑝𝑟𝑁𝑆𝑅𝔗2 (𝔜2 ) = {𝑔1 , 𝑔3 , 𝑔5 },
𝑎𝑝𝑟𝑁𝑆𝑅 (𝔜2 ) = {𝑔1 , 𝑔2 , 𝑔3 , 𝑔5 },
𝔗2
𝑎𝑝𝑟𝑁S𝑅𝔗2 (𝔜3 ) = {𝑔4 , 𝑔5 },
𝑎𝑝𝑟𝑁𝑆𝑅 (𝔜3 ) = {𝑔1 , 𝑔2 , 𝑔4 , 𝑔5 }
𝔗2
and
𝑎𝑝𝑟𝑁𝑆𝑅𝔗3 (𝔜1 ) = {𝑔1 , 𝑔2 , 𝑔4 },
𝑎𝑝𝑟𝑁𝑆𝑅 (𝔜1 ) = {𝑔1 , 𝑔2 , 𝑔3 , 𝑔4 },
𝔗3
𝑎𝑝𝑟𝑁𝑆𝑅𝔗3 (𝔜2 ) = {𝑔1 , 𝑔3 , 𝑔5 },
𝑎𝑝𝑟𝑁𝑆𝑅 (𝔜2 ) = {𝑔1 , 𝑔3 , 𝑔4 , 𝑔5 },
𝔗3
𝑎𝑝𝑟𝑁𝑆𝑅𝔗3 (𝔜3 ) = {𝑔2 , 𝑔5 },
𝑎𝑝𝑟𝑁𝑆𝑅 (𝔜3 ) = {𝑔1 , 𝑔2 , 𝑔4 , 𝑔5 }
𝔗3
𝑆𝑡𝑒𝑝4: Neutrosophic soft lower approximation matrix and neutrosophic soft upper approximation
matrix are obtained as follows:
(1,1,0,1,0)
𝑎 = ((0,0,0,1,0)
(1,1,0,1,0)
(1,0,1,0,1)
(1,0,1,0,1)
(1,0,1,0,1)
(0,0,0,1,1)
(0,0,0,1,1))
(0,1,0,0,0)
(7)
(1,1,1,1,0)
𝑎 = ((1,1,0,1,1)
(1,1,1,1,0)
(1,1,1,0,1)
(1,1,1,0,1)
(1,0,1,1,1)
(1,1,1,1,1)
(1,1,0,1,1))
(1,1,0,1,1)
(8)
𝑆𝑡𝑒𝑝5: Using Eqs. 7 and 8, neutrosophic soft lower approximation vector and neutrosophic soft
upper approximation vector are obtained as follows:
𝑛 = (5,3,3,5,5)
𝑛 = (9,8,6,7,7)
𝑆𝑡𝑒𝑝6: Decision vector is obtained as 𝑛 ⊕ 𝑛 = (14,11,9,12,12).
𝑆𝑡𝑒𝑝7: Since 𝑚𝑎𝑥𝑖∈𝐼𝑛 𝑤𝑖 = 𝑤1 = 14, optimal clothing brand is 𝑔1 = 𝐽𝑜𝑟.
M. Riaz, F. Smarandache, F. Karaaslan M.R. Hashmi, I. Nawaz, Neutrosophic Soft Topology and its Applications to MultiCriteria Decision Making
Neutrosophic Sets and Systems, Vol. 35, 2020
211
5 NSR-topology in multi-criteria decision-making
In this section, we use the concept of NSR-topology in multi-criteria decision-making. The idea of
core in the picking of attributes to the rough set was introduced by Thivagar in [45]. In the following
definition, we develop this idea of core to the NSR-set.
Definition 5.1 Let 𝑈 be the set of objects, 𝐾 = (𝛷, 𝔄) is the neutrosophic soft set and 𝐺 = (𝑈, 𝐾) is
the the corresponding neutrosophic soft approximation space. Let ℜ be an indiscernibility relation.
Let 𝜏𝑁𝑆𝑅 be an NSR-topology on 𝑈 and 𝛽𝑁𝑆𝑅 be the basis defined for 𝜏𝑁𝑆𝑅 . Let 𝔑 be the subset of
𝔄, is said to be core of ℜ if 𝛽𝔑 ≠ 𝛽𝑁𝑆𝑅−(𝑠) for each '𝑠' in 𝔑. i.e. a core of ℜ is the subset of attributes
with the condition that if we remove any element from 𝔑 it will affect the classification power of the
attributes.
Algorithm 2 for neutrosophic soft rough topology:
Input
Step-1: Consider initial universe 𝑈, set of attributes 𝔄 which can be classified into division 𝔻 of
decision attributes, ℂ of condition attributes and an indiscernibility relation ℜ on 𝑈. Construct the
neutrosophic soft set in tabular form corresponding to ℂ condition attributes and a subset 𝔜 of 𝑈.
The columns indicate the elements of universe, rows represent the attributes and entries of table give
attribute values.
Output
Step-2: Classify set 𝔜 and find the NSR-approximation subsets (ℜ𝐺 (𝔜), ℜ𝐺 (𝔜)) 𝑎𝑛𝑑 𝐵𝐺 (𝔜)
w.r.t ℜ.
Step-3: Define Neutrosophic Soft Rough Topology 𝜏ℜ on 𝑈 and find basis 𝛽𝑁𝑆𝑅 .
Step-4: By removing an attribute 𝜉 from ℂ , find again the NSR-approximation subsets
(ℜ𝐺 (𝔜), ℜ𝐺 (𝔜)), 𝐵𝐺 (𝔜)) w.r.t ℜ𝑜𝑛ℂ − (𝜉).
Step-5: Generate 𝑁𝑆𝑅 − 𝑡𝑜𝑝𝑜𝑙𝑜𝑔𝑦 𝜏𝑁𝑆𝑅−(𝜉) on 𝑈,define its basis 𝛽𝑁𝑆𝑅−(𝜉) .
Step-6: Repeat step 4 and step 5 for each attribute in ℂ.
Step-7: The attributes for which 𝛽𝑁𝑆𝑅−(𝜉) ≠ 𝛽𝑁𝑆𝑅 gives the 𝑐𝑜𝑟𝑒(ℜ).
The flow chart diagram of proposed algorithm 2 is represented as Figure 2.
M. Riaz, F. Smarandache, F. Karaaslan M.R. Hashmi, I. Nawaz, Neutrosophic Soft Topology and its Applications to MultiCriteria Decision Making
Neutrosophic Sets and Systems, Vol. 35, 2020
212
Fig 2: The flow chart diagram of algorithm 2 for NSR-topology.
Example 5.2 Here we consider the problem of Crime rate in developing countries of Asia, Crime is
an unlawful act punishable by a state or other authority. In other words, we can say that a crime is
an act harmful not only to some individual but also to a community, society or the state. A developing
country is a country with a less developed industrial base and a low Human Development Index
(HDI) relative to other countries. Developing countries are facing so many issues including high
crime rate. This is the fundamental reason of emerging questions in our mind, that why the crime
rate is higher in developing countries?
We apply the concept of NSR-topology in Crime rate of developing countries of Asia.
Consider the following information table which shows data about 5 developing countries. The rows
of the table represent the objects(countries). Let 𝑈 = {𝑔1 = 𝐵𝑎𝑛𝑔𝑙𝑎𝑑𝑒𝑠ℎ, 𝑔2 = 𝐴𝑓𝑔ℎ𝑎𝑛𝑖𝑠𝑡𝑎𝑛, 𝑔3 =
𝑆𝑟𝑖𝐿𝑎𝑛𝑘𝑎, 𝑔4 = 𝑁𝑒𝑝𝑎𝑙, 𝑔5 = 𝑃𝑎𝑘𝑖𝑠𝑡𝑎𝑛} be the set of developing countries and 𝔄 = {𝜉1 , 𝜉2 , 𝜉3 , 𝜉4 } ,
where 𝜉1 stands for `corruption', 𝜉2 stands for `poverty ', 𝜉3 stands for `self actualization' and 𝜉4
stands for `lack of education'. Let 𝐾 = (𝛷, 𝔄) is the neutrosophic soft set over 𝑈 shown by Table
8,corresponding soft approximation space 𝐺 = (𝑈, 𝐾).
𝐾
𝜉1
𝜉2
𝜉3
𝜉4
Crime Rate
𝑔1
(0.6,0.6,0.2)
(0.8,0.4.0.3)
(0.7,0.4,0.3)
(0.8,0.6,0.4)
High
𝑔2
(0.4,0.6,0.6)
(0.6,0.2,0.4)
(0.6,0.4,0.3)
(0.7,0.6,0.6)
Medium
𝑔3
(0.6,0.4,0.2)
(0.8,0.1,0.3)
(0.7,0.2,0.5)
(0.7,0.6,0.4)
Medium
𝑔4
(0.6,0.3,0.3)
(0.8,0.2,0.2)
(0.5,0.2,0.6)
(0.7,0.5,0.6)
High
𝑔5
(0.8,0.2,0.3)
(0.8,0.3,0.2)
(0.7,0.3,0.4)
(0.9,0.5,0.7)
High
Table 8: Neutrosophic soft set 𝐾 = (𝛷, 𝔄)
The tabular representation of neutrosophic right neighborhoods of 𝐾 w.r.t set 𝔄 is given Table 9.
Neighborhoods of 𝐾
𝑔1 ]𝔄
{𝑔1 }
𝑔2 ]𝔄
{𝑔1 , 𝑔2 }
𝑔3 ]𝔄
{𝑔1 , 𝑔3 }
M. Riaz, F. Smarandache, F. Karaaslan M.R. Hashmi, I. Nawaz, Neutrosophic Soft Topology and its Applications to MultiCriteria Decision Making
Neutrosophic Sets and Systems, Vol. 35, 2020
213
𝑔4 ]𝔄
{𝑔4 }
𝑔5 ]𝔄
{𝑔5 }
Table 9: Neutrosophic right neighborhoods of 𝐾 w.r.t set 𝔄
For 𝔜 = {𝑔1 , 𝑔3 , 𝑔5 } and indiscernibility relation 'Crime rate' we have ℜ𝐺 (𝔜) = {𝑔1 , 𝑔3 , 𝑔5 } ,
ℜ𝐺 (𝔜) = {𝑔1 , 𝑔2 , 𝑔3 , 𝑔5 } and 𝐵𝐺 (𝔜) = {𝑔2 }.
So we define NSR-topology as 𝜏𝑁𝑆𝑅 (𝔜) = {𝑈, ∅, {𝑔1 , 𝑔3 , 𝑔5 }, {𝑔1 , 𝑔2 , 𝑔3 , 𝑔5 }, {g 2 }} and its basis 𝛽𝑁𝑆𝑅 =
{𝑈, {𝑔1 , 𝑔3 , 𝑔5 }, {𝑔2 }}.
If we remove the attribute `Corruption', then the tabular representation of neutrosophic
right neighborhoods of 𝐾 w.r.t set 𝔄 − 𝜉1 is given Table 10.
Neighborhoods of 𝐾
𝑔1 ]𝔄−𝜉1
{𝑔1 }
𝑔2 ]𝔄−𝜉1
{𝑔1 , 𝑔2 }
𝑔3 ]𝔄−𝜉1
{𝑔1 , 𝑔3 }
𝑔4 ]𝔄−𝜉1
{𝑔4 }
𝑔5 ]𝔄−𝜉1
{𝑔5 }
Table 10: Neutrosophic right neighborhoods of 𝐾 w.r.t set 𝔄 − 𝜉1
we have
𝜏𝑁𝑆𝑅−𝜉1 (𝔜) = {𝑈, ∅, {𝑔1 , 𝑔3 , 𝑔5 }, {𝑔1 , 𝑔2 , 𝑔3 , 𝑔5 }, {𝑔2 }}
is a NSR-topology and its basis is
𝛽𝑁𝑆𝑅 − 𝜉1 = {𝑈, {𝑔1 , 𝑔3 , 𝑔5 }, {𝑔2 }} = 𝛽𝑁𝑆𝑅 .
If we remove the attribute `poverty', then the tabular representation of neutrosophic right
neighborhoods of 𝐾 w.r.t set 𝔄 − 𝜉2 is given Table 11.
Neighborhoods of 𝐾
𝑔1 ]𝔄−𝜉2
{𝑔1 }
𝑔2 ]𝔄−𝜉2
{𝑔1 , 𝑔2 }
𝑔3 ]𝔄−𝜉2
{𝑔1 , 𝑔3 }
𝑔4 ]𝔄−𝜉2
{𝑔1 , 𝑔3 , 𝑔4 }
𝑔5 ]𝔄−𝜉2
{𝑔5 }
Table 11: Neutrosophic right neighborhoods of 𝐾 w.r.t set 𝔄 − 𝜉2
We have an NSR-topology and its base as follows:
𝜏𝑁𝑆𝑅−𝜉2 (𝑌) = {𝑈, ∅, {𝑔1 , 𝑔3 , 𝑔5 }, {𝑔2 , 𝑔4 }}
𝛽𝑁𝑆𝑅 − 𝜉2 = {𝑈, {𝑔1 , 𝑔3 , 𝑔5 }, {𝑔2 , 𝑔4 }} ≠ 𝛽𝑁𝑆𝑅 ,
and
respectively.
If we remove the attribute 'self actualization', then the tabular representation of
neutrosophic right neighborhoods of 𝐾 w.r.t set 𝔄 − 𝜉3 is given Table 12.
Neighborhoods of 𝐾
𝑔1 ]𝔄−𝜉3
{𝑔1 }
𝑔2 ]𝔄−𝜉3
{𝑔1 , 𝑔2 }
M. Riaz, F. Smarandache, F. Karaaslan M.R. Hashmi, I. Nawaz, Neutrosophic Soft Topology and its Applications to MultiCriteria Decision Making
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214
𝑔3 ]𝔄−𝜉3
{𝑔1 , 𝑔3 }
𝑔4 ]𝔄−𝜉3
{𝑔4 }
𝑔5 ]𝔄−𝜉3
{𝑔5 }
Table 12: Neutrosophic right neighborhoods of 𝐾 w.r.t set 𝔄 − 𝜉3
We have an NSR-topology and its base as follows:
𝜏𝑁𝑆𝑅−𝜉3 (𝑌) = {𝑈, ∅, {𝑔1 , 𝑔3 , 𝑔5 }, {𝑔1 , 𝑔2 , 𝑔3 , 𝑔5 }, {𝑔2 }}
and
𝛽𝑁𝑆𝑅 − 𝜉3 = {𝑈, ∅, {𝑔1 , 𝑔3 , 𝑔5 }, {𝑔2 } = 𝛽𝑁𝑆𝑅 },
respectively.
If we remove the attribute `lack of education', then the tabular representation of
neutrosophic right neighborhoods of 𝐾 w.r.t set 𝔄 − 𝜉4 is given Table 13.
Neighborhoods of 𝐾
𝑔1 ]𝔄−𝜉4
{𝑔1 }
𝑔2 ]𝔄−𝜉4
{𝑔1 , 𝑔2 }
𝑔3 ]𝔄−𝜉4
{𝑔1 , 𝑔3 }
𝑔4 ]𝔄−𝜉4
{𝑔4 }
𝑔5 ]𝔄−𝜉4
{𝑔5 }
Table 13: Neutrosophic right neighborhoods of 𝐾 w.r.t set 𝔄 − 𝜉4
We have an NSR-topology and its base as follows:
𝜏𝑁𝑆𝑅−𝜉4 (𝑌) = {𝑈, ∅, {𝑔1 , 𝑔3 , 𝑔5 }, {𝑔1 , 𝑔2 , 𝑔3 , 𝑔5 }, {𝑔2 }}
and
𝛽𝑁𝑆𝑅 − 𝜉4 = {𝑈, ∅, {𝑔1 , 𝑔3 , 𝑔5 }, {𝑔2 } = 𝛽𝑁𝑆𝑅 },
respectively. Thus, 𝐶𝑂𝑅𝐸(𝑁𝑆𝑅) = {𝜉2 }, i.e., `poverty' is the deciding attributes of the Crime Rate in
developing countries of Asia.
Discussion and comparitive analysis 5.3 In this section, we discuss our results obtained from both
numerical examples and present a comparative analysis of proposed topological space to some
existing topological spaces. Table 14 describes the comparison of both proposed algorithms based on
NSR-sets and NSR-topology. The algorithm 1 is used to find the optimal decision about the set of
alternatives and establish the ranking order between them. We can choose the best and worst
alternative from the given input information. While algorithm 2 is used to choose the most relevant
and significant attribute to which one can observe the specific characteristic of the alternatives. This
is called the CORE of the problem, which is an essential part of the decision-making difficulty. Both
algorithms have their own merits and can be used to solve decision-making problems in medical,
artificial intelligence, business, agriculture, engineering, etc.
Proposed Algorithms
Choice values
Algorithm 1 (NSR-sets)
𝑔1 ≻ 𝑔4 ≻ 𝑔5 ≻ 𝑔2 ≻ 𝑔3
𝑔1
𝐶𝑂𝑅𝐸(𝑁𝑆𝑅) = {𝜉2 }
𝜉2 = poverty
Algorithm 2 (NSR-topology)
Final Decision
Selection criteria
Based on alternatives
Based on attributes
Table 14: Comparison of prooposed algorithms
Now we present a soft comparative analysis of proposed approach with some existing approaches.
In Table 15, we describe the comparison and discuss about their advantages and limitations.
M. Riaz, F. Smarandache, F. Karaaslan M.R. Hashmi, I. Nawaz, Neutrosophic Soft Topology and its Applications to MultiCriteria Decision Making
Neutrosophic Sets and Systems, Vol. 35, 2020
Set theories
215
Informa-
Upper and
Parameter-
tion
lower
izations
about
approxi-
Indeter-
mations
minacy
with
part
boundary
Advantages
Limitations
region
Fuzzy sets [1]
No
No
No
Deal
with
the
hesitations.
Do not collect
any
information about the
indeterminacy of input
data.
Neutrosophic
Yes
No
No
sets [4, 5]
Deal with the data
Do not deal with the
having
roughness
indeterminacy
parameterizations.
and
information.
Rough sets
No
Yes
No
[2, 3]
Deal
with
the
Do
not
give
any
roughness of input
information about the
information
parameterizations.
and
create upper, lower
and
boundary
regions.
Soft sets [6]
No
No
Yes
Deal
with
the
uncertainity
Soft rough sets
No
Yes
Yes
[17]
with
Yes
Yes
No
not
provide
information about the
parameterizations.
roughness of data.
Deal
with
Do not give information
and
about the indeterminacy
uncertainities
Rough
Do
roughness of data.
part of problem.
Deal
Do not deal with the
with
the
neutrosophic
roughness
having
sets [47]
indeterminacy
parameterizations.
information.
Neutrosophic
Yes
Provide the data of
Effective
soft rough sets
indeterminacy
calculations
and
and
topology
(proposed)
Yes
Yes
roughness
part
remove
under
compared
but
heavy
as
to
above
existing theories.
parameterizations
without any loss of
information.
Table 15: Comparitive analysis of proposed approach with some exsting theories.
M. Riaz, F. Smarandache, F. Karaaslan M.R. Hashmi, I. Nawaz, Neutrosophic Soft Topology and its Applications to MultiCriteria Decision Making
Neutrosophic Sets and Systems, Vol. 35, 2020
216
6. Conclusion
Most of the issues in decision-making problems are associated with uncertain, imprecise and,
multipolar information, which cannot be tackled properly through the fuzzy set. So to overcome this
particular deficiency rough set was introduced by Pawlak, which deals with the vagueness of input
data. This research implies the novel approach of neutrosophic soft rough set (NSR-set) with
neutrosophic soft rough topology (NSR-topology). We presented various topological structures of
NSR-topology named as NSR-interior, NSR-closure, NSR-exterior, NSR-neighborhood, NSR-limit
point and, NSR-bases with numerous examples. We established two novel algorithms to deal with
multi-criteria decision-making (MCDM) problems under NSR-data. One is based on NSR-sets and
the other is based on NSR-topology with NSR-bases. This research is more efficient and flexible than
the other approaches. The proposed algorithms are simple and easy to understand which can be
applied easily on whatever type of alternatives and measures. Both algorithms are flexible and easily
altered according to the different situations, inputs and, outputs. In the future, we will extend our
work to solve the MCDM problems by using TOPSIS, AHP, VIKOR, ELECTRE family and,
PROMETHEE family using different hybrid structures of fuzzy and rough sets.
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Received: Apr 22, 2020. Accepted: July 12, 2020
M. Riaz, F. Smarandache, F. Karaaslan M.R. Hashmi, I. Nawaz, Neutrosophic Soft Topology and its Applications to MultiCriteria Decision Making
Neutrosophic Sets and Systems, Vol. 35, 2020
University of New Mexico
Neutrosophic Sets and Systems, Vol. 35, 2020
221
However, crisp graphs do not represent any system because the world is now full of imprecise data.
The idea of fuzziness was used first to define the fuzzy graph [2] by Kaufmann (1973).
Fuzzy graph [3] theory was developed by Rosenfeld (1975). In the same time, Yeh and Bang (1975)
introduced various connectedness concepts in fuzzy graphs [4].
Also,
of a path and
in a fuzzy graph [3] was introduced by Rosenfeld (1975). Hence Bhattacharya (1987)
introduced the idea of eccentricity and centre in the fuzzy graph [5] using
properties of
. Also, the
[6] were developed by Sunitha and Vijayakumar (1998). Bhutani and
Rosenfeld (2003) introduced the concepts of
in fuzzy graphs [7, 8] and eccentricity,
centre etc. [9] were also developed. There were further studies on
[10] by Linda and
Sunitha (2012).
Day to day, there were developments on fuzzy graphs. Akram (2011) introduced bipolar fuzzy
graphs [11] and the interval-valued fuzzy graph [12] were introduced by Akram and Dudek (2011).
Samanta and Pal (2013, 2015) introduced fuzzy k-competition graphs, p-competition graphs [13] and
also introduced fuzzy planar graph [14]. Tom and Sunitha (2015) introduced a new definition of the
length of a path and strong sum distance in fuzzy graphs [15]. There are many research works on
fuzzy graphs.
But in all these fuzzy graphs, edge membership value is less than its vertex
membership values. To remove this limitation, Samanta and Sarkar (2016) introduced a generalized
fuzzy graph [16].
As a generalization of fuzzy set and intuitionistic set theory, Smarandache (1998) introduced the
concepts of neutrosophic set [17] that consist of a degree of truth membership, falsity membership
and indeterminacy membership. In reality, every uncertainty has some possibility, some risk and
some neutral factors. Neutrosophic graphs include all three notions properly. Thus any uncertainty/
ambiguity of networks can be represented by neutrosophic graphs. Broumi et al. (2016) introduced
the notion of a single-valued neutrosophic graph [18] as a generalization of fuzzy graphs. After that,
there are several research works on neutrosophic graphs [19,20]. Akram and Siddique (2017)
introduced the neutrosophic competition graphs [21]. Hence Das et al. (2020) proposed generalized
neutrosophic competition graphs [22] with applications to economic competitions among some
countries.
Abdel-Basset (2019) utilized the neutrosophic theory to solve the transition difficulties of IoT-based
enterprises [23]. Also, there are many real-life applications including evaluation of the green supply
chain management practices [24], evaluation Hospital medical care systems based on pathogenic sets
[25], decision-making approach with quality function deployment for selecting supply chain
sustainability metrics [26], intelligent medical decision support model based on soft computing and
IoT [27]. Chakraborty (2020) introduced pentagonal neutrosophic number in shortest path problem
[28] and a new score function of the pentagonal neutrosophic number and its application in
networking problem [29]. Das and Edalatpanah (2020) proposed a new ranking function of the
triangular neutrosophic number and its use in integer programming [30]. The remaining study can be
found in [31-40].
The rest of the paper is organized as follows. In Section 2, we discuss the contribution of the study. In
section 3, we study some preliminaries related to graph theory. In Section 4, we introduce the sum
distance in a neutrosophic graph with some properties. In Section 5, we introduce eccentricity, radius
K. Das, S. Samanta, S.K. Khan, U. Naseem and K. De; A Study on Discrete Mathematics: Sum Distance in
Neutrosophic Graphs with application.
Neutrosophic Sets and Systems, Vol. 35, 2020
222
and diameter in a neutrosophic graph with properties. In Section 6, we discuss an application to a
travelling salesman problem. In section 7, we conclude the study with future directions.
The gist of contributions of authors (Table 1) are arranged below.
Authors
Year
Rosenfeld
1975
Contributions
Introduce
of
a
path
and
in a fuzzy graph.
Bhattacharya
1987
Introduce eccentricity and centre in the fuzzy
graph.
Bhutani and Rosenfeld
2003
in fuzzy graphs and developed
eccentricity, centre etc.
Linda and Sunitha
2011
Studied on
in fuzzy graphs.
Tom and Sunitha
2015
Introduce length of a path and strong sum
distance in fuzzy graphs.
Das et al.
This
Introduce sum distance in neutrosophic graph
paper
and eccentricity, radius etc. are studied. An
application is illustrated.
Table 1. Contributions of authors
2.
Major contributions of the study
The neutrosophic graph is a generalization of the fuzzy graph. The contributions of the study are
below.
This study introduces the concepts of the weight of edges of a neutrosophic graph and
weighted sum distance in neutrosophic graph.
Also the eccentricity, diameter and radius are defined with some properties.
At last, an application of sum distance in the neutrosophic graph to a travelling salesman
problem is illustrated.
3.
Preliminaries
Definition 1. A graph is an ordered pair (
set of edges between vertices. A path of length
are distinct vertices and
) such that
is the set of vertices and
is a sequence
where
are distinct edges. The distance between the vertices
minimum length of the path between
is the
,
and
is the
and . The eccentricity of a vertex is the maximum distance to
any vertex in the graph. The radius of a graph is the minimum eccentricities of all vertices, and the
diameter of a graph is the maximum eccentricities of vertices.
K. Das, S. Samanta, S.K. Khan, U. Naseem and K. De; A Study on Discrete Mathematics: Sum Distance in
Neutrosophic Graphs with application.
Neutrosophic Sets and Systems, Vol. 35, 2020
is a triplet (
Definition 2.[3] A fuzzy graph
,
(
223
- such that (
)
( )
) in which
∑
(
in a connected fuzzy graph
( ) where ( ) represents membership values of edges
Definition 4.[15] The strong sum distance between vertices
paths between vertices
and
)
Definition 3.[15] Length ( ) of a path
( )
- and
( ) where ( ) represents the membership value of
) represents the membership value of edge (
given by
,
is the set of vertices,
) is
.
is the minimum length of all
.
Figure 1. Example of a fuzzy graph
Example 1. The fuzzy graph (Fig.1) has four vertices with five edges. There are three paths from
vertex
to vertex
( )
( )
. The paths are
( )
Definition 5.[18] A graph
i)
(V, ) where
,
there exist functions
( )
( )
-
,
( )
,
(
there exist functions
(
)
and
is
-
( )
,
- such that
(
for all
)
( ) denote the degree of true membership, degree of falsity membership and
and
where
. Then
is said to be neutrosophic graph if
degree of indeterminacy membership of the vertex
ii)
,
the strong sum distance between vertices
( )
where
,
(
)
-
respectively.
)
,
,
(
)
[
( )
( )-
(
)
[ ( )
( )-
(
)
(
( )
,
( )-
)
- such that
(
)
for all (
(
) denote the degree of true membership, degree of falsity
membership and degree of indeterminacy membership of the edge (
)
)
respectively.
K. Das, S. Samanta, S.K. Khan, U. Naseem and K. De; A Study on Discrete Mathematics: Sum Distance in
Neutrosophic Graphs with application.
Neutrosophic Sets and Systems, Vol. 35, 2020
224
Figure 2. A neutrosophic graph
4.
Weighted sum distance in the neutrosophic graph
In the neutrosophic graph, membership values of edges are in neutrosophic nature. So we cannot
compare among edges in a neutrosophic graph. To overcome it, we define weight function that maps
from the membership value of edges to a crisp value lies between 0 and 1.
,
Definition 6. Consider a function
(
)
(
)
-
,
-
,
-
,
- defined by
where
,
are the numbers
-
) in a neutrosophic graph is a number between 0 and 1 which is obtained
The weight of an edge (
from the image of the function
of the edge and it is denoted by
for corresponding membership value .
(
)
(
)
(
)/
.
Note: This function indicates the overall impression of true , falsity and indeterminacy values.
must be higher
Suppose, in one network, generally predictions are always true of some facts. Then
value and close to 1. Similarly for the other cases.
Example 2. Weight
of edge (
) in the neutrosophic graph (Fig.2)
where
and
.
Definition 7. Let
the length of the path
be any path in a neutrosophic graph
(V, ). Then
is the sum of the weights of the edges of the path in
(V, ).
( )
where
Example 3. Length of the path
and
∑
,
is the weight of edge between vertices
and
.
in the neutrosophic graph (Fig.2) is
where
.
Definition 8. Let
(V, ) be a neutrosophic graph and P be the collection of all paths between two
+. Then the weighted distance between the nodes
nodes
P =*
is denoted by (
) and is defined by
K. Das, S. Samanta, S.K. Khan, U. Naseem and K. De; A Study on Discrete Mathematics: Sum Distance in
Neutrosophic Graphs with application.
Neutrosophic Sets and Systems, Vol. 35, 2020
where
225
(
)
( ) is the length of the path
*
.
+,
( )
Example 4. Sum distance between the nodes
.
is
Theorem 1. Let
any two nodes
(
i)
(
ii)
(
iii)
(
iv)
and
(
(V, ) be a neutrosophic graph and
. Then
)
)
)
(
)
)
(
)
(
).
(
Proof. (i) It clears from the definition that
(ii) It clears from the definition that
(
)
in the neutrosophic graph (Fig.2)
) be weighted sum distance between
.
)
.
)denotes the strong sum distance from to Then there exists a path whose length is
(iii) (
minimum among all the path between to . Hence the length should be the same from to . So
(
)
(
).
( )
(
) and be a path
(iv) Let be a path
such that
(
). Then
is a walk and it is a strong path whose length is at most
)
(
)
(
).
Thus (
5.
(
such that
)
(
( )
).
Eccentricity, Radius and Diameter
The parameters eccentricity, radius and diameter are crucial in graph theory. We studied these
important parameters in neutrosophic graph considering the concepts of sum distance. The relations
among radius, diameters, eccentricity and distance are studied as follows.
Definition 9. The eccentricity
neutrosophic graph . Thus
( ) of a node,
( )
Example 5.
is the distance from
*
(
)
to the furthest node in the
+
Consider a neutrosophic graph (Fig.3). The eccentricity
calculated by the following:
( )
*
(
*
)
(
)
(
( ) of the vertex
)+
+
K. Das, S. Samanta, S.K. Khan, U. Naseem and K. De; A Study on Discrete Mathematics: Sum Distance in
Neutrosophic Graphs with application.
is
Neutrosophic Sets and Systems, Vol. 35, 2020
226
Fig. 3. An example of a neutrosophic graph
Theorem 2. Let
| ( )
( )|
(V, ) be a connected neutrosophic graph and
(
).
be any two nodes of
Then
( ) and
Proof. Let
be two nodes such that ( )
be anode such that ( )
(
) . Then
(
)
(
)
(
) , by theorem 3.7 (iv). Also
(
)
( ) . Thus
( )
(
)
(
)
( ), this implies
( )
( )
(
) Similarly, if we take
( )
( ), we will get
(
)
( )
( ) Thus | ( )
( )|
(
)
Definition 10. The radius
nodes. Thus
( ) of a neutrosophic graph,
( )
*
is the minimum among all eccentricity of
( )
+.
( ) of the graph
Example 6. Consider the neutrosophic graph (Fig. 3). The radius
the following:
( )
* ( ) ( ) ( )
*
+
Definition 11. The diameter
of nodes. Thus
is calculated by
( )+
( ) of a neutrosophic graph, is the maximum among all eccentricity
( )
*
( )
+.
Example 7. Consider the neutrosophic graph (Fig.3). The diameter ( ) of graph
the following:
( )
* ( ) ( ) ( ) ( )+
*
+
is calculated by
Definition 12. A node in a neutrosophic graph is called a central node if its eccentricity is equal to the
radius of the graph. Thus for a central node
( )
( ).
Example 8. Consider the neutrosophic graph (Fig. 3). The node
the eccentricity ( )
is a central node, since
( )
K. Das, S. Samanta, S.K. Khan, U. Naseem and K. De; A Study on Discrete Mathematics: Sum Distance in
Neutrosophic Graphs with application.
Neutrosophic Sets and Systems, Vol. 35, 2020
227
Definition 13. A node in a neutrosophic graph is called a peripheral node if its eccentricity is equal to
the diameter of the graph. Thus for a peripheral node
( )
( ).
Example 9. Consider the neutrosophic graph (Fig. 3). The nodes and
( )
( ).
the eccentricity ( )
Theorem 3. Let
(V, ) be a connected neutrosophic graph with
( )
( )
diameter respectively, then ( )
Proof. From the definition, it follows that ( )
( ) and
i.e. ( )
be peripheral node i.e.
(
) , by theorem (iv). This implies
( )
( )
( )
Therefore, ( )
6.
( ) Let
( )
( )
( )
( )
are peripheral nodes, since
( ) and
( ) be the radius and
such that be central node
( ). Now (
)
(
)
( ) Thus
( )
( )
Application to travelling salesman problem
Suppose there are few places in a city and roads connect the places. Hence the places and roads
together form a network. But the problem is to find a way that a salesman can visit all the planes once
with the lowest travelling cost. Now the travelling cost is directly proportional to the road distance
travel by salesman. But all the roads are not in the same smooth conditions to measure road distance
in practical. So the real travelling distance with cost may be effected the bad road, non-pucca roads,
water path etc. Thus to calculate the path distance, it is generally ignored the current condition of the
paths. The true value indicates the expected distance on good road. The falcity indicates the current
false parameter like general traffic on the routes, muddied on road. Indeterminacy includes delay due
to road construction, political movement and any other factors like water path. Therefore Travelling
salesman problem should be presented by neutrosophic environment. Hence the travelling distance
between the places should be taken as neutrosophic value.
6.1. Steps to find the sum distance of the travelling salesman problem.
To find the minimum travelling cost in travelling salesman problem in the neutrosophic environment,
all the necessary steps are given below as an algorithm.
Step-1: Input all edge membership values between the places.
Step-2: Evaluate the weight of edges.
Step-3: Find all the Hamiltonian cycles between the requird places.
Step-4: Evaluate length of the said cycles.
Step-5: Find the minimum length among the cycles.
6.2. Numerical example
K. Das, S. Samanta, S.K. Khan, U. Naseem and K. De; A Study on Discrete Mathematics: Sum Distance in
Neutrosophic Graphs with application.
Neutrosophic Sets and Systems, Vol. 35, 2020
228
Suppose there are four places and six roads are connecting the places in a city. A salesman wants to
visit all the places once and returning back to the starting place. The problem is to find a cycle with
minimum cost of travelling.
The edge membership values (Table 2) between the places are given in the figure 4 where the
membership value (
) between two places and represent that distance of good road
between
and
is
, distance of bad road between
and
is
and distance of nonconstructed road between and is
and similar for the other values.
Places
Distance between places
(
)
(
)
(
)
(
)
(
)
(
)
Table 2. Distace between two places
Figure 4: A graph among four cities.
The weight
between the places
(
(
(
and
)
)
)
are given below:
,
0.44,
,
(
(
(
) 0.24,
)
,
)
.
There are four possible cycles to visit all the places once from starting point to that point. These
cycles are:
K. Das, S. Samanta, S.K. Khan, U. Naseem and K. De; A Study on Discrete Mathematics: Sum Distance in
Neutrosophic Graphs with application.
Neutrosophic Sets and Systems, Vol. 35, 2020
The length
229
( ) travelled by the salesman for the above cycles
( )
,
( )
,
Since the value 1.48 is minimum length for the cycles
( )
and
are:
,
( )
, hence these cycles give the minimum
travelling cost to the salesman.
7.
Conclusions
In this article, sum distance, eccentricity, radius etc. in a neutrosophic graph has been developed.
Some definitions, examples and theorems give a clear idea about the proposed study. A neutrosophic
graph is recently a very important topic. There are many scopes to research on that topic. One can
develop this study to the generalized neutrosophic graph. The real application inthe travelling
salesman problem has been illustrated with a numerical example. This idea also gives us to develop
future research in neutrosophic graphs.
Funding: This research received no external funding.
Conflicts of Interest: The authors declare no conflict of interest.
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practices: A novel neutrosophic approach. Computers in Industry, 108, 210-220, 2019.
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Hospital medical care systems based on plithogenic sets. Artificial intelligence in medicine, 100,
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decision-making approach with quality function deployment for selecting supply chain
sustainability metrics. Symmetry, 11(7), 903, 2019.
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Support Model Based on Soft Computing and IoT. IEEE Internet of Things Journal, 2019.
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Received: Apr 23, 2020. Accepted: July 13, 2020
K. Das, S. Samanta, S.K. Khan, U. Naseem and K. De; A Study on Discrete Mathematics: Sum Distance in
Neutrosophic Graphs with application.
Neutrosophic Sets and Systems, Vol. 35, 2020
University of New Mexico
Exploration of the Factors Causing Autoimmune Diseases using
Fuzzy Cognitive Maps with Concentric Neutrosophic
Hypergraphic Approach
Nivetha Martin1, Florentine Smarandache2, I.Pradeepa3, N.Ramila Gandhi4, P.Pandiammal5
1,3
Department of Mathematics, Arul Anandar College(Autonomous), Karumathur, India, nivetha.martin710@gmail.com
2
Department of Mathematics, University of New Mexico, USA, smarand@unm.edu
4
5
Department of Mathematics, PSNA CET,Dindigul, India,satrami@gmail.com
Department of Mathematics, GTN Arts College (Autonomous), Dindigul, India,pandiammal1981@gmail.com
*Correspondence: nivetha.martin710@gmail.com
Abstract: Neutrosophic sets are comprehensively used in decision making environment. The
manifestation of neutrosophic sets in concentric hypergraphs is proposed in this research work.
The intention of developing a decision making model using the combination of Fuzzy Cognitive
Maps and concentric neutrosophic hypergraph is to rank the core factors of decision making
problem and find the inter relational impacts. This proposed model is validated with the
exploration of the causative factors of autoimmune diseases. The proposed model is highly
compatible as it assists in determining the core factors and their inter association. This model will
certainly benefit the decision maker at all managerial levels to design optimal decisions.
Keywords: Autoimmune disease, fuzzy cognitive maps, neutrosophic hypergraphs, optimal decision making
1. Introduction
Westernization the cause of modernization has unlocked the portals of cultural, behavioural
and environmental changes of the people which greatly influence the biological system of human
and this also lays the core reason for the outbreak of novel diseases. Presently the people of the
world are characterized by multicultural and multi technological adoption. The integration and the
association between people of varied culture have brought diverse implications on the external and
internal environment of the human. Not just the social interactions contribute to such modifications;
also the technological advancement and the work space of an individual cause a varied range of
changes in the mankind. The tendency of manhood repelling from indigenous practices is the
gateway for several health woes. The health system of the human is getting affected by several
factors and especially the vulnerable target group is the women. In recent days, the people are
chained by diseases of various kinds, even the economy of the nation face huge falls due to the effect
of epidemic diseases, amidst such miserable situations, the immunity of the human is the only
armed force against these viruses, but if the immune system fails to be defensive in nature and if it
joins hand with the external invaders the entire human health system collapses and it ends in
fatality. This is the characterization of auto immune diseases and the women are greatly affected by
these diseases. It is highly a dreadful circumstance to tackle the consequences of these selfdestructing diseases. The autoimmune diseases predominantly affecting the women are Rheumatoid
Arthritis, Multiple Sclerosis, Systemic lupus Erythematosus, Grave’s disease, Hashimoto’s
thyroiditis and Myasthenia gravis. Presently the rate of occurrences of such diseases is at its pinnacle
and the medical experts are investigating the ways and means of its mitigation. [1]
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Generally the women are highly susceptible to these autoimmune diseases as the immune system
gets weakened during pre and post pregnancy stages. This scenario has gained the medical concerns
and medical researchers are on their study, to render support to it, this paper aims to underlie the
core factors contributing to autoimmune diseases in women and to find the inter association
between the core factors. Optimal decisions can be made by applying scientific methods in the
process of decision making process. The entire scenario of decision making must be modeled based
on decisive factors of the study. One of the realistic tools of decision making is fuzzy cognitive maps
(FCM), introduced by Kosko [2], later several academicians extended this FCM tool based on the
requirements. FCM is a directed graph representing the casual relationship between factors
considered for study. The nodes and the edges of the graph represent the study factors and their
association. The weights [-1,1] represent the nature of the association. The integration of FCM with
other graphic structures was initiated by Nivetha and Pradeepa [3]. The hypergraphic and fuzzy
hypergraphic approaches with FCM unlocked the construction of concentric fuzzy hypergraphs and
its integration with FCM [4,5]. This field of integrated FCM with fuzzy hypergraphs has made the
researchers explore by introducing various types of concentric fuzzy hypergraphs.
In this research work, a fuzzy cognitive map with concentric neutrosophic hypergraphic approach is
introduced. The notion of neutrosophic fuzzy sets and neutrosophic logic was first coined by
Smarandache [6] and presently many researchers are highly interested to carry out their research in
this field, the concepts of neutrosophic is applied in almost all types of decision making tools.
Neutrosophic sets, play significant role in making decisions in uncertain environment as it provides
space for the pragmatic representation of the expert’s opinion. Abdel Basset et al [7]developed a
decision making model for evaluating the framework for smart disaster response system in an
uncertain environment, neutrosophic sets are used for uncertainty assessments of linear time-cost
tradeoffs [8]; resource levelling problem[9] in construction project was modeled under neutrosophic
environment. The concept of neutrosophic sets was extended to bipolar neutrosophic representation
[10] and it is used in multi criteria decision making framework for professional selection. Das et al
[11] developed neutrosophic fuzzy matrices and algebraic operation that had some utility in
decision making. Plithogenic sets, the extension of neutrosophic sets are used in solving supply
chain problem with the development of a novel plithogenic model [12]. Such massive applications of
neutrosophic sets in decision making and its robust nature triggered the idea of integrating
neutrosophic sets to concentric hypergraphs. To the best of our knowledge, the integration of
neutrosophic concentric fuzzy hypergraphs with FCM has not been instituted and so this is a new
arena of research towards optimal decision making.
Fuzzy Cognitive Maps are more useful in determining the association between study factors, if
the number of study factors is less, FCM’s are highly compatible, but if the number of factors is
more, then comparative analysis between the factors is difficult and tedious, to resolve such crisis,
the core factors of the problem are to be decided and then the inter association between the core
factors can be determined easily. To find the core factors, the intervention of various experts is
mandatory, based on which the factors can be ranked and the core factors are decided based on the
rank positions of the factors. This eases the process of making decisions as it helps in filtering the
non- core factors. Generally in medicinal environment, the medical experts analyze the factors
contributing to diseases, initially the causative factors taken for study will be more in number, but
the factors have to drop at each stage of their research to find the prime causative factors. In the
process of factor filtration, the expert’s opinions play a vital role. The role of each causative factor of
a disease cannot be certainly express but representation using neutrosophic sets makes it possible
and more meaningful. Thus the integration of FCM with concentric neutrosophic hypergraph will
help to tackle the difficulties in handling large number of study factors.
Nivetha et al Exploration of the Factors Causing Autoimmune Diseases using Fuzzy Cognitive Maps with Concentric
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The paper is structured as follows: section 2consists of the methodology in which the algorithm of
finding optimal decision is presented; section 3 comprises of the adaptation of the proposed model
to the decision making problem; section 4 discusses the results and the last section summarizes the
research work.
2. Methodology and its application
The steps in making optimal decisions is presented as an algorithm as follows,
Step 1: The expert’s opinion of the study factors are represented by concentric fuzzy hypergraphs
with neutrosophic fuzzy sets representations of the envelope.
Step 2: The score values of the neutrosophic fuzzy sets are determined.
Step 3: The factors are ranked based on the score values.
Step 4: The core factors are determined based on the ranking positions.
Step5: The inter association between the core factors is obtained based on the conventional FCM
procedure.
The case histories of patients belonging to women gender suffering from autoimmune diseases are
taken as the source of data collection and the factors contributing to the occurrence of auto immune
disease in women [13] are presented below based on the medical expert’s opinion and data obtained
from questionnaire.
F1. Excess presence of VGLL3 (Vestigial Like Family Member 3) in skin cells
F2. Changes in the gene system
F3. Exposure to ultraviolet radiation from sunlight
F4. Acquaintance with organic mercury
F5. Alteration in food habits
F6. Gene-Environment interface
F7. Fluctuations in sex hormones
F8. Modifications in Nutritional diet
F9. Post pregnancy impacts
F10. Genetic vulnerability
F11. Genetic differences in immunity
Fig.3.1.Concentric Neutrosophic Fuzzy Hypergraphic representation
Nivetha et al Exploration of the Factors Causing Autoimmune Diseases using Fuzzy Cognitive Maps with Concentric
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The concentric neutrosophic fuzzy hyper envelopes with neutrosophic representations of the
expert’s opinion are presented below in Table 3.1.
Table 3.1 Representations of Expert’s opinion
Experts
F1
F2
F3
F4
F5
F6
F7
F8
F9
F10
F11
E1
(0.3,0.2,
0.8)
(0.5,0.2,
0.3)
(0.4,0.1,
0.5)
(0.3,0.4,
0.6)
(0.8,0.1,
0.2)
(0.7,0.2,
0.3)
(0.7,0.3,
0.4)
(0.7,0.2,
0.3)
(0.3,0.2,
0.8)
(0.5,0.2,
0.3)
(0.5,0.2,
0.3)
E2
(0.2,0.2,
0.9)
(0.4,0.3,
0.5)
(0.5,0.2,
0.3)
(0.2,0.2,
0.9)
(0.7,0.2,
0.3)
(0.6,0.2,
0.3)
(0.7,0.5,
0.4)
(0.6,0.2,
0.3)
(0.4,0.3,
0.5)
(0.6,0.2,
0.3)
(0.8,0.3,
0.2)
E3
(0.3,0.4,
0.6)
(0.3,0.5,
0.6)
(0.4,0.3,
0.5)
(0.3,0.2,
0.8)
(0.8,0.3,
0.2)
(0.9,0.2,
0.3)
(0.9,0.1,
0.3)
(0.6,0.2,
0.3)
(0.3,0.5,
0.6)
(0.7,0.3,
0.4)
(0.6,0.2,
0.3)
E4
(0.5,0.2,
0.3)
(0.2,0.2,
0.9)
(0.5,0.2,
0.3)
(0.4,0.4,
0.6)
(0.7,0.1,
0.2)
(0.7,0.3,
0.4)
(0.6,0.2,
0.3)
(0.7,0.1,
0.2)
(0.2,0.2,
0.9)
(0.6,0.2,
0.3)
(0.4,0.3,
0.5)
E5
(0.2,0.5
,0.6)
(0.3,0.2,
0.8)
(0.6,0.2,
0.3)
(0.5,0.2,
0.3)
(0.6,0.2,
0.3)
(0.8,0.1,
0.2)
(0.6,0.2,
0.3)
(0.9,0.2,
0.3)
(0.4,0.4,
0.6)
(0.5,0.2,
0.3)
(0.7,0.3,
0.4)
The score values of the factors are presented in Table 3.2 and it is represented graphically in Fig.3.2
Table 3.2 Score values of the Factors
F1
F2
F3
F4
F5
F6
F7
F8
F9
F10
F11
0.571
0.571
0.546
0.538
0.667
0.783
0.756
0.573
0.445
0.636
0.667
7
7
8
9
5
1
2
6
10
3
4
Ranking of the Factors
1
0.8
0.6
0.4
0.2
0
F1
F2
F3
Fig.3.2
F4
F5
F6
F7
F8
F9
Based on the scores, the following factors are considered as the core factors and their inter
association is expressed as linguistic variables, which then later quantified by heptagonal fuzzy
numbers.
HP1. Alteration in food habits
HP2. Gene-Environment interface
Nivetha et al Exploration of the Factors Causing Autoimmune Diseases using Fuzzy Cognitive Maps with Concentric
Neutrosophic Hypergraphic Approach
F10
F11
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236
HP3. Fluctuations in sex hormones
HP4. Genetic vulnerability
HP5. Genetic differences in immunity
The connection matric between the factors, based on the expert’s opinion
HP1
HP2
HP3
HP4
HP5
HP1
0
M
H
L
L
HP2
L
0
M
H
H
HP3
L
M
0
M
L
HP4
L
M
H
0
M
HP5
L
M
M
H
0
The modified matrix based on the values of quantification in Table 3.3
Linguistic
Heptagonal Weight
Membership
Variable
value
Low
(0,0.1,0.2,0.3,0.35,0.4,0.45)
0.26
Medium
(0.4,0.45,0.5,0.55,0.6,0.65,0.7)
0.55
High
(0.65,0.7,0.8,0.9,1,1,1)
0.86
HP1
HP2
HP3
HP4
HP5
HP1
0
0.55
0.86
0.26
0.26
HP2
0.26
0
0.55
0.86
0.86
HP3
0.26
0.55
0
0.55
0.26
HP4
0.26
0.55
0.86
0
0.55
HP5
0.26
0.55
0.55
0.86
0
The interrelationship between the factors is determined by the similar application of FCM
methodology [9-10] and it is presented graphically in Fig 3.2
HP2
HP5
1
HP1
HP3
HP4
Fig.3.2 FCM representation of the inter association of the core factors
Nivetha et al Exploration of the Factors Causing Autoimmune Diseases using Fuzzy Cognitive Maps with Concentric
Neutrosophic Hypergraphic Approach
Neutrosophic Sets and Systems, Vol. 35, 2020
4.
237
Results and Discussion
Fig. 3.2 clearly states the factor, fluctuations in sex hormone is the core causative factor of auto
immune diseases. The findings of this research will certainly assist the medical experts to ascertain
the causes of the auto immune disease in women and give treatment in accordance to it. Hormonal
imbalance is quite common in the life of the women as they undergo various stages of puberty,
maternity, menopause, but still proper medications has to be given to avoid the risks of such fatal
diseases. The representation of the imprecise data in the form neutrosophic sets is the pragmatic
reflection of the expert’s opinion, as the factors contributing to the diseases are quite uncertain. The
degree of truth values, indeterminacy and false values are indeed very essential in making optimal
decisions.
5.
Conclusion
The proposed decision making tool with the integration of FCM and concentric neutrosophic fuzzy
hypergraphs is a highly feasible tool to obtain optimal decisions. The difficulty in handling several
factors in FCM is reduced and this integrated approach facilitate the determination of inter
association between the factors. This method of decision making can be extended to other kinds of
concentric fuzzy hypergraphs with various representations. Plithogenic sets representation is the
future extension of this proposed research work.
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Received: Apr 24, 2020. Accepted: July 14, 2020
Nivetha et al Exploration of the Factors Causing Autoimmune Diseases using Fuzzy Cognitive Maps with Concentric
Neutrosophic Hypergraphic Approach
Neutrosophic Sets and Systems, Vol. 35, 2020
University of New Mexico
Neutrosophy Logic and its Classification: An Overview
Aiman Muzaffar 1, Md Tabrez Nafis 2 and Shahab Saquib Sohail 3,*
1,2,3 Department
of Computer Science and Engineering, SEST, Jamia Hamdard, New Delhi, India.
1aimanmuzaffar14@gmail.com , 2tabrez.nafis@gmail.com
*Correspondence: shahabssohail@jamiahamdard.ac.in
Abstract: Over the past few years, neutrosophy has gained an exponential growth and has attracted
a good number of researchers especially those who focus on soft computing based uncertainty
computation. This paper presents the various techniques in neutrosophy. The various techniques
are discussed lucidly which help a naïve researcher in this field to understand the on-going
researches and establish a strong base. We have summarized the previous work carried out in the
field of neutrosophic logic, set, measure, and also classification techniques in neutrosophy and the
relevant research work has been discussed. Further, various applications in the field of neutrosophy
are elaborated. The major contributions of the existing research in neutrosophy is reviewed and
presented from different perspectives. The development of newer algorithms for solving the
problems of neutrosophy will provide impetus to the existing research in this field.
Keywords: Neutrosophy, indeterminacy, neutrosophic logic
1. Introduction
Neutrosophy, having emerged as a generalization to fuzzy logic is being used in the research area
in a number of fields like logics, set theory and others. Florentin Smarandache, in 1980, introduced
this new field of philosophy which deals with the uncertainties and indeterminacy in the data. He
defines neutrosophy as the science which deals with neutralities. This field takes into consideration
the dawn, kind and scope of such neutralities and how they interact with various ideational spectra.
The fundamentals of the study of the logic of neutrosophy, probability in neutrosophy, sets in
neutrosophy and the statistics is given by neutrosophy. Various researchers have incorporated the
idea of Neutrosophic Logic (NL), Neutrosophic Cognitive Maps (NCM) and other technologies in
areas such as Information system application, IT, Decision Support System Application, Physics,
Healthcare, Social Sciences etc. In 2019, F. Smarandache, introduced the concept of
Neutrosociology[1], which is the amalgamation of sociology and neutrosophic methods. In [3], an
improved method using clustering using k-means was incorporated for performing image
segmentation using neutrosophic logic. In [4], the authors presented a way of correcting the
uncertainties that arise in discursive analysis by applying Neutrosophy Theory in relation with
sentiment analysis. In [5], the authors gave a framework to see how mental models could be
analyzed using neutrosophic logic. In [6], [9], [10], [11], [15], [16] and [17], Neutrosophy was used
to deal with the uncertainties and indeterminacy in situation analysis. In [25], the evaluation of the
smart disaster response systems in times of ambiguity has been done using a framework. The
A muzaffar, T Nafis, S S Sohail, Neutrosophy Logic and its Classification: An Overview
240
degrees of contradiction in the evaluation criteria have been addressed with the help of plithogenic
set theory which checks the uncertainty environment. In [26], to tackle time scheduling in projects,
a framework has been given to minimize the cost of projects in environments which are ambiguous.
Neutrosophic theory has been used to consider the dynamic features of all parameters. In [29], a
resource levelling model to minimize the costs of daily resource fluctuations is given, using
neutrosophic set, with the aim of tackling the issues of uncertainty in the problems of the real
world. In [30], the authors have given a framework for professional selection by making use of
neutrosophic multi-criteria decision making, in an attempt to check the vagueness and ambiguity
in the selection process. In [31], a case study of Thailand’s sugar industry has been done to validate
the model proposed, using the plithogenic decision making perspective for evaluating supply
chain sustainability. In this paper, we have reviewed the neutrosophic technologies that have been
incorporated in various researches all over the world. The figure 1 depicts the workflow .
Figure 1. Block diagram for the process of the research carried out in the manuscript
2. Background Study
Florentin Smarandache [2019] in his book, Introduction to Neutrosophic Sociology
(Neutrosociology) discussed Sociological Forecasting, Neutrosophic Social norms and situations
which cannot be solved in the classical way. He discussed neutrosophic Grand Theories to find
abstract ideas about concrete facts in large social groups. He has also discussed Neutrosophic Big
Data, IoT and Neutrosophic Microsociology in this book. [1]
Aasim Zafar, Mohd Anas Wajid [2019] used the concept of neutrosophy to study the reasons of
criminal behavior in Nigeria. They found that out of various factor taken by the researchers, some
were excluded because they were found to be indeterminate. To show how such factors did actually
contribute to the criminal behavior, they modelled the situation mathematically using FCM’s and
NCM’s, where the former stands for Fuzzy cognitive Maps and the latter stands for Neutrosophic
A muzaffar, T Nafis, S S Sohail, Neutrosophy Logic and its Classification: An Overview
241
Cognitive Maps. They further conclude how NCM is more effective than FCM in dealing with
uncertainties and indeterminacy in situation analysis. They further concluded that if the
indeterminate factors were taken, it could improve the efficiency and accuracy of the mathematical
models using the concept of Neutrosophic Cognitive Maps. [6]
V Christiano, F Smarandache [2019] reviewed the seven applications of Neutrosophic Logic.
They have used logical analysis based on Neutrosophic Logic. They further suggest that NL theory
could be applied in Psychology pertaining to different cultures, forming theories in the field of
economics, resolution of conflicts, philosophy of science and other fields like applied mathematics,
economics and physics. [7]
Nancy El-Hefenawy, et al. [2016] reviewed the application of Neutrosophic Sets. They suggest
that there exist a number of application in fields such as in decision making systems, IT, various
information systems. This paper presented some important areas of neutrosophic sets, logic in
neutrosophy, neutrosophy related measures and a neutrosophic set of a single value (SVNS). They
further suggest that these could produce a new algorithm for tackling any neutrosophic problem.
These can help also to solve any fuzzy problem using neutrosophic algorithm. [8]
S Pramanik, S Chackrabarti [2013], studied the issues which were faced by the construction
workers in West Bengal and used the technique- neutrosophic cognitive maps in order to find the
solutions for it. They discussed the major problems faced by the workers and based on the opinions
of the experts and after considering the indeterminacy factor, they formulated the NCM. [9]
Anne-Laure Jousselme, et al. [2003], proposed a discussion on how uncertainty plays a role in
situation analysis. They gave an overall understanding of the principal typologies of uncertainty
which were found in the literature of the recent times. They discuss that besides richness and
ambiguity of the language which is natural is the reason for varied uncertainty conceptions, it is also
a result of the not-so-simple physical nature of the information. They further define some concepts
to better understand uncertainty and the benefits that are sought. [10]
Vasantha K, W. B.; Smarandache, Florentin [2004], used NCM to study and analyze the social
aspects of laborers who had migrated from different place and were suffering from HIV/AIDS in the
rural areas of Tamil Nadu. They made use of the Relational Maps in neutrosophy (NRM) and defined
some new neutrosophic tools which they adopted in the study and analysis of this issue. They further
gave a sketch of some sixty laborers who were infected with HIV/AIDS. [11]
K Pérez-Teruel, M Leyva-Vázquez [2014], gave a structure with the help of which they analyzed
the mental models and did their elicitation using neutrosophic logic. To show the applicability of the
project, they showed an illustrative example. They discuss a framework for the processing of
indeterminacy and uncertainty in mental models. [12]
Mustafa Mamat et al. [2012], used an approach based on fuzzy linear programming for the
planning of a balanced diet. They discussed the causes of disease-related lifestyle and eating
disorders which are critical issues in the world. They calculated the nutrient amount in food to be
taken by the Fuzzy Linear Programming Approach and considered it to estimate the nutritional
requirements for an individual on a daily basis. They further suggest that this planning could help in
preventing the eating disorders and certain disease-related lifestyle. [13].
Igor Bagány and Márta Takács [2017] discussed the correlations in a number of factors involved
in education system in a way that the functionality could be modelled. They do so to examine the
education system in an effective manner. They further employed the fuzzy cognitive map (FCM)
A muzaffar, T Nafis, S S Sohail, Neutrosophy Logic and its Classification: An Overview
242
technology, because it helps in determination of qualitative description of the given parameters and
relationships [14].
S No.
Author
Primary Contribution
References
and Year
1.
2.
Florentin
Smarandache
(2019)
Sociological Forecasting, Neutrosophic Social norms
Neutrosophic Grand Theories
Neutrosophic Big Data, IoT and Neutrosophic
Victor
Christiano and
Microsociology.
Applications of neutrosophy in :
Psychology with respect to cultures
F Smarandache
3.
4.
5.
6.
[1]
[7]
Forming theories in economics
(2019)
Nancy
El-Hefenawy et
Resolving conflicts.
Decision support system, IT, information system
Some important notions pertaining to Neutrosophy.
al. (2016)
Aasim Zafar
and Mohd
Anas Wajid
NCM to model the criminal behavior in Nigeria.
Indeterminate factors, if considered improve the
accuracy and efficiency of the model.
[6]
NCM for the issues related to laborers in West
Bengal.
[9]
(2019)
Surapati
Pramanik and
Sourendranath
Chackrabarti
(2013)
Anne-Laure
Role of uncertainty in situation analysis
[8]
[10]
Jousselme
(2003)
Vasantha K, W.
B. and
Smarandache, F
Analyzing the social aspects, using NCM, of those
laborers who had migrated and suffer from
HIV-AIDS.
[11]
8.
(2004)
KPTeruel and
ML Vázquez
A framework for the analysis of mental models
based on NL (neutrosophic logic).
[12]
9.
(2014)
M Mamat et al.
(2012)
An approach incorporating FLP for a balanced diet
planning.
[13]
Igor Bagány
and Márta
FCM for finding correlations in a number of factors
involved in education system in a way that the
[14]
Takács (2017)
Shuqi Xue et al.
functionality could be modelled.
The information processing model which focuses on
[15]
(2014)
the behavior of the human brain, with respect to
cognition.
NCM for analyzing the risk factors for Breast Cancer
[16]
7.
10.
11.
12.
13.
Dr.M.Albert
William et al.
(2013)
K Mondal and
S Pramanik
NCM for analyzing the issues faced by Hijra
community in West Bengal.
A muzaffar, T Nafis, S S Sohail, Neutrosophy Logic and its Classification: An Overview
[17]
243
14.
Abdel-Basset
(2020)
et
al.
Smart disaster response systems in uncertainty
environments
Plithogenic Decision Making approach (Supply Chain
Sustainability)
[25],
Bipolar Neutrosophic Multi-Criteria Decision Making
[29],
Framework (Professional Selection)
[31].
Neutrosophic Set for assessing uncertainty of linear timecost tradeoffs
Resource levelling model based on neutrosophic set
Shuqi Xue et al. [2014], described the information processing model which is based on the
behavior of the human brain, with respect to cognition. They proposed that the two methods of
modelling a situation cognitively are representing and reasoning about situation analysis with the
help of Ontology and the use of FCM, in order to formulate a Situation analysis framework. The
presented approach of FCM is for a systematic analysis of the Situation Analysis theory; it provides
an understanding of how the working of its elements. [15]
Dr.M.Albert William et al. [2013] analyzed the risk factors for breast cancer using NCMs. Based
on the expert’s opinion, they had chosen certain factors as the main nodes for obtaining a
neutrosophic directed graph. They had analyzed the risk factors and their solutions and discussed
how certain factors are crucial for the development of the disease [16]. However few softcomputing
approaches have been used in [27, 28] K Mondal and S Pramanik [2014] have studied the situation of
the hijra community in West Bengal and addressed their issues using NCMs. On the basis of the
experts’ opinion as well as the idea of indeterminacy, they have formulated the NCM [17].
.
3. Classification of Neutrosophic Techniques:
Various researchers have studied the concept of neutrosophy and applied various techniques to
address different problems of indeterminacy. Some techniques are given below:
a) Neutrosophic Cognitive Maps
b) Neutrosophic Logic
c)
Neutrosophic Set
d) Neutrosophic Measure
e) Single Valued Neutrosophic Set
3.1. Neutrosophic Cognitive Maps (NCM):
Florentin Smarandache introduced the idea of NCM. They are considered to be an addendum
of the Fuzzy Cognitive Maps with the difference being in the fact that, the values of indeterminacy
are included. Various real life situations contain the factor of indeterminacy which cannot be modeled
using existing methods. To show how indeterminacy affects the situation under consideration, NCMs
have proven to be an important tool.
Definition:
A muzaffar, T Nafis, S S Sohail, Neutrosophy Logic and its Classification: An Overview
[26],
[30],
244
It is a directed graph which has concepts (as in, any policy/event) and causalities where the
former is for nodes and the latter is for the edges. It is a representation of a relationship between
concepts. A simple NCM can be defined as those which have edge weights or causalities from the set
{-1, 0, 1, I}.
Let the two nodes of the NCM be denoted by Ai and Aj. The effect of one node on the other is
represented with the help of a directed edge from Ai to Aj, which is called connections. The weightage
is assigned to each edge with a number in the set {-1, 1, 0, I}. We assume that eij is the weight assigned
to the directed edge Ai Aj, eij belongs to {-1, 0, 1, I}. The following table II shows the value of eij and
the effect it has on the corresponding edges:
Table II: Value of eij and its effect on corresponding edges
Many researchers have incorporated the concept of NCMs in their work. NCMs are an effective
way to deal with uncertainties and indeterminacy in Situation Analysis. They have shown how
indeterminate factors if taken into consideration could enhance the efficiency and accuracy of the
mathematical models using the concept of Neutrosophic Cognitive Maps.
Dr. M. Albert William et al. (2013) have analyzed the risk factors of Breast Cancer and their
solutions with the help of Neutrosophic cognitive maps (NCMs). They have taken some twelve
factors as the main nodes for their study. With the help of corresponding adjacency matrix related to
the neutrosophic directed graph, they model the situation with the help of certain mathematical
calculations.
Dr A. Kalaichelvi and L. Gomathy (2011) have analyzed the issues that the girl students had to face who got
married while studying, with the help of Neutrosophic Cognitive Maps (NCM’s). they collected the data
from some hundred students in different courses in various colleges in Tamil Nadu, India. They
identified certain factors on the basis of the generated opinions by those who were considered. In
this way, they assessed what the effect of one factor would be on the other.
Surapati Pramanik et al. studied the issues faced by the laborers in the construction industry
in West Bengal on the basis of NCM’s. They identified some major problems and on the basis of
the opinion of the expert and the factor of indeterminacy, they formulated the NCM. Then, they
studied how the state vectors would affect the two matrices i.e; the connection matrix and
neutrosophic adjacency matrix.
Aasim Zafar and M Anas Wajid studied the various factors which led to criminal behavior in
Nigeria. They analyzed the situation of crime there and found out that the prominent researchers
who had been monitoring the situation there cited certain causes like family breakdown,
A muzaffar, T Nafis, S S Sohail, Neutrosophy Logic and its Classification: An Overview
245
corruption, poverty etc as the reasons for criminal behavior. However, they do not take into account
factors like inadequate equipment, NGOs, underemployment because these are considered to be
indeterminate factors. They used NCMs to shows that these indeterminate factors were actually
related to the crime in Nigeria. They further conclude that the accuracy and efficiency of
mathematical models can be enhanced using NCMs if indeterminate factors are taken into
consideration.
3.2. Neutrosophic Logic (NL):
It is also called Smarandache logic. The fuzzy logic is generalized on the basis of Neutrosophy
and it gives rise to something called Neutrosophic logic. It says that a proposition could be take three
values: true (t), false (f) and indeterminate (I) and each of these are the values from the range of [T, I,
F]. There is an introduction of a certain idea of ‘indeterminacy’ because of the parameters which are
not expected and therefore, concealed in some statements. NL is the analysis of the partition in a
triad. It includes the membership degrees of truthfulness T, falsity F and indeterminacy I. Figure 2
illustrates the following.
Figure 2. Neutrosophic logic and its relationship with intuitionistic logic
Florentin Smarandache in 2003 has written a paper to give an understanding of the Neutrosophic
Logic (NL). He has also pointed out the differences between the Intuitionistic Fuzzy set and the
neutrosophic set. [20]
Karina Pérez-Teruel and Maikel Leyva-Vázquez have analyzed the mental models and did their
elicitation using NL. To show the applicability of the project, they showed an illustrative example.
They discuss a model for the understanding the effect of indeterminacy and uncertainty in such
models. [5]
A muzaffar, T Nafis, S S Sohail, Neutrosophy Logic and its Classification: An Overview
246
Florentin Smarandache and Luige Vlâdâreanu in 2011, have introduced the concept of NL and
set operators. They have described the dynamics of a robot mathematically and how neutrosophic
science is applicable to robotics [8].
3.3. Neutrosophic Set (NS):
Neutrosophic set is defined as the area of neutrosophy that is associated with the study of the dawn,
scope and type of neutralities, and how they interact with various analytical spectra. [8]
Smarandache defined neutrosophic set as: Let the space of points be denoted by (Y). Let the
general element in (Y) be denoted by (y). A NS (B) in (Y) has three membership functions (MF):
truth MF -T B(y), an indeterminacy MF- I B(y) and a falsity MF- F B(y). The functions TB(y), I B(y),
and F B(y) are real subsets of [0−, 1+] (they could be real standard or nonstandard).
That is:
There is no limiting condition on the sum of T B(y), I B(y) and F B(y), so 0− ≤ sup T B(y) +sup
I B(y) +sup F B(y) ≤ 3+.
Neutrosophic Sets have been used in various research works. Some examples are:
F. Smarandache, in [7] wrote about the Schrödinger’s Cat Theory. He said that at one moment, the
photon’s quantum state could be in more than one place. It meant that one particular element might
or might not belong to a set or a place at one time. It also refers to the fact that an element (a
quantum state) has a possibility of belonging to two contrasting sets (or places) at one time.
In [26], to tackle time scheduling in projects, a framework has been given to minimize the cost
of projects in environments which are ambiguous. Neutrosophic set theory has been used to
consider the dynamic features of all parameters.
In K. Atanassov, Fuzzy Sets and Systems (2005), neutrosophic sets could also be used to relate
an image with information that is not certain, using a new tool; the information could have been
applied to some technique wherein the processing of images takes place. The examples are in the
field of image segmentation, thresholding and removing the noise. Neutrosophic sets find their real
life example in terms of philosophical application. They could also be used to calculate the truthvalue in some theories of philosophy of Zen doctrine .
3.4. Neutrosophic Measure (NM):
The classical measure is generalized for such a case where the space has some factor of uncertainty
or indeterminacy. The imprecise probabilities and the classical ones are generalized with the help of
neutrosophic probability. There are a number of rules of the classical probability that are defined in
the way that they are in unison with those of neutrosophy [8].
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247
Let an item be defined as <B>. <B> could be any thought, feature, hypothesis, concept etc. Let <anti
B> be the inverse of <B>; while <neut B> be none of the two: <B> and <anti B>, having some sense of
neutrality (or indeterminacy) in relation to <B>. For example, if <B> = rain, then <anti B>= no rain,
while <neut B> = no idea. Let <B> represent the truth value of a notion, then <anti B> represents its
falsehood, while <neut B> represents its degree of indeterminacy.
If <B> = it will rain tomorrow, <anti B> = it will not rain tomorrow, while <neut B> = not knowing if
it will rain or not/cloudy/humid day. We think of the measure to be null {m (anti B) =0} when the
case does not prevail. When <neut B> does not prevail, the measure is written as null {m (neutB) =
0} [8].
3.5. Single Valued Neutrosophic Sets (SVNS) :
It is the instance of a NS which gives an additional possibility for the representation of uncertainty
or indeterminacy, imprecision, incompleteness or inconsistency in some details which is present in
the real world. The use of information that is not determinate and consistent could be suitably used
in applications which include scientific and engineering domains. [9][10]
Let X define the space of points (objects). Let the collective elements in X be denoted by x
(Wang et al., 2010). A Single Valued Neutrosophic Set, A in X is described by three membership
functions (MF): truth MF TA(x), falsity MF FA(x) and an indeterminacy MF IA(x).
For every point x in X, the three MF’s: TA(x), IA(x), FA(x) belong to the interval [0, 1].
SVNS, when continuous is written mathematically as [9,10]:
SVNS, when discrete is written mathematically as:
Jun Ye, in [25], has presented the correlation and correlation coefficient of SVNSs, based on
the extension of the connection of intuitionistic fuzzy sets (IFS’s). Further, the use of correlation
coefficient or similarity measure in cosine (both weighted) is suggested for the decision-making
method. The options are evaluated on the basis of the criteria with the help of the membership
degrees of truth, falsehood and indeterminacy under the SVNS environment.
M Abdel-Basset et al. in their paper, have analyzed the role of SVNS’s and rough sets in smart city.
They have proposed a framework for dealing with information that is incomplete and imperfect with
the help of theories of SVNS and rough set. This combination of these two sets will take into account
A muzaffar, T Nafis, S S Sohail, Neutrosophy Logic and its Classification: An Overview
248
all aspects of uncertainty, imprecision of data and information and make lives of the citizens of the
smart cities better with the introduction of services and decisions. They have focused mainly on
making a framework of all kinds of imperfection that could possibly happen in smart cities [24].
4. Application Areas of Neutrosophy:
1.
2.
3.
Cultural Psychology
Socio-economic theorizing
Information System Application
4.
5.
Decision Support Systems
IT Application
6.
Healthcare and related areas
7.
8.
Situation Analysis
Sociological Forecasting
9.
Supply chain Sustainability
10. Project Management
•
In cultural psychology, NL theory can be used to reconcile the issues in socio-economic
theorizing (collectivism vs individualism).
•
In socio-economic theorizing, the conflicts arising out of human tensions could be
reconciled, as in the conflicts between the two different perspectives i.e.; fermions and bosons,
capitalism and socialism.
•
In the deep problem of philosophy of science, NL theory can be implemented wherein it
suggests that whenever there are two sides which oppose each other, a choice is always there to
find the part that is neutral, so that the two opposite sides could be reconciled.
•
In the field of cosmology, the NL analyses the underlying cause of changes of neutralities
and opposites. It concludes that there is a possibility that there had been some start, in addition to
some lasting background also, which they could be the ‘primordial fluid’.
•
In American football game, an attempt to score a goal involves an infinite sort of events
that could happen. So, there is a possibility that NL could be expanded some states which could be
more than three.
•
In gravitation, this perspective could help find a middle-course between the two kinds
of forces (pull and push), by keeping in view the fact that both the forces are in action. [11]
So, many fields of science are being improved with the help of the theory of neutrosophic
logic. This theory is applicable in different research areas as well- in applied mathematics,
social sciences, economics, and physics.
More Applications:
•
In Information System Application (Neutrosophic Database, Analysis of the
social
A muzaffar, T Nafis, S S Sohail, Neutrosophy Logic and its Classification: An Overview
249
networks, systems which deal with e-learning, in finding the middle course in the information of
financial markets).
•
In Information Technology Application (Neutrosophic Security, NCM’s for
Situation
Analysis, In Robotics).
•
Decision Support System (in markets related to finances, management of risks, expert
systems related to neutrosophy, linguistic variables in neutrosophy).
In short, it has applications in any field related to science or even human-centered, where
inconsistency, incompleteness, indeterminacy is present. In general terms, where <neut A> (i.e;
sense of neutrality in relation to item <A>) occurs [11].
.
5. Conclusion and Future work:
Neutrosophy is an important field of research nowadays as it deals with uncertainties which cannot
be taken into consideration using conventional modeling methods. There is indeterminacy in almost all
aspects of this world; neutrosophy is doing its bit to make sense of the unknown. This paper presents
a review of the technologies used in neutrosophy and the researches which have incorporated these
concepts as well. Various applications of neutrosophy in many fields such as information system,
information technology, decision support system and others are given. The future work holds the
potential to develop newer algorithms for solving any problem of neutrosophy, which can also help
in solving any fuzzy problems. The algorithms in the multi-criteria decision making problems which
are based on neutrosophic theory are being used to solve practical applications in other areas such as
medical diagnosis, financial market information, robotics, security, information fusion system, expert
system and bioinformatics.
Conflicts of Interest: “The authors declare no conflict of interest.”
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neutrosophic environment. J Supercomput 76, 964–988 (2020). https://doi.org/10.1007/s11227-01903055-6
30.
Abdel-Basset, M.; Gamal, A.; Son, L.H.; Smarandache, F. A Bipolar Neutrosophic Multi Criteria
Decision Making Framework for Professional Selection. Appl. Sci. 2020, 10, 1202.
31.
Abdel-Basset, M., Mohamed, R., Zaied, A. E. N. H., Gamal, A., & Smarandache, F. (2020). Solving
the supply chain problem using the best-worst method based on a novel Plithogenic model. In
Optimization Theory Based on Neutrosophic and Plithogenic Sets (pp. 1-19). Academic Press.
32.
Ye, Jun. (2013). Multicriteria decision-making method using the correlation coefficient under singlevalue
neutrosophic
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International
Journal
of
10.1080/03081079.2012.761609.
Received: Apr 25, 2020. Accepted: July 16, 2020
A muzaffar, T Nafis, S S Sohail, Neutrosophy Logic and its Classification: An Overview
General
Systems.
42.
Neutrosophic Sets and Systems, Vol.35, 2020
University of New Mexico
MADM Using m-Generalized q-Neutrosophic Sets
1*
2
3
Abhijit Saha , Florentin Smarandache , Jhulaneswar Baidya and Debjit Dutta
1
2
3
4
4
Faculty of Mathematics, Techno College of Engineering Agartala, , Tripura, India, Pin-799004; Email: abhijit84.math@gmail.com
Faculty of Maths & Science Div. ,University of New Mexico, Gallup, New Mexico 87301, USA; Email: fsmarandache@gmail.com
Research Scholar, Dept. of Basic and Applied Sciences, NIT Arunachal Pradesh, Pin-791112, India; Email: baidyajhulan@gmail.com
Faculty of Basic and Applied Sciences, NIT Arunachal Pradesh, Arunachal Pradesh-791112, India; Email: debjitdutta.math@gmail.com
*Correspondence: abhijit84.math@gmail.com
Abstract: Although the single valued neutrosophic sets (SVNSs) are effective tool to express uncertain information and
are superior to the fuzzy sets, intuitionistic fuzzy sets, Pythagorean fuzzy sets and q-rung orthopair fuzzy sets, there is not yet
reported an operation which can provide desirable generality and flexibility under single valued neutrosophic environment,
although many operations have been developed earlier to meet above such eventualities. So, the primary aim of this paper is
to propose the concept of m-generalized q-neutrosophic sets (mGqNSs) as a further generalization of fuzzy sets, intuitionistic
fuzzy sets, Pythagorean fuzzy sets and q-rung orthopair fuzzy sets, single valued neutrosophic sets, n-hyperspherical
neutrosophic sets and single valued spherical neutrosophic sets. Under the m-generalized q-neutrosophic environment, we
develop some new operational laws and study their properties. Using these operations, we define m-generalized
q-neutrosophic weighted aggregation operators. The distinguished features of these proposed weighted aggregation operators
are studied in detail. Furthermore, based on these proposed operators, a MADM (multi-attribute decision making) approach is
developed. Finally, an illustrative example is provided to show the feasibility and effectiveness of the proposed approach.
Keywords: Single valued neutrosophic set, m-generalized q-neutrosophic set, m-generalized q-neutrosophic
weighted averaging aggregation operator (mGqNWAA), m-generalized q-neutrosophic weighted geometric
aggregation operator (mGqNWGA), score value, decision making.
___________________________________________________________________________________________________
1. Introduction
Multi-attribute decision making (MADM) is basically a process of selecting an optimal alternative from a set
of chosen ones. In our daily life, we come across various types of multi-attribute decision making problems.
Therefore, all of us need to learn the techniques to make decisions. The area of decision making problems has
attracted the interest of many researchers. Many authors have worked in this field by utilizing various approaches.
All the traditional decision making processes involve crisp data set but in many real life problems, data may not be
in crisp form always. Fuzzy set theory is one such extremely useful tool that helps us to deal with non-crisp data. In
1965, Lotfi A. Zadeh [1] first published the famous research paper on fuzzy sets that originated due to mainly the
inclusion of vague human assessments in computing problems and it can deal with uncertainty, vagueness,
partially trueness, impreciseness, Sharpless boundaries etc. Basically, the theory of fuzzy set is founded on the
concept of relative graded membership which deals with the partial belongings of an element in a set in order to
process inexact information. Later on, fuzzy sets have been generalized to intuitionistic fuzzy sets [2] by adding a
non-membership function by Atanassov in 1986 in order to deal with problems that possess incomplete
information. In the context of fuzzy sets or intuitionistic fuzzy sets, it is known that the membership (or
non-membership) value of an element in a set admits a unique value in the closed interval [0,1]. However, the
application range of intuitionistic fuzzy set is narrow because it has the constraint that sum of membership degree
Abhijit Saha, Florentin Smarandache, Jhulaneswar Baidya and Debjit Dutta, MADM USING m-GENERALIZED
q-NEUTROSOPHIC SETS
253
Neutrosophic Sets and Systems,Vol. 35, 2020
and non-membership degree of an element is not greater than one. But, in complex decision‐making problems,
decision makers/experts may choose the preferences in such a way that the above condition gets violated. For
instance, if an expert gives his preference with membership degree 0.8 and non-membership degree 0.7, then
clearly their sum is 1.5, which is greater than 1. Therefore, this situation can’t be not properly handled by the
intuitionistic fuzzy sets. To solve this problem, Yager [3, 4] introduced the nonstandard fuzzy set named as
Pythagorean fuzzy sets with membership degree ζ and non-membership degree ϑ with the condition ζ2 + ϑ2 ≤ 1.
Obviously, the Pythagorean fuzzy sets accommodate more uncertainties than the intuitionistic fuzzy sets. Yager
[5] defined q-rung orthopair fuzzy sets (q-ROFSs) by enlarging the scope of Pythagorean fuzzy sets. The q-rung
orthopair fuzzy sets allows the result of the qth power of the membership grade plus the qth power of the
non-membership grade to be limited in interval [0,1]. If q=1, the q-rung orthopair fuzzy set transforms into the
intuitionistic fuzzy set; if q=2, the q-rung orthopair fuzzy set transforms into the Pythagorean fuzzy set, which
means that the q-rung orthopair fuzzy sets are extensions of intuitionistic fuzzy sets and Pythagorean fuzzy sets.
In 1999, Smarandache [6] introduced the notion neutrsophic set as a generalization of the classical set,
fuzzy set, intuitionistic fuzzy set, Pythagorean fuzzy set and q-rung orthopair fuzzy set. The characterization of this
neutrosophic set is explicitly done by truth-membership function, indeterminacy membership function and falsity
membership function. The concept of single valued neutrosophic set was developed by Wang et al. [7] as an
extension of fuzzy sets, Pythagorean fuzzy sets, q-rung orthopair fuzzy sets, intuitionistic fuzzy sets, single valued
spherical neutrosophic sets [8], n-hyperspherical neutrosophic sets [8]. The possible applications of neutrosophic
sets and single valued neutrosophic sets on image segmentation have been studied in Gou and Cheng [9], Gou and
Sensur [10]. Also, we find their probable infliction on clustering analysis in Karaaslan [11] and on medical
diagnosis problems in Ansari et al. [12] respectively. Furthermore, the subject of the neutrosophic set theory has
been practiced in Wang et al. [13], Gou et al. [14], Ye [15], Sun et al. [16], Ye [17-19] and Abdel Basset et al. [20,
21]. Some recent studies on this area can be found in [22-37].
The growing capacity of decision complexity induces the real-life decision-making problems that indulge
both generality and flexibility of the operations used. Some of the basic operations of single valued spherical
neutrosophic sets fail to generalize the basic operations of fuzzy sets, intuitionistic fuzzy sets, Pythagorean fuzzy
sets and q-rung orthopair fuzzy sets. Getting inspired and provoked with this fact, in this paper, we have tried to
propose a new concept called “m-generalized q-neutrosophic sets (mGqNSs)” and develop some aggregation
operators in m-generalized q-neutrosophic environment to deal with MADM problems. The aims in this article
are pursued below:
(1) To propose the concept of m-generalized q-neutrosophic sets (mGqNSs) as a further generalization of fuzzy
sets, Pythagorean fuzzy sets, q-rung orthopair fuzzy sets, intuitionistic fuzzy sets, single valued neutrosophic sets,
n-hyperspherical neutrosophic sets and single valued spherical neutrosophic sets.
(2) To define few operations between the m-generalized q-neutrosophic numbers.
(3) To develop the weighted aggregation operators such as m-generalized q-neutrosophic weighted averaging
aggregation operator (mGqNWAA) and m-generalized q-neutrosophic weighted geometric aggregation operator
(mGqNWGA) and study their properties.
(4) To propose a multi-attribute decision making method based on the m-generalized q-neutrosophic weighted
aggregation operators.
To do so, the rest of the article is arranged as follows:
In section 2, we review some basic concepts. In Section 3, we first define m-generalized q-neutrosophic sets
(mGqNSs) and m-generalized q-neutrosophic numbers (mGqNNs) and then propose few operations between the
mGqNNs. Furthermore, we introduce the score of a mGqNN to ranking the mGqNNs. In section 4, we propose two
types of m-generalized q-neutrosophic weighted aggregation operators to aggregate the m-generalized
q-neutrosophic information. In section 5, based on the m-generalized q-neutrosophic weighted aggregation
operators and score of mGqNNs, we develop a multi attribute decision making approach, in which the evaluation
values of alternatives on the attribute are represented in terms of mGqNNs and the alternatives are ranked
according to the values of the score of mGqNNs to select the best (most desirable) one. Also, we present a
practical example to demonstrate the application and effectiveness of the proposed method. In final section, we
present the conclusion of the study.
2. Preliminaries:
In this section, first we recall some basic notions that are relevant to our study.
2.1 Definition: [7] A single-valued neutrosophic set on the universe set 𝑈 is given by
{ x, ( x), ( x), ( x) : x U }
Abhijit Saha, Florentin Smarandache, Jhulaneswar Baidya and Debjit Dutta, MADM USING m-GENERALIZED
q-NEUTROSOPHIC SETS
254
Neutrosophic Sets and Systems,Vol. 35, 2020
, , :U
where the functions
[0,1]
satisfy the condition 0
( x)
( x)
( x)
3 for every
𝑥 ∈ 𝑈. The functions
( x), ( x), ( x) define the degree of truth-membership, indeterminacy-membership
and falsity-membership, respectively of 𝑥 ∈ 𝑈 .
2.2 Definition: [7] Suppose
and
be two single-valued neutrosophic sets on 𝑈 and are given by
{ x, ( x), ( x), ( x) : x U } and
if and onlyif ( x)
(i)
c
(ii)
{ x, ( x),1
{ x, ( x),
( x), ( x)
( x), ( x)
( x),
( x) : x
( x) x
U } . Then
U.
( x), ( x) : x U }
(iii)
={<x, max( ( x), ( x)), min( ( x),
( x)), min( ( x),
( x)) : x
U}.
(iv)
={<x, min( ( x), ( x)), max( ( x),
( x)), max( ( x),
( x)) : x
U }.
3. m-GENERALIZED q-NEUTROSOPHIC SETS:
In this section first we define a m-generalized q-neutrosophic set as a further generalization of fuzzy set,
Pythagorean fuzzy set, q-rung orthopair fuzzy set, intuitionistic fuzzy set, single valued neutrosophic set, single
valued n-hyperspherical neutrosophic set and single valued spherical neutrosophic set. Then we present few
operations between the m-generalized q-neutrosophic numbers.
3.1 Definition: Suppose U is a universe set and
described as:
, , :U
where
0
(
qm
( x)) 3
Here
(
[0, r ] (o
qm
( x)) 3
(
x U . A m-generalized q-neutrosophic set (mGqNs) in U is
{ x, ( x), ( x), ( x) : x U }
r 1) are functions such that 0
qm
( x)) 3
3
m
(m, q
( x), ( x), ( x)
1 and
1) .
( x), ( x), ( x) represent m-generalized truth membership, m-generalized indeterminacy
membership and m-generalized falsity membership respectively of
x U . The triplet
, ,
is
termed as m-generalized q-neutrosophic number (mGqNN for short).
In particular,
(i) when m=r=1 and q=3,
reduces to a single valued neutrosophic set [7].
reduces to an intuitionistic fuzzy set [2].
( x) 0 x U ,
reduces to a fuzzy set [1].
(iii) when m=3, r=q=1 and ( x)
( x) 0 x U ,
reduces to a q-Rung orthopair fuzzy set [5].
(iv) when m=3, r= 1 and ( x) 0 x U ,
reduces to a Pythagorean fuzzy set [3, 4].
(v) when m=3, r= 1, q=2 and ( x) 0 x U ,
(ii) when m=3, r=q=1 and
(vi) For
r
(vii) For
r
n
3 , m=1 and q=3n (n 1) ,
3 , m=1 and q=6,
reduces to a single valued n-hyperspherical neutrosophic set [8].
reduces to a single valued spherical neutrosophic set [8].
Next we define few operations between m-generalized q-neutrosophic numbers.
3.2 Definition: Suppose
numbers defined on U and
1
1, 1, 1
and
2
2, 2, 2
be two m-generalized q-neutrosophic
be any real number >0. We define
Abhijit Saha, Florentin Smarandache, Jhulaneswar Baidya and Debjit Dutta, MADM USING m-GENERALIZED
q-NEUTROSOPHIC SETS
255
Neutrosophic Sets and Systems,Vol. 35, 2020
(i)
1
(ii)
3
m
2
1
3
m
1
(iv)
1
1
3
m
,
3
m
1
(ii)
2
1
2
1
1
2
qm
3
3
m
,
1
3
qm
qm
3
,
2
2
(
1)
(
2)
(iv)
(
1
2)
(
1)
(
2)
1
(
1)
(
(vi)
(
1
2)
3
qm
qm
3
3
m
,
3
m
1
qm
3
3
m
2
qm
3
3
qm
1
3
m
3
m
1
3
qm
qm
3
2, 2, 2
2
be two m-generalized q-neutrosophic
1
2)
2)
1 2, 1 2
1
1
1
,
2 be any three real numbers >0. Then
(
(
2
and
(iii)
(v)
,
1
1, 1, 1
, 1,
3
qm
qm
3
3
qm
qm
3
1
numbers defined on U and
3
m
3
m
3
m
3.3 Theorem: Suppose
(i)
1
3
m
1 2,
2
(iii)
3
m
qm
3
(
1
1
1
1)
(
2
2
1)
1)
Proof: (i), (ii) are straight forward.
(iii) We have,
1
2
3
m
3
m
1
qm
3
3
m
2
qm
3
3
qm
,
1 2, 1 2
.
Abhijit Saha, Florentin Smarandache, Jhulaneswar Baidya and Debjit Dutta, MADM USING m-GENERALIZED
q-NEUTROSOPHIC SETS
256
Neutrosophic Sets and Systems,Vol. 35, 2020
(
2)
1
3
m
3
m
3
m
3
m
3
m
1
3
m
qm
3
1
qm
3
3
m
2
3
m
2
3
qm
qm
3
,(
1 2)
,(
1 2)
3
qm
qm
3
,
1
2
,
1
2
On the other hand, we have,
(
1)
(
3
m
3
m
3
m
2)
3
m
1
3
m
3
m
3
m
1
(
Thus, we get,
qm
3
3
qm
,
3
m
qm
3
1
3
m
1
qm
3
3
m
2)
1
,
1
2
3
m
3
m
2
3
m
2
,
2
,
2
1
2
3
qm
qm
3
,
,
1
2
3
qm
qm
3
(
3
m
3
qm
qm
3
,
1)
1
(
2
,
1
2
2) .
(iv) Similar to (iii)
(v) We have,
3
(
1
2)
1
3
m
3
m
1
qm
3
1
2 qm
,
1
1
2
,
1
1
2
On the other hand,
Abhijit Saha, Florentin Smarandache, Jhulaneswar Baidya and Debjit Dutta, MADM USING m-GENERALIZED
q-NEUTROSOPHIC SETS
257
Neutrosophic Sets and Systems,Vol. 35, 2020
(
1)
1
(
1)
2
3
3
m
3
m
3
m
1
3
m
3
1 qm
qm
3
,
3
m
3
m
1
1
1
,
1
3
m
1
1
qm
3
3
m
3
m
3
m
1
3
m
1
2 qm
qm
3
,
,
2
1
3
qm
2
qm
3
2
1
,
1
1
1
2
,
1
1
1
2
3
3
m
3
m
Thus we get,
(
1
,
2)
1
2 qm
1
qm
3
1
(
1
1
1
2
,
1)
1
(
1
2
1) .
2
(vi) Similar to (v).
, ,
3.4 Definition: The score of the mGqNN
2
S( )
is defined as:
3
The ranking method for ranking the mGqNNs is given below:
If
and
be two mGqNNs, then
, ,
, ,
(I) if S ( )
S(
(II) if S ( )
.
) , then
S(
) , then
4. m-GENERALIZED q-NEUTROSOPHIC WEIGHTED AGGREGATION OPERATORS:
In this section first we define m-generalized q-neutrosophic weighted averaging aggregation operator
(mGqNWAA) and m-generalized q-neutrosophic weighted geometric aggregation operator (mGqNWGA) and study
their basic properties.
4.1 Definition: Suppose
k
k, k, k
(k
1, 2,3,......., n)
be a collection of mGqNNs defined on
the universe set U. Then a m-generalized q-neutrosophic weighted averaging aggregation operator (mGqNWAA for
short) is given as
mGqNWAA :
mGqNWAA( 1,
where
n
and is defined as:
(w1
2 , 3 ,........, n )
1)
( w2
( w3
2)
is the collection of all mGqNNs defined on the universe set U,
weight vector of
( 1,
2,
3 ,........,
n)
such that
wk
0(k
w
......... ( wn
3)
n)
(w1, w2 , w3 ,......., wn )T is the
1, 2,3,...., n)
n
and
k 1
wk
1.
On the basis of the operational rules of the mGqNNs, we can get the aggregation result as described as
Theorem 4.2.
4.2 Theorem: Suppose
universe set U and
w
k
k, k, k
(k
1, 2,3,......., n)
be a collection of mGqNNs defined on the
(w1, w2 , w3 ,......., wn )T is the weight vector of ( 1,
2 , 3 ,........, n ) such that
Abhijit Saha, Florentin Smarandache, Jhulaneswar Baidya and Debjit Dutta, MADM USING m-GENERALIZED
q-NEUTROSOPHIC SETS
258
Neutrosophic Sets and Systems,Vol. 35, 2020
wk
0(k
n
1, 2,3,...., n)
and
k 1
1 . Then mGqNWAA( 1,
wk
2 , 3 ,........, n ) is also a mGqNN.
Moreover, we have,
mGqNWAA( 1 ,
2,
3 ,........,
n
3
m
n)
k 1
3
m
k
3
qm wk qm
3
,
n
k 1
k
wk
,
n
k 1
k
wk
.
Proof:
The first part of the theorem can be proved easily. To show the rest part, let us use the method of mathematical
induction on n.
Step-1: For n=1, the proof is straight forward. So first take n=2.
Then,
mGqNWAA( 1 ,
( w1
3
m
3
m
3
m
3
m
2)
( w2
1)
3
m
1
3
m
k 1
3
w
1
qm
qm
3
3
m
3
m
2
2)
1
3
m
qm w1
3
3
m
k
3
3
m
w1
w1
1 , 1
,
1
qm w1
3
3
m
2
3
qm wk qm
3
3
m
3
m
3
qm w2 qm
3
,
2
k 1
k
wk
3
m
3
m
2
qm w2
3
,
w2
w2
1 , 1
,
w1 w2
w1 w2
1
1 , 1
1
3
qm
w1 w2
w1 w2
1
1 , 1
1
,
2
,
2
qm
3
2 qm
k 1
k
wk
.
Thus the result is true for n=2.
Step-2: Suppose that the result is true for n=p i.e;
mGqNWAA( 1 ,
2,
3 ,........,
p)
3
m
p
k 1
3
m
k
3
qm wk qm
3
,
p
k 1
k
wk
,
p
k 1
k
wk
Step-3: Take n=p+1.Then we have,
Abhijit Saha, Florentin Smarandache, Jhulaneswar Baidya and Debjit Dutta, MADM USING m-GENERALIZED
q-NEUTROSOPHIC SETS
259
Neutrosophic Sets and Systems,Vol. 35, 2020
mGqNWAA( 1 ,
(( w1
p
k 1
3
m
3
m
k
k 1
p
k 1
p 1
k 1
k 1
qm wk
3
k
3
m
p
,
3
m
k
k
p
,
k 1
wp 1
,
p 1
,
p
k
k 1
p 1)
1
wk
wp 1
p 1
3
m
p 1
p 1
,
k
(wp
p ))
3
m
3
m
p 1
qm w p 1
3
3
qm
,
wk
3
m
3
w
k
qm
qm
3
wk
qm wk
3
k
wp 1
p 1
,
3
m
......... ( w p
3)
3
w
1
p
qm
qm
3
p
3
m
wk
( w3
k 1
p 1
p
p 1)
3
w
k
qm
qm
3
k
3
m
wp 1
p 1
3
m
2)
3
m
3
m
3
m
3 ,........,
( w2
1)
3
m
2,
k
k 1
3
w
p
1
qm
qm
3
wk
,
p 1
k 1
k
,
wp 1
p 1
p
k 1
k
wk
,
wp 1
p 1
p
k 1
k
wk
wk
Thus the result is true for n=p+1 also. Hence, by the method of induction, the result is true for all n.
Let us explore some more results related to mGqNWAA operator in the form of theorems 4.3-4.6.
4.3 Theorem: Suppose
the universe set U and
that
wk
collection
0(k
n
and
k 1
all
0
1, 0
1, 2,3,......., n)
be a collection of m-Gq-NNs defined on
(w1, w2 , w3 ,......., wn )T is the weight vector of ( 1,
1, 2,3,...., n)
of
mGqNWAA(
.
w
(k
k, k, k
k
mGqNNs
2, 0
wk
defined
1 . Then for
on
3 ,........, 0
0
the
n)
0 , 0, 0
universe
0
2 , 3 ,........, n ) such
set
mGqNWAA( 1,
(where
U),
is the
we
2 , 3 ,........,
have
n)
Proof:
Abhijit Saha, Florentin Smarandache, Jhulaneswar Baidya and Debjit Dutta, MADM USING m-GENERALIZED
q-NEUTROSOPHIC SETS
260
Neutrosophic Sets and Systems,Vol. 35, 2020
3
m
k
0
3
m
mGqNWAA(
n
3
m
k 1
0
1,
0
3
m
qm
3
3
m
2,
0
3
m
3
m
0
k
3
m
3
m
0
3
m
0
On the other hand,
0,
3
m
3
m
0,
wk
qm
k 1
3
qm
3
3
m
qm
3
n
k 1
0
0
0
3
m
0
3
m
k 1
qm
3
qm
3
3
m
k
k
n
k 1
n
k 1
n
k 1
3
m
3
m
3
m
1
3
m
k
k
0 k, 0 k
3
wk qm
n
(
wk
k) ,
0
0
0
k
k 1
k
k 1
wk
wk
n
,
0
n
k
k 1
,
0
(
k 1
n
wk n
k 1
n
k 1
0 k)
wk n
k 1
k
wk
wk
wk
2 , 3 ,........, n )
n
,
k 1
k 1
1, 2,3,........, n)
k 1
,
3
w
k
qm
qm
3
n
n
,
3
qm
,
(k
n)
0
qm
3
qm wk
3
3
w
k
qm
qm
3
mGqNWAA( 1,
3
m
3
m
n
,
3 ,........,
0
n
3
m
3
qm
qm
3
3
m
3
qm wk qm
3
,
1
k 1
k
wk
,
n
k 1
k
3
qm
qm wk
3
n
0
k
wk
,
,
n
0
n
0
k 1
wk
k
k 1
k
wk
,
n
0
k 1
k
wk
wk
Hence,
mGqNWAA(
.
0
1, 0
2, 0
4.4 Theorem: (Idempotency) Suppose
3 ,........, 0
k
m-Gq-NNs defined on the universe set U and
n)
0
k, k, k
(k
w
mGqNWAA( 1,
1, 2,3,......., n)
2 , 3 ,........, n )
be a collection of
(w1, w2 , w3 ,......., wn )T is the weight vector of
Abhijit Saha, Florentin Smarandache, Jhulaneswar Baidya and Debjit Dutta, MADM USING m-GENERALIZED
q-NEUTROSOPHIC SETS
261
Neutrosophic Sets and Systems,Vol. 35, 2020
( 1,
2,
3 ,........,
n)
such
0, 0, 0
0
k
k
0
3
m
n
k 1
n
k 1
mGqNWAA( 1,
3
m
k
3
m
0
qm wk
3
3
m
4.5
3
m
0
3
m
0
qm
3
Theorem:
3
qm
,
n
k
3
qm
,
n
0
k 1
wk
qm
k 1
3
3
qm
,
and
k 1
wk
1
.
If
3
qm
n
,
2 , 3 ,........, n )
0.
2 , 3 ,........, n )
k 1
qm wk
3
n
1, 2,3,...., n)
is the collection of all mGqNNs defined on the universe set U) such that
(where
n
3
m
0(k
1, 2,3,......., n , then we have mGqNWAA( 1,
Proof: We have,
3
m
wk
that
0
k 1
wk
n
,
k 1
wk
wk
n
,
,
wk
wk
n
n
0,
(Monotonocity)
0
k 1
k 1
0, 0
k
0
k 1
wk
0, 0
0
Suppose
k, k, k
k
and
k, k, k
k
(k
1, 2,3,......., n) be two collections of mGqNNs defined on the universe set U and
w
(w1, w2 , w3 ,......., wn )T
( 1,
2 , 3 ,........, n )
k, k
k
k, k
is
such
the
2 , 3 ,........, n ) .
Proof: Since
k, k
n
so
k 1
3
m
k
qm wk
3
k, k
n
k 1
3
m
0(k
k
k
( 1,
of
2 , 3 ,........, n )
n
1, 2,3,...., n)
and
k 1
1, 2,3,......., n)
mGqNWAA( 1,
k
vector
wk
that
(k
k
weight
,
mGqNWAA( 1,
then
wk
as
1
well
as
.
If
2 , 3 ,........, n )
for all k ,
qm wk n
3
,
k 1
k
wk
n
k 1
k
wk
,
n
k 1
k
wk
n
k 1
k
wk
Abhijit Saha, Florentin Smarandache, Jhulaneswar Baidya and Debjit Dutta, MADM USING m-GENERALIZED
q-NEUTROSOPHIC SETS
262
Neutrosophic Sets and Systems,Vol. 35, 2020
n
3
m
3
m
k 1
n
k
k 1
n
wk
3
m
k 1
n
k
k 1
k
3
w
k
qm
qm
3
k
k 1
n
3
m
k 1
n
3
m
k 1
3
m
k
3
w
k
qm
qm
3
n
,
k
k 1
n
wk
k 1
k
wk
,
wk
n
wk
3
m
2
k
k 1
n
3
m
k
3
w
k
qm
qm
3
wk
k
n
3
m
3
m
k 1
k
3
w
k
qm
qm
3
n
k 1
k
n
wk
k 1
k
wk
0
3
qm wk qm
3
n
k
k 1
n
wk
k
k 1
wk
3
2
n
3
m
k 1
3
m
k
3
w
k
qm
qm
3
n
k 1
3
S (mGqNWAA( 1 , 2 , 3 ,........,
k
n
wk
k 1
k
wk
0
S ( mGqNWAA( 1 ,
n ))
Hence by definition of score value and ranking method, we have,
mGqNWAA( 1,
3 ,........,
min
by:
1 k n
Then,
Then we have,
3
m
n)
k,
such that
max
1 k n
k,
mGqNWAA( 1,
max
3
m
1 k n
k
max
1 k n
qm wk
3
k
qm
3
3
m
wk
max
1 k n
0(k
2 , 3 ,........, n )
k
n
1, 2,3,...., n)
,
k
k
qm wk
3
qm
3
max
1 k n
n)
3
m
3
m
be a collection of
(w1, w2 , w3 ,......., wn )T is the weight vector of
w
and
k,
min
1 k n
wk
k,
1 . Let us define two mGqNNs
min
1 k n
k
.
2 , 3 ,........, n )
3
m
1, 2,3,......., n)
k 1
2 , 3 ,........,
mGqNWAA( 1,
Proof: Suppose
n ))
mGqNWAA( 1,
(k
k, k, k
k
mGqNNs defined on the universe set U and
2,
3 ,........,
2 , 3 ,........, n ) .
4.6 Theorem: (Boundedness) Suppose
( 1,
2,
, ,
min
k
1 k n
min
1 k n
k
.
qm
3
qm wk
3
Abhijit Saha, Florentin Smarandache, Jhulaneswar Baidya and Debjit Dutta, MADM USING m-GENERALIZED
q-NEUTROSOPHIC SETS
263
Neutrosophic Sets and Systems,Vol. 35, 2020
n
3
m
3
m
k 1
min
k
1 k n
3
qm wk qm
3
3
m
min
n
1 k n
n
( min
k)
1 k n
k)
k 1
n
wk
k
Similarly, we can get, min
1 k n
max
1 k n
2
S(
min
)
max
wk
( max
k)
max
k
max
k
1 k n
k 1
max
k
1 k n
wk
qm
k 1
3
wk
max
min
k
max
k
1 k n
k
1 k n
max
k
1 k n
min
max
k
1 k n
min
k
1 k n
2
2
k
1 k n
1 k n
min
1 k n
3
2,
3 ,........,
of
n)
k
k
min
1 k n
k
3
S(
n ))
score
k, k, k
k
k 1
wk
k.
1 k n
2 , 3 ,........,
4.7 Definition: Suppose
k)
1 k n
definition
mGqNWAA( 1,
1 k n
( max
1 k n
k
1 k n
by
3
m
k 1
3
S (mGqNWAA( 1,
Therefore
n
wk
max
k
1 k n
k
1 k n
3
m
3
m
3
qm
n
n
n
k 1
Hence min k
k
k 1
wk
k 1
k
n
3
m
3
qm wk qm
3
k
1 k n
( min
k 1
k
wk
qm
k 1
3
max
k
1 k n
Again,
min
1 k n
3
m
3
qm
n
3
m
n
3
m
3
qm wk qm
3
value
)
and
ranking
method,
we
have,
.
(k
1, 2,3,......., n)
be a collection of mGqNNs defined on
the universe set U. Then a m-generalized q-neutrosophic weighted geometric aggregation operator (mGqNWGA
for short) is given as
mGqNWGA :
mGqNWGA( 1,
where
n
and is defined as:
(w1
2 , 3 ,........, n )
1)
( w2
( w3
2)
is the collection of all mGqNNs defined on the universe set U,
weight vector of
( 1,
2,
3 ,........,
n)
such that
wk
0(k
w
......... ( wn
3)
n)
(w1, w2 , w3 ,......., wn )T is the
1, 2,3,...., n)
n
and
k 1
wk
1.
On the basis of the operational rules of the mGqNNs, we can get the aggregation result as described as
Theorem 4.2.
4.8 Theorem: Suppose
universe set U and
wk
0(k
w
(k
1, 2,3,......., n)
be a collection of mGqNNs defined on the
(w1, w2 , w3 ,......., wn )T is the weight vector of ( 1,
1, 2,3,...., n)
Moreover, we have,
k, k, k
k
n
and
k 1
wk
mGqNWGA( 1,
1 . Then mGqNWGA( 1,
2 , 3 ,........,
2 , 3 ,........, n ) such that
2 , 3 ,........, n ) is also a mGqNN.
n)
Abhijit Saha, Florentin Smarandache, Jhulaneswar Baidya and Debjit Dutta, MADM USING m-GENERALIZED
q-NEUTROSOPHIC SETS
264
Neutrosophic Sets and Systems,Vol. 35, 2020
n
k
k 1
wk
n
3
,
m
k 1
3
m
k
3
w
k
qm
qm
3
3
,
m
n
3
m
k 1
k
3
w
k
qm
qm
3
.
Proof: Similar to theorem 4.2.
4.9 Theorem: Suppose
universe set U and
wk
0(k
(k
k, k, k
k
1, 2,3,......., n)
be a collection of mGqNNs defined on the
(w1, w2 , w3 ,......., wn )T is the weight vector of ( 1,
w
n
1, 2,3,...., n)
and
k 1
wk
1 . Then for
2 , 3 ,........, n ) such that
0, 0, 0
0
(where
is the
collection of all mGqNNs defined on the universe set U), we have
mGqNWGA(
1, 0
0
2, 0
3 ,........, 0
n)
0
mGqNWGA( 1,
2 , 3 ,........, n ) .
1, 2,3,......., n)
be a collection of
Proof: Similar to theorem 4.3.
4.10 Theorem: (Idempotency) Suppose
2,
3 ,........,
0, 0, 0
0
k
n)
k
0
such
wk
that
(w1, w2 , w3 ,......., wn )T is the weight vector of
w
mGqNNs defined on the universe set U and
( 1,
(k
k, k, k
k
0(k
n
1, 2,3,...., n)
and
k 1
wk
1
.
If
is the collection of all mGqNNs defined on the universe set U) such that
(where
1, 2,3,......., n , then we have mGqNWGA( 1,
2 , 3 ,........, n )
0.
Proof: Similar to theorem 4.4.
4.11
Theorem:
(Monotonocity)
Suppose
k, k, k
k
and
k, k, k
k
(k
1, 2,3,......., n) be two collections of mGqNNs defined on the universe set U and
w
(w1, w2 , w3 ,......., wn )T
( 1,
k
2,
3 ,........,
k, k
n)
k, k
mGqNWGA( 1,
is
such
weight
wk
that
(k
k
the
vector
0(k
( 1,
of
2 , 3 ,........, n )
n
1, 2,3,...., n)
and
k 1
1, 2,3,......., n)
,
mGqNWGA( 1,
then
wk
as
1
well
as
.
If
2 , 3 ,........, n )
2 , 3 ,........, n ) .
Proof: Similar to theorem 4.5.
4.12 Theorem: (Boundedness) Suppose
w
mGqNNs defined on the universe set U and
( 1,
by:
Then
2,
3 ,........,
min
1 k n
n)
k,
such that
max
1 k n
k,
mGqNWGA( 1,
wk
max
1 k n
0(k
(k
k, k, k
k
1, 2,3,......., n)
(w1, w2 , w3 ,......., wn )T is the weight vector of
1, 2,3,...., n)
n
and
k 1
k
,
2 , 3 ,........, n )
be a collection of
max
1 k n
k,
min
1 k n
wk
k,
1 . Let us define two mGqNNs
min
1 k n
k
.
Abhijit Saha, Florentin Smarandache, Jhulaneswar Baidya and Debjit Dutta, MADM USING m-GENERALIZED
q-NEUTROSOPHIC SETS
265
Neutrosophic Sets and Systems,Vol. 35, 2020
Proof: Similar to theorem 4.6.
5. MULTI ATTRIBUTE DECISION MAKING:
Consider a multi-attribute decision making problem which consists of m different alternatives A1, A2, .........,
Al which are evaluated under the set of n different attributes C1, C2, ......, Cn. Assume that an expert evaluates the
given alternatives Ai (i 1, 2,..., l ) under the attribute C j ( j 1, 2,..., n) and the evaluation result is presented
ij ij ,ij , ij such that 0 ij ,ij , ij 1 and
by the form of m-generalized q-neutrosophic numbers
0 ij
qm
3
ij
qm
3
w j ( j 1, 2,..., n) is
ij
qm
3
3
m
where
i 1, 2,..., l; j 1, 2,..., n
the weight of the attribute such
C j such
that
.
Further
w j 0( j 1, 2,..., n)
assume
and
that
n
w
j 1
j
1.
Then to determine the most desirable alternative (s), the proposed operators are utilized to develop a multi-attribute
decision making with m-generalized q-neutrosophic information, which involves the following steps:
Step-1 Arrange the rating values of the expert in the form of decision matrix D ij
l n
ij ,ij , ij
l n
.
Step-2: Construct aggregated m-generalized q-neutrosophic decision matrix. In order to do that, the proposed
operators can be utilized as follows:
Let R Ri
l 1
be the aggregated m-generalized q-neutrosophic decision matrix, where
Ri mGqNWAA i1 , i 2 ,....., in w1 i1 w2 i 2 ........... wn in
OR
Ri mGqNWGA i1 , i 2 ,....., in w1 i1 w2 i 2 ........... wn in
Step-3: Calculate the score values
S Ri (i 1, 2,..., l ) of m-generalized q-neutrosophic numbers
Ri (i 1, 2,..., m) .
Step-4: Rank all the alternatives Ai (i 1, 2,..., l ) and hence select the most desirable alternative(s).
CASE STUDY:
We consider a multi attribute decision making problem adapted from [15, 17, 18, 19] to demonstrate the
application of the proposed decision making method.
“Suppose there is an investment company that wants to invest a sum of money in the best option available.
There is a panel with four possible alternatives in which to invest the money: (i) A1 is a car company, (ii) A2 is a
food company, (iii) A3 is a computer company and (iv) A4 is an arms company. The investment company must take
a decision according to the following attributes:
(1) C1 is the risk,
(2) C2 is the growth and
(3) C3 is the environmental impact.
The attribute weight vector is given as: w=(0.35, 0.25, 0.40)T. The four alternatives Ai (i 1, 2,3, 4) are to
be evaluated using the m-generalized q-neutrosophic information by some decision makers or experts under the
attributes
C j ( j 1, 2,3) ”.
Abhijit Saha, Florentin Smarandache, Jhulaneswar Baidya and Debjit Dutta, MADM USING m-GENERALIZED
q-NEUTROSOPHIC SETS
266
Neutrosophic Sets and Systems,Vol. 35, 2020
Step-1: The rating values of the expert(s) are given in the form of the following decision matrix
𝑐1
𝑐2
𝑐3
A1
<0.3, 0.1, 0.4>
<0.5, 0.3, 0.4>
<0.3, 0.2, 0.6>
A2
<0.8, 0.2, 0.3>
<0.7, 0.1, 0.3>
<0.7, 0.2 0.2>
A3
<0.5, 0.4, 0.3>
<0.6, 0.3, 0.4>
<0.5, 0.1, 0.3>
A4
<0.6, 0.1, 0.2>
<0.7, 0.1, 0.2>
<0.3, 0.2, 0.3>
D:
Step-2: Using the operator mGqNWAA , we construct the aggregated m-generalized q-neutrosophic decision
matrix
R given below (taking m=3 and q=3):
A1
<0.374405104, 0.173657007, 0.470431609>
A2
<0.741650663, 0.168179283, 0.2550849>
A3
<0.529784239, 0.213796854, 0.322237098>
A4
<0.56691263, 0.131950791, 0.235215805>
Step-3: The score values of the alternatives are calculated as:
S(A1)=0.5767, S(A2)=0.7727, S(A3)=0.6645, S(A4)=0.7332
Step-4: The ranking order of the alternatives are:
A2
A4
A3
A1
which coincides with the ranking order
determined by Jun Ye [15, 17, 18, 19] and hence the most desirable alternative is
A2 .
Now if we want to utilize the mGqNWGA operator instead of mGqNWAA operator, then the steps for
solving the multi attribute decision making problem are as follows:
Step-1: The rating values of the expert(s) are given in the form of the following decision matrix
𝑐1
𝑐2
𝑐3
A1
<0.3, 0.1, 0.4>
<0.5, 0.3, 0.4>
<0.3, 0.2, 0.6>
A2
<0.8, 0.2, 0.3>
<0.7, 0.1, 0.3>
<0.7, 0.2 0.2>
A3
<0.5, 0.4, 0.3>
<0.6, 0.3, 0.4>
<0.5, 0.1, 0.3>
A4
<0.6, 0.1, 0.2>
<0.7, 0.1, 0.2>
<0.3, 0.2, 0.3>
D:
Step-2: Using the operator mGqNWGA , we construct the aggregated m-generalized q-neutrosophic decision
matrix
R given below (taking m=3 and q=3):
Abhijit Saha, Florentin Smarandache, Jhulaneswar Baidya and Debjit Dutta, MADM USING m-GENERALIZED
q-NEUTROSOPHIC SETS
267
Neutrosophic Sets and Systems,Vol. 35, 2020
A1
<0.34086581, 0.2179417, 0.504033104>
A2
<0.73349173, 0.184246871, 0.268903022>
A3
<0.52331757, 0.310497774, 0.331365356>
A4
<0.472580665, 0.156129905, 0.250101937>
Step-3: The score values of the alternatives are calculated as:
S(A1)=0.5396, S(A2)=0.7601, S(A3)=0.6271, S(A4)=0.6887
Step-4: The ranking order of the alternatives are:
A2
A4
A3
A1
which also coincides with the ranking
order determined by Jun Ye [15, 17, 18, 19] and hence the most desirable alternative is still
A2 .
6. CONCLUSIONS:
In this paper, the notion of m-generalized q- neutrosophic sets is proposed and the basic properties of
m-generalized q- neutrosophic numbers (mGqNNs for short) are presented. Also, various types of operations
between the mGqNNs are discussed. Then, two types of m-generalized q- neutrosophic weighted aggregation
operators are proposed to aggregate the m-generalized q- neutrosophic information. Furthermore, score of a
mGqNN is proposed to ranking the mGqNNs. Utilizing the m-generalized q- neutrosophic weighted aggregation
operators and score of a mGqNN, a multi attribute decision making method is developed, in which the evaluation
values of alternatives on the attribute are represented in terms of mGqNNs and the alternatives are ranked
according to the values of the score of mGqNNs to select the most desirable one. Finally, a practical example for
investment decision making is presented to demonstrate the application and effectiveness of the proposed method.
The advantage of the proposed method is that it is more suitable for solving multi attribute decision making
problems because m-generalized q-neutrosophic sets (mGqNSs) are extensions of fuzzy sets, Pythagorean fuzzy
sets, q-rung orthopair fuzzy sets, intuitionistic fuzzy sets, single valued neutrosophic sets, n-hyperspherical
neutrosophic sets and single valued spherical neutrosophic sets.
FUNDING: This research received no external funding.
ACKNOWLEDGEMENTS: Nil.
CONFLICTS OF INTEREST: The authors declare no conflict of interest.
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Received: April 10, 2020.
Accepted: July 1, 2020
Abhijit Saha, Florentin Smarandache, Jhulaneswar Baidya and Debjit Dutta, MADM USING m-GENERALIZED
q-NEUTROSOPHIC SETS
Neutrosophic Sets and Systems, Vol. 35, 2020
University of New Mexico
HESITANT Triangular Neutrosophic Numbers and Their
Applications to MADM
1*
2
Abhijit Saha , Irfan Deli , and Said Broumi
1
3
Faculty of Mathematics, Techno College of Engineering Agartala, , Tripura, India, Pin-799004; Email: abhijit84.math@gmail.com
2
3
Muallim Rıfat Faculty of Education, Kilis 7 Aralık University, 79000 Kilis, Turkey. E-mail: irfandeli@kilis.edu.tr
Faculty of Science, University of Hassan II, B.P 7955, Sidi Othman, Casablanca, Morocco; Email: broumisaid78@gmail.com
*Correspondence: abhijit84.math@gmail.com
ABSTRACT: Hesitant neutrosophic sets can accomodate more uncertainty compare to hesitant fuzzy sets and hesitant
intuitionistic sets. On the other hand, triangular neutrosophic numbers are often used by the decision makers to evaluate their
opinion in multi-attribute group decision making problems. Based on the combination of triangular neutrosophic numbers and
hesitant neutrosophic sets, in this paper, we propose hesitant triangular neutrosophic numbers. Also, we discuss various types
of operations between them including some properties. Then, we propose various types of hesitant triangular neutrosophic
weighted aggregation operators to aggregate the hesitant triangular neutrosophic information. Furthermore, we introduce
score of hesitant triangular neutrosophic numbers to ranking the hesitant triangular neutrosophic numbers. Based on the
hesitant triangular neutrosophic weighted aggregation operators and score of hesitant triangular neutrosophic numbers, we
develop a multi attribute decision making (MADM) approach, in which the evaluation values of alternatives on the attribute
are represented in terms of hesitant triangular neutrosophic numbers and the alternatives are ranked according to the values of
the score of hesitant triangular neutrosophic numbers to select the most desirable one. Finally, we give a practical example,
including a comparision study with the other existing method, for enterprise resource planning system selection to verify the
application and effectiveness of the proposed method.
Keywords: Neutrosophic sets, hesitant triangular neutrosophic numbers, aggregation operators, score value, decision
making.
_____________________________________________________________________________________________________
1. INTRODUCTION
In our real life, most of the mathematical problems do not contain exact or complete information about the
given mathematicalmodeling. Therefore, fuzzy set theory by introduced Zadeh [01] is a proper tool to process
inexact information because it allows the partial belongings of an element in a set with a membership function.
Atanassov [02] generalized fuzzy sets to intuitionistic fuzzy sets by adding a non-membership function to
overcome problems that contain incomplete information. In case of fuzzy sets and intuitionistic fuzzy sets, the
membership (or non-membership) value of an element in a set is a unique value in the closed interval [0, 1]. But
since 2009, researchers begin to investigate, what if the membership (non-membership) value of an element in a
set is a discrete finite subset of [0, 1]. In order to tackle this situation, Torra [03] proposed the concept of a
hesitant fuzzy set, which as an extension of a fuzzy set arises from our hesitation among a few different values
lying between the number 0 and 1. Thus the hesitant fuzzy set can more accurately reflect the people’s hesitancy
in stating their preferences over objectives compared to the fuzzy set and its classical extensions. Beg and Rashid
[04] introduced the concept of intuitionistic hesitant fuzzy sets by merging the concept of intuitionistic fuzzy sets
and hesitant fuzzy sets.Various researchers have analyzed the decision making problems under fuzzy, hesitant
fuzzy, intuitionistic fuzzy and intuitionistic hesitant fuzzy environment in Li [05], Ye [06], Xia and Xu [07], Xu
and Xia [08], Wei et al. [09], Xu and Xia [10], Xu and Xia [11], Xu and Zhang [12], Chen et al. [13], Qian et al.
[14], Yu [15], Yu [16], Ye [17], Shi et al. [18], Pathinathan and Johnson [19], Joshi and Kumar [20], Liu [21],
Nehi [22], Zhang [23], Chen and Huang [24], Yang et al. [25], Lan et al. [26] and Zhang et al. [27].
Although intuitionistic fuzzy sets naturally include hesitancy degree to handle uncertain information, it cannot
manage indeterminate information properly because it is dependent on memebership and non-membership
degrees. To handle this situation, Smarandache [28] introduced the neutrosophic set which is basically a
powerful general formal framework that generalizes the concept of the classical set, fuzzy set, intuitionistic fuzzy
set. A neutrosophic set is characterized explicitly by truth-membership function, indeterminacy-membership
Abhijit Saha, Irfan Deli, and Said Broumi, HESITANT Triangular Neutrosophic Numbers and Their Applications
to MADM
Neutrosophic Sets and Systems, Vol. 35, 2020
270
function and falsity membership function and it has applications on image segmentation in Gou and Cheng [29],
Gou and Sensur [30], on clustering analysis in Karaaslan [31], on medical diagnosis problem in Ansari et al. [32]
etc.The neutrosophic set theory have also studied in Wang et al. [33], Wang et al. [34], Gou et al. [35], Ye [36],
Sun et al. [37], Ye [38] and Abdel Basset et al. [39]. The neutrosophic set cannot represent uncertain, imprecise,
incomplete and inconsistent information with a few different values assigned by truth-membership degree,
indeterminacy-membership degree and falsity-membership degree due to doubts of decision maker. In such a
situation, all the decision making algorithms based on neutrosophic sets are difficult to use for such a decision
making problem with three kinds of hesitancy information that exists in the real world. To overcome this
situation, Ye [40] introduced the concept of hesitant neutrosophic sets which is characterized by three
membership degrees, namely-truth membership degrees, indeterminacy membership degrees and falsity
membership degrees which is a few different values lying between the number 0 and 1.
Aggregation operators play a vital role in many fields such as decision making, supply chain, personnel
evaluation and financial investment to solve multi-criteria group decision making problems. A series of
aggregation operatorsin Xia et al. [41], Wang et al. [42], Zhao et al. [43], and Peng [44] were developed based
on fuzzy and hesitant fuzzy information and those were applied in solving decision-making problems. Xu [45],
Wan and Dong [46], Wan et al. [47] and Xu and Yager [48] presented an averaging and geometric aggregation
operators for aggregating the different intuitionistic fuzzy sets based information. Wang and Liu [49] proposed
some Einstein weighted geometric operators for intuitionistic fuzzy sets. Liu et al. [50] proposed some
generalized neutrosophic number Hamacher aggregation operators. Liu and Wang [51] defined few neutrosophic
normalized, weighted Bonferroni mean operators.Chen and Ye [52] used single-valued neutrosophic dombi
weighted aggregation operators for solving a multiple attribute decision-making problem. Some more
aggregation operators on neutrosophic environment can be found in Zhao et al. [53], Liu and Shi [54] and Liu
and Tang [55].
Since Smarandache put forward the concept of neutrosophic sets, the neutrosophic number is given by Şubaş
[56] subsequently, and it has been made much deeper by many authors in Abdel-Basset [57]. As a special
neutrosophic number,Şubaş gave two special forms of single valued neutrosophic numbers such as single valued
trapezoidal neutrosophic numbers and single valued triangular neutrosophic numbers on the real number set R.
Now the theory of neutrosphic number has become the fundamental of neutrosophic decision making. For
example; Deli and Şubaş [58] introduced the concepts of cut sets of neutrosophic numbers and also they applied
to single valued trapezoidal neutrosophic numbers and triangular neutrosophic numbers. Finally they presented a
ranking method by defining the values and ambiguities ofneutrosophic numbers. Also, by using the value and
ambiguity index, Biswas et al. [59] presented a multi-attribute decision making method. Broumi et al. [60] gave
an application shortest path problem under triangular fuzzy neutrosophic numbers. Deli and Şubaş [61]
developed an approach to handle multicriteriadecision making problems under the single valued triangular
neutrosophic numbers. Also, they presented some new geometric operators including weighted geometric
operator, ordered weighted geometric operator and ordered hybrid weighted geometric operator. Ye [62],
Biswas et al. [63] and Deli [64] proposed some weighted arithmetic operators and weighted geometric operators
to present some multi attribute decision making methods. Karaaslan [65] introduced Gaussiansingle valued
neutrosophic numbers and applied to a multi attribute decision making. Öztürk [66] and Deli and Öztürk [67, 68]
initiated concept of distance measure based on cut sets, magnitude function, 1. and 2. centroid point and 1. and 2.
score function. Deli [69] defined concept of centroid point based on single valued trapezoidal neutrosophic
numbers and examine several useful properties. Also, he developed hamming ranking value and Euclidean
ranking value of single valued trapezoidal neutrosophic numbers. Chakraborty et al. [70] presented a decision
making method by introducing different forms of triangular neutrosophic numbers including deneutrosophication techniques. Fan et al. [71] defined linguistic neutrosophic number Einstein sum, linguistic
neutrosophic number Einstein product, and linguistic neutrosophic number Einstein exponentiation operations
based on the Einstein operation and used them to develop some MADM problems. Garg and Nancy [72]
introduced some linguistic single valued neutrosophic power aggregation operators and presented their
applications to group decision making process. Zhao et al. [73] developed induced choquet integral aggregation
operators with single valued neutrosophic uncertain linguistic numbers. Recently, Deli and Karaaslan [74]
defined generalized trapezoidal hesitant fuzzy numbers and Deli [75] presented a TOPSIS method formulticriteria decision making problems by using the numbers. Some more trapezoidal/triangular hesitant fuzzy
numbers can be found in Zhang et al. [76] and Ye [77].
Motivated by the idea of triangular neutrosophic number, hesitant neutrosophic set and aggregation operators,
the aim of this present article is:
(1) To present the idea of hesitant triangular neutrosophic numbers.
(2) To define few operations between hesitant triangular neutrosophic numbers and study their basic
properties.
Abhijit Saha, Irfan Deli, and Said Broumi, HESITANT Triangular Neutrosophic Numbers and Their Applications
to MADM
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Neutrosophic Sets and Systems, Vol. 35, 2020
(3) To develop a few weighted aggregation operators such as hesitant triangular neutrosophic weighted
arithmetic aggregation operator of type-1, hesitant triangular neutrosophic weighted arithmetic aggregation
operator of type-2, hesitant triangular neutrosophic weighted geometric aggregation operator of type-1 and
hesitant triangular neutrosophic weighted geometric aggregation operator of type-2.
(4) To propose a decision making method based on the hesitant triangular neutrosophic weighted
aggregation operators to handle multicriteria decision making problems with hesitant triangular neutrosophic
information.
To do so, the rest of the article is arranged as follows:
In section 2, we review some basic concepts. In Section 3, we propose hesitant triangular neutrosophic number
and illustrate it with an example. Also, we discuss various types of operations between them including some
properties. In section 4, we propose various types of hesitant triangular neutrosophic weighted aggregation
operators to aggregate the hesitant triangular neutrosophic information. Furthermore, we introduce the score of a
hesitant triangular neutrosophic number to ranking the hesitant triangular neutrosophic numbers. In section 5,
based on the hesitant triangular neutrosophic weighted aggregation operators and score of hesitant triangular
neutrosophic numbers, we develop a multi attribute decision making approach, in which the evaluation values of
alternatives on the attribute are represented in terms of hesitant triangular neutrosophic numbers and the
alternatives are ranked according to the values of the score of hesitant triangular neutrosophic numbers to select
the best (most desirable) one. Also, we present a practical example for enterprise resource planning system
selection to demonstrate the application and effectiveness of the proposed method. Section 6 is devoted for
comparative study. In final section, we present the conclusion of the study.
2. PRELIMINARIES:
A neutrosophic set is a part of neutrosophy which studies the origin, nature and scope of neutralities as well as
their interactions with different ideational spectra and is a powerful general formal framework that generalizes
the traditional mathematical tools such as fuzzy sets and intuitionistic fuzzy sets.
Definition 1: [34] A single-valued neutrosophic set A on universe set E is given by
A=
x, TA x , IA x , FA x : x ∈ E
where TA : E → 0,1 , IA : E → 0,1 , and FA : E → 0,1 satisfy the condition 0 ≤ TA x + IA x + FA x ≤ 3, for
every x ∈ E. The functions TA , IA , and FA define the degree of truth-membership function, indeterminacymembership function and falsity-membership function, respectively.
Definition 2: [52] A = x, TA x , IA x , FA x : x ∈ E and B =
valued neutrosophic sets and λ ≠ 0. Then,
1
1. A B { x,1
T ( x)
TB ( x)
1 A
1
T
(
x
)
A
1 TB ( x)
p
p
1
p
x, TB x , IB x , FB x : x ∈ E be two single1
,
1
,
p
p
1 I ( x)
1 I B ( x)
A
1
I B ( x)
I A ( x)
p
1
: x E}
1
1 F ( x) p 1 F ( x) p
p
A
B
1
FA ( x)
FB ( x )
1
1
2. A B { x,
, 1
,
1
1
p
p
p
p
p
p
1 TB ( x)
I B ( x)
1 TA ( x)
I ( x)
1
1 A
1 I A ( x)
1 I B ( x)
TA ( x)
TB ( x)
1
1
: x E}
1
p
p
p
FB ( x )
FA ( x)
1
1
(
)
F
x
A
1 FB ( x)
3. . A { x,1
1
T ( x)
1 A
1 TA ( x)
p
1
p
,
1
1 I A ( x)
1
I A ( x)
p
1
p
,
1
1
: x E}
p
p
1 FA ( x)
1
FA ( x)
Abhijit Saha, Irfan Deli, and Said Broumi, HESITANT Triangular Neutrosophic Numbers and Their Applications
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Neutrosophic Sets and Systems, Vol. 35, 2020
1
4. A { x,
1 T ( x) p
A
1
T
(
x)
A
1
p
1
, 1
I ( x) p
1 A
1 I A ( x)
1
p
1
, 1
1
: x E}
F ( x) p p
1 A
1 FA ( x)
By combining single-valued neutrosophic sets and hesitant fuzzy sets, Ye (2015a) introduced the singlevalued neutrosophic hesitant fuzzy set as a further generalization of the concepts of fuzzy set, intuitionistic fuzzy
set, single-valued neutrosophic set. He also developed single-valued neutrosophic hesitant fuzzy weighted
averaging operator and single-valued neutrosophic hesitant fuzzy weighted geometric operator and applied them
to solve a multiple-attribute decision- making problem.
Definition 3: [40] A hesitant neutrosophicset on universe set E is given by
N=
x, TN x , IN x , FN x : x ∈ E
in whichTN (x), IN (x) and FN (x)are three sets of some values in [0,1], denoting the possible truth-membership
hesitant degrees, indeterminacy-membership hesitant degrees, and falsity-membership hesitant degrees of the
element x ∈ E to the set N, respectively, with the conditions 0 ≤ δ , γ , η ≤ 1and 0 ≤ δ++γ+ + η+ ≤ 3,
where
+
δ ∈ TN x , γ ∈ IN x , η ∈ FN x , δ+ ∈ TN+ x =
δ∈T N x max δ , γ ∈
IN+ (x) =
{γ},
γ∈I N (x) max
andη+ ∈ FN+ (x) =
{η
η ∈F N (x) max
For N1 = x, TN 1 x , IN 1 x , FN 1 x : x ∈ E and N2 =
neutrosophicsets and λ ≠ 0. Then,
}, for x ∈ E.
x, TN 2 x , IN 2 x , FN 2 x : x ∈ E
be
two
hesitant
1. N1 N1 { x, TN1 ( x) TN2 ( x), I N1 ( x) I N2 ( x), FN1 ( x) FN2 ( x) : x X }
1 TN1 ( x ), 2 TN2 ( x ), 1 I N1 ( x ), 2 I N2 ( x ),1 FN1 ( x ),2 FN2 ( x )
{ x,{1 2 1 . 2 },{ 1 . 2 },{12 } : x X }
2. N1 N1 { x, TN1 ( x) TN2 ( x), I N1 ( x) I N2 ( x), FN1 ( x) FN2 ( x) : x X }
1 TN1 ( x ), 2 TN2 ( x ), 1I N1 ( x ), 2 I N2 ( x ),1FN1 ( x ),2 FN2 ( x )
3. .N1
4. N1
Definition
1TN1 ( x ),1I N1 ( x ),1FN1 ( x )
1TN1 ( x ),1I N1 ( x ),1FN1 ( x )
4:
[56]
Let
{ x,{1 . 2 },{ 1 2 1 . 2 },{1 2 12 } : x X }
{ x,{1 (1 1 ) },{ 1 },{1 } : x X }( 0)
{ x,{1 },{1 (1 1 ) },{1 (1 1 ) } : x X }( 0)
a1 b1 c1
such
that
a1 , b1 , c1 R.
A
triangular
neutrosophic
number
A (a1 , b1 , c1 ); wA , u A , y A is a special neutrosophic set on the real number set R, whose truth-membership
function
A : R 0, wA
membership function
,
indeterminacy-membership function
A : R y A ,1
( x a1 ) wA
, a1 x b1
b1 a1
(c x) wA
A ( x) 1
, b1 x c1
c1 b1
0,
otherwise
A : R u A ,1
and falsity-
are given as follows;
b1 x u A x a1
,
b1 a1
x b1 u A c1 x
, v A ( x)
,
c1 b1
1,
a1 x b1
b1 x c1
otherwise
Abhijit Saha, Irfan Deli, and Said Broumi, HESITANT Triangular Neutrosophic Numbers and Their Applications
to MADM
273
Neutrosophic Sets and Systems, Vol. 35, 2020
b1 x y A x a1
, a1 x b1
b1 a1
x b y c x
1
1
A
A ( x)
, b1 x c1
c
b
1
1
1,
otherwise
Since triangular neutrosophic numbers ([56], [58]) is a special case of trapozidial neutrosophic numbers
(Ye 2017), operations of trapozidial neutrosophic numbers (Ye 2015b, 2017) based on algebraic sum and
algebraic product for triangular neutrosophic numbers can be given as;
If A a1 , b1 , c1 ; wA , u A , y A and B a2 , b2 , c2 ; wB , uB , yB be two triangular neutrosophic numbers
and γ ≠ 0, then we have
1. A B a1 a1 , b1 b2 , c1 c2 ; wA wB wA .wB , u A .uB , y A . yB
2. A.B a1a2 , b1b2 , c1c2 ; wA .wB , u A uB u A .uB , y A yB y A . yB
3. A
4. A
a1, b1, c1 ;1 (1 wA ) , uA , yA
a
1
, b1 , c1 ; wA ,1 (1 u A ) ,1 (1 y A )
Definition 5: [56] Let A a, b, c ; wA , u A , y A be atriangular neutrosophic number. Then,score function of
A, is denoted by SY A , is defined as:
SY A =
Definition 6: [61] Let Aj =
numbers. Then,
1.
1
a + b + c × 2 + μA − νA − γA
8
a j , bj , cj , ; wA j , uA j , yA j
j = 1,2, … , n be a collection of triangular neutrosophic
Triangular neutrosophic weighted arithmetic operatoris defined as;
n
Nao A1 , A2 , … , An =
2.
wj Aj
j=1
Triangular neutrosophic weighted geometric operatoris defined as;
n
Ngo A1 , A2 , … , An =
Aj
wj
j=1
where, w = w1 , w2 , … , wn is a weight vector associated with the Nao or Ngo operator, for every j ( j =
1,2, … , n) and wj ∈ 0,1 with nj=1 wj = 1.
T
3. HESITANT TRIANGULAR NEUTROSOPHIC NUMBERS:
In this section, the concept of a hesitant triangular neutrosophic number is presented on the basis of the
combination of triangular neutrosophic numbers and hesitant fuzzy sets as a further generalization of the concep
ttriangular neutrosophic numbers. A hesitant triangular neutrosophic number is a special hesitant neutrosophic
set on the real number R, whose truth-membership function, indeterminacy-membership function and falsitymembership function are expressed by several possible functions.
Definition
7.
Let
a1 b1 c1
such
a1 , b1 , c1 R,
that
wai [0,1](i I m {1, 2,..., m}),
uai [0,1](i I n {1, 2,..., n}) and yai [0,1](i I k {1, 2,..., k}). A hesitant triangular neutrosophic
number a
(a1 , b1 , c1 );{wai : i
I m },{uaj : j
I n },{ yal : l
real number R, whose truth-membership functions
Ik }
HTRI
a
is a special hesitant neutrosophic set on the
: R 0, wai (i I m ),
indeterminacy-membership
function aHTRI : R 0, uaj ( j I n ) and falsity-membership function aHTRI : R 0, yal (l I k ) are given as
follows;
Abhijit Saha, Irfan Deli, and Said Broumi, HESITANT Triangular Neutrosophic Numbers and Their Applications
to MADM
274
Neutrosophic Sets and Systems, Vol. 35, 2020
( x a1 )
},
b1 a1
a1
x
{wai : i I m }}, x
(c x )
{ 1
}, b1
c1 b1
i
{ : {wa :i I m }}
b1
{
{ :
{wai :i I m }}
{ :
HTRI
a
b1
,
x
c1
},
a1
{0}, otherwise
{
{ :
b1
x
b1
{uaj : j I n }}
(x
a1
{uaj : j
x b1
{
c1
j
{ua : j I n }}
I n }}, x b1
(c1 x)
}, b1
b1
{ :
HTRI
a
{ :
a1 )
x
b1
x
c1
{1}, otherwise
{
{ :
b1
x
b1
{ yal :l I k }}
(x
a1
{ yal : l
x b1
{
c1
l
{ : { ya :l I k }}
{ :
HTRI
a
a1 )
},
a1
I k }}, x
(c1 x)
},
b1
b1
b1
x
b1
x
c1
{1}, otherwise
Example 8. a
(1, 2,5);{0.8,0.9},{0.4,0.5,0.6},{0.4} is a hesitant triangular neutrosophic number whose
truth membership function, indeterminacy membership function and falsity membership functionare given
respectively by:
{0.8( x 1), 0.9( x 1)}, 1 x
x 2
{0.8, 0.9},
HTRI
( x)
a
{0.8
x)
(5
3
, 0.9
{0},
x)
(5
3
2
x
}, 2
otherwise
{1.6 0.6 x,1.5 0.5 x,1.4 0.4 x}, 1
{0.4, 0.5, 0.6}, x 2
HTRI
( x)
a
{
HTRI
( x)
a
,
5
{1.6 0.6 x}, 1 x 2
x 2
{0.4},
{0.2 x}, 2 x 3
{1}, otherwise
0.6 x
5 x 0.5 0.4 x 1
, 0.
,
},
3
3
3
{1},
x
x
2
2
5
otherwise
4. OPERATIONS ON HESITANT TRIANGULAR NEUTROSOPHIC NUMBERS:
In this section, we introduce various operations between hesitant triangular neutrosophic numbers and
demonstrate their basic properties.
Definition 9. Let a
b
and
1. a
(a2 , b2 , c2 );{wbi : i
0 , then
b
(a1
{
1 2
:
I m1 },{uaj : j
(a1 , b1 , c1 );{wai : i
a2 , b1
1
I m2 },{ubj : j
b2 , c1
{uaj : j
I n2 },{ ybl : l
c2 );{
I n1 },
2
1
2
{ubj : j
I n1 },{ yal : l
I k2 }
1 2
:
I n2 }}, {
I k1 }
and
be two hesitant triangular neutrosophic numbers
1
{wai : i
1 2
:
1
I m1 },
{ yal : l
2
{wbi : i
I k1 },
2
I m2 }},
{ ybl : l
I k2 }}
Abhijit Saha, Irfan Deli, and Said Broumi, HESITANT Triangular Neutrosophic Numbers and Their Applications
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2. a
b
(a1a2 , b1b2 , c1c2 );{
{uaj : j
1
I n1 },
a
( a, b, c);{1 (1
4.
a
(a , b , c );{
b
c
(a3 , b3 , c3 );{wci : i
and
, 1,
I m2 },{ubj : j
I m3 },{ucj : j
:i
{wbi : i
2
2
:
1 2
{ yal : l
1
I m1 }},{
I m1 }},{1 (1
I m2 }},{
1
I k1 },
{uaj : j
:
:
1 2
{ ybl : l
2
I n1 }},{
{uaj : j
) :
2
I k2 }}
{ yal : l
:
I k1 }}
I n1 }},
I k1 }}
:l
I m1 },{uaj : j
(a1 , b1 , c1 );{wai : i
(a2 , b2 , c2 );{wbi
1
{wai : i
{wai : i
{ yal
I m1 },
I n2 }}, {
) :
:
) :
Theorem 10. Let a
{wai : i
1
{ubj : j
2
3.
{1 (1
:
1 2
I n2 },{ ybl
I n1 },{ yal : l
I k1 }
,
:l
I k2 }
and
I n },{ ycl : l
Ik }
be three hesitant triangular neutrosophic numbers
3
3
0 , then
2
1. a
b
b
a
5.
(a
b)
(
2. a
b
b
a
6.
(a
b)
(
a)
(
b)
7.
8. (
(a
b)
a
(
(
a)
a)
(
(
2
b)
a)
2
{wbi : i
3. .a
(b
c)
(a
b)
c
4. a
(b
c)
(a
b)
c
2)
1
a)
1
b)
(
Proof:1-2 straight forward.
3. a
(b
c)
( a1, b1, c1 ); wa , ua , ya
3
{wci
3
{ ycl : l
I k3 }}
(a1
(a2
a3 ), b1
{wai
:i
I n2 },
3
1
j
(a1
I m3 }},{
2
a3 , b1
:i
2
:j
I n2 },
3
{ ycl
:l
I k3 }}
2
b3 ), c1
:i
{ucj
3
:j
I n2 },
(c2
c3 )); {
{wci
3
1
{wci
3
I n3 }},{
{ucj
3
(
1
:i
c3 ); {
I m2 },
c3 ;{
3
I n3 }}, {
2
3
:i
3
{ yal
1
I k1 },
I k2 },
:l
:j
{ ybl
:j
:l
2 31 ) :
I n1 },
{ ycl : l
3
I m2 },
I k2 },
3
{ubj :
2
I k3 }}
1 2 3) :
1 3
{uaj
1
2
2
{uaj
1 2
:
{ ybl
1( 2
2 3) : 1
1 2 3
:l
:
2 3)
2 3
I m3 }},{
2 3
{ ybl : l
2
:
2 3
:j
I k1 },
2
:
1 2 3
2
I m3 }},{ 1 (
{ yal : l
2 3) : 1
c2
:i
b3 , c2
:j
I m2 },
b3 , c1
{wbi
2
a3 , b2
{ubj
I n3 }},{ 1 (
b2
I m1 },
:
{wbi
{ucj : j
{ubj
2 3
(b2
I m1 },
a2
{wai
1
:i
a2
I n1 },
I k2 },
(1)
and
(a
b)
c
(a1
a2 , b1
b2 , c1
c2 );{
{uaj
: j I n1 }, 2 {ubj
1 2: 1
(a3 , b3 , c3 );{wci : i I m3 },{ucj
{
((a1
a3 ,(b1
{wai
:i
I m1 },
I n2 }
3
{ucj : j
1
j
a2 )
(a1
a2
a3 , b1
1
{wai
2
{ubj
3
{ ycl : l
:i
:j
I m1 },
I n2 },
b2 )
{wbi
2
:i
b3 , c1
2
{wbi
3
{ucj
:i
:j
2
1 2
:
I n2 }}, {
1 2
:j
I n },{ ycl
3
:l
c2 )
I m2 },
3
1 2) 3: 1
c2
I m2 },
3
I n3 }},{
:i
1
{wci
1 2 3
1
1
2
{wbi : i
I m2 }},
{ ybl
I k2 }}
2
:i
3
I k1 },
{ ybl : l
2
3
:l
{ yal
:l
1 2 3
I k1 },
:
1
(
1
{uaj :
I k2 },
2 3
I m3 }},{
1
1 2)
1 2) 3
I k3 }}
Hence from eq. 1-2, we have, a
4. Proof is similar to 3
I k1 },
I m3 }},{(
2
:
:l
2
3
{ yal : l
c3 ); {
:
{ yal
I m1 },
Ik }
c3 ); {(
{wci
{wai : i
1
:j
b3 ,(c1
I n3 }},{(
b2
1
1 2
:
1
2
3
1 3
{uaj : j
{ ybl
:l
1 1) 3
1
j
I n1 },
{ ycl : l
2
:
{ubj :
I k3 }}
1 2 3) :
I n1 },
I k2 },
(2)
(b
c)
(a
b)
c.
Abhijit Saha, Irfan Deli, and Said Broumi, HESITANT Triangular Neutrosophic Numbers and Their Applications
to MADM
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(a
5.
b)
(a1
{
{uaj
:
1 2
2
a)
I m2 }},{(
1 2)
1
{ yal : l
I k1 },
{ ybl : l
a2 , b1
:
I n1 },
:
1
{
:
2
{uaj
2
1)
:j
a
2
1
{ yal : l
{uaj
{
(
1
1
1
:i
2
{
1
1
:
{ yal
1
( 1a1
1
1
1)
2
:
( 1a1
{
1
1
b
and
(a2 , b2 , c2 );{wbi
0 , then
:
1
:
1
:
:l
2 a1 , 1b1
2
:
{uaj
1
1
1)
I m2 },{ubj
I k1 },
2
:
{wai : i
1
1 2)
:l
I m2 }},
I k2 }}
I m1 },
:
:
1
1
{ yal
1
1
1)
1
)
2
1)
I n1 },
{ ybl : l
I k2 }}
(3)
{uaj : j
1
:
I n1 }},
{wbi : i
2
I m2 }},
(1 (1
:l
2)
{ubj
)
I m2 }},{
I k1 },
(1
2
:i
2)
{ ybl
2
:
:j
I n2 }},{
1
I k2 }}
:i
1
:
2
:l
{wai
1
1
2
I m1 },
:
b) .
(1
1)
(1
:
1
1)
{ yal
:
2
{wai : i
1
I m1 }},{
1
1
:
{wai : i
1
{ yal
(1
1)
2
2
:
:
wa},
1
(5)
I m1 }},{
(1
2
1
:
:
(1 (1
1)
1
1)
1
1
{uaj : j
1
{wai
1)
2)
I n1 }},
:
I k1 }}
:l
I m1 }},{
1)
I k1 }}
:l
2b1 , 2c1 );{1
2
1
1 (1
{wai : i
1
2c1 );{(1
1
I m2 }},
I k1 }}
:
1
2
2)
{wbi
{ yal
(
2
1
{wbi : i
(4)
:l
1
I n1 }},{
2 ):
1)
I m1 },
:j
2
I m1 },
I k1 },
I m1 }},{
:
2
2 c1 );{1
1
1
2
:
1
{uaj : j
:
1
I n1 }},
I k1 }}
:j
I n1 }},{
2)
a =(
(1
2c1 );{1
2b1 , 1c1
:j
{ ybl
I k2 }}
:i
1
{uaj
( 2 a1,
:j
{wbi : i
2
I n2 } }, {(
{wai : i
1
{ yal : l
c2 );{1 (1
1
2b1 , 1c1
(1
:
1
:l
2 )c1 );{1
I n1 }},{
(a1 , b1 , c1 );{wai : i
:i
I n2 }},{
2b1 , 1c1
{uaj
{ yal
{wai
1
2 )b1 ,( 1
I k1 }}
2
2
a)
:j
:
{ ybl
(
2
:l
{ubj : j
2
I m1 },
1 2 ))
2
c2 );{(1 (1
):
2
b)
:l
1 )(1
1
Hencefrom eq. 5-6, we have (
8. Proof is similar to 7
Definition 11. Let a
:
2
:j
1
{ yal
1
1
2)
{wai : i
1
I k2 }}
2 a1 , 1b1
(1 (1
1
1
2
{ ybl : l
{uaj
1
1
I k1 },
1
{wai : i
1
( a2 , b2 , c2 );{1 (1
( 1a1, 1b1, 1c1 );{1 (1
I m1 }},{
i
{
:
:
(1
1
2
:
1 2
I n1 },
1)
I m2 }}, {
I n1 }},{
:j
1)
b2 , c1
2 a1 , 1b1
a)
2
2
2 ) a1 ,( 1
( 1a1
a)
I n2 }},{
2)
{ubj
(a
1
1
1
I n2 }}, {
b2 , c1
)(1 (1
:
1 2
I k2 }}
I n2 }},{
a2 , b1
{wbi
((
:j
I n1 },
Hence from eq. 3-4, we have
6. Proof is similar to 5
(
2
{uaj : j
I k1 }}
a2 , b1
( a1
And
:j
1
{ubj : j
2
{ubj
(1 (1
2)
1
c2 );{1 (1
{ yal : l
1
( a1
1
:
( a1, b1, c1 );{1 (1
{
7. (
2
b2 , c1
{uaj : j
1
b)
(
2
{wbi : i
and
(
I n1 },
c2 );{
{ubj
2
( a1
1
:j
b2 , c1
(a1 a2 ), (b1 b2 ), (c1 c2 ));{1 (1 (
(
{
1
a2 , b1
1
1
1
2
:
I m1 },{uaj : j
I n2 },{ ybl
{ yal
1
a)
1
:l
1)
(
:l
2
1 (1
2
I k1 }}
wa},
(6)
a) .
I n1 },{ yal : l
I k2 }
1)
I k1 }
and
be two hesitant triangular neutrosophic numbers
Abhijit Saha, Irfan Deli, and Said Broumi, HESITANT Triangular Neutrosophic Numbers and Their Applications
to MADM
277
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1. a
b
(a1
a2 , b1
1
c2 );{1
b2 , c1
2
1
1
{
1
1
2
2
2
1
2
1
2
2
2
1
1
1
1
2
2
1
b
2
1
2 2
2
1
a
2
2
2
1
1
2
1
1
2
2
1
1
1
1
a
1
2 2
2
2
2
1
{ yal : l
1
1
1
1
Theorem 12. Let a
c
and
, 1,
2
1
2 2
2
(a3 , b3 , c3 );{wci
I m2 }},
2
{ubj : j
2
{wbi : i
I n2 }},
{ ybl : l
2
I k2 }}
:
1
2 2
2
{wai : i
1
I m1 },
{wbi : i
2
I m2 }},
2
{uaj : j
1
:
I n1 },
2
{ubj : j
I n2 }},
{ yal : l
1
I k1 },
2
{ ybl : l
I k2 }}}
1
2 2
:
1
{wai : i
1
I m1 }},{
1
1
2
1
2 2
2
:
1
{uaj : j
I n1 }},
2
I k1 }}
2
{1
b
2
2
2
1
1
(a2 , b2 , c2 );{wbi
:
1
2 2
2
(a1 , b1 , c1 );{
1
2
I m1 },
2
1
:
2
2
2
( a1, b1, c1 );{1
{
1
1
2 2
2
2
1
I k1 },
1
1
4.
2
2
1
1
1
I n1 },
{ yal : l
1
2
{1
3.
:
1
1
{uaj : j
2
1
1
1
{1
2
2
{wai : i
1
2
1
1
1
1
(a1a2 , b1b2 , c1c2 );{
1
1
:
2
1
{
2. a
1
:
1
2 2
2
2
1
2 2
:
1
1
2 2
2
:
1
{wai : i
1
2
{ yal : l
:i
1
2 2
2
:
1
{uaj : j
I n1 }},
2
1
I k1 }}
2
(a1 , b1 , c1 );{wai : i I m1 },{uaj : j
:i
1
I m1 }},{1
I m2 },{ubj : j
I m3 },{ucj : j
I n1 },{ yal : l
I k1 }
,
I n2 },{ ybl
:l
I k2 }
and
I n },{ ycl
3
:l
Ik }
be three hesitant triangular neutrosophic numbers
3
0 , then
Abhijit Saha, Irfan Deli, and Said Broumi, HESITANT Triangular Neutrosophic Numbers and Their Applications
to MADM
278
Neutrosophic Sets and Systems, Vol. 35, 2020
1. a
b
b
a
2. a
b
b
a
3. a
(b
c)
(a
b)
c
4. a
(b
c)
(a
b)
c
5.
(a
b)
(
a)
(
6.
7. (
8. (
(a
1
1
b)
b) (
a) (
b)
a)
a ( 1 a) ( 2
2)
a ( 1 a) ( 2 a)
2)
Proof:1.-2. Straight forward.
3. a
(b
c)
(a1, b1, c1 );{wai
I m1 },{uaj : j
:i
1
{1
1
2
1
2
2
2
2
1
{
1
1
2
2
3
1
2
2
1
3
2
3
1
1
2
2
(a1
(a2
2
2
3
3
a3 ), b1
{wbi : i
2
1
2 2
2
(b2
:
2
{ubj : j
I n2 },
:
2
{ ybl : l
I k2 },
b3 ), c1
(c2
1
1
1
3
{wci : i
b3 , c2
c3 );
I m3 }},
3
{ucj : j
I n3 }},
{ ycl : l
3
I k3 }}
c3 ));
1
1
2 2
1
1
1
2
I m2 },
a3 , b2
2
{1
2
( a2
2
1
2
:
I k1 }
2
1
2 2
2
1
{
1
2 2
2
3
I n1 },{ yal : l
2
1
2
2
2
2
3
1
:
1
{wai : i
I m1 },
2
{wbi : i
I m2 },
3
{wci : i
I m3 }},
1
2 2
2
3
2
2
1
1
1
1
1
1
2
2
2
2
2
1
3
1
2 2
2
3
2
Abhijit Saha, Irfan Deli, and Said Broumi, HESITANT Triangular Neutrosophic Numbers and Their Applications
to MADM
279
Neutrosophic Sets and Systems, Vol. 35, 2020
1
{
1
1
1
1
2
1
2
2
1
2
1
2
2
2
2
1
3
3
2
1
1
2
2
2
2
2
1
3
3
2
2
1
1
2
1
1
2
2
1
2
2
1
3
3
2
2
1
1
1
a2
2
2
1
(a1
1
2 2
1
1
a3 , b1
2
b2
2
2
b3 , c1
2
2
1
3
c2
{wai : i
I m1 },
2
1
{
1
1
2
2
2
1
{wbi : i
3
2
1
1
2
2
2
2
2
1
3
3
2
1
1
2
2
2
1
1
1
(a
2
b)
(a1
2
2
2
2
1
3
a2 , b1
b2 , c1
2
2
2
1
1
1
2
2
1
2 2
2
1
{ucj : j
I n3 }},
{ yal : l
1
I k1 },
{ ybl : l
I k2 },
3
{ ycl : l
I k3 }}
2
2
1
1
2
2
1
1
2
2
2
2
1
3
1
2 2
2
3
:
2
I m3 }},
:
:
2
{uaj : j
1
{ yal : l
1
I n1 },
I k1 },
{ubj : j
2
2
I n2 },
{ ybl : l
I k2 },
{wai : i
I m1 },
3
{ucj : j
{ ycl : l
3
I n3 }},
I k3 }}
(7)
2
1
2
2
1
2
2
1
1
2
1
I n1 },
1
:
1
{uaj : j
:
1
{ yal : l
2
1
2 2
2
2
2
:
1
2
{wbi : i
I m2 }},
2
{ubj : j
I n2 }},
2
1
{
1
3
1
1
c2 );{1
1
{
1
1
2 2
2
I n2 },
c
1
1
3
2
1
2 2
2
{ubj : j
1
2 2
2
{wci : i
3
1
{
:
c3 ); {1
I m2 },
2
1
2 2
1
1
I n1 },
1
2 2
2
1
{
{uaj : j
1
1
2 2
2
1
1
:
1
2 2
1
2
2
1
2 2
2
I k1 },
2
{ ybl : l
I k2 }}
a3 , b3 , c3 ; wc , uc , yc
2
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((a1
a2 )
a3 ,(b1
b2 )
b3 ,(c1
c2 )
1
{1
c3 );
1
1
1
3
1
1
2
2
3
2
2
1
2
1
1
2
1
2
2
1
1
2
2
1
1
2
3
2
2
1
2
1
1
1
2
1
2
2
2
2
1
2
1
2
2
1
2
1
1
2
1
2
3
3
2
2
a2
1
2
2
2
2
1
2
a3 , b1
b2
1
2
2
1
2
1
1
2
b3 , c1
c2
2
2
{wbi : i
I m2 },
3
{wci : i
{uaj : j
I n1 },
2
{ubj : j
3
{ yal : l
I k1 },
2
{ ybl : l
I k2 },
Hence from eq. 7-8, we have, a
4. Proof is similar to 3.
{ yal : l
1
3
(b
{ubj : j
2
I n2 },
{ucj : j
3
I n3 }},
I k1 },
2
{ ybl : l
I k2 },
3
{ ycl : l
I k3 }}
1
2 2
2
1
1
3
2
2
3
2
2
2
1
2
1
1
1
1
1
3
3
{ucj : j
2
2
1
2
{ ycl : l
c)
2
2
1
2
1
1
2
1
3
3
2
2
2
I k3 }}
(a
2
2
2
1
2 2
2
2
1
2 2
2
:
1
1
2
1
2
1
{wai : i
I m1 },
2
:
1
I n3 },{
1
1
I n1 },
1
2 2
I m3 }},{
I n2 },
:
c3 ); {1
1
1
{uaj : j
1
1
2 2
1
2
:
1
2 2
2
2
1
2
1
1
1
2
2
1
(a1
I m3 }},
1
2 2
2
1
1
1
{wci : i
3
2
1
{
1
I m2 },
1
2 2
2
1
2 2
1
1
3
{wbi : i
2
2
1
{
1
I m1 },
1
2 2
2
1
1
1
1
{wai : i
1
2
1
2
:
1
2 2
2
1
2
2
2
1
2 2
2
:
(8)
b)
c.
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(a
5.
b)
(a1
a2 , b1
1
c2 );{1
b2 , c1
1
1
{
1
1
2
2
2
1
1
2
1
2
1
1
2
2
2
1
1
2
1
( a1
2
a2 , b1
1
1
2
1
1
:
1
{uaj : j
I n1 },
:
1
{ yal : l
I k1 },
2
2
2
{ubj : j
2
1
2 2
2
2
{ ybl : l
1
1
1
2 2
1
1
1
2
1
1
2 2
1
1
2
1
1
1
2
1
2 2
1
1
2
2
2
2
2
1
1
2
1
1
1
2
1
1
1
2
1
2 2
1
:
1
1
2 2
2 2
1
2
2
1
2
1
1
2
1
1
{uaj : j
2
1
1
{wai : i
I m1 },
2
1
I n1 },
1
2
2
2
1
2 2
2
{ubj : j
2
1
2 2
2
2
2
:
2
I n2 }},
1
2 2 2
1
1
I m2 }},
1
2 2 2
1
{
2 2
1
1
1
{wbi : i
I k2 }}
1
1
{
2
I n2 }},
1
c2 );{1
b2 , c1
I m2 }},
I m1 },
2
1
{wbi : i
{wai : i
1
2
1
2
:
2
1
{
1
2 2
2
2
2
1
2 2
1
2
2
2
2
2
:
1
{ yal : l
I k1 },
2
{ ybl : l
I k2 }}
1
2 2 2
1
2 2 2
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( a1
a2 , b1
b2 , c1
1
c2 );{1
1
2
{wbi : i
1
1
1
1
1
( a1
2
1
2 2
1
1
2
a2 , b1
1
1
I m2 }},{
{
2
1
2
2
2 2
2
1
2
1
1
2
b2 , c1
:
2
1
2
2
2
{ yal : l
1
2
I k1 },
2
1
:
1
2
2
2
1
{ ybl : l
{wbi : i
{ubj : j
1
1
{ ybl : l
1
1
1
1
2
1
2
1
1
2 2
2
1
1
2 2
2
2
2
2
2
1
1
1
1
2
1
1
2 2 2
1
1
1
2
2
1
2
2
2
2
1
2 2 2
1
2 2 2
1
2
1
1
1
2
1
:
1
{wai : i
1
1
1
2
1
1
2 2 2
1
1
1
2
1
I m1 }},{
1
1
1
2 2
2
1
2
1
:
1
:
2 2
1
1
1
2 2 2
1
:
2 2
1
1
1
2 2 2
1
I k2 }}
( a1, b1, c1 );{1
2
1
2
1
1
2 2
2
1
1
2
2
1
:
2 2
1
1
1
2 2
I n2 }},{
1
I k1 },
2
1
1
1
{ yal : l
1
1
2
1
1
1
I n2 }},
1
1
2
{ubj : j
2
2
1
I m2 }},{
1
I n1 },
2
1
2
1
1
1
{uaj : j
I n1 },
1
2 2
1
1
1
I m1 },
2
2
1
1
2
{wai : i
1
I k2 }}
1
1
I m1 },
2
:
1
1
{wai : i
2
c2 );{1
1
1
2
{uaj : j
1
2
2
1
2
2
2
2
2
1
2 2 2
1
2 2 2
{uaj : j
I n1 }},
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1
{
1
1
1
{
1
1
2
2
2
a)
(
:
1
2 2
2
1
2
1
{ yal : l
1
I k1 }}
1
:
1
2 2
2
{ubj : j
2
1
I n2 }},{
1
1
2
2
2
1
2 2
2
2 ) a1 , ( 1
1
2 )b1 , ( 1
1
(
1
((
2)
1
1
2
1
{uaj : j
1
1
2 2
2
(
2 )b1 , ( 1
1
1
1
2
1
1
(
2
1
1
1
2
1
2
1
{wai : i
1
I m1 }},
1
2 2
2
1
1
:
1
{ yal : l
I k1 }
2
1
:
1
{wai : i
I m1 }},
1
2 2
2
1
1
2
1
2
2
1
2 2 2
1
2
1
1
1
2 2
2
1
1
2 2
1
1
1
1
2
1
1
:
2
1
2 2
2
1
1
2 2
1
1
1
2 2
1
2)
1
1
1
I m2 }},
1
2
1
1
1
1
2
1
1
{
1
{wbi : i
2
I k2 }}
1
1
1
2 2
2
1
1
1
1
1
:
2 )c1 );
1
1
1
1
{1
1
2
2
{ ybl : l
2
2)
I n1 }},{
1
2 ) a1 , ( 1
1
2
1
1
2 )c1 );{1
1
{
:
2
1
2 2
b)
(
6. Proof is similar to 5
7. ( 1 2 ) a
((
1
a2 , b2 , c2 ;{1
:
1
{uaj : j
I n1 }},
1
2 2 2
1
2
1
1
1
2 2 2
1
2
1
1
2
1
1
2 2 2
1
2
1
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1
{
1
1
1
1
1
1
1
1
1
1
2
1
1
1
1
1
2
:
1
1
1
I k1 }}
1
2 2 2
1
2
1
:
{wai : i
1
1
I m1 }},{
1
2
I k1 }}
( 2 a1 ,
2
1
1
2
1
a)
(
2
:
1
2 2
2
1
2
1
{uaj : j
1
1
I n1 }},{
1
1
2
1
2 2
2
2
1
1
I n1 }},
:
2
1
:
1
2 2
2
1
1
{uaj : j
1
1
2 b1 , 2 c1 );{1
1
I m1 }}, {
:
1
2
2
2
1
1
1
{ yal : l
1
1
(
1
{ yal : l
1
2 2 2
1
2
1
1
1
2 2
1
1
{wai : i
1
1
2 2
2
1
2
1
1
1
1
2
1
1
2 2 2
1
2
1
( 1a1 , 1b1 , 1c1 );{1
{
1
1
1
2 2 2
1
2
1
:
1
2 2
1
{ yal : l
I k1 }}
2
1
a)
2
8. Proof is similar to 7.
5. HESITANT TRIANGULAR NEUTROSOPHIC WEIGHTED AGGREGATION OPERATORS:
This section deals with various types of hesitant triangular neutrosophic weighted aggregation operators along
with their basic properties.
Definition 13: Let a j
(a j , b j , c j );{wai j : i
I m j },{uar j : r
I n j },{ yal j : l
Ik j }
(j=1, 2, 3,….., n) be a
collection of hesitant triangular neutrosophic numbers. Then the hesitant triangular neutrosophic weighted
arithmetic aggregation operator of type-1 ( HTNWAAOT1 for short) is defined as:
HTNWAAOT1 a1 , a2 , a3 ,......, an
where w j is the weight of
Theorem 14: Let
aj
aj
a1 )
(w1
(j=1,2,3,……, n) such that
(a j , b j , c j );{wai j : i
a2 )
(w2
w j 0 and
I m j },{uar j : r
collection of hesitant triangularneutrosophic numbers. Then
triangular neutrosophic number and
HTNWAAOT1 (a1 , a2 , a3 ,......, an )
j
{wai j : i
(
n
j 1
I m j }},{
wj a j ,
n
j 1
jm
wj
n
j 1
:
wjbj ,
j
(w3
n
j 1
a3 )
........
wj
1.
I n j },{ yal j : l
Ik j }
( wn
an )
(j=1, 2, 3,….., n) be a
HTNWAAOT1 a1 , a2 , a3 ,......, an is a hesitant
n
j 1
w j c j ),{1
{uar j : r
I n j }},{
n
(1
jk )
j 1
n
j 1
jr
wj
:
wj
j
:
{ yal j : l
I k j }}
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aj
where w j is the weight of
Proof:
Let
( w1
a1 )
us
prove
the
HTNWAOT1 (a1 , a2 )
w1
1
{
2
:
w2
{ yal 1 : l
1
:
( w1a1
1)
( w1a1
(
2
2
j
j 1
I n2 }},{
2
:
w2
2)
w2
1
:
{wai 1 : i
I m1 }},{
{ yal 2 : l
2
):
{uar2 : j
2
wj a j ,
wj
:
w1
I m2 }},{
I k1 },
2
w1
j 1
1
1)
{wai 1 : i
1
w1
I n2 }},{
2
w2
{ yal 2 : l
2
2
wjbj ,
j 1
{ yal j : l
j
of
wj
1.
mathematical
induction.
For
n=2,
w1
1
:
1
2
:
{uar1 : r
1
{uar1 : r
1
2)
w2
:
I n1 }},
{wai 2 : i
2
I m2 }},
I k 2 }}
w1
I m1 },
w2
:
w1
2
2)
{wai 2 : i
{ yal 1 : l
1)
I n1 },
) (1 (1
2
1
w2 a2 , w1b1 w2b2 , w1c1 w2 c2 );{1 (1
{ yal 1 : l
j 1
method
( w2 a2 , w2b2 , w2 c2 );{1 (1
) (1 (1
I n1 },
{wai 2 : i
1
{
w1
{uar1 : r
2
the
w2 a2 , w1b1 w2b2 , w1c1 w2 c2 ); (1 (1
{(1 (1
1
1)
I k1 }}
{uar2 : j
2
using
j 1
a2 )
( w2
( w1a1 , w1b1 , w1c1 );{1 (1
{
result
n
w j 0 and
(j=1,2,3,……,n) such that
(1
)
w1
I m2 }},{
I k1 },
2)
w2
w2
2
w2
{ yal 2 : l
2
:
1
{wai 1 : i
1
w1
{uar2 : j
I n2 }},{
{wai j : i
I m j }},{
1
:
I k 2 }}
I m1 },
2
w2
:
wj
:
I k 2 }}
2
w j c j );{1
(1
j)
j 1
wj
:
j
2
j 1
j
j
{uar j : r
I n j }},
I k j }}
Thus the result is true for n=2. Let us assume that the result is true for n=s. Then HTNWAOT1 (a1 , a2 , a3 ,......, as )
(
s
j 1
s
{
j
j 1
s
wj a j ,
wj
:
j 1
s
w j bj ,
j 1
{ yal j : l
j
s
w j c j );{1
(1
j)
j 1
wj
:
{wai j : i
j
I m j }},{
s
j
j 1
wj
:
j
{uar j : r
I n j }},
I k j }}
Now for n=s+1, we have, HTNWAOT1 (a1, a2 , a3 ,......, as 1)
(
s
j 1
s
{
j 1
(
{{
j
s
j 1
s
:
s
j 1
wj
j 1
(1
j 1
w jb j ,
s
j 1
{ yal j : l
s
j 1
:
s 1
wja j
s
s
j
wja j ,
{wai
s 1
{(1
wj
j
j 1
(
wja j ,
w jb j ,
j 1
:i
Im
s 1
wj
)
}}{
s
j 1
(1
ws 1
s 1
j)
(1
j)
j 1
:
{wai j : i
j
I m j }},
wj
:
{wai j : i
j
I m j }},
ws 1as 1, ws 1bs 1, ws 1cs
:
s 1
{uar
w jb j ws 1bs 1,
(1
wj
( s 1) )
s
j
j 1
wj
:{
j
{uar j : r
I n j }},
as 1 )
1
I k j }}
ws 1as 1,
j)
s
w j c j ),{1
{ yal j : l
j
(1
j 1
( ws
I k j }}
s
s
w j c j );{1
s 1
s
j 1
ws 1
)
:r
In
w jc j
(1
s 1
}}, {
ws 1
s 1
1
:
s
j 1
j
wj
:
;{1 (1
s 1
{ yal
{uar j : r
j
( s 1) )
s 1
:l
I n j }},
ws 1
:
I ks
1
}}
ws 1cs 1 );
s
j 1
(1
j)
wj
) (1 (1
( s 1) )
ws 1
):
j
{wai j : i
I m j },
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{wai
s 1
{(
(
s
j
j 1
s
j 1
(
{
s 1
wj
{wai
s
j
j 1
s 1
j 1
j
j 1
:i
s 1
wj
wj
s 1
s 1
w
) s s1 1 :
s 1
j 1
:
j 1
s
j 1
s
{ yal j : l
s 1
j 1
w
) s s1 1 :
I k j },
j 1
j
s 1
wj
{uar j : r
j
{ yal
s
j 1
s 1
w
s 1
(1
j)
j 1
I n j },
I ks
1
s 1
wj
:
{uar
s 1
ws 1cs 1 );{1 ((
:l
j
s 1
:r
In
s 1
}},
}}
{uar j : r
j
{ yal
s 1
:l
wjc j
) s s1 1 :
I k j },
w j c j ),{1
{ yal j : l
j
wj
w jb j ws 1bs 1,
}}, {(
j
w jb j ,
j
{ yal j : l
j
Im
s
}},{(
ws 1as 1,
wja j ,
s 1
Im
w
) s s1 1 :
wja j
s 1
{(
:i
I ks
I n j },
1
s 1
s
(1
j)
j 1
{uar
s 1
wj
:r
w
) s s 11 ) :
In
s 1
{wai j : i
j
I m j },
}},
}}
{wai j : i
I m j }},{
s 1
j
j 1
wj
:
j
{uar j : r
I n j }},
I k j }}
Thus the result is true for n=s+1 also. Hence by the principle of mathematical induction, the result is true for any
natural number n.
aj
Theorem 15: Let
(a j , b j , c j );{wai j : i
I m j },{uar j : r
I n j },{ yal j : l
(j=1, 2, 3,….., n) be a
Ik j }
collection of hesitant triangular neutrosophic numbers. Then for any hesitant triangular neutrosophic number
we have,
(i) HTNWAAOT
a1 ,
a2 ,
a3 ,......,
an
HTNWAAOT a1 , a2 , a3 ,......, an
1
,
1
(ii) HTNWAAOT1 a1 , a2 , a3 ,......, an
if a j
for each j
Proof: Straight Forward.
aj
Definition 16: Let
(a j , b j , c j );{wai j : i
I m j },{uar j : r
I n j },{ yal j : l
Ik j }
(j=1, 2, 3,….., n) be a
collection of hesitant triangular neutrosophic numbers. Then the hesitant triangular neutrosophic weighted
geometric aggregation operator of type-1 ( HTNWGAOT1 for short) is defined as:
HTNWGAOT1 (a1, a2 , a3 ,......, an ) (w1 a1 )
where w j is the weight of
aj
Theorem 17: Let a j
(j=1,2,3,……,n) such that
(a j , b j , c j );{wai j : i
a2 )
( w2
w j 0 and
I m j },{uar j : r
collection of hesitant triangularneutrosophic numbers. Then
a3 )
( w3
n
j 1
........
wj
(wn
an )
1.
I n j },{ yal j : l
Ik j }
(j=1,2,3,…..,n) be a
HTNWGAOT1 (a1 , a2 , a3 ,......, an ) is a hesitant
triangular neutrosophic number and HTNWGAOT1 (a1, a2 , a3 ,......, an )
(
n
j 1
{1
a jwj ,
n
j 1
(1
n
j 1
n
bj w j ,
j)
wj
:
j 1
j
c j w j );
n
j 1
{ yal j : l
where w j is the weight of
aj
j
wj
:
j
{wai j : i
I m j }}, {1
n
j 1
(1
j)
wj
:
j
{uar j : r
I n j }},
I k j }}
(j=1,2,3,……,n) such that
w j 0 and
n
j 1
wj
1.
Proof: Similar to the proof of Theorem 14.
Theorem 18: Let a j
(a j , b j , c j );{wai j : i
I m j },{uar j : r
I n j },{ yal j : l
Ik j }
(j=1, 2, 3,….., n) be a
collection of hesitant triangular neutrosophic numbers. Then for any hesitant triangular neutrosophic number
we have,
Abhijit Saha, Irfan Deli, and Said Broumi, HESITANT Triangular Neutrosophic Numbers and Their Applications
to MADM
,
287
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(i) HTNWGAOT
1
a1 ,
a2 ,
a3 ,......,
(ii) HTNWGAOT1 a1 , a2 , a3 ,......, an
an
if a j
HTNWGAOT1 a1 , a2 , a3 ,......, an
for each j
Proof: Straight forward .
Definition 19: Let a j
(a j , b j , c j );{wai j : i
I m j },{uar j : r
I n j },{ yal j : l
(j=1, 2, 3,….., n) be a
Ik j }
collection of hesitant triangular neutrosophic numbers. Then the hesitant triangular neutrosophic weighted
arithmetic aggregation operator of type-2 is denoted by HTNWAAOT2 (a1 , a2 , a3 ,......, an ) and is defined by:
HTNWAAOT2 (a1 , a2 , a3 ,......, an ) (w1
aj
where w j is the weight of
Theorem 20: Let
aj
a1 )
(j=1,2,3,……,n) such that
(a j , b j , c j );{wai j : i
a2 )
( w2
n
w j 0 and
I m j },{uar j : r
a3 )
( w3
j 1
( wn
........
an )
wj 1 .
I n j },{ yal j : l
Ik j }
(j=1, 2, 3,….., n) be a
HTNWAAOT2 (a1 , a2 , a3 ,......, an ) is a hesitant
collection of hesitant triangular neutrosophic numbers. Then
triangular neutrosophic number and HTNWAAOT2 (a1, a2 , a3 ,......, an )
(
n
j 1
wja j ,
n
j 1
w jb j ,
n
j 1
1
w j c j ),{1
n
1
{uar j : r
j
j 1
1
I n j }}, {
n
1
j 1
where w j is the weight of
wj
aj
:
1
2 2
2
1
2
j
wj
1
j
1
2 2
:
{wai j : i
j
1
I m j }}, {
n
1
2
j
{ yal j : l
j 1
wj
1
2 2
2
1
2
j
:
j
I k j }}
j
2
j
(j=1,2,3,……,n) such that
n
w j 0 and
j 1
wj 1 .
Proof:For n=2, we have,
HTNWAAOT2 (a1, a2 )
a1 )
( w1
a2 )
( w2
1
( w1a1, w1b1, w1c1 );{1
1
{
w1
1
{
2
1
1
2 2
2
2
1
{wai 2 : i
:
1
w1
1
{ yal 1 : l
1
2
1
1
2 2
:
1
{wai 1 : i
1
I m1 }},
w1
1
2
1
1
1
w2
1
1
2 2
2
2
2
2
:
2
1
ua2 },{
1
w2
1
1
2 2
2
2
2
:{
1
w2
:
{uar1 : r
I n1 }},
1
1
1
1
2
1
( w2 a2 , w2b2 , w2c2 );{1
I k1 }}
1
I m2 }}, {
1
2 2
2
2
1
2
2
1
2 2
:
2
2
{ yal 2 : l
I k2 }}
2
Abhijit Saha, Irfan Deli, and Said Broumi, HESITANT Triangular Neutrosophic Numbers and Their Applications
to MADM
288
Neutrosophic Sets and Systems, Vol. 35, 2020
( w1a1
w2 a2 , w1b1
w2b2 , w1c1
w2c2 );
1
{1
1
1
w1
1
1
2
2
1
1
w1
1
2
1k
1
w2
1
2
1
1
1 1
1
1
1
2 2
1
w2
1
2
2
1
1
{
2
1
1
1
w1
1
1
2
1
1
w1
1
1
2
2
1
1
2 2
2
1m
1
w2
1
2
1
1
1
w1
1
1
2
1
1
w1
1
( w1a1
1
w2 a2 , w1b1
2
2
1
1
2 2 2
1
2
1
1
w2
w2b2 , w1c1
1
w2
1
2
2
{wai 2 : i
w1
1
1
{
w1
1
(
2
j 1
1
2
1
wja j ,
2
1
2
j 1
2
w2
w jb j ,
1
2
j 1
2
2
1
2
1
2
1
1
2 2
2
2
2
w2
:
1
1
2
2
I k1 },
j 1
wj
2
j
1
2
1
1
1
2 2
2
2
1
2
{uar1 : r
I n1 },
:
1
{ yal 1 : l
I k1 },
{uar2 : j
2
I n2 }},
{ yal 2 : l
2
I k2 }}
1
2 2 2
2
w1
{ yal 1 : l
w j c j );{1
1
1
1
1
I m2 }},{
:
1
2 2 2
2
w2c2 );{1
1
2
2
2
1
2 2
1
1
I m2 }},
2
2m
1
1
2 2 2
1
{wai 2 : i
1
2 2 2
2
1
{
2
1
2 2 2
2m
1
w2
1
2
1m
I m1 },
1
2 2
1
2 2
1
1
1
2 2 2
1
{wai 1 : i
1
2
2
1
2
1
:
1
2 2
2
2
1 1
1
2 2
1
2 2
:
2
w2
2
1
{uar1 : r
1
1
2 2
2
j
2
{ yal 2 : l
:
j
1
:
1
2 2
2
2
1
{wai 1 : i
I m1 },
2
2
I n1 },
2
{uar2 : j
I n2 }},
I k2 }}
{wai j : i
1
I m j }},{
1
2
j 1
wj
1
2 2
2
1
2
j
:
j
Abhijit Saha, Irfan Deli, and Said Broumi, HESITANT Triangular Neutrosophic Numbers and Their Applications
to MADM
289
Neutrosophic Sets and Systems, Vol. 35, 2020
{uar j : r
j
1
I n j }}, {
2
1
1
wj
j 1
:
1
2 2
2
{ yal j : l
j
I k j }}
j
2
j
Thus the result is true for n=2. Let us assume that the result is true for n=s.
Then we have, HTNWAAOT2 (a1, a2 , a3 ,......, as )
(
s
j 1
wja j ,
s
j 1
s
w jb j ,
j 1
1
w j c j ),{1
s
1
{uar j : r
j
j 1
1
I n j }}, {
s
1
j 1
1
2 2
2
1
wj
2
j
2
j
wj
:
:
1
2 2
{wai j : i
j
1
I m j }},{
s
1
1
2
j
j
{ yal j : l
j 1
wj
1
2 2
2
1
2
j
:
j
I k j }}
j
Now for n=s+1, we have, HTNWAOT2 (a1, a2 , a3 ,......, as 1)
(
s
j 1
s
wja j ,
j 1
w jb j ,
s
j 1
1
w j c j );{1
s
1
1
{
s
1
j 1
1
wj
:
1
2 2
2
2
j
j
j 1
{uar j : r
ws
1
( w1a1
1
2
( s 1)
1
:
1
2 2
{uar
s 1
s 1
:r
1
In
1
s 1
w2b2 , w1c1
s
1
j 1
wj
2
jk
1
1
2 2
1
s
j 1
wj
2
j
1
:
s 1
{wai
1
ws
1
1
ws
1
1
2 2
2
j
{ yal j : l
1
1
2
( s 1)
s 1
:i
1
2 2
1
1
2 2
1
ws
1
1
Im
:
s 1
s 1
}},
{ yal
s 1
:l
I ks
1
}}
:
j
{wai j : i
I m j },
s 1
{wai
s 1
:i
Im
s 1
1
2 2
2
( s 1)
2
( s 1)
1
1 1
I k j }}
2
( s 1)
1
2
j
1
1 1
j
w2c2 ),
1
2
1
1
1
2 2
}}, {
1
w2a2 , w1b1
:
j
2
j
2
( s 1)
2
( s 1)
2
( s 1)
{1
1
j 1
1
2 2
2
1
wj
1
ws
I m j }},
2
j
s
;{1
1
{wai j : i
j
1
1
1
1
1
I n j }},{
j
ws 1as 1, ws 1bs 1, ws 1cs
{
2
j
wj
:
1
2 2
1
2 2
2
( s 1)
2
( s 1)
Abhijit Saha, Irfan Deli, and Said Broumi, HESITANT Triangular Neutrosophic Numbers and Their Applications
to MADM
}},
290
Neutrosophic Sets and Systems, Vol. 35, 2020
1
{
2
1
1
s
1
j 1
1
1
wj
2
j
1
s
1
j 1
wj
2
j
1
1
2
1
1
s
1
j 1
1
1
wj
s
j 1
:r
In
s 1
}},
:
{ yal j : l
j
I k j },
{ yal
s 1
s 1
:
1
2
( s 1)
1
1
2 2 2
j
1
wj
s 1
1
2 2
2
( s 1)
1
ws
1
2
j
1
1
1
2 2
1
1
1
2 2 2
j
{uar
s 1
2
( s 1)
1
{
I n j },
1
2 2
2
( s 1)
1
ws
{uar j : r
j
2
( s 1)
1
1
:
1
2 2
2
( s 1)
1
ws
1
1
2 2 2
j
1
1
1
1
2 2 2
j
1
2 2
ws
1
2
j
1
2 2
2
( s 1)
1
1
2
( s 1)
I k s 1 }}
l
(
s 1
j 1
wj a j ,
s 1
j 1
s 1
w jbj ,
j 1
1
w j c j ),{1
s
1
j 1
j
{wai j : i
I m j },
{wai
s 1
s 1
:i
Im
s 1
1
{
s
1
j 1
1
wj
2
j
2
j
2
ws
1
s
1
j 1
(
s 1
j 1
1
wj
2
j
wj a j ,
s 1
j 1
2
j
2
w jbj ,
ws
s 1
j 1
1
1
1
j
s 1
j 1
wj
{ yal j : l
1
2 2
2
1
2
j
:
s 1
:
:
j
1
1
{uar j : r
I n j },
j
{ yal j : l
I k j },
1
w j c j );{1
j
2
j
2
( s 1)
2
( s 1)
s 1
s 1
s 1
{uar j : r
j 1
I n j },
wj
s 1
1
2 2
2
j
s 1
s 1
:r
In
s 1
}},
{ yal
s 1
I ks 1 }}
:l
j
{wai j : i
:l
I ks
1
I m j },
s 1
{wai
s 1
:i
Im
s 1
}},
2
j
1
{uar
:
s 1
:r
In
s 1
1
}},{
1
{ yal
{uar
2
( s 1)
j
I k j },
1
2 2
2
( s 1)
1
1
{
1
ws
2
( s 1)
1
{
2
j
:
1
2 2
}},
1
2 2
2
( s 1)
1
wj
2
s 1
j 1
wj
1
2 2
2
1
2
j
:
j
}}
Abhijit Saha, Irfan Deli, and Said Broumi, HESITANT Triangular Neutrosophic Numbers and Their Applications
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Thus the result is true for n=s+1 also. Hence by the principle of mathematical induction, the result is true for any
natural number n.
Theorem 21: Let
aj
(a j , b j , c j );{wai j : i
I m j },{uar j : r
I n j },{ yal j : l
(j=1, 2, 3,….., n) be a
Ik j }
collection of hesitant triangular neutrosophic numbers. Then for any hesitant triangular neutrosophic number
we have,
(i)
HTNWAAOT2
a1,
a2 ,
a3 ,......,
(ii) HTNWAAOT2 a1 , a2 , a3 ,......, an
if a j
an
,
HTNWAAOT2 a1, a2 , a3 ,......, an
for each j
Proof: Straight forward .
Definition 22: Let
aj
(a j , b j , c j );{wai j : i
I m j },{uar j : r
I n j },{ yal j : l
Ik j }
(j=1, 2, 3,….., n) be a
collection of hesitant triangular neutrosophic numbers. Then the hesitant triangular neutrosophicweighted
geometric aggregation operator of type-2 is denoted by HTNWGAOT2 (a1 , a2 , a3 ,......, an ) and is defined by:
HTNWGAOT2 (a1 , a2 , a3 ,......, an )
where w j is the weight of
Theorem 23: Let
aj
aj
a1 )
(w1
a2 )
( w2
w j 0 and
(j=1,2,3,……, n) such that
(a j , b j , c j );{wai j : i
I m j },{uar j : r
collection of hesitant triangular neutrosophic numbers. Then
triangular neutrosophic number and
a3 )
( w3
n
j 1
( wn
........
an )
wj 1 .
I n j },{ yal j : l
Ik j }
(j=1, 2, 3,….., n) be a
HTNWGAOT2 (a1 , a2 , a3 ,......, an ) is a hesitant
HTNWGAOT2 (a1, a2 , a3 ,......, an )
(
n
j 1
a jwj ,
n
j 1
bj w j ,
n
j 1
1
c j w j );{
n
1
j
{uar j : r
j 1
1
wj
2
j
1
I n j }}, {1
n
1
where w j is the weight of
j 1
aj
wj
2
j
1
1
2 2
2
:
j
{wai j : i
j
1
2 2
:
j
1
I m j }},{1
n
1
{ yal j : l
j 1
2
j
wj
1
1
2 2
:
2
j
Ik j }
2
j
(j=1,2,3,……,n) such that
w j 0 and
n
j 1
wj 1 .
Proof: Similar to the proof of Theorem 20.
Theorem 24: Let
aj
(a j , b j , c j );{wai j : i
I m j },{uar j : r
I n j },{ yal j : l
(j=1, 2, 3,….., n) be a
Ik j }
collection of hesitant triangular neutrosophic numbers. Then for any hesitant triangular neutrosophic number
we have,
(i) HTNWGAOT
a1,
a2 ,
a3 ,......,
an
HTNWGAOT a1, a2 , a3 ,......, an
2
,
2
(ii) HTNWGAOT2 a1 , a2 , a3 ,......, an
if a j
for each j
Proof: Straight forward .
Definition
25:
Let
a
(a1 , b1 , c1 );{wai : i
neutrosophic number. Then the score of
a
I m },{uaj : j
I n },{ yal : l
Ik }
be
a
hesitant
triangular
is defined by:
Abhijit Saha, Irfan Deli, and Said Broumi, HESITANT Triangular Neutrosophic Numbers and Their Applications
to MADM
292
Neutrosophic Sets and Systems, Vol. 35, 2020
S (a)
Where
If a j
{wai j : i
j
1
(a
3 max{a1 , b1 , c1} 1
I m j },
j
(a j , b j , c j );{wai j : i
{uar j : r
I m j },{uar j : r
c1 )
b1
I n j },
j
2
1
m
{ yal j : l
I n j },{ yal j : l
Ik j }
m
j 1
j
1
n
n
1
k
j
j 1
k
j 1
j
Ik j }
(j
1, 2)
be
two
hesitant
triangular
neutrosophic numbers, then, the comparison method is given as;
I. If S (a1 ) S (a2 ) then a1
a2
II. If S (a1 ) S (a2 ) then a1 a2
5. APPLICATION OF HESITANT TRAPEZOIDAL NEUTROSOPHIC NUMBERS:
In this section, we apply the weighted aggregation operators and the score function of hesitant triangular
neutrosophic numbers to the multi-attribute decision-making problem with hesitant triangular neutrosophic
information.
Let X {A1, A2 , A3 ,....., Am} be a set of alternatives, A {c1, c2 , c3 ,....., cn } be a set of attributes and
w {w1 , w2 , w3 ,....., wn } be a set of weights ( w j is the weight of attribute c j (j=1,2,3,……,n) such that
w j 0 and
n
j 1
w j 1 .) In this case, the characteristic of the alternative Ai (i 1, 2,..., m) on attribute
cj ( j
1, 2,..., n) is represented by the following form of a hesitant triangular neutrosophic number:
Aij
(aij , bij , cij );{wap : p
I mij },{uarij : r
ij
I nij },{ yal ij : l
I kij }
.
Now, we construct a multi-attribute decision making method by the following algorithm:
ALGORITHM:
Step-1: Express the evaluation results of the expert based on the alternative
Ai (i 1, 2,..., m) on attribute
cj (1,2, … , 𝑛) in terms of hesitant triangular neutrosophic numbers 𝑥𝑖𝑗 as a mn Table.
TA
Step-2: Compute the aggregation values gi k (i
1, 2,..., m) (k
Ai (i 1, 2,..., m) as;
A
giTk
TG
1, 2) or gi k (i
1, 2,..., m) (k
1, 2) of
1, 2,..., m) (k
1, 2) of
HTNWAAOTk ( Ai1 , Ai 2 ,..., Ain ) (i 1, 2,..., m) (k 1, 2)
or
TG
gi k
HTNWGAOTk ( Ai1 , Ai 2 ,..., Ain ) (i 1, 2,..., m) (k 1, 2)
Step-3: Calculate the score values of
Ai (i 1, 2,..., m) based on Definition 25.
TA
gi k (i
1, 2,..., m) (k
TG
1, 2) or gi k (i
Step-4: Rank the alternatives by using definition 25.
Example 22:
Let us consider a decision making problem adapted from Wei et al. (2017). Suppose an organisation plans to
implement enterprise resource planning (ERP) system. The first step is to form a project team that consists of
CIO and two senior representatives from user departments. By collecting all possible information about ERP
vendors and systems, project team chooses five potential ERP systems Ai (i=1, 2, 3, 4, 5) as candidates. The
company employs some external professional organizations (or experts) to aid this decision making. The project
team selects four attributes to evaluate the alternatives: function and technology 𝑐1 , strategic fitness 𝑐2 , vendor’s
ability 𝑐3 and vendor’s reputation 𝑐4 . The five possible ERP systems Ai (i=1, 2, 3, 4, 5) are to be evaluate during
the hesitant triangular neutrosophic numbers by the decision makers under the above four attributes whose
weighting vector is 𝑤 = 0.3, 0.3, 0.2, 0.2 .
Abhijit Saha, Irfan Deli, and Said Broumi, HESITANT Triangular Neutrosophic Numbers and Their Applications
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Step-1: We express the initial evaluation results of the expertforfivepossiblealternativesbased on
fourattributesbythe form of hesitant triangular neutrosophic numbers, as shownin Table 1.
Table 1:The evaluation result by the expert is shown in the below table
TA
Step-2:We compute the aggregationvalues gi 1 (i
A
g1T1
1, 2,..., 5) of Ai (i
1, 2,...,5) as;
HTNWAAOT1 ( A11, A12 , A13 , A14 )
<(0.30,0.51,0.52); {0.494124,0.522409, 0.526880, 0.553333}, {0.299254,0.308624,0.319973,0.329992,0.416080,
0.429108, 0.444888,0.458818},{0.262529, 0.301566,0.278077,0.319426, 0.323211, 0.371272,
0.342353, 0.393260,0.310519,0.356693, 0.328909,0.377818,0.382294, 0.439141,0.412567,0.465148}>
A
g 2T1
HTNWAAOT1 ( A21, A22 , A23 , A24 )
<(0.32,0.47,0.58); {0.468719,0.481089,0.617891,0.626787},{0.376740,0.393934, 0.399052,0.417264,0.416754,
{0.389321,0.447212,0.403779, 0.435774,0.441436,0.461583},0.463821,0.430672,0.494712, 0.446666,0.513085}>
A
g3T1
HTNWAAOT1 ( A31, A32 , A33 , A34 )
<(0.39,0.44,0.51);{0.419636, 0.457406,0.434930,0.471704,0.467623,0.502270,0.481653, 0.515387,0.344568,
0.387223, 0.361840,0.403371,0.398762, 0.437891, 0.414606,0.452704},{0.456007,0.505058,0.494527,
0.547722,0.497111, 0.550583, 0.539102,0.597092},{0.389321, 0.448274,0.479310,0.460489,
0.416275,0.530219,0.403779, 0.464922,0.497111,0.477590, 0.431735,0.549910}>
A
g 4T1
HTNWAAOT1 ( A41, A42 , A43 , A44 )
<(0.39,0.44,0.51);{0.398762,0.437891, 0.476592,0.510656,0.419636,0.457406, 0.494764,0.527644},{0.439833,
0.451737,0.505235,0.518910,0.520235,0.534315, 0.597593,0.613767,0.496724,0.510168, 0.570586,0.586030,0.587525,
0.603427, 0.674889,0.693156},{0.232461,0.267027,0.282842,0.246228,0.306007,0.351510, 0.372328,0.324130}>
A
g5T1
HTNWAAOT1 ( A51, A52 , A53 , A54 )
<(0.39,0.49,0.53);{0.365425,0.388147,0.427055, 0.447570,0.426353,0.446894,
0.482066,0.500612,0.463497,0.482708,0.515603,0.532948,0.515010, 0.532376,0.562112,0.577792},{0.328749,
0.356519,0.377634,0.409533,0.351510,0.381202,0.403779,0.437887,0.375847, 0.407595,0.431735,0.468204},
{0.267027,0.275388,0.282842,0.332644,0.343059,0.352345, 0.301566,0.311008,0.319426,0.375670, 0.387433,
0.397920,0.328749,0.339042,0.348219, 0.409533,0.422356,0.433787,0.371272,0.382897, 0.393260,0.462505,
0.476986,0.489897}>
Abhijit Saha, Irfan Deli, and Said Broumi, HESITANT Triangular Neutrosophic Numbers and Their Applications
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Neutrosophic Sets and Systems, Vol. 35, 2020
Step-3:
S ( A1 )
Wecalculate
(0.30
the
score
0.51 0.52)
[2
3 0.52
values
1
(0.494124
4
of
TA
gi k (i
0.522409
1, 2,..., 5)
0.526880
Ai (i 1, 2,...,5)
of
as
0.553333)
1
(0.299254 0.308624 0.319973 0.329992 0.416080 0.429108 0.444888 0.458818)
8
1
(0.262529 0.301566 0.278077 0.319426 0.323211 0.371272 0.342353 0.393260
16
0.310519 0.356693 0.328909 0.377818 0.382294 0.439141 0.412567 0.465148)
1.33
1.56
2
0.524186 0.375842 0.354048
1.5297,
Similarly, we have;
S ( A2 )
Step-4: Since S ( A1 )
1.3244, S ( A3 )
S ( A5 )
Thus we conclude that
A1
S ( A4 )
1.2687, S ( A4 )
S ( A2 )
S ( A3 ) , So A1
1.4110, S ( A5 )
A5
A4
A2
1.5235.
A3 .
is the best (most desirable) ERP system. On the other hand, if we apply the other
proposed weighted aggregation operators such as
HTNWGAOT1 , HTNWAAOT2 , HTNWGAOT2 for
computing the best alternative(s), then step 2 of the proposed approach has been executed for each weighted
aggregation operators and hence their corresponding hesitant triangular neutrosophic number has been
constructed. Finally, based on these, the score values of the aggregated hesitant triangular neutrosophic numbers
are computed and ranking has been done which are summarized in table-2. We can conclude from table-2 that
although the ranking orders of the alternatives are slightly different; the best (most desirable) alternative is still
A1 in all cases.
Table-2: Ranking order of alternatives
6. COMPARATIVE STUDY:
In order to compare the performance of the proposed method with some existing methods (Ye 2013a, Ye 2014,
Ye 2015a, Ye 2015b, Liu 2016, Abdel-Basset et al. 2017, Wei et al. 2017), a comparative study is presented and
their corresponding final ranking are summarized in table 3. From table-3, it is clear that although the ranking
order of the alternatives are slightly different, but the best (most desirable) alternative is the same as found in the
existing approaches (Ye 2013a, Ye 2014, Ye 2015a, Ye 2015b, Liu 2016, Abdel-Basset et al. 2017). Thus, our
proposed method can be suitably utilized to solve the multi attribute decision making problems than the other
existing methods due to the fact that more fuzziness and uncertainties are involved in our proposed approach.
Abhijit Saha, Irfan Deli, and Said Broumi, HESITANT Triangular Neutrosophic Numbers and Their Applications
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295
Neutrosophic Sets and Systems, Vol. 35, 2020
Table 3: Comparative study
Existing
approach
Ye [36]
Ranking
Our proposed
method
HTNWAAOT1
A2
A4
A3
A1
HTNWGAOT1
HTNWAAOT2
HTNWGAOT2
HTNWAAOT1
Ye [17]
A2
A4
A3
A1
HTNWGAOT1
HTNWAAOT2
HTNWGAOT2
HTNWAAOT1
Ye [40]
A2
A4
A3
A1
HTNWGAOT1
HTNWAAOT2
HTNWGAOT2
HTNWAAOT1
Ye [38]
A4
A2
A3
A1
HTNWGAOT1
HTNWAAOT2
HTNWGAOT2
HTNWAAOT1
Liu [21]
A4
A2
A3
A1
HTNWGAOT1
HTNWAAOT2
HTNWGAOT2
HTNWAAOT1
AbdelBasset et al.
[57]
A4
A2
A3
A1
HTNWGAOT1
HTNWAAOT2
HTNWGAOT2
Ranking
A2
A2
A2
A2
A2
A2
A2
A2
A2
A2
A2
A2
A4
A4
A4
A4
A4
A4
A4
A4
A4
A4
A4
A4
A4
A3
A4
A1
A4
A3
A4
A1
A4
A3
A4
A1
A2
A3
A2
A1
A2
A3
A2
A1
A2
A3
A2
A1
A3
A4
A1
A4
A3
A4
A1
A4
A3
A4
A1
A4
A3
A2
A1
A2
A3
A2
A1
A2
A3
A2
A1
A2
A1
A1
A3
A3
A1
A1
A3
A3
A1
A1
A3
A3
A1
A1
A3
A3
A1
A1
A3
A3
A1
A1
A3
A3
Best
alternative
A2
A2
A2
A2
A2
A2
A2
A2
A2
A2
A2
A2
A4
A4
A4
A4
A4
A4
A4
A4
A4
A4
A4
A4
7. CONCLUSION
In this paper, hesitant triangular neutrosophic numbers and their basic properties are presented. Also, various
types of operations between the hesitant triangular neutrosophic numbers are discussed. Then, various types of
hesitant triangular neutrosophic weighted aggregation operators are proposed to aggregate the hesitant triangular
neutrosophic information. Furthermore, score of hesitant triangular neutrosophic numbers is proposed to ranking
the hesitant triangular neutrosophic numbers.Based on the hesitant triangular neutrosophic weighted aggregation
operators and score of hesitant triangular neutrosophic numbers, a multi attribute decision making method is
developed, in which the evaluation values of alternatives on the attribute are represented in terms of hesitant
triangular neutrosophic numbers and the alternatives are ranked according to the values of the score of hesitant
triangular neutrosophic numbers to select the most desirable one. Finally, a practical example for enterprise
resource planning (ERP) system selection is presented to demonstrate the application and effectiveness of the
proposed method. The advantage of the proposed method is that it is more suitable for solving multi attribute
Abhijit Saha, Irfan Deli, and Said Broumi, HESITANT Triangular Neutrosophic Numbers and Their Applications
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Neutrosophic Sets and Systems, Vol. 35, 2020
296
decision making problems with hesitant triangular neutrosophic information because hesitant triangular
neutrosophic numbers can handle indeterminate and inconsistent information and are the extensions of hesitant
triangular fuzzy numbers, hesitant triangular intuitionistic fuzzy numbersas well as triangular neutrosophic
numbers.
In the future, we will develop another approach called linguistic hesitant triangular neutrosophic number as a
further generalization of it and this will be applied in different practical problems.
FUNDING: This research received no external funding.
ACKNOWLEDGEMENTS: Nil.
CONFLICTS OF INTEREST: The authors declare no conflict of interest.
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Received: April 11, 2020. Accepted: July 2, 2020
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Neutrosophic Sets and Systems, Vol. 35, 2020
University of New Mexico
Translative and Multiplicative Interpretation of Neutrosophic
Cubic Set
MOHSIN KHALID 1, FLORENTIN SMARANDACHE 2 NEHA ANDALLEB KHALID 3, SAID
BROUMI 4, *
1
Department of Mathematics, The University of Lahore, Lahore, Pakistan
E-mail: mk4605107@gmail.com
2
Department of Mathematics, Lahore Collage for Women University, Lahore, Pakistan
E-mail: nehakhalid97@gmail.com
3
Department of Mathematics, University of New Mexico Mathematics Department 705 Gurley Ave., Gallup, NM 87301,
USA
E-mail: fsmarandache@gmail.com, smarand@unn.edu
4
Laboratory of Information Processing, Faculty of Science Ben M’Sik, University Hassan II, Casablanca, Morocco
E-mail: s.broumi@flbenmsik.ma
* Correspondence; Mohsin Khalid; mk4605107@gmail.com
Abstract: In this paper, we introduce the idea of neutrosophic cubic translation (NCT) and neutrosophic cubic
multiplication (NCM) and provide entirely new type of conditions for neutrosophic cubic translation and
neutrosophic cubic multiplication on BF-algebra. This is the new kind of approach towards translation and
multiplication which involves the indeterminacy membership function. We also define neutrosophic cubic
magnified translation (NCMT) on BF-algebra which handles the neutrosophic cubic translation and
neutrosophic cubic multiplication at the same time on membership function, indeterminacy membership
function and non-membership function. We present the examples for better understanding of neutrosophic cubic
translation, neutrosophic cubic multiplication, and neutrosophic cubic magnified translation, and investigate
significant results of BF-ideal and BF-subalgebra by applying the ideas of NCT, NCM and NCMT. Intersection
and union of neutrosophic cubic BF-ideals are also explained through this new type of translation and
multiplication.
Keywords: BF-algebra, neutrosophic cubic translation, neutrosophic cubic multiplication, neutrosophic cubic
BF ideal, neutrosophic cubic BF subalgebra, neutrosophic cubic magnified translation.
1. Introduction
Zadeh [1] presented the theory of fuzzy set in 1965. Fuzzy idea has been applied to different algebraic structures
like groups, rings, modules, vector spaces and topologies. In this way, Iseki and Tanaka [2] introduced the idea
of BCK-algebra in 1978. Iseki [3] introduced the idea of BCI-algebra in 1980 and it is obvious that the class of
BCK-algebra is a proper sub class of the class of BCI-algebra. Lee et al. [4] studied the fuzzy translation,
Mohsin khalid,Florentin Smarandache,Neha Andaleeb khalid and Said Broumi, Translative And Multiplicative
Interpretation of Neutrosophic Cubic Set
Neutrosophic Sets and Systems, Vol. 35, 2020
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(normalized, maximal) fuzzy extension and fuzzy multiplication of fuzzy subalgebra in BCK/BCI-algebra. Link
among fuzzy translation, (normalized, maximal) fuzzy extension and fuzzy multiplication are also discussed.
Ansari and Chandramouleeswaran [5] introduced the idea of fuzzy translation, fuzzy extension and fuzzy
multiplication of fuzzy β ideal of β-algebra and investigated some of their properties. Satyanarayana [6]
introduced the concepts of fuzzy ideals, fuzzy implicative ideals and fuzzy p-ideals in BF-algebras and
investigate some of its properties. Andrzej [7] defined the BF-algebra. Lekkoksung [8] focused on fuzzy
magnified translation in ternary hemirings, which is a extension of BCI / BCK/Q / KU / d-algebra. Senapati et
al. [9] have done much work on intuitionistic fuzzy H-ideal in BCK/BCI-algebra. Jana et al. [10] wrote on
intuitionistic fuzzy G-algebra. Senapati et al. [11] studied fuzzy translations of fuzzy H-ideals in BCK/BCIalgebra. Atanassov [12] introduced the intuitionistic fuzzy sets. Senapati [13] investigated the relationship
among intuitionistic fuzzy translation, intuitionistic fuzzy extension and intuitionistic fuzzy multiplication in
B-algebra. Kim and Jeong [14] defined the intuitionistic fuzzy structure of B-algebra. Senapati et al. [15]
introduced the cubic subalgebras and cubic closed ideals of B-algebras. Senapati et al. [16] discussed the fuzzy
dot subalgebra and fuzzy dot ideal of B-algebras. Priya and Ramachandran [17] worked on fuzzy translation
and fuzzy multiplication in PS-algebra. Chandramouleeswaran et al. [18] worked on fuzzy translation and fuzzy
multiplication in BF/BG-algebra. Jana et al. [19] studied the cubic G-subalgebra of G-algebra. Smarandache
[20,21] extended the intuitionistic fuzzy set, paraconsistent set, and intuitionistic set to the neutrosophic set
through Several examples. Jun et al. [22] studied the Cubic set and apply the idea of cubic sets in group and
gave the definition of cubic subgroups. Saeid and Rezvani [23] introduced and studied fuzzy BF-algebra, fuzzy
BF-subalgebras, level subalgebras,fuzzy topological BF-algebra. Jun et al. [24] defined the neutrosophic cubic
set introduced truth-internal and truth-external and discuss the many properties. Jun et al. [25] investigated the
commutative falling neutrosophic ideals in BCK-algebra. C. H. Park [26] defined the neutrosophic ideal in
subtraction algebra and studied it through several properties, he also discussed conditions for a neutrosophic set
to be a neutrosophic ideal along with properties of neutrosophic ideal. Khalid et al. [27] investigated the
neutrosophic soft cubic subalgebra through significant characteristic like P-union, R-intersection etc. Khalid et
al. [28] interestinly investigated the intuitionistic fuzzy translation and multiplication through subalgebra and
ideals. Khalid et al. [29] defined the T-neutrosophic cubic set and studied this set through ideals and subalgebras
and investigated many results.
The purpose of this paper is to introduce the idea of neutrosophic cubic translation (NCT), neutrosophic cubic
multiplication (NCM) and neutrosophic cubic magnified translation (NCMT) on BF-algebra. In second section
we discuss some fundamental definitions which are used to develop the paper. In third’s first subsection we
discuss the neutrosophic cubic translation (NCT) and neutrosophic cubic multiplication (NCM) of BF
subalgebra. In second subsection we discuss the neutrosophic cubic translation (NCT) and neutrosophic cubic
multiplication (NCM) of BF ideal. In third subsection we discuss the neutrosophic cubic magnified translation
(NCMT) of BF ideal and BF subalgebra.
2 Preliminaries
First we discuss some definitions which are used to present this paper.
Mohsin khalid,Florentin Smarandache,Neha Andaleeb khalid and Said Broumi, Translative And Multiplicative
Interpretation of Neutrosophic Cubic Set
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301
Definition 2.1 [3] An algebra (Y,∗ ,0) of type (2,0) is called a BCI-algebra if it satisfies the following
conditions:
i) (t1 ∗ t 2 ) ∗ (t1 ∗ t 3 ) ≤ (t 3 ∗ t 2 ),
ii) t1 ∗ (t1 ∗ t 2 ) ≤ t 2 ,
iii) t1 ≤ t1 ,
iv) t1 ≤ t 2 and t 2 ≤ t1 ⇒ t1 = t 2 ,
v) t1 ≤ 0 ⇒ t1 = 0, where t1 ≤ t 2 is defined by t1 ∗ t 2 = 0, for all t1 , t 2 , t 3 ∈ Y.
Definition 2.2 [1] An algebra (Y,∗ ,0) of type (2,0) is called a BCK-algebra if it satisfies the following
conditions:
i) (t1 ∗ t 2 ) ∗ (t1 ∗ t 3 ) ≤ (t 3 ∗ t 2 ),
ii) t1 ∗ (t1 ∗ t 2 ) ≤ t 2 ,
iii) t1 ≤ t1 ,
iv) t1 ≤ t 2 and t 2 ≤ t1 ⇒ t1 = t 2 ,
v) 0 ≤ t1 ⇒ t1 = 0, where t1 ≤ t 2 is defined by t1 ∗ t 2 = 0, for all t1 , t 2 , t 3 ∈ Y.
Definition 2.3 [7] A nonempty set Y with a constant 0 and a binary operation ∗ is called BF–algebra when it
fulfills these axioms.
i) t1 ∗ t1 = 0
ii) t1 ∗ 0 = 0
iii) 0 ∗ (t1 ∗ t 2 ) = t 2 ∗ t1 for all t1 , t 2 ∈ Y.
A BF-algebra is denoted by (Y,∗ ,0).
Definition 2.4 [7] Let S be a nonempty subset of BF-algebra Y, then S is called a BF-subalgebra of Y if t1 ∗
t 2 ∈ S, for all t1 , t 2 ∈ S.
Definition 2.5 [6] Let Y ba a BF-algebra and I is a subset of Y, then I is called a BF ideal of Y if it satisfies
the following conditions:
i) 0 ∈ I,
ii) t 2 ∗ t1 ∈ I and t 2 ∈ I → t1 ∈ I.
Definition 2.6 [6] Let Y be a BF-algebra. A fuzzy set B of Y is called a fuzzy BF ideal of Y if it satisfies the
following conditions:
i) κ(0) ≥ κ(t1 ),
ii) κ(t1 ) ≥ min{κ(t 2 ∗ t1 ), κ(t 2 )}, for all t1 , t 2 ∈ Y.
Definition 2.7 [1] Let Y be a group of objects denoted generally by t1 . Then a fuzzy set B of Y is defined as
B = {< t1 , κB (t1 ) > |t1 ∈ Y}, where κB (t1 ) is called the membership value of t1 in B and κB (t1 ) ∈ [0,1].
Definition 2.8 [23] A fuzzy set B of BF-algebra Y is called a fuzzy PS subalgebra of Y if κ(t1 ∗ t 2 ) ≥
min{κ(t1 ), κ(t 2 )}, for all t1 , t 2 ∈ Y.
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Definition 2.9 [4,5] Let a fuzzy subset B of Y and α ∈ [0,1 − sup{κB (t1 )|t1 ∈ Y}]. A mapping (κB )Tα |Y ∈
[0,1] is said to be a fuzzy α translation of κB if it satisfies (κB )Tα (t1 ) = κB (t1 ) + α, for all t1 ∈ Y.
Definition 2.9 [4,5] Let a fuzzy subset B of Y and α ∈ [0,1]. A mapping (κB )M
α : Y → [0,1] is said to be a
fuzzy α multiplication of B if it satisfies (κB )M
α (t1 ) = α. (κB )(t1 ), for all t1 ∈ Y.
Definition 2.10 [12] An intuitionistic fuzzy set (IFS) B over Y is an object having the form B =
{〈t1 , κB (t1 ), υB (t1 )〉|t1 ∈ Y}, where κB (t1 ): Y → [0,1] and υB (t1 )|Y → [0,1], with the condition 0 ≤
κB (t1 ) + υB (t1 ) ≤ 1, for all t1 ∈ Y. κB (t1 ) and υB (t1 ) represent the degree of membership and the degree
of non-membership of the element t1 in the set B respectively.
Definition 2.11 [12] Let B = {〈t1 , κB (t1 ), υB (t1 )〉|t1 ∈ Y} and B = {〈t1 , κB (t1 ), υB (t1 )〉|t1
∈ Y} be two IFSs
on Y. Then intersection and union of A and B are indicated by A ∩ B and A ∪ B
respectively and are given by
A ∩ B = {〈t1 , min(κA (t1 ), κB (t1 )), max(υA (t1 ), υB (t1 ))〉|t1 ∈ Y},
A ∪ B = {〈t1 , max(κA (t1 ), κB (t1 )), min(υA (t1 ), υB (t1 ))〉|t1 ∈ Y}.
Definition 2.12 [14] An IFS B = {〈t1 , κB (t1 ), υB (t1 )〉|t1 ∈ Y} of Y is called an IFSU of Y if it satisfies these
two conditions:
(i) κB (t1 ∗ t 2 ) ≥ min{κB (t1 ), κB (t 2 )},
(ii) υB (t1 ∗ t 2 ) ≤ max{υB (t1 ), υB (t 2 )}, for all t1 , t 2 ∈ Y.
Definition 2.13 An IFS B = {〈t1 , κB (t1 ), υB (t1 )〉|t1 ∈ Y} of Y is said to be an IFID of Y if it satisfies these
three conditions:
(i) κB (0) ≥ κB (t1 ), υB (0) ≤ υB (t1 ),
(ii) κB (t1 ) ≥ min{κB (t1 ∗ t 2 ), κB (t 2 )},
(iii) υB (t1 ) ≤ max{υB (t1 ∗ t 2 ), υB (t 2 )}, for all t1 , t 2 ∈ Y.
T
Definition 2.14 [8] Let κ be a fuzzy subset of Y, α ∈ [0,T] and β ∈ [0,1]. A mapping κM
β α |Y →[0,1] is said
T
to be a fuzzy magnified βα translation of κ if it satisfies: κM
β α (t1 ) = β. κ(t1 ) + α for all t1 ∈ Y.
Jun et al. [22,24]introduced neutrosophic cubic set and investigated several properties.
Definition 2.15 [24] Suppose X be a nonempty set. A neutrosophic cubic set in X is pair 𝒞 = (κ, σ) where
κ = {〈t1 ; κE (t1 ), κI (t1 ), κN (t1 )〉 |t1 ∈ X}
is
an
interval
neutrosophic
set
in
X
and
σ=
{〈t1 ; σE (t1 ), σI (t1 ), σN (t1 )〉 |t1 ∈ X} is a neutrosophic set in X.
Definition 2.16 [15] Let C = {〈t1 , κ(t1 ), σ(t1 )〉} be a cubic set, where κ(t1 ) is an interval-valued fuzzy set in
X, σ(t1 ) is a fuzzy set in X. Then C is cubic subalgebra under binary operation " ∗”, if following axioms are
satisfied:
i)
κ(t1 ∗ t 2 ) ≥ rmin{κ(t1 ), κ(t 2 )},
ii)
σ(t1 ∗ t 2 ) ≤ max{σ(t1 ), σ(t 2 )} ∀ t1 , t 2 ∈ X.
Mohsin khalid,Florentin Smarandache,Neha Andaleeb khalid and Said Broumi, Translative And Multiplicative
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Definition 2.17 [28] Let A = (κA , υA ) be an IFS of G-algebra and let α ∈ [0, ¥]. An object of the form ATα =
((κA )Tα , (υA )Tα ) is called an intuitionistic fuzzy α-translation (IFAT) of A when (κA )Tα (t1 ) = κA (t1 ) + α and
(υA )Tα (t1 ) = υA (t1 ) − α for all t1 ∈ Y.
3
Translative and Multiplicative Interpretation of Neutrosophic Cubic Set
For our simplicity, we use the notation B = (κT,I,F , υT,I,F ) for the NCS B = {⟨t1 , κT,I,F (t1 ), υT,I,F (t1 )⟩|t1 ∈ Y}.
In
this
paper,
we
used
ℸ = [1,1] − rsup{κ{T,I} (t1 )|t1 ∈ Y} ,
¥ = rinf{κF (t1 )|t1 ∈ Y} ,
Γ=1−
sup{υ{T,I} (t1 )|t1 ∈ Y}, £ = inf{υF (t1 )|t1 ∈ Y for any NCS B = (κT,I,F , υT,I,F ) of Y.
3.1 Translative and Multiplicative Interpretation of Neutrosophic Cubic Subalgebra
Definition 3.1.1 Let B = (κT,I,F , υT,I,F ) be a NCS of Y and for κT,I,F , α, β ∈ [[0,0], ℸ] and γ ∈ [[0,0], ¥],
T
where for υT,I,F , α, β ∈ [0, Γ] and γ ∈ [0, £]. An object of the form Bα,β,γ
= ((κT,I,F )Tα,β,γ , (υT,I,F )Tα,β,γ ) is
called a NCT of B, when (κT )Tα (t1 ) = κB (t1 ) + α , (κI )Tβ (t1 ) = κB (t1 ) + β , (κF )Tγ (t1 ) = κF (t1 ) − γ and
(υT )Tα (t1 ) = υT (t1 ) + α, (υI )Tβ (t1 ) = υB (t1 ) + β, (υF )Tγ (t1 ) = υB (t1 ) − γ for all t1 ∈ Y.
Example 3.1.1 Let Y = {0,1,2} be a BF-algebra with the following Cayley table:
*
0
1
2
0
0
1
2
1
0
0
1
2
0
2
0
Let B = (κT,I,F , υT,I,F ) be a NCS of Y is defined as
κT (t1 ) = {
[0.1, 0.3]
[0.4, 0.7]
if t1 = 0
if otherwise
κI (t1 ) = {
[0.2, 0.4]
[0.5, 0.7]
if t1 = 0
if otherwise
κF (t1 ) = {
[0.4, 0.6]
[0.3, 0.8]
if t1 = 0
if otherwise
and
υT (t1 ) = {
0.1
0.4
if t1 = 0
if otherwise
υI (t1 ) = {
0.2
0.3
if t1 = 0
if otherwise
Mohsin khalid,Florentin Smarandache,Neha Andaleeb khalid and Said Broumi, Translative And Multiplicative
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υF (t1 ) = {
0.5
0.7
if t1 = 0
if otherwise.
Then B is a neutrosophic cubic subalgebra. Here we choose for υT,I,F , α = 0.01, β = 0.02, γ = 0.03, and for
κT,I,F , α = [0.1,0.2], β = [0.2,0.25], γ = [0.2,0.3] then the mapping B T : Y → [0,1] is given by
(κT )T[0.1,0.2] (t1 ) = {
(κI )T[0.2,0.25] (t1 ) = {
(κF )T[0.2,0.3] (t1 ) = {
[0.2, 0.5]
[0.5, 0.9]
if t1 = 0
if otherwise
[0.4, 0.7]
[0.7, 0.95]
[0.2, 0.3]
[0.1, 0.5]
if t1 = 0
if otherwise
if t1 = 0
if otherwise
and
(υT )T0.01 (t1 ) = {
0.11
0.41
if t1 = 0
if otherwise.
(υI )T0.02 (t1 ) = {
0.22
0.32
if t1 = 0
if otherwise.
(υF )T0.03 (t1 ) = {
0.47
0.67
if t1 = 0
if otherwise,
which
imply
(κT )T[0.1,0.2] (t1 ) = κT (t1 ) + [0.1,0.2] ,
(κI )T[0.2,0.25] (t1 ) = κI (t1 ) + [0.2,0.25 ] ,
(κF )T[0.2,0.3] (t1 ) = κF (t1 ) − [0.2,0.3] and (υT )T0.01 (t1 ) = υT (t1 ) + 0.01 , (υI )T0.02 (t1 ) = υI (t1 ) + 0.02 ,
(υF )T0.03 (t1 ) = υF (t1 ) − 0.03 for all t1 ∈ Y. Hence B T is a neutrosophic cubic translation.
Theorem 3.1.1 Let B be a NCSU of Y and for κT,I,F , α, β ∈ [[0,0], ℸ] and γ ∈ [[0,0], ¥], where for υT,I,F ,
T
α, β ∈ [0, Γ] and γ ∈ [0, £]. Then NCT Bα,β,γ
of B is a NCSU of Y.
Proof. Assume t1 , t 2 ∈ Y. Then
(κT )Tα (t1 ∗ t 2 ) = κT (t1 ∗ t 2 ) + α
≥ rmin{κT (t1 ), κT (t 2 )} + α
= rmin{κT (t1 ) + α, κT (t 2 ) + α}
(κT )Tα (t1 ∗ t 2 ) = rmin{(κT )Tα (t1 ), (κT )Tα (t 2 )},
(κI )Tβ (t1 ∗ t 2 ) = κI (t1 ∗ t 2 ) + β
≥ rmin{κI (t1 ), κI (t 2 )} + β
= rmin{κI (t1 ) + β, κI (t 2 ) + β}
Mohsin khalid,Florentin Smarandache,Neha Andaleeb khalid and Said Broumi, Translative And Multiplicative
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305
(κI )Tβ (t1 ∗ t 2 ) = rmin{(κI )Tβ (t1 ), (κI )Tβ (t 2 )},
(κF )Tγ (t1 ∗ t 2 ) = κF (t1 ∗ t 2 ) − γ
≥ rmin{κF (t1 ), κF (t 2 )} − γ
= rmin{κF (t1 ) − γ, κF (t 2 ) − γ}
(κF )Tγ (t1 ∗ t 2 ) = rmin{(κF )Tγ (t1 ), (κF )Tγ (t 2 )}
and
(υT )Tα (t1 ∗ t 2 ) = υT (t1 ∗ t 2 ) + α
≤ max{υT (t1 ), υT (t 2 )} + α
= max{υT (t1 ) + α, υT (t 2 ) + α}
(υT )Tα (t1 ∗ t 2 ) = max{(υT )Tα (t1 ), (υT )Tα (t 2 )},
(υI )Tβ (t1 ∗ t 2 ) = υI (t1 ∗ t 2 ) + β
≤ max{υI (t1 ), υI (t 2 )} + β
= max{υI (t1 ) + β, υI (t 2 ) + β}
(υI )Tβ (t1 ∗ t 2 ) = max{(υI )Tβ (t1 ), (υI )Tβ (t 2 )},
(υF )Tγ (t1 ∗ t 2 ) = υF (t1 ∗ t 2 ) − γ
≤ max{υF (t1 ), υF (t 2 )} − γ
= max{υF (t1 ) − γ, υF (t 2 ) − γ}
(υF )Tγ (t1 ∗ t 2 ) = max{(υF )Tγ (t1 ), (υF )Tγ (t 2 )}.
T
Hence NCT Bα,β,γ
of B is a NCSU of Y.
T
Theorem 3.1.2 Let B be a NCS of Y such that NCT Bα,β,γ
of B is a NCSU of Y for some κT,I,F , α, β ∈
[[0,0], ℸ] and γ ∈ [[0,0], ¥], where for υT,I,F , α, β ∈ [0, Γ] and γ ∈ [0, £]. Then B is a NCSU of Y.
T
Proof. Let Bα,β,γ
= ((κT,I,F )Tα,β,γ , (υT,I,F )Tα,β,γ ) be a NCSU of Y for some κT,I,F , α, β ∈ [[0,0], ℸ] and γ ∈
[[0,0], ¥], where for υT,I,F , α, β ∈ [0, Γ] and γ ∈ [0, £] and t1 , t 2 ∈ Y. Then
κT (t1 ∗ t 2 ) + α = (κT )Tα (t1 ∗ t 2 )
≥ rmin{(κT )Tα (t1 ), (κT )Tα (t 2 )}
= rmin{κT (t1 ) + α, κT (t 2 ) + α}
κT (t1 ∗ t 2 ) + α = rmin{κT (t1 ), κT (t 2 )} + α,
κI (t1 ∗ t 2 ) + β = (κI )Tβ (t1 ∗ t 2 )
≥ rmin{(κI )Tβ (t1 ), (κI )Tβ (t 2 )}
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= rmin{κI (t1 ) + β, κI (t 2 ) + β}
κI (t1 ∗ t 2 ) + β = rmin{κI (t1 ), κI (t 2 )} + β,
κF (t1 ∗ t 2 ) − γ = (κF )Tγ (t1 ∗ t 2 )
≥ rmin{(κF )Tγ (t1 ), (κF )Tγ (t 2 )}
= rmin{κF (t1 ) − γ, κF (t 2 ) − γ}
κF (t1 ∗ t 2 ) − γ = rmin{κF (t1 ), κF (t 2 )} − γ
and
υT (t1 ∗ t 2 ) + α = (υT )Tα (t1 ∗ t 2 )
≤ max{(υT )Tα (t1 ), (υT )Tα (t 2 )}
= max{υT (t1 ) + α, υB (t 2 ) + α}
υT (t1 ∗ t 2 ) + α = max{υT (t1 ), υT (t 2 )} + α,
υI (t1 ∗ t 2 ) + β = (υI )Tβ (t1 ∗ t 2 )
≤ max{(υI )Tβ (t1 ), (υI )Tβ (t 2 )}
= max{υI (t1 ) + β, υB (t 2 ) + β}
υI (t1 ∗ t 2 ) + β = max{υI (t1 ), υI (t 2 )} + β,
υF (t1 ∗ t 2 ) − γ = (υF )Tγ (t1 ∗ t 2 )
≤ max{(υF )Tγ (t1 ), (υF )Tγ (t 2 )}
= max{υF (t1 ) − γ, υB (t 2 ) − γ}
υF (t1 ∗ t 2 ) − γ = max{υF (t1 ), υF (t 2 )} − γ,
which
imply
κT (t1 ∗ t 2 ) ≥ rmin{κT (t1 ), κT (t 2 )} ,
κI (t1 ∗ t 2 ) ≥ rmin{κI (t1 ), κI (t 2 )} ,
κF (t1 ∗ t 2 ) ≥
rmin{κF (t1 ), κF (t 2 )} , and υT (t1 ∗ t 2 ) ≤ max{υT (t1 ), υT (t 2 )} , υI (t1 ∗ t 2 ) ≤ max{υI (t1 ), υI (t 2 )} , υF (t1 ∗
t 2 ) ≤ max{υF (t1 ), υF (t 2 )}, for all t1 , t 2 ∈ Y. Hence B is a NCSU of Y.
Definition 3.1.2 Let B be a NCS of Y and δ
M
M
M
M
M
(((κT )M
δ , (κI )δ , (κF )δ ), ((υT )δ , (υI )δ , (υF )δ ))
is
∈
called
M
δ. κT (t1 ) , (κI )M
δ (t1 ) = δ. κI (t1 ) , (κF )δ (t1 ) = δ. κF (t1 )
[0,1]. An object having the form BδM =
a
and
NCM
of
B,
when
(υT )M
δ (t1 ) = δ. υT (t1 ) ,
(κT )M
δ (t1 ) =
(υI )M
δ (t1 ) =
δ. υI (t1 ),(υF )M
δ (t1 ) = δ. υF (t1 ) for all t1 ∈ Y.
Example 3.1.2 Let Y = {0,1,2} be a BF-algebra with the following Cayley table:
Mohsin khalid,Florentin Smarandache,Neha Andaleeb khalid and Said Broumi, Translative And Multiplicative
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Neutrosophic Sets and Systems, Vol. 35, 2020
*
0
1
2
0
0
1
2
1
0
0
1
2
0
2
0
Let B = (κT,I,F , υT,I,F ) be a NCS of Y is defined as
[0.1, 0.3] if t1 = 0
κT (t1 ) = (
[0.4, 0.7] if otherwise
[0.2, 0.4]
κI (t1 ) = (
[0.5, 0.7]
κF (t1 ) = (
if t1 = 0
if otherwise
[0.4, 0.6] if t1 = 0
[0.3, 0.8] if otherwise
and
0.1
0.4
if t1 = 0
if otherwise
0.2
0.3
if t1 = 0
if otherwise
0.5
0.7
if t1 = 0
if otherwise.
υT (t1 ) = (
υI (t1 ) = (
υF (t1 ) = (
Then B is a neutrosophic cubic subalgebra, choose δ = 0.01 for υ and δ = [0.1,0.2] for κ then the mapping
BδM |Y → [0,1] is given by
[0.01, 0.06] if t1 = 1
(κT )M
[0.1,0.2] (t1 ) = ([0.04, 0.14] if otherwise,
[0.02, 0.08] if t1 = 1
(κI )M
[0.1,0.2] (t1 ) = ([0.05, 0.14] if otherwise,
[0.04,0.12]
(κF )M
[0.1,0.2] (t1 ) = ([0.03, 0.16]
if t1 = 1
if otherwise
and
0.001
(υT )M
0.01 (t1 ) = (
0.004
if t1 = 0
if otherwise,
0.002
(υI )M
0.01 (t1 ) = (
0.003
if t1 = 0
if otherwise,
(υF )M
0.01 (t1 ) = (
0.005
0.007
if t1 = 0
if otherwise,
Mohsin khalid,Florentin Smarandache,Neha Andaleeb khalid and Said Broumi, Translative And Multiplicative
Interpretation of Neutrosophic Cubic Set
Neutrosophic Sets and Systems, Vol. 35, 2020
308
M
M
which imply (κT )M
[0.1,0.2] (t1 ) = κT (t1 ). [0.1,0.2] , (κI )[0.1,0.2] (t1 ) = κI (t1 ). [0.1,0.2] , (κF )[0.1,0.2] (t1 ) =
κF (t1 ). [0.1,0.2] and (υT )M
(υI )M
(υF )M
0.01 (t1 ) = υT (t1 ). (0.01) ,
0.01 (t1 ) = υI (t1 ). (0.01) ,
0.01 (t1 ) =
M
υF (t1 ). (0.01) for all t1 ∈ Y. Hence Bδ is a neutrosophic cubic multiplication.
Theorem 3.1.3 Let B be a NCS of Y such that NCM BδM of B is a NCSU of Y for some δ ∈ [0,1]. Then B
is a NCSU of Y.
Proof. Assume BδM of B is a NCSU of Y for some δ ∈ [0,1]. Now for all t1 , t 2 ∈ Y, we have
κT (t1 ∗ t 2 ). δ = (κT )M
δ (t1 ∗ t 2 )
M
≥ rmin{(κT )M
δ (t1 ), (κT )δ (t 2 )}
= rmin{κT (t1 ). δ, κT (t 2 ). δ}
κT (t1 ∗ t 2 ). δ = rmin{κT (t1 ), κT (t 2 )}. δ,
κI (t1 ∗ t 2 ). δ = (κI )M
δ (t1 ∗ t 2 )
M
≥ rmin{(κI )M
δ (t1 ), (κI )δ (t 2 )}
= rmin{κI (t1 ). δ, κI (t 2 ). δ}
κI (t1 ∗ t 2 ). δ = rmin{κI (t1 ), κI (t 2 )}. δ,
κF (t1 ∗ t 2 ). δ = (κF )M
δ (t1 ∗ t 2 )
M
≥ rmin{(κF )M
δ (t1 ), (κF )δ (t 2 )}
= rmin{κF (t1 ). δ, κF (t 2 ). δ}
κF (t1 ∗ t 2 ). δ = rmin{κF (t1 ), κF (t 2 )}. δ
and
υT (t1 ∗ t 2 ). δ = (υT )M
δ (t1 ∗ t 2 )
M
≤ max{(υT )M
δ (t1 ), (υT )δ (t 2 )}
= max{υT (t1 ). δ, υT (t 2 ). δ}
υT (t1 ∗ t 2 ). δ = max{υT (t1 ), υT (t 2 )}. δ,
υI (t1 ∗ t 2 ). δ = (υI )M
δ (t1 ∗ t 2 )
M
≤ max{(υI )M
δ (t1 ), (υI )δ (t 2 )}
= max{υI (t1 ). δ, υI (t 2 ). δ}
υI (t1 ∗ t 2 ). δ = max{υI (t1 ), υI (t 2 )}. δ,
υF (t1 ∗ t 2 ). δ = (υF )M
δ (t1 ∗ t 2 )
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309
M
≤ max{(υF )M
δ (t1 ), (υF )δ (t 2 )}
= max{υF (t1 ). δ, υF (t 2 ). δ}
υF (t1 ∗ t 2 ). δ = max{υF (t1 ), υF (t 2 )}. δ,
which
imply
κT (t1 ∗ t 2 ) ≥ rmin{κT (t1 ), κT (t 2 )} , κI (t1 ∗ t 2 ) ≥ rmin{κI (t1 ), κI (t 2 )} , κF (t1 ∗ t 2 ) ≥
rmin{κF (t1 ), κF (t 2 )} and υT (t1 ∗ t 2 ) ≤ max{υT (t1 ), υT (t 2 )} , υI (t1 ∗ t 2 ) ≤ max {υI (t1 ), υI (t 2 )} , υF (t1 ∗
t 2 ) ≤ max{υF (t1 ), υF (t 2 )} for all t1 , t 2 ∈ Y. Hence B is a NCSU of Y.
Theorem 3.1.4 Let B be a NCSU of Y for δ ∈ [0,1]. Then NCM BδM of B is a NCSU of Y.
Proof. Assume t1 , t 2 ∈ Y. Then
(κT )M
δ (t1 ∗ t 2 ) = δ. κT (t1 ∗ t 2 )
≥ δ. rmin{(κT )(t1 ), (κT )(t 2 )}
= rmin{δ. κT (t1 ), δ. κT (t 2 )}
M
= rmin{(κT )M
δ (t1 ), (κT )δ (t 2 )}
M
M
(κT )M
δ (t1 ∗ t 2 ) ≥ rmin{(κT )δ (t1 ), (κT )δ (t 2 )},
(κI )M
δ (t1 ∗ t 2 ) = δ. κI (t1 ∗ t 2 )
≥ δ. rmin{(κI )(t1 ), (κI )(t 2 )}
= rmin{δ. κI (t1 ), δ. κI (t 2 )}
M
= rmin{(κI )M
δ (t1 ), (κI )δ (t 2 )}
M
M
(κI )M
δ (t1 ∗ t 2 ) ≥ rmin{(κI )δ (t1 ), (κI )δ (t 2 )},
(κF )M
δ (t1 ∗ t 2 ) = δ. κF (t1 ∗ t 2 )
≥ δ. rmin{(κF )(t1 ), (κF )(t 2 )}
= rmin{δ. κF (t1 ), δ. κF (t 2 )}
M
= rmin{(κF )M
δ (t1 ), (κF )δ (t 2 )}
M
M
(κF )M
δ (t1 ∗ t 2 ) ≥ rmin{(κF )δ (t1 ), (κF )δ (t 2 )}
and
(υT )M
δ (t1 ∗ t 2 ) = δ. υT (t1 ∗ t 2 )
≤ δ. max{(υT )(t1 ), (υT )(t 2 )}
= max{δ. υT (t1 ), δ. υT (t 2 )}
M
= max{(κB )M
δ (t1 ), (κB )δ (t 2 )}
M
M
(υT )M
δ (t1 ∗ t 2 ) ≤ max{(υT )δ (t1 ), (υT )δ (t 2 )},
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(υI )M
δ (t1 ∗ t 2 ) = δ. υI (t1 ∗ t 2 )
≤ δ. max{(υI )(t1 ), (υI )(t 2 )}
= max{δ. υI (t1 ), δ. υI (t 2 )}
M
= max{(κB )M
δ (t1 ), (κB )δ (t 2 )}
M
M
(υI )M
δ (t1 ∗ t 2 ) ≤ max{(υI )δ (t1 ), (υI )δ (t 2 )},
(υF )M
δ (t1 ∗ t 2 ) = δ. υF (t1 ∗ t 2 )
≤ δ. max{(υF )(t1 ), (υF )(t 2 )}
= max{δ. υF (t1 ), δ. υF (t 2 )}
M
= max{(κB )M
δ (t1 ), (κB )δ (t 2 )}
M
M
(υF )M
δ (t1 ∗ t 2 ) ≤ max{(υF )δ (t1 ), (υF )δ (t 2 )},
which
imply
κT (t1 ∗ t 2 ) ≥ rmin{κT (t1 ), κT (t 2 )} ,
κI (t1 ∗ t 2 ) ≥ rmin{κI (t1 ), κI (t 2 )} ,
κF (t1 ∗ t 2 ) ≥
rmin{κF (t1 ), κF (t 2 )} and υT (t1 ∗ t 2 ) ≤ max{υT (t1 ), υT (t 2 )} , υI (t1 ∗ t 2 ) ≤ max{υI (t1 ), υI (t 2 )} , υF (t1 ∗
t 2 ) ≤ max{υF (t1 ), υF (t 2 )} for all t1 , t 2 ∈ Y. Hence BδM is a NCSU of Y.
3.2
Translative and Multiplicative Interpretation of Neutrosophic Cubic Ideal
In this section, neutrosophic cubic translation of NCID, neutrosophic cubic multiplication of NCID, union and
intersection of neutrosophic cubic translation of NCID are investigated through some results.
T
Theorem 3.2.1 If NCT Bα,β,γ
of B is a neutrosophic cubic BF ideal, then it fulfills the conditions (κT )Tα (t1 ∗
T
T
(t 2 ∗ t1 )) ≥ (κT )α (t 2 ), (κI )β (t1 ∗ (t 2 ∗ t1 )) ≥ (κI )Tβ (t 2 ),(κF )Tγ (t1 ∗ (t 2 ∗ t1 )) ≥ (κF )Tγ (t 2 ) and (υT )Tα (t1 ∗
(t 2 ∗ t1 )) ≤ (υT )Tα (t 2 ), (υI )Tβ (t1 ∗ (t 2 ∗ t1 )) ≤ (υI )Tβ (t 2 ), (υF )Tγ (t1 ∗ (t 2 ∗ t1 )) ≤ (υF )Tγ (t 2 ).
T
Proof. Let NCT Bα,β,γ
of B be a neutrosophic cubic BF ideal. Then
(κT )Tα (t1 ∗ (t 2 ∗ t1 )) = κT (t1 ∗ (t 2 ∗ t1 )) + α
≥ rmin{κT (t 2 ∗ (t1 ∗ (t 2 ∗ t1 ))) + α, κT (t 2 ) + α}
= rmin{κT (0) + α, κT (t 2 ) + α}
= rmin{(κT )Tα (0), (κT )Tα (t 2 )}
(κT )Tα (t1 ∗ (t 2 ∗ t1 )) = (κT )Tα (t 2 ),
(κI )Tα (t1 ∗ (t 2 ∗ t1 )) = κI (t1 ∗ (t 2 ∗ t1 )) + β
≥ rmin{κI (t 2 ∗ (t1 ∗ (t 2 ∗ t1 ))) + β, κI (t 2 ) + β}
= rmin{κI (0) + β, κI (t 2 ) + β}
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= rmin{(κI )Tβ (0), (κI )Tβ (t 2 )}
(κI )Tβ (t1 ∗ (t 2 ∗ t1 )) = (κI )Tβ (t 2 ),
(κF )Tγ (t1 ∗ (t 2 ∗ t1 )) = κF (t1 ∗ (t 2 ∗ t1 )) − γ
≥ rmin{κF (t 2 ∗ (t1 ∗ (t 2 ∗ t1 ))) − γ, κF (t 2 ) − γ}
= rmin{κF (0) − γ, κF (t 2 ) − γ}
= rmin{(κF )Tγ (0), (κF )Tγ (t 2 )}
(κF )Tγ (t1 ∗ (t 2 ∗ t1 )) = (κF )Tγ (t 2 )
and
(υT )Tα (t1 ∗ (t 2 ∗ t1 )) = υT (t1 ∗ (t 2 ∗ t1 )) + α
≤ max{υT (t 2 ∗ (t1 ∗ (t 2 ∗ t1 ))) + α, υT (t 2 ) + α}
= max{υT (0) + α, υT (t 2 ) + α}
= max{(υT )Tα (0), (υT )Tα (t 2 )}
(υT )Tα (t1 ∗ (t 2 ∗ t1 )) = (υT )Tα (t 2 ),
(υI )Tα (t1 ∗ (t 2 ∗ t1 )) = υI (t1 ∗ (t 2 ∗ t1 )) + β
≤ max{υI (t 2 ∗ (t1 ∗ (t 2 ∗ t1 ))) + β, υI (t 2 ) + β}
= max{υI (0) + β, υI (t 2 ) + β}
= max{(υI )Tβ (0), (υI )Tβ (t 2 )}
(υI )Tβ (t1 ∗ (t 2 ∗ t1 )) = (υI )Tβ (t 2 ),
(υF )Tγ (t1 ∗ (t 2 ∗ t1 )) = υF (t1 ∗ (t 2 ∗ t1 )) − γ
≤ max{υF (t 2 ∗ (t1 ∗ (t 2 ∗ t1 ))) − γ, υF (t 2 ) − γ}
= max{υF (0) − γ, υF (t 2 ) − γ}
= max{(υF )Tγ (0), (υF )Tγ (t 2 )}
(υF )Tγ (t1 ∗ (t 2 ∗ t1 )) = (υF )Tγ (t 2 ).
Hence
(κT )Tα (t1 ∗ (t 2 ∗ t1 )) ≥ (κT )Tα (t 2 ) ,
(κI )Tβ (t1 ∗ (t 2 ∗ t1 )) ≥ (κI )Tβ (t 2 ) ,
(κF )Tγ (t1 ∗ (t 2 ∗ t1 )) ≥
(κF )Tγ (t 2 ) and (υT )Tα (t1 ∗ (t 2 ∗ t1 )) ≤ (υT )Tα (t 2 ) , (υI )Tβ (t1 ∗ (t 2 ∗ t1 )) ≤ (υI )Tβ (t 2 ) , (υF )Tγ (t1 ∗ (t 2 ∗
t1 )) ≤ (υF )Tγ (t 2 ).
Theorem 3.2.2 Let B be a NCID of Y and for κT,I,F , α, β ∈ [[0,0], ℸ] and γ ∈ [[0,0], ¥], where for υT,I,F ,
T
α, β ∈ [0, Γ] and γ ∈ [0, £]. Then NCT Bα,β,γ
of B is a NCID of Y.
Proof. Let B be a NCID of Y and for κT,I,F , α, β ∈ [[0,0], ℸ] and γ ∈ [[0,0], ¥], where for υT,I,F , α, β ∈
[0, Γ] and γ ∈ [0, £]. Then (κT )Tα (0) = κT (0) + α ≥ κT (t1 ) + α = (κT )Tα (t1 ) , (κI )Tβ (0) = κI (0) + β ≥
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κI (t1 ) + β = (κI )Tβ (t1 ) , (κF )Tγ (0) = κF (0) − γ ≥ κF (t1 ) − γ = (κF )Tγ (t1 ) and (υT )Tα (0) = υT (0) + α ≤
υT (t1 ) + α = (υT )Tα (t1 ) , (υI )Tβ (0)
= υI (0) + β ≤ υI (t1 ) + β = (υI )Tβ (t1 ) ,
(υF )Tγ (0) = υF (0) − γ ≤
υF (t1 ) − γ = (υF )Tγ (t1 ). So
(κT )Tα (t1 ) = κT (t1 ) + α
≥ rmin{κT (t1 ∗ t 2 ), κT (t 2 )} + α
= rmin{κT (t1 ∗ t 2 ) + α, κT (t 2 ) + α}
(κT )Tα (t1 ) = rmin{(κT )Tα (t1 ∗ t 2 ), (κT )Tα (t 2 )},
(κI )Tβ (t1 ) = κI (t1 ) + β
≥ rmin{κI (t1 ∗ t 2 ), κI (t 2 )} + β
= rmin{κI (t1 ∗ t 2 ) + β, κI (t 2 ) + β}
(κI )Tβ (t1 ) = rmin{(κI )Tβ (t1 ∗ t 2 ), (κI )Tβ (t 2 )},
(κF )Tα (t1 ) = κF (t1 ) − γ
≥ rmin{κF (t1 ∗ t 2 ), κF (t 2 )} − γ
= rmin{κF (t1 ∗ t 2 ) − γ, κF (t 2 ) − γ}
(κF )Tγ (t1 ) = rmin{(κF )Tγ (t1 ∗ t 2 ), (κF )Tγ (t 2 )}
and
(υT )Tα (t1 ) = υT (t1 ) + α
≤ max{υT (t1 ∗ t 2 ), υT (t 2 )} + α
= max{υT (t1 ∗ t 2 ) + α, υT (t 2 ) + α}
(υT )Tα (t1 ) = max{(υT )Tα (t1 ∗ t 2 ), (υT )Tα (t 2 )},
(υI )Tβ (t1 ) = υI (t1 ) + β
≤ max{υI (t1 ∗ t 2 ), υI (t 2 )} + β
= max{υI (t1 ∗ t 2 ) + β, υI (t 2 ) + β}
(υI )Tβ (t1 ) = max{(υI )Tβ (t1 ∗ t 2 ), (υI )Tβ (t 2 )},
(υF )Tγ (t1 ) = υF (t1 ) − γ
≤ max{υF (t1 ∗ t 2 ), υF (t 2 )} − γ
= max{υF (t1 ∗ t 2 ) − γ, υF (t 2 ) − γ}
(υF )Tγ (t1 ) = max{(υF )Tγ (t1 ∗ t 2 ), (υF )Tγ (t 2 )},
Mohsin khalid,Florentin Smarandache,Neha Andaleeb khalid and Said Broumi, Translative And Multiplicative
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for all t1 , t 2 ∈ Y and for κT,I,F , α, β ∈ [[0,0], ℸ] and γ ∈ [[0,0], ¥], where for υT,I,F , α, β ∈ [0, Γ] and γ ∈
T
[0, £]. Hence Bα,β,γ
of B is a NCID of Y.
T
Theorem 3.2.3 Let B be a neutrosophic cubic set of Y such that NCT Bα,β,γ
of B is a NCID of Y for all
κT,I,F , α, β ∈ [[0,0], ℸ] and γ ∈ [[0,0], ¥], where for υT,I,F , α, β ∈ [0, Γ] and γ ∈ [0, £]. Then B is a NCID of
Y.
T
Proof. Suppose Bα,β,γ
is a NCID of Y, where for κT,I,F , α, β ∈ [[0,0], ℸ] and γ ∈ [[0,0], ¥], and for υT,I,F ,
α, β ∈ [0, Γ] and γ ∈ [0, £] and t1 , t 2 ∈ Y. Then
κT (0) + α = (κT )Tα (0) ≥ (κT )Tα (t1 ) = κT (t1 ) + α,
κI (0) + β = (κI )Tβ (0) ≥ (κI )Tβ (t1 ) = κI (t1 ) + β,
κF (0) − γ = (κF )Tγ (0) ≥ (κF )Tγ (t1 ) = κF (t1 ) − γ,
and
υT (0) + α = (υT )Tα (0) ≤ (υT )Tα (t1 ) = υT (t1 ) + α,
υI (0) + β = (υI )Tβ (0) ≤ (υI )Tβ (t1 ) = υI (t1 ) + β
υF (0) − γ = (υF )Tγ (0) ≤ (υF )Tγ (t1 ) = υF (t1 ) − γ,
which imply κT (0) ≥ κT (t1 ), κI (0) ≥ κI (t1 ), κF (0) ≥ κF (t1 ) and υT (0) ≤ υT (t1 ), υI (0) ≤ υI (t1 ),
υF (0) ≤ υF (t1 ), now
κT (t1 ) + α = (κT )Tα (t1 ) ≥ rmin{(κT )Tα (t1 ∗ t 2 ), (κT )Tα (t 2 )}
= rmin{κT (t1 ∗ t 2 ) + α, κT (t 2 ) + α}
κT (t1 ) + α = rmin{κT (t1 ∗ t 2 ), κT (t 2 )} + α,
κI (t1 ) + β = (κI )Tβ (t1 ) ≥ rmin{(κI )Tβ (t1 ∗ t 2 ), (κI )Tβ (t 2 )}
= rmin{κI (t1 ∗ t 2 ) + β, κI (t 2 ) + β}
κI (t1 ) + β = rmin{κI (t1 ∗ t 2 ), κI (t 2 )} + β,
κF (t1 ) − γ = (κF )Tγ (t1 ) ≥ rmin{(κF )Tγ (t1 ∗ t 2 ), (κF )Tγ (t 2 )}
= rmin{κF (t1 ∗ t 2 ) − γ, κF (t 2 ) − γ}
κF (t1 ) − γ = rmin{κF (t1 ∗ t 2 ), κF (t 2 )} − γ,
and
υT (t1 ) + α = (υT )Tα (t1 ) ≤ max{(υT )Tα (t1 ∗ t 2 ), (υT )Tα (t 2 )}
= max{υT (t1 ∗ t 2 ) + α, υT (t 2 ) + α}
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υT (t1 ) + α = max{υT (t1 ∗ t 2 ), υT (t 2 )} + α,
υI (t1 ) + β = (υI )Tβ (t1 ) ≤ max{(υI )Tβ (t1 ∗ t 2 ), (υI )Tβ (t 2 )}
= max{υI (t1 ∗ t 2 ) + β, υI (t 2 ) + β}
υI (t1 ) + β = max{υI (t1 ∗ t 2 ), υI (t 2 )} + β,
υF (t1 ) − γ = (υF )Tγ (t1 ) ≤ max{(υF )Tγ (t1 ∗ t 2 ), (υF )Tγ (t 2 )}
= max{υF (t1 ∗ t 2 ) − γ, υF (t 2 ) − γ}
υF (t1 ) − γ = max{υF (t1 ∗ t 2 ), υF (t 2 )} − γ,
which imply κT (t1 ) ≥ rmin{κT (t1 ∗ t 2 ), κT (t 2 )}, κI (t1 ) ≥ rmin{κI (t1 ∗ t 2 ), κI (t 2 )}, κF (t1 ) ≥ rmin{κF (t1 ∗
t 2 ), κF (t 2 )}
and
υT (t1 ) ≤ max{υT (t1 ∗ t 2 ), υT (t 2 )} ,
υI (t1 ) ≤ max{υI (t1 ∗ t 2 ), υI (t 2 )} ,
υF (t1 ) ≤
max{υF (t1 ∗ t 2 ), υF (t 2 )} for all t1 , t 2 ∈ Y. Hence B is a NCID of Y.
Theorem 3.2.4 Let B be a NCID of Y for some κT,I,F , α, β ∈ [[0,0], ℸ] and γ ∈ [[0,0], ¥], where for υT,I,F ,
T
α, β ∈ [0, Γ] and γ ∈ [0, £]. Then NCT Bα,β,γ
of B is a NCSU of Y.
Proof. Assume t1 , t 2 ∈ Y. Then
(κT )Tα (t1 ∗ t 2 ) = κT (t1 ∗ t 2 ) + α
≥ rmin{κT (t 2 ∗ (t1 ∗ t 2 )), κT (t 2 )} + α
= rmin{κT (0), κT (t 2 )} + α
≥ rmin{κT (t1 ), κT (t 2 )} + α
= rmin{κT (t1 ) + α, κT (t 2 ) + α}
(κT )Tα (t1 ∗ t 2 ) = rmin{(κT )Tα (t1 ), (κT )Tα (t 2 )}
(κT )Tα (t1 ∗ t 2 ) ≥ rmin{(κT )Tα (t1 ), (κT )Tα (t 2 )},
(κI )Tβ (t1 ∗ t 2 ) = κI (t1 ∗ t 2 ) + β
≥ rmin{κI (t 2 ∗ (t1 ∗ t 2 )), κI (t 2 )} + β
= rmin{κI (0), κI (t 2 )} + β
≥ rmin{κI (t1 ), κI (t 2 )} + β
= rmin{κI (t1 ) + β, κI (t 2 ) + β}
(κI )Tβ (t1 ∗ t 2 ) = rmin{(κI )Tβ (t1 ), (κI )Tβ (t 2 )}
(κI )Tβ (t1 ∗ t 2 ) ≥ rmin{(κI )Tβ (t1 ), (κI )Tβ (t 2 )},
(κF )Tγ (t1 ∗ t 2 ) = κF (t1 ∗ t 2 ) − γ
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≥ rmin{κF (t 2 ∗ (t1 ∗ t 2 )), κF (t 2 )} − γ
= rmin{κF (0), κF (t 2 )} − γ
≥ rmin{κF (t1 ), κF (t 2 )} − γ
= rmin{κF (t1 ) − γ, κF (t 2 ) − γ}
(κF )Tγ (t1 ∗ t 2 ) = rmin{(κF )Tγ (t1 ), (κF )Tγ (t 2 )}
(κF )Tγ (t1 ∗ t 2 ) ≥ rmin{(κF )Tγ (t1 ), (κF )Tγ (t 2 )}
and
(υT )Tα (t1 ∗ t 2 ) = υT (t1 ∗ t 2 ) + α
≤ max{υT (t 2 ∗ (t1 ∗ t 2 )), υT (t 2 )} + α
= max{υT (0), υT (t 2 )} + α
≤ max{υT (t1 ), υT (t 2 )} + α
= max{υT (t1 ) + α, υT (t 2 ) + α}
(υT )Tα (t1 ∗ t 2 ) = max{(υT )Tα (t1 ), (υT )Tα (t 2 )}
(υT )Tα (t1 ∗ t 2 ) ≤ max{(υT )Tα (t1 ), (υT )Tα (t 2 )},
(υI )Tβ (t1 ∗ t 2 ) = υI (t1 ∗ t 2 ) + β
≤ max{υI (t 2 ∗ (t1 ∗ t 2 )), υI (t 2 )} + β
= max{υI (0), υI (t 2 )} + β
≤ max{υI (t1 ), υI (t 2 )} + β
= max{υI (t1 ) + β, υI (t 2 ) + β}
(υI )Tβ (t1 ∗ t 2 ) = max{(υI )Tβ (t1 ), (υI )Tβ (t 2 )}
(υI )Tβ (t1 ∗ t 2 ) ≤ max{(υI )Tβ (t1 ), (υI )Tβ (t 2 )},
(υF )Tγ (t1 ∗ t 2 ) = υF (t1 ∗ t 2 ) − γ
≤ max{υF (t 2 ∗ (t1 ∗ t 2 )), υF (t 2 )} − γ
= max{υF (0), υF (t 2 )} − γ
≤ max{υF (t1 ), υF (t 2 )} − γ
= max{υF (t1 ) − γ, υF (t 2 ) − γ}
(υF )Tγ (t1 ∗ t 2 ) = max{(υF )Tγ (t1 ), (υF )Tγ (t 2 )}
(υF )Tγ (t1 ∗ t 2 ) ≤ max{(υF )Tγ (t1 ), (υF )Tγ (t 2 )}.
T
Hence Bα,β,γ
is a NCSU of Y.
Mohsin khalid,Florentin Smarandache,Neha Andaleeb khalid and Said Broumi, Translative And Multiplicative
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316
T
Theorem 3.2.5 If NCT Bα,β,γ
of B is a NCID of Y for some κT,I,F , α, β ∈ [[0,0], ℸ] and γ ∈ [[0,0], ¥], and
for υT,I,F , α, β ∈ [0, Γ] and γ ∈ [0, £]. Then B is a NCSU of Y.
T
Proof. Suppose Bα,β,γ
of B is a NCID of Y. Then
(κT )(t1 ∗ t 2 ) + α = (κT )Tα (t1 ∗ t 2 )
≥ rmin{(κT )Tα (t 2 ∗ (t1 ∗ t 2 )), (κT )Tα (t 2 )}
= rmin{(κT )Tα (0), (κT )Tα (t 2 )}
≥ rmin{(κT )Tα (t1 ), (κT )Tα (t 2 )}
= rmin{κT (t1 ) + α, κT (t 2 ) + α}
(κT )(t1 ∗ t 2 ) + α = rmin{κT (t1 ), κT (t 2 )} + α,
(κI )(t1 ∗ t 2 ) + β = (κI )Tβ (t1 ∗ t 2 )
≥ rmin{(κI )Tβ (t 2 ∗ (t1 ∗ t 2 )), (κI )Tβ (t 2 )}
= rmin{(κI )Tβ (0), (κI )Tβ (t 2 )}
≥ rmin{(κI )Tβ (t1 ), (κI )Tβ (t 2 )}
= rmin{κI (t1 ) + β, κI (t 2 ) + β}
(κI )(t1 ∗ t 2 ) + β = rmin{κI (t1 ), κI (t 2 )} + β,
(κF )(t1 ∗ t 2 ) − γ = (κF )Tγ (t1 ∗ t 2 )
≥ rmin{(κF )Tγ (t 2 ∗ (t1 ∗ t 2 )), (κF )Tγ (t 2 )}
= rmin{(κF )Tγ (0), (κF )Tγ (t 2 )}
≥ rmin{(κF )Tγ (t1 ), (κF )Tγ (t 2 )}
= rmin{κF (t1 ) − γ, κF (t 2 ) − γ}
(κF )(t1 ∗ t 2 ) − γ = rmin{κF (t1 ), κF (t 2 )} − γ
⇒ κT (t1 ∗ t 2 ) ≥ rmin{κT (t1 ), κT (t 2 )}, κI (t1 ∗ t 2 ) ≥ rmin{κI (t1 ), κI (t 2 )} and κF (t1 ∗ t 2 )
≥ rmin{κF (t1 ), κF (t 2 )} and now
(υT )(t1 ∗ t 2 ) + α = (υT )Tα (t1 ∗ t 2 )
≤ max{(υT )Tα (t 2 ∗ (t1 ∗ t 2 )), (υT )Tα (t 2 )}
= max{(υT )Tα (0), (υT )Tα (t 2 )}
≤ max{(υT )Tα (t1 ), (υT )Tα (t 2 )}
= max{υT (t1 ) + α, υT (t 2 ) + α}
(υT )(t1 ∗ t 2 ) + α = max{υT (t1 ), υT (t 2 )} + α,
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(υI )(t1 ∗ t 2 ) + β = (υI )Tβ (t1 ∗ t 2 )
≤ max{(υI )Tβ (t 2 ∗ (t1 ∗ t 2 )), (υI )Tβ (t 2 )}
= max{(υI )Tβ (0), (υI )Tβ (t 2 )}
≤ max{(υI )Tβ (t1 ), (υI )Tβ (t 2 )}
= max{υI (t1 ) + β, υI (t 2 ) + β}
(υI )(t1 ∗ t 2 ) + β = max{υI (t1 ), υI (t 2 )} + β,
(υF )(t1 ∗ t 2 ) − γ = (υF )Tγ (t1 ∗ t 2 )
≤ max{(υF )Tγ (t 2 ∗ (t1 ∗ t 2 )), (υF )Tγ (t 2 )}
= max{(υF )Tγ (0), (υF )Tγ (t 2 )}
≤ max{(υF )Tγ (t1 ), (υF )Tγ (t 2 )}
= max{υF (t1 ) − γ, υF (t 2 ) − γ}
(υF )(t1 ∗ t 2 ) − γ = max{υF (t1 ), υF (t 2 )} − γ
⇒ υT (t1 ∗ t 2 ) ≤ max{υT (t1 ), υT (t 2 )}, υI (t1 ∗ t 2 ) ≤ max{υI (t1 ), υI (t 2 )}
and
υF (t1 ∗ t 2 ) ≤
max{υF (t1 ), υF (t 2 )}. Hence B is a NCSU of Y.
Theorem 3.2.6 Intersection of any two neutrosophic cubic translations of a neutrosophic cubic BF ideals B of
Y is a neutrosophic cubic BF ideal of Y.
T
Proof. Suppose Bα,β,γ
and BαT′ ,β′ ,γ′ are two neutrosophic cubic translations of neutrosophic cubic BF ideal B
T
and C of Y respectively, where for Bα,β,γ
, for κT,I,F , α, β ∈ [[0,0], ℸ], γ ∈ [[0,0], ¥], for υT,I,F , α, β ∈ [0, Γ],
T
γ ∈ [0, £] and for Bα′,β′,γ′
, for κT,I,F α′, β′ ∈ [[0,0], ℸ], γ′ ∈ [[0,0], ¥], for υT,I,F , α′, β′ ∈ [0, Γ], γ′ ∈ [0, £]
T
and α ≤ α′ , β ≤ β′ , γ ≤ γ′ as we know that, Bα,β,γ
and BαT′ ,β′ ,γ′ are neutrosophic cubic BF ideals of Y. So
((κT )Tα ∩ (κT )Tα′ )(t1 ) = rmin{(κT )Tα (t1 ), (κT )Tα′ (t1 )}
= rmin{κT (t1 ) + α, κT (t1 ) + α′ }
= κT (t1 ) + α
((κT )Tα ∩ (κT )Tα′ )(t1 ) = (κT )Tα (t1 ),
((κI )Tβ ∩ (κI )Tβ′ )(t1 ) = rmin{(κI )Tβ (t1 ), (κI )Tβ′ (t1 )}
= rmin{κI (t1 ) + β, κI (t1 ) + β′ }
= κI (t1 ) + β
((κI )Tβ ∩ (κI )Tβ′ )(t1 ) = (κI )Tβ (t1 ),
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((κF )Tγ ∩ (κF )Tγ′ )(t1 ) = rmin{(κF )Tγ (t1 ), (κF )Tγ′ (t1 )}
= rmin{κF (t1 ) − γ, κF (t1 ) − γ′ }
= κF (t1 ) − γ′
((κF )Tγ ∩ (κF )Tγ′ )(t1 ) = (κF )Tγ′ (t1 )
and
((υT )Tα ∩ (υT )Tα′ )(t1 ) = max{(υT )Tα (t1 ), (υT )Tα′ (t1 )}
= max{υT (t1 ) + α, υT (t1 ) + α′ }
= υT (t1 ) + α′
((υT )Tα ∩ (υT )Tα′ )(t1 ) = (υT )Tα′ (t1 ),
((υI )Tβ ∩ (υI )Tβ′ )(t1 ) = max{(υI )Tβ (t1 ), (υI )Tβ′ (t1 )}
= max{υI (t1 ) + β, υI (t1 ) + β′ }
= υI (t1 ) + β′
((υI )Tβ ∩ (υI )Tβ′ )(t1 ) = (υI )Tβ′ (t1 ),
((υF )Tγ ∩ (υF )Tγ′ )(t1 ) = max{(υF )Tγ (t1 ), (υF )Tγ′ (t1 )}
= max{υF (t1 ) − γ, υF (t1 ) − γ′ }
= υF (t1 ) − γ
((υF )Tγ ∩ (υF )Tγ′ )(t1 ) = (υF )Tγ (t1 ).
T
Hence Bα,β,γ
∩ BαT′,β′ ,γ′ is a neutrosophic cubic BF ideal of Y.
Theorem 3.2.7 Union of any two neutrosophic cubic translations of a neutrosophic cubic BF ideals B of Y is
a neutrosophic cubic BF ideal of Y.
T
Proof. Suppose Bα,β,γ
and BαT′ ,β′ ,γ′ are two neutrosophic cubic translations of neutrosophic cubic BF ideal B
T
of Y respectively, where for Bα,β,γ
, for κT,I,F , α, β ∈ [[0,0], ℸ], γ ∈ [[0,0], ¥], for υT,I,F , α, β ∈ [0, Γ], γ ∈
T
[0, £] and for Bα′,β′,γ′ , for κT,I,F α′, β′ ∈ [[0,0], ℸ], γ′ ∈ [[0,0], ¥], for υT,I,F , α′, β′ ∈ [0, Γ], γ′ ∈ [0, £] and
T
α ≥ α′ , β ≥ β′ , γ ≥ γ′ as we know that, Bα,β,γ
and BαT′,β′ ,γ′ are neutrosophic cubic BF ideals of Y. Then
((κT )Tα ∪ (κT )Tα′ )(t1 ) = rmax{(κT )Tα (t1 ), (κT )Tα′ (t1 )}
= rmax{κT (t1 ) + α, κT (t1 ) + α′ }
= κT (t1 ) + α
((κT )Tα ∪ (κT )Tα′ )(t1 ) = (κT )Tα (t1 ),
((κI )Tβ ∪ (κI )Tβ′ )(t1 ) = rmax{(κI )Tβ (t1 ), (κI )Tβ′ (t1 )}
= rmax{κI (t1 ) + β, κI (t1 ) + β′ }
= κI (t1 ) + β
((κI )Tβ ∪ (κI )Tβ′ )(t1 ) = (κI )Tβ (t1 ),
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((κF )Tγ ∪ (κF )Tγ′ )(t1 ) = rmax{(κF )Tγ (t1 ), (κF )Tγ′ (t1 )}
= rmax{κF (t1 ) − γ, κF (t1 ) − γ′ }
= κF (t1 ) − γ′
((κF )Tγ ∪ (κF )Tγ′ )(t1 ) = (κF )Tγ′ (t1 )
and
((υT )Tα ∪ (υT )Tα′ )(t1 ) = min{(υT )Tα (t1 ), (υT )Tα′ (t1 )}
= min{υT (t1 ) + α, υT (t1 ) + α′ }
= υT (t1 ) + α′
((υT )Tα ∪ (υT )Tα′ )(t1 ) = (υT )Tα′ (t1 ),
((υI )Tβ ∪ (υI )Tβ′ )(t1 ) = min{(υI )Tβ (t1 ), (υI )Tβ′ (t1 )}
= min{υI (t1 ) + β, υI (t1 ) + β′ }
= υI (t1 ) + β′
((υI )Tβ ∪ (υI )Tβ′ )(t1 ) = (υI )Tβ (t1 ),
((υF )Tγ ∪ (υF )Tγ′ )(t1 ) = min{(υF )Tγ (t1 ), (υF )Tγ′ (t1 )}
= min{υF (t1 ) − γ, υF (t1 ) − γ′ }
= υF (t1 ) − γ
((υF )Tγ ∪ (υF )Tγ′ )(t1 ) = (υF )Tγ (t1 )
T
Hence Bα,β,γ
∪ BαT′ ,β′ ,γ′ is a neutrosophic cubic BF ideal of Y.
Theorem 3.2.8 Let B be a NCS of Y such that NCM BδM of B is a NCID of Y for δ ∈ (0,1] then B is a
NCID of Y.
Proof. Suppose that BδM is a NCID of Y for δ ∈ (0,1] and t1 , t 2 ∈ Y. Then δ. κT (0) = (κT )M
δ (0) ≥
M
M
M
(κT )δ (t1 ) = δ. κT (t1 ), so κT (0) ≥ κT (t1 ),δ. κI (0) = (κI )δ (0) ≥ (κI )δ (t1 ) = δ. κI (t1 ), so κI (0) ≥ κI (t1 ),
M
M
M
δ. κF (0) = (κF )M
δ (0) ≥ (κF )δ (t1 ) = δ. κF (t1 ), so κF (0) ≥ κF (t1 ) and δ. υT (0) = (υT )δ (0) ≤ (υT )δ (t1 )
= δ. υT (t1 ), so υT (0) ≤ υT (t1 ) , δ. υI (0) = (υI )M
≤ (υI )M
= δ. υI (t1 ), so υI (0) ≤ υI (t1 ) ,
δ (0)
δ (t1 )
M
M
δ. υF (0) = (υF )δ (0) ≤ (υF )δ (t1 ) = δ. υF (t1 ), so υF (0) ≤ υF (t1 ). Now
δ. κT (t1 ) = (κT )M
δ (t1 )
M
≥ rmin{(κT )M
δ (t1 ∗ t 2 ), (κT )δ (t 2 )}
= rmin{δ. κT (t1 ∗ t 2 ), δ. κT (t 2 )}
δ. κT (t1 ) = δ. rmin{κT (t1 ∗ t 2 ), κT (t 2 )},
δ. κI (t1 ) = (κI )M
δ (t1 )
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M
≥ rmin{(κI )M
δ (t1 ∗ t 2 ), (κI )δ (t 2 )}
= rmin{δ. κI (t1 ∗ t 2 ), δ. κI (t 2 )}
δ. κI (t1 ) = δ. rmin{κI (t1 ∗ t 2 ), κI (t 2 )},
δ. κF (t1 ) = (κF )M
δ (t1 )
M
≥ rmin{(κF )M
δ (t1 ∗ t 2 ), (κF )δ (t 2 )}
= rmin{δ. κF (t1 ∗ t 2 ), δ. κF (t 2 )}
δ. κF (t1 ) = δ. rmin{κF (t1 ∗ t 2 ), κF (t 2 )},
so
κT (t1 ) ≥ rmin{κT (t1 ∗ t 2 ), κT (t 2 )}, κI (t1 ) ≥ rmin{κI (t1 ∗ t 2 ), κI (t 2 )}
and
κF (t1 ) ≥ rmin{κF (t1 ∗
and
υF (t1 ) ≤ max{υF (t1 ∗
t 2 ), κF (t 2 )} and also
δ. υT (t1 ) = (υT )M
δ (t1 )
M
≤ max{(υT )M
δ (t1 ∗ t 2 ), (υT )δ (t 2 )}
= max{δ. υT (t1 ∗ t 2 ), δ. υT (t 2 )}
δ. υT (t1 ) = δ. max{υT (t1 ∗ t 2 ), υT (t 2 )},
δ. υI (t1 ) = (υI )M
δ (t1 )
M
≤ max{(υI )M
δ (t1 ∗ t 2 ), (υI )δ (t 2 )}
= max{δ. υI (t1 ∗ t 2 ), δ. υI (t 2 )}
δ. υI (t1 ) = δ. max{υI (t1 ∗ t 2 ), υI (t 2 )},
δ. υF (t1 ) = (υF )M
δ (t1 )
M
≤ max{(υF )M
δ (t1 ∗ t 2 ), (υF )δ (t 2 )}
= max{δ. υF (t1 ∗ t 2 ), δ. υF (t 2 )}
δ. υF (t1 ) = δ. max{υF (t1 ∗ t 2 ), υF (t 2 )},
so
υT (t1 ) ≤ max{υT (t1 ∗ t 2 ), υT (t 2 )}, υI (t1 ) ≤ max{υI (t1 ∗ t 2 ), υI (t 2 )}
t 2 ), υF (t 2 )}. Hence B is a NCID of Y.
Theorem 3.2.9 If B is a NCID of Y, then NCM BδM of B is a NCID of Y, for all δ ∈ (0,1].
M
Proof. Let B be a NCID of Y and δ ∈ (0,1]. Then we have (κT )M
δ (0) = δ. κT (0) ≥ δ. κT (t1 ) →(κT )δ (0) =
(κT )M
δ (t1 ),
M
M
(κI )M
( t1 ),
δ (0) = δ. κI (0) ≥ δ. κI (t1 ) → (κI )δ (0) = (κI )δ
M
δ. κF (t1 ) → (κF )M
δ (0) = (κF )δ (t1 )
and
(κF )M
δ (0) = δ. κF (0) ≥
M
M
(υT )M
δ (0) = δ. υT (0) ≤ δ. υT (t1 ) → (υT )δ (0) = (υT )δ (t1 ),
Mohsin khalid,Florentin Smarandache,Neha Andaleeb khalid and Said Broumi, Translative And Multiplicative
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321
M
M
M
M
(υI )M
δ (0) = δ. υI (0) ≤ δ. υI (t1 ) → (υI )δ (0) = (υI )δ (t1 ), (υF )δ (0) = δ. υF (0) ≤ δ. υF (t1 ) → (υF )δ (0) =
(υF )M
δ (t1 ).
Now
(κT )M
δ (t1 ) = δ. κT (t1 )
≥ δ. rmin{κT (t1 ∗ t 2 ), κT (t 2 )}
= rmin{δ. κT (t1 ∗ t 2 ), δ. κT (t 2 )}
M
M
(κT )M
δ (t1 ) = rmin{(κT )δ (t1 ∗ t 2 ), (κT )δ (t 2 )}
M
M
(κT )M
δ (t1 ) ≥ rmin{(κT )δ (t1 ∗ t 2 ), (κT )δ (t 2 )},
(κI )M
δ (t1 ) = δ. κI (t1 )
≥ δ. rmin{κI (t1 ∗ t 2 ), κI (t 2 )}
= rmin{δ. κI (t1 ∗ t 2 ), δ. κI (t 2 )}
M
M
(κI )M
δ (t1 ) = rmin{(κI )δ (t1 ∗ t 2 ), (κI )δ (t 2 )}
M
M
(κI )M
δ (t1 ) ≥ rmin{(κI )δ (t1 ∗ t 2 ), (κI )δ (t 2 )},
(κF )M
δ (t1 ) = δ. κF (t1 )
≥ δ. rmin{κF (t1 ∗ t 2 ), κF (t 2 )}
= rmin{δ. κF (t1 ∗ t 2 ), δ. κF (t 2 )}
M
M
(κF )M
δ (t1 ) = rmin{(κF )δ (t1 ∗ t 2 ), (κF )δ (t 2 )}
M
M
(κF )M
δ (t1 ) ≥ rmin{(κF )δ (t1 ∗ t 2 ), (κF )δ (t 2 )}
and
(υT )M
δ (t1 ) = δ. υT (t1 )
≤ δ. max{υT (t1 ∗ t 2 ), υT (t 2 )}
= max{δ. υT (t1 ∗ t 2 ), δ. υT (t 2 )}
M
M
(υT )M
δ (t1 ) = max{(υT )δ (t1 ∗ t 2 ), (υT )δ (t 2 )}
M
M
(υT )M
δ (t1 ) ≤ max{(υT )δ (t1 ∗ t 2 ), (υT )δ (t 2 )},
(υI )M
δ (t1 ) = δ. υI (t1 )
≤ δ. max{υI (t1 ∗ t 2 ), υI (t 2 )}
= max{δ. υI (t1 ∗ t 2 ), δ. υI (t 2 )}
M
M
(υI )M
δ (t1 ) = max{(υI )δ (t1 ∗ t 2 ), (υI )δ (t 2 )}
M
M
(υI )M
δ (t1 ) ≤ max{(υI )δ (t1 ∗ t 2 ), (υI )δ (t 2 )},
Mohsin khalid,Florentin Smarandache,Neha Andaleeb khalid and Said Broumi, Translative And Multiplicative
Interpretation of Neutrosophic Cubic Set
Neutrosophic Sets and Systems, Vol. 35, 2020
(υF )M
δ (t1 ) = δ. υF (t1 )
≤ δ. max{υF (t1 ∗ t 2 ), υF (t 2 )}
= max{δ. υF (t1 ∗ t 2 ), δ. υF (t 2 )}
M
M
(υF )M
δ (t1 ) = max{(υF )δ (t1 ∗ t 2 ), (υF )δ (t 2 )}
M
M
(υF )M
δ (t1 ) ≤ max{(υF )δ (t1 ∗ t 2 ), (υF )δ (t 2 )}.
Hence BδM of B is a NCID of Y, for all δ ∈ (0,1].
Theorem 3.2.10 Let B be a NCID of Y and δ ∈ [0,1] then NCM BδM of B is a NCSU of Y.
Proof. Suppose t1 , t 2 ∈ Y. Then
(κT )M
δ (t1 ∗ t 2 ) = δ. κT (t1 ∗ t 2 )
≥ δ. rmin{κT (t 2 ∗ (t1 ∗ t 2 )), κT (t 2 )}
= δ. rmin{κT (0), κT (t 2 )}
≥ δ. rmin{κT (t1 ), κT (t 2 )}
= rmin{δ. κT (t1 ), δ. κT (t 2 )}
M
M
(κT )M
δ (t1 ∗ t 2 ) = rmin{(κT )δ (t1 ), (κT )δ (t 2 )}
M
M
(κT )M
δ (t1 ∗ t 2 ) ≥ rmin{(κT )δ (t1 ), (κT )δ (t 2 )},
(κI )M
δ (t1 ∗ t 2 ) = δ. κI (t1 ∗ t 2 )
≥ δ. rmin{κI (t 2 ∗ (t1 ∗ t 2 )), κI (t 2 )}
= δ. rmin{κI (0), κI (t 2 )}
≥ δ. rmin{κI (t1 ), κI (t 2 )}
= rmin{δ. κI (t1 ), δ. κI (t 2 )}
M
M
(κI )M
δ (t1 ∗ t 2 ) = rmin{(κI )δ (t1 ), (κI )δ (t 2 )}
M
M
(κI )M
δ (t1 ∗ t 2 ) ≥ rmin{(κI )δ (t1 ), (κI )δ (t 2 )},
(κF )M
δ (t1 ∗ t 2 ) = δ. κF (t1 ∗ t 2 )
≥ δ. rmin{κF (t 2 ∗ (t1 ∗ t 2 )), κF (t 2 )}
= δ. rmin{κF (0), κF (t 2 )}
≥ δ. rmin{κF (t1 ), κF (t 2 )}
= rmin{δ. κF (t1 ), δ. κF (t 2 )}
M
M
(κF )M
δ (t1 ∗ t 2 ) = rmin{(κF )δ (t1 ), (κF )δ (t 2 )}
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323
M
M
(κF )M
δ (t1 ∗ t 2 ) ≥ rmin{(κF )δ (t1 ), (κF )δ (t 2 )}
and
(υT )M
δ (t1 ∗ t 2 ) = δ. υT (t1 ∗ t 2 )
≤ δ. max{υT (t 2 ∗ (t1 ∗ t 2 )), υT (t 2 )}
= δ. max{υT (0), υT (t 2 )}
≤ δ. max{υT (t1 ), υT (t 2 )}
= max{δ. υT (t1 ), δ. υT (t 2 )}
M
M
(υT )M
δ (t1 ∗ t 2 ) = max{(υT )δ (t1 ), (υT )δ (t 2 )}
M
M
(υT )M
δ (t1 ∗ t 2 ) ≤ max{(υT )δ (t1 ), (υT )δ (t 2 )},
(υI )M
δ (t1 ∗ t 2 ) = δ. υI (t1 ∗ t 2 )
≤ δ. max{υI (t 2 ∗ (t1 ∗ t 2 )), υI (t 2 )}
= δ. max{υI (0), υI (t 2 )}
≤ δ. max{υI (t1 ), υI (t 2 )}
= max{δ. υI (t1 ), δ. υI (t 2 )}
M
M
(υI )M
δ (t1 ∗ t 2 ) = max{(υI )δ (t1 ), (υI )δ (t 2 )}
M
M
(υI )M
δ (t1 ∗ t 2 ) ≤ max{(υI )δ (t1 ), (υI )δ (t 2 )},
(υF )M
δ (t1 ∗ t 2 ) = δ. υF (t1 ∗ t 2 )
≤ δ. max{υF (t 2 ∗ (t1 ∗ t 2 )), υF (t 2 )}
= δ. max{υF (0), υF (t 2 )}
≤ δ. max{υF (t1 ), υF (t 2 )}
= max{δ. υF (t1 ), δ. υF (t 2 )}
M
M
(υF )M
δ (t1 ∗ t 2 ) = max{(υF )δ (t1 ), (υF )δ (t 2 )}
M
M
(υF )M
δ (t1 ∗ t 2 ) ≤ max{(υF )δ (t1 ), (υF )δ (t 2 )}.
Hence BδM is a NCSU of Y.
Theorem 3.2.11 If the NCM BδM of B is a NCID of Y, for δ ∈ (0,1]. Then B is a neutrosophic cubic BFsubalgebra of Y.
Proof. Assume BδM of B is a NCID of Y. Then
δ. (κT )(t1 ∗ t 2 ) = (κT )M
δ (t1 ∗ t 2 )
M
≥ rmin{(κT )M
δ (t 2 ∗ (t1 ∗ t 2 )), (κT )δ (t 2 )}
M
= rmin{(κT )M
δ (0), (κT )δ (t 2 )}
Mohsin khalid,Florentin Smarandache,Neha Andaleeb khalid and Said Broumi, Translative And Multiplicative
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M
≥ rmin{(κT )M
δ (t1 ), (κT )δ (t 2 )}
= rmin{δ. κT (t1 ), δ. κT (t 2 )}
δ. (κT )(t1 ∗ t 2 ) = δ. rmin{κT (t1 ), κT (t 2 )}
⇒ κT (t1 ∗ t 2 ) ≥ rmin{κT (t1 ), κT (t 2 )},
δ. (κI )(t1 ∗ t 2 ) = (κI )M
δ (t1 ∗ t 2 )
M
≥ rmin{(κI )M
δ (t 2 ∗ (t1 ∗ t 2 )), (κI )δ (t 2 )}
M
= rmin{(κI )M
δ (0), (κI )δ (t 2 )}
M
≥ rmin{(κI )M
δ (t1 ), (κI )δ (t 2 )}
= rmin{δ. κI (t1 ), δ. κI (t 2 )}
δ. (κI )(t1 ∗ t 2 ) = δ. rmin{κI (t1 ), κI (t 2 )}
⇒ κI (t1 ∗ t 2 ) ≥ rmin{κI (t1 ), κI (t 2 )},
δ. (κF )(t1 ∗ t 2 ) = (κF )M
δ (t1 ∗ t 2 )
M
≥ rmin{(κF )M
δ (t 2 ∗ (t1 ∗ t 2 )), (κF )δ (t 2 )}
M
= rmin{(κF )M
δ (0), (κF )δ (t 2 )}
M
≥ rmin{(κF )M
δ (t1 ), (κF )δ (t 2 )}
= rmin{δ. κF (t1 ), δ. κF (t 2 )}
δ. (κF )(t1 ∗ t 2 ) = δ. rmin{κF (t1 ), κF (t 2 )}
⇒ κF (t1 ∗ t 2 ) ≥ rmin{κF (t1 ), κF (t 2 )}
and
δ. (υT )(t1 ∗ t 2 ) = (υT )M
δ (t1 ∗ t 2 )
M
≤ max{(υT )M
δ (t 2 ∗ (t1 ∗ t 2 )), (υT )δ (t 2 )}
M
= max{(υT )M
δ (0), (υT )δ (t 2 )}
M
≤ max{(υT )M
δ (t1 ), (υT )δ (t 2 )}
= max{δ. υT (t1 ), δ. υT (t 2 )}
δ. (υT )(t1 ∗ t 2 ) = δ. max{υT (t1 ), υT (t 2 )}
⇒ υT (t1 ∗ t 2 ) ≤ max{υT (t1 ), υT (t 2 )},
δ. (υI )(t1 ∗ t 2 ) = (υI )M
δ (t1 ∗ t 2 )
M
≤ max{(υI )M
δ (t 2 ∗ (t1 ∗ t 2 )), (υI )δ (t 2 )}
M
= max{(υI )M
δ (0), (υI )δ (t 2 )}
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325
M
≤ max{(υI )M
δ (t1 ), (υI )δ (t 2 )}
= max{δ. υI (t1 ), δ. υI (t 2 )}
δ. (υI )(t1 ∗ t 2 ) = δ. max{υI (t1 ), υI (t 2 )}
⇒ υI (t1 ∗ t 2 ) ≤ max{υI (t1 ), υI (t 2 )},
δ. (υF )(t1 ∗ t 2 ) = (υF )M
δ (t1 ∗ t 2 )
M
≤ max{(υF )M
δ (t 2 ∗ (t1 ∗ t 2 )), (υF )δ (t 2 )}
M
= max{(υF )M
δ (0), (υF )δ (t 2 )}
M
≤ max{(υF )M
δ (t1 ), (υF )δ (t 2 )}
= max{δ. υF (t1 ), δ. υF (t 2 )}
δ. (υF )(t1 ∗ t 2 ) = δ. max{υF (t1 ), υF (t 2 )}
⇒ υF (t1 ∗ t 2 ) ≤ max{υF (t1 ), υF (t 2 )}.
Hence B is a NCSU of Y.
Theorem 3.2.12 Intersection of any two neutrosophic cubic multiplications of a NCID B of Y is a NCID of
Y.
Proof. Suppose BδM and BδM′ are neutrosophic cubic multiplications of NCID B of Y, where δ, δ′ ∈ (0,1]
and δ ≤ δ′ , as we know that BδM and BδM′ are NCIDs of Y. Then
M
M
M
((κT )M
δ ∩ (κT )δ′ )(t1 ) = rmin{(κT )δ (t1 ), (κT )δ′ (t1 )}
= rmin{κT (t1 ). δ, κT (t1 ). δ′ }
= κT (t1 ). δ
M
M
((κT )M
δ ∩ (κT )δ′ )(t1 ) = (κT )δ (t1 ),
M
M
M
((κI )M
δ ∩ (κI )δ′ )(t1 ) = rmin{(κI )δ (t1 ), (κI )δ′ (t1 )}
= rmin{κI (t1 ). δ, κI (t1 ). δ′ }
= κI (t1 ). δ
M
M
((κI )M
δ ∩ (κI )δ′ )(t1 ) = (κI )δ (t1 ),
M
M
M
((κF )M
δ ∩ (κF )δ′ )(t1 ) = rmin{(κF )δ (t1 ), (κF )δ′ (t1 )}
= rmin{κF (t1 ). δ, κF (t1 ). δ′ }
= κF (t1 ). δ
M
M
((κF )M
δ ∩ (κF )δ′ )(t1 ) = (κF )δ (t1 )
and
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326
M
M
M
((υT )M
δ ∩ (υT )δ′ )(t1 ) = max{(υT )δ (t1 ), (υT )δ′ (t1 )}
= max{υT (t1 ). δ, υT (t1 ). δ′ }
= υT (t1 ). δ′
M
M
((υT )M
δ ∩ (υT )δ′ )(t1 ) = (υT )δ′ (t1 ),
M
M
M
((υI )M
δ ∩ (υI )δ′ )(t1 ) = max{(υI )δ (t1 ), (υI )δ′ (t1 )}
= max{υI (t1 ). δ, υI (t1 ). δ′ }
= υI (t1 ). δ′
M
M
((υI )M
δ ∩ (υI )δ′ )(t1 ) = (υI )δ′ (t1 ),
M
M
M
((υF )M
δ ∩ (υF )δ′ )(t1 ) = max{(υF )δ (t1 ), (υF )δ′ (t1 )}
= max{υF (t1 ). δ, υF (t1 ). δ′ }
= υF (t1 ). δ′
M
M
((υF )M
δ ∩ (υF )δ′ )(t1 ) = (υF )δ′ (t1 ).
Hence BδM ∩ BδM′ is NCID of Y.
Theorem 3.2.13 Union of any two neutrosophic cubic multiplications of a NCID B of Y is a NCID of Y.
Proof. Suppose BδM and BδM′ are neutrosophic cubic multiplications of NCID B of Y, where δ, δ′ ∈ (0,1]
and δ ≤ δ′ , as we know that BδM and BδM′ are NCIDs of Y. Then
M
M
M
((κT )M
δ ∪ (κT )δ′ )(t1 ) = rmax{(κT )δ (t1 ), (κT )δ′ (t1 )}
= rmax{κT (t1 ). δ, κT (t1 ). δ′ }
= κT (t1 ). δ′
M
M
((κT )M
δ ∪ (κT )δ′ )(t1 ) = (κT )δ′ (t1 ),
M
M
M
((κI )M
δ ∪ (κI )δ′ )(t1 ) = rmax{(κI )δ (t1 ), (κI )δ′ (t1 )}
= rmax{κI (t1 ). δ, κI (t1 ). δ′ }
= κI (t1 ). δ′
M
M
((κI )M
δ ∪ (κI )δ′ )(t1 ) = (κI )δ′ (t1 ),
M
M
M
((κF )M
δ ∪ (κF )δ′ )(t1 ) = rmax{(κF )δ (t1 ), (κF )δ′ (t1 )}
= rmax{κF (t1 ). δ, κF (t1 ). δ′ }
= κF (t1 ). δ′
M
M
((κF )M
δ ∪ (κF )δ′ )(t1 ) = (κF )δ′ (t1 )
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and
M
M
M
((υT )M
δ ∪ (υT )δ′ )(t1 ) = min{(υT )δ (t1 ), (υT )δ′ (t1 )}
= min{υT (t1 ). δ, υT (t1 ). δ′ }
= υT (t1 ). δ
M
M
((υT )M
δ ∪ (υT )δ′ )(t1 ) = (υT )δ (t1 ),
M
M
M
((υI )M
δ ∪ (υI )δ′ )(t1 ) = min{(υI )δ (t1 ), (υI )δ′ (t1 )}
= min{υI (t1 ). δ, υI (t1 ). δ′ }
= υI (t1 ). δ
M
M
((υI )M
δ ∪ (υI )δ′ )(t1 ) = (υI )δ (t1 ),
M
M
M
((υF )M
δ ∪ (υF )δ′ )(t1 ) = min{(υF )δ (t1 ), (υF )δ′ (t1 )}
= min{υF (t1 ). δ, υF (t1 ). δ′ }
= υF (t1 ). δ
M
M
((υF )M
δ ∪ (υF )δ′ )(t1 ) = (υF )δ (t1 ).
Hence BδM ∪ BδM′ is NCID of Y.
3.3 Magnified Translative Interpretation of Neutrosophic Cubic Subalgebra and Neutrosophic Cubic
Ideal
In this section, we define the notion of neutrosophic cubic magnified translation NCMT and investigate some
results.
Definition 3.3.1 Let B = (κT,I,F , υT,I,F ) be a NCS of Y and for κT,I,F , α, β ∈ [[0,0], ℸ] and γ ∈ [[0,0], ¥],
T
where for υT,I,F , α, β ∈ [0, Γ] and γ ∈ [0, £] and for all δ ∈ [0,1]. An object having the form BδMα,β,γ
=
T
MT
MT
MT
{(κT,I,F )M
δ α,β,γ , (υT,I,F )δ α,β,γ } is said to be a NCMT of B, when (κT )δ α (t1 ) = δ. κT (t1 ) + α,(κI )δ β (t1 ) =
T
MT
MT
δ. κI (t1 ) + β , (κF )M
δ γ (t1 ) = δ. κF (t1 ) - γ and (υT )δ α (t1 ) = δ. υT (t1 ) + α , (υI )δ β (t1 ) = δ. υI (t1 ) + β ,
T
(υF )M
δ γ (t1 ) = δ. υF (t1 )-γ for all t1 ∈ Y.
Example 3.3.1 Let Y = {0,1,2} be a BF-algebra as defined in Example 3.2.1. A NCS B = (κT,I,F , υT,I,F ) of Y
is defined as
[0.1, 0.3] if t1 = 0
κT (t1 ) = (
[0.4, 0.7] if otherwise
[0.2,0.4]
κI (t1 ) = (
[0.5, 0.7]
κF (t1 ) = (
if t1 = 0
if otherwise
[0.4,0.6] if t1 = 0
[0.5, 0.8] if otherwise
and
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0.1
0.4
if t1 = 0
if otherwise
0.2
0.3
if t1 = 0
if otherwise
0.5
υF (t1 ) = (
0.7
if t1 = 0
if otherwise.
υT (t1 ) = (
υI (t1 ) = (
Then B is a neutrosophic cubic subalgebra, for υT,I,F choose δ = 0.1, α = 0.02, β = 0.03, γ = 0.04 and for
κT,I,F choose
δ = [0.1,0.4], α = [0.03,0.07], β = [0.04,0.08], γ = [0.02,0.06] then the mapping
MT
B(0.1) (α,β,γ) |Y → [0,1] is given by
[0.04, 0.19] if t1 = 1
(κT )M
[0.1,0.4] [0.03,0.07] (t1 ) = ([0.07, 0.35] if otherwise
T
(κI )M
[0.1,0.4] [0.04,0.08] (t1 ) = (
[0.06, 0.24] if t1 = 1
[0.09, 0.36] if otherwise
T
(κF )M
[0.1,0.4] [0.02,0.06] (t1 ) = (
[0.02, 0.18]
[0.03, 0.26]
if t1 = 1
if otherwise
and
0.03
0.06
if t1 = 1
if otherwise
0.05
T
(υI )M
0.1,0.03 (t1 ) = (
0.06
if t1 = 1
if otherwise
0.01
T
(υF )M
0.1.0.04 (t1 ) = (
0.03
if t1 = 1
if otherwise,
(υT )M
0.1,0.02 (t1 ) = (
which
imply
T
(κT )M
[0.1,0.4][0.03,0.07] (t1 ) = [0.1,0.4]. κT (t1 ) + [0.03,0.07]
[0.1,0.4]. κT (t1 ) + [0.04,0.08]
T
(υT )M
(0.1)(0.02) (t1 )
,
,
T
(κI )M
[0.1,0.4][0.04,0.08] (t1 ) =
T
(κF )M
[0.1,0.4][0.02,0.06] (t1 ) = [0.1,0.4]. κF (t1 ) − [0.02,0.06]
= (0.1). υT (t1 ) + 0.02 ,
T
(υI )M
(0.1)(0.03) (t1 )
MT
(0.1). νF (t1 ) − 0.04 for all t1 ∈ Y. Hence B
= (0.1). υT (t1 ) + 0.03 ,
and
T
(νF )M
(0.1) (0.04) (t1 )
=
is a neutrosophic cubic magnified translation.
Theorem 3.3.1 Let B be a neutrosophic cubic subset of Y such that for κT,I,F , α, β ∈ [[0,0], ℸ] and γ ∈
T,I,F
[[0,0], ¥], where for υT,I,F , α, β ∈ [0, Γ] and γ ∈ [0, £] and δ ∈ [0,1] and a mapping BδMα,β,γ
|Y → [0,1] be a
T,I,F
NCMT of B. If B is NCSU of Y, then BδMα,β,γ
is a NCSU of Y.
Proof. Let B be a neutrosophic cubic subset of Y such that for κT,I,F , α, β ∈ [[0,0], ℸ] and γ ∈ [[0,0], ¥],
T,I,F
where for υT,I,F , α, β ∈ [0, Γ] and γ ∈ [0, £] and δ ∈ [0,1] and a mapping BδMα,β,γ
|Y → [0,1] be a NCMT of
B. Suppose B is a NCSU of Y. Then κT (t1 ∗ t 2 ) ≥ rmin{κT (t1 ), κT (t 2 )}, κI (t1 ∗ t 2 ) ≥
rmin{κI (t1 ), κI (t 2 )}, κF (t1 ∗ t 2 ) ≥ rmin{κF (t1 ), κF ( t 2 ) } and υT (t1 ∗ t 2 ) ≤ max{υT (t1 ), υT (t 2 )}, υI (t1 ∗
t 2 ) ≤ max{υI (t1 ), υI (t 2 )}, υF (t1 ∗ t 2 ) ≤ max{υF (t1 ), υF (t 2 )}. Now
T
(κT )M
δ α (t1 ∗ t 2 ) = δ. κT (t1 ∗ t 2 ) + α
≥ δ. rmin{κT (t1 ), κT (t 2 )} + α
Mohsin khalid,Florentin Smarandache,Neha Andaleeb khalid and Said Broumi, Translative And Multiplicative
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= rmin{δ. κT (t1 ) + α, δ. κT (t 2 ) + α}
T
MT
MT
(κT )M
δ α (t1 ∗ t 2 ) = rmin{(κT )δ α (t1 ), (κT )δ α (t 2 )}
T
MT
MT
(κT )M
δ α (t1 ∗ t 2 ) ≥ rmin{(κT )δ α (t1 ), (κT )δ α (t 2 )},
T
(κI )M
δ β (t1 ∗ t 2 ) = δ. κI (t1 ∗ t 2 ) + β
≥ δ. rmin{κI (t1 ), κI (t 2 )} + β
= rmin{δ. κI (t1 ) + β, δ. κI (t 2 ) + β}
T
MT
MT
(κI )M
δ β (t1 ∗ t 2 ) = rmin{(κI )δ β (t1 ), (κI )δ β (t 2 )}
T
MT
MT
(κI )M
δ β (t1 ∗ t 2 ) ≥ rmin{(κI )δ β (t1 ), (κI )δ β (t 2 )},
T
(κF )M
δ γ (t1 ∗ t 2 ) = δ. κF (t1 ∗ t 2 ) − γ
≥ δ. rmin{κF (t1 ), κF (t 2 )} − γ
= rmin{δ. κF (t1 ) − γ, δ. κF (t 2 ) − γ}
T
MT
MT
(κF )M
δ γ (t1 ∗ t 2 ) = rmin{(κF )δ γ (t1 ), (κF )δ γ (t 2 )}
T
MT
MT
(κF )M
δ γ (t1 ∗ t 2 ) ≥ rmin{(κF )δ γ (t1 ), (κF )δ γ (t 2 )}
and
T
(υT )M
δ α (t1 ∗ t 2 ) = δ. υT (t1 ∗ t 2 ) + α
≤ δ. max{υT (t1 ), υT (t 2 )} + α
= max{δ. υT (t1 ) + α, δ. υT (t 2 ) + α}
T
MT
MT
(υT )M
δ α (t1 ∗ t 2 ) = max{(υT )δ α (t1 ), (υT )δ α (t 2 )}
T
MT
MT
(υT )M
δ α (t1 ∗ t 2 ) ≤ max{(υT )δ α (t1 ), (υT )δ α (t 2 )},
T
(υI )M
δ β (t1 ∗ t 2 ) = δ. υI (t1 ∗ t 2 ) + β
≤ δ. max{υI (t1 ), υI (t 2 )} + β
= max{δ. υI (t1 ) + β, δ. υI (t 2 ) + β}
T
MT
MT
(υI )M
δ β (t1 ∗ t 2 ) = max{(υI )δ β (t1 ), (υI )δ β (t 2 )}
T
MT
MT
(υI )M
δ β (t1 ∗ t 2 ) ≤ max{(υI )δ β (t1 ), (υI )δ β (t 2 )},
T
(υF )M
δ γ (t1 ∗ t 2 ) = δ. υF (t1 ∗ t 2 ) − γ
≤ δ. max{υF (t1 ), υF (t 2 )} − γ
= max{δ. υF (t1 ) − γ, δ. υF (t 2 ) − γ}
T
MT
MT
(υF )M
δ γ (t1 ∗ t 2 ) = max{(υF )δ γ (t1 ), (υF )δ γ (t 2 )}
T
MT
MT
(υF )M
δ γ (t1 ∗ t 2 ) ≤ max{(υF )δ γ (t1 ), (υF )δ γ (t 2 )}.
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T
Hence NCMT BδMα,β,γ
is a NCSU of Y.
Theorem 3.3.2 Let B be a NCS of Y such that and for κT,I,F , α, β ∈ [[0,0], ℸ] and γ ∈ [[0,0], ¥], where for
T
υT,I,F , α, β ∈ [0, Γ] and γ ∈ [0, £] and δ ∈ [0,1] and a mapping BδMα,β,γ
: Y → [0,1] be a NCMT of B. If
T
BδMα,β,γ
is NCSU of Y. Then B is a NCSU of Y.
Proof. Let B be a neutrosophic cubic subset of Y, where α, β, γ ∈ [0, ¥], δ ∈ [0,1] and a mapping
T,I,F
M T,I,F
T
T
BδMα,β,γ
: Y → [0,1] be a NCMT of B. Suppose BδMα,β,γ
= {(κB )M
δ α,β,γ , (υB )δ α,β,γ } is a NCSU of Y, then
T
δ. κT (t1 ∗ t 2 ) + α = (κT )M
δ α (t1 ∗ t 2 )
T
MT
≥ rmin{(κT )M
δ α (t1 ), (κT )δ α (t 2 )}
= rmin{δ. κT (t1 ) + α, δ. κT (t 2 ) + α}
δ. κT (t1 ∗ t 2 ) + α = δ. rmin{κT (t 2 ), κT (t1 )} + α,
T
δ. κI (t1 ∗ t 2 ) + β = (κI )M
δ β (t1 ∗ t 2 )
T
MT
≥ rmin{(κI )M
δ β (t1 ), (κI )δ β (t 2 )}
= rmin{δ. κI (t1 ) + β, δ. κI (t 2 ) + β}
δ. κI (t1 ∗ t 2 ) + β = δ. rmin{κI (t 2 ), κI (t1 )} + β,
T
δ. κF (t1 ∗ t 2 ) − γ = (κF )M
δ γ (t1 ∗ t 2 )
T
MT
≥ rmin{(κF )M
δ γ (t1 ), (κF )δ γ (t 2 )}
= rmin{δ. κF (t1 ) − γ, δ. κF (t 2 ) − γ}
δ. κF (t1 ∗ t 2 ) − γ = δ. rmin{κF (t 2 ), κF (t1 )} − γ,
and
T
δ. υT (t1 ∗ t 2 ) + α = (υT )M
δ α (t1 ∗ t 2 )
T
MT
≤ max{(υT )M
δ α (t1 ), (υT )δ α (t 2 )}
= max{δ. υT (t1 ) + α, δ. υT (t 2 ) + α}
δ. υT (t1 ∗ t 2 ) + α = δ. max{υT (t 2 ), υT (t1 )} + α,
T
δ. υI (t1 ∗ t 2 ) + β = (υI )M
δ β (t1 ∗ t 2 )
T
MT
≤ max{(υI )M
δ β (t1 ), (υI )δ β (t 2 )}
= max{δ. υI (t1 ) + β, δ. υI (t 2 ) + β}
δ. υI (t1 ∗ t 2 ) + β = δ. max{υI (t 2 ), υI (t1 )} + β,
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331
T
δ. υF (t1 ∗ t 2 ) − γ = (υF )M
δ γ (t1 ∗ t 2 )
T
MT
≤ max{(υF )M
δ γ (t1 ), (υF )δ γ (t 2 )}
= max{δ. υF (t1 ) − γ, δ. υF (t 2 ) − γ}
δ. υF (t1 ∗ t 2 ) − γ = δ. max{υF (t 2 ), υF (t1 )} − γ,
which
imply
κT (t1 ∗ t 2 ) ≥ rmin{κT (t1 ), κT (t 2 )} , κI (t1 ∗ t 2 ) ≥ rmin{κI (t1 ), κI (t 2 )} , κF (t1 ∗ t 2 ) ≥
rmin{κF (t1 ), κF (t 2 )} and υT (t1 ∗ t 2 ) ≤ max{υT (t1 ), υT (t 2 )},υI (t1 ∗ t 2 ) ≤ max{υI (t1 ), υI (t 2 )},υF (t1 ∗ t 2 ) ≤
max{υF (t1 ), υF (t 2 )} for all t1 , t 2 ∈ Y. Hence B is a NCSU of Y.
T
Theorem 3.3.3 If B is a NCID of Y. Then NCMT BδMα,β,γ
of B is a NCID of Y for all κT,I,F , α, β ∈
[[0,0], ℸ] and γ ∈ [[0,0], ¥], where for υT,I,F , α, β ∈ [0, Γ] and γ ∈ [0, £] and δ ∈ (0,1].
Proof. Suppose B = (κT,I,F , υT,I,F ) is a NCID of Y. Then
T
(κT )M
δ α (0) = δ. κT (0) + α
≥ δ. κT (t1 ) + α
T
MT
(κT )M
δ α (0) = (κT )δ α (t1 ),
T
(κI )M
δ β (0) = δ. κI (0) + β
≥ δ. κI (t1 ) + β
T
MT
(κI )M
δ β (0) = (κI )δ β (t1 ),
T
(κF )M
δ γ (0) = δ. κF (0) − γ
≥ δ. κF (t1 ) − γ
T
MT
(κF )M
δ γ (0) = (κF )δ γ (t1 )
and
T
(υT )M
δ α (0) = δ. υT (0) + α
≤ δ. υT (t1 ) + α
T
MT
(υT )M
δ α (0) = (υT )δ α (t1 ),
T
(υI )M
δ β (0) = δ. υI (0) + β
≤ δ. υI (t1 ) + β
T
MT
(υI )M
δ β (0) = (υI )δ β (t1 ),
T
(υF )M
δ γ (0) = δ. υF (0) − γ
≤ δ. υF (t1 ) − γ
T
MT
(υF )M
δ γ (0) = (υF )δ γ (t1 )
Mohsin khalid,Florentin Smarandache,Neha Andaleeb khalid and Said Broumi, Translative And Multiplicative
Interpretation of Neutrosophic Cubic Set
Neutrosophic Sets and Systems, Vol. 35, 2020
Now
T
(κT )M
δ α (t1 ) = δ. κT (t1 ) + α
≥ δ. rmin{κT (t1 ∗ t 2 ), κT (t 2 )} + α
= rmin{δ. κT (t1 ∗ t 2 ) + α, δ. κT (t 2 ) + α}
T
MT
MT
(κT )M
δ α (t1 ) = rmin{(κT )δ α (t1 ∗ t 2 ), (κT )δ α (t 2 )}
T
MT
MT
⇒ (κT )M
δ α (t1 ) ≥ rmin{(κT )δ α (t1 ∗ t 2 ), (κT )δ α (t 2 )},
T
(κI )M
δ β (t1 ) = δ. κI (t1 ) + β
≥ δ. rmin{κI (t1 ∗ t 2 ), κI (t 2 )} + β
= rmin{δ. κI (t1 ∗ t 2 ) + β, δ. κI (t 2 ) + β}
T
MT
MT
(κI )M
δ β (t1 ) = rmin{(κI )δ β (t1 ∗ t 2 ), (κI )δ β (t 2 )}
T
MT
MT
⇒ (κI )M
δ β (t1 ) ≥ rmin{(κI )δ β (t1 ∗ t 2 ), (κI )δ β (t 2 )},
T
(κF )M
δ γ (t1 ) = δ. κF (t1 ) − γ
≥ δ. rmin{κF (t1 ∗ t 2 ), κF (t 2 )} − γ
= rmin{δ. κF (t1 ∗ t 2 ) − γ, δ. κF (t 2 ) − γ}
T
MT
MT
(κF )M
δ γ (t1 ) = rmin{(κF )δ γ (t1 ∗ t 2 ), (κF )δ γ (t 2 )}
T
MT
MT
⇒ (κF )M
δ γ (t1 ) ≥ rmin{(κF )δ γ (t1 ∗ t 2 ), (κF )δ γ (t 2 )}
and
T
(υT )M
δ α (t1 ) = δ. υT (t1 ) + α
≤ δ. max{υT (t1 ∗ t 2 ), υT (t 2 )} + α
= max{δ. υT (t1 ∗ t 2 ) + α, δ. υT (t 2 ) + α}
T
MT
MT
(υT )M
δ α (t1 ) = max{(υT )δ α (t1 ∗ t 2 ), (υT )δ α (t 2 )}
T
MT
MT
⇒ (υT )M
δ α (t1 ) ≤ max{(υT )δ α (t1 ∗ t 2 ), (υT )δ α (t 2 )},
T
(υI )M
δ β (t1 ) = δ. υI (t1 ) + β
≤ δ. max{υI (t1 ∗ t 2 ), υI (t 2 )} + β
= max{δ. υI (t1 ∗ t 2 ) + β, δ. υI (t 2 ) + β}
T
MT
MT
(υI )M
δ β (t1 ) = max{(υI )δ β (t1 ∗ t 2 ), (υI )δ β (t 2 )}
Mohsin khalid,Florentin Smarandache,Neha Andaleeb khalid and Said Broumi, Translative And Multiplicative
Interpretation of Neutrosophic Cubic Set
332
Neutrosophic Sets and Systems, Vol. 35, 2020
333
T
MT
MT
⇒ (υI )M
δ β (t1 ) ≤ max{(υI )δ β (t1 ∗ t 2 ), (υI )δ β (t 2 )},
T
(υF )M
δ γ (t1 ) = δ. υF (t1 ) − γ
≤ δ. max{υF (t1 ∗ t 2 ), υF (t 2 )} − γ
= max{δ. υF (t1 ∗ t 2 ) − γ, δ. υF (t 2 ) − γ}
T
MT
MT
(υF )M
δ γ (t1 ) = max{(υF )δ γ (t1 ∗ t 2 ), (υF )δ γ (t 2 )}
T
MT
MT
⇒ (υF )M
δ γ (t1 ) ≤ max{(υF )δ γ (t1 ∗ t 2 ), (υF )δ γ (t 2 )},
for all t1 , t 2 ∈ Y and all for κT,I,F , α, β ∈ [[0,0], ℸ] and γ ∈ [[0,0], ¥], where for υT,I,F , α, β ∈ [0, Γ] and γ ∈
T
[0, £] and δ ∈ (0,1]. Hence BδMα,β,γ
of B is a NCID of Y.
T
Theorem 3.3.3 If B is a neutrosophic cubic set of Y such that NCMT BδMα,β,γ
of B is a NCID of Y for all for
κT,I,F , α, β ∈ [[0,0], ℸ] and γ ∈ [[0,0], ¥], where for υT,I,F , α, β ∈ [0, Γ] and γ ∈ [0, £] and δ ∈ (0,1], then
B is a NCID of Y.
T
Proof. Suppose NCMT BδMα,β,γ
is a NCID of Y for some κT,I,F , α, β ∈ [[0,0], ℸ] and γ ∈ [[0,0], ¥], where
for υT,I,F , α, β ∈ [0, Γ] and γ ∈ [0, £] and δ ∈ (0,1] and t1 , t 2 ∈ Y. Then
T
δ. κT (0) + α = (κT )M
δ α (0)
T
≥ (κT )M
δ α (t1 )
δ. κT (0) + α = δ. κT (t1 ) + α,
T
δ. κI (0) + β = (κI )M
δ β (0)
T
≥ (κI )M
δ β (t1 )
δ. κI (0) + β = δ. κI (t1 ) + β,
T
δ. κF (0) − γ = (κF )M
δ γ (0)
T
≥ (κF )M
δ γ (t1 )
δ. κF (0) − γ = δ. κF (t1 ) − γ,
and
T
δ. υT (0) + α = (υT )M
δ α (0)
T
≤ (υT )M
δ α (t1 )
δ. υT (0) + α = δ. υT (t1 ) + α,
T
δ. υI (0) + β = (υI )M
δ β (0)
T
≤ (υI )M
δ β (t1 )
δ. υI (0) + β = δ. υI (t1 ) + β,
Mohsin khalid,Florentin Smarandache,Neha Andaleeb khalid and Said Broumi, Translative And Multiplicative
Interpretation of Neutrosophic Cubic Set
Neutrosophic Sets and Systems, Vol. 35, 2020
334
T
δ. υF (0) − γ = (υF )M
δ γ (0)
T
≤ (υF )M
δ γ (t1 )
δ. υF (0) − γ = δ. υF (t1 ) − γ,
which imply κT (0) ≥ κT (t1 ),κI (0) ≥ κI (t1 ),κF (0) ≥ κF (t1 ) and υT (0) ≤ υT (t1 ), υI (0) ≤ υI (t1 ), υF (0) ≤
υF (t1 ). Now, we have
T
δ. κT (t1 ) + α = (κT )M
δ α (t1 )
T
MT
≥ rmin{(κT )M
δ α (t1 ∗ t 2 ), (κT )δ α (t 2 )}
= rmin{δ. κT (t1 ∗ t 2 ) + α, δ. κT (t 2 ) + α}
δ. κT (t1 ) + α = δ. rmin{κT (t1 ∗ t 2 ), κT (t 2 )} + α,
T
δ. κI (t1 ) + β = (κI )M
δ β (t1 )
T
MT
≥ rmin{(κI )M
δ β (t1 ∗ t 2 ), (κI )δ β (t 2 )}
= rmin{δ. κI (t1 ∗ t 2 ) + β, δ. κI (t 2 ) + β}
δ. κI (t1 ) + β = δ. rmin{κI (t1 ∗ t 2 ), κI (t 2 )} + β,
T
δ. κF (t1 ) − γ = (κF )M
δ γ (t1 )
T
MT
≥ rmin{(κF )M
δ γ (t1 ∗ t 2 ), (κF )δ γ (t 2 )}
= rmin{δ. κF (t1 ∗ t 2 ) − γ, δ. κF (t 2 ) − γ}
δ. κF (t1 ) − γ = δ. rmin{κF (t1 ∗ t 2 ), κF (t 2 )} − γ
and
T
δ. υT (t1 ) + α = (υT )M
δ α (t1 )
T
MT
≤ max{(υT )M
δ α (t1 ∗ t 2 ), (υT )δ α (t 2 )}
= max{δ. υT (t1 ∗ t 2 ) + α, δ. υT (t 2 ) + α}
δ. υT (t1 ) + α = δ. max{υT (t1 ∗ t 2 ), υT (t 2 )} + α,
T
δ. υI (t1 ) + β = (υI )M
δ β (t1 )
T
MT
≤ max{(υI )M
δ β (t1 ∗ t 2 ), (υI )δ β (t 2 )}
= max{δ. υI (t1 ∗ t 2 ) + β, δ. υI (t 2 ) + β}
δ. υI (t1 ) + β = δ. max{υI (t1 ∗ t 2 ), υI (t 2 )} + β,
T
δ. υF (t1 ) − γ = (υF )M
δ γ (t1 )
T
MT
≤ max{(υF )M
δ γ (t1 ∗ t 2 ), (υF )δ γ (t 2 )}
= max{δ. υF (t1 ∗ t 2 ) − γ, δ. υF (t 2 ) − γ}
Mohsin khalid,Florentin Smarandache,Neha Andaleeb khalid and Said Broumi, Translative And Multiplicative
Interpretation of Neutrosophic Cubic Set
Neutrosophic Sets and Systems, Vol. 35, 2020
335
δ. υF (t1 ) − γ = δ. max{υF (t1 ∗ t 2 ), υF (t 2 )} − γ
which imply κT (t1 ) ≥ rmin{κT (t1 ∗ t 2 ), κT (t 2 )} , κI (t1 ) ≥ rmin{κI (t1 ∗ t 2 ), κI (t 2 )} , κF (t1 ) ≥ rmin{κF (t1 ∗
t 2 ), κF (t 2 )} and υT (t1 ) ≤ max{υT (t1 ∗ t 2 ), υT (t 2 )},υI (t1 ) ≤ max{υI (t1 ∗ t 2 ), υI (t 2 )},υF (t1 ) ≤ max{υF (t1 ∗
t 2 ), υF (t 2 )} for all t1 , t 2 ∈ Y. Hence B is a NCID of Y.
Theorem 3.3.4 Intersection of any two NCMT of a NCID B of Y is a NCID of Y.
T
MT
Proof. Suppose BδMα,β,γ
and BδM′ αT′ ,β′ ,γ′ are two NCMTs of NCID B ofY, where for Bα,β,γ
, for κT,I,F , α, β ∈
[[0,0], ℸ], γ ∈ [[0,0], ¥], for υT,I,F , α, β ∈ [0, Γ], γ ∈ [0, £] and for BαT′ ,β′ ,γ′ , for κT,I,F α′ , β′ ∈ [[0,0], ℸ], γ′ ∈
[[0,0], ¥], for υT,I,F , α′ , β′ ∈ [0, Γ], γ′ ∈ [0, £]. Assume α ≤ α′, β ≤ β′, γ ≤ γ′ and δ = δ′. Then by Theorem
T
MT
3.3.3, BδMα,β,γ
and Bδ′
α′,β′,γ′ are NCIDs of Y. So
T
MT
MT
MT
((κT )M
δ α ∩ (κT )δ′ α′ )(t1 ) = rmin{(κT )δ α (t1 ), (κT )δ′ α′ (t1 )}
= rmin{δ. κT (t1 ) + α, δ′. κT (t1 ) + α′}
= δ. κT (t1 ) + α
T
MT
MT
((κT )M
δ α ∩ (κT )δ′ α′ )(t1 ) = (κT )δ α (t1 ),
T
MT
MT
MT
((κI )M
δ β ∩ (κI )δ′ β′ )(t1 ) = rmin{(κI )δ β (t1 ), (κI )δ′ β′ (t1 )}
= rmin{δ. κI (t1 ) + β, δ′. κI (t1 ) + β′}
= δ. κI (t1 ) + β
T
MT
MT
((κI )M
δ β ∩ (κI )δ′ β′ )(t1 ) = (κI )δ β (t1 ),
T
MT
MT
MT
((κF )M
δ γ ∩ (κF )δ′ γ′ )(t1 ) = rmin{(κF )δ γ (t1 ), (κF )δ′ γ′ (t1 )}
= rmin{δ. κF (t1 ) − γ, δ′. κF (t1 ) − γ′}
= δ′. κF (t1 ) − γ′
T
MT
MT
((κF )M
δ γ ∩ (κF )δ′ γ′ )(t1 ) = (κF ),δ′ γ′ (t1 )
and
T
MT
MT
MT
((υT )M
δ α ∩ (υT )δ′ α′ )(t1 ) = max{(υT )δ α (t1 ), (υT )δ′ α′ (t1 )}
= max{δ. υT (t1 ) + α, δ′. υT (t1 ) + α′}
= δ′. υT (t1 ) + α′
MT
T
MT
((υT )M
δ α ∩ (υT )δ′ α′ )(t1 ) = (υT )δ′ α′ (t1 ),
T
MT
MT
MT
((υI )M
δ β ∩ (υI )δ′ β′ )(t1 ) = max{(υI )δ β (t1 ), (υI )δ′ β′ (t1 )}
= max{δ. υI (t1 ) + β, δ′. υI (t1 ) + β′}
= δ′. υI (t1 ) + β′
MT
MT
T
((υI )M
δ β ∩ (υI )δ′ β′ )(t1 ) = (υI )δ′ β′ (t1 ),
Mohsin khalid,Florentin Smarandache,Neha Andaleeb khalid and Said Broumi, Translative And Multiplicative
Interpretation of Neutrosophic Cubic Set
Neutrosophic Sets and Systems, Vol. 35, 2020
336
MT
MT
T
MT
((υF )M
δ γ ∩ (υF )δ′ γ′ )(t1 ) = max{(υF )δ γ (t1 ), (υF )δ′ γ′ (t1 )}
= max{δ. υF (t1 ) − γ, δ′. υF (t1 ) − γ′}
= δ. υF (t1 ) − γ
T
MT
MT
((υF )M
δ γ ∩ (υF )δ′ γ′ )(t1 ) = (υF )δ γ (t1 ).
T
Hence BδMα,β,γ
∩ BδM′ αT′ ,β′ ,γ′ is NCID of Y.
T
Theorem 3.3.5 Union of any two NCMT BδMα,β,γ
of a NCID B of Y is a NCID of Y.
T
MT
Proof. Suppose BδMα,β,γ
and BδM′ αT′ ,β′ ,γ′ are two NCMTs of NCID B of Y , where for Bα,β,γ
, for κT,I,F , α, β ∈
[[0,0], ℸ], γ ∈ [[0,0], ¥], for υT,I,F , α, β ∈ [0, Γ], γ ∈ [0, £] and for BαT′ ,β′ ,γ′ , for κT,I,F α′ , β′ ∈ [[0,0], ℸ], γ′ ∈
[[0,0], ¥], for υT,I,F , α′ , β′ ∈ [0, Γ], γ′ ∈ [0, £]. Assume α ≥ α′, β ≥ β′, γ ≥ γ′ and δ = δ′. Then by Theorem
T
3.3.3, BδMα,β,γ
and BδM′ Tα′ ,β′ ,γ′ are NCIDs of Y. So
MT
T
MT
MT
((κT )M
δ α ∪ (κT )δ′ α′ )(t1 ) = rmax{(κT )δ α (t1 ), (κT )δ′ α′ (t1 )}
= rmax{δ. κT (t1 ) + α, δ′. κT (t1 ) + α′}
= δ. κT (t1 ) + α
MT
T
MT
((κT )M
δ α ∪ (κT )δ′ α′ )(t1 ) = (κT )δ α (t1 ),
MT
MT
T
MT
((κI )M
δ β ∪ (κI )δ′ β′ )(t1 ) = rmax{(κI )δ β (t1 ), (κI )δ′ β′ (t1 )}
= rmax{δ. κI (t1 ) + β, δ′. κI (t1 ) + β′}
= δ. κI (t1 ) + β
T
MT
MT
((κI )M
δ β ∪ (κI )δ′ β′ )(t1 ) = (κI )δ β (t1 ),
MT
T
MT
MT
((κF )M
δ γ ∪ (κF )δ′ γ′ )(t1 ) = rmax{(κF )δ γ (t1 ), (κF )δ′ γ′ (t1 )}
= rmax{δ. κF (t1 ) − γ, δ′. κF (t1 ) − γ′}
= δ′. κF (t1 ) − γ′}
MT
MT
T
((κF )M
δ γ ∪ (κF )δ′ γ′ )(t1 ) = (κF )δ′ γ′ (t1 )
and
T
MT
MT
MT
((υT )M
δ α ∪ (υT )δ′ α′ )(t1 ) = min{(υT )δ α (t1 ), (υT )δ′ α′ (t1 )}
= min{δ. υT (t1 ) + α, δ′. υT (t1 ) + α′}
= δ′ . υT (t1 ) + α′
MT
T
MT
((υT )M
δ α ∪ (υT )δ′ α′ )(t1 ) = (υT )δ′ α′ (t1 ),
T
MT
MT
MT
((υI )M
δ β ∪ (υI )δ′ β′ )(t1 ) = min{(υI )δ β (t1 ), (υI )δ′ β′ (t1 )}
= min{δ. υI (t1 ) + β, δ′. υI (t1 ) + β′}
= δ′. υI (t1 ) + β′
Mohsin khalid,Florentin Smarandache,Neha Andaleeb khalid and Said Broumi, Translative And Multiplicative
Interpretation of Neutrosophic Cubic Set
Neutrosophic Sets and Systems, Vol. 35, 2020
337
MT
T
MT
((υI )M
δ β ∪ (υI )δ′ β′ )(t1 ) = (υI )δ′ β′ (t1 ),
T
MT
MT
MT
((υF )M
δ γ ∪ (υF )δ′ γ′ )(t1 ) = min{(υF )δ γ (t1 ), (υF )δ′ γ′ (t1 )}
= min{δ. υF (t1 ) − γ, δ′. υF (t1 ) − γ′}
= δ. υF (t1 ) − γ
T
MT
MT
((υF )M
δ γ ∪ (υF )δ′ γ′ )(t1 ) = (υF )δ γ (t1 ).
T
MT
Hence BδMα,β,γ
∪ Bδ′
α′,β′,γ′ is NCID of Y.
4. Conclusion
In this paper, we defined neutrosophic cubic translation,, neutrosophic cubic multiplication and neutrosophic
cubic magnified translation for neutrosophic cubic set on BF-algebra. We provided the new sort of different
conditions for neutrosophic cubic translation, neutrosophic cubic multiplication and neutrosophic cubic
magnified translation and proved with examples. Moreover, for better understanding we investigated many
results for NCT, NCM and NCMT using the subalgebra and ideals. For future work, translation and
multiplication can be applied on neutrosophic cubic soft set and T-neutrosophic cubic set.
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(2015),8-41.
25. Y. B. Jun, F. Smarandache, M. A. Ozturk, Commutative falling neutrosophic ideals in BCK-algebras,
Neutrosophic Sets and Systems, vol. 20, pp. 44-53, 2018, http://doi.org/10.5281 /zenodo .1235351.
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and Systems, 28, (2019), 259-272. 10.5281/zenodo.3382552
Mohsin khalid,Florentin Smarandache,Neha Andaleeb khalid and Said Broumi, Translative And Multiplicative
Interpretation of Neutrosophic Cubic Set
Neutrosophic Sets and Systems, Vol. 35, 2020
339
28. M. Khalid, R. Iqbal, S. Zafar, H. Khalid, Intuitionistic fuzzy translation and multiplication of Galgebra,The Journal of Fuzzy Mathematics, 27, (3), (2019), 17.
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Received: Apr 13, 2020. Accepted: July 4 2020
Mohsin khalid,Florentin Smarandache,Neha Andaleeb khalid and Said Broumi, Translative And Multiplicative
Interpretation of Neutrosophic Cubic Set
Neutrosophic Sets and Systems, Vol. 35, 2020
University of New Mexico
Neutrosophic Generalized Homeomorphism
Md. Hanif PAGE1 and Qays Hatem Imran2*
1Department
of Mathematics, KLE Technological University, Hubballi-580031, Karnataka, India.
E-mail: mb_page@kletech.ac.in (hanif01@yahoo.com)
2Department of Mathematics, College of Education for Pure Science, Al-Muthanna University, Samawah, Iraq.
E-mail: qays.imran@mu.edu.iq
* Correspondence: qays.imran@mu.edu.iq
Abstract: The idea of neutrosophic generalized homeomorphism is presented in neutrosophic
topological spaces. In addition to this, neutrosophic generalized closed and open mappings are also
presented. At the same time, their characterizations are discussed by establishing their related
attributes.
Keywords: GN-closed set, GN-closed map, GN-open map, Neutrosophic g-homeomorphism,
Neutrosophic g*-homeomorphism.
1. Introduction
Neutrosophic sets were initially established as a generality of intuitionistic fuzzy sets [10] by
Smarandache [18] such that the membership, the non-membership, and the indeterminacy degrees
are considered. In analogy with more unsure philosophy, the neutrosophic set discharge agreement
with an indeterminacy condition. The neutrosophic conception has a broad scope of real-time
requests in the fields of [1-9] Artificial Intelligence, Computer Science, Information Systems,
Decision Making, Uncertainty assessments of linear time-cost tradeoffs, Applied Mathematics, and
solving the supply chain problem. Salama et al. [15, 16] adapted the notion of the neutrosophic set
to operate in neutrosophic topological spaces (NTSs in short) and pioneered generalized
neutrosophic set and topological spaces. In [11], generalized neutrosophic closed set (in short,
GNCS) is defined and using this generalized neutrosophic continuous (GN-continuous), and
generalized neutrosophic irresolute (in short, GN-irresolute) functions are defined. Recently in [12,
13],
the
perception
of
generalized
α-contra
continuous
and
neutrosophic
almost
α-contra-continuous functions are introduced. Parimala M et al. [14] introduced and studied the
thought of Neutrosophic homeomorphism and Neutrosophic αψ homeomorphism in Neutrosophic
topological spaces. This paper aspires to overly enunciate the thought of neutrosophic generalized
homeomorphism (in short, neutrosophic g-homeomorphism) in NTSs by utilizing GN-continuous
function and study some of their properties. We have also provided the idea of generalized
neutrosophic closed and open mappings by establishing some of their characterizations. Besides,
neutrosophic g*-homeomorphism is also presented and establish its relation with the neutrosophic
g-homeomorphism.
2. Preliminaries
Definition 2.1 [15]: A neutrosophic topology (in short,N-topology) on 𝑋 ≠ ∅ is a family 𝜉 of N-sets
in 𝑋 satisfying the laws given below:
(i) 0𝑁 , 1𝑁 ∈ 𝜉,
Md. Hanif PAGE and Qays Hatem Imran, Neutrosophic Generalized Homeomorphism
Neutrosophic Sets and Systems, Vol. 35, 2020
341
(ii) 𝑊1 ⋂𝑊2 ∈ 𝜉 being 𝑊1 , 𝑊2 ∈ 𝜉,
(iii) ⋃𝑊𝑖 ∈ 𝜉 for arbitrary family {𝑊𝑖 |𝑖 ∈ 𝛬} ⊆ 𝜉.
In this situation the ordered pair (𝑋, 𝜉) or simply 𝑋 is termed as NTS and each NS in 𝜉 is named as
neutrosophic open set (in short, NOS). The complement Λ of an N-open set Λ in 𝑋 is known as
neutrosophic closed set (briefly, NCS) in 𝑋.
Definition 2.2 [15]: Let Λ be an NS in an NTS (𝑋, 𝜉). Thereupon
(i)
𝑁𝑖𝑛𝑡(Λ) = ⋃{𝐺|𝐺 is a NOS in 𝑋 and 𝐺 ⊆ Λ} is termed as neutrosophic interior (in brief 𝑁𝑖𝑛𝑡)
of Λ;
(ii) 𝑁𝑐𝑙(Λ) = ⋂{𝐺|𝐺 is an NCS in 𝑋 and 𝐺 ⊇ Λ} is termed as neutrosophic closure (shortly 𝑁𝑐𝑙) of
Λ.
Definition 2.3 [11]: Allow (𝑋, 𝜉) be a NTS. A NS Λ in (𝑋, 𝜉) is termed as generalized neutrosophic
closed set (in short GNCS) if 𝑁𝑐𝑙(Λ) ⊆ Γ whenever Λ ⊆ Γ and Γ is a NOS. The complement of a
GNCS is generalized neutrosophic open set (in short GNOS).
Definition 2.4 [11]: Let (𝑋, 𝜉) be NTS and 𝐵 be a NS in 𝑋. Then neutrosophic generalized closure is
defined as, 𝐺𝑁𝑐𝑙(𝐵) = ⋂{𝐺: 𝐺 is a GNCS in 𝑋 and 𝐵 ⊆ 𝐺}.
Definition 2.5 [11, 17]: A map 𝜂: 𝑋 → 𝑌 is said to be
(i)
neutrosophic closed (in short, NC-map) if the image of every NCS in X is a NCS in Y.
(ii) neutrosophic continuous (in short, N-continuous) if inverse image of every NCS in 𝑌 is a NCS
i𝑛 𝑋.
(iii) generalized neutrosophic continuous (in short, GN-continuous) if inverse image of every NCS
in 𝑌 is a GNCS in 𝑋.
(iv) generalized neutrosophic irresolute (in short, GN-irresolute) if inverse image of every GNCS in
𝑌 is a GNCS in 𝑋.
Definition 2.6 [14]: A bijection g: 𝑋 → 𝑌 is called a neutrosophic homeomorphism if g and g −1 are
neutrosophic continuous.
3. Neutrosophic Generalized Homeomorphism
Definition 3.1: A bijection 𝜂: 𝑋 → 𝑌 is named as neutrosophic generalized homeomorphism (in
short neutrosophic g-homeomorphism) if 𝜂 and 𝜂 −1 are GN-continuous.
Proposition 3.2: Every neutrosophic homeomorphism is a neutrosophic g-homeomorphism.
Proof: Consider a bijection mapping 𝜂: 𝑋 → 𝑌 be a neutrosophic homeomorphism, in which 𝜂 as
well as 𝜂 −1 are N-continuous. We have each N-continuous mapping is GN-continuous, so 𝜂 and
𝜂 −1 are GN-continuous. Hence, 𝜂 is neutrosophic g-homeomorphism.
Md. Hanif PAGE and Qays Hatem Imran, Neutrosophic Generalized Homeomorphism
Neutrosophic Sets and Systems, Vol. 35, 2020
342
Remark 3.3: The next illustration makes clear that the opposite of the above proposition is not valid.
Example 3.4: Let 𝑋 = {𝑝, 𝑞, 𝑟}, 𝜉 = {0𝑁 , 𝒜1 , 𝒜2 , 𝒜3 , 𝒜4 , 1𝑁 } be a N-topology on 𝑋.
𝒜1 = 〈𝑥, (0.2,0.1,0.1), (0.2,0.1,0.1), (0.3,0.5,0.5)〉, 𝒜2 = 〈𝑥, (0.1,0.2,0.2), (0.4,0.3,0.3), (0.3,0.3,0.3)〉,
𝒜3 = 〈𝑥, (0.2,0.2,0.2), (0.2,0.1,0.1), (0.3,0.3,0.3)〉, 𝒜4 = 〈𝑥, (0.1,0.1,0.1), (0.4,0.3,0.3), (0.3,0.5,0.5)〉,
and let 𝑌 = {𝑝, 𝑞, 𝑟}, 𝜎 = {0𝑁 , ℬ1 , ℬ2 , ℬ3 , ℬ4 , 1𝑁 } be a neutrosophic topology on 𝑌.
ℬ1 = 〈𝑦, (0.3,0.3,0.3), (0.2,0.1,0.1), (0.2,0.2,0.2)〉, ℬ2 = 〈𝑦, (0.2,0.2,0.2), (0.1,0.1,0.1), (0.3,0.3,0.3)〉,
ℬ3 = 〈𝑦, (0.3,0.3,0.3), (0.1,0.1,0.1), (0.2,0.1,0.1)〉, ℬ4 = 〈𝑦, (0.2,0.2,0.2), (0.2,0.1,0.1), (0.3,0.3,0.3)〉.
Define
𝜂: (𝑋, 𝜉) → (𝑌, 𝜎)
by
𝜂(𝑝) = 𝑝, 𝜂(𝑞) = 𝑞
and
𝜂(𝑟) = 𝑟 .
Then
𝜂
is
neutrosophic
g-homeomorphism but not neutrosophic homeomorphism.
Definition 3.5: A mapping 𝜂: 𝑋 → 𝑌 is generalized neutrosophic closed (in short, GNC-map) if the
image 𝜂(𝑄) is GNCS in 𝑌 for every NCS 𝑄 in 𝑋.
Definition 3.6: A mapping 𝜂: 𝑋 → 𝑌 is generalized neutrosophic open (in short, GNO-map) if the
image 𝜂(𝑅) is GNOS in 𝑌 for every NOS 𝑅 in 𝑋.
Proposition 3.7: Every NC-mapping is a GNC-mapping.
Proof: Consider 𝜂: 𝑋 → 𝑌 is a NC-mapping, so as 𝑄 is an NCS in 𝑋. As 𝜂 is NC- mapping, 𝜂(𝑄) is
NCS in 𝑌. Since each NCS is GNCS. Therefore, 𝜂(𝑄) is a GNCS in 𝑌. Hence, 𝜂 is GNC-mapping.
Remark 3.8: The opposite of the above proposition is not valid as indicated.
Example 3.9: Let 𝑋 = {𝑝, 𝑞, 𝑟}, 𝜉 = {0𝑁 , 𝒜1 , 𝒜2 , 𝒜3 , 𝒜4 , 1𝑁 } be a N-topology on 𝑋.
𝒜1 = 〈𝑥, (0.2,0.1,0.1), (0.2,0.1,0.1), (0.3,0.5,0.5)〉, 𝒜2 = 〈𝑥, (0.1,0.2,0.2), (0.4,0.3,0.3), (0.3,0.3,0.3)〉,
𝒜3 = 〈𝑥, (0.2,0.2,0.2), (0.2,0.1,0.1), (0.3,0.3,0.3)〉, 𝒜4 = 〈𝑥, (0.1,0.1,0.1), (0.4,0.3,0.3), (0.3,0.5,0.5)〉,
and let 𝑌 = {𝑝, 𝑞, 𝑟}, 𝜎 = {0𝑁 , ℬ1 , ℬ2 , ℬ3 , ℬ4 , 1𝑁 } be a neutrosophic topology on 𝑌.
ℬ1 = 〈𝑦, (0.3,0.3,0.3), (0.2,0.1,0.1), (0.2,0.2,0.2)〉, ℬ2 = 〈𝑦, (0.2,0.2,0.2), (0.1,0.1,0.1), (0.3,0.3,0.3)〉,
ℬ3 = 〈𝑦, (0.3,0.3,0.3), (0.1,0.1,0.1), (0.2,0.1,0.1)〉, ℬ4 = 〈𝑦, (0.2,0.2,0.2), (0.2,0.1,0.1), (0.3,0.3,0.3)〉.
Define 𝜂: (𝑋, 𝜉) → (𝑌, 𝜎) by 𝜂(𝑝) = 𝑝, 𝜂(𝑞) = 𝑞 and 𝜂(𝑟) = 𝑟 . Then 𝜂 is GNC-mapping but not
NC-mapping.
Proposition 3.10: A map 𝜂: 𝑋 → 𝑌 is a GNC-mapping if the image of each NOS in 𝑋 is GNOS in 𝑌.
Proof: Let 𝑅 be a NOS in 𝑋. Hence 𝑅 is a NCS in 𝑋. As 𝜂 is GNC-mapping, 𝜂(𝑅) is a GNCS in 𝑌.
Since 𝜂(𝑅) = (𝜂(𝑅)), 𝜂(𝑅) is a GNOS in 𝑌.
Proposition 3.11: Let 𝜂: 𝑋 → 𝑌 be a bijective mapping, then the next assertions are same:
(i)
𝜂 is GNO-mapping.
(ii) 𝜂 is GNC-mapping.
(iii) 𝜂−1 is GN-continuous.
Md. Hanif PAGE and Qays Hatem Imran, Neutrosophic Generalized Homeomorphism
Neutrosophic Sets and Systems, Vol. 35, 2020
343
Proof: (𝑖) → (𝑖𝑖). Suppose that 𝜂 is GNO-mapping. Then, 𝑃 is a NOS in 𝑋, then image 𝜂(𝑃) is
GNOS in 𝑌. Here, 𝑃 is NCS in 𝑋, then 𝑋 − 𝑃 is a NOS in 𝑋. By prediction, 𝜂(𝑋 − 𝑃) is a GNOS in
𝑌. Hence, 𝑌 − 𝜂(𝑋 − 𝑃) is a GNCS in 𝑌. Hence, 𝜂 is a GNC-mapping.
(𝑖𝑖) → (𝑖𝑖𝑖). Let 𝑅 be an NCS in 𝑋. By (ii), 𝜂(𝑅) is GNCS in 𝑌. Therefore, 𝜂(𝑅) = (𝜂−1 )−1 (𝑅), so
𝜂 −1 is a GNCS in 𝑌. Hence, 𝜂−1 is a GN-continuous.
(𝑖𝑖𝑖) → (𝑖). Let 𝑄 be a NOS in 𝑋. By (iii), (𝜂 −1 )−1 (𝑄) = 𝜂(𝑄) is GNO-mapping.
Proposition 3.12: Let 𝜂: (𝑋, 𝜉) → (𝑌, 𝜎) be a bijective mapping. If 𝜂 is GN-continuous, thereupon
the declarations are identical:
(i)
𝜂 is GNC-mapping.
(ii) 𝜂 is GNO-mapping.
(iii) 𝜂−1 is neutrosophic g-homeomorphism.
Proof:
(𝑖) → (𝑖𝑖) . Presume that 𝜂 is bijective as well as a GNC-mapping. So, 𝜂 −1 is a
GN-continuous mapping. As we have every NOS is GNOS in 𝑌. Hence, 𝜂 is GNO-mapping.
(𝑖𝑖) → (𝑖𝑖𝑖). Consider a bijective NO-mapping 𝜂. Furthermore, 𝜂 −1 is a GN-continuous mapping.
Accordingly, 𝜂 and 𝜂 −1 are GN-continuous. Hence, 𝜂 is neutrosophic g-homeomorphism.
(𝑖𝑖𝑖) → (𝑖). Let 𝜂 be neutrosophic g-homeomorphism, then 𝜂 and 𝜂 −1 are GN-continuous. As each
NCS in 𝑋 is a GNCS in 𝑌, therefore 𝜂 is a GNC-mapping.
Definition 3.13 [19]: Let (𝑋, 𝜉) be an NTS said to be a as neutrosophic-T1/2 (in short N-T1/2 ) space if
every GNCS is NCS in 𝑋.
Proposition 3.14: Let 𝜂: (𝑋, 𝜉) → (𝑌, 𝜎) be neutrosophic g-homeomorphism, then 𝜂 is neutrosophic
homoemorphism if 𝑋 and 𝑌 are N-T1/2 space.
Proof: Consider that 𝐷 is an NCS in 𝑌, then 𝜂 −1 (𝐷) is a GNCS in 𝑋 due to the assumption. Since
𝑋 is N - T1/2 space, 𝜂 −1 (𝐷) is NCS in 𝑋 . Then, 𝜂 is GN-continuous. By hypothesis 𝜂 −1 is
GN-continuous. Let 𝐻 be a NCS in 𝑋. (𝜂 −1 )−1 (𝐻) = 𝜂(𝐻) is a NCS in 𝑌, by preassumption. As 𝑌 is
N-T1/2 space, 𝜂(𝐻) is a NCS in 𝑌. Hence, 𝜂 −1 is N-continuous. Therefore, 𝜂 is a neutrosophic
homeomorphism.
Proposition 3.15: Let 𝜂: 𝑋 → 𝑌 and 𝜇: 𝑌 → 𝑍 be GNC-mappings where 𝑋 and 𝑍 are NTSs and 𝑌 is
N-T1/2 space, then (𝜇 𝑜 𝜂) is GNC-mapping.
Proof: Let 𝑅 be a NCS in 𝑋. As 𝜂 is GNC-map and 𝜂(𝑅) is a GNCS in 𝑌, by assumption, 𝜂(𝑅) is a
NCS in 𝑌. Since 𝜇 is GNC-map, then 𝜇(𝜂(𝑅)) is a GNCS in 𝑋 and 𝑍 and 𝜇(𝜂(𝑅)) = (𝜇 𝑜 𝜂)(𝑅).
Therefore, (𝜇 𝑜 𝜂) is GNC-map.
Proposition 3.16: Let 𝜇: 𝑋 → 𝑌 and 𝜆: 𝑌 → 𝑍 be NTSs, then the following hold:
(i)
If (𝜆 𝑜 𝜇) is GNO-map and 𝜇 is N-continuous, then 𝜆 is GNO-map.
(ii) If (𝜆 𝑜 𝜇) is GNO-map and 𝜇 is GN-continuous, then 𝜆 is GNO-map.
Md. Hanif PAGE and Qays Hatem Imran, Neutrosophic Generalized Homeomorphism
Neutrosophic Sets and Systems, Vol. 35, 2020
344
Proof: (i) Let 𝐾 be NOS in 𝑌 . Then, 𝜇 −1 (𝐾) is a NOS in 𝑋 . Since (𝜆 𝑜 𝜇) GNO-map and
(𝜆 𝑜 𝜇)𝜇 −1 (𝐾) = 𝜆 (𝜇(𝜇 −1 (𝐾))) = 𝜆(𝐾) is GN-open in 𝑍, hence 𝜆 is GN-open map.
(ii) Let 𝐾 be NOS in 𝑋. Then, 𝜆 ( 𝜇(𝐾) ) is a NOS in 𝑍. Hence, 𝜆−1 (𝜆 (𝜇(𝐾)) = 𝜇(𝐾) is GNOS in 𝑌.
Therefore 𝜇 is GNO- map.
4. Neutrosophic g*-Homeomorphism
Definition 4.1: A bijection 𝜇: 𝑋 → 𝑌 is called neutrosophic g*-homeomorphism if 𝜇 and 𝜇 −1 are
GN-irresolute mappings.
Proposition 4.2: Every neutrosophic g*-homeomorphism is a neutrosophic g-homeomorphism.
Proof: A map 𝜇 is a neutrosophic g*-homeomorphism. Predict that 𝐾 is a NCS in 𝑌. So it is a
GNCS in 𝑌. By pressumption, 𝜇−1 (𝐾) is a GNCS in 𝑋. Accordingly, 𝜇 is GN-continuous mapping.
Therefore,
𝜇
and
𝜇 −1
are
GN-continuous
mappings.
Henec,
𝜇
is
a
neutrosophic
g-homeomorphism.
Remark 4.3: The example is given to show that the reverese of the above proposition is not possible.
Example 4.4: Let 𝑋 = {𝑝, 𝑞, 𝑟}, 𝜉 = {0𝑁 , 𝒜1 , 𝒜2 , 𝒜3 , 𝒜4 , 1𝑁 } be a N-topology on 𝑋.
𝒜1 = 〈𝑥, (0.2,0.1,0.1), (0.2,0.1,0.1), (0.3,0.5,0.5)〉, 𝒜2 = 〈𝑥, (0.1,0.2,0.2), (0.4,0.3,0.3), (0.3,0.3,0.3)〉,
𝒜3 = 〈𝑥, (0.2,0.2,0.2), (0.2,0.1,0.1), (0.3,0.3,0.3)〉, 𝒜4 = 〈𝑥, (0.1,0.1,0.1), (0.4,0.3,0.3), (0.3,0.5,0.5)〉,
and let 𝑌 = {𝑝, 𝑞, 𝑟}, 𝜎 = {0𝑁 , ℬ1 , ℬ2 , ℬ3 , ℬ4 , 1𝑁 } be a neutrosophic topology on 𝑌.
ℬ1 = 〈𝑦, (0.3,0.3,0.3), (0.2,0.1,0.1), (0.2,0.2,0.2)〉, ℬ2 = 〈𝑦, (0.2,0.2,0.2), (0.1,0.1,0.1), (0.3,0.3,0.3)〉,
ℬ3 = 〈𝑦, (0.3,0.3,0.3), (0.1,0.1,0.1), (0.2,0.1,0.1)〉, ℬ4 = 〈𝑦, (0.2,0.2,0.2), (0.2,0.1,0.1), (0.3,0.3,0.3)〉.
Define
𝜂: (𝑋, 𝜉) → (𝑌, 𝜎)
by
𝜂(𝑝) = 𝑝, 𝜂(𝑞) = 𝑞
and
𝜂(𝑟) = 𝑟 .
Then
𝜂
is
neutrosophic
g-homeomorphism but not neutrosophic g*-homeomorphism.
Proposition 4.5: If 𝜇: 𝑋 → 𝑌 and 𝜆: 𝑌 → 𝑍 are neutrosophic g*-homeomorphisms, then (𝜆 𝑜 𝜇) is a
neutrosophic g*-homeomorphism.
Proof: Consider
𝜇 and 𝜆 as neutrosophic g*-homeomorphisms. Predict 𝐾 is a GNCS in 𝑍 .
Thereupon, by the presumption, 𝜆−1 (𝐾) is a GNCS in 𝑌. Hence, by hypothesis, 𝜇 −1 (𝜆−1 (𝐾)) is a
GNCS in 𝑋. Hence, (𝜆 𝑜 𝜇) is a GN-irresolute mapping. Now, consider 𝐻 be a GNCS in 𝑋. Then, by
the presumption, 𝜇(𝐻) is a GNCS in 𝑌. So, by hypothesis, 𝜆( 𝜇(𝐻)) is a GNCS in 𝑍. This implies
that (𝜆 𝑜 𝜇) is a GN-irresolute mapping. Therefore, (𝜆 𝑜 𝜇) is neutrosophic g*-homeomorphism.
Proposition 4.6: If 𝜇: 𝑋 → 𝑌 is a neutrosophic g*-homeomorophism, then 𝑁𝐺𝑐𝑙(𝜇−1 (𝐾)) =
𝜇 −1 (𝑁𝐺𝑐𝑙(𝐾)) for each NS 𝐾 in 𝑌.
Proof: As 𝜇 is neutrosophic g*-homeomorphism, then 𝜇 is GN-irresolute mapping. Let 𝐾 be a NS
in 𝑌. Clearly, 𝑁𝐺𝑐𝑙(𝐾) is GNCS in 𝑋. This proves that 𝐺𝑁𝑐𝑙(𝐾) is GNCS in 𝑋. Since 𝜇−1 (𝐾) ⊆
Md. Hanif PAGE and Qays Hatem Imran, Neutrosophic Generalized Homeomorphism
Neutrosophic Sets and Systems, Vol. 35, 2020
345
𝜇 −1 (𝐺𝑁𝑐𝑙(𝐾)), then 𝐺𝑁𝑐𝑙(𝜇 −1 (𝐾)) ⊆ 𝐺𝑁𝑐𝑙 (𝜇 −1 (𝐺𝑁𝑐𝑙(𝐾))) = 𝜇−1 (𝐺𝑁𝑐𝑙(𝐾)). Therefore,
𝐺𝑁𝑐𝑙(𝜇 −1 (𝐾)) ⊆ 𝜇 −1 (𝐺𝑁𝑐𝑙(𝐾)).
Let 𝜇 be neutrosophic g*-homeomorphism. 𝜇 −1 is a GN-irresolute mapping. Consider NS 𝜇−1 (𝐾)
in 𝑋, which implies that 𝐺𝑁𝑐𝑙(𝜇 −1 (𝐾)) is GNCS in 𝑋. Therefore, 𝐺𝑁𝑐𝑙(𝜇−1 (𝐾)) is a GNCS in 𝑋.
This implies that (𝜇−1 )−1 (𝐺𝑁𝑐𝑙(𝜇 −1 (𝐾)) = 𝜇(𝐺𝑁𝑐𝑙(𝜇−1 (𝐾))) is a GNCS in 𝑌. This proves that 𝐾 =
(𝜇−1 )−1 (𝜇 −1 (𝐾)) ⊆ (𝜇−1 )−1 (𝐺𝑁𝑐𝑙(𝜇 −1 (𝐾))) = 𝜇(𝐺𝑁𝑐𝑙(𝜇−1 (𝐾))) ,
since
𝜇 −1
is
GN-irresolute
mapping. Hence, 𝜇−1 (𝐺𝑁𝑐𝑙(𝐾)) ⊆ 𝜇 −1 (𝜇 (𝐺𝑁𝑐𝑙(𝜇 −1 (𝐾)))) = 𝐺𝑁𝑐𝑙(𝜇−1 (𝐾)).
That is, 𝜇 −1 (𝐺𝑁𝑐𝑙(𝐾)) ⊆ 𝐺𝑁𝑐𝑙(𝜇 −1 (𝐾)). Hence, 𝐺𝑁𝑐𝑙(𝜇 −1 (𝐾)) = 𝜇−1 (𝐺𝑁𝑐𝑙(𝐾)).
5. Conclusions
We have introduced neutrosophic generalized homeomorphism in neutrosophic topological
space using GN-contiuous functions. Some characterizations have been provided to illustrate how
far topological structures are conserved by the new neutrosophic notion defined. Furthermore,
neutrosophic g*-homeomorphism, neutrosophic generalized open and closed mappings are also
studied. The study demonstrated neutrosophic g*-homeomorphisms and also proved some of their
related attributes. Also, the relation between generalized neutrosophic closed mappings and other
existed Neutrosophic closed mappings in Neutrosophic topological spaces were established and
derived some of their related attributes. Examples are given wherever necessary.
In future, we can carry out the further rsearch on neutrosophic g-compactness, neutrosophic
g-connectedness and neutrosophic almost g-contra continuous functions.
Funding: This research received no external funding.
Acknowledgments: The authors are highly grateful to the Referees for their constructive
suggestions.
Conflicts of Interest: The authors declare no conflict of interest.
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Received: Apr 15, 2020.
Accepted: July 3 2020
Md. Hanif PAGE and Qays Hatem Imran, Neutrosophic Generalized Homeomorphism
Neutrosophic Sets and Systems, Vol. 35, 2020
University of New Mexico
Triangular Neutrosophic Based Production Reliability Model of
Deteriorating Item with Ramp Type Demand under Shortages and
Time Discounting
Shilpi Pal¹, Avishek Chakraborty1,2*
¹Department of Basic Science & Humanities (Mathematics), Narula Institute of Technology, Kolkata, India.
E-mail: shilpi88pal@gmail.com
²Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, India.
E-mail: tirtha.avishek93@gmail.com
*Corresponding
email: tirtha.avishek93@gmail.com
Abstract: An economic production quantity model with triangular neutrosophic environment has been
developed for deteriorating items with ramp type demand rate and reliability dependent unit
production. The main objective of this paper is to determine the most cost effective production to
generate better quality items under time discounting. Additionally, it is considered that the deterioration
function deals with three parameters Weibull's distribution under finite time horizon. Moreover, it also
considered the effect of shortages which are partially backordered and partially lost in sale. Here the
reliability of the production process along with the production period is considered as decision variables.
A numerical example is studied in both crisp and neutrosophic environment and a comparative analysis
is performed here. It is observed that the model performs better in triangular neutrosophic arena rather
than crisp domain. Finally, a sensitivity analysis of optimal solution is observed for some parameters and
some crucial decision is taken with managerial insight.
Keywords: Ramp-type demand, Finite time horizon, Time-value of money, Reliability, Triangular
Neutrosophic number.
1.
Introduction
In market economy system, for a single product, many items are produced by the different
manufacturing companies. The manufacturers are trying to give wide variety of option to the customer to
gain competitive advantages over their competitors. But customers choose those items which have high
reliability i.e. better in quality, and lower in cost. The companies require advanced planning many years
prior to the sale target date in order to minimize the total cost and maximize the profit. Thus the facts like
variation in the reliability of the production process, demand rate of an item, deterioration and shortages
are in growing interest. In case of classical EPQ model the basic assumptions are that the production
set-up cost is fixed and the item produced are of perfect quality. All the manufacturing sectors want to
Shilpi Pal, Avishek Chakraborty; Triangular Neutrosophic Based Production Reliability Model of Deteriorating Item
with Ramp Type Demand under Shortages and Time Discounting
Neutrosophic Sets and Systems, Vol. 35, 2020
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produce perfect quality item, but in reality the product quality are not always perfect because there may
be machine breakdown, labor problem, etc. The product quality is directly affected by the reliability of
production process. In addition to that, the classical models also consider an ideal case that the demand
and quality of the items remains unaffected by time and replenishment is done instantaneously.
However in reality these assumptions do not hold. The inventories are often replenished periodically at
certain production rate. Even if the items are purchased it takes days to sell the item so the items
remained stored and hence the item deteriorates and their value reduces with time. Cheng [1] proposed a
general equation for relationship between production set up cost and process reliability and flexibility.
Later it was used by (Leung [2]; Bag et al. [3]) in their respective models studied on fuzzy random
demand with flexibility and reliability on production process. Sarkar [4] analyzed an EMQ model with
reliability in an imperfect production process. Many researchers (like Gomez et al. [5]; Cai et al. [6])
worked for production quality, tracking production control, etc. Pan and Li [7] worked with stochastic
production system for deteriorating item with some environmental constrains. Rathore [8] explored a
production reliability model with advertisement related demand. The paper considers reliability in unit
production cost in order to identify the product quality with minimum total cost.
Traditionally in inventory models, the researchers have assumed constant demand pattern in their
deterministic models, but in reality demand has specific patterns which depicts the real scenarios in
market. There are various types of demand rates such as linear or quadratic function of time,
exponentially increasing or decreasing, price and stock dependent, etc. If the demand is linearly
dependent on time i.e., demand as well as the vending increases and decreases in growth and decline
phase respectively. Researchers have manifested these demands in their respective papers (Hariga [9],
Bose et al. [10], etc). Demand of the item depending on price and stocking amount of the items with
optimal replenishment policy for non-instantaneous deteriorating items with partial backlogging was
discussed by Wu et al. [11]. Alfares [12] worked on stock dependent demand. Chung and Wee [13]
organized an inventory model for stock dependent selling rate with deterioration under replenishment
plan. Pal et al. [14] has developed a inventory model with price and stock depended demand rate for
deteriorating item under inflation and delay in payment. In this field, some remarkable researches were
done by Yang et al. [15]. It was observed that for seasonal and fashionable products the nature of demand
is increasing-steady-decreasing. But for newly launched fashion goods and cosmetics, garments, etc. the
demand rate increases linearly with time and then it become constant. Thus to understand the concept of
such a demand, the ramp type function of time was introduced. (Skouri et al. [16], Luo [17], Manna and
Chaudhari [18]) worked with ramp type demand rate with time dependent deterioration. Pal et al. [19]
considered the EOQ model with ramp type demand under finite time horizon.
As the effect of deterioration cannot be ignored so many researchers worked on it (Skouri et al. [20], Jaggi
et al. [21], etc.). Generally, deterioration means spoilage or damage obsolescence, etc. which cannot be
used further for its original purpose. Medicine, blood banks, etc. are difficult to preserve and they have
some expiry date i.e., products maximum life time is time bounded. Electronic products become obsolete
as technology changes; new fashion depreciates the clothing value over time; all these are also considered
as deterioration. It has been observed that the delinquency in the life expectancy drugs, deterioration of
Shilpi Pal, Avishek Chakraborty; Triangular Neutrosophic Based Production Reliability Model of Deteriorating Item
with Ramp Type Demand under Shortages and Time Discounting
Neutrosophic Sets and Systems, Vol. 35, 2020
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roasted ground coffee, corn seeds, frozen food, pasteurized milk, refrigerated meat, ice creams, and
leakage failure of the batteries can be expressed in terms of Weibull's distribution. Wu [22] presented an
inventory model with ramp type demand and Weibull's distribution deterioration under partial
backlogging. Many researcher such as Skouri et al. [23], Sharma and Chaudhury [24], etc. worked with
this type of deterioration. Mandal [25] discussed an inventory model with Weibull's distributed
deterioration with ramp type demand rate. A common characteristic in most of these models are that they
does not allows shortages. Widyadana et al. [26] developed an EOQ model for deteriorating items with
planned backorder level. Wee et al. al. [27] worked with shortages and finite time horizon for
deteriorating items. Yang [28] developed an inventory model with deterioration as three parameter
Weibull’s distribution in two ware house system. Recently Pal and Chakraborty [29] have worked on
non- instantaneous deteriorating items under shortage, Rahaman et al. [30] worked on arbitrary ordered
generalized EPQ model with and without deterioration. In this paper shortages is also considered where
the part of the unsatisfied demand are backordered and part of the sales are lost.
As the amount of the money available at the present time is worth more than that of the same amount in
the future due to its potential earning capacity. So it is necessary to consider the effect of time value of
money in today's inventory where forecasting is required. To consider the effect of time value of money, a
finite time horizon for planning the replenishment cycle is considered. From the last few decades we have
observed that the economic situation of most countries has changes so it would be unrealistic to ignore
the effect of time value of money. Hariga [31] developed the effect of inflation and time value of money
for time dependent demand. Hou [32] considered a model for deteriorating items and stock-dependent
demand rate with shortages and time discounting. Dash et al. [33] worked on EPQ model for declined
quadratic demand with time value of money and shortages. Thus the paper considers time value of
money specially when investment and forecasting are considered.
In this current century, vagueness theory plays a crucial role in different Öeld of mathematical modeling
and engineering problems. The theory of impreciseness was first invented by Zadeh [34]. Difference
between crisp set and fuzzy set is shown briefly in this article by considering membership gradation and
its formulation. Demonstration of triangular [35], trapezoidal [36], pentagonal [37] fuzzy number has
already been developed by the researchers. In 1983 and later in 1986 Attasonov [38, 39] manifested a
remarkable idea of intuitionistic fuzzy set where membership and non-membership functions are both
considered together. Further, triangular intuitionistic [40, 41], trapezoidal intuitionistic [42] number has
been introduced in this intuitionistic fuzzy research arena. After that, in 1998 Smarandache [43]
established an amazing concept of neutrosophic fuzzy set where three disjunctive kinds of membership
functions has been considered namely i) truthness ii) falseness iii) indeterminacy. Due to the presence of
hesitation factor in fuzzy arena, neutrosophic number becomes more logical and scientific significance in
research work. In this current era, researchers from different arena are focusing on neutrosophic concept
and developed lots of interesting articles in this domain. Illustration of triangular, trapezoidal
neutrosophic number has been introduced day by day and recently in 2018 Chakraborty et.al [44, 45]
classifies different form of triangular and trapezoidal neutrosophic number and de-neutrosophication
technique for crispification. Further, bipolarization of triangular bipolar number has been developed by
Shilpi Pal, Avishek Chakraborty; Triangular Neutrosophic Based Production Reliability Model of Deteriorating Item
with Ramp Type Demand under Shortages and Time Discounting
Neutrosophic Sets and Systems, Vol. 35, 2020
350
Chakraborty et.al [46] and also Maity et.al [47] manifested the concept of heptagonal dense fuzzy number
related EOQ based model in 2018. Recently, Mullai [48] introduced EOQ model in neutrosophic domain
and Mondal et.al [49] manifested optimization of EOQ Model with limited storage capacity by
neutrosophic Geometric Programming application. Also, Majumdar et.al [50] focused on EPQ Model of
deteriorating Items under partial trade credit financing and demand declining market in neutrosophic
environment. Some useful articles [51-58] are also developed by the researchers in the neutrosophic arena
recently. As developments goes on, some researchers [59-62] have extended the idea of neutrosophic set
into plithogenic set and applied it in MCDM, MADM and optimization technique supply chain based
model. Currently, several researchers from distinct fields focused on triangular neutrosophic number
related to operation research models. As uncertainty prevails in various parameters such as inflation,
holding cost, purchase cost so we have developed an EPQ under ramp type demand and considered the
hesitation in those parameters by considering those parameter as neutrosophic number. Finally we
compare the model in crisp and neutrosophic domain and observe that the model works better in
neutrosophic arena.
Previously the researchers have worked on ramp type demand with two parameter Weibull’s
distribution as deterioration. But in this paper we have considered ramp type demand with three
parameter Weibull’s distribution. In addition the model assumes that the product qualities are never
perfect and it is the function of reliability of the production process so the production of items depend on
the reliability of the items i.e., if the items are highly reliable then there is more demand in the market and
hence its production should be more in order to fulfill the demand. In this model we also have considered
finite planning horizon to observe the effect of time value of money under shortage. The shortage items
are partially backlogged or partially lost in sales, which cannot be ignored. Also under this complicated
scenario no work has been done by considering holding cost, purchase cost and inflation as triangular
neutrosophic number.
The rest of the paper is organized as follows: In section 2 we have presented some assumptions and
notations and some definition of neutrosophic number that we have used in this paper. In this section we
have defined few terminologies related to triangular neutrosophic number and also have formulated the
model. In section 3 we have analyzed and optimized of the model. In Section 4 we have discussed the
de-neutrosophication of the triangular neutrosophic number. In section 5 we present the numerical
example and its mathematical analysis which is shown graphically. It is observed that the model works
better in neutrosophic domain. In section 6 we present sensitivity analysis of some parameters. Finally in
section 7 a concluding remark is stated along with its future extension.
2. Mathematical formulation of the inventory model
In this model we have considered ramp type demand with deterioration as three parameter Weibull
distributions, shortages, lost in sales under the influence of time discounting in finite planning horizon.
The finite time horizon has been considered to evaluate the effect of inflation on the total cost for a finite
period. The paper also considered reliability in production of items. The proposed model is graphically
shown in figure-1.
Shilpi Pal, Avishek Chakraborty; Triangular Neutrosophic Based Production Reliability Model of Deteriorating Item
with Ramp Type Demand under Shortages and Time Discounting
Neutrosophic Sets and Systems, Vol. 35, 2020
351
The production process starts from t=0 and ends t=t1. The production has occurred along with the
demand in the market and at t=t1 the inventory level is maximum, Qm. From t=t1 to t=t2 the inventory level
decreases and at time t=t2, the inventory level reaches zero. Now during [t2,t3] the model undergoes
shortage with partial backlog and partial lost in sales. Only the backlogged items are replaced by the next
replenishment. During [t3,T1] production resumes to overcome the shortage (i.e., for backlogged items).
Thus the total number of backlogged items is replaced in the next replenishment and the cycle repeats.
Notations
The notations used in this paper are as follows:
G
Demand rate,
P
Production rate,
p
Unit production cost,
ρ(t)
Time distribution for deterioration of the item,
k
Discount rate,
h
Inventory carrying cost per unit item per unit time,
d
Deterioration cost per unit per unit time,
S
Set-up cost for one replenishment cycle.
c1
Purchase cost per unit item,
c2
Shortage cost,
c3
Penalty cost of a lost sale including loss of profit,
r
Production process reliability (a decision variable)
B
Fraction of backorder (0<B≤1),
T
Replenishment cycle,
H
Finite Planning horizon,
m
No. of replenishment during the planning horizon i.e., m=(H/T),
Tj
Time between start and end of jth replenishment cycle i.e., T0=0,T1=T,T2=2T,....,Tm=mT=H,
Qm
Maximum quantity of inventory,
Qs
Maximum quantity of inventory after shortage.
Shilpi Pal, Avishek Chakraborty; Triangular Neutrosophic Based Production Reliability Model of Deteriorating Item
with Ramp Type Demand under Shortages and Time Discounting
Neutrosophic Sets and Systems, Vol. 35, 2020
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Assumptions
The assumptions which are considered in this model are as follows.
1.
A ramp type demand rate G=f(t) is a function of time𝑓(𝑡) = 𝑅[𝑡 − (𝑡 − 𝜇)𝐻(𝑡 − 𝜇)], 𝑅 > 0 𝑎𝑛𝑑 𝐻(𝑡)
1
is a Heaviside function 𝐻(𝑡 − 𝜇) = {
0
2.
𝑖𝑓 𝑡 ≥ 𝜇
𝑖𝑓 𝑡 < 𝜇
A function of three parameter Weibull's distribution of time is used to represent deterioration of the
item is 𝜌(𝑡) = 𝛼𝛽(𝑡 − 𝛾)𝛽−1 , 0 < 𝛼 < 1, 𝛽 ≥ 1, −∞ < 𝛾 < ∞ actually in this model
𝑇𝑗 < 𝛾 < 𝑇𝑗+1 , 𝑖 =
0,1,2, . . . , 𝑚, 𝑤ℎ𝑒𝑟𝑒 𝛼 (0 < 𝛼 < 1) is a scaling parameter, β is the shape parameter and γ is the location
parameter i.e., items shelf-time and t is the time of deterioration.
3. Deterioration begins as it reaches the inventory.
4. One item is considered in the prescribed time cycle.
5. Demand during shortage is partially lost and partially backordered.
6. Time discounting effect is considered under finite time horizon.
7 Production rate is greater than demand rate so P=σf(t) is the production rate where σ >1.
8. μ is less than production time.
9. The unit production cost is inversely proportional to the demand rate (G) and directly proportional
to production reliability (r), so the unit production cost is 𝑝 = 𝑎𝐺 −𝑏 𝑟 𝑐 , where b(>1) is called price
elasticity and a,c (>0) are scaling parameters.
10. The reliability r means, r% of all the item produced are of acceptable quality that can fulfill the
demand.
Few assumptions taken above are the basic assumption used in classical inventory model for
deteriorating item with shortages. The first assumption states that the demand rate linearly increases
with time when t<μ and then become steady i.e., constant at and after t≥μ. We can see this type of
demand in newly launched items like fashionable products, electronic items, etc. The demand increases
with time during the initial stage i.e., [0,μ]. After some time the demand become constant, this continues
for some period i.e., in the time interval [μ,T1]. Then the cycle ends. Again the next cycle starts with
another new brand item and it will follow the same pattern of demand and production i.e., increasing
and then steady and then stops. The finite time horizon has been considered to evaluate the effect of the
time value of money on the total cost. Thus to understand the concept of value of future money in present
date (which actually decreases due to time discounting rate) we need to consider a finite time horizon
where its effect will be observed. The last assumption is mainly based on the unit variable production
which is dependent on demand and process reliability. When the demand of an item increases then the
production/purchase cost per unit item decreases and hence the unit production cost reduces which is
inversely proportional to demand. Again the reliability of the produced items increases by using high
quality raw material, technologically advanced machinery, quality control inspections, etc. Thus to
produce high reliable product the production cost per unit item increases.
Shilpi Pal, Avishek Chakraborty; Triangular Neutrosophic Based Production Reliability Model of Deteriorating Item
with Ramp Type Demand under Shortages and Time Discounting
Neutrosophic Sets and Systems, Vol. 35, 2020
353
3. Neutrosophic number and its De-neutrosophication technique
Definition 3.1 (Neutrosophic Set [5]) A set 𝑆̃ in the universal discourse X, it is said to be a neutrosophic
set if 𝑆̃ = {〈𝑥; [𝜋𝑆̃ (𝑥), 𝜃𝑆̃ (𝑥), 𝜂𝑆̃ (𝑥)]〉: 𝑥 ∈ 𝑋}, 𝑤ℎ𝑒𝑟𝑒 𝜋𝑆̃ (𝑥): 𝑋 →] − 0,1 + [ is called the truth membership
function, 𝜃𝑆̃ (𝑥): 𝑋 →] − 0,1 + [ is called the hesitation membership function, and 𝜂𝑆̃ (𝑥): 𝑋 →] − 0,1 + [
is called the false membership function of the decision maker, where 𝜋𝑆̃ (𝑥), 𝜃𝑆̃ (𝑥), 𝜂𝑆̃ (𝑥) satisfies the
following condition: 0 ≤ 𝑆𝑢𝑝{𝜋𝑆̃ (𝑥)} + 𝑆𝑢𝑝{𝜃𝑆̃ (𝑥)} + 𝑆𝑢𝑝{𝜂𝑆̃ (𝑥)} ≤ 3.
Definition 3.2 (Single-Valued Neutrosophic Set) A Neutrosophic set 𝑆̃ in the above definition 2.1 is also
known as single-Valued Neutrosophic Set sig(𝑆̃) if x is a single-valued independent variable.
𝑠𝑖𝑔(𝑆̃) = {< 𝑥; [π𝑠𝑖𝑔(𝑆̃) (x), θ𝑠𝑖𝑔(𝑆̃) (x), η𝑠𝑖𝑔(𝑆̃) (x)]〉: x ∈ X}, where π𝑠𝑖𝑔(𝑆̃) (x), θ𝑠𝑖𝑔(𝑆̃) (x), η𝑠𝑖𝑔(𝑆̃) (x) represent the
concept of truth, hesitation and falsity memberships function respectively.
Definition 3.2.1: (Neutro-normal) Let us consider three points, for which p,q,r for which, π𝑠𝑖𝑔(𝑆̃) (p) = 1,
(r) = 1 then the sig(𝑆̃) is defined as neutro-normal.
θ𝑠𝑖𝑔(𝑆̃) (q) = 1, η
𝑠𝑖𝑔(𝑆̃)
Definition 3.2.2: (Neutro-convex) A sig(𝑆̃) is called neutro-convex if the following condition holds:
(𝑖)𝜋𝑠𝑖𝑔(𝑆̃) (𝜆𝛼 + (1 − 𝜆)𝛽) ≥ 𝑚𝑖𝑛(𝜋𝑠𝑖𝑔(𝑆̃) (𝛼), 𝜋𝑠𝑖𝑔(𝑆̃) (𝛽))
(𝑖𝑖)𝜃𝑠𝑖𝑔(𝑆̃) (𝜆𝛼 + (1 − 𝜆)𝛽) ≥ 𝑚𝑖𝑛(𝜃𝑠𝑖𝑔(𝑆̃) (𝛼), 𝜃𝑠𝑖𝑔(𝑆̃) (𝛽)),
(𝑖𝑖𝑖)𝜂𝑠𝑖𝑔(𝑆̃) (𝜆𝛼 + (1 − 𝜆)𝛽) ≥ 𝑚𝑖𝑛(𝜂𝑠𝑖𝑔(𝑆̃) (𝛼), 𝜂𝑠𝑖𝑔(𝑆̃) (𝛽))
𝑤ℎ𝑒𝑟𝑒 𝛼, 𝛽 ∈ 𝑅, 𝑎𝑛𝑑 𝜆 ∈ [0,1]
Definition 3.3 (Triangular Single Valued Neutrosophic Number) A triangular Single Valued Neutrosophic
Number ( 𝑆̃ ) is defined as 𝑆̃ =< (𝑚₁, 𝑚₂, 𝑚₃: 𝜇), (𝑛₁, 𝑛₂, 𝑛₃: 𝜗), (𝑝₁, 𝑝₂, 𝑝₃: 𝜁) >, 𝑤ℎ𝑒𝑟𝑒 𝜇, 𝜗, 𝜁 ∈ [0,1]. Here the
truth membership function𝜋𝑆̃ : R → [0, μ], the hesitation membership function θ𝑆̃ : R → [ϑ, 1] and the falsity
membership function η𝑆̃ : R → [ζ, 1] are defined as follows:
𝛿𝑆̃𝑙 (𝑥),
𝜇,
𝜋𝑆̃ (𝑥) = {
𝛿𝑆̃𝑟 (𝑥),
0,
𝑙𝑆̃𝑙 (𝑥), 𝑝₁ ≤ 𝑥 < 𝑝₂
𝑚1 ≤ 𝑥 < 𝑚2
𝜀𝑆̃𝑙 (𝑥), 𝑛₁ ≤ 𝑥 < 𝑛₂
𝑥 = 𝑚2
𝜗,
𝑥 = 𝑝₂
𝜗,
𝑥 = 𝑛₂
𝜃̃ = {
η𝑆̃ (𝑥) = {
𝜀𝑆̃𝑟 (𝑥), 𝑛₂ < 𝑥 ≤ 𝑛₃
𝑚2 < 𝑥 ≤ 𝑚3 𝑆(𝑥)
𝑙𝑆̃𝑟 (𝑥), 𝑝₂ < 𝑥 ≤ 𝑝₃
1,
𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
1,
𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
De-neutrosophication of triangular single valued neutrosophic number: In this model we have applied
removal area technique to evaluate the de-neutrosophication value of triangular single valued neutrosophic
number
𝑆̃ =< (𝑚₁, 𝑚₂, 𝑚₃: 𝜇), (𝑛₁, 𝑛₂, 𝑛₃: 𝜗), (𝑝₁, 𝑝₂, 𝑝₃: 𝜁) > as done by (Chakraborty, et. al.). The de-neutrosophic form
𝑚 +2𝑚2 +𝑚3 +𝑛1 +2𝑛2 +𝑛3 +𝑝1 +2𝑝2 +𝑝3
of 𝑆̃ is given as 𝑛𝑒𝑢𝐷𝑆̃ = ( 1
)
12
4.
Proposed model
Thus the inventory level for the proposed model at any time t over [0,T] is described mathematically by
the following equations:
Shilpi Pal, Avishek Chakraborty; Triangular Neutrosophic Based Production Reliability Model of Deteriorating Item
with Ramp Type Demand under Shortages and Time Discounting
Neutrosophic Sets and Systems, Vol. 35, 2020
𝑑𝑄(𝑡)
𝑑𝑡
𝑑𝑄(𝑡)
𝑑𝑡
𝑑𝑄(𝑡)
𝑑𝑡
𝑑𝑄(𝑡)
𝑑𝑡
𝑑𝑄(𝑡)
𝑑𝑡
+ 𝜌(𝑡)𝑄(𝑡) = 𝑟𝑃 − 𝐺 = (𝑟𝜎 − 1)𝑅𝑡,
354
0≤𝑡≤𝜇
(1)
+ 𝜌(𝑡)𝑄(𝑡) = (𝑟𝜎 − 1)𝑅𝜇,
𝜇 ≤ 𝑡 ≤ 𝑡₁
(2)
+ 𝜌(𝑡)𝑄(𝑡) = −𝐺 = −𝑅𝜇,
𝑡₁ ≤ 𝑡 ≤ 𝑡₂
(3)
= −𝐵𝐺 = −𝐵𝑅𝜇,
𝑡2 ≤ 𝑡 ≤ 𝑡3
(4)
= 𝑟𝑃 − 𝐺 = 𝑟𝐾 − 𝑅𝜇 = (𝑟𝜎 − 1)𝑅𝜇, 𝑡₃ ≤ 𝑡 ≤ 𝑇₁
(5)
with boundary conditions
𝑄(0) = 0, 𝑄(𝜇) = 𝐼, 𝑄(𝑡₁) = 𝑄𝑚 , 𝑄(𝑡₂) = 0, 𝑄(𝑡₃) = −𝑄𝑠 𝑎𝑛𝑑 𝑄(𝑇₁) = 0,
𝑤ℎ𝑒𝑟𝑒 𝐼 = (𝑟𝜎 − 1)𝑅[(
(−1)𝛽 𝛾 𝛽+2
𝜇2
𝛼𝛾
𝛼
)+(
) (𝜇 − 𝛾)𝛽+1 + (
) (𝜇 − 𝛾)𝛽+2 +
]
(𝛽 + 1)(𝛽 + 2)
2
𝛽+1
𝛽+2
4.1 Mathematical Analysis of the proposed model
From the above differential equations [1, 2, 3, 4, 5] and using the assumptions and the boundary conditions
we obtain the inventory level of the proposed inventory model as follows:
𝛽+2
(−1){𝛽}𝛾
𝑡2
𝛼𝑡 2
𝛼𝛾
𝛼
𝑄(𝑡) = (𝑟𝜎 − 1)𝑅[( ) − (
) (𝑡 − 𝛾)𝛽 + (
) (𝑡 − 𝛾)𝛽+1 + (
) (𝑡 − 𝛾)𝛽+2 + (
)]
(𝛽 + 1)(𝛽 + 2)
2
2
𝛽+1
𝛽+2
𝜇2
𝑡−𝛾
2
𝛽+1
𝑄(𝑡) = (𝑟𝜎 − 1)𝑅[𝑡𝜇 − ( ) + 𝜇𝛼(𝑡 − 𝛾)𝛽 ((
𝑄(𝑡) = 𝑅𝜇[𝑡₁ − 𝑡 + (
𝛼
𝛽+1
𝜇
𝛼
) − 𝑡 + ( )) − ((𝛽+1)(𝛽+2)) {(𝜇 − 𝛾)𝛽+2 − (−1)𝛽 𝛾 𝛽+2 }]
(6)
2
) {(𝑡1 − 𝛾)𝛽+1 − (𝑡 − 𝛾)𝛽+1 } + 𝛼(𝑡 − 𝑡1 )(𝑡 − 𝛾)𝛽 ] + 𝑄𝑚 (1 − 𝛼(𝑡 − 𝛾)𝛽 + 𝛼(𝑡1 − 𝛾)𝛽 ), 𝑡₁ ≤ 𝑡 ≤ 𝑡₂
(7)
𝑄(𝑡) = −𝐵𝑅𝜇(𝑡 − 𝑡₂), 𝑡₂ ≤ 𝑡 ≤ 𝑡₃
(8)
𝑄(𝑡) = (𝑟𝜎 − 1)𝑅𝜇(𝑡 − 𝑡₃) − 𝑄𝑠 , 𝑡₃ ≤ 𝑡 ≤ 𝑇₁
(9)
Now using Q(t₂)=0 and eq.(6) we get the maximum amount inventory Q m,
𝑄𝑚 = 𝑅𝜇[𝑡₂ − 𝑡₁ + (
𝛼
𝛽+1
) (𝑡2 − 𝛾)𝛽+1 − 𝛼(𝑡1 − 𝛾)𝛽 ((
𝑡 1 −𝛾
𝛽+1
) + 𝑡₂ − 𝑡₁)]
(10)
Now using eq.(8), eq.(9) and the relation Q(t₃)=-Qs we get the maximum shortages in the inventory level,
𝑄𝑠 = 𝐵𝑅𝜇(𝑡₃ − 𝑡₂)
(11)
Inventory carrying cost or holding cost:
𝑡1
𝜇
𝑡2
𝐻𝐶 = ℎ [∫ 𝑄(𝑡)𝑑𝑡 + ∫ 𝑄(𝑡)𝑑𝑡 + ∫ 𝑄(𝑡)𝑑𝑡]
0
𝜇
= ℎ[(𝑟𝜎 − 1)𝑅{(
𝑡1
𝜇4
𝛼𝛽𝜇(𝜇 − 𝛾)𝛽+3
𝛾(𝛽 + 5)
((−1)𝛽 𝛼𝜇𝛾 𝛽+2 )
𝛾
)−(
) ((
)+𝜇+(
) (𝜇 − (
))
(𝛽 + 1)(𝛽 + 2)
6
2(𝛽 + 2)(𝛽 + 3)
𝛽+1
𝛽+3
+(
(𝑡2 − 𝑡1 )2
𝛼𝜇𝛾 2 (𝜇 − 𝛾)𝛽+1
𝜇𝑡1
𝛼𝛽𝜇(𝑡1 − 𝛾)𝛽+2
) + ( ) (𝑡₁ − 𝜇) − (
) + 𝑅𝜇{− (
)
(𝛽 + 1)(𝛽 + 2)
2(𝛽 + 1)
2
2
Shilpi Pal, Avishek Chakraborty; Triangular Neutrosophic Based Production Reliability Model of Deteriorating Item
with Ramp Type Demand under Shortages and Time Discounting
Neutrosophic Sets and Systems, Vol. 35, 2020
𝜇
+(
(
𝛼𝜇(( 2 )−𝛾)
𝛽+1
355
(𝛼(𝜇−𝛾)𝛽+2 )
𝑡1
𝛽+2
𝛽+1
) [(𝑡1 − 𝛾)𝛽+1 − (𝜇 − 𝛾)𝛽+1 ] + (
𝛼(𝑡1 −𝛾)(𝑡2 −𝛾)
𝛽+1
) (𝜇 − (
) [(𝑡1 − 𝛾)𝛽 − (𝑡2 − 𝛾)𝛽 ] + 𝑄𝑚 (𝑡2 − 𝑡1 − (
𝛼(𝑡2 −𝛾)𝛽+1
𝛽+1
)) + (
(−1)𝛽 𝛼𝛾(𝛽+2)
(𝛽+1)(𝛽+2)
) + 𝛼(𝑡1 − 𝛾)𝛽 ((
) (𝑡₁ − 𝜇)} +
𝑡1 −𝛾
𝛽+1
) − 𝑡2 − 𝑡1 ))}] (12)
Production cost: The unit production cost depends on demand and process reliability. When the demand of
an item increases then the production/purchase cost of the item decreases hence the unit production cost
reduces i.e., production / purchase cost varies inversely with demand. The process reliability level r means
only r% of the produced items is of acceptable quality which can be used to meet demand.
The unit production cost 𝑝 = 𝑎𝐷 −𝑏 𝑟 𝑐 𝑤ℎ𝑒𝑟𝑒 𝑎, 𝑏, 𝑐 > 0 𝑎𝑛𝑑 𝑏 ≠ 2.
The cost of production in [𝑡, 𝑡 + 𝑑𝑡] 𝑖𝑠 𝐾𝑝𝑑𝑡 = 𝜎𝐷. 𝑎𝐷 −𝑏 𝑟 𝑐 𝑑𝑡 = (
𝜎𝑎𝑟 𝑐
𝐷𝑏−1
) 𝑑𝑡.
Since the production occurs [0,t₁] and [t₃,T₁] so the production cost (PDC) is given as follows.
𝜇
Production cost (PDC)= ∫0 (
𝜇
𝜎𝑎𝑟 𝑐
𝐷𝑏−1
𝑡
) 𝑑𝑡 + ∫𝜇 1 (
𝑡
𝜎𝑎𝑟 𝑐
𝐷𝑏−1
𝑇
𝜎𝑎𝑟 𝑐
3
𝐷𝑏−1
) 𝑑𝑡 + ∫𝑡 1 (
) 𝑑𝑡
𝑇
= 𝜎𝑎𝑟 𝑐 [∫0 (𝑅𝑡)1−𝑏 𝑑𝑡 + ∫𝜇 1(𝑅𝜇)1−𝑏 𝑑𝑡 + ∫𝑡 1(𝑅𝜇)1−𝑏 𝑑𝑡]
3
=(
𝜎𝑎𝑟 𝑐 𝑅 1−𝑏
2−𝑏
) [(𝑏 − 1)𝜇 2−𝑏 + (2 − 𝑏)𝜇1−𝑏 (𝑡₁ + 𝑇₁ − 𝑡₃)], 𝑏 ≠ 2
(13)
Deterioration cost: The total no. of deteriorated items in [0,T₁] is same as deterioration in [0,t₂] as there is no
deterioration of items during the period [t₂,T₁].
D₁=Total no. of deteriorated items in [0,t₂]
=r×Production in [0,μ]+r×Production in [μ,t₁]-Demand in [0,μ]-Demand in [μ,t₂]
𝑡1
𝜇
𝑡2
𝜇
= 𝑟𝜎 ∫ 𝑅𝑡𝑑𝑡 + 𝑟𝜎 ∫ 𝑅𝜇𝑑𝑡 − ∫ 𝑅𝑡𝑑𝑡 − ∫ 𝑅𝜇𝑑𝑡
0
𝜇
0
𝜇
1
1
= ( ) 𝑅𝑟𝜇𝜎(2𝑡₁ − 𝜇) − ( ) 𝑅𝜇(2𝑡₂ − 𝜇)
2
2
∴ 𝐷𝑒𝑡𝑒𝑟𝑖𝑜𝑟𝑎𝑡𝑖𝑜𝑛 𝑐𝑜𝑠𝑡 (𝐷𝐶) = (𝑑𝐷₁) = (
𝑅𝜇𝑑
2
) (𝑟𝜎(2𝑡₁ − 𝜇) − (2𝑡₂ − 𝜇))
(14)
Purchase cost: Since there is shortages in our model so the producer has to purchase raw material not only
during [0,t₁] but also in [t₃, T₁]. So we have to calculate purchase cost during the above two period.
𝜇
𝑡
𝑇
𝜇
𝑃𝐶 = 𝑐₁𝜎(∫0 𝑅𝑡𝑑𝑡 + 𝑟𝜎 ∫𝜇 1 𝑅𝜇𝑑𝑡 + ∫𝑡 1 𝑅𝜇𝑑𝑡) = 𝑐₁𝜎𝑅𝜇(𝑡₁ + 𝑇₁ − 𝑡₃ − ( ))
2
3
(15)
Shortage cost: Since the model undergoes shortages so we observe shortages during [t₂, T₁].
𝑡
𝑇
𝑐 2 𝑅𝜇
2
3
2
𝑆𝐶 = 𝑐₂ ∫𝑡 3 −𝑄(𝑡)𝑑𝑡 + 𝑐₂ ∫𝑡 1 −𝑄(𝑡)𝑑𝑡 = (
) [𝐵(𝑡₃ − 𝑡₂)² + (𝑟𝜎 − 1)(𝑇₁ − 𝑡₃)²]
(16)
Lost cost: Due to urgency of demand the consumer opt to another shop so there is a chance for loss in sale
during the shortages period [t₂, t₃]. Thus the lost cost for one replenishment interval is (LC).
𝑡
𝐿𝐶 = 𝑐₃(1 − 𝐵) ∫𝑡 3 𝑅𝜇𝑑𝑡 = 𝑐₃(1 − 𝐵)𝑅𝜇(𝑡₃ − 𝑡₂)
2
(17)
Shilpi Pal, Avishek Chakraborty; Triangular Neutrosophic Based Production Reliability Model of Deteriorating Item
with Ramp Type Demand under Shortages and Time Discounting
Neutrosophic Sets and Systems, Vol. 35, 2020
356
The present value of total cost is (TC):
𝑚
𝑇𝐶 = (𝐷𝐶 + 𝑃𝐶 + 𝐻𝐶 + 𝐿𝐶 + 𝑃𝐷𝐶 + 𝑆𝐶) ∑ 𝑒 −(𝑖−1)𝑘𝑇 ≈ (𝐷𝐶 + 𝑃𝐶 + 𝐿𝐶 + 𝑆𝐶 + 𝑃𝐷𝐶 + 𝐻𝐶) (
𝑖=1
𝑑
𝜇
𝑐
2
2
2
1 − 𝑒 −𝑘𝑚𝑇
)
1 − 𝑒 −𝑘𝑇
= 𝑅𝜇 [( ) (𝑟𝜎(2𝑡1 − 𝜇) − (2𝑡2 − 𝜇)) + 𝑐1 𝜎 (𝑡1 + 𝑇1 – 𝑡3 − ) + 𝑐3 (1 − 𝐵)(𝑡3 − 𝑡2 ) + ( 2) [𝐵(𝑡3 − 𝑡2 )2 +
(𝑟𝜎 − 1)(𝑇1 − 𝑡3 )2 ] + (
(
𝜎𝑎𝑟 𝑐 𝑅 −𝑏
2−𝑏
2
𝛼{𝜇𝛽(𝑡1 −𝛾)𝛽+2 −𝑡1 (𝜇−𝛾)𝛽+2 +(−1)(𝛽) 𝛼𝛾𝛽+2 𝑡1 }
𝜇(𝛽+1)(𝛽+2)
𝑄𝑚 (𝑡2 − 𝑡1 − (
𝛼(𝑡2 −𝛾)𝛽+1
𝛽+1
𝑡
) [(𝑏 − 1)𝜇1−𝑏 + (2 − 𝑏)𝜇 −𝑏 (𝑡1 + 𝑇1 − 𝑡3 )] + ℎ [(𝑟𝜎 − 1) {𝜉 + ( 1) (𝑡1 − 𝜇) −
𝜇
)+
(𝛼( 2 −𝛾))(𝑡1 −𝛾)𝛽+1
) + 𝛼(𝑡1 − 𝛾)𝛽 ((
𝛽+1
𝑡1 −𝛾
𝛽+1
}−(
(𝑡2 −𝑡1 )2
) − 𝑡2 − 𝑡1 ))]] (
2
𝛼[(𝑡1 −𝛾)𝛽+1 (𝑡2 −𝛾)−(𝑡1 −𝛾)(𝑡2 −𝛾)𝛽+1 ]
)+(
1−𝑒 −𝑘𝑚𝑇
1−𝑒 −𝑘𝑇
𝛽+1
)
)+
(18)
𝑊ℎ𝑒𝑟𝑒
𝑏 ≠ 2,
𝜉=
(−1)𝛽 𝛼𝛾 𝛽+2
𝜇3
𝛼𝛽(𝜇 − 𝛾)𝛽+3 𝛾(𝛽 + 5)
𝛾
−
(
+ 𝜇) +
(𝜇 −
− 1)
(𝛽 + 1)(𝛽 + 2)
6 2(𝛽 + 2)(𝛽 + 3) 𝛽 + 1
𝛽+3
+
𝛼(𝜇 − 𝛾)𝛽+1 (𝛾 2 + 2𝛾 − 𝜇) 𝛼(𝜇 − 𝛾)𝛽+2
+
2(𝛽 + 1)
𝛽+2
We observe that TC is a function of t₁,t₂,t₃ and m. But for the sake of simplicity we simplified t₂ and t₃ in terms
of t₁ and r.
Considering eq.(7), eq.(8) and the condition Q(t₁)=Qm we get t₂ in terms of t₁, and r. Expanding the
exponential terms and neglecting the second and higher order terms of α and after simplifying the above two
equations we get,
𝜇
𝛼
2
𝜇(𝛽+1)(𝛽+2)
𝑡₂ = (𝑟𝜎 − 1)[ −
{(𝜇 − 𝛾)𝛽+2 − (−1)𝛽 𝛾 𝛽+2 }] + 𝑟𝜎[𝑡₁ +
𝛼(𝑡1 −𝛾)𝛽+1
𝛽+1
]
(19)
Also considering (11), and Q(T₁)=0, we get t₃ in terms of t₁,and r.
𝐵𝑃𝜇(𝑡₃ − 𝑡₂) = (𝛾 − 1)𝑅𝜇(𝑇₁ − 𝑡₃)
𝑡₃ =
1
𝐵+𝑟𝜎−1
((𝑟𝜎 − 1)𝑇₁ + 𝐵𝑡₂)
(20)
Thus the total cost TC is function of t₁, r and m.
Optimization process
The following technique is derived to obtain the optimal value of t₁, r and m.
Step 1: Start by choosing a discrete value of m, a positive integer number.
Step 2: Take the partial derivative of total cost TC(t₁, r, m) with respect to t₁ and r and equate it to zero,
the necessary condition for optimality is
𝜕𝑇𝐶(𝑡1 ,𝑟,𝑚)
𝜕𝑡1
= 0 𝑎𝑛𝑑
𝜕𝑇𝐶(𝑡1 ,𝑟,𝑚)
𝜕𝑟
=0.
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with Ramp Type Demand under Shortages and Time Discounting
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357
Step 3: For different values of m, Obtain the optimum value of the time taken t₁ * and reliability r* from
the above two equation. Then substituting the value of t₁*, r* and m in equation [18] and obtain TC(t₁*,r*,m)
Step 4: Repeat step 2 and step 3 for different values of m and obtain the TC(t₁ *,r*,m). The minimum value
of TC is obtained for optimum value of m*. Thus (t₁*,r*,m*) and TC(t₁*,r*,m*) are the optimal solution of our
model. It satisfies the following condition:
𝛥𝑇𝐶(t1∗ , r ∗ , m∗ − 1) < 0 < 𝛥𝑇𝐶(t1∗ , r ∗ , m∗ + 1)
Where 𝛥𝑇𝐶(t1∗ , r ∗ , m∗ ) = 𝑇𝐶(t1∗ , r ∗ , m∗ + 1) − 𝑇𝐶(t1∗ , r ∗ , m∗ )
Step 5: To confirm that the objective function is convex, the derived value of TC(t1∗ , r ∗ , m∗ ) must satisfy
the sufficient condition:
(
𝜕2 𝑇𝐶(𝑡1 ,𝑟)
𝜕2 𝑇𝐶(𝑡1 ,𝑟)
𝜕𝑡12
𝜕𝑟𝜕𝑡1
𝜕2 𝑇𝐶(𝑡1 ,𝑟)
𝜕𝑟 2
𝜕2 𝑇𝐶(𝑡1 ,𝑟)
𝜕𝑡1 𝜕𝑟
) > 0 𝑎𝑛𝑑
𝜕2 𝑇𝐶(𝑡1 ,𝑟)
𝜕𝑡12
> 0 𝑜𝑟
𝜕2 𝑇𝐶(𝑡1 ,𝑟)
𝜕𝑟 2
>0
(21)
Since TC* is very complicated with high powers so it is not possible to show the analytic validity of
eq.(21). For this reason the above inequality is assessed by a numerical example.
4.2
Effect of Neutrosophication of parameter in proposed inventory model
Neutrosophic number actually deals with the conception of three different kinds of membership
function related with real life scenario. It consists of truth, hesitation and falseness of an imprecise number.
In this model we have considered purchase cost (c₁), holding cost (h) and inflation (k) as neutrosophic fuzzy
number since in reality all the parameters are uncertain and contains a dilemma in decision maker's mind. So
we try to manifest the model by introducing neutrosophication in the above cost and rates, and thus observe
the effect of the above by comparing it with crisp model. The neutrosophic form of holding cost, purchase
cost and inflation are represented by ℎ̃, 𝑐₁
̃ and 𝑘̃. Thus
ℎ̃ = < (ℎ₁ − 𝜀₁, ℎ₁, ℎ₁ + 𝜀₂: 𝜇), (ℎ₂ − 𝜀₁, ℎ₂, ℎ₂ + 𝜀₂: 𝜗), (ℎ₃ − 𝜀₁, ℎ₃, ℎ₃ + 𝜀₂: 𝜁) > ,
𝑐₁
̃ =< (𝑐₁₁ − 𝜀₁, 𝑐₁₁, 𝑐₁₁ + 𝜀₂: 𝜇), (𝑐₁₂ − 𝜀₁, 𝑐₁₂, 𝑐₁₂ + 𝜀₂: 𝜗), (𝑐₁₃ − 𝜀₁, 𝑐₁₃, 𝑐₁₃ + 𝜀₂: 𝜁) >,
𝑘̃ =< (𝑘₁ − 𝜀₁, 𝑘₁, 𝑘₁ + 𝜀₂: 𝜇), (𝑘₂ − 𝜀₁, 𝑘₂, 𝑘₂ + 𝜀₂: 𝜗), (𝑘₃ − 𝜀₁, 𝑘₃, 𝑘₃ + 𝜀₂: 𝜁) >
𝑤ℎ𝑒𝑟𝑒 𝜇, 𝜗, 𝜁 ∈ [0,1] 𝑎𝑛𝑑 0 < 𝜀₁, 𝜀₂ < 1.
This neutrosophic fuzzy number is implemented in this model and thus the total cost obtain using this
neutrosophic number is
𝑑
𝜇
𝑇𝐶𝑛𝑒𝑢 (ℎ̃, 𝑐₁
̃ , 𝑘̃) = 𝑅𝜇 [( ) (𝑟𝜎(2𝑡1 − 𝜇) − (2𝑡2 − 𝜇)) + 𝑐₁
̃ 𝜎 (𝑡1 + 𝑇1 – 𝑡3 − ) + 𝑐3 (1 − 𝐵)(𝑡3 − 𝑡2 ) +
2
2
𝑐
𝜎𝑎𝑟 𝑐 𝑅 −𝑏
2
2−𝑏
( 2 ) [𝐵(𝑡3 − 𝑡2 )2 + (𝑟𝜎 − 1)(𝑇1 − 𝑡3 )2 ] + (
) [(𝑏 − 1)𝜇1−𝑏 + (2 − 𝑏)𝜇 −𝑏 (𝑡1 + 𝑇1 − 𝑡3 )] + ℎ̃ [(𝑟𝜎 − 1) {𝜉 +
𝑡1
𝛼{𝜇𝛽(𝑡1 −𝛾)𝛽+2 −𝑡1 (𝜇−𝛾)𝛽+2 +(−1)(𝛽) 𝛼𝛾𝛽+2 𝑡1 }
2
𝜇(𝛽+1)(𝛽+2)
( ) (𝑡1 − 𝜇) − (
𝜇
)+
(𝛼( 2 −𝛾))(𝑡1 −𝛾)𝛽+1
𝛽+1
}−(
(𝑡2 −𝑡1 )2
2
)+
Shilpi Pal, Avishek Chakraborty; Triangular Neutrosophic Based Production Reliability Model of Deteriorating Item
with Ramp Type Demand under Shortages and Time Discounting
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(
𝛼[(𝑡1 −𝛾)𝛽+1 (𝑡2 −𝛾)−(𝑡1 −𝛾)(𝑡2 −𝛾)𝛽+1 ]
𝛽+1
358
) + 𝑄𝑚 (𝑡2 − 𝑡1 − (
𝛼(𝑡2 −𝛾)𝛽+1
𝛽+1
) + 𝛼(𝑡1 − 𝛾)𝛽 ((
𝑡1 −𝛾
𝛽+1
̃
1−𝑒 −𝑘𝑚𝑇
) − 𝑡2 − 𝑡1 ))]] (
̃
1−𝑒 −𝑘𝑇
)
(22)
Using removal area technique (Chakraborty et. al. [3]) the de- neutrosophic numbers are
ℎ̃
𝑛𝑒𝑢𝐷 =
ℎ1 + ℎ2 + ℎ3 𝜀1 + 𝜀2
𝑐11 + 𝑐12 + 𝑐13 𝜀1 + 𝜀2
𝑘1 + 𝑘2 + 𝑘3 𝜀1 + 𝜀2
)𝑛𝑒𝑢𝐷 =
−
, (𝑐1̃
−
, 𝑎𝑛𝑑 𝑘̃
−
.
𝑛𝑒𝑢𝐷 =
3
4
3
4
3
4
So we substitute the value of
hneuD , (c1 )neuD and k neuD and obtain the total cost in neutrosophic
domain.
Thus by de-neutrosophication we get
𝑑
𝜇
𝑐
)𝑛𝑒𝑢𝐷 𝜎 (𝑡1 + 𝑇1 – 𝑡3 − ) + 𝑐3 (1 − 𝐵)(𝑡3 − 𝑡2 ) + ( 2 ) [𝐵(𝑡3 −
𝑇𝐶𝑛𝑒𝑢 (ℎ̃, 𝑐̃1 , 𝑘̃ ) = 𝑅𝜇 [( ) (𝑟𝜎(2𝑡1 − 𝜇) − (2𝑡2 − 𝜇)) + (𝑐1̃
2
2
𝑡2 )2 + (𝑟𝜎 − 1)(𝑇1 − 𝑡3 )2 ] + (
(
𝜎𝑎𝑟 𝑐 𝑅−𝑏
2−𝑏
𝑡1
) [(𝑏 − 1)𝜇1−𝑏 + (2 − 𝑏)𝜇−𝑏 (𝑡1 + 𝑇1 − 𝑡3 )] + ℎ̃
𝑛𝑒𝑢𝐷 [(𝑟𝜎 − 1) {𝜉 + ( ) (𝑡1 − 𝜇) −
𝛼{𝜇𝛽(𝑡1 −𝛾)𝛽+2 −𝑡1 (𝜇−𝛾)𝛽+2 +(−1)(𝛽) 𝛼𝛾𝛽+2 𝑡1 }
𝜇(𝛽+1)(𝛽+2)
𝑄𝑚 (𝑡2 − 𝑡1 − (
𝛼(𝑡2 −𝛾)𝛽+1
𝛽+1
2
2
𝜇
)+
(𝛼( −𝛾))(𝑡1 −𝛾)𝛽+1
2
) + 𝛼(𝑡1 − 𝛾)𝛽 ((
𝛽+1
𝑡1 −𝛾
𝛽+1
}−(
(𝑡2 −𝑡1 )2
2
)+(
𝛼[(𝑡1 −𝛾)𝛽+1 (𝑡2 −𝛾)−(𝑡1 −𝛾)(𝑡2 −𝛾)𝛽+1 ]
𝛽+1
)+
̃
) − 𝑡2 − 𝑡1 ))]] (
1−𝑒 −𝑘𝑛𝑒𝑢𝐷𝑚𝑇
̃
1−𝑒 −𝑘𝑛𝑒𝑢𝐷𝑇
)
(23)
5. Numerical Example
The model is illustrated by an example. A new brand item follows the demand rate as ramp type function of
time where the produced items are directly affected by reliability(r) of production process. The manufacturer
maintains the production rate 1.3 times the demand rate where demand factor is considered as 12 unit per
cycle. Also the items deteriorate with time is in the form of αβ(t − γ)β−1 , (where γ = 0.6 unit and α=0.001,β=1)
which cost 1$ per unit time. The purchase cost of the raw material of the item is 3.5$ per unit item and 100$ is
used for setting up for the production cycle. To hold the item in store the retailer has to pay 0.4$ per unit
item. During shortages, which cost 3.2$, let 0.75 fraction of stock demand get backordered as the rest sales are
lost. The cost for penalty (lost in sell) is 15$. The model is considered under 15 years of planning horizon with
various replenishment cycle i.e., m=2,3,4,5 and discounting rate of inflation as 12%.
Therefore, the data considered to illustrate the models are as follows:
𝑐₁ = 3.5, 𝑐₂ = 3.2, 𝑐₃ = 15, ℎ = 0.4, 𝑑 = 1, 𝐵 = 0.75, 𝐻 = 15, 𝑇 = 𝐻/𝑚, 𝜇 = 1.2, 𝜎 = 1.3, 𝛼 = 0.001, 𝛽 = 1, 𝛾
= 0.6, 𝑎 = 3, 𝑏 = 0.8, 𝑐 = 2, 𝑘 = 0.12, 𝑅 = 12, 𝑆 = 100.
Table 1: Optimal solution of inventory model for different replenishment
m
T in year
t1* in year
t2* in year
t3* in year
reliability (r*)
TC*
2
7.5
7.175
7.452
7.454
0.799
806.54
3
5
4.571
4.883
4.894
0.828
738.13
4
3.75
3.249
3.579
3.603
0.864
717.58
5*
3*
2.451*
2.789*
2.83*
0.909*
715.26*
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with Ramp Type Demand under Shortages and Time Discounting
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From the table 1 it is observed that the optimal solution is obtained (i.e., total cost is minimum) if we consider
short replenishment cycles. This is realistic because if we decrease the time of the production then it
produces less items and hence the total cost of the inventory decreases. It is also observed the better quality
items are produced at shorter replenishment cycle i.e., the reliability (r) of the items increases in shorter
production or replenishment cycle. This occurs because if we take small cycle then at the end of each cycle
their is maintenance in production system happens regularly and thus the reliability of the items increases.
Figure 2: Graphical presentation of production
cycle vs total cost
Figure 3: Graphical presentation of reliability
vs total cost.
We observe from the figure 2 that for smaller production cycle (i.e., for large value of m), the optimal total
cost (TC) decreases with optimal cost at m = 5.
In figure 3 we observe that the as reliability (r) increases then the total cost (TC) decreases. This holds because
as reliability increases the demand of the item in the market increases as a result the cost per unit item
decreases and hence the total cost decreases.
The above result is desirable because in the competitive market the business strategies of the manufacturer is
to work in small cycle and producing highly reliable items at less cost.
Figure 4: Graphical representation of total cost vs reliability and production time
Shilpi Pal, Avishek Chakraborty; Triangular Neutrosophic Based Production Reliability Model of Deteriorating Item
with Ramp Type Demand under Shortages and Time Discounting
Neutrosophic Sets and Systems, Vol. 35, 2020
360
Figure 4 gives the 3-dimensional plot of the total cost, reliability and no. of replenishment cycle in crisp
model. In this figure we observe that reliability (r) increases for large value of m where the total cost (TC)
decreases, i.e., highly reliable items are produced during small replenishment cycle at less cost, which is
desirable in producer-oriented EPQ model. This is obvious as, in small cycle, the machinery gets upgraded
and ameliorated eventually at the end of each cycle, and hence better quality of items are produced at much
faster rate and thus cost per unit items decreases and hence the total costing of the inventory decreases.
In reality few parameters are uncertain and thus there is a dilemma in decision maker's mind. Thus instead
of considering the model in crisp domain let us consider the model in neutrosophic domain and examine the
same example as above. Here we have considered purchase cost (c₁), holding cost (h) and inflation (k) as
triangular neutrosophic fuzzy number. Thus the neutrosophic numbers of the above parameters are
𝑘₁ =
0.125, 𝑘₂ = 0.118, 𝑘₃ = 0.132, ℎ₁ = 0.38, ℎ₂ = 0.4, ℎ₃ = 0.42, 𝑐₁₁ = 2.5, 𝑐₁₂ = 2.45. 𝑐₁₃ = 2.55, 𝜀₁ = 0.005, 𝜀₂ =
0.007.
𝑇ℎ𝑒𝑛, 𝑐₁
̃ =< (2.495,2.5,2.507), (2.445,2.45,2.457), (2.545,2.55,2.557) >,
̃ℎ =< (0.375,0.38,0.387), (0.395,0.4,0.407), (0.415,0.42,0.427) > 𝑎𝑛𝑑
̃𝑘 =< (0.12,0.125,0.132), (0.113,0.118,0.125), (0.127,0.132,0.139) >.
Thus we obtained table 2 under neutrosophic arena for the optimal solution of the model for different
replenishment cycle.
Table 2: Optimal time and cost of inventory model under neutrosophic domain
m
T in year
t1* in year
t2* in year
t3* in year
reliability (r*)
TC*
2
7.5
7.172
7.448
7.451
0.799
802.45
3
5
4.569
4.886
4.897
0.829
733
3.605
0.865
711.87
2.83
0.91
709.11*
4
5
*
3.75
3.247
3
2.449
*
3.58
*
2.789
*
*
*
Thus if we compare table 1 and table 2 it is observed that the total cost (TC) decreases if we consider the
model in neutrosophic arena. This is desirable as few parameters has hesitation factor in decision maker's
mind and thus this model under neutrosophic domain gives us better result.
6.
Sensitivity Analysis
The retailer should be aware of the effect in the total cost for any changes in the parameter. In order to
examine the implications of these changes, the sensitivity analysis will be helpful for decision-making. Using
the numerical example as given in the preceding section, we perform the sensitivity analysis by changing
few crisp parameters by -10%, -5%, 5% and 10% by taking one parameter at time and keeping the other
parameter fixed. As per Table 1 we observe that optimal solution is obtained when we consider small
replenishment cycle. So we perform the sensitivity analysis for m=5.
Shilpi Pal, Avishek Chakraborty; Triangular Neutrosophic Based Production Reliability Model of Deteriorating Item
with Ramp Type Demand under Shortages and Time Discounting
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Table 3. Sensitivity analysis of some parameters
Parameters
Change
t1* in year
t2*in year
t3* in year
(%)
c₁
h
S
σ
μ
k
R
reliability
TC*
% change
(r )
of TC*
*
-10
2.642
2.933
2.44
0.878
677.95
-5.5
-5
2.557
2.872
2.894
0.892
696.78
-2.65
5
2.313
2.677
2.748
0.932
733.29
2.46
10
2.122
2.517
2.64
0.968
750.71
4.72
-10
2.398
2.776
2.825
0.93
711.92
-0.47
-5
2.425
2.761
2.808
0.91
713.67
-0.22
5
2.475
2.796
2.834
0.9
716.7
0.2
10
2.498
2.803
2.838
0.892
718.01
0.38
-10
2.451
2.789
2.83
0.909
687.65
-4.02
-5
2.451
2.789
2.83
0.909
701.45
-1.97
5
2.451
2.789
2.83
0.909
729.06
1.89
10
2.451
2.789
2.83
0.909
742.87
3.72
-10
2.617
2.903
2.918
0.975
681.84
-4.9
-5
2.533
2.844
2.871
0.939
698.61
-2.38
5
2.364
2.728
2.786
0.883
731.72
2.25
10
2.264
2.66
2.742
0.865
747.95
4.37
-10
2.398
2.771
2.819
0.923
681.47
-4.96
-5
2.425
2.781
2.825
0.916
698.63
-2.38
5
2.475
2.798
2.836
0.903
731.35
2.2
10
2.498
2.805
2.841
0.897
746.92
4.24
-10
2.451
2.789
2.83
0.909
750.84
4.74
-5
2.451
2.789
2.83
0.909
732.63
2.37
5
2.451
2.789
2.83
0.909
698.69
-2.37
10
2.451
2.789
2.83
0.909
682.87
-4.74
-10
2.41
2.781
2.828
0.926
671.72
-6.48
-5
2.431
2.785
2.829
0.917
693.48
-3.14
5
2.471
2.794
2.832
0.901
737.05
2.96
10
2.49
2.799
2.834
0.894
758.86
5.74
From the above table 3 it is observed that the model is highly sensitive to purchase cost, demand rate factor
(R), moderately sensitive to setup cost, σ, μ, inflation (k) and less sensitive to holding cost. It is also noted
that the model is insensitive to the shortage cost, lost in sale cost and deterioration cost. That means
deterioration is not going to affect the model as much.
(i) The model is highly sensitive to purchase cost i.e., if we increase purchase cost (c₁), the total cost increases.
It is also noted that as the purchase cost increases, the reliability increases and production time decreases
which means if we buy good quality raw material then we have better quality of finished good at less
manufacturing time. Again the total cost TC increases with increase in demand factor R. This is obvious
Shilpi Pal, Avishek Chakraborty; Triangular Neutrosophic Based Production Reliability Model of Deteriorating Item
with Ramp Type Demand under Shortages and Time Discounting
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362
because if demand increases means more items are produced and hence the production time and production
cost also increases which leads to increase in total cost.
(ii) The model is moderately sensitive to set up cost (S), σ, μ, inflation (k). Investing more money for
upgradation of machineries, i.e., by increasing in set up cost (S), the total cost increases. It is noted that in our
model the set up cost does not depends on reliability and production time. Again with the increase in
production rate (σ) and production time (μ), the total cost increases. This is true because, if production time
increases then more items are produced also if we increase the production rate then we have more finished
good at less manufacturing time and thus in both the case the total cost increases. Also the toal cost decreases
with increase in inflation (k). This is obvious because with the increase in inflation the time value of money
increases and thus the total cost decreases in present day.
(iii) It is noticed that as the holding cost (h) is a less sensitive parameter. With the increase in holding cost, the
total cost increases. It is also observed that the production time also increases with increase in holding cost. It
means that the items has to be held for longer time with high value of holding cost then obviously the total
cost will increase.
It has been observed that there are various parameters which are very less sensitive hence it is not included
in the table.
7.
Concluding remarks
This paper developed an EPQ model for deteriorating item with reliability in production process and ramp
type demand rate under crisp and neutrosophic domain. The model also considers shortages where part of
the items gets backlogged and part of the sales are lost. The model coincides with practical situations since
we have considered the effect of time value of money under finite time horizon. Also the model optimizes by
considering the reliability of production process, as the reliability of production process increases, the total
cost decreases. This model is cost effective because highly reliable items are obtained at less cost and which is
desirable in managerial point of view. It is also observed that the highly reliable items are produced in small
cycles. The paper also compares the model under two different environment, crisp and neutrosophic, and it
is observed that the model works better in neutrosophic domain as compare to crisp environment. In this
paper we have done sensitivity analysis in crisp environment to illustrate our example and we have noted
that the minimum value of total cost is obtained for short replenishment cycle. This work could be extended
by considering multi-layer supply chain lot sizing model with manufacturer end, retailer end under
neutrosophic environment. Also we can extend this same model and can compare the model with
neutrosophic number and hybrid plithogenic decision-making method.
Further, in the forthcoming research, people can fruitfully execute and apply the idea of triangular
neutrosophic into distinct research arenas like structural modeling, diagnostic problems, realistic
modeling, recruitment based problems, pattern recognition etc.
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Received: Apr 18, 2020
Accepted: July 12, 2020.
Shilpi Pal, Avishek Chakraborty; Triangular Neutrosophic Based Production Reliability Model of Deteriorating Item
with Ramp Type Demand under Shortages and Time Discounting
Neutrosophic Sets and Systems, Vol. 35, 2020
University of New Mexico
Operations on Neutrosophic Vague Graphs
S. Satham Hussain1*, Saeid Jafari2, Said Broumi3 and N. Durga4
1Department
of Mathematics, Jamal Mohamed College, Trichy, Tamil Nadu, India.
E-mail: sathamhussain5592@gmail.
2College of Vestsjaelland South, Slagelse, Denmark and Mathematical and Physical Science Foundation, Slagelse, Denmark.
E-mail: jafaripersia@gmail.com
3Laboratory of Information Processing, Faculty of Science Ben MSik, University Hassan II, Casablanca, Morocco.
E-mail: broumisaid78@gmail.com
4Department of Mathematics, The Gandhigram Rural Institute (Deemed to be University), Gandhigram, Tamil Nadu, India.
E-mail: durga1992mdu@gmail.com
*Correspondence: S. Satham Hussain (sathamhussain5592@gmail.com)
Abstract: Neutrosophic graph is a mathematical tool to hold with imprecise and unspecified data. In
this manuscript, the operations on neutrosophic vague graphs are introduced. Moreover, Cartesian
product, lexicographic product, cross product, strong product and composition of neutrosophic
vague graphs are investigated. The proposed concepts are demonstrated with suitable examples.
Keywords: Neutrosophic vague graph, Operations of neutrosophic vague graph, Cartesian product,
Cross product, Strong product
1. Introduction
In a classical graph, any vertex or edge have two situations, namely, it is either in the graph or it
is not in the graph and it is not sufficient to model uncertain optimization problems. Therefore,
real-life problems are not suitable to model using classical graphs. Hence the fuzzy set arises, which
is an extension of classical set; here the objects have varying membership degrees. Vague sets are
regarded as a special case of context-dependent fuzzy sets. At first, vague set theory was
investigated by Gau and Buehrer [36] that is an extension of fuzzy set theory. The classical fuzzy set
handles only the membership degree, but intuitionistic fuzzy handles independent membership
degree and non-membership degree for any element with the only requirement is that the sum of
non-membership and membership degree values is not greater than one [16].
On the other hand, to hold this indeterminate and inconsistent information, the neutrosophic
set is introduced by F. Smarandache and has been studied extensively (see [31]-[35]). Neutrosophic
set and related notions have weird applications in many different fields. In the definition of
neutrosophic set, the indeterminacy value is quantified explicitly and truth-membership,
false-membership and indeterminacy-membership are stated as exactly independent provided sum
of these values belonging to 0 and 3. Neutrosophic soft rough graphs with applications are
established in [10]. Neutrosophic soft relations and neutrosophic refined relations with their
properties are studied in [15, 20]. Single valued neutrosophic graph are studied in [17, 18]. Some
types of neutrosophic graphs and co-neutrosophic graphs are discussed in [23]. Neutrosophic vague
set is first initiated in [11]. Al-Quran and Hassan in [7] introduced the notion of neutrosophic vague
soft expert set as a generalization of neutrosophic vague set and soft expert set in order to revise the
application in decision-making in real-life problems. Intuitionistic bipolar neutrosophic set and its
application to graphs are established in [28]. Further, neutrosophic vague graphs are investigated in
S. Satham Hussain , Saeid Jafari , Said Broumi and N. Durga “Operations on Neutrosophic Vague Graphs”
Neutrosophic Sets and Systems, Vol. 35, 2020
369
[27]. Motivated by the articles [11, 27, 28, 29], we introduce the concept of operations on
neutrosophic vague graphs. The main contributions in this manuscript are given below:
Operations on neutrosophic vague graphs are established. In Section 2, basic definitions
regarding to neutrosophic vague graphs are explained with an example.
In Section 3, Cartesian product, lexicographic product, cross product, strong product
and composition of neutrosophic vague graph are illustrated with examples. Finally, a
conclusion is elaborated with future direction.
2. Preliminaries
In this section, basic definitions and example are given, which is used to prove the main results.
Definition 2.1 [36] A vague set 𝔸 on a non empty set
𝕏 is a pair (𝕋𝔸 , 𝔽𝔸 ), where 𝕋𝔸 : 𝕏 → [0,1]
and 𝔽𝔸 : 𝕏 → [0,1] are true membership and false membership functions, respectively, such that
0 ≤ 𝕋𝔸 (x) + 𝔽𝔸 (x) ≤ 1 for every x ∈ 𝕏.
𝕏 and 𝕐 be two non-empty sets. A vague relation R of 𝕏 to 𝕐 is a vague set R on 𝕏 × 𝕐
that is R = (𝕋R , 𝔽R ), where 𝕋R : 𝕏 × 𝕐 → [0,1], 𝔽R : 𝕏 × 𝕐 → [0,1] and satisfies the condition:
0 ≤ 𝕋R (x, y) + 𝔽R (x, y) ≤ 1 for any x, y ∈ 𝕏.
Let
Definition 2.2 [12] Let 𝔾∗ = (𝕍, 𝔼) be a graph. A pair 𝔾 = (𝕁, 𝕂) is called a vague graph on 𝔾∗ ,
where 𝕁 = (𝕋𝕁 , 𝔽𝕁 ) is a vague set on 𝕍 and 𝕂 = (𝕋𝕂 , 𝔽𝕂 ) is a vague set on 𝔼 ⊆ 𝕍 × 𝕍 such that for
each xy ∈ 𝔼,
𝕋𝕂 (xy) ≤ min{𝕋𝕁 (x), 𝕋𝕁 (y)} and 𝔽𝕂 (xy) ≥ max{𝔽𝕁 (x), 𝔽𝕁 (y)}.
Definition 2.3 [31] A Neutrosophic set 𝔸 is contained in another neutrosophic set 𝔹, (i.e) 𝔸 ⊆ 𝔹 if
∀x ∈ 𝕏, 𝕋𝔸 (x) ≤ 𝕋𝔹 (x), 𝕀𝔸 (x) ≥ 𝕀𝔹 (x)and 𝔽𝔸 (x) ≥ 𝔽𝔹 (x).
Definition 2.4 [20, 31] Let 𝕏 be a space of points (objects), with generic elements in 𝕏 denoted by x.
A single valued neutrosophic set 𝔸 in 𝕏 is characterised by truth-membership function 𝕋𝔸 (x),
indeterminacy-membership function 𝕀𝔸 (x) and falsity-membership-function 𝔽𝔸 (x),
For each point x in 𝕏, 𝕋𝔸 (x), 𝕀𝔸 (x), 𝔽𝔸 (x) ∈ [0,1]. Also
𝔸 = {〈x, 𝕋𝔸 (x), 𝕀𝔸 (x), 𝔽𝔸 (x)〉} and 0 ≤ 𝕋𝔸 (x), +𝕀𝔸 (x) + 𝔽𝔸 (x) ≤ 3.
Definition 2.5 [6, 18] A neutrosophic graph is defined as a pair 𝔾∗ = (𝕍, 𝔼) where
(i) 𝕍 = {v1 , v2 , . . , vn } such that 𝕋1 ∶ 𝕍 → [0,1], 𝕀1 ∶ 𝕍 → [0,1] and 𝔽1 ∶ 𝕍 → [0,1] denote the
degree of truth-membership function, indeterminacy function and falsity-membership function,
respectively, and
0 ≤ 𝕋1 (v) + 𝕀1 (v) + 𝔽1 (v) ≤ 3,
(ii) 𝔼 ⊆ 𝕍 × 𝕍 where 𝕋2 ∶ 𝔼 → [0,1], 𝕀2 ∶ 𝔼 → [0,1] and 𝔽2 ∶ 𝔼 → [0,1] are such that
𝕋2 (uv) ≤ min{𝕋1 (u), 𝕋1 (v)},
𝕀2 (uv) ≤ min{𝕀1 (u), 𝕀1 (v)},
𝔽2 (uv) ≤ max{𝔽1 (u), 𝔽1 (v)}
and 0 ≤ 𝕋2 (uv) + 𝕀2 (uv) + 𝔽2 (uv) ≤ 3, ∀uv ∈ 𝔼.
Definition 2.6 [11] A Neutrosophic Vague Set 𝔸NV (NVS in short) on the universe of discourse 𝕏
written as
̂ 𝔸 (x), 𝕀̂𝔸 (x), 𝔽
̂ 𝔸 (x)〉, x ∈ 𝕏},
𝔸NV = {〈x, 𝕋
NV
NV
NV
whose truth-membership, indeterminacy membership and falsity-membership function are defined
as
S. Satham Hussain, Saeid Jafari, Said Broumi and N. Durga “Operations on Neutrosophic Vague Graphs”
Neutrosophic Sets and Systems, Vol. 35, 2020
370
̂ 𝔸 (x) = [𝔽− (x), 𝔽+ (x)],
̂ 𝔸 (x) = [𝕋− (x), 𝕋+ (x)], ̂𝕀𝔸 (x) = [𝕀− (x), 𝕀+ (x)]and 𝔽
𝕋
NV
NV
NV
where 𝕋+ (x) = 1 − 𝔽− (x), 𝔽+ (x) = 1 − 𝕋− (x), and 0 ≤ 𝕋− (x) + 𝕀− (x) + 𝔽− (x) ≤ 2.
Definition 2.7 [11] The complement of NVS 𝔸NV is denoted by 𝔸cNV and it is defined by
̂ c𝔸 (x) = [1 − 𝕋+ (x),1 − 𝕋− (x)],
𝕋
NV
𝕀̂c𝔸NV (x) = [1 − 𝕀+ (x),1 − 𝕀− (x)],
̂ c𝔸 (x) = [1 − 𝔽+ (x),1 − 𝔽− (x)].
𝔽
NV
Definition 2.8 [11] Let 𝔸NV and 𝔹NV be two NVSs of the universe 𝕌. If for all ui ∈ 𝕌,
̂ 𝔹 (ui ), ̂𝕀𝔸 (ui ) ≥ 𝕀̂𝔹 (ui ), 𝔽
̂ 𝔸 (ui ) ≥ 𝔽
̂ 𝔹 (ui ),
̂ 𝔸 (ui ) ≤ 𝕋
𝕋
NV
NV
NV
NV
NV
NV
then the NVS, 𝔸NV are included in 𝔹NV , denoted by 𝔸NV ⊆ 𝔹NV where 1 ≤ i ≤ n.
Definition 2.9 [11] The union of two NVSs , 𝔸NV and 𝔹NV , is a NVSs, 𝔻NV , written as 𝔻NV =
𝔸NV ∪ 𝔹NV whose
truth-membership
function,
indeterminacy-membership
function
and
false-membership function are related to those of 𝔸NV and 𝔹NV by
−
+
+
̂ 𝔻 (x) = [max(𝕋−
𝕋
𝔸NV (x), 𝕋𝔹NV (x)), max(𝕋𝔸NV (x), 𝕋𝔹NV (x))]
NV
−
+
+
̂𝕀𝔻 (x) = [min(𝕀−
𝔸NV (x), 𝕀𝔹NV (x)), min(𝕀𝔸NV (x), 𝕀𝔹NV (x))]
NV
−
+
+
̂ 𝔻 (x) = [min(𝔽−
𝔽
𝔸NV (x), 𝔽𝔹NV (x)), min(𝔽𝔸NV (x), 𝔽𝔹NV (x))].
NV
Definition 2.10 [11] The intersection of two NVSs, 𝔸NV and 𝔹NV is a NVSs, 𝔻NV , written as 𝔻NV =
𝔸NV ∩ 𝔹NV ,
whose truth-membership function, indeterminacy-membership function and
false-membership function are related to those of 𝔸NV and 𝔹NV by
−
+
+
̂ 𝔻 (x) = [min(𝕋−
𝕋
𝔸NV (x), 𝕋𝔹NV (x)), min(𝕋𝔸NV (x), 𝕋𝔹NV (x))]
NV
−
+
+
𝕀̂𝔻NV (x) = [max(𝕀−
𝔸NV (x), 𝕀𝔹NV (x)), max(𝕀𝔸NV (x), 𝕀𝔹NV (x))]
−
+
+
̂ 𝔻 (x) = [max(𝔽−
𝔽
𝔸NV (x), 𝔽𝔹NV (x)), max(𝔽𝔸NV (x), 𝔽𝔹NV (x))].
NV
Definition 2.11 [27] Let G∗ = (R, S) be a graph. A pair 𝔾 = (𝔸, 𝔹) is called a neutrosophic vague
̂ 𝔸 , ̂𝕀𝔸 , 𝔽
̂ 𝔸 ) is a neutrosophic vague
graph (NVG) on G∗ or a neutrosophic vague graph where 𝔸 = (𝕋
̂ 𝔹 )is a neutrosophic vague set S ⊆ R × R where
̂ 𝔹 , 𝕀̂𝔹 , 𝔽
set on R and 𝔹 = (𝕋
(1)
−
R = {v1 , v2 , . . . , vn } such that 𝕋𝔸− : R → [0,1], 𝕀−
𝔸 : R → [0,1], 𝔽𝔸 : R → [0,1] which satisfies the
+
condition 𝔽−
𝔸 = [1 − 𝕋𝔸 ],
+
+
+
−
𝕋+
𝔸 : R → [0,1], 𝕀𝔸 : R → [0,1], 𝔽𝔸 : R → [0,1] which satisfies the condition 𝔽𝔸 = [1 − 𝕋𝔸 ]
denotes the degree of truth membership function, indeterminacy membership and falsity
membership of the element vi ∈ R, and
−
−
0 ≤ 𝕋−
𝔸 (vi ) + 𝕀𝔸 (vi ) + 𝔽𝔸 (vi ) ≤ 2
+
+
0 ≤ 𝕋+
𝔸 (vi ) + 𝕀𝔸 (vi ) + 𝔽𝔸 (vi ) ≤ 2.
(2) S ⊆ R × R where
−
−
𝕋−
𝔹 : R × R → [0,1], 𝕀𝔹 : R × R → [0,1], 𝔽𝔹 : R × R → [0,1]
+
+
𝕋+
𝔹 : R × R → [0,1], 𝕀𝔹 : R × R → [0,1], 𝔽𝔹 : R × R → [0,1]
denotes the degree of truth membership function, indeterminacy membership and falsity
membership of the element vi vj ∈ S, respectively and such that,
−
−
0 ≤ 𝕋−
𝔹 (vi vj ) + 𝕀𝔹 (vi vj ) + 𝔽𝔹 (vi vj ) ≤ 2
0 ≤ 𝕋𝔹+ (vi vj ) + 𝕀𝔹+ (vi vj ) + 𝔽+
𝔹 (vi vj ) ≤ 2.
such that
S. Satham Hussain, Saeid Jafari, Said Broumi and N. Durga “Operations on Neutrosophic Vague Graphs”
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𝕋𝔹− (vi vj ) ≤ min{𝕋𝔸− (vi ), 𝕋𝔸− (vj )}
−
−
𝕀−
𝔹 (vi vj ) ≤ min{𝕀𝔸 (vi ), 𝕀𝔸 (vj )}
−
−
𝔽−
𝔹 (vi vj ) ≤ max{𝔽𝔸 (vi ), 𝔽𝔸 (vj )}
and similarly
𝕋𝔹+ (vi vj ) ≤ min{𝕋𝔸+ (vi ), 𝕋𝔸+ (vj )}
+
+
𝕀+
𝔹 (vi vj ) ≤ min{𝕀𝔸 (vi ), 𝕀𝔸 (vj )}
+
+
𝔽+
𝔹 (vi vj ) ≤ max{𝔽𝔸 (vi ), 𝔽𝔸 (vj )}.
Example 2.12 Consider a neutrosophic vague graph G = (R, S) such that 𝔸 = {a, b, c} and 𝔹 =
{ab, bc, ca} are defined by
â = T[0.5,0.6], I[0.4,0.3], F[0.4,0.5],
b̂ = T[0.4,0.6], I[0.7,0.3], F[0.4,0.6],
ĉ = T[0.4,0.4], I[0.5,0.3], F[0.6,0.6]
a− = (0.5,0.4,0.4), b− = (0.4,0.7,0.4), c − = (0.4,0.5,0.6)
𝐚+ = (𝟎. 𝟔, 𝟎. 𝟑, 𝟎. 𝟓), 𝐛+ = (𝟎. 𝟔, 𝟎. 𝟑, 𝟎. 𝟔), 𝐜 + = (𝟎. 𝟒, 𝟎. 𝟑, 𝟎. 𝟔).
𝐹𝑖𝑔𝑢𝑟𝑒 1: NEUTROSOPHIC VAGUE GRAPH
3. Operations on Neutrosophic Vague Graphs
In this section, the results on operations of neutrosophic vague graphs with example are established.
Definition 3.1 The Cartesian product of two NVGs G1 and G2 is denoted by the pair G1 × G2 =
(R1 × R 2 , S1 × S2 ) and defined as
TA−1×A2 (kl) = TA−1 (k) ∧ TA−2 (l)
IA−1×A2 (kl) = IA−1 (k) ∧ IA−2 (l)
FA−1×A2 (kl) = FA−1 (k) ∨ FA−2 (l)
TA+1×A2 (kl) = TA+1 (k) ∧ TA+2 (l)
IA+1×A2 (kl) = IA+1 (k) ∧ IA+2 (l)
FA+1×A2 (kl) = FA+1 (k) ∨ FA+2 (l),
S. Satham Hussain, Saeid Jafari, Said Broumi and N. Durga “Operations on Neutrosophic Vague Graphs”
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for all (k, l) ∈ R1 × R 2 .
The membership value of the edges in G1 × G2 can be calculated as,
(1) TB−1×B2 (kl1 )(kl2 ) = TA−1 (k) ∧ TB−2 (l1 l2 )
TB+1×B2 (kl1 )(kl2 ) = TA+1 (k) ∧ TB+2 (l1 l2 ),
(2) IB−1×B2 (kl1 )(kl2 ) = IA−1 (k) ∧ IB−2 (l1 l2 )
IB+1 ×B2 (kl1 )(kl2 ) = IA+1 (k) ∧ IB+2 (l1 l2 ),
(3) FB−1×B2 (kl1 )(kl2 ) = FA−1 (k) ∨ FB−2 (l1 l2 )
FB+1×B2 (kl1 )(kl2 ) = FA+1 (k) ∨ FB+2 (l1 l2 ),
for all k ∈ R1 , l1 l2 ∈ S2 .
(4) TB−1×B2 (k1 l)(k 2 l) = TA−2 (l) ∧ TB−2 (k1 k 2 )
TB+1×B2 (k1 l)(k 2 l) = TA+2 (l) ∧ TB+2 (k1 k 2 ),
(5) IB−1 ×B2 (k1 l)(k 2 l) = IA−2 (l) ∧ IB−2 (k1 k 2 )
IB+1 ×B2 (k1 l)(k 2 l) = IA+2 (l) ∧ IB+2 (k1 k 2 ),
(6) FB−1×B2 (k1 l)(k 2 l) = FA−2 (l) ∨ FB−2 (k1 k 2 )
FB+1×B2 (k1 l)(k 2 l) = FA+2 (l) ∨ FB+2 (k1 k 2 ),
for all k1 k 2 ∈ S1 , l ∈ R 2 .
Example 3.2 Consider G1 = (R1 , S1 ) and G2 = (R 2 , S2 ) are two NVGs of G = (R, S), as represented
in Figure 2, now we get G1 × G2 as follows see Figure 3.
k̂1 = T[0.5,0.6], I[0.6,0.4], F[0.4,0.5], k̂ 2 = T[0.4,0.6], I[0.7,0.3], F[0.4,0.6],
k̂ 3 = T[0.6,0.4], I[0.3,0.7], F[0.6,0.4],k̂ 4 = T[0.4,0.4], I[0.4,0.6], F[0.6,0.6]
̂l1 = T[0.4,0.4], I[0.5,0.3], F[0.6,0.6], ̂l2 = T[0.5,0.6], I[0.4,0.3], F[0.4,0.5],
̂l3 = T[0.4,0.6], I[0.7,0.3], F[0.4,0.6]
−
−
k1− = (0.5,0.6,0.4), k −
2 = (0.4,0.7,0.4), k 3 = (0.6,0.3,0.6),k 4 = (0.4,0.4,0.6)
−
+
k1+ = (0.6,0.4,0.5), k +
2 = (0.6,0.3,0.6), k 3 = (0.4,0.7,0.4),k 4 = (0.4,0.6,0.6)
−
l1− = (0.4,0.5,0.6), l−
2 = (05,0.4,0.4), l3 = (0.4,0.7,0.4)
+
+
𝐥+
𝟏 = (𝟎. 𝟒, 𝟎. 𝟑, 𝟎. 𝟔), 𝐥𝟐 = (𝟎. 𝟔, 𝟎. 𝟑, 𝟎. 𝟓), 𝐥𝟑 = (𝟎. 𝟔, 𝟎. 𝟑, 𝟎. 𝟔).
S. Satham Hussain, Saeid Jafari, Said Broumi and N. Durga “Operations on Neutrosophic Vague Graphs”
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𝑮𝟏
𝑮𝟐
Figure 2: NEUTROSOPHIC VAGUE GRAPH
S. Satham Hussain, Saeid Jafari, Said Broumi and N. Durga “Operations on Neutrosophic Vague Graphs”
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Figure 3: CARTESIAN PRODUCT OF NEUTROSOPHIC VAGUE GRAPH
Theorem 3.3 The Cartesian product G1 × G2 = (R1 × R 2 , S1 × S2 ) of two NVG G1 and G2 is also the
NVG of G1 × G2 .
Proof. We consider two cases.
Case 1: for k ∈ R1 , l1 l2 ∈ S2 ,
̂A (k) ∧ T
̂B (l1 l2 )
̂(B ×B ) ((kl1 )(kl2 )) = T
T
1
2
1
2
̂A (l1 ) ∧ T
̂A (l2 )]
̂A (k) ∧ [T
≤T
1
2
2
̂A (l1 )] ∧ [T
̂A (k) ∧ T
̂A (l2 )]
̂A (k) ∧ T
= [T
1
2
1
2
̂(A ×A ) (k, l2 )
̂(A ×A ) (k, l1 ) ∧ T
=T
1
2
1
2
Î(B1×B2 ) ((kl1 )(kl2 )) = ÎA1 (k) ∧ ÎB2 (l1 l2 )
≤ ÎA1 (k) ∧ [ÎA2 (l1 ) ∧ ÎA2 (l2 )]
= [ÎA1 (k) ∧ ÎA2 (l1 )] ∧ [ÎA1 (k) ∧ ÎA2 (l2 )]
= Î(A1×A2) (k, l1 ) ∧ Î(A1×A2) (k, l2 )
F̂(B1×B2 ) ((kl1 )(kl2 )) = F̂A1 (k) ∨ F̂B2 (l1 l2 )
≤ F̂A1 (k) ∨ [F̂A2 (l1 ) ∨ F̂A2 (l2 )]
= [F̂A1 (k) ∨ F̂A2 (l1 )] ∨ [F̂A1 (k) ∨ F̂A2 (l2 )]
= F̂(A1×A2) (k, l1 ) ∨ F̂(A1×A2) (k, l2 )
for all kl1 , kl2 ∈ G1 × G2 .
Case 2: for k ∈ R 2 , l1 l2 ∈ S1 .
̂A (k) ∧ T
̂B (l1 l2 )
̂(B ×B ) ((l1 k)(l2 k)) = T
T
1
2
2
1
̂A (l1 ) ∧ T
̂A (l2 )]
̂A (k) ∧ [T
≤T
2
1
1
̂A (l1 )] ∧ [T
̂A (k) ∧ T
̂A (l2 )]
̂A (k) ∧ T
= [T
2
1
2
1
̂(A ×A ) (l2 , k)
̂(A ×A ) (l1 , k) ∧ T
=T
1
2
1
2
Î(B1×B2 ) ((l1 k)(l2 k)) = ÎA2 (k) ∧ ÎB1 (l1 l2 )
S. Satham Hussain, Saeid Jafari, Said Broumi and N. Durga “Operations on Neutrosophic Vague Graphs”
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375
≤ ÎA2 (k) ∧ [ÎA1 (l1 ) ∧ ÎA1 (l2 )]
= [ÎA2 (k) ∧ ÎA1 (l1 )] ∧ [ÎA2 (k) ∧ ÎA1 (l2 )]
= Î(A1×A2) (l1 , k) ∧ Î(A1×A2) (l2 , k)
F̂(B1×B2 ) ((l1 k)(l2 k)) = F̂A2 (k) ∨ F̂B1 (l1 l2 )
≤ F̂A2 (k) ∨ [F̂A1 (l1 ) ∨ F̂A1 (l2 )]
= [F̂A2 (k) ∨ F̂A1 (l1 )] ∨ [F̂A2 (k) ∨ F̂A1 (l2 )]
= F̂(A1×A2) (l1 , k) ∨ F̂(A1×A2) (l2 , k)
for all l1 k, l2 k ∈ G1 × G2 .
Definition 3.4 The Cross product of two NVGs G1 and G2 is denoted by the pair G1 ⋆ G2 = (R1 ⋆
R 2 , S1 ⋆ S2 ) and is defined as
(i)TA−1⋆A2 (kl) = TA−1 (k) ∧ TA−2 (l)
IA−1⋆A2 (kl) = IA−1 (k) ∧ IA−2 (l)
FA−1 ⋆A2 (kl) = FA−1 (k) ∨ FA−2 (l)
TA+1 ⋆A2 (kl) = TA+1 (k) ∧ TA+2 (l)
IA+1⋆A2 (kl) = IA+1 (k) ∧ IA+2 (l)
FA+1 ⋆A2 (kl) = FA+1 (k) ∨ FA+2 (l),
for all k, l ∈ R1 ⋆ R 2.
−
(ii)T(B
(k1 l1 )(k 2 l2 ) = TB−1 (k1 k 2 ) ∧ TB−2 (l1 l2 )
1 ⋆B2 )
−
(k1 l1 )(k 2 l2 ) = IB−1 (k1 k 2 ) ∧ IB−2 (l1 l2 )
I(B
1 ⋆B2 )
−
(k1 l1 )(k 2 l2 ) = FB−1 (k1 k 2 ) ∨ FB−2 (l1 l2 )
F(B
1 ⋆B2 )
+
(k1 l1 )(k 2 l2 ) = TB+1 (k1 k 2 ) ∧ TB+2 (l1 l2 )
(iii)T(B
1 ⋆B2 )
+
(k1 l1 )(k 2 l2 ) = IB+1 (k1 k 2 ) ∧ IB+2 (l1 l2 )
I(B
1 ⋆B2 )
+
(k1 l1 )(k 2 l2 ) = FB+1 (k1 k 2 ) ∨ FB+2 (l1 l2 ),
F(B
1 ⋆B2 )
for all k1 k 2 ∈ S1 , l1 l2 ∈ S2 .
Example 3.5 Consider 𝐆𝟏 = (𝐑 𝟏 , 𝐒𝟏 ) and 𝐆𝟐 = (𝐑 𝟐 , 𝐒𝟐 ) as two NVG of 𝐆 = (𝐑, 𝐒) respectively, (see
Figure 2). We obtain the cross product of 𝐆𝟏 ⋆ 𝐆𝟐 as follows (see Figure 4).
Figure 4: CROSS PRODUCT OF NEUTROSOPHIC VAGUE GRAPH
S. Satham Hussain, Saeid Jafari, Said Broumi and N. Durga “Operations on Neutrosophic Vague Graphs”
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Theorem 3.6 The cross product G1 ⋆ G2 = (R1 ⋆ R 2 , S1 ⋆ S2 ) of two NVG G1 and G2 is an the NVG
of G1 ⋆ G2 .
Proof. For all k1 l1 , k 2 l2 ∈ G1 ⋆ G2
̂B (k1 k 2 ) ∧ T
̂B (l1 l2 )
̂(B ⋆B ) ((k1 l1 )(k 2 l2 )) = T
T
1 2
1
2
̂A (k 2 )] ∧ [T
̂A (l1 ) ∧ T
̂A (l2 )]
̂A (k1 ) ∧ T
≤ [T
1
1
2
2
̂A (l1 )] ∧ [T
̂A (k 2 ) ∧ T
̂A (l2 )]
̂A (k1 ) ∧ T
= [T
1
2
1
2
̂(A ⋆A ) (k 2 , l2 )
̂(A ⋆A ) (k1 l1 ) ∧ T
=T
1 2
1 2
Î(B1⋆B2 ) ((k1 l1 )(k 2 l2 )) = ÎB1 (k1 k 2 ) ∧ ÎB2 (l1 l2 )
≤ [ÎA1 (k1 ) ∧ ÎA1 (k 2 )] ∧ [ÎA2 (l1 ) ∧ ÎA2 (l2 )]
= [ÎA1 (k1 ) ∧ ÎA2 (l1 )] ∧ [ÎA1 (k 2 ) ∧ ÎA2 (l2 )]
= Î(A1⋆A2) (k1 l1 ) ∧ Î(A1⋆A2) (k 2 , l2 )
F̂(B1⋆B2 ) ((k1 l1 )(k 2 l2 )) = F̂B1 (k1 k 2 ) ∨ F̂B2 (l1 l2 )
≤ [F̂A1 (k1 ) ∨ F̂A1 (k 2 )] ∨ [F̂A2 (l1 ) ∨ F̂A2 (l2 )]
= [F̂A1 (k1 ) ∨ F̂A2 (l1 )] ∨ [F̂A1 (k 2 ) ∨ F̂A2 (l2 )]
= F̂(A1⋆A2) (k1 l1 ) ∨ F̂(A1⋆A2) (k 2 , l2 ).
This completes the proof.
Definition 3.7 The lexicographic product of two NVGs G1 and G2 is denoted by the pair G1 • G2 =
(R1 • R 2 , S1 • S2 ) and defined as
−
(i)T(A
(kl) = TA−1 (k) ∧ TA−2 (l)
1 •A2 )
−
(kl) = IA−1 (k) ∧ IA−2 (l)
I(A
1 •A2 )
−
(kl) = FA−1 (k) ∨ FA−2 (l)
F(A
1 •A2 )
+
(kl) = TA+1 (k) ∧ TA+2 (l)
T(A
1 •A2 )
+
(kl) = IA+1 (k) ∧ IA+2 (l)
I(A
1 •A2 )
+
(kl) = FA+1 (k) ∨ FA+2 (l),
F(A
1 •A2 )
for all kl ∈ R1 × R 2
−
(ii)T(B
(kl1 )(kl2 ) = TA−1 (k) ∧ TB−2 (l1 l2 )
1 •B2 )
−
(kl1 )(kl2 ) = IA−1 (k) ∧ IB−2 (l1 l2 )
I(B
1 •B2 )
−
(kl1 )(kl2 ) = FA−1 (k) ∨ FB−2 (l1 l2 )
F(B
1 •B2 )
+
(kl1 )(kl2 ) = TA+1 (k) ∧ TB+2 (l1 l2 )
T(B
1 •B2 )
+
(kl1 )(kl2 ) = IA+1 (k) ∧ IB+2 (l1 l2 )
I(B
1 •B2 )
+
(kl1 )(kl2 ) = FA+1 (k) ∨ FB+2 (l1 l2 ),
F(B
1 •B2 )
for all k ∈ R1 , l1 l2 ∈ S2 .
−
(iii)T(B
(k1 l1 )(k 2 l2 ) = TB−1 (k1 k 2 ) ∧ TB−2 (l1 l2 )
1 •B2 )
−
(k1 l1 )(k 2 l2 ) = IB−1 (k1 k 2 ) ∧ IB−2 (l1 l2 )
I(B
1 •B2 )
−
(k1 l1 )(k 2 l2 ) = FB−1 (k1 k 2 ) ∨ FB−2 (l1 l2 )
F(B
1 •B2 )
+
(k1 l1 )(k 2 l2 ) = TB+1 (k1 k 2 ) ∧ TB+2 (l1 l2 )
T(B
1 •B2 )
+
(k1 l1 )(k 2 l2 ) = IB+1 (k1 k 2 ) ∧ IB+2 (l1 l2 )
I(B
1 •B2 )
+
F(B
(k1 l1 )(k 2 l2 ) = FB+1 (k1 k 2 ) ∨ FB+2 (l1 l2 ), for all k1 k 2 ∈ S1 , l1 l2 ∈ S2 .
1 •B2 )
S. Satham Hussain, Saeid Jafari, Said Broumi and N. Durga “Operations on Neutrosophic Vague Graphs”
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Example 3.8 The lexicographic product of NVG G1 = (R1 , S1 ) and G2 = (R 2 , S2 ) shown in Figure 2
is defined as G1 • G2 = (R1 • R 2 , S1 • S2 ) and is presented in Figure 5.
Figure 5: LEXICOGRAPHIC PRODUCT OF NEUTROSOPHIC VAGUE GRAPH
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Theorem 3.9 The lexicographic product G1 • G2 = (R1 • R 2 , S1 • S2 ) of two NVG G1 and G2 is the
NVG of G1 • G2 .
Proof. We have two cases.
Case 1: For k ∈ R1 , l1 l2 ∈ S2 ,
̂A (k) ∧ T
̂B (l1 l2 )
̂(B •B ) ((kl1 )(kl2 )) = T
T
1 2
1
2
̂A (l1 ) ∧ T
̂A (l2 )]
̂A (k) ∧ [T
≤T
1
2
2
̂A (l1 )] ∧ [T
̂A (k) ∧ T
̂A (l2 )]
̂A (k) ∧ T
= [T
1
2
1
2
̂(A •A ) (k, l2 )
̂(A •A ) (k, l1 ) ∧ T
=T
1 2
1 2
Î(B1•B2) ((kl1 )(kl2 )) = ÎA1 (k) ∧ ÎB2 (l1 l2 )
≤ ÎA1 (k) ∧ [ÎA2 (l1 ) ∧ ÎA2 (l2 )]
= [ÎA1 (k) ∧ ÎA2 (l1 )] ∧ [ÎA1 (k) ∧ ÎA2 (l2 )]
= Î(A1•A2) (k, l1 ) ∧ Î(A1•A2) (k, l2 )
F̂(B1•B2 ) ((kl1 )(kl2 )) = F̂A1 (k) ∨ F̂B2 (l1 l2 )
≤ F̂A1 (k) ∨ [F̂A2 (l1 ) ∨ F̂A2 (l2 )]
= [F̂A1 (k) ∨ F̂A2 (l1 )] ∨ [F̂A1 (k) ∨ F̂A2 (l2 )]
= F̂(A1•A2) (k, l1 ) ∨ F̂(A1•A2) (k, l2 )
for all kl1 , kl2 ∈ S1 × S2 .
Case 2: For all k1 l1 ∈ S1 , k 2 l2 ∈ S2 ,
̂B (k1 k 2 ) ∧ T
̂B (l1 l2 )
̂(B •B ) ((k1 l1 )(k 2 l2 )) = T
T
1 2
1
2
̂A (k 2 )] ∧ [T
̂A (l1 ) ∧ T
̂A (l2 )]
̂A (k1 ) ∧ T
≤ [T
1
1
2
2
̂A (l1 )] ∧ [T
̂A (k 2 ) ∧ T
̂A (l2 )]
̂A (k1 ) ∧ T
= [T
1
2
1
2
̂(A •A ) (k 2 , l2 )
̂(A •A ) (k1 l1 ) ∧ T
=T
1 2
1 2
Î(B1•B2) ((k1 l1 )(k 2 l2 )) = ÎB1 (k1 k 2 ) ∧ ÎB2 (l1 l2 )
≤ [ÎA1 (k1 ) ∧ ÎA1 (k 2 )] ∧ [ÎA2 (l1 ) ∧ ÎA2 (l2 )]
= [ÎA1 (k1 ) ∧ ÎA2 (l1 )] ∧ [ÎA1 (k 2 ) ∧ ÎA2 (l2 )]
= Î(A1•A2) (k1 l1 ) ∧ Î(A1•A2) (k 2 , l2 )
F̂(B1•B2 ) ((k1 l1 )(k 2 l2 )) = F̂B1 (k1 k 2 ) ∨ F̂B2 (l1 l2 )
≤ [F̂A1 (k1 ) ∨ F̂A1 (k 2 )] ∨ [F̂A2 (l1 ) ∨ F̂A2 (l2 )]
= [F̂A1 (k1 ) ∨ F̂A2 (l1 )] ∨ [F̂A1 (k 2 ) ∨ F̂A2 (l2 )]
= F̂(A1•A2) (k1 l1 ) ∨ F̂(A1•A2) (k 2 , l2 )
for all k1 , l1 ∈ k 2 , l2 ∈ R1 • R 2 .
Definition 3.10 The strong product of two NVG G1 and G2 is denoted by the pair G1 ⊠ G2 = (R1 ⊠
R 2 , S1 ⊠ S2 ) and defined as
−
(i)T(A
(kl) = TA−1 (k) ∧ TA−2 (l)
1 ⊠A2 )
−
(kl) = IA−1 (k) ∧ IA−2 (l)
I(A
1 ⊠A2 )
−
F(A
(kl) = FA−1 (k) ∨ FA−2 (l)
1 ⊠A2 )
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+
T(A
(kl) = TA+1 (k) ∧ TA+2 (l)
1 ⊠A2 )
+
(kl) = IA+1 (k) ∧ IA+2 (l)
I(A
1 ⊠A2 )
+
(kl) = FA+1 (k) ∨ FA+2 (l)
F(A
1 ⊠A2 )
for all kl ∈ R1 ⊠ R 2
−
(ii)T(B
(kl1 )(kl2 ) = TA−1 (k) ∧ TB−2 (l1 l2 )
1 ⊠B2 )
−
(kl1 )(kl2 ) = IA−1 (k) ∧ IB−2 (l1 l2 )
I(B
1 ⊠B2 )
−
(kl1 )(kl2 ) = FA−1 (k) ∨ FB−2 (l1 l2 )
F(B
1 ⊠B2 )
+
(kl1 )(kl2 ) = TA+1 (k) ∧ TB+2 (l1 l2 )
T(B
1 ⊠B2 )
+
(kl1 )(kl2 ) = IA+1 (k) ∧ IB+2 (l1 l2 )
I(B
1 ⊠B2 )
+
(kl1 )(kl2 ) = FA+1 (k) ∨ FB+2 (l1 l2 ),
F(B
1 ⊠B2 )
for all k ∈ R1 , l1 l2 ∈ S2 .
(iii)TB−1 ⊠B2 (k1 l)(k 2 l) = TA−2 (l) ∧ TB−2 (k1 k 2 )
IB−1 ⊠B2 (k1 l)(k 2 l) = IA−2 (l) ∧ IB−2 (k1 k 2 )
FB−1⊠B2 (k1 l)(k 2 l) = FA−2 (l) ∨ FB−2 (k1 k 2 )
TB+1⊠B2 (k1 l)(k 2 l) = TA+2 (l) ∧ TB+2 (k1 k 2 )
IB+1 ⊠B2 (k1 l)(k 2 l) = IA+2 (l) ∧ IB+2 (k1 k 2 )
FB+1⊠B2 (k1 l)(k 2 l) = FA+2 (l) ∨ FB+2 (k1 k 2 ),
for all k1 k 2 ∈ S1 , l ∈ R 2 .
−
(iv)T(B
(k1 l1 )(k 2 l2 ) = TB−1 (k1 k 2 ) ∧ TB−2 (l1 l2 )
1 ⊠B2 )
−
(k1 l1 )(k 2 l2 ) = IB−1 (k1 k 2 ) ∧ IB−2 (l1 l2 )
I(B
1 ⊠B2 )
−
(k1 l1 )(k 2 l2 ) = FB−1 (k1 k 2 ) ∨ FB−2 (l1 l2 )
F(B
1 ⊠B2 )
+
(k1 l1 )(k 2 l2 ) = TB+1 (k1 k 2 ) ∧ TB+2 (l1 l2 )
T(B
1 ⊠B2 )
+
(k1 l1 )(k 2 l2 ) = IB+1 (k1 k 2 ) ∧ IB+2 (l1 l2 )
I(B
1 ⊠B2 )
+
(k1 l1 )(k 2 l2 ) = FB+1 (k1 k 2 ) ∨ FBN2 (l1 l2 ),
F(B
1 ⊠B2 )
for all k1 k 2 ∈ S1 , l1 l2 ∈ S2 .
Example 3.11 The strong product of NVG G1 = (R1 , S1 ) and G2 = (R 2 , S2 ) shown in Figure 2 is
defined as G1 ⊠ G2 = (S1 ⊠ S2 , T1 ⊠ T2 ) and is presented in Figure 6.
S. Satham Hussain, Saeid Jafari, Said Broumi and N. Durga “Operations on Neutrosophic Vague Graphs”
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Figure 6: STRONG PRODUCT OF NEUTROSOPHIC VAGUE GRAPH
Theorem 3.12 The strong product G1 ⊠ G2 = (R1 ⊠ R 2 , S1 ⊠ S2 ) of two NVG G1 and G2 is a NVG
of G1 ⊠ G2 .
Proof. There are three cases:
Case 1: for k ∈ R1 , l1 l2 ∈ S2 ,
̂A (k) ∧ T
̂B (l1 l2 )
̂(B ⊠B ) ((kl1 )(kl2 )) = T
T
1
2
1
2
̂A (l1 ) ∧ T
̂A (l2 )]
̂A (k) ∧ [T
≤T
1
2
2
̂A (l1 )] ∧ [T
̂A (k) ∧ T
̂A (l2 )]
̂A (k) ∧ T
= [T
1
2
1
2
̂(A ⊠A ) (k, l2 )
̂(A ⊠A ) (k, l1 ) ∧ T
=T
1
2
1
2
Î(B1⊠B2 ) ((kl1 )(kl2 )) = ÎA1 (k) ∧ ÎB2 (l1 l2 )
≤ ÎA1 (k) ∧ [ÎA2 (l1 ) ∧ ÎA2 (l2 )]
= [ÎA1 (k) ∧ ÎA2 (l1 )] ∧ [ÎA1 (k) ∧ ÎA2 (l2 )]
= Î(A1⊠A2) (k, l1 ) ∧ Î(A1⊠A2) (k, l2 )
F̂(B1⊠B2) ((kl1 )(kl2 )) = F̂A1 (k) ∨ F̂B2 (l1 l2 )
≤ F̂A1 (k) ∨ [F̂A2 (l1 ) ∨ F̂A2 (l2 )]
= [F̂A1 (k) ∨ F̂A2 (l1 )] ∨ [F̂A1 (k) ∨ F̂A2 (l2 )]
= F̂(A1⊠A2) (k, l1 ) ∨ F̂(A1⊠A2) (k, l2 ),
for all kl1 , kl2 ∈ R1 ⊠ R 2 .
Case 2: for k ∈ R 2 , l1 l2 ∈ S1 ,
̂(B ⊠B ) ((l1 k)(l2 k)) = T
̂A (k) ∧ T
̂B (l1 l2 )
T
1
2
2
1
S. Satham Hussain, Saeid Jafari, Said Broumi and N. Durga “Operations on Neutrosophic Vague Graphs”
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̂A (k) ∧ [T
̂A (l1 ) ∧ T
̂A (l2 )]
≤T
2
1
1
̂A (l1 )] ∧ [T
̂A (k) ∧ T
̂A (l2 )]
̂A (k) ∧ T
= [T
2
1
2
1
̂(A ⊠A ) (l2 , k)
̂(A ⊠A ) (l1 , k) ∧ T
=T
1
2
1
2
Î(B1⊠B2 ) ((l1 k)(l2 k)) = ÎA2 (k) ∧ ÎB1 (l1 l2 )
≤ ÎA2 (k) ∧ [ÎA1 (l1 ) ∧ ÎA1 (l2 )]
= [ÎA2 (k) ∧ ÎA1 (l1 )] ∧ [ÎA2 (k) ∧ ÎA1 (l2 )]
= Î(A1⊠A2) (l1 , k) ∧ Î(A1⊠A2) (l2 , k)
F̂(B1⊠B2) ((l1 k)(l2 k)) = F̂A2 (k) ∨ F̂B1 (l1 l2 )
≤ F̂A2 (k) ∨ [F̂A1 (l1 ) ∨ F̂A1 (l2 )]
= [F̂A2 (k) ∨ F̂A1 (l1 )] ∨ [F̂A2 (k) ∨ F̂A1 (l2 )]
= F̂(A1⊠A2) (l1 , k) ∨ F̂(A1⊠A2) (l2 , k)
for all l1 k, l2 k ∈ R1 ⊠ R 2 .
Case 3: for k1 , k 2 ∈ S1 , l1 l2 ∈ S2
̂B (k1 k 2 ) ∧ T
̂B (l1 l2 )
̂(B ⊠B ) ((k1 l1 )(k 2 l2 )) = T
T
1
2
1
2
̂A (k 2 )] ∧ [T
̂A (l1 ) ∧ T
̂A (l2 )]
̂A (k1 ) ∧ T
≤ [T
1
1
2
2
̂A (l1 )] ∧ [T
̂A (k 2 ) ∧ T
̂A (l2 )]
̂A (k1 ) ∧ T
= [T
1
2
1
2
̂(A ⊠A ) (k 2 , l2 )
̂(A ⊠A ) (k1 l1 ) ∧ T
=T
1
2
1
2
Î(B1⊠B2 ) ((k1 l1 )(k 2 l2 )) = ÎB1 (k1 k 2 ) ∧ ÎB2 (l1 l2 )
≤ [ÎA1 (k1 ) ∧ ÎA1 (k 2 )] ∧ [ÎA2 (l1 ) ∧ ÎA2 (l2 )]
= [ÎA1 (k1 ) ∧ ÎA2 (l1 )] ∧ [ÎA1 (k 2 ) ∧ ÎA2 (l2 )]
= Î(A1⊠A2) (k1 l1 ) ∧ Î(A1⊠A2) (k 2 , l2 )
F̂(B1⊠B2) ((k1 l1 )(k 2 l2 )) = F̂B1 (k1 k 2 ) ∨ F̂B2 (l1 l2 )
≤ [F̂A1 (k1 ) ∨ F̂A1 (k 2 )] ∨ [F̂A2 (l1 ) ∨ F̂A2 (l2 )]
= [F̂A1 (k1 ) ∨ F̂A2 (l1 )] ∨ [F̂A1 (k 2 ) ∨ F̂A2 (l2 )]
= F̂(A1⊠A2) (k1 l1 ) ∨ F̂(A1⊠A2) (k 2 , l2 ),
for all l1 k1 , l2 k1 ∈ R1 ⊠ R 2 .
Definition 3.13 The composition of two NVG G1 and G2 is denoted by the pair G1 ∘ G2 = (R1 ⊠
R 2 , S1 ∘ S2 ) and defined as
−
(i)T(A
(kl) = TA−1 (k) ∧ TA−2 (l)
1 ∘A2 )
−
(kl) = IA−1 (k) ∧ IA−2 (l)
I(A
1 ∘A2 )
−
(kl) = FA−1 (k) ∨ FA−2 (l)
F(A
1 ∘A2 )
+
(kl) = TA+1 (k) ∧ TA+2 (l)
T(A
1 ∘A2 )
+
(kl) = IA+1 (k) ∧ IA+2 (l)
I(A
1 ∘A2 )
+
(kl) = FA+1 (k) ∨ FA+2 (l)
F(A
1 ∘A2 )
for all kl ∈ R1 ∘ R 2 .
−
(kl1 )(kl2 ) = TA−1 (k) ∧ TB−2 (l1 l2 )
(ii)T(B
1 ∘B2 )
S. Satham Hussain, Saeid Jafari, Said Broumi and N. Durga “Operations on Neutrosophic Vague Graphs”
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−
I(B
(kl1 )(kl2 ) = IA−1 (k) ∧ IB−2 (l1 l2 )
1 ∘B2 )
−
(kl1 )(kl2 ) = FA−1 (k) ∨ FB−2 (l1 l2 )
F(B
1 ∘B2 )
+
(kl1 )(kl2 ) = TA+1 (k) ∧ TB+2 (l1 l2 )
T(B
1 ∘B2 )
+
(kl1 )(kl2 ) = IA+1 (k) ∧ IB+2 (l1 l2 )
I(B
1 ∘B2 )
+
(kl1 )(kl2 ) = FA+1 (k) ∨ FB+2 (l1 l2 ),
F(B
1 ∘B2 )
for all k ∈ R1 , l1 l2 ∈ S2 .
(iii)TB−1 ∘B2 (k1 l)(k 2 l) = TA−2 (l) ∧ TB−2 (k1 k 2 )
IB−1∘B2 (k1 , l)(k 2 , l) = IA−2 (l) ∧ IB−2 (k1 k 2 )
FB−1∘B2 (k1 , l)(k 2 , l) = FA−2 (l) ∨ FB−2 (k1 k 2 )
TB+1∘B2 (k1 , l)(k 2 , l) = TA+2 (l) ∧ TB+2 (k1 k 2 )
IB+1∘B2 (k1 , l)(k 2 , l) = IA+2 (l) ∧ IB+2 (k1 k 2 )
FB+1∘B2 (k1 , l)(k 2 , l) = FA+2 (l) ∨ FB+2 (k1 k 2 ),
for all k1 k 2 ∈ S1 , l ∈ R 2 .
−
(iv)T(B
(k1 l1 )(k 2 l2 ) = TB−1 (k1 k 2 ) ∧ TA−2 (l1 ) ∧ TA−2 (l2 )
1 ∘B2 )
−
(k1 l1 )(k 2 l2 ) = IB−1 (k1 k 2 ) ∧ IA−2 (l1 ) ∧ IA−2 (l2 )
I(B
1 ∘B2 )
−
(k1 l1 )(k 2 l2 ) = FB−1 (k1 k 2 ) ∨ FA−2 (l1 ) ∨ FA−2 (l2 )
F(B
1 ∘B2 )
+
(k1 l1 )(k 2 l2 ) = TB−1 (k1 k 2 ) ∧ TA+2 (l1 ) ∧ TA+2 (l2 )
T(B
1 ∘B2 )
+
(k1 l1 )(k 2 l2 ) = IB+1 (k1 k 2 ) ∧ IA+2 (l1 ) ∧ IA+2 (l2 )
I(B
1 ∘B2 )
+
(k1 l1 )(k 2 l2 ) = FB+1 (k1 k 2 ) ∨ FA+2 (l1 ) ∨ FA+2 (l2 ),
F(B
1 ∘B2 )
for all k1 k 2 ∈ S1 , l1 l2 ∈ S2 .
Example 3.14 The composition of NVG G1 = (R1 , S1 ) and G2 = (R 2 , S2 ) shown in Figure 2 is
defined as G1 ∘ G2 = (R1 ∘ R 2 , S1 ∘ S2 ) and is presented in Figure 7.
S. Satham Hussain, Saeid Jafari, Said Broumi and N. Durga “Operations on Neutrosophic Vague Graphs”
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Figure 7: COMPOSITION OF NEUTROSOPHIC VAGUE GRAPH
Theorem 3.15 Composition G1 ∘ G2 = (R1 ∘ R 2 , S1 ∘ S2 ) of two NVG G1 and G2 is the NVG of G1 ∘
G2 .
Proof. We divide the proof into three cases:
Case:1 For k ∈ R1 , l1 l2 ∈ S2 ,
̂A (k) ∧ T
̂B (l1 l2 )
̂(B ∘B ) ((kl1 )(kl2 )) = T
T
1 2
1
2
̂A (l1 ) ∧ T
̂A (l2 )]
̂A (k) ∧ [T
≤T
1
2
2
̂A (l1 )] ∧ [T
̂A (k) ∧ T
̂A (l2 )]
̂A (k) ∧ T
= [T
1
2
1
2
̂(A ∘A ) (k, l2 )
̂(A ∘A ) (k, l1 ) ∧ T
=T
1 2
1 2
Î(B1∘B2) ((kl1 )(kl2 )) = ÎA1 (k) ∧ ÎB2 (l1 l2 )
≤ ÎA1 (k) ∧ [ÎA2 (l1 ) ∧ ÎA2 (l2 )]
= [ÎA1 (k) ∧ ÎA2 (l1 )] ∧ [ÎA1 (k) ∧ ÎA2 (l2 )]
= Î(A1∘A2) (k, l1 ) ∧ Î(A1∘A2) (k, l2 )
F̂(B1∘B2 ) ((kl1 )(kl2 )) = F̂A1 (k) ∨ F̂B2 (l1 l2 )
≤ F̂A1 (k) ∨ [F̂A2 (l1 ) ∨ F̂A2 (l2 )]
= [F̂A1 (k) ∨ F̂A2 (l1 )] ∨ [F̂A1 (k) ∨ F̂A2 (l2 )]
= F̂(A1∘A2) (k, l1 ) ∨ F̂(A1∘A2) (k, l2 )
for all kl1 , kl2 ∈ R1 ∘ R 2 .
Case 2: for k ∈ R 2 , l1 l2 ∈ S1 ,
̂A (k) ∧ T
̂B (l1 l2 )
̂(B ∘B ) ((l1 k)(l2 k)) = T
T
1 2
2
1
̂A (l1 ) ∧ T
̂A (l2 )]
̂A (k) ∧ [T
≤T
2
1
1
̂A (l1 )] ∧ [T
̂A (k) ∧ T
̂A (l2 )]
̂A (k) ∧ T
= [T
2
1
2
1
̂(A ∘A ) (l2 , k)
̂(A ∘A ) (l1 , k) ∧ T
=T
1 2
1 2
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Î(B1∘B2) ((l1 k)(l2 k)) = ÎA2 (k) ∧ ÎB1 (l1 l2 )
≤ ÎA2 (k) ∧ [ÎA1 (l1 ) ∧ ÎA1 (l2 )]
= [ÎA2 (k) ∧ ÎA1 (l1 )] ∧ [ÎA2 (k) ∧ ÎA1 (l2 )]
= Î(A1∘A2) (l1 , k) ∧ Î(A1∘A2) (l2 , k)
F̂(B1∘B2 ) ((l1 k)(l2 k)) = F̂A2 (k) ∨ F̂B1 (l1 l2 )
≤ F̂A2 (k) ∨ [F̂A1 (l1 ) ∨ F̂A1 (l2 )]
= [F̂A2 (k) ∨ F̂A1 (l1 )] ∨ [F̂A2 (k) ∨ F̂A1 (l2 )]
= F̂(A1∘A2) (l1 , k) ∨ F̂(A1∘A2) (l2 , k), for all l1 k, l2 k ∈ R1 ∘ R 2 .
Case 3: For k1 k 2 ∈ S1 , l1 , l2 ∈ R 2 such that l1 ≠ l2,
̂B (k1 , k 2 ) ∧ T
̂A (l1 ) ∧ T
̂A (l2 )
̂(B ∘B ) ((k1 l1 )(k 2 l2 )) = T
T
1 2
1
2
2
̂A (k 2 )] ∧ [T
̂A (l1 ) ∧ T
̂A (l2 )]
̂A (k1 ) ∧ T
≤ [T
1
1
2
2
̂A (l1 )] ∧ [T
̂A (k 2 ) ∧ T
̂A (l2 )]
̂A (k1 ) ∧ T
= [T
1
2
1
2
̂(A ∘A ) (k 2 l2 )
̂(A ∘A ) (k1 l1 ) ∧ T
=T
1 2
1 2
Î(B1∘B2) ((k1 l1 )(k 2 l2 )) = ÎB1 (k1 , k 2 ) ∧ ÎA2 (l1 ) ∧ ÎA2 (l2 )
≤ [ÎA1 (k1 ) ∧ ÎA1 (k 2 )] ∧ [ÎA2 (l1 ) ∧ ÎA2 (l2 )]
= [ÎA1 (k1 ) ∧ ÎA2 (l1 )] ∧ [ÎA1 (k 2 ) ∧ ÎA2 (l2 )]
= Î(A1∘A2) (k1 l1 ) ∧ Î(A1∘A2) (k 2 l2 )
F̂(B1∘B2 ) ((k1 l1 )(k 2 l2 )) = F̂B1 (k1 , k 2 ) ∨ F̂A2 (l1 ) ∨ F̂A2 (l2 )
≤ [F̂A1 (k1 ) ∨ F̂A1 (k 2 )] ∨ [F̂A2 (l1 ) ∨ F̂A2 (l2 )]
= [F̂A1 (k1 ) ∨ F̂A2 (l1 )] ∨ [F̂A1 (k 2 ) ∨ F̂A2 (l2 )]
= F̂(A1∘A2) (k1 l1 ) ∨ F̂(A1∘A2) (k 2 l2 ), for all k1 l1 , k 2 l2 ∈ R1 ∘ R 2 .
Conclusion
Graph theory is an extremely useful tool in studying and modeling several applications in
computer science, engineering, genetics, decision-making, economics, etc. An extension of
intuitionistic fuzzy graph is regarded as a single-valued neutrosophic graph which is very useful to
formulate the appropriate real life situation. In this research article, the operations on neutrosophic
vague graphs have been established. Moreover, Cartesian product, lexicographic product, cross
product, strong product and composition of neutrosophic vague graph have been investigated and
the given concepts are demonstrated through examples. Furthermore, in future, we are able to
investigate the domination number and isomorphic properties of the NVGs.
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Received: Apr 10, 2020.
Accepted: July 2 2020
S. Satham Hussain, Saeid Jafari, Said Broumi and N. Durga “Operations on Neutrosophic Vague Graphs”
Neutrosophic Sets and Systems, Vol. 35, 2020
University of New Mexico
Partner selection in Virtual enterprises using the Interval
Neutrosophic fuzzy approach
Hanieh Shambayati 1, Mohsen Shafiei Nikabadi *,2 , Seyed Mohammad Ali Khatami Firouzabadi 3 and
Mohammad Rahmanimanesh4
Ph.D. candidate in Industrial Management, Faculty of Economics, Management and Administrative Sciences, Semnan
University, IRAN. h.shambayati@semnan.ac.ir
2 Associate Professor, Faculty of Economic, Management and Administrative Sciences, Semnan University, IRAN.
shafiei@semnan.ac.ir
3. Professor, Dept. of Industrial Management, Faculty of Management and Accounting, Allameh Tabataba'i University, Tehran,
IRAN. a.khatami@atu.ac.ir
4. Assistant Professor, Electrical & Computer Engineering Faculty, Semnan University, IRAN. rahmanimanesh@semnan.ac.ir
* Correspondence: shafiei@semnan.ac.ir; Tel.: (0982331532579)
1
Abstract: With the rapid development of the Internet, information technology, and globalization of
the economy, Some small and medium-sized companies know that they cannot compete with their
limited capacity alone. As a result, they are beginning to seek collaboration and a collective
approach to meet the dynamic needs of customers and increase their power for competition in the
market. Virtual enterprise is a temporary platform for working with different companies that share
their core tasks to meet customer’s demand. Partner selection is a major issue in the formation of a
virtual organization. This is especially difficult due to the uncertainties regarding information,
market dynamics, customer expectations, and rapidly changing technology, with highly random
decision making. As a generalization of fuzzy sets and intuitionistic fuzzy sets, Neutrosophic sets
are created to show the uncertain, and inconsistent information available in the real world. The
main purpose of this paper is to identify and select partners in the formation of Virtual Enterprises
under uncertainty and contradictory factors using the extended VIKOR group decision making
technique using the Interval Neutrosophic fuzzy approach. For this purpose, after identifying the
factors affecting partner selection, the factors are weighted using the Maximizing deviation method
and the partners are ranked using this method. Finally, a sensitivity analysis for assessing the
validity of the method is also presented. The results show that the Willingness to share information
criterion is the most important partner selection criterion in this enterprise.
Keywords: Virtual Enterprise, Partner Selection, Interval Neutrosophic Numbers, Group Decision
Making, Uncertainty, VIKOR.
1. Introduction
With the globalization of the market and the economy, the rapid development of the use of the
Internet and information technologies, faster product updates and market needs have become more
uncertain and personalized [1]. Globally, companies are increasingly in need of the competence of
other companies to meet growing customers’ demands [2]. Therefore, it is difficult to adapt the
traditional business model to the new market environment. At the same time, companies need to
maintain lower costs and shorter delivery cycles, that this challenges old organizational form [3]. In
fact, a enterprise cannot meet the rapid market changes by integrating internal resources and
Hanieh Shambayati, Mohsen Shafiei Nikabadi, Seyed Mohammad Ali Khatami Firouzabadi and Mohammad
Rahmanimanesh, Partner Selection in Virtual Enterprises using the Interval Neutrosophic Fuzzy Approach
Neutrosophic Sets and Systems, Vol. 35, 2020
388
competencies alone [1]. As such, many companies are attracting partners to absorb opportunities in
emerging markets to share costs, reduce development time, and utilize the effectiveness of design,
production, and marketing skills within and outside companies [4]. With the rapid growth of
competition in the global industry, a dynamic virtual enterprise (VE) approach will be needed to
meet market needs for quality, responsiveness and customer satisfaction [5]. VE is created to address
a specific opportunity in a fast-paced and simultaneous market, creating a collaborative work
environment for managing and using a set of resources provided by companies. Business partners
are all connected to share their skills, and take advantage of the rapidly changing opportunities in a
dynamic network [6]. In fact, through the VE framework, each VE partner brings its expertise for
implementing the original project, [7] and each partner focuses on its own core competence. This
increases the ability of the organization to meet the unpredictable demands of customers [8].
Therefore, by maintaining the agility of the entire structure, this collaboration will deliver high
quality products based on customer’s specific needs [7]. In this alliance, the links are made easier by
computer technology, [4] and eventually when the market opportunity is over, the VE will be
dissolved [5].
Compared to the traditional organizational form, VE is considered a low cost, high responsive and
adaptive organization and members of this alliance can share cost, risk, technology, and key
competition with each other, through which members can gain win-win policy. However, many
issues arise throughout the life cycle of a VE, including how we can find the right partners, which is
a key issue for the core enterprise in the VE development phase, and this issue has been considered
by many researchers [3]. As the VE environment continues to grow in size and complexity, the
importance of managing such complexities increases [5]. In a virtual enterprise (VE), choosing a
partner is very important because of the short life of these organizations (temporary alliances) and
the absence of formal mechanisms (contracts) to ensure participants' responsibility [9].
The complexity of the partner selection process is reinforced by the fact that there are several
centralized internal and external organizational factors that have both tangible and intangible
characteristics and should be incorporated into the decision analysis for this selection process [8].
Like all decision-making issues, partner selection involves tangible and intangible paradox
specifications under conflicting or incomplete information [10]. Therefore, it is important to select
the most appropriate companies while there may be dozens of volunteer companies involved in the
project [7]. The multitude of factors that are considered when choosing partners for a business
opportunity such as cost, quality, trust and delivery time cannot be expressed by the same size or
scale [11]. In practice, partner selection should consider higher levels of uncertainty and risk as a
way of addressing uncontrolled factors: such as price or demand fluctuations, lack of enough
knowledge sharing among VE members, resource constraints, and incomplete information about
candidates and their performance [12].
The multi-attribute group decision making (MAGDM) approach is to provide a comprehensive
solution by evaluating and ranking alternatives based on contrasting features based on decision
makers’ (DM) preferences [13]. Decision-making is often about the optimal choice between a set of
options, considering the impact of many criteria. In the past five decades, Multi criteria decision
making method (MCDM) has become one of the most important and key ways of solving complex
decision problems, despite of various criteria and options. In MCDM problems, the characteristics of
Hanieh Shambayati, Mohsen Shafiei Nikabadi, Seyed Mohammad Ali Khatami Firouzabadi and Mohammad
Rahmanimanesh, Partner Selection in Virtual Enterprises using the Interval Neutrosophic Fuzzy Approach
Neutrosophic Sets and Systems, Vol. 35, 2020
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dependence, opposition, and interaction are ambiguous between decision criteria, which obscures
the degree of membership [14]. In fact, it is difficult for DMs due to the uncertainty of the
information and the many constraints such as time pressure, lack of awareness, and problems of
data extraction and so on to express their preferences numerically in many complex realities [15].
The fuzzy set theories or the intuitionistic fuzzy theories are used to overcome this obstacle.
However, these sets are not always suitable [14]. The fuzzy set (FS) has only one member and cannot
display complex information and the intuitionistic fuzzy set, which includes membership and
non-membership degree, can only manage incomplete information, and cannot deal with
inconsistent information, and degree of indeterminate membership at IFS has always been ignored
[16]. Smarandache recommended Neutrosophic set (NS) by adding an indefinite membership
function based on IFS. In NS, the degree of accuracy, lack of reliability, and the degree of inaccuracy
are completely independent [17].
The Neutrosophic set is becoming a scientific tool and has attracted the attention of many scientists
and academic researchers to develop and improve the Neutrosophic method [14]. Abdel-Basset et al.
(2020) considered inventory location problem, They applied the best-worst method (BWM) to find
the weight of these criteria and propose a combination of plithogenic aggregation operations, and
the BWM to solve MCDM problems [18]. Veerappan et al (2020) considered Multi-Aspect
Decision-Making Process in Equity Investment Using Neutrosophic Soft Matrices [19]. Abdel-Basset
and Mohamed (2020) proposed a combination of plithogenic multi-criteria decision-making
approach based on the TOPSIS and Criteria Importance Through Inter-criteria Correlation (CRITIC)
methods for sustainable supply chain risk management [20]. Abdel-Basset et al. (2020) provided a
new hybrid neutrosophic MCDM framework that employs a collection of neutrosophic ANP, and
TOPSIS under bipolar neutrosophic numbers for professional selection [21]. Edalatpanah and
Smarandache proposed an input-oriented DEA model with simplified neutrosophic numbers and
present a new strategy to solve it [22]. Abdel-Basset et al. (2020) applied a combination of quality
function deployment (QFD) with plithogenic aggregation operations for Selecting Supply Chain
Sustainability Metrics [23].
In this paper, we combine the Interval Neutrosophic Numbers (NS) set and the VIKOR method to
select a partner in a virtual enterprise. One of the best ways to solve decision problems with
inconsistent and unbelievable criteria is the VIKOR approach. VIKOR can be an effective tool for
decision making when the decision maker is unable to identify and express the superiority of a
problem at the time it is started and designed [24]. For this purpose, the criteria for selecting the
partner were first identified by the experts and then their opinions about each of the candidate
partners were collected according to the effective factors. Finally, partner rating and selection are
performed using the VIKOR method, which is based on the concurrent planning of multivariate
decision problems and evaluates issues with inappropriate, and incompatible criteria, in the Interval
Neutrosophic environment. The innovation of this paper is that the Interval Neutrosophic set is used
to express the evaluation of information, and partner selection in virtual enterprise will be
implemented under an Interval Neutrosophic environment. Since the weight of the criteria varies
with the mental state, and no specific information is available, in this paper the weight of the criteria
is determined using the Maximizing deviation method under the Interval Neutrosophic
environment.
Hanieh Shambayati, Mohsen Shafiei Nikabadi, Seyed Mohammad Ali Khatami Firouzabadi and Mohammad
Rahmanimanesh, Partner Selection in Virtual Enterprises using the Interval Neutrosophic Fuzzy Approach
Neutrosophic Sets and Systems, Vol. 35, 2020
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2. Research literature and Related studies
The widespread development of Internet technologies in the late twentieth century has led to the
dramatic formation and enhancement of the virtual environment in the employment sector, and
virtual enterprises, virtual sectors and a series of virtual businesses have expanded. Information on
so-called virtual companies was first provided in the early 1990s by Steven L. Goldman, Rocer N.
Nagel and David B. Greenberger, and William H. Davidow and Michael S. Malone. The innovative
technology market enables companies to form temporary partnerships, and the creation of such
links through the Internet leads to the formation of Virtual Enterprises [25]. Member companies in
such a virtual enterprise, rather than being independent companies and focusing on their own
business goals, work together to share their information about their capabilities, programs and cost
structures, to improve their technical, logistical, financial and other activities in order to compete [4].
The short-term goal of a VE is primarily to increase productivity, reduce inventory and total cycle
time. The long-term goal is to increase customer satisfaction, market share, and profit levels for all
members. Failure to cooperate may result in a delay in delivery, poor customer service, and
inventory creation, and so on [26]. The success of this mission depends on all the organizations that
work together as a unit. Because everyone gives its own core strengths or competencies to the virtual
enterprise. In other words, the competitive advantage gained by a virtual enterprise depends on
each other and their ability to integrate with each other. The key factor in forming a virtual
enterprise is choosing agile, competent and consistent partners [27]. The life cycle of a VE consists of
four stages: creation, operation, evolution, and dissolution [28]. In the creation phase, when an
organization wins a large contract project and is unable to complete it with its proper capacity, it
seeks out potential partners and negotiates with them through its information infrastructures and
VE will be created. At the operation stage, after signing contracts between the partners, VE manages
the process of production or execution of the project. At the development stage, the VE is configured
to meet the resource requirements when the project is changed, and at the dissolution stage, when
the project is completed, the VE will be eventually dissolved [29]. Obviously, the first step, namely
the selection of partners, is crucial to the success of the VE [30]. The main difference between a
regular supplier selection issue and a partner selection issue in a VE is the expected duration of the
relationship. In fact, companies in a VE rarely have the time to implement, and develop all the
features needed for successful relationships. They therefore emphasize on the fact that partner
selection is definitely an important step in VE development [12]. Determining the right criteria and
evaluating all of the influencing factors in partner selection is difficult. There are many factors that
must be considered during decision making. Some are qualitative, such as friendship, credibility,
and reliability, and others are quantitative, such as cost, and delivery time. It is very costly and
time-consuming to evaluate each partner and identify the most desirable ones [26].
There is an extensive literature on partner selection in VE, each offering a new approach for
evaluating and selecting the most appropriate partners among the set of organizations. Sha and Che
(2004) develop a partner selection and production distribution planning problem with a new partner
selection Model based on Analytical Hierarchy Process (AHP), multi-attribute utility theory
(MAUT), and integer programming (IP), for Virtual integration (VI) with multiple criteria. The AHP
and MAUT methods are used to evaluate and weight each partner's candidate, and the IP model
Hanieh Shambayati, Mohsen Shafiei Nikabadi, Seyed Mohammad Ali Khatami Firouzabadi and Mohammad
Rahmanimanesh, Partner Selection in Virtual Enterprises using the Interval Neutrosophic Fuzzy Approach
Neutrosophic Sets and Systems, Vol. 35, 2020
391
applies this weigh to find the best potential partners and provide the right distribution plan for the
selected partners [31]. Sarkis et al. (2007) present a practical paradigm that can be used by
organizations to help form agile virtual companies using ANP method [8]. Ye and Li (2009)
proposed two group decision models for spatial decision making to solve the problem of partner
selection under incomplete information. The first model is a technique for preferring the order with
similarity with ideal Solution (TOPSIS) for group decision making based on degree of deviation. The
second approach is TOPSIS group decision-making based on risk factor [28]. Crispim and Sousa
(2009) propose an exploratory process to help the decision maker to acquire knowledge about the
network in order to identify the criteria and companies that provide the needs of a project very well.
This process involves a multi-objective meta-heuristic search algorithm designed to find a good
approximation of the PARETO front and a fuzzy TOPSIS algorithm to rank the configuration of VE
options. Preliminary computational results clearly showed the potential of this approach for
practical applications [9]. Ye (2010) investigated the problem of partner selection in partial and
uncertain information environments and used the extended TOPSIS technique for group decision
making with intuitive fuzzy numbers with interval values for problem solving [32].
Liu et al (2016) proposed a partner selection method based on distance multipliers preferences with
approximate compatibility. In this paper, using a (n - 1) pairwise comparison, a new partner
selection method is proposed, which introduces a new concept of approximate compatibility for
multidimensional
preferential
relationships
[27].
Nikghadam
et
al.
(2016)
designed
a
customer-based algorithm to select a partner in a virtual enterprise. In this study, customers were
classified into three categories: passive, standard and assertive. Three different approaches; fuzzy
logic-FAHP TOPSIS and ideal programming were used for each type of customer, respectively. The
results confirm that adopting this algorithm not only helps VE to select the most appropriate
partners based on customer preferences, but also adapts its model to each customer's attitude. As a
result, the overall flexibility of the system significantly improves [7]. Polyantchikov et al. (2017)
performed virtual enterprise formation in the context of a sustainable partner network using
methodologies such as Analytical Hierarchy Process (AHP), fuzzy AHP approach and TOPSIS
method [33]. Huang et al. (2018) studied the problem of partner selection for virtual production
companies facing an uncertain environment and using the gray system theory studied uncertainty at
the start of a project, in the completion time, in shipping time, and also studied the cost. They used
the chaotic particle swarm optimization (CPSO) algorithm to solve the problem [30].
Meng et al. (2019) in their paper presented Interval Neutrosophic Preferred Relations and examined
its application with numerical examples in virtual partner selection. The algorithm presented in this
paper is based on group decision-making based on INPRs which can be applied to address
incomplete and inconsistent INPRs [3]. Chen and Goh (2019) sought a cooperative partner selection
mechanism from the perspective of dual-factor theory. They proposed a new framework for
problem solving and cooperative partner selection. This framework uses the degree of compatibility
of the triangular fuzzy soft set (TFSS) to measure the level of participation, and a broad TODIM
based on TFSS to measure the degree of influence on the individual level [34]. Ionescu (2020) reviews
the most prominent approaches to solving partner selection problems and discuss some of the most
documented methods and algorithms for VO creation and reconfiguration [35]. Zhao et al. (2020)
studied a multi-objective virtual enterprise partner selection model with relative superiority
Hanieh Shambayati, Mohsen Shafiei Nikabadi, Seyed Mohammad Ali Khatami Firouzabadi and Mohammad
Rahmanimanesh, Partner Selection in Virtual Enterprises using the Interval Neutrosophic Fuzzy Approach
Neutrosophic Sets and Systems, Vol. 35, 2020
392
parameter in fuzzy environment. In this paper, the completion time and delivery time were fuzzily
processed [36] . Wan and Dong (2020) applied the group decision making (GDM) problems with
interval-valued Atanassov intuitionistic fuzzy preference relations (IV-AIFPRs) and developed a
novel method for solving a virtual enterprise partner selection problem [37].
These papers use different methods and techniques to select partners in virtual enterprises. Many of
these studies make use of fixed weights of the criteria, and consider a limited set of uncertainties.
They do not make sensitivity analysis to examine solutions, and are, in general, very
time-consuming or too complex to be understood by the DM. However, in practice, there are
multiple uncertainties in the VE partner selection problem and to assign precise weights to criteria
becoming more critical when the number of criteria increases and when the VE life cycle is rather
short. In this paper, the weight of the criteria is determined using the maximizing deviation method
under the Interval Neutrosophic environment. and combine the Interval Neutrosophic Numbers
(NS) set and the VIKOR method have considerable potential to this problem. Neutrosophic sets are
very powerful and successful in overcoming situations and cases in uncertainty, vagueness, and
imprecision. This model is easy to understand and use, and flexible, and tolerant with inconsistent
and inaccurate information. Additionally, the procedure proposed in this work overcomes some of
the shortcomings of decision-support tools and provides automatic sensitivity analysis on the
results.
On the other hand, many factors should be taken into consideration when selecting partners of a VE
By studying the research literature, the most important factors influencing partner selection in
Virtual Enterprises can be classified according to Table 1. These factors are the most popular and
most influential factors in choosing a partner in a virtual enterprise.
Table 1. Criteria for partner selection
Criteria
Reference
Cost
[28], [9], [32],[12], [4],[27], [30], [10], [38], [2], [39]
Time
[28], [32], [12], [10], [2]
Trust
[28], [32], [10], [34], [3]
Risk
[28], [32], [12], [9] ,[10], [40]
Quality
[28], [9], [33], [27], [10], [26], [38], [39], [6]
Productivity & Performance history
[9], [33], [7], [26], [2]
Market entrance capability
[9], [12]
Knowledge and managerial experience
[9], [33], [34]
Age of the organisation
[9], [12]
Competency & technical expertise
[9], [33], [3]
Information and communication
[9], [33]
technology resources
Price
[12], [33], [7], [26], [6]
Delivery
[12], [33], [7], [30], [26], [39]
Customer service
[12], [7], [27], [26], [38], [2], [6]
Geographical location
[33], [26], [34]
The financial stability
[27], [34], [38], [6]
Hanieh Shambayati, Mohsen Shafiei Nikabadi, Seyed Mohammad Ali Khatami Firouzabadi and Mohammad
Rahmanimanesh, Partner Selection in Virtual Enterprises using the Interval Neutrosophic Fuzzy Approach
Neutrosophic Sets and Systems, Vol. 35, 2020
393
Willingness to share information
[12], [34]
Tardiness penalty
[4], [27]
Technology capability
[34], [26], [38], [34],
Reputation and position in industry
[3], [26], [38], [33]
IT infrastructure
[38], [26]
3. Methedology
This research is applied in terms of purpose and quantitative in terms of variables. In the partner
selection process, decision makers are usually unsure of their preferences [41]. Because information
about candidates and their performance is incomplete and unclear. In terms of data collection,
selecting and evaluating partners is difficult due to the complex interactions between different
entities, and because of their preferences they may be inaccessible based on incomplete or partial
information. To address this issue under a multi-criteria perspective, several types of information
(numerical, interval, qualitative and binary) are used to facilitate the expression of preferences or the
evaluation of stakeholders in decision making [12]. In this paper, Interval Neutrosophic numbers are
used to express the preferences of experts. In this regard, First, the effective criteria influencing the
choice of partner are selected, and then experts express their opinion about candidates with the
competence of linguistic terms according to the effectiveness criteria. After converting the experts'
opinions to Interval Neutrosophic numbers, the weight of the criteria is calculated using the
maximum deviation method. In the second step, expert opinions on each company integrate using
the interval neutrosophic weighted average operator. Finally, rankings of companies perform by
using the Vikor fuzzy interval neutrosophic method. The general framework of proposed method
presented in Fig 1.
Consider the criteria, alternatives and experts
Determine linguistic preference of criteria and alternatives
Convert DMs opinions to Interval Neutrosophic numbers
Calculate weight for each criteria using the maximum deviation method
Aggregate DMs opinions using the interval neutrosophic weighted average
operator
Rank the alternatives using Vikor fuzzy interval neutrosophic
method
Sensitivity analysis of the value β
Fig 1. A general framework of proposed method
3.1. Interval Neutrosophic fuzzy set
In the real world, decision information is often incomplete, uncertain, and inconsistent. In order to
process this type of information, Smarandache introduced Neutrosophic set (NS) from a
philosophical perspective by adding independent indeterminacy-membership, which is an
Hanieh Shambayati, Mohsen Shafiei Nikabadi, Seyed Mohammad Ali Khatami Firouzabadi and Mohammad
Rahmanimanesh, Partner Selection in Virtual Enterprises using the Interval Neutrosophic Fuzzy Approach
Neutrosophic Sets and Systems, Vol. 35, 2020
394
extension of the fuzzy set (FS), the fuzzy set with interval values, the intuitionistic fuzzy set, and so
on [42]. Smarandache believed that these types of sets not only had the degree of membership and
the degree of non-membership, but also consider the degree of non-determination and lack of
compatibility [16]. The new theory of Neutrosophic sets allows to work with the "Knowledge of
neural thought". In fact, Neutrosophic sets are generalizations of fuzzy logic and allow to deal with
more complex uncertainty models. In "classical" fuzzy sets, each element is defined by a degree of
membership, and the available methods are controlled by fuzzy sets [43]. The fuzzy set cannot
express neutral state, meaning neither support nor opposition. To overcome this defect, Atanassov
introduced the concept of the Intuitionistic Fuzzy Set (IFS). Compared to The fuzzy set, the
intuitionistic fuzzy set can simultaneously express three modes of support, opposition, and
neutrality. Although the FS and IFS have been developed and publicized, they cannot address the
uncertain and inconsistent issues of real decision-making. To solve this problem, Neutrosophic (NS)
sets have been suggested [44]. Unlike The intuitionistic fuzzy sets, which depend on the degree of
uncertainty on membership and non-membership, by the Neutrosophic logic the value of the
indeterminate membership is independent of the degree of truth and falsehood [43]. Neutrosophic
logic is flexible and tolerant with inconsistent and inaccurate data. This logic is based on natural
language and is made up of specialized knowledge. The concept of the Neutrosophic set provides an
alternative approach in the case of inaccuracies in the decisions made by deterministic sets or
traditional fuzzy sets, and where the information provided is inadequate for finding it inaccurate
[45]. Neutrosophic sets are powerful and successful in overcoming situations and in an inadequate
information environment, uncertainty, ambiguity and inaccuracy [14]. A Neutrosophic set A with an
A value in X is expressed by 1.
{
)
)
}
))|
(1)
With Neutrosophic set logic, every aspect of the problem is represented by the degree of the truth
membership (TA(x)), the degree of the indeterminate membership (I A(x)) and the degree of the false
membership (FA(x)) according to 1.
)
For each x,
)
)
[
] and the sum of these memberships is less than or equal to
three [46]. Thus, Neutrosophic sets provide a means of expressing DM preferences and priorities,
and fully determine membership performance in situations where DM comments are subject to the
indeterminate membership or lack of information [14].
)
)
)
(2)
Sometimes the degree of truth, falsehood, and uncertainty of a particular sentence is not precisely
defined in real terms, but is determined by several possible interval values [47]. Thus, the Interval
Neutrosophic Set (INS) was introduced by Wang et al (2005). [48]. As a special case of Neutrosophic
sets, the Interval Neutrosophic Set (INS) can be used to address uncertain and inconsistent
information in decision making [3]. Wang et al showed Interval Neutrosophic (INS) assemblages
with distance membership, the degree of non-membership, and degree of hesitant (The
indeterminate membership) as follows.
[
][
][
])
(3)
The Interval Neutrosophic set can be simpler to express incomplete, uncertain, and contradictory
information [49], and is flexible and practical for dealing with decision problems. Compared to other
Hanieh Shambayati, Mohsen Shafiei Nikabadi, Seyed Mohammad Ali Khatami Firouzabadi and Mohammad
Rahmanimanesh, Partner Selection in Virtual Enterprises using the Interval Neutrosophic Fuzzy Approach
Neutrosophic Sets and Systems, Vol. 35, 2020
395
fuzzy set expansions, INS has the following advantages. (A) Compared to The fuzzy set, INS can
simultaneously express positive, negative, and hesitant judgments of DM using membership degree,
non-membership degree, and degree of hesitation. (B) Compared with The Intuitionistic fuzzy sets,
INS independently express the degree of positive, negative, and uncertain judgments. That DMs
have more flexibility to express their uncertain and contradictory information [3].
3.2. Interval Neutrosophic Fuzzy VIKOR Method
VIKOR is an effective decision making method that selects the optimal option with group utility
maximization and individual regret minimization. And it is used as one of the applied MCDM
techniques to solve a discrete decision problem with disproportionate criteria with different and
conflicting units of measurement [50]. This method was proposed by Opricovic (1998) to solve the
problem of multi-criteria decision making in an incompatible and inconsistent criteria environment
[43]. VIKOR is an efficient tool for finding the compromise solution from a set of conflicting criteria.
Where compromise means an agreement made with mutual concessions [51]. That can help decision
makers to make a final decision [52]. The VIKOR method is based on the specific property of being
close to the ideal solution. One of the features of this method is that the options are evaluated
according to all defined criteria (performance matrix) and the stability analysis of the intervals
shows the stability of the weight [53]. The effectiveness of this approach becomes more apparent
when the decision maker is not able to express his/her preferences and uses agreed solutions to solve
the problems. An agreed solution is a justified solution that is close to the ideal solution and that
decision makers accept because of the maximum utility of the group [50].
}.
{
{
} is given as fij with respect to criteria of
{
Suppose the rating of options
} is the weight vector of the criteria. The formula for measuring
distance on Pi options is based on equation (4).
(4)
(∑ (
where
Let
) )
are the ideal and anti-ideal points, respectively [53].
and
{
} be the weight of the criteria,
} and the weight of decision makers be
{
decision makers be
and ∑
. If the set of
{
}
and ∑
Suppose that ̃
̃ )
([
)
of Interval Neutrosophic Numbers,
[
)
)
)
][
)
] [
)
)
] [
)
)
])
is the Matrix of decision
.
)
)
][
)
)
)
]
[
]
)
(5)
(6)
(7)
The steps of the VIKOR method for multi-criteria group decision-making problems of the Interval
Neutrosophic set are as follows [13].
Step 1. Convert Evaluation Information to the Interval Neutrosophic Number Set
Hanieh Shambayati, Mohsen Shafiei Nikabadi, Seyed Mohammad Ali Khatami Firouzabadi and Mohammad
Rahmanimanesh, Partner Selection in Virtual Enterprises using the Interval Neutrosophic Fuzzy Approach
Neutrosophic Sets and Systems, Vol. 35, 2020
396
Step 2. Calculate the weight of the criteria
Since the weight of the benchmarks may be completely unknown, the benchmark weight is
calculated using the Maximizing deviation method. According to this view, if the criterion values of
all alternatives to a particular attribute are quantitative deviations, quantitative weight can be
assigned to this criterion. Otherwise, the criterion that causes the deviation to be greater should be
weightier. In particular, if the criterion values of all the different options are equal to a given
property, the weight of such a criterion may be zero [49]. The weight of the criteria is thus calculated
using the equation (8) [48].
∑
∑
|
|
|
∑
∑
|
∑
∑
|
)
(8)
∑
|
)
|
|
|
|
|
|
(9)
Step 3. Using ̃ and calculating the interval neutrosophic number weighted averaging (INNWA)
operator [47]
)
〈[
∏
[∏
(
(
)
∏
)
∏
(
(
) ] [∏
(
) ]
(10)
∏
)
(
) ]〉
Step 4. Define the solution of positive and negative ideals (R+ and R-)
̃
([
̃
][
([
][
][
])
][
(11)
])
(12)
For positive and incremental criteria
([
][
][
])
([
([
][
(13)
][
][
][
])
])
([
(14)
][
][
])
For negative and decreasing criteria
([
][
][
])
(15)
([
([
][
][
][
][
])
])
(16)
([
][
][
])
Step 5. Calculate the indicators of maximum group utility (Γi) and minimum individual regret (Zi)
∑
(([
(([
][
][
][
][
]) ([
]) ([
][
][
][
][
]))
]))
Hanieh Shambayati, Mohsen Shafiei Nikabadi, Seyed Mohammad Ali Khatami Firouzabadi and Mohammad
Rahmanimanesh, Partner Selection in Virtual Enterprises using the Interval Neutrosophic Fuzzy Approach
(17)
Neutrosophic Sets and Systems, Vol. 35, 2020
{
)
(([
(([
][
][
|
|
397
][
][
|
|
]) ([
][
]) ([
|
|
][
|
|
][
]))
][
|
|
]))
|
(18)
}
|)
(19)
Step 6. Calculate VIKOR Index (Qi)
)
)
)
)
)
(20)
(21)
(22)
Step 7. Rank the options based on Qi, Γi and Zi in accordance with the classic VIKOR ranking rule
Step 8. The compromise solution must meet one of the following conditions:
(A) Acceptable advantage in the sense that a compromise solution must be significantly different
from its next solution:
)
)
Where
and
are the first and second
choices in the ordered list and m is the number of options.
(B) Acceptable consistency in the decision-making process means that the adaptive solution chosen
must have Group utility maximization and at least individual impact: A 1 should be the best rank in
Γi and Zi. This is the compromise solution throughout the decision-making process.
If the above conditions for a compromise solution are not met, a set of adaptation strategies is
provided instead of one.
Step 9. A set of compromise solutions is obtained if one of the conditions is not satisfied.
and
are compromise solutions if only condition 2 is not met. Or
compromise solutions if condition 1 is not fulfilled, by the constraint
,
)
and ... AM are
)
decides
for maximum M [54].
4. Case study
A company in the online sales of various products has been selected as the numerical example of this
research. The company supplies products to various suppliers and sends them to its customers. Due
to limited resources and limited resources, the company cannot independently complete the entire
project. Therefore, the company intends to select an optimal partner from the project candidates for
the transport sector of the company and create a dynamic virtual enterprise alliance to collectively
complete the entire project. In the issue of partner selection, first by studying the research literature,
the most important criteria affecting partner selection in different domains were identified in
accordance with Table (1). Then, 8 experts from the company with expertise in virtual enterprise and
partner selection and with over 5 years' experience were selected 13 criteria of the most important
partner selection criteria in the transport sector of the company according to Table (2).
Table 2. Criteria linguistic assessments for partner selection by experts
Hanieh Shambayati, Mohsen Shafiei Nikabadi, Seyed Mohammad Ali Khatami Firouzabadi and Mohammad
Rahmanimanesh, Partner Selection in Virtual Enterprises using the Interval Neutrosophic Fuzzy Approach
Neutrosophic Sets and Systems, Vol. 35, 2020
398
Partner 1
Criteria
Partner 2
E1
E2
E3
E4
E1
E2
E3
E4
Cost
C1
VH
H
VH
VH
H
VH
VH
VH
Delivery
C2
H
L
H
M
L
M
L
M
Trust
C3
VH
H
VH
M
H
H
VH
M
Risk
C4
L
VL
M
VL
M
M
M
VL
Quality
C5
H
H
H
VH
H
H
VH
H
Reputation and position in
C6
H
VH
VH
H
VH
H
VH
VH
Customer service
C7
H
M
M
VH
H
VH
M
VH
Knowledge and managerial
C8
H
H
M
H
H
L
M
L
Technology capability
C9
M
VL
L
VL
M
VL
L
L
Information and
C10
VH
H
VH
H
H
VH
VH
H
C11
M
M
L
VL
M
VL
H
L
C12
M
H
M
M
VH
H
VH
H
C13
VH
H
H
H
VH
M
H
M
industry
experience
communication technology
resources
Willingness to share
information
Competency & technical
expertise
IT infrastructure
After defining effective criteria in the Partner selection of the transport sector, 4 experts of the
company expressed their opinion about the 4 candidates with the competence of linguistic terms
according to the effective criteria. Table (2) gives some examples of expert opinions.
4.1. Findings
After gathering the experts' opinions in the form of linguistic terms, they first converted to Interval
Neutrosophic numbers using Table 3.
Table 3. Transformations between numerical ratings and INSs
Linguistic terms
INSs
VH
{[0.9,1],[0,0.1],[0,0.1]
H
{[0.75,0.85],[0.05,0.15],[0.15,0.25]}
M
{[0.55,0.65],[0.15,0.25],[0.35,0.45]}
L
{[0.35,0.45],[0.25,0.35],[0.55,0.65]}
VL
{[0.15,0.25],[0.35,0.45],[0.75,0.85]
Next, using these observations, the weighting of the criteria was calculated using the maximum
deviation and correlation technique (8) according to Table (4). The results show that the Willingness
to share information criterion with a weight of 0.113 is the most important partner selection criterion
in this company. This illustrates the importance of the quality of information shared. As such, it is
important for Virtual Enterprises to collaborate effectively with the information sharing
organization for optimal collaboration. And keeping in touch with other partners, such as finding
Hanieh Shambayati, Mohsen Shafiei Nikabadi, Seyed Mohammad Ali Khatami Firouzabadi and Mohammad
Rahmanimanesh, Partner Selection in Virtual Enterprises using the Interval Neutrosophic Fuzzy Approach
Neutrosophic Sets and Systems, Vol. 35, 2020
399
out where and when to deliver the goods, and keeping the customer informed of the delivery and
planning process of the company to ship other products will ultimately lead to better overall
company performance. Competency & technical expertise is ranked second and reflects the
importance of technical and practical expertise from the point of view of company experts in
choosing a virtual partner. Reputation and position in the industry are of third importance for the
company and the background, reputation and position of the company in the industry and among
the competitors can be an effective choice. The notable point in this company is that the cost criterion
(lowest-weighted) is the last priority. This indicates the importance of other criteria for cost, and the
company tends to be more costly in choosing the optimal partner.
Table 4. Weight of criteria
Criteria
C1
C2
C3
C4
C5
C6
C7
C8
C9
C10
C11
C12
C13
Weight
0.048
0.095
0.051
0.086
0.049
0.096
0.084
0.085
0.056
0.067
0.113
0.098
0.071
Given the group decision-making of choosing a virtual partner, it is necessary to integrate expert
opinions on each company. For this purpose, using the Interval Neutrosophic Weighted Average
Operator for each candidate company, the relation of 10 decision matrices of consensus of expert
opinions is calculated. The Consensus Decision Matrix of Business Partner 1 is in the form of Interval
Neutrosophic Numbers as shown in Table 5. The same applies to other business partners.
Table 5. The decision matrix ̃
T+
T-
I+
I-
F+
F-
C1
0.874257
1
0
0.110668
0
0.125743
C2
0.632293
0.743461
0.098399
0.210643
0.256539
0.367707
C3
0.816858
1
0
0.139158
0
0.183142
C4
0.321982
0.426361
0.260341
0.364845
0.573639
0.678018
C5
0.801182
1
0
0.13554
0
0.198818
C6
0.841886
1
0
0.122474
0
0.158114
C7
0.733258
1
0
0.174982
0
0.266742
C8
0.710427
0.81461
0.065804
0.170433
0.18539
0.289573
C9
0.321982
0.426361
0.260341
0.364845
0.573639
0.678018
C10
0.841886
1
0
0.122474
0
0.158114
C11
0.421652
0.525878
0.210643
0.314985
0.474122
0.578348
C12
0.611497
0.716813
0.113975
0.220028
0.283187
0.388503
C13
0.801182
1
0
0.13554
0
0.198818
Finally, the VIKOR fuzzy Interval Neutrosophic method and the equations of 10 to 23 rankings of
the four transport companies were performed. After calculating the performance and distance from
the ideal level of options and obtaining the indicators of maximum group utility (G i) and minimum
individual regret (Zi) and the value of VIKOR index (Qi), the final ranking of options was done
according to Table 5. Accordingly, the least Q value is chosen as the best option.
Table 6. Sorting results
Hanieh Shambayati, Mohsen Shafiei Nikabadi, Seyed Mohammad Ali Khatami Firouzabadi and Mohammad
Rahmanimanesh, Partner Selection in Virtual Enterprises using the Interval Neutrosophic Fuzzy Approach
Neutrosophic Sets and Systems, Vol. 35, 2020
400
Partner
Partner 1
Partner 2
Partner 3
Partner 4
The ranking order
Гi
0.383959
0.355709
0.471911
0.666381
P2>P1>P3> P4
Zi
0.100536
0.085379
0.113432
0.098186
P2>P4> P1>P3
Qi
0.31562
0
0.687017
0.728261
P2>P1>P3> P4
Thus Business Partner 2 with Q2 = 0 is selected as the best virtual partner. This result is now
examined by two conditions.
Hence the first condition is not
applicable. Since option A2 has the best rank in Gi and Zi (β = 0.5), so the second condition holds.
Since only the second condition is in place, the options are rated P 2 ~ P1> P3> P4, and both A2 and A1
are eventually selected and get top rankings.
In the relationships of the Neutrosophic fuzzy VIKOR method, β is defined as the weight of most
criteria strategy, or most group utility, and is usually considered to be 0.5. However, the β value may
affect the value of the VIKOR index. For this purpose, calculations for different values of β are
performed according to Table 7, and the applicability and stability of the proposed method are
investigated.
Table 6. Sensitivity analysis of the value β
Rank order
𝛃
0
0.540309
0
1
0.456522
P2> P4>P1>P3
0.2
0.450433
0
0.874807
0.565217
P2>P1 >P4>P3
0.4
0.360558
0
0.749613
0.673913
P2>P1> P4>P3
0.5
0.31562
0
0.687017
0.728261
P2>P1>P3> P4
0.6
0.270682
0
0.62442
0.782609
P2>P1>P3> P4
0.8
0.180807
0
0.499227
0.891304
P2>P1>P3> P4
1
0.090931
0
0.374034
1
P2>P1>P3> P4
Weight sensitivity analysis of the majority (β) strategy indicates that the firm manager can select the
appropriate group (β) value to reflect the decision maker priority. If the manager prefers to eliminate
Group utility maximization, it supports β = 1 and uses the G marker. Conversely, if the decision
maker pays more attention to regret thinking, then β = 0 and the value of Z is accepted. Figure (2)
shows the effect of changing β on Qi. In different values of β, trading partner 2 and 1 are ranked first
and second, respectively, with values below 0.5 third partner and values above 0.5 partner 4 last.
Hanieh Shambayati, Mohsen Shafiei Nikabadi, Seyed Mohammad Ali Khatami Firouzabadi and Mohammad
Rahmanimanesh, Partner Selection in Virtual Enterprises using the Interval Neutrosophic Fuzzy Approach
Neutrosophic Sets and Systems, Vol. 35, 2020
401
1/2
1
0/8
Q1
0/6
Q2
0/4
Q3
0/2
Q4
0
0
0/2
0/4
0/6
0/8
1
1/2
Fig 2. Sensitivity analysis of the value β for each alternative with INSs
Figure 3 shows the spider diagram of the sensitivity analysis and the effect of the β parameter
change on the VIKOR index. Partner rankings in this chart are centered outward, and Partner 2 in
the chart is ranked first in all β values, and Partner 2 is not second only to value β = 0. This chart
shows the gap between the partners. At point β = 0, business partner 4 ranks second. While in other
values of β the first partner is at this rank. The spider diagram shows that the distance between these
two partners is very small at this point, and the Q value of Partner 1 is only slightly different from
Partner 4, and the stability of this ratio can be confirmed. But for the third and fourth partner the
subject is slightly different and when the β value is greater than 0.5 the rating changes and the
distance between the two graphs is noticeable indicating the influence of individual views of the
group. Accordingly, when the group views are more important, the third partner is ranked third and
in the smaller values of β the individual opinions are more important. The fourth partner ranks
third. The impact of the importance of group versus individual views on this ranking is clearly
illustrated by the decrease and increase in the distance between the third and fourth partner graphs
in Figure 3.
1
0
0/2
1
0/5
Q1
Q2
Q3
0
0/8
0/4
Q4
0/6
0/5
Fig 3. Spider chart of the value β for each alternative with INSs
Sensitivity analysis showed that the value of the parameter β did not significantly influence the
results of the selection of the best partner. Therefore, the ranking results obtained using the
proposed method for INS are reliable and effective.
Hanieh Shambayati, Mohsen Shafiei Nikabadi, Seyed Mohammad Ali Khatami Firouzabadi and Mohammad
Rahmanimanesh, Partner Selection in Virtual Enterprises using the Interval Neutrosophic Fuzzy Approach
Neutrosophic Sets and Systems, Vol. 35, 2020
402
5. Conclusions
In today's business environment, competition is focused on innovation, speed, and flexibility. A new
business model is needed to help companies gain competitive benefit in the volatile market [55].
Increasing complexity has allowed any business to reconfigure itself to meet its needs, and
opportunities and remain in a highly competitive environment, because they do not have all the
skills and resources needed to meet new market demand. Virtual enterprise (VE) has been proposed
as a new organizational approach to meet the requirements of low cost, high quality, fast
responsiveness, and greater customer satisfaction to be adapted with this rapidly changing
environment [56]. The criteria for choosing a partner in Virtual Enterprises vary depending on the
type of activity. In this paper, firstly, by studying the research literature and using the experts'
opinions, 13 criteria affecting the selection of a partner in the transport sector of virtual enterprise
were identified. How to choose the right partners for success in Virtual Enterprises (VE) is very
important and has received a great deal of attention from researchers and experts. Given the
different types of uncertainty in the real environment, decision makers are usually not sure when
choosing a partner because the information on the candidates is incomplete and unclear. In addition,
some of the features of decision making are subjective and qualitative. In many cases decision
makers are unable to express their decisions about candidates in precise quantities. For this purpose,
in the second step, the partner selection problem with VIKOR method is used to form a VE under
Interval Neutrosophic environment. The VIKOR method considers the boundary rationality of
decision makers, and makes more rational decisions. Interval Neutrosophic Numbers are used to
address problems with uncertain, incomplete and inconsistent information. This method helps to
reduce the mentality of decision makers. In this paper, the method of weighting the maximum
deviation in the Neutrosophic environment is used in the absence of benchmark information, which
can be very useful in deciding issues with inconsistent and uncertain criteria. The Partner Selection
Process In this paper, we have designed a new combination and comprehensive classification of
partner evaluation criteria in the context of the virtual enterprise. The proposed approach can
effectively reduce the subjectivity and uncertainty of the multi-criteria decision-making problem and
rely on the underlying data to make the evaluation result more objective and reliable. Also, by
improving the existing method of weight calculation, the Maximizing deviation method can
effectively guarantee the consistency of the judgments and simplify the weighting function in cases
where the information is incomplete or there is no metric weight information. Expanding the VIKOR
method to Interval Neutrosophic numbers can effectively counteract uncertainty assessment
information. Without increasing mental states, it retains more decision information and makes
Partner selection in the virtual enterprise more scientific. The results of the weight sensitivity
analysis of the group utility strategy (β) show that the business firm is selected as the best partner for
all β values according to the identified effective factors. Ranked second in trading partner 1 for all
values of β, with only zero for trading partner 4. The β parameter is determined by the degree of
agreement of the decision maker, and the larger the β, the greater the group's views (too much
agreement) and the smaller the β, the greater the individual's opinions (little agreement). In this
paper, the rankings are slightly different for the smaller β values as illustrated in Fig. 2, with the
Hanieh Shambayati, Mohsen Shafiei Nikabadi, Seyed Mohammad Ali Khatami Firouzabadi and Mohammad
Rahmanimanesh, Partner Selection in Virtual Enterprises using the Interval Neutrosophic Fuzzy Approach
Neutrosophic Sets and Systems, Vol. 35, 2020
403
trading partner 4 being ranked second and the trading partner 3 last. But one still remains the top
partner.
In the actual decision making, there is much qualitative information that can be expressed by
uncertain linguistic variables. Interval Neutrosophic numbers can easily express uncertain and
contradictory information in the real world, and by combining multi-criteria decision-making
techniques to make the paradoxical features more scientific and reasonable. In this paper, the VIKOR
method is developed to deal with uncertain linguistic information in the Interval Neutrosophic
environment. In this method, the criterion values are presented as Interval Neutrosophic numbers.
Neutrosophic set with interval value is used to express incomplete knowledge of the expert group
and to prevent loss of information. However, the approach proposed for selecting the best partner in
Virtual Enterprises has advantages in terms of selection criteria. But the main limitation is the lack of
quantitative data and the limited number of respondents in the study. With increasing awareness of
Virtual Enterprises, effective benchmarks should be developed according to the field of business
activity, and other weighting techniques such as AHP, ANP and artificial intelligence techniques can
be used in combination with VIKOR. Other ranking methods such as AHP and TOPSIS can be used
in combination with the Neutrosophic environment. Optimization techniques can also be applied to
partner selection in Virtual Enterprises. The proposed model can be applied to other
decision-making issues such as supplier selection, risk assessment. Also, comparison of model
results with other uncertainty modeling techniques can be suggested. Finally, the robustness of the
proposed model can be tested through scenario analysis and uncertainty analysis.
Funding: This research received no external funding
Conflicts of Interest: The authors declare no conflict of interest.
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Received: Apr 25, 2020. Accepted: July 15 2020
Hanieh Shambayati, Mohsen Shafiei Nikabadi, Seyed Mohammad Ali Khatami Firouzabadi and Mohammad
Rahmanimanesh, Partner Selection in Virtual Enterprises using the Interval Neutrosophic Fuzzy Approach
Neutrosophic Sets and Systems, Vol. 35, 2020
University of New Mexico
Basic operations on hypersoft sets and hypersoft point
Mujahid Abbas 1, Ghulam Murtaza 2,* and Florentin Smarandache 3
1
Department of Mathematics, Government College University, Lahore 54000, Pakistan and Department of
Mathematics and Applied Mathematics,, University of Pretoria, Lynnwood Road, Pretoria 0002, South Africa;
abbas.mujahid@gmail.com,
2Department of Mathematics, University of Management Technology, Lahore, 54000, Pakistan;
ghulammurtaza@umt.edu.pk
2 University of New Mexico, USA; fsmarandache@gmail.com; smarand@unm.edu
* Correspondence: ghulammurtaza@umt.edu.pk
Abstract: The aim of this paper is to initiat formal study of hypersoft sets. We first, present basic
operations like union, intersection and difference of hypersoft sets; basic ingrediants for topological
structures on the collection of hypersoft sets. Moreover we introduce hypersoft points in different
envorinments like fuzzy hypersoft set, intuitionistic fuzzy hypersoft set, neutrosophic hypersoft,
plithogenic hypersoft set, and give some basic properties of hypersoft points in these
envorinments. We expect that this will constitue an appropriate framework of hypersoft functions
and the study of hypersoft function spaces. Examples are provided to explain the newly defined
concepts.
Keywords: soft set; hypersoft set; set operations on hypersoft sets; hypersoft point; fuzzy hypersoft
set; intuitionistic fuzzy hypersoft set; neutrosophic hypersoft; plithogenic hypersoft set.
1. Introduction
Molodtsov [16] defined soft set as a mathematical tool to deal with uncertainties associated
with real world problems. Soft set theory has application in decision making, demand analysis,
forecasting, information sciences and other disciplines (see for example, [ 13, 14, 15, 17, 18, 19, 20, 21,
22, 23]). Plithogenic and neutrosophic hypersoft sets theory is being applied successfully in decision
making problems (see, [2, 3, 4, 5, 6, 7, 8, 9,10,11,12]).
By definition, a soft set can be identified by a pair (𝐹, 𝐴), where 𝐹 stands for a multivalued
function defined on the set of parameters 𝐴.
Smarandache [1] extended the notion of a soft set to the hypersoft set by replacing the
function 𝐹 with a multi-argument function defined on the Cartesian product of 𝑛 different set of
parameters. This concept is more flexible than soft set and more suitable in the context of decision
making problems.
We expect the notion of hypersoft set will attract the attention of researchers working on soft
set theory and its diverse applications. The purpose of this paper is to initiate a formal investigation
in this new area of research.
As a first step, we present the basic operations like union, intersection and difference of
hypersoft sets. Moreover we introduce hypersoft points and some basic properties of these points
Mujahid Abbas, Ghulam Murtaza, and Florentin Smarandache, Basic operations on hypersoft sets and hypersoft point
Neutrosophic Sets and Systems, Vol. 35, 2020
408
which may provide the foundation for the hypersoft functions and hence the hypersoft fixed point
theory.
2. Operations on hypersoft sets
In this section, we define basic operations on hypersoft sets. Smarandache defined the
hypersoft set in the following manner:
Definition 1 [1] Let 𝑈 be a universe of discourse, 𝑃(𝑈) the power set 𝑈 and 𝐸1 , 𝐸2 , … , 𝐸𝑛 the pairwise
disjoint sets of parameters. Let 𝐴𝑖 be the nonempty subset of 𝐸𝑖 for each 𝑖 = 1,2, . . . , 𝑛. A hypersoft set can be
identified by the pair (𝐹, 𝐴1 × 𝐴2 × ⋯ × 𝐴𝑛 ), where:
𝐹: 𝐴1 × 𝐴2 × ⋯ × 𝐴𝑛 → 𝑃(𝑈).
For sake of simplicity, we write the symbols 𝐄 for 𝐸1 × 𝐸2 × ⋯ × 𝐸𝑛 , 𝐀 for 𝐴1 × 𝐴2 × ⋯ × 𝐴𝑛 and
𝛂 for an element of the set 𝐀. We also suppose that none of the set 𝐴𝑖 is empty.
Definition 2 [1] A hypersoft set;
on a crisp universe of discourse 𝑈𝐶 is called Crisp Hypersoft set (or simply "hypersoft set");
on a fuzzy universe of discourse 𝑈𝐹 is called Fuzzy Hypersoft set.
on a Intuitionistic Fuzzy universe of discourse 𝑈𝐼𝐹 is called Intuitionistic Fuzzy Hypersoft set;
on a Neutrosophic universe of discourse 𝑈𝑁 is called Neutrosophic Hypersoft Set;
on a Plithogenic universe of discourse 𝑈𝑃 is called Plithogenic Hypersoft Set.
The nature of 𝐹(𝛂) is determined by the nature of universe of discourse. Therefore 𝑃(𝑈)
depends upon the nature of universe. We denote ℋ(𝑈∗ , 𝐄) by the family of all *-hypersoft sets over
(𝑈∗ , 𝐄), where ∗ can take any value in the set {𝐶, 𝐹, 𝐼𝐹, 𝑁, 𝑃}, where symbols 𝐶, 𝐹, 𝐼𝐹, 𝑁, 𝑃 denote
Crisp, Fuzzy, Intuitionistic Fuzzy, Neutrosophic, and Plithogenic sets, respectively.
The following are the basic operations on *-hypersoft set.
Definition 3 Let 𝑈∗ be a universe of discourse and 𝑨 a subset of 𝑬. Then (𝐹, 𝑨) is called
1. a null *-hypersoft set if for each parameter 𝜶 ∈ 𝑨, 𝐹(𝜶) is an 0∗ . We will denote it by 𝛷𝑨 .
̃𝑨 .
2. an absolute *-hypersoft set if for each parameter 𝜶 ∈ 𝑨, 𝐹(𝜶) = 𝑈∗ . We will denote it by 𝑈
𝑥
𝑥
0
<0,1>
Remark 1 We consider 0𝐶 = ⌀ for empty set, 0𝐹 = { , 𝑥 ∈ 𝑈𝐹 } for null fuzzy set, 0𝐼𝐹 = {
for null intuitionistic fuzzy set, 0𝑁 = {
𝑥
<0,1,1>
, 𝑥 ∈ 𝑈𝐼𝐹 }
, 𝑥 ∈ 𝑈𝑁 } for null neutrosophic set. However, in case of
plithogenic set, we have the following notations:
• Null plithgenic crisp set
0𝑃𝐶 = {𝑥(0,0, . . . ,0), forall𝑥 ∈ 𝑈𝑃 }.
• Universal plithgenic crisp set
1𝑃𝐶 = {𝑥(1,1, . . . ,1), forall𝑥 ∈ 𝑈𝑃 }.
Note that null plithgenic fuzzy set will be same as null plithgenic crisp set and universal plithgenic fuzzy set
will be the same as universal plithgenic crisp set.
Mujahid Abbas, Ghulam Murtaza, and Florentin Smarandache, Basic operations on hypersoft sets and hypersoft point
Neutrosophic Sets and Systems, Vol. 35, 2020
409
• Null plithgenic intuitionistic fuzzy set
0𝑃𝐼𝐹 = {𝑥((0,1), (0,1), . . . , (0,1)), forall𝑥 ∈ 𝑈𝑃 }.
• Universal plithgenic intuitionistic fuzzy set
1𝑃𝐼𝐹 = {𝑥((1,0), (1,0), . . . , (1,0)), forall𝑥 ∈ 𝑈𝑃 }.
• Null plithgenic neutrosophic set
0𝑃𝑁 = {𝑥((0,1,1), (0,1,1), . . . , (0,1,1)), forall𝑥 ∈ 𝑈𝑃 }.
• Universal plithgenic neutrosophic set
1𝑃𝑁 = {𝑥((1,0,0), (1,0,0), . . . , (1,0,0)), forall𝑥 ∈ 𝑈}.
Definition 4 Let (𝐹, 𝑨) and (𝐺, 𝑩) be two *-hypersoft sets over 𝑈∗ . Then union of (𝐹, 𝑨) and (𝐺, 𝑩) is
̃ (𝐺, 𝑩) with 𝑪 = 𝐶1 × 𝐶2 × ⋯ × 𝐶𝑛 , where 𝐶𝑖 = 𝐴𝑖 ∪ 𝐵𝑖 for 𝑖 = 1,2, . . . , 𝑛 ,
denoted by (𝐻, 𝑪) = (𝐹, 𝑨) ∪
and 𝐻 is defined by
𝐹(𝛂),
if𝛂 ∈ 𝐀 − 𝐁
𝐺(𝛂),
if𝛂 ∈ 𝐁 − 𝐀
𝐻(𝛂) = (
𝐹(𝛂) ∪∗ 𝐺(𝛂), if𝛂 ∈ 𝐀 ∩ 𝐁,
else,
0∗ ,
where 𝛂 = (𝑐1 , 𝑐2 , . . . , 𝑐𝑛 ) ∈ 𝐶.
Remark 2 Note that, in the case of union of two hypersoft sets the set of parameters is a Cartesian product of
sets of parameters whereas in the case of union of two soft sets the set of parameter is just the union of sets of
parameters.
Definition 5 Let (𝐹, 𝑨) and (𝐺, 𝑩) be two *-hypersoft sets over 𝑈∗ . Then intersection of (𝐹, 𝑨) and (𝐺, 𝑩)
̃ (𝐺, 𝑩) , where 𝑪 = 𝐶1 × 𝐶2 × ⋯ × 𝐶𝑛 is such that 𝐶𝑖 = 𝐴𝑖 ∩ 𝐵𝑖 for 𝑖 =
is denoted by (𝐻, 𝑪) = (𝐹, 𝑨) ∩
1,2, . . . , 𝑛 and 𝐻 is defined as
𝐻(𝛂) = 𝐹(𝛂) ∩∗ 𝐺(𝛂),
̃ (𝐺, 𝑩) is defined to be a null
where 𝜶 = (𝑐1 , 𝑐2 , . . . , 𝑐𝑛 ) ∈ 𝑪. If 𝐶𝑖 is an empty set for some 𝑖, then (𝐹, 𝑨) ∩
*-hypersoft set.
Definition 6 Let (𝐹, 𝑨) and (𝐺, 𝑩) be two *-hypersoft sets over 𝑈∗ . Then (𝐹, 𝑨) is called a *-hypersoft
̃ (𝐺, 𝑩). Thus (𝐹, 𝑨)
subset of (𝐺, 𝑩) if 𝑨 ⊆ 𝑩, and 𝐹(𝜶) ⊆∗ 𝐺(𝜶) for all 𝜶 ∈ 𝑨. We denote this by (𝐹, 𝑨) ⊆
̃ (𝐺, 𝑩) and (𝐹, 𝑨) ⊇
̃ (𝐺, 𝑩).
and (𝐺, 𝑩) are said to equal if (𝐹, 𝑨) ⊆
Definition 7 Let (𝐹, 𝑨) and (𝐺, 𝑩) be two *-hypersoft sets over 𝑈∗ . Then *-hypersoft difference of (𝐹, 𝑨)
and (𝐺, 𝑩), denoted by (𝐻, 𝑪) = (𝐹, 𝑨)\̃(𝐺, 𝑩), where 𝑪 = 𝐶1 × 𝐶2 × ⋯ × 𝐶𝑛 is such that 𝐶𝑖 = 𝐴𝑖 ∩ 𝐵𝑖 for
𝑖 = 1,2, . . . , 𝑛, and 𝐻 is defined by
𝐻(𝛂) = 𝐹(𝛂)\∗ 𝐺(𝛂),
where 𝛂 = (𝑐1 , 𝑐2 , . . . , 𝑐𝑛 ) ∈ 𝐂. If 𝐶𝑖 is an empty set for some 𝑖 then (𝐹, 𝐀)\̃(𝐺, 𝐁) is defined to be
(𝐹, 𝐀).
Definition 8 The complement of a *-hypersoft set (𝐹, 𝑨) is denoted as (𝐹, 𝑨)𝑐 and is defined by (𝐹, 𝑨)𝑐 =
(𝐹 𝑐 , 𝑨) where 𝐹 𝑐 (𝜶) is the *-complemet of 𝐹(𝜶) for each 𝜶 ∈ 𝑨.
Example 1 Let U = {x1 , x2 , x3 , x4 }. Define the attributes sets by:
Mujahid Abbas, Ghulam Murtaza, and Florentin Smarandache, Basic operations on hypersoft sets and hypersoft point
Neutrosophic Sets and Systems, Vol. 35, 2020
410
𝐸1 = {𝑎11 , 𝑎12 }, 𝐸2 = {𝑎21 , 𝑎22 }, 𝐸3 = {𝑎31 , 𝑎32 }.
Suppose that
𝐴1 = {𝑎11 , 𝑎12 }, 𝐴2 = {𝑎21 , 𝑎22 }, 𝐴3 = {𝑎31 }, and
𝐵1 = {𝑎11 }, 𝐵2 = {𝑎21 , 𝑎22 }, 𝐵3 = {𝑎31 , 𝑎32 }
that is,. 𝐴𝑖 , 𝐵𝑖 ⊆ 𝐸𝑖 for each 𝑖 = 1,2,3.
Let the crisp hypersoft sets (𝐹, 𝐀) and (𝐺, 𝐁) be defined by
(𝐹, 𝐀) = {((𝑎11 , 𝑎21 , 𝑎31 ), {𝑥1 , 𝑥2 }), ((𝑎11 , 𝑎22 , 𝑎31 ), {𝑥2 }),
((𝑎12 , 𝑎21 , 𝑎31 ), {𝑥3 , 𝑥4 }), ((𝑎12 , 𝑎22 , 𝑎31 ), {𝑥1 , 𝑥4 })}.
and
(𝐺, 𝐁) = {((𝑎11 , 𝑎21 , 𝑎31 ), {𝑥2 , 𝑥3 }), ((𝑎11 , 𝑎22 , 𝑎31 ), {𝑥2 }),
((𝑎11 , 𝑎21 , 𝑎32 ), {𝑥1 , 𝑥4 }), ((𝑎11 , 𝑎22 , 𝑎32 ), {𝑥3 , 𝑥4 })}.
We have excluded those 𝛂 ∈ 𝐀 for which 𝐹(𝛂) is an empty set (similarly for those 𝛃 ∈ 𝐁 for which
𝐺(𝛃) is an empty set).
Then the union and intersections of (𝐹, 𝐀) and (𝐺, 𝐁) are given by:
̃ (𝐺, 𝐁) = {((𝑎11 , 𝑎21 , 𝑎31 ), {𝑥1 , 𝑥2 , 𝑥3 }), ((𝑎11 , 𝑎22 , 𝑎31 ), {𝑥2 }),
(𝐹, 𝐀) ∪
((𝑎12 , 𝑎21 , 𝑎31 ), {𝑥3 , 𝑥4 }), ((𝑎12 , 𝑎22 , 𝑎31 ), {𝑥1 , 𝑥4 }),
((𝑎11 , 𝑎21 , 𝑎32 ), {𝑥1 , 𝑥4 }), ((𝑎11 , 𝑎22 , 𝑎32 ), {𝑥3 , 𝑥4 }),
((𝑎12 , 𝑎21 , 𝑎32 ), 0𝐶 ), ((𝑎12 , 𝑎22 , 𝑎32 ), 0𝐶 )};
and
̃ (𝐺, 𝐁) = {((𝑎11 , 𝑎21 , 𝑎31 ), {𝑥2 }), ((𝑎11 , 𝑎22 , 𝑎31 ), {𝑥2 })}.
(𝐹, 𝐀) ∩
The differences (𝐹, 𝐀)\̃(𝐺, 𝐁) and (𝐺, 𝐁)\̃(𝐹, 𝐀) are the following
(𝐹, 𝐀)\̃(𝐺, 𝐁) = {((𝑎11 , 𝑎21 , 𝑎31 ), {𝑥1 }), ((𝑎11 , 𝑎22 , 𝑎31 ), 0𝐶 )};
(𝐺, 𝐁)\̃(𝐹, 𝐀) = {((𝑎11 , 𝑎21 , 𝑎31 ), {𝑥3 }), ((𝑎11 , 𝑎22 , 𝑎31 ), 0𝐶 )}.
Example 2 Let U = {x1 , x2 , x3 , x4 }. Define the attributes sets by:
𝐸1 = {𝑎11 , 𝑎12 }, 𝐸2 = {𝑎21 , 𝑎22 }, 𝐸3 = {𝑎31 , 𝑎32 }.
Suppose that
𝐴1 = {𝑎11 , 𝑎12 }, 𝐴2 = {𝑎21 , 𝑎22 }, 𝐴3 = {𝑎31 }, and
𝐵1 = {𝑎11 }, 𝐵2 = {𝑎21 , 𝑎22 }, 𝐵3 = {𝑎31 , 𝑎32 }
are subsets of 𝐸𝑖 for each 𝑖 = 1,2,3, that is,. 𝐴𝑖 , 𝐵𝑖 ⊆ 𝐸𝑖 for each 𝑖.
Let the fuzzy hypersoft sets (𝐹, 𝐀) and (𝐺, 𝐁) be defined by
𝑥
𝑥
𝑥
(𝐹, 𝐀) = {((𝑎11 , 𝑎21 , 𝑎31 ), { 1 , 2 }), ((𝑎11 , 𝑎22 , 𝑎31 ), { 2 }),
((𝑎12 , 𝑎21 , 𝑎31 ), {
𝑥3
,
𝑥4
0.8 0.9
0.5 0.7
}), ((𝑎12 , 𝑎22 , 𝑎31 ), {
𝑥1
,
𝑥4
0.5 0.4
0.3
})}.
and
(𝐺, 𝐁) = {((𝑎11 , 𝑎21 , 𝑎31 ), {
((𝑎11 , 𝑎21 , 𝑎32 ), {
𝑥1
,
𝑥4
0.4 0.7
𝑥2
,
𝑥3
0.2 0.9
𝑥
}), ((𝑎11 , 𝑎22 , 𝑎31 ), { 2 }),
}), ((𝑎11 , 𝑎22 , 𝑎32 ), {
𝑥3
,
𝑥4
0.2 0.8
0.6
})}.
We have excluded those 𝛂 ∈ 𝐀 for which 𝐹(𝛂) is a null fuzzy set (similarly for those 𝛃 ∈ 𝐁 for
which 𝐺(𝛃) is a null fuzzy set).
Then the union and intersections of (𝐹, 𝐀) and (𝐺, 𝐁) are given by:
Mujahid Abbas, Ghulam Murtaza, and Florentin Smarandache, Basic operations on hypersoft sets and hypersoft point
Neutrosophic Sets and Systems, Vol. 35, 2020
411
𝑥1 𝑥2 𝑥3
𝑥
, , }), ((𝑎11 , 𝑎22 , 𝑎31 ), { 2 }),
0.5 0.7 0.9
0.6
𝑥
𝑥
𝑥
𝑥
((𝑎12 , 𝑎21 , 𝑎31 ), { 3 , 4 }), ((𝑎12 , 𝑎22 , 𝑎31 ), { 1 , 4 }),
0.8 0.9
0.5 0.4
𝑥
𝑥
𝑥
𝑥
((𝑎11 , 𝑎21 , 𝑎32 ), { 1 , 4 }), ((𝑎11 , 𝑎22 , 𝑎32 ), { 3 , 4 }),
0.4 0.7
0.2 0.8
̃ (𝐺, 𝐁) = {((𝑎11 , 𝑎21 , 𝑎31 ), {
(𝐹, 𝐀) ∪
((𝑎12 , 𝑎21 , 𝑎32 ), 0𝐹 ), ((𝑎12 , 𝑎22 , 𝑎32 ), 0𝐹 )};
and
𝑥
𝑥
0.2
0.3
̃ (𝐺, 𝐁) = {((𝑎11 , 𝑎21 , 𝑎31 ), { 2 }), ((𝑎11 , 𝑎22 , 𝑎31 ), { 2 })}.
(𝐹, 𝐀) ∩
The differences (𝐹, 𝐀)\̃(𝐺, 𝐁) and (𝐺, 𝐁)\̃(𝐹, 𝐀) are the following
𝑥
𝑥
(𝐹, 𝐀)\̃(𝐺, 𝐁) = {((𝑎11 , 𝑎21 , 𝑎31 ), { 1 , 2 }), ((𝑎11 , 𝑎22 , 𝑎31 ), 0𝐹 )};
0.5 0.5
𝑥
𝑥
(𝐺, 𝐁)\̃(𝐹, 𝐀) = {((𝑎11 , 𝑎21 , 𝑎31 ), { 3 }), ((𝑎11 , 𝑎22 , 𝑎31 ), { 2 })}.
0.9
0.3
Example 3 Let U = {x1 , x2 , x3 , x4 }. Define the attributes sets by:
𝐸1 = {𝑎11 , 𝑎12 }, 𝐸2 = {𝑎21 , 𝑎22 }, 𝐸3 = {𝑎31 , 𝑎32 }.
Suppose that
𝐴1 = {𝑎11 , 𝑎12 }, 𝐴2 = {𝑎21 , 𝑎22 }, 𝐴3 = {𝑎31 }, and
𝐵1 = {𝑎11 }, 𝐵2 = {𝑎21 , 𝑎22 }, 𝐵3 = {𝑎31 , 𝑎32 }
that is,. 𝐴𝑖 , 𝐵𝑖 ⊆ 𝐸𝑖 for each 𝑖 = 1,2,3.
Let the intuitionistic fuzzy hypersoft sets (𝐹, 𝐀) and (𝐺, 𝐁) be defined by
𝑥1
(𝐹, 𝐀) = {((𝑎11 , 𝑎21 , 𝑎31 ), {
𝑥2
,
<0.5,0.3> <0.7,0.2>
((𝑎12 , 𝑎21 , 𝑎31 ), {
𝑥3
𝑥4
,
<0.8,0.1> <0.1,0.5>
}), ((𝑎11 , 𝑎22 , 𝑎31 ), {
}), ((𝑎12 , 𝑎22 , 𝑎31 ), {
𝑥1
𝑥2
<0.3,0.5>
,
𝑥4
<0.5,0.3> <0.4,0.2>
}),
})}.
and
𝑥2
(𝐺, 𝐁) = {((𝑎11 , 𝑎21 , 𝑎31 ), {
𝑥3
,
<0.2,0.6> <0.8,0.1>
((𝑎11 , 𝑎21 , 𝑎32 ), {
𝑥1
,
𝑥4
<0.4,0.5> <0.7,0.2>
}), ((𝑎11 , 𝑎22 , 𝑎31 ), {
}), ((𝑎11 , 𝑎22 , 𝑎32 ), {
𝑥3
,
𝑥2
}),
<0.6,0.3>
𝑥4
<0.4,0.2> <0.1,0.8>
})}.
We have excluded all those 𝛂 ∈ 𝐀 for which 𝐹(𝛂) is a null intuitionistic fuzzy set (similarly for
those 𝛃 ∈ 𝐁 for which 𝐺(𝛃) is a null intuitionistic fuzzy set).
The union and intersections of (𝐹, 𝐀) and (𝐺, 𝐁) are given by:
̃ (𝐺, 𝐁)
(𝐹, 𝐀) ∪
= {((𝑎11 , 𝑎21 , 𝑎31 ), {
𝑥1
,
𝑥2
,
𝑥3
<0.5,0.3> <0.7,0.2> <0.8,0.1>
((𝑎12 , 𝑎21 , 𝑎31 ), {
𝑥3
,
𝑥4
<0.8,0.1> <0.1,0.5>
((𝑎11 , 𝑎21 , 𝑎32 ), {
𝑥1
,
𝑥4
<0.4,0.5> <0.7,0.2>
}), ((𝑎11 , 𝑎22 , 𝑎31 ), {
}), ((𝑎12 , 𝑎22 , 𝑎31 ), {
𝑥1
,
𝑥2
<0.6,0.3>
𝑥4
<0.5,0.3> <0.4,0.2>
}), ((𝑎11 , 𝑎22 , 𝑎32 ), {
𝑥3
,
𝑥4
<0.4,0.2> <0.1,0.8>
}),
}),
}),
((𝑎12 , 𝑎21 , 𝑎32 ), 0𝐼𝐹 ), ((𝑎12 , 𝑎22 , 𝑎32 ), 0𝐼𝐹 )};
and
̃ (𝐺, 𝐁) = {((𝑎11 , 𝑎21 , 𝑎31 ), {
(𝐹, 𝐀) ∩
𝑥2
<0.2,0.6>
}), ((𝑎11 , 𝑎22 , 𝑎31 ), {
𝑥2
<0.3,0.5>
})}.
Mujahid Abbas, Ghulam Murtaza, and Florentin Smarandache, Basic operations on hypersoft sets and hypersoft point
Neutrosophic Sets and Systems, Vol. 35, 2020
412
The differences (𝐹, 𝐀)\̃(𝐺, 𝐁) and (𝐺, 𝐁)\̃(𝐹, 𝐀) are the following
𝑥1
𝑥2
(𝐹, 𝐀)\̃(𝐺, 𝐁) = {((𝑎11 , 𝑎21 , 𝑎31 ), {
<0.5,0.3> <0.6,0.2>
(𝐺, 𝐁)\̃(𝐹, 𝐀) = {((𝑎11 , 𝑎21 , 𝑎31 ), {
<0.2,0.7> <0.8,0.1>
𝑥2
,
,
𝑥3
𝑥2
}), ((𝑎11 , 𝑎22 , 𝑎31 ), {
<0.3,0.6>
𝑥2
}), ((𝑎11 , 𝑎22 , 𝑎31 ), {
<0.5,0.3>
})};
})}.
Example 4 Let U = {x1 , x2 , x3 , x4 }. Define the attributes sets by:
𝐸1 = {𝑎11 , 𝑎12 }, 𝐸2 = {𝑎21 , 𝑎22 }, 𝐸3 = {𝑎31 , 𝑎32 }.
Suppose that
𝐴1 = {𝑎11 , 𝑎12 }, 𝐴2 = {𝑎21 , 𝑎22 }, 𝐴3 = {𝑎31 }, and
𝐵1 = {𝑎11 }, 𝐵2 = {𝑎21 , 𝑎22 }, 𝐵3 = {𝑎31 , 𝑎32 }
that is,. 𝐴𝑖 , 𝐵𝑖 ⊆ 𝐸𝑖 for each 𝑖 = 1,2,3.
Let the neutrosophic hypersoft sets (𝐹, 𝐀) and (𝐺, 𝐁) be defined by
𝑥1
(𝐹, 𝐀) = {((𝑎11 , 𝑎21 , 𝑎31 ), {
𝑥2
,
<0.5,0.2,0.3> <0.7,0.3,0.2>
𝑥3
((𝑎12 , 𝑎21 , 𝑎31 ), {
𝑥4
,
<0.8,0.4,0.1> <0.1,0.5,0.5>
}), ((𝑎11 , 𝑎22 , 𝑎31 ), {
𝑥2
<0.3,0.2,0.5>
𝑥1
}), ((𝑎12 , 𝑎22 , 𝑎31 ), {
}),
𝑥4
,
<0.5,0.2,0.3> <0.4,0.3,0.2>
})}.
and
𝑥2
(𝐺, 𝐁) = {((𝑎11 , 𝑎21 , 𝑎31 ), {
𝑥3
,
<0.2,0.5,0.6> <0.8,0.6,0.1>
𝑥1
((𝑎11 , 𝑎21 , 𝑎32 ), {
𝑥4
,
<0.4,0.3,0.5> <0.7,0.3,0.2>
𝑥2
}), ((𝑎11 , 𝑎22 , 𝑎31 ), {
<0.6,0.2,0.3>
𝑥3
}), ((𝑎11 , 𝑎22 , 𝑎32 ), {
,
}),
𝑥4
<0.4,0.4,0.2> <0.1,0.3,0.8>
})}.
We have excluded those 𝛂 ∈ 𝐀 for which 𝐹(𝛂) is a null intuitionistic fuzzy set (similarly for those
𝛃 ∈ 𝐁 for which 𝐺(𝛃) is a null intuitionistic fuzzy set).
The union and intersections of (𝐹, 𝐀) and (𝐺, 𝐁) are given by:
̃ (𝐺, 𝐁)
(𝐹, 𝐀) ∪
𝑥1
= {((𝑎11 , 𝑎21 , 𝑎31 ), {
𝑥2
,
,
𝑥3
<0.5,0.2,0.3> <0.7,0.3,0.2> <0.8,0.6,0.1>
𝑥3
((𝑎12 , 𝑎21 , 𝑎31 ), {
,
𝑥4
<0.8,0.4,0.1> <0.1,0.5,0.5>
𝑥1
((𝑎11 , 𝑎21 , 𝑎32 ), {
,
𝑥4
<0.4,0.3,0.5> <0.7,0.3,0.2>
}), ((𝑎11 , 𝑎22 , 𝑎31 ), {
𝑥1
}), ((𝑎12 , 𝑎22 , 𝑎31 ), {
,
𝑥2
<0.6,0.2,0.3>
𝑥4
<0.5,0.2,0.3> <0.4,0.3,0.2>
𝑥3
}), ((𝑎11 , 𝑎22 , 𝑎32 ), {
,
𝑥4
<0.4,0.4,0.2> <0.1,0.3,0.8>
}),
}),
}),
((𝑎12 , 𝑎21 , 𝑎32 ), 0𝑁 ), ((𝑎12 , 𝑎22 , 𝑎32 ), 0𝑁 )};
and
̃ (𝐺, 𝐁)
(𝐹, 𝐀) ∩
= {((𝑎11 , 𝑎21 , 𝑎31 ), {
𝑥2
<0.2,0.5,0.6>
}), ((𝑎11 , 𝑎22 , 𝑎31 ), {
𝑥2
<0.3,0.2,0.5>
})}.
The differences (𝐹, 𝐀)\̃(𝐺, 𝐁) and (𝐺, 𝐁)\̃(𝐹, 𝐀) are the following
(𝐹, 𝐀)\̃(𝐺, 𝐁)
Mujahid Abbas, Ghulam Murtaza, and Florentin Smarandache, Basic operations on hypersoft sets and hypersoft point
Neutrosophic Sets and Systems, Vol. 35, 2020
= {((𝑎11 , 𝑎21 , 𝑎31 ), {
413
𝑥1
𝑥2
,
<0.5,0.2,0.3> <0.6,0.15,0.2>
}), ((𝑎11 , 𝑎22 , 𝑎31 ), {
𝑥2
<0.3,0.4,0.6>
})};
(𝐺, 𝐁)\̃(𝐹, 𝐀)
= {((𝑎11 , 𝑎21 , 𝑎31 ), {
𝑥2
,
𝑥3
<0.2,0.15,0.7> <0.8,0.6,0.1>
}), ((𝑎11 , 𝑎22 , 𝑎31 ), {
𝑥2
<0.5,0.4,0.3>
})}.
Remark 3 There are four types of plithogenic hypersoft sets namely: plithogenic crisp hypersoft set,
plithogenic fuzzy hypersoft set, plithogenic intuitionistic fuzzy hypersoft set, plithogenic neutrosophic
hypersoft set. Here we discuss only plithogenic crisp hypersoft point whereas examples for other types of sets
can be constructed in the similar way.
Example 5 Let U = {x1 , x2 , x3 , x4 }. Define the attributes sets by:
𝐸1 = {𝑎11 , 𝑎12 }, 𝐸2 = {𝑎21 , 𝑎22 }, 𝐸3 = {𝑎31 , 𝑎32 }.
Suppose that
𝐴1 = {𝑎11 , 𝑎12 }, 𝐴2 = {𝑎21 , 𝑎22 }, 𝐴3 = {𝑎31 }, and
𝐵1 = {𝑎11 }, 𝐵2 = {𝑎21 , 𝑎22 }, 𝐵3 = {𝑎31 , 𝑎32 }
that is,. 𝐴𝑖 , 𝐵𝑖 ⊆ 𝐸𝑖 for each 𝑖 = 1,2,3.
Let the plithogenic crisp hypersoft sets (𝐹, 𝐀) and (𝐺, 𝐁) be defined by
(𝐹, 𝐀) = {((𝑎11 , 𝑎21 , 𝑎31 ), {𝑥1 (1,0,1), 𝑥2 (1,1,1)}), ((𝑎11 , 𝑎22 , 𝑎31 ), {𝑥2 (0,0,1)}),
((𝑎12 , 𝑎21 , 𝑎31 ), {𝑥3 (1,1,0), 𝑥4 (1,1,1)}), ((𝑎12 , 𝑎22 , 𝑎31 ), {𝑥1 (1,0,1), 𝑥4 (0,1,0)})}.
and
(𝐺, 𝐁) = {((𝑎11 , 𝑎21 , 𝑎31 ), {𝑥2 (1,1,1), 𝑥3 (1,1,0)}), ((𝑎11 , 𝑎22 , 𝑎31 ), {𝑥2 (0,1,0)}),
((𝑎11 , 𝑎21 , 𝑎32 ), {𝑥1 (0,1,1), 𝑥4 (1,1,1)}), ((𝑎11 , 𝑎22 , 𝑎32 ), {𝑥3 (1,1,1), 𝑥4 (1,1,1)})}.
We have excluded all those 𝛂 ∈ 𝐀 for which 𝐹(𝛂) is a null plithogenic crisp set (similarly for those
𝛃 ∈ 𝐁 for which 𝐺(𝛃) is a null plithogenic crisp set).
The union and intersections of (𝐹, 𝐀) and (𝐺, 𝐁) are given by:
̃ (𝐺, 𝐁) = {((𝑎11 , 𝑎21 , 𝑎31 ), {𝑥1 (1,0,1), 𝑥2 (1,1,1), 𝑥3 (1,1,0)}),
(𝐹, 𝐀) ∪
((𝑎11 , 𝑎22 , 𝑎31 ), {𝑥2 (0,1,1)}), ((𝑎12 , 𝑎21 , 𝑎31 ), {𝑥3 (1,1,0), 𝑥4 (1,1,1)}),
((𝑎12 , 𝑎22 , 𝑎31 ), {𝑥1 (1,0,1), 𝑥4 (0,1,0)}),
((𝑎11 , 𝑎21 , 𝑎32 ), {𝑥1 (0,1,1), 𝑥4 (1,1,1)}), ((𝑎11 , 𝑎22 , 𝑎32 ), {𝑥3 (1,1,1), 𝑥4 (1,1,1)}),
((𝑎12 , 𝑎21 , 𝑎32 ), 0𝑃𝐶 ), ((𝑎12 , 𝑎22 , 𝑎32 ), 0𝑃𝐶 )};
and
̃ (𝐺, 𝐁) = {((𝑎11 , 𝑎21 , 𝑎31 ), {𝑥2 (1,1,1)}), ((𝑎11 , 𝑎22 , 𝑎31 ), 0𝑃𝐶 )}.
(𝐹, 𝐀) ∩
The differences (𝐹, 𝐀)\̃(𝐺, 𝐁) and (𝐺, 𝐁)\̃(𝐹, 𝐀) are the following
(𝐹, 𝐀)\̃(𝐺, 𝐁) = {((𝑎11 , 𝑎21 , 𝑎31 ), {𝑥1 (1,0,1)}), ((𝑎11 , 𝑎22 , 𝑎31 ), {𝑥2 (0,0,1)})};
(𝐺, 𝐁)\̃(𝐹, 𝐀) = {((𝑎11 , 𝑎21 , 𝑎31 ), {𝑥3 (1,1,0)}), ((𝑎11 , 𝑎22 , 𝑎31 ), {𝑥2 (0,1,0)})}.
Proposition 1 Let (𝐹, 𝑨) be a *-hypersoft set over 𝑈∗ . Then the following holds;
̃ Φ𝐀 = (𝐹, 𝐀);
1. (𝐹, 𝐀) ∪
Mujahid Abbas, Ghulam Murtaza, and Florentin Smarandache, Basic operations on hypersoft sets and hypersoft point
Neutrosophic Sets and Systems, Vol. 35, 2020
414
̃ Φ𝐀 = Φ𝐀 ;
2. (𝐹, 𝐀) ∩
̃𝐀 ;
̃𝐀 = 𝑈
̃𝑈
3. (𝐹, 𝐀) ∪
̃𝐀 = (𝐹, 𝐀);
̃𝑈
4. (𝐹, 𝐀) ∩
̃𝐀 \̃(𝐹, 𝐀) = (𝐹, 𝐀)𝑐 ;
5. 𝑈
̃𝐀 ;
̃ (𝐹, 𝐀)𝑐 = 𝑈
6. (𝐹, 𝐀) ∪
̃ (𝐹, 𝐀)𝑐 = Φ𝐀 .
7. (𝐹, 𝐀) ∩
Proof. We will prove only (i), (ii) and (v) and proofs of remaining are similar.
(i) By the definition of union, we have
̃ Φ𝐀 = (𝐻, 𝐂),
(𝐹, 𝐀) ∪
where 𝐂 = 𝐀 and 𝐻(𝛂) = 𝐹(𝛂) ∪∗ 0∗ = 𝐹(𝛂) for all 𝛂 ∈ 𝐂. Hence (𝐻, 𝐂) = (𝐹, 𝐀).
(ii) By the definition of intersection, we obtain that
̃ Φ𝐀 = (𝐻, 𝐂),
(𝐹, 𝐀) ∩
where 𝐂 = 𝐀 and 𝐻(𝛂) = 𝐹(𝛂) ∩∗ 0∗ = 0∗ for all 𝛂 ∈ 𝐂. Hence (𝐻, 𝐂) = Φ𝐀 .
(v) By the definition of difference, we get
̃𝐀 \̃(𝐹, 𝐀) = (𝐻, 𝐂),
𝑈
where 𝐂 = 𝐀 and 𝐻(𝛂) = 𝑈\∗ 𝐹(𝛂) = 𝐹 𝑐 (𝛂) for all 𝛂 ∈ 𝐂. Hence (𝐻, 𝐂) = (𝐹, 𝐀)𝑐 .
3. Hypersoft point
In this section, we define hypersoft point in different frameworks and study some basic
properties of such points in each setup.
3.1 Crisp hypersoft point
Definition 9 Let 𝑨 ⊆ 𝑬, 𝜶 ∈ 𝑨, and 𝑥 ∈ 𝑈. A hypersoft set (𝐹, 𝑨) is said to be a hypersoft point if 𝐹(𝜶′ ) is
an empty set for every 𝜶′ ∈ 𝑨\{𝜶} and 𝐹(𝜶) is a singleton set. We will denote hypersoft point (𝐹, 𝑨) simply
by 𝑃(𝜶,𝑥) .
Definition 10 A hypersoft set (𝐹, 𝑨) is said to be an empty hypersoft point if 𝐹(𝜶) is an empty set for each
𝜶 ∈ 𝑨. We will denote an empty hypersoft set, corresponding to 𝜶, by 𝑃(𝜶,⌀) .
As a matter of fact if (𝐹, 𝐀) is a null hypersoft set then for every 𝛂 ∈ 𝐀 it may be regarded
as empty hypersoft set 𝑃(𝛂,⌀) .
̃ (𝐺, 𝑨). We write
Definition 11 A hypersoft point 𝑃(𝜶,𝑥) is said to belong to a hypersoft set (𝐺, 𝑨) if 𝑃 (𝜶,𝑥) ⊆
̃ (𝐺, 𝑨).
it as 𝑃(𝜶,𝑥) ∈
It is straightforward to check that the hypersoft union of hypersoft points of a hypersoft set
(𝐺, 𝐀) returns the hypersoft set (𝐺, 𝐀), that is,
̃ (𝐺, 𝐀)}.
̃ {𝑃 (𝛂,𝑥) : 𝑃 (𝛂,𝑥) ∈
(𝐺, 𝐀) =∪
We illustrate the above observation through the following example.
Mujahid Abbas, Ghulam Murtaza, and Florentin Smarandache, Basic operations on hypersoft sets and hypersoft point
Neutrosophic Sets and Systems, Vol. 35, 2020
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Example 6 Let U = {x1 , x2 , x3 , x4 }, and (F, 𝐀) be as given in the example 1. Then the hypersoft points
of (F, 𝐀) are the following:
𝑃1
((𝑎11 ,𝑎21 ,𝑎31 ),𝑥1 )
= {((𝑎11 , 𝑎21 , 𝑎31 ), {𝑥1 })};
((𝑎 ,𝑎 ,𝑎 ),𝑥 )
𝑃2 11 21 31 2
((𝑎 ,𝑎 ,𝑎 ),𝑥 )
𝑃3 11 22 31 2
((𝑎 ,𝑎 ,𝑎 ),𝑥 )
𝑃4 12 21 31 3
((𝑎 ,𝑎 ,𝑎 ),𝑥 )
𝑃5 12 21 31 4
((𝑎 ,𝑎 ,𝑎 ),𝑥 )
𝑃6 12 22 31 1
((𝑎 ,𝑎 ,𝑎 ),𝑥 )
𝑃7 12 22 31 4
= {((𝑎11 , 𝑎21 , 𝑎31 ), {𝑥2 })};
= {((𝑎11 , 𝑎22 , 𝑎31 ), {𝑥2 })};
= {((𝑎12 , 𝑎21 , 𝑎31 ), {𝑥3 })};
= {((𝑎12 , 𝑎21 , 𝑎31 ), {𝑥4 })};
= {((𝑎12 , 𝑎22 , 𝑎31 ), {𝑥1 })};
= {((𝑎12 , 𝑎22 , 𝑎31 ), {𝑥4 })}.
Moreover
((𝑎11 ,𝑎21 ,𝑎31 ),𝑥1 )
(𝐹, 𝐀) = 𝑃1
̃ 𝑃2((𝑎11,𝑎21,𝑎31 ),𝑥2 ) ∪
̃ 𝑃3((𝑎11,𝑎22 ,𝑎31),𝑥2 )
∪
̃ 𝑃5((𝑎12,𝑎21,𝑎31 ),𝑥4) ∪
̃ 𝑃6((𝑎12,𝑎22 ,𝑎31),𝑥1 ) ∪
̃ 𝑃7((𝑎12 ,𝑎22,𝑎31),𝑥4 ) .
̃ 𝑃4((𝑎12 ,𝑎21,𝑎31),𝑥3) ∪
∪
Proposition 2 Let (𝐹, 𝑨), (𝐹1 , 𝑨) and (𝐹2 , 𝑨) be hypersoft sets over 𝑈. Then the following hold:
1. If (𝐹, 𝑨) is not a null hypersoft set, then (𝐹, 𝑨) contains at least one nonempty hypersoft point.
̃ (𝐹2 , 𝑨) if and only if 𝑃 (𝜶,𝑥) ∈
̃ (𝐹1 , 𝑨) implies that 𝑃 (𝜶,𝑥) ∈
̃ (𝐹2 , 𝑨).
2. (𝐹1 , 𝑨) ⊆
̃ (𝐹1 , 𝑨) ∪
̃ (𝐹2 , 𝑨) if and only if 𝑃 (𝜶,𝑥) ∈
̃ (𝐹1 , 𝑨) or 𝑃(𝜶,𝑥) ∈
̃ (𝐹2 , 𝑨).
3. 𝑃(𝜶,𝑥) ∈
̃ (𝐹1 , 𝑨) ∩
̃ (𝐹2 , 𝑨) if and only if 𝑃 (𝜶,𝑥) ∈
̃ (𝐹1 , 𝑨) and 𝑃(𝜶,𝑥) ∈
̃ (𝐹2 , 𝑨).
4. 𝑃(𝜶,𝑥) ∈
̃ (𝐹2 , 𝑨).
̃ (𝐹1 , 𝑨)\̃(𝐹2 , 𝑨) if and only if 𝑃 (𝜶,𝑥) ∈
̃ (𝐹1 , 𝑨) and 𝑃(𝜶,𝑥) ∉
5. 𝑃(𝜶,𝑥) ∈
Proof. We will prove (1), (2) and (3). Proofs of (4) and (5) are similar to that of (3).
(1) Suppose that (𝐹, 𝐀) is not a null hypersoft set, that is, 𝐹(𝛂) ≠ ⌀ for some 𝛂 ∈ 𝐀. Now if 𝛂0 ∈ 𝐀
is such that 𝐹(𝛂0 ) ≠ ⌀, then for 𝑥 ∈ 𝐹(𝛂0 ), there will be a hypersoft point 𝑃 (𝛂0,𝑥) such that
̃ (𝐹, 𝐀).
𝑃 (𝛂0,𝑥) ∈
̃ (𝐹2 , 𝐀) and 𝑃 (𝛂,𝑥) ∈
̃ (𝐹1 , 𝐀). By the definition 11, we have
(2) Suppose that (𝐹1 , 𝐀) ⊆
̃ (𝐹1 , 𝐀).
𝑃 (𝛂,𝑥) ⊆
Thus
̃ (𝐹1 , 𝐀) ⊆
̃ (𝐹2 , 𝐀)
𝑃 (𝛂,𝑥) ⊆
̃ (𝐹2 , 𝐀).
implies that 𝑃 (𝛂,𝑥) ∈
̃ (𝐹1 , 𝐀) which implies that 𝑃 (𝛂,𝑥) ∈
̃ (𝐹2 , 𝐀). By the
Conversely suppose that 𝑃(𝛂,𝑥) ∈
definition 11, we obtain that
̃ (𝐹2 , 𝐀)forall𝑃 (𝛂,𝑥) ∈
̃ (𝐹1 , 𝐀).
𝑃 (𝛂,𝑥) ⊆
Thus we have
̃ (𝐹2 , 𝐀).
̃ {𝑃 (𝛂,𝑥) : 𝑃 (𝛂,𝑥) ∈
̃ (𝐺, 𝐀)} ⊆
(𝐹1 , 𝐀) =∪
̃ (𝐹1 , 𝐀) ∪
̃ (𝐹2 , 𝐀). It follows from the definition 11 that
(3) Suppose that 𝑃 (𝛂,𝑥) ∈
̃ (𝐹1 , 𝐀) ∪
̃ (𝐹2 , 𝐀),
𝑃(𝛂,𝑥) ⊆
which implies that 𝑥 ∈ 𝐹1 (𝛂) ∪𝐶 𝐹2 (𝛂). Thus 𝑥 ∈ 𝐹1 (𝛂) or 𝐹2 (𝛂). Hence we have
̃ (𝐹1 , 𝐀)or𝑃(𝛂,𝑥) ∈
̃ (𝐹2 , 𝐀).
𝑃 (𝛂,𝑥) ∈
̃ (𝐹1 , 𝐀) or 𝑃 (𝛂,𝑥) ∈
̃ (𝐹2 , 𝐀). This implies that 𝑥 ∈ 𝐹1 (𝛂) or 𝐹2 (𝛂).
Conversely suppose that 𝑃(𝛂,𝑥) ∈
Thus 𝑥 ∈ 𝐹1 (𝛂) ∪𝐶 𝐹2 (𝛂) and so we have
̃ (𝐹1 , 𝐀) ∪
̃ (𝐹2 , 𝐀).
𝑃 (𝛂,𝑥) ⊆
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Neutrosophic Sets and Systems, Vol. 35, 2020
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3.2 Fuzzy hypersoft point
Definition 12 Let 𝑨 ⊆ 𝑬, 𝜶 ∈ 𝑨, and 𝑥 ∈ 𝑈𝐹 . A fuzzy hypersoft set (𝐹, 𝑨) is said to be a fuzzy hypersoft
point if 𝐹(𝜶′ ) is a null fuzzy set for every 𝜶′ ∈ 𝑨\{𝜶} and 𝐹(𝜶)(𝑦) = 0 for all 𝑦 ≠ 𝑥. We will denote
(𝐹, 𝑨) simply by 𝐹𝑃 (𝜶,𝑥) .
Definition 13 A fuzzy hypersoft set (𝐹, 𝑨) is said to be a null fuzzy hypersoft point if 𝐹(𝜶) is a null fuzzy
set for each 𝜶 ∈ 𝑨. We denote a null fuzzy hypersoft set, corresponding to 𝜶, by 𝐹𝑃 (𝜶,0𝐹) .
Note that if (𝐹, 𝐀) is a null fuzzy hypersoft set then for every 𝛂 ∈ 𝐀, it can be regarded as
null fuzzy hypersoft set 𝐹𝑃(𝛂,0𝐹) .
Definition 14 A fuzzy hypersoft point 𝐹𝑃 (𝜶,𝑥) is said to belong to a fuzzy hypersoft set (𝐺, 𝑨) if
̃ (𝐺, 𝑨). We write it as 𝐹𝑃 (𝜶,𝑥) ∈
̃ (𝐺, 𝑨).
𝐹𝑃(𝜶,𝑥) ⊆
It is straightforward to check that the fuzzy hypersoft union of fuzzy hypersoft points of a
fuzzy hypersoft set (𝐺, 𝐀) returns the fuzzy hypersoft set (𝐺, 𝐀), that is,
̃ (𝐺, 𝐀)}.
̃ {𝐹𝑃 (𝛂,𝑥) : 𝐹𝑃 (𝛂,𝑥) ∈
(𝐺, 𝐀) =∪
We illustrate this observation through the following example.
Example 7 Let U = {x1 , x2 , x3 , x4 }, and (F, 𝐀) be as given in the example 2. Then some of the fuzzy
hypersoft points of (F, 𝐀) are given as:
((𝑎11 ,𝑎21 ,𝑎31 ),𝑥1 )
𝐹𝑃1
𝑥
= {((𝑎11 , 𝑎21 , 𝑎31 ), { 1 })} ;
0.5
((𝑎11 ,𝑎21 ,𝑎31 ),𝑥1 )
𝐹𝑃2
𝑥
= {((𝑎11 , 𝑎21 , 𝑎31 ), { 1 })} ;
0.2
((𝑎11 ,𝑎21 ,𝑎31 ),𝑥2 )
𝐹𝑃3
𝑥
= {((𝑎11 , 𝑎21 , 𝑎31 ), { 2 })} ;
0.7
((𝑎11 ,𝑎22 ,𝑎31 ),𝑥2 )
𝐹𝑃4
𝑥
= {((𝑎11 , 𝑎22 , 𝑎31 ), { 2 })} ;
0.3
((𝑎12 ,𝑎21 ,𝑎31 ),𝑥3 )
𝐹𝑃5
𝑥
= {((𝑎12 , 𝑎21 , 𝑎31 ), { 3 })} ;
0.8
((𝑎12 ,𝑎21 ,𝑎31 ),𝑥4 )
𝐹𝑃6
𝑥
= {((𝑎12 , 𝑎21 , 𝑎31 ), { 4 })} ;
0.6
((𝑎12 ,𝑎21 ,𝑎31 ),𝑥4 )
𝐹𝑃7
𝑥
= {((𝑎12 , 𝑎21 , 𝑎31 ), { 4 })} ;
0.9
((𝑎12 ,𝑎22 ,𝑎31 ),𝑥1 )
𝐹𝑃8
𝑥
= {((𝑎12 , 𝑎22 , 𝑎31 ), { 1 })} ;
0.5
((𝑎12 ,𝑎22 ,𝑎31 ),𝑥4 )
𝐹𝑃9
𝑥
= {((𝑎12 , 𝑎22 , 𝑎31 ), { 4 })}.
0.4
Moreover we have
((𝑎11 ,𝑎21 ,𝑎31 ),𝑥1 )
(𝐹, 𝐀) = 𝐹𝑃1
̃ 𝐹𝑃2((𝑎11,𝑎21 ,𝑎31),𝑥1 ) ∪
̃ 𝐹𝑃3((𝑎11,𝑎21,𝑎31 ),𝑥2)
∪
̃ 𝐹𝑃5((𝑎12,𝑎21 ,𝑎31),𝑥3 ) ∪
̃ 𝐹𝑃6((𝑎12 ,𝑎21,𝑎31),𝑥4)
̃ 𝐹𝑃4((𝑎11,𝑎22,𝑎31),𝑥2) ∪
∪
̃ 𝐹𝑃8((𝑎12,𝑎22 ,𝑎31),𝑥1 ) ∪
̃ 𝐹𝑃9((𝑎12 ,𝑎22,𝑎31),𝑥4) .
̃ 𝐹𝑃7((𝑎12,𝑎21,𝑎31),𝑥4) ∪
∪
Proposition 3 Let (𝐹, 𝑨), (𝐹1 , 𝑨) and (𝐹2 , 𝑨) be fuzzy hypersoft sets over 𝑈. Then the following hold:
1.
If (𝐹, 𝐀) is not a null fuzzy hypersoft set, then (𝐹, 𝐀) contains at least one nonnull fuzzy
hypersoft point.
̃ (𝐹2 , 𝐀) if and only if 𝐹𝑃 (𝛂,𝑥) ∈
̃ (𝐹1 , 𝐀) implies that 𝐹𝑃(𝛂,𝑥) ∈
̃ (𝐹2 , 𝐀).
2. (𝐹1 , 𝐀) ⊆
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Neutrosophic Sets and Systems, Vol. 35, 2020
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̃ (𝐹1 , 𝐀) ∪
̃ (𝐹2 , 𝐀) if and only if 𝐹𝑃 (𝛂,𝑥) ∈
̃ (𝐹1 , 𝐀) or 𝐹𝑃(𝛂,𝑥) ∈
̃ (𝐹2 , 𝐀).
3. 𝐹𝑃(𝛂,𝑥) ∈
̃ (𝐹1 , 𝐀) ∩
̃ (𝐹2 , 𝐀) if and only if 𝐹𝑃 (𝛂,𝑥) ∈
̃ (𝐹1 , 𝐀) and 𝐹𝑃 (𝛂,𝑥) ∈
̃ (𝐹2 , 𝐀).
4. 𝐹𝑃(𝛂,𝑥) ∈
̃ (𝐹2 , 𝐀).
̃ (𝐹1 , 𝐀)\̃(𝐹2 , 𝐀) if and only if 𝐹𝑃 (𝛂,𝑥) ∈
̃ (𝐹1 , 𝐀) and 𝐹𝑃 (𝛂,𝑥) ∉
5. 𝐹𝑃(𝛂,𝑥) ∈
The proof of above proposition is similar as in the case of crisp hypersoft point.
3.3 Intuitionistic fuzzy hypersoft point
Definition 15 Let 𝑨 ⊆ 𝑬, 𝜶 ∈ 𝑨, and 𝑥 ∈ 𝑈𝐼𝐹 . An intuitionistic fuzzy hypersoft set (𝐹, 𝑨) is said to be an
intuitionistic fuzzy hypersoft point if 𝐹(𝜶′ ) is a null intuitionistic fuzzy set for every 𝜶′ ∈ 𝑨\{𝜶} and
𝐹(𝜶)(𝑦) =< 0,1 > for all 𝑦 ≠ 𝑥. We will denote (𝐹, 𝑨) simply by 𝐼𝐹𝑃 (𝜶,𝑥) .
Definition 16 An intuitionistic fuzzy hypersoft set (𝐹, 𝑨) is said to be a null intuitionistic fuzzy hypersoft
point if 𝐹(𝜶) is a null intuitionistic fuzzy set for each 𝜶 ∈ 𝑨. We will denote a null intuitionistic fuzzy
hypersoft set, corresponding to 𝜶, by 𝐼𝐹𝑃(𝜶,0𝐼𝐹) .
If (𝐹, 𝐀) is a null intuitionistic fuzzy hypersoft set, then for every 𝛂 ∈ 𝐀 it can be regarded as
null intuitionistic fuzzy hypersoft set 𝐼𝐹𝑃 (𝛂,0𝐼𝐹) .
Definition 17 An intuitionistic fuzzy hypersoft point 𝐼𝐹𝑃 (𝜶,𝑥) is said to belong to an intuitionistic fuzzy
̃ (𝐺, 𝑨). We write it as 𝐼𝐹𝑃(𝜶,𝑥) ∈
̃ (𝐺, 𝑨).
hypersoft set (𝐺, 𝑨) if 𝐼𝐹𝑃 (𝜶,𝑥) ⊆
It is straightforward to check that the intuitionistic fuzzy hypersoft union of intuitionistic fuzzy
hypersoft points of an intuitionistic fuzzy hypersoft set (𝐺, 𝐀) gives the intuitionistic fuzzy
hypersoft set (𝐺, 𝐀), that is,
̃ (𝐺, 𝐀)}.
̃ {𝐼𝐹𝑃(𝛂,𝑥) : 𝐼𝐹𝑃 (𝛂,𝑥) ∈
(𝐺, 𝐀) =∪
We illustrate this observation through the following example.
Example 8 Let U = {x1 , x2 , x3 , x4 }, and (F, 𝐀) be as given in the example 3. Then some of the
intuitionistic fuzzy hypersoft points of (F, 𝐀) are the following:
((𝑎11 ,𝑎21 ,𝑎31 ),𝑥1 )
𝐼𝐹𝑃1
= {((𝑎11 , 𝑎21 , 𝑎31 ), {
((𝑎11 ,𝑎21 ,𝑎31 ),𝑥1 )
𝐼𝐹𝑃2
= {((𝑎11 , 𝑎21 , 𝑎31 ), {
𝑥1
<0.2,0.3>
((𝑎11 ,𝑎21 ,𝑎31 ),𝑥2 )
𝐼𝐹𝑃3
= {((𝑎11 , 𝑎21 , 𝑎31 ), {
𝑥2
<0.7,0.2>
((𝑎11 ,𝑎22 ,𝑎31 ),𝑥2 )
𝐼𝐹𝑃4
= {((𝑎11 , 𝑎22 , 𝑎31 ), {
𝑥2
<0.3,0.5>
((𝑎12 ,𝑎21 ,𝑎31 ),𝑥3 )
𝐼𝐹𝑃5
= {((𝑎12 , 𝑎21 , 𝑎31 ), {
𝑥3
<0.8,0.1>
((𝑎12 ,𝑎21 ,𝑎31 ),𝑥4 )
𝐼𝐹𝑃6
= {((𝑎12 , 𝑎21 , 𝑎31 ), {
𝑥4
<0.1,0.6>
(𝑎12 ,𝑎21 ,𝑎31 ),𝑥4 )
𝐼𝐹𝑃𝐼𝐼𝐼7
((𝑎12 ,𝑎22 ,𝑎31 ),𝑥1 )
𝐼𝐹𝑃8
((𝑎12 ,𝑎22 ,𝑎31 ),𝑥4 )
𝐼𝐹𝑃9
𝑥1
<0.5,0.3>
= {((𝑎12 , 𝑎21 , 𝑎31 ), {
= {((𝑎12 , 𝑎22 , 𝑎31 ), {
})} ;
})} ;
})} ;
})} ;
})} ;
𝑥4
<0.1,0.5>
𝑥1
<0.5,0.3>
= {((𝑎12 , 𝑎22 , 𝑎31 ), {
})} ;
𝑥4
<0.4,0.2>
})} ;
})} ;
})}.
Mujahid Abbas, Ghulam Murtaza, and Florentin Smarandache, Basic operations on hypersoft sets and hypersoft point
Neutrosophic Sets and Systems, Vol. 35, 2020
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Moreover we have
((𝑎11 ,𝑎21 ,𝑎31 ),𝑥1 )
(𝐹, 𝐀) = 𝐼𝐹𝑃1
̃ 𝐼𝐹𝑃2((𝑎11 ,𝑎21,𝑎31),𝑥1 ) ∪
̃ 𝐼𝐹𝑃3((𝑎11 ,𝑎21,𝑎31),𝑥2 )
∪
̃ 𝐼𝐹𝑃5((𝑎12,𝑎21 ,𝑎31),𝑥3 ) ∪
̃ 𝐼𝐹𝑃6((𝑎12 ,𝑎21,𝑎31),𝑥4 )
̃ 𝐼𝐹𝑃4((𝑎11,𝑎22 ,𝑎31),𝑥2 ) ∪
∪
̃ 𝐼𝐹𝑃8((𝑎12,𝑎22 ,𝑎31),𝑥1 ) ∪
̃ 𝐼𝐹𝑃9((𝑎12 ,𝑎22,𝑎31),𝑥4 ) .
̃ 𝐼𝐹𝑃7((𝑎12,𝑎21 ,𝑎31),𝑥4 ) ∪
∪
Proposition 4 Let (𝐹, 𝑨), (𝐹1 , 𝑨) and (𝐹2 , 𝑨) be intuitionistic fuzzy hypersoft sets over 𝑈 . Then the
following hold:
1. If (𝐹, 𝐀) is not a null intuitionistic fuzzy hypersoft set then (𝐹, 𝐀) contains at least one nonnull
intuitionistic fuzzy hypersoft point.
̃ (𝐹2 , 𝑨) if and only if 𝐼𝐹𝑃 (𝜶,𝑥) ∈
̃ (𝐹1 , 𝑨) implies that 𝐼𝐹𝑃(𝜶,𝑥) ∈
̃ (𝐹2 , 𝑨).
2. (𝐹1 , 𝑨) ⊆
̃ (𝐹1 , 𝑨) ∪
̃ (𝐹2 , 𝑨) if and only if 𝐼𝐹𝑃 (𝜶,𝑥) ∈
̃ (𝐹1 , 𝑨) or 𝐼𝐹𝑃 (𝜶,𝑥) ∈
̃ (𝐹2 , 𝑨).
3. 𝐼𝐹𝑃(𝜶,𝑥) ∈
̃ (𝐹1 , 𝑨) ∩
̃ (𝐹2 , 𝑨) if and only if 𝐼𝐹𝑃 (𝜶,𝑥) ∈
̃ (𝐹1 , 𝑨) and 𝐼𝐹𝑃 (𝜶,𝑥) ∈
̃ (𝐹2 , 𝑨).
4. 𝐼𝐹𝑃(𝜶,𝑥) ∈
(𝜶,𝑥) ̃
(𝜶,𝑥)
(𝜶,𝑥)
̃ (𝐹2 , 𝑨).
̃ (𝐹1 , 𝑨) and 𝐼𝐹𝑃
∈ (𝐹1 , 𝑨)\̃(𝐹2 , 𝑨) if and only if 𝐼𝐹𝑃
∈
∉
5. 𝐼𝐹𝑃
The proof of above proposition is similar as in the case of crisp hypersoft point.
3.4 Neutrosophic hypersoft point
Definition 18 Let 𝑨 ⊆ 𝑬 and 𝜶 ∈ 𝑨 , 𝑥 ∈ 𝑈𝑁 . A neutrosophic hypersoft set (𝐹, 𝑨) is said to be a
neutrosophic fuzzy hypersoft point if 𝐹(𝜶′ ) is a null neutrosophic set for every 𝜶′ ∈ 𝑨\{𝜶} and 𝐹(𝜶)(𝑦) =<
0,1,1 > for all 𝑦 ≠ 𝑥. We will denote (𝐹, 𝑨) simply by 𝑁𝑃 (𝜶,𝑥) .
Definition 19 A neutrosophic hypersoft set (𝐹, 𝑨) is said to be a null neutrosophic hypersoft point if 𝐹(𝜶) is
a null neutrosophic set for each 𝜶 ∈ 𝑨. We will denote a null neutrosophic hypersoft set, corresponding to 𝜶,
by 𝑁𝑃(𝜶,0𝑁) .
Its a matter of fact that if (𝐹, 𝐀) is a null neutrosophic hypersoft set then for every 𝛂 ∈ 𝐀 it
can be regarded as null neutrosophic hypersoft set 𝑁𝑃(𝛂,0𝑁) .
Definition 20 A neutrosophic hypersoft point 𝑁𝑃 (𝜶,𝑥) is said to belong to a neutrosophic hypersoft set (𝐺, 𝑨)
̃ (𝐺, 𝑨). We write it as 𝑁𝑃(𝜶,𝑥) ∈
̃ (𝐺, 𝑨).
if 𝑁𝑃(𝜶,𝑥) ⊆
It is straightforward to check that the neutrosophic hypersoft union of neutrosophic
hypersoft points of a neutrosophic hypersoft set (𝐺, 𝐀) returns the neutrosophic hypersoft set
(𝐺, 𝐀), that is,
̃ (𝐺, 𝐀)}.
̃ {𝑁𝑃 (𝛂,𝑥) : 𝑁𝑃(𝛂,𝑥) ∈
(𝐺, 𝐀) =∪
We illustrate this observation through the following example.
Example 9 Let U = {x1 , x2 , x3 , x4 }, and (F, 𝐀) be as given in the example 4. Some of the neutrosophic
hypersoft points of (F, 𝐀) are the following:
((𝑎11 ,𝑎21 ,𝑎31 ),𝑥1 )
𝑁𝑃1
((𝑎11 ,𝑎21 ,𝑎31 ),𝑥1 )
𝑁𝑃2
((𝑎11 ,𝑎21 ,𝑎31 ),𝑥2 )
𝑁𝑃3
((𝑎11 ,𝑎22 ,𝑎31 ),𝑥2 )
𝑁𝑃4
𝑥1
= {((𝑎11 , 𝑎21 , 𝑎31 ), {
<0.5,0.2,0.3>
= {((𝑎11 , 𝑎21 , 𝑎31 ), {
<0.2,0.2,0.3>
= {((𝑎11 , 𝑎21 , 𝑎31 ), {
<0.7,0.3,0.2>
= {((𝑎11 , 𝑎22 , 𝑎31 ), {
<0.3,0.2,0.5>
𝑥1
𝑥2
𝑥2
})} ;
})} ;
})} ;
})} ;
Mujahid Abbas, Ghulam Murtaza, and Florentin Smarandache, Basic operations on hypersoft sets and hypersoft point
Neutrosophic Sets and Systems, Vol. 35, 2020
((𝑎12 ,𝑎21 ,𝑎31 ),𝑥3 )
𝑁𝑃5
((𝑎12 ,𝑎21 ,𝑎31 ),𝑥4 )
𝑁𝑃6
((𝑎12 ,𝑎22 ,𝑎31 ),𝑥1 )
𝑁𝑃7
((𝑎12 ,𝑎22 ,𝑎31 ),𝑥4 )
𝑁𝑃8
419
𝑥3
= {((𝑎12 , 𝑎21 , 𝑎31 ), {
<0.8,0.4,0.1>
= {((𝑎12 , 𝑎21 , 𝑎31 ), {
<0.1,0.5,0.5>
= {((𝑎12 , 𝑎22 , 𝑎31 ), {
<0.5,0.2,0.3>
= {((𝑎12 , 𝑎22 , 𝑎31 ), {
<0.4,0.3,0.2>
𝑥4
𝑥1
𝑥4
})} ;
})} ;
})} ;
})}.
Moreover we have
((𝑎11 ,𝑎21 ,𝑎31 ),𝑥1 )
(𝐹, 𝐀) = 𝑁𝑃1
̃ 𝑁𝑃2((𝑎11,𝑎21 ,𝑎31),𝑥1 ) ∪
̃ 𝑁𝑃3((𝑎11 ,𝑎21,𝑎31 ),𝑥2 )
∪
̃ 𝑁𝑃5((𝑎12,𝑎21,𝑎31 ),𝑥3 ) ∪
̃ 𝑁𝑃6((𝑎12 ,𝑎21,𝑎31),𝑥4 )
̃ 𝑁𝑃4((𝑎11 ,𝑎22,𝑎31),𝑥2) ∪
∪
̃ 𝑁𝑃8((𝑎12,𝑎22,𝑎31 ),𝑥4 ) .
̃ 𝑁𝑃7((𝑎12 ,𝑎22,𝑎31),𝑥1) ∪
∪
Proposition 5 Let (𝐹, 𝑨), (𝐹1 , 𝑨) and (𝐹2 , 𝑨) be neutrosophic hypersoft sets over 𝑈. Then the following
hold:
1. If (𝐹, 𝑨) is not a null neutrosophic hypersoft set then (𝐹, 𝑨) contains at least one nonnull neutrosophic
hypersoft point.
̃ (𝐹2 , 𝑨) if and only if 𝑁𝑃 (𝜶,𝑥) ∈
̃ (𝐹1 , 𝑨) implies that 𝑁𝑃 (𝜶,𝑥) ∈
̃ (𝐹2 , 𝑨).
2. (𝐹1 , 𝑨) ⊆
̃ (𝐹1 , 𝑨) ∪
̃ (𝐹2 , 𝑨) if and only if 𝑁𝑃 (𝜶,𝑥) ∈
̃ (𝐹1 , 𝑨) or 𝑁𝑃(𝜶,𝑥) ∈
̃ (𝐹2 , 𝑨).
3. 𝑁𝑃(𝜶,𝑥) ∈
̃ (𝐹1 , 𝑨) ∩
̃ (𝐹2 , 𝑨) if and only if 𝑁𝑃 (𝜶,𝑥) ∈
̃ (𝐹1 , 𝑨) and 𝑁𝑃(𝜶,𝑥) ∈
̃ (𝐹2 , 𝑨).
4. 𝑁𝑃(𝜶,𝑥) ∈
̃ (𝐹2 , 𝑨).
̃ (𝐹1 , 𝑨)\̃(𝐹2 , 𝑨) if and only if 𝑁𝑃 (𝜶,𝑥) ∈
̃ (𝐹1 , 𝑨) and 𝑁𝑃(𝜶,𝑥) ∉
5. 𝑁𝑃(𝜶,𝑥) ∈
The proof of above proposition is similar as in the case of crisp hypersoft point.
3.5 Plithogenic hypersoft point
There may be four types of plithogenic hypersoft points namely: plithogenic crisp hypersoft
point, plithogenic fuzzy hypersoft point, plithogenic intuitionistic fuzzy hypersoft point, plithogenic
neutrosophic hypersoft point. But in this section we discuss only plithogenic crisp hypersoft point
whereas other concepts and examples can be given in the similar way.
Definition 21 Let 𝑨 ⊆ 𝑬, 𝜶 ∈ 𝑨, and 𝑥 ∈ 𝑈𝑃 . A plithogenic crisp hypersoft set (𝐹, 𝑨) is said to be a
plithogenic crisp hypersoft point if 𝐹(𝜶′ ) is a null plithogenic crisp set for every 𝜶′ ∈ 𝑨\{𝜶} and
𝐹(𝜶)(𝑦)(𝟎) for all 𝑦 ≠ 𝑥. We will denote (𝐹, 𝑨) simply by 𝑃𝑐 𝑃 (𝜶,𝑥) .
Definition 22 A plithogenic crisp hypersoft set (𝐹, 𝑨) is said to be a null plithogenic crisp hypersoft point if
𝐹(𝜶) is a null plithogenic crisp set for each 𝜶 ∈ 𝑨. We will denote a null plithogenic crisp hypersoft set,
corresponding to 𝜶, by 𝑃𝑐 𝑃(𝜶,0𝑃𝐶) .
Note that if (𝐹, 𝐀) is a null plithogenic crisp hypersoft set, then for every 𝛂 ∈ 𝐀 it can be
regarded as a null plithogenic crisp hypersoft set 𝑃𝑐 𝑃(𝛂,0𝑃𝐶) .
Definition 23 A plithogenic crisp hypersoft point 𝑃𝑐 𝑃 (𝜶,𝑥) is said to belong to a plithogenic crisp hypersoft set
̃ (𝐺, 𝑨). We write it as 𝑃𝑐 𝑃(𝜶,𝑥) ∈
̃ (𝐺, 𝑨).
(𝐺, 𝑨) if 𝑃𝑐 𝑃(𝜶,𝑥) ⊆
Mujahid Abbas, Ghulam Murtaza, and Florentin Smarandache, Basic operations on hypersoft sets and hypersoft point
Neutrosophic Sets and Systems, Vol. 35, 2020
420
It is straightforward to check that the plithogenic crisp hypersoft union of plithogenic crisp
hypersoft points of a plithogenic crisp hypersoft set (𝐺, 𝐀) gives back the plithogenic crisp
hypersoft set (𝐺, 𝐀), that is,
̃ (𝐺, 𝐀)}.
̃ {𝑃𝑐 𝑃 (𝛂,𝑥) : 𝑃𝑐 𝑃 (𝛂,𝑥) ∈
(𝐺, 𝐀) =∪
We illustrate this observation through the following example.
Example 10 Let U = {x1 , x2 , x3 , x4 }, and (F, 𝐀) be as given in the example 5. Then some of the
plithogenic crisp hypersoft points of (F, 𝐀) are the following:
𝑃𝑐 𝑃1
((𝑎11 ,𝑎21 ,𝑎31 ),𝑥1 )
= {((𝑎11 , 𝑎21 , 𝑎31 ), {𝑥1 (1,0,1)})};
((𝑎 ,𝑎 ,𝑎 ),𝑥 )
𝑃𝑐 𝑃2 11 21 31 2
((𝑎 ,𝑎 ,𝑎 ),𝑥 )
𝑃𝑐 𝑃3 11 22 31 2
((𝑎 ,𝑎 ,𝑎 ),𝑥 )
𝑃𝑐 𝑃4 12 21 31 3
((𝑎 ,𝑎 ,𝑎 ),𝑥 )
𝑃𝑐 𝑃5 12 21 31 4
((𝑎 ,𝑎 ,𝑎 ),𝑥 )
𝑃𝑐 𝑃6 12 22 31 1
((𝑎 ,𝑎 ,𝑎 ),𝑥 )
𝑃𝑐 𝑃7 12 22 31 4
= {((𝑎11 , 𝑎21 , 𝑎31 ), {𝑥2 (1,1,1)})};
= {((𝑎11 , 𝑎22 , 𝑎31 ), {𝑥2 (0,0,1)})};
= {((𝑎12 , 𝑎21 , 𝑎31 ), {𝑥3 (1,1,0)})};
= {((𝑎12 , 𝑎21 , 𝑎31 ), {𝑥4 (1,1,1)})};
= {((𝑎12 , 𝑎22 , 𝑎31 ), {𝑥1 (1,0,1)})};
= {((𝑎12 , 𝑎22 , 𝑎31 ), {𝑥4 (0,1,0)})}.
Moreover we have
((𝑎11 ,𝑎21 ,𝑎31 ),𝑥1 )
(𝐹, 𝐀) = 𝑃𝑐 𝑃1
̃ 𝑃𝑐 𝑃2((𝑎11,𝑎21 ,𝑎31),𝑥2 )
∪
̃ 𝑃𝑐 𝑃4((𝑎12,𝑎21 ,𝑎31),𝑥3 ) ∪
̃ 𝑃𝑐 𝑃5((𝑎12 ,𝑎21,𝑎31),𝑥4)
̃ 𝑃𝑐 𝑃3((𝑎11,𝑎22,𝑎31),𝑥2) ∪
∪
̃ 𝑃𝑐 𝑃7((𝑎12,𝑎22 ,𝑎31),𝑥4 ) .
̃ 𝑃𝑐 𝑃6((𝑎12,𝑎22,𝑎31),𝑥1) ∪
∪
Proposition 6 Let (𝐹, 𝑨), (𝐹1 , 𝑨) and (𝐹2 , 𝑨) be plithogenic crisp hypersoft sets over 𝑈. Then the following
hold:
1. If (𝐹, 𝑨) is not a null plithogenic crisp hypersoft set then (𝐹, 𝑨) contains at least one nonnull plithogenic
crisp hypersoft point.
̃ (𝐹2 , 𝑨) if and only if 𝑃𝑐 𝑃 (𝜶,𝑥) ∈
̃ (𝐹1 , 𝑨) implies that 𝑃𝑐 𝑃 (𝜶,𝑥) ∈
̃ (𝐹2 , 𝑨).
2. (𝐹1 , 𝑨) ⊆
̃ (𝐹1 , 𝑨) ∪
̃ (𝐹2 , 𝑨) if and only if 𝑃𝑐 𝑃(𝜶,𝑥) ∈
̃ (𝐹1 , 𝑨) or 𝑃𝑐 𝑃 (𝜶,𝑥) ∈
̃ (𝐹2 , 𝑨).
3. 𝑃𝑐 𝑃 (𝜶,𝑥) ∈
̃ (𝐹1 , 𝑨) ∩
̃ (𝐹2 , 𝑨) if and only if 𝑃𝑐 𝑃 (𝜶,𝑥) ∈
̃ (𝐹1 , 𝑨) and 𝑃𝑐 𝑃(𝜶,𝑥) ∈
̃ (𝐹2 , 𝑨).
4. 𝑃𝑐 𝑃 (𝜶,𝑥) ∈
(𝜶,𝑥) ̃
(𝜶,𝑥)
(𝜶,𝑥)
̃ (𝐹2 , 𝑨).
̃ (𝐹1 , 𝑨) and 𝑃𝑐 𝑃
∈ (𝐹1 , 𝑨)\̃(𝐹2 , 𝑨) if and only if 𝑃𝑐 𝑃
∈
∉
5. 𝑃𝑐 𝑃
The proof of above proposition is similar as in the case of crisp hypersoft point.
4. Conclusions
In this paper, we have initiated the concept of hypersoft point that will lead to define
Cartesian product and then function on *-hypersoft sets. As a future work, one may carry out the
study of *-hypersoft topological spaces. Once the functions on *-hypersoft sets are defined, this may
lead to the study of fixed point results in this new framework.
Funding: This research received no external funding.
Conflicts of Interest: The authors declare no conflict of interest.
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Received: Apr 15, 2020.
Accepted: July 5 2020
Mujahid Abbas, Ghulam Murtaza, and Florentin Smarandache, Basic operations on hypersoft sets and hypersoft point
Neutrosophic Sets and Systems, Vol. 35, 2020
University of New MeSico
Neutrosophic ℵ −bi-ideals in semigroups
K. Porselvi1, B. Elavarasan2*, F. Smarandache3 and Y. B. Jun4
1,2Department
of Mathematics, Karunya Institute of Technology and Sciences, Coimbatore - 641 114, Tamilnadu, India.
E-mail: porselvi94@yahoo.co.in; porselvi@karunya.edu.
E-mail: belavarasan@gmail.com;elavarasan@karunya.edu.
3 Mathematics Department, University of New Mexico, 705 Gurley Ave., Gallup, NM 87301, USA.
E-mail:fsmarandache@gmail.com; smarand@unm.edu.
4 Department of Mathematics, Gyeongsang National University, Jinju 52828, Korea.
E-mail: skywine@gmail.com
* Correspondence: belavarasan@gmail.com
Abstract: In this paper, we introduce the notion of neutrosophic ℵ-bi-ideal for a semigroup. We
infer different semigroups using neutrosophic ℵ -bi-ideal structures. Moreover, for regular
semigroups, neutrosophic ℵ-product and intersection of neutrosophic ℵ-ideals are identical.
Keywords: Semigroup, ideal, bi-ideal, neutrosophic ℵ − ideals, neutrosophic ℵ -bi-ideals,
neutrosophic ℵ −product.
1.
Introduction
In 1965, Zadeh [16] introduced the idea of fuzzy sets for modeling the ambiguous theories in the
globe. In 1986, Atanassov [1] generalized fuzzy set and named as intuitionistic fuzzy set, and
discussed it. Also from his view point, there are two degrees for any object in the world. They are
degree of membership to a vague subset and degree of non-membership to that given subset.
Smarandache generalized fuzzy and intuitionistic fuzzy set, and referred as Neutrosophic set
(see [2, 3, 6, 13-15]). It is identified by a truth, a falsity and an indeterminacy membership function.
These sets are applied to many branches of mathematics to overcome the complexities arising from
uncertain data. Neutrosophic set can distinguish between absolute membership and relative
membership. Smarandache used this in non-standard analysis such as result of sport games
(winning/defeating/tie), decision making and control theory, etc. This area has been studied by
several authors (see [5, 10-12]).
In [8], M. Khan et al. presented and discussed the concepts of neutrosophic ℵ −subsemigroup
of semigroup. In [5], Gulistan et al. have studied the idea of complex neutrosophic subsemigroups.
They have introduced the notion of characteristic function of complex neutrosophic sets, direct
product of complex neutrosophic sets.
In [4], B. Elavarasan et al. introduced the concepts of neutrosophic ℵ −ideal of semigroup and
explored its properties. Also, the conditions are given for neutrosophic ℵ −structure becomes
neutrosophic ℵ −ideal. Further, presented the notion of characteristic neutrosophic ℵ −structure
over semigroup.
Throughout this article, 𝑋 denotes a semigroup. Recall that for any subsets 𝐴 and 𝐵 of 𝑋,
𝐴𝐵 = {𝑢𝑤|𝑢 ∈ 𝐴 𝑎𝑛𝑑 𝑤 ∈ 𝐵}, the multiplication of A and B.
For a semigroup X,
(i) ∅ ≠ 𝑈 ⊆ 𝑋 is a subsemigroup of 𝑋 if 𝑈 2 ⊆ 𝑈.
K. Porselvi, B. Elavarasan, F. Smarandache and Y. B. Jun, Neutrosophic ℵ-bi-ideals in semigroups
Neutrosophic Sets and Systems, Vol. 35, 2020
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(ii) A subsemigroup 𝑈 of X is left (resp., right) ideal if 𝑋𝑈 ⊆ 𝑈 (resp., 𝑈𝑋 ⊆ 𝑈). 𝑈 is an ideal of 𝑋
if 𝑈 is both left and right ideal of 𝑋.
(iii) 𝑋 is left (resp., right) regular if for each 𝑠 ∈ 𝑋, there exists 𝑥 ∈ 𝑋 such that 𝑠 = 𝑥𝑠 2 (resp., 𝑠 =
𝑠 2 𝑥) [7].
(iv) 𝑋 is regular if for each 𝑠 ∈ 𝑋, there exists 𝑥 ∈ 𝑋 such that 𝑠 = 𝑠𝑥𝑠 [9].
(v) 𝑋 is intra-regular if for every𝑠 ∈ 𝑋, there exist 𝑥, 𝑦 ∈ 𝑋 such that 𝑠 = 𝑥𝑠 2 𝑦 [9].
2
(vi) A subsemigroup 𝑌 of 𝑋 is bi-ideal if 𝑌𝑋𝑌 ⊆ 𝑌. For any 𝑟′ ∈ 𝑋, 𝐵(𝑟 ′ ) = {𝑟′, 𝑟 ′ , 𝑟′𝑋𝑟′} is the
principal bi-ideal of 𝑋 generated by 𝑟 ′ .
2. Basics of neutrosophic ℵ – structures
In this section, we present the required basic definitions of neutrosophic ℵ −structures of 𝑋 that
we need in the sequel.
The collection of functions from a set 𝑋 to [−1, 0] is denoted by ℑ(𝑋, [−1, 0]). Note that
f
∈ ℑ(𝑋, [−1, 0]) is a negative-valued function from 𝑋 to [−1, 0] (briefly, ℵ −function on 𝑋). Here
ℵ −structure means (𝑋, 𝑓) of 𝑋.
Definition 2.1. [8] A neutrosophic ℵ − structure of 𝑋 is defined to be the structure:
𝑋𝑁 : =
𝑋
(𝑇𝑁 ,𝐼𝑁 , 𝐹𝑁 )
= {
𝑥
𝑇𝑁 (𝑥), 𝐼𝑁 (𝑥), 𝐹𝑁 (𝑥)
|𝑥 ∈ 𝑋}
where 𝑇𝑁 is the negative truth membership function on X, 𝐼𝑁 is the negative indeterminacy
membership function on X and 𝐹𝑁 is the negative falsity membership function on X.
Note that for any 𝑥 ∈ 𝑋, 𝑋𝑁 satisfies the condition −3 ≤ 𝑇𝑁 (𝑥) + 𝐼𝑁 (𝑥) + 𝐹𝑁 (𝑥) ≤ 0.
Definition 2.2. [8] A neutrosophic ℵ −structure 𝑋𝑁 of 𝑋 is called a neutrosophic ℵ −subsemigroup
of 𝑋 if the below condition is valid:
𝑇𝑁 (𝑔𝑖 ℎ𝑗 ) ≤ 𝑇𝑁 (𝑔𝑖 ) ˅ 𝑇𝑁 (ℎ𝑗 )
(∀ 𝑔𝑖 , ℎ𝑗 ∈ 𝑋) ( 𝐼𝑁 (𝑔𝑖 ℎ𝑗 ) ≥ 𝐼𝑁 (𝑔𝑖 ) ˄ 𝐼𝑁 (ℎ𝑗 ) ).
𝐹𝑁 (𝑔𝑖 ℎ𝑗 ) ≤ 𝐹𝑁 (𝑔𝑖 ) ˅ 𝐹𝑁 (ℎ𝑗 )
Let 𝑋𝑁 be a neutrosophic ℵ − structure of 𝑋 and let 𝜆, 𝛿, ε ∈ [−1, 0] with −3 ≤ 𝜆 + 𝛿 + ε ≤
0. Then the set 𝑋𝑁 (𝜆, 𝛿, ε) ≔ {𝑥 ∈ 𝑋|𝑇𝑁 (𝑥) ≤ 𝜆, 𝐼𝑁 (𝑥) ≥ 𝛿, 𝐹𝑁 (𝑥) ≤ ε} is called a (λ, 𝛿, ε) – level set
of XN.
Definition 2.3. [4] A neutrosophic ℵ −structure 𝑋𝑁 of 𝑋 is called a neutrosophic ℵ −left (resp.,
right) ideal of 𝑋 if it satisfies:
𝑇𝑁 (𝑔𝑖 ℎ𝑗 ) ≤ 𝑇𝑁 (ℎ𝑗 ) (𝑟𝑒𝑠𝑝. , 𝑇𝑁 (𝑔𝑖 ℎ𝑗 ) ≤ 𝑇𝑁 (𝑔𝑖 ))
(∀ 𝑔𝑖 , ℎ𝑗 ∈ 𝑋) ( 𝐼𝑁 (𝑔𝑖 ℎ𝑗 ) ≥ 𝐼𝑁 (ℎ𝑗 ) (𝑟𝑒𝑠𝑝., 𝐼𝑁 (𝑔𝑖 ℎ𝑗 ) ≥ 𝐼𝑁 (𝑔𝑖 )) ).
𝐹𝑁 (𝑔𝑖 ℎ𝑗 ) ≤ 𝐹𝑁 (ℎ𝑗 ) (𝑟𝑒𝑠𝑝., 𝐹𝑁 (𝑔𝑖 ℎ𝑗 ) ≤ 𝐹𝑁 (𝑔𝑖 ))
If 𝑋𝑁 is both neutrosophic ℵ −left and neutrosophic ℵ −right ideal of X, then it is called a
neutrosophic ℵ −ideal of X.
Definition 2.4. A neutrosophic ℵ −subsemigroup 𝑋𝑁 of 𝑋 is a neutrosophic ℵ −bi-ideal of 𝑋 if
the following condition is valid:
K. Porselvi, B. Elavarasan, F. Smarandache and Y. B. Jun, Neutrosophic ℵ-bi-ideals in semigroups
Neutrosophic Sets and Systems, Vol. 35, 2020
424
𝑇𝑁 (𝑟𝑠𝑡) ≤ 𝑇𝑁 (𝑟)˅ 𝑇𝑁 (𝑡)
(∀ 𝑟, 𝑠, 𝑡 ∈ 𝑋) ( 𝐼𝑁 (𝑟𝑠𝑡) ≥ 𝐼𝑁 (𝑟)˄ 𝐼𝑁 (𝑡) ).
𝐹𝑁 (𝑟𝑠𝑡) ≤ 𝐹𝑁 (𝑟)˅ 𝐹𝑁 (𝑡)
Clearly any neutrosophic ℵ − left (resp., right) ideal is neutrosophic ℵ −bi-ideal, but the
neutrosophic ℵ −bi-ideal is not necessary to be a neutrosophic ℵ −left (resp., right) ideal.
Example 2.5. Consider the semigroup 𝑋 = {0, 𝑎, 𝑏, 𝑐} with binary operation as follows:
.
0
a
b
c
Then 𝑋𝑁 = {
0
(−0.9,−0.1,−0.7)
,
𝑎
,
0
0
0
0
b
a
0
0
0
b
𝑏
b
0
0
0
b
c
0
b
b
c
𝑐
,
} is a neutrosophic ℵ −bi-ideal of
(−0.8,−0.2,−0.5) (−0.7,−0.3,−0.3) (−0.5,−0.4,−0.1)
𝑋, but 𝑋𝑁 is not neutrosophic ℵ −left ideal as well as neutrosophic ℵ −right ideal of 𝑋.
□
Definition 2.6. [8] For Φ ≠ A ⊆ 𝑋, the characteristic neutrosophic ℵ −structure of 𝑋 is denoted by
𝜒𝐴 (𝑋𝑁 ) and is defined to be neutrosophic ℵ −structure
𝜒𝐴 (𝑋𝑁 ) =
𝑋
(𝜒𝐴 (𝑇)𝑁 , 𝜒𝐴 ( 𝐼)𝑁 , 𝜒𝐴 (𝐹)𝑁 )
where
−1 𝑖𝑓 𝑥 ∈ 𝐴
𝜒𝐴 (𝑇)𝑁 : X→ [−1, 0], 𝑥 → {
0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒,
𝜒𝐴 (𝐼)𝑁 : X→ [−1, 0], 𝑥 → {
0 𝑖𝑓 𝑥 ∈ 𝐴
−1 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒,
−1 𝑖𝑓 𝑥 ∈ 𝐴
𝜒𝐴 (𝐹)𝑁 : X→ [−1, 0], 𝑥 → {
0 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒.
Definition 2.7. [8] Let 𝑋𝑁 : =
𝑋
(𝑇𝑁 , 𝐼𝑁 , 𝐹𝑁 )
and 𝑋𝑀 : =
𝑋
(𝑇𝑀 , 𝐼𝑀 , 𝐹𝑀 )
.
𝑋𝑀 is called a neutrosophic ℵ − substructure of 𝑋𝑁 over 𝑋, denoted by 𝑋𝑁 ⊆ 𝑋𝑀 , if
𝑇𝑁 (𝑡) ≥ 𝑇𝑀 (𝑡), 𝐼𝑁 (𝑡) ≤ 𝐼𝑀 (𝑡), 𝐹𝑁 (𝑡) ≥ 𝐹𝑀 (𝑡) ∀t ∈ 𝑋.
If 𝑋𝑁 ⊆ 𝑋𝑀 and 𝑋𝑀 ⊆ 𝑋𝑁 , then we say that 𝑋𝑁 = 𝑋𝑀 .
(ii) The neutrosophic ℵ − product of 𝑋𝑁 and 𝑋𝑀 is defined to
be a neutrosophic ℵ −structure of 𝑋,
(i)
𝑋𝑁 ʘ 𝑋𝑀 ∶=
𝑋
(𝑇𝑁∘𝑀 , 𝐼𝑁∘𝑀 , 𝐹𝑁∘𝑀 )
= {
ℎ
𝑇𝑁∘𝑀 (ℎ), 𝐼𝑁∘𝑀 (ℎ), 𝐹𝑁∘𝑀 (ℎ)
| ℎ ∈ 𝑋},
where
⋀ {𝑇𝑁 (𝑟) ˅ 𝑇𝑀 (𝑠)} 𝑖𝑓 ∃ 𝑟, 𝑠 ∈ 𝑋 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 ℎ = 𝑟𝑠
(𝑇𝑁 ∘ 𝑇𝑀 )(ℎ) = 𝑇𝑁∘𝑀 (ℎ) = {ℎ=𝑟𝑠
0
𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒,
⋁ {𝐼𝑁 (𝑟) ˄ 𝐼𝑀 (𝑠)} 𝑖𝑓 ∃ 𝑟, 𝑠 ∈ 𝑋 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 ℎ = 𝑟𝑠
(𝐼𝑁 ∘ 𝐼𝑀 )(ℎ) = 𝐼𝑁∘𝑀 (ℎ) = {ℎ=𝑟𝑠
−1
𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒,
⋀ {𝐹𝑁 (𝑟) ˅ 𝐹𝑀 (𝑠)} 𝑖𝑓 ∃ 𝑟, 𝑠 ∈ 𝑋 𝑠𝑢𝑐ℎ 𝑡ℎ𝑎𝑡 ℎ = 𝑟𝑠
(𝐹𝑁 ∘ 𝐹𝑀 )(ℎ) = 𝐹𝑁∘𝑀 (ℎ) = {ℎ=𝑟𝑠
0
𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒.
K. Porselvi, B. Elavarasan, F. Smarandache and Y. B. Jun, Neutrosophic ℵ-bi-ideals in semigroups
Neutrosophic Sets and Systems, Vol. 35, 2020
(iii) For t ∈ X, the element
425
t
(TN∘M (t), IN∘M (t), FN∘M (t))
is simply denoted by
(XN ʘ XM )(t) = (TN∘M (t), IN∘M (t), FN∘M (t)) for the sake of convenience.
(iv) The union of 𝑋𝑁 and 𝑋𝑀 is a neutrosophic ℵ −structure over 𝑋 is defined as
𝑋𝑁 ∪ 𝑋𝑀 = 𝑋𝑁∪𝑀 = (𝑋; 𝑇𝑁∪𝑀, 𝐼𝑁∪𝑀, 𝐹𝑁∪𝑀 ),
where
(𝑇𝑁 ∪ 𝑇𝑀 )(ℎ𝑖 ) = 𝑇𝑁∪𝑀 (ℎ𝑖 ) = 𝑇𝑁 (ℎ𝑖 ) ˄ 𝑇𝑀 (ℎ𝑖 ),
(𝐼𝑁 ∪ 𝐼𝑀 )(ℎ𝑖 ) = 𝐼𝑁∪𝑀 (ℎ𝑖 ) = 𝐼𝑁 (ℎ𝑖 ) ˅ 𝐼𝑀 (ℎ𝑖 ),
(𝐹𝑁 ∪ 𝐹𝑀 )(ℎ𝑖 ) = 𝐹𝑁∪𝑀 (ℎ𝑖 ) = 𝐹𝑁 (ℎ𝑖 ) ˄ 𝐹𝑀 (ℎ𝑖 ) ∀ℎ𝑖 ∈ 𝑋.
(v) The intersection of 𝑋𝑁 and 𝑋𝑀 is a neutrosophic ℵ −structure over 𝑋 is defined as
𝑋𝑁 ∩ 𝑋𝑀 = 𝑋𝑁∩𝑀 = (𝑋; 𝑇𝑁∩𝑀, 𝐼𝑁∩𝑀, 𝐹𝑁∩𝑀 ),
where
(𝑇𝑁 ∩ 𝑇𝑀 )(ℎ𝑖 ) = 𝑇𝑁∩𝑀 (ℎ𝑖 ) = 𝑇𝑁 (ℎ𝑖 ) ˅ 𝑇𝑀 (ℎ𝑖 ),
(𝐼𝑁 ∩ 𝐼𝑀 )(ℎ𝑖 ) = 𝐼𝑁∩𝑀 (ℎ𝑖 ) = 𝐼𝑁 (ℎ𝑖 ) ˄ 𝐼𝑀 (ℎ𝑖 ),
(𝐹𝑁 ∩ 𝐹𝑀 )(ℎ𝑖 ) = 𝐹𝑁∩𝑀 (ℎ𝑖 ) = 𝐹𝑁 (ℎ𝑖 ) ˅ 𝐹𝑀 (ℎ𝑖 ) ∀ ℎ𝑖 ∈ 𝑋.
3. Neutrosophic ℵ −bi-ideals of semigroups
In this section, we examine different properties of neutrosophic ℵ −bi-ideals of 𝑋.
Theorem 3.1. For Φ ≠ B ⊆ 𝑋, the following assertions are equivalent:
(i)
χB (XN ) is a neutrosophic ℵ −bi-ideal of X,
(ii)
𝐵 is a bi-ideal of X.
Proof: Suppose 𝜒𝐵 (𝑋𝑁 ) is a neutrosophic ℵ −bi-ideal of X. Let r, t ∈ 𝐵 and 𝑠 ∈ 𝑋. Then
𝜒𝐵 (𝑇)𝑁 (𝑟𝑠𝑡) ≤ 𝜒𝐵 (𝑇)𝑁 (𝑟) ∨ 𝜒𝐵 (𝑇)𝑁 (𝑡) = −1,
𝜒𝐵 (𝐼)𝑁 (𝑟𝑠𝑡) ≥ 𝜒𝐵 (𝐼)𝑁 (𝑟) ∧ 𝜒𝐵 (𝐼)𝑁 (𝑡) = 0,
𝜒𝐵 (𝐹)𝑁 (𝑟𝑠𝑡) ≤ 𝜒𝐵 (𝐹)𝑁 (𝑟) ∨ 𝜒𝐵 (𝐹)𝑁 (𝑡) = −1.
Thus 𝑟𝑠𝑡 ∈ 𝐵 and hence B is a bi-ideal of X,
Conversely, assume
𝐵 is a bi-ideal of 𝑋. Let 𝑟, 𝑠, 𝑡 ∈ 𝑋.
If 𝑟 ∈ 𝐵 and 𝑡 ∈ 𝐵, then 𝑟𝑠𝑡 ∈ 𝐵. Now
𝜒𝐵 (𝑇)𝑁 (𝑟𝑠𝑡) = −1 = 𝜒𝐵 (𝑇)𝑁 (𝑟) ∨ 𝜒𝐵 (𝑇)𝑁 (𝑡),
𝜒𝐵 (𝐼)𝑁 (𝑟𝑠𝑡) = 0 = 𝜒𝐵 (𝐼)𝑁 (𝑟) ∧ 𝜒𝐵 (𝐼)𝑁 (𝑡),
𝜒𝐵 (𝐹)𝑁 (𝑟𝑠𝑡) = −1 = 𝜒𝐵 (𝐹)𝑁 (𝑟) ∨ 𝜒𝐵 (𝐹)𝑁 (𝑡).
If 𝑟 ∉ 𝐵 or t∉ 𝐵, then
𝜒𝐵 (𝑇)𝑁 (𝑟𝑠𝑡) ≤ 0 = 𝜒𝐵 (𝑇)𝑁 (𝑟) ∨ 𝜒𝐵 (𝑇)𝑁 (𝑡),
𝜒𝐵 (𝐼)𝑁 (𝑟𝑠𝑡) ≥ −1 = 𝜒𝐵 (𝐼)𝑁 (𝑟) ∧ 𝜒𝐵 (𝐼)𝑁 (𝑡)
𝜒𝐵 (𝐹)𝑁 (𝑟𝑠𝑡) ≤ 0 = 𝜒𝐵 (𝐹)𝑁 (𝑟) ∨ 𝜒𝐵 (𝐹)𝑁 (𝑡).
Therefore 𝜒𝐵 (𝑋𝑁 ) is a neutrosophic ℵ −bi-ideal of 𝑋.
□
Theorem 3.2. Let 𝜆, 𝛿, ε ∈ [−1, 0] be such that −3 ≤ 𝜆 + 𝛿 + ε ≤ 0. If 𝑋𝑁 is a neutrosophic ℵ −biideal, then (𝜆, 𝛿, ε) −level set of 𝑋𝑁 is a neutrosophic bi- ideal of 𝑋 whenever 𝑋𝑁 (𝜆, 𝛿, ε) ≠ ∅.
Proof: Suppose 𝑋𝑁 ( 𝜆, 𝛿, ε) ≠ ∅ for 𝜆, 𝛿, ε ∈ [−1, 0] with −3 ≤ 𝜆 + 𝛿 + ε ≤ 0. Let 𝑋𝑁 be a
neutrosophic ℵ −bi-ideal and let 𝑥, 𝑦, 𝑧 ∈ 𝑋𝑁 (𝜆, 𝛿, ε). Then
𝑇𝑁 (𝑥𝑦𝑧) ≤ 𝑇𝑁 (𝑥)⋁𝑇𝑁 (𝑧) ≤ 𝜆,
𝐼𝑁 (𝑥𝑦𝑧) ≥ 𝐼𝑁 (𝑥)⋀ 𝐼𝑁 (𝑧) ≥ 𝛿,
K. Porselvi, B. Elavarasan, F. Smarandache and Y. B. Jun, Neutrosophic ℵ-bi-ideals in semigroups
Neutrosophic Sets and Systems, Vol. 35, 2020
426
𝐹𝑁 (𝑥𝑦𝑧) ≤ 𝐹𝑁 (𝑥)⋁𝐹𝑁 (𝑧) ≤ ε
which imply 𝑥𝑦𝑧 ∈ 𝑋𝑁 (𝜆, 𝛿, ε). Therefore 𝑋𝑁 (𝜆, 𝛿, ε) is a neutrosophic ℵ −bi-ideal of 𝑋.
□
Theorem 3.3. Let 𝑋𝑀 be a neutrosophic ℵ − structure of 𝑋. Then the equivalent assertions are:
(i)
𝑋𝑀 ʘ 𝑋𝑀 ⊆ 𝑋𝑀 and 𝑋𝑀 ⨀𝜒𝑋 (𝑋𝑁 ) ʘ 𝑋𝑀 ⊆ 𝑋𝑀 for any neutrosophic ℵ − structure 𝑋𝑁 ,
(ii)
𝑋𝑀 is a neutrosophic ℵ −bi-ideal of 𝑋.
Proof: Suppose (i) holds. Then 𝑋𝑀 is neutrosophic ℵ − subsemigroup of 𝑋 by Theorem 4.6 of [8].
Let 𝑟, 𝑠, 𝑡 ∈ 𝑋 and let 𝑎 = 𝑟𝑠𝑡. Then
(𝑇𝑀 )(𝑟𝑠𝑡) ≤ (𝑇𝑀 ∘ 𝜒𝑋 (𝑇)𝑁 ∘ 𝑇𝑀 )(𝑟𝑠𝑡) = ⋀ {(𝑇𝑀 ∘ 𝜒𝑋 (𝑇)𝑁 ) (𝑟𝑠) ˅ 𝑇𝑀 (𝑡)}
𝑎=𝑟𝑠𝑡
= ⋀ { ⋀ {(𝑇𝑀 (𝑟) ˅ 𝜒𝑋 (𝑇)𝑁 (𝑠)} ˅ 𝑇𝑀 (𝑡)}
𝑎=𝑏𝑡 𝑏=𝑟𝑠
≤ ⋀{𝑇𝑀 (𝑟) ∨ 𝑇𝑀 (𝑡)} ≤ 𝑇𝑀 (𝑟) ∨ 𝑇𝑀 (𝑡),
𝐼𝑀 (𝑟𝑠𝑡) ≥ (𝐼𝑀 ∘ 𝜒𝑋 (𝐼)𝑁 ∘ 𝐼𝑀 )(𝑟𝑠𝑡) = ⋁ {(𝐼𝑀 ∘ 𝜒𝑋 (𝐼)𝑁 )(𝑟𝑠) ˄ 𝐼𝑀 (𝑡)}
𝑎=𝑟𝑠𝑡
= ⋁ { ⋁ { 𝐼𝑀 (𝑟)˄ 𝜒𝑋 (𝐼)𝑁 (𝑠)} ˄ 𝐼𝑀 (𝑡)}
𝑎=𝑏𝑡 𝑏=𝑟𝑠
≥ ⋁ { 𝐼𝑀 (𝑟)˄ 𝐼𝑀 (𝑡)} ≥ 𝐼𝑀 (𝑟)˄ 𝐼𝑀 (𝑡),
𝑎=𝑟𝑠𝑡
(𝐹𝑀 )(𝑟𝑠𝑡) ≤ (𝐹𝑀 ∘ 𝜒𝑋 (𝐹)𝑁 ∘ 𝐹𝑀 )(𝑟𝑠𝑡) = ⋀ {(𝐹𝑀 ∘ 𝜒𝑋 (𝐹)𝑁 ) (𝑟𝑠) ˅ 𝐹𝑀 (𝑡)}
𝑎=𝑟𝑠𝑡
= ⋀ { ⋀ {(𝐹𝑀 (𝑟) ˅ 𝜒𝑋 (𝐹)𝑁 (𝑠)} ˅ 𝐹𝑀 (𝑡)}
𝑎=𝑏𝑡 𝑏=𝑟𝑠
≤ ⋀ {𝐹𝑀 (𝑟) ˅ 𝐹𝑀 (𝑡)} ≤ 𝐹𝑀 (𝑟)˅ 𝐹𝑀 (𝑡).
𝑎=𝑟𝑠𝑡
Therefore 𝑋𝑀 is a neutrosophic ℵ − bi-ideal of 𝑋.
For converse, suppose (ii) holds. Then 𝑋𝑀 ʘ 𝑋𝑀 ⊆ 𝑋𝑀 by Theorem 4.6 of [8].
Let 𝑥 ∈ 𝑋. If 𝑥 = 𝑟𝑏 and r= 𝑠𝑡 for some r, 𝑏, 𝑠, 𝑡 ∈ 𝑋, then
(𝑇𝑀 ∘ 𝜒𝑋 (𝑇)𝑁 ∘ 𝑇𝑀 )(𝑥) = ⋀ {(𝑇𝑀 ∘ 𝜒𝑋 (𝑇)𝑁 ) (𝑟) ˅ 𝑇𝑀 (𝑏)}
𝑥=𝑟𝑏
= ⋀ {⋀{𝑇𝑀 (𝑠) ˅ 𝜒𝑋 (𝑇)𝑁 (𝑡)} ˅ 𝑇𝑀 (𝑏)}
𝑥=𝑟𝑏 𝑟=𝑠𝑡
= ⋀ {⋀{(𝑇𝑀 (𝑠)} ˅ 𝑇𝑀 (𝑏)}
𝑥=𝑟𝑏 𝑟=𝑠𝑡
= ⋀𝑥=𝑟𝑏{𝑇𝑀 (𝑠𝑖 ) ˅ 𝑇𝑀 (𝑏)} for some 𝑠𝑖 ∈ 𝑋 and r= 𝑠𝑖 𝑡𝑖
≥ ⋀ 𝑇𝑀 (𝑠𝑖 𝑡𝑖 𝑏) = 𝑇𝑀 (𝑥),
𝑥=𝑠𝑖 𝑡𝑖 𝑏
(𝐼𝑀 ∘ 𝜒𝑋 (𝐼)𝑁 ∘ 𝐼𝑀 )(𝑥) = ⋁ {(𝐼𝑀 ∘ 𝜒𝑋 (𝐼)𝑁 )(𝑟) ˄ 𝐼𝑀 (𝑏)}
𝑥=𝑟𝑏
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= ⋁ { ⋁ { 𝐼𝑀 (𝑠)˄ 𝜒𝑋 (𝐼)𝑁 (𝑡)} ˄ 𝐼𝑀 (𝑏)}
𝑥=𝑟𝑏 𝑟=𝑝𝑞
= ⋁ {⋁{ 𝐼𝑀 (𝑠)} ˄ 𝐼𝑀 (𝑏)}
𝑥=𝑟𝑏 𝑟=𝑠𝑡
= ⋁𝑥=𝑎𝑏{𝐼𝑀 (𝑠𝑖 ) ˄ 𝐼𝑀 (𝑏)}, for some 𝑠𝑖 ∈ 𝑋 and 𝑟 = 𝑠𝑖 𝑡𝑖
≤ ⋁ 𝐼𝑀 (𝑠𝑖 𝑡𝑖 𝑏) = 𝐼𝑀 (𝑥),
𝑥=𝑠𝑖 𝑡𝑖 𝑏
(𝐹𝑀 ∘ 𝜒𝑋 (𝐹)𝑁 ∘ 𝐹𝑀 )(𝑥) = ⋀ {(𝐹𝑀 ∘ 𝜒𝑋 (𝐹)𝑁 ) (𝑟) ˅ 𝐹𝑀 (𝑏)}
𝑥=𝑟𝑏
= ⋀ {⋀ {(𝐹𝑀 (𝑠) ˅ 𝜒𝑋 (𝐹)𝑁 (𝑡)} ˅ 𝐹𝑀 (𝑏)}
𝑥=𝑟𝑏 𝑎=𝑠𝑡
= ⋀ {⋀{(𝐹𝑀 (𝑠)} ˅ 𝐹𝑀 (𝑏)}
𝑥=𝑟𝑏 𝑟=𝑠𝑡
= ⋀𝑥=𝑟𝑏{𝐹𝑀 (𝑠𝑖 ) ˅ 𝐹𝑀 (𝑏)} for some 𝑠𝑖 ∈ 𝑋 and 𝑎 = 𝑠𝑖 𝑡𝑖
≥ ⋀ 𝐹𝑀 (𝑠𝑖 𝑡𝑖 𝑏) = 𝐹𝑀 (𝑥).
𝑥=𝑠𝑖 𝑡𝑖 𝑏
Otherwise 𝑥 ≠ 𝑟𝑏 or 𝑎 ≠ 𝑠𝑡 for all r, 𝑏, 𝑠, 𝑡 ∈ 𝑋. Then
(𝑇𝑀 ∘ 𝜒𝑋 (𝑇)𝑁 ∘ 𝑇𝑀 )(𝑥) = 0 ≥ 𝑇𝑀 (𝑥),
(𝐼𝑀 ∘ 𝜒𝑋 (𝐼)𝑁 ∘ 𝐼𝑀 )(𝑥) = −1 ≤ 𝐼𝑀 (𝑥),
(𝐹𝑀 ∘ 𝜒𝑋 (𝐹)𝑁 ∘ 𝐹𝑀 )(𝑥) = 0 ≥ 𝐹𝑀 (𝑥).
Therefore 𝑋𝑀 ⨀𝜒𝑋 (𝑋𝑁 ) ʘ 𝑋𝑀 ⊆ 𝑋𝑀 for any neutrosophic ℵ − structure 𝑋𝑁 over 𝑋.
□
Definition 3.4. A semigroup 𝑋 is called neutrosophic ℵ − left (resp., right) duo if every
neutrosophic ℵ −left (resp., right) ideal is neutrosophic ℵ −ideal of 𝑋.
If 𝑋 is both neutrosophic ℵ − left duo and neutrosophic ℵ − right duo, then 𝑋 is called
neutrosophic ℵ −duo
Theorem 3.5. If 𝑋 is regular left duo (resp., duo, right duo), then the equivalent assertions are:
(i) 𝑋𝑀 in X is neutrosophic ℵ −bi- ideal,
(ii) 𝑋𝑀 in X is neutrosophic ℵ −right ideal (resp., ideal, left ideal).
Proof: (𝒊) ⟹ (𝒊𝒊) Suppose 𝑋𝑀 is a neutrosophic ℵ −bi- ideal and 𝑔, ℎ ∈ 𝑋. As 𝑋 is regular, we get
𝑔 = 𝑔𝑡𝑔 ∈ 𝑔𝑋 ∩ 𝑋𝑔 for some 𝑡 ∈ 𝑋 which gives 𝑔ℎ ∈ (𝑔𝑋 ∩ 𝑋𝑔)𝑋 ⊆ 𝑔𝑋 ∩ 𝑋𝑔 as 𝑋 is left duo. So
𝑔ℎ = 𝑔𝑠 and 𝑔ℎ = 𝑠′𝑔 for some 𝑠, 𝑠 ′ ∈ 𝑋. As 𝑋 is regular, ∃𝑟 ∈ 𝑋 : 𝑔ℎ = 𝑔ℎ𝑟𝑔ℎ = 𝑔𝑠𝑟𝑠 ′ 𝑔 =
𝑔(𝑠𝑟𝑠 ′ )𝑔. Since 𝑋𝑀 is neutrosophic ℵ −bi- ideal, we have
𝑇𝑀 (𝑔ℎ) = 𝑇𝑀 (𝑔(𝑠𝑟𝑠 ′ )𝑔) ≤ 𝑇𝑀 (𝑔) ∨ 𝑇𝑀 (𝑔) = 𝑇𝑀 (𝑔),
𝐼𝑀 (𝑔ℎ) = 𝐼𝑀 (𝑔(𝑠𝑟𝑠 ′ )𝑔) ≥ 𝐼𝑀 (𝑔) ∧ 𝐼𝑀 (𝑔) = 𝐼𝑀 (𝑔),
𝐹𝑀 (𝑔ℎ) = 𝐹𝑀 (𝑔(𝑠𝑟𝑠 ′ )𝑔) ≤ 𝐹𝑀 (𝑔) ∨ 𝐹𝑀 (𝑔) = 𝐹𝑀 (𝑔).
Therefore 𝑋𝑀 is neutrosophic ℵ −right ideal.
(𝒊𝒊) ⟹ (𝒊) Suppose 𝑋𝑀 is neutrosophic ℵ −right ideal and let 𝑥, 𝑦, 𝑧 ∈ 𝑋. Then
𝑇𝑀 (𝑥𝑦𝑧) ≤ 𝑇𝑀 (𝑥) ≤ 𝑇𝑀 (𝑥) ∨ 𝑇𝑀 (𝑧),
𝐼𝑀 (𝑥𝑦𝑧) ≥ 𝐼𝑀 (𝑥) ≥ 𝐼𝑀 (𝑥) ∧ 𝐼𝑀 (𝑧),
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𝐹𝑀 (𝑥𝑦𝑧) ≤ 𝐹𝑀 (𝑥) ≤ 𝐹𝑀 (𝑥) ∨ 𝐹𝑀 (𝑧).
Therefore 𝑋𝑀 is a neutrosophic ℵ −bi-ideal.
□
Theorem 3.6. If 𝑋 is regular, then the equivalent assertions are:
(i) 𝑋 is left duo (resp., right duo, duo),
(ii) 𝑋 is neutrosophic ℵ −left duo (resp., right duo, duo).
Proof: (𝒊) ⟹ (𝒊𝒊) Let r, s ∈ 𝑋, we have 𝑟𝑠 ∈ (𝑟𝑋𝑟)𝑠 ⊆ 𝑟(𝑋𝑟)𝑋 ⊆ 𝑋𝑟 as 𝑋𝑟 is left ideal. Since 𝑋 is
regular, we have 𝑟𝑠 = 𝑡𝑟 for some 𝑡 ∈ 𝑋.
If 𝑋𝑀 is neutrosophic ℵ −left ideal, then 𝑇𝑀 (𝑟𝑠) = 𝑇𝑀 (𝑡𝑟) ≤ 𝑇𝑀 (𝑟), 𝐼𝑀 (𝑟𝑠) = 𝐼𝑀 (𝑡𝑟) ≥ 𝐼𝑀 (𝑟) and
𝐹𝑀 (𝑟𝑠) = 𝐹𝑀 (𝑡𝑟) ≤ 𝐹𝑀 (𝑟). Thus 𝑋𝑀 is neutrosophic ℵ −right ideal and therefore 𝑋 is neutrosophic
ℵ −left duo.
(𝒊𝒊) ⟹ (𝒊) Let 𝐴 be a left ideal of 𝑋. Then 𝜒𝐴 (𝑋𝑀 ) is a neutrosophic ℵ −left ideal by Theorem
3.5 of [4]. By assumption, 𝜒𝐴 (𝑋𝑀 ) is neutrosophic ℵ −ideal. Thus 𝐴 is a right ideal of 𝑋.
□
Theorem 3.7. If 𝑋 is regular, then the equivalent assertions are:
(i) Every neutrosophic ℵ −bi-ideal is a neutrosophic ℵ −right (resp., left ideal, ideal) ideal,
(ii) Every bi-ideal of X is a right ideal (resp., left ideal, ideal).
Proof: (𝒊) ⟹ (𝒊𝒊) Let 𝐴 be a bi-ideal of 𝑋 . Then by Theorem 3.1 𝜒𝐴 (𝑋𝑀 ) is neutrosophic
ℵ −bi-ideal for a neutrosophic ℵ −structure 𝑋𝑀 . Now by assumption, 𝜒𝐴 (𝑋𝑀 ) is neutrosophic
ℵ −right ideal. So by Theorem 3.5 of [4], 𝐴 is right ideal.
(𝒊𝒊) ⟹ (𝒊) Let 𝑋𝑀 be a neutrosophic ℵ −bi-ideal and let 𝑟, 𝑠 ∈ 𝑋. Then we get r𝑋𝑟 is a bi-ideal
of 𝑋. By hypothesis, we can have r𝑋𝑟 is right ideal. Since 𝑋 is regular, we can get r∈ 𝑟𝑋𝑟. So 𝑟𝑠 ∈
(𝑟𝑋𝑟)𝑋 ⊆ 𝑟𝑋𝑟 implies 𝑟𝑠 = 𝑟𝑥𝑟 for some 𝑥 ∈ 𝑋. Now,
𝑇𝑀 (𝑟𝑠) = 𝑇𝑀 (𝑟𝑥𝑟) ≤ 𝑇𝑀 (𝑟) ∨ 𝑇𝑀 (𝑟) = 𝑇𝑀 (𝑟),
𝐼𝑀 (𝑟𝑠) = 𝐼𝑀 (𝑟𝑥𝑟) ≥ 𝐼𝑀 (𝑟) ∧ 𝐼𝑀 (𝑟) = 𝐼𝑀 (𝑟)
𝐹𝑀 (𝑟𝑠) = 𝐹𝑀 (𝑟𝑥𝑟) ≤ 𝐹𝑀 (𝑟) ∨ 𝐹𝑀 (𝑟) = 𝐹𝑀 (𝑟).
Thus 𝑋𝑀 is a neutrosophic ℵ −right ideal of 𝑋.
□
Theorem 3.8. For any 𝑋, the equivalent conditions are:
(i)
𝑋 is regular,
(ii) 𝑋𝑀 ∩ 𝑋𝑁 = 𝑋𝑀 ʘ𝑋𝑁 ⨀𝑋𝑀 for every neutrosophic ℵ − bi-ideal 𝑋𝑀 and neutrosophic ℵ −
ideal 𝑋𝑁 of 𝑋.
Proof: (𝒊) ⇒ (𝒊𝒊) Suppose 𝑋is regular, 𝑋𝑀 is a neutrosophic ℵ − bi-ideal and 𝑋𝑁 is a neutrosophic
ℵ − ideal of X. Then by Theorem 3.3, we have 𝑋𝑀 ʘ𝑋𝑁 ⨀𝑋𝑀 ⊆ 𝑋𝑀 and 𝑋𝑀 ʘ𝑋𝑁 ⨀𝑋𝑀 ⊆ 𝑋𝑁 . So
𝑋𝑀 ʘ𝑋𝑁 ⨀𝑋𝑀 ⊆ 𝑋𝑀 ∩ 𝑋𝑁 .
Let 𝑟 ′ ∈ 𝑋. As 𝑋 is regular, there is 𝑝 ∈ 𝑋 such that 𝑟 ′ = 𝑟′𝑝𝑟′ = 𝑟′𝑝𝑟′𝑝𝑟′. Now
𝑇𝑀∘𝑁∘𝑀 (𝑟′) = ⋀ {𝑇𝑀 (𝑑) ∨ 𝑇𝑁∘𝑀 (𝑒)}
𝑟′=𝑑𝑒
= ⋀ {𝑇𝑀 (𝑟′) ∨ { ⋀ {𝑇𝑁 (𝑝𝑟′𝑝) ∨ 𝑇𝑀 (𝑟′)}}
𝑟′=𝑟′𝑒
𝑣=𝑝𝑟′𝑝𝑟′
≤ ⋀ {𝑇𝑀 (𝑟′) ∨ 𝑇𝑁 (𝑟′)} ≤ 𝑇𝑀 (𝑟′) ∨ 𝑇𝑁 (𝑟′) = 𝑇𝑀∩𝑁 (𝑟′),
𝑟′=𝑟′𝑒
𝐼𝑀∘𝑁∘𝑀 (𝑟′) = ⋁𝑟′=𝑑𝑒 {𝐼𝑀 (𝑑) ˄ 𝐼𝑁∘𝑀 (𝑒)}
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= ⋁ {𝐼𝑀 (𝑟 ′ )˄{ ⋁ { 𝐼𝑁 (𝑝𝑟 ′ 𝑝)˄ 𝐼𝑀 (𝑟′)}}
𝑟′=𝑟′𝑒
𝑣=𝑝𝑟 ′𝑝𝑟 ′
≥ ⋁ {𝐼𝑀 (𝑟′) ˄ 𝐼𝑁 (𝑟′)} ≥ 𝐼𝑀 (𝑟′) ˄ 𝐼𝑁 (𝑟′) = 𝐼𝑀∩𝑁 (𝑟′),
𝑟′=𝑟′𝑒
𝐹𝑀∘𝑁∘𝑀 (𝑟′) = ⋀ {𝐹𝑀 (𝑑) ∨ 𝐹𝑁∘𝑀 (𝑒)}
𝑟′=𝑑𝑒
= ⋀ {𝐹𝑀 (𝑟′) ∨ { ⋀ {𝐹𝑁 (𝑝𝑟′𝑝) ∨ 𝐹𝑀 (𝑟′)}}
𝑟′=𝑟′𝑒
𝑣=𝑝𝑟′𝑝𝑟′
≤ ⋀ {𝐹𝑀 (𝑟′) ∨ 𝐹𝑁 (𝑟′)} ≤ 𝐹𝑀 (𝑟′) ∨ 𝐹𝑁 (𝑟′) = 𝐹𝑀∩𝑁 (𝑟′).
𝑟′=𝑟′𝑒
Thus 𝑋𝑀∩𝑁 ⊆ 𝑋𝑀 ʘ 𝑋𝑁 ⨀ 𝑋𝑀 and hence 𝑋𝑀∩𝑁 = 𝑋𝑀 ʘ𝑋𝑁 ⨀𝑋𝑀 .
(𝒊𝒊) ⇒ (𝒊) Suppose (ii) holds. Then 𝑋𝑀 ∩ 𝜒𝑋 (𝑋𝑁 ) = 𝑋𝑀 ⨀𝜒𝑋 (𝑋𝑁 )⨀𝑋𝑀 . But 𝑋𝑀 ∩ 𝜒𝑋 (𝑋𝑁 ) =
𝑋𝑀 , so 𝑋𝑀 = 𝑋𝑀 ⨀𝜒𝑋 (𝑋𝑁 )⨀𝑋𝑀 for every neutrosophic ℵ − bi-ideal 𝑋𝑀 of 𝑋.
Let 𝑢′ ∈ 𝑋. Then 𝜒𝐵(𝑢′) (𝑋𝑀 ) is neutrosophic ℵ − bi-ideal by Theorem 3.1.
By assumption, we have
𝜒𝐵(𝑢′) (𝑇)𝑀 =𝜒𝐵(𝑢′) (𝑇)𝑀 ∘ 𝜒𝑋 (𝑇)𝑁 ∘ 𝜒𝐵(𝑢′) (𝑇)𝑀 = 𝜒𝐵(𝑢′)𝑋𝐵(𝑢′) (𝑇)𝑀 ,
𝜒𝐵(𝑢′) (𝐼)𝑀 =𝜒𝐵(𝑢′) (𝐼)𝑀 ∘ 𝜒𝑋 (𝐼)𝑁 ∘ 𝜒𝐵(𝑢′) (𝐼)𝑀 = 𝜒𝐵(𝑢′)𝑋𝐵(𝑢′) (𝐼)𝑀 ,
𝜒𝐵(𝑢′) (𝐹)𝑀 =𝜒𝐵(𝑢′) (𝐹)𝑀 ∘ 𝜒𝑋 (𝐹)𝑁 ∘ 𝜒𝐵(𝑢′) (𝐹)𝑀 = 𝜒𝐵(𝑢′)𝑋𝐵(𝑢′) (𝐹)𝑀 .
Since 𝑢′ ∈ 𝐵(𝑢′), we have
𝜒𝐵(𝑢′)𝑋𝐵(𝑢′) (𝑇)𝑀 (𝑢′) = 𝜒𝐵(𝑢′) (𝑇)𝑀 (𝑢′) = −1,
𝜒𝐵(𝑢′)𝑋𝐵(𝑢′) (𝐼)𝑀 (𝑢′) = 𝜒𝐵(𝑢′) (𝐼)𝑀 (𝑢′) = 0,
𝜒𝐵(𝑢′)𝑋𝐵(𝑢′) (𝐹)𝑀 (𝑢′) = 𝜒𝐵(𝑢′) (𝐹)𝑀 (𝑢′) = −1
Thus u’∈ 𝐵(𝑢′)𝑋𝐵(𝑢′) and hence 𝑋 is regular.
□
Theorem 3.9. For any 𝑋, the below statements are equivalent:
(i) 𝑋 is regular,
(ii) 𝑋𝑀 ∩ 𝑋𝑁 = 𝑋𝑀 ʘ𝑋𝑁 for every neutrosophic ℵ − bi-ideal 𝑋𝑀 and neutrosophic ℵ − left
ideal 𝑋𝑁 of 𝑋.
Proof:(𝒊) ⇒ (𝒊𝒊) Let 𝑋𝑀 and 𝑋𝑁 be neutrosophic ℵ − bi-ideal and neutrosophic ℵ −left ideal of 𝑋
respectively. Let r ∈ 𝑋. Then ∃𝑥 ∈ 𝑋 : r= 𝑟𝑥𝑟. Now
𝑇𝑀∘𝑁 (𝑟) = ⋀ {𝑇𝑀 (𝑢) ˅ 𝑇𝑁 (𝑣)} ≤ 𝑇𝑀 (𝑟)˅ 𝑇𝑁 (𝑥𝑟) ≤ 𝑇𝑀 (𝑟)˅ 𝑇𝑁 (𝑟) = 𝑇𝑀∩𝑁 (𝑟),
𝑟=𝑢𝑣
𝐼𝑀∘𝑁 (𝑟) = ⋁ {𝐼𝑀 (𝑢) ˄ 𝐼𝑁 (𝑣)} ≥ 𝐼𝑀 (𝑟) ˄ 𝐼𝑁 (𝑥𝑟) ≥ 𝐼𝑀 (𝑟) ˄ 𝐼𝑁 (𝑟) = 𝐼𝑀∩𝑁 (𝑟),
𝑟=𝑢𝑣
𝐹𝑀∘𝑁 (𝑟) = ⋀ {𝐹𝑀 (𝑢) ˅ 𝐹𝑁 (𝑣)} ≤ 𝐹𝑀 (𝑟)˅ 𝐹𝑁 (𝑥𝑟) ≤ 𝐹𝑀 (𝑟)˅ 𝐹𝑁 (𝑟) = 𝐹𝑀∩𝑁 (𝑟).
𝑟=𝑢𝑣
Therefore 𝑋𝑀∩𝑁 ⊆ 𝑋𝑀 ʘ𝑋𝑁 .
(𝒊𝒊) ⇒ (𝒊) Suppose (ii) holds, and let 𝑋𝑀 and 𝑋𝑁 be neutrosophic ℵ − right ideal and
neutrosophic ℵ − left ideal of X respectively. Since every neutrosophic ℵ − right ideal is
neutrosophic ℵ − bi-ideal, 𝑋𝑀 is neutrosophic ℵ − bi-ideal. Then by assumption, 𝑋𝑀∩𝑁 ⊆
𝑋𝑀 ʘ𝑋𝑁 . By Theorem 3.8 and Theorem 3.9 of [4], we can get 𝑋𝑀 ⨀𝑋𝑁 ⊆ 𝑋𝑁 and 𝑋𝑀 ⨀𝑋𝑁 ⊆ 𝑋𝑀 and
so 𝑋𝑀 ⨀𝑋𝑁 ⊆ 𝑋𝑀 ∩ 𝑋𝑁 = 𝑋𝑀∩𝑁 . Therefore 𝑋𝑀 ⨀𝑋𝑁 = 𝑋𝑀∩𝑁 .
Let 𝐾 and 𝐿 be right and left ideals of 𝑋 respectively, and r ∈ 𝐾 ∩ 𝐿. Then
𝜒𝐾 (𝑋𝑀 )⨀𝜒𝐿 (𝑋𝑀 ) = 𝜒𝐾 (𝑋𝑀 ) ∩ 𝜒𝐿 (𝑋𝑀 ) which implies 𝜒𝐾𝐿 (𝑋𝑀 ) = 𝜒𝐾∩𝐿 (𝑋𝑀 ). Since r ∈ 𝐾 ∩ 𝐿, we have
K. Porselvi, B. Elavarasan, F. Smarandache and Y. B. Jun, Neutrosophic ℵ-bi-ideals in semigroups
Neutrosophic Sets and Systems, Vol. 35, 2020
430
𝜒𝐾∩𝐿 (𝑇)𝑀 (𝑟) = −1 = 𝜒𝐾𝐿 (𝑇)𝑀 (𝑟), 𝜒𝐾∩𝐿 (𝐼)𝑀 (𝑟) = 0 = 𝜒𝐾𝐿 (𝐼)𝑀 (𝑟)
and
𝜒𝐾∩𝐿 (𝐹)𝑀 (𝑟) = −1 =
𝜒𝐾𝐿 (𝐹)𝑀 (𝑟) which imply r ∈ 𝐾𝐿.Thus 𝐾 ∩ 𝐿 ⊆ 𝐾𝐿 ⊆ 𝐾 ∩ 𝐿. So 𝐾 ∩ 𝐿 = 𝐾𝐿. Thus 𝑋 is regular.
□
Theorem 3.10. For any 𝑋, the equivalent conditions are:
(i)
𝑋 is regular,
(ii)
𝑋𝑀 ∩ 𝑋𝑁 ⊆ 𝑋𝑀 ʘ𝑋𝑁 for every neutrosophic ℵ − right ideal 𝑋𝑁 and neutrosophic ℵ −
bi-ideal 𝑋𝑀 of 𝑋.
Proof: It is same as Theorem 3.9.
□
Theorem 3.11. For any 𝑋, the equivalent assertions are:
(i)
𝑋 is regular,
(ii) 𝑋𝐿 ∩ 𝑋𝑀 ∩ 𝑋𝑁 ⊆ 𝑋𝐿 ⨀𝑋𝑀 ʘ𝑋𝑁 for every neutrosophicℵ − right ideal 𝑋𝐿, neutrosophic ℵ −
bi-ideal 𝑋𝑀 and neutrosophic ℵ − left ideal 𝑋𝑁 of 𝑋.
Proof: (𝒊) ⇒ (𝒊𝒊) Suppose 𝑋 is regular, and let 𝑋𝐿 , 𝑋𝑀 , 𝑋𝑁 be neutrosophic ℵ − right, bi-ideal, left
ideals of 𝑋 respectively. Let r ∈ 𝑋. Then there is 𝑥 ∈ 𝑋 with r= 𝑟𝑥𝑟 = 𝑟𝑥𝑟𝑥𝑟. Now
𝑇𝐿∘𝑀∘𝑁 (𝑟) = ⋀ {𝑇𝐿 (𝑢) ˅ 𝑇𝑀∘𝑁 (𝑣)} ≤ 𝑇𝐿 (𝑟𝑥)˅ 𝑇𝑀∘𝑁 (𝑟𝑥𝑟) ≤ 𝑇𝐿 (𝑟)˅{𝑇𝑀 (𝑟)˅ 𝑇𝑁 (𝑥𝑟)}
𝑟=𝑢𝑣
≤ 𝑇𝐿 (𝑟)˅𝑇𝑀 (𝑟)˅ 𝑇𝑁 (𝑟) = 𝑇𝐿∩𝑀∩𝑁 (𝑟),
𝐼𝐿∘𝑀∘𝑁 (𝑟) = ⋁ {𝐼𝐿 (𝑢) ˄ 𝐼𝑀∘𝑁 (𝑣)} ≥ 𝐼𝐿 (𝑟𝑥) ˄ 𝐼𝑀∘𝑁 (𝑟𝑥𝑟) ≥ 𝐼𝐿 (𝑟)˄{𝐼𝑀 (𝑟) ˄ 𝐼𝑁 (𝑥𝑟)}
𝑟=𝑢𝑣
≥ 𝐼𝐿 (𝑟)˄𝐼𝑀 (𝑟) ˄ 𝐼𝑁 (𝑟) = 𝐼𝐿∩𝑀∩𝑁 (𝑟),
𝐹𝐿∘𝑀∘𝑁 (𝑟) = ⋀ {𝐹𝐿 (𝑢) ˅ 𝐹𝑀∘𝑁 (𝑣)} ≤ 𝐹𝐿 (𝑟𝑥)˅ 𝐹𝑀∘𝑁 (𝑟𝑥𝑟) ≤ 𝐹𝐿 (𝑟)˅𝐹𝑀 (𝑟)˅ 𝐹𝑁 (𝑥𝑟)
𝑟=𝑢𝑣
≤ 𝐹𝐿 (𝑟)˅𝐹𝑀 (𝑟)˅ 𝐹𝑁 (𝑟) = 𝐹𝐿∩𝑀∩𝑁 (𝑟).
Therefore 𝑋𝐿∩𝑀∩𝑁 ⊆ 𝑋𝐿 ⨀𝑋𝑀 ʘ𝑋𝑁 .
(𝒊𝒊) ⇒ (𝒊) Suppose (ii) holds, and let 𝑋𝐿 and 𝑋𝑁 be neutrosophic ℵ − right and neutrosophic
ℵ − left ideal of X respectively, and 𝑋𝑀 a neutrosophic ℵ −bi-ideal of 𝑋. Then 𝜒𝑋 (𝑋𝑀 ) is a
neutrosophic ℵ − bi-ideal by Theorem 3.1. Now 𝑋𝐿 ∩ 𝑋𝑁 = 𝑋𝐿 ∩ 𝜒𝑋 (𝑋𝑀 ) ∩ 𝑋𝑁 ⊆ 𝑋𝐿 ʘ𝜒𝑋 (𝑋𝑀 )⨀𝑋𝑁 ⊆
𝑋𝐿 ⨀𝑋𝑁 . Again by Theorem 3.8 and Theorem 3.9 of [4], we can get 𝑋𝐿 ⨀𝑋𝑁 ⊆ 𝑋𝐿 ∩ 𝑋𝑁 and so 𝑋𝐿 ⨀𝑋𝑁 =
𝑋𝐿 ∩ 𝑋𝑁 .
Let 𝐾 and L be right and left ideals of 𝑋 respectively. Then 𝜒𝐾 (𝑋𝑀 )⨀𝜒𝐿 (𝑋𝑀 ) = 𝜒𝐾 (𝑋𝑀 ) ∩
𝜒𝐿 (𝑋𝑀 ). By Theorem 3.6 of [4], we have 𝜒𝐾𝐿 (𝑋𝑀 ) = 𝜒𝐾∩𝐿 (𝑋𝑀 ). Let r ∈ 𝐾 ∩ 𝐿. Then
𝜒𝐾𝐿 (𝑇)𝑀 (𝑟) = 𝜒𝐾∩𝐿 (𝑇)𝑀 (𝑟) = −1,
𝜒𝐾𝐿 (𝐼)𝑀 (𝑟) = 𝜒𝐾∩𝐿 (𝐼)𝑀 (𝑟) = 0,
𝜒𝐾𝐿 (𝐹)𝑀 (𝑟) = 𝜒𝐾∩𝐿 (𝐹)𝑀 (𝑟) = −1.
So r ∈ 𝐾𝐿. Thus 𝐾 ∩ 𝐿 ⊆ 𝐾𝐿 ⊆ 𝐾 ∩ 𝐿. Hence 𝐾 ∩ 𝐿 = 𝐾𝐿. Therefore 𝑋 is regular.
□
Theorem 3.12. For any 𝑋, the equivalent conditions are:
(i) 𝑋 is regular and intra- regular,
(ii) 𝑋𝑀 ∩ 𝑋𝑁 ⊆ 𝑋𝑀 ʘ𝑋𝑁 for every neutrosophic ℵ − bi-ideals 𝑋𝑀 , 𝑋𝑁 of 𝑋.
Proof: (𝒊) ⇒ (𝒊𝒊) Let 𝑋𝑀 and 𝑋𝑁 be neutrosophic ℵ − bi-ideals. Let ℎ ∈ 𝑋. Then by regularity of
𝑋 , h = ℎ𝑥ℎ = ℎ𝑥ℎ𝑥ℎ for some x ∈ 𝑋. Since 𝑋 is intra-regular, ∃𝑦, 𝑧 ∈ 𝑋 : h = 𝑦ℎ2 𝑧. Then
ℎ = ℎ𝑥𝑦ℎℎ𝑧𝑥ℎ. Now
K. Porselvi, B. Elavarasan, F. Smarandache and Y. B. Jun, Neutrosophic ℵ-bi-ideals in semigroups
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431
𝑇𝑀∘𝑁 (ℎ) = ⋀ {𝑇𝑀 (𝑟) ˅ 𝑇𝑁 (𝑡)} ≤ 𝑇𝑀 (ℎ𝑥𝑦ℎ)˅ 𝑇𝑁 (ℎ𝑧𝑥ℎ) ≤ 𝑇𝑀 (ℎ)˅ 𝑇𝑁 (ℎ)
= 𝑇𝑀∩𝑁 (ℎ),
ℎ=𝑟𝑡
𝐼𝑀∘𝑁 (ℎ) = ⋁ {𝐼𝑀 (𝑟) ˄ 𝐼𝑁 (𝑡)} ≥ 𝐼𝑀 (ℎ𝑥𝑦ℎ) ˄ 𝐼𝑁 (ℎ𝑧𝑥ℎ) ≥ 𝐼𝑀 (ℎ) ˄ 𝐼𝑁 (ℎ) = 𝐼𝑀∩𝑁 (ℎ),
ℎ=𝑟𝑡
𝐹𝑀∘𝑁 (ℎ) = ⋀ {𝐹𝑀 (𝑟) ˅ 𝐹𝑁 (𝑡)} ≤ 𝐹𝑀 (ℎ𝑥𝑦ℎ)˅ 𝐹𝑁 (ℎ𝑧𝑥ℎ) ≤ 𝐹𝑀 (ℎ)˅ 𝐹𝑁 (ℎ) = 𝐹𝑀∩𝑁 (ℎ).
ℎ=𝑟𝑡
Therefore 𝑋𝑀 ∩ 𝑋𝑁 ⊆ 𝑋𝑀 ʘ𝑋𝑁 for every neutrosophic ℵ − bi-ideals 𝑋𝑀 and 𝑋𝑁 .
(𝒊𝒊) ⇒ (𝒊) Suppose (ii) holds, and let 𝑋𝑀 and 𝑋𝑁 be neutrosophic ℵ − right and left ideal of X
respectively. Then 𝑋𝑀 and 𝑋𝑁 are neutrosophic ℵ − bi-ideals. By assumption, 𝑋𝑀∩𝑁 ⊆ 𝑋𝑀 ʘ𝑋𝑁 .
By Theorem 3.8 and Theorem 3.9 of [4], we can get 𝑋𝑀 ⨀𝑋𝑁 ⊆ 𝑋𝑁 and 𝑋𝑀 ⨀𝑋𝑁 ⊆ 𝑋𝑀 and so
𝑋𝑀 ⨀𝑋𝑁 ⊆ 𝑋𝑀 ∩ 𝑋𝑁 = 𝑋𝑀∩𝑁 . Therefore 𝑋𝑀 ⨀𝑋𝑁 = 𝑋𝑀∩𝑁 .
Let 𝐾, 𝐿 be right, left ideals of 𝑋 respectively. Then 𝜒𝐾 (𝑋𝑀 )⨀𝜒𝐿 (𝑋𝑀 ) = 𝜒𝐾 (𝑋𝑀 ) ∩ 𝜒𝐿 (𝑋𝑀 ).
By Theorem 3.6 of [4], 𝜒𝐾𝐿 (𝑋𝑀 ) = 𝜒𝐾∩𝐿 (𝑋𝑀 ). Let 𝑟 ∈ 𝐾 ∩ 𝐿. Then
𝜒𝐾∩𝐿 (𝑇)𝑀 (𝑟) = −1 =
𝜒𝐾𝐿 (𝑇)𝑀 (𝑟), 𝜒𝐾∩𝐿 (𝐼)𝑀 (𝑟) = 0 = 𝜒𝐾𝐿 (𝐼)𝑀 (𝑟) and 𝜒𝐾∩𝐿 (𝐹)𝑀 (𝑟) = −1 = 𝜒𝐾𝐿 (𝐹)𝑀 (𝑟) which imply 𝑟 ∈
𝐾𝐿. Thus 𝐾 ∩ 𝐿 ⊆ 𝐾𝐿 ⊆ 𝐾 ∩ 𝐿 and hence 𝐾 ∩ 𝐿 = 𝐾𝐿. Therefore 𝑋 is regular.
Also, for 𝑟 ∈ 𝑋, 𝜒𝐵(𝑟) (𝑋𝑀 ) ∩ 𝜒𝐵(𝑟) (𝑋𝑀 ) = 𝜒𝐵(𝑟) (𝑋𝑀 )⨀𝜒𝐵(𝑟) (𝑋𝑀 ). By Theorem 3.8 and Theorem 3.9
of [4], we get 𝜒𝐵(𝑟) (𝑋𝑀 ) = 𝜒𝐵(𝑟)𝐵(𝑟) (𝑋𝑀 ).since𝜒𝐵(𝑟) (𝑇)𝑀 (𝑟) = −1 = 𝜒𝐵(𝑟) (𝐹)𝑀 (𝑟)and 𝜒𝐵(𝑟) (𝐼)𝑀 (𝑟) =
0, we get 𝜒𝐵(𝑟)𝐵(𝑟) (𝑇)𝑀 (𝑟) = −1 = 𝜒𝐵(𝑟)𝐵(𝑟) (𝐹)𝑀 (𝑟) and 𝜒𝐵(𝑟)𝐵(𝑟) (𝐼)𝑀 (𝑟) = 0 which imply 𝑟 ∈
𝐵(𝑟)𝐵(𝑟). Thus 𝑋 is intra-regular.
□
Theorem 3.13. For any 𝑋, the equivalent conditions are:
(i) 𝑋 is intra-regular and regular,
(ii) 𝑋𝑀 ∩ 𝑋𝑁 ⊆ (𝑋𝑀 ʘ𝑋𝑁 ) ∩ (𝑋𝑁 ʘ𝑋𝑀 ) for every neutrosophic ℵ − bi-ideals 𝑋𝑀 and 𝑋𝑁 of 𝑋.
Proof:(𝒊) ⇒ (𝒊𝒊) Suppose 𝑋 is regular and intra- regular, and let 𝑋𝑀 and 𝑋𝑁 be neutrosophic ℵ −
bi-ideals of 𝑋. Then by Theorem 3.12, 𝑋𝑀 ʘ𝑋𝑁 ⊇ 𝑋𝑀 ∩ 𝑋𝑁 . Similarly we can prove that 𝑋𝑁 ʘ𝑋𝑀 ⊇
𝑋𝑁 ∩ 𝑋𝑀 .Therefore (𝑋𝑀 ʘ𝑋𝑁 ) ∩ (𝑋𝑁 ʘ𝑋𝑀 ) ⊇ 𝑋𝑀 ∩ 𝑋𝑁 for every neutrosophic ℵ − bi-ideals 𝑋𝑀 and
𝑋𝑁 of 𝑋.
(𝒊𝒊) ⇒ (𝒊) Let 𝑋𝑀 and 𝑋𝑁 be neutrosophic ℵ − bi-ideals of 𝑋. Then 𝑋𝑀 ∩ 𝑋𝑁 ⊆ 𝑋𝑀 ʘ𝑋𝑁 gives
𝑋 is intra-regular and regular by Theorem 3.12.
□
Theorem 3.14. For any 𝑋, the equivalent assertions are:
(i) 𝑋 is intra-regular and regular,
(ii) 𝑋𝑀 ∩ 𝑋𝑁 ⊆ 𝑋𝑀 ʘ𝑋𝑁 ⨀𝑋𝑀 for every neutrosophic ℵ − bi-ideals 𝑋𝑀 and 𝑋𝑁 of 𝑋.
Proof:(𝒊) ⇒ (𝒊𝒊) Let 𝑋𝑀 and 𝑋𝑁 be neutrosophic ℵ − bi-ideals, and 𝑎 ∈ 𝑋. As 𝑋 is regular, 𝑎 =
𝑎𝑥𝑎 = 𝑎𝑥𝑎𝑥𝑎𝑥𝑎 for some 𝑥 ∈ 𝑋. Since 𝑋 is intra-regular, 𝑎 = 𝑦𝑎2 𝑧 for some 𝑦, 𝑧 ∈ 𝑋. Then 𝑎 =
(𝑎𝑥𝑦𝑎)(𝑎𝑧𝑥𝑦𝑎)(𝑎𝑧𝑥𝑎). Now
𝑇𝑀∘𝑁∘𝑀 (𝑎) = ⋀ {𝑇𝑀 (𝑘) ˅ 𝑇𝑁∘𝑀 (𝑚)}
𝑎=𝑘𝑚
=
⋀
{𝑇𝑀 (𝑎𝑥𝑦𝑎) ˅ { ⋀{ 𝑇𝑁 (𝑟) ∨ 𝑇𝑀 (𝑡)}}
𝑎=(𝑎𝑥𝑦𝑎)𝑣
𝑣=𝑟𝑡
≤ 𝑇𝑀 (𝑎𝑥𝑦𝑎)˅ 𝑇𝑁 (𝑎𝑧𝑥𝑦𝑎)˅ 𝑇𝑀 (𝑎𝑧𝑥𝑎)
≤ 𝑇𝑀 (𝑎)˅ 𝑇𝑁 (𝑎)˅ 𝑇𝑀 (𝑎)
= 𝑇𝑀∩𝑁 (𝑎),
𝐼𝑀∘𝑁∘𝑀 (𝑎) = ⋁ {𝐼𝑀 (𝑘) ˄ 𝐼𝑁∘𝑀 (𝑚)}
𝑎=𝑘𝑚
=
⋁
𝑎=(𝑎𝑥𝑦𝑎)𝑣
{𝐼𝑀 (𝑎𝑥𝑦𝑎) ˄ {⋁{ 𝐼𝑁 (𝑟) ∧ 𝐼𝑀 (𝑡)}}
𝑣=𝑟𝑡
K. Porselvi, B. Elavarasan, F. Smarandache and Y. B. Jun, Neutrosophic ℵ-bi-ideals in semigroups
Neutrosophic Sets and Systems, Vol. 35, 2020
432
≥ 𝐼𝑀 (𝑎𝑥𝑦𝑎) ˄ 𝐼𝑁 (𝑎𝑧𝑥𝑦𝑎)˄ 𝐼𝑀 (𝑎𝑧𝑥𝑎)
≥ 𝐼𝑀 (𝑎) ˄ 𝐼𝑁 (𝑎) ∧ 𝐼𝑀 (𝑎) = 𝐼𝑀∩𝑁 (𝑎),
and
𝐹𝑀∘𝑁∘𝑀 (𝑎) = ⋀ {𝐹𝑀 (𝑘) ˅ 𝐹𝑁∘𝑀 (𝑚)}
𝑎=𝑘𝑚
=
⋀
{𝐹𝑀 (𝑎𝑥𝑦𝑎) ˅ {⋀{ 𝐹𝑁 (𝑟) ∨ 𝐹𝑀 (𝑡)}}
𝑎=(𝑎𝑥𝑦𝑎)𝑣
𝑣=𝑟𝑡
≤ 𝐹𝑀 (𝑎𝑥𝑦𝑎)˅ 𝐹𝑁 (𝑎𝑧𝑥𝑦𝑎)˅ 𝐹𝑀 (𝑎𝑧𝑥𝑎)
≤ 𝐹𝑀 (𝑎)˅ 𝐹𝑁 (𝑎)˅ 𝐹𝑀 (𝑎) = 𝐹𝑀∩𝑁 (𝑎).
for every neutrosophic ℵ − bi-ideals 𝑋𝑀 and 𝑋𝑁 of 𝑋.
Therefore 𝑋𝑀 ∩ 𝑋𝑁 ⊆ 𝑋𝑀 ʘ𝑋𝑁 ʘ𝑋𝑀
(𝒊𝒊) ⇒ (𝒊) Let ℎ𝑗 ∈ 𝑋. Then
𝜒𝐵(ℎ𝑗 ) (𝑋𝑀 ) ⊆ 𝜒𝐵(ℎ𝑗 ) (𝑋𝑀 ) ∩ 𝜒𝐵(ℎ𝑗 ) (𝑋𝑀 ) ⊆ 𝜒𝐵(ℎ𝑗 ) (𝑋𝑀 )⨀ 𝜒𝐵(ℎ𝑗 ) (𝑋𝑀 ) ⨀𝜒𝐵(ℎ𝑗 ) (𝑋𝑀 ).
So
𝜒𝐵(ℎ𝑗 ) (𝑇)𝑀 (ℎ𝑗 ) ≥ 𝜒𝐵(ℎ𝑗 )𝐵(ℎ𝑗 )𝐵(ℎ𝑗 ) (𝑇)𝑀 (ℎ𝑗 ),
𝜒𝐵(ℎ𝑗 ) (𝐼)𝑀 (ℎ𝑗 ) ≤ 𝜒𝐵(ℎ𝑗 )𝐵(ℎ𝑗 )𝐵(ℎ𝑗 ) (𝐼)𝑀 (ℎ𝑗 ),
𝜒𝐵(ℎ𝑗 ) (𝐹)𝑀 (ℎ𝑗 ) ≥ 𝜒𝐵(ℎ𝑗 )𝐵(ℎ𝑗 )𝐵(ℎ𝑗 ) (𝐹)𝑀 (ℎ𝑗 ).
Since
𝜒𝐵(ℎ𝑗 ) (𝑇)𝑀 (ℎ𝑗 ) = −1 = 𝜒𝐵(ℎ𝑗 ) (𝐹)𝑀 (ℎ𝑗 )
𝜒𝐵(ℎ𝑗 )𝐵(ℎ𝑗 )𝐵(ℎ𝑗 ) (𝑇)𝑀 (ℎ𝑗 ) = −1 = 𝜒𝐵(ℎ𝑗 )𝐵(ℎ𝑗 )𝐵(ℎ𝑗 ) (𝐹)𝑀 (ℎ𝑗 )
and
and
𝜒𝐵(ℎ𝑗 ) (𝐼)𝑀 (ℎ𝑗 ) = 0,
we
𝜒𝐵(ℎ𝑗 )𝐵(ℎ𝑗 )𝐵(ℎ𝑗 ) (𝐼)𝑀 (ℎ𝑗 ) = 0
get
which
imply ℎ𝑗 ∈ 𝐵(ℎ𝑗 )𝐵(ℎ𝑗 )𝐵(ℎ𝑗 ). Therefore 𝑋 is intra-regular and regular.
□
Theorem 3.15. For any 𝑋, the equivalent assertions are:
(i)
𝑋 is intra-regular,
(ii) For each neutrosophic ℵ −ideal 𝑋𝑀 of 𝑋, 𝑋𝑀 (𝑎) = 𝑋𝑀 (𝑎2 )
∀𝑎 ∈ 𝑋.
Proof: (𝒊) ⇒ (𝒊𝒊) Let 𝑎 ∈ 𝑋. Then 𝑎 = 𝑦𝑎2 𝑧 for some 𝑦, 𝑧 ∈ 𝑋. For a neutrosophic ℵ −ideal 𝑋𝑀 ,
we have
𝑇𝑀 (𝑎) = 𝑇𝑀 (𝑦𝑎2 𝑧) ≤ 𝑇𝑀 (𝑎2 𝑧) ≤ 𝑇𝑀 (𝑎2 ) ≤ 𝑇𝑀 (𝑎),
𝐼𝑀 (𝑎) = 𝐼𝑀 (𝑦𝑎2 𝑧) ≥ 𝐼𝑀 (𝑎2 𝑧) ≥ 𝐼𝑀 (𝑎2 ) ≥ 𝐼𝑀 (𝑎),
𝐹𝑀 (𝑎) = 𝐹𝑀 (𝑦𝑎2 𝑧) ≤ 𝐹𝑀 (𝑎2 𝑧) ≤ 𝐹𝑀 (𝑎2 ) ≤ 𝐹𝑀 (𝑎),
so 𝑇𝑀 (𝑎) = 𝑇𝑀 (𝑎2 ); 𝐼𝑀 (𝑎) = 𝐼𝑀 (𝑎2 ) and 𝐹𝑀 (𝑎) = 𝐹𝑀 (𝑎2 ) for all 𝑎 ∈ 𝑋. Therefore 𝑋𝑀 (𝑎) = 𝑋𝑀 (𝑎2 )
(𝒊𝒊) ⇒ (𝒊) Let 𝑎 ∈ 𝑋. Then 𝐼(𝑎2 ) is an ideal of 𝑋. Thus 𝜒𝐼(𝑎2 ) (𝑋𝑀 ) is neutrosophic ℵ −ideal
by Theorem 3.5 of [4].
By assumption, 𝜒𝐼(𝑎2 ) (𝑋𝑀 )(𝑎) = 𝜒𝐼(𝑎2 ) (𝑋𝑀 )(𝑎2 ).
Since 𝜒𝐼(𝑎2 ) (𝑇)𝑀 (𝑎2 ) =
−1 = 𝜒𝐼(𝑎2 ) (𝐹)𝑀 (𝑎2 ) and 𝜒𝐼(𝑎2 ) (𝐼)𝑀 (𝑎2 ) = 0, we get 𝜒𝐼(𝑎2 ) (𝑇)𝑀 (𝑎) = −1 = 𝜒𝐼(𝑎2 ) (𝐹)𝑀 (𝑎) and
𝜒𝐼(𝑎2 ) (𝐼)𝑀 (𝑎) = 0 imply 𝑎 ∈ 𝐼(𝑎2 ). Thus 𝑋 is intra-regular.
□
Theorem 3.16. For any 𝑋, the equivalent assertions are:
(i)
𝑋 is left (resp., right) regular,
(ii)
For each neutrosophic ℵ −left (resp., right) ideal 𝑋𝑀 of 𝑋, 𝑋𝑀 (𝑎) = 𝑋𝑀 (𝑎2 )
Proof: (𝒊) ⇒ (𝒊𝒊) Suppose 𝑋 is left regular. Then 𝑎 = 𝑦𝑎2 for some 𝑦 ∈ 𝑋
∀𝑎 ∈ 𝑋.
Let 𝑋𝑀 be
neutrosophic ℵ − left ideal. Then 𝑇𝑀 (𝑎) = 𝑇𝑀 (𝑦𝑎2 ) ≤ 𝑇𝑀 (a2 ) and so 𝑇𝑀 (𝑎) = 𝑇𝑀 (𝑎2 ), 𝐼𝑀 (𝑎) =
K. Porselvi, B. Elavarasan, F. Smarandache and Y. B. Jun, Neutrosophic ℵ-bi-ideals in semigroups
Neutrosophic Sets and Systems, Vol. 35, 2020
433
𝐼𝑀 (𝑦𝑎2 ) ≥ 𝐼𝑀 (𝑎) and so 𝐼𝑀 (𝑎) = 𝐼𝑀 (𝑎2 ), and 𝐹𝑀 (𝑎) = 𝐹𝑀 (𝑦𝑎2 ) ≤ 𝐹𝑀 (𝑎) and so 𝐹𝑀 (𝑎) = 𝐹𝑀 (𝑎2 ).
Therefore 𝑋𝑀 (𝑎) = 𝑋𝑀 (𝑎2 ) for all 𝑎 ∈ 𝑋.
(𝒊𝒊) ⇒ (𝒊) Let 𝑋𝑀 be neutrosophic ℵ −left ideal. Then for any 𝑎 ∈ 𝑋, we have 𝜒𝐿(𝑎2 ) (𝑇)𝑀 (𝑎) =
𝜒𝐿(𝑎2 ) (𝑇)𝑀 (𝑎2 ) = −1, 𝜒𝐿(𝑎2 ) (𝐼)𝑀 (𝑎) = 𝜒𝐿(𝑎2 ) (𝐼)𝑀 (𝑎2 ) = 0 and 𝜒𝐿(𝑎2 ) (𝐹)𝑀 (𝑎) = 𝜒𝐿(𝑎2 ) (𝐹)𝑀 (𝑎2 ) = −1
imply 𝑎 ∈ 𝐿(𝑎2 ). Thus 𝑋 is left regular.
□
Corollary 3.17. Let 𝑋 be a regular right duo (resp., left duo). Then the equivalent conditions are:
(i)
𝑋 is left regular,
(ii)
For each neutrosophic ℵ −bi- ideal 𝑋𝑀 of 𝑋, we have 𝑋𝑀 (𝑎) = 𝑋𝑀 (𝑎2 ) for all 𝑎 ∈ 𝑋.
Proof: It is evident from Theorem 3.5 and Theorem 3.16.
□
Conclusions
In this paper, we have presented the concept of neutrosophic ℵ − bi −ideals of semigroups and
explored their properties, and characterized regular semigroups, intra-regular semigroups and
semigroups using neutrosophic ℵ-bi-ideal structures. We have also shown that the neutrosophic
ℵ-product of ideals and the intersection of neutrosophic ℵ-ideals are identical for a regular
semigroup. In future, we will focus on the idea of neutrosophic ℵ −prime ideals of semigroups and
its properties.
Acknowledgments: The authors express their gratitude to the referees for valuable comments and
suggestions which improve the article a lot.
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Received: Apr 28, 2020. Accepted: July 17, 2020
K. Porselvi, B. Elavarasan, F. Smarandache and Y. B. Jun, Neutrosophic ℵ-bi-ideals in semigroups
Neutrosophic Sets and Systems, Vol. 35, 2020
University of New Mexico
A Novel Approach for Pairwise Separation Axioms on Bi-Soft
Topology Using Neutrosophic Sets and An Output Validation in
Real Life Application
Chinnadurai V1, Sindhu M P2
1
2
Department of Mathematics, Annamalai University, Annamalainagar, Tamilnadu, India; kv.chinnaduraimaths@gmail.com
Department of Mathematics, Karpagam College of Engineering, Coimbatore, Tamilnadu, India; sindhu.mp@kce.ac.in
Correspondence: kv.chinnaduraimaths@gmail.com
Abstract: The set that lightens the vagueness stage more energetically than fuzzy sets are
neutrosophic sets. Bi-soft topological space is a space which goes for two different topologies with
certain parameters. This work carries out, construction of such type of topology on neutrosophic.
Besides by means of this, separation axioms are extended to pairwise separation axioms by using
neutrosophic and to analyze the relationship among the class of such spaces. Here some of their
properties are discussed with illustrative examples. In addition to it, we initiate the matrix form of
neutrosophic soft sets in such space. Here problems deal to take a decision in life by the choice of
two different groups. The aim of this decision making problem is to determine the unique thing or
person from the universe by giving marks depending on parameters. Step by step process of
solving the problem is explained in algorithm, also formulae given to determine their values with
illustrative examples.
Keywords: Neutrosophic sets (NSs); neutrosophic soft sets (NSSs); neutrosophic soft topological
spaces (NSTSs); neutrosophic soft Ti 0,1, 2, 3, 4 -spaces (NS Ti 0,1, 2, 3, 4 -spaces); neutrosophic
bitopological spaces (NBTSs); neutrosophic bi-soft topological spaces (NBSTSs); pairwise
neutrosophic soft Ti 0,1, 2, 3, 4 -spaces (pairwise NS Ti 0,1, 2, 3, 4 -spaces); decision making (DM).
1. Introduction
Zadeh [54], evaluated a fuzzy set (1965) to explore the situations like risky, unclear, erratic and
distortion occurs in our life cycle. Fuzzy sets simplify classical sets and are unique cases of the
membership functions. It has been used in a spacious collection of domains. This set extended to
develop intuitionistic fuzzy set (IFS) theory (1986) by Atanassov [47]. Smarandache [7] originated a
set which forecast the indeterminancy part along with truth and false statements, called NS (1998),
such as blending of network arises to unpredictable states. It is a dynamic structure which postulates
the concept of all other sets introduced before. Later, he generalized the NS on IFS [8] and recently
proposed his work on attributes valued set, plithogenic set (PS) [9]. In day-to-day life decisions taken
to diagnostic the problems either positive or negative even not both. Such types of problems are key
role in all fields and so most of the researchers studied DM problem. In recent times various works
Chinnadurai V and Sindhu M P, A Novel Approach for Pairwise Separation Axioms on Bi-Soft Topology Using
Neutrosophic Sets and An Output Validation in Real Life Application
Neutrosophic Sets and Systems, Vol. 35, 2020
436
have been done on these sets by Salma and Alblowi [33] and on extension of neutrosophic analysis
on DM by Abdel et al. [1-6].
Soft set (1999) is a broad mathematical gadget which accord with a group of objects based on
fairly accurate descriptions with orientation to elements of a parameter set was projected by
Molodtsov [46]. Topological structure on this set explored by Shabir & Naz [38] as soft topological
spaces (2011). Anon this thought were developed by Ali et al. [35, 40], Bayramov and Gunduz [22,
29], Cagman et al. [37], Chen [43], Feng et al. [41], Hussain and Ahmad [36], Maji et al. [44, 45], Min
[39], Nazmul and Samanta [32], Pie and Miao [42], Tantawy et al. [26], Varol and Aygun [31],
Zorlutuna et al. [34]. Maji [30] presented the binding of neutrosophic with soft set termed as NSSs
(2013). Bera & Mahapatra [23] defined such type of set on topological structure, named as NSTSs
(2017). Using these concepts, Deli & Broumi [27], Bera & Mahapatra [10, 24], on separation axioms by
Cigdem et al. [20, 21] have done some research works. Mostly DM applied on these sets related to
fuzzy with multicriteria by Chinnadurai et al. [13, 14 & 19], Abishek et al. [12], Muhammad et al.
[16], Mehmood et al. [17], Evanzalin Ebenanjar et al. [18] and Faruk [25].
Kelly [55] imported the approach of a set equipped with two topologies, named as bitopological
space (BTS) (1963), which is the generic system of topological space. Also it was carried out by Lane
[53], Patty [52], Kalaiselvi and Sindhu [15] and pairwise concepts by Kim [51], Singal and Asha [50],
Lal [48], Reilly [49]. Naz, Shabir and Ali [28] introduced bi-soft topological spaces (BSTSs) (2015) and
studied the separation axioms on it. Taha and Alkan [11] presented BTS on neutrosophic structure as
NBTSs (2019) which is engaged with two neutrosophic topologies (NTs).
The intension of this paper is to initiate the idea of NS on BSTS and to study some essential
properties of such spaces. Also, the pairwise concept on separation axioms implemented in NBSTS.
In addition, the NSSs referred as matrix form on NBSTS. As real life application, decisions made to
select the one among the universe based on its parameters by considering two different groups as
neutrosophic soft topologies (NSTs).
The arrangements made in this paper are as follows. Some basic definitions related to NS are in
segment 2. The results of NBSTS are proved and disproved by counter examples in segment 3. The
bonding among the pairwise separation axioms on NBSTS are stated with illustrative examples in
segment 4. In segment 5, the method of evaluating DM problems are described in algorithm and
formula specified to calculate the scores of universe set, to choose the best among them with
illustrative examples. Finally, concluded with future work in segment 6.
2. Preamble
In this segment, we evoke few primary definitions associated to NSS, NSTS, BSTS and NBSTS.
Definition 2.1 [30] Let V be the set of universe and E be a set of parameters. Let NS(V) denote the set
of all NSs of V. Then a estimated function of NSS K over V is a set defined by a mapping
f K : E NS (V ) . The NSS is a parameterized family of the set NS(V) which can be written as a set of
ordered pairs,
K e, v, T f K (e) (v), I f K (e) (v), Ff K (e) (v) : v V
: e E
Chinnadurai V and Sindhu M P, A Novel Approach for Pairwise Separation Axioms on Bi-Soft Topology Using
Neutrosophic Sets and An Output Validation in Real Life Application
Neutrosophic Sets and Systems, Vol. 35, 2020
T f K (e) (v), I f K (e) (v), Ff K (e) (v) [0,1]
where
437
respectively
called
the
truth-membership,
indeterminacy-membership and false-membership functions of
f K (e)
and the inequality
0 T f K (e) (v) I f K (e) (v) Ff K (e) (v) 3 is obvious.
Definition 2.2 [23] Let NSS(V) denote the set of all NSSs of V through all e E and u NSS (V , E )
. Then u is called NST on (V, E) if the following conditions are satisfied.
(i) u ,1u u , where null NSS u e, v, 0, 0,1) : v V : e E and absolute NSS
1u e, v,1,1, 0) : v V : e E .
(ii) the intersection of any finite number of members of u belongs to u .
(iii) the union of any collection of members of u belongs to u .
Then the triplet (V, E, u ) is called a NSTS.
Every member of u is called u -open NSS, whose complement is u -closed NSS.
Definition 2.3 [21] Let NSS(V, E) denote the family of all NSSs over V. The NSS ue( , , ) is called a
NSP, for every u V , 0 , , 1, e E and is defined as follows:
( , , ), if e e and v u
u((e), , ) (e)(v)
(0, 0,1), if e e and v u
Obviously, every NSS is the union of its NSPs.
Definition 2.4 [11] Let (V, u1 ) and (V, u 2 ) be the two different NTs on V. Then (V, u1 , u 2 ) is called a
NBTS.
3. NBSTS
In this segment, the conception of NBSTS is defined and some key resources of topology are
studied on it. The theoretical results are supported by some significant descriptive examples.
Definition 3.1 The quadruple (V, E, u1 , u 2 ) is called a NBSTS over (V, E), where u1 and u 2 are
NSTs independently satisfy the axioms of NSTS.
The elements of u1 are u1 -neutrosophic soft open sets ( u1 -NSOSs) and the complement of it are
u1 -neutrosophic soft closed sets ( u1 -NSOSs).
Example 3.2 Let V = v1,v2 , E = e1,e2 and u1 = { u ,1u , K1 } and u 2 = { u ,1u , L1, L2 } where K1, L1, L2
are NSSs over (V, E), defined as follows
f K1 (e1) { v1, (1,1, 0) , v2 , (0, 0,1) } ,
f K1 (e2 ) { v1, (0, 0,1) , v2 , (1, 0,1) }
and
f L1 (e1) { v1, (1, 0,1) , v2 , (0, 0,1) } ,
Chinnadurai V and Sindhu M P, A Novel Approach for Pairwise Separation Axioms on Bi-Soft Topology Using
Neutrosophic Sets and An Output Validation in Real Life Application
Neutrosophic Sets and Systems, Vol. 35, 2020
438
f L1 (e2 ) { v1, (0,1, 0) , v2 , (1,1, 0) } ;
f L2 (e1) { v1, (0,1, 0) , v2 , (1,1, 0) } ,
f L2 (e2 ) { v1, (1, 0,1) , v2 , (0, 0,1) } .
Thus u1 and u 2 are NSTs on (V, E) and so (V, E, u1 , u 2 ) is a NBSTS over (V, E).
Example 3.3 Let the neutrosophic soft indiscrete (trivial) topology u1 = {u ,1u } and neutrosophic
soft discrete topology u 2 NSS (V , E ) .
Then (V, E, u1 , u 2 ) is a NBSTS over (V, E).
Definition 3.4 Let (V, E, u1 , u 2 ) be a NBSTS over (V, E), where u1 and u 2 are NSTs on (V, E) and
P, Q NSS (V , E)
be
any
two
arbitrary
NSSs.
Suppose
v1 P Ki / Ki u1
and
v 2 Q Li / Li u 2 . Then v1 and v 2 are also NSTs on (V, E). Thus (V, E, v1 , v 2 ) is a NBST
subspace of (V, E, u1 , u 2 ).
Theorem 3.5 Let (V, E, u1 , u 2 ) be a NBSTS over (V, E), where u1 (e) and u 2 (e) are defined as
u1(e) { f K (e) / K u1}
u 2 (e) { f L (e) / L u 2} for each e E .
Then (V, E, u1 (e) , u 2 (e) ) is a NBTS over (V, E).
Proof. Follows from the fact that u1 and u 2 are NTs on V.
Example 3.6 Let V = v1, v2 , v3 , E = e1,e2 and u1 = { u ,1u , K1, K2 } and u 2 = { u ,1u , L1, L2 , L3 , L4 }
where K1, K2 , L1, L2 , L3 , L4 are NSSs over (V, E), defined as follows
f K1 (e1) { v1, (1, .5, .4) , v2 , (.6, .6, .6) , v3 , (.3, .4, .9) } ,
f K1 (e2 ) { v1, (.8, .4, .5) , v2 , (.7, .7, .3) , v3 , (.7, .5, .6) } ;
f K 2 (e1) { v1, (.8, .5, .1) , v2 , (.8, .6, .5) , v3 , (.5, .6, .4) } ,
f K 2 (e2 ) { v1, (.9, .7, .1) , v2 , (.9, .9, .2) , v3 , (.8, .6, .3) }
and
f L1 (e1) { v1, (.3, .7, .6) , v2 , (.4, .3, .8) , v3 , (.6, .4, .5) } ,
f L1 (e2 ) { v1, (.4, .6, .8) , v2 , (.3, .7, .2) , v3 , (.3, .3, .7) } ;
f L2 (e1) { v1, (.6, .6, .8) , v2 , (.2, .9, .3) , v3 , (.1, .2, .4) } ,
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f L2 (e2 ) { v1, (.7, .9, .5) , v2 , (.4, .2, .3) , v3 , (.5, .5, .4) } ;
f L3 (e1) { v1, (.6, .7, .6) , v2 , (.4, .9, .3) , v3 , (.6, .4, .4) } ,
f L3 (e2 ) { v1, (.7, .9, .5) , v2 , (.4, .7, .2) , v3 , (.5, .5, .4) } ;
f L4 (e1) { v1, (.3, .6, .8) , v2 , (.2, .3, .8) , v3 , (.1, .2, .5) } ,
f L4 (e2 ) { v1, (.4, .6, .8) , v2 , (.3, .2, .3) , v3 , (.3, .3, .7) }
Thus u1 and u 2 are NSTs on (V, E) and so (V, E, u1 , u 2 ) is a NBSTS over (V, E).
Now,
,1 , v1, (1, .5, .4) , v2 , (.6, .6, .6) , v3 , (.3, .4, .9) ,
u1 (e1 ) u u
,
v1, (.8, .5, .1) , v2 , (.8, .6, .5) , v3 , (.5, .6, .4)
u ,1u , v1, (.3, .7, .6) , v2 , (.4, .3, .8) , v3 , (.6, .4, .5) ,
v , (.6, .6, .8) , v2 , (.2, .9, .3) , v3 , (.1, .2, .4) ,
u 2 (e1 ) 1
v1, (.6, .7, .6) , v2 , (.4, .9, .3) , v3 , (.6, .4, .4) ,
v1, (.3, .6, .8) , v2 , (.2, .3, .8) , v3 , (.1, .2, .5)
and
,1 , v1, (.8, .4, .5) , v2 , (.7, .7, .3) , v3 , (.7, .5, .6) ,
u1 (e2 ) u u
,
v1, (.9, .7, .1) , v2 , (.9, .9, .2) , v3 , (.8, .6, .3)
u ,1u , v1, (.4, .6, .8) , v2 , (.3, .7, .2) , v3 , (.3, .3, .7) ,
v , (.7, .9, .5) , v2 , (.4, .2, .3) , v3 , (.5, .5, .4) ,
u 2 (e2 ) 1
v
,
(.
7
,
.
9
,
.
5
)
,
v
,
(.
4
,
.
7
,
.
2
)
,
v
,
(.
5
,
.
5
,
.
4
)
,
2
3
1
v1, (.4, .6, .8) , v2 , (.3, .2, .3) , v3 , (.3, .3, .7)
are NTs on V.
Thus (V, E, u1 (e) , u 2 (e) ) is a NBTS over (V, E).
Definition 3.7 Let (V, E, u1 , u 2 ) be a NBSTS over (V, E). Then the supremum NST is u1 u 2 ,
which is the smallest NST on V that contains u1 u 2 .
Example 3.8 Let us consider 3.5 example, where u1 and u 2 are NSTs on (V, E).
Then,
f P (e1 ) { v1, (.3, .7, .4) , v2 , (.6, .6, .6) , v3 , (.6, .4, .5) }
K1 L1 P
f P (e2 ) { v1, (.8, .6, .5) , v2 , (.7, .7, .2) , v3 , (.7, .5, .6) }
and
f Q (e1 , e1 ) { v1 , (.3, .7, .4) , v2 , (.6, .6, .6) , v3 , (.6, .4, .5) }
f Q (e1 , e2 ) { v1 , (.8, .4, .5) , v2 , (.7, .7, .3) , v3 , (.7, .5, .4) }
K1 L1 Q
f Q (e2 , e1 ) { v1 , (1, .6, .4) , v2 , (.6, .7, .2) , v3 , (.3, .4, .7) }
f (e , e ) { v , (.8, .6, .5) , v , (.7, .7, .2) , v , (.7, .5, .6) }
1
2
3
Q 2 2
Thus K1 L1 is the smallest NSS on V that contains K1 L1 .
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Theorem 3.9 If (V, E, u1 , u 2 ) is a NBSTS over (V, E), then u1 u 2 is a NST over (V, E).
Proof. Let (V, E, u1 , u 2 ) be a NBSTS over (V, E).
(i) Since u ,1u u1 and u ,1u u1 , it follows that u ,1u u1 u 2 .
(ii) Suppose that Ki / i I is a family of NSSs in u1 u 2 .
Then Ki u1 and Ki u 2 for all i I .
Thus iI K i u1 and iI K i u 2 .
Therefore iI K i u1 u 2 .
(iii) Let K , L u1 u 2 .
Then K , L u1 and K , L u 2 .
Since K L u1 and K L u 2 , we have K L u1 u 2 .
Hence u1 u 2 is a NST over (V, E).
Remark 3.10 If (V, E, u1 , u 2 ) is a NBSTS over (V, E), then u1 u 2 need not be a NST over (V, E).
Example 3.11 Let us consider 3.5 example where u1 and u 2 are NSTs on (V, E).
Then,
f P (e1 ) { v1, (0.3, 0.7, 0.4) , v2 , (0.6, 0.6, 0.6) , v3 , (0.6, 0.4, 0.5) }
K1 L1 P
f P (e2 ) { v1, (0.8, 0.6, 0.5) , v2 , (0.7, 0.7, 0.2) , v3 , (0.7, 0.5, 0.6) }
Thus K1 L1 u1 u 2 .
Hence u1 u 2 is not a NST over (V, E).
4. Neutrosophic bi-soft separation axioms
In this segment, the separation of NBSTS is explored. The pairwise NS Ti 0,1, 2, 3, 4 -spaces on
NBSTS are introduced and the relationships among them are examined with relevant examples.
Definition 4.1 A NBSTS (V, E, u1 , u 2 ) over (V, E) is called a pairwise NS T0 -space, if u((e), , ) and
v((e), , ) are distinct NSPs then there exist u1 -NSOS K and u 2 -NSOS L such that
u((e), , ) K ; u((e), , ) L u
or v((e), , ) L ; v((e), , ) K u .
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Example 4.2 Consider neutrosophic soft indiscrete (trivial) topology u1 = {u ,1u } and neutrosophic
soft discrete topology u 2 NSS (U , E ) .
Thus (V, E, u1 , u 2 ) is a pairwise NS T0 -space.
Theorem 4.3 Let (V, E, u1 , u 2 ) be a NBSTS over (V, E). If (V, E, u1 , u 2 ) is a pairwise NS T0 -space
then (V, E, u1 u 2 ) is a NS T0 -space.
Proof. Let (V, E, u1 , u 2 ) be a NBSTS over (V, E).
Suppose that (V, E, u1 , u 2 ) is a pairwise NS T0 -space.
Let u((e), , ) and v((e), , ) be any two distinct NSPs.
Then there exist u1 -NSOS K and u 2 -NSOS L such that
u((e), , ) K ; u((e), , ) L u
or v((e), , ) L ; v((e), , ) K u
In either case K , L u1 u 2 .
Hence (V, E, u1 u 2 ) is a NS T0 -space.
Theorem 4.4 Let (V, E, u1 , u 2 ) be a NBSTS over (V, E). If (V, E, u1 , u 2 ) is a pairwise NS T0 -space
then (V, E, v1 , v 2 ) is also a pairwise NS T0 -space.
Proof. Let (V, E, u1 , u 2 ) be a NBSTS over (V, E).
Let u((e), , ) and v((e), , ) be any two distinct NSPs and P, Q NSS (U , E ) .
Suppose that (V, E, u1 , u 2 ) is a pairwise NS T0 -space.
Then there exist u1 -NSOS K and u 2 -NSOS L such that
u((e), , ) K ; u((e), , ) L u
or v((e), , ) L ; v((e), , ) K u
Now u((e), , ) P and u((e), , ) K .
Then u((e), , ) P K , where K u1 .
Consider u((e), , ) L u .
u((e), , ) L Q u Q .
u((e), , ) (Q L) u .
Thus u((e), , ) P K ; u((e), , ) (Q L) u , where P K v1 , Q L v 2 .
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Or if v((e), , ) L ; v((e), , ) K u , it can be proved that
v((e), , ) Q L ; v((e), , ) (Q L) u , where P K v1 , Q L v 2 .
Hence (V, E, v1 , v 2 ) is also a pairwise NS T0 -space.
Definition 4.5 A NBSTS (V, E, u1 , u 2 ) over (V, E) is called pairwise NS T1 -space, if u((e), , ) and
v((e), , ) are distinct NSPs then there exist u1 -NSOS K and u 2 -NSOS L such that
u((e), , ) K ;
and v((e), , ) L ;
u((e), , ) L u
v((e), , ) K u .
Example 4.6 Let V = v1,v2 , E = {e}, and v1(.(2e,).3 ,.7 ) and v2(.(9e,).4 ,.1) be NSPs. Let u1 {u ,1u , K} and
u 2 {u ,1u , L} where K and L are NSSs over (V, E), defined as
K v1(.(2e,).3, .7) f K (e) { v1, (.2, .3, .7) , v2 , (0, 0,1) }
and
L v2(.(9e,).4, .1) f L (e) { v1, (0, 0,1) , v2 , (.9,.4, .1) } .
Thus (V, E, u1 , u 2 ) is a NBSTS over (V, E).
Hence (V, E, u1 , u 2 ) is a pairwise NS T1 -space, also a pairwise NS T0 -space.
Theorem 4.7 Every pairwise NS T1 -space is also a pairwise NS T0 -space.
Proof. Follows from the Definitions 4.1 and 4.3.
Remark 4.8 The converse of the 4.7 theorem is not true, which is shown in the following example.
Example 4.9 Let V = v1,v2 , E = e1,e2 , and v1(.(2e, .)5 ,.7) , v1(.(2e, .)8 ,.2) , v2(.(2e, .7) ,.5) and v2(.(1e, .1),.9) be NSPs. Let
1
2
1
2
u1 = { u ,1u , K1, K2 , K3 } and u 2 = { u ,1u , L1, L2 } where K1, K 2 , K3 , L1, L2 are NSSs over (V, E), defined
as
f K1 (e1 ) { v1, (.2, .5, .7) , v2 , (0, 0,1) }
K1 v1(.(2e, .)5 ,.7)
1
f K1 (e2 ) { v1, (0, 0,1) , v2 , (0, 0,1) }
;
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f K 2 (e1 ) { v1, (0, 0,1) , v2 , (0, 0,1) }
K 2 v2(.(1e, .1),.9)
2
f K 2 (e2 ) { v1, (0, 0,1) , v2 , (.1, .1, .9) }
;
K3 K1 K 2
and
f L1 (e1 ) { v1, (0, 0,1) , v2 , (.2, .7, .5) }
L1 v2(.(2e, .7) ,.5)
1
f L1 (e2 ) { v1, (0, 0,1) , v2 , (0, 0,1) }
;
f L2 (e1 ) { v1, (.2, .5, .7) , v2 , (.2, .7, .5) }
L2 v1(.(2e, .)5 ,.7) , v11((.e2,).8 ,.2) , v2(.(2e, .7) ,.5) , v2(.(1e, .1),.9)
1
2
1
2
f L2 (e2 ) { v1, (.2, .8, .2) , v2 , (.1, .1, .9) }
Thus (V, E, u1 , u 2 ) is a NBSTS over (V, E).
Hence (V, E, u1 , u 2 ) is a pairwise NS T0 -space, but not a pairwise NS T1 -space since for NSPs
v1(.(2e, .)5 ,.7) and v2(.(1e, .1),.9) , (V, E, u1 , u 2 ) is not a pairwise NS T1 -space.
1
2
Theorem 4.10 Let (V, E, u1 , u 2 ) be a NBSTS over (V, E). If (V, E, u1 ) or (V, E, u 2 ) is not a NS T0-space,
then (V, E, u1 , u 2 ) is a pairwise NS T0 -space but not a pairwise NS T1 -space.
Proof. Let K u1 and L u 2 , also u((e), , ) and v((e), , ) be any two distinct NSPs.
Suppose (V, E, u1 ) is a NS T0 -space and (V, E, u 2 ) is not a NS T0 -space.
Then, u((e), , ) K ;
and v((e), , ) L ;
u((e), , ) L u
v((e), , ) K u
Thus by Definitions 4.1 and 4.3,
(V, E, u1 , u 2 ) is a pairwise NS T0 -space but not a pairwise NS T1 -space.
Theorem 4.11 Let (V, E, u1 , u 2 ) be a NBSTS over (V, E). Then (V, E, u1 , u 2 ) is a pairwise NS
T1 -space if and only if (V, E, u1 ) and (V, E, u 2 ) are NS T1 -spaces.
Proof. Let (V, E, u1 , u 2 ) be a NBSTS over (V, E).
Let u((e), , ) and v((e), , ) be any two distinct NSPs.
Suppose that (V, E, u1 ) and (V, E, u 2 ) are NS T1 -spaces.
Then there exist u1 -NSOS K and u 2 -NSOS L such that
u((e), , ) K ; u((e), , ) L u
and v((e), , ) L ; v((e), , ) K u
In either case the result follows immediately.
Thus (V, E, u1 , u 2 ) is a pairwise NS T1 -space.
Conversely, assume that (V, E, u1 , u 2 ) is a pairwise NS T1 -space.
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Then there exist some u1 -NSOS K1 and u 2 -NSOS L1 such that
u((e), , ) K1 ; u((e), , ) L1 u
and v((e), , ) L1 ; v((e), , ) K1 u
Also there exist some u1 -NSOS K 2 and u 2 -NSOS L2 such that
u((e), , ) K 2 ; u((e), , ) L2 u
and v((e), , ) L2 ; v((e), , ) K 2 u
Hence (V, E, u1 ) and (V, E, u 2 ) are NS T1 -spaces.
Theorem 4.12 Let (V, E, u1 , u 2 ) be a NBSTS over (V, E). If (V, E, u1 , u 2 ) is a pairwise NS T1 -space
then (V, E, u1 u 2 ) is a NS T1 -space.
Proof. Let (V, E, u1 , u 2 ) be a NBSTS over (V, E).
Suppose that (V, E, u1 , u 2 ) is a pairwise NS T1 -space.
Let u((e), , ) and v((e), , ) be any two distinct NSPs.
Then there exist u1 -NSOS K and u 2 -NSOS L such that
u((e), , ) K ; u((e), , ) L u
and v((e), , ) L ; v((e), , ) K u
In either case K , L u1 u 2 .
Hence (V, E, u1 u 2 ) is a NS T1 -space.
Theorem 4.13 Let (V, E, u1 , u 2 ) be a NBSTS over (V, E). If (V, E, u1 , u 2 ) is a pairwise NS T1 -space
then (V, E, v1 , v 2 ) is also a pairwise NS T1 -space.
Proof. Let (V, E, u1 , u 2 ) be a NBSTS over (V, E).
Let u((e), , ) and v((e), , ) be any two distinct NSPs and P, Q NSS (U , E ) .
Suppose that (V, E, u1 , u 2 ) is a pairwise NS T1 -space.
Then there exist u1 -NSOS K and u 2 -NSOS L such that
u((e), , ) K ; u((e), , ) L u
and v((e), , ) L ; v((e), , ) K u
Now u((e), , ) P and u((e), , ) K .
Then u((e), , ) P K , where K u1 .
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Consider u((e), , ) L u .
u((e), , ) L Q u Q .
u((e), , ) (Q L) u .
Thus u((e), , ) P K ; u((e), , ) (Q L) u , where P K v1 , Q L v 2 .
Further if v((e), , ) L ; v((e), , ) K u , it can be proved that
v((e), , ) Q L ; v((e), , ) (Q L) u , where P K v1 , Q L v 2 .
Hence (V, E, v1 , v 2 ) is also a pairwise NS T1 -space.
Theorem 4.14 Let (V, E, u1 , u 2 ) be a NBSTS over (V, E). For each pair of distinct NSPs u((e), , ) and
v((e), , ) , u((e), , ) is a u 2 -NSCS and v((e), , ) is a u1 -NSCS, then (V, E, u1 , u 2 ) is a pairwise NS
T1 -space.
Proof. Let (V, E, u1 , u 2 ) be a NBSTS over (V, E).
Suppose that for each pair of distinct NSPs u((e), , ) and v((e), , ) , u((e), , ) is a u 2 -NSCS.
Then u((e), , )
c
is a u 2 -NSOS.
Let u((e), , ) and v((e), , ) be any two distinct NSPs.
(i.e.,) u((e), , ) v((e), , ) u .
Thus
v((e), , ) u((e), , )
c
and u((e), , ) u((e), , )
c
u
(1)
u
(2)
Similarly assume that for each NSP, v((e), , ) is a u1 -NSCS.
Then v((e), , )
c
is a u1 -NSOS such that
u((e), , ) v((e), , )
c
and v((e), , ) v((e), , )
c
From (1) and (2),
u((e), , ) v((e), , )
and v((e), , ) u((e), , )
c
; u((e), , ) u((e), , )
c
; v((e), , ) v((e), , )
c
c
u
u
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Hence (V, E, u1 , u 2 ) is a pairwise NS T1 -space.
Definition 4.15 A NBSTS (V, E, u1 , u 2 ) over (V, E) is called pairwise NS T2 -space or pairwise NS
Hausdorff space, if u((e), , ) and v((e), , ) are distinct NSPs then there exist u1 -NSOS K and
u 2 -NSOS L such that
u((e), , ) K , v((e), , ) L and K L u .
Example 4.16 Let V = v1,v2 , E = e1,e2 , and v1(.(2e, .)5 ,.7) , v1(.(2e, .)8 ,.2) , v2(.(2e, .7) ,.5) and v2(.(1e, .1),.9) be NSPs. Let
1
2
1
2
u1 = { u ,1u , K1, K2 , K3 } and u 2 = { u ,1u , L1, L2 , L3 } where K1, K 2 , K3 , L1, L2 , L3 are NSSs over (V, E),
defined as follows
f K1 (e1 ) { v1, (.2, .5, .7) , v2 , (0, 0,1) }
K1 v1(.(2e, .)5 ,.7)
1
f K1 (e2 ) { v1, (0, 0,1) , v2 , (0, 0,1) }
;
f K 2 (e1 ) { v1, (0, 0,1) , v2 , (0, 0,1) }
K 2 v2(.(1e, .1),.9)
2
f K 2 (e2 ) { v1, (0, 0,1) , v2 , (.1, .1, .9) }
;
K3 K1 K 2
and
f L1 (e1 ) { v1, (0, 0,1) , v2 , (.2, .7, .5) }
L1 v2(.(2e, .7) ,.5)
1
f L1 (e2 ) { v1, (0, 0,1) , v2 , (0, 0,1) }
;
f L2 (e1 ) { v1, (0, 0,1) , v2 , (0, 0,1) }
;
L2 v1(.(2e, .)8 ,.2)
2
f L2 (e2 ) { v1, (.2, .8, .2) , v2 , (0, 0,1) }
L3 L1 L2
Then (V, E, u1 , u 2 ) is a NBSTS over (V, E).
Hence (V, E, u1 , u 2 ) is a pairwise NS T2 -space.
Theorem 4.17 Let (V, E, u1 , u 2 ) be a NBSTS over (V, E). If (V, E, u1 , u 2 ) is a pairwise NS T2 -space
then (V, E, u1 u 2 ) is a NS T2 -space.
Proof. Let (V, E, u1 , u 2 ) be a NBSTS over (V, E).
Suppose that (V, E, u1 , u 2 ) is a pairwise NS T2 -space.
Let u((e), , ) and v((e), , ) be any two distinct NSPs.
Then there exist u1 -NSOS K and u 2 -NSOS L such that
u((e), , ) K , v((e), , ) L and K L u .
In either case K , L u1 u 2 .
Hence (V, E, u1 u 2 ) is a NS T2 -space.
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Theorem 4.18 Let (V, E, u1 , u 2 ) be a NBSTS over (V, E). If (V, E, u1 , u 2 ) is a pairwise NS T2 -space
then (V, E, v1 , v 2 ) is also a pairwise NS T2 -space.
Proof. Let (V, E, u1 , u 2 ) be a NBSTS over (V, E).
Let u((e), , ) and v((e), , ) be any two distinct NSPs and P, Q NSS (U , E ) .
Suppose that (V, E, u1 , u 2 ) is a pairwise NS T2 -space.
Then there exist u1 -NSOS K and u 2 -NSOS L such that
u((e), , ) K , v((e), , ) L and K L u .
Now u((e), , ) P and u((e), , ) K ; v((e), , ) Q and v((e), , ) L
Then u((e), , ) P K , v((e), , ) Q L where K u1 , L u 2 .
Consider K L u .
P K L Q P u Q .
P K Q L u . .
Thus u((e), , ) P K , v((e), , ) Q L and P K Q L u .
Hence (V, E, v1 , v 2 ) is also a pairwise NS T2 -space.
Theorem 4.19 Every pairwise NS T2 -space is also a pairwise NS T1 -space.
Proof. Follows from Definitions 4.3 and 4.15.
Theorem 4.20 Let (V, E, u1 , u 2 ) be a NBSTS over (V, E). (V, E, u1 , u 2 ) is a pairwise NS T2 -space if
and only if for any two distinct NSPs u((e), , ) and v((e), , ) , there exist u1 -NSOS K containing
u((e), , ) but not v((e), , ) such that v((e), , ) K .
Proof. Let (V, E, u1 , u 2 ) be a NBSTS over (V, E).
Let u((e), , ) and v((e), , ) be any two distinct NSPs.
Suppose that (V, E, u1 , u 2 ) is a pairwise NS T2 -space.
Then there exist u1 -NSOS K and u 2 -NSOS L such that
u((e), , ) K , v((e), , ) L and K L u .
Since u((e), , ) v((e), , ) u and K L u , v((e), , ) K .
Thus v((e), , ) K .
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Conversely, assume that for any two distinct NSPs u((e), , ) and v((e), , ) , there exist u1 -NSOS K
containing u((e), , ) but not v((e), , ) such that v((e), , ) K .
c
Then v((e), , ) K .
K c
Thus K and
are disjoint u1 -NSOS and u1 -NSOS containing u((e), , ) and v((e), , )
respectively.
Theorem 4.21 Let (V, E, u1 , u 2 ) be a NBSTS over (V, E) and (V, E, u1 , u 2 ) be a pairwise NS T1 -space
for every NSP u((e), , ) K u1 . If there exist u 2 -NSOS L such that u((e), , ) L L K , then
(V, E, u1 , u 2 ) is a pairwise NS T2 -space.
Proof. Let (V, E, u1 , u 2 ) be a NBSTS over (V, E) and let it be a pairwise NS T1 -space
Suppose that u((e), , ) v((e), , ) u .
Let u((e), , ) be a u1 -NSCS and v((e), , ) be a u 2 -NSCS.
Then v((e), , )
c
is a u 2 -NSOS such that
u((e), , ) v((e), , )
c
u2
Then there exist a u 2 -NSOS L such that
c
u((e), , ) L L v((e), , ) .
Thus v((e), , )
L
c
c
c u .
, u((e), , ) L and L L
Hence (V, E, u1 , u 2 ) is a pairwise NS T2 -space.
Remark 4.22 Let (V, E, u1 , u 2 ) be a NBSTS over (V, E). For any NSS K over (V, E), K
u2
denotes the
NS closure of K with respect to u 2 -NST over (V, E).
Theorem 4.23 Let (V, E, u1 , u 2 ) be a NBSTS over (V, E). Then the following are equilavent:
(1) (V, E, u1 , u 2 ) is a pairwise NS Hausdorff space over (V, E).
(2) If u((e), , ) and v((e), , ) are distinct NSPs, there exist u1 -NSOS K such that
u((e), , ) K and v((e), , ) K
u2
c
.
Proof. (1) (2). Suppose that (V, E, u1 , u 2 ) is a pairwise NS Hausdorff space over (V, E).
Then there exist u1 -NSOS K and u 2 -NSOS L such that
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u((e), , ) K , v((e), , ) L and K L u .
So that K L c .
Since K
u2
is the smallest u 2 -NSCS that contains K and L c is a u 2 -NSCS, then K
L K
Thus v((e), , ) L K
Hence v((e), , ) K
u2
u2
u2
Lc
c
.
c
.
c
.
u2
(2) (1). Let u((e), , ) and v((e), , ) be any two distinct NSPs.
By assumption, there exist u1 -NSOS K such that u((e), , ) K and v((e), , ) K
As K
u2
is a u 2 -NSCS so L K
u2
u2
c
.
c
.
u2
Now u((e), , ) K , v((e), , ) L and
K L K K
u2
.
K K
u2
c
( K K
u2
)
u .
Thus K L u .
Hence (V, E, u1 , u 2 ) is a pairwise NS Hausdorff space over (V, E).
Definition 4.24 Let NSS(V, E) be the family of all NSSs over the universe V and u V . Then uE( , , )
denotes the NSS over (V, E) for which u((e), , ) u ( , , ) , for all
e E .
Corollary 4.25 Let (V, E, u1 , u 2 ) be a pairwise NS T2 -space over (V, E). Then for each NSP u((e), , ) ,
u E( , , ) K
u2
: u((e), , ) K u1 .
Proof. Let (V, E, u1 , u 2 ) be a pairwise NS T2 -space over (V, E) and u((e), , ) be a NSP.
Then there exist a NSOS u((e), , ) K u1 .
If u((e), , ) and v((e), , ) are distinct NSPs, by 4.24 theorem, there exist u1 -NSOS K such that
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u((e), , ) K and v((e), , ) K
v((e), , ) f
u2
450
c
.
K u 2 (e ) .
v((e), , ) u ( , , ) K u1 f u 2 (e)
(e)
K
Thus
for all e E .
( , , )
u2
K u1 u E( , , )
K : u(e)
( , , )
K K
Also it is obvious that u(e)
u2
.
Thus
u E( , , ) K
Hence from (1) and (2),
u E( , , ) K
u2
(1)
u2
: u((e), , ) K u1
(2)
: u((e), , ) K u1 .
Corollary 4.26 Let (V, E, u1 , u 2 ) be a pairwise NS T2 -space over (V, E). Then for each NSP u((e), , ) ,
u
( , , ) c
E
ui for i = 1, 2.
Proof. Let (V, E, u1 , u 2 ) be a pairwise NS T2 -space over (V, E) and u((e), , ) be a NSP.
By 4.25 corollary,
u
K
( , , ) c
E
Since K
u2
u2
c
: u ( , , ) K
u1
(e)
is a u 2 -NSCS, then K
u2
c
.
u2
By the axioms of a NS topological space,
u2
( , , )
K u1 u 2 .
K : u(e)
c
Thus u E( , , )
c
u 2 .
Similarly it can be proved that, uE( , , )
Hence uE( , , )
c
c
u1 .
ui for i = 1, 2.
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Definition 4.27 A NBSTS (V, E, u1 , u 2 ) over (V, E) is called pairwise NS regular space, if K is a u1
-NSCS and u((e), , ) K u then there exist u 2 -NSOSs L1 and L2 such that u((e), , ) L1 , K L2
and L1 L2 u .
A NBSTS (V, E, u1 , u 2 ) over (V, E) is called pairwise NS T3 -space, if it is both a pairwise NS regular
space and a pairwise NS T1 -space.
Theorem 4.28 Let (V, E, u1 , u 2 ) be a NBSTS over (V, E). Then (V, E, u1 , u 2 ) is a pairwise NS
T3 -space if and only if for every
u((e), , ) K u1 , there exists
L u 2
such that
u((e), , ) L L K .
Proof. Let (V, E, u1 , u 2 ) be a NBSTS over (V, E).
Suppose that (V, E, u1 , u 2 ) is a pairwise NS T3 -space and
u((e), , ) K u1 .
Since (V, E, u1 , u 2 ) is a pairwise NS T3 -space for the NSP u((e), , ) and u1 -NSCS K c , there exist
u 2 -NSOSs L1 and L2 such that u((e), , ) L1 , K c L2 and L1 L2 u .
Thus u((e), , ) L1 ( L2 ) c K .
Since ( L2 ) c is a u 2 -NSCS, L1 ( L2 ) c .
Hence u((e), , ) L1 L1 K .
Conversely, let u((e), , ) K u and K be a u1 -NSCS.
Thus u((e), , ) K c .
From the condition of the theorem,
u((e), , ) L L K c .
Then u((e), , ) L , K L c and L L c u .
Hence (V, E, u1 , u 2 ) is a pairwise NS T3 -space.
Definition 4.29 A NBSTS (V, E, u1 , u 2 ) over (V, E) is called pairwise NS normal space, if for every
pair of disjoint u1 -NSCSs K1 and K 2 , there exists disjoint u 2 -NSOSs L1 and L2 such that
K1 L1 and K 2 L2 .
A NBSTS (V, E, u1 , u 2 ) over (V, E) is called pairwise NS T4 -space, if it is both a pairwise NS normal
space and a pairwise NS T1 -space.
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Theorem 4.30 Let (V, E, u1 , u 2 ) be a NBSTS over (V, E). Then (V, E, u1 , u 2 ) is a pairwise NS
T4 -space if and only if for each u1 -NSCS K and u1 -NSOS L with K L , there exists a u 2 -NSOS
P such that K P P L .
Proof. Let (V, E, u1 , u 2 ) be a NBSTS over (V, E).
Suppose that (V, E, u1 , u 2 ) is a pairwise NS T4 -space and K be u1 -NSCS and K L u1 .
Then L c is a u1 -NSCS and K Lc u .
Since (V, E, u1 , u 2 ) is a pairwise NS T4 -space, there exist u 2 -NSOSs P1 and P2 such that
K P1 , M c P2 and P1 P2 u .
Thus K P1 ( P2 ) c L .
Since ( P2 ) c is a u 2 -NSCS, P1 ( P2 ) c .
Hence K P1 P1 L .
Conversely, let K1 and K 2 be any two disjoint u1 -NSCSs.
Then K1 ( K 2 ) c .
From the condition of the theorem, there exists a u 2 -NSOS P such that K1 P P ( K2 ) c .
Thus P and ( P) c are u 2 -NSOSs.
Then K1 P , K 2 ( P ) c and P ( P ) c u .
Hence (V, E, u1 , u 2 ) is a pairwise NS T4 -space.
Example 4.31 Let V = v1,v2 , E = e1, e2 , e3 , and v1(.(2e, .)4 ,.3) , v1(.(2e, .)8 ,.2) , v1(.(2e, .)5 ,.7) , v2(.(1e, .2) ,.5) , v2(.(2e, .7) ,.5) and
1
2
3
1
2
v2(.(1e, .1),.9) be NSPs.
3
Then u1 = u ,1u , K1, K2 , K3 , K4 , K5 , K6 , K7 and u 2 = u ,1u , L1, L2 , L3 , L4 , L5 , L6 , L7 where K1, K 2 ,
K3 , K4 , K5 , K6 , K7 , L1, L2 , L3 , L4 , L5 , L6 , L7 are NSSs over (V, E), defined as follows
f K (e1 ) { v1, (.2, .4, .3) , v2 , (1,1, 0) }
1
;
K1 f K1 (e2 ) { v1, (1,1, 0) , v2 , (1,1, 0) }
f K1 (e3 ) { v1, (1,1, 0) , v2 , (1,1, 0) }
f K (e1 ) { v1, (1,1, 0) , v2 , (1,1, 0) }
2
K 2 f K 2 (e2 ) { v1, (1,1, 0) , v2 , (.2, .7 ,.5) }
f K 2 (e3 ) { v1, (1,1, 0) , v2 , (1,1, 0) }
;
f K (e1 ) { v1, (1,1, 0) , v2 , (1,1, 0) }
3
K 3 f K 3 (e2 ) { v1, (1,1, 0) , v2 , (1,1, 0) }
f K 3 (e3 ) { v1, (.2, .5 ,.7) , v2 , (1,1, 0) }
;
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K 4 K1 K 2 ;
K5 K1 K3 ;
K 6 K 2 K3 ;
K7 K1 K2 K3
and
f L (e1 ) { v1, (.5, .7, .1) , v2 , (0, 0 ,1) }
1
L1 f L1 (e2 ) { v1, (0, 0,1) , v2 , (0, 0,1) }
f L1 (e3 ) { v1, (0, 0,1) , v2 , (0, 0,1) }
;
f L (e1 ) { v1, (0, 0,1) , v2 , (0, 0,1) }
2
L2 f L2 (e2 ) { v1, (0. 0 ,1) , v2 , (.8, .6, .1) }
f L2 (e3 ) { v1, (0, 0,1) , v2 , (0, 0,1) }
;
f L (e1 ) { v1, (0, 0,1) , v2 , (0, 0,1) }
3
L3 f L3 (e2 ) { v1, (0, 0,1) , v2 , (0, 0,1) }
f L3 (e3 ) { v1, (.7, .5, .2) , v2 , (0, 0 ,1) }
;
L4 L1 L2 ;
L5 L1 L3 ;
L6 L2 L3 ;
L7 L1 L2 L3 ;
Thus (V, E, u1 , u 2 ) is a NBSTS over (V, E).
Consider ( u1 ) c = { u ,1u , ( K1) c , ( K2 ) c , ( K3 ) c , ( K4 ) c , ( K5 ) c , ( K6 ) c , ( K7 ) c }
where ( K1) c , ( K2 ) c , ( K3 ) c , ( K4 ) c , ( K5 ) c , ( K6 ) c , ( K7 ) c are u1 -NSCSs over (V, E), defined as follows
f
c (e ) { v1 , (.3, .6, .2) , v2 , (0, 0,1) }
( K1 ) 1
( K1 ) c f ( K ) c (e2 ) { v1 , (0, 0,1) , v2 , (0, 0,1) }
1
f
c (e ) { v1 , (0, 0,1) , v2 , (0, 0,1) }
( K1 ) 3
;
f
c (e ) { v1 , (0, 0,1) , v2 , (0, 0,1) }
(K2 ) 1
c
( K 2 ) f ( K ) c (e2 ) { v1 , (0, 0,1) , v2 , (.5, .3 ,.2) }
2
f ( K ) c (e3 ) { v1 , (0, 0,1) , v2 , (0, 0,1) }
2
;
f
c (e ) { v1 , (0, 0,1) , v2 , (0, 0,1) }
(K3 ) 1
c
( K 3 ) f ( K ) c (e2 ) { v1 , (0, 0,1) , v2 , (0, 0,1) }
3
f ( K ) c (e3 ) { v1 , (.7, .5 ,.2) , v2 , (0, 0,1) }
3
;
( K4 ) c ( K1) c ( K2 ) c ;
( K5 ) c ( K1) c ( K3 ) c ;
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( K6 ) c ( K2 ) c ( K3 ) c ;
( K7 ) c ( K1) c ( K2 ) c ( K3 ) c
Hence (V, E, u1 , u 2 ) is a pairwise NS T4 -space, also a pairwise NS T3 -space.
5. DM Problem in NBSTS
In this segment, measured the output of problem and evaluated the decision on NBSTS.
Definition 5.1 Let V be the set of universal set, E be its parameter and u1 u ,1u , P and
u 2 u ,1u , Q be two NSTs. Then NSSs P and Q in NBSTS (V, E, u1 , u 2 ) over (V, E) are defined by
k l matrix where every entries are marks of vk based on each parameters el .
Pkl
T f P (e1 ) (v1 ), F f P (e1 ) (v1 ), I f P (e1 ) (v1 ) T f P (e2 ) (v1 ), F f P (e2 ) (v1 ), I f P (e2 ) (v1 ) T f P (el ) (v1 ), F f P (el ) (v1 ), I f P (el ) (v1 )
T
T f P (e2 ) (v2 ), F f P (e2 ) (v2 ), I f P (e2 ) (v2 ) T f P (el ) (v2 ), F f P (el ) (v2 ), I f P (el ) (v2 )
f P ( e1 ) (v2 ), F f P ( e1 ) (v2 ), I f P ( e1 ) ( 2 )
(
),
(
),
(
)
(
),
(
),
(
)
(
),
(
),
(
)
T
v
F
v
I
v
T
v
F
v
I
v
T
v
F
v
I
v
f P ( e1 ) k
f P ( e1 ) k
f P ( e2 ) k
f P ( e2 ) k
f P ( e2 ) k
f P ( el ) k
f P ( el ) k
f P ( el ) k
f P (e1 ) k
and
Qkl
T fQ (e1 ) (v1 ), F fQ (e1 ) (v1 ), I fQ (e1 ) (v1 ) T fQ (e2 ) (v1 ), F fQ (e2 ) (v1 ), I fQ (e2 ) (v1 ) T fQ (el ) (v1 ), F fQ (el ) (v1 ), I fQ (el ) (v1 )
T
T fQ (e2 ) (v2 ), F fQ (e2 ) (v2 ), I fQ (e2 ) (v2 ) T fQ (el ) (v2 ), F fQ (el ) (v2 ), I fQ (el ) (v2 )
fQ ( e1 ) (v2 ), F fQ ( e1 ) (v2 ), I fQ ( e1 ) ( 2 )
T fQ (e1 ) (vk ), F fQ (e1 ) (vk ), I fQ (e1 ) (vk ) T fQ (e2 ) (vk ), F fQ (e2 ) (vk ), I fQ (e2 ) (vk ) T fQ (el ) (vk ), F fQ (el ) (vk ), I fQ (el ) (vk )
where v1, v2 , , vk V and e1, e2 , , el E .
Clearly u1 u ,1u , Pk l and u 2 u ,1u , Qk l are also NSTs in NBSTS (V, E, u1 , u 2 ) over
(V, E).
Thus the outcome result (OR) of v V is given by the formula
T f (e) (v) F f Q (e) (v) T f Q (e) (v) F f P (e) (v)
OR(v)e P
2
1 I
f P ( e ) (v ) I f Q ( e ) (v )
2
(5.1.1)
where e E .
The Net Result (NR) of each v1, v2 , , vk V is
NR (vi )
ej
l
R(vi )
j 1
ej
(5.1.2)
for all i = 1 to k.
Example 5.2 Let V = v1 , v2 , E = e1,e2 and u1 = u ,1u , K1 22 , K 2 22 , K3 22 , K 4 22 and
u 2 = u ,1u , L1 22 , L2 22 where K1 2 2 , K 2 2 2 , K 3 2 2 , K 4 2 2 , L1 2 2 , L2 2 2 are NSSs over (V, E),
defined as follows
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.1, .2, .3 .4, .5, .6
,
.9, .4, .7 .2, .6 .3
K1 2 2
.2, .3, .1 .4, .7, .2
,
.5, .7, .6 .3, .5 .2
K 2 22
.2, .3, .1 .4, .7, .2
,
.9, .7, .6 .3, .6 .2
K 3 22
.1, .2, .3 .4, .5, .6
.
.5, .4, .7 .2, .5 .3
K 4 22
and
.5, .8, .1 .3, .4, .2
,
.3, .1, .2 .6, .7 .2
L1 2 2
.4, .7, .2 .2, .3, .5
.
.2, .1, .6 .3, .6 .9
L2 2 2
Thus (V, E, u1 , u 2 ) is a NBSTS over (V, E).
Algorithm
Step 1: List the set of things or person v V with their parameters e E .
Step 2: Go through the records of the particulars.
Step 3: Collect the data for each v V according to all e E .
Step 4: Define NSSs.
Step 5: Define two different topologies u1 and u 2 where each satisfies the condition of NST and
so (V, E, u1 , u 2 ) is a NBSTS over (V, E).
Step 6: Form NSSs ui u1, u 2 matrix with collected data where vk as rows and el as columns.
Step 7: Calculate the OR for all v V .
Step 8: Calculate the NR for all v V .
Step 9: Select a highest value among all the calculated NR.
Step 10: If two or more NR are identical, add one more parameter and repeat the process.
Step 11: End the process while we acquire the unique NR of vk .
Problem 5.3 Let us suppose that there are two groups of women. First group consists of young age
women (YAW, aging 20-25), say u1 , and second group consists of middle age women (MAW, aging
30-35), say u 2 . Our aim is to insist both groups of women to select a saree together according to
their desire and choice.
1. Let V = sr1, sr2 , sr3 , sr4 , sr5 be the set of sample sarees and selection done by the set of parameters
let it be E = {c, q, d, p} where is c = colour, q = quality, d = design and p = price.
2. Both groups are analyzing the sarees collections.
3. Data are collected for each sarees according to its paramaters given.
4. Convert these data as NSSs, say YAW and MAW.
5. Let u1 u ,1u , YAW and u 2 u ,1u , MAW be two NSTs and so (V, E, u1 , u 2 ) is a NBSTS over
(V, E).
6. The matrix form of NSSs YAW and MAW are as follows:
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YAW 54
456
.5,.7,.2
.9,.3,.4
.6,.3,.9
.2,.4,.6
.7,.5,.2
.7,.2,.2 .4,.2,.7 .8,.4,.5
.3,.4,.2 .8,.7,.5 .9,.3,.6
.7,.4,.8 .2,.7,.1 .3,.2,.5
.4,.6,.8 .5,.3,.1 .7,.1,.3
.1,.2,.3 .3,.6,.9 .1,.3,.4
.6,.4,.3
.8,.4,.2
.1,.9,.2
.2,.6,.7
.3,.1,.2
.7,.3,.2 .3,.6,.2 .4,.7,.3
.8,.4,.2 .5,.9,.4 .4,.4,.4
.6,.7,.3 .8,.2,.1 .5,.1,.2
.3,.4,.5 .7,.2,.1 .2,.9,.1
.2,.1,.2 .3,.5,.2 .2,.4,.8
and
MAW 54
7. The Table 5.3.1 is obtained by using the formula (5.1.1),
Table 5.3.1. OR table.
sr1
sr2
sr3
sr4
sr5
c
.105
.3575
.08
.225
.21
q
.375
.21
.045
.15
.085
d
.06
.04
.22
.45
.1125
p
.09
.0975
.0425
.125
.2925
8. The Table 5.3.2 is obtained by using the formula (5.1.2),
Table 5.3.2. NR table.
sr1
sr2
sr3
sr4
sr5
c
.105
.3575
.08
.225
.21
q
.375
.21
.045
.15
.085
d
.06
.04
.22
.45
.1125
p
.09
.0975
.0425
.125
.2925
NR
.63
.705
.2275
.2
.28
Thus the second saree has selected by both the categories of women.
Problem 5.4 Consider the situation of problem 5.3.
1. Let V = sr1, sr2 , sr3 , sr4 , sr5 be the set of sample sarees and selection done by the set of parameters
let it be E = {c, q, d, p} where is c = colour, q = quality, d = design and p = price.
2. Both groups are analyzing the sarees collections.
3. Data are collected for each sarees according to its paramaters given.
4. Convert these data as NSSs, say YAW and MAW.
5. Let u1 u ,1u , YAW and u 2 u ,1u , MAW be two NSTs and so (V, E, u1 , u 2 ) is a NBSTS over
(V, E).
6. The matrix form of NSSs YAW and MAW are as follows:
Chinnadurai V and Sindhu M P, A Novel Approach for Pairwise Separation Axioms on Bi-Soft Topology Using
Neutrosophic Sets and An Output Validation in Real Life Application
Neutrosophic Sets and Systems, Vol. 35, 2020
YAW 54
457
.5,.7,.2
.9,.3,.4
.1,.3,.4
.2,.4,.6
.7,.5,.2
.5,.3,.1 .4,.2,.7 .8,.4,.5
.3,.4,.2 .8,.7,.5 .9,.3,.6
.7,.4,.8 .2,.7,.1 .3,.2,.5
.4,.6,.8 .7,.2,.2 .7,.1,.3
.1,.2,.3 .3,.6,.9 .6,.3,.9
.6,.4,.3
.8,.4,.2
.2,.4,.8
.2,.6,.7
.3,.1,.2
.7,.2,.1 .3,.6,.2 .4,.7,.3
.8,.4,.2 .5,.9,.4 .4,.4,.4
.6,.7,.3 .8,.2,.1 .5,.1,.2
.3,.4,.5 .7,.3,.2 .2,.9,.1
.2,.1,.2 .3,.5,.2 .1,.9,.2
and
MAW 54
7. The Table 5.4.1 is obtained by using the formula (5.1.1),
Table 5.4.1. OR table.
sr1
sr2
sr3
sr4
sr5
c
.105
.3575
.2925
.225
.21
q
.45
.21
.045
.15
.085
d
.06
.04
.22
.375
.1125
p
.09
.0975
.0425
.125
.08
8. The Table 5.4.2 is obtained by using the formula (5.1.2),
Table 5.4.2. NR table.
sr1
sr2
sr3
sr4
sr5
c
.105
.3575
.2925
.225
.21
q
.45
.21
.045
.15
.085
d
.06
.04
.22
.375
.1125
p
.09
.0975
.0425
.125
.08
NR
.705
.705
.015
.125
.0675
Thus first and second sarees have selected by both categories of women.
In this situation, we just add a parameter f = fabric in E and repeat the process.
4. After adding one more parameter, convert these data as NSSs, say YAW and MAW .
5. Let u1 u ,1u , YAW and u 2 u ,1u , MAW be two NSTs and so (V, E, u1 , u 2 ) is a NBSTS
over (V, E).
6. The matrix form of NSSs YAW and MAW are as follows:
Chinnadurai V and Sindhu M P, A Novel Approach for Pairwise Separation Axioms on Bi-Soft Topology Using
Neutrosophic Sets and An Output Validation in Real Life Application
Neutrosophic Sets and Systems, Vol. 35, 2020
YAW
55
458
.5,.7,.2
.9,.3,.4
.1,.3,.4
.2,.4,.6
.7,.5,.2
.5,.3,.1 .4,.2,.7 .8,.4,.5 .6,.7,.2
.3,.4,.2 .8,.7,.5 .9,.3,.6 .5,.1,.3
.7,.4,.8 .2,.7,.1 .3,.2,.5 .4,.5,.2
.4,.6,.8 .7,.2,.2 .7,.1,.3 .7,.8,.4
.1,.2,.3 .3,.6,.9 .6,.3,.9 .1,.3,.6
.6,.4,.3
.8,.4,.2
.2,.4,.8
.2,.6,.7
.3,.1,.2
.7,.2,.1 .3,.6,.2 .4,.7,.3 .9,.6,.3
.8,.4,.2 .5,.9,.4 .4,.4,.4 .7,.8,.1
.6,.7,.3 .8,.2,.1 .5,.1,.2 .6,.5,.4
.3,.4,.5 .7,.3,.2 .2,.9,.1 .2,.3,.4
.2,.1,.2 .3,.5,.2 .1,.9,.2 .6,.2,.7
and
MAW
55
7. The Table 5.4.3 is obtained by using the formula (5.1.1),
Table 5.4.3. OR table after adding a parameter.
sr1
sr2
sr3
sr4
sr5
c
.105
.3575
.2925
.225
.21
q
.45
.21
.045
.15
.085
d
.06
.04
.22
.375
.1125
p
.09
.0975
.0425
.125
.08
f
.175
.22
.1
.0225
.225
8. The Table 5.4.4 is obtained by using the formula (5.1.2),
Table 5.4.4. NR table after adding a parameter.
sr1
sr2
sr3
sr4
sr5
c
.105
.3575
.2925
.225
.21
q
.45
.21
.045
.15
.085
d
.06
.04
.22
.375
.1125
p
.09
.0975
.0425
.125
.08
f
.175
.22
.1
.0225
.225
NR
.88
.925
.115
.1475
.2925
Thus the second saree has selected by both categories of women.
Problem 5.5 Consider the situation that there are six students on the main stage for Quiz Finale.
There are two teams, each team consists of three students, one is Winner (W) and other is Runner (R).
Let FA1 and FA2 be two final authorities to judge the event. Our problem is to find the best player in
the winning team whose teammates are not mentioned here.
1. Let V =
st1, st2 , st3 , st4 , st5 , st6
be the set of students and judgement is based on the set of
parameters let it be E = {ra, eff, ca, mr, gp} where ra = right answers, eff = effectiveness, ca = complex
analysis, mr = memory, gp = grasping power.
Chinnadurai V and Sindhu M P, A Novel Approach for Pairwise Separation Axioms on Bi-Soft Topology Using
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459
2. First of all these final authorities will go through the records of the students.
3. They will collect student’s data according to their paramaters given.
4. These data are converted into two different NSSs, say FA1 and FA2.
5. Let u1 u ,1u , FA1 and u 2 u ,1u , FA2 be two NSTs and so (V, E, u1 , u 2 ) is a NBSTS over
(V, E).
6. The matrix form of NSSs FA1 and FA2 are as follows:
FA165
.4, .2, .7
.7, .3, .2
.3, .6, .6
.2, .6, .3
.6, .5, .4
.7, .3, .4
.6, .3, .1 .2, .4, .8 .2, .9, .1
.6, .7, .2 .8, .9, .6 .3, .5, .4
.4, .7, .3
.5, .1, .2
.7, .8, .1
.2, .6, .7
.5, .9, .4
.8, .4, .5
.2, .3, .4 .5, .7, .2 .3, .4, .2 .2, .7, .1
.6, .7, .3 .9, .3, .4 .7, .4, .8 .7, .2, .2
.9, .3, .6 .1, .3, .4 .4, .6, .8 .3, .6, .9
.7, .4, .8 .2, .4, .6 .1, .2, .3 .8, .4, .5
.1, .2, .3 .7, .5, .2 .4, .2, .7 .3, .2, .5
.9, .6, .3 .5, .3, .1 .1, .3, .4 .7, .3, .2
.8, .6, .1 .5, .4, .3 .9, .7, .2
.3, .5, .4 .6, .4, .2 .1, .2, .3
.7, .5, .4 .8, .6, .1 .4, .2, .7
.9, .2, .1 .7, .3, .4 .3, .5, .4
.6, .5, .3
.2, .7, .5
.5, .4, .6
.7, .3, .4
.4, .1, .4
.3, .5, .2
and
FA265
7. The Table 5.5.1 is obtained by using the formula (5.1.1),
Table 5.5.1. OR table.
st1
st2
st3
st4
st5
st6
ra
.11
.32
.045
.12
.045
.195
eff
.105
.175
.24
.055
.24
.175
ca
.0675
.2275
.0325
.075
.24
.12
mr
.035
.135
.018
.02
.13
.24
gp
.08
.055
.175
.195
.085
.18
8. The Table 5.5.2 is obtained by using the formula (5.1.2),
Table 5.5.2. NR table.
st1
st2
st3
st4
st5
st6
ra
.11
.32
.045
.12
.045
.195
eff
.105
.175
.24
.055
.24
.175
ca
.0675
.2275
.0325
.075
.24
.12
mr
.035
.135
.018
.02
.13
.24
gp
.08
.055
.175
.195
.085
.18
NR
.3975
.9125
.0375
.005
.31
.91
Here both st 2 and st 6 got high score from judges, so they both does not belongs to R.
Chinnadurai V and Sindhu M P, A Novel Approach for Pairwise Separation Axioms on Bi-Soft Topology Using
Neutrosophic Sets and An Output Validation in Real Life Application
Neutrosophic Sets and Systems, Vol. 35, 2020
460
Case (i). If st 2 W and st 6 R , then the best player award goes to st 2 .
Case (ii). If st 2 R and st 6 W , then the best player award goes to st 6 .
Case (ii). If st 2 W and st 6 W , then we just add a parameter ld = leadership.
4. After adding one more parameter, convert these data as NSSs, say FA1 and FA2 .
5. Let u1 u ,1u , FA1 and u 2 u ,1u , FA2 be two NSTs and so (V, E, u1 , u 2 ) is a NBSTS over
(V, E).
6. The matrix form of NSSs FA1 and FA2 are as follows:
FA1
66
.4, .2, .7
.7, .3, .2
.3, .6, .6
.2, .6, .3
.6, .5, .4
.7, .3, .4
.6, .3, .1
.2, .4, .8 .2, .9, .1
.6, .7, .2 .8, .9, .6 .3, .5, .4
.4, .7, .3
.5, .1, .2
.7, .8, .1
.2, .6, .7
.5, .9, .4
.8, .4, .5
.2, .3, .4 .5, .7, .2 .3, .4, .2 .2, .7, .1 .7, .5, .4
.6, .7, .3 .9, .3, .4 .7, .4, .8 .7, .2, .2 .6, .9, .1
.9, .3, .6 .1, .3, .4 .4, .6, .8 .3, .6, .9 .8, .4, .4
.7, .4, .8 .2, .4, .6 .1, .2, .3 .8, .4, .5 .3, .4, .5
.1, .2, .3 .7, .5, .2 .4, .2, .7 .3, .2, .5 .7, .3, .2
.9, .6, .3 .5, .3, .1 .1, .3, .4 .7, .3, .2 .5, .9, .4
.8, .6, .1 .5, .4, .3 .9, .7, .2
.3, .5, .4 .6, .4, .2 .1, .2, .3
.7, .5, .4 .8, .6, .1 .4, .2, .7
.9, .2, .1 .7, .3, .4 .3, .5, .4
.6, .5, .3 .3, .2, .4
.2, .7, .5 .9, .1, .1
.5, .4, .6 .7, .5, .3
.7, .3, .4 .8, .2, .1
.4, .1, .4 .3, .1, .5
.3, .5, .2 .6, .2, .6
and
FA2
66
7. The Table 5.5.3 is obtained by using the formula (5.1.1),
Table 5.5.3. OR table after adding a parameter.
st1
st2
st3
st4
st5
st6
ra
.11
.32
.045
.12
.045
.195
eff
.105
.175
.24
.055
.24
.175
ca
.0675
.2275
.0325
.075
.24
.12
mr
.035
.135
.018
.02
.13
.24
gp
.08
.055
.175
.195
.085
.18
ld
.065
.352
.055
.175
.12
.0675
8. The Table 5.5.4 is obtained by using the formula (5.1.2),
Table 5.5.4. NR table after adding a parameter.
st1
st2
st3
st4
st5
st6
ra
.11
.32
.045
.12
.045
.195
eff
.105
.175
.24
.055
.24
.175
ca
.0675
.2275
.0325
.075
.24
.12
mr
.035
.135
.018
.02
.13
.24
gp
.08
.055
.175
.195
.085
.18
Chinnadurai V and Sindhu M P, A Novel Approach for Pairwise Separation Axioms on Bi-Soft Topology Using
Neutrosophic Sets and An Output Validation in Real Life Application
Neutrosophic Sets and Systems, Vol. 35, 2020
461
ld
.065
.352
.055
.175
.12
.0675
NR
.4625
1.2375
.0925
.18
.43
.9775
Thus the best player award goes to st 2 .
6. Conclusion
The main involvement of this paper is to preface the definition of NBSTSs and the study of
some important properties of such spaces including separation axioms and the relationship between
Ti 0,1, 2, 3, 4 -spaces. The key of this paper is to apply NBSTS in real life problems to take a decision,
which might be positive or negative. In our problems two different types of NSTs are combined
together to choose a unique decision according to the algorithm and calculation made by the
formulae given here. Subsequently, NBSTS can be built up to pairwise NS separated sets, pairwise
NS connected spaces, pairwise NS connected sets, pairwise NS disconnected spaces, pairwise NS
disconnected sets and so on. We look forward to encourage this type of NBSTS will find a way to
other types of topological structures. In future, some case studies which we mention in this paper
need to develop on multicriteria DM also.
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Received: Apr 22, 2020.
Accepted: July 12 2020
Chinnadurai V and Sindhu M P, A Novel Approach for Pairwise Separation Axioms on Bi-Soft Topology Using
Neutrosophic Sets and An Output Validation in Real Life Application
Neutrosophic Sets and Systems, Vol. 35, 2020
University of New Mexico
Ranking of Pentagonal Neutrosophic Numbers and its
Applications to solve Assignment Problem
K.Radhika1 and K.Arun Prakash2
1
Department of Mathematics, Kongu Engineering College, Perundurai; radhikavisu@gmail.com
Department of Mathematics, Kongu Engineering College, Perundurai; arunfuzzy@gmail.com
2
Abstract: Introduction of Neutrosophic sets and Neutrosophic numbers paves a way to handle
uncertainty more effectively. In this paper we propose a new approach for ranking neutrosophic
number by using its magnitude. We develop an algorithm for the solution of neutrosophic
assignment problems involving pentagonal neutrosophic number. The proposed method is easy to
understand and to apply for finding solution of neutrosophic assignment problems occurring in real
life situations. To show the proposed strategy numerical models are given and the acquired results
are analyzed.
Keywords:
Neutrosophic
sets,
Neutrosophic
number,
Pentagonal
neutrosophic
number,
Neutrosophic Assignment Problem, Optimal Solution.
1. Introduction
This section gives a survey of research work carried out so far to handle uncertainity. The
novelty of present work, motivation behind it and structure of the remaining sections were also
provided.
1.1: Literature survey
Smarandache [1] introduced neutrosophic sets having three components truthiness,
indeterminacies, and falseness. Wang et al [3] introduced a single valued neutrosophic set, which is
a subclass of a neutrosophic set presented by Smarandache [1]. Introduction of neutrosophic
measure, neutrosophic integral, and neutrosophic probability by Smarandache [2,4] gave notation
and many examples for neutrosophic measure, and consequently, the neutrosophic integral and
neutrosophic probability are also defined. Many researchers have applied the neutrosophic logic in
various fields.
To develop an optimization problem and its solution procedure in uncertain environment, the
study of fuzzy number, intuitionistic fuzzy number, neutrosophic number and their ranking is
necessary. Several researchers paid attention to fuzzy and intuitionistic fuzzy optimization methods
by adopting various ranking techniques. But ranking of neutrosophic number is a risk task. To
handle optimization problems having indeterminacy, ranking of neutrosophic numbers plays a
vital role. S.Subasri and K.Selvakumari [5] ranked triangular neutrosophic number and applied the
same to solve travelling salesman problems. Avishek Chakraborty [6], [7] gave a new ranking
method to rank pentagonal neutrosophic number. Chakraborty A, Mondal SP, Ahmadian A, Senu
N, Alam S,Salahshour S in 2018 [8] formatted Different forms of triangular neutrosophic numbers,
K.Radhika, K.Arun Prakash, Ranking of Pentagonal Neutrosophic Numbers and itsApplications in Assignment Problem
Neutrosophic Sets and Systems, Vol. 35, 2020
465
and introduced de-neutrosophication techniques, and
applied in critical path analysis.
Smarandache [9] in 2019 approached TOPSIS technique for developing supplier selection with
group decision making under type-2 neutrosophic number. Nabeeh NA, Abdel-Basset M,
El-Ghareeb HA, Aboelfetouh A in 2019[10] developed multi-criteria decision making approach for
IoT-based enterprises using neutrosophic numbers. Neutrosophic functions and neutrosophic
calculus, was defined by Florentin Smarandache [11].Neutrosophic ordinary differential equation of
first order via neutrosophic numbers is epitomized by Sumathi IR, MohanaPriya V [12], Differential
equations in neutrosophic environment are explored, and solution of second-order linear differential
equation with trapezoidal neutrosophic numbers as boundary conditions is discussed by R. Sumathi
[13].Minimal spanning tree is one of the important fact in the field of graph theory. Single valued
neutrosophic minimal spanning tree and clustering method was solved by Ye [14] in 2014. Mandal &
Basu [15] solved similarity measure to find spanning tree related with neutrosophic arena. Mullai
et.al [16] formulated minimum spanning tree problem in bipolar neutrosophic number. Broumi et.al
[17] formulated shortest path problem on single valued neutrosophic graphs. Kandasamy [18]
developed double-valued neutrosophic sets and their application in minimum spanning tree
problems. Broumi et.al [19] formulated neutrosophic shortest path for solving Dijkstra’s algorithm in
graph theory. Mohamed Abdel-Basset [20] introduced bipolar neutrosophic number and applied in
decission making problems he also proposed a model in [21] to evaluate the supply chain
sustainability metrics based on a combination of quality function deployment and plithogenic
aggregation operations. Assignment problems plays an important role in optimization. Many
researchers have handled assignment problems in fuzzy and intuitionistic fuzzy environment but in
neutrosophic environment, only few articles were published, that too involving other forms
neutrosophic numbers. This was the first attempt to discuss assignment problems in neutrosophic
environments involving pentagonal neutrosophic numbers.
1.2. Motivation
For the past few years the ambiguous data were handled by fuzzy sets, intuitionistic fuzzy sets,
interval valued fuzzy sets and many such structures. Recently, the introduction of neutrosophic sets
proves to be more suited to handle vagueness than existing set theoretical structure. Fuzzy number
can measure only uncertainty, intuitionistic and interval valued intuitionistic fuzzy number can
measure uncertainty and vagueness not hesitation. Only neutrosophic number can measure all the
three parameters effectively. Thus pentagonal neutrosophic number attracts more attention and
paves path for new research.
1.3. Novelties
From its inception, a few research articles had just distributed in various journals in
neutrosophic field. Only a countable amount of articles had dealt with pentagonal neutrosophic
number in that other types neutrosophic number can be generalized from pentagonal neutrosophic
number. Neutrosophic assignment problem is an area in which focus on the de-neutrosophication
technique applied to solve neutrosophic assignment problem.
1.4. Contribution
In this research article, symmetric pentagonal neutrosophic fuzzy numbers are considered.
These numbers are converted into crisp values by means of ranking approach by magnitude. There
are many ranking procedures which rank uncertainty and vagueness separately. Here our ranking
procedure converts all the three parts of pentagonal neutrosophic number into crisp number. Lastly,
the proposed ranking was applied to solve neutrosophic assignment problem. Section-1 throws an
introduction to neutrosophic number and literature survey in the field. Section-2 gives the
preliminaries Section-3 covers representation, definition and ranking of pentagonal neutrosophic
K.Radhika, K.Arun Prakash, Ranking of Pentagonal Neutrosophic Numbers and itsApplications in Assignment Problem
Neutrosophic Sets and Systems, Vol. 35, 2020
466
number. Section-4 provides mathematical formation of neutrosophic assignment problem, algorithm
to solve it and numerical example illustrating the procedure. The last section gives the conclusion
and scope of the future work.
2. Preliminaries
X be a universe set. A neutrosophic set A on X is defined as
Definition 2.1: [1] Let
A TA x , I A x , FA x : x X , where
TA x , I A x , FA x : X 0,1
represents
the
degree of membership, degree of indeterministic, and degree of non-membership respectively of the
element x X , such that 0 TA x I A x FA x 3 .
Definition 2.2: [12] , , cut: The , , cut neutrosophic set is denoted by F , , ,
where , , 0,1 and
are
fixed
numbers,
such
that
3
is
defined
as
F , , TA x , I A x , FA x : x X , TA x , I A x , FA x .
Definition 2.3: [12] A neutrosophic set A defined on the universal set of real numbers R is said to
be neutrosophic number if it has the following properties. (i) A is normal if there exist
x0 R, , such
that TA ( x0 ) 1, I A ( x0 ) FA x0 0. (ii) A is convex set for the truth function TA x , i.e.,
TA x1 1 x2 min TA x1 , TA x2 , x1, x2 R, 0,1 . (iii) A is concave set for the
indeterministic
function
and
false
fuunction
FA x1 1 x2 max FA x1 , FA x2 , x1, x2 R, 0,1 .
I A x1 1 x2 max I A x1 , I A x2 , x1, x2 R, 0,1 ,
I A x and FA x ,
i.e.,
3 Ranking of pentagonal Neutrosophic number
This section gives the definition of symmetric pentagonal neutrosphic number and a method of
ranking it by means of magnitude. Numerical examples were illustrated to explain the proposed
ranking procedure.
Definition: Symmetric Pentagonal neutrosophic number: A is a subset of neutrosophic number
in R with the following truth function, indeterministic function, and falsity function which is given
by the following:
A
a1, a2 , a3 , a4 , a5 , b1, b2 , b3 , b4 , b5 , c1, c2 , c3 , c4 , c5 ; p, q, r,
The accuracy membership function A x :
A x :
where p, q, r 0,1 .
0,1 , the indeterminacy membership function
0,1 and the falsity membership function A x :
0,1 are defined as follows:
K.Radhika, K.Arun Prakash, Ranking of Pentagonal Neutrosophic Numbers and itsApplications in Assignment Problem
Neutrosophic Sets and Systems, Vol. 35, 2020
467
0, x a1
x a1
p a a , a1 x a2
1
2
p 1 x a3
1
, a2 x a3
a3 a2
A x 1, x a3
1 p 1 x a3 , a x a
2
3
a3 a2
a5 x
p
, a4 x a5
a5 a4
0, x a
5
1, x b1
q 1 x b1
, b1 x b2
1
b2 b1
b3 x
q
, b2 x b3
b3 b2
A x 0, x b3
q x b3 , b x b
4
b4 b3 3
1 q x b4
, b4 x b5
q
b5 b4
1, x b
5
1, x c1
r 1 x c1
, c1 x c2
1
c2 c1
c3 x
r
, c2 x c3
c3 c2
A x 0, x c3
r x c3 , c x c
4
c4 c3 3
1 r x c4
, c4 x c5
r
c5 c4
1, x c
5
K.Radhika, K.Arun Prakash, Ranking of Pentagonal Neutrosophic Numbers and itsApplications in Assignment Problem
Neutrosophic Sets and Systems, Vol. 35, 2020
468
1
p
q
r
a1
b1
c1
a2
b2
c2
a3
b3
c3
a4
b4
c4
a5
b5
c5
Figure 1.Representation of Symmetric Pentagonal neutrosophic number
3.1 Magnitude of a Pentagonal Neutrosophic Number
Let A
a1, a2 , a3 , a4 , a5 , b1, b2 , b3 , b4 , b5 , c1, c2 , c3, c4 , c5 ; p, q, r, where p, q, r 0,1
be a
symmetric pentagonal neutrosophic number whose accuracy membership function is given by
0, x a1
L1
z A , a1 x a2
z L2 , a x a
A
2
3
A ( x) 1, x a3
R1
z A , a3 x a4
z R2 , a x a
4
5
A
0,
x
a
5
indeterminacy membership function is given by
1, x b1
L1
k A , b1 x b2
k L 2 , b x b
3
A 2
A ( x) 0, x b3
R1
k A , b3 x b4
k R 2 , b x b
4
5
A
1, x b5
and
and falsity membership function is given by
K.Radhika, K.Arun Prakash, Ranking of Pentagonal Neutrosophic Numbers and itsApplications in Assignment Problem
Neutrosophic Sets and Systems, Vol. 35, 2020
469
1, x c1
L1
mA , c1 x c2
m L 2 , c x c
A
2
3
A ( x) 0, x c3
R1
mA , c3 x c4
m R 2 , c x c
4
5
A
1,
x
c
5
Here
z AL1 ( x) :[a1 , a2 ] [0, p], z AL 2 ( x) :[a2 , a3 ] [p,1], z AR1 ( x) :[a3 , a4 ] [p,1], z RA2 (x) :[a 4 , a 5 ] [0, p],
L1
L2
R1
R2
where z A ( x), z A (x) are non-decreasing left accuracy functions and z A ( x), z A ( x) are non-increasing
right
accuracy
functions
of
symmetric
Pentagonal
neutrosophic
number.
Also
k AL1 ( x) :[b1, b2 ] [q,1], k AL 2 ( x) :[b2, b3 ] [0, q], k AR1 ( x) :[b3, b4 ] [0, q], k AR 2 ( x) :[b4, b5 ] [q,1], where k AL1 (x), k LA2 (x)
R1
R2
are non-increasing left indeterminacy membership functions and k A ( x), k A ( x) are non-decreasing
right indeterminacy membership functions of symmetric pentagonal neutrosophic number. Similarly
the functions that occur in falsity membership function were defined as follows:
mAL1 ( x) :[c1, c2 ] [r,1], mLA2 ( x) :[c2, c3 ] [0, r], mRA1 ( x) :[c3, c4 ] [0, r], m RA2 ( x) :[c4, c5 ] [r,1], where mAL1 ( x), mLA2 (x)
R1
R2
are non-increasing left falsity membership function and mA ( x), m A ( x) are non-decreasing right falsity
L1
L2
membership function of symmetric Pentagonal neutrosophic number. It is clear that z A ( x), z A (x)
z AR1 ( x), z AR 2 ( x) k AL1 ( x), k LA2 (x) k AR1 ( x), k RA2 ( x) , mAL1 ( x), m LA2 (x) mAR1 ( x), m RA2 ( x) are one to one and inverse
,
,
,
exist.
The inverse functions of left and right accuracy, indeterminacy and falsity functions are defined as
follows:
f AL1 :[0, p] R, f AL 2 :[ p,1] R, f AR1 :[ p,1] R, f AR 2 :[0, p] R, g AL1 :[1, q] R, g AL 2 :[0, q] R,
g AR1 :[0, q] R, g AR 2 :[1, q] R, hAL1 :[1, q] R, hAL 2 :[0, q] R, hAR1 :[0, q] R, hAR 2 :[1, q] R,
where
(a2 a1 )
, 0 y p
p
(a a2 )( y p )
f AL 2 ( y ) a2 3
, p y 1
1 p
(a a3 )( y 1)
f AR1 ( y ) a3 4
, p y 1
p 1
(a a5 )
f AR 2 ( y ) a5 y 4
, 0 y p
p
f AL1 ( y ) a1 y
K.Radhika, K.Arun Prakash, Ranking of Pentagonal Neutrosophic Numbers and itsApplications in Assignment Problem
Neutrosophic Sets and Systems, Vol. 35, 2020
470
(b2 b1 )( y 1)
, q y 1
q 1
(b b ) y
g AL 2 ( y ) b3 3 2 , 0 y q
q
b
b )y
(
g AR1 ( y ) b3 4 3 , 0 y q
q
(b b )( y q)
g AR 2 ( y ) b4 5 4
, q y 1
1 q
(c c )( y 1)
hAL1 ( y ) c1 2 1
, r y 1
r 1
(c c ) y
hAL 2 ( y ) c3 3 2 , 0 y r
r
c
c )y
(
hAR1 ( y ) c3 4 3 , 0 y r
r
(
c
c )( y r )
hAR 2 ( y ) c4 5 4
, r y 1
1 r
g AL1 ( y ) b1
The magnitude denoted by Mag(A) of a symmetric pentagonal neutrosophic number
A
a1, a2 , a3 , a4 , a5 , b1, b2 , b3 , b4 , b5 , c1, c2 , c3 , c4 , c5 ; p, q, r,
Mag ( A)
is determined as follows:
1
( f AL1 ( ) f AL 2 ( ) f AR1 ( ) f AR 2 ( ) 2 a 3 g AL1 ( ) g AL 2 ( ) g AR1 ( ) g AR 2 ( ) 2 b 3 hAL1 ( ) hAL 2 ( )
2
hAR1 ( ) hAR 2 ( ) 2 c3 ) t ( ) d .
1 2
[ p ( a1 a5 ) (1 p )( a2 a4 ) ( 2 p 2 2 p 10)a3 ( q 2 q 2)(b1 b5 ) (1 q )(b2 b4 )
12
( 2q 2 6)b3 ( r 2 r 2)(c1 c5 ) (1 r )(c2 c4 ) ( 2r 2 6)c3 ].
(1)
where the function t(𝛼)is a weighted function and is a non-negative and increasing function on [0,1]
1
with t(0)=0,t(1)=1and
1
t ( )d 2 we choose
t(𝛼) = 𝛼.The scalar value Mag(A) is used to rank
0
Pentagonal neutrosophic number.
Remark:
When 𝑝 = 0, 𝑞 = 1, 𝑟 = 1 pentagonal neutrosophic number becomes triangular neutrosophic number.
Then the magnitude of A defined in equation (1) will be transformed into
Mag ( A)
1
[(a2 a4 ) 10a3 2(b2 b4 ) 8b3 2(c2 c4 ) 8c3 ]
12
3.2 Ranking Procedure
K.Radhika, K.Arun Prakash, Ranking of Pentagonal Neutrosophic Numbers and itsApplications in Assignment Problem
Neutrosophic Sets and Systems, Vol. 35, 2020
471
Using the magnitude of symmetric pentagonal neutrosophic number defined above, the ordering
of pentagonal neutrosophic numbers is explained in this section.
Let A
B
a1, a2 , a3 , a4 , a5 , b1, b2 , b3 , b4 , b5 , c1, c2 , c3 , c4 , c5 ; p, q, r, and
d1, d2 , d3 , d4 , d5 , e1, e2 , e3 , e4 , e5 , i1, i2 , i3 , i4 , i5 ; u, v, w
be
any
two
arbitrary
Pentagonal
neutrosophic numbers. Then the ranking procedure is as follows:
Step 1: Compute Mag(A) , Mag(B),any one of the following cases prevail.
Step 2:
(i)
If Mag ( A) Mag(B), then A B
(ii)
If Mag ( A) Mag(B) , then A B
(iii)
If Mag ( A) Mag(B), then A B
3.3 Numerical examples
The ordering procedure in the previous section is illustrated by numerical examples.
Consider the following sets of Pentagonal neutrosophic numbers.
Set 1:A={(.5,1.5,2.5,3.5,4.5)(0.3,1.3,2.3,3.3,4.3)(1.8,2.8,3.8,4.8,5.8);0.5,0.5,0.5}
B={(.7,1.7,2.5,3.5,4.7)(.5,1.5,2.2,3.2,4)(1.7,2.7,3.,4.7,5.7);0.5,0.5,0.5}
C={(1,4,7,10,13)(0.5,3.5,6.5,9.5,12.5)(4.5,7.5,9,12,14.5);0.5,0.5,0.5}
Set 2:A={(10,15,20,25,30)(0,3,5,7,10)(0,1,2,3,4,);0.5,0.5,0.5}
B={(5,10,15,20,25)(1,2,3,4,5)(1,1.5,2,2.5,3);0.5,0.5,0.5}
C={(10,20,30,40,50)(1,4,7,8,10)(1,1.5,2,2.5,3);0.5,0.5,0.5}
The table.1 gives the comparison of proposed ranking of Pentagonal neutrosophic numbers with
the existing methods.
Author name and method
Proposed method
Set 1
Set 2
A=8.6
A=27
B=8.4
B=20
C=22.79
C=38.5
Result: C > A > B
Result:
A=2.86
A=9
De-Neutrosophication
B=2.66
B=6.66
value [6]
C=7.66
C=12.70
Result: C > A = B
Result:
A= -.533
A=5
B= -.45
B=4
C= -2.33
C=8
Result: B > A >C
Result:
Avishek
Avishek
Chakraborty’s
Chakraborty’s
accuracy function value [7]
C>A>B
C>A>B
C>A>B
Table.1 Comparison table for ranking Pentagonal neutrosophic numbers
The table 2 gives the numerical example of the De-Neutrosophication value of Triangular
neutrosophic numbers.
K.Radhika, K.Arun Prakash, Ranking of Pentagonal Neutrosophic Numbers and itsApplications in Assignment Problem
Neutrosophic Sets and Systems, Vol. 35, 2020
472
Sl.No
Triangular neutrosophic numbers
proposed method of ranking
1
A={(1,2,3)(0.5,1.5,2.5)(1.2,2.7,3.5)}
6.083
B={(.5,1.5,2.5)(.3,1.3,2.2)(.7,1.7,2.2}
3.723
Result
A>B
Table.2 De-Neutrosophication value Triangular neutrosophic numbers
4. Application of Ranking of Pentagonal Neutrosophic Number in solving Neutosphic
Assignment Problem
In this section neutrosophic assignment problem with pentagonal neutrosophic numbers as
parameters was formulated, algorithm for identifying the optimal solution to neutrosophic
assignment problem was stated. Finally a numerical example was produced to explain the
proposed algorithm.
Need for Pentagonal neutrosophic numbers
Suppose there are n facilities and n jobs it is clear that in this case, there will be n
assignments. Each facility or say worker can perform each job, one at a time. But there should be
certain procedure by which assignment should be made so that the profit is maximized or the
cost or time is minimized. But in our real life applications the times taken to complete the job
undergo uncertainty, hesitation and vagueness. In such cases we cannot have the parameter as a
real value. So we have to use some other representation of the parameter with which the
uncertainty, hesitation and vagueness can be measured. The below discussion justify the need
for selecting the cost parameter in the terms of Pentagonal neutrosophic number.
If the parameter is a real value - uncertainty hesitation and vagueness cannot be handled
If the parameter is a fuzzy value- uncertainty but hesitation and vagueness cannot be
handled
If the parameter is an Intuitionistic Fuzzy value - uncertainty and hesitation can be
handled but vagueness cannot be handled.
If the parameter is a Pentagonal neutrosophic value - uncertainty, hesitation and
vagueness (i.e) all the components can be handled.
From the above discussion, it is clear that only pentagonal neutrosophic environment
can tackle the impreciseness, hesitation and truthiness in a membership function of an
uncertain number, which is more reliable, logical and realistic for a decision maker.
Pentagonal neutrosophic numbers enabled to meet the imprecise parameters as well, which
is approvingly the advantageous for the decision makers to analyze the result in a more
precise manner. Moreover Pentagonal neutrosophic numbers generalize other types of
neutrosophic numbers.
Pentagonal neutrosophic assignment problem may be formulated as follows:
K.Radhika, K.Arun Prakash, Ranking of Pentagonal Neutrosophic Numbers and itsApplications in Assignment Problem
Neutrosophic Sets and Systems, Vol. 35, 2020
473
Consider the assignment problem with cost function as Pentagonal neutrosophic number.
Minimize z
n
n
cij xij ,
subject to
i 1 j 1
n
n
i 1
j 1
xij 1, j 1, 2,3,......n and xij
1, j 1, 2, 3,......n,
xij 1 or 0 for al i, j, where cij is a
Pentagonal neutrosophic number and the total cost for performing all the activity is given by
n
n
cij xij .
i 1 j 1
Fundamental Theorems of a Pentagonal neutrosophic Assignment Problem
The solution of a Pentagonal neutrosophic assignment problem is fundamentally based on the
following two theorems:
Theorem 1:
In a Pentagonal neutrosophic assignment problem, if we add or subtract an Pentagonal
neutrosophic number to every element of any row (or column) of the Pentagonal neutrosophic
parameter matrix [ 𝑐𝑖𝑗 ], then an assignment that minimizes the total Pentagonal neutrosophic
parameter on one matrix also minimizes the total Pentagonal neutrosophic parameter on the other
matrix.
n
n
n
n
Minimize Z cij xij withn n xij 1, j 12....n, xij 1, i 1, 2..n xij 0or1for every i, j
*
*
*
*
i
j
j
i
then xij also minimize Z cij xij where cij cij ui v j for all i, j 1, 2...n for all i, j=1,
i
j
2,……n are some real valued Pentagonal neutrosophic number.
Proof:
Z * cij* xij
n
n
(cij u i v j )xij
j
i
n
n
i
j
n
n
i
j
n
n
j
i
cij xij u i xij v j xij
n
n
n
n
i
j
i
j
Z u i v j sin ce xij 1 and xij 1
This shows that the minimization of the new objective function 𝑍 ∗ yields the same solution as the
minimization of original objective function Z
Theorem 2:
In a pentagonal neutrosophic assignment problem with parameter function
𝑛 𝑛
the feasible solution which x ij satisfies ∑𝑖 ∑𝑗 𝑐𝑖𝑗 𝑥𝑖𝑗 = 0 is an optimal solution.
c ij
if all
cij 0
K.Radhika, K.Arun Prakash, Ranking of Pentagonal Neutrosophic Numbers and itsApplications in Assignment Problem
Neutrosophic Sets and Systems, Vol. 35, 2020
Proof: Since all cij 0 all xij 0. The objective function Z
474
n
n
j
i
cij xij cannot be negative the
minimum
n
n possible that Z can have is 0.Therefore any feasible solution
Z cij xij 0 will be optimal.
j
xij obtained
that satisfies
i
Algorithm to solve Pentagonal neutrosophic assignment problem:
We, now introduce a new algorithm called the Pentagonal neutrosophic Hungarian method
for finding a Pentagonal neutrosophic optimal assignment for Pentagonal neutrosophic assignment
problem.
Step 1: Determine the Pentagonal neutrosophic parameter table from the given problem.
Step 2: Convert the given Pentagonal neutrosophic assignment matrix to crisp by using the
magnitude method.
Step 3: Subtract the row minimum from each row entry of that row. Subtract the column minimum
of the resulting matrix from each column entry of that column. Each column and row now has at
least one zero.
Step 4: In the modified assignment table obtained in step 3, search for optimal assignment as
follows.
Examine the rows successively until a row with a single zero is found. Assign the zero and cross off
all other zeros in its column. Continue this for all the rows. Repeat the procedure for each column of
reduced assignment table. If a row and / or column have two or more zeros assign arbitrary any one
of these zeros and cross off all other zeros of that row/column. Repeat the above process successively
until the chain of assigning or cross ends.
Step 5: If the number of assignments is equal to n, the order of the parameter matrix, optimal
solution is reached. If the number of assignments is less than n, parameter matrix, go to the step 6.
Step 6: Draw the minimum number of horizontal and / or vertical lines to cover all the zeros of the
reduced assignment matrix. This can be done by using the following:
(i)Mark rows that do not have any assigned zero. (ii)Mark columns that have zeros in the marked
rows. (iii)Mark rows that do have zeros in the marked columns. Repeat (ii) and (iii) of the above until
the chain of marking is completed. Draw lines through all the unmarked rows and marked columns.
This gives the desired minimum number of lines.
Step 7: Develop the new revised reduced parameter matrix as follows: Find the smallest entry of the
reduced matrix not covered by any of the lines. Subtract this entry from all the uncovered entries
and add the same to all the entries lying at the intersection of any two lines.
Step 8: Repeat step 5 to step 7 until optimal solution to the given assignment problem is attained.
Numerical example:
Suppose we want to assign jobs A, B, C, D to machine M1, M2, M3, and M4. Our aim is to
find the minimum time so that the job is completed so that each machine is assigned only one job,
K.Radhika, K.Arun Prakash, Ranking of Pentagonal Neutrosophic Numbers and itsApplications in Assignment Problem
Neutrosophic Sets and Systems, Vol. 35, 2020
475
The time parameter may not be a real value since the time taken to complete a job depend on the
facts such as (i) working condition of the machine (ii) climatic condition and so on. So we represent
the time parameter
4 as
4 Pentagonal neutrosophic number. The problem can be considered as follows.
Minimize
Z cij xij
j
Where
i
c11 = {(8,13,19,24,30)(7,10,15,22,27)(10,16,23,25,32);0.5,0.5,0.5}
c12 = {(7,12,18,24,30)(6,10,14,20,25)(10,15,20,25,35);0.6,0.4,0.3}
c13 = {(3,8,14,20,26)(2,7,12,18,22)(5,10,15,24,30);0.4,0.3,0.4}
c14 = {(10,15,20,26,32)(7,12,18,22,26)(12,16,22,28,35);0.6,0.4,0.3}
c 21 = {(8,14,20,26,32)(6,12,18,22,28)(10,18,24,28,35);0.4,0.5,0.4}
c 22 = {(6,10,15,20,25)(4,8,12,18,22)(8,14,20,24,30);0.6,0.6,0.5}
c 23 = {(9,14,20,25,30)(6,12,16,21,24)(12,15,23,28,35);0.5,0.4,0.3}
c 24 = {(11,15,19,23,27)(8,12,17,21,24)(14,18,22,26,30);0.6,0.6,0.4}
c 31 = {(7,10,13,16,20)(6,8,12,15,18)(10,14,18,22,25);0.7,0.4,0.5}
c 32 = {(12,15,18,24,26)(5,9,13,17,21)(10,14,18,22,26);0.8,0.6,0.2}
c 33 = {(6,11,15,18,24,26)(9,13,17,20,23)(14,18,22,25,29);0.6,0.4,0.3}
c 34 = {(7,11,14,17,26)(4,8,13,19,23)(9,14,19,24,28);0.6,0.5,0.4}
c 41 = {( 4,9,15,21,27)(3,8,13,19,23)(6,11,16,25,31);0.6,0.6,0.4}
c 42 = {(11,14,17,23,25)(7,11,16,20,23)(13,17,21,25,29);0.5,0.3,0.4}
c 43 = {(6,11,15,18,24,26)(7,11,15,18,20)(13,17,21,24.27);0.7,0.3,0.4}
c 44 = {(10,14,18,22,26)(5,9,13,19,23)(9,15,21,25,31);0.5,0.4,0.3}
4
Subject to
x
j 1
ij
1, i 1, 2, 3, 4.
4
x
ij
i
1, j 1, 2, 3, 4. , xij 1or 0, for all i, j.
Pentagonal neutrosophic assignment matrix in the crisp form
A
C ij =
B
C
D
M1 56.5
53.52 42.18
60.0
M 2 60.9
47.0
58.54 57.69
M3 42.82 49.55 54.12 46.25
M 4 45.23 54.98 49.99 51.98
Applying step 3, 4 the following time parameter matrix is obtained is
A
B
M1 14.32 11.34
C ij =
M 2 13.9
0
C
0
D
17.82
11.54 10.69
M3
0
6.73
11.3
3.43
M4
0
9.75
4.76
6.75
Applying step 5 we the following result
K.Radhika, K.Arun Prakash, Ranking of Pentagonal Neutrosophic Numbers and itsApplications in Assignment Problem
Neutrosophic Sets and Systems, Vol. 35, 2020
A
B
C
D
0
14.39
0
11.54
6.26
M1 14.32 11.34
C ij =
M 2 13.9
476
M3
0
6.73
11.3
0
M4
0
8.81
4.76
3.32
Number of assignment is equal to the order of the matrix. Therefore the optimal assignment is
A M 4, B M 2, C M 1, D M 3
the
minimum
time
to
complete
the
job
is
45.23 + 47 + 42.18 + 46.25 = 180.66. .
5. Conclusions
In this research article the de-Neutrosophication Pentagonal neutrosophic number into a real
number has been introduced by means of magnitude approach. The resulted ranking has been
applied to solve neutrosophic assignment problems. The algorithm stated in this paper is simple to
use and applicable to solve neutrosophic assignment problems in short time. Also it produces
accurate result. There is much scope for future work in this field. This ranking can be applied to
solve linear, non-linear and transportation problems involving pentagonal neutrosophic number.
Further image processing multi-criteria decision making problems can also make use of this
ranking method for smart computation.
Funding: “This research received no external funding”
Acknowledgments: In this section we acknowledge the Chief Editor and reviewers for their
valuable suggestions.
Conflicts of Interest: “The authors declare no conflict of interest.”
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Received: Apr 10, 2020.
Accepted: July 13 2020
K.Radhika, K.Arun Prakash, Ranking of Pentagonal Neutrosophic Numbers and itsApplications in Assignment Problem
Neutrosophic Sets and Systems, Vol. 35, 2020
University of New Mexico
A new distance measure for trapezoidal fuzzy neutrosophic
numbers based on the centroids
Broumi said 1*, Malayalan Lathamaheswari2, Ruipu Tan3, Deivanayagampillai Nagarajan2, Talea
Mohamed1 , Florentin Smarandache4, Assia Bakali5
1*
Laboratory of Information Processing, Faculty of Science Ben M’Sik, University Hassan II, B.P 7955, Sidi Othman,
Casablanca, Morocco; broumisaid78@gmail.com; s.broumi@flbenmsik.ma; taleamohamed@yahoo.fr
2 Department of Mathematics, Hindustan Institute of Technology and Science, Chennai-603 103, India;
lathamax@gmail.com
3 College of Electronics and Information Science, Fujian Jiangxia University, Fuzhou 350108, China;
4 Department of Mathematics, University of New Mexico, 705 Gurley Avenue, Gallup, NM 87301, USA
5
Ecole Royale Navale, Boulevard Sour Jdid, B.P 16303 Casablanca, Morocco,Email: assiabakali@yahoo.fr
* Correspondence: broumisaid78@gmail.com; s.broumi@flbenmsik.ma
Abstract: Distance measure is a numerical measurement of the distance between any two objects.
The aim of this paper is to propose a new distance measure for trapezoidal fuzzy neutrosophic
numbers based on the centroids with graphical representation. In addition, the metric properties of
the proposed measure are examined in detail. A decision making problem also has been solved
using the proposed distance measure for a software selection process. comparative analysis has been
done with the existing methods to show the potential of the proposed distance measure and various
forms of trapezoidal fuzzy neutrosophic number have been listed out to show the uniqueness of the
proposed graphical representation. Further, advantages of the proposed distance measure have
been given.
Keywords: trapezoidal fuzzy neutrosophic numbers; centroids; distance measure
1-Introduction
Zadeh introduced a mathematical frame work called fuzzy set [43] which plays a very significant role
in many aspects of science. Intuitionistic fuzzy set is the generalization of the Zadeh’s fuzzy set which
was presented by Atanassov [3]. Later, triangular intuitionistic fuzzy sets was developed by Liu and
Yuan [22] which is based on the combination of triangular fuzzy numbers and intuitionistic fuzzy
sets. The fundamental characteristic of the triangular intuitionistic fuzzy set is that the values of its
membership function and non-membership function are triangular fuzzy numbers rather than exact
numbers. Furthermore, Ye [38] extended the triangular intuitionistic fuzzy set to the trapezoidal
intuitionistic fuzzy set, where its fundamental characteristic is that the values of its membership
function and non-membership function are trapezoidal fuzzy numbers rather than triangular fuzzy
numbers, and proposed the trapezoidal intuitionistic fuzzy prioritized weighted averaging
(TIFPWA) operator and trapezoidal intuitionistic fuzzy prioritized weighted geometric (TIFPWG)
operator and their multi-criteria decision-making method, in which the criteria are in different
Said Broumi, Malayalan Lathamaheswari, Ruipu Tan, Deivanayagampillai Nagarajan, Talea Mohamed, Florentin
Smarandache and Assia Bakali, A new distance measure for trapezoidal fuzzy neutrosophic numbers based on the
centroids
Neutrosophic Sets and Systems, Vol. 35, 2020
479
priority level. Recently, Wang et al. [35] introduced a single-valued neutrosophic set, which is a
subclass of a neutrosophic set presented by Smarandache [30], as a generalization of the classic set,
fuzzy set and intuitionistic fuzzy set. The single-valued neutrosophic set can independently express
truth-membership degree, indeterminacy-membership degree and falsity-membership degree and
deal with incomplete, indeterminate and inconsistent information. All the factors described by the
single-valued neutrosophic set are very suitable for human thinking due to the imperfection of
knowledge that human receives or observes from the external world. For example, for a given
proposition ‘‘Movie X would be hit,’’ in this situation human brain certainly cannot generate precise
answers in terms of yes or no, as indeterminacy is the sector of unawareness of a proposition’s value
between truth and falsehood. Obviously, the neutrosophic components are best fit in the
representation of indeterminacy and inconsistent information, while the intuitionistic fuzzy set
cannot represent and handle indeterminacy and inconsistent information. Hence, the single-valued
neutrosophic set has been a rapid development and a wide range of applications [39, 40]. Ye [42]
introduced the trapezoidal neutrosophic set and its application to multiple attribute decision-making.
Cui and Ye [10], Donghai et al. [16], Ebadi et al. [17], Guha and Chakraborty [18], Hajjari [19],
Nayagam et al. [25], Rouhparvar et al. [29], Wu [37], Ye [40], Zou et al. [45] and more researchers have
shown interest on decision making problem using distance measures.
Weighted projection
measure, the combination of angle cosine and weighted projection measure,similarity measure,
hybrid vector similarity measure of single valued neutrosophic set and interval valued neutrosophic
set, outranking strategy, complete ranking, new ranking function have been introduced so far under
fuzzy, intuitionistic fuzzy and neutrosophic environments and applied in decision making problem.
The rest of the paper is organized as follows. In section 2, literature review is given. In section 3, basic
concepts are presented for better understanding. In section 4, proposed a new distnace measure and
its graphical representation, and derived its properties in detail. In section 5, new methodology is
described for a decision making process using the proposed measure. In section 6, a numerical
example is using the proposed methodology to choose the best software system. In section 7,
comparative analysis has been done with the existing methods and various forms of trapezoidal
fuzzy neutrosophic numbers have been listed out to ahow the uniqueness of the proposed graphical
representation. In section 8, advantages of the proposed measure are given. In section 9, conclusion
of the present work is given with the future direction.
2-Literature Review
The authors of, Ahmad et al. [1] proposed a similarity measure based on the distance and set theory
for generalized trapezoidal fuzzy numbers. Allahviranloo et al. [2] contributed a new distance
measure and ranking method for generalized trapezoidal fuzzy numbers. Atanassov [3] introduced
intuitionistic fuzzy sets. Azman and Abdullah [4] proposed a novel centroid method for trapezoidal
fuzzy numbers for ranking. Biswas et al. [6] solved a decision making problem using expected value
of neutrosophic trapezoidal numbers. Biswas et al. [6] solved a decision making problem using
distance measure under interval trapezoidal neutrosophic numbers. Bolos et al. [7] designed the
performance indicators of financial assets using neutrosophic fuzzy numbers. Bora and Gupta [8]
studied the reaction of distance measure on the work of K-Means algorithm Matlab. Chakraborty et
al. [9] presented
different forms of trapezoidal neutrosophic number and deneutrosophication
Said Broumi, Malayalan Lathamaheswari, Ruipu Tan, Deivanayagampillai Nagarajan, Talea Mohamed, Florentin
Smarandache and Assia Bakali, A new distance measure for trapezoidal fuzzy neutrosophic numbers based on the centroids
Neutrosophic Sets and Systems, Vol. 35, 2020
480
techniques. Cui and Ye [10] proposed logarithmic similarity measure and applied in medical
diagnosis under dynamic neutrosophic cubic setting. Darehmiraki [11] introduced a new ranking
methodology to solve linear programming problem. Das and De [12] introduced a new distance
measure for the ranking IFNs. Das and Guha [13] introduced a ranking method for IFN using the
point of centroid. Deli and Oztaurk [14] introduced a defuzzification method and applied in a
decision-making problem for single valued trapezoidal neutrosophic numbers. Dhar et al. [15]
indicated square neutrosophic fuzzy matrices. Donghai et al. [16] proposed a new similarity measure
and distance measure between hesitant linguisticterm sets and applied the proposed concepts in a
decision making problem. Ebadi et al. [17] proposed a novel distance measure for trapezoidal fuzzy
numbers. Guha and Chakraborty [18] contributed a theoretical development of distance measure for
intuitionistic fuzzy numbers (IFNs). Hajjari [19] conferred a new distance measure for Trapezoidal
fuzzy numbers. Huang and Wu [20] presented equivalent forms of the triangle inequalities in fuzzy
metric spaces. Liang et al. [21] proposed an integrated approach under a single valued trapezoidal
neutrosophic environment. Liu and Yuan [22] prospected fuzzy number of intuitionistic fuzzy set.
Llopis and Micheli [23] rectified a state of conflict in the sequence of input images. Minculete and
Paltanea [24] introduced an enhanced estimates for the triangle inequality. Nayagam et al. [25]
contributed a complete ranking of IFNs. Pardha Saradhi et al. [26] presented ordering of IFNs using
centroids of centroids. Ravi Shankar et al. [27] developed a new ranking formula using centroid of
centroids for fuzzy numbers and applied in a fuzzy critical path method. Rezvani [28] proposed a
new ranking exponential formula using median value for trapezoidal fuzzy numbers. Rouhparvar et
al. [29] introduced a novel fuzzy distance measure. Uppada [31] examined clustering algorithm using
centroid clearly. Varghese and Kuriakose [32] proposed a formula to find the centroid of the fuzzy
number. Wang [33] introduced geometric aggregation operator and applied in a decision making
problem under intuitionistic fuzzy environment. Wang [34] proposed arithmetic aggregation
operators. Wang et al. [35] introduced single valued neutrosophic sets. Wei et al. [36] introduced
some persuaded aggregation operators under intuinistic fuzzy setting and applied in a group
decision making problem. Wu [37] explained about distance metrics and their role in data
transformations.Ye [38] proposed prioritized aggregation operators based on trapezoidal
intuitionistic fuzzy concept and applied in a multi-criteria decision making problem. Ye [39] solved
minimum spanning tree problem under single valued neutrosophic setting and its clustering method.
Ye [40] proposed single valued neutrosophic cross entropy measure and applied in a decision making
problem. Ye [41] introduced the expected Dice similarity measure and applied in a decision making
problem. Ye [42] projected trapezoidal neutrosophic set and applied in a multiple attribute decision
making. Zhang et al. [44] introduced interval neutrosophic sets and used in multi criteria decision
making problem. Zou et al. [45] introduced a distance measure between neutrosophic sets as an
evidential approach. From the literature, it is found that distance measure for trapezoidal
neutrosophic numbers using centroids with its properties has not yet been studied so far. Hence the
motivation of the present study.
Hence, in this paper a new distance measure for trapezoidal fuzzy neutrosophic numbers based on
centroids has been proposed with its metric properties in detail. Also the graphical representation is
presented for trapezoidal fuzzy neutrosophic number. Comparative study also have been made with
Said Broumi, Malayalan Lathamaheswari, Ruipu Tan, Deivanayagampillai Nagarajan, Talea Mohamed, Florentin
Smarandache and Assia Bakali, A new distance measure for trapezoidal fuzzy neutrosophic numbers based on the centroids
Neutrosophic Sets and Systems, Vol. 35, 2020
481
the existing cases for both proposed distance measure and proposed graphical representation.
Further advantages of the proposed distance measure are presented.
3-Preliminaries
Definition 1. [38] Let
defined as: B
X
y ,
B
be a space of discourse, a trapezoidal intuitionistic fuzzy set
y , B y
y X
, where y 0,1
B
and
B
in
B y 0,1
B y B1 y , B2 y , B3 y , B4 y :Y 0,1
two trapezoidal fuzzy numbers
B y B1 y , B2 y , B3 y , B4 y :Y 0,1
with
X
the
condition
is
are
and
that
0 B4 y B4 y 1, y Y .
For Convenience, let
B y a, b , c , d
B y e , f , g , h
and
be two trapezoidal
fuzzy numbers, thus a trapezoidal intuitionistic fuzzy number (TrIFN) can be denoted by
j a, b , c , d , e , f , g , h , which is basic element in a trapezoidal intuitionistic fuzzy set.
If
b c
and f g hold in a TrIFN j , which is a special case of the TrIFN.
Definition 2. [38]
Let j1 a1 , b1 , c1 , d 1 , e1 , f 1 , g 1 , h1 and
j 2 a2 , b2 , c 2 , d 2 , e 2 , f 2 , g 2 , h2 , be two TrIFNs. Then there are the following operational
rules:
1.
2.
3.
j1 j 2
a1a2 , b1b2 , c1c 2 , d 1d 2 ,
e1 e 2 e1e 2 , f 1 f 2 f 1f 2 , g 1 g 2 g 1g 2 , h1 h2 h1h2
j1 j 2
j1
1 1 a
4.
1
e
1
m1
a1 a2 a1a2 , b1 b2 b1b2 , c1 c 2 c1c 2 , d 1 d 2 d 1d 2 ,
e1e 2 , f 1f 2 , g 1g 2 , h1h2
a
1
,1 1 b1 ,1 1 c1 ,1 1 d 1
, f 1 , g 1 , h1
,
, 0;
, b1 , c1 , d 1 , 1 1 e1 ,1 1 f 1 ,1 1 g 1 ,1 1 h1 ,
1 1 i ,1 1 j ,1 1 k ,1 1 l
1
1
1
, 0
1
Definition 3. [30] From philosophical point of view, Smarandache [30] originally presented the
concept of a neutrosophic set B in a universal set Y , which is characterized independently by a
Said Broumi, Malayalan Lathamaheswari, Ruipu Tan, Deivanayagampillai Nagarajan, Talea Mohamed, Florentin
Smarandache and Assia Bakali, A new distance measure for trapezoidal fuzzy neutrosophic numbers based on the centroids
Neutrosophic Sets and Systems, Vol. 35, 2020
482
truth-membership function TB y , an indeterminacy membership function I B y and a falsitymembership function FB y . The function TB y , I B y and FB y in Y are real standard or
nonstandard
subsets
] 0,1 [,
of
such
FB y : Y ] 0,1 [ .Then, the sum of
TB y : Y ] 0,1 [, I B y : Y ] 0,1 [,
that
TB y , I B y
and
FB y
and
satisfies the condition
0 sup TB y sup I B y sup FB y 3 . Obviously, it is difficult to apply the neutrosophic set
to practical problems. To easily apply it in science and engineering fields, Wang et al. [35] introduced
the concept of a single-valued neutrosophic set as a subclass of the neutrosophic set and gave the
following definition.
Definition 4. [35] A single-valued neutrosophic set B in a universal set Y is characterized by a
truth-membership function TB y , an indeterminacy-membership function I B y and a falsity-
B
membership function FB y . Then, a single-valued neutrosophic set
B
y, T
B
y , I B y , FB y
y Y
can be denoted by
where, TB y , I B y , FB y 0,1 for each y Y . Therefore, the sum of TB y , I B y and
FB y satisfies 0 TB y I B y FB y 3 .
Let M
y, T
M
y , I M y , FM y
y Y
and N
y, T
N
y , I N y , FN y
y Y
be two single-
valued neutrosophic sets, then we the following relations [8,11]:
y, F
y ,1 I M y , TM y
1.
Complement: M C
2.
Inclusion: M N if and only if TM y TN y , I M y I N y and FM y FN y for
M
y Y ;
any y Y ;
3.
Equality: M N if and only if M N and N M ;
4.
Union: M
5.
Intersection: M
N
y, T
M
N
y TN y , I M y I N y , FM y FN y
y, T
M
y Y ;
y TN y , I M y I N y , FM y FN y
y Y ;
Said Broumi, Malayalan Lathamaheswari, Ruipu Tan, Deivanayagampillai Nagarajan, Talea Mohamed, Florentin
Smarandache and Assia Bakali, A new distance measure for trapezoidal fuzzy neutrosophic numbers based on the centroids
Neutrosophic Sets and Systems, Vol. 35, 2020
483
6.
y, TM y TN y TM y TN y , I M y I N y ,
Addition: M N
y Y ;
FM y FN y
7.
y, TM y TN y , I M y I N y I M y I N y ,
Multiplication: M N
y Y .
FM y FN y FM y FN y
Definition 5. [42] Let Y be a space of discourse, a trapezoidal neutrosophic set H in Y is defined
as follow:
H
y, T
H
y Y , where TH y 0,1 , I H y 0,1 and FH y 0,1 are
y , I H y , FH y
three trapezoidal fuzzy numbers TH y t1H y , t H2 y , t H3 y , t H4 y : Y 0,1 ,
I H y i H1 y , i H2 y , i H3 y , i H4 y :Y 0,1 and
FH y f H1 y , f H2 y , f H3 y , f H4 y :Y 0,1 with the condition
0 t H4 y i H4 y f H4 y 3, y Y .
For convenience, the three trapezoidal fuzzy numbers are denoted by
TH y a, b, c, d , I H y e, f , g, h and FH y i, j, k , l . Thus, a trapezoidal neutrosophic
numbers is denoted by m a, b, c, d , e, f , g, h , i, j, k , l , which is a basic element in the
trapezoidal neutrosophic set.
If
b c, f g
and j k hold in a trapezoidal neutrosophic number j 1 , it reduces to the
triangular neutrosophic number, which is considered as a special case of the trapezoidal neutrosophic
number.
Definition 6. [42] Let m1
a1 , b1 , c1 , d1 , e1 , f1 , g1 , h1 , i1 , j1 , k1 , l1
, and
m2 a2 , b2 , c2 , d 2 , e2 , f 2 , g 2 , h2 , i2 , j2 , k2 , l2 be two trapezoidal neutrosophic numbers. Then
there are the following operational rules:
1.
m1 m2
a1 a2 a1a2 , b1 b2 b1b2 , c1 c2 c1c2 , d1 d2 d1d 2 ,
e1e2 , f1 f 2 , g1 g2 , h1h2 , i1i2 , j1 j2 , k1k2 , l1l2
,
Said Broumi, Malayalan Lathamaheswari, Ruipu Tan, Deivanayagampillai Nagarajan, Talea Mohamed, Florentin
Smarandache and Assia Bakali, A new distance measure for trapezoidal fuzzy neutrosophic numbers based on the centroids
Neutrosophic Sets and Systems, Vol. 35, 2020
2.
484
a1a2 , b1b2 , c1c2 , d1d2 ,
e1 e2 e1e2 , f1 f2 f1 f 2 , g1 g2 g1 g2 , h1 h2 h1h2 ,
i1 i2 i1i2 , j1 j2 j1 j2 , k1 k2 k1k2 , l1 l2 l1l2
m1 m2
1 1 a ,1 1 b ,1 1 c ,1 1 d ,
3.
m1
e , f
1
a
1
4.
m1
1
1
1
;
1
1
, g1 , h1 , i1 , j1 , k1 , l1
, 0;
, b1 , c1 , d1 ,
1 1 e ,1 1 f ,1 1 g ,1 1 h ,
1 1 i ,1 1 j ,1 1 k ,1 1 l
1
1
1
1
1
1
1
, 0
1
Definition 7. [18] Let P and Q be the intuitionistic fuzzy sets with membership functions
P x , Q x , non-membership functions P x , Q x and hesitation degree P x , Q x . Then
the normalized Hamming distance is
D P, Q
1 n
P xi Q xi P xi Q xi P xi Q xi
2n i 1
And the normalized Euclidean distance is
2
2
2
1 n
P xi Q xi P xi Q xi P xi Q xi
2n i 1
DE P, Q
Definition 8. [17] Consider the real values ri , i 1, 2, 3,..., 6 and if r1 r2 , r3 r4 , r5 r6 then the
following results are true.
1. max r1 , r3 , r5 max r2 , r4 , r6
2. max r1 r2 , r3 r4 , r5 r6 max r1 , r3 , r5 max r2 , r4 , r6
Definition 9. [34] For any real numbers ri , si 0, i 1,2,..., d , the Euclidean distance is defined as,
D r, s
d
ri si
i 1
2
1 p
d
p
ri si
and satisfies the condition that
i 1
1 p
d
p
ri
i 1
1 p
d
p
si
i 1
.
Said Broumi, Malayalan Lathamaheswari, Ruipu Tan, Deivanayagampillai Nagarajan, Talea Mohamed, Florentin
Smarandache and Assia Bakali, A new distance measure for trapezoidal fuzzy neutrosophic numbers based on the centroids
Neutrosophic Sets and Systems, Vol. 35, 2020
Definition 10. [42] Let mp
a
p
485
, bp , c p , d p , e p , f p , g p , hp , i p , j p , k p , l p , p 1, 2,3,..., n be the
trapezoidal fuzzy neutrosophic numbers then the trapezoidal fuzzy neurosophic weighted geometric
operator is defined by
TFNWG(m1 , m2 ,..., mn ) m11 m22 m33 ... mnn
n n n n
a p p , bp p , c p p , d p p
p 1
p 1
p 1
p 1
n
n
n
n
, 1 1 e p ,1 1 f p ,1 1 g p ,1 1 hp ,
p 1
p 1
p 1
p 1
p
p
p
p
n
n
n
n
p
p
p
p
1 1 i p ,1 1 j p ,1 1 k p ,1 1 l p
p 1
p 1
p 1
p 1
where, 1 , 2 ,..., n are the weight vectors and the sum of the weight vectors is 1.
Definition 11. [9] Graphical representation of trapezoidal neutrosophic number
Figure 1. Graphical representation of Trapezoidal neutrosophic number
Figure 1 shows that graphical representation of trapezoidal fuzzy neutrosophic number can be
done in different ways. It is a linear trapezoidal neutrosophic number.
4-Proposed Distance Measure for Trapezoidal Fuzzy Neutrosophic Number
Here we propose a new distance measure for trapezoidal fuzzy neutrosophic number based on
centroids. Firstly, individual graphical representation proposed measure is presented here with the
individual representation of truth, indeterminacy, falsity membership functions and trapezoidal
fuzzy neutrosophic fuzzy number described by Figure 2-Figure 6.
Centre point of the object is called centroid. It should lie inside the object. At this point, the three
medians of the triangle intersect and is termed point of intersection. Centroid is the average of
coordinate points in X axis and Y axis of each vertex of the triangle. Centroid is the fixed point of all
linear transformation which maintains length in translation, rotation, glides and reflection.
The centroid of the truth, indeterminacy and falsity trapezoid is treated as a balance point for the
trapezoid. The centroid of each part are estimated using the calculation of centroid and the simple
area and this combination will generate a triangle. Also the distance is measured from the centroid
of all the parts to X axis and Y axis. Here the area of all the parts are multiplied by the distance and
Said Broumi, Malayalan Lathamaheswari, Ruipu Tan, Deivanayagampillai Nagarajan, Talea Mohamed, Florentin
Smarandache and Assia Bakali, A new distance measure for trapezoidal fuzzy neutrosophic numbers based on the centroids
Neutrosophic Sets and Systems, Vol. 35, 2020
486
find their sum to get the total value. And the sum of the products of the area and distances is divided
by the total area and obtain the centroid of circumcentre described by x and y point. Since centroid
based distance measure may be derived using Euclidean measure, here it is obtained from the
circumcentre of the centroids and the authentic point for the trapezoidal fuzzy neutrosophic number.
Tn ( x )
1
fTL
fTR
OT
0
a1
a2
a3
a4
1
x
Figure 2. Truth membership function of trapezoidal fuzzy neutrosophic set with centroid
x
1
a4
gTR
a3
a2
a1
gTL
1
0
Tn ( x )
Figure 3. Truth membership function of trapezoidal fuzzy neutrosophic set
Suppose n
a1 , a2 , a3 , a4 , b1 , b2 , b3 , b4 , c1, c2 , c3 , c4
be a trapezoidal fuzzy neutrosophic
number. Based on the literature (Y. M. Wang et al. On the centroids of fuzzy numbers), we can get the
centroid point
OT ( xoT (n), yoT (n))
of the truth membership function of trapezoidal fuzzy
neutrosophic number n .
Said Broumi, Malayalan Lathamaheswari, Ruipu Tan, Deivanayagampillai Nagarajan, Talea Mohamed, Florentin
Smarandache and Assia Bakali, A new distance measure for trapezoidal fuzzy neutrosophic numbers based on the centroids
Neutrosophic Sets and Systems, Vol. 35, 2020
xoT (n)
a2
a3
a2
a1
T
o
y
a4
xfTL dx x 1dx xfTR dx
a1
1
487
a2
a3
a3
a4
a2
a3
fTL dx 1dx fTR dx
a4 a3 a1a2
1
[a1 a2 a3 a4
],
3
(a4 a3 ) (a1 a2 )
y( g g )dy 1 [1
( n)
( g g )dy 3 (a
0
L
T
1
0
R
T
L
T
a3 a2
].
4 a3 ) ( a1 a2 )
R
T
I n ( x)
1
f IL
0
f IR
OI
b1
b2
b3
b4
1
x
Figure 4. Indeterminate membership function of trapezoidal fuzzy neutrosophic set with centroid
x
1
b4
g IR
b3
b2
b1
g IL
0
1
I n ( x)
Figure 5. Indeterminate membership function of trapezoidal fuzzy neutrosophic number
Said Broumi, Malayalan Lathamaheswari, Ruipu Tan, Deivanayagampillai Nagarajan, Talea Mohamed, Florentin
Smarandache and Assia Bakali, A new distance measure for trapezoidal fuzzy neutrosophic numbers based on the centroids
Neutrosophic Sets and Systems, Vol. 35, 2020
we can get the centroid point
488
OI ( xoI (n), yoI (n))
of indeterminacy membership function of
trapezoidal fuzzy neutrosophic number n .
x ( n)
I
o
b2
b3
b4
xf IL dx x 1dx xf IR dx
b1
b2
b2
b3
b3
b4
b2
b3
f IL dx 1dx f IR dx
b1
1
b4b3 b1b2
1
[b1 b2 b3 b4
],
3
(b4 b3 ) (b1 b2 )
y( g g )dy 1 [1
y ( n)
( g g )dy 3 (b
I
o
0
L
I
1
L
I
0
R
I
b3 b2
].
4 b3 ) (b1 b2 )
R
I
Similarly, we can get the centroid point
O F ( xoF (n), yoF (n))
of trapezoidal fuzzy neutrosophic number
xoF
( n)
c2
c3
c1
c2
c1
1
F
o
y
c4
xf FL dx x 1dx xf FR dx
c2
c3
c3
c4
c2
c3
f FL dx 1dx f FR dx
of falsity membership function
n.
c4c3 c1c2
1
[c1 c2 c3 c4
],
3
(c4 c3 ) (c1 c2 )
y( g g )dy 1 [1
( n)
( g g )dy 3 (c
0
L
F
1
L
F
0
R
F
c3 c2
].
4 c3 ) (c1 c2 )
R
F
TN ( x), I N ( x),
FN ( x)
1
OF
OI
OT
0
a1 c1
c2 a2 b1 c3 b2 a3
c4 b3
a4 b4
1
x
Figure 6. Trapezoidal fuzzy neutrosophic number with circumcentre of Centroids
Said Broumi, Malayalan Lathamaheswari, Ruipu Tan, Deivanayagampillai Nagarajan, Talea Mohamed, Florentin
Smarandache and Assia Bakali, A new distance measure for trapezoidal fuzzy neutrosophic numbers based on the centroids
Neutrosophic Sets and Systems, Vol. 35, 2020
489
In the above figure 5, the red dot represents the center of gravity of the triangle consisting of O T , O I
, and O F . According to the coordinate formula of the center of gravity of the triangle, we can get the
coordinates of red dots O ( x(n), y (n)) .
xoT (n) xoI (n) xoF (n)
x ( n)
3
a4 a3 a1a2
1
3 [a1 a2 a3 a4 (a a ) (a a ) ]
1
4
3
1
2
b4b3 b1b2
c4c3 c1c2
1
3 1
[b1 b2 b3 b4
] [c1 c2 c3 c4
]
3
(
b
b
)
(
b
b
)
3
(
c
c
)
(
c
c
)
4
3
1
2
4
3
1
2
4
4
a4 a3 a1a2
b4b3 b1b2
c4c3 c1c2
1 4
[ ai bi ci
]
9 i 1
(a4 a3 ) (a1 a2 ) (b4 b3 ) (b1 b2 ) (c4 c3 ) (c1 c2 )
i 1
i 1
yoT (n) yoI (n) yoF (n)
y ( n)
3
a3 a2
b3 b2
c3 c2
1
1
1
[1
] [1
] [1
]
3
(a4 a3 ) (a1 a2 ) 3
(b4 b3 ) (b1 b2 ) 3
(c4 c3 ) (c1 c2 )
3
a3 a2
b3 b2
c3 c2
1
[3
]
9
(a4 a3 ) (a1 a2 ) (b4 b3 ) (b1 b2 ) (c4 c3 ) (c1 c2 )
Definition1: Let n1
a1 , a2 , a3 , a4 , b1, b2 , b3 , b4 , c1, c2 , c3 , c4
and
n2 e1 , e2 , e3 , e4 , f1 , f 2 , f3 , f 4 , g1 , g2 , g3 , g4 be two trapezoidal fuzzy neutrosophic
numbers, and their centroids are O1 ( x(n1 ), y (n1 )) , O2 ( x(n2 ), y (n2 )) respectively, then the
distance between n1 and n2 is
4
4
4
4
4
4
i 1
i 1
i 1
i 1
i 1
i 1
[ ai bi ci ei f i g i (
D(n1 , n2 )
1
9
(
b4b3 b1b2
f 4 f3 f1 f 2
c4c3 c1c2
g 4 g 3 g1 g 2
)]2
)(
(b4 b3 ) (b1 b2 ) ( f 4 f3 ) ( f1 f 2 ) (c4 c3 ) (c1 c2 ) ( g 4 g3 ) ( g1 g 2 )
[(
(
a4 a3 a1a2
e4 e3 e1e2
)
(a4 a3 ) (a1 a2 ) (e4 e3 ) (e1 e2 )
a4 a3 a1a2
e4 e3 e1e2
b4b3 b1b2
f 4 f 3 f1 f 2
)(
)
(a4 a3 ) (a1 a2 ) (e4 e3 ) (e1 e2 ) (b4 b3 ) (b1 b2 ) ( f 4 f3 ) ( f1 f 2 )
c4c3 c1c2
g 4 g3 g1 g 2
)]2
(c4 c3 ) (c1 c2 ) ( g 4 g 3 ) ( g1 g 2 )
Theorem 1: This distance
D n1 , n2
of n1 and n2 fulfills the following properties:
Said Broumi, Malayalan Lathamaheswari, Ruipu Tan, Deivanayagampillai Nagarajan, Talea Mohamed, Florentin
Smarandache and Assia Bakali, A new distance measure for trapezoidal fuzzy neutrosophic numbers based on the centroids
Neutrosophic Sets and Systems, Vol. 35, 2020
1.
0 D n1 , n2 1 ;
2.
D n1 , n2 0
3.
D n1 , n2 D n2 , n1 .
4. If
n1 , n2 & n3
490
if and only if n1 n2 , i.e., ai ei , bi f i and ci gi hold for i 1, 2, 3, 4 ;
are the trapezoidal fuzzy neutrosophic numbers then
D n1 , n3 D n1 , n2 D n2 , n3
Proof
1. It is easy to prove 0 D n1 , n2 . In addition, it can be seen from figure 1, the maximum distance is
the distance between the point (0, 0) and the point (1,1) , or the point (0,1) and the point (1, 0) ,
assume the coordinates of centroids of n1 and n2 are O1 and O2 , and O1 (0,1) and
O2 (1, 0) , or O1 (1, 0) and O2 (0,1) , or O1 (0, 0) and O2 (1,1) , or O1 (1,1) and
O2 (0, 0) , then the D n1 , n2 1 , otherwise, D n1 , n2 1 , thus
0 D n1 , n2 1 .
2. if n1 n2 , i.e., ai ei , bi f i and ci gi , then
Said Broumi, Malayalan Lathamaheswari, Ruipu Tan, Deivanayagampillai Nagarajan, Talea Mohamed, Florentin
Smarandache and Assia Bakali, A new distance measure for trapezoidal fuzzy neutrosophic numbers based on the centroids
Neutrosophic Sets and Systems, Vol. 35, 2020
491
4
4
4
4
4
4
i 1
i 1
i 1
i 1
i 1
i 1
[ ai bi ci ai bi ci (
D(n1 , n2 )
(
1
9
b4b3 b1b2
b4b3 b1b2
c4c3 c1c2
c4c3 c1c2
)(
)]2
(b4 b3 ) (b1 b2 ) (b4 b3 ) (b1 b2 ) (c4 c3 ) (c1 c2 ) (c4 c3 ) (c1 c2 )
[(
(
a4 a3 a1a2
a4 a3 a1a2
)
(a4 a3 ) (a1 a2 ) (a4 a3 ) (a1 a2 )
a4 a3 a1a2
a4 a3 a1a2
b4b3 b1b2
b4b3 b1b2
)(
)
(a4 a3 ) (a1 a2 ) (a4 a3 ) (a1 a2 )
(b4 b3 ) (b1 b2 ) (b4 b3 ) (b1 b2 )
c4c3 c1c2
c4c3 c1c2
)]2
(c4 c3 ) (c1 c2 ) (c4 c3 ) (c1 c2 )
if
0.
D n1 , n2 0 , then
4
4
4
4
4
4
i 1
i 1
i 1
i 1
i 1
i 1
[ ai bi ci ei fi gi (
(
b4b3 b1b2
f 4 f3 f1 f 2
c4c3 c1c2
g 4 g3 g1 g 2
)(
)]2
(b4 b3 ) (b1 b2 ) ( f 4 f3 ) ( f1 f 2 )
(c4 c3 ) (c1 c2 ) ( g 4 g3 ) ( g1 g 2 )
[(
(
a4 a3 a1a2
e4e3 e1e2
)
(a4 a3 ) (a1 a2 ) (e4 e3 ) (e1 e2 )
a4 a3 a1a2
e4e3 e1e2
b4b3 b1b2
f 4 f 3 f1 f 2
)(
)
(a4 a3 ) (a1 a2 ) (e4 e3 ) (e1 e2 )
(b4 b3 ) (b1 b2 ) ( f 4 f3 ) ( f1 f 2 )
c4c3 c1c2
g 4 g3 g1 g 2
)]2 0,
(c4 c3 ) (c1 c2 ) ( g 4 g3 ) ( g1 g 2 )
Thus,
4
4
4
4
4
4
a b c e f g
i
i 1
(
i 1
i
i 1
i
i 1
i
i 1
i
i 1
i
(
a4 a3 a1a2
e4 e3 e1e2
)
(a4 a3 ) (a1 a2 ) (e4 e3 ) (e1 e2 )
b4b3 b1b2
f 4 f 3 f1 f 2
c4c3 c1c2
g 4 g 3 g1 g 2
)
)(
(b4 b3 ) (b1 b2 ) ( f 4 f 3 ) ( f1 f 2 )
(c4 c3 ) (c1 c2 ) ( g 4 g 3 ) ( g1 g 2 )
0,
(
a4 a3 a1a2
e4 e3 e1e2
b4b3 b1b2
f 4 f3 f1 f 2
)(
)
(a4 a3 ) (a1 a2 ) (e4 e3 ) (e1 e2 )
(b4 b3 ) (b1 b2 ) ( f 4 f 3 ) ( f1 f 2 )
(
c4 c3 c1c2
g 4 g3 g1 g 2
)
(c4 c3 ) (c1 c2 ) ( g 4 g3 ) ( g1 g 2 )
0,
thus
Said Broumi, Malayalan Lathamaheswari, Ruipu Tan, Deivanayagampillai Nagarajan, Talea Mohamed, Florentin
Smarandache and Assia Bakali, A new distance measure for trapezoidal fuzzy neutrosophic numbers based on the centroids
Neutrosophic Sets and Systems, Vol. 35, 2020
492
a4 a3 a1a2
e4 e3 e1e2
0,
(a4 a3 ) (a1 a2 ) (e4 e3 ) (e1 e2 )
b4b3 b1b2
f 4 f3 f1 f 2
0,
(b4 b3 ) (b1 b2 ) ( f 4 f3 ) ( f1 f 2 )
c4 c3 c1c2
g 4 g3 g1 g 2
0,
(c4 c3 ) (c1 c2 ) ( g 4 g3 ) ( g1 g 2 )
thus
ai ei , bi f i , ci gi ,
that is
n1 n2 .
3. Since,
4
4
4
4
4
4
i 1
i 1
i 1
i 1
i 1
i 1
[ ai bi ci ei f i gi (
(
b4b3 b1b2
f 4 f 3 f1 f 2
c4c3 c1c2
g 4 g 3 g1 g 2
)(
)]2
(b4 b3 ) (b1 b2 ) ( f 4 f3 ) ( f1 f 2 )
(c4 c3 ) (c1 c2 ) ( g 4 g3 ) ( g1 g 2 )
[(
(
a4 a3 a1a2
e4 e3 e1e2
)
(a4 a3 ) (a1 a2 ) (e4 e3 ) (e1 e2 )
a4 a3 a1a2
e4 e3 e1e2
b4b3 b1b2
f 4 f 3 f1 f 2
)(
)
(a4 a3 ) (a1 a2 ) (e4 e3 ) (e1 e2 )
(b4 b3 ) (b1 b2 ) ( f 4 f 3 ) ( f1 f 2 )
c4 c3 c1c2
g 4 g3 g1 g 2
)]2
(c4 c3 ) (c1 c2 ) ( g 4 g3 ) ( g1 g 2 )
4
4
4
4
4
4
i 1
i 1
i 1
i 1
i 1
i 1
[ ei f i gi ai bi ci (
(
f 4 f3 f1 f 2
b4b3 b1b2
g 4 g 3 g1 g 2
c4c3 c1c2
)(
)]2
( f 4 f 3 ) ( f1 f 2 ) (b4 b3 ) (b1 b2 )
( g 4 g3 ) ( g1 g 2 ) (c4 c3 ) (c1 c2 )
[(
(
e4 e3 e1e2
a4 a3 a1a2
)
(e4 e3 ) (e1 e2 ) (a4 a3 ) (a1 a2 )
e4 e3 e1e2
a4 a3 a1a2
f 4 f3 f1 f 2
b4b3 b1b2
)(
)
(e4 e3 ) (e1 e2 ) (a4 a3 ) (a1 a2 )
( f 4 f 3 ) ( f1 f 2 ) (b4 b3 ) (b1 b2 )
g 4 g3 g1 g 2
c4c3 c1c2
)]2
( g 4 g3 ) ( g1 g 2 ) (c4 c3 ) (c1 c2 )
then
D n1 , n2 D n2 , n1 .
4. Using Def. 8, we can prove (4).
Let n1
a1 , a2 , a3 , a4 , b1, b2 , b3 , b4 , c1, c2 , c3 , c4
,
n2 e1 , e2 , e3 , e4 , f1 , f 2 , f3 , f 4 , g1 , g2 , g3 , g4 and
Said Broumi, Malayalan Lathamaheswari, Ruipu Tan, Deivanayagampillai Nagarajan, Talea Mohamed, Florentin
Smarandache and Assia Bakali, A new distance measure for trapezoidal fuzzy neutrosophic numbers based on the centroids
Neutrosophic Sets and Systems, Vol. 35, 2020
n3
493
j1 , j2 , j3 , j4 , k1, k 2 , k3 , k 4 , l1, l2 , l3 , l4
numbers then
are the three trapezoidal fuzzy neutrosophic
D n1 , n3 D n1 , n2 D n2 , n3
Using the results we have,
D(n1 , n3 )
4
4
4
4
4
4
i 1
i 1
i 1
i 1
i 1
i 1
[ ai bi ci ji ki li (
1
9
(
b4b3 b1b2
k4 k3 k1k2
c4 c3 c1c2
l4l3 l1l2
)]2
)(
(b4 b3 ) (b1 b2 ) (k 4 k3 ) (k1 k2 )
(c4 c3 ) (c1 c2 ) (l4 l3 ) (l1 l2 )
[(
(
a4 a3 a1a2
j4 j3 j1 j2
)
(a4 a3 ) (a1 a2 ) (j4 j3 ) (j1 j2 )
a4 a3 a1a2
j4 j3 j1 j2
b4b3 b1b2
k4 k3 k1k 2
)(
)
(a4 a3 ) (a1 a2 ) (j4 j3 ) (j1 j2 )
(b4 b3 ) (b1 b2 ) (k 4 k3 ) (k1 k2 )
c4 c3 c1c2
l4l3 l1l2
)]2
(c4 c3 ) (c1 c2 ) (l4 l3 ) (l1 l2 )
Said Broumi, Malayalan Lathamaheswari, Ruipu Tan, Deivanayagampillai Nagarajan, Talea Mohamed, Florentin
Smarandache and Assia Bakali, A new distance measure for trapezoidal fuzzy neutrosophic numbers based on the centroids
Neutrosophic Sets and Systems, Vol. 35, 2020
494
4
4
4
4
4
4
4
4
4
4
4
4
i 1
i 1
i 1
i 1
i 1
i 1
i 1
i 1
i 1
i 1
i 1
i 1
{ ai bi ci ei f i gi ei f i gi ji ki li
a4 a3 a1a2
e4 e3 e1e2
e4 e3 e1e2
j4 j3 j1 j2
[(
)(
)]
(a4 a3 ) (a1 a2 ) (e 4 e3 ) (e1 e2 )
(e 4 e3 ) (e1 e2 ) (j4 j3 ) (j1 j2 )
[(
b4 b3 b1b2
f 4 f 3 f1 f 2
f 4 f 3 f1 f 2
k 4 k3 k1k 2
)(
)]
(b4 b3 ) (b1 b2 ) (f 4 f 3 ) (f1 f 2 )
(f 4 f 3 ) (f1 f 2 ) (k 4 k3 ) (k1 k2 )
c4 c3 c1c2
g 4 g 3 g1 g 2
g 4 g3 g1 g 2
l4 l3 l1l2
1
[(
)(
)]}2
9
(c4 c3 ) (c1 c2 ) (g 4 g3 ) (g1 g 2 )
(g 4 g3 ) (g1 g 2 ) (l4 l3 ) (l1 l2 )
{[(
1
9
a4 a3 a1a2
e4 e3 e1e2
e4 e3 e1e2
j4 j3 j1 j2
)(
)]
(a4 a3 ) (a1 a2 ) (e 4 e3 ) (e1 e2 )
(e 4 e3 ) (e1 e2 ) (j4 j3 ) (j1 j2 )
[(
b4 b3 b1b2
f 4 f 3 f1 f 2
f 4 f 3 f1 f 2
k4 k3 k1k2
)(
)]
(b4 b3 ) (b1 b2 ) (f 4 f 3 ) (f1 f 2 )
(f 4 f 3 ) (f1 f 2 ) (k 4 k3 ) (k1 k2 )
[(
c4 c3 c1c2
g 4 g 3 g1 g 2
g 4 g3 g1 g 2
l4 l3 l1l2
)(
)]}2
(c4 c3 ) (c1 c2 ) (g 4 g3 ) (g1 g 2 )
(g 4 g3 ) (g1 g 2 ) (l4 l3 ) (l1 l2 )
{ ai bi ci ei f i gi ei f i gi ji ki li
i 1
i 1
i 1
i 1
i 1
i 1
i 1
i 1
i 1
i 1
i 1
i 1
a4 a3 a1a2
e4 e3 e1e2
e4 e3 e1e2
j4 j3 j1 j2
[(
)(
)]
(a4 a3 ) (a1 a2 ) (e 4 e3 ) (e1 e2 )
(e 4 e3 ) (e1 e2 ) (j4 j3 ) (j1 j2 )
b4 b3 b1b2
f 4 f 3 f1 f 2
f 4 f 3 f1 f 2
k4 k3 k1k2
[(
)(
)]
(b4 b3 ) (b1 b2 ) (f 4 f 3 ) (f1 f 2 )
(f 4 f 3 ) (f1 f 2 ) (k 4 k3 ) (k1 k2 )
c4 c3 c1c2
g 4 g 3 g1 g 2
g 4 g 3 g1 g 2
l4l3 l1l2
2
[(
)(
)]}
(c4 c3 ) (c1 c2 ) (g 4 g3 ) (g1 g 2 )
(g 4 g 3 ) (g1 g 2 ) (l4 l3 ) (l1 l2 )
a4 a3 a1a2
e4 e3 e1e2
e4 e3 e1e2
j4 j3 j1 j2
{[(
)(
)]
(a4 a3 ) (a1 a2 ) (e 4 e3 ) (e1 e2 )
(e 4 e3 ) (e1 e2 ) (j4 j3 ) (j1 j2 )
b4 b3 b1b2
f 4 f 3 f1 f 2
f 4 f 3 f1 f 2
k4 k3 k1k2
[(
)(
)]
(b4 b3 ) (b1 b2 ) (f 4 f 3 ) (f1 f 2 )
(f 4 f 3 ) (f1 f 2 ) (k 4 k3 ) (k1 k2 )
c4 c3 c1c2
g 4 g 3 g1 g 2
g 4 g 3 g1 g 2
l4l3 l1l2
[(
)(
)]}2
(c4 c3 ) (c1 c2 ) (g 4 g3 ) (g1 g 2 )
(g 4 g3 ) (g1 g 2 ) (l4 l3 ) (l1 l2 )
4
4
4
4
4
4
4
4
4
4
4
4
2
12
4
4
4
4
4
a4 a3 a1a2
e4 e3 e1e2
4
[ ai bi ci ei fi gi ( (a a ) (a a ) (e e ) (e e ) )
i 1
i 1
i 1
i 1
i 1
4
3
1
2
4
3
1
2
i 1
b4b3 b1b2
f 4 f3 f1 f 2
c4 c3 c1c2
g 4 g3 g1 g 2
)(
)]2
(
1 (b4 b3 ) (b1 b2 ) ( f 4 f3 ) ( f1 f 2 ) (c4 c3 ) (c1 c2 ) ( g 4 g3 ) ( g1 g 2 )
a4 a3 a1a2
e4 e3 e1e2
b4b3 b1b2
f 4 f3 f1 f 2
9
)(
)
[(
(a4 a3 ) (a1 a2 ) (e4 e3 ) (e1 e2 ) (b4 b3 ) (b1 b2 ) ( f 4 f 3 ) ( f1 f 2 )
c4 c3 c1c2
g 4 g3 g1 g 2
2
(
(c c ) (c c ) ( g g ) ( g g ) )]
4
3
1
2
4
3
1
2
Said Broumi, Malayalan Lathamaheswari, Ruipu Tan, Deivanayagampillai Nagarajan, Talea Mohamed, Florentin
Smarandache and Assia Bakali, A new distance measure for trapezoidal fuzzy neutrosophic numbers based on the centroids
Neutrosophic Sets and Systems, Vol. 35, 2020
495
12
4
4
4
4
4
e4 e3 e1e2
j4 j3 j1 j2
4
[
e
f
g
j
k
li (
)
i
i
i
i
i
(e4 e3 ) (e1 e2 ) (j4 j3 ) (j1 j2 )
i 1
i 1
i 1
i 1
i 1
i 1
f 4 f3 f1 f 2
k4 k3 k1k2
g 4 g3 g1 g 2
l4l3 l1l2
)]2
)(
(
( g 4 g3 ) ( g1 g 2 ) (l4 l3 ) (l1 l2 )
1 ( f 4 f3 ) ( f1 f 2 ) (k 4 k3 ) (k1 k2 )
e4 e3 e1e2
j4 j3 j1 j2
f 4 f3 f1 f 2
k4 k3 k1k2
9
)(
)
[(
( f 4 f3 ) ( f1 f 2 ) (k 4 k3 ) (k1 k2 )
(e4 e3 ) (e1 e2 ) (j4 j3 ) (j1 j2 )
g 4 g3 g1 g 2
l4l3 l1l2
2
(
)]
( g g ) ( g g ) (l l ) (l l )
4
3
1
2
4
3
1
2
Using Def.9 we have,
4
4
4
4
4
4
i 1
i 1
i 1
i 1
i 1
i 1
[ ai bi ci ei fi gi (
1
9
(
b4b3 b1b2
f 4 f 3 f1 f 2
c4 c3 c1c2
g 4 g 3 g1 g 2
)(
)]2
(b4 b3 ) (b1 b2 ) ( f 4 f3 ) ( f1 f 2 )
(c4 c3 ) (c1 c2 ) ( g 4 g3 ) ( g1 g 2 )
[(
(
a4 a3 a1a2
e4 e3 e1e2
b4 b3 b1b2
f 4 f 3 f1 f 2
)(
)
(a4 a3 ) (a1 a2 ) (e4 e3 ) (e1 e2 )
(b4 b3 ) (b1 b2 ) ( f 4 f 3 ) ( f1 f 2 )
c4 c3 c1c2
g 4 g3 g1 g 2
)]2
(c4 c3 ) (c1 c2 ) ( g 4 g3 ) ( g1 g 2 )
4
4
4
4
4
4
i 1
i 1
i 1
i 1
i 1
i 1
[ ei fi gi ji ki li (
1
9
(
e4 e3 e1e2
j4 j3 j1 j2
)
(e4 e3 ) (e1 e2 ) (j4 j3 ) (j1 j2 )
f 4 f3 f1 f 2
k4 k3 k1k2
g 4 g3 g1 g 2
l4l3 l1l2
)(
)]2
( f 4 f3 ) ( f1 f 2 ) (k 4 k3 ) (k1 k2 )
( g 4 g3 ) ( g1 g 2 ) (l4 l3 ) (l1 l2 )
[(
(
a4 a3 a1a2
e4 e3 e1e2
)
(a4 a3 ) (a1 a2 ) (e4 e3 ) (e1 e2 )
e4 e3 e1e2
j4 j3 j1 j2
f 4 f3 f1 f 2
k4 k3 k1k2
)(
)
(e 4 e3 ) (e1 e2 ) (j4 j3 ) (j1 j2 )
( f 4 f3 ) ( f1 f 2 ) (k 4 k3 ) (k1 k2 )
g 4 g3 g1 g 2
l4l3 l1l2
)]2
( g 4 g3 ) ( g1 g 2 ) (l4 l3 ) (l1 l2 )
D n1 , n2 D n2 , n3 and hence the result (4).
5- Decision Making method based on new distance measure based on centroids
In this section, we establish an approach based an trapezoidal fuzzy neutrosophic number weighted
geometric arithmetic operator and a new distance measure based on centroid to deal with trapezoidal
fuzzy neutrosophic information. The proposed approach is described as follows.
Step 1: Apply trapezoidal fuzzy neutrosophic number weighted geometric arithmetic operator [39]
to find the aggregated trapezoidal fuzzy neutrosophic numbers for all the alternatives.
Step 2: Use the proposed distance measure, find the distances between all the alternatives and the
ideal trapezoidal fuzzy neutrooshic number
Said Broumi, Malayalan Lathamaheswari, Ruipu Tan, Deivanayagampillai Nagarajan, Talea Mohamed, Florentin
Smarandache and Assia Bakali, A new distance measure for trapezoidal fuzzy neutrosophic numbers based on the centroids
Neutrosophic Sets and Systems, Vol. 35, 2020
496
Step 3: Rank the alternatives in which smaller value of distance indicate the best one.
Step 4: End
6- Numerical Example for the application of the proposed distance measure
In this section, a numerical example of a software selection problem and the aggregation operator
called trapezoidal neutrosophic number weighted geometric averaging operator are get used from
Ye [39] for a multiple attribute decision making problem is contributed to exhibit the application and
effectiveness of the proposed distance measure under trapezoidal fuzzy neutrosophic environment.
For a software selection process, consider candidate software systems are given as the set of five
alternatives
S1, S2 , S3 , S4 , S5
and the investment company need to take a decision according to four
criteria: (i). the contribution to organization performance, (ii). The effort totranform from current
system, (iii). The costs of hardware/software investment, (iv). The outsourcing software deneloper
reliability denoted by
C1, C2 , C3 , C4
respectively with the weight vector
0.25,0.25,0.3,0.2
T
. The experts
evaluate the five alternatives with repect to the four criteions under trapezoidal fuzzy neutrosophic
environment and thus we can form the trapezoidal fuzzy neutrosophic decision matrix:
Table 1: Decision matrix using trapezoidal fuzzy neutrosophic numbers
D
0.4,0.5,0.6,0.7 , 0.0,0.1,0.2,0.3 , 0.1,0.1,0.1,0.1
0.3,0.4,0.5,0.5 , 0.1,0.2,0.3,0.4 , 0.0,0.1, 0.1,0.1
0.1,0.1,0.1,0.1 , 0.1,1.1,0.1,0.1 , 0.6,0.7,0.8,0.9
0.7,0.7,0.7,0.7 , 0.0,0.1,0.2,0.3 , 0.1,0.1,0.1,0.1
0.0,0.1,0.2,0.2 , 0.1,0.1,0.1,0.1 , 0.5,0.6,0.7,0.8
0.0,0.1,0.2,0.3 , 0.0,0.1,0.2,0.3 , 0.2,0.3,0.4,0.5
0.2,0.3,0.4,0.5 , 0.0,0.1,0.2,0.3 , 0.0,0.1,0.2,0.3
0.0,0.1,0.1,0.2 , 0.0,0.1,0.2,0.3 , 0.3,0.4,0.5,0.6
0.4,0.5,0.6,0.7 , 0.1,0.1,0.1,0.1 , 0.0,0.1,0.2,0.2
0.4,0.4,0.4,0.4 , 0.0,0.1,0.2,0.3 , 0.0,0.1,0.2,0.3
0.3,0.4,0.5,0.6 , 0.0,0.1,0.2,0.3 , 0.1,0.1,0.1,0.1
0.0,0.1,0.1,0.2 , 0.1,0.1,0.1,0.1 , 0.5,0.6,0.7,0.8
0.2,0.3,0.4,0.5 , 0.0,0.1,0.2,0.3 , 0.1,0.2,0.2,0.3
0.2,0.3,0.4,0.5 , 0.0,0.1,0.2,0.3 , 0.1,0.2,0.3,0.3
0.6,0.7,0.7,0.8 , 0.1,0.1,0.1,0.1 , 0.0,0.1,0.1,0.2
0.3,0.4,0.5,0.6 , 0.1,0.1,0.1,0.1 , 0.1,0.2,0.3,0.4
0.3,0.4,0.5,0.6 , 0.0,0.1,0.2,0.3 , 0.1,0.1,0.1,0.2
0.1,0.2,0.3,0.4 , 0.1,0.1,0.1,0.1 , 0.3,0.4,0.5,0.6
0.1,0.2,0.3,0.4 , 0.1,0.1,0.1,0.1 , 0.1,0.1,0.1,0.1
0.1,0.2,0.3,0.3 , 0.1,0.2,0.3,0.4 , 0.2,0.3,0.4,0.5
Here we used the developed method to obtain the best software system(s) and it is described as
follows:
Step 1: Using trapezoidal fuzzy neutrosophic weighted geometric operator in Definition 10, get the
aggregated trapezoidal fuzzy neutrosophic numbers of
ni , i 1, 2, 3, 4, 5 for the software system
Si , i 1, 2, 3, 4, 5 as follows:
Said Broumi, Malayalan Lathamaheswari, Ruipu Tan, Deivanayagampillai Nagarajan, Talea Mohamed, Florentin
Smarandache and Assia Bakali, A new distance measure for trapezoidal fuzzy neutrosophic numbers based on the centroids
Neutrosophic Sets and Systems, Vol. 35, 2020
497
n1 0.0000,0.2985,0.4162,0.5244 , 0.0209,0.1003,0.1809,0.2639 , 0.1261,0.1745,0.2266,0.2836
n2 0.0000,0.2458,0.2919,0.3798 , 0.0563,0.1262,0.1984,0.2739 , 0.1879,0.2944,0.3717,0.4743
n3 0.0000,0.1599,0.1888,0.2545 , 0.0464,0.1000,0.1566,0.2162 , 0.3437,0.4502,0.5424,0.6655
n4 0.2833,0.3885,0.4807,0.5658 , 0.0464,0.1000,0.1566,0.2162 , 0.1480,0.2276,0.3109,0.3109
n5 0.0000,0.2912,0.3756,0.3910 , 0.0760,0.1210,0.1690,0.2208 , 0.1958,0.3012,0.3877,0.5020
Step 2: Use the proposed distance measure and find the distance between all ni , i 1, 2,3, 4,5 and
the ideal trapezoidal fuzzy neutrosophic number nIdeal
1,1,1,1 , 0, 0, 0, 0 , 0, 0, 0, 0
.
The obtained distances are as follows:
D n1 , I 0.1712 D1
D n2 , I 0.1276 D2
D n3 , I 0.1000 D3
D n4 , I 0.1280 D4
D n5 , I 0.1246 D5
Step 3: Find the best alternative by considering the smaller value of the distance as the smaller value
of distance indicates the best one.
Using step 2 it is found that, D3 D5 D2 D4 D1 and from the ranking order, S 3 is the best is
the best software system.
7- Comparative analysis for the proposed distance measure and graphical representation
In this section, a comparative study is made to show the effectiveness of the proposed distance
measure with the existing methods and to show the uniqueness of the proposed graphical
representation.
Table 2: Comparative analysis with the existing methods
Existing
Methods
[6]
[16]
[42]
[45]
Score/ distance values
Ranking
D1
D2
D3
D4
0.6092
0.4512
0.6039
0.6121
0.2788
0.6553
0.7716
0.6790
0.5779
0.7798
0.9394
0.5069
0.7349
0.6564
0.6835
0.8124
D5
0.6321
S2 S3 S1 S4 S5
0.4014
S3 S2 S4 S5 S1
0.5904
S4 S1 S5 S2 S3
0.8201
S3 S1 S2 S4 S5
From the Table 2, it is found that, the third software system is the best one among the five alternatives.
The results in the existing methods overlaps the proposed result. Theresore the proposed
methodology using the proposed under trapezoidal fuzzy neutrosophic environment to solve the
decision making problem suitably in comparision with the existing methods.
Said Broumi, Malayalan Lathamaheswari, Ruipu Tan, Deivanayagampillai Nagarajan, Talea Mohamed, Florentin
Smarandache and Assia Bakali, A new distance measure for trapezoidal fuzzy neutrosophic numbers based on the centroids
Neutrosophic Sets and Systems, Vol. 35, 2020
498
Table 3 represents the various forms of trapezoidal fuzzy neutrosophic numbers (TrFNN) have been
listed out and it shows the uniqueness of the proposed graphical representation among the existing
graphical representations.
Table 3: Comparative analysis with the existing graphical representation
Trapezoidal fuzzy neutrosophic forms
Graphical representation
𝑇𝐴 (𝑥), 𝐼𝐴 (𝑥), 𝐹𝐴 (𝑥)
Darehmiraki [11]; A is a TrFNN,
a1'' , a1 , a1' , a2 , a3 , a4' , a4 , a4'' R such that
a1'' a1 a1' a2 a3 a4' a4 a4''
A a1'' , a1 , a1' , a2 , a3 , a4' , a4 , a4'' , TA , I A , FA
a1''
a1
a1'
a2
a3
a4'
a4
x
a4''
Liang [21]; A is a TrFNN,
a1, a2 , a3 , a4 [0,1] such that
0 a1 a2 a3 a4 1
1
TA
A a1, a2 , a3 , a4 , TA , I A , FA
IA
A ( x)
A ( x)
A ( x)
FA
0
a1
a2
b11 a11
c21 b21 a21
a3
x
a4
Biswas [5]; A is a TpFNN,
a41 ,a 21 ,a 31 ,a 41 , b41 , b 21 , b31 , b 41 ,
c41 ,c21 ,c31 ,c 41 R
1
such that
c11 b11 a11 c 21 b 21 a 21
a 31 b31 c31 a 41 b 41 c 41
and
A a11 ,a 21 ,a 31 ,a 41 , b11 ,b21 ,b31 ,b41 ,
c11 ,c21 ,c31 ,c41
0
c11
a31 b31 c31 a41 b41
c41
Said Broumi, Malayalan Lathamaheswari, Ruipu Tan, Deivanayagampillai Nagarajan, Talea Mohamed, Florentin
Smarandache and Assia Bakali, A new distance measure for trapezoidal fuzzy neutrosophic numbers based on the centroids
Neutrosophic Sets and Systems, Vol. 35, 2020
499
8-Advantages of the proposed measure
An efficient distance measure boosts the performance of task analysis or clustering. Also centroid
method is specific and location based one and acquire the best geographical location in consideration
of the distance between all the competences. Though the existing methods namely Euclidean
measure, Manhattan measure Minkowski measure and Hamming distance measure have been
applied in many real time problems they could not provide good results for the indeterminate data.
Hence in this paper, we proposed a new distance measure for trapezoidal neutrosophic fuzzy
numbers based on centroids and the significant advantages of the proposed measure are given as
follows.
(i). Trapezoidal fuzzy neutrosophic number is a simple design of arithmetic operations and easy and
perceptive interpretation as well. Therefore the proposed measure is an easy and effective one under
neutrosophic environment.
(ii). Distance measure can be estimated with simple algorithm and significant level of accuracy can
be acquired as well.
(iii). While taking the important decision of choosing the method to measure a distance it can be used
due its simplicity.
(iv). The proposed distance measure is based on centroid and hence estimation of the distance
between all objects of the data set is possible and indeterminacy also can be addressed.
(v). It is derived using Euclidean distance and hence it is very useful in correlation analysis.
(vi). Also it can be applied in location planning, operations management, Neutrosophic Statistics,
clustering, medical diagnosis, Optimization and image processing to get more accurate results
without any computational complexity.
9-Conclusion and Future Research
The concept of distance measure of trapezoidal fuzzy neutrosophic number has sufficient scope of
utilization in different studies in various domain. In this paper, we proposed a new distance measure
for the trapezoidal fuzzy neutrosophic number based on centroid with the graphical representation.
Also, the properties of the proposed measure have been derived in detail. In addition, a decision
making problem has been solved using the proposed measure as a numerical example. Further,
comparative analysis has been done with the existing methods to show the potential of the proposed
distance measure and various forms of trapezoidal fuzzy neutrosophic number have been listed and
shown the uniqueness of the proposed graphical representation. Furthermore, advantages of the
proposed measure are given. In future, the present work may be extended to other special types of
neutrosophic set like pentagonal neutrosophic set, neutrosophic rough set, interval valued
neutrosophic set and plithogeneic environments.
Said Broumi, Malayalan Lathamaheswari, Ruipu Tan, Deivanayagampillai Nagarajan, Talea Mohamed, Florentin
Smarandache and Assia Bakali, A new distance measure for trapezoidal fuzzy neutrosophic numbers based on the centroids
Neutrosophic Sets and Systems, Vol. 35, 2020
500
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Received: Apr 17, 2020.
Accepted: July 3 2020
Said Broumi, Malayalan Lathamaheswari, Ruipu Tan, Deivanayagampillai Nagarajan, Talea Mohamed, Florentin
Smarandache and Assia Bakali, A new distance measure for trapezoidal fuzzy neutrosophic numbers based on the centroids
Neutrosophic Sets and Systems, Vol. 35, 2020
University of New Mexico
Introduction to Combined Plithogenic Hypersoft Sets
Nivetha Martin* 1, Florentin Smarandache2
Department of Mathematics, Arul Anandar College (Autonomous), Karumathur, India,
Email:nivetha.martin710@gmail.com
Department of Mathematics, University of New Mexico, Gallup, NM 87301, USA, Email: smarand@unm.edu
1
2
* Correspondence: nivetha.martin710@gmail.com
Abstract: Plithogenic Hypersoft sets was introduced by Florentin Smarandache, who has extended
crisp sets, fuzzy sets, intuitionistic sets, neutrosophic sets to plithogenic sets. The plithogenic sets
considers the degree of appurtenance of the elements with respect to the attribute system.
Smarandache has presented the classification of the plithogenic hypersoft sets and the applications
of plithogenic fuzzy whole hypersoft sets in multi attribute decision making. Inspired by these
research works, the concept of combined plithogenic hypersoft sets is introduced in this article. The
representations of the degree of appurtenance of the elements determines the type of plithogenic
hypersoft set, if it takes a combination of values then the new archetype of plithogenic hypersoft
sets gets emerged into decision making scenario. This research work is put forth to project the
realistic perception of the experts in the construction process of optimal conclusions.
Keywords: Plithogenic hypersoft set, combined plithogenic hypersoft set, decision making, multi
attribute system.
1. Introduction
Classical set theory deals with the sets consisting of elements with membership values 0 or 1. The
degree of belongingness of an element in a set has been extended to [0,1] by Zadeh [1] in the name of
fuzzy sets, which is gaining momentum since its introduction. Sets comprising of quantitative
elements can be defined by conventional concepts of sets, but the elements of qualitative nature can
be treated only by fuzzy concepts and its membership value states the degree of confidence of its
presence in the set. Atanassov [2] investigated on the degree of its absence in the set, by defining
non-membership values. This paved way for the intuitionistic fuzzy sets, which consists of degree of
membership, non-membership and hesitation. Fuzzy sets and intuitionistic fuzzy sets are
extensively applied in decision making process. But still the human perception was not completely
reflected in these two kinds of sets. This gap was filled by Florentine Smarandache [3-5] who
introduced neutrosophic fuzzy sets, comprising of degree of truth membership, indeterminacy and
degree of false membership. Smarandache has represented each of the three function in a more
generalized and independent manner. Neutrosophic sets have extensive application in decision
making at recent times. Abdel- Basset et al [6-7] has developed neutrosophic decision making
models to solve transition difficulties of IoT-based enterprises and to evaluate green supply chain
management practices.
Smarandache also extended the classical sets, fuzzy sets, intuitionistic fuzzy sets and neutrosophic
fuzzy sets to plithogenic sets which is a quintuple (P, a, V, d, c) consisting of a set P, the attribute a,
Nivetha Martin and Florentin Smarandache, Introduction to Combined Plithogenic Hypersoft Sets
Neutrosophic Sets and Systems, Vol. 35, 2020
504
the range of attribute values V, degree of appurtenance d, and the degree of contradiction c. The
nature of d determines the type of plithogenic sets. Smarandache presented an elaborate discussion
on the genesis of plithogenic sets in his research work [8]. Abdel-Basset et al [9-11] has developed
decision making models with incorporation of plithogenic sets to evaluate green supply chain
management practices and intelligent Medical Decision Support Model Based on Soft Computing
and IoT was also built; a hybrid plithogenic decision-making approach with quality function
deployment for selecting supply chain sustainability metrics was also framed. These plithogenic
decision making models are highly robust and feasible.
Molodtsov [12] introduced and applied soft sets in decision making which was extended
to fuzzy soft sets predominantly by Maji [13]. The comprehensive outlook of hypersoft sets was
made by Smarandache [14] which took the different forms of fuzzy sets in the course of time. Shazia
Rana et al [15] in their recent work on application of plithogenic fuzzy whole hypersoft set in multi
attribute decision making introduced the matrix representation of plithogenic hypersoft set and
plithogenic fuzzy whole hypersoft set which adds to the compatibility of this decision making
technique. The validation of the proposed decision making model with a numerical example in this
work has inspired to introduce combined plithogenic hypersoft set.
The paper is organized as follows; section 2 presents a brief description of combined plithogenic
hypersoft sets; section 3 comprises the application of combined plithogenic hypersoft sets in decision
making based on the technique of Shazia Rana et al [15]; section 4 discusses the results and the last
section concludes with the future extension of the proposed concept.
2. Combined plithogenic hypersoft sets
This section comprises of the direct narration and representation of the combined plithogenic
hypersoft sets based on the preliminaries discussed by Smarandache [14] and Shazia Rana et al [15]
to avoid the repetition of the elementary definitions. Smarandache presented the classification of
plithogenic hypersoft sets and the categorization was purely based on the nature of degree of
appurtenance. Based on his discussion, let us consider the following example to explain the need of
combined plithogenic hypersoft sets
Let U be the universe of discourse that consists of pollution mitigation methods say
U = {M1, M2, M3, M4, M5} and the set ℳ = {M1, M4} ⊂ U.
The attributes are 𝑎1 = Cost efficiency, 𝑎2 = Eco-compatibility, 𝑎3 = Time efficacy, 𝑎4 = Profit yield. If
the pollution abatement methods are supposed to fulfill these attributes, then in realistic perspective
the possible attribute values are taken as follows,
Cost efficiency = A1 = {low, medium, high}, Eco-compatibility = A 2 = {very high, high}, Time efficacy
= A3 = {less, more}, Profit yield = A4 ={maximum, minimum}.
Suppose a manufacturing firm has decided to implement a pollution control method, then the
above attributes and its values are considered for making optimal decision with the possible range of
values of attributes. By usual consideration,
Let the function be: G: A1 × A2 × A3 × A4 ⟶ P(U)
Let’s assume: G ({low, high, more, maximum}) = {M1, M4}.
Nivetha Martin and Florentin Smarandache, Introduction to Combined Plithogenic Hypersoft Sets
Neutrosophic Sets and Systems, Vol. 35, 2020
505
The degree of appurtenance of an element x to the set ℳ, with respect to each attribute value a is
𝑑𝑥0(a) that is the deciding factor of the nature of plithogenic hypersoft set.
In the context of decision making with the expert’s opinion, then 𝑑𝑥0(a) is the resultant of the expert’s
perception. Sometimes the expert’s outlook may be a combination of certain, fuzzy, intuitionistic
and neutrosophic, which is expressed as follows
G({low, high, more, maximum}) = { M 1 (1,0.8,0.7,(0.4,0.5)),
M4 (1,0.9,(0.8,0.1,0.1),(0.5,0.6)) }.
This is the pragmatic reflection of the person’s perception in decision making process and this is the
point of origin of combined plithogenic hypersoft sets. Thus a combined plithogenic hypersoft sets is
a plithogenic hypersoft set in which the degree of appurtenance of an element x to the set ℳ, with
respect to each attribute value is a combination of either crisp, fuzzy, intuitionistic or neutrosophic.
Combined plithogenic hypersoft sets can be classified into completely combined plithogenic
hypersoft sets and partially combined plithogenic hypersoft sets based on the nature and
combination of values taken by 𝑑𝑥0(a). In the above stated example G({low, high, more, maximum}) =
{ M1 (1,0.8,0.7,(0.4,0.5)), M4 (1,0.9,(0.8,0.1,0.1),(0.5,0.6))} is a completely combined plithogenic
hypersoft sets as 𝑑𝑥0(a) takes all possible types of values. Suppose G({low, high, more, maximum}) =
{ M1 (0.9,0.8,0.7,(0.4,0.5)), M4 (0.8,0.9,0.6,(0.5,0.6))} then this combined plithogenic hypersoft set is
partial in nature as 𝑑𝑥0(a) takes only a combination of two types of values. Thus a combined
plithogenic hypersoft set which is not complete is partial in its nature.
It is very evident that combined plithogenic hypersoft sets are highly rational in nature
and it will certainly play a vital role in receiving the expert’s opinion, which is very significant in any
multi attribute decision making process. Also the need of such new types of plithogenic hypersoft
sets are very essential, because in the manufacturing firms and in business sectors the
implementation of certain methods and installation of certain mechanisms and machinery may not
be characterized by only crisp or fuzzy values with regard to the degree of appurtenance as the
possibility aspect has some extent of participation in it. To handle such situations the combined
plithogenic hypersoft sets may lend a helping hand to the decision makers.
3. Application of Combined Plithogenic Hypersoft set in Multi Attribute Decision Making
The previous section presented an elaborate portrayal of combined plithogenic hypersoft set,
the significant feature is the realistic representation, but it has certain difficulties in computations as
the degree of appurtenance varies for each attribute. To handle such crisis, all the values of 𝑑𝑥0(a)
must be similar in nature, i.e. either all the values must be fuzzy values which is the lower level of
fuzzy representation or it must be neutrosophic values, the higher level of fuzzy representation.
A manufacturing sector has decided to enhance its production rate by installing new kinds
of machinery. The ultimate aim of incorporating such a change in the production mechanism is
quality production and customer satisfaction. The market is flooded with several varieties of well
equipped, modern machines and since the manufacturing sector makes huge investment, the
decision making process takes place in two phases based on the expert’s opinion and advice. In the
first phase, ten machines were selected by the manufacturing sector and in the next phase five were
selected based on the feedback of the users. The decision making problem begins here, as the
company has to purchase only three out of five based on the extent of satisfaction of the attributes by
these machines.
Nivetha Martin and Florentin Smarandache, Introduction to Combined Plithogenic Hypersoft Sets
Neutrosophic Sets and Systems, Vol. 35, 2020
506
Let U = { M1, M2, M3, M4, M5, M6, M7,M8, M9, M10} be the university of discourse and set
T = {M1, M3, M6, M7, M9} ⊂ U.
The attribute system is represented as follows A = { (A 1)Maintenance Cost {Maximum in the initial
years of utility(A11), Maximum in the latter years of utility(A 12)}, (A2)Reliability {High with
additional expenditure(A21), Moderate with no extra expense(A22)}, (A3)Flexibility {Single task
oriented(A31), Multi task oriented(A32)}, (A4)Durability {Very high in the beginning years of
service(A41), High in the latter years of service(A42), }, (A5)Profitability {Moderate in the initial
years(A51), Maximum in the latter years(A52)}}.
The attributes are quite common, but the attribute values are more realistic as it mirror the actual
aspects involved in making decision.
Let the function be: G: A11 × A22 × A32 × A41 × A52 ⟶P(U). Based on the Expert’s opinion, the degree
of appurtenance of the elements with respect to the attribute values is represented as follows
G( A11, A22 , A32 , A41, A52) =
{M1(0.9,(0.7,0.1),0.8,(0.6,0.2),0.5),M3((0.6,0.3),0.5,(0.4,0.1,0.3),0.8,0.7),
M6(0.8,0.7,0.6,(0.5,0.2),(0.6,0.1,0.1)),M7((0.7,0.2,0.1),(0.7,0.1),0.9,(0.7,0.2),0.8),M9(1,0.9,0.5,0.8,(0.6,0.1,0.
2))}.
The modified lower and higher fuzzy values of the degree of appurtenance of the elements with
respect to the attribute values are denoted as GL(A11, A22 , A32 , A41, A52) and GH(A11, A22 , A32 , A41,
A52)
GL(A11, A22 , A32 , A41, A52) = {M1(0.9,0.875,0.8,0.75,0.5),M3(0.67,0.5,0.4,0.8,0.7),M6(0.8,0.7,0.6,0.7,0.5),
M7(0.67,0.875,0.9,0.78,0.8), M9(1,0.9,0.5,0.8,0.47)}
GH(A11, A22 , A32 , A41, A52) =
{M1(0.9,0.1,0.1),(0.7,0.2,0.1),(0.8,0.1,0.1),(0.6,0.3,0.2),(0.5,0.2,0.7)),M3((0.6,0.3,0.3),
(0.5,0.2,0.7),(0.4,0.1,0.3),(0.8,0.1,0.1),(0.7,0.2,0.1)),M6((0.8,0.1,0.1),(0.7,0.2,0.1),(0.6,0.2,0.3),(0.5,0.3,0.2),(
0.6,0.1,0.1)),M7((0.7,0.2,0.1),(0.7,0.1,0.1),(0.9,0.1,0.1),(0.7,0.1,0.2),(0.8,0.1,0.1)),M9((1,0,0),(0.9,0.1,0.1),(0.
5,0.2,0.7),(0.8,0.1,0.1),(0.6,0.1,0.2))}
The lower and higher fuzzy values of the degree of appurtenance correspond to single fuzzy value
and neutrosophic values. The matrix representation C of the degree of appurtenance of the elements
with respect to the attribute values in combined plithogenic hypersoft sets is
A11
A22
A32
A41
A52
M1
0.9
(0.7,0.1)
0.8
(0.6,0.2)
0.5
M3
(0.6,0.3)
0.5
(0.4,0.1,0.3)
0.8
0.7
M6
0.8
0.7
0.6
(0.5,0.2)
(0.6,0.1,0.1)),
M7
(0.7,0.2,0.1)
(0.7,0.1)
0.9
(0.7,0.2)
0.8
M9
1
0.9
0.5
0.8
(0.6,0.1,0.2)
Nivetha Martin and Florentin Smarandache, Introduction to Combined Plithogenic Hypersoft Sets
Neutrosophic Sets and Systems, Vol. 35, 2020
507
The intuitionistic and neutrosophic values are transformed to the above fuzzy values by the
methods of imprecision and Defuzzification [16]
Method I (Imprecision membership): Any neutrosophic fuzzy set NA = (
,
) including
neutrosophic fuzzy values are transformed into intuitionistic fuzzy values or vague values as (A)
= (
,
) where
is estimated the formula stated below which is called as Impression
membership method.
=
Method II (Defuzzification): After Method I (Median membership), intuitionistic (vague),fuzzy
values of the form (A)= (
,
) are transformed into fuzzy set including fuzzy values
>.
as <Δ(A)>= <
The matrix representation CL of the lower fuzzy values of the degree of appurtenance of the
elements with respect to the attribute values in combined plithogenic hypersoft sets is
A11
A22
A32
A41
A52
M1
0.9
0.875
0.8
0.75
0.5
M3
0.67
0.5
0.4
0.8
0.7
M6
0.8
0.7
0.6
0.7
0.5
M7
0.67
0.875
0.9
0.78
0.8
M9
1
0.9
0.5
0.8
0.47
By using the procedure of ranking as discussed by Shazia Rana et. al [15] the machines are ranked by
considering the values of CL.
The frequency matrix FL representing the ranking of the machines is
R1
R2
R3
R4
R5
M1
1
2
0
0
0
M3
0
0
0
1
2
M6
0
1
0
2
0
M7
2
0
1
0
0
M9
1
1
1
0
0
Nivetha Martin and Florentin Smarandache, Introduction to Combined Plithogenic Hypersoft Sets
Neutrosophic Sets and Systems, Vol. 35, 2020
508
The percentage measure of authenticity of ranking is presented below in Table 3.1
Table 3.1
R1
M7
50%
R2
M1
50%
R3
M9
50%
R4
M6
67%
R5
M3
100%
The matrix representation CH of higher fuzzy values (neutrosophic representations) of the degree of
appurtenance of the elements with respect to the attribute values in combined plithogenic hypersoft
sets is
A11
A22
A32
A41
A52
M1
(0.9,0.1,0.1)
(0.7,0.2,0.1)
(0.8,0.1,0.1)
(0.6,0.3,0.2)
(0.5,0.2,0.7)
M3
(0.6,0.3,0.3)
(0.5,0.2,0.7)
(0.4,0.1,0.3)
(0.8,0.1,0.1)
(0.7,0.2,0.1)
M6
(0.8,0.1,0.1)
(0.7,0.2,0.1)
(0.6,0.2,0.3)
(0.5,0.3,0.2)
(0.6,0.1,0.1)
M7
(0.7,0.2,0.1)
(0.7,0.1,0.1)
(0.9,0.1,0.1)
(0.7,0.1,0.2)
(0.8,0.1,0.1)
M9
(1,0,0)
(0.9,0.1,0.1)
(0.5,0.2,0.7)
(0.8,0.1,0.1)
(0.6,0.1,0.2)
To make the ranking of the machines based on the higher values in C H the score
values K(A) of the single valued neutrosophic representations [say A = (a,b,c)] are determined by
K(A) =
[17]
A11
A22
A 32
A41
A52
M1
0.8
0.6
0.75
0.4
0.2
M3
0.35
0.2
0.45
0.75
0.6
M6
0.75
0.6
0.45
0.35
0.65
M7
0.6
0.7
0.8
0.65
0.75
M9
1
0.8
0.2
0.75
0.6
The frequency matrix FH representing the ranking of machines is
R1
R2
R3
R4
R5
M1
1
0
1
1
0
M3
0
0
1
1
1
M6
0
1
1
1
0
M7
3
0
0
0
0
M9
1
1
1
0
0
Nivetha Martin and Florentin Smarandache, Introduction to Combined Plithogenic Hypersoft Sets
Neutrosophic Sets and Systems, Vol. 35, 2020
509
The percentage measure of authenticity of ranking is presented below in Table 3.2
Table 3.2
R1
M7
60%
R2
M9
50%
R3
M6
25%
R4
M1
33%
R5
M3
100%
4. Discussion
The combined plithogenic hypersoft set representations are so deliberate in nature. The resultant
of computations in making decisions in two ways is represented in Table 3.1 and 3.2. The machines
M7 and M3 occupy first and fifth rank respectively in both kinds of representation of degree of
appurtenance. Also by making inferences from the table values M 1, M3 and M6 can be ranked in
second ,third and fourth positions respectively. It is very evident that the transformation of
combined attribute values to lower order fuzzy values yields best results in ranking the machines,
but still the higher order values will also yield optimum results based on the selection of the score
functions. The methods of converting combined attribute value to the values of similar fashion have
to be constituted in the upcoming research works to attain feasible solutions to the decision making
problems.
5. Conclusions
This research work encompasses the discussion of the new concept of combined plithogenic
hypersoft set and its application in multi attribute decision making. Besides these types of
appurtenance degrees, others can be used under the plithogenic umbrella, such as: Pythagorean,
picture fuzzy, spherical fuzzy, spherical neutrosophic, etc. and even the most general one, refined
neutrosophic type of appurtenance degree. The combined plithogenic hypersoft set can be extended
to interval-valued combined plithogenic hypersoft sets and it can be converted to simple fuzzy
values using score functions. The matrix representations of degree of appurtenance in combined
plithogenic hypersoft set has induced the author to extend the proposed theoretical
conceptualization to plithogenic concentric hypergraphs, most probably the scope and future
research work.
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Received: Apr 22, 2020.
Accepted: July 2 2020
Nivetha Martin and Florentin Smarandache, Introduction to Combined Plithogenic Hypersoft Sets
Neutrosophic Sets and Systems, Vol. 35, 2020
University of New Mexico
On Neutrosophic Generalized Alpha Generalized Continuity
Qays Hatem Imran1*, R. Dhavaseelan2, Ali Hussein Mahmood Al-Obaidi3, and Md. Hanif PAGE4
1Department
of Mathematics, College of Education for Pure Science, Al-Muthanna University, Samawah, Iraq.
E-mail: qays.imran@mu.edu.iq
2Department of Mathematics, Sona College of Technology, Salem-636005, Tamil Nadu, India.
E-mail: dhavaseelan.r@gmail.com
3Department of Mathematics, College of Education for Pure Science, University of Babylon, Hillah, Iraq.
E-mail: aalobaidi@uobabylon.edu.iq
4Department of Mathematics, KLE Technological University, Hubballi-580031, Karnataka, India.
E-mail: mb_page@kletech.ac.in
* Correspondence: qays.imran@mu.edu.iq
Abstract: This article demonstrates a further class of neutrosophic closed sets named neutrosophic
generalized αg-closed sets and discuss their essential characteristics in neutrosophic topological
spaces. Moreover, we submit neutrosophic generalized αg-continuous functions with their elegant
features.
Keywords: neutrosophic generalized α g-closed sets, neutrosophic generalized α g-continuous
functions, and neutrosophic generalized αg-irresolute functions.
1. Introduction
Smarandache [1,2] originally handed the theory of “neutrosophic set”. Recently, Abdel-Basset
et al. discussed a novel neutrosophic approach [3-8] in several fields, for a few names, information
and communication technology. Salama et al. [9] gave the clue of neutrosophic topological space (or
simply 𝑁𝑇𝑆). Arokiarani et al. [10] added the view of neutrosophic α-open subsets of neutrosophic
topological spaces. Imran et al. [11] presented the idea of neutrosophic semi--open sets in neutrosophic
topological spaces. Dhavaseelan et al. [12] presented the idea of neutrosophic α𝑚 -continuity. Our aim
is to introduce a new idea of neutrosophic generalized αg-closed sets and examine their vital merits
in neutrosophic topological spaces. Additionally, we propose neutrosophic generalized
αg-continuous functions by employing neutrosophic generalized αg-closed sets and emphasizing
some of their primary characteristics.
2. Preliminaries
Everywhere of these following sections, we assume that 𝑁𝑇𝑆s (𝒰, 𝜉), (𝒱, 𝜚) and (𝒲, 𝜇) are
briefly denoted as 𝒰, 𝒱, and 𝒲, respectively. Let 𝒞 be a neutrosophic set in 𝒰, and we are easily
symbolized it by 𝑁𝑆, then the complement of 𝒞 is basically given by 𝒞̅ . If 𝒞 is a neutrosophic
open set in 𝒰 and shortly indicated by Ne-OS. Then, 𝒞̅ is termed a neutrosophic closed set in 𝒰
and simply referred by Ne-CS. The neutrosophic closure and the neutrosophic interior of 𝒞 are
merely signified by Ne-𝑐𝑙(𝒞) and Ne-𝑖𝑛𝑡(𝒞), correspondingly.
Definition 2.1 [10]: A 𝑁𝑆 𝒞 in a 𝑁𝑇𝑆 𝒰 is named a neutrosophic α-open set and simply written as
Ne-αOS if 𝒞 ⊆Ne-𝑖𝑛𝑡(Ne-𝑐𝑙(Ne-𝑖𝑛𝑡(𝒞))). Besides, if Ne-𝑐𝑙(Ne-𝑖𝑛𝑡(Ne-𝑐𝑙(𝒞))) ⊆ 𝒞, then 𝒞 is called a
neutrosophic α-closed set, and we are shortly given it as Ne-αCS. The collection of all such these
Qays Hatem Imran, R. Dhavaseelan, Ali Hussein Mahmood and Md. Hanif PAGE, On Neutrosophic Generalized Alpha
Generalized Continuity
Neutrosophic Sets and Systems, Vol. 35, 2020
512
Ne-αOSs (correspondently, Ne-αCSs) in 𝒰 is referred to Ne-αO(𝒰) (correspondently, Ne-αC(𝒰)).
The intersection of all Ne-αCSs that contain 𝒞 is called the neutrosophic α-closure of 𝒞 in 𝒰 and
represented by Ne-α𝑐𝑙(𝒞).
Definition 2.2 [13]: A 𝑁𝑆 𝒞 in 𝑁𝑇𝑆 𝒰 is so-called a neutrosophic generalized closed set and
denoted by Ne-gCS if for any Ne-OS ℳ in 𝒰 such that 𝒞 ⊆ ℳ, then Ne-𝑐𝑙(𝒞) ⊆ ℳ. Moreover, its
complement is named a neutrosophic generalized open set and referred to Ne-gOS.
Definition 2.3 [14]: A 𝑁𝑆 𝒞 in 𝑁𝑇𝑆 𝒰 is so-called a neutrosophic αg-closed set and indicated by
Ne-αgCS if for any Ne-OS ℳ in 𝒰 such that 𝒞 ⊆ ℳ , then Ne-α𝑐𝑙(𝒞) ⊆ ℳ . Furthermore, its
complement is named a neutrosophic αg-open set and symbolized by Ne-αgOS.
Definition 2.4 [15]: A 𝑁𝑆 𝒞 in 𝑁𝑇𝑆 𝒰 is so-called a neutrosophic gα-closed set and signified by
Ne-g α CS if far any Ne- α OS ℳ in 𝒰 such that 𝒞 ⊆ ℳ , then Ne- α𝑐𝑙(𝒞) ⊆ ℳ . Besides, its
complement is named a neutrosophic gα-open set and briefly written as Ne-gαOS.
Theorem 2.5 [10,13-15]: For any 𝑁𝑇𝑆 𝒰, the next declarations valid and but not vice versa:
(i) for all Ne-OSs (correspondingly, Ne-CSs) are Ne-αOSs (correspondingly, Ne-αCSs).
(ii) for all Ne-OSs (correspondingly, Ne-CSs) are Ne-gOSs (correspondingly, Ne-gCSs).
(iii) for all Ne-gOSs (correspondingly, Ne-gCSs) are Ne-αgOSs (correspondingly, Ne-αgCSs).
(iv) for all Ne-αOS (correspondingly, Ne-αCSs) are Ne-gαOSs (correspondingly, Ne-gαCSs).
(v) for all Ne-gαOSs (correspondingly, Ne-gαCSs) are Ne-αgOSs (correspondingly, Ne-αgCSs).
Definition 2.6: Let (𝒰, 𝜉) and (𝒱, 𝜚) be NTSs and 𝜂: (𝒰, 𝜉) ⟶ (𝒱, 𝜚) be a mapping, we have
(i) if for each Ne-OS (correspondingly, Ne-CS) 𝒦 in 𝒱 , 𝜂 −1 (𝒦) is a Ne-OS (correspondingly,
Ne-CS) in 𝒰, then 𝜂 is known as neutrosophic continuous and denoted by Ne-continuous. [16]
(ii) if for each Ne-OS (correspondingly, Ne-CS) 𝒦 in 𝒱, 𝜂 −1 (𝒦) is a Ne-αOS (correspondingly,
Ne-αCS) in 𝒰, then 𝜂 is known as neutrosophic α-continuous and referred to Ne-α-continuous. [10]
(iii) if for each Ne-OS (correspondingly, Ne-CS) 𝒦 in 𝒱, 𝜂−1 (𝒦) is a Ne-gOS (correspondingly,
Ne-gCS) in 𝒰, then 𝜂is known as neutrosophic g-continuous and signified by Ne-g-continuous. [17]
Remark 2.7 [17,10]: Let 𝜂: (𝒰, 𝜉) ⟶ (𝒱, 𝜚) be a map, the next declarations valid and but not vice
versa:
(i) For all Ne-continuous functions are Ne-α-continuous.
(ii) For all Ne-continuous functions are Ne-g-continuous.
3. Neutrosophic Generalized 𝛂g-Closed Sets
Qays Hatem Imran, R. Dhavaseelan, Ali Hussein Mahmood and Md. Hanif PAGE, On Neutrosophic Generalized Alpha
Generalized Continuity
Neutrosophic Sets and Systems, Vol. 35, 2020
513
The neutrosophic generalized 𝛂g-closed sets and their features are studied and discussed in
this part of the paper.
Definition 3.1: Let 𝒞 be a 𝑁𝑆 in 𝑁𝑇𝑆 𝒰, then it called a neutrosophic generalized αg-closed set
and denoted by Ne-gαgCS if for any Ne-αgOS ℳ in 𝒰 such that 𝒞 ⊆ ℳ, then Ne-𝑐𝑙(𝒞) ⊆ ℳ.We
indicated the collection of all Ne-gαgCSs in 𝑁𝑇𝑆 𝒰 by Ne-gαgC(𝒰).
Definition 3.2: Let 𝒞 be a 𝑁𝑆 in 𝑇𝑆 𝒰, then its neutrosophic gαg-closure is the intersection of each
Ne-gαgCS in 𝒰 including 𝒞, and we are shortly written it as Ne-gαg𝑐𝑙(𝒞) . In other words,
Ne-gαg𝑐𝑙(𝒞) = ⋂{𝒟: 𝒞 ⊆ 𝒟, 𝒟 is a Ne-gαgCS}.
Theorem 3.3: The subsequent declarations are valid in any 𝑇𝑆 𝒰:
(i) for all Ne-CSs are Ne-gαgCSs.
(ii) for all Ne-gαgCSs are Ne-gCSs.
(iii) for all Ne-gαgCSs are Ne-αgCSs.
(iv) for all Ne-gαgCSs are Ne-gαCSs.
Proof:
(i) Suppose a Ne-CS 𝒞 is in 𝑇𝑆 𝒰. For any Ne-αgOS ℳ, including 𝒞, we have ℳ ⊇ 𝒞 =Ne-𝑐𝑙(𝒞).
Therefore, 𝒞 stands a Ne-gαgCS.
(ii) Suppose Ne-gαgCS 𝒞 is in 𝑇𝑆 𝒰. For any Ne-OS ℳ, including 𝒞, we have theorem (2.5) states
that ℳ stands a Ne-αgOS in 𝒰. Because a Ne-gαgCS 𝒞 satisfying this fact Ne-𝑐𝑙(𝒞) ⊆ ℳ. As a
result, 𝒞 considers a Ne-gCS.
(iii) Assume Ne-gαgCS 𝒞 is in 𝑇𝑆 𝒰. For any Ne-OS ℳ, including 𝒞, we have theorem (2.5) states
that ℳ remains a Ne- α gOS in 𝒰 . Subsequently, Ne-g α gCS 𝒞 satisfying this statement
Ne-α𝑐𝑙(𝒞) ⊆Ne-𝑐𝑙(𝒞) ⊆ ℳ. Therefore, 𝒞 becomes a Ne-αgCS.
(iv) Assume Ne-gαgCS 𝒞 is in 𝑇𝑆 𝒰. For any Ne-αOS ℳ including 𝒞, we have theorem (2.5) states
that ℳ remains a Ne- α gOS in 𝒰 . Subsequently, Ne-g α gCS 𝒞 satisfying this statement
Ne-α𝑐𝑙(𝒞) ⊆Ne-𝑐𝑙(𝒞) ⊆ ℳ. Therefore, 𝒞 considers a Ne-gαCS.
The opposite conditions for this previous theorem do not look accurate by the next obvious
examples.
Example 3.4: Suppose 𝒰 = {𝑝, 𝑞} and let 𝜉 = {0𝑁 , 𝒜, ℬ, 1𝑁 } , such that we have the sets 𝒜 =
〈𝓊, (0.6,0.7), (0.1,0.1), (0.4,0.2)〉 and ℬ = 〈𝓊, (0.1,0.2), (0.1,0.1), (0.8,0.8)〉, so that (𝒰, 𝜉) is a 𝑁𝑇𝑆 .
However, the 𝑁𝑆 𝒞 = 〈𝓊, (0.2,0.2), (0.1,0.1), (0.6,0.7)〉 is a Ne-gαgCS but not a Ne-CS.
Example 3.5: Suppose 𝒰 = {𝑝, 𝑞, 𝑟} and let 𝜉 = {0𝑁 , 𝒜, ℬ, 1𝑁 }, where such that we have the sets
𝒜 = 〈𝓊, (0.5,0.5,0.4), (0.7,0.5,0.5), (0.4,0.5,0.5)〉 and ℬ = 〈𝓊, (0.3,0.4,0.4), (0.4,0.5,0.5), (0.3,0.4,0.6)〉 ,
so that (𝒰, 𝜉) is a 𝑁𝑇𝑆 . However, the 𝑁𝑆 𝒞 = 〈𝓊, (0.4,0.6,0.5), (0.4,0.3,0.5), (0.5,0.6,0.4)〉 is a
Ne-gCS but not a Ne-gαgCS.
Example 3.6: Suppose 𝒰 = {𝑝, 𝑞} and let 𝜉 = {0𝑁 , 𝒜, ℬ, 1𝑁 }, where such that we have the sets 𝒜 =
〈𝓊, (0.5,0.6), (0.3,0.2), (0.4,0.1)〉 and ℬ = 〈𝓊, (0.4,0.4), (0.4,0.3), (0.5,0.4)〉 , so that (𝒰, 𝜉) is a 𝑁𝑇𝑆 .
Qays Hatem Imran, R. Dhavaseelan, Ali Hussein Mahmood and Md. Hanif PAGE, On Neutrosophic Generalized Alpha
Generalized Continuity
Neutrosophic Sets and Systems, Vol. 35, 2020
514
However, the 𝑁𝑆 𝒞 = 〈𝓊, (0.5,0.4), (0.4,0.4), (0.4,0.5)〉 is a Ne-αgCS and hence Ne-gαCS but not a
Ne-gαgCS.
Definition 3.7: Let 𝒞 be any 𝑁𝑆 in 𝑇𝑆 𝒰, then it is called a neutrosophic generalized αg-open set
and referred to by Ne-gαgOS iff the set 𝒰 − 𝒞 is a Ne-gαgCS. The collection of the whole Ne-gαgOSs
in 𝑁𝑇𝑆 𝒰 indicated by Ne-gαgO(𝒰).
Definition 3.8: The union of the whole Ne-g α gOSs in 𝑁𝑇𝑆 𝒰 included in 𝑁𝑆 𝒞 is termed
neutrosophic gαg-interior of 𝒞 and symbolized by Ne-gαg𝑖𝑛𝑡(𝒞). In symbolic form, we have this
thing Ne-gαg𝑖𝑛𝑡(𝒞) = ⋃{𝒟: 𝒞 ⊇ 𝒟, 𝒟 is a Ne-gαgOS}.
Proposition 3.9: For any 𝑁𝑆 ℳ in 𝑇𝑆 𝒰, the subsequent features stand:
(i) Ne-gαg𝑖𝑛𝑡(ℳ) = ℳ iff ℳ is a Ne-gαgOS.
(ii) Ne-gαg𝑐𝑙(ℳ) = ℳ iff ℳ is a Ne-gαgCS.
(iii) Ne-gαg𝑖𝑛𝑡(ℳ ) is the biggest Ne-gαgOS included in ℳ.
(iv) Ne-gαg𝑐𝑙(ℳ) is the littlest Ne-gαgCS, including ℳ.
Proof: the features (i-iv) are understandable.
Proposition 3.10: For any 𝑁𝑆 ℳ in 𝑇𝑆 𝒰, the subsequent features stand:
̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
̅ ) = (Ne
− gαg𝑐𝑙(ℳ )),
(i) Ne-gαg𝑖𝑛𝑡(ℳ
̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
̅ ) = (Ne
− gαg𝑖𝑛𝑡(ℳ )).
(ii) Ne-gαg𝑐𝑙( ℳ
Proof:
(i) The proof will be evident by symbolic definition, Ne-gαg𝑐𝑙(ℳ) = ⋂{𝒟: ℳ ⊆ 𝒟, 𝒟 is a Ne-gαgCS}
̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
̅ ⊆𝐷
̅: ℳ
̅, 𝐷
̅ is a Ne-g𝛼gCS}
(Ne − gαg𝑐𝑙(ℳ)) = ⋂{𝐷
̅ ⊆𝒟
̅: ℳ
̅, 𝒟
̅ is a Ne-gαgCS}
= ⋃{𝒟
= ⋃{𝒩: ℳ ⊇ 𝒩, 𝒩is a Ne-gαgOS}
̅ ).
= Ne-gαg𝑖𝑛𝑡(ℳ
(ii) This feature has undeniable proof analogous to feature (i).
Theorem 3.11: For any Ne-OS 𝒞 in 𝑇𝑆 𝒰, then this set is a Ne-gαgOS.
Proof: Suppose Ne-OS 𝒞 in 𝑇𝑆 𝒰, so we obtain that 𝒞̅ is a Ne-CS. Therefore, 𝒞̅ is a Ne-gαgCS via
the previous theorem (3.3), part (i). Consequently, 𝒞 is a Ne-gαgOS.
Theorem 3.12: For any Ne-gαgOS 𝒞 in 𝑇𝑆 𝒰, then this set is a Ne-gOS.
Proof: Suppose Ne-gαgOS 𝒞 in 𝑇𝑆 𝒰, so we obtain that 𝒞̅ is a Ne-gαgCS. Therefore, 𝒞̅ is a Ne-gCS
via the previous theorem (3.3), part (ii). Consequently, 𝒞 is a Ne-gOS.
Lemma 3.13: For any Ne-gαgOS 𝒞 in 𝑇𝑆 𝒰, then this set is Ne-αgOS (correspondingly, Ne-gαOS).
Proof: The proof of this lemma is similar to one of the previous theorem.
Proposition 3.14: For any two Ne-gαgCSs 𝒞 and 𝒟 in 𝑇𝑆 𝒰, then the set 𝒞⋃𝒟 is a Ne-gαgCS.
Qays Hatem Imran, R. Dhavaseelan, Ali Hussein Mahmood and Md. Hanif PAGE, On Neutrosophic Generalized Alpha
Generalized Continuity
Neutrosophic Sets and Systems, Vol. 35, 2020
515
Proof: Suppose any two Ne-gαgCSs 𝒞 and 𝒟 in 𝑁𝑇𝑆 𝒰 and ℳ is a Ne-αgOS, including 𝒞 and
𝒟. In other words, we have 𝒞⋃𝒟 ⊆ ℳ. So, 𝒞, 𝒟 ⊆ ℳ. Because 𝒞 and 𝒟 are Ne-gαgCSs, we get that
Ne-𝑐𝑙(𝒞), Ne-𝑐𝑙(𝒟) ⊆ ℳ. Therefore, Ne-𝑐𝑙(𝒞⋃𝒟) = Ne-𝑐𝑙(𝒞)⋃Ne-𝑐𝑙(𝒟) ⊆ ℳ. Then Ne-𝑐𝑙(𝒞⋃𝒟) ⊆
ℳ. Thus, 𝒞⋃𝒟 stands a Ne-gαgCS.
Proposition 3.15: For any two Ne-gαgOSs 𝒞 and 𝒟 in 𝑇𝑆 𝒰, then the set 𝒞⋂𝒟 is a Ne-gαgOS.
̅ are
Proof: Suppose any two Ne-gαgOSs 𝒞 and 𝒟 in 𝑇𝑆 𝒰. Subsequently, we have that 𝒞̅ and 𝒟
̅ is a Ne-gαgCS by proposition (3.14). Because 𝒞̅ ⋃𝒟
̅=
Ne-gαgCSs. So, we reach to this fact 𝒞̅ ⋃𝒟
̅̅̅̅̅̅̅̅̅, we obtain to our final result 𝒞⋂𝒟 is a Ne-gαgOS.
(𝒞⋂𝒟)
Proposition 3.16: Let Ne-gαgCS 𝒞 be in 𝑇𝑆 𝒰, then Ne-𝑐𝑙(𝒞) − 𝒞 does not include non-empty
Ne-CS in 𝒰.
Proof: Assume we have Ne-gαgCS 𝒞 and Ne-CS ℱ in 𝑁𝑇𝑆 𝒰 so as ℱ ⊆ Ne-𝑐𝑙(𝒞) − 𝒞. Because 𝒞
̅̅̅̅̅̅̅̅̅̅̅̅̅̅
stands a Ne-g α gCS, this gives us the fact Ne- 𝑐𝑙(𝒞) ⊆ ℱ̅ . The latter means ℱ ⊆ Ne
− 𝑐𝑙(𝒞) .
̅̅̅̅̅̅̅̅̅̅̅̅̅̅
Subsequently, we arrive to ℱ ⊆ Ne-𝑐𝑙(𝒞)⋂(Ne − 𝑐𝑙(𝒞)) = 0𝑁 . Therefore, ℱ = 0𝑁 and so, we reach
to our final result Ne-𝑐𝑙(𝒞) − 𝒞 does not include non-empty Ne-CS.
Proposition 3.17: Let Ne-gαgCS 𝒞 be in 𝑁𝑇𝑆 𝒰 iff Ne-𝑐𝑙(𝒞) − 𝒞 does not include non-empty
Ne-αgCS in 𝒰.
Proof: Assume we have Ne-gαgCS 𝒞 and Ne-αgCS 𝒢 in 𝑁𝑇𝑆 𝒰 so as 𝒢 ⊆ Ne-𝑐𝑙(𝒞) − 𝒞. Because
̅̅̅̅̅̅̅̅̅̅̅̅̅̅
𝒞 considers a Ne-gαgCS, this gives us the fact Ne-𝑐𝑙(𝒞) ⊆ 𝒢̅ . The latter means 𝒢 ⊆ Ne
− 𝑐𝑙(𝒞) .
̅̅̅̅̅̅̅̅̅̅̅̅̅̅
Subsequently, we arrive to 𝒢 ⊆ Ne-𝑐𝑙(𝒞)⋂(Ne − 𝑐𝑙(𝒞)) = 0𝑁 . Therefore, 𝒢 is empty.
On The Other Hand, let us assume that Ne-𝑐𝑙(𝒞) − 𝒞 does not include non-empty Ne-αgCS in 𝒰.
Suppose ℳ is Ne-αgOS so as 𝒞 ⊆ ℳ. If we have this truth Ne-𝑐𝑙(𝒞) ⊆ ℳ but then we get this fact
̅ ) is non-empty. Meanwhile, we know that Ne-𝑐𝑙(𝒞) is Ne-CS and at the same time,
Ne-𝑐𝑙(𝒞)⋂( ℳ
̅ is Ne-αgCS, so Ne-𝑐𝑙(𝒞)⋂(ℳ
̅ ) is non-empty Ne-αgCS included Ne-𝑐𝑙(𝒞) − 𝒞. This
we have ℳ
leads us to a contradiction. Consequently Ne-𝑐𝑙(𝒞) ⊈ ℳ. Therefore, 𝒞 considers a Ne-gαgCS.
Theorem 3.18: Let Ne-αgOS and Ne-gαgCS 𝒞 be in 𝑇𝑆 𝒰, then 𝒞 considers a Ne-CS in 𝒰.
Proof: Assume we have Ne-αgOS and Ne-gαgCS 𝒞 is in 𝑇𝑆 𝒰, so we get that Ne-𝑐𝑙(𝒞) ⊆ 𝒞 and
subsequently, we reach to 𝒞 ⊆ Ne-𝑐𝑙(𝒞). Consequently, Ne-𝑐𝑙(𝒞) = 𝒞. Therefore, 𝒞 stands a Ne-CS.
Theorem 3.19: Let Ne-gαgCS 𝒞 be in 𝑁𝑇𝑆 𝒰 so as 𝒞 ⊆ 𝒟 ⊆ Ne-cl(𝒞), but then again 𝒟 considers a
Ne-gαgCS in 𝒰.
Proof: Assume we have Ne-gαgCS 𝒞 and Ne-αgOS ℳ are in 𝑁𝑇𝑆 𝒰 so as 𝒟 ⊆ ℳ. Later, 𝒞 ⊆ ℳ.
Subsequently, 𝒞 stands a Ne-gαgCS; this fact pursues Ne-𝑐𝑙(𝒞) ⊆ ℳ . So, 𝒟 ⊆ Ne-𝑐𝑙(𝒞) infers
Ne- 𝑐𝑙(𝒟) ⊆ Ne- 𝑐𝑙( Ne- 𝑐𝑙(𝒞)) = Ne- 𝑐𝑙(𝒞) . Consequently, Ne- 𝑐𝑙(𝒟) ⊆ ℳ . Therefore, 𝒟 exists a
Ne-gαgCS.
Theorem 3.20: Let Ne-gαgOS 𝒞 be in 𝑁𝑇𝑆 𝒰 so as Ne-𝑖𝑛𝑡(𝒞) ⊆ 𝒟 ⊆ 𝒞, but then again 𝒟 considers
a Ne-gαgOS in 𝒰.
Qays Hatem Imran, R. Dhavaseelan, Ali Hussein Mahmood and Md. Hanif PAGE, On Neutrosophic Generalized Alpha
Generalized Continuity
Neutrosophic Sets and Systems, Vol. 35, 2020
516
Proof: Assume we have Ne-gαgOS 𝒞 is in 𝑁𝑇𝑆 𝒰 so as Ne-𝑖𝑛𝑡(𝒞) ⊆ 𝒟 ⊆ 𝒞 . After that, 𝒰 −
̅ ⊆ Ne-𝑐𝑙( 𝒞̅ ). But then again, we depend on theorem (3.19) to
𝒞 stands a Ne-gαgCS as well as 𝒞̅ ⊆ 𝒟
get 𝒰 − 𝒟 is a Ne-gαgCS. Therefore, 𝒟 exists a Ne-gαgOS.
Theorem 3.21: A 𝑁𝑆 𝒞 is Ne-gαgOS iff 𝒫 ⊆ Ne-𝑖𝑛𝑡(𝒞) so as 𝒫 ⊆ 𝒞 and 𝒫 considers a Ne-gαgCS.
Proof: Assume we have that Ne-gαgCS 𝒫 satisfying 𝒫 ⊆ 𝒞 and 𝒫 ⊆ Ne-𝑖𝑛𝑡(𝒞). Afterward, ̅𝒞 ⊆
Ne − 𝑖𝑛𝑡(𝒞) ⊆
𝒫̅ and we have by lemma (3.13), 𝒫̅ remains a Ne-αgOS. Accordingly, Ne-𝑐𝑙( ̅𝒞 ) = ̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅
𝒫̅ . Subsequently, 𝒞̅ stands a Ne-gαgCS. Therefore, 𝒞 stands a Ne-gαgOS.
On the contrary, we assume Ne-gαgOS 𝒞 and Ne-gαgCS 𝒫 is so as 𝒫 ⊆ 𝒞. Subsequently, 𝒞̅ ⊆ 𝒫̅ .
While 𝒞̅ exists a Ne-gαgCS and 𝒫̅ remains a Ne-αgOS, we reach to that Ne-𝑐𝑙( ̅𝒞 ) ⊆ 𝒫̅. Therefore,
𝒫 ⊆ Ne-𝑖𝑛𝑡(𝒞).
Remark 3.22: The next illustration demonstrates the relative among the distinct kinds of Ne-CS:
Ne-αCS
Ne-gαCS
Ne-αgCS
Ne-CS
Ne-gαgCS
Ne-gCS
Ne-αgOS
+
Fig. 3.1
4. Neutrosophic Generalized 𝛂g-Continuous Functions
In this part of this paper, the neutrosophic generalized 𝛂g-continuous functions are performed
and examined their fundamental features.
Definition 4.1: Let 𝜂: (𝒰, 𝜉) ⟶ (𝒱, 𝜚) be a map so as 𝒰 and 𝒱 are 𝑁𝑇𝑆s, then:
(i) 𝜂 is named a neutrosophic αg-continuous and signified by Ne-αg-continuous if for every Ne-OS
(correspondingly, Ne-CS) 𝒦 in 𝒱, 𝜂−1 (𝒦) is a Ne-αgOS (correspondingly, Ne-αgCS) in 𝒰.
(ii) 𝜂 is named a neutrosophic gα-continuous and signified by Ne-gα-continuous if for every Ne-OS
(correspondingly, Ne-CS) 𝒦 in 𝒱, 𝜂 −1 (𝒦) is a Ne-gαOS (correspondingly, Ne-gαCS) in 𝒰.
Theorem 4.2: Let 𝜂 be a function on 𝑁𝑇𝑆 𝒰 and valued in 𝑇𝑆 𝒱. So, we have the following:
(i) all Ne-g-continuous functions are Ne-αg-continuous.
(ii) all Ne-α-continuous functions are Ne-gα-continuous.
(iii) all Ne-gα-continuous functions are Ne-αg-continuous.
Proof:
(i) Let Ne-CS 𝒦 be in 𝑁𝑇𝑆 𝒱 and Ne-g-continuous function 𝜂 defined on 𝑁𝑇𝑆 𝒰 and valued in
𝑇𝑆 𝒱. By definition of Ne-g-continuous, 𝜂−1 (𝒦) remains a Ne-gCS in 𝒰. So, we have 𝜂 −1 (𝒦) is a
Ne-αgCS in 𝒰 because of theorem (2.5) part (iii). As a result, 𝜂 stands a Ne-αg-continuous.
Qays Hatem Imran, R. Dhavaseelan, Ali Hussein Mahmood and Md. Hanif PAGE, On Neutrosophic Generalized Alpha
Generalized Continuity
Neutrosophic Sets and Systems, Vol. 35, 2020
517
(ii) Let Ne-CS 𝒦 be in 𝑁𝑇𝑆 𝒱 and Ne-α-continuous function 𝜂 defined on 𝑁𝑇𝑆 𝒰 and valued in
𝑁𝑇𝑆 𝒱. By definition of Ne-α-continuous, 𝜂 −1 (𝒦) remains a Ne-αCS in 𝒰. So, we have 𝜂 −1 (𝒦) is a
Ne-gαCS in 𝒰 because of theorem (2.5) part (iv). As a result, 𝜂 stands a Ne-gα-continuous.
(iii) Let Ne-CS 𝒦 be in 𝑁𝑇𝑆 𝒱 and Ne-gα-continuous function 𝜂 defined on 𝑁𝑇𝑆 𝒰 and valued
in 𝑇𝑆 𝒱. So, we have 𝜂 −1 (𝒦) is a Ne-gαCS and then 𝜂 −1 (𝒦) is a Ne-αgCS in 𝒰 because of
theorem (2.5) part (v). Therefore, 𝜂 stands a Ne-αg-continuous.
The reverse of the beyond proposition does not become valid as shown in the next examples.
Example 4.3: (i) Assume 𝒰 = {𝑝, 𝑞} and 𝜉 = {0𝑁 , 𝒜, ℬ, 1𝑁 } and 𝜚 = {0𝑁 , ℬ, 𝒞, 1𝑁 } ,
〈𝓊, (0.6,0.7), (0.4,0.3), (0.5,0.2)〉
,
ℬ = 〈𝓊, (0.5,0.5), (0.5,0.4), (0.6,0.5)〉
where 𝒜 =
and
𝒞=
〈𝓊, (0.5,0.5), (0.6,0.4), (0.7,0.5)〉 are the neutrosophic sets, then (𝒰, 𝜉) and (𝒰, 𝜚) are NTSs. Define
𝜂: (𝒰, 𝜉) ⟶ (𝒰, 𝜚) as a 𝜂(𝑝) = 𝑞 and 𝜂(𝑞) = 𝑝. Then 𝜂 is Ne- α g- continuous. But 𝒞̅ =
〈𝓊, (0.7,0.5), (0.6,0.4), (0.5,0.5)〉 is a Ne-CS
in (𝒰, 𝜚), 𝜂 −1 (𝒞̅ ) is not a Ne-gCS in (𝒰, 𝜉).
Thus
𝜂 is not a Ne-g-continuous.
(ii)
Let
𝒰 = {𝑝, 𝑞}
and
𝜉 = {0𝑁 , 𝒜, ℬ, 1𝑁 }
let
〈𝓊, (0.6,0.7), (0.4,0.3), (0.5,0.2)〉
,
and
𝜚 = {0𝑁 , ℬ, 𝒞, 1𝑁 } ,
ℬ = 〈𝓊, (0.5,0.5), (0.5,0.4), (0.6,0.5)〉
where
and
𝒜=
𝒞=
〈𝓊, (0.5,0.5), (0.5,0.5), (0.4,0.5)〉 are the neutrosophic sets, then (𝒰, 𝜉) and (𝒰, 𝜚) are NTSs. Define
𝜂: (𝒰, 𝜉) ⟶ (𝒰, 𝜚) as a 𝜂(𝑝) = 𝑝 and 𝜂(𝑞) = 𝑞. Then 𝜂 is Ne-g α - continuous . But 𝒞̅ =
〈𝓊, (0.4,0.5), (0.5,0.5), (0.5,0.5)〉 is a Ne-CS in (𝒰, 𝜚), 𝜂 −1 (𝒞̅ ) is not a Ne-𝛼CS in (𝒰, 𝜉). Thus 𝜂 is
not a Ne-α-continuous.
(iii)
Let
𝒰 = {𝑝, 𝑞}
and
〈𝓊, (0.6,0.7), (0.4,0.3), (0.5,0.2)〉
𝜉 = {0𝑁 , 𝒜, ℬ, 1𝑁 }
let
,
and
𝜚 = {0𝑁 , ℬ, 𝒞, 1𝑁 } ,
ℬ = 〈𝓊, (0.5,0.5), (0.5,0.4), (0.6,0.5)〉
where
and
𝒜=
𝒞=
〈𝓊, (0.5,0.5), (0.6,0.4), (0.7,0.5)〉 are the neutrosophic sets, then (𝒰, 𝜉) and (𝒰, 𝜚) are NTSs. Define
𝜂: (𝒰, 𝜉) ⟶ (𝒰, 𝜚) as a 𝜂(𝑝) = 𝑞 and 𝜂(𝑞) = 𝑝 . Then 𝜂 is Ne- α g- continuous . But 𝒞̅ =
〈𝓊, (0.5,0.5), (0.5,0.5), (0.6,0.4)〉 is a Ne-CS in (𝒰, 𝜚), 𝜂 −1 (𝒞̅ ) is not a Ne-𝑔𝛼CS in (𝒰, 𝜉). Thus 𝜂 is
not a Ne-gα-continuous.
Definition 4.4: Let 𝜂 be a function on 𝑁𝑇𝑆 𝒰 and valued in 𝑇𝑆 𝒱 . Then, we named 𝜂 as
neutrosophic generalized αg-continuous and shortly wrote it as Ne-gαg-continuous if for each
Ne-CS 𝒦 in 𝒱, 𝜂 −1 (𝒦) is a Ne-gαgCS in 𝒰.
Theorem 4.5: Let 𝜂 be a function on 𝑁𝑇𝑆 𝒰 and valued in 𝑇𝑆 𝒱 . Afterward, 𝜂 remains a
Ne-gαg-continuous function iff for each Ne-OS 𝒦 in 𝒱, 𝜂 −1 (𝒦) is a Ne-gαgOS in 𝒰.
−1 (𝒦)) remains a Ne-gαgCS
̅̅̅̅̅̅̅̅̅̅̅
̅ ) = (𝜂
̅ are in 𝒱. Therefore, 𝜂−1 (𝒦
Proof: Let Ne-OS 𝒦 and Ne-CS 𝒦
in 𝒰. Consequently, 𝜂 −1 (𝒦) exists a Ne-gαgOS in 𝒰. The reverse proof is evident.
Proposition 4.6: For all Ne-gαg-continuous functions are Ne-αg-continuous.
Proof: Let Ne-CS 𝒦 be in 𝑁𝑇𝑆 𝒱 and Ne-gαg-continuous function 𝜂 defined on 𝑁𝑇𝑆 𝒰 and
valued in 𝑇𝑆 𝒱. By definition of Ne-gαg-continuous function, 𝜂 −1 (𝒦) stands a Ne-gαgCS in 𝒰. So,
we have 𝜂 −1 (𝒦) remains a Ne-αgCS in 𝒰 because of theorem (3.3) part (iii). As a result, 𝜂 exists a
Ne-αg-continuous.
Qays Hatem Imran, R. Dhavaseelan, Ali Hussein Mahmood and Md. Hanif PAGE, On Neutrosophic Generalized Alpha
Generalized Continuity
Neutrosophic Sets and Systems, Vol. 35, 2020
518
Proposition 4.7: For all Ne-gαg-continuous functions are Ne-gα-continuous.
Proof: Let Ne-CS 𝒦 be in 𝑁𝑇𝑆 𝒱 and Ne-gαg-continuous function 𝜂 defined on 𝑁𝑇𝑆 𝒰 and
valued in 𝑇𝑆 𝒱. By definition of Ne-gαg-continuous function, 𝜂 −1 (𝒦) stands a Ne-gαgCS in 𝒰.
So, we have 𝜂 −1 (𝒦) remains a Ne-gαCS in 𝒰 because of theorem (3.3) part (iv). As a result, 𝜂
exists a Ne-gα-continuous.
The reverse of the beyond proposition does not become valid as shown in the next examples.
Example 4.8: Let 𝒰 = {𝑝, 𝑞} and let 𝜉 = {0𝑁 , 𝒜, ℬ, 1𝑁 } and 𝜚 = {0𝑁 , 𝒞, 1𝑁 } ,
〈𝓊, (0.5,0.6), (0.3,0.2), (0.4,0.1)〉
,
ℬ = 〈𝓊, (0.4,0.4), (0.4,0.3), (0.5,0.4)〉
where 𝒜 =
and
𝒞=
〈𝓊, (0.5,0.4), (0.4,0.4), (0.4,0.5)〉 are the neutrosophic sets, then (𝒰, 𝜉) and (𝒰, 𝜚) are NTSs. Define
𝜂: (𝒰, 𝜉) ⟶ (𝒰, 𝜚)
as a 𝜂(𝑝) = 𝑞 and 𝜂(𝑞) = 𝑝.
〈𝓊, (0.4,0.5), (0.4,0.4), (0.5,0.4)〉 is a Ne-CS in (𝒰, 𝜚), 𝜂
Then 𝜂 is Ne- α g- continuous . But 𝒞 =
−1 (𝒞̅ )
is a Ne-αgCS but not a Ne-gαgCS in
(𝒰, 𝜉). Thus 𝜂 is not a Ne-gαg-continuous.
Example 4.9: Let 𝒰 = {𝑝, 𝑞} and let 𝜉 = {0𝑁 , 𝒜, ℬ, 1𝑁 } and 𝜚 = {0𝑁 , 𝒞, 1𝑁 } , where 𝒜 =
〈𝓊, (0.5,0.6), (0.3,0.2), (0.4,0.1)〉
,
ℬ = 〈𝓊, (0.4,0.4), (0.4,0.3), (0.5,0.4)〉
and
𝒞=
〈𝓊, (0.5,0.4), (0.4,0.4), (0.4,0.5)〉 are the neutrosophic sets, then (𝒰, 𝜉) and (𝒰, 𝜚) are NTSs. Define
𝜂: (𝒰, 𝜉) ⟶ (𝒰, 𝜚) as a 𝜂(𝑝) = 𝑞 and 𝜂(𝑞) = 𝑝.
Then 𝜂 is Ne-g α - continuous . But 𝒞 =
〈𝓊, (0.4,0.5), (0.4,0.4), (0.5,0.4)〉 is a Ne-CS in (𝒰, 𝜚), 𝜂 −1 (𝒞̅ ) is a Ne-g𝛼CS but not a Ne-gαgCS in
(𝒰, 𝜉). Thus 𝜂 is not a Ne-gαg-continuous.
Definition 4.10: Let 𝜂 be a function on 𝑁𝑇𝑆 𝒰 and valued in 𝑇𝑆 𝒱 . Then, we named 𝜂 as
neutrosophic generalized α g-irresolute and shortly wrote it as Ne-g α g-irresolute if for each
Ne-gαgCS 𝒦 in 𝒱, 𝜂 −1 (𝒦) is a Ne-gαgCS in 𝒰.
Theorem 4.11: Let 𝜂 be a function on 𝑁𝑇𝑆 𝒰 and valued in 𝑇𝑆 𝒱 . Afterward, 𝜂 remains a
Ne-gαg-irresolute function iff for each Ne-gαgOS 𝒦 in 𝒱, 𝜂−1 (𝒦) is a Ne-gαgOS in 𝒰.
−1 (𝒦)) remains a
̅̅̅̅̅̅̅̅̅̅̅
̅ ) = (𝜂
̅ are in 𝒱. Therefore, 𝜂−1 (𝒦
Proof: Let Ne-gαgOS 𝒦 and Ne-gαgCS 𝒦
Ne-gαgCS in 𝒰. Consequently, 𝜂 −1 (𝒦) exists a Ne-gαgOS in 𝒰. The reverse proof is evident.
Proposition 4.12: For all Ne-gαg-irresolute functions are Ne-gαg-continuous.
Proof: Let Ne-CS 𝒦 be in 𝑁𝑇𝑆 𝒱 and Ne-g α g-irresolute function 𝜂 defined on 𝑁𝑇𝑆 𝒰 and
valued in 𝑇𝑆 𝒱. So, we have 𝒦 stands a Ne-gαgCS in 𝒱 by theorem (3.3) part (i). By definition of
Ne-g α g-irresolute function, 𝜂−1 (𝒦) stands a Ne-g α gCS in 𝒰 . As a result, 𝜂 exists a
Ne-gαg-continuous.
The subsequent example explains that the inverse of the overhead proposition does not work.
Example 4.13: Suppose 𝒰 = {𝑝, 𝑞} and let 𝜉 = {0𝑁 , ℬ, 1𝑁 } and 𝜚 = {0𝑁 , 𝒜, ℬ, 1𝑁 } , where 𝒜 =
〈𝓊, (0.6,0.7), (0.4,0.3), (0.5,0.2)〉 and ℬ = 〈𝓊, (0.5,0.5), (0.5,0.4), (0.6,0.5)〉 are the neutrosophic sets,
then (𝒰, 𝜉) and (𝒰, 𝜚) are NTSs. Define 𝜂: (𝒰, 𝜉) ⟶ (𝒰, 𝜚) as a 𝜂(𝑝) = 𝑞 and 𝜂(𝑞) = 𝑝. Then 𝜂 is
Ne-gαg-continuous. But 𝒞 = 〈𝓊, (0.5,0.5), (0.6,0.4), (0.5,0.7)〉 is a Ne-gαgCS in (𝒰, 𝜚), 𝜂 −1 (𝒞) is not
a Ne-gαgCS in (𝒰, 𝜉). Thus 𝜂 is not a Ne-gαg-irresolute.
Qays Hatem Imran, R. Dhavaseelan, Ali Hussein Mahmood and Md. Hanif PAGE, On Neutrosophic Generalized Alpha
Generalized Continuity
Neutrosophic Sets and Systems, Vol. 35, 2020
519
Definition 4.14: We called a 𝑁𝑇𝑆 𝒰 with a neutrosophic T1 -space if for each Ne-gCS in 𝒰 is a
2
Ne-CS and we denoted it by Ne-T1 -space.
2
Definition 4.15: We called a 𝑁𝑇𝑆 𝒰 with a neutrosophic Tgαg -space if for each Ne-gαgCS in 𝒰 is a
Ne-CS and we denoted by Ne-Tgαg -space.
Proposition 4.16: Every Ne-T1 -space stands a Ne-Tgαg -space.
2
Proof: Let 𝒞 be a Ne-gαgCS in Ne-T1 -space 𝒰. By theorem (3.3) part (ii), we obtain 𝒞 is a Ne-gCS.
2
By definition of Ne- T1 -space, we reach to that 𝒞 is a Ne-CS in 𝒰 . Therefore, 𝒰 endures a
2
Ne-Tgαg -space.
Theorem 4.17: Let 𝜂1 be a Ne-gαg-continuous function on 𝑁𝑇𝑆 𝒰 and valued in 𝑁𝑇𝑆 𝒱 and let 𝜂2
be a Ne-g-continuous function on 𝑁𝑇𝑆 𝒱 and valued in 𝑇𝑆 𝒲. If 𝒱 is a Ne-T1 -space, then 𝜂2 ∘ 𝜂1
2
is a Ne-gαg-continuous function.
Proof: Assume Ne-CS 𝒦 is in 𝒲. Meanwhile, we have a Ne-g-continuous function 𝜂2 defined on a
Ne-T1 -space 𝒱, then 𝜂2 −1 (𝒦) stands a Ne-CS in 𝒱. Subsequently, we also see a Ne-gαg-continuous
2
function 𝜂1 defined on 𝒰, then 𝜂1 −1 (𝜂2 −1 (𝒦)) stands a Ne-gαgCS in 𝒰. Therefore, 𝜂2 ∘ 𝜂1 stands
a Ne-gαg-continuous.
Theorem 4.18: Let 𝜂 be a function on 𝑁𝑇𝑆 𝒰 and valued in 𝑇𝑆 𝒱, we have the following results:
(i) If 𝑁𝑇𝑆 𝒰 stands a Ne-T1 -space then the function 𝜂 becomes a Ne-g-continuous iff it considers a
2
a Ne-gαg-continuous.
(ii) If 𝑁𝑇𝑆 𝒰 stands a Ne-Tgαg -space then the function 𝜂 becomes a Ne-continuous iff it considers
a Ne-gαg-continuous.
Proof:
(i) Let Ne-CS 𝒦 be in 𝒱 and 𝜂 be a Ne-g-continuous function. By definition of Ne-g-continuous,
𝜂 −1 (𝒦) is a Ne-gCS in 𝒰. Besides, the definition of Ne-T1 -space states 𝜂 −1 (𝒦) is a Ne-CS. So,
2
𝜂 −1 (𝒦) is a Ne-gαgCS in 𝒰 by theorem (3.3) part (i). Therefore, 𝜂 is a Ne-gαg-continuous.
On the contrary, let Ne-CS 𝒦 be in 𝒱 and let 𝜂 be a Ne-g α g-continuous. By definition of
Ne-gαg-continuous, 𝜂 −1 (𝒦) is a Ne-gαgCS in 𝒰. Besides, we have 𝜂 −1 (𝒦) is a Ne-gCS in 𝒰 by
theorem (3.3) part (ii). Therefore, 𝜂 is a Ne-g-continuous.
Qays Hatem Imran, R. Dhavaseelan, Ali Hussein Mahmood and Md. Hanif PAGE, On Neutrosophic Generalized Alpha
Generalized Continuity
Neutrosophic Sets and Systems, Vol. 35, 2020
520
(ii) Let Ne-CS 𝒦 be in 𝒱 and let 𝜂 be a Ne-continuous. By definition of Ne-continuous, 𝜂 −1 (𝒦) is
a Ne-CS in 𝒰. So, we have 𝜂 −1 (𝒦) is a Ne-gαgCS in 𝒰 by theorem (3.3) part (i). Therefore, 𝜂 is a
Ne-gαg-continuous.
On the contrary, let Ne-CS 𝒦 be in 𝒱 and let 𝜂 be a Ne-gαg-continuous. Besides, we have 𝜂−1 (𝒦)
is a Ne-gαgCS in 𝒰. Furthermore, the definition of Ne-Tgαg -space gives 𝜂−1 (𝒦) is a Ne-CS in 𝒰.
Therefore, 𝜂 is a Ne-continuous.
Remark 4.19: The subsequent illustration indicates the relative among the various kinds of
Ne-continuous functions:
Ne-α-continuous
Ne-gα-continuous
Ne-αg-continuous
Ne-continuous
Ne-gαg-continuous
Ne-g-continuous
𝒰 is Ne-Tgαg -space
𝒰 is Ne-T1 -space
2
+
+
Fig. 4.1
5. Conclusion
The class of Ne-gαgCS described employing Ne-αgCS structures a neutrosophic topology and
deceptions between the classes of Ne-CS and Ne-gCS. We as well illustration Ne-gαg-continuous
functions by applying Ne-gαgCS. The Ne-gαgCS know how to be developed to establish another
neutrosophic homeomorphism.
Funding: This work does not obtain any external grant.
Acknowledgments: The authors are highly grateful to the Referees for their constructive
suggestions.
Conflicts of Interest: The authors declare no conflict of interest.
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Abdel-Basset, M., Mohamed, M., Elhoseny, M., Chiclana, F., & Zaied, A. E. N. H., Cosine similarity
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Abdel-Basset, M., Gamal, A., Son, L. H., & Smarandache, F., A Bipolar Neutrosophic Multi Criteria
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A. Salama and S. A. Alblowi, Neutrosophic set and neutrosophic topological spaces. IOSR Journal of
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Received: Apr 25, 2020.
Accepted: July 5 2020
Qays Hatem Imran, R. Dhavaseelan, Ali Hussein Mahmood and Md. Hanif PAGE, On Neutrosophic Generalized Alpha
Generalized Continuity
Neutrosophic Sets and Systems, Vol. 35, 2020
University of New Mexico
Generalized neutrosophic b-open sets in neutrosophic topological
space
Suman Das1, and Surapati Pramanik2,*
1Department
2Department
of Mathematics, Tripura University, Agartala , 799022, Tripura, India.
Email: suman.mathematics@tripurauniv.in
of Mathematics, Nandalal Ghosh B. T. College, Narayanpur, 743126, West Bengal, India.
Email: sura_pati@yahoo.co.in
* Correspondence: sura_pati@yahoo.co.in Tel.: (+91-9477035544))
Abstract: The purpose of the study is to introduce the notion of generalized neutrosophic b-open set
in neutrosophic topological space. We define generalized neutrosophic b-open set, generalized
neutrosophic b-interior, generalized neutrosophic b-closure and investigate some of their properties.
By defining generalized neutrosophic b-open set, we prove some theorems on neutrosophic
topological spaces. We also furnish some suitable examples.
Keywords: Neutrosophic set; neutrosophic b-open set; generalized neutrosophic b-open set;
generalized neutrosophic b-interior; generalized neutrosophic b-closure
1. Introduction
Smarandache (1998) grounded the Neutrosophic Set (NS) in 1998. From then it became very
popular and attracted many researchers' attention for theoretical and practical researches (Broumi et
al., 2018; Khalid, 2020; Peng & Dai, 2018; Pramanik, 2013; 2016a; 2016b; 2020; Pramanik & Mallick,
2018; 2019; Pramanik & Mondal, 2016; Pramanik & Roy, 2014; Smarandache & Pramanik, 2016; 2018,
Biswas, Pramanik & Giri, 2014; 2016a; 2016b; Dalapati et al., 2017; Dey, Pramanik, & Giri, 2016a;
2016b; Pramanik, Mallick, & Dasgupta, 2018; Mondal & Pramanik, 2015; Pramanik & Dalapati, 2018,
Pramanik, Dey, & Smarandache, 2018; Pramanik, Mondal, & Smarandache, 2016a; 2016b).
Salama and Alblowi (2012a) grounded the “Neutrosophic Topological Space” (NTS). Salama
and Alblowi (2012b) also presented generalized NS and generalized NTSs. Salama, Smarandache,
& Alblowi (2014) studied the concept of neutrosophic crisp topological space. Arokiarani,
Dhavaseelan, Jafari, and Parimala (2017) defined neutrosophic semi-open functions and established
relation between them. Iswaraya and Bageerathi (2016) studied neutrosophic semi-closed set and
neutrosophic semi-open set.
Rao and Srinivasa (2017) introduced neutrosophic pre-open set
and pre-closed set. Dhavaseelan and Jafari (2018) studied generalized neutrosophic closed sets.
Pushpalatha and Nandhini (2019) defined the neutrosophic generalized closed sets in NTSs. Shanthi,
Chandrasekar, Safina, and Begam (2018) presented the neutrosophic generalized semi closed sets in
Suman Das, Surapati Pramanik, Generalized neutrosophic b-open sets in neutrosophic topological space
Neutrosophic Sets and Systems, Vol. 35, 2020
523
NTSs. Ebenanjar, Immaculate, and Wilfred (2018) studied neutrosophic b -open sets in NTSs.
Maheswari, Sathyabama, and Chandrasekar (2018) studied the neutrosophic generalized b- closed
sets in NTSs.
Research gap: No investigation on neutrosophic generalized b-open set has been reported in the
recent literature.
Motivation: In order to fill the research gap, we introduce neutrosophic generalized b-open set.
Remaining of the paper is designed as follows:
Section 2 recalls of NTS, neutrosophic b- closed sets and a theorem. Section 3 introduces
neutrosophic generalized b-open set and proofs of some theorems on neutrosophic b-open sets.
Section 4 presents concluding remarks.
2. Preliminaries and some properties
Definition 2.1 Assume that (W , ) is an NTS. Then , an NS over W is said to be a Neutrosophic
b-Open (N-b-open) set (Ebenanjar, Immaculate, & Wilfred, 2018) if and only if (iff)
⊆Nint(Ncl(
))∪ Ncl(Nint( )).
Definition 2.2 In an NTS (W , ) , an NS is said to be a Neutrosophic b-Closed (N-b-closed) set
(Ebenanjar, Immaculate, & Wilfred, 2018) iff ⊇ Nint(Ncl( ))∩ Ncl(Nint( )).
Remark 2.1 An NS over W is said to be an N-b-closed set (Ebenanjar, Immaculate, & Wilfred,
2018) in (W , ) iff c is a N-b-open set in (W , ) .
In 2018, Ebenanjar, Immaculate, and Wilfred (2018) studied the concept of N-b-open set in NTS
but they did not check whether the union or intersection of two N-b-open sets (N-b-closed sets) is
again an N-b-open set (N-b-closed set) or not. In this paper we show some results on the intersection
and union of neutrosophic b-closed sets.
Theorem 2.1 The intersection of any two N-b-closed sets is again an N-b-closed set.
Proof. Assume that E, F be any two N-b-closed sets in an NTS (W , ) . Then we have
E ⊇ Nint(Ncl(E)) ∩ Ncl(Nint(E))
(1)
and F ⊇ Nint(Ncl(F)) ∩ Ncl(Nint(F))
(2)
For any two NSs E and F We know that E∩F ⊆ Eand E∩F ⊆ 𝐹.
Now E∩F ⊆ E⟹Nint(E∩F) ⊆ Nint(E) ⟹Ncl(Nint(E∩F)) ⊆ Ncl(Nint(E))
(3)
E∩F ⊆ E⟹Ncl(E∩F) ⊆ Ncl(E) ⟹Nint(Ncl(E∩F)) ⊆ Nint(Ncl(E))
(4)
E∩F ⊆ F⟹Nint(E∩F) ⊆ Nint(F) ⟹Ncl(Nint(E∩F)) ⊆ Ncl(Nint(F))
(5)
E∩F⊆ F⟹Ncl(E∩F) ⊆ Ncl(F) ⟹Nint(Ncl(E∩F)) ⊆ Nint(Ncl(F))
(6)
From (1) and (2) we have,
E∩F ⊇ Nint(Ncl(E)) ∩ Ncl(Nint(E)) ∩ Nint(Ncl(F)) ∩ Ncl(Nint(F))
⊇Nint(Ncl(E∩F)) ∩ Ncl(Nint(E∩F)) ∩ Nint(Ncl(E∩F)) ∩ Ncl(Nint(E∩F))
[ by eqs (3), (4), (5) & (6)]
= Nint(Ncl(E∩F)) ∩ Ncl(Nint(E∩F))
⟹E∩F ⊇ Ncl(Nint(E∩F)) ∩ Nint(Ncl(E∩F)).
Therefore E∩F is an N-b-closed set.
Suman Das, Surapati Pramanik, Generalized neutrosophic b-open sets in neutrosophic topological space
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Hence the intersection of any two N-b-closed sets is again an N-b-closed set.
Remark 2.2: The union of any two N-b-closed sets may not be an N-b-closed set. This is proved as
follows:
Example 2.1: Assume that W { p1 , p2 } and 𝜏 = {0N, 1N, {( p1 , 0.5, 0.2, 0.4), ( p2 , 0.6, 0.1, 0.3)}, {( p1 , 0.3,
0.5, 0.6), ( p2 , 0.4, 0.4, 0.5)}} be the family of some NSs over W . Then 𝜏 is an NT on W . Now it can
be verified that E= {(a, 0.6, 0.5, 0.6), (b, 0.5, 0.6, 0.7)}, F={(a,1, 0, 1), (b, 0.9, 0.1, 0.1)} are two N-b-closed
sets in (𝑊, 𝜏). But their union E∪F = {(a, 1, 0, 0.6), (b, 0.9, 0.1, 0.1)} is not an N-b-closed set.
Definition 2.3 Assume that (W , ) is an NTS and is an NS over W . Then the Neutrosophic
b-Closure (Nbcl) and Neutrosophic b-Interior (Nbint) (Ebenanjar, Immaculate & Wilfred, 2018) of
are defined by
Nbcl( ) = ∩{ : is an N-b-closed set in (W , ) and ⊆ };
Nbint( ) = ∪{ : is an N-b-open set in (W , ) and ⊆ }.
Remark 2.3 Clearly Nbint( ) is the largest N-b-open set (Ebenanjar, Immaculate, & Wilfred, 2018) in
(W , ) which is contained in and Nbcl( ) is the smallest N-b-closed set in (W , ) which contains .
Definition 2.4 Assume that (W , ) is an NTS. A neutrosophic subset E of (W , ) is said to be a
Neutrosophic Generalized Closed Set (NGCS) (Dhavaseelan & Jafari, 2018) if Ncl(E)⊆F whenever
E⊆F and F is an NOS. A subset K of (W , ) is called Neutrosophic Generalized Open Set (NGOS)
iff Kc is an NGCS in (W , ).
3. Generalized neutrosophic b-open set
Definition 3.1 Assume that (W , ) is an NTS. An NS G over W is called a Generalized
Neutrosophic b-Open (g-N-b-open) set if ∃ an N-b-closed set H (except 1N) with G⊆H such that
G ⊆ Nint(H). A neutrosophic subset K in (W , ) is called a Generalized Neutrosophic b-Closed
(g-N-b-closed) set iff Kc is a g-N-b-open set in (W , ) .
Example 3.1 Assume that W { p1 , p2 } and 𝜏={0N, 1N, {( p1 , 0.5, 0.6, 0.7), ( p2 ,0.6, 0.7, 0.8)}, {( p1 ,0.6,
0.5, 0.6), ( p2 ,0.7, 0.6, 0.7)}} are the collection of some NSs over W . Then (W , ) is clearly an NTS.
Here K = {( p1 , 0.6, 0.7, 0.8), ( p2 , 0.5, 0.8, 0.8)} is a g-N-b-open set, because there exists an N-b-closed
set G = { p1 , 0.7, 0.3, 0.4), ( p2 , 0.8, 0.3, 0.4)} in (W , ) with K ⊆ G such that K ⊆ Nint(G).
Proposition 3.1 In an NTS (W , ) , 0N is a g-N-b-open set but 1N is not a g-N-b-open set.
Proof. Assume that (W , ) is an NTS. Since a Neutrosophic Open Set (NOS) is an N-b-open set, so 1N
is an N-b-open set. Therefore, 0N is an N-b-closed set (since it is the complement of N-b-open set 1N).
Now 0N ⊆ 0N and 0N ⊆ Nint(0N)= 0N.
Thus there exist an N-b-closed set 0N (except 1N) with 0N ⊆0N such that 0N ⊆ Nint(0N). Hence 0N is a
g-N-b-open set in (W , ).
But in case of NS 1N, we cannot find any neutrosophic b-closed set H (except 1N) with 1N⊆H such that
1N⊆ Nint(H). Hence 1N is not a g-N-b-open set in (W , ).
Proposition 3.2 Assume that is a g-N-b-open set in an NTS (W , ). Then, every NS contained
in is a g-N-b-open set.
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Proof. Assume that be a g-N-b-open set in an NTS (W , ) and be any arbitrary NS over W
which is contained in . Since is a g-N-b-open set, so there exists an N-b-closed set (except
1N) with ⊆ such that ⊆Nint( ).
Now is contained in A, so
⊆
⟹ ⊆ ⊆ & ⊆ ⊆Nint( ).
Therefore there exists an N-b-closed set (except 1N) with ⊆ such that ⊆Nint( ). Hence
is a g-N-b-open set. Thus each NS contained in is again a g-N-b-open set in (W , ) .
Definition 3.2 Assume that (W , ) is an NTS and be an NS over W . Then the Generalized
Neutrosophic b-Interior (g-Nbint) and Generalized Neutrosophic b- Closure (g-Nbcl) of are defined
by
g-Nbint( ) = ∪{ : is a g-N-b-open set and ⊆ };
g-Nbcl( ) = ∩{ : is a g-N-b-closed set and ⊆ }.
Theorem 3.1 Assume that (W , ) is an NTS. Then each neutrosophic open subset of (W , ) is a
g-N-b-open set.
Proof. Assume that be an arbitrary NOS in an NTS (W , ) . So = Nint( ). Since each
neutrosophic closed set is an N-b-closed set so Ncl( ) is an N-b-closed set. Also we know that
⊆ Ncl( ).
Now ⊆ Ncl( )
⟹Nint( ) ⊆ Nint(Ncl( ))
⟹ = Nint( ) ⊆ Nint(Ncl( ))
⟹ ⊆ Nint(Ncl( ))
Therefore there exists an N-b-closed set Ncl( ) with ⊆ Ncl( ) such that ⊆ Nint(Ncl( )). Hence
is a g-N-b-open set in (W , ) . Thus each neutrosophic open subset of (W , ) is again a g-N-b-open
set.
Remark 3.1 The converse of the theorem 3.1 is not true. This can be shown by the example 3.2.
Example 3.2 In example 3.1, it can be easily seen that K = {(a, 0.6, 0.7, 0.8), (b, 0.5, 0.8, 0.8)} is a
g-N-b-open set in (W , ) but it is not an NOS.
Theorem 3.2 Assume that (W , ) is an NTS. Then each Neutrosophic Pre-Open Set (NPOS) in
(W , ) is a g-N-b-open set.
Proof. Assume that (W , ) is an NTS and is an NPOS. Then ⊆ Nint(Ncl( )). Since for any NS
, Ncl( ) is an N-b-closed set and ⊆ Ncl( ). Therefore there exists an N-b-closed set Ncl( ) with
⊆ Ncl( ) such that ⊆ Nint(Ncl( )). Hence is a g-N-b-open set in (W , ) . Thus each NPOS in
(W , ) is again a g-N-b-open set.
Theorem 3.3 If is both NOS and Neutrosophic Semi-Open Set (NSOS) in an NTS (W , ) then it is
a g-N-b-open set.
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Proof. Assume that (W , ) is an NTS and is both NSOS and NOS. Since is an NOS, so
= Nint( ). Again since is an NSOS, so ⊆ Ncl(Nint( )). It can be verified that Ncl(Nint( )) is an
N-b-closed set (since it is an NCS).
Now ⊆ Ncl(Nint( ))
⟹Nint( ) ⊆ Nint(Ncl(Nint( )))
⟹ = Nint( ) ⊆ Nint(Ncl(Nint( )))
[ since ⊆ 𝛿 ⟹ Nint( ) ⊆ Nint(𝛿) ]
[ since = Nint( ) ]
⟹ ⊆ Nint(Ncl(Nint( )))
Therefore there exists an N-b-closed set Ncl(Nint( )) with ⊆ Ncl(Nint( )) in (W , ) such that
⊆ Nint(Ncl(Nint( ))). Hence is a g-N-b-open set.
Theorem 3.4 Assume that (W , ) is an NTS and is both neutrosophic 𝛼-open and neutrosophic
open set. Then is again a g-N-b-open set.
Proof. Assume that is an arbitrary NS which is both neutrosophic 𝛼-open set and NOS. Since
is an NOS so = Nint( ). Again since is a neutrosophic 𝛼-open set, so ⊆ Nint(Ncl(Nint( ))).
Hence, it is clear that Ncl(Nint( )) is an N-b-closed set (since it is an NCS) in (W , ) .
Now = Nint( )
⟹ = Nint( ) ⊆ Ncl(Nint( ))
⟹ ⊆ Ncl(Nint( ))
Therefore there exists an N-b-closed set Ncl(Nint( )) with ⊆ Ncl(Nint( )) such that ⊆
Nint(Ncl(Nint( ))). Hence is a generalized N-b-open set in (W , ) .
Theorem 3.5 The intersection of any two g-N-b-open sets in an NTS (W , ) is again a g-N-b-open
set.
Proof. Let and be any two g-N-b-open sets in an NTS (W , ) . Then there exist two N-b-closed
sets K, L with ⊆ K, ⊆L such that ⊆ Nint(K) and ⊆ Nint(L).
Here ∩ ⊆ K∩L.
We know that the intersection of two N-b-closed sets is again an N-b-closed set. So K∩L is an
N-b-closed set in (W , ) .
Now ∩ ⊆Nint(K) ∩ Nint(L) [since ⊆ Nint(K), ⊆ Nint(L)]
=Nint(K ∩ L)
⟹ ∩ ⊆ Nint(K ∩ L).
Therefore there exists an N-b-closed set K∩L with ∩ ⊆K∩L such that ∩ ⊆ Nint(K ∩ L).
Hence ∩ is a g-N-b-open set in (W , ) . Thus the intersection of any two g-N-b-open sets in
(W , ) is again a g-N-b-open set.
Theorem 3.6 The union of two g-N-b-open sets is a g-N-b-open set if one is contained in the other.
Proof. Let , are any two g-N-b-open sets in (W , ) such that ⊆ . Since and are
g-N-b-open sets, so there exist two N-b-closed sets G1, G2 with ⊆ G1 and ⊆ G2 such that ⊆
Nint(G1) and ⊆ Nint(G2).
Now ∪ ⊆ [since ⊆ ]
⊆G2
⟹ ∪ ⊆G2
Again ∪ ⊆ ⊆ Nint(G2), where G2 is an N-b-closed set in (𝑋, 𝜏).
Suman Das, Surapati Pramanik, Generalized neutrosophic b-open sets in neutrosophic topological space
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Therefore there exists an N-b-closed set G2 with ∪ ⊆G2 in (𝑋, 𝜏) such that ∪ ⊆ Nint(G2).
Hence the union of two g-N-b-open sets is again a g-N-open set if one is contained in the other.
Definition 3.3 An NS is called a g-N-b-open set relative to an NS if there exists an N-b-closed
set with ⊆ ∩ such that ⊆ Nint( ∩ ).
Theorem 3.7 Assume that (W , ) is an NTS. If is a g-N-b-open set relative to and is a
g-N-b-open set relative to then is a g-N-b-open set relative to .
Proof.
Since is a g-N-b-open set relative to so there exists an N-b-closed set K with
⊆ ∩K such that ⊆Nint( ∩K). Similarly, since is a g-N-b-open set relative to then
there exists an N-b-closed set L with ⊆ ∩L such that ⊆ Nint( ∩L).
We know that the intersection of two N-b-closed sets is again an N-b-closed set. So K∩L is an
N-b-closed set.
Now ⊆ ∩K ⊆ ∩L∩K
= ∩(L∩K)
= ∩G , where G = K∩L is an N-b-closed set.
Again ⊆ Nint( ∩K)
⊆ Nint( ∩G).
Therefore there exists an N-b-closed set G with G such that Nint ( G)
Hence is a g-N- b-open relative to .
4. Conclusion
In this article, we introduce generalized neutrosophic b-open set, generalized neutrosophic
b-interior, generalized neutrosophic b-closure and investigate some of their properties. By defining
generalized neutrosophic b-open set, we prove some theorems on NTSs and few illustrative
examples are provided. In the future, we hope that based on these notions in NTSs, many new
investigations can be carried out.
.
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Received: Apr 20, 2020.
Accepted: July 15 2020
Suman Das, Surapati Pramanik, Generalized neutrosophic b-open sets in neutrosophic topological space
Neutrosophic Sets and Systems, Vol. 35, 2020
University of New Mexico
Neutrosophic Soft Fixed Points
Madad Khan
2
1
1
, Muhammad Zeeshan
1
, Saima Anis
1
, Abdul Sami Awan
1
and Florentin Smarandache
2
Department of Mathematics, COMSATS University Islamabad, Abbottabad Campus Pakistan
Department of Mathematics and Sciences, University of New Mexico, 705 Gurley Ave., Gallup, NM 87301, USA
madadkhan@cuiatd.edu, muhammadzeeshan@ciit.net.pk, saimaanis@cuiatd.edu.pk, abdulsamiawan@ciit.net.pk,
smarand@unm.edu
Abstract. In a wide spectrum of mathematical issues, the presence of a fixed point (FP) is equal to the presence
of a appropriate map solution. Thus in several fields of math and science, the presence of a fixed point is important. Furthermore, an interesting field of mathematics has been the study of the existence and uniqueness
of common fixed point (CFP) and coincidence points of mappings fulfilling the contractive conditions. Therefore, the existence of a FP is of significant importance in several fields of mathematics and science. Results of
the FP, coincidence point (CP) contribute conditions under which maps have solutions. The aim of this paper
is to explore these conditions (mappings) used to obtain the FP, CP and CFP of a neutrosophic soft set. We study
some of these mappings (conditions) such as contraction map, L-lipschitz map, non-expansive map, compatible
map, commuting map, weakly commuting map, increasing map, dominating map, dominated map of a neutrosophic soft set. Moreover we introduce some new points like a coincidence point, common fixed point and
periodic point of neutrosophic soft mapping. We establish some basic results, particular examples on these
mappings and points. In these results we show the link between FP and CP. Moreover we show the importance
of mappings for obtaining the FP, CP and CFP of neutrosophic soft mapping.
Keyword. Neutrosophic set, fuzzy neutrosophic soft mapping, fixed point, coincidence point.
_________________________________________________________________________________
1.
Introduction
It is well known fact that fuzzy sets (FS) [1], complex fuzzy sets (CFS) [2], intuitionistic fuzzy sets (IFSs),
the soft sets [3], fuzzy soft sets (FSS) and the fuzzy parameterized fuzzy soft sets (FPFS-sets) [4], [5] have been
used to model the real life problems in various fields like in medical science, environments, economics, engineering, quantum physics and psychology etc.
In 1965, L. A. Zadeh [1] introduced a FS, which is the generalization of a crisp set. A grade value of a crisp set
is either 1 or 0 but a grade value of fuzzy set has all the values in closed interval [0,1]. A FS plays a central
role in modeling of real world problems. There are a lot of applications of FS theory in various branches of
science such as in engineering, economics, medical science, mathematical chemistry, image processing, nonequilibrium thermodynamics etc. The concept for IFSs is provided in [3] which are generalizations of FS. An
IFS
P can be expressed as P { , , P ( ), P ( ) : X },
where P (v) represents the degree of mem-
bership, P (v) represents the degree of non-membership of the element X . FPFS-sets is the extension of
a FS and soft set proposed in [4], [5] . FPFS-sets maintain a proper degree of membership to both elements and
parameters.
The notion of a complex CFS, the extension of the fuszy set, was introduced by Ramot et, al., [2]. A CFS membership function has all the values in the unit disk. A complex fuzzy set is used for representing two-dimensional phenomena and plays an important role in periodic phenomena. Complex fuzzy set is used in signals
and systems to identify a reference signal out of large signals detected by a digital receiver. Moreover it is used
for expressing complex fuzzy solar activity (solar maximum and solar minimum) through the average number
Madad Khan, Muhammad Zeeshan, Saima Anis, Abdul Sami Awan and Florentin Smarandache, Neutrosophic Soft Fixed
Points
Neutrosophic Sets and Systems, Vol. 35, 2020
532
of sunspot.
Smarandache [6], [7] has given the notion of a neutrosophic set (NS). A NS is the extension of a crisp set, FS and
IFS. In NS, truth membership (TM), falsity membership (FM) and indeterminacy membership (IM) are independent. In decision-making problems, the indeterminacy function is very significant. A NS and its extensions
plays a vital role in many fields such as decision making problems, educational problems, image processing,
medical diagnosis and conflict resolution. Moreover the field of neutrosophic probability, statistics, measures
and logic have been developed in [8]. The generalization of fuzzy logic (FL) has been suggested by Smarandache
in [8] and is termed as neutrosophic logic (NL). A proposition in NL is true (t ), indeterminate (i ) and false
( f ) are real values from the ranges T , I , F . T , I , F and also the sum of t , i , f are not restricted. In neutrosophic logic, there is indeterminacy term, which have no other logics, such as intuitionistic logic (IL), FL, boolean logic (BL) etc. Neutrosophic probability (NP) [8] is the extension of imprecise probability and classical probability. In NP, the chance occurs by an event is t % true, i% indeterminate and f % false where t , i , f varies
in the subsets T , I and
F
respectively. Dynamically these subsets are functions based on parameters, but they
are subsets on a static basis. In NP
n _ sup 3 ,
while in classical probability n _ sup 1. The extension of
classical statistics is neutrosophic statistics [8] which is the analysis of events described by NP. There are twenty
seven new definitions derived from NS, neutrosophic statistics and a neutrosophic probability. Each of these
are independent. The sets derived from NS are intuitionistic set, paradoxist set, paraconsistent set, nihilist set,
faillibilist set, trivialist set, and dialetheist set. Intuitionistic probability and statistics, faillibilist probability and
statistics,tautological probability and statistics, dialetheist probabilityand statistics, paraconsistent probability
and statistics, nihilist probability and statistics and trivialist probability and statistics are derived from neutosophic probability and statistics. N. A. Nabeeh [9] suggested a technique that would promote a personal selection process by integrating the neutrosophic analytical hierarchy process to show the ideal solution among
distinct options with order preference tevhnique similar to an ideal solution (TOPSIS). M. A. Baset [10] introduced a new type of neutrosophy technique called type 2 neutrosophic numbers. By combining type 2 neutrosophic number and TOPSIS, they suggested a novel method T2NN-TOPSIS which is very useful in group decision making. They researched a multi criteria group decision making technique of the analytical network process method and Visekriterijusmska Optmzacija I Kommpromisno Resenje method under neutrosophic environment that deals high order imprecision and incomplete information [11]. M. A. Baet suggested a new strategy for estimating the smart medical device selecting process in a GDM in a vague decision environment. Neutrosophic with TOPSIS strategy is used in decision-making processes to deal with incomplete information,
vagueness and uncertainty, taking into account the decision requirements in the information gathered by decision-makers [12]. They suggested the robust ranking method with NS to manage supply chain management
(GSCM) performance and methods that have been widely employed to promote environmental efficiency and
gain competitive benefits. The NS theory was used to manage imprecise understanding, linguistic imprecision,
vague data and incomplete information [13]. Moreover M. A. Baset [14] et, al., used NS for assessment technique
and decision-making to determine and evaluate the factors affecting supplier selection of supply chain management. T. Bera [15] et, al., defined a neutrosophic norm on a soft linear space known as neutrosophic soft
linear space. They also modified the concept of neutrosophic soft (Ns) prime ideal over a ring. They presented
the notion of Ns completely semi prime ideals, Ns completely prime ideals and Ns prime K-ideals [16]. Moreover T. Bera [17] introduced the concept of compactness and connectedness on Ns topological space along with
their several characteristics. R. A. Cruz [18] et, al., discussed P-intersection, P- union, P-AND and P-OR of neutrosophic cubic sets and their related properties. N. Shah [19] et, al., studied neutrosophic soft graphs. They
presented a link between neutorosophic soft sets and graphs. Moreover they also discussed the notion of strong
neutosophic soft graphs.
Smarandache [20] discussed the idea of a single valued neutrosophic set (SVNS). A SVNS defined as for any
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Madad Khan, Muhammad Zeeshan, Saima Anis, Abdul Sami Awan and Florentin Smarandache, Neutrosophic Soft Fixed Points
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Neutrosophic Sets and Systems, Vol. 35, 2020
space of points set
U'
with 𝑢 in U ' , a SVNS
W
in U ' , the truth membership, false memebership and inde-
terminac membership functions denoetd as TA , FA and I A respectively with TA , FA , I A [0,1] for each
in
U '. A SVNS W is expressed as W X TW ( ), I W ( ), FW ( ) / , X ,
n
crete case, a SVNS can be expressed as W T (i ), I (i ), F (i ) / i,
i 1
when
X
u
is continous. For a dis-
i X . Later, Maji [21] gave a new
concept neutrosophic soft set (NSS). For any initial universal set W and any parameters set E with A E
and P (W ) represents all the NS of W . The order set ( , A) is said to be the soft NS over W where
: A P(W ). Arockiarani et al., [22] introduced fuzzy neutrosophic soft topological space and presents
main results of fuzzy neutrosophic soft topological space. Later on the researchers linked the above theories
with different field of sciences.
The purpose of this paper is to study the mappings such as contraction mapping, expansive mapping, nonexpansive mapping, commuting mapping, and weakly commuting mapping used to attain the FP, CP and CFP
of a neutrosophic soft set. We present some basic resultsnd particular examples of fixed points, coincidence
points, common fixed points in which contraction mapping, expansive mapping, non-expansive mapping, commuting mapping, and weakly commuting mapping are used.
2. Preliminaries
We will discuss here the basic notions of NS and neutrosophic soft sets. We will also discuss some new
neutrosophic soft mappings such as contraction mapping, increasing mapping, dominated mapping, dominating mapping, K-lipschitz mapping, non-expansive mapping, commuting mapping, weakly compatible mapping. Moreover we will study periodic point, common fixed point, coinciding point of neutrosophic soft-map
ping. Here N S (U E ) is the collection of all neutrosophic soft points.
Definition 2.1 [7] Let
U
be any universal set, with generic element
N { , T ( ), I ( ), F ( ) , U }, where
N
N
N
T , I , F : U 0,1
U .
and
0 T ( ) I ( ) F ( ) 3 .
N
N
N
T ( ), I ( ) and F ( ) denote TM, IM and FM functions respectively. In
N
N
N
it's non-standard part and
1 is it's standard part. Likely 0 0 ,
A NS N is defined by
0,1 ,1
1 ,
where
is it's non-standard part and
is
0 is it's
standard part. It is difficult to employ these values in real life applications. Hence we take all the values of
neutrosophic set from subset [0,1].
Definition 2.2 [23] Let 𝐸 and 𝑊 be the set of parameters and initial universal set respectively. Let the power set
of 𝑊 is denoted by 𝑃(𝑊). Then a pair ( , A) is called soft set (SS) over 𝑊, where A E and
: A P(W ).
Definition 2.3 [21] Let 𝐸 and 𝑊 be the set of parameters and initial universal set respectively. Suppose that the
set of all neutrosophic soft set (NSS) is denoted as N S (W ) . Then for E , a pair ( , ) is called a N SS
over
W , where : N S (W )
is a mapping.
Definition 2.4 [24] Let 𝐸 and 𝑊 be the set of parameters and initial universal set respectively. Suppose that the
set of all NSS is denoted as N S (W ) . A NSS
N over W is a set which defined by a set valued function
N
representing a mapping : E N S (W ). is known as approximate function of the N S (W ). The
N
N
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Madad Khan, Muhammad Zeeshan, Saima Anis, Abdul Sami Awan and Florentin Smarandache, Neutrosophic Soft Fixed Points
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Neutrosophic Sets and Systems, Vol. 35, 2020
neutrosophic soft set can be written as:
N {(e,{ , T ( e ) ( ), I ( e ) ( ), F ( e) ( )
N
N
: W }) : e E}
N
where TN ( ), I N ( ), FN ( ) represents the
TM, IM and FM functions of (e) respectively and has valN
ues in [0,1]. Also
0 T ( e ) ( ), I ( e ) ( ), F ( e ) ( ) 3.
Definition 2.5 [22] Let
where
U
N
N
N
be any universal set. The fuzzy neutrosophic set (fn-s)
N { , TN ( ), I N ( ), FN ( ) , X }
TN ( ), I N ( ), FN ( )
T , I , F : N [0,1]. Also
represents
the
TM,
IM
and
FM
N
is defined as
functions
respectively
and
0 TN ( ) I N ( ) FN ( ) 3.
Definition 2.6 [22] Let 𝐸 and 𝑊 be the set of parameters and initial universal set respectively. Suppose that the
set of all fuzzy neutrosophic soft set (FNS-set) is denoted as FN S (U E ) . Then for E , a pair ( , ) is said
to be a FNS-set over
W , where : N S (W )
is a mapping.
Definition 2.7 [25] Let A , B be two fuzzy neutrosophic soft set. An fuzzy neutrosophic soft (FNS) relation
from A to B is known as FNS mapping if the two conditions are fulfilled.
i
. For every A , there exists B , where , are FNS elements.
A
B
A
B
1
1
1
1
ii
. For empty fuzzy FNS element in A , the ( A ) is also empty FNS element.
Definition 2.8 [25] Let A FNS (W , R) be a FNS-set and : A A an FNS-mapping. A fuzzy neutrosophic element
A
is called a fixed point of if
(A ) A .
Criterion [26], [27] Let N S (W ) be the set of all neutrosophic points over (W , E ). Then the neutrosophic soft
metric on based of neutrosophic points is defined as d : N S (W E ) N S (W E ) having the following properties.
M 1 ).
d (A , B ) 0 for all
M 2 ).
d (A , B ) 0 A B .
A , B N S (W E ).
M 3 ). d (A , B ) d (B , A ).
M 4 ).
d (A , B ) d (A , C ) d (C , B ).
Then
( N S (U E ), d )
is
said
to
be
neutrosophic
T T , I I and F F .
A
B
A
B
A
B
soft
metric
space.
Here
A B
implies
3.
Mappings on Neutrosophic Soft Set
Here, we introduced some new neutrosophic soft mappings such as contraction mapping, increasing
mapping, dominated mapping, dominating mapping, K-lipschitz mapping, non-expansive mapping, commuting mapping, weakly compatible mapping. Also we introduced periodic point, common fixed point, coinciding
point of neutrosophic soft-mapping. Here N S (U E ) is the collection of all neutrosophic soft points.
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Madad Khan, Muhammad Zeeshan, Saima Anis, Abdul Sami Awan and Florentin Smarandache, Neutrosophic Soft Fixed Points
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Neutrosophic Sets and Systems, Vol. 35, 2020
Definition 3.1 Let be a mapping from N S (U E ) to N S (U E ). Then is called neutrosophic soft contraction if
d ( (A ), (B )) kd (A , B )
for all A , B F N S (U E ) and k [0,1). Where k is called
contraction factor.
Example 3.1 Let
and
B
U {1 ,1 , 3 } be any initial universal set and
A
R A B {1 , 2 } . Define a NSS
as below:
A {( 1 ,{ 1 , 0.8, 0.1, 0.3 , 2 , 0.6, 0.7, 0.4 , 3 ,1, 0.2, 0.4 }),
( 2 ,{ 1 , 0.3, 0.7, 0.6 , 2 , 0.1, 0.9, 0.3 , 3 , 0.1, 0.8, 0.7 })}
and
B {( 1 ,{ 1 , 0.9, 0.7, 0.1 , 2 ,1, 0.8, 0.6 , 3 ,1, 0.2, 0.4 }}),
( 2 ,{ 1 , 0.1, 0.3, 0.6 , 2 , 0.2, 0.3, 0.9 , 3 , 0.1, 0.8, 0.7 })}.
The distance defined [27] as
1
d ( (A1 ), (A2 )) min{(| T1 ( i ) T 2 ( i ) | p | I 1 ( i ) I 2 ( i ) | p | T1 ( i ) T 2 ( i ) | p ) p }
i
( p 1).
B
B
B
B
B
B
In this example, we take p 1, now
d ( (A1 ), ((A2 )) min{| T1 ( i ) T 2 ( i ) | | I 1 ( i ) I 2 ( i ) |
1
1
i
B
B
B
B
| F1 ( i ) F 2 ( i ) |}
B
B
| T1 ( 2 ) T 2 ( 2 ) | | I 1 ( 2 ) I 2 ( 2 ) |
B
B
B
B
| F1 ( 2 ) F 2 ( 2 ) |
B
B
| 1 0.2 | | 0.8 0.3 | | 0.6 0.9 |
0.8 0.5 0.3
0.16
(0.2)(0.8)
0.2d (A1 , A2 ).
1
1
Here k 0.2, so is a contraction.
Definition 3.2 Let be a mapping from N S (W E ) to F N S (W E ). Then is called neutrosophic soft non-
d ( (A ), (B )) kd (A , B ) for all A , B N S (WE ) and k 1.
Example 3.2 Let W {1 , 2 , 3 } and R A B {1 , 2 } . Define a neutrosophic soft sets A and B
expansive mapping if
as follows:
A {( 1 ,{ 1 ,1, 0.1, 0.2 , 2 , 0.6, 0.7, 0.4 , 3 , 0.2, 0.4, 0.6 }),
( 2 ,{ 1 , 0.3, 0.7, 0.6 , 2 , 0.1, 0.9, 0.3 , 3 , 0.4, 0.6, 0.7 })}
and
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B {( 1 ,{ 1 ,1, 0.5, 0.2 , 2 ,1, 0.5, 0.6 , 3 , 0.2, 0.4, 0.6 }}),
( 2 ,{ 1 , 0.1, 0.3, 0.6 , 2 , 0.2, 0.3, 0.9 , 3 , 0.4, 0.6, 0.7 })}.
d ( (A1 ), ((A2 )) min{| T1 ( i ) T 2 ( i ) | | I 1 ( i ) I 2 ( i ) |
1
1
xi
B
B
B
B
| F1 ( i ) F 2 ( i ) |}
B
B
| T1 ( 3 ) T 2 ( 3 ) | | I 1 ( 3 ) I 2 ( 3 ) |
B
B
A
B
| F1 ( 3 ) F 2 ( 3 ) |
A
B
| 0.2 0.4 | | 0.4 0.6 | | 0.6 0.7 |
0.2 0.2 0.1
0.5
(1)(0.5)
1d (A1 , A2 ).
1
1
Here k 1, so is non-expansive.
Definition 3.3 Let be a mapping from N S (W E ) to N S (W E ). Then is called neutrosophic soft k-Lip-
d ( (A ), (B )) kd (A , B ) for all A , B F N S (WE ) and k 0.
Example 3.3 Let W {1 , 2 , 3 } and R A B {1 , 2 } . Define a NSS A and B as below:
A {( 1 ,{ 1 , 0.3, 0.4, 0.3 , 2 , 0.6, 0.7, 0.4 , 3 , 0.2, 0.4, 0.6 }),
schitz mapping if
( 2 ,{ 1 , 0.5, 0.6, 0.4 , 2 , 0.1, 0.9, 0.3 , 3 , 0.4, 0.6, 0.7 })}
and
B {( 1 ,{ 1 ,1, 0.4, 0.3 , 2 ,1, 0.6, 0.3 , 3 , 0.2, 0.4, 0.6 }}),
( 2 ,{ 1 , 0.5, 0.7, 0.5 , 2 , 0.3, 0.2, 0.9 , 3 ,1, 0.3, 0.9 })}.
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d ( (A1 ), ((A2 )) min{| T1 ( i ) T 2 ( i ) | | I 1 ( i ) I 2 ( i ) |
1
?i
1
B
B
B
B
| F1 ( i ) F 2 ( i ) |}
B
B
| T1 (1 ) T 2 (1 ) | | I 1 (1 ) I 2 (1 ) |
B
B
A
B
| F1 (1 ) F 2 (1 ) |
A
B
| 1 0.5 | | 0.4 0.7 | | 0.3 0.5 |
0.5 0.3 0.2
1
(2)(0.5)
2d (A1 , A2 ).
1
1
Here k 2, so is k-lipschitz.
Note: Every neutrosophic soft contraction mapping is neutrosophic soft K-lipschitz mapping but its converse
does not hold.
Definition 3.4 Let be a mapping from N S (W E ) to N S (W E ). Then is said to be neutrosophic soft kanan
contraction if
d ( (A ), (B )) k[d (A , (A )) d (B , (B ))] for
k [0, 12 ). Where k
all A , B N S (W E ) and
is called contraction factor.
Definition 3.5 Let and be two mappings from N S (U E ) to N S (U E ) . Then and are called neu
trosophic soft commuting mapping if
( (A )) ( (A )) for all
A N S (U E ).
Definition 3.6 Let and be two mappings from N S (U E ) to N S (U E ) . Then and are called neu
trosophic soft weakly commuting mapping if
d ( ( (A )), ( (A ))) d ( (A ), (A ))
for all
A N S (U E ).
Definition 3.7 Let and be two mappings from N S (U E ) to N S (U E ) . If for
( A ) A
n
0
as
n and A , A
n
0
( A ) A
n
0
and
N S (U E ). Then it is called neutrosophic soft compatible map-
ping if lim d ( ( (A )), ( (A ))) 0.
n
Definition 3.8 Let , : N S (U E ) N S (U E ) be two mappings. If there is A N S (U E ) such that
(A ) (A ) A , then A N S (U E ) is called common fixed point neutrosophic soft mappings.
k
Definition 3.9 If A is a fixed point of : N S (U E ) N S (U E ), then A is also a fixed point that is
k (A ) A for all A N S (U E ). So A is called periodic point of neutrosophic soft mapping and
k is called period of .
Remark Every fixed point of neutrosophic soft mapping is a periodic point but every periodic point of neutrosophic soft mapping is not a fixed point.
Definition 3.9 Let , be two mappings from N S (U E ) to N S (U E ). If
(A ) (A ) B ,
for all
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A , B F N S (U E ). Then
dence for and .
A
is called coincidence point of and and
B
is called point of coinci-
Definition 3.10 Let : N S (U E ) N S (U E ) be a mapping. Then is said to be neutrosophic soft increasing map if for any
A B
(A ) (B ) for all A , B N S (U E ).
implies
Definition 3.11 Let : N S (U E ) N S (U E ) be a mapping. Then is said to be neutrosophic soft dominated map if
(A ) A
for all A N S (U E ).
Definition 3.12 Let : N S (U E ) N S (U E ) be a mapping. Then is said to be neutrosophic soft dominating map if
A (A )
for all A N S (U E ).
4. Main Results
Banach Contraction Theorem
Proposition 1 Let N S (U E ) be a non-empty set of neutrosophic points and ( N S (U E ), d ) be a complete neu
trosophic soft metric space. Suppose is a mapping from N S (U E ) to N S (U E ) be contraction. Then fixed
point of exists and unique.
Proof Let
form
A N S (U E ) be arbitrary. Define A ( A ) and by continuing we have a sequence in the
1
0
0
A ( A ). Now
n 1
n
d ( A , A ) d ( ( A ), ( A ))
n 1
n
n
n 1
kd ( A , A )
n
n 1
kd ( ( A ), ( A ))
n 1
2
n2
k d ( A , A )
n 1
n2
k d ( ( A ), ( A ))
2
n2
3
n 3
k d ( A , A )
n2
n 3
.
.
.
k n d ( A , A ).
1
Now for
m, n n0 ,
0
we have
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d ( A , A ) d ( A , A ) d ( A , A ) ... d ( A , A )
n 1
n
n
n
n 1
n 1
k d ( A , A ) k
1
n 1
0
n
So
d ( A , A ) ... k
2
m n 1
m 1
0
m
d ( A , A )
1
0
]d ( A , A )
1
0
n
k
d ( A , A )
1
0
1 k
, A ) 0 as n .
n 1
m 1
1
k [1 k k ... k
d ( A
n2
n
A is a cauchy sequence in ( N S (U E ), d ), but ( N S (U E ), d ) is complete, so there exists
n
A N S (U E ) such that d ( A , A ) 0 as
n
n . Now
d ( A , ( A )) d ( ( A ), ( A ))
n 1
n
kd ( A , A ).
On taking limit as
n ,
n
we get
d ( (A ), A ) 0.
But
d ( (A ), A ) 0.
So
d ( ( A ), A ) 0
( A ) A .
So
A
is the FP of .
Now we have to show that
A
is unique. Suppose there exists another FP B N S (U E ) such that
(B ) B . Now
d (A , B ) d ( (A ), (B ))
kd (A , B )
(1 k )d (A , B ) 0.
Here (1 k )
0 , so
d (A? , B? ) 0.
But
d ( A , B ) 0
d ( A , B ) 0.
Hence
A B ,
so the fixed point is unique.
Proposition 2 Let ( N S (U E ), d ) be a complete neutrosophic soft metric space. Suppose be a mapping from
m
m
F N S (U E ) to F N S (U E ) satisfies the contraction d ( ( A ), ( B )) kd ( A , B ) for all
1
A , B N S (U E ), where k [0,1) and
1
1
m
1
1
1
is any natural number. Then has a FP.
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Proof It follows from banach contraction theorem that
m
has unique a FP that is
( ( A ))
m
m 1
1
m (A ) A . Now
1
1
( A )
1
( ( A ))
m
1
( A ).
1
By the uniqueness of FP, we have
( A ) A .
1
1
Proposition 3 Let ( N S (U E ), d ) be a complete neutrosophic soft metric space. Suppose , satisfy
d ( ( A ), ( B )) d ( A , ( A )) d ( B , ( B )) [d ( A , ( B )) d ( B , ( A )] for all
1
1
1
1
1
1
1
1
1
1
A , B F N S (U E ) with , , are non-negative and 1. Then and have a unique FP.
1
1
Proof Let
Now
A N S (U E ) be a fixed point of that is ( A ) A . We need to show that ( A ) A .
1
1
1
1
1
d ( A , ( A )) d ( ( A ), ( A ))
1
1
1
1
d ( A , ( A )) d ( A , ( A )) [d ( A , ( A )) d ( A , ( A ))]
1
1
1
1
1
1
1
1
d ( A , A ) d ( A , ( A )) [d ( A , ( A )) d ( A , A )]
1
1
1
1
1
1
1
1
d ( A , ( A ) d ( A , ( A ))
1
1
1
1
(1 )d ( A , ( A )) 0
1
1
Since (1 )
0, so
d (A , (A )) 0.
1
But
1
d (A , (A )) 0
1
hence
1
d (A , ( A )) 0.
1
1
Thus ( A ) A .
1
1
Proposition 4 Let N S (U E ) be a non-empty set of neutrosophic points and ( N S (U E ), d ) be a complete neu
trosophic soft metric space. Suppose is a mapping from N S (U E ) to N S (U E ) be kanan contraction. Then
fixed point of exists and unique.
Proof Let
form
A N S (U E ) be arbitrary. Define A ( A ) and by continuing we have a sequence in the
1
0
0
A ( A ). Now
n 1
n
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d ( A , A ) d ( ( A ), ( A ))
n 1
n
n
n 1
k[d ( A , ( A )) d ( A , ( A ))
n
n
n 1
n 1
k[d ( A , A ) d ( A , A )]
n
n 1
n 1
n
kd ( A , A ) kd ( A , A )
n
n 1
n 1
n
(1 k )d ( A , A ) kd ( A , A )
n 1
n
n
n 1
k
d ( A , A )
d ( A , A )
n 1
n
n
n 1
1 k
hd ( A , A )
n
for
h
n 1
k
1k
d ( A , A ) hd ( A , A )
n 1
n
n
2
3
n 1
h d ( A , A )
n 1
n2
h d ( A , A )
n2
n 3
.
.
.
For
h n d ( A , A ).
1
mn
0
d ( A , A ) d ( A , A ) d ( A , A ) ... d ( A , A )
n
m
n
n
n 1
n 1
h d ( A , A ) h
1
n 1
0
n
2
h [1 h h ... h
n2
m 1
d ( A , A ) ... h
1
m n 1
0
m 1
m
d ( A , A )
1
0
]d ( A , A )
1
0
1
)d ( A , A )
hn (
1
0
1 h
d ( A , A ) 0 as n .
n
m
The sequence
A is a cauchy sequence in ( N S (U E ), d ) . Since ( N S (U E ), d ) is complete, so A converges
n
n
to any A N S (U E ). Now
d ( ( A ), A ) d ( ( A ), ( A ))
n 1
n
Taking limit as
n ,
h[d ( A , ( A )) d ( A , ( A ))].
n
n
we have
d ( (A ), A ) h[d (A , (A )) d (A , (A ))]
2hd (A , (A ))
(1 2h)d ( (A ), A ) 0
As (1 2h) 0, so
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d ( (A ), A ) 0
but
d ( (A ), A ) 0
thus
d ( (A ), A ) 0.
Hence A N S (U E ) is a FP of .
Suppose B N S (U E ) be another FP. Now
d (A , B ) d ( (A ), (B ))
h[d (A , (A )) d (B , (B ))]
h[d (A , A ) d (B , B )]
d (A , B ) 0
(1)
d (A , B ) 0.
(2)
but
From (1) and ( 2) we have
d (A , B ) 0.
A B .
Hence
Proposition 5 Let , : N S (U E ) N S (U E ) be weakly compatible maps. If and have unique coin-
cidence point. Then and have unique common fixed point (CFP).
Proof Suppose there is
compatible, so
A N S (U E ) such that ( A ) ( A ) B . Since and are weakly
1
1
1
1
( ( A )) ( ( A ))
1
1
A N S (U E ). Now
for all
1
( B ) ( B ) ( (A )) ( (A )).
1
So
1
1
1
B is also coincidence point (CP) of and , but A is the unique CP of and , so
1
1
(A ) (A ) (B ) (B )
1
1
1
1
B (B ) (B ).
1
1
1
So
B N S (U E ) is CFP.
1
Proposition 6 Let ( N S (U E ), d ) be a complete metric space and : N S (U E ) N S (U E ) be a mapping
satisfies d ( ( A ), ( A )) kd ( ( A ), A ) for all
2
1
1
1
1
A N S (U E ) and k [0,1). Then fixed point of
1
is singleton.
Proof Let
A N S (U E ) be arbitrary and defines A n ( A ) ( A ). Now
n
0
0
n 1
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d ( n 1 ( A ), n ( A )) kd ( n ( A ), n 1 ( A ))
0
0
0
k d (
n 1
k d (
n 3
2
3
0
( A ),
n2
( A ))
( A ),
n4
( A ))
0
0
0
0
.
.
.
k n d ( ( A ), A ).
0
Now for
0
mn
d ( m ( A ), n ( A )) d ( n ( A ), n 1 ( A )) d ( n 1 ( A ), n 2 ( A ))
0
0
0
... d (
0
m 1
0
0
0
k d ( ( A ), A ) k
n
0
... k
m 1
0
( A ), ( A ))
m
n 1
0
d ( ( A ), A )
0
0
d ( ( A ), A )
0
n
0
2
k [1 k k ... k
m n 1
]d ( ( A ), A )
0
0
n
k
d ( ( A ), A )
0
0
1 k
m
n
d ( ( A ), ( A )) 0 as n .
0
So
n (A )
0
0
is a cauchy sequence in ( N S (U E ), d ), but ( N S (U E ), d ) is complete, so every cauchy sequence
is convergent that is
taking limit as
n (A ) A
n ,
0
0
as
n . Now
d ( n1 (A ), (A )) kd ( n (A ), A )
0
0
0
0
we have
d ( A , ( A )) kd ( A , A )
0
0
0
0
d ( A , ( A )) 0
but
0
0
d ( A , ( A )) 0
0
0
d ( , ( )) 0
A0
A0
( A ) A .
0
0
Hence
A N S (U E ) is the FP of .
0
Now suppose
B N S (U E ) is another FP of with ( B ) B , then
0
0
0
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d ( B , A ) d ( ( B ), ( A ))
0
0
0
0
d ( ( ( B )), ( A ))
0
0
kd ( ( B )), ( A ))
0
0
kd ( B , A )
0
0
(1 k )d ( B , A ) 0.
0
0
As (1 k ) 0, so
d ( B? , A? ) 0
(1)
d ( B , A ) 0.
(2)
0
but
0
0
0
From (1) and ( 2) we have
d ( B , A ) 0
0
0
B A .
0
0
Hence the FP is unique.
Proposition 7 Let , : N S (U E ) N S (U E ) be commuting maps. If and have unique coincidence
point. Then and have unique common fixed point.
Proof Suppose there is
incidence point, so let
A N S (U E ) such that ( ( A )) ( ( A )). Since and have unique co1
1
1
( A ) ( A ) B . Now
1
1
1
(B ) ( (A )) ( (A )) (B ).
1
1
1
Here
1
B N S (U E ) is also a coincidence point, but A N S (U E ) is unique coincidence point, so
1
1
( B ) ( B ) ( A ) (A ) B .
1
1
1
1
1
Hence
B N S (U E ) is also a fixed point.
1
Proposition 8 Every neutrosophic soft identity map is non-expansive.
from N S (U E ) to N S (U E ) be a neutrosophic soft identity map such that
Proof Suppose that I
I ( A ) A for all A N S (U E ). Now
1
1
1
d ( I ( A ), I ( B )) d ( A , B )
Here k 1, so
I
1
1
1
1
is non-expansive map.
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545
5.
Conclusion
In this paper, we have discussed some new mappings of NSS and some basic results and particular
examples. Like fixed point, here also present some new concepts of points that is coincidence point, periodic
point and CFP.
FP theory has a lot of applications in control and communicating system. FP theory is an important mathematical instrument used to demonstrate the existence of a solution in mathematical economics and game theory. So
the notion of a neutrosophic soft fixed point can be used in these areas. For stabilization of dynamic systems,
neutrosophic soft fixed point can be used. In addition, dynamic programming may employ the notion of presence and uniqueness of the common solution of neutrosophic soft set.
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Madad Khan, Muhammad Zeeshan, Saima Anis, Abdul Sami Awan and Florentin Smarandache, Neutrosophic Soft Fixed Points
Neutrosophic Sets and Systems, Vol. 35, 2020
546
[25] M. Riaz and M. R. Hashmi, “Fixed points of fuzzy neutrosophic soft mapping with decision-making,” Fixed Point Theory
Appl., vol. 2018, no. 1, p. 7, 2018.
[26] T. Bera and N. K. Mahapatra, On neutrosophic soft metric space. Infinite Study, 2018.
[27] T. Bera and N. K. Mahapatra, “Compactness and Continuity On Neutrosophic Soft Metric Space.”
Received: Apr 22, 2020. Accepted: July 10 2020
____________________________________________________________________________________________________________
Madad Khan, Muhammad Zeeshan, Saima Anis, Abdul Sami Awan and Florentin Smarandache, Neutrosophic Soft Fixed Points
Neutrosophic Sets and Systems, Vol. 35, 2020
University of New Mexico
Selection of Alternative under the Framework of Single-Valued Neutrosophic
Sets
M. Sarwar Sindhu1 , Tabasam Rashid2,∗ and Agha Kashif3
1
Department of Mathematics University of Management and Technology, Lahore - 54770, Pakistan;
sarwartajdin@gmail.com
2
Department of Mathematics University of Management and Technology, Lahore - 54770, Pakistan;
tabasam.rashid@umt.edu.pk
3
Department of Mathematics University of Management and Technology, Lahore - 54770, Pakistan;
kashif.khan@umt.edu.pk
∗
Correspondence: tabasam.rashid@umt.edu.pk
Abstract. The multiple criteria decision making (MCDM) problems indicate the alternatives which have more
or less resemblance to each other. An important mathematical tool used by decision-makers (DMs) to quantify these resemblances is the similarity measure (SM). SM is a powerful tool that measures the resemblance
more accurately. Mostly, fuzzy sets (FSs) and its extensions handle the vague and uncertain information by
considering the membership, non-membership, and indeterminacy degrees whose sum always lies in the interval
[0, 1]. However, single-valued neutrosophic sets (SVNSs) and interval-valued neutrosophic sets (IVNSs) have
information whose sum is bounded in [0, 3]. In the present work, we extended the SM presented by William and
Steel for SVNSs and IVNSs by using the concept of Euclidean distance. The weights of criteria indicate much
influence for the selection of the best alternative, sometimes DMs feel hesitation to allocate the weights to the
criteria. We applied the linear programming (LP) model to evaluate the weights of the criteria to reduce the
hesitancy. Later on, SM is utilized to establish an MCDM model for the selection of the best option. Moreover,
the Spearman’s rank correlation coefficient is implemented to analyze the ranking order. Finally, a medical
diagnosis example is illustrated for the feasibility and effectiveness of the proposed model.
Keywords: picture fuzzy sets; fuzzy sets; similarity measure; neutrosophic sets; linear programming model.
—————————————————————————————————————————-
1. Introduction
Most of the information provided to the experts or decision makers (DMs) are ambiguous
and uncertain. DMs handle such information precisely by using the fuzzy sets (FSs) theory
presented by Zadeh [31] in 1965. FSs contain a single value in its specification, called a membership degree (MDg) which is always bounded in the closed interval [0, 1]. FSs have been
broadly used in different fields, for example, medical diagnosis, image processing, etc. [12, 17].
Sindhu et al., Selection of Alternative under the Framework of SVNSs
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548
In various ambiguous decision making problems, the MDg is assumed not exactly as a numerical value but as an interval. Therefore, Zadeh [32] introduced the interval-valued fuzzy sets
(IVFSs), an augmentation of FSs. Though, the FSs and IVFSs only have the MDg , and they
cannot designate the non membership degree (NMDg) of the element belonging to the set.
Consider that in a competition of university’s postgraduate students, a board of seven experts
evaluate the efficiency of a student. According to three experts a student can be accepted
for admission, according to two experts he or she is rejected and the remaining two experts
remained impartial. In such circumstances, FSs and IVFSs could not handle the vagueness
and uncertainty precisely. Atanassov [6] further extended the notion of FSs into intuitionistic
fuzzy sets (IFSs) to cope such problems which comprise both MDg and NMDg in its structure
so that, 0 ≤ M Dg + N M Dg ≤ 1. Most rapidly, IFSs become an important device to deal with
the imprecise and ambiguous information than the FSs and IVFSs.
In spite of the fact that, IFSs have been successfully implemented in distinct fields, however,
IFSs were not covering the human’s attitude perfectly. Casting of vote is an excellent example
of such type of attitude, we may divide the voters into four groups: vote for, vote against,
neutral and refusal of voting. When a person refuses to vote, we can say that the person is
not anxious about the general election. Cuong [11] focused such types of human’s attitude by
presenting the idea of picture fuzzy sets Pc F Ss, the generalized form of IFSs. Pc F Ss have
three components in its formation called, MDg , NMDg and of degree refusal (Dg R) such that,
0 ≤ M Dg + N M Dg + Dg R ≤ 1. But Pc F Ss also have some limitations to express the decision
information. For instance, three groups of decision makers (DMs) assess the advantages of
a new business. First group predicts that the business will be profitable is 0.7, according to
second group the possibility of loss is 0.2 and the third group is not sure whether the business
will be profitable is 0.4. In this scenario, Pc F Ss cannot handle the information because,
0.7 + 0.2 + 0.3 = 1.2 > 1.
Therefore, to handle such situations Wang et al. [22] introduced an amazing concept of singlevalued neutrosophic sets (SVNSs) that consists of three degrees, the truth-membership (Tn (x))
degree, indeterminacy-membership (In (x)) degree, and falsity membership (Fn (x)) degree in
the closed interval [0, 1] so that it satisfy the condition, 0 ≤ Tn (x) + In (x) + Fn (x) ≤ 3. Later
on, Wang [23] described these three degrees in the form of an interval, called an intervalvalued neutrosophic sets (IVNSs). Nowadays, NSs have become the center of the eye of the
researcher due to its innovation. Many researchers are trying to print it for example, AbdelBasset et.al [1–4] used the score and accuracy functions of trapezoidal neutrosophic numbers to
minimize the cost of projects under uncertain environmental conditions, in order to tackle the
ambiguity and uncertainty present in the data for MCDM problems, utilized the plithogenic
Sindhu et al., Selection of Alternative under the Framework of SVNSs
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549
set, a generalization of NSs, a novel hybrid neutrosophic MCDM model is presented on the
basis of TOPSIS by using bipolar neutrosophic numbers and resolve the supply chain issues
with the help of best-worst method (evaluating weights) and plithogenic set, respectively.
SM is one of the vital and powerful tools that measures the level of resemblance among
the objects. In order to show the preference strength among the alternatives, the similarity
measures have achieved more attention from the DMs since the previous few decades. Various
DMs have presented a number of similarity measures for MCDM problems to select the most
favorable alternative from the various options having identical features under the certain criteria. For example, Beg and Ashraf discussed the various characteristic of similarity measures
under the framework of FSs [7]. Ye [28–30] introduced the cosine similarity measures (vector
similarity) and implemented it to pattern recognition and medical diagnosis under the environments of simplified neutrosophic sets, interval neutrosophic sets and IFSs. Intarapaiboon [14]
applied two new similarity measures to pattern recognition in IFSs situations. Moreover, Song
and Hu [20] established two measures of similarity between hesitant fuzzy linguistic term sets
and used it for MCDM problems. Recently, Wei and Gao [26] developed the generalized Dice
similarity measures for Pc F Ss and implemented for pattern recognition. Consequently, Wang
et al. [24] presented the generalized Dice similarity measures for Pythagorean fuzzy sets and
used it in multiple attribute group decision making.
The linear programming (LP) model introduced by Vanderbei [21], permits some target function to be minimized or maximized inside the system of given situational limitations. LP is a
computational technique that enables DMs to solve the problems which they face in decisionmaking model. It encourages the DMs to deal with constrained ideal conditions which they
need to make the best of their resources. Various experts utilized LP model in MCDM for
different extensions of FSs [5, 10, 13, 18, 25]. Recently, Sindhu et al. [19] implemented the LP
methodology with extended TOPSIS (technique for order of preference by similarity to ideal
solution) for picture fuzzy sets. The weights of criteria appear to specify that the DMs identify
the significance of people views and its influence on attaining the objective. Sometimes DMs
hesitate or confused to allocate the weights to criteria. Thereby, we applied TOPSIS to get the
objective function and then find out the weights of criteria under some constraints by using
LP model. The novelty of this article is concerned about proposing the SM to overcome the
shortcoming present in the existing technique. The following are the major contributions of
this study:
• William and Steel SM is extended on the basis of novel distance measure.
• Evaluate the objective function by using TOPSIS.
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• Weights of criteria are calculated with the help of LP model.
• An MCDM model is developed on the basis of SM and implemented it for medical
diagnosis under the framework of SVNSs and IVNSs.
• Spearman’s rank-correlation coefficient and the critical value are applied to strength
the proposed MCDM model.
Rest of the article is organized as: Section 2 encloses some preliminaries regarding SVNSs
and IVNSs. Various pre-existing similarity measures of SVNSs, IVNSs and their shortcoming
are elaborated in Section 3. The modified similarity measures for SVNSs and IVNSs are
described in Section 4. An MCDM model is proposed in Section 5 and the developed model
is then applied on an example of medical diagnosis in Section 6 to elaborate the validity and
effectiveness. A comprehensive comparative analysis based on Spearman’s rank correlation
coefficient is penned in Section 7. Conclusions and future work are highlighted in Section 8.
2. Preliminaries
A brief introduction of the notions F Ss, Pc F Ss, SV N S and IV N S and the LP model is
presented in this section.
Definition 2.1. [31] Let X = {x1 , x2 , ..., xn } be a discourse set. A fuzzy set (FS) A on X is
represented in terms of a functions m : X → [0, 1] such that
A = {hxi , mA (xi )i |xi ∈ X}.
Definition 2.2. [11] Let X = {x1 , x2 , ..., xn } be a fixed set. A picture fuzzy set Pc on X is
defined as:
Pc = {hxi , αPc (xi ), γPc (xi ), βPc (xi )i |xi ∈ X, i = 1, 2, ..., n},
where αPc (xi ), βPc (xi ), γPc (xi ) ∈ [0, 1] are called the acceptance membership, neutral and
rejection membership degrees of xi ∈ X to the set Pc , respectively and αPc (xi ), γPc (xi ) and
βPc (xi ) fulfil the condition: 0 ≤ αPc (xi ) + γPc (xi ) + βPc (xi ) ≤ 1, for all xi ∈ X. Also
ζPc (xi ) = 1−αPc (xi )−γPc (xi )−βPc (xi ), then ζPc (xi ) is said to be a degree of refusal membership
of xi ∈ X in Pc . For our convenience, we can write pi = (αPc (xi ), βPc (xi ), γPc (xi )) as the
picture fuzzy numbers (Pc F N s) over a set Pc , where i = 1, 2, ..., n.
Definition 2.3. [22] Let X = {x1 , x2 , ..., xn } be a fixed set. A SVNS Ns on X is defined as:
Ns = {hxi , αNs (xi ), γNs (xi ), βNs (xi )i |xi ∈ X, i = 1, 2, ..., n},
where αNs (xi ), γNs (xi ), βNs (xi ) ∈ [0, 1] are called the truth-membership, indeterminacy and
falsity- membership degrees of xi ∈ X to the set Ns , respectively and αNs (xi ), γNs (xi ) and
Sindhu et al., Selection of Alternative under the Framework of SVNSs
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551
βNs (xi ) fulfil the condition:
for all xi ∈ X then, 0 ≤ αNs (xi ) + γNs (xi ) + βNs (xi ) ≤ 3. Let Ns1 and Ns2 be two SVNS, then
following conditions hold:
(1) Ns1 ⊆ Ns2 iff αNs1 (xi ) ≤ αNs2 (xi ), βNs1 (xi ) ≥ βNs2 (xi ) and γNs1 (xi ) ≥ γNs2 (xi ),
(2) Ns1 = Ns2 iff Ns1 ⊆ Ns2 and Ns2 ⊆ Ns1 .
Definition 2.4. [23] Let X = {x1 , x2 , ..., xn } be a fixed set. An ISVNS Ñs on X is defined
as:
D
E
Ñs = { xi , αÑs (xi ), γÑs (xi ), βÑs (xi ) |xi ∈ X, i = 1, 2, ..., n},
l (x ), αu (x )] ⊆ [0, 1], γ (x ) = [γ l (x ), γ u (x )] ⊆ [0, 1], β (x ) =
where αÑs (xi ) = [αÑ
i
i
i
i
Ñs i
Ñs i
Ñ
Ñ
Ñ
s
s
s
s
l (x ), β u (x )] ⊆ [0, 1] are called the truth-membership, indeterminacy and falsity- mem[βÑ
i
i
Ñ
s
s
bership degrees of xi ∈ X to the set Ñs , respectively and satisfy the condition:
u (x ) + γ u (x ) + β u (x ) ≤ 3. Let Ñ 1 and Ñ 2 be two SVNS, then
for all xi ∈ X then, 0 ≤ αÑ
i
i
i
s
s
Ñ
Ñ
s
s
s
following conditions hold:
l (x ) ≤ αl (x ), αu (x ) ≤ αu (x ), β l (x ) ≥ β l (x ), β u (x ) ≥
(1) Ñs1 ⊆ Ñs2 iff αN
i
1
N2 i
N1 i
N2 i
N1 i
N2 i
N1 i
s
s
s
s
s
s
s
u (x ), γ l (x ) ≥ γ l (x ) and γ u (x ) ≥ γ u (x ),
βN
i
2
N1 i
N2 i
N1 i
N2 i
s
s
s
s
s
(2) Ñs1 = Ñs2 iff Ñs1 ⊆ Ñs2 and Ñs2 ⊆ Ñs1 .
Definition 2.5. [21]. The linear programming model is constructed as:
Maximize: Z = c1 t1 + c2 t2 + c3 t3 + ... + cn tn
Subject to: a11 t1 + a12 t2 + a13 t3 + ... + a1n tn ≤ b1
a21 t1 + a22 t2 + a23 t3 + ... + a2n tn ≤ b2
..
.
am1 t1 + am2 t2 + am3 t3 + ... + amn tn ≤ bm
t1 , t2 , ..., tn ≥ 0,
where m and n denotes the cardinalities of the constraints and decision variables t1 , t2 , ..., tn ,
respectively. A solution (t1 , t2 , ..., tn ) is called feasible point if it fulfils all of the restrictions.
LP model is used to find the optimal solution of the decision variables to maximize or minimize
the linear function Z.
3. Some existing similarity measures for SVNSs and IVNSs
Similarity measure is a most widely used tool to evaluate the relationship between two sets.
Two sets are said to be perfectly similar if similarity measure between them is exactly 1. The
following are the compulsory axioms for the sets (SVNSs or IVNSs) to be perfectly similar:
Sindhu et al., Selection of Alternative under the Framework of SVNSs
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Definition 3.1. Let X = {x1 , x2 , ..., xn } be a universal set and Ns1 = {< xi , αNs1 (xi ), γNs1 (xi ),
βNs1 (xi )} and Ns2 = {< xi , αNs1 (xi ), γNs2 (xi ), βNs2 (xi ) >} be two SVNS, where, i = 1, 2, ..., n.
Then,
(1) 0 ≤ S(Ns1 , Ns2 ) ≤ 1,
(2) S(Ns1 , Ns2 ) = S(Ns2 , Ns1 ),
(3) S(Ns1 , Ns2 ) = 1 if and only if Ns1 = Ns2 .
A cosine similarity measure S(Ns1 , Ns2 ) of SVNS presented by Ye [29] is given as:
S(Ns1 , Ns2 ) =
(αN 1 (xi ))(αN 2 (xi ))+(γN 1 (xi ))(γN 2 (xi ))+(βN 1 (xi ))(βN 2 (xi ))
s
s
s
s
s
q
q s
.
[ (αN 1 (xi ))2 +(γN 1 (xi ))2 +(βN 1 (xi ))2 ][ (αN 2 (xi ))2 +(γN 2 (xi ))2 +(βN 2 (xi ))2 ]
s
s
s
s
s
s
Suppose that Ns1 = (x, 0.4, 0.2, 0.6) and Ns2 = (x, 0.2, 0.1, 0.3) are two SVNSs, the Definition
2.3 shows that Ns1 6= Ns2 . However, by using cosine similarity measure presented by Ye [29],
we see that, S(Ns1 , Ns2 ) = 1, show the contradiction of the property 3 of Definition 3.1 which
describe that S(Ns1 , Ns2 ) = 1 if and only if Ns1 = Ns2 . Similarly, if we take, αNs1 (xi ) =
(k + 1)αNs2 (xi ), γNs1 (xi ) = (k + 1)γNs2 (xi ) and βNs1 (xi ) = (k + 1)βNs2 (xi ), where k ≥ 1, then
according to cosine similarity measure, its value is:
S(Ns1 , Ns2 ) =
(αN 1 (xi ))(αN 2 (xi ))+(γN 1 (xi ))(γN 2 (xi ))+(βN 1 (xi ))(βN 2 (xi ))
s
s
s
s
s
q s
q
,
[ (αN 1 (xi ))2 +(γN 1 (xi ))2 +(βN 1 (xi ))2 ][ (αN 2 (xi ))2 +(γN 2 (xi ))2 +(βN 2 (xi ))2 ]
s
S(Ns1 , Ns2 )
=
s
s
s
S(Ns1 , Ns2 )
=
s
s
s
((k+1)αN 2 (xi ))(αN 2 (xi ))+((k+1)γN 2 (xi ))(γN 2 (xi ))+((k+1)βN 2 (xi ))(βN 2 (xi ))
s
s
s
s q
s
s
q
,
[ ((k+1)αN 2 (xi ))2 +((k+1)γN 2 (xi ))2 +((k+1)βN 2 (xi ))2 ][ (αN 2 (xi ))2 +(γN 2 (xi ))2 +(βN 2 (xi ))2 ]
s
s
(k+1)(αN 2 (xi ))2 +(γN 2 (xi ))2 +(βN 2 (xi ))2 )
s
s
s
s
s
s
(k+1)((αN 2 (xi ))2 +(γN 2 (xi ))2 +(βN 2 (xi ))2 )
s
s
s
=1, which again opposes the property 3 of
Definition 3.1.
Further, if Ns1 = (0, 0, 0) and Ns2 = (0, 0, 0) are two SVNS then according to Jaccrd and Dice
similarity measures presented in [29] become undefined or meaningless.
Same as, if Ñs1 = (y, [0.3, 0.4], [0.2, 0.3], [0.4, 0.5]) and
Ñs1 = (y, [0.6, 0.8], [0.4, 0.6], [0.8, 1]) are two IVNSs, then according to Definition 2.4, Ñs1 6= Ñs2 ,
but the similarity measure presented by Ye [30] gives that, S(Ñs1 , Ñs2 ) = 1, that is, Ñs1 = Ñs2
which again presents a contradiction with property 3 of Definition 3.1. Also for two IVNSs,
Ñs1 = [0, 0] and Ñs2 = [0, 0], we get the meaningless or undefined results by using Equation 9
presented in [15]. So the similarity measures presented in [15, 29, 30] have a deficiency.
Hence, from the above discussion, it is clear that the existing similarity measures have some
drawbacks and cannot be able to select the best alternative. Consequently, there is a need to
improve the similarity measure which satisfy the axiom of Definition 3.1.
4. Proposed similarity measures for SVNSs and IVNSs
In order to overcome the deficiencies present in the above discussed similarity measures, we
extend a similarity measure presented by William and Steel [27] for the SVNSs (IVNSs) based
on the novel distance measure as:
Sindhu et al., Selection of Alternative under the Framework of SVNSs
Neutrosophic Sets and Systems, Vol. 35,2020
n
D(Ns1 , Ns2 )
553
!
αNs1 (xi ) − αNs2 (xi ) + γNs1 (xi ) − γNs2 (xi ) + βNs1 (xi ) − βNs2 (xi ) +
,
max αNs1 (xi ) − αNs2 (xi ) , γNs1 (xi ) − γNs2 (xi ) , βNs1 (xi ) − βNs2 (xi )
1 X
=
3n
i=1
(1)
i
Sm
(Ns1 , Ns2 ) = e
1
−n
D(Ns1 ,Ns2 )
,
(2)
where n is the number of alternatives and 1 ≤ i ≤ n.
Similarly for the IVNSs the distance and similarity measures are:
l (x ) − αl (x )| + |αu (x ) − αu (x )|+
[|αÑ
i
1
Ñ 2 i
Ñ 1 i
Ñ 2 i
s
s
s
s
|γ l (xi ) − γ l (xi )| + |γ u (xi ) − γ u (xi )|+
1
2
1
2
Ñs
Ñs
Ñs
Ñs
n |β l (x ) − β l (x )| + |β u (x ) − β u (x )|]+
X
1
Ñs1 i
Ñs2 i
Ñs1 i
Ñs2 i
D̃(Ñs1 , Ñs2 ) =
l
l
u
max[|α 1 (xi ) − α 2 (xi )|, |α 1 (xi ) − αu 2 (xi )|,
3n
i=1
Ñs
Ñs
Ñs
Ñs
l
l
u
u
|γÑ 1 (xi ) − γÑ 2 (xi )|, |γÑ 1 (xi ) − γÑ 2 (xi )|
s
s
s
s
l (x ) − β l (x )|, |β u (x ) − β u (x )|]
, |βÑ
i
i
i
1
2
1
Ñ
Ñ
Ñ 2 i
s
s
i
(Ñs1 , Ñs2 )
S̃m
=e
1
−n
D̃(Ñs1 ,Ñs2 )
s
,
s
.
i (N 1 , N 2 ) defined in Equation (2) amongst N 1
4.1. The SM Sm
s
s
s
Theorem
(3)
(4)
=
{ xi , αNs1 (xi ), γNs1 (xi ), βNs1 (xi ) } and Ns2 = { xi , αNs2 (xi ), γNs2 (xi ), βNs2 (xi ) } satisfies the
given properties:
i (N 1 , N 2 ) = 1 if and only if N 1 = N 2 ,
(1) Sm
s
s
s
s
i (N 1 , N 2 ) = S i (N 2 , N 1 ),
(2) Sm
s
s
m
s
s
i (N 1 , N 2 ) ≤ 1.
(3) 0 ≤ Sm
s
s
Proof
(1) Suppose that, Ns1 = Ns2 that is, αNs1 (xi ) = αNs2 (xi ), γNs1 (xi ) = γNs2 (xi ) and
βNs1 (xi ) = βNs2 (xi ), then by using Equation (2), we have
i
Sm
(Ns1 , Ns2 ) = e0 = 1.
1
1
2
i (N 1 , N 2 ) = e− n D(Ns ,Ns )
(2) Consider Sm
s
s
=
−
1
3n2
Pn
i=1
e
=
−
1
3n2
e
=e
Pn
i=1
αNs1 (xi ) − αNs2 (xi ) + γNs1 (xi ) − γNs2 (xi ) + βNs1 (xi ) − βNs2 (xi ) +
max αNs1 (xi ) − αNs2 (xi ) , γNs1 (xi ) − γNs2 (xi ) , βNs1 (xi ) − βNs2 (xi )
αNs2 (xi ) − αNs1 (xi ) + γNs2 (xi ) − γNs1 (xi ) + βNs2 (xi ) − βNs1 (xi ) +
max αNs2 (xi ) − αNs1 (xi ) , γNs2 (xi ) − γNs1 (xi ) , βNs2 (xi ) − βNs1 (xi )
1
−n
D(Ns2 ,Ns1 )
i (N 2 , N 1 ),
= Sm
s
s
Sindhu et al., Selection of Alternative under the Framework of SVNSs
,
,
Neutrosophic Sets and Systems, Vol. 35,2020
554
i (N 1 , N 2 ) ≤ 1 and it become zero
(3) From Equations (1) and (2), it is obvious that, Sm
s
s
i (N 1 , N 2 ) = 0 only when the distance between N 1 and N 2 is very large.
i.e., Sm
s
s
s
s
Example 4.2. Let Ns1 = (x, 0.4, 0.2, 0.6) and Ns2 = (x, 0.2, 0.1, 0.3) be two SVNSs, then by
i (N 1 , N 2 ) = 0.7408.
using Equations (1) and (2), the similarity measure is, Sm
s
s
Example 4.3. Let Ñs1 = (x, [0.3, 0.4], [0.2, 0.3], [0.4, 0.5]) and Ñs2 = (x, [0.6, 0.8], [0.4, 0.6],
i (Ñ 1 ,
[0.8, 1]) be two IVNSs, then by using Equations (3) and (4), the similarity measure is, Sm
s
Ñs2 ) = 0.3679.
i (Ñ 1 , Ñ 2 ) defined in Equation (4) amongst Ñ 1
=
Theorem 4.4. The SM S̃m
s
s
s
E
D
E
D
{ xi , αÑ 1 (xi ), γÑ 1 (xi ), βÑ 1 (xi ) } and Ñs2 = { xi , αÑ 2 (xi ), γÑ 2 (xi ), βÑ 2 (xi ) } satisfies the
s
s
s
s
s
s
given properties:
i (Ñ 1 , Ñ 2 ) = 1 if and only if Ñ 1 = Ñ 2 ,
(1) S̃m
s
s
s
s
i (Ñ 1 , Ñ 2 ) = S̃ i (Ñ 2 , Ñ 1 ),
(2) S̃m
s
s
m
s
s
i (Ñ 1 , Ñ 2 ) ≤ 1.
(3) 0 ≤ S̃m
s
s
Proof The proof of this Theorem is obvious.
4.1. Proposed weighted similarity measures (WSM) for SVNSs and IVNSs
Since the weights of the criteria have a great impact in making decision process therefore we
can further extend the proposed similarity measures into the WSM. Let w = (w1 , w2 , ..., wm )T
P
iw
1
2
be a weight vector of the m criteria with m
j=1 wj = 1. In order to get WSM Sm (Ns , Ns ) for
SVNSs, we first define the weighted distance as:
D
w
(Ns1 , Ns2 )
=
n X
m
X
i=1 j=1
!
αNs1 (xi ) − αNs2 (xi ) + γNs1 (xi ) − γNs2 (xi ) + βNs1 (xi ) − βNs2 (xi ) +
,
max αNs1 (xi ) − αNs2 (xi ) , γNs1 (xi ) − γNs2 (xi ) , βNs1 (xi ) − βNs2 (xi )
wj
(5)
and
1
iw
(Ns1 , Ns2 ) = e− n D
Sm
w (N 1 ,N 2 )
s
s
.
(6)
iw (Ñ 1 , Ñ 2 ) on the basis of weighted distance D̃ w (Ñ 1 , Ñ 2 ) for
In the similar way, a WSM S̃m
s
s
s
s
IVNSs is obtained as:
l (x ) − αl (x )| + |αu (x ) − αu (x )|+
[|αÑ
i
1
Ñ 2 i
Ñ 1 i
Ñ 2 i
s
s
s
s
|γ l (xi ) − γ l (xi )| + |γ u (xi ) − γ u (xi )|+
Ñs1
Ñs2
Ñs1
Ñs2
l
l
u
n
m
X X |β 1 (xi ) − β 2 (xi )| + |β 1 (xi ) − β u 2 (xi )|]+
Ñs
Ñs
Ñs
Ñs
wj
D̃w (Ñs1 , Ñs2 ) =
max[|αl 1 (xi ) − αl 2 (xi )|, |αu 1 (xi ) − αu 2 (xi )|,
i=1 j=1
Ñs
Ñs
Ñs
Ñs
l (x ) − γ l (x )|, |γ u (x ) − γ u (x )|
|γÑ
i
1
Ñs2 i
Ñs1 i
Ñs2 i
s
l
l
u
u (x )|]
, |βÑ 1 (xi ) − βÑ 2 (xi )|, |βÑ 1 (xi ) − βÑ
i
2
s
s
s
Sindhu et al., Selection of Alternative under the Framework of SVNSs
s
,
(7)
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555
and
1
iw
(Ñs1 , Ñs2 ) = e− n D̃
S̃m
w (Ñ 1 ,Ñ 2 )
s
s
.
(8)
Theorem 4.5. Let Ns1 = {< xi , αNs1 (xi ), γNs1 (xi ), βNs1 (xi ) >} and Ns2 = {< xi , αNs2 (xi ),
γNs2 (xi ), βNs2 (xi ) >} be two SVNSs (IVNSs) , then the WSM presented in Equation (6) (Equation (8)) between two SVNSs (IVNSs) satisfies the following properties:
iw (N 1 , N 2 ) ≤ 1,
(1) 0 ≤ Sm
s
s
iw (N 1 , N 2 ) = S iw (N 2 , N 1 ),
(2) Sm
s
s
m
s
s
iw (N 1 , N 2 ) = 1 if and only if N 1 = N 2 .
(3) Sm
s
s
s
s
Proof It is obvious as Theorem 4.1.
Example 4.6. Let Ns1 = {x, (0.3, 0.2, 0.5), (0.4, 0.6, 0.0)} and Ns2 = {x, (0.1, 0.1, 0.8),
(0.2, 0.1, 0.7)} be two SVNSs and w = (0.7, 0.3)T the weight vector, then the WSM for SVNSs
iw (N 1 , N 2 ) = 0.9162.
is: Sm
s
s
Example 4.7. Let Ñs1 = {x, ([0.4, 0.6], [0.2, 0.3], [0.3, 0.4]), ([0.5, 0.8], [0.1, 0.4], [0.1, 0.3])} and
Ñs2 = {x, ([0.7, 0.9], [0.1, 0.2], [0.1, 0.2]), ([0.3, 0.6], [0.1, 0.3], [0.4, 0.7])} be two IVNSs and w =
iw (N 1 , N 2 ) =
(0.6, 0.4)T the weight vector, then the weighted similarity measure for IVNSs is: Sm
s
s
0.8781.
5. Decision making model under SVNSs (IVNSs)
The model for MCDM problems is presented on the basis of proposed weighted similarity
measure in this section. Suppose that Q = {Q1 , Q2 , ..., Qn } is a discrete set of alternatives and
G = {G1 , G2 , ..., Gm } is another discrete set of criteria. If the DMs gave the various values for
the alternative Qi (i = 1, 2, ..., n) under the criteria Gj (j = 1, 2, ..., m), and form a neutrosofic
decision matrix N = [bij ]n×m . The concept of optimal solution assists the DMs to identify the
best alternative from the decision set in MCDM framework. In spite of the fact that the perfect option does not exist in actual, it provides a valuable paradigm to appraise alternatives.
Hence, we can find the ideal options N ? from the given information as N ? = max([bij ]n×m ).
Since the weights of the criteria have an excessive impact, thereby a weighing vector of criteria
P
is provided as w = (w1 , w2 , w3 , ..., wm )T , where m
j=1 wj = 1 and wj > 0, can be evaluated
by using the LP model presented in Definition 2.5. The model based on proposed weighted
similarity measure described by Equation (6) (Equation (8)) has the following steps.
Step 1. Based on the information provided by DMs, form a single valued neutrosophic decision matrix (SVNDM) denoted by N = [bij ]n×m .
Step 2. Find the optimal solution N ? from the SVNDM.
Step 3. On the basis of TOPSIS, an objective function is obtained and then calculate the
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556
weights of criteria by using LP model as described in Definition 2.5.
Step 4. With the aid of weights evaluated in Step 3, calculate the similarity measures amongst
the alternative Qi (i = 1, 2, ..., n) and the optimal alternative N ? by using Equation (6) (Equation (8)).
Step 5. Rank all the alternatives Qi (i = 1, 2, ..., n) from highest to lowest values of similarity
measures obtained in Step 4 and choose the alternative having highest value of the similarity
measure.
6. Practical examples
In this section, a medical diagnosis decision problem is considered to see the validity and
effectiveness of the proposed MCDM model.
Example 1. For parents, it is significant to be aware of the most updated treatment process
so you can be certain about your kids are getting the superlative care possible. According
to the child specialist, some common childhood sicknesses and their appropriate symptoms
are listed. Suppose a collection of diagnoses, chest infections (C), malaria (M ), typhoid (T ),
sore throat (S) and bronchitis (B) are examined on the basis of some symptoms, fever (S1 ),
headache (S2 ), breathlessness (S3 ), cough (S4 ) and chest pain (S5 ). All the information is
given in the form of neutrosophic decision matrix (NDM) N = [bij ]n×m . Assume that patient
K1 = N ? has all the symptoms in the diagnosis process, all the information collected about
the kids Ki (i = 1, 2, ..., n) is provided in the form of SVNS in Table 1.
Maximize: Z = 0.2175w1 + 0.2350w2 + 0.2200w3 + 0.1950w4 + 0.1850w5
Subject to: 10w1 + 8w2 + 12w3 + 10w4 + 15w5 ≥ 10,
10w1 + 8w2 + 12w3 + 10w4 + 15w5 ≤ 10.5,
8w1 + 11w2 + 7w3 + 10w4 + 10w5 ≥ 8,
8w1 + 11w2 + 7w3 + 10w4 + 10w5 ≤ 8.5,
12w1 + 15w2 + 12w3 + 10w4 + 6w5 ≥ 12,
12w1 + 15w2 + 12w3 + 10w4 + 6w5 ≤ 12.5,
w1 + w2 + w3 + w4 + w5 = 1,
w1 , w2 , ..., w5 ≥ 0.
Table 1. Neutrosophic decision matrix NDM
Daignosis
C
M
T
S
B
S1
<
<
<
<
<
0.4, 0.6, 0.0
0.7, 0.3, 0.0
0.3, 0.4, 0.3
0.1, 0.2, 0.7
0.1, 0.1, 0.8
S2
>
>
>
>
>
<
<
<
<
<
0.3, 0.2, 0.5
0.2, 0.2, 0.6
0.6, 0.3, 0.1
0.2, 0.4, 0.4
0.0, 0.2, 0.8
S3
>
>
>
>
>
<
<
<
<
<
0.1, 0.3, 0.7
0.0, 0.1, 0.9
0.2, 0.1, 0.7
0.8, 0.2, 0.0
0.2, 0.0, 0.8
S4
>
>
>
>
>
<
<
<
<
<
0.4, 0.3, 0.3
0.7, 0.3, 0.0
0.2, 0.2, 0.6
0.2, 0.1, 0.7
0.2, 0.0, 0.8
S5
>
>
>
>
>
<
<
<
<
<
0.1, 0.2, 0.7
0.1, 0.1, 0.8
0.1, 0.0, 0.9
0.2, 0.1, 0.7
0.8, 0.1, 0.1
>
>
>
>
>
Step 1. Based on the information provided by the professional, form a SVNDM N = [nij ]5×5 .
Step 2. Assume that a kid K1 = {(0.8, 0.2, 0.1), (0.9, 0.3, 0.2), (0.2, 0.1, 0.8), (0.6, 0.5, 0.1),
(0.1, 0.4, 0.6)} has all the symptoms in the process of diagnosis.
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557
Step 3. By using TOPSIS an objective function is obtained and then calculate the weights
of criteria by applying the LP model as described in Definition 2.5.
Step 4. The values of the weighted similarity measure calculated with the help of Equation
1w = 0.7774, S 2w = 0.7675, S 3w = 0.7969,
(6) amongst the diagnoses and the kid K1 are: Sm
m
m
4w = 0.6353 and S 5w = 0.6127.
Sm
m
Step 5. According to values obtained in Step 4, we get the ranking order as: T C M
B S. Figure 1 indicates the ranking order presented in [8, 9, 16, 29] and the proposed model
graphically.
Figure 1. Ranking order of alternatives
Example 2. Consider the same scenario as Example 1 with interval-valued data provided in
Table 2. Assume that another Kid K2 suffers from all the symptoms, which can be expressed
by the following IVNS data.
Step 1. Based on the information given by the professional form an interval-valued neutrosofic decision matrix (INDM) denoted by Ñ = [ñij ]5×5 .
Step 2. Assume a kid K2 = {([0.3, 0.5], [0.2, 0.3], [0.4, 0.5]), ([0.7, 0.9], [0.1, 0.2], [0.1, 0.2]),
([0.4, 0.6], [0.2, 0.3], [0.3, 0.4]), ([0.3, 0.6], [0.1, 0.3], [0.4, 0.7]), ([0.5, 0.8], [0.1, 0.4], [0.1, 0.3])} has
all the symptoms in the process of diagnosis.
Step 3. Use the same weights for the symptoms which are evaluated in Example 1.
Step 4. The values of the weighted similarity measure calculated with the help of Equation
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558
Table 2. Neutrosofic decision matrix NDM
Daignosis
S1
S2
S3
C
M
T
S
B
([0.4, 0.4], [0.6, 0.6], [0.0, 0.0])
([0.7, 0.7], [0.3, 0.3], [0.0, 0.0])
([0.3, 0.3], [0.4, 0.4], [0.3, 0.3])
([0.1, 0.1], [0.2, 0.2], [0.7, 0.7])
([0.1, 0.1], [0.1, 0.1], [0.8, 0.8])
([0.3, 0.3], [0.2, 0.2], [0.5, 0.5])
([0.2, 0.2], [0.2, 0.2], [0.6, 0.6])
([0.6, 0.6], [0.3, 0.3], [0.1, 0.1])
([0.2, 0.2], [0.4, 0.4], [0.4, 0.4])
([0.0, 0.0], [0.2, 0.2], [0.8, 0.8])
([0.1, 0.1], [0.3, 0.3], [0.7, 0.7])
([0.0, 0.0], [0.1, 0.1], [0.9, 0.9])
([0.2, 0.2], [0.1, 0.1], [0.7, 0.7])
([0.8, 0.8], [0.2, 0.2], [0.0, 0.0])
([0.2, 0.2], [0.0, 0.0], [0.8, 0.8])
Daignosis
S4
S5
C
M
T
S
B
([0.4, 0.4], [0.3, 0.3], [0.3, 0.3])
([0.7, 0.7], [0.3, 0.3], [0.0, 0.0])
([0.2, 0.2], [0.2, 0.2], [0.6, 0.6])
([0.2, 0.2], [0.1, 0.1], [0.7, 0.7])
([0.2, 0.2], [0.0, 0.0], [0.8, 0.8])
([0.1, 0.1], [0.2, 0.2], [0.7, 0.7])
([0.1, 0.1], [0.1, 0.1], [0.8, 0.8])
([0.1, 0.1], [0.0, 0.0], [0.9, 0.9])
([0.2, 0.2], [0.1, 0.1], [0.7, 0.7])
([0.8, 0.8], [0.1, 0.1], [0.1, 0.1])
1w = 0.6445, S̃ 2w = 0.5760, S̃ 3w = 0.7222,
(8) amongst the diagnoses and the kid K1 are: S̃m
m
m
4w = 0.6668 and S̃ 5w = 0.5884.
S̃m
m
Step 5. The ranking order obtained by using the values calculated in Step 4 is: T C
M B S. A graphical representation of ranking order presented in [8, 9, 16, 29] and the
proposed model is shown in Figure 2.
Figure 2. Ranking order of alternatives
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559
7. Comparative analysis with the existing techniques
Various DMs have applied the SMs for medical diagnosis in the environment of SVNSs and
IVNSs [8, 9, 16, 29]. In order to portray the usefulness and validation of the proposed SMs, we
apply it for the same problem and the results are shown in the Tables 3 and 4. According to
the results obtained by applying our proposed MCDM model, we see that the Kids K1 and K2
suffered in the disease typhoid (T ) under the observations of five symptoms Sj (j = 1, 2, ..., 5).
The results obtained by proposed and existing methods are different because of assigning the
weights to the criteria, These results are further analyzed by using Spearman’s correlation
coefficient.
Table 3. Results obtained by proposed SVNS’s SM
SMs
C
M
T
S
B
Ranking
Proposed 0.7774 0.7675 0.7969 0.6353 0.6127 T C M B S
[8]
0.9443 0.9571 0.9264 0.8214 0.7650 M C T S B
[9]
0.7941 0.8094 0.4568 0.5851 0.5517 M C S B T
[16]
0.5385 0.6282 0.6206 0.3336 0.3154 M T C S B
[28]
0.8505 0.8661 0.8185 0.5148 0.4244 M C T S B
Table 4. Results obtained by proposed IVNS’s SM
SMs
C
M
T
S
B
Ranking
Proposed 0.6445 0.5760 0.7222 0.6668 0.5884 T C M B S
[8]
0.9443 0.9571 0.9264 0.8214 0.7650 M C T S B
[9]
0.7941 0.8094 0.4568 0.5851 0.5517 M C S B T
[16]
0.5385 0.6282 0.6206 0.3336 0.3154 M T C S B
[28]
0.8505 0.8661 0.8185 0.5148 0.4244 M C T S B
7.1. Ranking analysis with Spearman’s rank-correlation coefficient
. The ranking preference of the diagnosis obtained by our and existing techniques are
different and presented in Tables 3 and 4. In order to compare the diagnosis further, we use
the Spearman’s rank-correlation coefficient (ρs ) and the critical value Z, where, ρs and Z can
be calculated with the formulae given below:
ρs = 1 − 6
i−1=k
X
l=1
(4l )2
,
n(n − 1)
and
√
Z = ρs n − 1.
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Here, 4l is the difference between two sets of ranking. The values of ρs are always bounded in
the closed interval [−1, 1]. The values of ρs which are nearer to ±1 show the perfect relationship
amongst two ranking orders. Moreover,the critical value Z is compared with a pre-estimated
degree of significance value η. The critical value Z corresponding to the degree of significance
value η = 0.05 for the examples (n = 5) is, Z0.05 = 0.9. If the critical value Z more than 0.9,
it indicates that there exist a strong relationship between two rankings. On the other hand,
the two rankings can be considered as dissimilar or have weaker relationship.
There are five collections of preference rankings obtained by the proposed method and [8, 9,
16, 28], represented by X, Y, V, T and U , respectively and their ranking order can be seen in
Tables 3 and 4. In order to compare these ranking orders, ρs and Z evaluated in Table 5. The
analysis of the results is summarized in Table 5 as follows:
The results obtained by the proposed model with those obtained in [8] and [28], the critical
Table 5. Comparison with existing methods
Daignosis
X Y
V
T
U
X-Y X-V X-T X-U
C
2
2
2
3
2
0
0
-1
0
M
3
1
1
1
1
2
2
2
2
T
1
3
5
2
3
-2
-4
-1
-2
S
5
4
3
4
4
1
2
1
1
B
4
5
4
5
5
-1
0
-1
-1
Spearman’s rank-correlation coefficient ρs
0.5
-0.2
0.6
0.5
Critical value Z
1
-0.4
1.2
1
value Z = 1 > 0.9, shows that there is a positive relationship between the ranking of the
proposed model (X), the ranking [8] (Y ) and [28] (U ). Also, the results obtained by the
proposed model (X) with those obtained in [16] (T ), the critical value Z = 1.2 > 0.9 indicates
that there is a strongly positive relationship between the ranking X and T . However, the
ranking X of the proposed model is significantly dissimilar to the ranking [9] (V ) because the
critical value Z = −0.4 is smaller than 0.9.
8. Conclusions
The similarity measures are extensively utilized in MCDM problems from the last few
decades. This paper suggested a novel technique to develop the similarity measures on the basis
of Euclidean distance measure for SVNSs and IVNSs, respectively. However, the similarity
measures presented in [15, 29, 30] have some shortcoming. On the other hand the suggested
similarity measures satisfy all the axioms of the similarity measure. Moreover, we used the
suggested similarity measures to medical diagnosis decision problems. A practical example is
used to exemplify the practicability and efficiency of the proposed similarity measure, which are
then compared to other existing similarity measures. We will emphasize to apply the proposed
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similarity measure in pattern recognition and supply chain problems under the framework of
SVNSs and IVNSs in future.
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Received: 23 March 2020 / Accepted: 22 May 2020
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University of New Mexico
Application of Single Valued Trapezoidal Neutrosophic Numbers
in Transportation Problem
Rajesh Kumar Saini* 1 Atul Sangal 2 Manisha3
1, 3Deptt.
of Mathematical Sciences and Computer Applications, Bundelkhand University, Jhansi, India,
prof.rksaini@bujhansi.ac.in*
2Deptt. of Management Studies, Sharda University, Greater Noida, Uttar Pradesh, India
atul.sangal@sharda.ac.in
* Correspondence: prof.
prof.rksaini@bujhansi.ac.in, Mob. 9412322576(091)
Abstract: In the present paper, we introduced the concept of single valued trapezoidal neutrosophic
number, which is generalization of single valued neutrosophic number. A generalization of crisp, fuzzy
and intuitionistic fuzzy sets represents as neutrosophic sets, which have uncertainty, inconsistent, and
incompleteness
leteness information in real world problem. De
De-neutrosophication
neutrosophication is a process to convert
neutrosophic number into a crisp number for practical applications. For unbalanced neutrosophic
transportation problem, we also use here minimum row column method and set a comparison among
crisp and neutrosophic optimal solutions. Here we use two models of transportation problems to
understand the applications in neutrosophic environment.
De-neutro
neutrosophication,
Keywords: Fuzzy Number, Single valued trapezoidal neutrosophic number, De
neutrosophic transportation problem
problem.
1. Introduction
In present scenario the classical theory of mathematics can’t be handling the different kind of
uncertainties, vagueness or impreciseness of mathematical problem
problems.
s. Many researchers around the
world define many approaches to understand or define it. In 1965, Zadeh [37]] first time introduce the
mathematical
cal formulation of a fuzzy set (FS) as a set with its membership function or membership
grade. Sometimes the membership
rship function in FS was not suitable one to describe the ambiguity of a
problem.
After development of FS theory in various fields of uncertainty, Atanassov [1] in 1986, believe
about the belongingness and non-belongingness
belongingness in fuzzy set and present it’s extension as intuitionistic
intu
fuzzy set (IFS) theory, which included the degree of membership and degree of non-membership
function of each element in the set. More development of IFS theory in decision problems plays key role
in recent scenario [17, 20]. In real life decision making problems, the theory of FS and IFS is much
applicable,, IFS approach in the solution of transportation problems used by many researchers [15, 22,
23].
The basic theme of a transportation problem is to find a direct connection between source and
destination in minimum time with minimum cost. Hitchcock [[12]] was first, who originally developed
the basic results of transportation problem by simplex method, which was recognized as special
mathematical module. Sincee in early stage th
the transportation parameters like transportation cost,
demand and supply were on the crisp values. In present time the real life transportation problems have
uncertain, uncontrolled factors as the transportation cost, supply and demand are in fuzzy values.
In that period many research problems related tto fuzzy transportation problem (FTP
FTP) were solved,
in which some are partial fuzzy and some are fully fuzzy. A FTP in which cost demand and supply are
Rajesh Kumar Saini* 1 Atul Sangal 2 and Manisha3, Application of Single Valued Trapezoidal Neutrosophic Numbers
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as fuzzy number is called fully FTP while in case of either cost, demand or supply are in fuzzy number,
then it is FTP see [24, 7]. In a fuzzy solid transportation problem the parameters are trapezoidal fuzzy
number (TrFN), introduced by Jim´enez and Verdegay [16] in 1999. For more research work about FTP,
see [18, 19, 22, 25].
In current scenario, due to uncertainty, unawareness, vagueness, ambiguity in the constraints or
some poor handling of data, the indeterminacy exists in transportation problems. The IFS theory can
handle the problems of incomplete information but not the indeterminate and inconsistent information
exists in transportation modal.
The problems with inconsistent information or indeterminate cannot be handled by any evocation
of fuzzy set, so to overpower of such problems, Smarandache [27] introduced the neutrosophic set (NS)
in 1988, which was an extension of classical set, FS and IFS. The well applicable fundamentals of NS, to
represent the indeterminacy and inconsistent information are truth-membership degree, indeterminacy
membership degree, and falsity-membership degree. The NS becomes the IFS, if indeterminacy
membership degree I(𝑥) of NS is equal to hesitancy degree h(𝑥) of IFS. For practical applications and
some technical references in NS, Wang et. al. [31] in 2010 introduced the idea of single valued
neutrosophic set (SVNS). The notion of SVNS is more suitable and effective in solving many real life
problems of decision making and supply chain management. For more applications of FS, IFS and NS
in some different fields see [1- 10, 14, 21, 29- 32, 34, 36].
Since the study of transportation models with optimal and effective cost play a key role in every
real life situation. Many researchers formulated efficient mathematical models in various uncertain
environments. For practical application, two models of neutrosophic transportation problem (NTP)
with all entries such as cost, demand, and supply are as single valued trapezoidal neutrosophic number
(SVTrNN). Here we also use minimum row column method (MRCM) for balance the unbalance crisp
transportation problem (CTP) and NTP with some existing method.
The main features of the paper are obtaining the optimal solutions of CTP and NTP after balancing
with different methods and to compare the results. The paper is well organized in seven sections. In
section first, introduction of the present paper with some earlier research are given. In second section,
the basics concepts of FS, IFS and NS are discussed and reviewed. In third section, introduce the deneutrosophication as score function to convert neutrosophic values into crisp values. Section fourth
composes the classification and mathematical formulation of CTP & NTP of type-2 & 3. In fifth section,
we introduce the procedures for solutions of CTP & NTP. In section six, seven and eight, we introduce
two models of transportation with their solutions in different tables, their comparison, result and
discussion. The conclusion and future aspects of research work exhibit in last section of the paper.
2. Preliminaries
2.1. Some basic definitions and examples
{ x, ( x) / x X} where
Definition 2.1.1. (FS [37]): A FS A of a non empty set X is defined as A
A
A ( x) is called the membership function such that A (x) : X [0,1] .
Definition 2.1.2. (FN): A convex, normalized fuzzy set A is called fuzzy number on the universal set of
real numbers R, if the membership function of A has the following belongingness:
A
(i)
μA : X 0,1 is continuous
(ii)
A ( x) 0, for all x , a d ,
(iii)
A ( x) is strictly increasing on a , b and strictly decreasing on c , d
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A ( x) 1, for all x b, c , where a b c d.
(iv)
(a, b, c , d) , with its
Definition 2.1.3. (TrFN[19]): A trapezoidal fuzzy number (TrFN) denoted as A
membership function A ( x) , on R, is given by
x - a b - a ,
1,
μà x =
d - x d - c ,
0,
for a x < b
for b x < c
for c < x d
otherwise
If b = c in TrFN A ( a , b , c , d ) , then it becomes TFN A ( a , b , d) .
Definition
2.1.4.
An
IFS
in
a
non-empty
set
X
is
denoted
I and
by A
defined
as
x, I , I : x X , where I , I : X [0,1] , such that 0 I , I 1, x X. The degree of
A
A
A
A
A
A
A
I
I . The degree of
membership A I and degree of non-membership A I are functions from X to [0,1] in A
I .
hesitation is defined as h( x) 1 A I A I 1, x X in A
Definition 2.1.5. (ITrFN [20]): An Intutinistic trapezoidal fuzzy number (ITrFN) is denoted by
I = (a ,a ,a ,a )(a ,a ,a ,a ) where a a a a a a with membership function I and nonA
1
1
2
3
4
4
3 4
1 2 3 4
1 2
A
membership function A I defined by
0 ,
x - a1 ,
a2 - a1
μ A I (x) = 1,
a -x
4
,
a4 - a3
0 ,
for x < a1 ,
for a1 x a2 ,
for a2 x a3 ,
for a3 x a4 ,
for x > a4 .
for x < a1 ,
1,
x - a1 , for a x a ,
1
2
a2 - a1
for a2 x a3 ,
ν A I (x) = 0,
a - x
4
, for a3 x a4 ,
a4 - a3
for x > a4 .
1,
I
If a2 a3 then ITrFN becomes ITFN denoted as A (a1 , a2 , a3 )(a1 , a2 , a3 ) where a1 a1 a2 a3 a3 .
Definition 2.1.6. ([4]): Let x be a generic element of a non empty set X. A neutrosophic number A N in
N { x , T N ( x), I N ( x), F N ( x) / x X}, T N ( x), I N ( x) and F N ( x) ] 0,1 [ where
X is defined as A
A
A
A
A
A
A
TA N : X ]0 ,1 [ ,
I A N : X ]0 ,1 [
and
FA N : X ]0 ,1 [
are functions of truth-membership,
indeterminacy membership and falsity-membership in A N respectively also there is no restrictions on
the sum of TA N ( x), I A N ( x) and FA N ( x) so that 0 TA N (x) I A N (x) FA N (x) 3 .
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For the practical applications it is difficult to apply directly NS theory, hence the notion of SVNS as
well as single valued neutrosophic numbers [SVNN] introduced by Deli I., S¸uba Y[8] in 2014 .
Definition 2.1.7. (SVNS [8]): Let x be the generic point of a non-empty space X. A SVNS is denoted and
N { x, T N ( x), I N ( x), F N ( x) / x X} , where for each point x in X, T N ( x), called truth
defined as A
A
A
A
A
membership I A N ( x) called indeterminacy membership and FA N ( x) called falsity membership function
in [0, 1] and 0 TA N ( x) I A N ( x) FA N (x) 3 .
N can be written as
For continuous SVNS A
N
A
TA N ( x), I A N ( x) , FA N ( x) / x , x X
N
A
N can be written as
When X is discrete, a SVNS A
n
A N TA N ( xi ), I A N ( xi ) , FA N ( xi ) / xi , xi X
i 1
Example 2.1.1. Let X be a space with capability x1 , trustworthiness x2 and price x3 in [0,1]. If expert
wants “degree of good services”, “degree of indeterminacy” and degree of poor services”, then a SVNS
N of X is defined as
A
A N 0.7 , 0.1, 0.3 / x1 0.4, 0.2, 0.7 / x 2 0.5, 0.1, 0.6 / x 3 .
N , a crisp subset of R is defined by
Definition 2.1.8. An ( , , ) cut set of SVNS A
N { x : T ( x) , I ( x) , F ( x) }
A
, ,
A
A
A
where 0 1, 0 1,0 1 and 0 3.
N { x , T N ( x), I N ( x), F N ( x) : x X} is called neut-normal, if there exist at
Definition 2.1.9. A SVNS A
A
A
A
least three points x1 , x2 , x 3 X such that TA N ( x1 ) 1, I A N ( x2 ) 1, FA N ( x3 ) 1.
N { x , T N ( x), I N ( x), F N ( x) : x X} is called neut-convex set on the real
Definition 2.1.10. A SVNS A
A
A
A
line; if the following conditions are satisfied x1 , x 2 , x 3 R and [0,1]
(i)
TA N ( x1 (1 )x2 ) min(TA N ( x1 ),TA N ( x2 ))
(ii)
I A N ( x1 (1 )x2 ) max( I A N ( x1 ), I A N ( x2 ))
(iii)
FA N ( x1 (1 )x2 ) max( FA N ( x1 ), FA N ( x2 ))
Definition 2.1.11. (SVTrNN [8]):
A single valued trapezoidal neutrosophic number (SVTrNN)
a (a1 , a2 , a3 , a4 ); wa , ua , va is a special NS on the real line R, whose truth-membership Ta N ( x) ,
N
indeterminacy-membership I a N ( x) , and a falsity-membership Fa N ( x ) are given as follows:
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( ( x - a1 ) wa N
, for a1 x a2 ,
a2 - a1
w N ,
for a2 x a3 ,
Ta N ( x) = a
(a4 - x) wa N
, for a3 x a4 ,
a -a
4
3
0,
for x > a4 and x < a1
a2 - x + ( x - a1 )ua N
, for a1 x a2 ,
a2 - a1
u N ,
for a2 x a3 ,
I a N ( x) = a
x - a3 + (a4 - x)ua N
, for a3 x a4 ,
a4 - a3
1,
for x > a4 and x < a1
a2 - x+ ( x - a1 )va N
, for a1 x a2 ,
a2 - a1
v N ,
for a2 x a3 ,
Fa N ( x) = a
x - a3 + (a4 - x)va N
, for a3 x a4 ,
a4 - a3
1,
for x > a4 and x < a1
where w a , ua , and va denotes the maximum truth-membership degree, minimum-indeterminacy
membership degree and minimum falsity-membership degree in [0,1] respectively and a1 , a2 , a3 , a4 R
such that a1 a2 a3 a4 . When a1 0, a ( a1 , a2 , a3 , a4 ); w a , ua , va is called positive SVTrNN, denoted
by a 0 , and if a 4 0, then a ( a1 , a2 , a3 , a4 ); w a , ua , va becomes a negative SVTrNN, denoted by a 0.
If
0 a1 a2 a3 a4 1 ,
w a , ua , v a [0,1] ,
then a
called
a
normalized
SVTrNN.
When
I a N = 1- Ta N Fa N , then SVTrNN reduces as TIFN. If a2 a3 , then SVTrNN is reduces single valued
triangular neutrosophic number (SVTNN), denoted as a ( a1 , a3 , a4 ); w a , ua , va .
Definition 2.1.12. A single valued trapezoidal neutrosophic number (SVTrNN) with twelve
components is defined and denoted as:
A N ( p1 , p 2 , p3 , p4 ); ( q1 , q 2 , q 3 , q 4 ); ( r1 , r2 , r3 , r4 ); w A N , u A N , v A N
where r1 q1 p1 r2 q 2 p2 r3 q 3 p 3 r4 q 4 p4 in which the quantity of the truth membership,
indeterminacy membership and falsity membership are not dependent and is defined as follows:
(x - p1 )wA N
,
p2 - p1
w N ,
TA N ( x) = A
(p4 - x)wA N
p -p ,
4 3
0,
for p1 x p2 ,
for p2 x p3 ,
for p3 x p4 ,
for x > p4 and x < p1
Rajesh Kumar Saini* 1 Atul Sangal 2 and Manisha3, Application of Single Valued Trapezoidal Neutrosophic Numbers
in Transportation Problem
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q2 - x ( x - q1 )u A N
, for q1 x q2 ,
q2 - q1
u N ,
for q2 x q3 ,
I A N ( x) = A
x q3 (q4 x)u A N
, for q3 x q4 ,
q 4 - q3
for x > q4 and x < q1
1,
r2 - x ( x - r1 ) A N
, for r1 x r2 ,
r2 - r1
N ,
for r2 x r3 ,
FA N ( x) = A
x r3 (r4 x) A N
, for r3 x r4 ,
r4 - r3
for x > r4 and x < r1
1,
N.
where 0 TA N (x) I A N (x) FA N ( x) 3, x A
N of SVTrNN for some 0 1, 0 1,0 1 and
Definition 2.1.13. The parametric form A
N)
N ( ), I A N ( ), FA N ( ), FA N ( )] ,
0 3 is defined as ( A
N ( ), TA
N ( ), I A
, , [TA
TA N ( ) p1
where
I A N ( )
FA N ( )
wA N
TAN ( ) p4
( p2 p1 ) ,
q1uA N q2 (q2 q1 )
uA N 1
r1 A N r2 (r2 r1 )
A 1
,
N
,
I A N ( )
FAN ( )
wA N
( p4 p3 )
q3 q4 uA N (q4 q3 )
1 uA N
r3 r4 A N (r4 r3 )
1 A N
Example 2.1.2. let us take A N (7,12,16, 22),(6,11,15, 20),(5,10,14,19); 0.4,0.6,0.6 . The parametric
( ) 7 12.5 ,
( ) 22 15 ,
representation is
T0.4
T0.4
( ) 18.5 12.5 ,
I 0.6
( ) 7.5 12.5 ,
I 0.6
( ) 17.5 12.5 ,
F0.6
( ) 6.5 12.5
F0.6
For different values of , , the degree of truthfulness, degree of indeterminacy and degree of
falsity shown in table 1 and their graphical representation in figure 2:
Table 1
α, β,γ
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
8.25
9.50 10.75 12.00 13.25 14.50 15.75
TA N ( ) 7.00
0.8
17.00
0.9
18.25
1.0
19.50
TA N ( )
22.00
20.50
19.5
17.50
16.00
14.50
13.00
11.50
10.00
8.50
7.00
I A N ( )
18.5
17.25
16.00
14.75
13.50
12.25
11.00
9.75
8.50
7.25
6
I A N ( )
7.50
8.75
10.00
11.25
12.50
13.75
15.00
16.25
17.50
18.75
20.00
FA N ( )
17.5
16.25
15.00
13.75
12.50
11.25
10.00
8.75
7.50
6.25
5.00
FA N ( )
6.50
7.75
9.00
10.25
11.50
12.75
14.00
15.25
16.50
17.75
19.50
Rajesh Kumar Saini* 1 Atul Sangal 2 and Manisha3, Application of Single Valued Trapezoidal Neutrosophic Numbers
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Figure 2: Graphical representation of
N
A ( p1 , p 2 , p3 , p 4 ); ( q1 , q 2 , q 3 , q 4 ); ( r1 , r2 , r3 , r4 ); w A N , u A N , v A N
where r1 q1 p1 r2 q 2 p2 r3 q 3 p 3 r4 q 4 p4
2.2. Operational Laws on SVTrNN
N and B N are two SVTrNN with twelve components having truth-membership
Definition 2.2. 1. If A
TA N ( x) , TBN ( x) , indeterminacy-membership I A N ( x), I B N ( x) and falsity-membership FA N ( x), FBN ( x)
respectively and three real numbers in [0.1], such as
A N ( p1 , p 2 , p3 , p 4 ); ( q1 , q 2 , q 3 , q 4 ); ( r1 , r2 , r3 , r4 ); w A N , uA N , v A N
B N ( p1, p2 , p3 , p 4 ); ( q1, q2 , q 3 , q 4 ); ( r1, r2, r3, r4); w B N , uB N , v B N
Addition of SVTrNN:
Negative of SVTrNN:
N B N ( p p, p p , p p , p p )( q q , q q , q q , q q );
C N A
1
1
2
2
3
3
4
4
1
1
2
2
3
3
4
4
(r1 r1, r2 r2, r3 r3, r4 r4); wA N wB N , uA N uB N , vA N vB N
If A N ( p1 , p 2 , p3 , p4 ); ( q1 , q 2 , q 3 , q 4 ); ( r1 , r2 , r3 , r4 ); w A N , u A N , v A N , then
A N ( p4 , p3 , p 2 , p1 ); ( q 4 , q 3 , q 2 , q1 ); ( r4 , r3 , r2 , r1 ); w A N , uA N , v A N
Subtraction of SVTrNN:
N B N ( p p , p p , p p , p p)( q q , q q , q q , q q);
A
4
2
3
3
2
4
1
1
4
2
3
3
2
4
1
1
(r1 r4, r2 r3, r3 r2, r4 r1); wA N wB N , uA N uB N , vA N vB N
Multiplication of SVTrNN:
( p1 p1, p2 p2 , p3 p3 , p4 p4 );( q1 q1, q 2 q2 , q 3 q3 , q4 q 4 );
( r1 r1, r2 r2, r3 r3, r4 r4); w A N w B N , uA N uB N , vA N vB N
if p4 0, p4 0; q 4 0, q4 0; r4 0, r4 0
( p1 p4 , p2 p3 , p3 p2 , p4 p1);( q1 q4 , q2 q3 , q3 q2 , q4 q1);
N
N
A B ( r1 r4, r2 r3, r3 r2, r4 r1); w A N w B N , uA N uB N , vA N vB N
if p4 0, p4 0; q4 0, q 4 0; r4 0, r4 0
( p4 p4 , p3 p3 , p2 p2 , p1 p1);( q4 q4 , q3 q3 , q2 q2 , q1 q1);
( r4 r4, r3 r3, r2 r2, r1 r1); w A N w B N , uA N uB N , vA N vB N
if p4 0, p4 0; q4 0, q4 0; r4 0, r4 0
Rajesh Kumar Saini* 1 Atul Sangal 2 and Manisha3, Application of Single Valued Trapezoidal Neutrosophic Numbers
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Scalar multiplication of SVTrNN:
( kp1 , kp2 , kp3 , kp4 );( kq1 , kq2 , kq3 , kq4 ); ( kr1 , kr2 , kr3 , kr4 ); w A N , uA N , v A N , if k 0,
N
kA
( kp4 , kp3 , kp2 , kp1 );( kq4 , kq3 , kq2 , kq1 ); ( kr4 , kr3 , kr2 , kr1 ); w A N , uA N , v A N , if k 0.
Inverse of SVTrNN:
1 1 1 1
1 1
( p , p , p , p );( q , q
4
3
4 3 2 1
1
N )1
(A
N 1 1 1 1
A
( , , , );( 1 , 1
p1 p2 p3 p4 q1 q2
,
1 1
1 1 1 1
, );( , , , ); w A N , uA N , vA N ,
q2 q1 r4 r3 r2 r1
if p' s 0, q' s 0, r' s 0,
,
1 1
1 1 1 1
, );( , , , ); w A N , uA N , vA N ,
q3 q4 r1 r2 r3 r4
if p' s 0, q' s 0, r' s 0.
Division of SVTrNN:
p1 p2 p3 p4 q1 q2 q3 q4 r1 r2 r3 r4
( , , , );( , , , );( , , , ); w A N wB N , uA N uB N , vA N vB N
p4 p3 p2 p1 q4 q3 q2 q1 r4 r3 r2 r1
if p4 0, p4 0; q4 0, q4 0; r4 0, r4 0
p p p p q q q q r r r r
N ( 4 , 3 , 2 , 1 );( 4 , 3 , 2 , 1 );( 4 , 3 , 2 , 1 ); w N w N , u N u N , v N v N
A
B
A
B
A
B
p4 p3 p2 p1 q4 q3 q2 q1 r4 r3 r2 r1 A
B N
if p4 0, p4 0; q4 0, q4 0; r4 0, r4 0
p4 p3 p2 p1 q4 q3 q2 q1 r4 r3 r2 r1
( , , , );( , , , );( , , , ); w A N wB N , uA N uB N , vA N vB N
p1 p2 p3 p4 q1 q2 q3 q4 r1 r2 r3 r4
if p4 0, p4 0; q4 0, q4 0; r4 0, r4 0
Example 2.2.1. let A N (7 ,11,16, 21),(6,10,15, 20),(5, 9,14,19); 0.4, 0.6,0.6 and
B N (6,11,13, 20),(5,10,12,18),(3,8,11,16); 0.3, 0.6, 0.7 be two SVTrNN, then
A N B N (13, 22, 29, 41),(11, 20, 27 , 38),(8,17 , 25, 35); 0.4,0.6,0.6
A N B N ( 13, 2, 5,15),( 12, 2, 5,15),( 11, 2, 6,16); 0.4,0.6,0.6
A N .B N (42,121, 208, 420),(30,100,180, 360),(15, 72,154, 304); 0.4, 0.6,0.6
A N / B N (0.35, 0.85,1.45, 3.50),(0.33,0.83,1.50, 4.00,(0.31, 0.81,1.75,6.33); 0.4, 0.6, 0.6
N (35, 55, 80,105),(30, 50,75,100),(25, 45,70, 95); 0.4,0.6, 0.6
5A
3. De-Neutrosophication by using score function
We use the score and accuracy functions of a SVTrNN, is defined by an expert [31] to compare any
two SVTrNN. So that the score function is defined as
N ) p1 p2 p3 p4 q1 q2 q3 q4 r1 r2 r3 r4 2 w N u N v N
S( A
A
A
A
12
and accuracy function is
N ) p1 p2 p3 p4 q1 q2 q3 q4
A( A
4
Example 3.1. Let
A( A N ) 0.7
2 w A N uA N vA N
A N (7 ,11,16, 21),(6,10,15, 20),(5,10,14,19); 0.4,0.6, 0.6
then S( A N ) 4.4
and
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Definition 3.1. (Comparison of SVTrNN). Let A N and B N be any two SVTrNN, then one has the
following:
(a) S( A N ) S( B N ) A N B N
N ) S( B N ) and if
(b) If S( A
S
S
(i) A( A N ) A( B N ) then A N B N
(ii) A( A N ) A( B N ) then A N B N
(iii) A( A N ) A( B N ) then A N B N
Example 3.2. Let A N (6,10,16, 20),(5, 9,14,19),(3, 8,12,18); 0.3, 0.6, 0.7 and
B N (7 ,11,16, 21),(6,15,14, 20),(5,10,14,19); 0.3, 0.6, 0.7
C N (8,11,16, 22),(6,15,14, 21),(5,10,14, 20); 0.3,0.6, 0.7 be two SVTrNN, then
N ) 3.00 , A( A N ) 1.25 , S( B N ) 0.4 , A( B N ) 0.0 , and S(C N ) 0.4 , A(C N ) 0.25 ,
S( A
which implies that if
S( A N ) S( B N ) then A N B N
Also
S( B N ) S(C N ) and A( B N ) A(C N ) then B N C N .
4. Neutrosophic Transportation Problem (NTP) and its Mathematical formulation
4.1. Classification of NTP
Definition 4.1.1. In a TP, if atleast one parameter such as cost, demand or supply is in form of
neutrosophic numbers, the TP is termed as NTP.
Definition 4.1.2. A NTP having neutrosophic availabilities and neutrosophic demand but crisp cost, is
classified as NTP of type-1.
Definition 4.1.3. The NTP having crisp availabilities and crisp demand but neutrosophic cost, is
classified as NTP of type-2.
Definition 4.1.4. If all the specifications of TP such as cost, demand and availabilities are combination
of crisp, triangular or trapezoidal neutrosophic numbers, then NTP classified as NTP of type-3.
Definition 4.1.5. If all the specifications of TP must be in neutrosophic numbers form, then TP is said to
be NTP of type-4 or fully NTP.
4.2. Mathematical Formulation of NTP
The TP is very important for transporting goods from one source to another destination. In TP if
ambiguity occurs in cost, demand or supply then it is more difficult to solve it. To handle this type of
impreciseness in cost to transferred product from ith sources to jth destination or uncertainty in supply
and demand the decision maker introduce NTP of SVTrNN .
Here we consider two models in which the decision maker is unsettled about the specific values i.e.
cost from ith sources to jth destination and also certainty or uncertainty in supply or demand of the
product, so that a new type of TP is introduced namely NTP with parameters like cost, demand and
supply as SVTrNN. The NTP with assumptions and constraints is defined as the number of unites xijN
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and the neutrosophic cost cijN are transported from ith sources to jth destination. For balance NTP
m
a
i 0
N
i
n
b jN i.e. total supply is equal to total demand.
j 0
For the formulation of NTP the following assumptions and constraints are required:
m
Total number of source point
n
Total number of destination point
i
Table of source (for all m)
j
Table of destination (for all n)
x ijN
Number of transported neutrosophic unites from ith source to jth destination
cijN
Neutrosophic cost of one unit transported from ith source to jth destination
aiN
Available neutrosophic supply quantity from ith source
b jN
Required neutrosophic demand quantity to jth destination
cij
Crisp cost of one unit quantity
xij
Number of transported crisp unites from ith source to jth destination
ai
Available crisp supply quantity from ith source
bj
Required crisp demand quantity to jth destination
Modal I
In NTP the objective is to minimize the cost of transported product from source to destination. The
mathematical formulation of NTP with uncertain transported units and transportation cost, demand
and supply is:
m
n
Minimum Z N x ijN cijN
i 0 j 0
(P1)
Subject to
n
N
ij
a iN, i sources 1, 2, 3, . . . , m,
N
ij
b Nj , j destination 1, 2, 3, . . . , n ,
x
j 0
m
x
i 0
N
ij
x 0 , i 1, 2, 3, . . . , m, j 1, 2, 3, . . . , n.
Modal II
The mathematical formulation of NTP with uncertain transported units and transportation cost but
curtained about demand and supply is termed a NTP of type-2 is:
m
n
Minimum Z N xij cijN
i 0 j 0
(P2)
Subject to
n
x
j 0
ij
m
x
i 0
ij
ai , i sources 1, 2, 3, . . . , m ,
bj , j destination 1, 2, 3, . . . , n ,
xij 0, i 1, 2, 3, . . . , m , j 1, 2, 3, . . . , n.
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5.
Procedure for Proposed Algorithms for solution of CTP and NTP
5.1. Basic Assumptions of the Proposed Algorithms
The total transportation cost does not depends on the mode of transportation and distance, also the
framework of the problem will be denoted by either crisp or SVTrNN.
m
n
i 0
j 0
If a iN or b jN , i , j , then first one can make sure to balance the TP as
m
a
i0
N
i
n
b jN , i , j ,
j0
5.2. Steps for solution of CTP after balancing by existing method
Step5.2.1.
To change the each neutrosophic cost cij , neutrosophic supply aiN and neutrosophic
demand b jN of NTP in cost matrix to crisp values, we use here score function method i.e.
N
we convert these by using S( A N ) .
Step5.2.2.
For balance TP, verify that the sum of demands is equal to the sum of supply i.e.
m
n
a b , i , j .
i0
Step5.2.3.
Step5.2.4.
Step5.2.5.
Step5.2.6.
Step5.2.7.
Step5.2.8.
i
j 0
j
If
m
a
i0
n
i
or b j , i , j , then first one can make sure to balance the TP
j 0
by adding a row or column with zero entries in cost matrix [30].
After conversion of NTP into TP, choose the minimum entry in each row and subtract it to
all other entries in that row. The same way is applicable in each column to find minimum
one zero in each row and each column in TP matrix. For better (see table 4 and table 6).
Verify that the sum of demands is greater than the supply in each row and the sum of
supplies are greater than the demand in each column, if ok go on step 5.2.6, otherwise go
on step 5.2.5.
Draw the horizontal and vertical lines that cover all the zeros and equal to minimum
number of order of matrix or reduced table. Now if number of lines is less than to the
minimum number, revise table by choose the least element which is not under horizontal
or vertical line and add it to the entry at the cross point of the lines. Again go to step 5.2.3 to
check the condition.
To allot the maximum possible units of supply or demand in the cost cell, choose a cell of
maximum cost in the reduced cost matrix. If the maximum cost exists more than one place,
choose any one cell of maximum supply or demand.
If none cell occur for the maximum cost then go for next maximum. If such cell does not
occur for any value, then choose any cell at random, whose reduced cost is zero.
From the reduced table omit the row which are fully exhausted or column which are fully
satisfied, then repeats steps and again. Repeat the procedure until all the demand units and
all the supply units are fully received respectively.
The procedure for the solution of NTP by using existing method is same as steps in 5.2, while the
cost, demand, supply and solution vales are in SVTrNN.
5.3. Steps for solution of CTP after balancing by MRCM
For balance the unbalance CTP, we use MRCM which is generalization of method in [27]. We use
the following steps for solution of CTP by MRCM:
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Step5.3.1.
N
Convert neutrosophic cost cij , neutrosophic supply aiN and neutrosophic demand b jN of
NTP in cost matrix to crisp values by using score function S( A N ) i.e NTP convert into CTP.
Step5.3.2
If CTP is unbalance then to make it balance one by applying the steps of MRCM that is if
sum of supply is less then to the sum of demand, then add a row of minimum costs in each
row with a supply equal to sum of supplies and add a column of minimum costs in each
column with demand equal to the difference value from sum of all supplies differ to sum of
demand. The same is applicable when sum of demand is less than the sum of supply. i.e.
m
n
i 0
j 0
n
m
j 0
i 1
am1 ai and bn1 a j excess supply,
bn1 b j and am1 bi excess demand.
or
The unit transportation costs are taken as follows:
ci( n1) min cij , 1 i m,
1 j n
c( m1) j min cij , 1 j n,
1i m
cij c ji , 1 i m, 1 j n, and c( m1)( n1) 0.
Step5.3.3
Obtain optimal solution of converted CTP after balance it by existing method using excel
solver. Let the crisp optimal solution be x ij , 1 i m 1, 1 j n 1.
Step5.3.4
'm1 0 and using the relation
By assuming
'i 'j 'ij for basic variables, find the
'i , 1 i m and 'j , 1 j n 1,
values of all the dual variables
Step5.3.5.
'i
i and 'j j for 1 i m ,1 j n , obtain only central rank
According to MRCM,
zero duals. After that in terms of original supply S i and demand M j find the neutrosophic
optimal solution of the problem.
5.4. Steps for solution of NTP after balancing by MRCM
Step5.4.1.
N
Convert neutrosophic cost cij , neutrosophic supply aiN and neutrosophic demand b jN of
NTP in cost matrix to crisp values by using score function S( A N ) to check either it is balance
Step5.4.2
or unbalance.
If NTP is unbalance than same procedure as in 5.3 is applicable. i.e.
m
n
i 0
j 0
n
m
j 0
i 1
amN1 aiN and bnN1 a Nj excess supply,
bnN1 b jN and amN1 biN excess demand.
or
The unit transportation costs are taken as follows:
ciN(n1) min cijN, 1 i m,
1 j n
c(Nm1) j min cijN, 1 j n,
c c , 1 i m, 1 j n, and c
N
ij
Step5.4.3
Step5.4.4
N
ji
1i m
N
( m 1)( n 1)
0.
Obtain optimal solution of NTP by excel solver. Let the neutrosophic optimal solution
obtained be x ijN , 1 i m 1, 1 j n 1.
N
'iN 'jN 'ijN for basic variables, find the
'm
0 and using the relation
By assuming
1
'iN , 1 i m and 'jN , 1 j n 1,
values of all the dual variables
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'N
iN and 'N
According to MRCM,
Nj for 1 i m ,1 j n , obtain only central rank
i
j
Step5.4.5.
zero duals.
6.
Numerical Example
6.1. Modal I (NTP of type-3)
Let us consider a NTP with three sources say S1 , S 2 , S 3 in which wheat are initially stored and
ready to transport in three flour mills namely M 1 , M 2 , M 3 with unit transportation cost, demand and
supply are as SVTrNN. The input data of SVTrNN -TP given in table 2 as follows:
Table 2
M2
M1
M3
Supply
S1
(3,5,7.5,11) 0.2
(2,4,7 ,10) ;0.4
(1,3.5,6,9) 0.8
(2,4.5,10,15) 0.3
(0.5,3.5,8,14) ;0.5
(0,2.5,6,12) 0.8
(1,5,9,14.5) 0.2
( 3,3.5,8,12) ;0.5
( 4,2,7 ,11) 0.7
(9,17,26,36) 0.4
(6,14,23,33) ;0.7
(3,11,20,30) 0.7
S2
(1,7 ,11.5,16) 0.4
( 1,5,10,14) ;0.5
( 3,3,8.5,12) 0.7
(1,6,9.5,12) 0.5
( 1,4,8.5,11) ;0.7
( 2,2,8,10) 0.8
( 1,4,8,15) 0.3
( 2,3,6,12) ;0.6
( 3,2,5,11) 0.6
(7,17,25,31) 0.3
(3,12,22.5,29) ;0.6
(1,10,19.5,27) 0.7
S3
(3,6,9,12) 0.2
(2,5,8,11) ;0.4
(1,4,7 ,10) 0.8
( 1,3.5,9,12) 0.2
( 2,2.5,7 ,11) ;0.4
( 4,1,5,10) 0.8
(0,5,8,14) 0.2
( 2,3,7 ,12) ;0.6
( 4,1,6,10) 0.6
(9,16,22,31) 0.3
(5,14,20,27) ;0.6
(1,12,18,23) 0.7
Demand
(12,21,30,37) 0.3
(9,19,28,34) ;0.6
(6,16,25,33) 0.7
(10,16,22,27) 0.4
(5,14,20,25) ;0.7
(0,12,18,23) 0.7
(7,12,19,27) 0.2
(4,11,18,24) ;0.6
(1,9,15,21) 0.6
6.2. Neutrosophic optimal solution with score function method
One can use score function to convert SVTrNN cost, demand and supply to obtain the crisp
numbers in TP of table 2 as follows:
N ) p1 p2 p3 p4 q1 q2 q3 q4 r1 r2 r3 r4 2 w N u N v N
S( A
A
A
A
12
Here
S c
N
11
(3,5,7.5,11) 0.2
(2,4,7,10) ;0.4
(1,3.5,6,9) 0.8
3 5 7.5 11 2 4 7 10 1 3.5 6 9
0.2 0.6 0.2 1.33
12
S c 0.50 , S c 0.50 , S a 4.33 , S a 3.67 , S a 3.50 , S b 5.83 ,
S b 3.50 , S b 3.17 .
N
N
N
N
S c12N 1.25 , S c13N = -0.58 , S c 21
1.08 , S c22
1.00 , S c23
0.67 , S c 31
1.50 ,
N
32
N
33
N
2
N
3
N
1
N
2
N
3
N
1
After converting cost, demand and supply of NTP from SVTrNN to the crisp numbers by using
score function method, the unbalance CTP cost matrix is given in table 3:
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Table 3
M1
M2
M3
Supply
S1
-1.33
-1.25
-0.58
-4.33
S2
-1.08
-1.00
-0.67
-3.67
S3
-1.50
-0.50
-0.50
-3.50
Demand
-5.83
-3.50
-3.17
By using the steps in 5.2, the optimal crisp solution of CTP and their allotment of demand and
supply in cost matrix shown in table 4:
Table 4
M1
M2
M3
Supply
S1
-1.33(-2.16)
-1.25(-2.17)
-0.58
-4.33
S2
-1.08(-3.67)
-1.00
-0.67
-3.67
S3
-1.50
-0.50(-0.33)
-0.50(-3.17)
-3.50
S4
0
0(-1.00)
0
-1.00
Demand
-5.83
-3.50
-3.17
The complete solution of CTP is x11 = -2.16, x12 = -2.17, x 21 = -3.67, x 32 = -0.33, x 33 = -3.17, x 42 = -1.00,
and Z = 11.30 . The corresponding optimal solution of NTP with allotment of SVTrNN is shown in table
5 as follows:
Table 5
M1
M2
M3
Supply
S1
(-19,-4,13,30) 0.3
(-20,-3.5,16,31) ;0.6
(-21,-3.5,15,32) 0.7
(-21,4,30,55) 0.3
(-25,-2,26.5,53) ;0.6
(-29,-4,23.5,51) 0.7
-
-
S2
(7,17,25,31) 0.3
(3,12,22.5,29) ;0.6
(1,10,19.5,27) 0.7
-
-
-
-
(-18,-3,10,24) 0.2
(-19,-4,9,23) ;0.6
(-20,-3,9,22) 0.6
(7,12,19,27) 0.2
(4,11,18,24) ;0.6
(1,9,15,21) 0.6
-
-
(-69,-24,21,66) 0.3
(-71,-21.5,26,69) ;0.6
(-73,-20.5,25,72) 0.7
-
-
-
-
-
-
S3
S4
Demand
Z N
i.e.
(3,5,7.5,11) 0.2
(2,4,7 ,10) ;0.4
(1,3.5,6,9) 0.8
(1,7 ,11.5,16) 0.4
( 1,5,10,14) ;0.5
( 3,3,8.5,12) 0.7
(-19,-4,13,30) 0.3
(2,4.5,10,15) 0.3
(-20,-3.5,16,31)
;0.6
(0.5,3.5,8,14) ;0.5
(-21,-3.5,15,32) 0.7
(0,2.5,6,12) 0.8
( 1,3.5,9,12) 0.2
(7,17,25,31) 0.3
(3,12,22.5,29)
;0.6
( 2,2.5,7,11) ;0.4
(1,10,19.5,27) 0.7
( 4,1,5,10) 0.8
(0,5,8,14) 0.2
( 2,3,7 ,12) ;0.6
( 4,1,6,10) 0.6
(0,0,0,0) 0.2
(7,12,19,27) 0.2
(4,11,18,24)
;0.6
(0,0,0,0) ;0.6
(1,9,15,21) 0.6
(0,0,0,0) 0.6
(-21,4,30,55) 0.3
(-25,-2,26.5,53) ;0.6
(-29,-4,23.5,51) 0.7
(-18,-3,10,24) 0.2
(-19,-4,9,23) ;0.6
(-20,-3,9,22) 0.6
(-69,-24,21,66) 0.3
(-71,-21.5,26,69) ;0.6
(-73,-20.5,25,72) 0.7
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Z N =
(-74,187.5,927,2317) 0.4
(-25.5,62,738,1999) ;0.4
(52,13.75,531.75,1654) 0.6
= -194.54
Now for application of MRCM, we use steps in 5.3 to balance the unbalance CTP of table 2 as
follows in table 6:
Table 6
M1
M2
M3
M4
Supply
S1
(3,5,7.5,11) 0.2
(2,4,7 ,10) ;0.4
(1,3.5,6,9) 0.8
(2,4.5,10,15) 0.3
(0.5,3.5,8,14) ;0.5
(0,2.5,6,12) 0.8
(1,5,9,14.5) 0.2
( 3,3.5,8,12) ;0.5
( 4,2,7 ,11) 0.7
(1,5,9,14.5) 0.2
( 3,3.5,8,12) ;0.5
( 4,2,7 ,11) 0.7
(9,17,26,36) 0.4
(6,14,23,33) ;0.7
(3,11,20,30) 0.7
S2
(1,7,11.5,16) 0.4
( 1,5,10,14) ;0.5
( 3,3,8.5,12) 0.7
(1,6,9.5,12) 0.5
( 1,4,8.5,11) ;0.7
( 2,2,8,10) 0.8
( 1,4,8,15) 0.3
( 2,3,6,12) ;0.6
( 3,2,5,11) 0.6
( 1,4,8,15) 0.3
( 2,3,6,12) ;0.6
( 3,2,5,11) 0.6
(7,17,25,31) 0.3
(3,12,22.5,29) ;0.6
(1,10,19.5,27) 0.7
S3
(3,6,9,12) 0.2
(2,5,8,11) ;0.4
(1,4,7,10) 0.8
( 1,3.5,9,12) 0.2
( 2,2.5,7 ,11) ;0.4
( 4,1,5,10) 0.8
(0,5,8,14) 0.2
( 2,3,7,12) ;0.6
( 4,1,6,10) 0.6
(0,5,8,14) 0.2
( 2,3,7,12) ;0.6
( 4,1,6,10) 0.6
(9,16,22,31) 0.3
(5,14,20,27) ;0.6
(1,12,18,23) 0.7
S4
(1,7,11.5,16) 0.4
( 1,5,10,14) ;0.5
( 3,3,8.5,12) 0.7
( 1,3.5,9,12) 0.2
( 2,2.5,7 ,11) ;0.4
( 4,1,5,10) 0.8
(0,5,8,14) 0.2
( 2,3,7 ,12) ;0.6
( 4,1,6,10) 0.6
(0,0,0,0) 0.2
(0,0,0,0) ;0.6
(0,0,0,0) 0.6
(25,50,73,98) 0.3
(14,40,65.5,89) ;0.6
(5,33,57.5,80) 0.7
D
e
m
a
n
d
(12,21,30,37) 0.3
(9,19,28,34) ;0.6
(6,16,25,33) 0.7
(10,16,22,27) 0.4
(5,14,20,25) ;0.7
(0,12,18,23) 0.7
(7,12,19,27) 0.2
(4,11,18,24) ;0.6
(1,9,15,21) 0.6
(-41,29,97,167) 0.3
(-55,14,87,160) ;0.6
(-67,8,78,153) 0.7
After converting cost, demand and supply of NTP in table 6 from SVTrNN to the crisp numbers by
using score function method, the balance CTP cost matrix is given in table 7:
Table 7
M1
M2
M3
M4
Supply
S1
-1.33
-1.25
-0.58
-0.58
-4.33
S2
-1.08
-1.00
-0.67
-0.67
-3.67
S3
-1.50
-0.50
-0.50
-0.50
-3.50
S4
-1.08
-0.50
-0.50
0
-11.50
Demand
-5.83
-3.50
-3.17
-10.50
The complete allotment of demand and supply in cost matrix of CTP shown in table 8:
Table 8
M1
M2
M3
M4
Supply
S1
-1.33(-1.16)
-1.25
-0.58(-3.17)
-0.58
-4.33
S2
-1.08(-3.67)
-1.00
-0.67
-0.67
-3.67
S3
-1.50
-0.50(-3.50)
-0.50
-0.50
-3.50
S4
-1.08(-1.00)
-0.50
-0.50
0(-10.50)
-11.50
Demand
-5.83
-3.50
-3.17
-10.50
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The optimal crisp solution and minimum cost of balance CTP of table 8 is x11 1.16 , x13 3.17,
x21 3.67, x 32 3.50, x41 1.00, x44 10.50 and Z 10.18 .
Similarly after balance the unbalance NTP by MRCM, the corresponding optimal solution of
balance NTP with allotment of SVTrNN is shown in table 9 as follows:
M1
S1
(-18,-2,14,29) 0.4
(-18,-4,12,29) ;0.6
(-18,-4,11,29) 0.6
S2
(7,17,25,31) 0.3
(3,12,22.5,29) ;0.6
(1,10,19.5,27) 0.7
S3
-
S4
(-142,-47,44,139) 0.3
(-146,-47,51.5,144) ;0.6
(-148,-45,49.5,147) 0.7
Demand
Z N
Table 9
M2
-
(9,16,22,31) 0.3
(5,14,20,27) ;0.6
(1,12,18,23) 0.7
M3
M4
Supply
(7,12,19,27) 0.2
(4,11,18,24) ;0.6
(1,9,15,21) 0.6
-
-
-
-
-
-
-
-
-
-
-
-
(-41,29,97,167) 0.3
(-55,14,87,160) ;0.6
(-67,8,78,153) 0.7
-
-
(-18,-2,14,29) 0.4
(1,5,9,14.5) 0.2
(3,5,7.5,11) 0.2
(7,12,19,27) 0.2
.
.
(-18,-4,12,29)
;0.6
(2,4,7
,10)
;0.4
(
3,3.5,8,12)
;0.5
(4,11,18,24) ;0.6
(1,9,15,21) 0.6
(1,3.5,6,9) 0.8
(-18,-4,11,29) 0.6
( 4,2,7,11) 0.7
(1,7,11.5,16) 0.4
(7,17,25,31) 0.3
( 1,3.5,9,12) 0.2
(9,16,22,31) 0.3
( 1,5,10,14) ;0.5 . (3,12,22.5,29) ;0.6 ( 2,2.5,7 ,11) ;0.4 . (5,14,20,27) ;0.6
(1,12,18,23) 0.7
( 3,3,8.5,12) 0.7
(1,10,19.5,27) 0.7
( 4,1,5,10) 0.8
(1,7 ,11.5,16) 0.4
(-142,-47,44,139) 0.3
(-41,29,97,167) 0.3
(0,0,0,0) 0.2
( 1,5,10,14) ;0.5 . (-146,-47,51.5,144) ;0.6 (0,0,0,0) ;0.6 . (-55,14,87,160) ;0.6
( 3,3,8.5,12) 0.7
(-148,-45,49.5,147) 0.7
(0,0,0,0) 0.6
(-67,8,78,153) 0.7
this implies
(-191,-104,1267.5, 3802.5) 0.4
Z N = (85,-117.5,1108, 3297)
; 0.4 417.77
(415,-89,847.5,2810)
0.6
6.3. Model II (NTP of type-2)
For solution of NTP of type-2 i.e. a problem in which costs are in SVTrNN while demand and
supply are given in crisp numbers. Here we are taking the problem in table 2 in which costs are in
SVTrNN while demand and supply are as crisp numbers given as follows in table 10:
Rajesh Kumar Saini* 1 Atul Sangal 2 and Manisha3, Application of Single Valued Trapezoidal Neutrosophic Numbers
in Transportation Problem
579
Neutrosophic Sets and Systems, Vol. 35, 2020
Table 10
M2
M1
M3
Supply
S1
(3,5,7.5,11) 0.2
(2,4,7 ,10) ;0.4
(1,3.5,6,9) 0.8
(2,4.5,10,15) 0.3
(0.5,3.5,8,14) ;0.5
(0,2.5,6,12) 0.8
(1,5,9,14.5) 0.2
( 3,3.5,8,12) ;0.5
( 4,2,7,11) 0.7
-4.33
S2
(1,7 ,11.5,16) 0.4
( 1,5,10,14) ;0.5
( 3,3,8.5,12) 0.7
(1,6,9.5,12) 0.5
( 1,4,8.5,11) ;0.7
( 2,2,8,10) 0.8
( 1,4,8,15) 0.3
( 2,3,6,12) ;0.6
( 3,2,5,11) 0.6
-3.67
S3
(3,6,9,12) 0.2
(2,5,8,11) ;0.4
(1,4,7,10) 0.8
( 1,3.5,9,12) 0.2
( 2,2.5,7,11) ;0.4
( 4,1,5,10) 0.8
(0,5,8,14) 0.2
( 2,3,7 ,12) ;0.6
( 4,1,6,10) 0.6
-3.50
Demand
-5.83
-3.50
-3.17
The optimal crisp solution of NTP type-2 is shown in table 11 as follows:
Table 11
M1
M2
M3
Supply
S1
-2.16
-2.17
-
-4.33
S2
-3.67
-
-
-3.67
S3
-
-0.33
-3.17
-3.50
S4
-
-1.00
-
-3.50
Demand
-5.83
-3.50
-3.17
The corresponding neutrosophic solution of NTP type-2 is:
(3,5,7.5,11) 0.2
Z tN2 2.16 (2,4,7,10) ; 0.4 2.17
(1,3.5,6,9) 0.8
( 1,3.5,9,12) 0.2
0.33 ( 2, 2.5,7,11) ; 0.4 3.17
( 4,1, 5,10) 0.8
(2, 4.5,10,15) 0.3
(0.5, 3.5,8,14) ; 0.5 3.67
(0, 2.5,6,12) 0.8
(1,7,11.5,16) 0.4
( 1, 5,10,14) ; 0.5
( 3, 3, 8.5,12) 0.7
(0, 5,8,14) 0.2
( 14.16, 63.27, 108.44, 163.37) 0.2
( 2, 3,7,12) ; 0.6 (5.26, 44.93, 93.68, 145.07)
; 0.6 15.89
( 4,1, 6,10) 0.6
(22.85, 27.5, 77.87, 124.52)
0.6
Similarly after balance the unbalance NTP of type-2 by MRCM, the corresponding optimal
neutrosophic solution of balance NTP of type-2 with allotment is as follows:
(3, 5,7.5,11) 0.2
Z tN2 1.16 (2, 4,7,10) ; 0.4 3.67
(1, 3.5,6,9) 0.8
(1, 5,9,14.5) 0.2
( 3, 3.5,8,12) ; 0.5 3.17
( 4,2,7,11) 0.7
(1,7,11.5,16) 0.4
1.00 ( 1, 5,10,14) ; 0.5 10.50
( 3, 3,8.5,12) 0.7
(1,7,11.5,16) 0.4
( 1, 5,10,14) ; 0.5 3.50
( 3,3,8.5,12) 0.7
( 1, 3.5,9,12) 0.2
( 2, 2.5,7,11) ; 0.4
( 4,1, 5,10) 0.8
(0,0,0,0) 0.2
7.82,65.59,121.85,174.70 0.2
(0,0,0,0)
;
0.6
19.86,47.09,103.68,152.52 ; 0.6 14.35
(0,0,0,0) 0.6
40.03,27.41, 85.60,135.85 0.6
Rajesh Kumar Saini* 1 Atul Sangal 2 and Manisha3, Application of Single Valued Trapezoidal Neutrosophic Numbers
in Transportation Problem
580
Neutrosophic Sets and Systems, Vol. 35, 2020
Comparative Study
Real life application of single valued trapezoidal neutrosophic numbers in transportation problem
have been solved by some existing and proposed MRCM methods. In present paper, the minimum cost
obtained through proposed method with some existing method discussed in [30] have been
summarized in table 12. From the table it is clear that minimum cost obtained by using MRCM is better
than to the existing method in both either crisp or in neutrosophic environment. Figure 3 shows the
graphical representation of the minimum crisp or neutrosophic cost degree of satisfaction by different
approaches.
7.
Model
I
Comparison
Balance by existing method
Balance by MRCM
Crisp cost of The neutrosophic cost of
Crisp cost of
The neutrosophic cost of
CTP
NTP
CTP
NTP
10.18
0.4
Z = 11.30
Z
( 191,104,1267.5,3802.5) 0.4
(-74,187.5,927,2317)
Z N =
Z N
(-25.5,62,738,1999) ;0.4
(52,13.75,531.75,1654) 0.6
The neutrosophic cost of NTP
(14.16, 63.27, 108.44, 163.37) 0.2
Z tN2 (5.26, 44.93, 93.68, 145.07)
; 0.6
(22.85, 27.5, 77.87, 124.52)
0.6
corresponding Crisp cost of NTP
Z N 15.89
t2
;0.4
0.6
corresponding Crisp cost of
NTP
N
Z 417.77
corresponding Crisp cost
of NTP
Z N = -194.54
Model
II
(85,117.5,1108,3297)
(415,89,847.5,2810)
The neutrosophic cost of NTP
Z tN2
7.82,65.59,121.85,174.70 0.2
19.86,47.09,103.68,152.52 ;0.6
40.03,27.41,85.60,135.85 0.6
corresponding Crisp cost of NTP
Z N 14.35
t2
Figure 3: Comparision of results with proposed MRCM and existing method
Rajesh Kumar Saini* 1 Atul Sangal 2 and Manisha3, Application of Single Valued Trapezoidal Neutrosophic Numbers
in Transportation Problem
581
Neutrosophic Sets and Systems, Vol. 35, 2020
8.
Result and discussion
In this present study the optimal transportation crisp cost and optimal transportation neutrosophic
cost of unbalanced NTP using MRCM is minimum than the existing method in [30]. It is also verified
that in de-neutrosophication, the crisp values before and after conversion from neutrosophic to crisp
and crisp to neutrosophic are different in score function method.
For the real life applications one can find the degree of result. The best of minimum neutrosophic
(-191,-104,1267.5, 3802.5) 0.4
; 0.4
(415,-89,847.5, 2810)
0.6
cost of unbalanced NTP is Z N = (85,-117.5,1108, 3297)
i.e. total minimum transportation cost
lies between -191 to 3802.5 for level of truthfulness or acceptance, 85 to 3297 for level of indeterminacy
and 415 to 2810 for level of falsity. The degree of truthfulness or acceptance, degree of indeterminacy
and degree of falsity is defined as wT N ( x) 100 , uI N ( x) 100 and vF N ( x) 100 respectively, where x
Z
Z
Z
denotes the total cost and
0.4( x 191)
191 104 , for -191 x 104 ,
for - 104 x 1267.5,
0.4,
wT N ( x)
Z
0.4(3802.5 x) , for 1267.5 x 3802.5,
3802.5 1267.5
0,
for otherwise.
( 117.5 x) 0.4( x 85)
, for -117.5 x 85 ,
62 25.5
for -117.5 x 1108,
0.4,
uI N ( x )
Z
x
x
(
1108)
0.4(3297
)
, for 1108 x 3297 ,
3297 1108
0,
for otherwise.
( 89 x) 0.6( x 415)
,
13.75 52
0.6,
vF N ( x )
Z
( x 847) 0.6(2810 x) ,
2810 847
0,
for - 89 x 415,
for - 89 x 847 ,
for 847 x 2810 ,
for otherwise.
x
Degree
-100
0
500
1000
2000
3000
wT N 100
40.0
40.0
40.0
40.0
30.0
12.6
uT N 100
40.0
40.0
40.0
40.0
64.4
91.8
vT N 100
60.0
60.0
60.0
63.1
83.4
0
Z
Z
Z
With the help of degree of truthfulness, degree of indeterminacy and degree of falsity, we can
conclude the total neutrosophic cost from the range of -191 to 3802.5 for truthfulness, 85 to 3297 for
indeterminacy and 415 to 2810 for falsity to scheduled the transportation and budget allocation.
Rajesh Kumar Saini* 1 Atul Sangal 2 and Manisha3, Application of Single Valued Trapezoidal Neutrosophic Numbers
in Transportation Problem
582
Neutrosophic Sets and Systems, Vol. 35, 2020
9. Conclusions
In recent scenario the applied mathematical modeling with uncertainty or vagueness is necessity of
the society. Nowadays the concept of neutrosophic number is very popular to handle such type of
problems. In this research paper, we study of unbalance NTP and introduced a new approach MRCM
for optimal solution with the concept of single valued trapezoidal neutrosophic number of twelve
components from different viewpoints. Also the optimal neutrosophic solution and minimum cost
obtained by using MRCM is better than by using some existing methods. The proposed method
provides the more practical structure and considers the various characteristics of transportation
problems in neutrosophic environment. In future the proposed MRCM is applied to the unbalance
multi-attribute transportation problem, assignment problems and multilevel programming problem in
neutrosophic environment. The present research will be a mile stone for transportation problems with
generalization of the pick value of truth, indeterminacy and falsity functions by considering, which are
very important for uncertainty theory.
Acknowledgement : Prof. Yanhui Guo, University of Illinois at Springfield, One University Plaza,
Springfield, IL 62703, United States,
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Received: July 26, 2020. Accepted: Jun 24, 2020
Rajesh Kumar Saini* 1 Atul Sangal 2 and Manisha3, Application of Single Valued Trapezoidal Neutrosophic Numbers
in Transportation Problem
Neutrosophic Sets and Systems (NSS) is an academic journal, published quarterly online and on
paper, that has been created for publications of advanced studies in neutrosophy, neutrosophic set,
neutrosophic logic, neutrosophic probability, neutrosophic statistics etc. and their applications in any
field.
All submitted papers should be professional, in good English, containing a brief review of a problem
and obtained results.
It is an open access journal distributed under the Creative Commons Attribution License that
permits unrestricted use, distribution, and reproduction in any medium, provided the original work
is properly cited.
ISSN (print): 2331-6055, ISSN (online): 2331-608X
Impact Factor: 1.739
NSS has been accepted by SCOPUS. Starting with Vol. 19, 2018, all NSS articles are indexed in
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NSS is also indexed by Google Scholar, Google Plus, Google Books, EBSCO, Cengage Thompson
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(Tongfang Knowledge Network Technology Co., Beijing, China), etc.
Google Dictionary has translated the neologisms "neutrosophy" (1) and "neutrosophic" (2), coined
in 1995 for the first time, into about 100 languages.
FOLDOC Dictionary of Computing (1, 2), Webster Dictionary (1, 2), Wordnik (1), Dictionary.com,
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DOI numbers are assigned to all published articles.
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Recently, NSS was also approved for Emerging Sources Citation Index (ESCI) available on
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(GLWRUsLQ&KLHI
Prof. Dr. Florentin Smarandache
Department of Mathematics and Science
University of New Mexico
705 Gurley Avenue
Gallup, NM 87301, USA
E-mail: smarans@unm.edu
Dr. Mohamed Abdel-Basset
Department of Operations Research
Faculty of Computers and Informatics
Zagazig University
Zagazig, Ash Sharqia 44519, Egypt
E-mail:mohamed.abdelbasset@fci.zu.edu
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