V olum e 3 2 , 2 0 2 0
Neutrosophic Sets and Systems
An International Journal in Information Science and Engineering
<A> <neutA> <antiA>
Florentin Smarandache . Mohamed Abdel-Basset
Editors-in-Chief
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ISSN 2331-6055 (print)
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Neutrosophic
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University of New Mexico
ISSN 2331-6055 (print)
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University of New Mexico
Neutrosophic Sets and Systems
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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.
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Copyright © Neutrosophic Sets and Systems, 2020
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Contents
Abhijit Saha and Said Broumi, Parameter Reduction of Neutrosophic Soft Sets and Their
Applications…………………………………………………………………………………………….1
Huda E. Khalid, Neutrosophic Geometric Programming (NGP) Problems Subject to (⋁,.) Operator;
the Minimum Solution……………………………………………………………………………..…15
K. Ramesh, Ngpr Homeomorphism in Neutrosophic Topological Spaces…………………..…......25
A. Mehmood, F. Nadeem, G. Nordo, M. Zamir,C. Park, H. Kalsoom, S. Jabeen and M. I. KHAN,
Generalized Neutrosophic Separation Axioms in Neutrosophic Soft Topological Spaces………38
T. Nanthini and A. Pushpalatha, Interval Valued Neutrosophic Topological Spaces………………...52
Avishek Chakraborty,Baisakhi Banik, Sankar Prasad Mondal and Shariful Alam, Arithmetic
and\Geometric Operators of Pentagonal Neutrosophic Number and its Application in Mobile
Communication Service Based MCGDM Problem ......................................................................................61
Majdoleen Abu Qamar, Abd Ghafur Ahmad and Nasruddin Hassan, On Q-Neutrosophic Soft Fields
…………………………………………………………………………………………………………80
Binu R. and P. Isaac, Neutrosophic projective G-submodules……………………………………….94
M.Mohanasundari and K.Mohana, Quadripartitioned Single valued Neutrosophic Dombi Weighted
Aggregation Operators for Multiple Attribute Decision Making………………………………….107
Rajab Ali Borzooei, Florentin Smarandache and Young Bae Jun, Polarity of generalized neutrosophic
subalgebras in BCK/BCI-algebras…………………………………………………………………123
Muhammad Riaz, Khalid Naeem, Iqra Zareef and Deeba Afzal, Neutrosophic N-Soft Sets with TOPSIS
method for Multiple Attribute Decision Making ………..…………………………………………146
R. Narmada Devi, A Novel of neutrosophic T-Structure Ring ExtB and ExtV Spaces……………171
D. Ajay, Said Broumi and J. Aldring, An MCDM Method under Neutrosophic Cubic Fuzzy Sets with
Geometric Bonferroni Mean Operator……………………………………………………………...187
Abdelkrim Latreche, Omar Barkat, Soheyb Milles and Farhan Ismail, Single valued neutrosophic
mappings defined by single valued neutrosophic relations with applications……………………203
M. Aslam Malik, Hossein Rashmanlou, Muhammad Shoaib, R. A. Borzooei and Morteza Taheri and Said
Broumi, A Study on Bipolar Single-Valued Neutrosophic Graphs With Novel Application……..221
Huda E. Khalid, Neutrosophic Geometric Programming (NGP) with (max-product) Operator, An
Innovative Model…………………………………………………………………………………….269
Abhijit Saha, Said Broumi, and Florentin Smarandache, Neutrosophic Soft Sets Applied on Incomplete
Data…………………………………………………………………………………………………..282
Copyright © Neutrosophic Sets and Systems, 2020
ISSN 2331-6055 (print)
ISSN 2331-608X (online)
University of New Mexico
Muhammad Saqlain, Sana Moin, Muhammad Naveed Jafar, Muhammad Saeed, Florentin Smarandache,
Aggregate Operators of Neutrosophic Hypersoft Set……………………………………………..294
Muhammad Saqlain, Muhammad Naveed Jafar and Muhammad Riaz, A New Approach of Neutrosophic
Soft Set with Generalized Fuzzy TOPSIS in Application of Smart Phone Selection……………..307
Muhammad Saqlain, Naveed Jafar, Sana Moin, Muhammad Saeed and Said Broumi, Single and Multivalued Neutrosophic Hypersoft set and Tangent Similarity Measure of Single valued
Neutrosophic Hypersoft Sets………………………………………………………………………..317
Hamiden Abd El- Wahed Khalifa, Pavan Kumar and Florentin Smarandache, On Optimizing
Neutrosophic Complex Programming Using Lexicographic Order……………………………...330
Maikel Leyva-Vázquez, Miguel A. Quiroz-Martínez, Yoenia Portilla-Castell, Jesús R. HechavarríaHernández and Erick González-Caballero, A New Model for the selection of Information Technology
Project in a Neutrosophic Environment……………………………………………………………344
S. Broumi, M.Lathamaheswari, A. Bakali, M. Talea, F. Smarandache, D. Nagarajan, Kavikumar and
Guennoun Asmae, Analyzing Age Group and Time of the Day Using Interval Valued Neutrosophic
Sets …………………………………………………………………………………………………...361
Majid Khan, Muhammad Gulistan, Nasruddin Hassan and Abdul Muhaimin Nasruddin, Air Pollution
Model using Neutrosophic Cubic Einstein Averaging Operators………………………………...372
T.RajeshKannan and S.Chandrasekar, Neutrosophic 𝛼-Irresolute Multifunction in Neutrosophic
Topological Spaces…………………………………………………………………………………..390
Rakhal Das, Florentin Smarandache and Binod Chandra Tripathy, Neutrosophic Fuzzy Matrices and
Some Algebraic Operations………………………………………………………………………....401
Anjan Mukherjee and Rakhal Das, Neutrosophic Bipolar Vague Soft Set and Its Application to
Decision Making Problems…………………………………………………………………………410
Qays Hatem Imran, Ali Hussein Mahmood Al-Obaidi and Florentin Smarandache, On Some Types of
Neutrosophic Topological Groups with Respect to Neutrosophic Alpha Open Sets…………….425
D. Sasikala and K.C. Radhamani, A Contemporary Approach on Neutrosophic Nano Topological
Spaces………………………………………………………………………………………………...435
Copyright © Neutrosophic Sets and Systems, 2020
Neutrosophic Sets and Systems, Vol.32, 2020
University of New Mexico
Parameter Reduction of Neutrosophic Soft Sets and Their
Applications
Abhijit Saha1 and Said Broumi2
1Faculty
of Mathematics, Techno College of Engineering Agartala, , Tripura, India, Pin-799004; Email: abhijit84.math@gmail.com
of Science, University of Hassan II, B.P 7955, Sidi Othman, Casablanca, Morocco; Email: broumisaid78@gmail.com
2Faculty
* Correspondence: abhijit84.math@gmail.com
Abstract: Parameter reduction can be treated as an effective tool in many fields, including pattern
recognition. Many reduction techniques have been reported so far for soft sets, fuzzy soft sets and bipolar fuzzy soft sets to solve decision-making problems. However, there is almost no attention to the parameter reduction of neutrosophic soft sets. In this present paper we focus our discussion on the parameter reduction of neutrosophic soft sets as an extension of parameter reduction of soft sets and
fuzzy soft sets. To do that, using the concept of indiscernibility relation, we first define the terms ‘dispensable set’ and ‘indispensible set’. We utilize these definitions to define the terms ‘decision partition’,
‘parameter reduction’ and ‘degree of importance of a parameter’ with a suitable example. Next we present an algorithm based on the concept of degree of importance and parameter reduction of a neutrosophic soft set. An illustrative example is employed to show the feasibility and validity of our proposed
algorithm based on parameter reduction of neutrosophic soft sets in real life decision making problem.
Keywords: Neutrosophic set, neutrosophic soft set, parameter reduction, decision making.
1. Introduction
Molodstov [31] initiated the concept of soft set theory as a fundamental mathematical tool for modelling uncertainty, vague concepts and not clearly defined objects. Although various traditional tools,
including but not limited to rough set theory [33], fuzzy set theory [41], intuitionistic fuzzy set theory
[10] etc. have been used by many researchers to extract useful information hidden in the uncertain data, but there are inherent complications connected with each of these theories.
Additionally, all these approaches lack in parameterizations of the tools and hence they
couldn’t be applied effectively in real life problems, especially in areas like environmental, economic
and social problems. Soft set theory is standing uniquely in the sense that it is free from the above
A. Saha and S. Broumi; Parameter reduction of Neutrosophic soft sets and their applications
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2
mentioned impediments and obliges approximate illustration of an object from the beginning, which
makes this theory a natural mathematical formalism for approximate reasoning.
The Theory of soft set has excellent potential for application in various directions some of
which are reported by Molodtsov [31] in his pioneer work. Later on Maji et al. [27] introduced some
new annotations on soft sets such as subset, complement, union and intersection of soft sets and discussed in detail its applications in decision making problems. Ali et al. [7] defined some new operations on soft sets and shown that De Morgan's laws holds in soft set theory with respect to these newly
defined operations. Atkas and Cagman [6] compared soft sets with fuzzy sets and rough sets to show
that every fuzzy set and every rough set may be considered as a soft set. Jun [24] connected soft sets
to the theory of BCK/BCI-algebra and introduced the concept of soft BCK/BCI-algebras. Feng et al.[21]
characterized soft semi rings and a few related notions to establish a relation between soft sets and
semi rings.
Chen et al. [15] introduced the concept of parameter reduction of soft sets in 2005. In 2008, Z.
Kong et al [25] introduced the definition of normal parameter reduction in soft sets and presented a
heuristic algorithm of normal parameter reduction. The soft sets mentioned above are based on complete information. However, incomplete information widely exists in various real life problems. H.
Qin et al [34] studied the data filling approach of incomplete soft sets. Y. Zou et al [42] investigated data analysis approaches of soft sets under incomplete information. In 2001, Maji et al. [28] defined the
concept of fuzzy soft set by combining of fuzzy sets [41] and soft sets [31]. Roy and Maji [35] proposed
a fuzzy soft set based decision making method.
Xiao et al. [39] presented a combined forecasting method based on fuzzy soft set. Feng et al.
[22] discussed the validity of the Roy-Maji method [35] and presented an adjustable decision-making
method based on fuzzy soft set. Yang et al. [40] initiated the idea of interval valued fuzzy soft set
(IVFS-set) and analyzed a decision making method using the IVFS-sets. The notion of intuitionistic
fuzzy set (IFS) was initiated by Atanassov [10] as a significant generalization of fuzzy set [41]. Intuitionistic fuzzy sets are very useful in situations when description of a problem by a linguistic variable,
given in terms of a membership function only, seems too complicated. Recently intuitionistic fuzzy
sets have been applied to many fields such as logic programming, medical diagnosis, decision making
problems etc.
A. Saha and S. Broumi; Parameter reduction of Neutrosophic soft sets and their applications
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3
Smarandache [38] introduced the concept of neutrosophic set which is a mathematical tool for
handling problems involving imprecise, indeterminacy and inconsistent data. Maji [30] introduced the
concept of neutrosophic soft set and established some operations on these sets. Mukherjee et al [32] introduced the concept of interval valued neutrosophic soft sets and studied their basic properties. In
2013, Broumi and Smarandache [12, 13] combined the intuitionistic neutrosophic and soft set which
lead to a new mathematical model called” intuitionistic neutrosophic soft set”. They studied the notions of intuitionistic neutrosophic soft set union, intuitionistic neutrosophic soft set intersection,
complement of intuitionistic neutrosophic soft set and several other properties of intuitionistic neutrosophic soft set along with examples and proofs of certain results.
Also, in [11] S. Broumi presented the concept of “generalized neutrosophic soft set” by combining the generalized neutrosophic sets [11] and soft set models, studied some properties on it, and
presented an application of generalized neutrosophic soft set [11] in decision making problem. Recently, Deli [17] introduced the concept of interval valued neutrosophic soft set as a combination of interval neutrosophic set and soft set. In 2014, S. Broumi et al. [14] initiated the concept of relations on interval valued neutrosophic soft sets.I. Deli [18] proposed a new notation called expansion and reduction of the neutrosophic classical soft sets that are based on the linguistic modifiers. Saha et al. [36]
proposed the concept of data filling of neutrosophic soft sets having incomplete/missing data. Few
more works on neutrosophic soft sets can be found in [9, 19, 23, 37].
Parameter reduction can be treated an effective tool in many fields, including pattern recognition.
Many reduction techniques [8, 15, 16, 20, 25, 26 ] have been reported so far for soft sets, fuzzy soft sets
and bipolar fuzzy soft sets to solve decision-making problems. However, there is almost no attention
to the parameter reduction of neutrosophic soft sets. In this present paper we focus our discussion on
the parameter reduction of neutrosophic soft sets as an extension of parameter reduction of soft sets
and fuzzy soft sets.
This present paper is organized as follows:
Section-2 presents some basic definitions related to fuzzy set theory with their generalizations and soft
set theory with their generalizations. In section-3, we first present the concept of indiscernibility relations and then based on it, we define the terms ‘dispensable set’, ‘indispensible set’, ‘decision partition’, ‘parameter reduction’, ‘degree of importance of a parameter’ with a suitable example in neutrosophic soft environment. In the next section (section-4), we have presented an algorithm based on the
concept of degree of importance and parameter reduction supported by an illustrative example to
show the feasibility and validity of our algorithm.
A. Saha andS. Broumi; Parameter reduction of Neutrosophic soft sets and their applications
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2. Preliminaries:
2.1 Definition: [41] Let U be a non empty set. Then afuzzy set τ on U is a set having the form
τ
x, μ
τ
x :x U
where the function μ τ :U [0, 1] is called the membership function and
μ τ x represents the degree ofmembership of each element x U .
2.2 Definition: [10] Let U be a non empty set. Then an intuitionistic fuzzy set (IFS for short)
object
having
the
form
τ
x, μ
τ
x , γτ x
: xU
where
the
τ
is an
functions
μ τ :U [0, 1] and γ τ :U [0, 1] are called membership function and non-membership function
respectively.
μ τ x and γ τ x represent the
degree ofmembership and the degree of non-membership
respectively of each element x U and 0 μ τ x + γ τ x 1 for each x U. We denote the class of
all intuitionistic fuzzy sets on U by IFSU.
2.3 Definition: [31] Let U be a universe set and E be a set of parameters. Let P U denotes the
power set of U and AE. Then the pair F, A is called a soft set over
U , where
F is a mapping
given by F: A P U .
In other words, the soft set is not a kind of set, but a parameterized family of subsets of U . For eA,
F e U may be considered as the set of e-approximate elements of the soft set F, A .
2.4 Definition: [28] Let U be a universe set, E be a set of parameters and
A E . Then the pair
2.5 Definition: [29] Let U be a universe set, E be a set of parameters and
A E . Then the pair
F, A is called a fuzzy soft setover U , where F is a mapping given by F: A FSU .
F, A is called an intuitionistic fuzzy soft set over U , where F is a mapping given by F: A IFSU .
For e A , F e is an intuitionistic fuzzy subset of U and is called the intuitionistic fuzzy value set
of the parameter ‘e’.
Let us denote μ F e x by the membership degree that object ‘x’ holds parameter ‘e’ and γ F e x by
the membership degree that object ‘x’ doesn’t hold parameter ‘e’ , where eA and x U . Then
F e can be written as an intuitionistic fuzzy set such that F e = x, μ F e x , γ F e x : x U .
2.6 Definition: [38] A neutrosophicset A on the universe of discourse U is defined as
A x, A x , A x , A x x U , where A , A , A U 0,1 are functions such that the
condition: x U , 0 A x A x A x 3 is satisfied.
Here A x , A x , A x represent the truth-membership, indeterminacy-membership and falsity-
membership (hesitancy membership) respectively of the element x U .
A. Saha and S. Broumi; Parameter reduction of Neutrosophic soft sets and their applications
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Neutrosophic Sets and Systems, Vol. 32, 2020
Smarandache [25] applied neutrosophic sets in many directions after giving examples of neutrosophic
sets. Then he introduced the neutrosophic set operations namely-complement, union, intersection,
difference, Cartesian product etc.
2.7 Definition: [30] Let U be an initial universe, E be a set of parameters and A E . Let NP U
denotes the set of all neutrosophic sets of
f , A
A NP U .
U . Then the pair
neutrosophicsoftset over U , where f is a mapping given by f
2.8 Example: Let us consider a neutrosophic soft set
is termed to be the
f , A which describes the “attractiveness of the
house”. Suppose U = {u1 , u2 , u3 , u4 , u5 , u6 } be the set of six houses under consideration and
E = {e1 (beautiful), e2 (expensive), e3 (cheap), e4 (good location), e5 (wooden)}be the set of parameters. Then
a neutrosophic soft set
e1
e2
e3
e4
e5
(0.8,0.5,0.2)
(0.3,0.4,0.6)
(0.1,0.6,0.4)
(0.7,0.3,0.6)
(0.3,0.4,0.6)
(0.4,0.1,0.7)
(0.8,0.2,0.4)
(0.4,0.1,0.7)
(0.2,0.4,0.4)
(0.1,0.1,0.3)
(0.2,0.6,0.4)
(0.5,0.5,0.5)
(0.8,0.1,0.7)
(0.5,0.3,0.5)
(0.5,0.5,0.5)
(0.3,0.4,0.4)
(0.1,0.3,0.3)
(0.3,0.4,0.4)
(0.6,0.6,0.6)
(0.1,0.1,0.5)
(0.1,0.1,0.7)
(0.2,0.6,0.7)
(0.4,0.2,0.1)
(0.8,0.6,0.1)
(0.6,0.7,0.7)
(0.5,0.3,0.9)
(0.3,0.6,0.6)
(0.1,0.5,0.5)
(0.3,0.6,0.5)
(0.4,0.4,0.4)
U
u1
u2
u3
u4
u5
u6
f , A over U can be given by:
3. Parameter reduction of neutrosophic soft sets:
Suppose U = {x1 , x2 , x3 ,.., x}n be the universe set of objects and E = {e1 , e2 , e3 ,. , e}m be the set of
parameters.
f ( e) =
Consider
a
{ x, mf (e) ( x), g f (e) ( x), d f (e) ( x)
%
f E ( xi ) =
å
j
(m
f ( e j ) ( xi ) +
neutrosophic
:xÎ U
}
soft
set
( f , E)
given
by
for e Î E . Let us define a function %
f E given by:
)
g f ( e j ) ( xi ) + d f ( e j ) ( xi ) , xi Î U .
We use %
f e j ( xi ) to denote mf (e j ) ( xi ) + g f (e j ) ( xi ) + d f (e j ) ( xi ) .
3.1 Definition: For any subset of parameters B Í E , an indiscernibility relation INDB is defined as:
INDB =
f B ( xi ) =
{(xi , x j )Î U ´ U : %
%
f B ( x j )}.
U
x}j
For the neutrosophic soft set ( f , E ) , we denote CE = {{x1 , x2 , x3 ,.., }xi x{1 , xi +, 1 xi +,.....,
2
{xk , xk + 1 ,.., x}n
xs
} as a partition of objects in U
x2 ,.
......
which partitions and ranks the objects according to
the value of %
f E ( xi ) based on the indiscernibility relation INDE . CEU is called the decision partition ,
where the sub classes are: {x1 , x2 , x3 ,....., xi },{xi+ 1, xi+ 2 ,....., x j },......,{ x,k xk +,.....,
x}n where s is the
1
number of sub-classes, and x1 ³ x2 ³ x3 ³ . .
³s x .
A. Saha andS. Broumi; Parameter reduction of Neutrosophic soft sets and their applications
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f ( x )ù= éê%
f ( x )ù= ........ = éê%
f ( x )ù= xq , where [. ]
For any sub-class {xz , xz + 1 ,....., xz + h }xq , éê%
ëE z ú
û ë E z +1 ú
û
ë E z+h ú
û
denotes the greatest integer function. Thus objects from U with the same value of %
f E (.) are included
into a same class.
3.2 Example:Let U = {x1 , x2 , x3 ,....., x6 } be the set of six houses and E = {e1 , e2 , e3 ,....., e6 } be the set of
parameters where the parameters e1 , e2 , e3 , e4 , e5 , e6 represents ‘beautiful’, ‘in the main town’,
‘expensive’, ‘concrete’, ‘in green surroundings’, ‘wooden’ respectively. Consider the neutrosophic soft
set ( f , E ) which describes the attractiveness and physical trait of the houses given by the following
table (table-1).
Table-1
U
e1
e2
e3
e4
e5
e6
%
f E (.)
x1
(0.3,0.7,0.4)
(0.4,0.5,0.1)
(0.2,0.2,0.4)
(0.6,0.3,0.4)
(0.1,0.1,0.3)
(0.2,0.4,0.6)
6.2
x2
(0.4,0.5,0.5)
(0.2,0.2,0.6)
(0.5,0.5,0.1)
(0.2,0.8,0.3)
(0.4,0.3,0.2)
(0.6,0.3,0.4)
7.0
x3
(0.2,0.5,0.7)
(0.3,0.2,0.5)
(0.8,0.2,0.4)
(0.5,0.5,0.3)
(0.2,0.4,0.2)
(0.9,0.6,0.6)
8.0
x4
(0.5,0.3,0.6)
(0.6,0.3,0.1)
(0.2,0.5,0.6)
(0.4,0.4,0.5)
(0.7,0.3,0.2)
(0.5,0.5,0.8)
8.0
x5
(0.3,0.5,0.6)
(0.4,0.4,0.2)
(0.3,0.3,0.5)
(0.6,0.1,0.6)
(0.7,0.8,0.1)
(0.4,0.6,0.6)
8.0
x6
(0.7,0.3,0.4)
(0.3,0.5,0.2)
(0.4,0.8,0.5)
(0.5,0.3,0.5)
(0.1,0.2,0.3)
(0.4,0.4,0.2)
7.0
In this case, CEU = {{x3 , x4 , x5 }x1 ,{x2 , x6 }x2 ,{x1}x3 } as %
f E ( x1 ) = 6.2, %
f E ( x2 ) = 7.0, %
f E ( x3 ) = 8.0, %
f E ( x4 ) = 8.0,
%
f E ( x5 ) = 8.0, %
f E ( x6 ) = 7.0; where x1 = 8, x2 = 7, x3 = 6 .
3.3 Definition:For a neutrosophic soft set ( f , E ) with E = {e1 , e2 , e3 ,....., em } , if there exists a subset
f A ( x1 ) = %
f A ( x2 ) = %
f A ( x3 ) = ............ = %
f A ( xn ) , then we say that A
A= {e1¢, e2¢, e3¢,....., e¢p } Ì E satisfying %
is dispensable, otherwise A is indispensable. Roughly speaking, A Ì E is dispensable means that the
difference between among all objects according to the parameters in A doesn’t influence the final
decision. A Ì E is called a parameter reduction of E if A is indispensible and %
f E- A ( x1 ) = %
f E- A ( x2 )
%
%
f E- A ( . )
= f E- A ( x3 ) = ............ = f E- A ( xn ) i.e; E-A is the maximal subset of E that keeps the value %
constant.
Clearly after the parameter reduction of E, we have fewer parameters although the partition of
objects have not been changed. In the above definition, %
f A ( x1 ) = %
f A ( x2 ) = %
f A ( x3 ) = ............ = %
f A ( xn )
implies CEU = CEU- A .
A. Saha and S. Broumi; Parameter reduction of Neutrosophic soft sets and their applications
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Neutrosophic Sets and Systems, Vol. 32, 2020
f{e1 ,e2 ,e4 }( x1 ) = %
f{e ,e
3.4 Example:Using table-1, we have, %
( x2 ) = %
f{e1 ,e2 ,e4 }( x3 ) = %
f{e1 ,e2 ,e4 }( x4 ) =
1 2 ,e4 }
%
f{e1 ,e2 ,e4 }( x5 ) = %
f{e1 ,e2 ,e4 }( x6 ) = 3.7 . Hence the neutrosophic soft set ( f , E ) given by Table-1 has a
parameter reduction {e3 , e5 , e6 } and the corresponding neutrosophic soft set ( f , A) is displayed in
table-2 given below:
U
Table-1
shows
Table-2
e5
e3
%
f A (.)
e6
x1
(0.2,0.2,0.4)
(0.1,0.1,0.3)
(0.2,0.4,0.6)
2.5
x2
(0.5,0.5,0.1)
(0.4,0.3,0.2)
(0.6,0.3,0.4)
3.3
x3
(0.8,0.2,0.4)
(0.2,0.4,0.2)
(0.9,0.6,0.6)
4.3
x4
(0.2,0.5,0.6)
(0.7,0.3,0.2)
(0.5,0.5,0.8)
4.3
x5
(0.3,0.3,0.5)
(0.7,0.8,0.1)
(0.4,0.6,0.6)
4.3
x6
(0.4,0.8,0.5)
(0.1,0.2,0.3)
(0.4,0.4,0.2)
3.3
that
%
f E ( x1 ) = 6.2, %
f E ( x2 ) = %
f E ( x6 ) = 7, %
f E ( x3 ) = %
f E ( x4 ) = %
f E ( x5 ) = 8
and
so
x3 or x4 or x5 is the optimal choice, x2 or x6 is the sub optional choice and x1 is the inferior choice.
Again according to Table-2, %
f A ( x1 ) = 2.5, %
f A ( x2 ) = %
f A ( x6 ) = 3.3, %
f A ( x3 ) = %
f A ( x4 ) = %
f A ( x5 ) = 4.3 and so
in this case also x3 or x4 or x5 is the optimal choice, x2 or x6 is the sub optional choice and x1 is the
inferior choice. Thus parameter reduction gives the same result as the original one.
We also have CU
E- {e ,e ,e } = {{x3 , x4 , x5 }4 ,{x2 , x6 }3 ,{x1}2 }.
1 2 4
For
the
neutrosophic
soft
set ( f , E ) ,
E = {e1 , e2 , e3 ,....., em } is the parameter set and
U = {x1 , x2 , x3 ,....., xn } is the set of objects, CEU = {{x1 , x2 , x3 ,....., xi }x1 ,{xi + 1 , xi + 2 ,....., x j }x2 ,........
{xk , xk + 1 ,....., xn }xs }is a decision partition of objects in U. Now deleting the parameter ei from E, we
get a new decision partition deleted ei denoted by C U
E - {ei } , which is given by:
{
}
. CU
E- {ei } = {x1¢, x2¢, x3¢,....., xi ¢}x1¢ ,{xi+ 1¢, xi+ 2¢,....., x j ¢}x2¢ ,........ , {xk ¢, xk + 1¢,....., xn ¢}xs ¢ .
For sake of convenience we denote:
{
²
²
²
CEU = {Ex1 , Ex2 ,........ Exs }and CU
E- {ei } = E - {ei }x1¢ , E - {ei }x2¢ ,........, E -
{ei }x
A. Saha andS. Broumi; Parameter reduction of Neutrosophic soft sets and their applications
s¢
} where
Neutrosophic Sets and Systems, Vol. 32, 2020
8
Ex1 = {x1 , x2 , x3 ,....., xi }x1 ,
Ex2 = {xi + 1 , xi + 2 ,....., x j }x2 ,
..........................................,
Exs = {xk , xk + 1 ,....., xn }xs ,
²- {e } = {x , x , x ,....., x } ,
E
i x
1¢ 2 ¢ 3¢
i ¢ x1¢
1¢
²- {e } = {x , x ,....., x } ,
E
i x
i + 1¢ i + 2 ¢
j ¢ x2 ¢
2¢
........................................................,
²- {e } = {x , x ,....., x } .
E
k¢
i x
s¢
n ¢ xs ¢
k + 1¢
3.5 Definition:The degree of importance of er for the decision partition is denoted by
1 s
defined by Im(er ) =
å Wq,er where
U q= 1
Im(er ) and is
ìï
²
ïïï Exq - E - {er }x y ¢ , if $ y ¢ such that xq = x y ¢,1 £ y ¢£ s ¢,1 £ q £ s
Wq,er = í
ïï
Exq , otherwise
ïï
î
U
3.6 Definition:For A= {e1¢, e2¢, e3¢,....., e¢
p } Ì E , the decision partition deleted A is denoted by CE-
U
and is given by CE-
A
² - Ax } .
²- Ax , E
²- Ax ,........, E
= {E
1¢
2¢
s¢
A
The degree of importance of A for the decision partition is defined by:
Im( A) =
1
U
s
å
q= 1
Wq,A where
ìï E - E
ïï xq ²- Ax y ¢ , if $ y ¢ such that xq = x y ¢,1 £ y ¢£ s ¢,1 £ q £ s
Wq,A = ïí
ïï
Exq , otherwise
ïïî
3.7 Example:Consider the neutrosophic soft set given in example 3.2. Then we have:
CEU = {{x3 , x4 , x5 }8 ,{x2 , x6 }7 ,{x1}6 }, s=3 and CU
E- {e1} = {{x3 , x4 , x5 }6 ,{x2 , x6 }5 ,{x1}4 } .
\ W1,e1 = {x1 }- {x1 } = 0, W2,e1 = {x2 , x6 } = 2, W3,e1 = {x3 , x4 , x5 } = 3. So Im(e1 ) =
3.8
Proposition:For
the
neutrosophic
soft
set
( f , E)
where
0 £ Im(er ) £ 1, r = 1,2,..., m .
A. Saha and S. Broumi; Parameter reduction of Neutrosophic soft sets and their applications
1
(0 + 2 + 3) = 0.833.
6
E = {e1 , e2 ,....., em }
,
9
Neutrosophic Sets and Systems, Vol. 32, 2020
Proof:
²- {e }
If $ y ¢ such that xq = x y ¢,1 £ y ¢£ s ¢,1 £ q £ s , then Wq ,er = Exq - E
r x
otherwise.
\ Im(er ) =
1
U
s
å
q= 1
Wq,er £
1
U
Again it is easy to verify that
s
å
q= 1
Exq =
1
U
{Ex
1
+ Ex2 + ...... + Exs
}=
y¢
£ Exq
and Wq,er = Exq ,
1
´ U = 1.
U
Im(er ) ³ 0 . Thus we have 0 £ Im(er ) £ 1 .
4. Decision making problem solving based on parameter reductionof neutrosophic soft set:
In this section we first develop an algorithm using parameter reduction of neutrosophic soft set and
then we illustrate this with a real life application.
Algorithm:
Step-1: Input the neutrosophic soft set ( f , E ) .
Step-2: Choose a parameter reduction A of E.
Step-3: Compute the choice value of the object xi Î U using the formula given below:
ci = å Im(e j )´ %
fe j ( xi ) where e j Î A .
j
Step-4: Find k for which ck = max ci .
i
Then ck is the optimal choice object. If k has more than one values, then any one of them can be
chosen by the decision maker.
An Illustrative example: Consider the neutrosophic soft set given in example 3.2. Now suppose
that Mr. John is interested to buy a house on the basis of his choice parameters e1 , e2 , e3 ,....., e6 ,
which means that out of the available houses in U, he will select that house that qualifies with all
or maximum number of parameters in E.
A. Saha andS. Broumi; Parameter reduction of Neutrosophic soft sets and their applications
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10
Step-1: The neutrosophic soft set ( f , E ) is given below:
%
f E (.)
U
e1
e2
e3
e4
e5
e6
x1
(0.3,0.7,0.4)
(0.4,0.5,0.1)
(0.2,0.2,0.4)
(0.6,0.3,0.4)
(0.1,0.1,0.3)
(0.2,0.4,0.6)
6.2
x2
(0.4,0.5,0.5)
(0.2,0.2,0.6)
(0.5,0.5,0.1)
(0.2,0.8,0.3)
(0.4,0.3,0.2)
(0.6,0.3,0.4)
7.0
x3
(0.2,0.5,0.7)
(0.3,0.2,0.5)
(0.8,0.2,0.4)
(0.5,0.5,0.3)
(0.2,0.4,0.2)
(0.9,0.6,0.6)
8.0
x4
(0.5,0.3,0.6)
(0.6,0.3,0.1)
(0.2,0.5,0.6)
(0.4,0.4,0.5)
(0.7,0.3,0.2)
(0.5,0.5,0.8)
8.0
x5
(0.3,0.5,0.6)
(0.4,0.4,0.2)
(0.3,0.3,0.5)
(0.6,0.1,0.6)
(0.7,0.8,0.1)
(0.4,0.6,0.6)
8.0
x6
(0.7,0.3,0.4)
(0.3,0.5,0.2)
(0.4,0.8,0.5)
(0.5,0.3,0.5)
(0.1,0.2,0.3)
(0.4,0.4,0.2)
7.0
Step-2: A parameter reduction of E is A = {e3 , e5 , e6 }. The corresponding neutrosophic soft set is given
below:
U
e3
e5
e6
%
f A (.)
x1
(0.2,0.2,0.4)
(0.1,0.1,0.3)
(0.2,0.4,0.6)
2.5
x2
(0.5,0.5,0.1)
(0.4,0.3,0.2)
(0.6,0.3,0.4)
3.3
x3
(0.8,0.2,0.4)
(0.2,0.4,0.2)
(0.9,0.6,0.6)
4.3
x4
(0.2,0.5,0.6)
(0.7,0.3,0.2)
(0.5,0.5,0.8)
4.3
x5
(0.3,0.3,0.5)
(0.7,0.8,0.1)
(0.4,0.6,0.6)
4.3
x6
(0.4,0.8,0.5)
(0.1,0.2,0.3)
(0.4,0.4,0.2)
3.3
Step-3: CUA = {{x3 , x4 , x5}4 ,{x2 , x6 }3 ,{ x1}2 } and s = 3.
CUA- {e } = {{x5 }3 ,{x4 }3 ,{x3}2 ,{x2 }2 ,{x1}1 ,{x6 }1 },
3
CUA- {e } = {{x3}3 ,{x4 }3 ,{x5 , x6 }2 ,{x2 }2 ,{x1}2 },
5
CUA- {e } = {{x5 }2 ,{x4 }2 ,{x6 }2 ,{x3}2 ,{x2 }2 ,{x1}1 }.
6
A. Saha and S. Broumi; Parameter reduction of Neutrosophic soft sets and their applications
11
Neutrosophic Sets and Systems, Vol. 32, 2020
\ W1,e3 = {x3 , x4 , x5 } = 3, W2,e3 = {x2 , x6 } = 2, W3,e3 = {x1 } = 1;
W1,e5 = {x3 , x4 , x5 } = 3, W2,e5 = {x2 , x6 } = 2, W3,e5 = {x1 }- {x1 } = 0;
W1,e6 = {x3 , x4 , x5 } = 3, W2,e6 = {x2 , x6 } = 2, W3,e6 = {x1 } = 1.
Hence Im(e3 ) =
Im(e6 ) =
1
U
1
U
3
å
q= 1
3
å
q= 1
Wq ,e3 =
Wq ,e6 =
1
6
1
6
(3 + 2 + 1) = 1, Im(e5 ) =
1
U
3
å
q= 1
Wq ,e5 =
1
6
(3 + 2 + 0) = 0.83,
(3 + 2 + 1) = 1.
The computation table for obtaining the choice values is given by:
U
x1
x2
e5
e3
(0.2,0.2,0.4)
(0.5,0.5,0.1)
(0.1,0.1,0.3)
(0.4,0.3,0.2)
ci
e6
(0.2,0.4,0.6)
c1 =(0.2+0.2+0.4)1+(0.1+0.1+0.3)0.83
(0.6,0.3,0.4)
+(0.2+0.4+0.6)1=2.415
c2 =(0.5+0.5+0.5)1+(0.4+0.3+0.2)0.83
+(0.6+0.3+0.4)1=3.147
x3
(0.8,0.2,0.4)
(0.2,0.4,0.2)
(0.9,0.6,0.6)
c3 =(0.8+0.2+0.4)1+(0.2+0.4+0.2)0.83
+(0.9+0.6+0.6)1=4.164
x4
(0.2,0.5,0.6)
(0.7,0.3,0.2)
(0.5,0.5,0.8)
c4 =(0.2+0.5+0.6)1+(0.7+0.3+0.2)0.83
+(0.5+0.5+0.8)1=4.096
x5
(0.3,0.3,0.5)
(0.7,0.8,0.1)
(0.4,0.6,0.6)
c5 =(0.3+0.3+0.5)1+(0.7+0.8+0.1)0.83
+(0.4+0.6+0.6)1=3.828
x6
(0.4,0.8,0.5)
(0.1,0.2,0.3)
(0.4,0.4,0.2)
c6 =(0.4+0.8+0.5)1+(0.1+0.2+0.3)0.83
+(0.4+0.4+0.2)1=3.198
Step-4: Since the choice value c3 is maximum, so house x3 is the best option for Mr. John.
Conclusion
In this paper we have proposed the concept of parameter reduction for neutrosophic soft sets
and we have used it to solve a decision making problem by developing an algorithm based on degree
of importance of parameters. The experimental results prove that our proposed parameter reduction
techniques delete the irrelevant parameters while keeping definite decision-making choices
unchanged. The parameter reduction presented in this paper may play an important role in some
knowledge discovery problem. Using the concept presented in this paper, one can think of parameter
reduction of
interval valued neutrosophic soft sets, hesitant neutrosophic soft sets and hesitant
interval valued neutrosophic soft sets.
A. Saha andS. Broumi; Parameter reduction of Neutrosophic soft sets and their applications
Neutrosophic Sets and Systems, Vol. 32, 2020
12
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Received: Oct 26, 2019. Accepted: Mar 21, 2020
A. Saha and S. Broumi; Parameter reduction of Neutrosophic soft sets and their applications
Neutrosophic Sets and Systems, Vol. 32, 2020
University of New Mexico
Neutrosophic Geometric Programming (NGP) Problems
Subject to (⋁, . ) Operator; the Minimum Solution
Huda E. Khalid
Telafer University, Head of the Scientific Affairs and Cultural Relations Department, Telafer, Iraq.
E-mail: hodaesmail@yahoo.com
Abstract. This paper comes as a second step serves the purpose of constructing a
neutrosophic optimization model for the relation geometric programming problems subject
to (max, product) operator in its constraints. This essay comes simultaneously with my
previous paper entitled (Neutrosophic Geometric Programming (NGP) with (max-product)
Operator, An Innovative Model) which contains the structure of the maximum solution. The
purpose of this article is to set up the minimum solution for the (RNGP) problems, the author
faced many difficulties, where the feasible region for this type of problems is already nonconvex; furthermore, the negative signs of the exponents with neutrosophic variables 𝑥𝑗 ∈
[0,1] ∪ 𝐼 . A new technique to avoid the divided by the indeterminacy component (𝐼) was
introduced; Separate the neutrosophic geometric programming into two optimization
models, introducing two new matrices named as the distinguishing matrix and the
facilitation matrix. All these notions were important for finding the minimum solution of the
program. Finally, two numerical examples were presented to enable the reader to understand
this work.
Keyword: Relational Neutrosophic Geometric Programming (RNGP); (⋁, . ) Operator;
Neutrosophic Relation Equations; Distinguishing Matrix; Facilitation Matrix; Minimum
Solution; Incompatible Problem.
1. Introduction
As of 1995 so far, dozens of mathematicians and researchers in many fields of
sciences trying to study and understand the neutrosophic theory, the first mathematician who
set up and put forward the neutrosophic theory was Smarandache F. at 1995 [2,11], he is in
the neutrosophic theory as Lotfi A. Zadeh [12] in fuzzy theory and as K. Atanasov [10] in
intuitionistic fuzzy theory. The importance of the neutrosophic logic comes from its ability to
deal with the indeterminacy component (𝐼), this component makes scholars generalize the
fuzzy and intuitionistic fuzzy logics, give them the ability to put the paradoxes in a new
framework, and it makes the researchers deal with contradicted information in more
relaxation. This paper comes as an establishing article in the relational neutrosophic
programming problems (RNGP) with (⋁, . ) in its constraints. This kind of problems has many
applications in real-world problems, like communication system, civil engineering,
mechanical engineering, structural design and optimization, business management …etc. The
author published previous articles [1,3,4,6,7,9] to expand the fuzzy theory to be fit with
neutrosophic theory, this essay was one of the series of these articles.
This publication includes three original sections, despite the second section goes to
the basic concepts, but these pure concepts were originated by the author at the
Huda E. Khalid, Neutrosophic Geometric Programming (NGP) Problems Subject to (⋁, . ) Operator; the Minimum
Solution
Neutrosophic Sets and Systems, Vol. 32, 2020
16
simultaneously published paper, which focused on the form of the maximum solution in the
(RNGP) with (∨, . ) operator, the third section was dedicated to many unprecedented
mathematical formulas such as pre-distinguishing matrices, pre-facilitation matrices, a new
technique to separate the optimization model into two models depending upon the sign of
terms powers in the objective function, and a technique to filter all minimum solutions, the
forth section was for two numerical examples, they are the same examples that presented in
the article [8] which assigned to the maximum solution, the last section includes the
conclusion.
2. Basic Concepts
We call
𝛾
𝛾
𝛾
min 𝑓(𝑥) = (𝑐1 . 𝑥1 1 ) ∨ (𝑐2 . 𝑥22 ) ∨ … ∨ (𝑐𝑛 . 𝑥𝑛𝑛 )
}
𝑠. 𝑡.
𝐴𝑜𝑥 = 𝑏
𝑥𝑗 ∈ [0,1] ∪ 𝐼,
1≤𝑗≤𝑛
(1)
A ( ∨, . ) (max- product) neutrosophic geometric programming, where 𝐴 = (𝑎𝑖𝑗 ), 1 ≤
𝑖 ≤ 𝑚 , 1 ≤ 𝑗 ≤ 𝑛, 𝑎𝑖𝑗 ∈ [0,1] is (𝑚 × 𝑛) dimensional neutrosophic matrix, 𝑥 = (𝑥1 , 𝑥2 , … , 𝑥𝑛 )𝑇
an n-dimensional variable vector, 𝑏 = (𝑏1 , 𝑏2 , … , 𝑏𝑚 )𝑇 (𝑏𝑖 ∈ [0,1] ∪ 𝐼) an m- dimensional
constant vector, 𝑐 = (𝑐1 , 𝑐2 , … , 𝑐𝑛 )𝑇 (𝑐𝑗 ≥ 0) an n- dimensional constant vector, 𝛾𝑗 is an
arbitrary real number, and the composition operator ‘’𝑜’’ is ( ∨, . ) , i.e. ⋁𝑛𝑗=1(𝑎𝑖𝑗 . 𝑥𝑗 ) = 𝑏𝑖 . Note
that the program (1) is undefined and has no minimal solution in the case of 𝛾𝑗 < 0 with all
𝑥𝑗 ′𝑠 taking indeterminacy value.
2.1. Definition [8]
𝑏𝑖
,
𝑛𝐼
,
𝑎𝑖𝑗
𝑎𝑖𝑗 ⋈ 𝑏𝑖 = { 1,
1,
𝑎𝑖𝑗 Θ𝑏𝑖 =
𝑎𝑖𝑗
𝑖𝑓 𝑎𝑖𝑗 > 𝑏𝑖 , 𝑎𝑖𝑗 ∈ [0,1], 𝑏𝑖 ∈ [0,1]
𝑖𝑓 𝑎𝑖𝑗 ≤ 𝑏𝑖 , 𝑎𝑖𝑗 ∈ [0,1], 𝑏𝑖 ∈ [0,1]
𝑖𝑓
𝑎𝑖𝑗 ∈ [0,1], 𝑏𝑖 = 𝑛𝐼, 𝑛 ∈ (0,1]
𝑖𝑓 𝑎𝑖𝑗 > 𝑛 , 𝑎𝑖𝑗 ∈ [0,1], 𝑏𝑖 = 𝑛𝐼, 𝑛 ∈ (0,1]
1,
𝑖𝑓 𝑎𝑖𝑗 ≤ 𝑛 , 𝑎𝑖𝑗 ∈ [0,1], 𝑏𝑖 = 𝑛𝐼, 𝑛 ∈ (0,1]
𝑛𝑜𝑡 𝑐𝑜𝑚𝑝. 𝑖𝑓
𝑎𝑖𝑗 = 𝑚𝐼 , 𝑚 ∈ (0,1] , 𝑏𝑖 ∈ [0,1] ∪ 𝐼
1
𝑖𝑓
𝑎𝑖𝑗 , 𝑏𝑖𝑗 ∈ [0,1]
{
Where ⋈ is an operator defined at [0,1], while the operator Θ is defined at [0,1] ∪ 𝐼. Let
𝑥̂𝑗 = ⋀𝑚
𝑖=1(𝑎𝑖𝑗 ⋈ 𝑏𝑖 ),
(1 ≤ 𝑗 ≤ 𝑛)
be the components of the pre maximum solution 𝑥̂𝑣1 .(i.e. 𝑥̂𝑣1 = (𝑥̂1 , 𝑥̂2 , … , 𝑥̂𝑛 ))
Let 𝑥̂𝑗 = ⋀𝑚
𝑖=1(𝑎𝑖𝑗 Θ𝑏𝑖 ),
(1 ≤ 𝑗 ≤ 𝑛) ,
(2)
(3)
(4)
(5)
be the components of the pre maximum solution 𝑥̂𝑣2 . (i.e. 𝑥̂𝑣2 = (𝑥̂1 , 𝑥̂2 , … , 𝑥̂𝑛 ))
Now the following question will be raised,
Which one 𝑥̂𝑣1 or 𝑥̂𝑣2 should be the exact maximum solution?
Neither 𝑥̂𝑣1 nor 𝑥̂𝑣2 will be the exact solution! The exact solution is integrated between them.
Before solving 𝐴𝑜𝑥̂ = 𝑏, we first define the matrices 𝐴𝑣1 , 𝐴𝑣2 .
Let 𝐴𝑣1 be a matrix has the same dimension and the same rows elements of 𝐴 except for those
rows of the indexes 𝑖 = 𝑖𝑜 corresponding to those indexes of 𝑏𝑖𝑜 = 𝑛𝐼, those special rows of
Huda E. Khalid, Neutrosophic Geometric Programming (NGP) Problems Subject to (⋁, . ) Operator; the Minimum
Solution
Neutrosophic Sets and Systems, Vol. 32, 2020
17
𝐴𝑣1 will be zeros. Let 𝐴𝑣2 be a matrix has the same dimension and the same rows elements of
𝐴 except for those rows of the indexes 𝑖 = 𝑖𝑜 corresponding to those indexes of 𝑏𝑖𝑜 ∈ [0,1],
those special rows of 𝐴𝑣2 will be zeros. Consequently,
𝐴𝑜𝑥̂ = 𝑏 = (𝐴𝑣1 𝑜𝑥̂𝑣1 ) + (𝐴𝑣2 𝑜𝑥̂𝑣2 )
(6)
The formula (6) is the greatest solution in 𝑋(𝐴, 𝑏).
The maximum value of the objective function 𝑓(𝑥̂) = 𝑓(𝑥̂𝑣1 ) ∨ 𝑓(𝑥̂𝑣2 ).
2.2. Theorem [8]
If 𝛾𝑗 < 0 (1 ≤ 𝑗 ≤ 𝑛), then the greatest solution to the problem (1) is an optimal
solution.
2.3. Definition [5]
If there exists a solution to 𝑥 = 𝑏 , it's called compatible. Suppose 𝑋(𝐴, 𝑏) = {(𝑥1 , 𝑥2 , … , 𝑥𝑚 )𝑇 ∈
[0,1]𝑛 ∪ 𝐼, 𝐼 𝑛 = 𝐼 , 𝑛 > 0|𝑥𝜊𝐴 = 𝑏, 𝑥𝑖 ∈ [0,1] ∪ 𝐼} is a solution set of 𝐴𝑜𝑥 = 𝑏 , we define 𝑥 1 ≤
𝑥 2 ⟺ 𝑥𝑗1 ≤ 𝑥𝑗2 (1 ≤ 𝑗 ≤ 𝑛), ∀ 𝑥 1 , 𝑥 2 ∈ 𝑋(𝐴, 𝑏). Where " ≤ " is a partial order relation on 𝑋(𝐴, 𝑏).
̆.
3. The Structure of the Minimum Solution 𝒙
The feasible region of the solution domain for the neutrosophic geometric
programming (NGP) problems subject to (max-product) operator in its constraints is a
solution to 𝐴𝑜𝑥 = 𝑏 , therefore the definition of the solution set 𝑋(𝐴, 𝑏) and the shape of the
maximum and the minimum solutions are very important to optimize the (NGP) model.
The structure of the maximum solution was introduced by Huda E. Khalid in [8].
The definition (2.3) was constructed by Huda E. Khalid at 2016 [5], this definition was
dedicated for (RNGP) problems subject to (max-min) operator, this definition is also
appropriate for (RNGP) problems with (max, product) operator.
3.1. Definition
If there exists a minimum solution in the solution set 𝑋(𝐴, 𝑏), then the numbers of the
minimum solutions are not lonesome such as the maximum solution. If we denote all
minimum elements by 𝑋̌ (𝐴, 𝑏), then another version of 𝑋(𝐴, 𝑏) can be presented depending
upon the minimum and the maximum solutions as follows:
𝑋(𝐴, 𝑏) = ∪𝑥̆∈𝑋̌(𝐴,𝑏)
{𝑥 ⎸𝑥̌ ≤ 𝑥 ≤ 𝑥̂, 𝑥 ∈ 𝑋}
(7)
The following definitions introduce some important new matrices that were constructed by
the author for using them in the filtering rule for finding the minimum solution.
3.2. Definition
Let 𝑆1 = (𝑠𝑖𝑗 1 )𝑚×n , 𝑆2 = (𝑠𝑖𝑗 2 )𝑚×n be two pre - distinguishing matrices of 𝐴, where
𝑎𝑖𝑗 ,
𝑠𝑖𝑗 1 = {
0,
𝑎𝑖𝑗 . 𝑥̂𝑗 = 𝑏𝑖
𝑎𝑖𝑗 . 𝑥̂𝑗 ≠ 𝑏𝑖
(8)
Huda E. Khalid, Neutrosophic Geometric Programming (NGP) Problems Subject to (⋁, . ) Operator; the Minimum
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In (8), the 𝑥̂𝑗 ’s are the components of the pre - maximum solution 𝑥̂𝑣1 which supports the
fuzzy part of the problem, while the elements 𝑎𝑖𝑗 are the elements of the matrix 𝐴𝑣1 .
𝑠𝑖𝑗 2 = {
𝑎𝑖𝑗 ,
0,
𝑎𝑖𝑗 . 𝑥̂𝑗 = 𝑏𝑖
𝑎𝑖𝑗 . 𝑥̂𝑗 ≠ 𝑏𝑖
(9)
In (9), the 𝑥̂𝑗 ’s are the components of the pre - maximum solution 𝑥̂𝑣2 which supports the
neutrosophic part of the problem, while 𝑎𝑖𝑗 are the elements of the matrix 𝐴𝑣2 .
Let
𝑆 = (𝑠𝑖𝑗 )𝑚×𝑛 = (𝑠𝑖𝑗 1 )𝑚×n + (𝑠𝑖𝑗 2 )𝑚×n = 𝑆1 + 𝑆2
(10)
The matrix 𝑆 is called the distinguishing matrix of 𝐴. It is obvious that the constraints system
𝐴𝑜𝑥 = 𝑏 has a solution if and only if the distinguishing matrix 𝑆 of 𝐴 has non zero rows (i.e. 𝑆
has at least a nonzero element in each row).
3.3. Definition
Let 𝐹1 = (𝑓𝑖𝑗 1 )𝑚×n , 𝐹2 = (𝑓𝑖𝑗 2 )𝑚×n be two pre - facilitation matrices of 𝐴, where
𝑥̂𝑖𝑗 ,
𝑓𝑖𝑗 1 = {
0,
𝑎𝑖𝑗 . 𝑥̂𝑗 = 𝑏𝑖
𝑎𝑖𝑗 . 𝑥̂𝑗 ≠ 𝑏𝑖
(11)
In (11), the 𝑥̂𝑗 ’s are the components of the pre- maximum solution 𝑥̂𝑣1 which supports the
fuzzy part of the problem, while the elements 𝑎𝑖𝑗 are the entries of 𝐴𝑣1 .
𝑥̂𝑖𝑗 ,
𝑓𝑖𝑗 2 = {
0,
𝑎𝑖𝑗 . 𝑥̂𝑗 = 𝑏𝑖
𝑎𝑖𝑗 . 𝑥̂𝑗 ≠ 𝑏𝑖
(12)
In (12), the 𝑥̂𝑗 ’s are the components of the pre - maximum solution 𝑥̂𝑣2 which supports the
neutrosophic part of the problem,
Let
𝐹 = (𝑓𝑖𝑗 )𝑚×𝑛 = (𝑓𝑖𝑗 1 )𝑚×n + (𝑓𝑖𝑗 2 )𝑚×n = 𝐹1 + 𝐹2
(13)
The matrix 𝐹 is called the Facilitation matrix of 𝐴.
Both matrices 𝑆 𝑎𝑛𝑑 𝐹 are first introduced in this paper and they have a key role in finding
the set of all quasi-minimum solutions and then the optimal solution for NGP problems.
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3.4 The Filtration Method for Finding Minimum Solutions
1. Delete the 𝑖 − 𝑡ℎ row of F, for which 𝑏𝑖 = 0
2.
At 𝑏𝑖 > 0, find an index 𝑧 ∈ {1,2, … , 𝑚} such that 𝑧 > 𝑖, if for all 𝑗 = 1,2, … , 𝑛
we find 𝑓𝑧𝑗 ≠ 0 ⟺ 𝑓𝑖𝑗 ≠ 0, then delete the 𝑖 − 𝑡ℎ row of F.
3. Denote 𝐹̃ for the matrix that gained from the above steps (i.e steps 1&2).
4. To each row of 𝐹̃ , in each time, the only nonzero value is selected in every
row with all entries of the rest seen as zero, perhaps all of the matrices are
denoted by 𝐹̃1 , 𝐹̃2 , … . , 𝐹̃𝑝 .
5. To each column of 𝐹̃𝑘 (1 ≤ 𝑘 ≤ 𝑝), the maximum element is selected, a quasiminimum solution 𝑥̌𝑗 can be obtained through such a method
The set composed of all 𝑥̌𝑗 is called a quasi-minimum solution, and it includes all
minimum solutions to 𝐴𝑜𝑥 = 𝑏. Delete all repeated solutions, and then all minimum
solutions 𝑋̌(𝐴, 𝑏) can be obtained.
As an integrated study for all cases of the exponents (𝛾𝑗 ) of the terms in the
objective function 𝑓(𝑥), we saw that the theorem (2.2) covered the negative
exponents, while the following theorem will cover the positive exponents for the
terms of 𝑓(𝑥).
3.5 Theorem
If 𝛾𝑗 ≥ 0 (1 ≤ 𝑗 ≤ 𝑛), then a certain minimum solution 𝑥̌ to 𝐴𝑜𝑥 = 𝑏 is an optimal one
to the program (1).
Proof
Since 𝛾𝑗 ≥ 0 (1 ≤ 𝑗 ≤ 𝑛), then
𝛾
𝛾𝑗
𝑑(𝑥𝑗 )
𝑑𝑥𝑗
𝛾 −1
= 𝛾𝑗 𝑥𝑗 𝑗
≥ 0.
We have 𝑥𝑗 ∈ [0,1] ∪ 𝐼, so 𝑥𝑗 𝑗 is a monotone increasing function concerning 𝑥𝑗 , so is
𝛾
𝑐𝑗 𝑥𝑗 𝑗 concerning 𝑥𝑗 . Hence, ∀ 𝑥 ∈ 𝑋(𝐴, 𝑏), depending on formula (7), then there exists
𝛾
𝛾
𝑥̌ ∈ 𝑋̌(𝐴, 𝑏), such that 𝑥 ≥ 𝑥̌ (i.e. 𝑥𝑗 ≥ 𝑥̌𝑗 ) ⟹ 𝑐𝑗 . 𝑥𝑗 𝑗 ≥ 𝑐𝑗 . 𝑥̌𝑗 𝑗 (1 ≤ 𝑗 ≤ 𝑛) ⟹ 𝑓(𝑥) ≥
𝑓(𝑥̌), this means that the optimal solution to the program (1) must exist in
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𝑋̌(𝐴, 𝑏).𝑓(𝑥̌ ∗ ) = min { 𝑓(𝑥̌) ⎸𝑥̌ ∈ 𝑋̌(𝐴, 𝑏)}. Then ∀ 𝑥 ∈ 𝑋(𝐴, 𝑏), there exists 𝑓(𝑥) ≥ 𝑓(𝑥̌ ∗ ),
so 𝑥̌ ∗ ∈ 𝑋̌(𝐴, 𝑏) is an optimal solution to the program (1).
3.6 Two Optimization Models Based on the Sign of 𝜸𝒋
Let 𝑀1 = {𝑗 ⎸𝛾𝑗 < 0, 1 < 𝑗 < 𝑛}, 𝑀2 = {𝑗 ⎸𝛾𝑗 > 0, 1 < 𝑗 < 𝑛}, then 𝑀1 ∩ 𝑀2 = ∅, 𝑀1 ∪ 𝑀2 = 𝐽 ,
here 𝐽 = {1,2, … , 𝑛}. It is evident that the terms of the objective function 𝑓(𝑥) in the program
(1) having negative powers is
𝛾𝑗
𝑓1 (𝑥) =∨𝑗∈𝑀1 {(𝑐𝑗 . 𝑥𝑗 )}
While the terms of 𝑓(𝑥) that having positive exponents is
𝛾𝑗
𝑓2 (𝑥) =∨𝑗∈𝑀2 {(𝑐𝑗 . 𝑥𝑗 )}
(14)
(15)
Based on (14) and (15), we have the following two optimization models,
min 𝑓1 (𝑥)
𝑠. 𝑡. 𝐴𝑜𝑥 = 𝑏
𝑥𝑗 ∈ [0,1] ∪ 𝐼
min 𝑓2 (𝑥)
𝑠. 𝑡. 𝐴𝑜𝑥 = 𝑏
𝑥𝑗 ∈ [0,1] ∪ 𝐼
(16)
(17)
Using theorem (2.2), 𝑥̂ is an optimal solution for (16). By theorem (3.5), there exists
𝑥̌ ∗ ∈ 𝑋̌(𝐴, 𝑏) , where 𝑥̌ ∗ is an optimal solution for (17).
3.7 Important Notes
1.
In this type of problems, the first step is to search for the maximum solution which is
lonesome for every problem. If the purpose of the program (1) is to optimize it, with
the restriction that all powers of the variables 𝑥𝑗 are negative, then the greatest
2.
solution is the optimal one {i.e. 𝑓(𝑥 ∗ ) = 𝑓(𝑥̂) = 𝑓(𝑥̂𝑣1 )⋀𝑓(𝑥̂𝑣2 )}.
The second step is to search for the minimum solution which is the set of all minimal
solutions 𝑋̌(𝐴, 𝑏). When the purpose of the program (1) is to optimize it, with the
restriction that some of the exponents are negative and others are positive, then
𝑓(𝑥 ∗ ) = 𝑓1 (𝑥̂)⋀𝑓2 (𝑥̌).
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3.
It should be noticed that the components of 𝑥̂𝑣2 containing indeterminate values (I)
raised to the negative powers of 𝑓(𝑥) must be neglected, otherwise, it will be
undefined program.
The upcoming section covering numerical examples, those examples are the same
that discussed in [8] for its maximal solution, we could not be remote far away from the paper
[8], present paper regarded as the complement of [8] which contained the formula of the
maximum solution, while this present paper introduces the set of all minimum solutions.
4 Numerical examples
We now gaze the (max, product) neutrosophic relation geometric programming examples as
follows
3.1 Example
Solve
1
1
min 𝑓(𝑥) = (0.3. 𝑥12 )⋁(1.8𝐼 . 𝑥23 )⋁(𝐼 . 𝑥34 )
s. t. 𝐴𝑜𝑥 = 𝑏
𝑥𝑗 ∈ [0,1]⋃𝐼
(1 ≤ 𝑗 ≤ 𝑛)
.6
1 1
Where 𝑏 = (1, 3 𝐼, 5 𝐼)𝑇 , 𝐴 = (. 5
.3
Solution:
1 .2
. 2 . 1) .
. 5 . 1 3×3
2
2
𝑇
𝑥̂𝑣1 = (𝑥̂1 , 𝑥̂2 , 𝑥̂3 )𝑇 = (1,1,1)𝑇 , 𝑥̂𝑣2 = (𝑥̂1 , 𝑥̂2 , 𝑥̂3 )𝑇 = (3 𝐼, 5 𝐼, 1) ,
.6
𝐴𝑣1 = ( 0
0
1 .2
0 0),
0 0
0 0
𝐴𝑣2 = (. 5 . 2
.3 .5
0
. 1),
.1
It is easy to notice that all exponents of 𝑓(𝑥) terms are positive. Therefore
there will not be a need to separate 𝑓(𝑥) into 𝑓1 and 𝑓2.
𝑓(𝑥̂) = 𝑓(𝑥̂𝑣1 )⋁𝑓(𝑥̂𝑣2 ) = 1.8𝐼 is the maximum solution.
Using theorem (3.5), it is essential to find the set of all minimum solutions for
𝑓(𝑥), where the optimal solution occurs at the minimal solution.
0 1
𝑆1 = [0 0
0 0
0
0
0 0
0
1 0
0], 𝑆1 = [0.5 0 0] , 𝑆 = [0.5 0 0].
0
0.3 0.5 0
0.3 0.5 0
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0
𝐹1 = [0
0
0
1 0
2
0 0] , 𝐹2 = [3 𝐼
2
0 0
𝐼
3
22
1
0
0
0
2
0], 𝐹 = [ 𝐼
3
2
0
𝐼
2
𝐼
5
1
0
0
0].
2
𝐼
5
3
0
Using the filtration rule stated in section (3.4),
𝐹̃ =
2
𝐼
[23
𝐼
3
0
0
2
𝐼
5
0
] ⟹ 𝐹̃1 =
2
𝐼
[32
𝐼
3
0 0
0 0
2
𝐼
3
] , 𝐹̃2 = [
0
0
2
𝐼
5
0
0
],
2
so the minimum solutions that related to 𝐹̃1 and 𝐹̃2 are 𝑥̌1 = [ 𝐼, 0,0], 𝑥̌2 =
𝑓(𝑥̌1 ) = 𝑓(𝑥̌2 ) =
2
𝐼
15
3
is the minimum solution.
2
2
[3 𝐼, 5 𝐼, 0].
3.2 Example
−
1
2
1
Let min 𝑓(𝑥) = (0.2𝐼. 𝑥1 3 ) ⋁ (1.3. 𝑥23 ) ⋁ (𝐼 . 𝑥32 ) ⋁ (0.35. 𝑥4−2 )
s. t. 𝐴𝑜𝑥 = 𝑏
𝑥𝑗 ∈ [0,1]⋃𝐼
(1 ≤ 𝑗 ≤ 𝑛)
.2 .3
Where 𝑏 = (0.3, 0.7𝐼, 0.5, 0.2𝐼) , 𝐴 = (. 3 . 2
1 0
0 .5
𝑇
Solution
3
.4
.9
.1
1
.6
. 8) .
1
0 4×4
𝑇
𝑥̂𝑣1 = (𝑥̂1 , 𝑥̂2 , 𝑥̂3 , 𝑥̂4 )𝑇 = (0.5,1, 4 , 0.5) , 𝑥̂𝑣2 = (𝑥̂1 , 𝑥̂2 , 𝑥̂3 , 𝑥̂4 )𝑇 =
𝑇
2
(5 𝐼, 1,0.2𝐼, 0.875𝐼) ,
The greatest solution for this problem is 𝑓(𝑥̂) = 𝑓(𝑥̂𝑣1 ) ⋁ 𝑓(𝑥̂𝑣2 ) = 1.3.
The following calculations are for finding the minimum solution.
𝐴𝑣1
.2
= (0
1
0
0 .3
𝑆1 = (0 0
1 0
0 0
0
𝐹1 = 0
.5
[0
1
0
0
0
0 0
.3 .4 .6
0
0 0 ) , 𝐴 = (. 3 . 2
𝑣2
0 0
0 .1 1
0 .5
0
0 0
.4
0
0
0
3
4
0 0
.6
0 ) , 𝑆 = (0 0
2
1
0 0
0
0 0
0 0
0 0 , 𝐹2 = [0 0
0 0
0 .5
0 1
0 0]
.5
0
0
0
1
0
.9
0
1
0
. 8).
0
0
0 .3 .4 .6
0
. 8), ⟹ 𝑆 = (0 0 0 . 8 ).
0
1 0
0 1
0
0 0
1 0
0
0
0
0 . 875𝐼 ], ⟹ 𝐹 = 0
0
0
.5
. 2𝐼
0
[0
1
0
0
1
3
4
.5
0 . 875𝐼 .
0
.5
. 2𝐼
0]
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0
𝐹̃ = [. 5
0
0
0
1
0 . 875𝐼
0
. 5 ],
. 2𝐼
0
0
𝐹̃1 = [. 5
0
0
0
1
0 . 875𝐼
0
0 ] ⟹ 𝑥̌1 = (.5,1, .2𝐼, .875𝐼)𝑇 ,
. 2𝐼
0
0
𝐹̃2 = [0
0
0
𝐹̃3 = [. 5
0
0
𝐹̃4 = [. 5
0
0
𝐹̃5 = [. 5
0
0
0
1
0
0
1
0
0
0
0
0
1
0 . 875𝐼
0
. 5 ] ⟹ 𝑥̌2 = (0,1, .2𝐼, .875𝐼)𝑇 ,
. 2𝐼
0
0 . 875𝐼
0
. 5 ] ⟹ 𝑥̌3 = (.5,1,0, .875𝐼)𝑇 ,
0
0
0 . 875𝐼
0
. 5 ] ⟹ 𝑥̌4 = (.5,0, .2𝐼, .875𝐼)𝑇 ,
. 2𝐼
0
0 . 875𝐼
0
0 ] ⟹ 𝑥̌5 = (.5,1,0, .875𝐼)𝑇 ,
0
0
0
𝐹̃6 = [0
0
0
0
1
0
0
0
0
𝐹̃8 = [0
0
0
0
0
0 . 875𝐼
0
. 5 ] ⟹ 𝑥̌8 = (0,0, .2𝐼, .875𝐼)𝑇 .
. 2𝐼
0
0
𝐹̃7 = [. 5
0
0
0
0
23
. 875𝐼
. 5 ] ⟹ 𝑥̌6 = (0,1,0, .875𝐼)𝑇 ,
0
0 . 875𝐼
0
0 ] ⟹ 𝑥̌7 = (.5,0, .2𝐼, .875𝐼)𝑇 ,
. 2𝐼
0
It is clear that there are two repeated solution,
𝑥̌5 = (.5,1,0, .875𝐼)𝑇 = 𝑥̌3 , and 𝑥̌7 = (.5,0, .2𝐼, .875𝐼)𝑇 = 𝑥̌4 , after deleting all
repeated solutions, the set of all quasi- minimum solutions 𝑋̌(𝐴, 𝑏) =
{𝑥̌1 , 𝑥̌2 , 𝑥̌3 , 𝑥̌4 , 𝑥̌6 , 𝑥̌8 }.
Since the powers of some terms in 𝑓(𝑥) are positive while others are negative,
we separate the objective function 𝑓(𝑥) into
−
2
1
1
𝑓1 (𝑥) = (0.2𝐼. 𝑥1 3 ) ⋁ (0.35 . 𝑥4−2 ) , 𝑓2 (𝑥) = (1.3. 𝑥23 ) ⋁ (𝐼 . 𝑥32 ),
First, solve for optimizing
min 𝑓1 (𝑥)
𝑠. 𝑡. 𝐴𝑜𝑥 = 𝑏
𝑥𝑗 ∈ [0,1] ∪ 𝐼
By theorem (2.2), we have f1 (𝑥 ∗ ) = f1 (𝑥̂) = 𝑓1 (𝑥̂𝑣1 ) ⋀ 𝑓1 (𝑥̂𝑣2 ) = 1.4, take care
of those terms of 𝑥̂𝑣2 that holding indeterminate components must be
neglected and avoid apply them in the terms of f1 (x).
Huda E. Khalid, Neutrosophic Geometric Programming (NGP) Problems Subject to (⋁, . ) Operator; the Minimum
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Second, solve for optimizing
min 𝑓2 (𝑥)
𝑠. 𝑡. 𝐴𝑜𝑥 = 𝑏
𝑥𝑗 ∈ [0,1] ∪ 𝐼
𝑓2 (𝑥̌1 ) = 1.3, 𝑓2 (𝑥̌2 ) = 1.3, 𝑓2 (𝑥̌3 ) = 1.3, 𝑓2 (𝑥̌4 ) = .447𝐼, 𝑓2 (𝑥̌6 ) = 1.3,
𝑓2 (𝑥̌8 ) = 0.447𝐼,
𝑥̌4 , 𝑥̌8 are the optimal for 𝑓2 (𝑥), (i.e. 𝑓2 (𝑥 ∗ ) = 0.447𝐼).
∴ 𝒇(𝒙∗ ) = 𝐟𝟏 (𝒙∗ ) ⋀ 𝒇𝟐 (𝒙∗ ) = 𝟎. 𝟒𝟒𝟕𝑰
5 Conclusion
The importance of this work comes from the unprecedented notions that were firstly
introduced in this article which are essential mathematical tools to establish the structure of
neutrosophic geometric programming (NGP) problems with (∨, . ) operator. Any optimization
problem needs to specify its minimum and maximum solution, in this article the author
introduced an effective technique to find the set of all quasi- minimum solution 𝑋̌(𝐴, 𝑏), side
by side with the structure of the maximum solution 𝑥̂. This work contains the theoretical rules
with two numerical examples to enable the readers to understand the pure mathematical
concepts.
Reference
[1] F. Smarandache & Huda E. Khalid "Neutrosophic Precalculus and Neutrosophic Calculus". Second
enlarged edition, Pons asbl 5, Quai du Batelage, Brussels, Belgium, European Union, 2018.
[2] F. Smarandache, H. E. Khalid & A. K. Essa, “Neutrosophic Logic: the Revolutionary Logic in Science
and Philosophy”, Proceedings of the National Symposium, EuropaNova, Brussels, 2018.
[3] F. Smarandache, H. E. Khalid, A. K. Essa, M. Ali, “The Concept of Neutrosophic Less Than or Equal
To: A New Insight in Unconstrained Geometric Programming”, Critical Review, Volume XII, 2016, pp.
72-80.
[4] H. E. Khalid, “An Original Notion to Find Maximal Solution in the Fuzzy Neutrosophic Relation
Equations (FNRE) with Geometric Programming (GP)”, Neutrosophic Sets and Systems, vol. 7, 2015, pp.
3-7.
[5] H. E. Khalid, “The Novel Attempt for Finding Minimum Solution in Fuzzy Neutrosophic Relational
Geometric Programming (FNRGP) with (max, min) Composition”, Neutrosophic Sets and Systems, vol.
11, 2016, pp. 107-111.
[6]. H. E. Khalid, F. Smarandache, & A. K. Essa, (2018). The Basic Notions for (over, off, under)
Neutrosophic Geometric Programming Problems. Neutrosophic Sets and Systems, 22, 50-62.
[7] H. E. Khalid, (2020). Geometric Programming Dealt with a Neutrosophic Relational Equations Under
the (𝑚𝑎𝑥 − 𝑚𝑖𝑛) Operation. Neutrosophic Sets in Decision Analysis and Operations Research, chapter
four. IGI Global Publishing House.
[8] H. E. Khalid, “Neutrosophic Geometric Programming (NGP) with (max-product) Operator, An
Innovative Model”, Neutrosophic Sets and Systems, vol. 32, 2020.
[9] H. E. Khalid, F. Smarandache, & A. K. Essa, (2016). A Neutrosophic Binomial Factorial Theorem with
their Refrains. Neutrosophic Sets and Systems, 14, 50-62.
[10] K. Atanassov KT (1986) Intuitionistic fuzzy sets. Fuzzy Sets System 20 (1):87–96
[11] V. Kandasamy, F. Smarandache, “Fuzzy Relational Maps and Neutrosophic Relational Maps”,
American Research Press, Rehoboth,2004.
[12] Zadeh LA (1965) Fuzzy sets. Inf. control 8(3):338–353.
Received: 17 Feb, 2020. Accepted: 20 Mar, 2020
Huda E. Khalid, Neutrosophic Geometric Programming (NGP) Problems Subject to (⋁, . ) Operator; the Minimum
Solution
Neutrosophic Sets and Systems, Vol. 32, 2020
University of New Mexico
Ngpr Homeomorphism in Neutrosophic Topological Spaces
K. Ramesh
Department of Mathematics, Government Arts College, Udumalpet - 642126, Tamilnadu, India.E-mail:
ramesh251989@gmail.com
* Correspondence: ramesh251989@gmail.com
Abstract: As a generalization of Fuzzy sets introduced by Zadeh [21] in 1965 and Intuitionistic
Fuzzy sets introduced by Atanassav [8] in 1983, the Neutrosophic set had been introduced and
developed by Smarandache. A Neutrosophic set is characterized by a truth value (membership), an
indeterminacy value and a falsity value (non-membership). Salama and Alblowi [17] introduced
the new concept of neutrosophic topological space (NTS) in 2012, which had been investigated
recently. In 2018, Parimala M et al. introduced and studied the concept of Neutrosophic
homeomorphism and Neutrosophic αψ homeomorphism in Neutrosophic topological spaces. The
impact of this article is to introduce and study the concepts of Ngpr homeomorphism and Nigpr
homeomorphism in Neutrosophic topological space. Further, the work is extended to Ngpr open
mappings, Ngpr closed mappings, Nigpr closed mappings and some of their properties are
explored in Neutrosophic topological space.
Keywords: Neutrosophic generalized pre regular closed set, Ngpr open mappings, Ngpr closed
mappings, Ngpr homeomorphism and Nigpr homeomorphism.
1. Introduction
Zadeh [21] introduced the concept of fuzzy set in 1965 and Chang C. L. [9] introduced fuzzy
topological spaces in 1968. Later, Atanassov [8] proposed the concept of intuitionistic fuzzy sets in
1986, where the degree of membership and degree of non-membership are discussed. Intuitionistic
fuzzy topological spaces was introduced by Coker [10] in 1997 using intuitionistic fuzzy sets. As a
generalization of Fuzzy sets and Intuitionistic Fuzzy sets, Neutrosophic set have been introduced
and developed by Florentin Smarandache [12]. He also defined the Neutrosophic set on three
components, namely Truth (membership) (T), Indeterminacy (I) and Falsehood (non-membership)
(F).
Neutrosophic concept has wide range of real time applications in the fields of [1 - 6] Information
Systems, Computer Science, Artificial Intelligence, Applied Mathematics and Decision Making,
Uncertainty assessments of linear time-cost tradeoffs and solving the supply chain problem.
In 2012, Salama A. A and Alblowi [17] introduced the concept of Neutrosophic topological
space by using Neutrosophic sets. Salama A. A. [18] introduced Neutrosophic closed set and
Neutrosophic continuous function in Neutrosophic topological spaces and their properties are
studied by various authors [7 & 11]. Since, Neutrosophic homeomorphism plays an important role
in Neutrosophic topology. Parimala M et al. [14] introduced and studied the concept of
Neutrosophic homeomorphism and Neutrosophic αψ homeomorphism in Neutrosophic topological
spaces. In this article, introduce and study few properties of Ngpr open mappings, Ngpr closed
mappings, Nigpr closed mappings, Ngpr homeomorphism and Nigpr homeomorphism in
Neutrosophic topological space. The present study demonstrates some of the related theorems,
results and properties.
2. Preliminaries
K. Ramesh, Ngpr Homeomorphism in Neutrosophic Topological Spaces
Neutrosophic Sets and Systems, Vol. 32, 2020
26
2.1. Definition: [17] Let X be a non-empty fixed set. A Neutrosophic set (NS for short) A in X is an
object having the form A = {〈x, µA(x), σA(x), νA(x)〉: x ∈ X} where the functions µA(x), σA(x) and νA(x)
represent the degree of membership, degree of indeterminacy and the degree of non-membership
respectively of each element x ∈ X to the set A.
2.2 Remark: [17] A Neutrosophic set A = {〈x, µA(x), σA(x), νA(x) 〉: x ∈ X} can be identified to an
ordered triple A = 〈x, µA(x), σA(x), νA(x)〉 in non-standard unit interval ]-0, 1+[ on X.
2.3 Remark: [17] For the sake of simplicity, we shall use the symbol A = 〈x, µA, σA, νA〉 for the
neutrosophic set A = {〈x, µA(x), σA(x), νA(x)〉: x ∈ X}.
2.4 Example: [17] Every IFS A is a non-empty set in X is obviously on NS having the form
A = {〈x, µA(x), 1 – (µA(x) + νA(x)), νA(x)〉: x ∈ X}. Since our main purpose is to construct the tools for
developing Neutrosophic set and Neutrosophic topology, we must introduce the NS 0 N and 1N in X
as follows:
0N may be defined as:
(01) 0N = {〈x, 0, 0, 1〉: x ∈ X}
(02) 0N = {〈x, 0, 1, 1〉: x ∈ X}
(03) 0N = {〈x, 0, 1, 0〉: x ∈ X}
(04) 0N = {〈x, 0, 0, 0〉: x ∈ X}
1N may be defined as:
(11) 1N = {〈x, 1, 0, 0〉: x ∈ X}
(12) 1N = {〈x, 1, 0, 1〉: x ∈ X}
(13) 1N = {〈x, 1, 1, 0〉: x ∈ X}
(14) 1N = {〈x, 1, 1, 1〉: x ∈ X}
2.5 Definition: [17] Let A = 〈µA, σA, νA〉 be a NS on X, then the complement of the set A [C(A) for
short] may be defined as three kind of complements:
(C1) C(A) = {〈x, 1-µA(x), 1-σA(x), 1-νA(x)〉: x ∈ X }
(C2) C(A) = {〈x, νA(x), σA(x), µA(x)〉: x ∈ X}
(C3) C(A) = {〈x, νA(x), 1-σA(x), µA(x)〉: x ∈ X}
2.6 Definition: [17] Let X be a non-empty set and Neutrosophic sets A and B in the form A = {〈x,
µA(x), σA(x), νA(x)〉: x ∈ X} and B = {〈x, µB(x), σB(x), νB(x)〉: x ∈ X}. Then we may consider two possible
definitions for subsets (A ⊆ B).
(1) A ⊆ B ⇔ µA(x) ≤ µB(x), σA(x) ≤ σB(x) and µA(x) ≥ µB(x) ∀ x ∈ X
(2) A ⊆ B ⇔ µA(x) ≤ µB(x), σA(x) ≥ σB(x) and µA(x) ≥ µB(x) ∀ x ∈ X
2.7 Proposition: [17] For any Neutrosophic set A, the following conditions hold:
0N ⊆ A, 0N ⊆ 0N
A ⊆ 1N, 1N ⊆1N
2.8 Definition: [17] Let X be a non-empty set and A = {〈x, µA(x), σA(x), νA(x)〉: x ∈ X}, B = {〈x, µB(x),
σB(x), νB(x)〉: x ∈ X} are NSs. Then A∩B may be defined as:
(I1) A∩B = 〈x, µA(x)∧µB(x), σA(x)∧σB(x) and νA(x)∨νB(x)〉
(I2) A∩B = 〈x, µA(x)∧µB(x), σA(x)∨σB(x) and νA(x)∨νB(x)〉
A∪B may be defined as:
(U1) A∪B = 〈x, µA(x)∨µB(x), σA(x)∨σB(x) and νA(x)∧νB(x)〉
(U2) A∪B = 〈x, µA(x)∨µB(x), σA(x)∧σB(x) and νA(x)∧νB(x)〉
K. Ramesh, Ngpr Homeomorphism in Neutrosophic Topological Spaces
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2.9 Definition: [17] A Neutrosophic topology [NT for short] is a non-empty set X is a family τ of
Neutrosophic subsets in X satisfying the following axioms:
(NT1) 0N, 1N ∈ τ,
(NT2) G1∩G2 ∈ τ for any G1, G2 ∈ τ,
(NT3) ∪Gi ∈ τ for every {Gi : i ∈ J} ⊆ τ.
Throughout this paper, the pair (X, τ) is called a Neutrosophic topological space (NTS for short).
The elements of are called Neutrosophic open sets [NOS for short]. A complement C(A) of a NOS
A in NTS (X, τ) is called a Neutrosophic closed set [NCS for short] in X.
2.10 Definition: [17] Let (X, τ) be NTS and A = {〈x, µA(x), σA(x), νA(x)〉: x ∈ X} be a NS in X. Then the
Neutrosophic closure and Neutrosophic interior of A are defined by
NCl(A) =∩{K : K is a NCS in X and A ⊆ K}
NInt(A) =∪{G : G is a NOS in X and G ⊆ A}
It can be also shown that NCl(A) is NCS and NInt(A) is a NOS in X.
a) A is NOS if and only if A = NInt(A),
b) A is NCS if and only if A = NCl(A).
2.11 Definition: [13] A NS A = {〈x, µA(x), σA(x), νA(x)〉: x ∈ X} in a NTS (X, τ) is said to be
(i) Neutrosophic regular closed set (NRCS for short) if A = NCl(NInt(A)),
(ii) Neutrosophic regular open set (NROS for short) if A = NInt(NCl(A)),
(iii) Neutrosophic pre closed set (NPCS for short) if NCl(NInt(A)) ⊆ A,
(iv) Neutrosophic pre open set (NPOS for short) if A ⊆ NInt(NCl(A)),
(v) Neutrosophic α- closed set (NSCS for short) if NCl(NInt(NCl(A))) ⊆ A,
(vi) Neutrosophic α- open set (NSOS for short) if A ⊆ NInt(NCl(NInt(A))).
2.12 Definition: [19] Let (X, τ) be NTS and A = {〈x, µA(x), σA(x), νA(x)〉: x ∈ X} be a NS in X. Then the
Neutrosophic pre closure and Neutrosophic pre interior of A are defined by
NPCl(A) = ∩{K : K is a NPCS in X and A ⊆ K},
NPInt(A) = ∪{G : G is a NPOS in X and G ⊆ A}.
2.13 Definition: [15] A NS A = {〈x, µA(x), σA(x), νA(x)〉: x ∈ X} in a NTS (X, τ) is said to be a
Neutrosophic generalized closed set (NGCS for short) if NCl(A) ⊆ U whenever A ⊆ U and U is a
NOS in (X, τ). A NS A of a NTS (X, τ) is called a Neutrosophic generalized open set (NGOS for short)
if C(A) is a NGCS in (X, τ).
2.14 Definition: [20] A NS A = {〈x, µA(x), σA(x), νA(x)〉: x ∈ X} in a NTS (X, τ) is said to be a
Neutrosophic generalized pre closed set (NGPCS for short) if NPCl(A) ⊆ U whenever A ⊆ U and U
is a NOS in (X, τ). A NS A of a NTS (X, τ) is called a Neutrosophic generalized pre open set (NGPOS
for short) if C(A) is a NGPCS in (X, τ).
2.15 Definition: [13] A NS A = {〈x, µA(x), σA(x), νA(x)〉: x ∈ X} in a NTS (X, τ) is said to be a
Neutrosophic generalized pre regular closed set (NGPRCS for short) if NPCl(A) ⊆ U whenever A ⊆
U and U is a NROS in (X, τ). The family of all NGPRCSs of a NTS(X, τ) is denoted by NGPRC(X). A
NS A of a NTS (X, τ) is called a Neutrosophic generalized pre regular open set (NGPROS for short) if
C(A) is a NGPRCS in (X, τ).
Every NRCS, NCS, NWCS, NαCS, NGCS, NPCS, NαGCS, NGPCS, NRαGCS, NRGCS is an
NGPRCS but the converses are not true in general.
2.16 Definition: [13] A Neutrosophic topological space (X, τ) is called a Neutrosophic pre regular T 1/2
(NPRT1/2 for short) space if every NGPRCS in (X, τ) is NPCS in (X, τ).
K. Ramesh, Ngpr Homeomorphism in Neutrosophic Topological Spaces
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2.17 Definition: [13] A Neutrosophic topological space (X, τ) is called a Neutrosophic pre regular T*1/2
(NPRT*1/2 for short) space if every NGPRCS in (X, τ) is NCS in (X, τ).
2.18 Definition: [16] Let (X, τ) and (Y, σ) be two NTSs. A mapping f: (X, τ) → (Y, σ) is called Ngpr
continuous (resp. NG continuous, NGP continuous) mapping if f-1(B) is NGPRCS (resp. NGCS,
NGPCS) in (X, τ) for every NCS B of (Y, σ).
Every Neutrosophic continuous, NG continuous, NGP continuous is a Ngpr continuous
mapping but the converses are not true in general.
2.19 Definition: [16] Let (X, τ) and (Y, σ) be two NTSs. A mapping f: (X, τ) → (Y, σ) is called Ngpr
irresolute mapping if f-1(A) is NGPRCS in (X, τ) for every NGPRCS A of (Y, σ).
2.20 Definition: [14] Let (X, τ) and (Y, σ) be two NTSs. A mapping f: (X, τ) → (Y, σ) is called
Neutrosophic closed mapping (resp. Neutrosophic open mapping) (NCM (resp. NOM) for short) if
the image of every Neutrosophic closed set (resp. Neutrosophic open set) in (X, τ) is a Neutrosophic
closed set (resp. Neutrosophic open set) in (Y, σ).
2.21 Definition: [14] Let (X, τ) and (Y, σ) be two NTSs. A bijection f: (X, τ) → (Y, σ) is called a
Neutrosophic homeomorphism if f and f-1 are Neutrosophic continuous mapping.
3. Ngpr open mappings and Ngpr closed mappings
In this section introduce Ngpr open mapping, Ngpr closed mapping and Nigpr closed
mapping in the Neutrosophic topological space and study some of their properties. Also established
the relation between the newly introduced mappings and already existing mappings.
3.1 Definition: Let (X, τ) and (Y, σ) be two NTSs. A mapping f: (X, τ) → (Y, σ) is called
(i) Neutrosophic generalized open mapping (NGOM for short) if f(A) is NGOS in (Y, σ) for
every NOS A of (X, τ).
(ii) Neutrosophic α open mapping (NαOM for short) if f(A) is NαOS in (Y, σ) for every NOS A
of (X, τ).
(iii) Neutrosophic pre-open mapping (NPOM for short) if f(A) is NPOS in (Y, σ) for every NOS
A of (X, τ).
(iv) Neutrosophic generalized pre-open mapping (NGPOM for short) if f(A) is NGPOS in (Y, σ)
for every NOS A of (X, τ).
3.2 Definition: Let (X, τ) and (Y, σ) be two NTSs. A mapping f: (X, τ) → (Y, σ) is called Ngpr open
mapping (NGPROM for short) if f(A) is NGPROS in (Y, σ) for every NOS A of (X, τ).
3.3 Definition: Let (X, τ) and (Y, σ) be two NTSs. A mapping f: (X, τ) → (Y, σ) is called
(i) Neutrosophic generalized closed mapping (NGCM for short) if f(A) is NGCS in (Y, σ) for
every NCS A of (X, τ).
(ii) Neutrosophic α closed mapping (NαCM for short) if f(A) is NαCS in (Y, σ) for every NCS
A of (X, τ).
(iii) Neutrosophic pre-closed mapping (NPCM for short) if f(A) is NPCS in (Y, σ) for every NCS
A of (X, τ).
(iv) Neutrosophic generalized pre-closed mapping (NGPCM for short) if f(A) is NGPCS in
(Y, σ) for every NCS A of (X, τ).
3.4 Definition: Let (X, τ) and (Y, σ) be two NTSs. A mapping f: (X, τ) → (Y, σ) is called Ngpr closed
mapping (NGPRCM for short) if f(A) is NGPRCS in (Y, σ) for every NCS A of (X, τ).
K. Ramesh, Ngpr Homeomorphism in Neutrosophic Topological Spaces
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3.5 Example: Let X= {a, b} and Y = {u, v}. Then τ = {0 N, U, 1N} and σ = {0N, V1, V2, 1N} are Neutrosophic
topologies on X and Y respectively, where U = 〈x, (0.4, 0.4, 0.5), (0.6, 0.3, 0.4)〉 and V1 = 〈y, (0.7, 0.5,
0.3), (0.8, 0.4, 0.2)〉 and V2 = 〈y, (0.6, 0.4, 0.4), (0.7, 0.3, 0.3)〉. Define a mapping f: (X, τ) → (Y, σ) by f(a) =
u and f(b) = v. Here the Neutrosophic set Uc = 〈x, (0.5, 0.6, 0.4), (0.4, 0.7, 0.6)〉 is a Neutrosophic closed
set in X. Then f(Uc) = 〈y, (0.5, 0.6, 0.4), (0.4, 0.7, 0.6)〉 is a NGPRCS in (Y, σ) as f(Uc) ⊆1N implies
Npcl(f(Uc)) = f(Uc) ⊆ 1N where 1N is a NROS in Y. Therefore f is a Ngpr closed mapping.
3.6 Proposition: Every Neutrosophic closed mapping is Ngpr closed mapping but not conversely in
general.
Proof: Let f: (X, τ) → (Y, σ) be a Neutrosophic closed mapping. Let A be a NCS in X. Then f(A) is a
NCS in Y. Since every NCS is a NGPRCS in Y, f(A) is a NGPRCS in Y. Hence f is a Ngpr closed
mapping.
3.7 Example: Let X= {a, b} and Y = {u, v}. Then τ = {0 N, U, 1N} and σ = {0N, V1, V2, 1N} are Neutrosophic
topologies on X and Y respectively, where U = 〈x, (0.4, 0.4, 0.5), (0.6, 0.3, 0.4)〉 and V1 = 〈y, (0.7, 0.5,
0.3), (0.8, 0.4, 0.2)〉 and V2 = 〈y, (0.6, 0.4, 0.4), (0.7, 0.3, 0.3)〉. Define a mapping f: (X, τ) → (Y, σ) by f(a) =
u and f(b) = v. Here the Neutrosophic set Uc = 〈x, (0.5, 0.6, 0.4), (0.4, 0.7, 0.6)〉 is a NCS in X. Then f(Uc)
= 〈y, (0.5, 0.6, 0.4), (0.4, 0.7, 0.6)〉 is a NGPRCS in (Y, σ) as f(Uc) ⊆1N implies Npcl(f(Uc)) = f(Uc) ⊆ 1N
where 1N is a NROS in Y. Therefore f is a Ngpr closed mapping. But f is not a Neutrosophic closed
mapping since Uc is NCS in X but f(Uc) is not a NCS in Y as Ncl(f (Uc)) = 1N ≠ f(Uc).
3.8 Proposition: Every Neutrosophic generalized closed mapping is Ngpr closed mapping but not
conversely in general.
Proof: Let f: (X, τ) → (Y, σ) be a Neutrosophic generalized closed mapping. Let A be a NCS in X.
Then f(A) is a NGCS in Y. Since every NGCS is a NGPRCS in Y, f(A) is a NGPRCS in Y. Hence f is a
Ngpr closed mapping.
3.9 Example: Let X= {a, b} and Y = {u, v}. Then τ = {0 N, U, 1N} and σ = {0N, V, 1N} are Neutrosophic
topologies on X and Y respectively, where U = 〈x, (0.3, 0.5, 0.4), (0.2, 0.5, 0.3) 〉 and V = 〈y, (0.6, 0.5,
0.2), (0.4, 0.5, 0.2)〉. Define a mapping f: (X, τ) → (Y, σ) by f(a) = u and f(b) = v. Here the Neutrosophic
set Uc = 〈x, (0.4, 0.5, 0.3), (0.3, 0.5, 0.2) 〉 is a NCS in X. Then f(Uc) = 〈y, (0.4, 0.5, 0.3), (0.3, 0.5, 0.2)〉 is a
NGPRCS in (Y, σ) as f(Uc) ⊆1N implies Npcl(f(Uc)) = f(Uc) ⊆ 1N where 1N is a NROS in Y. Therefore f is
a Ngpr closed mapping. But f is not a Neutrosophic generalized closed mapping since U c is NCS in X
but f(Uc) is not a NGCS in Y as f(Uc) ⊆ V implies Ncl(f(Uc)) = 1N ⊈ V.
3.10 Proposition: Every Neutrosophic α closed mapping is Ngpr closed mapping but not conversely
in general.
Proof: Let f: (X, τ) → (Y, σ) be a Neutrosophic α closed mapping. Let A be a NCS in X. Then f(A) is a
NαCS in Y. Since every NαCS is a NGPRCS in Y, f(A) is a NGPRCS in Y. Hence f is a Ngpr closed
mapping.
3.11 Example: Let X= {a, b} and Y = {u, v}. Then τ = {0 N, U, 1N} and σ = {0N, V, 1N} are Neutrosophic
topologies on X and Y respectively, where U = 〈x, (0.4, 0.5, 0.4), (0.2, 0.5, 0.3)〉 and V = 〈y, (0.7, 0.5, 0.2),
(0.3, 0.5, 0.2)〉. Define a mapping f: (X, τ) → (Y, σ) by f(a) = u and f(b) = v. Here the Neutrosophic set Uc
= 〈x, (0.4, 0.5, 0.4), (0.3, 0.5, 0.2)〉 is a NCS in X. Then f(Uc) = 〈y, (0.4, 0.5, 0.4), (0.3, 0.5, 0.2)〉 is a
NGPRCS in (Y, σ) as f(Uc) ⊆1N implies Npcl(f(Uc)) = f(Uc) ⊆ 1N where 1N is a NROS in Y. Therefore f is
a Ngpr closed mapping. But f is not a Neutrosophic α closed mapping since Uc is NCS in X but f(Uc)
is not a NαCS in Y as Ncl(Nint(Ncl(f(Uc)))) = 1N ⊈ f(Uc).
K. Ramesh, Ngpr Homeomorphism in Neutrosophic Topological Spaces
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3.12 Proposition: Every Neutrosophic pre-closed mapping is Ngpr closed mapping but not
conversely in general.
Proof: Let f: (X, τ) → (Y, σ) be a Neutrosophic pre-closed mapping. Let A be a NCS in X. Then f(A) is
a NPCS in Y. Since every NPCS is a NGPRCS in Y, f(A) is a NGPRCS in Y. Hence f is a Ngpr closed
mapping.
3.13 Example: Let X= {a, b} and Y = {u, v}. Then τ = {0 N, U, 1N} and σ = {0N, V, 1N} are Neutrosophic
topologies on X and Y respectively, where U = 〈x, (0.4, 0.5, 0.6), (0.2, 0.5, 0.3)〉 and V = 〈y, (0.3, 0.5, 0.7),
(0.3, 0.5, 0.4)〉. Define a mapping f: (X, τ) → (Y, σ) by f(a) = u and f(b) = v. Here the Neutrosophic set Uc
= 〈x, (0.6, 0.5, 0.4), (0.3, 0.5, 0.2)〉 is a NCS in X. Then f(Uc) = 〈y, (0.6, 0.5, 0.4), (0.3, 0.5, 0.2)〉 is a
NGPRCS in (Y, σ) as f(Uc) ⊆1N implies Npcl(f(Uc)) = 〈y, (0.7, 0.5, 0.3), (0.4, 0.5, 0.2)〉 ⊆ 1N where 1N is a
NROS in Y. Therefore f is a Ngpr closed mapping. But f is not a Neutrosophic pre-closed mapping
since Uc is NCS in X but f(Uc) is not a NPCS in Y as Ncl(Nint(f(Uc))) = Vc ⊈ f(Uc).
3.14 Proposition: Every Neutrosophic generalized pre-closed mapping is Ngpr closed mapping but
not conversely in general.
Proof: Let f: (X, τ) → (Y, σ) be a Neutrosophic generalized pre-closed mapping. Let A be a NCS in X.
Then f(A) is a NGPCS in Y. Since every NGPCS is a NGPRCS in Y, f(A) is a NGPRCS in Y. Hence f is
a Ngpr closed mapping.
3.15 Example: Let X= {a, b} and Y = {u, v}. Then τ = {0 N, U, 1N} and σ = {0N, V, 1N} are Neutrosophic
topologies on X and Y respectively, where U = 〈x, (0.3, 0.8, 0.5), (0.4, 0.7, 0.6)〉 and V = 〈y, (0.5, 0.2, 0.3),
(0.6, 0.3, 0.4)〉. Define a mapping f: (X, τ) → (Y, σ) by f(a) = u and f(b) = v. Here the Neutrosophic set Uc
= 〈x, (0.5, 0.2, 0.3), (0.6, 0.3, 0.4)〉 is a NCS in X. Then f(Uc) = 〈y, (0.5, 0.2, 0.3), (0.6, 0.3, 0.4)〉 is a
NGPRCS in (Y, σ) as f(Uc) ⊆1N implies Npcl(f(Uc)) = 1N ⊆ 1N where 1N is a NROS in Y. Therefore f is a
Ngpr closed mapping. But f is not a Neutrosophic generalized pre-closed mapping since Uc is NCS
in X but f (Uc) is not a NGPCS in Y as f(Uc) ⊆V implies Npcl(f (Uc)) = 1N ⊈ V where V is a NOS in Y.
3.16 Definition: Let (X, τ) and (Y, σ) be two NTSs. A mapping f: (X, τ) → (Y, σ) is called Nigpr open
mapping (NiGPROM for short) if f(A) is NGPROS in (Y, σ) for every NGPROS A of (X, τ).
3.17 Definition: Let (X, τ) and (Y, σ) be two NTSs. A mapping f: (X, τ) → (Y, σ) is called Nigpr closed
mapping (NiGPRCM for short) if f(A) is NGPRCS in (Y, σ) for every NGPRCS A of (X, τ).
3.18 Example: Let X= {a, b} and Y = {u, v}. Then τ = {0 N, U, 1N} and σ = {0N, V, 1N} are Neutrosophic
topologies on X and Y respectively, where U = 〈x, (0.5, 0.4, 0.3), (0.7, 0.8, 0.2)〉 and V = 〈y, (0.7, 0.4, 0.5),
(0.8, 0.5, 0.5)〉. Define a mapping f: (X, τ) → (Y, σ) by f(a) = u and f(b) = v. Hence f(A) is NGPRCS in
(Y, σ) for every NGPRCS A of (X, τ). Therefore f is a Nigpr closed mapping.
3.19 Proposition: Every Nigpr closed mapping is Ngpr closed mapping but not conversely in
general.
Proof: Let f: (X, τ) → (Y, σ) be a Nigpr closed mapping. Let A be a NCS in X. Since every NCS is a
NGPRCS in X, A is a NGPRCS in X. Then f(A) is a NGPRCS in Y. Hence f is a Ngpr closed mapping.
3.20 Example: Let X= {a, b} and Y = {u, v}. Then τ = {0 N, U, 1N} and σ = {0N, V, 1N} are Neutrosophic
topologies on X and Y respectively, where U = 〈x, (0.2, 0.5, 0.7), (0.3, 0.5, 0.6)〉 and V = 〈y, (0.3, 0.5, 0.6),
(0.4, 0.5, 0.5)〉. Define a mapping f: (X, τ) → (Y, σ) by f(a) = u and f(b) = v. Here the Neutrosophic set Uc
= 〈x, (0.7, 0.5, 0.2), (0.6, 0.5, 0.3)〉 is a NCS in X. Then f(Uc) = 〈y, (0.7, 0.5, 0.2), (0.6, 0.5, 0.3)〉 is a
NGPRCS in (Y, σ) as f(Uc) ⊆1N implies Npcl(f(Uc)) = f(Uc) ⊆ 1N where 1N is a NROS in Y. Therefore f is
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a Ngpr closed mapping. But f is not a Nigpr closed mapping since W = 〈x, (0.3, 0.5, 0.6), (0.4, 0.5, 0.5)〉
is NGPRCS in X but f(W) is not a NGPRCS in Y as f(W) ⊆ V implies Npcl(f(W)) = Vc ⊈ V where V is a
NROS in Y. Therefore f is not a Nigpr closed mapping.
The relation between various types of Neutrosophic closed mappings is given by
NαCM
NCM
NGPCM
NiGPRCM
NGPRCM
NPCM
NGCM
Fig.3.1.1 The reverse implications of Fig.3.1.1 are not true in general in the above diagram.
3.21 Theorem: A mapping f: (X, τ) → (Y, σ) is Ngpr closed mapping if and only if Ngprcl(f(A)) ⊆
f(Ncl(A)).
Proof: Let A ⊆ X and f: (X, τ) → (Y, σ) be a Ngpr closed mapping, then f(Ncl(A)) is NGPRCS in Y
which implies Ngprcl(f(Ncl(A))) = f(Ncl(A)). Since f(A) ⊆ f(Ncl(A)), Ngprcl(f(A)) ⊆ Ngprcl(f(Ncl(A)))
= f(Ncl(A)) for every NS A of X.
Conversely, let A be any NCS in (X, τ). Then A = Ncl(A) and so f(A) = f(Ncl(A)) ⊇ Ngprcl(f(A)), by
hypothesis. Since f(A) ⊆ Ngprcl(f(A)), therefore f(A) = Ngprcl(f(A)). i.e., f(A) is NGPRCS in Y and
hence f is Ngpr closed mapping.
3.22 Theorem: If f: (X, τ) → (Y, σ) is Ngpr open mapping iff for every NS A of (X, τ), f(Nint(A)) ⊆
Ngprint(f(A)).
Proof: Necessity: Let A be a NOS in X and f: (X, τ) → (Y, σ) be a Ngpr open mapping then f(Nint(A))
is NGPROS in Y. Since f(Nint(A)) ⊆ f(A) which implies Ngprint(f(Nint(A))) ⊆ Ngprint(f(A)). Since
f(Nint(A)) is NGPROS in Y, we have f(Nint(A)) ⊆ Ngprint(f(A)).
Sufficiency: Assume A is a NOS of (X, τ). Then f(A) = f(Nint(A)) ⊆ Ngprint(f(A)). But Ngprint(f(A))
⊆ f(A). So f(A) = Ngprint(f(A)) which implies f(A) is a NGPROS in (Y, σ) and hence f is a Ngpr open
mapping.
3.23 Theorem: If f: (X, τ) → (Y, σ) is a Ngpr open mapping then Nint(f-1(A)) ⊆ f-1(Ngprint(A)) for
every NS A of (Y, σ).
Proof: Let A be a NS in (Y, σ). Then Nint(f-1(A)) is a NOS of (X, τ). Since f is Ngpr open mapping
which implies f(Nint(f-1(A))) is Neutrosophic gpr open in (Y, σ) and hence f(Nint(f -1(A))) ⊆
Ngprint(f(f-1(A))) ⊆ Ngprint(A). Thus Nint(f-1(A)) ⊆ f-1(Ngprint(A)).
3.24 Theorem: A mapping f: (X, τ) → (Y, σ) is Ngpr open mapping iff for each NS A of (Y, σ) and for
each NCS B of (X, τ) containing f-1(A) there is a NGPRCS C of (Y, σ) such that A ⊆ C and f-1(C) ⊆ B.
Proof: Necessity: Assume f: (X, τ) → (Y, σ) is Ngpr open mapping. Let A be the NS of (Y, σ) and B be
a NCS of (X, τ) such that f-1(A) ⊆ B. Then C = (f(Bc))c is NGPRCS of (Y, σ) such that f-1(C) ⊆ B.
Sufficiency: Assume D is a NOS of (X, τ). Then f-1((f(D))c ⊆ Dc and Dc is NCS in (X, τ). By hypothesis
there is a NGPRCS C of (Y, σ) such that (f(D))c ⊆C and f-1(C) ⊆ Dc. Therefore D ⊆ (f-1(C))c. Hence Cc ⊆
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f(D) ⊆ f((f-1(C))c) ⊆ Cc which implies f(D) = Cc. Since Cc is NGPROS of (Y, σ). Hence f(D) is
Neutrosophic gpr open in (Y, σ) and thus f is Ngpr open mapping.
3.25 Theorem: A mapping f: (X, τ) → (Y, σ) is Ngpr open mapping iff f-1(Ngprcl(A)) ⊆ Ncl(f-1(A)) for
every NS A of (Y, σ).
Proof: Necessity: Assume f is a Ngpr open mapping. For any NS A of (Y, σ), f -1(A) ⊆ Ncl(f-1(A)).
Therefore by Theorem 3.24., there exists a NGPRCS C in (Y, σ) such that A ⊆ C and f-1(C) ⊆
Ncl(f-1(A)). Therefore we obtain f-1(Ngprcl(A)) ⊆ f-1(C) ⊆ Ncl(f-1(A)).
Sufficiency: Assume A is a NS of (Y, σ) and B is a NCS of (X, τ) containing f -1(A). Put C = Ncl(A),
then A ⊆ C and C is NGPRCS, since f-1(C) ⊆ Ncl(f-1(A)) ⊆ B. Then by Theorem 3.24., f is Ngpr open
mapping.
3.26 Theorem: If f: (X, τ) → (Y, σ) and g: (Y, σ) → (Z, η) be two Neutrosophic mappings and gof: (X, τ)
→ (Z, η) Ngpr open mapping. If g is Ngpr irresolute mapping then f is Ngpr open mapping.
Proof: Let A be a NOS of (X, τ). Then gof(A) is NGPROS in (Z, η) because gof is Ngpr open mapping.
Since g is Ngpr irresolute mapping and gof(A) is NGPROS of (Z, η) therefore g-1(gof(A)) = f(A) is
NGPROS in (Y, σ). Hence f is Ngpr open mapping.
3.27 Theorem: If f: (X, τ) → (Y, σ) is Neutrosophic open mapping and g: (Y, σ) → (Z, η) is Ngpr open
mapping then gof: (X, τ) → (Z, η) is Ngpr open mapping.
Proof: Let A be a NOS of (X, τ). Then f(A) is a NOS in (Y, σ) because f is a Neutrosophic open
mapping. Since g is Ngpr open mapping, g(f(A)) = gof(A) is NGPROS in (Z, η). Hence gof is Ngpr
open mapping.
3.28 Theorem: Let f: (X, τ) → (Y, σ) be a bijective mapping then the following statements are
equivalent:
(i) f is a Ngpr open mapping.
(ii) f is a Ngpr closed mapping.
(iii) f-1 is Neutrosophic continuous mapping.
Proof: (i) ⇒ (ii): Let us assume that f is a Ngpr open mapping. By definition, A is a NOS in (X, τ),
then f(A) is a NGPROS in (Y, σ). Here A is NCS of (X, τ), then X-A is a NOS of (X, τ). By assumption,
f(X-A) is a NGPROS in (Y, σ). Hence, Y-f(X-A) is a NGPRCS in (Y, σ). Therefore, f is a Ngpr closed
mapping.
(ii) ⇒ (iii): Let A be a NCS in (X, τ). By (ii), f(A) is a NGPRCS in (Y, σ). Hence, f(A) = (f-1)-1(A), so f-1 is a
NGPRCS in (Y, σ). Therefore, f-1 is Neutrosophic continuous mapping.
(iii) ⇒ (iv): Let A be a NOS in (X, τ). By (iii), (f-1)-1(A) = f(A) is a Ngpr open mapping.
3.29 Theorem: Let f: (X, τ) → (Y, σ) be a mapping. Then the following statements are equivalent if Y is
a NPRT1/2 space:
(i) f is a Ngpr closed mapping.
(ii) Npcl(f(A)) ⊆ f(Ncl(A)) for each NS A of X.
Proof: (i) ⇒ (ii): Let A be a NS in X. Then Ncl(A) is a NCS in X. By (i) implies that f(Ncl(A)) is a
NGPRCS in Y. Since Y is a NPRT1/2 space, f(Ncl(A)) is a NPCS in Y. Therefore Npcl(f(Ncl(A))) =
f(Ncl(A)). Now Npcl(f(A)) ⊆ Npcl(f(Ncl(A))) = f(Ncl(A)). Hence Npcl(f(A)) ⊆ f(Ncl(A)) for each NS
A of X.
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(ii) ⇒ (i): Let A be any NCS in X. Then Ncl(A) = A. By (ii) implies that Npcl(f(A)) ⊆ f(Ncl(A)) = f(A).
But f(A) ⊆ Npcl(f(A)). Therefore Npcl(f(A)) = f(A). This implies f(A) is a NPCS in Y. Since every
NPCS is NGPRCS in Y, f(A) is NGPRCS in Y. Hence f is a Ngpr closed mapping.
3.30 Theorem: If f: (X, τ) → (Y, σ) is a mapping where X and Y are NPRT1/2 space. Then the following
statements are equivalent:
(i) f is a Nigpr closed mapping.
(ii) f(A) is a NGPROS in Y for every NGPROS A in X.
(iii) f(Npint(B)) ⊆ Npint(f(B)) for each NS B of X.
(iv) Npcl(f(B)) ⊆ f(Npcl(B)) for each NS B of X.
Proof: (i) ⇒ (ii): is obvious by definition of Nigpr closed mapping.
(ii) ⇒ (iii): Let B be any NS in X. Since Npint(B) is a NPOS, it is a NGPROS in X. Then by hypothesis,
f(Npint(B)) is a NGPROS in Y. Since Y is NPRT1/2 space, f(Npint(B)) is a NPOS in Y. Therefore,
f(Npint(B)) = Npint(f(Npint(B))) ⊆ Npint(f(B)).
(iii) ⇒ (iv) is obvious by taking complement in (iii).
(iv) ⇒ (i) Let B be a NGPRCS in X. By Hypothesis, Npcl(f(B)) ⊆ f(Npcl(B)). Since X is a NPRT1/2 space,
B is a NPCS in X. Therefore, Npcl(f(B)) ⊆ f(Npcl(B)) = f(B) ⊆ Npcl(f(B)) implies f(B) is NPCS in Y and
hence f(B) is a NGPRCS in Y. Thus f is Nigpr closed mapping.
4. Ngpr homeomorphism and Nigpr homeomorphism
4.1 Definition: A bijection f: (X, τ) → (Y, σ) is called Ngpr homeomorphism (resp. NG
homeomorphism, NGP homeomorphism) if f and f -1 are Ngpr continuous (resp. NG continuous,
NGP continuous) mapping.
4.2 Example: Let X= {a, b} and Y = {u, v}. Then τ = {0N, U1, U2, 1N} and σ = {0N, V, 1N} are Neutrosophic
topologies on X and Y respectively, where U1 = 〈x, (0.3, 0.5, 0.6), (0.5, 0.5, 0.5)〉, U2 = 〈x, (0.2, 0.4, 0.7),
(0.4, 0.5, 0.6)〉 and V = 〈y, (0.2, 0.4, 0.7), (0.4, 0.3, 0.6)〉. Define a mapping f: (X, τ) → (Y, σ) by f(a) = u
and f(b) = v. Here Vc = 〈y, (0.7, 0.6, 0.2), (0.6, 0.7, 0.4)〉 is a Neutrosophic closed set in (Y, σ). Then
f-1(Vc) is a NGPRCS in (X, τ). Therefore f is Ngpr continuous mapping. Here U1c = 〈x, (0.6, 0.5, 0.3),
(0.5, 0.5, 0.5)〉 is a Neutrosophic closed set in (X, τ). Then f(U 1c) is a NGPRCS in (Y, σ). Therefore f-1 is
a Ngpr continuous mapping. Hence, f and f-1 are Ngpr continuous mapping then it is a Ngpr
homeomorphism.
4.3 Theorem: Each Neutrosophic homeomorphism is Ngpr homeomorphism but not conversely in
general.
Proof: Let a bijection mapping f: (X, τ) → (Y, σ) be Neutrosophic homeomorphism, in which f and f -1
are Neutrosophic continuous mapping. Since every Neutrosophic continuous mapping is Ngpr
continuous mapping. Hence f and f-1 are Ngpr continuous mapping. Therefore, f is Ngpr
homeomorphism.
4.4 Example: Let X= {a, b} and Y = {u, v}. Then τ = {0 N, U1, U2, 1N} and σ = {0N, V, 1N} are Neutrosophic
topologies on X and Y respectively, where U1 = 〈x, (0.2, 0.5, 0.7), (0.5, 0.5, 0.5)〉, U2 = 〈x, (0.1, 0.4, 0.7),
(0.4, 0.5, 0.6)〉 and V = 〈y, (0.4, 0.3, 0.5), (0.3, 0.4, 0.7)〉. Define a mapping f: (X, τ) → (Y, σ) by f(a) = u
and f(b) = v. Here Vc = 〈y, (0.5, 0.7, 0.4), (0.7, 0.6, 0.3)〉 is a NCS in (Y, σ). Then f-1(Vc) is a NGPRCS in
(X, τ). Therefore f is Ngpr continuous mapping. Here U1c = 〈x, (0.7, 0.5, 0.2), (0.5, 0.5, 0.5)〉 is a NCS in
(X, τ). Then f(U1c) is a NGPRCS in (Y, σ). Therefore f-1 is a Ngpr continuous. Hence, f and f-1 are Ngpr
continuous mapping then it is a Ngpr homeomorphism. However, here Vc is a NCS in (Y, σ) but it is
not a NCS in (X, τ). Hence, f is not Neutrosophic continuous mapping. Therefore, f is not a
Neutrosophic homeomorphism.
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4.5 Theorem: Each NG homeomorphism is Ngpr homeomorphism but not conversely in general.
Proof: Let a bijection mapping f: (X, τ) → (Y, σ) be NG homeomorphism, in which f and f -1 are NG
continuous mapping. Since every NG continuous mapping is Ngpr continuous mapping. Hence f
and f-1 are Ngpr continuous mapping. Therefore, f is Ngpr homeomorphism.
4.6 Example: Let X= {a, b} and Y = {u, v}. Then τ = {0 N, U, 1N} and σ = {0N, V, 1N} are Neutrosophic
topologies on X and Y respectively, where U = 〈x, (0.4, 0.5, 0.6), (0.3, 0.4, 0.5)〉 and V = 〈y, (0.8, 0.5, 0.2),
(0.7, 0.7, 0.3)〉. Define a mapping f: (X, τ) → (Y, σ) by f(a) = u and f(b) = v. Here V c = 〈y, (0.2, 0.5, 0.8),
(0.3, 0.3, 0.7)〉 is a NCS in (Y, σ). Then f-1(Vc) is a NGPRCS in (X, τ). Therefore f is Ngpr continuous
mapping. Here Uc = 〈x, (0.6, 0.5, 0.4), (0.5, 0.6, 0.3)〉 is a NCS in (X, τ). Then f(Uc) is a NGPRCS in (Y, σ).
Therefore f-1 is a Ngpr continuous mapping. Hence, f and f-1 are Ngpr continuous mapping then it is
Ngpr homeomorphism. However, here Vc is a NCS in (Y, σ) but it is not a NGCS in (X, τ). Hence, f is
not Neutrosophic continuous mapping. Therefore, f is not a NG homeomorphism.
4.7 Theorem: Each NGP homeomorphism is a Ngpr homeomorphism but not conversely in general.
Proof: Let a bijection mapping f: (X, τ) → (Y, σ) be NGP homeomorphism, in which f and f-1 are NGP
continuous mapping. Since every NGP continuous mapping is Ngpr continuous mapping. Hence f
and f-1 are Ngpr continuous mapping. Therefore, f is Ngpr homeomorphism.
4.8 Example: Let X= {a, b} and Y = {u, v}. Then τ = {0N, U1, U2, U3, 1N} and σ = {0N, V, 1N} are
Neutrosophic topologies on X and Y respectively, where U 1 = 〈x, (0.3, 0.5, 0.7), (0.2, 0.5, 0.6)〉, U2 = 〈x,
(0.6, 0.5, 0.5), (0.7, 0.5, 0.5)〉, U3 = 〈x, (0.8, 0.5, 0.2), (0.7, 0.5, 0.1)〉 and V = 〈y, (0.3, 0.5, 0.7), (0.3, 0.5, 0.7)〉.
Define a mapping f: (X, τ) → (Y, σ) by f(a) = u and f(b) = v. Here Vc = 〈y, (0.7, 0.5, 0.3), (0.7, 0.5, 0.3)〉 is a
NCS in (Y, σ). Then f-1(Vc) is a NGPRCS in (X, τ). Therefore f is Ngpr continuous mapping. Here U1c =
〈x, (0.7, 0.5, 0.3), (0.6, 0.5, 0.2)〉 is a NCS in (X, τ). Then f(Uc) is a NGPRCS in (Y, σ). Therefore f-1 is a
Ngpr continuous mapping. Hence, f and f-1 are Ngpr continuous mapping then it is a Ngpr
homeomorphism. However, here Vc is a NCS in (Y, σ) but it is not a NGPCS in (X, τ). Hence, it is not
NGP continuous mapping. Therefore, it is not a NGP homeomorphism.
The relation between various types of Neutrosophic homeomorphisms is given by
N homeomorphism
NG homeomorphism
NGP homeomorphism
NGPR homeomorphism
Fig.4.1.1 The reverse implications of Fig.4.1.1 are not true in general in the above diagram.
4.9 Theorem: Let f:(X, τ) → (Y, σ) be a Ngpr homeomorphism, then f is a Neutrosophic
homeomorphism if X and Y are NPRT *1/2 space.
Proof: Let A be a NCS in (Y, σ), then f-1(A) is a NGPRCS in (X, τ). Since X is NPRT *1/2 space, f-1(A) is a
NCS in (X, τ). Therefore, f is Neutrosophic continuous mapping. By hypothesis, f-1 is Ngpr
continuous mapping. Let B be a NCS in (X, τ). Then (f -1)-1 (B) = f(B) is a NGPRCS in Y. Since Y is
NPRT*1/2 space, f(B) is NCS in Y. Hence f-1 is Neutrosophic continuous mapping. Hence f is a
Neutrosophic homeomorphism.
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4.10 Theorem: Let f:(X, τ) → (Y, σ) be a bijective mapping. If f is Ngpr continuous mapping then the
following statements are equivalent:
(i) f is a Ngpr closed mapping.
(ii) f is a Ngpr open mapping.
(iii) f is a Ngpr homeomorphism.
Proof: (i) ⇒ (ii): Let us assume that f be a bijective mapping and a Ngpr closed mapping. Hence f -1 is
Ngpr continuous mapping. Since each NOS in (X, τ) is a NGPROS in (Y, σ). Hence, f is a Ngpr open
mapping.
(ii) ⇒ (iii): Let f be a bijective mapping and a Ngpr open mapping. Furthermore, f-1 is a Ngpr
continuous mapping. Hence f and f-1 are Ngpr continuous mapping. Therefore, f is a Ngpr
homeomorphism.
(iii) ⇒ (i): Let f be a Ngpr homeomorphism. Then f and f -1 are Ngpr continuous mapping. Since each
NCS in (X, τ) is a NGPRCS in (Y, σ). Hence f is a Ngpr closed mapping.
4.11 Theorem: The composition of two Ngpr homeomorphisms need not be a Ngpr homeomorphism
in general.
4.12 Example: Let X = {a, b}, Y = {c, d} and Z = {e, f}. Then τ = {0 N, U, 1N}, σ = {0N, V, 1N} and ɳ = {0N, W,
1N} are Neutrosophic topologies on X and Y respectively, where U = 〈x, (0.2, 0.5, 0.8), (0.3, 0.3, 0.7)〉, V
= 〈y, (0.4, 0.5, 0.6), (0.3, 0.4, 0.5)〉, W = 〈z, (0.8, 0.5, 0.2), (0.7, 0.7, 0.3)〉. Define a mapping f: (X, τ) → (Y,
σ) by f(a) = c and f(b) = d and g: (Y, σ) → (Z, ɳ) by g(c) = e and g(d) = f. Then f and g are Ngpr
homeomorphisms but their composition g∘f: (X, τ) → (Z, ɳ) is not a Ngpr homeomorphism. Since Wc
is NCS in (Z, ɳ) but it is not NGPRCS in (X, τ).
4.13 Definition: A bijection f: (X, τ) → (Y, σ) is called Nigpr homeomorphism if f and f -1 are Ngpr
irresolute mappings.
4.14 Theorem: Each Nigpr homeomorphism is a Ngpr homeomorphism but not conversely in
general.
Proof: Let a bijection mapping f: (X, τ) → (Y, σ) be Nigpr homeomorphism. Assume that A is a NCS
in (Y, σ) implies A is a NGPRCS in (Y, σ). Since f is Ngpr irresolute mapping, f -1(A) is a NGPRCS in
(X, τ). Hence f is Ngpr continuous mapping. Therefore, f and f-1 are Ngpr continuous mapping.
Hence, f is Ngpr homeomorphism.
4.15 Example: Let X= {a, b} and Y = {u, v}. Then τ = {0N, U1, U2, 1N} and σ = {0N, V, 1N} are Neutrosophic
topologies on X and Y respectively, where U1 = 〈x, (0.2, 0.5, 0.7), (0.4, 0.5, 0.6)〉, U2 = 〈x, (0.2, 0.4, 0.8),
(0.3, 0.5, 0.7)〉 and V = 〈y, (0.5, 0.4, 0.5), (0.4, 0.5, 0.6)〉. Define a mapping f: (X, τ) → (Y, σ) by f(a) = u
and f(b) = v. Here Vc = 〈y, (0.5, 0.6, 0.5), (0.6, 0.5, 0.4)〉 is a NCS in (Y, σ). Then f-1(Vc) is a NGPRCS in
(X, τ). Therefore f is Ngpr continuous mapping. Here U1c = 〈x, (0.7, 0.5, 0.2), (0.6, 0.5, 0.4)〉 is a NCS in
(X, τ). Then f(U1c) is a NGPRCS in (Y, σ). Therefore f-1 is a Ngpr continuous mapping. Hence, f and f-1
are Ngpr continuous mapping then it is a Ngpr homeomorphism. However, here A = 〈y, (0.2, 0.4,
0.7), (0.3, 0.5, 0.6)〉 is a NGPRCS in (Y, σ) but it is not a NGPRCS in (X, τ). Hence, f is not
Neutrosophic irresolute mapping. Therefore, f is not a Nigpr homeomorphism.
4.16 Theorem: If f: (X, τ) → (Y, σ) is a Nigpr homeomorphism then Ngprcl(f -1(A)) ⊆ f-1(Npcl(A)) for
each NS A in (Y, σ).
Proof: Let A be a NS in (Y, σ). Then Npcl(A) is NPCS in (Y, σ) and since every NPCS is NGPRCS in
(Y, σ). Assuming f is Ngpr irresolute mapping, f-1(Npcl(A)) is a NGPRCS in (X, τ), then
Ngprcl(f-1(Npcl(A))) = f-1(Npcl(A)). Here, Ngprcl(f-1(A)) ⊆ Ngprcl(f-1(Npcl(A))) = f-1(Npcl(A)).
Therefore, Ngprcl(f-1(A)) ⊆ f-1(Npcl(A)) for each NS A in (Y, σ).
K. Ramesh, Ngpr Homeomorphism in Neutrosophic Topological Spaces
Neutrosophic Sets and Systems, Vol. 32, 2020
36
4.17 Theorem: If f: (X, τ) → (Y, σ) is a Nigpr homeomorphism then Npcl(f-1(A)) = f-1(Npcl(A)) for each
NS A in (Y, σ).
Proof: Given f is a Nigpr homeomorphism, then f is a Ngpr irresolute mapping. Let A be a NS in (Y,
σ). Clearly, Npcl(A) is a NPCS in (Y, σ). This shows that Npcl(A) is a NGPRCS in (Y, σ). Since f-1(A) ⊆
f-1(Npcl(A)), then Npcl(f-1(A)) ⊆ Npcl(f-1(Npcl(A))) = f-1(Npcl(A)). Therefore, Npcl(f-1(A)) ⊆
f-1(Npcl(A)).
Let f be a Nigpr homeomorphism, f-1 is a Ngpr irresolute mapping. Let us consider NS f-1(A) in (X, τ),
which bring out that Npcl(f-1(A)) is a NGPRCS in (X, τ). Hence Ngprcl(f -1(A)) is a NGPRCS in (X, τ).
This implies that (f-1)-1(Npcl(f-1(A))) = f(Npcl(f-1(A))) is a NPCS in (Y, σ). This proves A = (f-1)-1(f-1(A)) ⊆
(f-1)-1(Npcl(f-1(A))) = f(Npcl(f-1(A))). Therefore, Npcl(A) ⊆ Npcl(f(Npcl(f-1(A)))) = f(Npcl(f-1(A))), since
f-1 is a Ngpr irresolute mapping. Hence f-1(Npcl(A)) ⊆ f-1(f(Npcl(f-1(A)))) = Npcl(f-1(A)). That is
f-1(Npcl(A)) ⊆ Npcl(f-1(A)). Hence, Npcl(f-1(A)) = f-1(Npcl(A)).
4.18 Theorem: If f: (X, τ) → (Y, σ) and g: (Y, σ) → (Z, ɳ) are Nigpr homeomorphisms, then the
composition g∘f: (X, τ) → (Z, ɳ) is a Nigpr homeomorphism.
Proof: Let f and g be two Nigpr homeomorphisms. Assume C is a NGPRCS in (Z, ɳ). Then g-1(C) is a
NGPRCS in (Y, σ). Then by hypothesis, f-1(g-1(C)) is a NGPRCS in (X, τ). Hence g∘f is a Ngpr
irresolute mapping. Now, let A be a NGPRCS in (X, τ). By assumption, f(A) is a NGPRCS in (Y, σ).
Then by hypothesis, g(f(A)) is a NGPRCS in (Z, ɳ). This implies that g∘f is a Ngpr irresolute
mapping. Hence, g∘f is a Nigpr homeomorphism.
5. Conclusion
In this article, the new class of Neutrosophic homeomorphism namely, Ngpr homeomorphism
and Nigpr homeomorphism was defined and studied some of their properties in Neutrosophic
topological spaces. Furthermore, the work was extended as the Ngpr open mappings, Ngpr closed
mappings and Nigpr closed mappings and discussed some of their properties. Many results have
been established to show how far topological structures are preserved by this Ngpr
homeomorphism.
Also, the relation between Ngpr closed mappings and other existed Neutrosophic closed
mappings in Neutrosophic topological spaces were established and derived some of their related
attributes. Many examples are given to justify the results.
This concept can be used to drive few more new results of Ngpr connectedness and Ngpr
compactness in Neutrosophic topological spaces.
Acknowledgements
The author would like to thank the referees for their valuable suggestions to improve the paper.
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Received: Dec 29, 2019. Accepted: Mar 15, 2020.
K. Ramesh, Ngpr Homeomorphism in Neutrosophic Topological Spaces
Neutrosophic Sets and Systems, Vol. 32, 2020
University of New Mexico
Generalized Neutrosophic Separation Axioms in Neutrosophic Soft Topological
Spaces
A. Mehmood
1,∗
, F. Nadeem 2 , G. Nordo 3 , M. Zamir 4 ,
5
C. Park , H. Kalsoom 6 , S. Jabeen
1
7
and M. I. KHAN
8
Department of Mathematics and Statistics, Riphah International University, Sector I 14, Islamabad,
Pakistan; mehdaniyal@gmail.com
2
Department of Mathematics, University of Science and Technology, Bannu, Khyber Pakhtunkhwa, Pakistan;
fawadnadeem2@gmail.com
3
MIFT – Dipartimento di Scienze Matematiche e Informatiche, scienze Fisiche e scienze della Terra, Messina
University, Messina, Italy; giorgio.nordo@unime.it
4
Department of Mathematics, University of Science and Technology, Bannu, Khyber Pakhtunkhwa, Pakistan;
zamirburqi@ustb.edu.pk
5
Department of Mathematics, Research Institute for Natural Sciences, Hanyang University;
baak@hanyang.ac.kr
6
School of Mathematical Sciences, Zhejiang University, Hangzhou 310027, China; humaira87@zju.edu.cn
7
School of Mathematics and system Sciences Beihang University Beijing, Beijing, China;
shamoona011@buaa.edu.cn
8
Department of Epidemiology and Biostatistics, Anhui Medical University, China; ddqec@ustb.edu.pk
∗
Correspondence: mehdaniyal@gmail.com
Abstract. The idea of neutrosophic set was floated by Smarandache by considering a truth membership,
an indeterminacy membership and a falsehood or falsity membership functions. The engagement between
neutrosophic set and soft set was done by Maji. More over it was used effectively to model uncertainty in different
areas of application, such as control, reasoning, pattern recognition and computer vision. The first aim of this
paper leaks out the notion of neutrosophic soft p-open set,neutrosophic soft p-closed sets and their important
characteristics. Also the notion of neutrosophic soft p-neighborhood and neutrosophic soft p-separation axioms
in neutrosophic soft topological spaces are developed. Important results are examed marrying to these newly
defined notion relative to soft points. The notion of neutrosophic soft p-separation axioms of neutrosophic soft
topological spaces is diffused in different results concerning soft points. Furthermore, properties of neutrosophic
soft -P i -space (i = 0, 1, 2, 3, 4) and linkage between them is built up.
Keywords: neutrosophic soft set; neutrosophic soft point; neutrosophic soft p-open set; neutrosophic soft
p-neighborhood; neutrosophic soft p-separation axioms.)
A. Mehmood, F. Nadeem, G. Nordo, M. Zamir, C. Park, H. Kalsoom, S. Jabeen and M. I. Khan,
Generalized Neutrosophic Separation Axioms in Neutrosophic Soft Topological Spaces
Neutrosophic Sets and Systems, Vol. 32, 2020
39
1. Introduction
The outdated fuzzy sets is behaviorized by the membership worth or the grade of membership worth. Some times it may be very difficult to assign the membership worth for a fuzzy
sets.This gap was bridged with the introduction of interval valued fuzzy sets. In some real
life problems in expert system, belief system and so forth,we must take in account the truthmembership and the falsity-membership simultaneously for appropriate narration of an object
in uncertain,ambiguous atmosphere. Fuzzy sets and interval valued fuzzy sets are badly failed
to handle this situation. The importance of intuitionistic fuzzy sets is automatically come in
play in such a hazardous situation.The intuitionistic fuzzy sets can only handle the imperfect
information supposing both the truth-membership or association( or simply membership)and
falsity-membership( or non-membership )values. It fails to switch the indeterminate and inconsistent information which exists in belief system. Smarandache [14] bounced up conception
of neutrosophic set which is a mathematical technique for facing problems involving imprecise,
indeterminacy and inconsistent data.The words neutrosophy and neutrosophic were introduced
by Smarandache. Neutrosophy (noun) means knowledge of neutral thought, while neutrosophic
(adjective), means having the nature of or having the behavior of neutrosophy. This theory
is nothing but just generalization of ordinary sets, fuzzy set theory [15], intuitionistic fuzzy
set theory [1] etc. Some work have been supposed on neutrosophic sets by some mathematicians in many area of mathematics [4,12]. Many practical problems in economics, engineering,
environment,medical science social science etc.can not be treated by conventional methods,
because conventional methods have genetic complexities. These complexities may be taking
birth due to the insufficiency of the theories of paramertrization tools. Each of these theories has its transmissible difficulties, as was exposed by Molodtsov [11]. Molodtsov developed
an absolutely modern approach to cope with uncertainty and vagueness and applied it more
and more in different directions such as smoothness of functions, game theory, operations research, Riemann integration, perron integration, and so forth. Meticulously,theory of soft set
is free from the parameterization meagerness condition of fuzzy set theory, rough set theory.
probability theory for facing with uncertainty Shabir and Naz [13] floated the conception of
soft topological spaces, which are defined over an initial universe of discourse with a fixed
set of parameters, and showed that a soft topological space produces a parameterized family
of topological spaces. Theoretical studies of soft topological spaces were also done by some
authors in [2, 3, 6, 8]. Kattak et al. [9] leaked out the notion of some basic result in soft bi
topological spaces with respect to soft points.These results supposed the engagement of soft
limit point, soft interior point, soft neighborhood, the relation between soft weak structures
and soft weak closures. Moreover the authors also addressed soft sequences uniqueness of limit
in soft weak-Hausdorff spaces, the product of soft Hausdorff spaces with respect to soft points
A. Mehmood, F. Nadeem, G. Nordo, M. Zamir, C. Park, H. Kalsoom, S. Jabeen and M. I.
Khan, Generalized Neutrosophic Separation Axioms in Neutrosophic Soft Topological Spaces
Neutrosophic Sets and Systems, Vol. 32, 2020
40
in different soft weak open set and the marriage between soft Hausdorff space and the diagonal.
The combination of Neutrosophic set with soft sets was first introduced by Maji [10]. This
combination makes entirely a new mathematical model Neutrosophic Soft Set and later this
notion was improved by Deli and Broumi [7]. Work was progressively continue,later on mathematician came in action and defined a new mathematical structure known as neutrosophic
soft topological spaces. Neutrosophic soft topological spaces were presented by Bera in [5].
M. Abdel-Basset et al. [16] proposed some novel similarity measures for bipolar and intervalvalued bipolar neutrosophic set such as the cosine similarity measures and weighted cosine
similarity measures. The propositions of these similarity measures are examined, and two
multi-attribute decision making techniques are presented based on proposed measures. For
verifying the feasibility of proposed measures, two numerical examples are presented in comparison with the related methods for demonstrating the practicality of the proposed method.
Finally, the authors applied the proposed measures of similarity for diagnosing bipolar disorder
diseases significantly.
M. Abdel-Basset et al. [17] supposed the objective function of scheduling problem to minimize
the costs of daily resource fluctuations using the precedence relationships during the project
completion time. The authors designed a resource leveling model based on neutrosophic set
to overcome the ambiguity caused by the real-world problems. In this model, trapezoidal neutrosophic numbers are used to estimate the activities durations. The crisp model for activities
time is obtained by applying score and accuracy functions. The authors produced a numerical
example to illustrate the validation of the proposed model in this study.
Arif et al. [18] introduced the notion of most generalized neutrosophic soft open sets in neutrosophic soft topological structures relative to neutrosophic soft points. The authors leaked out
the concept of most generalized separation axioms in neutrosophic soft topological spaces with
respect to soft points. Gradually the study is extended to deliberate important results related
to these newly defined concepts with respect to soft points. Several related properties, structural characteristics have been investigated. The convergence of sequence in neutrosophic soft
topological space is defined and its uniqueness in neutrosophic soft most generalized-Hausdorff
space relative to soft points is reflected. The authors further studied and switched over neutrosophic monotonous soft function and its characteristics to multifarious results. The authors
lastly addressed neutrosophic soft product spaces under most generalized neutrosophic soft
open set with respect to crisp points.
The first aim of this article bounces the notion of neutrosophic soft p-open set,neutrosophic
soft p-neighborhood and neutrosophic soft p-separation axioms in neutrosophic soft topology
which is defined on neutrosophic soft sets. Later on the important results are discussed related
to these newly defined concepts with respect to soft points. Finally, the concept of p-separation
A. Mehmood, F. Nadeem, G. Nordo, M. Zamir, C. Park, H. Kalsoom, S. Jabeen and M. I.
Khan, Generalized Neutrosophic Separation Axioms in Neutrosophic Soft Topological Spaces
Neutrosophic Sets and Systems, Vol. 32, 2020
41
axioms of neutrosophic soft topological spaces is diffused in different results with respect to
soft points. Furthermore, properties of neutrosophic soft-P i -space (i = 0, 1, 2, 3, 4) and some
switch between them are discussed. We hope that these results will best fit for future study
on neutrosophic soft topology to carry out a general framework for practical applications.
2. Preliminaries
In this phase we now state certain useful definitions, theorems, and several existing results
for neutrosophic soft sets that we require in the next sections.
Definition 2.1. [14] A neutrosophic set A on the universe set X is defined as:
A = {hx, T A (x), I A (x), ̥A (x)i : x ∈ X},
where T, I, ̥ : X →]− 0, 1+ [ and − 0 ≦ T A (x) + I A (x) + ̥A (x) ≦ 3+ .
Definition 2.2. [11] Let X be an initial universe, E be a set of all parameters, and P(x)
denote the power set of X. A pair (̥, E) is called a soft set over X, where ̥ is a mapping
given by ̥ : E → P (X). In other words, the soft set is a parameterized family of subsets of
the set X. For λ ∈ E, ̥(λ) may be considered as the set of λ-elements of the soft set (̥, E),
or as the set of λ-approximate element of the set, i.e.
(̥, E) = {(λ, ̥(λ)) : λ ∈ E, ̥ : E → P (X)}.
After the neutrosophic soft set was defined by Maji [10], this concept was modified by Deli
and Broumi [7] as given below:
Definition 2.3. [7] Let X be an initial universe set and E be a set of parameters. Let P(X)
e E) over X is a set
denote the set of all neutrosophic sets of X. Then a neutrosophic soft set (̥,
e representing a mapping ̥
e : E → P (X), where ̥
e is called
defined by a set valued function ̥
e E). In other words, the neutrosophic
the approximate function of the neutrosophic soft set (̥,
soft set is a parameterized family of some elements of the set P(X) and therefore it can be
written as a set of ordered pairs:
e E) = {(λ, hx, T ̥e (λ) (x), I ̥e (λ) (x), ̥̥e (λ) (x)i : x ∈ X) : λ ∈ E},
(̥,
where T ̥e (λ) (x), I ̥e (λ) (x), ̥̥e (λ) (x) ∈ [0, 1] are respectively called the truth-membership,
e
indeterminacy-membership, and falsity-membership function of ̥(λ).
Since the supremum of
each T, I, F is 1, the inequality 0 ≦ T ̥e (λ) (x) + I ̥e (λ) (x) + ̥̥e (λ) (x) ≦ 3 is obvious.
e E) be a neutrosophic soft set over the universe set X. The
Definition 2.4. [5] Let (̥,
e E) is denoted by (̥,
e E)c and is defined by:
complement of (̥,
e E)c = {(λ, hx, ̥̥e (λ) (x), 1 − I ̥e (λ) (x), T ̥e (λ) (x)i : x ∈ X) : λ ∈ E}.
(̥,
e E)c )c = (̥,
e E).
It is obvious that ((̥,
e E) be two neutrosophic soft sets over the universe
e E) and (G,
Definition 2.5. [10] Let (̥,
e
e E) if T ̥e (λ) (x) ≦ T G(λ)
e E) is said to be a neutrosophic soft subset of (G,
(x), I ̥e (λ) (x)
set X. (̥,
A. Mehmood, F. Nadeem, G. Nordo, M. Zamir, C. Park, H. Kalsoom, S. Jabeen and M. I.
Khan, Generalized Neutrosophic Separation Axioms in Neutrosophic Soft Topological Spaces
Neutrosophic Sets and Systems, Vol. 32, 2020
42
e
e
e E) . (̥,
e E) j (G,
e E)
≦ I G(λ) (x), ̥̥e (λ) (x) ≧ ̥G(λ) (x),∀λ ∈ E, ∀x ∈ X. It is denoted by (̥,
e E) if (̥,
e E)
e E) is a neutrosophic soft subset of (G,
is said to be neutrosophic soft equal to (G,
e E) is a neutrosophic soft subset of (̥,
e E).
e E). It is denoted by (̥,
e E) = (G,
and (G,
3. Applications of Neutrosophic Soft Point and its Characteristics
e 2 , E) be two neutrosophic soft sets over universe set X.
e 1 , E) and (̥
Definition 3.1. Let (̥
e 2 , E) = (̥
e 3 , E) and is defined by:
e 1 , E) ⊔ (̥
Then their union is denoted by (̥
3
3
3
e 3 , E) = {(λ, hx, T ̥e (λ) (x), I ̥e (λ) (x), ̥̥e (λ) (x)i : x ∈ X) : λ ∈ E},
(̥
where T ̥e
I
e 3 (λ)
̥
3 (λ)
(x) = max {T ̥e
(x) = max {I
3
̥̥e (λ) (x)
= max
e 1 (λ)
̥
1 (λ)
(x), I
(x), T ̥e
e 2 (λ)
̥
2 (λ)
(x)},
(x)},
1
2
{̥̥e (λ) (x), ̥̥e (λ) (x)}.
e 1 , E) and (̥
e 2 , E) be two neutrosophic soft sets over the universe set
Definition 3.2. Let (̥
e 2 , E) = (̥
e 3 , E) and is defined by:
e 1 , E) ⊓ (̥
X. Then their intersection is denoted by (̥
where
e 3 , E) = min {T ̥e 1 (λ) (x), T ̥e 2 (λ) (x)}
T (̥
I ̥e
3 (λ)
(x) = max {I ̥e
3
̥̥e (λ) (x)
= max
1 (λ)
(x), I ̥e
2 (λ)
(x)},
1
2
{̥̥e (λ) (x), ̥̥e (λ) (x)}.
e E) over the universe set X is said to be a null
Definition 3.3. A neutrosophic soft set (̥,
neutrosophic soft set if T ̥e (λ) (x) = 0, I ̥e (λ) (x) = 0, ̥̥e (λ) (x) = 1; ∀λ ∈ E, ∀x ∈ X. It is
denoted by 0(X,E) .
e E) over the universe set X is said to be an
Definition 3.4. A neutrosophic soft set (̥,
absolute neutrosophic soft set if T ̥e (λ) (x) = 1, I ̥e (λ) (x) = 1, ̥̥e (λ) (x) = 0; ∀λ ∈ E, ∀x ∈ X.
It is denoted by 1(X,E) .
Clearly, 0c( X, E) = 1(X,E) and 1c( X, E) = 0(X,E) .
Definition 3.5. Let NSS(X, E) be the family of all neutrosophic soft sets over the universe
set X and ℑ ⊏ NSS(X, E). Then ℑ is said to be a neutrosophic soft topology on X if:
1. 0(X,E) and 1(X,E) belong to ℑ,
2. the union of any number of neutrosophic soft sets in ℑ belongs to ℑ,
3. the intersection of a finite number of neutrosophic soft sets in ℑ belongs to ℑ.
Then (X, ℑ, E) is said to be a neutrosophic soft topological space over X. Each member of
ℑ is said to be a neutrosophic soft open set.
e E) be a
Definition 3.6. Let (X, ℑ, E) be a neutrosophic soft topological space over X and (̥,
e E) is said to be a neutrosophic
subset of neutrosophic soft topological space over X. Then (̥,
soft closed set iff its complement is a neutrosophic soft open set.
A. Mehmood, F. Nadeem, G. Nordo, M. Zamir, C. Park, H. Kalsoom, S. Jabeen and M. I.
Khan, Generalized Neutrosophic Separation Axioms in Neutrosophic Soft Topological Spaces
Neutrosophic Sets and Systems, Vol. 32, 2020
43
e E) be a
Definition 3.7. Let (X, ℑ, E) be a neutrosophic soft topological space over X and (̥,
e E) is said to be a neutrosophic
subset of neutrosophic soft topological space over X. Then (̥,
e E) ⊆ N Sint(N Scl((̥,
e E)))
soft p-open (NSPO) set if (̥,
e E) be a
Definition 3.8. Let (X, ℑ, E) be a neutrosophic soft topological space over X and (̥,
e E) is said to be a neutrosophic
subset of neutrosophic soft topological space over X. Then (̥,
e E) ⊇ N Scl(N Sint((̥,
e E)))
soft p-closed (NSPC) set if (̥,
Definition 3.9. Let NS be the family of all neutrosophic sets over the universe set X and x
∈ X. The neutrosophic set x(α,β,γ) is called a neutrosophic point, for 0 < α, β, γ ≦ 1, and is
defined as follow:
x(α,β,γ) (y) =
(
(α, β, γ), if y = x
(1)
(0, 0, 1), if y 6= x.
It is clear that every neutrosophic set is the union of its neutrosophic points.
Definition 3.10. Suppose that X = {x1 , x2 }. Then neutrosophic set
A = {hx1 , 0.1, 0.3, 0.5i, hx2 , 0.5, 0.4, 0.7i}
is the union of neutrosophic points x1 (0.1, 0.3, 0.5)and x2 (0.5, 0.4, 0.7).
Now we define the concept of neutrosophic soft points for neutrosophic soft sets.
Definition 3.11. Let NSS(X, E) be the family of all neutrosophic soft sets over the universe
set X. Then neutrosophic soft set xλ (α, β, γ) is called a neutrosophic soft point, for every x ∈
X, 0 < α, β, γ ≦ 1, λ ∈ E, and is defined as follows:
(
(α, β, γ) if λ′ = λ and y = x
(α,β,γ)
xλ
(λ′ )(y) =
(0, 0, 1), if λ′ 6= λ or y 6= x.
(2)
Definition 3.12. Suppose that the universe set X is given by X = {x1 , x2 } and the set of
e E) over the universe
parameters by E = {λ1 , λ2 }. Let us consider neutrosophic soft sets (̥,
X as follows:
It
is
clear
e E) =
(̥,
(
λ1 = {hx1 , 0.3, 0.7, 0.6i, hx2 , 0.4, 0.3, 0.8i}
λ2 = {hx1 , 0.4, 0.6, 0.8i, hx2 , 0.3, 0.7, 0.2i}.
e E)
that(̥,
is
(0.3,0.7,0.6) 1 2 (0.4,0.6,0.8) 2 1
,x λ
,x λ ,
x1 λ1
x 1 λ1
(0.3,0.7,0.6)
1 2 (0.4,0.6,0.8)
x λ
=
=
union
of
its
)
(3)
neutrosophic
and
Here
(
)
λ1 = {hx1 , 0.3, 0.7, 0.6i, hx2 , 0, 0, 1i}
(
soft
point
(0.3,0.7,0.6)
x2 λ 2
.
(4)
λ2 = {hx1 , 0, 0, 1i, hx2 , 0, 0, 1i}.
λ1 = {hx1 , 0.3, 0.7, 0.6i, hx2 , 0, 0, 1i}
λ2 = {hx1 , 0.4, 0.6, 0.8i, hx2 , 0, 0, 1i}.
)
.
(5)
A. Mehmood, F. Nadeem, G. Nordo, M. Zamir, C. Park, H. Kalsoom, S. Jabeen and M. I.
Khan, Generalized Neutrosophic Separation Axioms in Neutrosophic Soft Topological Spaces
Neutrosophic Sets and Systems, Vol. 32, 2020
2 1 (0.4,0.3,0.8)
x λ
2 2 (0.3,0.7,0.2)
x λ
44
=
(
λ1 = {hx1 , 0, 0, 1i, hx2 , 0.4, 0.3, 0.8i}
=
(
λ1 = {hx1 , 0, 0, 1i, hx2 , 0, 0, 1i}
λ2 = {hx1 , 0, 0, 1i, hx2 , 0, 0, 1i}.
λ2 = {hx1 , 0, 0, 1i, hx1 , 0.3, 0.7, 0.2i}.
)
.
(6)
)
.
(7)
e E) be a neutrosophic soft set over the universe set X. We say
Definition 3.13. Let (̥,
(α,β,γ)
e E) read as belonging to the neutrosophic soft set (̥,
e E) whenever
∈ (̥,
that xλ
α ≦ T ̥e (λ) (x),β ≦ I ̥e (λ) (x) and γ ≧ ̥̥e (λ) (x).
Definition 3.14. Let (X, ℑ, E) be a neutrosophic soft topological space over X. A neutroe E) in (X, ℑ, E) is called a neutrosophic soft p-neighborhood of the neutrosophic soft set (̥,
(α,β,γ)
e E) such
e E) , if there exists a neutrosophic soft p-open set (G,
∈ (̥,
sophic soft point xλ
that xλ
(α,β,γ)
e E) ⊏ (̥,
e E).
∈ (G,
e E) be a neuTheorem 3.15. Let (X, ℑ, E) be a neutrosophic soft topological space and (̥,
e E) is a neutrosophic soft p-open set if and only if (̥,
e E)
trosophic soft set over X. Then (̥,
is a neutrosophic soft p-neighborhood of its neutrosophic soft points.
e E) be a neutrosophic soft p-open set and xλ (α,β,γ) ∈ (̥,
e E). Then xλ (α,β,γ) ∈
Proof. Let (̥,
e E) ⊏ (̥,
e E).
(̥,
(α,β,γ)
e E) is a neutrosophic soft p-neighborhood of xλ
.
Therefore, (̥,
e E) be a neutrosophic soft p-neighborhood of its neutrosophic soft points.
Conversely, let (̥,
Let xλ
(α,β,γ)
e E). Since (̥,
e E) is a neutrosophic soft p-neighborhood of the neutrosophic
∈ (̥,
e E) ∈ ℑ such that xλ (α,β,γ) ∈ (G,
e E) ⊏ (̥,
e E). Since (̥,
e E)
, there exists (G,
soft point x
(α,β,γ)
(α,β,γ)
λ
λ
e E) is a union of neutrosophic soft p-open
e E)}, it follows that (̥,
:x
∈ (̥,
= ⊔ {x
λ (α,β,γ)
e E) is a neutrosophic soft p-open set.
sets and hence (̥,
The p-neighborhood system of a neutrosophic soft point xλ
(x
λ (α,β,γ)
(α,β,γ)
, denoted by U
, E), is the family of all its p-neighborhoods.
Theorem 3.16. The neighborhood system U (xλ
(α,β,γ)
, E) at xλ
(α,β,γ)
in a neutrosophic soft
topological space (X, ℑ, E) has the following properties.
e E) ;
e E) ∈ U (xλ (α,β,γ) , E) , then xλ (α,β,γ) ∈ (̥,
1) If (̥,
e E) , then (H,
e E) ∈ U
e E) , then (H,
e E) ⊏ (H,
e E) ∈ U (xλ (α,β,γ) , E) and (̥,
2) If (̥,
(xλ
(α,β,γ)
, E) ;
e E) ∈ U (xλ (α,β,γ) , E) ;
e E) ∈ U (xλ (α,β,γ) , E) , then (̥,
e E) ⊓ (G,
e E) , If (G,
3) If (̥,
e E)
e E) ∈ U (xλ (α,β,γ) , E) such that (G,
e E) ∈ U (xλ (α,β,γ) , E) , then there exists a (G,
4) If (̥,
∈ U (y λ
′ (α′ ,β ′ ,γ ′ )
, E) , for each y λ
′ (α′ ,β ′ ,γ ′ )
e E).
∈ (G,
A. Mehmood, F. Nadeem, G. Nordo, M. Zamir, C. Park, H. Kalsoom, S. Jabeen and M. I.
Khan, Generalized Neutrosophic Separation Axioms in Neutrosophic Soft Topological Spaces
Neutrosophic Sets and Systems, Vol. 32, 2020
45
Proof. The proof of 1), 2), and 3) is obvious from definition 3.12.
e E) such
e E) ∈ U (xλ (α,β,γ) , E) , then there exists a neutrosophic soft p-open set (G,
4) If (̥,
(α,β,γ)
e E) ∈ U (xλ (α,β,γ) , E) , so for each
e E) ⊏ (̥,
e E). From Proposition 3.1, (G,
∈ (G,
that xλ
′
′
′
′
′
′
′ (α ,β ,γ )
e E) , (G,
e E) ∈ U (y λ′ (α ,β ,γ ) , E) is obtained.
yλ
∈ (G,
Definition 3.17. Let xλ
(α,β,γ)
λ (α,β,γ)
sophic soft points x
and y λ
and y
soft points are distinct points if
It is clear that xλ
6= y or
λ′
(α,β,γ)
′ (α′ ,β ′ ,γ ′ )
′ ′ ′
λ′ (α ,β ,γ )
(α,β,γ)
xλ
and y λ
over a common universe X, we say that neutrosophic
⊓ yλ
′ (α′ ,β ′ ,γ ′ )
be two neutrosophic soft points. For the neutro-
′ (α′ ,β ′ ,γ ′ )
= 0(X,E) .
are distinct neutrosophic soft points if and only if x
6= λ.
4. Neutrosophic Soft p-Separation Structures
In this phase, we suppose neutrosophic soft p-separation axioms and neutrosophic soft topological subspace consisting of distinct neutrosophic soft points of neutrosophic soft topological
space over X.
Definition 4.1. a) Let (X, ℑ, E) be a neutrosophic soft topological space over the crisp set
X , and xλ
(α,β,γ)
> yλ
′ (α′ ,β ′ ,γ ′ )
are neutrosophic soft points. If there exist neutrosophic soft
e E) such that
p-open sets (Fe , E) and (G,
(α,β,γ)
(α,β,γ)
e E) = 0(X,E) or
∈ (Fe , E) and xλ
⊓ (G,
xλ
′
′
′
′
′
′
(α
,β
,γ
)
(α
,β
,γ
)
′
e E) and y λ′
⊓ (Fe , E) = 0(X,E) ,
∈ (G,
yλ
(α,β,γ)
e E) such that
(G,
(α,β,γ)
(α,β,γ)
e E) = 0(X,E) or
⊓ (G,
∈ (Fe , E) , xλ
xλ
′ ,β ′ ,γ ′ )
′ ,β ′ ,γ ′ )
(α
(α
′
′
e E) , y λ
∈ (G,
⊓ (Fe , E) = 0(X,E) ,
yλ
(α,β,γ)
then (X, ℑ, E) is called a neutrosophic soft-P o -space.
>
b) Let (X, ℑ, E) be a neutrosophic soft topological space over the crisp set X and xλ
′ ,β ′ ,γ ′ )
(α
′
are neutrosophic soft points. If there exist neutrosophic soft p-open sets (Fe , E) and
yλ
then (X, ℑ, E) is called a neutrosophic soft-P 1 -space.
>
c) Let (X, ℑ, E) be a neutrosophic soft topological space over the crisp set X, and xλ
′ ,β ′ ,γ ′ )
(α
′
are neutrosophic soft points. If there exist neutrosophic soft p-open sets (Fe , E) and
yλ
e E) such that
(G,
′ (α′ ,β ′ ,γ ′ )
(α,β,γ)
e E) and (Fe , E) ⊓ (G,
e E) = 0(X,E) ,
∈ (G,
∈ (Fe , E) , y λ
xλ
then (X, ℑ, E) is called a neutrosophic soft-P 2 -space.
Example 4.2. Let X = {x1 , x2 } be a universe set, E= {λ1 , λ2 } be a parameters set, and
(x1 )λ1
(0.1,0.4,0.7)
, (x1 )λ2
(0.2,0.5,0.6)
, (x2 )λ1
(0.3,0.3,0.5)
(0.4,0.4,0.4)
, and (x2 )λ2
be neutrosophic soft
(X,E)
(X,E)
1
2
3
4
,1
, (Fe , E), (Fe , E), (Fe , E), (Fe , E),
points. Then the family ℑ = {0
(Fe 5 , E), (Fe 6 , E), (Fe 7 , E), (Fe 8 , E)} , where
A. Mehmood, F. Nadeem, G. Nordo, M. Zamir, C. Park, H. Kalsoom, S. Jabeen and M. I.
Khan, Generalized Neutrosophic Separation Axioms in Neutrosophic Soft Topological Spaces
Neutrosophic Sets and Systems, Vol. 32, 2020
46
λ1 (0.1,0.4,0.7)
},
(Fe 1 , E) = {x1
(0.2,0.5,0.6)
2
1
2
},
(Fe , E) = {(x )λ
(Fe 3 , E) = {(x2 )λ1 (0.3, 0.3, 0.5)} ,
(Fe 4 , E) = (Fe1 , E) ⊔ (Fe 2 , E) ,
(Fe 5 , E) = (Fe1 , E) ⊔ (Fe 3 , E) ,
(Fe 6 , E) = (Fe2 , E) ⊔ (Fe 3 , E) ,
(Fe 7 , E) = (Fe1 , E) ⊔ (Fe 2 , E) ⊔ (Fe 3 , E) ,
(0.1,0.4,0.7)
(0.2,0.5,0.6)
(0.3,0.3,0.5)
, (x1 )λ2
, (x2 )λ2
, (x2 )λ2 (0.4, 0.4, 0.4)} ,
(Fe 8 , E) = {(x1 )λ1
is a neutrosophic soft topology over X. Hence, (X, ℑ, E) is a neutrosophic soft topolog-
ical space over X. Also, (X, ℑ, E) is a neutrosophic soft- P 0 -space but not a neutrosophic
soft-P 1 -space because for neutrosophic soft points (x1 )λ1 (0.1, 0.4, 0.7) and (x2 )λ2 (0.4, 0.4, 0.4)
,(X, ℑ, E) is not a neutrosophic soft-P 1 -space.
Example 4.3. Let X = N be a natural numbers set and E = {λ} be a parameters set. Here
nλ
(αn,βn,γn)
are neutrosophic soft points. Here we can give (αn, βn, γn) appropriate values and
the neutrosophic soft points nλ
(αn,βn,γn)
,mλ
(αn,βn,γn)
are distinct neutrosophic soft points if
and only if n 6= m. It is clear that there is one-to-one compatibitily between the set of natural
numbers and the set of neutrosophic soft points N λ = {nλ
(αn,βn,γn)
}.
Then we give cofinite topology on this set. Then neutrosophic soft set (Fe , E) is a neutro-
sophic soft p-open set if and only if the finite neutrosophic soft point is discarded from N λ .
Hence, (X, ℑ, E) is a neutrosophic soft-P 1 -space but not a neutrosophic soft-P 2 -space.
Example 4.4. Let X = {x1 , x2 } be a universe set, E = {λ1 , λ2 } be a parameters set, and
x1 λ1
(0.1,0.4,0.7)
,x1 λ2
(0.3,0.3,0.5)
,
x2 λ 1
(0.2,0.5,0.6)
,
(0.4,0.4,0.4)
,
x2 λ 2
be neutrosophic soft points. Then the family
e 1 , E), (̥
e 2 , E), ..., (̥
e 15 , E)} , where
ℑ = {0(X,E) , 1(X,E) , (̥
e 1 , E) = {x1 λ1 (0.1,0.4,0.7) } ,
(̥
and
(0.2,0.5,0.6)
e 2 , E) = {(x1 )λ2
},
(̥
(0.3,0.3,0.5)
3
2
1
e , E) = {(x )λ
},
(̥
(0.4,0.4,0.4)
4
2
2
e , E) = {(x )λ
},
(̥
(Fe 5 , E) = (Fe1 , E) ⊔ (Fe 2 , E) ,
(Fe 6 , E) = (Fe1 , E) ⊔ (Fe 3 , E)
(Fe 7 , E) = (Fe1 , E) ⊔ (Fe 4 , E)
(Fe 8 , E) = (Fe2 , E) ⊔ (Fe 3 , E)
(Fe 9 , E) = (Fe2 , E) ⊔ (Fe 4 , E)
(Fe 10 , E)
=
(Fe3 , E)
⊔
,
,
,
,
(Fe 4 , E)
,
(Fe 11 , E) = (Fe1 , E) ⊔ (Fe 2 , E) ⊔ (Fe 3 , E) ,
A. Mehmood, F. Nadeem, G. Nordo, M. Zamir, C. Park, H. Kalsoom, S. Jabeen and M. I.
Khan, Generalized Neutrosophic Separation Axioms in Neutrosophic Soft Topological Spaces
Neutrosophic Sets and Systems, Vol. 32, 2020
47
(Fe 12 , E) = (Fe1 , E) ⊔ (Fe 2 , E) ⊔ (Fe 4 , E) ,
(Fe 13 , E) = (Fe2 , E) ⊔ (Fe 3 , E) ⊔ (Fe 4 , E) ,
(Fe 14 , E) = (Fe1 , E) ⊔ (Fe 3 , E) ⊔ (Fe 4 , E) ,
(0.1,0.4,0.7)
(0.2,0.5,0.6)
(0.3,0.3,0.5)
(0.4,0.4,0.4)
, (x1 )λ2
, (x2 )λ2
, (x2 )λ2
},
(Fe 15 , E) = {(x1 )λ1
is a neutrosophic soft topology over X. Hence, (X, ℑ, E) is a neutrosophic soft topological
space over X. Also, (X, ℑ, E) is a neutrosophic soft -P 2 -space.
Theorem 4.5. Let (X, ℑ, E) be a neutrosophic soft topological space over X. Then (X, ℑ, E)
is a neutrosophic soft-P 1 -space if and only if each neutrosophic soft point is a neutrosophic
soft p-closed set.
Proof. Let(X, ℑ, E) be a neutrosophic soft-P 1 -space and xλ
soft point. We show that
(x
λ
λ (α,β,γ)
or
) ; then x
λ′
λ (α,β,γ)
(α,β,γ)
)
(xλ
and y
λ
(α,β,γ)
be an arbitrary neutrosophic
is a neutrosophic soft p-open set. Let y λ
′ ′ ′
λ′ (α ,β ,γ )
′ (α′ ,β ′ ,γ ′ )
∈
are distinct neutrosophic soft points. Hence, x 6= y
6= λ.
Since (X, ℑ, E) is a neutrosophic soft-P 1 -space, there exists a neutrosophic soft p-open set
e E) such that
(G,
′ (α′ ,β ′ ,γ ′ )
e E) and xλ (α,β,γ) ⊓ (G,
e E) = 0(X,E) .
∈ (G,
′ ′ ′
λ
(α,β,γ)
e E) = 0(X,E) , we have y λ′ (α ,β ,γ ) ∈ (G,
e E) ⊏ (xλ (α,β,γ) ) . This
⊓ (G,
Then, since xλ
yλ
implies that (xλ
(α,β,γ) λ
) is a neutrosophic soft p-open set, i.e. xλ
(α,β,γ)
is a neutrosophic soft
p-closed set.
Suppose that each neutrosophic soft point xλ
(α,β,γ)
)
(xλ
yλ
λ
′ (α′ ,β ′ ,γ ′ )
(α,β,γ)
is a neutrosophic soft p-closed set. Then
is a neutrosophic soft p-open set. Let xλ
∈ (xλ
(α,β,γ) λ
) and xλ
(α,β,γ)
⊓ (xλ
(α,β,γ)
⊓ yλ
′ (α′ ,β ′ ,γ ′ )
= 0(X,E) . Thus
(α,β,γ) λ
) = 0(X,E) . Therefore, (X, ℑ, E) is a neu-
trosophic soft-P 1 -space over X.
Theorem 4.6. Let (X, ℑ, E) be a neutrosophic soft topological space over X. Then (X, ℑ, E)
(α,β,γ)
′ (α′ ,β ′ ,γ ′ )
is a neutrosophic soft-P 2 -space iff for distinct neutrosophic soft points xλ
and y λ
,
′
e E) containing xλ (α, β, γ) but not y λ (α′ , β ′ , γ ′ )
there exists a neutrosophic soft p-open set (̥,
′ ,β ′ ,γ ′ )
(α
′
e E).
does not belong to (̥,
such that y λ
Proof. Let xλ
(α,β,γ)
and y λ
′ (α′ ,β ′ ,γ ′ )
be two neutrosophic soft points in neutrosophic soft -P 2 -
space (X, ℑ, E).
e E) such that
e E) , (G,
Then there exist disjoint neutrosophic soft p-open set (̥,
′ ,β ′ ,γ ′ )
(α
′
(α,β,γ)
e E).
e E) , y λ
xλ
∈ (̥,
∈ (G,
(α,β,γ)
′ (α′ ,β ′ ,γ ′ )
e E) = 0(X,E) , y λ
e E) ⊓ (G,
= 0(X,E) and (̥,
′
′
′
e E).
e E) . It implies that y λ′ (α ,β ,γ ) does not belong to (̥,
belong to (̥,
Since xλ
⊓ yλ
′ (α′ ,β ′ ,γ ′ )
does not
A. Mehmood, F. Nadeem, G. Nordo, M. Zamir, C. Park, H. Kalsoom, S. Jabeen and M. I.
Khan, Generalized Neutrosophic Separation Axioms in Neutrosophic Soft Topological Spaces
Neutrosophic Sets and Systems, Vol. 32, 2020
48
(α,β,γ)
′ (α′ ,β ′ ,γ ′ )
Next suppose that, for distinct neutrosophic soft points xλ
, yλ
, there exists a
′ ,β ′ ,γ ′ )
(α
′ (α′ ,β ′ ,γ ′ )
′
(α,β,γ)
e E) containing xλ
such that y λ
but not y λ
neutrosophic soft p-open set (̥,
e E). Then y λ
does not belong to (̥,
′ (α′ ,β ′ ,γ ′ )
neutrosophic soft p-open sets containing xλ
e E))c , i.e. (̥,
e E))c are disjoint
e E) and ((̥,
∈ ((̥,
(α,β,γ)
, yλ
′ (α′ ,β ′ ,γ ′ )
respectively.
Theorem 4.7. Let (X, ℑ, E) be a neutrosophic soft-P 1 -space for every neutrosophic soft point
(α,β,γ)
e E) such that
e E) ∈ ℑ. If there exists a neutrosophic soft p-open set (G,
∈ (̥,
xλ
xλ
(α,β,γ)
e E) ⊏ (G,
e E) ⊏ (̥,
e E),
∈ (G,
then (X, ℑ, E) is a neutrosophic soft-P 2 -space.
Proof. Suppose that xλ
T 1 -space, xλ
(α,β,γ)
(α,β,γ)
and y λ
⊓ yλ
′ (α′ ,β ′ ,γ ′ )
= 0(X,E) . Since (X, ℑ, E) is a neutrosophic soft
′ (α′ ,β ′ ,γ ′ )
(α,β,γ)
are neutrosophic soft p-closed sets in ℑ. Thus xλ
∈
′
λ
′
′
′
c
e
(y (α , β , γ )) ∈ ℑ. Then there exists a neutrosophic soft p-open set (G, E) in ℑ such that
(α,β,γ)
e E) ⊏ (G,
e E) ⊏ (y λ′ (α′ , β ′ , γ ′ ))c .
∈ (G,
xλ
Hence, we have y λ
′ (α′ ,β ′ ,γ ′ )
e E), and (G,
e E) ⊓ ((G,
e E))c , xλ (α,β,γ) ∈ (G,
e E))c = 0(X,E) ,
∈ ((G,
i,e. (X, ℑ, E) is a neutrosophic soft P 2 -space.
Remark 4.8. Let (X, ℑ, E) be a neutrosophic soft-P 1 -space for i = 0, 1, 2. For each x
6= y, neutrosophic points x( α, β, γ) and y ( α′ , β ′ , γ ′ ) have neighborhoods satisfying conditions
of-P i -space in neutrosophic topological space (X, ℑλ ) for each λ ∈ E because xλ
′ (α′ ,β ′ ,γ ′ )
yλ
(α,β,γ)
and
are distinct neutrosophic soft points.
e E) be a
Definition 4.9. Let (X, ℑ, E) be a neutrosophic soft topological space over X, (̥,
(α,β,γ)
e E) = 0(X,E) . If there exist neutrosophic
⊓ (̥,
neutrosophic soft p-closed set , and xλ
e 2 , E) such that xλ (α,β,γ) ∈ (G
e 1 , E), (̥,
e 2 , E), and
e E) ⊏ (G
soft p-open open sets (e1 , E) and (G
e 2 , E) = 0(X,E) , then (X, ℑ, E) is called a neutrosophic soft b-regular space.
e 1 , E) ⊓ (G
(G
(X, ℑ, E) is said to be a neutrosophic soft-P 3 -space if is both a neutrosophic soft p-regular
and neutrosophic soft-P 1 -space.
Theorem 4.10. Let (X, ℑ, E) be a neutrosophic soft topological space over X, (X, ℑ, E) is a
(α,β,γ)
e E)
e E) ∈ ℑ , there exists (G,
∈ (̥,
neutrosophic soft-P 3 -space if and only if for every xλ
∈ ℑ such that xλ
(α,β,γ)
e E) ⊏ (G,
e E) ⊏ (̥,
e E) .
∈ (G,
Proof. Let (X, ℑ, E) be a neutrosophic soft P 3 -space and xλ
soft-P 3 -space
(α,β,γ)
e E) ∈ ℑ. Since
∈ (̥,
(α,β,γ)
for the neutrosophic soft point xλ
and neue 1 , E) , (G
e 2 , E) ∈ ℑ such that xλ (α,β,γ) ∈
e E)c , there exist (G
trosophic soft p-closed set (̥,
e 2 , E), and (G
e 1 , E) ⊓ (G
e 2 , E) = 0(X,E) . thus, we have xλ (α,β,γ) ∈ (G
e 1 , E)
e 1 , E) , (̥,
e E)c ⊏ (G
(G
(X, ℑ, E) is a neutrosophic
e 2 , E)c .
e 2 , E)c is a neutrosophic soft p-closed set, (G
e 2 , E)c ⊏ (̥,
e 1 , E) ⊏ (G
e E). Since (G
⊏ (G
A. Mehmood, F. Nadeem, G. Nordo, M. Zamir, C. Park, H. Kalsoom, S. Jabeen and M. I.
Khan, Generalized Neutrosophic Separation Axioms in Neutrosophic Soft Topological Spaces
Neutrosophic Sets and Systems, Vol. 32, 2020
49
(α,β,γ)
e E) be a neutrosophic soft p-closed set.
e E) = 0(X,E) and (H,
Conversely , let xλ
⊓ (H,
(α,β,γ)
e E)c and from the condition of the theorem, we have xλ (α,β,γ) ∈ (G,
e E)
Thus, xλ
∈ (H,
e E) ⊏ (H
e , E)c .
⊏ (G,
Then xλ
(α,β,γ)
e E) ⊓ (G,
e E) , (H,
e E) ⊏ ((G,
e E))c , and (G,
e E)c = 0(X,E) are satisfied,
∈ (G,
i.e. (X, ℑ, E) is a neutrosophic soft-P 3 -space.
Definition 4.11. A neutrosophic soft topological space (X, ℑ, E) over X is called a neutroe 1 , E) ,
sophic soft p-normal space if for every pair of disjoint neutrosophic soft b-closed set (̥
e 2 , E) such that (̥
e 1 , E) , (G
e 1 , E)
e 2 , E) , there exists disjoint neutrosophic soft p-open sets (G
(̥
e 2 , E).
e 1 , E) and (̥
e 2 , E) ⊏ (G
⊏ (G
(X, ℑ, E) is said to be a neutrosophic soft b-T 4 -space if it is both a neutrosophic soft p-
normal and neutrosophic soft-P 1 -space.
Theorem 4.12. Let (X, ℑ, E) be a neutrosophic soft topological space over X . Then (X, ℑ, E)
e E)
is a neutrosophic soft-P 4 -space if and only if, for each neutrosophic soft p-closed set (̥,
e E) with (̥,
e E) , there exists a neutrosophic soft
e E) ⊏ (G,
and neutrosophic soft p-open set (G,
e E) such that
p-open set (D,
e E) ⊏ (D,
e E) ⊏ (G,
e E).
e E) ⊏ (D,
(̥,
e E) be a neutrosophic soft p-closed
Proof. Let (X, ℑ, E) be a neutrosophic soft-P 4 -space, (̥,
e E) ∈ ℑ. Then (G,
e E)c is a neutrosophic soft p-closed set and (̥,
e E) ⊏ (G,
e E)
set and (̥,
e E)c = 0(X,E) . Since (X, ℑ, E) is a neutrosophic soft-P 4 -space, there exist neutrosophic
⊓ (G,
e E)c ⊏ (D
e 2 , E) , and
e 2 , E) such that (̥,
e 1 , E) , (G,
e 1 , E) and (D
e E) ⊏ (D
soft p-open sets (D
e 2 , E) = 0(X,E) . This implies that
e 1 , E) ⊓ (D
(D
e 2 , E)c ⊏ (G,
e E).
e 1 , E) ⊏ (D
e E) ⊏ (D
(̥,
e 2 , E)c is satisfied. Thus,
e 2 , E)c is a neutrosophic soft p-closed set and (D
e 1 , E) ⊏ (D
(D
e E)
e 1 , E) ⊏ (G,
e 1 , E) ⊏ (D
e E) ⊏ (D
(̥,
is obtained.
e 2 , E) be two disjoint neutrosophic soft p-closed sets. Then
e 1 , E) , (̥
Conversely, let (̥
e 2 , E)c . From the condition of theorem, there exists a neutrosophic soft p-open
e 1 , E) ⊏ (̥
(̥
e E) such that
set (D,
e E) ⊏ (D
e 1 , E) ⊏ (̥
e 2 , E)c .
e 1 , E) ⊏ (D,
(̥
e E) , ((D,
e E) , (̥
e E))c are neutrosophic soft p-open sets and (̥
e 2 , E) ⊏
e 1 , E) ⊏ (D,
Thus, (D,
e E) ⊓ ((D,
e E))c , and (D,
e E))c = 0(X,E) are obtained. Hence, (X, ℑ, E) is a neutrosophic
((D,
soft-P 4 -space.
A. Mehmood, F. Nadeem, G. Nordo, M. Zamir, C. Park, H. Kalsoom, S. Jabeen and M. I.
Khan, Generalized Neutrosophic Separation Axioms in Neutrosophic Soft Topological Spaces
Neutrosophic Sets and Systems, Vol. 32, 2020
50
e E) be
Definition 4.13. Let (X, ℑ, E) be a neutrosophic soft topological space over X and (̥,
e E) : (H,
e E) ∈ ℑ} is said to
e E) ⊓ (H,
an arbitrary neutrosophic soft set. Then ℑ(̥e ,E) = {(̥,
e E) and ((̥,
e E), ℑ(̥e ,E) , E) is called a neutrosophic soft
be neutrosophic soft topology on (̥,
topological subspace of (X, ℑ, E).
Theorem 4.14. Let (X, ℑ, E) be a neutrosophic soft topological space over X. If (X, ℑ, E) is a
e E), ℑ(̥e ,E) , E)
neutrosophic soft-P i -space, then the neutrosophic soft topological subspace ((̥,
is a neutrosophic soft-P i -space for i = 0, 1, 2, 3.
′ (α′ ,β ′ ,γ ′ )
′
′
′
e E), ℑ(̥e ,E) , E) such that xλ (α,β,γ) ⊓ y λ′ (α ,β ,γ ) = 0(X,E) .
∈ ((̥,
e 1 , E) and (̥
e 2 , E) satisfying the conditions
Thus , there exist neutrosophic soft p-open set (̥
′ ′ ′
(α,β,γ)
e 1 , E) , y λ′ (α ,β ,γ ) ∈ (̥
e 2 , E). Then
∈ (̥
of neutrosophic soft -P i -space such that xλ
Proof. Let xλ
(α,β,γ)
, yλ
′
′
′
e 1 , E) ⊓ (̥,
e E) and y λ′ (α ,β ,γ ) ∈ (̥
e 2 , E) ⊓ (̥,
e E) . Also, the neutrosophic soft
∈ (̥
e
e E) , (̥
e 2 , E) ⊓ (̥,
e E) in ℑ(̥,E) satisfy the conditions of neutrosophic
e 1 , E) ⊓ (̥,
p-open set (̥
xλ
(α,β,γ)
soft-P i -space for i = 0, 1, 2, 3.
Theorem 4.15. Let (X, ℑ, E) be a neutrosophic soft topological space over X. If (X, ℑ, E) is
e E) is a neutrosophic soft p-closed set in (X, ℑ, E), then
a neutrosophic soft-P 4 -space and (̥,
e E), ℑ(̥e ,E) , E) is a neutrosophic soft -P 4 -space.
((̥,
e E) be a neutrosophic soft pProof. Let (X, ℑ, E) be a neutrosophic soft P 4 -space and (̥,
e 2 , E) be two neutrosophic soft p-closed sets in
e 1 , E) and (̥
closed set in (X, ℑ, E). Let (̥
e E), ℑ(̥e ,E) , E) such that (̥
e 1 , E) ⊓ (̥
e E) is a neutrosophic
e 2 , E) = 0(X,E) . When (̥,
((̥,
e 1 , E) and (̥
e 2 , E) are neutrosophic soft p-closed sets in
soft p-closed set in (X, ℑ, E) , (̥
(X, ℑ, E). Since (X, ℑ, E) is a neutrosophic soft-P 4 -space, there exist neutrosophic soft pe 2 , E) such that (̥
e 1 , E) , (̥
e 2 , E) and (G
e 1 , E)
e 1 , E) and (G
e 2 , E) ⊏ (G
e 1 , E) ⊏ (G
open sets (G
e 1 , E) ⊓ (̥,
e 2 , E) ⊓ (̥,
e 2 , E) = 0(X,E) . Then (̥
e 1 , E) = (G
e E) , (̥
e 2 , E) = (G
e E) and
⊓ (G
e 2 , E) ⊓ (̥,
e 1 , E) ⊓ (̥,
e E)) ⊓ ((G
e E)) = 0(X,E) . This implies that ((̥,
e E), ℑ(̥e ,E) , E) is a
((G
neutrosophic soft-P 4 -space.
5. Conclusion
Neutrosophic soft p-separation structures are the most imperative and fascinating notions
in neutrosophic soft topology.We have introduced neutrosophic soft p-separation axioms in
neutrosophic soft topological structures with respect to soft points, which are defined over
an initial universe of discourse with a fixed set of parameters( data,decision variables). We
further investigated and scrutinized some essential features of the initiated neutrosophic soft
p-separation structures. It is supposed that these results will be very very useful for future
A. Mehmood, F. Nadeem, G. Nordo, M. Zamir, C. Park, H. Kalsoom, S. Jabeen and M. I.
Khan, Generalized Neutrosophic Separation Axioms in Neutrosophic Soft Topological Spaces
Neutrosophic Sets and Systems, Vol. 32, 2020
51
studies on neutrosophic soft topology to carry out a general framework for practical applications. Applications of neutrosophic soft p-separation structures in neutrosophic soft topological
spaces can be traced out in decision making problems.
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16. M. Abdel-Basset; M. Mohamed; M. Elhoseny; F. Chiclana; A.E.H. Zaied, Cosine similarity measures of
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most generalized neutrosophic soft open set, Communication in Mathematics and Applications, [In press]
Received: Oct 21, 2019. Accepted: Mar 20, 2020
A. Mehmood, F. Nadeem, G. Nordo, M. Zamir, C. Park, H. Kalsoom, S. Jabeen and M. I.
Khan, Generalized Neutrosophic Separation Axioms in Neutrosophic Soft Topological Spaces
Neutrosophic Sets and Systems, Vol. 32, 2020
University of New Mexico
Interval Valued Neutrosophic Topological Spaces
T. Nanthini 1,* and A. Pushpalatha 2
1
Research Scholar, Department of Mathematics, Government Arts College, Udumalpet-642 126, Tamil Nadu, India, Email:
tnanthinimaths@gmail.com
2 Assistant Professor, Department of Mathematics, Government Arts College, Udumalpet-642 126, Tamil Nadu, India,
Email:velu_pushpa@yahoo.co.in
* Correspondence: tnanthinimaths@gmail.com
Abstract: Within this paper, we present and research the definition of interval valued neutrosophic
topological space along with interval valued neutrosophic finer and interval valued neutrosophic
coarser topologies. We also describe interval valued neutrosophic interior and closer of an interval
valued neutrosophic set. Interval valued neutrosophic subspace topology also studied.
Some examples and theorems are presented concerning this concept.
Keywords: Interval valued neutrosophic topology, Interval valued neutrosophic subspace topology
1. Introduction
The notion of fuzzy set has invaded almost all branches of mathematics since its introduction by
Zadeh[20]. Fuzzy sets and fuzzy logic has been applied in many real applications to handle
uncertaintely fuzzy set theory is very successful in handling uncertainties arising from vagueness or
partial belongingness of an element in a set, it cannot model all type of uncertainties pre – veiling in
different real physical problems such as problems involving incomplete information. Turksen [18]
introducted the idea of interval valued fuzzy sets.
Later, Atanassov[10] introduced the concept generalization of fuzzy set, which is known as
intuitionistic fuzzy sets. Intuitionistic fuzzy sets take into account both the degree of membership
and non – membership. Further, intuitionistic fuzzy sets were extended to the interval valued
intuitionistic fuzzy sets[11]. The interval valued intuitionistic fuzzy set uses a pair of interval
[𝑡 − , 𝑡 + ], 0 ≤ 𝑡 − ≤ 𝑡 + ≤ 1 and
[𝑓 − , 𝑓 + ], 0 ≤ 𝑓 − ≤ 𝑓 + ≤ 1 with 𝑡 + + 𝑓 + ≤ 1,
to describe the
degree of true belief and false belief. Because of the restriction that 𝑡 + + 𝑓 + ≤ 1, intuitionistic fuzzy
sets and interval valued intuitionistic fuzzy sets can only handle incomplete information not the
indeterminate information and inconsistent information which exists commonly in belief systems.
As a generalization of fuzzy set and intuitionistic fuzzy set, neutrosophic set have been
introduced and developed by F. Smaramdache[15,16 & 17 ]. It is a logic in which each proposition is
calculated to have degree of truth(T), a degree of indeterminacy(I) and a degree of falsity(F).
Smarandache’s neutrosophic concept have wide range of real applications for many fields of
T.Nanthini and A.Pushpalatha, Interval valued Neutosophic Topological Pushpalatha Space
Neutrosophic Sets and Systems, Vol. 32, 2020
53
[1,2,3,4,5,6,7 & 8] information system, computer science, artificial intelligence, applied mathematics,
decision making, mechanics, electrical and electronics, medicine and management science etc.
Salama, Albloe[14] proposed the concept of neutrosophic topological space. Later, Wang,
Smarandache, Zhang and Sunderraman introduced the notion of interval valued neutrosophic
set[19]. An interval valued neutrosophic set 𝐴 defined on 𝑋, 𝑥 = 𝑥(𝑇, 𝐼, 𝐹) ∈ 𝐴 with 𝑇, 𝐼 and 𝐹
being the subinterval of [0,1]. Lupianez discusses the relation between interval value neutrosophic
sets and topology [12]
The purpose of this article is to propose the idea of interval valued neutrosophic topological
space and discuss the some of the basic properties.
2. Preliminaries
Definition 2.1[19] Let 𝑋 be a space of points (objects), with a generic element in 𝑋 denoted by 𝑥. An
interval valued neutrosophic set(𝐼𝑁𝑆) 𝐴 in 𝑋 is characterized by truth – membership function 𝑇𝐴 ,
indeterminacy – membership function 𝐼𝐴 and falsity – membership function 𝐹𝐴 . For each point 𝑥 in
𝑋, 𝑇𝐴 (𝑥), 𝐼𝐴 (𝑥), 𝐹𝐴 (𝑥) ⊆ [0,1].
Example 2.2[19] Suppose, 𝑋 = {𝑥1 , 𝑥2 , 𝑥3 }. The strength is 𝑥1 , the trust is 𝑥2 and the price is 𝑥3 .
The 𝑥1 , 𝑥2 and 𝑥3 values are given in [0,1] . They're obtained from some domain experts '
questionnaire, their choice could be degree of goodness, degree of indeterminacy, and degree of
poorness. 𝐴
and
𝐵
are
the
interval
neutrosophic
[0.2,0.4],[0.3,0.5],[0.3,0.5] [0.5,0.7],[0,0.2],[0.2,0.3] [0.6,0.8],[0.2,0.3],[0.2,0.3]
,
𝑥1
,
𝑥2
𝑥3
[0.5,0.7],[0.1,0.3],[0.1,0.3] [0.2,0.3],[0.2,0.4],[0.5,0.8] [0.4,0.6],[0,0.1],[0.3,0.4]
,
𝑥1
𝑥2
,
𝑥3
sets
of
>
𝑋
define
>
by
𝐴 =<
𝐵 =<
Definition 2.3[19] An interval neutrosophic set 𝐴 is empty 𝑖𝑓 𝑎𝑛𝑑 𝑜𝑛𝑙𝑦 𝑖𝑓 its inf 𝑇𝐴 (𝑥) =
sup 𝑇𝐴 (𝑥) = 0, inf 𝐼𝐴 (𝑥) = sup 𝐼𝐴 (𝑥) = 1 and inf 𝐹𝐴 (𝑥) = sup 𝐹𝐴 (𝑥) = 0, for all 𝑥 in 𝑋.
Definition 2.4(Containment) [19] An interval neutrosophic set 𝐴 is contained in the other interval
neutrosophic set 𝐵, 𝐴 ⊆ 𝐵, 𝑖𝑓 𝑎𝑛𝑑 𝑜𝑛𝑙𝑦 𝑖𝑓
inf 𝑇𝐴 (𝑥) ≤ inf 𝑇𝐵 (𝑥) , sup 𝑇𝐴 (𝑥) ≤ sup 𝑇𝐵 (𝑥)
for all 𝑥 in 𝑋.
inf 𝐼𝐴 (𝑥) ≥ inf 𝐼𝐵 (𝑥) , sup 𝐼𝐴 (𝑥) ≥ sup 𝑇𝐵 (𝑥)
inf 𝐹𝐴 (𝑥) ≥ inf 𝐹𝐵 (𝑥) , sup 𝐹𝐴 (𝑥) ≥ sup 𝐹𝐵 (𝑥)
Definition 2.5[19] Two interval neutrosophic sets 𝐴 and 𝐵 are equal, written as 𝐴 =
𝐵, 𝑖𝑓 𝑎𝑛𝑑 𝑜𝑛𝑙𝑦 𝑖𝑓 𝐴 ⊆ 𝐵 and 𝐵 ⊆ 𝐴. Let 0𝑁 =< 0,1,1 > and 1𝑁 =< 1,0,0 >.
Definition 2.6[19] The complement of an interval neutrosophic set 𝐴 is denoted by 𝐴̅ and is defined
by 𝑇𝐴̅ (𝑥) = 𝐹𝐴 (𝑥); inf 𝐼𝐴̅ (𝑥) = 1 − sup 𝐼𝐴 (𝑥) ;sup 𝐼𝐴̅ (𝑥) = 1 − inf 𝐼𝐴 (𝑥); 𝐹𝐴̅ (𝑥) = 𝑇𝐴 (𝑥) for all 𝑥 in 𝑋.
Example 2.7[19] Let 𝐴
be the interval neutrosophic set defined in Example 2.3, then
[0.3,0.5],[0.5,0.7],[0.3,0.4] [0.2,0.3],[0.8,0],[0.5,0.7] [0.2,0.3],[0.7,0.8],[0.6,0.8]
𝐴̅ =<
,
,
>
𝑥1
𝑥2
𝑥3
T.Nanthini and A.Pushpalatha, Interval valued Neutosophic Topological Pushpalatha Space
Neutrosophic Sets and Systems, Vol. 32, 2020
54
Definition 2.8 (Intersection) [19] The intersection of two interval neutrosophic sets 𝐴 and 𝐵 is an
interval neutrosophic set 𝐶 = 𝐴 ∩ 𝐵, whose truth-membership, indeterminacy – membership and
false – membership are related to those of 𝐴 and 𝐵 by
sup 𝑇𝐶 (𝑥) = min(sup 𝑇𝐴 (𝑥) , sup 𝑇𝐵 (𝑥))
inf 𝑇𝐶 (𝑥) = min(inf 𝑇𝐴 (𝑥) , inf 𝑇𝐵 (𝑥)),
sup 𝑇𝐶 (𝑥) = max(sup 𝐼𝐴 (𝑥) , sup 𝐼𝐵 (𝑥))
inf 𝐼𝐶 (𝑥) = max(inf 𝐼𝐴 (𝑥) , inf 𝐼𝐵 (𝑥)),
sup 𝑇𝐶 (𝑥) = max(sup 𝐹𝐴 (𝑥) , sup 𝐹𝐵 (𝑥))
inf 𝐹𝐶 (𝑥) = max(inf 𝐹𝐴 (𝑥) , inf 𝐹𝐵 (𝑥)),
for all 𝑥 in 𝑋.
Example 2.9[19] Let 𝐴 and 𝐵 be the interval neutrosophic sets defined in Example 2.3, then 𝐴 ∩ 𝐵 =
<
[0.2,0.4],[0.3,0.5],[0.3,0.5] [0.2,0.3],[0.2,0.4],[0.5,0.8] [0.4,0.6],[0.2,0.3],[0.3,0.4]
,
𝑥1
,
𝑥2
𝑥3
>.
Theorem 2.10[19] 𝐴 ∩ 𝐵 is the largest interval neutrosophic set contained in both 𝐴 and 𝐵.
Definition 2.11(Union) [19] The union of two interval neutrosophic sets 𝐴 and 𝐵 is an interval
neutrosophic set 𝐶, written as 𝐶 = 𝐴 ∪ 𝐵, whose truth – membership, indeterminacy – membership
and false membership are related to those of 𝐴 and 𝐵 by
inf 𝑇𝐶 (𝑥) = max(inf 𝑇𝐴 (𝑥) , inf 𝑇𝐵 (𝑥)),
inf 𝐼𝐶 (𝑥) = min(inf 𝐼𝐴 (𝑥) , inf 𝐼𝐵 (𝑥)),
inf 𝐹𝐶 (𝑥) = min(inf 𝐹𝐴 (𝑥) , inf 𝐹𝐵 (𝑥)),
for all 𝑥 in 𝑋.
sup 𝑇𝐶 (𝑥) = max(sup 𝑇𝐴 (𝑥) , sup 𝑇𝐵 (𝑥))
sup 𝑇𝐶 (𝑥) = min(sup 𝐼𝐴 (𝑥) , sup 𝐼𝐵 (𝑥))
sup 𝑇𝐶 (𝑥) = min(sup 𝐹𝐴 (𝑥) , sup 𝐹𝐵 (𝑥))
Example 2.12[19] Let 𝐴 and 𝐵 be the interval neutrosophic sets defined in Example 2.3, then 𝐴 ∪
𝐵 =<
[0.5,0.7],[0.1,0.3],[0.1,0.3] [0.5,0.7],[0,0.2],[0.2,0.3] [0.6,0.8],[0,0.1],[0.2,0.3]
𝑥1
,
𝑥2
,
𝑥3
>.
Theorem 2.13[19] 𝐴 ∪ 𝐵 is the smallest interval neutrosophic set containing both 𝐴 and 𝐵.
3. Interval Valued Neutrosophic Topological Spaces
With some examples and results, we give the concept of interval valued neutrosophic topologi
cal spaces.
Definition 3.1 An interval valued neutrosophic topological space of interval valued neutrosophic set
(In short 𝐼𝑉𝑁 topological space) is a pair (𝑋, 𝜏𝑁 ) where 𝑋 is a nonempty set and 𝜏𝑁 is a family of
𝐼𝑉𝑁 sets on 𝑋 satisfying the following axioms:
1.
2.
3.
0𝑁 , 1𝑁 ∈ 𝜏𝑁
𝐴, 𝐵 ∈ 𝜏𝑁 ⇒ 𝐴 ∩ 𝐵 ∈ 𝜏𝑁
𝐴𝑖 ∈ 𝜏𝑁 , 𝑖 ∈ 𝐼 ⇒∪𝑖∈𝐼 𝐴𝑖 ∈ 𝜏𝑁
𝜏𝑁 is called an interval valued neutrosophic topology on 𝑋. 𝜏𝑁 members are called interval valued
neutrosophic open sets (In Short 𝐼𝑉𝑁 open sets).
Example 3.2 Assume that 𝑋 = {𝑎, 𝑏}. Here 𝑎 is denoted by quality of Computers, 𝑏 is denoted by
Price of Computers. The value of 𝑎 and 𝑏 are in [0,1]. These are collected from some domain
expects questionnaire; their choices could be degree of excellence, degree of indeterminacy, degree
of poorness. The 𝐼𝑉𝑁 set are
T.Nanthini and A.Pushpalatha, Interval valued Neutosophic Topological Pushpalatha Space
Neutrosophic Sets and Systems, Vol. 32, 2020
55
0𝑁 = 〈[0,0], [1,1], [1,1]〉, 1𝑁 = 〈[1,1], [0,0], [0,0]〉, 𝐴 = 〈
〈
([0.1,0.4],[0.2,0.7],[0.4,0.6]) ([0.6,0.8],[0.2,0.3],[0.2,0.3])
([0.1,0.3],[0.3,0.8],[0.5,0.8]) ([0.2,0.7],[0.4,0.8],[0.3,0.7])
,
𝑎
(𝑋, 𝜏𝑁 ) is called an 𝐼𝑉𝑁𝑇𝑆.
([0.4,0.7],[0.5,0.7],[0.4,0.9]) ([0.2,0.3],[0.4,0.5],[0.7,0.9])
,
𝑎
〉, 𝐵 =
𝑏
〉 ,. 𝜏𝑁 = {0𝑁 , 1𝑁 , 𝐴, 𝐵} is called an 𝐼𝑉𝑁 topology on 𝑋 .
𝑏
Example3.3 Let 𝑋 = {𝑎, 𝑏} and the 𝐼𝑉𝑁 sets are
𝐶=〈
,
𝑎
〉
𝑏
𝐷=〈
,
([0.5,0.8],[0.3,0.5],[0.2,0.7]) ([0.5,0.7],[0.1,0.5],[0.3,0.7])
,
𝑎
𝑏
𝜏𝑁 = {0𝑁 , 1𝑁 , 𝐶, 𝐷} is called an 𝐼𝑉𝑁 topology on 𝑋. (𝑋, 𝜏𝑁 ) is called an 𝐼𝑉𝑁 topological space.
〉
.
Theorem 3.4 Let {𝜏𝑁𝑖 : 𝑖 ∈ 𝐼} be a family of 𝐼𝑉𝑁 topologies of 𝐼𝑉𝑁 sets on 𝑋. Then ∩𝑖 {𝜏𝑁𝑖 : 𝑖 ∈ 𝐼} is
also an 𝐼𝑉𝑁 topology of 𝐼𝑉𝑁 sets on 𝑋.
Proof: (i) 0𝑁 , 1𝑁 ∈ 𝜏𝑁𝑖 for each 𝑖 ∈ 𝐼, Hence
of 𝐼𝑉𝑁 sets where
0 N ,1N Ni .(ii) Let {𝐴𝑖 : 𝑖 ∈ 𝐼} be a arbitrary family
iI
Ai Ni for each 𝑖 ∈ 𝐼.Then for each 𝑖 ∈ 𝐼. Ai Ni for 𝑖 ∈ 𝐼 and since for
iI
each 𝑖 ∈ 𝐼, 𝜏𝑁𝑖 is a 𝐼𝑉𝑁 topology, Therefore
A
i
iI
Ni
for each 𝑖 ∈ 𝐼. Hence
A
iI
i
iI
Ni
But union of 𝐼𝑉𝑁 topologies as seen in the following example need not be an 𝐼𝑉𝑁 topology.
Example: 3.5 In example 3.2 and 3.3 the families 𝜏𝑁1 = {0𝑁 , 1𝑁 , 𝐴, 𝐵} and 𝜏𝑁2 = {0𝑁 , 1𝑁 , 𝐶, 𝐷} are
𝐼𝑉𝑁 topologies in 𝑋 . For 𝑋 , however their union 𝜏𝑁1 ∪ 𝜏𝑁2 = {0𝑁 , 1𝑁 , 𝐴, 𝐵, 𝐶, 𝐷} is not a 𝐼𝑉𝑁
topology.
Definition 3.6 Let (𝑋, 𝜏𝑁 ) be an 𝐼𝑉𝑁 topological space. An 𝐼𝑉𝑁 set 𝐴 of 𝑋 is called an interval
valued neutrosophic closed set (in short 𝐼𝑉𝑁 -closed set) if its complement 𝐴𝑐 is an 𝐼𝑉𝑁 open set
in 𝜏𝑁 .
Example 3.7 Let us consider the Example 3.2, the 𝐼𝑉𝑁 closed sets in (𝑋, 𝜏𝑁 )
〈
([0.4,0.6],[0.3,0.8],[0.1,0.4]) ([0.2,0.3],[0.7,0.8],[0.6,0.8])
,
𝑎
𝑏
〉, 𝐵𝑐 = 〈
([0.5,0.8],[0.2,0.7],[0.1,0.3]) ([0.3,0.7],[0.2,0.6],[0.2,0.7])
and 1𝑐𝑁 = 0𝑁 are the 𝐼𝑉𝑁 – closed sets in (𝑋, 𝜏𝑁 ).
,
𝑎
𝑏
are 𝐴𝑐 =
〉, 0𝑐𝑁 = 1𝑁
Theorem 3.8 Let (𝑋, 𝜏𝑁 ) be an 𝐼𝑉𝑁 topological space. Then (i) 0𝑁 , 1𝑁 are 𝐼𝑉𝑁 – closed sets. (ii)
Arbitrary intersection of 𝐼𝑉𝑁 – closed sets is 𝐼𝑉𝑁 – closed set. (iii) Finite union of 𝐼𝑉𝑁 – closed sets
is 𝐼𝑉𝑁 – closed set.
Proof: (i) since 0𝑁 , 1𝑁 ∈ 𝜏𝑁 , 0𝑐𝑁 = 1𝑁 and 1𝑐𝑁 = 0𝑁 , therefore 0𝑐𝑁 and 1𝑐𝑁 are 𝐼𝑉𝑁 – closed sets. (ii)
Let {𝐴𝑖 : 𝑖 ∈ 𝐼} be an arbitrary family of 𝐼𝑉𝑁 – closed sets in (𝑋, 𝜏𝑁 ) and let
c
c
A Ai Ai and Ac N for each 𝑖 ∈ 𝐼, hence
iI
iI
c
A
iI
i
c
A Ai Now
iI
N , therefore 𝐴𝑐 ∈ 𝜏𝑁 .
Thus 𝐴 is an 𝐼𝑉𝑁– closed set. (iii) Let {𝐴𝑘 : 𝑘 = 1,2, … … … 𝑛𝑛} be a family of 𝐼𝑉𝑁 – closed set in
T.Nanthini and A.Pushpalatha, Interval valued Neutosophic Topological Pushpalatha Space
Neutrosophic Sets and Systems, Vol. 32, 2020
(𝑋, 𝜏𝑁 ) and let G
n
, so
c
k
A
k 1
n
A
k
k 1
56
c
n
n
c
Ak Akc and Ak N for k 1,2,......... n
k 1
k 1
. Now G
c
N . Hence G c N , thus G is 𝐼𝑉𝑁 – closed set.
Definition 3.9 Let both (𝑋, 𝜏𝑁1 ) and (𝑋, 𝜏𝑁2 ) be two 𝐼𝑉𝑁𝑇𝑆. If each 𝐴 ∈ 𝜏𝑁2 implies 𝐴 ∈ 𝜏𝑁1 , then
𝜏𝑁1 is called interval valued neutrosophic finer topology than 𝜏𝑁2 and 𝜏𝑁2 is called interval valued
neutrosophic coarser topology than 𝜏𝑁1
Example 3.10 Let 𝑋 = {𝑎, 𝑏} and 𝐼𝑉𝑁 sets are 𝐴 = 〈
〈
〈
([0.3,0.7],[0.4,0.6],[0.3,0.8]) ([0.1,0.7],[0.3,0.8],[0.2,0.6])
,
𝑎
〉, 𝐶 = 〈
𝑏
([0.3,0.7],[0.4,0.6],[0.3,0.8]) ([0.1,0.7],[0.3,0.8],[0.4,0.7])
,
𝑎
([0.5,0.7],[0.3,0.6],[0.2,0.8]) ([0.4,0.6],[0.3,0.5],[0.4,0.7])
𝑎
,
𝑏
([0.5,0.7],[0.3,0.6],[0.2,0.8]) ([0.4,0.7],[0.3,0.5],[0.2,0.6])
𝑎
,
𝑏
〉, 𝐵 =
〉, 𝐷 =
〉. Let 𝜏𝑁1 = {0𝑁 , 1𝑁 , 𝐴, 𝐵, 𝐶, 𝐷} and 𝜏𝑁2 = {0𝑁 , 1𝑁 , 𝐴, 𝐶} be
𝑏
an 𝐼𝑉𝑁 topologies on 𝑋 and let (𝑋, 𝜏𝑁1 ) and (𝑋, 𝜏𝑁2 )be a 𝐼𝑉𝑁 topological spaces. If 𝜏𝑁1 is 𝐼𝑉𝑁
finer topology than 𝜏𝑁2 and𝜏𝑁2 is 𝐼𝑉𝑁 coarser topology than 𝜏𝑁1
Definition 3.11 Let (𝑋, 𝜏𝑁 ) be a 𝐼𝑉𝑁 topological space. A subcollection 𝔅 of 𝜏𝑁 is said to be base
of 𝜏𝑁 if every element of 𝜏𝑁 can be expressed as the arbitray 𝐼𝑉𝑁 union of some elements of
𝔅, then 𝔅 is called an 𝐼𝑉𝑁 basis for the 𝐼𝑉𝑁 topology 𝜏𝑁 .
Example 3.12 In Example 3.10, for the 𝐼𝑉𝑁 topology 𝜏𝑁1 = {0𝑁 , 1𝑁 , 𝐴, 𝐵, 𝐶, 𝐷}. The sub collection
𝔅 = {0𝑁 , 1𝑁 , 𝐴, 𝐵, 𝐶} of 𝛲(𝑋) is a 𝐼𝑉𝑁 basis for the 𝐼𝑉𝑁 topology 𝜏𝑁1 .
Definition 3.13 Let (𝑋, 𝜏𝑁 ) be a 𝐼𝑉𝑁 topological space and 𝐴 ∈ 𝐼𝑉𝑁𝑠(𝑋), the interior and closure of
𝐴 is denoted by 𝐼𝑉𝑁 𝐼𝑛𝑡(𝐴) and 𝐼𝑉𝑁 𝐶𝑙(𝐴) are defined as
𝐼𝑉𝑁 𝐼𝑛𝑡(𝐴) =∪ {𝐺 ∈ 𝜏𝑁 : 𝐺 ⊆ 𝐴} , 𝐼𝑉𝑁 𝐶𝑙(𝐴) =∩ {𝐾 ∈ 𝜏𝑁𝑐 : 𝐴 ⊆ 𝐾}
Example 3.14 Let us take an Example 3.3 and consider an 𝐼𝑉𝑁 set
𝐸=〈
([0.4,0.6],[0.4,0.7],[0.2,0.7]) ([0.3,0.5],[0.3,0.6],[0.3,0.5])
𝑎
,
𝑏
〉. Now 𝐼𝑉𝑁 𝐼𝑛𝑡(𝐸) = 0𝑁 and 𝐼𝑉𝑁 𝐶𝑙(𝐸) = 1𝑁 .
Theorem 3.15 Let (𝑋, 𝜏𝑁 ) be a 𝐼𝑉𝑁 topological space and 𝐴, 𝐵 ∈ 𝐼𝑉𝑁𝑠(𝑋) then the following
properties holds:
(i) 𝐼𝑉𝑁 𝐼𝑛𝑡(𝐴) ⊆ 𝐴
(ii) 𝐴 ⊆ 𝐵 ⇒ 𝐼𝑉𝑁 𝐼𝑛𝑡(𝐴) ⊆ 𝐼𝑉𝑁 𝐼𝑛𝑡(𝐵)
(iii) 𝐼𝑉𝑁 𝐼𝑛𝑡(𝐴) ∈ 𝜏𝑁
(iv) 𝐴 ∈ 𝜏𝑁 𝑖𝑓𝑓 𝐼𝑉𝑁 𝐼𝑛𝑡(𝐴) = 𝐴
(v) 𝐼𝑉𝑁 𝐼𝑛𝑡(𝐼𝑉𝑁 𝐼𝑛𝑡(𝐴)) = 𝐼𝑉𝑁 𝐼𝑛𝑡(𝐴)
(vi) 𝐼𝑉𝑁 𝐼𝑛𝑡(0𝑁 ) = 0𝑁 , 𝐼𝑉𝑁 𝐼𝑛𝑡(1𝑁 ) = 1𝑁
Proof:
(i) Straight forward.
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57
(ii) 𝐴 ⊆ 𝐵 ⇒ All of the 𝐼𝑉𝑁 open sets in 𝐴 that are also in 𝐵. Both 𝐼𝑉𝑁 open sets included
in 𝐴 also included in 𝐵.
𝑖𝑒. , {𝐾 ∈ 𝜏𝑁 : 𝐾 ⊆ 𝐴} ⊆ {𝐺 ∈ 𝜏𝑁 : 𝐺 ⊆ 𝐵}. 𝑖𝑒. ,∪ {𝐾 ∈ 𝜏𝑁 : 𝐾 ⊆ 𝐴} ⊆∪
{𝐺 ∈ 𝜏𝑁 : 𝐺 ⊆ 𝐵}. 𝑖𝑒. , 𝐼𝑉𝑁 𝐼𝑛𝑡(𝐴) ⊆ 𝐼𝑉𝑁𝐼𝑛𝑡(𝐵).
(iii) 𝐼𝑉𝑁 𝐼𝑛𝑡(𝐴) =∪ {𝐾 ∈ 𝜏𝑁 : 𝐾 ⊆ 𝐴}. It is clear that ∪ {𝐾 ∈ 𝜏𝑁 : 𝐾 ⊆ 𝐴} ∈ 𝜏𝑁 . So, 𝐼𝑉𝑁 𝐼𝑛𝑡(𝐴) ∈ 𝜏𝑁 .
(iv) Let 𝐴 ∈ 𝜏𝑁 , then by(i), 𝐼𝑉𝑁 𝐼𝑛𝑡(𝐴) ⊆ 𝐴. Now since 𝐴 ∈ 𝜏𝑁 and 𝐼𝑉𝑁 𝐼𝑛𝑡(𝐴) ⊆ 𝐴. Therefore
𝐴 ⊆∪ {𝐺 ∈ 𝜏𝑁 : 𝐺 ⊆ 𝐴} = 𝐼𝑉𝑁 𝐼𝑛𝑡(𝐴), 𝐴 ⊆ 𝐼𝑁𝑉 𝐼𝑛𝑡(𝐴). Thus 𝐼𝑉𝑁 𝐼𝑛𝑡(𝐴) = 𝐴. Conversely, let
𝐼𝑉𝑁 𝐼𝑛𝑡(𝐴) = 𝐴. Since by (iii), 𝐼𝑉𝑁 𝐼𝑛𝑡(𝐴) ∈ 𝜏𝑁 . Therefore 𝐴 ∈ 𝜏𝑁 .
(v) By (iii), 𝐼𝑉𝑁 𝐼𝑛𝑡(𝐴) ∈ 𝜏𝑁 . Therefore by (iv), 𝐼𝑉𝑁 𝐼𝑛𝑡(𝐼𝑉𝑁 𝐼𝑛𝑡(𝐴)) = 𝐼𝑉𝑁 𝐼𝑛𝑡(𝐴).
(vi) We know that 0𝑁 , 1𝑁 ∈ 𝜏𝑁 , by (iv), 𝐼𝑉𝑁 𝐼𝑛𝑡(0𝑁 ) = 0𝑁 , 𝐼𝑉𝑁 𝐼𝑛𝑡(1𝑁 ) = 1𝑁 .
Theorem 3.16 Let (𝑋, 𝜏𝑁 ) be a 𝐼𝑉𝑁𝑇𝑆 and 𝐴, 𝐵 ∈ 𝐼𝑉𝑁𝑠(𝑋) then possess the following properties:
(i) 𝐴 ⊆ 𝐼𝑉𝑁 𝐶𝑙(𝐴)
(ii) 𝐴 ⊆ 𝐵 ⇒ 𝐼𝑉𝑁 𝐶𝑙(𝐴) ⊆ 𝐼𝑉𝑁 𝐶𝑙(𝐵)
𝑐
(iii) (𝐼𝑉𝑁 𝐶𝑙(𝐴)) ∈ 𝜏𝑁
(iv) 𝐴𝑐 ∈ 𝜏𝑁 𝑖𝑓𝑓 𝐼𝑉𝑁 𝐶𝑙(𝐴) = 𝐴
(v) 𝐼𝑉𝑁 𝐶𝑙(𝐼𝑉𝑁 𝐶𝑙(𝐴)) = 𝐼𝑉𝑁 𝐶𝑙(𝐴)
(vi) 𝐼𝑉𝑁 𝐶𝑙(0𝑁 ) = 0𝑁 , 𝐼𝑉𝑁 𝐶𝑙(1𝑁 ) = 1𝑁
Proof:
Straight forward.
Theorem 3.17 Let (𝑋, 𝜏𝑁 ) be a 𝐼𝑉𝑁 topological space and 𝐴, 𝐵 ∈ 𝐼𝑉𝑁𝑠(𝑋)then hold the following
properties:
(i) 𝐼𝑉𝑁 𝐼𝑛𝑡(𝐴 ∩ 𝐵) = 𝐼𝑉𝑁 𝐼𝑛𝑡(𝐴) ∩ 𝐼𝑉𝑁 𝐼𝑛𝑡(𝐵)
(ii) 𝐼𝑉𝑁 𝐼𝑛𝑡(𝐴 ∪ 𝐵) ⊇ 𝐼𝑉𝑁𝐼𝑛𝑡(𝐴) ∪ 𝐼𝑉𝑁𝐼𝑛𝑡(𝐵)
(iii) 𝐼𝑉𝑁 𝐶𝑙(𝐴 ∪ 𝐵) = 𝐼𝑉𝑁 𝐶𝑙(𝐴) ∪ 𝐼𝑉𝑁𝐼𝑛𝑡(𝐵)
(iv) 𝐼𝑉𝑁 𝐶𝑙(𝐴 ∩ 𝐵) ⊆ 𝐼𝑉𝑁 𝐶𝑙(𝐴) ∩ 𝐼𝑉𝑁 𝐼𝑛𝑡(𝐵)
(v) (𝐼𝑉𝑁 𝐼𝑛𝑡(𝐴))𝑐 = 𝐼𝑉𝑁 𝐶𝑙 (𝐴𝑐 )
(vi) (𝐼𝑉𝑁 𝐶𝑙(𝐴))𝑐 = 𝐼𝑉𝑁 𝐼𝑛𝑡(𝐴𝑐 )
Proof:
(i) By Theorem 3.15(i), 𝐼𝑉𝑁 𝐼𝑛𝑡 (𝐴) ⊆ 𝐴 and 𝐼𝑉𝑁 𝐼𝑛𝑡(𝐵) ⊆ 𝐵. Thus 𝐼𝑉𝑁 𝐼𝑛𝑡(𝐴) ∩ 𝐼𝑉𝑁 𝐼𝑛𝑡(𝐵) ⊆
𝐴 ∩ 𝐵. Hence 𝐼𝑉𝑁 𝐼𝑛𝑡(𝐴) ∩ 𝐼𝑉𝑁 𝐼𝑛𝑡(𝐵) ⊆ 𝐼𝑉𝑁𝐼𝑛𝑡(𝐴 ∩ 𝐵) -----------(1)
Again since 𝐴 ∩ 𝐵 ⊆ 𝐴 , by Theorem 3.15(ii). 𝐼𝑉𝑁 𝐼𝑛𝑡(𝐴 ∩ 𝐵) ⊆ 𝐼𝑉𝑁𝐼𝑛𝑡(𝐴) . Similarly
𝐼𝑉𝑁 𝐼𝑛𝑡(𝐴 ∩ 𝐵) ⊆ 𝐼𝑉𝑁𝐼𝑛𝑡(𝐵).
Hence 𝐼𝑉𝑁 𝐼𝑛𝑡(𝐴 ∩ 𝐵) ⊆ 𝐼𝑉𝑁𝐼𝑛𝑡(𝐴) ∩ 𝐼𝑉𝑁𝐼𝑛𝑡(𝐵) --------(2) from (1) and (2) we get,
𝐼𝑉𝑁 𝐼𝑛𝑡(𝐴 ∩ 𝐵) = 𝐼𝑉𝑁𝐼𝑛𝑡(𝐴) ∩ 𝐼𝑉𝑁 𝐼𝑛𝑡(𝐵).
(ii) Since 𝐴 ⊆ 𝐴 ∪ 𝐵. 𝐼𝑉𝑁 𝐼𝑛𝑡(𝐴) ⊆ 𝐼𝑉𝑁𝐼𝑛𝑡(𝐴 ∪ 𝐵) by Theorem 3.15(ii). Similarly 𝐼𝑉𝑁 𝐼𝑛𝑡(𝐵) ⊆
𝐼𝑉𝑁𝐼𝑛𝑡(𝐴 ∪ 𝐵). Hence 𝐼𝑉𝑁 𝐼𝑛𝑡(𝐴) ∪ 𝐼𝑉𝑁 𝐼𝑛𝑡(𝐵) ⊆ 𝐼𝑉𝑁𝐼𝑛𝑡(𝐴 ∪ 𝐵).
(iii) By Theorem 3.16(i), 𝐴 ⊆ 𝐼𝑉𝑁 𝐶𝑙(𝐴) and 𝐵 ⊆ 𝐼𝑉𝑁 𝐶𝑙(𝐵) . Thus 𝐴 ∪ 𝐵 ⊆ 𝐼𝑉𝑁 𝐶𝑙(𝐴) ∪
𝐼𝑉𝑁𝐶𝑙(𝐵), 𝐼𝑉𝑁 𝐶𝑙(𝐴 ∪ 𝐵) ⊆ 𝐼𝑉𝑁 𝐶𝑙(𝐴) ∪ 𝐼𝑉𝑁𝐶𝑙(𝐵)-----------(1)
T.Nanthini and A.Pushpalatha, Interval valued Neutosophic Topological Pushpalatha Space
Neutrosophic Sets and Systems, Vol. 32, 2020
58
Again since 𝐴 ⊆ 𝐴 ∪ 𝐵 , by Theorem 3.16(ii). 𝐼𝑉𝑁 𝐶𝑙(𝐴) ⊆ 𝐼𝑉𝑁𝐶𝑙(𝐴 ∪ 𝐵) . Similarly
𝐼𝑉𝑁 𝐶𝑙(𝐵) ⊆ 𝐼𝑉𝑁𝐶𝑙(𝐴 ∪ 𝐵). Hence 𝐼𝑉𝑁 𝐶𝑙(𝐴) ∪ 𝐼𝑉𝑁 𝐶𝑙(𝐵) ⊆ 𝐼𝑉𝑁𝐶𝑙(𝐴 ∪ 𝐵)------(2) from (1)
and (2) we get 𝐼𝑉𝑁 𝐶𝑙(𝐴) ∪ 𝐼𝑉𝑁 𝐶𝑙(𝐵) = 𝐼𝑉𝑁𝐶𝑙(𝐴 ∪ 𝐵).
(iv) Since 𝐴 ∩ 𝐵 ⊆ 𝐴 , 𝐼𝑉𝑁 𝐶𝑙(𝐴 ∩ 𝐵) ⊆ 𝐼𝑉𝑁 𝐶𝑙(𝐴) by Theorem 3.16(ii), Similarly, 𝐼𝑉𝑁 𝐶𝑙(𝐴 ∩
𝐵) ⊆ 𝐼𝑉𝑁 𝐶𝑙(𝐵). Hence 𝐼𝑉𝑁 𝐶𝑙(𝐴 ∩ 𝐵) ⊆ 𝐼𝑉𝑁 𝐶𝑙(𝐴) ∩ 𝐼𝑉𝑁𝐶𝑙(𝐵).
(v) {𝐼𝑉𝑁 𝐼𝑛𝑡(𝐴)}𝑐 = [∪ {𝐺 ∈ 𝜏𝑁 : 𝐺 ⊆ 𝐴}]𝑐 =∩ {𝐺 ∈ 𝜏𝑁𝑐 : 𝐴𝑐 ⊆ 𝐺},
{𝐼𝑉𝑁 𝐼𝑛𝑡(𝐴)}𝑐 = 𝐼𝑉𝑁 𝐶𝑙(𝐴)𝑐 .
(vi) {𝐼𝑉𝑁 𝐶𝑙(𝐴)}𝑐 = [∩ {𝐺 ∈ 𝜏𝑁𝑐 : 𝐴𝑐 ⊆ 𝐺}]𝑐 =∪ {𝐺 ∈ 𝜏𝑁 : 𝐺 ⊆ 𝐴},
{𝐼𝑉𝑁 𝐶𝑙(𝐴)}𝑐 = 𝐼𝑉𝑁 𝐼𝑛𝑡(𝐴)𝑐 .
In theorem 3.17((ii) and (iv)), the equality does not hold. Let us display this by an example below
Example 3.18 Let 𝑋 = {𝑎, 𝑏} and the 𝐼𝑉𝑁 sets are 0𝑁 =<
1𝑁 =<
𝐵 =<
[1,1],[0,0],[0,0] [1,1],[0,0],[0,0]
,
𝑎
𝑏
>; 𝐴 =<
,
𝑏
consider two 𝐼𝑉𝑁 sets 𝐶 =<
,
𝑎
,
𝑏
𝑏
𝑏
>;
>;
>, 𝜏𝑁 = {0𝑁 , 1𝑁 , 𝐴, 𝐵} is an 𝐼𝑉𝑁 topology on 𝑋. Let us
[0.1,0.4],[0.3,0.7],[0.5,0.6] [0.4,0.8],[0.2,0.3],[0.2,0.3]
[0,0.3],[0.2,0.8],[0.4,0.9] [0.6,0.7],[0.3,0.6],[0.2,0.5]
𝑎
,
𝑎
[0.1,0.4],[0.2,0.7],[0.4,0.6] [0.6,0.8],[0.2,0.3],[0.2,0.3]
[0.1,0.3],[0.3,0.8],[0.5,0.8] [0.2,0.7],[0.4,0.8],[0.3,0.7]
𝑎
[0,0],[0,0],[1,1] [0,0],[0,0],[1,1]
,
𝑎
>; Now 𝐶 ∪ 𝐷 =<
𝑏
> and 𝐷 =<
[0.1,0.4],[0.2,0.7],[0.4,0.6] [0.6,0.8],[0.2,0.3],[0.2,0.3]
,
𝑎
𝑏
𝐴; 𝐼𝑉𝑁 𝐼𝑛𝑡(𝐶 ∪ 𝐷) = 𝐼𝑉𝑁 𝐼𝑛𝑡(𝐴) = 𝐴; 𝐼𝑉𝑁 𝐼𝑛𝑡(𝐶) = 0𝑁 , 𝐼𝑉𝑁 𝐼𝑛𝑡(𝐷) = 0𝑁 , 𝐼𝑉𝑁 𝐼𝑛𝑡(𝐶) ∪
𝐼𝑉𝑁 𝐼𝑛𝑡(𝐷) = 0𝑁 ;
Therefore 𝐼𝑉𝑁 𝐼𝑛𝑡(𝐶 ∪ 𝐷) ≠ 𝐼𝑉𝑁 𝐼𝑛𝑡(𝐶) ∪ 𝐼𝑉𝑁 𝐼𝑛𝑡(𝐷).
By Theorem 3.17(v), 𝐼𝑉𝑁 𝐶𝑙(𝐶)𝑐 = (𝐼𝑉𝑁 𝐼𝑛𝑡(𝐶))𝑐 = (0𝑁 )𝑐 = 1𝑁 ,
(0𝑁 )𝑐 = 1𝑁 ,
>=
𝐼𝑉𝑁 𝐶𝑙(𝐷)𝑐 = (𝐼𝑉𝑁 𝐼𝑛𝑡(𝐷))𝑐 =
𝐼𝑉𝑁 𝐶𝑙(𝐶 𝑐 ∩ 𝐷𝑐 ) = 𝐼𝑉𝑁 𝐶𝑙((𝐶 ∪ 𝐷)𝑐 ) = (𝐼𝑉𝑁 𝐼𝑛𝑡(𝐶 ∪
𝐼𝑉𝑁 𝐼𝑛𝑡(𝐶) ∩ 𝐼𝑉𝑁 𝐼𝑛𝑡(𝐷) = 1𝑁 ;
𝐷))𝑐 = (𝐼𝑉𝑁 𝐼𝑛𝑡(𝐴))𝑐 = 𝐴𝑐 ; 𝐼𝑉𝑁 𝐶𝑙(𝐶 𝑐 ∩ 𝐷𝑐 ) ≠ 𝐼𝑉𝑁 𝐶𝑙(𝐶 𝑐 ) ∪ 𝐼𝑉𝑁 𝐶𝑙(𝐷𝑐 ).
4. Interval Valued Neutrosophic Subspace Topology
In this section we present, along with some examples and findings, the definition of interval
valued neutrosophic subspace topology.
Theorem 4.1 Let (𝑋, 𝜏𝑁 ) be a 𝐼𝑉𝑁 topological space on 𝑋 and 𝑌 ∈ 𝑃(𝑋). Then the collection 𝜏𝑁𝑌 =
{𝑌 ∩ 𝐺: 𝐺 ∈ 𝜏𝑁 } is a 𝐼𝑉𝑁 topology on 𝑋.
Proof:
(i)
(ii)
Since 0𝑁 , 1𝑁 ∈ 𝜏𝑁 , therefore 𝑌 ∩ 0𝑁 = 0𝑁 ∈ 𝜏𝑁𝑌 and 𝑌 ∩ 1𝑁 = 𝑌 ∈ 𝜏𝑁𝑌 .
Let 𝑌𝑘 ∈ 𝜏𝑁𝑌 , ∀ 𝑘 ∈ 𝐼 , then 𝑌𝑘 = 𝑌 ∩ 𝐺𝑘 where 𝐺𝑘 ∈ 𝜏𝑁 for each 𝑘 ∈ 𝐼 . Now
Y (Y G ) Y G
kI
k
kI
k
kI
k
NY . Since
G
kI
k
N as each 𝐺𝑘 ∈ 𝜏𝑁 .
T.Nanthini and A.Pushpalatha, Interval valued Neutosophic Topological Pushpalatha Space
Neutrosophic Sets and Systems, Vol. 32, 2020
(iii)
59
Let 𝑌1 , 𝑌2 ∈ 𝜏𝑁𝑌 , 𝑌1 = 𝑌 ∩ 𝐺1 and 𝑌2 = 𝑌 ∩ 𝐺2 where 𝐺1 , 𝐺2 ∈ 𝜏𝑁 Now 𝑌1 ∩ 𝑌2 =
(𝑌 ∩ 𝐺1 ) ∩ (𝑌 ∩ 𝐺2 ) = 𝑌 ∩ (𝐺1 ∩ 𝐺2 ) ∈ 𝜏𝑁𝑌 , since 𝐺1 ∩ 𝐺2 ∈ 𝜏𝑁 as 𝐺1 , 𝐺2 ∈ 𝜏𝑁 .
Definition 4.2 Let (𝑋, 𝜏𝑁 ) be an 𝐼𝑉𝑁 topological space on 𝑋 and 𝑌 is a interval values neutrosophic
subset (In short 𝐼𝑉𝑁 subset) of 𝑋, the collection 𝜏𝑁𝑌 = {𝑌 ∩ 𝐺: 𝐺 ∈ 𝜏𝑁 } is called interval valued
neutrosophic subspace (In short 𝐼𝑉𝑁 subspace) of 𝑌. 𝑌 is called 𝐼𝑉𝑁 subspace of 𝑋.
Example 4.3 Let us consider the 𝐼𝑉𝑁 topology 𝜏𝑁1 = {0𝑁 , 1𝑁 , 𝐴, 𝐵, 𝐶, 𝐷} as in Example 3.10 and an
𝐼𝑉𝑁 set 𝑌 =<
[0.4,0.6],[0.3,0.7],[0.1,0.5] [0.5,0.9],[0.4,1],[0.2,0.6]
,
𝑎
𝐺1 = 𝑌 ∩ 𝐴, 𝐺1 =<
𝐺2 = 𝑌 ∩ 𝐵, 𝐺2 =<
𝐺3 = 𝑌 ∩ 𝐶, 𝐺3 =<
𝐺4 = 𝑌 ∩ 𝐷, 𝐺4 =<
𝑏
>, 0𝑁 = 𝑌 ∩ 0𝑁 = 0𝑁 ;
[0.4,0.6],[0.3,0.7],[0.2,0.8] [0.4,0.6],[0.4,1],[0.4,0.7]
𝑎
,
𝑏
[0.3,0.6],[0.4,0.7],[0.3,0.8] [0.1,0.7],[0.4,1],[0.2,0.6]
𝑎
,
𝑏
[0.4,0.6],[0.3,0.7],[0.2,0.8] [0.4,0.7],[0.4,1],[0.2,0.6]
𝑎
,
𝑏
[0.3,0.6],[0.4,0.7],[0.3,0.8] [0.1,0.7],[0.4,1],[0.4,0.7]
𝑎
,
𝑏
>;
>;
>;
>; Then 𝜏𝑁𝑌 = {0𝑁 , 1𝑁 , 𝐺1 , 𝐺2 , 𝐺3 } is an 𝐼𝑉𝑁
subspace topology for 𝜏𝑁1 and 𝜏𝑁𝑌 is called 𝐼𝑉𝑁 subspace of (𝑋, 𝜏𝑁1 ).
Theorem 4.4 Let (𝑋, 𝜏𝑁 ) be an 𝐼𝑉𝑁 topological space, 𝔅 be an 𝐼𝑉𝑁 basis for 𝜏𝑁 and 𝑌 is an 𝐼𝑉𝑁
subset of 𝑋. Then the family 𝔅𝑌 = {𝑌 ∩ 𝐺: 𝐺 ∈ 𝔅} is an 𝐼𝑉𝑁 basis for 𝐼𝑉𝑁 subspace topology 𝜏𝑁𝑌 .
Proof:
Let 𝑈 ∈ 𝜏𝑁𝑌 be arbitrary, then there exists an 𝐼𝑉𝑁 set 𝐺 ∈ 𝜏𝑁 such that 𝑈 = 𝑌 ∩ 𝐺. Since 𝔅 is an
𝐼𝑉𝑁 basis for 𝜏𝑁 , therefore there exists a sub collection {𝜒𝑖 : 𝑖 ∈ 𝐼} of 𝔅 such that
G i . Now,
iI
U Y G i Y i . Since 𝑌 ∩ 𝜒𝑖 ∈ 𝐵𝑌 , therefore 𝐵𝑌 is a 𝐼𝑉𝑁 basis for an 𝐼𝑉𝑁
iI
iI
subspace topology 𝜏𝑁𝑌 .
5. Conclusion
The concept of interval valued neutrosophic topological space, interval valued neutrosophic
interior and interval valued neutrosophic closure of an interval valued neutrosophic sets were
introduced. An interval valued neutrosophic subspace topology of interval valued neutrosophic sets
are also introduced. The newly introduced’ Interval Valued Neutrophic Topological Spaces’ is a
stronger version of ‘Neutrosophic Topological Spaces’.
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Received: Nov 15, 2019. Accepted: Mar 17, 2020.
T.Nanthini and A.Pushpalatha, Interval valued Neutosophic Topological Pushpalatha Space
Neutrosophic Sets and Systems, Vol. 32, 2020
University of New Mexico
Arithmetic and Geometric Operators of Pentagonal Neutrosophic
Number and its Application in Mobile Communication Service
Based MCGDM Problem
Avishek Chakraborty 1,2*,Baisakhi Banik 3,Sankar Prasad Mondal4 and Shariful Alam2
1Department
of Basic Science, Narula Institute of Technology, Agarpara, Kolkata-700109, India.
Email- avishek.chakraborty@nit.ac.in
2Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, Howrah-711103, India.
Email- salam50in@yahoo.co.in
3Department of Basic Science, Techno Engineering College (Banipur), Habra, W.Bengal-743233, India.
Email-baisakhibanik14@gmail.com.
4Department of Applied Science, MaulanaAbulKalam Azad University of Technology, West Bengal, Haringhata-721249,
Nadia,WestBengal, India. Email-sankar.res07@gmail.com
*Correspondence: avishek.chakraborty@nit.ac.in.
Abstract: In this paper, the theory of pentagonal neutrosophic number has been studied in a
disjunctive frame of reference. Moreover, the dependency and independency of the membership
functions for the pentagonal neutrosophic number are also classified here. Additionally, the
development of a new score function and its computation have been formulated in distinct rational
perspectives. Further, weighted arithmetic averaging operator and weighted geometric averaging
operator in the pentagonal neutrosophic environment are introduced here using an influx of
different logical & innovative thought. Also, a multi-criteria group decision-making problem
(MCGDM) in a mobile communication system is formulated in this paper as an application in the
pentagonal neutrosophic arena. Lastly, the sensitivity analysis portion reflects the variation of this
noble work.
Keywords: Pentagonal neutrosophic number, Weighted arithmetic and geometric averaging
operator, Score functions, MCGDM.
1. Introduction
1.1 Neutrosophic Sets
Handling the notion of vagueness and uncertainty concepts, fuzzy set theory is a dominant field,
was first presented by Zadeh [1] in his paper (1965).Vagueness theory has a salient feature for
solving engineering and statistical problem very lucidly. It has a great impact on social-science,
networking, decision making and numerous kinds of realistic problems. On the basis of ideas of
Zadeh’s research paper, Atanassov [2] invented the prodigious concept of intuitionistic fuzzy set
where he precisely interpreted the idea of membership as well as non membership function very
aptly. Further, researchers developed the formulation of triangular [3], trapezoidal [4], pentagonal
[5] fuzzy numbers in uncertainty arena. Also, Liu & Yuan [6] established the concept of the
triangular intuitionistic fuzzy set;Ye [7] put forth the basic idea of trapezoidal intuitionistic fuzzy set
Avishek Chakraborty, Baisakhi Banik ,Sankar Prasad Mondal and SharifulAlam; Arithmetic and Geometric Operators of
Pentagonal Neutrosophic Number and its Application in Mobile Communication Based MCGDM Problem
Neutrosophic Sets and Systems, Vol. 32, 2020
62
in the research field. Naturally, the question arises, how can we evolve the idea of uncertainty
concepts in mathematical modelling? Researchers have invented disjunctive kinds of methodologies
to define elaborately the concepts and have suggested some new kinds of ambivalent parameters. To
deal with those kinds of problems, the decision-makers’ choice varies in different areas. F.
Smarandache [8] in 1998 generated the concept of a neutrosophic set having three different
integrants namely, (i) truthiness, (ii) indeterminacies, and (iii) falseness. Each and every
characteristic of the neutrosophic set are very pertinent factors in our real-life models. Later, Wang et
al. [9] proceeded with the idea of a single typed neutrosophic set, which is very productive to sort
out the solution of any complicated kind of problem. Recently, Chakraborty et al. [10, 11]
conceptualized the dynamic idea of triangular and trapezoidal neutrosophic numbers in the
research domain and applied it in different real-life problem. Also, Maity et al. [12] built the
perception of ranking and defuzzification in a completely different type of attributes. To handle
human decision making procedure on the basis of positive and negative sides, Bosc and Pivert [13]
cultivated the notion of bipolarity. With that continuation, Lee [14] elucidated the perception of
bipolar fuzzy set in their research article. Further, Kang and Kang [15] broadened this concept into
semi-groups and group structures field. As research proceeded, Deli et al. [16] germinated the idea
of a bipolar neutrosophic set and used it as an implication to a decision-making related problem.
Broumi et al. [17] produced the idea of bipolar neutrosophic graph theory and, subsequently, Ali
and Smarandache [18] put forth the concept of the uncertain complex neutrosophic set. Chakraborty
[19] introduced the triangular bipolar number in different aspects. In succession; Wang et al. [20]
also introduced the idea of operators in a bipolar neutrosophic set and applied it in a
decision-making problem. The multi-criteria decision making (MCDM) problem is a supreme
interest to the researchers who deal with the decision scientific analysis. Presently, it is more
acceptable in such issues where a group of criteria is utilized. Such cases of problems relating to
multi-criteria group decision making (MCGDM) have shown its fervent influence. Also MCDM has
broad applications in disjunctive fields under various uncertainty contexts.We can find many
applications and development of neutrosophic theory in multi-criteria decision making problem in
the literature surveys presented in [21–25], graph theory [26-30], optimization techniques [31-33] etc.
In this current era, Basset [34-40] presented some worthy articles related to neutrosophic sphere and
applied
it
in
many
different
well-known
fields.Also,
K.Mondal [41,42]
successfully
applied the notion of neutrosophic number
in faculty recruitment MCDM problem in education purpose. Recently, the viewpoint of
plithogenic set is being constructed by Abdel [43] and it has an immense influential motivation in
impreciseness field in various sphere of research field. Also, Chakraborty [44] developed the
conception of cylindrical neutrosophic number is minimal tree problem.
Neutrosophic concept is very fruitful & vibrant in a realistic approach in the recent research field. R.
Helen [45] first germinated the idea of the pentagonal fuzzy number then Christi [46] utilized the
conception of pentagonal fuzzy number into pentagonal intuitionistic number and skillfully applied
it to solve a transportation problem. Additionally, Chakraborty [47, 48] put forward the notion of
pentagonal neutrosophicnumber and its different and disjunctive representation in transportation
problem and graph-theoretical research arenas. Subsequently, Karaaslan [51-56] put forth some
Avishek Chakraborty, Baisakhi Banik ,Sankar Prasad Mondal and SharifulAlam; Arithmetic and Geometric Operators of
Pentagonal Neutrosophic Number and its Application in Mobile Communication Based MCGDM Problem
Neutrosophic Sets and Systems, Vol. 32, 2020
63
innovative idea on multi-attribute decision making in neutrosophic domain. Also, Karaaslan [57-61]
presented the notion of soft set theory with the appropriate justification of neutrosophic fuzzy
number. Recently, Broumi et.al [62-66] manifested the conception of the graph-theoretical shortest
path problem under neutrosophic environment. Further, Broumi [67] implemented the concept of
neutrosophic membership functions using MATLAB programming. A few works [68-71] are also
established recently, based on impreciseness domain.
In this article, we mainly focus on the different representation of pentagonal neutrosophic number
and its dependency, independency portions. We generate a new logical score function for
crispification of pentagonal neutrosophic number. Additionally, we introduce two different logical
operators namely i) pentagonal neutrosophic weighted arithmetic averaging operator (PNWAA), ii)
pentagonal neutrosophic weighted geometric averaging operator (PNWGA) and established its
theoretical developments along with its different properties. Also, we discussed the utility of these
operators in real-life problems. Later, we consider a mobile communication based MCGDM problem
in neutrosophic domain and solve it using the established two operators & score function.Sensitivity
analysis of this problem is also addressed here which will show distinct results in different aspects.
Finally, comparison analysis is performed here with the established methods which give an
important impact in the research arena. This noble thought will help us to solve a plethora of daily
life problems in uncertainty arena.
1.2 Motivation for the study
With the advent of vagueness theory the arena of numerous realistic mathematical modeling,
engineering structural issues, multi-criteria problem have immensely achieved a productive and
impulsive effect.Naturally it is very intriguing to the researchers that if someone sheds light on the
pentagonal neutrosophic number then what will be it in the form of linearity and its classification?
Based on this perception we impose three components on a pentagonal neutrosophic number i.e.
truthiness, indeterminacy and falsity. Proceeding with the PNNWAA and PNNWGA operators and
based on the score function of pentagonal neutrosophic numbers, an MCGDM method is built up
and some interesting and worthy conclusions are tried to extract from this research article.
1.3 Novelties of the work
Recently, researchers are utmost persevere to develop theories connecting neutrosophic field and
constantly try to generate its distinct application in various sphere of neutrosophic arena. However,
justifying all the perspectives related to pentagonal neutrosophic fuzzy set theory; numerous
theories and problems are yet to be solved. In this research article our ultimate objective is to shed
light some unfocussed points in the pentagonal domain.
(1) Classification of Pentagonal Neutrosophic Number.
(2) Illustrative demonstration of aggregation operations and geometric operations on
Pentagonal Neutrosophic Number’s.
(3) Proposed new score function and its utility.
(4)
Execute the idea of Pentagonal Neutrosophic Number’s in MCGDM problem.
2. Preliminaries
Avishek Chakraborty, Baisakhi Banik ,Sankar Prasad Mondal and SharifulAlam; Arithmetic and Geometric Operators of
Pentagonal Neutrosophic Number and its Application in Mobile Communication Based MCGDM Problem
Neutrosophic Sets and Systems, Vol. 32, 2020
64
̃ = {(β, αà (β)): βϵA, αà (β)ϵ[0,1]} which is
Definition 2.1: Fuzzy Set: [1] Let 𝐴̃ be a set such that A
normally denoted by this ordered pair(β, αà (β)), here β is a member of the set𝐴and 0 ≤ αà (β) ≤ 1,
̃ is called a fuzzy set.
then set A
Definition 2.2: Neutrosophic Set:[8] A set 𝐴̃𝑁𝑒𝑢 in the domain of discourse 𝐴, most commonly
stated as ∈ is called a neutrosophic set if 𝐴̃𝑁𝑒𝑢 = {〈∈; [𝜑𝐴̃ (∈), 𝛾𝐴̃ (∈), 𝛿𝐴̃ (∈)]〉 ⋮∈
𝑁𝑒𝑢
𝑁𝑒𝑢
𝑁𝑒𝑢
(∈): 𝐴 →] − 0,1 + [ symbolizes the index of confidence, 𝛾𝐴̃
(∈): 𝐴 →] − 0,1 +
𝜖𝐴} ,where 𝜑𝐴̃
𝑁𝑒𝑢
𝑁𝑒𝑢
(∈): 𝐴 →] − 0,1 + [symbolizes the degree of falseness
[symbolizes the index of uncertainty and 𝛿𝐴̃
𝑁𝑒𝑢
(∈), 𝛾𝐴̃
(∈), 𝛿𝐴̃
(∈)] satisfies the in the equation
in the decision making procedure. Where,[𝜑𝐴̃
𝑁𝑒𝑢
𝑁𝑒𝑢
𝑁𝑒𝑢
(∈) ≤ 3 +.
(∈) + 𝛾𝐴̃
(∈) + 𝛿𝐴̃
−0 ≤ 𝜑𝐴̃
𝑁𝑒𝑢
𝑁𝑒𝑢
𝑁𝑒𝑢
Definition 2.3: Single Typed Neutrosophic Number: [8]Single Typed Neutrosophic Number (𝑛̃) is
denoted as 𝑛̃ = 〈[(𝑢1 , 𝑣1 , 𝑤 1 , 𝑥 1 ); 𝛼], [(𝑢2 , 𝑣 2 , 𝑤 2 , 𝑥 2 ); 𝛽], [(𝑢3 , 𝑣 3 , 𝑤 3 , 𝑥 3 ); 𝛾]〉where𝛼, 𝛽, 𝛾 ∈ [0,1],
where(𝜑𝑛̃ ): ℝ → [0, 𝛼], (𝛾𝑛̃ ): ℝ → [𝛽, 1] and (𝛿𝑛̃ ): ℝ → [𝛾, 1] is given as:
1
1
€𝑛𝑙
̃ (∈) 𝑤ℎ𝑒𝑛 𝑢 ≤∈≤ 𝑣
1
1
𝜑𝑛̃ (∈) = { 𝛼 𝑤ℎ𝑒𝑛 𝑣 1 ≤∈≤ 𝑤 1 ,
€𝑛𝑢
̃ (∈) 𝑤ℎ𝑒𝑛 𝑤 ≤∈≤ 𝑥
0
𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
𝛾𝑛𝑙
̃ (∈) 𝑤ℎ𝑒𝑛 𝑢 2 ≤∈≤ 𝑣 2
𝛽 𝑤ℎ𝑒𝑛 𝑣 2 ≤∈≤ 𝑤 2
£𝑛̃ (∈) = {
2
2
𝛾𝑛𝑢
̃ (∈) 𝑤ℎ𝑒𝑛 𝑤 ≤∈≤ 𝑥
𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
1
and
3
3
µ𝑛𝑙
̃ (∈) 𝑤ℎ𝑒𝑛 𝑢 ≤∈≤ 𝑣
𝛾 𝑤ℎ𝑒𝑛 𝑣 3 ≤∈≤ 𝑤 3
𝛿𝑛̃ (∈) = {
3
3
µ𝑛𝑢
̃ (∈) 𝑤ℎ𝑒𝑛 𝑤 ≤∈≤ 𝑥
𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
1
Definition 2.4: Single-Valued Neutrosophic Set:[9] A Neutrosophic set in the definition 2.2 is
𝐴̃𝑁𝑒𝑢 said to be a Single-Valued Neutrosophic Set (𝐴̃𝑁𝑒𝑢 ) if ∈ is a single-valued independent
variable. 𝐴̃𝑁𝑒𝑢 = {〈∈; [𝛼𝐴̃ (∈), 𝛽𝐴̃ (∈), γ𝐴̃ (∈)]〉 ⋮∈ 𝜖𝐴} , where 𝛼𝐴̃ (∈), 𝛽𝐴̃ (∈)&γ𝐴̃ (∈)
𝑁𝑒𝑢
𝑁𝑒𝑢
𝑁𝑒𝑢
𝑁𝑒𝑢
𝑁𝑒𝑢
𝑁𝑒𝑢
̃ is named
denote the idea of accuracy, ambiguity and falsity membership functions respectively.𝑆𝑛𝑆
̃ is a subset of R by satisfying the following criterion:
as neut-convex, which implies that 𝑆𝑛𝑆
i.
ii.
iii.
𝛼𝐴̃𝑁𝑒𝑢 〈𝛿𝑎1 + (1 − 𝛿)𝑎2 〉 ≥ 𝑚𝑖𝑛〈𝛼𝐴̃𝑁𝑒𝑢 (𝑎1 ), 𝛼𝐴̃𝑁𝑒𝑢 (𝑎2 )〉
𝛽𝐴̃𝑁𝑒𝑢 〈𝛿𝑎1 + (1 − 𝛿)𝑎2 〉 ≤ 𝑚𝑎𝑥〈𝛽𝐴̃𝑁𝑒𝑢 (𝑎1 ), 𝛽𝐴̃𝑁𝑒𝑢 (𝑎2 )〉
𝛾𝐴̃𝑁𝑒𝑢 〈𝛿𝑎1 + (1 − 𝛿)𝑎2 〉 ≤ 𝑚𝑎𝑥〈𝛾𝐴̃𝑁𝑒𝑢 (𝑎1 ), 𝛾𝐴̃𝑁𝑒𝑢 (𝑎2 )〉
where𝑎1 &𝑎2 𝜖ℝ𝑎𝑛d𝛿𝜖[0,1]
3. Single Type Linear Pentagonal Neutrosophic Number:
In this section we introduce different type single type linear pentagonal neutrosophic number. For
the help of the researchers we pictorially draw the following block diagram as follows:
Avishek Chakraborty, Baisakhi Banik ,Sankar Prasad Mondal and SharifulAlam; Arithmetic and Geometric Operators of
Pentagonal Neutrosophic Number and its Application in Mobile Communication Based MCGDM Problem
Neutrosophic Sets and Systems, Vol. 32, 2020
65
Figure 3.1: Block diagram for a different type of uncertain numbers and their categories
Definition 3.1: Single-Valued Pentagonal Neutrosophic Number: [47]A Single-Valued Pentagonal
̃
̃
Neutrosophic Number (𝑁
𝑃𝑒𝑛 ) is defined as𝑁𝑃𝑒𝑛 =
〈[(ℎ1 , ℎ2 , ℎ3 , ℎ4 , ℎ5 ); 𝜋], [(ℎ1 , ℎ2 , ℎ3 , ℎ4 , ℎ5 ); 𝜇], [(ℎ1 , ℎ2 , ℎ3 , ℎ4 , ℎ5 ); 𝜎]〉, where𝜋, 𝜇, 𝜎 ∈ [0,1]. The accuracy
membership function(𝜏𝑆̃ ): ℝ → [0, 𝜋], the ambiguity membership function (𝜗𝑆̃ ): ℝ → [𝜌, 1] and the
falsity membership function (𝜀𝑆̃ ): ℝ → [𝜎, 1] are defined by:
𝜋(𝑥−ℎ1 )
(ℎ2 −ℎ1 )
𝜋(𝑥−ℎ2 )
(ℎ3 −ℎ2 )
𝜋
𝜏𝑆̃ (𝑥) =
𝜋(ℎ4 −𝑥)
{
and
(ℎ4 −ℎ3 )
𝜋(ℎ4 −𝑥)
(ℎ5 −ℎ4 )
𝑤ℎ𝑒𝑛 ℎ1 ≤ 𝑥 ≤ ℎ2
𝑤ℎ𝑒𝑛 ℎ2 ≤ 𝑥 < ℎ3
𝑤ℎ𝑒𝑛 𝑥 = ℎ3
𝑤ℎ𝑒𝑛 ℎ3 < 𝑥 ≤ ℎ4
,
𝑤ℎ𝑒𝑛 ℎ4 ≤ 𝑥 ≤ ℎ5
0
𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
ℎ2 − 𝑥 + 𝜇(𝑥 − ℎ1 )
𝑤ℎ𝑒𝑛 ℎ1 ≤ 𝑥 ≤ ℎ2
(ℎ2 − ℎ1 )
ℎ3 − 𝑥 + 𝜇(𝑥 − ℎ2 )
𝑤ℎ𝑒𝑛 ℎ2 ≤ 𝑥 < ℎ3
(ℎ3 − ℎ2 )
𝑤ℎ𝑒𝑛 𝑥 = ℎ3
𝜇
𝜗𝑆̃ (𝑥) =
𝑥 − ℎ3 + 𝜇(ℎ4 − 𝑥)
𝑤ℎ𝑒𝑛 ℎ3 < 𝑥 ≤ ℎ4
(ℎ4 − ℎ3 )
𝑥 − ℎ4 + 𝜇(ℎ5 − 𝑥)
𝑤ℎ𝑒𝑛 ℎ4 ≤ 𝑥 ≤ ℎ5
(ℎ5 − ℎ4 )
{
1
𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
Avishek Chakraborty, Baisakhi Banik ,Sankar Prasad Mondal and SharifulAlam; Arithmetic and Geometric Operators of
Pentagonal Neutrosophic Number and its Application in Mobile Communication Based MCGDM Problem
Neutrosophic Sets and Systems, Vol. 32, 2020
66
ℎ2 − 𝑥 + 𝜎(𝑥 − ℎ1 )
(ℎ2 − ℎ1 )
ℎ3 − 𝑥 + 𝜎(𝑥 − ℎ2 )
(ℎ3 − ℎ2 )
𝜎
𝜀𝑆̃ (𝑥) =
𝑥 − ℎ3 + 𝜎(ℎ4 − 𝑥)
(ℎ4 − ℎ3 )
𝑥 − ℎ4 + 𝜎(ℎ5 − 𝑥)
(ℎ5 − ℎ4 )
{
1
4. Proposed Score Function:
𝑤ℎ𝑒𝑛 ℎ1 ≤ 𝑥 ≤ ℎ2
𝑤ℎ𝑒𝑛 ℎ2 ≤ 𝑥 < ℎ3
𝑤ℎ𝑒𝑛 𝑥 = ℎ3
𝑤ℎ𝑒𝑛 ℎ3 < 𝑥 ≤ ℎ4
𝑤ℎ𝑒𝑛 ℎ4 ≤ 𝑥 ≤ ℎ5
𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
Score function of a pentagonal neutrosophic number entirely depends on the value of truth
membership indicator degree, falsity membership indicator degree and uncertainty membership
indicator degree. The necessity of score function is to draw a comparison or transfer a pentagonal
neutrosophic fuzzy number into a crisp number. In this section, we will generate a score function as
follows.For any Pentagonal Single typed Neutrosophic Number (PSNN)
𝐴̃𝑃𝑡 = (𝑠1 , 𝑠2 , 𝑠3 , 𝑠4 , 𝑠5 ; 𝜋, 𝜇, 𝜎)
We define the score function as
𝑆𝑃𝑡 =
Here, 𝑆𝑃𝑡 belongs to [0,1].
1
(𝑠 + 𝑠2 + 𝑠3 + 𝑠4 + 𝑠5 ) × (2 + 𝜋 − 𝜎 − 𝜇)
15 1
4.1Relationship between any two pentagonal neutrosophic fuzzy numbers:
Let us consider any two pentagonal neutrosophic fuzzy number defined as follows
𝐴̃𝑃𝑡1 = (𝑠𝑃𝑡11 , 𝑠𝑃𝑡12 , 𝑠𝑃𝑡13 , 𝑠𝑃𝑡14 , 𝑠𝑃𝑡15 ; 𝜋𝑃𝑡1 , 𝜇𝑃𝑡1 , 𝜎𝑃𝑡1 )and𝐴̃𝑃𝑡2 =
(𝑠𝑃𝑡21 , 𝑠𝑃𝑡22 , 𝑠𝑃𝑡23 , 𝑠𝑃𝑡24 , 𝑠𝑃𝑡25 ; 𝜋𝑃𝑡2 , 𝜇𝑃𝑡2 , 𝜎𝑃𝑡2 )
The score function for the are
and
𝑆𝑃𝑡1 =
1
(𝑠
+ 𝑠𝑃𝑡12 + 𝑠𝑃𝑡13 + 𝑠𝑃𝑡14 + 𝑠𝑃𝑡15 ) × (2 + 𝜋𝑃𝑡1 − 𝜎𝑃𝑡1 − 𝜇𝑃𝑡1)
15 𝑃𝑡11
𝑆𝑃𝑡2 =
1
(𝑠
+ 𝑠𝑃𝑡22 + 𝑠𝑃𝑡23 + 𝑠𝑃𝑡24 + 𝑠𝑃𝑡25 ) × (2 + 𝜋𝑃𝑡2 − 𝜎𝑃𝑡2 − 𝜇𝑃𝑡2)
15 𝑃𝑡21
Then we can say the following
1) 𝐴̃𝑃𝑡1 > 𝐴̃𝑃𝑡2 if 𝑆𝑃𝑡1 > 𝑆𝑃𝑡2
2) 𝐴̃𝑃𝑡1 < 𝐴̃𝑃𝑡2if 𝑆𝑃𝑡1 < 𝑆𝑃𝑡2
3) 𝐴̃𝑃𝑡1 = 𝐴𝑃𝑡2 if 𝑆𝑃𝑡1 = 𝑆𝑃𝑡2
Table 4.1: Numerical Examples
̃ 𝑷𝒕 )
Pentagonal Neutrosophic Number (𝑨
̃ 𝑷𝒕𝟏 =< (𝟎. 𝟐, 𝟎. 𝟑, 𝟎. 𝟒, 𝟎. 𝟓, 𝟎. 𝟔; 𝟎. 𝟒, 𝟎. 𝟓, 𝟎. 𝟔) >
𝑨
̃ 𝑷𝒕𝟐 =< (𝟎. 𝟑𝟓, 𝟎. 𝟒, 𝟎. 𝟒𝟓, 𝟎. 𝟓, 𝟎. 𝟓𝟓; 𝟎. 𝟔, 𝟎. 𝟑, 𝟎. 𝟒) >
𝑨
̃ 𝑷𝒕𝟑 =< (𝟎. 𝟏𝟓, 𝟎. 𝟐, 𝟎. 𝟐𝟓, 𝟎. 𝟑, 𝟎. 𝟑𝟓; 𝟎. 𝟔, 𝟎. 𝟒, 𝟎. 𝟓) >
𝑨
̃ 𝑷𝒕𝟒 =< (𝟎. 𝟕, 𝟎. 𝟕𝟓, 𝟎. 𝟖, 𝟎. 𝟖𝟓, 𝟎. 𝟗; 𝟎. 𝟑, 𝟎. 𝟐, 𝟎. 𝟔) >
𝑨
Score Value (𝑺𝑷𝒕 )
0.17333
0.28500
0.14167
Ordering
𝐴𝑃𝑡4 > 𝐴𝑃𝑡2 > 𝐴𝑃𝑡1 > 𝐴𝑃𝑡3
0.40000
Avishek Chakraborty, Baisakhi Banik ,Sankar Prasad Mondal and SharifulAlam; Arithmetic and Geometric Operators of
Pentagonal Neutrosophic Number and its Application in Mobile Communication Based MCGDM Problem
Neutrosophic Sets and Systems, Vol. 32, 2020
67
4.1 Basic Operations for pentagonal neutrosophic fuzzy number:
̃2 = 〈(𝑑1 , 𝑑2 , 𝑑3 , 𝑑4 , 𝑑5 ); 𝜋𝑝̃2 , 𝜇𝑝̃2 , 𝜎𝑝̃2 〉 be two IPFNs and
Let ̃
𝑝1 = 〈(𝑐1 , 𝑐2 , 𝑐3 , 𝑐4 , 𝑐5 ); 𝜋𝑝̃1 , 𝜇𝑝̃1 , 𝜎𝑝̃1 〉 and 𝑝
𝛼 ≥ 0. Then the following operational relations hold:
4.1.1 Addition:
𝑝1 + 𝑝
̃
̃2 = 〈(𝑐1 + 𝑑1 , 𝑐2 + 𝑑2 , 𝑐3 + 𝑑3 , 𝑐4 + 𝑑4 , 𝑐5 + 𝑑5 ); 𝜋𝑝̃1 + 𝜋𝑝̃2 − 𝜋𝑝̃1 𝜋𝑝̃2 , 𝜇𝑝̃1 𝜇𝑝̃2 , 𝜎𝑝̃1 𝜎𝑝̃2 〉
4.1.2Multipliction:
𝑝1 ̃2 = 〈(𝑐1 𝑑1 , 𝑐2 𝑑2 , 𝑐3 𝑑3 , 𝑐4 𝑑4 , 𝑐5 𝑑5 ); 𝜋𝑝̃1 𝜋𝑝̃2 , 𝜇𝑝̃1 +, 𝜇𝑝̃2 − 𝜇𝑝̃1 𝜇𝑝̃2 , 𝜎𝑝̃1 + 𝜎𝑝̃2 − 𝜎𝑝̃1 𝜎𝑝̃2 〉
̃𝑝
4.1.3 Multiplication by scalar:
4.1.4 Power:
𝛼𝑝
̃1 = 〈(𝛼𝑐1 , 𝛼𝑐2 , 𝛼𝑐3 , 𝛼𝑐4 , 𝛼𝑐5 ); 1 − (1 − 𝜋𝑝̃1 )𝛼 , 𝜇𝑝̃1 𝛼 , 𝜎𝑝̃1 𝛼 )〉
̃1 𝛼 = 〈(𝑐1 𝛼 , 𝑐2 𝛼 , 𝑐3 𝛼 , 𝑐4 𝛼 , 𝑐5 𝛼 ); 𝜋𝑝̃1 𝛼 , (1 − 𝜇𝑝̃1 )𝛼 , (1 − 𝜎𝑝̃1 )𝛼 〉
𝑝
5. Arithmetic and Geometric Operators:
5.1 Two weighted aggregation operators of Pentagonal Neutrosophic Numbers
Aggregation operators are such pertinent tool for aggregating information to tactfully handle the
decision making procedure, this section generates a brief understanding between two weighted
aggregation operators to aggregate PNNs as a generalization of the weighted aggregation operators
for PNNs, which are broadly and aptly used in decision making.
5.1.1 Pentagonal neutrosophic weighted arithmetic averaging operator
Let 𝑝̃𝑗 = 〈(𝑐𝑗1 , 𝑐𝑗2 , 𝑐𝑗3 , 𝑐𝑗4 , 𝑐𝑗5 ); 𝜋𝑝̃1 , 𝜇𝑝̃1 , 𝜎𝑝̃1 〉(𝑗 = 1,2,3, … . , 𝑛) be a set of PNNs, then a PNWAA
operator is defined as follows:
𝑃𝑁𝑊𝐴𝐴 (𝑝̃1 , 𝑝̃2 , … . , 𝑝̃𝑛 ) = ∑𝑛𝑗=1 𝜔𝑗 𝑝̃𝑗
(5.1)
where𝜔𝑗 is the weight of 𝑝̃𝑗 (𝑗 = 1,2,3, … . , 𝑛) such that 𝜔𝑗 > 0 and ∑𝑛𝑗=1 𝜔𝑗 = 1.
In accordance with the result ofSection 4.1 and equation (5.1) we can introduce the following
theorems:
Theorem 5.1. Let 𝑝̃𝑗 = 〈(𝑐𝑗1 , 𝑐𝑗2 , 𝑐𝑗3 , 𝑐𝑗4 , 𝑐𝑗5 ); 𝜋𝑝̃1 , 𝜇𝑝̃1 , 𝜎𝑝̃1 〉(𝑗 = 1,2,3, … . , 𝑛) be a set of PNNs, then
according to Section 4.1 and equation (5.1) we can give the following PNWAA operator
𝑛
𝑃𝑁𝑊𝐴𝐴 (𝑝̃1 , 𝑝̃2 , … . , 𝑝̃𝑛 ) = ∑ 𝜔𝑗 𝑝̃𝑗
𝑗=1
=〈(∑𝑛𝐽=1 𝜔𝑗 𝑐𝑗1, ∑𝑛𝐽=1 𝜔𝑗 𝑐𝑗2, ∑𝑛𝐽=1 𝜔𝑗 𝑐𝑗3, ∑𝑛𝐽=1 𝜔𝑗 𝑐𝑗4, ∑𝑛𝐽=1 𝜔𝑗 𝑐𝑗5 ); 1 − ∏𝑛𝑗=1(1 −
𝜋𝑝̃𝑗 )𝜔𝑗 , ∏𝑛𝑗=1 𝜇𝑝̃𝑗 𝜔𝑗 , ∏𝑛𝑗=1 𝜎𝑝̃𝑗 𝜔𝑗 〉
Where 𝜔𝑗 is the weight of 𝑝̃𝑗 (𝑗 = 1,2,3, … . , 𝑛) such that 𝜔𝑗 > 0 and ∑𝑛𝑗=1 𝜔𝑗 = 1.
Theorem 5.1 can be proved with the help of mathematical induction.
Proof: When 𝑛 = 2 then,
𝜔1 𝑝̃1 = 〈(𝜔1 𝑐11 , 𝜔1 𝑐12 , 𝜔1 𝑐13 , 𝜔1 𝑐14 , 𝜔1 𝑐15 ); 1 − (1 − 𝜋𝑝̃1 )
𝜔2
𝜔1
𝜎𝑝̃2 𝜔2 , 𝜇𝑝̃1 𝜔1 , 𝜎𝑝̃1 𝜔1 〉
and 𝜔2 𝑝̃2 = 〈(𝜔2 𝑐21 , 𝜔2 𝑐22 , 𝜔2 𝑐23 , 𝜔2 𝑐24 , 𝜔2 𝑐25 ); 1 − (1 − 𝜋𝑝̃2 ) , 𝜇𝑝̃2 𝜔2 , 𝜎𝑝̃2 𝜔2 〉
Avishek Chakraborty, Baisakhi Banik ,Sankar Prasad Mondal and SharifulAlam; Arithmetic and Geometric Operators of
Pentagonal Neutrosophic Number and its Application in Mobile Communication Based MCGDM Problem
Neutrosophic Sets and Systems, Vol. 32, 2020
Thus, 𝑃𝑁𝑊𝐴𝐴 (𝑝
̃,
̃)
̃1 + 𝜔2 𝑝̃2
1 𝑝
2 = 𝜔1 𝑝
68
=〈(𝜔1 𝑐11 + 𝜔2 𝑐21 + 𝜔1 𝑐12 + 𝜔2 𝑐22 + 𝜔1 𝑐13 + 𝜔2 𝑐23 + 𝜔1 𝑐14 + 𝜔2 𝑐24 + 𝜔1 𝑐15 + 𝜔2 𝑐25 ); 1 −
(1 − 𝜋𝑝̃1 )
𝜔1
+ 1 − (1 − 𝜋𝑝̃2 )
𝜔2
(1 − (1 − 𝜋𝑝̃1 )
𝜔1
)(1 − (1 − 𝜋𝑝̃2 )
When applying 𝑛 = 𝑘, by applying equation (5.1) , we get
𝑃𝑁𝑊𝐴𝐴 (𝑝̃1 , 𝑝̃2 , … . , 𝑝̃𝑘 ) = ∑𝑘𝑗=1 𝜔𝑗 𝑝̃𝑗 (5.2)
𝜔2
), 𝜇𝑝̃1 𝜔1 𝜇𝑝̃2 𝜔2 , 𝜎𝑝̃1 𝜔1 𝜎𝑝̃2 𝜔2 〉
= 〈(∑𝑘𝐽=1 𝜔𝑗 𝑐𝑗1, ∑𝑘𝐽=1 𝜔𝑗 𝑐𝑗2, ∑𝑘𝐽=1 𝜔𝑗 𝑐𝑗3, ∑𝑘𝐽=1 𝜔𝑗 𝑐𝑗4, ∑𝑘𝐽=1 𝜔𝑗 𝑐𝑗5 ); 1 − ∏𝑘𝑗=1(1 −
𝜋𝑝̃𝑗 )𝜔𝑗 , ∏𝑘𝑗=1 𝜇𝑝̃𝑗 𝜔𝑗 , ∏𝑘𝑗=1 𝜎𝑝̃𝑗 𝜔𝑗 〉
When 𝑛 = 𝑘 + 1, by applying equations (5.1) and (5.2) we can yield
̃𝑗 (5.3)
𝑃𝑁𝑊𝐴𝐴 (𝑝̃1 , 𝑝̃2 , … . , 𝑝̃𝑘+1 ) = ∑𝑘+1
𝑗=1 𝜔𝑗 𝑝
𝑘
𝑘+1
𝑘+1
𝑘+1
𝑘+1
= 〈(∑𝑘+1
̃𝑗 )
𝐽=1 𝜔𝑗 𝑐𝑗1, ∑𝐽=1 𝜔𝑗 𝑐𝑗2, ∑𝐽=1 𝜔𝑗 𝑐𝑗3, ∑𝐽=1 𝜔𝑗 𝑐𝑗4, ∑𝐽=1 𝜔𝑗 𝑐𝑗5 ); 1 − ∏𝑗=1 (1 − 𝜋𝑝
(1 − 𝜋𝑝̃
)
𝑘+1
𝜔𝑘+1
𝜔𝑗
𝜔𝑗
, ∏𝑘+1
, ∏𝑘+1
̃𝑗 〉
̃𝑗
𝑗=1 𝜎𝑝
𝑗=1 𝜇𝑝
𝜔𝑗
+1−
𝑘+1
𝑘+1
𝑘+1
𝑘+1
𝑘+1
= 〈(∑𝑘+1
𝐽=1 𝜔𝑗 𝑐𝑗1, ∑𝐽=1 𝜔𝑗 𝑐𝑗2, ∑𝐽=1 𝜔𝑗 𝑐𝑗3, ∑𝐽=1 𝜔𝑗 𝑐𝑗4, ∑𝐽=1 𝜔𝑗 𝑐𝑗5 ); 1 − ∏𝑗=1 (1 −
𝜔𝑗
𝜔𝑗
, ∏𝑘+1
𝜋𝑝̃𝑗 )𝜔𝑗 , ∏𝑘+1
̃𝑗
̃𝑗 〉
𝑗=1 𝜇𝑝
𝑗=1 𝜎𝑝
This completes the proof.
Obviously, the 𝑃𝑁𝑊𝐴𝐴 operator satisfies the following properties:
i) Idempotency: Let 𝑝̃𝑗 (𝑗 = 1,2,3, … . , 𝑛) be a set of PNNs. If 𝑝̃𝑗 (𝑗 = 1,2,3, … . , 𝑛) is equal , i.e. 𝑝̃𝑗 = 𝑝̃
for j=1,2,3,….,n then 𝑃𝑁𝑊𝐴𝐴 (𝑝̃1 , 𝑝̃2 , … . , 𝑝̃𝑛 ) = 𝑝̃.
Proof: Since 𝑝̃𝑗 = 𝑝̃ for 𝑗 = 1,2,3, … . , 𝑛 we have,
𝑛
𝑃𝑁𝑊𝐴𝐴 (𝑝̃1 , 𝑝̃2 , … . , 𝑝̃𝑛 ) = ∑ 𝜔𝑗 𝑝̃𝑗
𝑗=1
=〈(∑𝑛𝐽=1 𝜔𝑗 𝑐𝑗1, ∑𝑛𝐽=1 𝜔𝑗 𝑐𝑗2, ∑𝑛𝐽=1 𝜔𝑗 𝑐𝑗3, ∑𝑛𝐽=1 𝜔𝑗 𝑐𝑗4, ∑𝑛𝐽=1 𝜔𝑗 𝑐𝑗5 ); 1 − ∏𝑛𝑗=1(1 −
𝜋𝑝̃𝑗 )𝜔𝑗 , ∏𝑛𝑗=1 𝜇𝑝̃𝑗 𝜔𝑗 , ∏𝑛𝑗=1 𝜎𝑝̃𝑗 𝜔𝑗 〉
= 〈(𝑐1 ∑𝑛𝐽=1 𝜔𝑗 , 𝑐2 ∑𝑛𝐽=1 𝜔𝑗 , 𝑐3 ∑𝑛𝐽=1 𝜔𝑗 , 𝑐4 ∑𝑛𝐽=1 𝜔𝑗 , 𝑐5 ∑𝑛𝐽=1 𝜔𝑗 ); (1 − (1 −
𝑛
𝑛
𝑛
𝜋𝑝̃𝑗 ))∑𝐽=1 𝜔𝑗 , 𝜇𝑝̃𝑗 ∑𝐽=1 𝜔𝑗 , 𝜎𝑝̃𝑗 ∑𝐽=1 𝜔𝑗 〉
=〈( 𝑐1 , 𝑐2 , 𝑐3 , 𝑐4 , 𝑐5 ); 1 − (1 − 𝜋𝑝̃1 ), 𝜇𝑝̃1 , 𝜎𝑝̃1 〉 = 𝑝̃
ii) Boundedness: Let 𝑝̃𝑗 (j=1,2,3,….,n) be a set of PNNs and let
𝑝̃ − = 〈(𝑚𝑖𝑛𝑗 (𝑐𝑗1 ), 𝑚𝑖𝑛𝑗 (𝑐𝑗2 ), 𝑚𝑖𝑛𝑗 (𝑐𝑗3 ), 𝑚𝑖𝑛𝑗 (𝑐𝑗4 ), 𝑚𝑖𝑛𝑗 (𝑐𝑗5 )) ; 𝑚𝑖𝑛𝑗 (𝜋𝑝̃𝑗 ) , 𝑚𝑎𝑥𝑗 (𝜇𝑝̃𝑗 ) , 𝑚𝑎𝑥𝑗 (𝜎𝑝̃𝑗 )〉
and
〈(𝑚𝑎𝑥𝑗 (𝑐𝑗1 ), 𝑚𝑎𝑥𝑗 (𝑐𝑗2 ), 𝑚𝑎𝑥𝑗 (𝑐𝑗3 ), 𝑚𝑎𝑥𝑗 (𝑐𝑗4 ), 𝑚𝑎𝑥𝑗 (𝑐𝑗5 )) ; 𝑚𝑎𝑥𝑗 (𝜋𝑝̃𝑗 ) , 𝑚𝑖𝑛𝑗 (𝜇𝑝̃𝑗 ) , 𝑚𝑖𝑛𝑗 (𝜎𝑝̃𝑗 )〉
𝑝̃ + =
Then 𝑝̃ − ≤ 𝑃𝑁𝑊𝐴𝐴 (𝑝̃1 , 𝑝̃2 , … . , 𝑝̃𝑛 ) ≤ 𝑝̃ + .
Avishek Chakraborty, Baisakhi Banik ,Sankar Prasad Mondal and SharifulAlam; Arithmetic and Geometric Operators of
Pentagonal Neutrosophic Number and its Application in Mobile Communication Based MCGDM Problem
Neutrosophic Sets and Systems, Vol. 32, 2020
69
Proof: Since the minimum PNN is 𝑝̃− and the maximum is 𝑝̃+ there is 𝑝̃ − ≤ 𝑝̃𝑗 ≤ 𝑝̃+ . Thus there is
∑𝑛𝐽=1 𝜔𝑗 𝑝̃− ≤ ∑𝑛𝐽=1 𝜔𝑗 𝑝̃𝑗 ≤ ∑𝑛𝐽=1 𝜔𝑗 𝑝̃+ .According to the above property (i) there is 𝑝̃ − ≤ ∑𝑛𝐽=1 𝜔𝑗 𝑝̃𝑗 ≤
𝑝̃+ ,
i.e.,𝑝̃− ≤ PNWAA (𝑝̃1 , 𝑝̃2 , … . , 𝑝̃𝑛 ) ≤ 𝑝̃+ .
iii) Monotonicity: Let 𝑝̃𝑗 (𝑗 = 1,2,3, … . , 𝑛) be a set of PNNs. If 𝑝̃𝑗 ≤ 𝑝̃𝑗 ∗ for j= j=1,2,3,….,n, then
Proof:
Since
PNWAA (𝑝̃1 , 𝑝̃2 , … . , 𝑝̃𝑛 ) ≤ PNWAA(𝑝̃1,∗ 𝑝̃2,∗ 𝑝̃3,∗ 𝑝̃4,∗ 𝑝̃5,∗ )
𝑝̃𝑗 ≤ 𝑝̃𝑗 ∗
for
𝑗 = 𝑗 = 1,2,3, … . , 𝑛
there
is
∑𝑛𝑗=1 𝜔𝑗 𝑝̃𝑗 ≤ ∑𝑛𝑗=1 𝜔𝑗 𝑝̃𝑗 ∗
i.e.
PNWAA (𝑝̃1 , 𝑝̃2 , … . , 𝑝̃𝑛 ) ≤ PNWAA(𝑝̃1,∗ 𝑝̃2,∗ 𝑝̃3,∗ 𝑝̃4,∗ 𝑝̃5∗ ).Thus we complete the proofs of all the properties.
5.2 Pentagonal neutrosophic weighted geometric averaging operator
Let 𝑝̃𝑗 = 〈(𝑐𝑗1 , 𝑐𝑗2 , 𝑐𝑗3 , 𝑐𝑗4 , 𝑐𝑗5 ); 𝜋𝑝̃1 , 𝜇𝑝̃1 , 𝜎𝑝̃1 〉(𝑗 = 1,2,3, … . , 𝑛) be a set of PNNs, then a PNWGAA
operator is defined as follows:
𝜔𝑗
̃
𝑃𝑁𝑊𝐺𝐴 (𝑝̃1 , 𝑝̃2 , … . , 𝑝̃𝑛 ) = ∏𝑛𝑗=1 𝑝
𝑗 (5.4)
where𝜔𝑗 is the weight of 𝑝̃𝑗 (𝑗 = 1,2,3, … . , 𝑛) such that 𝜔𝑗 > 0 and ∑𝑛𝑗=1 𝜔𝑗 = 1.
Theorem 5.2. Let 𝑝̃𝑗 =< (𝑐𝑗1 , 𝑐𝑗2 , 𝑐𝑗3 , 𝑐𝑗4 , 𝑐𝑗5 ); 𝜋𝑝̃1 , 𝜇𝑝̃1 , 𝜎𝑝̃1 > (𝑗 = 1,2,3, … . , 𝑛) be a set of PNNs, then
according to Section 4.1 and equation (5.4) we can give the following PNWGA operator
𝜔𝑗
̃
(5.5)
𝑃𝑁𝑊𝐺𝐴 (𝑝̃1 , 𝑝̃2 , … . , 𝑝̃𝑛 ) = ∏𝑛𝑗=1 𝑝
𝑗
= 〈(∏𝑛𝑗=1 𝑐𝑗1
𝜔𝑗
, ∏𝑛𝑗=1 𝑐𝑗2
∏𝑛𝑗=1(1 − 𝜎𝑝̃𝑗 )𝜔𝑗 〉
𝜔𝑗
, ∏𝑛𝑗=1 𝑐𝑗3
𝜔𝑗
, ∏𝑛𝑗=1 𝑐𝑗4
𝜔𝑗
, ∏𝑛𝑗=1 𝑐𝑗5
𝜔𝑗
; ∏𝑛𝑗=1 𝜋𝑝̃𝑗 𝜔𝑗 ,1 − ∏𝑛𝑗=1(1 − 𝜇𝑝̃𝑗 )𝜔𝑗 , 1 −
where𝜔𝑗 is the weight of 𝑝̃𝑗 (𝑗 = 1,2,3, … . , 𝑛) such that 𝜔𝑗 > 0 and ∑𝑛𝑗=1 𝜔𝑗 = 1.
By the similar proof manner of Theorem 5.1 we can prove the Theorem 5.2 which is not repeated
here.
Obviously, the PNWGA operator satisfies the following properties:
i) Idempotency: Let 𝑝̃𝑗 (𝑗 = 1,2,3, … . , 𝑛) be a set of PNNs.
If 𝑝̃𝑗 (𝑗 = 1,2,3, … . , 𝑛) is equal , i.e. 𝑝̃𝑗 = 𝑝̃ for 𝑗 = 1,2,3, … . , 𝑛 then PNWGA (𝑝̃1 ,𝑝̃2 ,….,𝑝̃𝑛 ) = 𝑝̃.
ii) Boundedness: Let 𝑝̃𝑗 (𝑗 = 1,2,3, … . , 𝑛) be a set of PNNs and let
𝑝̃− =〈(𝑚𝑖𝑛𝑗 (𝑐𝑗1 ), 𝑚𝑖𝑛𝑗 (𝑐𝑗2 ), 𝑚𝑖𝑛𝑗 (𝑐𝑗3 ), 𝑚𝑖𝑛𝑗 (𝑐𝑗4 ), 𝑚𝑖𝑛𝑗 (𝑐𝑗5 ); 𝑚𝑖𝑛𝑗 (𝜋𝑝̃𝑗 ) , 𝑚𝑎𝑥𝑗 (𝜇𝑝̃𝑗 ) , 𝑚𝑎𝑥𝑗 (𝜎𝑝̃𝑗 )〉
and
𝑝̃+ =〈(𝑚𝑎𝑥𝑗 (𝑐𝑗1 ), 𝑚𝑎𝑥𝑗 (𝑐𝑗2 ), 𝑚𝑎𝑥𝑗 (𝑐𝑗3 ), 𝑚𝑎𝑥𝑗 (𝑐𝑗4 ), 𝑚𝑎𝑥𝑗 (𝑐𝑗5 ); 𝑚𝑎𝑥𝑗 (𝜋𝑝̃𝑗 ) , 𝑚𝑖𝑛𝑗 (𝜇𝑝̃𝑗 ) , 𝑚𝑖𝑛𝑗 (𝜎𝑝̃𝑗 )〉
Then 𝑝̃ − ≤ PNWGA (𝑝̃1 ,𝑝̃2 ,….,𝑝̃𝑛 ) ≤ 𝑝̃ + .
iii) Monotonicity: Let 𝑝̃𝑗 (𝑗 = 1,2,3, … . , 𝑛) be a set of PNNs. If 𝑝̃𝑗 ≤ 𝑝̃𝑗 ∗ for 𝑗 = 𝑗 = 1,2,3, … . , 𝑛,
then
PNWGA (𝑝̃1 , 𝑝̃2 , … . , 𝑝̃𝑛 ) ≤ PNWGA(𝑝̃1,∗ 𝑝̃2,∗ 𝑝̃3,∗ 𝑝̃4,∗ 𝑝̃5∗ )
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As the proofs of these properties are similar to the proofs of the above properties, so we don’t repeat
them.
6. Multi-Criteria Group Decision Making Problem in Pentagonal Neutrosophic Environment
Multi-criteria group decision-making problem is one of the reliable, logistical and mostly used topics
in this current era. The main goal of this process is to find out the best alternatives among a finite
number of distinct alternatives based on finite different attribute values. Such decision-making
program may be raised powerfully by the methods of multi-criteria group decision analysis
(MCGDA) which is extremely beneficial to produce decision counselling and offers procedure
benefits in terms of upgraded decision attributes,delivers improvised communication techniques
and enrichesresolutions of decision-makers.The execution process is not so much easy to evaluate in
the pentagonal neutrosophic environment. Using some mathematical operators, score function
technique, we developed an algorithm to tackle this MCGDM problem.
In this section, we consider a multi-criteria group decision-making problem based on mobile
communication provider services in which we need to select the best service according to different
opinions from people. The developed algorithm is described briefly as follows:
6.1 Illustration of the MCGDM problem
We consider the problem as follows:
Suppose 𝐺 = { 𝐺1 , 𝐺2 , 𝐺3 … … … . . 𝐺𝑚 } is a distinctive alternative set and 𝐻 = { 𝐻1 , 𝐻2 , 𝐻3 … … … . . 𝐻𝑛 }
is the distinctive attribute set respectively. Let 𝜔 = { 𝜔1 , 𝜔2 , 𝜔3 … … … . . 𝜔𝑛 } be the corresponding
weight set attributes where each 𝜔 ≥0 and also satisfies the relation∑𝑛𝑖=1 𝜔𝑖 = 1. Thus we consider
the set of decision-maker 𝜆 = { 𝜆1 , 𝜆2 , 𝜆3 … … … . . 𝜆𝐾 } associated with alternatives whose weight
vector is stated as Ω = {Ω1 , Ω2 , Ω3 … … … . . Ω𝑘 } where each Ω𝑖 ≥ 0 and also satisfies the
relation∑𝑘𝑖=1 Ω𝑖 = 1, this weight vector will be chosen in accordance with the decision-makers
capability of judgment, experience, innovative thinking power etc.
6.2 Normalisation Algorithm of MCGDM Problem:
Step 1: Composition of Decision Matrices
Here, we construct all decision matrices proposed by the decision maker’s choice connected with
finite alternatives and finite attribute functions. The interesting fact is that the member’s 𝑠𝑖𝑗 for each
matrix are of pentagonal neutrosophic numbers. Thus, we finalize the matrix and is given as follows:
.
𝐺1
𝐺
𝐾
2
𝑋 =
𝐺3
.
(𝐺𝑚
𝐻1
𝑘
𝑠11
𝑘
𝑠21
.
..
𝑘
𝑠𝑚1
𝐻2
𝑘
𝑠12
𝑘
𝑠22
.
.
𝑘
𝑠𝑚2
Step 2: Composition of Single decision matrix
𝐻3
𝑘
𝑠13
𝑘
𝑠23
.
.
𝑘
𝑠𝑚3
. . . 𝐻𝑛
𝑘
. . . . 𝑠1𝑛
𝑘
. . . 𝑠2𝑛 (6.1)
. . .
.
. . .
.
𝑘
. . . 𝑠𝑚𝑛
)
Avishek Chakraborty, Baisakhi Banik ,Sankar Prasad Mondal and SharifulAlam; Arithmetic and Geometric Operators of
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For generating a single group decision matrix X we have promoted the logical pentagonal
neutrosophic weighted arithmetic averaging operator (PNWAA) as, 𝑠𝑖𝑗′ = ∑𝑛𝑗=1 𝜔𝑗 𝑠𝑖𝑗𝑘 , for individual
decision matrix 𝑋 𝑘 , where𝑘 = 1,2,3 … . 𝑛 . hence, we finalize the matrix and defined as follows:
.
𝐺1
𝐺2
𝑋=
𝐺3
.
(𝐺𝑚
𝐻1
′
𝑠11
′
𝑠21
.
..
′
𝑠𝑚1
𝐻2
′
𝑠12
′
𝑠22
.
.
′
𝑠𝑚2
𝐻3
′
𝑠13
′
𝑠23
.
.
′
𝑠𝑚3
.
.
.
.
.
.
. . 𝐻𝑛
′
. . . 𝑠1𝑛
′
. . 𝑠2𝑛
(6.2)
. . .
. . .
′
. . 𝑠𝑚𝑛
)
Step 3: Composition of leading matrix
To illustrate the single decision matrix we have promoted the logical pentagonal neutrosophic
𝜔𝑗
weighted geometric averaging operator (PNWGA ) as, 𝑠𝑖𝑗′′ = ∏𝑛𝑗=1 𝑠̃
for each individual column
𝑖𝑗
and finally, we construct the decision matrix as below,
Step 4: Ranking
. 𝐻1
′′
𝐺1 𝑠11
′′
𝐺2 𝑠21
𝑋=
(6.3)
.
.
.
.
′′
( 𝐺𝑚 𝑠𝑚1
)
Now, considering the score value and transforming the matrix (6.3) into crisp form, we can evaluate
the best substitute corresponding to the best attributes. We align the values as increasing order
according to their score values and then detect the best fit result. The best result will be the highest
magnitude and the worst ones will be the least one.
6.3.1 Flowchart:
Composition of Decision matrices
Composition of Single Decision matrix
Composition of leading matrix
ComputeRanking using Score Value.
Sensitivity Analysis
Figure 6.3.1: Flowchart for the problem
Avishek Chakraborty, Baisakhi Banik ,Sankar Prasad Mondal and SharifulAlam; Arithmetic and Geometric Operators of
Pentagonal Neutrosophic Number and its Application in Mobile Communication Based MCGDM Problem
Neutrosophic Sets and Systems, Vol. 32, 2020
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6.3 Illustrative Example:
Here, we consider a mobile communication service provider based problem in which there are three
different companies are accessible. Among those companies, our problem is to find out the best
mobile communication service provider in a logical and meaningful way. Normally, mobile
communication service providers mostly depend on attributes such as Service & Reliability, Price &
Availability, and Quality & Features of the system. Here, we also consider three different categories
of people i) youth age ii) adult age iii) old age people as a decision-maker. According to their
opinions we formulate the different decision matrices in the pentagonal neutrosophic environment
described below:
𝐺1 = 𝑀𝑜𝑏𝑖𝑙𝑒𝑐𝑜𝑚𝑚𝑢𝑛𝑖𝑐𝑎𝑡𝑖𝑜𝑛𝑠𝑒𝑟𝑣𝑖𝑐𝑒𝑝𝑟𝑜𝑣𝑖𝑑𝑒𝑟 1,
𝐺2 = 𝑀𝑜𝑏𝑖𝑙𝑒𝑐𝑜𝑚𝑚𝑢𝑛𝑖𝑐𝑎𝑡𝑖𝑜𝑛𝑠𝑒𝑟𝑣𝑖𝑐𝑒𝑝𝑟𝑜𝑣𝑖𝑑𝑒𝑟 2,
are the alternatives.
𝐺3 = 𝑀𝑜𝑏𝑖𝑙𝑒𝑐𝑜𝑚𝑚𝑢𝑛𝑖𝑐𝑎𝑡𝑖𝑜𝑛𝑠𝑒𝑟𝑣𝑖𝑐𝑒𝑝𝑟𝑜𝑣𝑖𝑑𝑒𝑟 3
Also
𝐻1 =Service & Reliability,
𝐻2 =Price & Availability,
arethe attributes.
𝐻3 = Quality & Features
Let, 𝐷1 = 𝑌𝑜𝑢𝑡ℎ𝑎𝑔𝑒𝑝𝑒𝑜𝑝𝑙𝑒 , 𝐷2 = 𝐴𝑑𝑢𝑙𝑡𝑎𝑔𝑒𝑝𝑒𝑜𝑝𝑙𝑒, 𝐷3 = 𝑆𝑒𝑛𝑖𝑜𝑟𝑎𝑔𝑒𝑝𝑒𝑜𝑝𝑙𝑒 having weight allocation
𝐷 = { 0.31, 0.35, 0.34 } and the weight allocation in different attribute function is ∆= {0.3,0.4,0.3}.A
verbal matrix is built up by the decision maker’s to assist the classification of the decision matrix.
Attribute vs. Verbal Phrase matrix is given below in Table 6.3.1. The total MCGDM problem is
graphically described as below:
Avishek Chakraborty, Baisakhi Banik ,Sankar Prasad Mondal and SharifulAlam; Arithmetic and Geometric Operators of
Pentagonal Neutrosophic Number and its Application in Mobile Communication Based MCGDM Problem
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Table 6.3.1: List of Verbal Phrase
Sl no.
Attribute
Verbal phrase
Quantitative Attributes
1
Service & Reliability
Very High (VH), High (L), Intermediate (I), Small (S), Very
small (VS)
2
Price & Availability
Very high (VH), High (H), Mid (M), Low (L),
Very low (VL)
3
Quality & Features
Very high (VH), High (H), Standard (SD), Low (L),
Very low (VL)
Table 6.3.2: Relationship between Verbal Phrase and PNN
Verbal Phase
Very Low (VL)
Low (L)
Moderate (M)
Little High (LH)
High (H)
Very High (VH)
Linguistic Pentagonal Neutrosophic Number (PNN)
< (0.1,0.1,0.1,0.1,0.1; 0.4,0.4,0.4) >
< (0.2,0.3,0.4,0.5,0.6; 0.5,0.3,0.3) >
< (0.4,0.5,0.6,0.7,0.8; 0.7,0.2,0.2) >
< (0.5,0.6,0.7,0.8,0.9; 0.75,0.18,0.18) >
< (0.6,0.7,0.8,0.9,1.0; 0.8,0.15,0.15) >
< (1.0,1.0,1.0,1.0,1.0; 0.95,0.05,0.05) >
Step 1
In accordance with finite alternatives and finite attribute functions the decision matrices are
constructed by the proposal of decision maker’s choice. The noteworthy fact is that the entity 𝑠𝑖𝑗 for
each matrix are of pentagonal neutrosophic numbers. Finally, the matrices are presented as follows:
𝐷1
.
𝐺1
=(
𝐺2
𝐺3
𝐻1
𝐻2
𝐻3
< 0.1,0.2,0.3,0.4,0.5; 0.5,0.6,0.7 >
< 0.3,0.4,0.5,0.6,0.7; 0.6,0.3,0.3 >
< 0.2,0.3,0.4,0.5,0.6; 0.4,0.6,0.5 >
)
< 0.15,0.25,0.35,0.45,0.5; 0.5,0.6,0.5 >
< 0.3,0.4,0.5,0.6,0.7; 0.7,0.3,0.5 >
< 0.4,0.5,0.55,0.6,0.7; 0.8,0.7,0.3 >
< 0.4,0.5,0.6,0.7,0.8; 0.6,0.4,0.3 >
< 0.25,0.3,0.35,0.4,0.45; 0.4,0.6,0.5 > < 0.35,0.4,0.45,0.5,0.55; 0.6,0.3,0.4 >
.
𝐺
𝐷2 = ( 1
𝐺2
𝐺3
.
𝐺1
𝐷 =(
𝐺2
𝐺3
3
𝑌𝑜𝑢𝑡ℎ′ 𝑠 𝑜𝑝𝑖𝑛𝑖𝑜𝑛
𝐻1
𝐻2
𝐻3
< 0.15,0.2,0.25,0.3,0.35; 0.6,0.4,0.5 > < 0.1,0.15,0.3,0.35,0.4; 0.7,0.5,0.3 > < 0.7,0.75,0.8,0.85,0.9; 0.3,0.2,0.6 >
)
< 0.2,0.25,0.3,0.35,0.4; 0.7,0.5,0.4 > < 0.2,0.25,0.3,0.4,0.45; 0.6,0.3,0.3 > < 0.4,0.5,0.55,0.6,0.7; 0.8,0.7,0.4 >
< 0.3,0.35,0.4,0.45,0.5; 0.7,0.5,0.3 >
< 0.5,0.55,0.6,0.7,0.8; 0.5,0.6,0.7 >
< 0.6,0.7,0.75,0.8,0.9; 0.6,0.5,0.6 >
𝐴𝑑𝑢𝑙𝑡 ′ 𝑠 𝑂𝑝𝑖𝑛𝑖𝑜𝑛
𝐻1
𝐻2
𝐻3
< 0.2,0.25,0.3,0.4,0.45; 0.6,0.3,0.3 > < 0.2,0.3,0.4,0.5,0.6; 0.4,0.6,0.5 >
< 0.7,0.75,0.8,0.85,0.9; 0.3,0.2,0.6 >
)
< 0.3,0.4,0.5,0.6,0.7; 0.7,0.3,0.5 >
< 0.6,0.7,0.75,0.8,0.9; 0.6,0.5,0.6 > < 0.7,0.75,0.8,0.85,0.9; 0.3,0.2,0.6 >
< 0.3,0.35,0.4,0.45,0.5; 0.7,0.5,0.3 > < 0.4,0.5,0.55,0.6,0.7; 0.8,0.7,0.3 > < 0.15,0.2,0.25,0.3,0.35; 0.6,0.4,0.5 >
𝑆𝑒𝑛𝑖𝑜𝑟 ′ 𝑠 𝑂𝑝𝑖𝑛𝑖𝑜𝑛
Step 2: Composition of Single decision matrix
Avishek Chakraborty, Baisakhi Banik ,Sankar Prasad Mondal and SharifulAlam; Arithmetic and Geometric Operators of
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Neutrosophic Sets and Systems, Vol. 32, 2020
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In this step we generate a single group decision matrix M and have incorporated the idea of logical
pentagonal neutrosophic weighted arithmetic averaging operator (PNWAA) as, 𝑠𝑖𝑗′ = ∑𝑛𝑗=1 𝜔𝑗 𝑠𝑖𝑗𝑘 , for
individual decision matrix 𝐷𝑘 , where𝑘 = 1,2,3 … . 𝑛 . Thus we finalize the matrix which is presented
as follows:
𝑀
.
𝐺1
=(
𝐺2
𝐺3
𝐻1
𝐻2
𝐻3
< 0.18,0.25,0.31,0.4,0.46; 1.00,0.41,0.42 > < 0.13,0.22,0.33,0.42,0.50; 0.99,0.56,0.46 > < 0.58,0.64,0.70,0.77,0.84; 0.98,0.23,0.43 >
)
< 0.22,0.30,0.38,0.47,0.53; 1.00,0.44,0.46 > < 0.38,0.45,0.52,0.60,0.68; 1.00,0.36,0.44 > < 0.42,0.48,0.53,0.58,0.65; 1.00,0.40,0.39 >
< 0.33,0.40,0.46,0.53,0.59; 1.00,0.47,0.30 > < 0.39,0.46,0.51,0.57,0.66; 1.00,0.41,0.47 > < 0.37,0.48,0.49,0.58,0.60; 1.00,0.40,0.50 >
Step 3: Composition of leading matrix
To define the single decision matrix we have employed the concept of the logical pentagonal
𝜔𝑗
neutrosophic weighted geometric averaging operator ( PNWGA ) as, 𝑠𝑖𝑗′′ = ∏𝑛𝑗=1 𝑠̃
for each
𝑖𝑗
individual column and finally, we present the decision matrix as below
Step 4: Ranking
〈0.26, 0.35, 0.44, 0.56, 0.60; 0.99,0.98,0.99〉
𝑀 = ( 〈0.33, 0.41, 0.48,0.55,0.62; 1.00,0.98,0.99〉 )
〈0.36,0.43,0.48,0.54,0.62; 1.00,0.99,0.99〉
Now, we examine the proposed score value for crispification of the PNN into a real number, thus we
get the ultimate decision matrix as
< 0.1503 >
𝑀 = (< 0.1641 >)
< 0.1652 >
Here, ordering is 0.1503 < 0.1641 < 0.1652. Hence, the ranking of the mobile communication
service provider is 𝐺3 > 𝐺2 > 𝐺1 .
6.4 Results and Sensitivity Analysis
To understand how the attribute weights of each criterion affect 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. 𝟑, 𝟎. 𝟑, 𝟎. 𝟒)>
< 0.1367 >
(< 0.1617 >)
< 0.1650 >
< 0.1387 >
(< 0.1641 >)
< 0.1666 >
< 0.1394 >
(< 0.1621 >)
< 0.1692 >
< 0.1415 >
(< 0.1641 >)
< 0.1699 >
< 0.1799 >
(< 0.1559 >)
< 0.1623 >
𝐺3 > 𝐺2 > 𝐺1
<(𝟎. 𝟑𝟑, 𝟎. 𝟑𝟓, 𝟎. 𝟑𝟐)>
<(𝟎. 𝟑, 𝟎. 𝟑𝟕, 𝟎. 𝟑𝟑)>
<(𝟎. 𝟒𝟓, 𝟎. 𝟐𝟓, 𝟎. 𝟑)>
<(𝟎. 𝟐𝟓, 𝟎. 𝟒𝟓, 𝟎. 𝟑)>
𝐺3 > 𝐺2 > 𝐺1
𝐺3 > 𝐺2 > 𝐺1
𝐺3 > 𝐺2 > 𝐺1
𝐺1 > 𝐺3 > 𝐺2
Avishek Chakraborty, Baisakhi Banik ,Sankar Prasad Mondal and SharifulAlam; Arithmetic and Geometric Operators of
Pentagonal Neutrosophic Number and its Application in Mobile Communication Based MCGDM Problem
Neutrosophic Sets and Systems, Vol. 32, 2020
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< 0.1367 >
(< 0.1669 >)
< 0.1680 >
< 0.1544 >
(< 0.1675 >)
< 0.1666 >
< 0.1503 >
(< 0.1641 >)
< 0.1652 >
<(𝟎. 𝟐𝟓, 𝟎. 𝟑, 𝟎. 𝟒𝟓)>
< (𝟎. 𝟒, 𝟎. 𝟑, 𝟎. 𝟑) >
< (𝟎. 𝟑, 𝟎. 𝟒, 𝟎. 𝟑) >
𝐺3 > 𝐺2 > 𝐺1
𝐺2 > 𝐺3 > 𝐺1
𝐺3 > 𝐺2 > 𝐺1
Weights
0.5
Service &
Reliability
0.4
0.3
Price &
Availability
0.2
0.1
Quality &
Features
0
1
2
3
4
5
6
7
8
Trials
Figure 6.4.1: Sensitivity analysis on attribute functions
0.2
0.15
G1
0.1
G2
0.05
G3
0
G3
1
2
3
4
5
G1
6
7
8
Figure 6.4.2: Best Alternative Mobile Communication Service
6.5 Comparison Table
This section actually contains a comparative study among the established work and proposed
work.Comparing with 49,50 , we find that the best service provider among those three and it is noticed
that in each case 𝐺3 becomes the best mobile communication service provider. The comparison table
is given as follows:
Avishek Chakraborty, Baisakhi Banik ,Sankar Prasad Mondal and SharifulAlam; Arithmetic and Geometric Operators of
Pentagonal Neutrosophic Number and its Application in Mobile Communication Based MCGDM Problem
Neutrosophic Sets and Systems, Vol. 32, 2020
76
Approach
Ranking
Deli
𝐺3 > 𝐺2 > 𝐺1
49
Garg50
Proposed method
𝐺3 > 𝐺1 > 𝐺2
𝐺3 > 𝐺2 > 𝐺1
7. Conclusion and future research scope
The idea of pentagonal neutrosophic number is intriguing, competent and has ample scope of
utilization in various research domains. In this research article, we vigorously erect the perception of
pentagonal neutrosophic number from different aspects. We also resort to the perception of
truthiness, falsity and ambiguity functions in case of pentagonal neutrosophic number when the
membership functions are interconnected to each other and a new score function is formulated here.
Also, two logical operators have been developed here theoretically as well as applied it in MCGDM
problem. Finally we perform a sensitivity analysis and also demonstrate a comparative study with
the other results derived from other research articles to enumerate our proposed work and conclude
that our result is pretty satisfactory as we consider the pentagonal neutrosophic value in the
problem of multi-criteria decision making.
Further, researchers can immensely apply this idea of neutrosophic number in numerous flourishing
research fields like an engineering problem, mobile computing problems, diagnoses problem,
realistic mathematical modelling, cloud computing issues, pattern recognition problems, an
architecture based structural modelling, image processing, linear programming, big data analysis,
neural network etc. Apart from these there is an immense scope of application basis works in
various fields which can be constructed by taking the help of pentagonal neutrosophic numbers.
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Received: Oct 13, 2019. Accepted: Mar 10, 2020.
Avishek Chakraborty, Baisakhi Banik ,Sankar Prasad Mondal and SharifulAlam; Arithmetic and Geometric Operators of
Pentagonal Neutrosophic Number and its Application in Mobile Communication Based MCGDM Problem
Neutrosophic Sets and Systems, Vol. 32 2020
University of New Mexico
On Q-Neutrosophic Soft Fields
Majdoleen Abu Qamar1,∗ , Abd Ghafur Ahmad2 , Nasruddin Hassan3
1,2,3
School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia,
Bangi 43600, Selangor, Malaysia.
1
E-mail: p90675@siswa.ukm.edu.my, mjabuqamar@gmail.com, 2 ghafur@ukm.edu.my, 3 nas@ukm.edu.my
∗
Correspondence: mjabuqamar@gmail.com
Abstract: As an extension of neutrosophic soft sets, Q-neutrosophic soft sets were established to deal with twodimensional indeterminate data. Different hybrid models of fuzzy sets were utilized to different algebraic structures,
for example groups, rings, fields and lie-algebras. A field is an essential algebraic structure, which is widely used
in algebra and several domains of mathematics. The motivation of the current work is to extend the thought of
Q-neutrosophic soft sets to fields. In this paper, we define the notion of Q-neutrosophic soft fields. Structural characteristics of it are investigated. Moreover, the concepts of homomorphic image and pre-image of Q-neutrosophic soft
fields are discussed. Finally, the Cartesian product of Q-neutrosophic soft fields is defined and some related properties
are discussed.
Keywords: Neutrosophic soft field, Neutrosophic soft set, Q-neutrosophic soft field, Q-neutrosophic soft set.
1
Introduction
Fuzzy sets were established by Zadeh [1] as a tool to deal with uncertain data. Since then, fuzzy logic has
been utilized in several real-world problems in uncertain environments. Consequently, numerous analysts
discussed many results using distinct directions of fuzzy-set theory, for instance, interval valued fuzzy set [2]
and intuitionistic fuzzy set [3]. These extensions can deal with uncertain real-world problems but it does not
cope with indeterminate data. Thus, Smarandache [4] initiated the neutrosophic idea to overcome this problem.
A neutrosophic set (NS) [5] is a mathematical notion serving issues containing inconsistent, indeterminate,
and imprecise data. Molodtsov [6] introduced the concept of soft sets as another way to handle uncertainty.
Since its initiation, a plenty of hybrid models of soft set have been produced, for example, fuzzy soft sets [7],
neutrosophic soft sets (NSSs) [8]. Accordingly, NSSs became an important notion for more deep discussions
[9–17]. NSSs were extended to Q-neutrosophic soft sets (Q-NSSs) [18] a new model that deals with twodimensional uncertain data. Q-NSSs were further investigated and their basic operations and relations were
discussed in [18, 19].
Different hybrid models of fuzzy sets and soft sets were utilized in different branches of mathematics,
including algebra. This was started by Rosenfeld in 1971 [20] when he established the idea of fuzzy subgroup.
Since then, the theories and approaches of fuzzy soft sets on different algebraic structures developed rapidly.
Mukherjee and Bhattacharya [21] studied fuzzy groups, Sharma [22] discussed intuitionistic fuzzy groups.
Recently, many researchers have applied different hybrid models of fuzzy sets and soft sets to several algebraic
structures such as groups, semigroups, rings, fields and BCK/BCI-algebras [23–32]. NSs and NSSs have
Majdoleen Abu Qamar, Abd Ghafur Ahmad and Nasruddin Hassan, On Q-Neutrosophic Soft Fields.
81
Neutrosophic Sets and Systems, Vol. 32 2020
received more attention in studying the algebraic structure of set theories dealing with uncertainty. Cetkin
and Aygun [33] established the concept of neutrosophic subgroup. Bera and Mahapatra introduced the notion
of neutrosophic soft group [34], neutrosophic soft fields [35]. Moreover, two-dimensional hybrid models
of fuzzy sets and soft sets were also applied to different algebraic structures. Solairaju and Nagarajan [36]
introduced the notion of Q-fuzzy groups. Thiruveni and Solairaju defined the concept of neutrosophic Q-fuzzy
subgroup [37], while Rasuli [38] established the notion of Q-fuzzy subring and anti Q-fuzzy subring. The
concept of Q-NSSs was also implemented in the theories of groups and rings [39, 40].
Inspired by the above works and to utilize Q-NSSs to different algebraic structures, in the current paper,
we continue the work presented in [41] about Q-neutrosophic soft fields (Q-NSFs) and investigate some of
its structural characteristics; we give some theorems that simplifies the main definition, also we discuss the
intersection and union of two Q-NSFs . The concepts of homomorphic image and pre-image of Q-NSFs are
investigated. Also, we discuss the Cartesian product of Q-NSFs and discuss some related properties.
2
Preliminaries
In this section, we recall the basic definitions related to this work.
Definition 2.1 ( [18]). Let X be a universal set, Q be a nonempty set and A ⊆ E be a set of parameters. Let
µl QN S(X) be the set of all multi Q-NSs on X with dimension l = 1. A pair (ΓQ , A) is called a Q-NSS over
X, where ΓQ : A → µl QN S(X) is a mapping, such that ΓQ (e) = φ if e ∈
/ A.
Definition 2.2 ( [19]). The union of two Q-NSSs (ΓQ , A) and (ΨQ , B) is the Q-NSS (ΛQ , C) written as
(ΓQ , A) ∪ (ΨQ , B) = (ΛQ , C), where C = A ∪ B and for all c ∈ C, (x, q) ∈ X × Q, the truth-membership,
indeterminacy-membership and falsity-membership of (ΛQ , C) are as follows:
if c ∈ A − B,
TΓQ (c) (x, q)
TΛQ (c) (x, q) = TΨQ (c) (x, q)
if c ∈ B − A,
max{TΓQ (c) (x, q), TΨQ (c) (x, q)} if c ∈ A ∩ B,
IΓQ (c) (x, q)
IΛQ (c) (x, q) = IΨQ (c) (x, q)
min{IΓQ (c) (x, q), IΨQ (c) (x, q)}
FΓQ (c) (x, q)
FΛQ (c) (x, q) = FΨQ (c) (x, q)
min{FΓQ (c) (x, q), FΨQ (c) (x, q)}
if c ∈ A − B,
if c ∈ B − A,
if c ∈ A ∩ B,
if c ∈ A − B,
if c ∈ B − A,
if c ∈ A ∩ B.
Definition 2.3 ( [19]). The intersection of two Q-NSSs (ΓQ , A) and (ΨQ , B) is the Q-NSS (ΛQ , C) written as
(ΓQ , A) ∩ (ΨQ , B) = (ΛQ , C), where C = A ∩ B and for all c ∈ C and (x, q) ∈ X × Q the truth-membership,
Majdoleen Abu Qamar, Abd Ghafur Ahmad and Nasruddin Hassan, On Q-Neutrosophic Soft Fields.
Neutrosophic Sets and Systems, Vol. 32 2020
82
indeterminacy-membership and falsity-membership of (ΛQ , C) are as follows:
TΛQ (c) (x, q) = min{TΓQ (c) (x, q), TΨQ (c) (x, q)},
IΛQ (c) (x, q) = max{IΓQ (c) (x, q), IΨQ (c) (x, q)},
FΛQ (c) (x, q) = max{FΓQ (c) (x, q), FΨQ (c) (x, q)}.
3
Q-Neutrosophic Soft Fields
In this section, we define the notion of Q-NSF and discuss several related properties.
Definition 3.1. Let (ΓQ , A) be a Q-NSS over a field (F, +, .). Then, (ΓQ , A) is said to be a Q-NSF over
(F, +, .) if for all e ∈ A, ΓQ (e) is a Q-neutrosophic subfield of (F, +, .), where ΓQ (e) is a mapping given by
ΓQ (e) : F × Q → [0, 1]3 .
Definition 3.2. Let (F, +, .) be a field and (ΓQ , A) be a Q-NSS over (F, +, .). Then, (ΓQ , A) is called a Q-NSF
over (F, +, .) if for all x, y ∈ F, q ∈ Q and e ∈ A it satisfies:
1. TΓQ (e) (x + y, q) ≥ min TΓQ(e) (x, q), TΓQ (e) (y, q) , IΓQ (e) (x + y, q) ≤ max IΓQ (e) (x, q), IΓQ (e) (y, q)
and FΓQ (e) (x + y, q) ≤ max FΓQ (e) (x, q), FΓQ (e) (y, q) .
2. TΓQ (e) (−x, q) ≥ TΓQ (e) (x, q), IΓQ (e) (−x, q) ≤ IΓQ (e) (x, q) and FΓQ (e) (−x, q) ≤ FΓQ (e) (x, q).
3. TΓQ (e) (x.y, q) ≥ min TΓQ (e) (x, q), TΓQ (e) (y, q) , IΓQ (e) (x.y, q) ≤ max IΓQ (e) (x, q), IΓQ (e) (y, q) and
FΓQ (e) (x.y, q) ≤ max FΓQ (e) (x, q), FΓQ (e) (y, q) .
4. TΓQ (e) (x−1 , q) ≥ TΓQ (e) (x, q), IΓQ (e) (x−1 , q) ≤ IΓQ (e) (x, q) and FΓQ (e) (x−1 , q) ≤ FΓQ (e) (x, q).
Example 3.3. Let F = (R, +, .) be the field of real numbers and A = N the set of natural numbers be the
parametric set. Define a Q-NSS (ΓQ , A) as follows for q ∈ Q, x ∈ R and m ∈ N
(
0
if x is rational
TΓQ (m) (x, q) = 1
,
if x is irrational
9m
(
1
1 − 3m
if x is rational
IΓQ (m) (x, q) =
,
0
if x is irrational
(
1 + m3 if x is rational
FΓQ (m) (x, q) =
.
0
if x is irrational
It is clear that (ΓQ , N) is a Q-NSF over F .
Proposition 3.4. Let (ΓQ , A) be a Q-NSF over (F, +, .). Then, for the additive identity 0F and the multiplicative identity 1F , for all x ∈ F, q ∈ Q and e ∈ A the following hold
1. TΓQ (e) (0F , q) ≥ TΓQ (e) (x, q), IΓQ (e) (0F , q) ≤ IΓQ (e) (x, q) and FΓQ (e) (0F , q) ≤ FΓQ (e) (x, q).
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2. TΓQ (e) (1F , q) ≥ TΓQ (e) (x, q), IΓQ (e) (1F , q) ≤ IΓQ (e) (x, q) and FΓQ (e) (1F , q) ≤ FΓQ (e) (x, q), for x 6= 0F .
3. TΓQ (e) (0F , q) ≥ TΓQ (e) (1F , q), IΓQ (e) (0F , q) ≤ IΓQ (e) (1F , q) and FΓQ (e) (0F , q) ≤ FΓQ (e) (1F , q).
Proof. ∀x ∈ F, q ∈ Q and e ∈ A
1. TΓQ (e) (0F , q) = TΓQ (e) (x − x, q) ≥ min
TΓQ (e) (x, q), TΓQ (e) (x, q) = TΓQ (e) (x, q),
IΓQ (e) (0F , q) = IΓQ (e) (x − x, q) ≤ max IΓQ (e) (x, q), IΓQ (e) (x, q) = IΓQ (e) (x, q),
FΓQ (e) (0F , q) = FΓQ (e) (x − x, q) ≤ max FΓQ (e) (x, q), FΓQ (e) (x, q) = FΓQ (e) (x, q).
2. TΓQ (e) (1F , q) = TΓQ (e) (x.x−1 , q) ≥ min
TΓQ (e) (x, q), TΓQ (e) (x, q) = TΓQ (e) (x, q),
−1
IΓQ (e) (1F , q) = IΓQ (e) (x.x , q) ≤ max IΓQ (e) (x, q), IΓQ (e) (x, q) = IΓQ (e) (x, q),
FΓQ (e) (1F , q) = FΓQ (e) (x.x−1 , q) ≤ max FΓQ (e) (x, q), FΓQ (e) (x, q) = FΓQ (e) (x, q).
3. Follows directly by applying 1.
Theorem 3.5. A Q-NSS (ΓQ , A) over the field (F, +, .) is a Q-NSF if and only if for all x, y ∈ F, q ∈ Q and
e∈A
1. TΓQ (e) (x − y, q) ≥ min TΓQ (e) (x, q), T
(y,
q)
,
I
(x
−
y,
q)
≤
max
IΓQ (e) (x, q),
Γ
(e)
Γ
(e)
Q
Q
IΓQ (e) (y, q) , FΓQ (e) (x − y, q) ≤ max FΓQ (e) (x, q), FΓQ (e) (y, q) .
−1
(y,
q)
,
I
(x.y
,
q)
≤
max
IΓQ (e) (x, q),
2. TΓQ (e) (x.y −1 , q) ≥ min TΓQ (e) (x, q), T
Γ
(e)
Γ
(e)
Q
Q
−1
IΓQ (e) (y, q) , FΓQ (e) (x.y , q) ≤ max FΓQ (e) (x, q), FΓQ (e) (y, q) .
Proof. Suppose that (ΓQ , A) is a Q-NSF over (F, +, .). Then,
TΓQ (e) (x − y, q) ≥ min TΓQ (e) (x, q), TΓQ (e) (−y, q) ≥ min TΓQ (e) (x, q), TΓQ (e) (y, q) ,
IΓQ (e) (x − y, q) ≤ max IΓQ (e) (x, q), IΓQ (e) (−y, q) ≤ max IΓQ (e) (x, q), IΓQ (e) (y, q) ,
FΓQ (e) (x − y, q) ≤ max FΓQ (e) (x, q), FΓQ (e) (−y, q) ≤ max FΓQ (e) (x, q), FΓQ (e) (y, q) .
Also,
TΓQ (e) (x.y −1 , q) ≥ min TΓQ (e) (x, q), TΓQ (e) (y −1 , q) ≥ min TΓQ (e) (x, q), TΓQ (e) (y, q) ,
IΓQ (e) (x.y −1 , q) ≤ max IΓQ (e) (x, q), IΓQ (e) (y −1 , q) ≤ max IΓQ (e) (x, q), IΓQ (e) (y, q) ,
FΓQ (e) (x.y −1 , q) ≤ max FΓQ (e) (x, q), FΓQ (e) (y −1 , q) ≤ max FΓQ (e) (x, q), FΓQ (e) (y, q) .
Conversely, Suppose that conditions 1 and 2 are satisfied. We show that for each e ∈ A, (ΓQ , A) is a
Q-neutrosophic subfield
TΓQ (e) (−x, q) = TΓQ (e) (0F − x, q) ≥ min TΓQ (e) (0F , q), TΓQ (e) (x, q)
≥ min TΓQ (e) (x, q), TΓQ (e) (x, q) = TΓQ (e) (x, q),
IΓQ (e) (−x, q) = IΓQ (e) (0F − x, q) ≤ max IΓQ (e) (0F , q), IΓQ (e) (x, q)
≤ max IΓQ (e) (x, q), IΓQ (e) (x, q) = IΓQ (e) (x, q),
FΓQ (e) (−x, q) = FΓQ (e) (0F − x, q) ≤ max FΓQ (e) (0F , q), FΓQ (e) (x, q)
≤ max FΓQ (e) (x, q), FΓQ (e) (x, q) = FΓQ (e) (x, q)
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also,
Next,
and
TΓQ (e) (x + y, q) = TΓQ (e) (x − (−y), q) ≥ min TΓQ (e) (x, q), TΓQ (e) (y, q) ,
IΓQ (e) (x + y, q) = IΓQ (e) (x − (−y), q) ≤ max IΓQ (e) (x, q), IΓQ (e) (y, q) ,
FΓQ (e) (x + y, q) = FΓQ (e) (x − (−y), q) ≤ max FΓQ (e) (x, q), FΓQ (e) (y, q) .
TΓQ (e) (x−1 , q) = TΓQ (e) (1F .x−1 , q) ≥ min TΓQ (e) (1F , q), TΓQ (e) (x, q)
≥ min TΓQ (e) (x, q), TΓQ (e) (x, q) = TΓQ (e) (x, q),
IΓQ (e) (x−1 , q) = IΓQ (e) (1F .x−1 , q) ≤ max IΓQ (e) (1F , q), IΓQ (e) (x, q)
≤ max IΓQ (e) (x, q), IΓQ (e) (x, q) = IΓQ (e) (x, q),
FΓQ (e) (x−1 , q) = FΓQ (e) (1F .x−1 , q) ≤ max FΓQ (e) (1F , q), FΓQ (e) (x, q)
≤ max FΓQ (e) (x, q), FΓQ (e) (x, q) = FΓQ (e) (x, q)
TΓQ (e) (x.y, q) = TΓQ (e) (x(y −1 )−1 , q) ≥ min TΓQ (e) (x, q), TΓQ (e) (y, q) ,
IΓQ (e) (x.y, q) = IΓQ (e) (x(y −1 )−1 , q) ≤ max IΓQ (e) (x, q), IΓQ (e) (y, q) ,
FΓQ (e) (x.y, q) = FΓQ (e) (x(y −1 )−1 , q) ≤ max FΓQ (e) (x, q), FΓQ (e) (y, q) .
This completes the proof.
Theorem 3.6. Let (ΓQ , A) and (ΨQ , B) be two Q-NSFs over (F, +, .). Then, (ΓQ , A) ∩ (ΨQ , B) is also Q-NSF
over (F, +, .).
Proof. Let (ΓQ , A) ∩ (ΨQ , B) = (ΛQ , A ∩ B). Now, ∀x, y ∈ F, q ∈ Q and e ∈ A ∩ B,
n
o
TΛQ (e) (x − y, q) = min TΓQ (e) (x − y, q), TΨQ (e) (x − y, q)
n
o
≥ min min TΓQ (e) (x, q), TΓQ (e) (y, q) , min TΨQ (e) (x, q), TΨQ (e) (y, q)
n
o
= min min TΓQ (e) (x, q), TΨQ (e) (x, q) , min TΓQ (e) (y, q), TΨQ (e) (y, q)
n
o
= min TΛQ (e) (x, q), TΛQ (e) (y, q) ,
also,
n
o
IΛQ (e) (x − y, q) = max IΓQ (e) (x − y, q), IΨQ (e) (x − y, q)
n
o
≤ max max IΓQ (e) (x, q), IΓQ (e) (y, q) , max IΨQ (e) (x, q), IΨQ (e) (y, q)
n
o
= max max IΓQ (e) (x, q), IΨQ (e) (x, q) , max IΓQ (e) (y, q), IΨQ (e) (y, q)
n
o
= max IΛQ (e) (x, q), IΛQ (e) (y, q) ,
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n
o
similarly, FΛQ (e) (x − y, q) ≤ max FΛQ (e) (x, q), FΛQ (e) (y, q) . Next,
o
TΛQ (e) (x.y , q) = min TΓQ (e) (x.y , q), TΨQ (e) (x.y , q)
n
o
≥ min min TΓQ (e) (x, q), TΓQ (e) (y, q) , min TΨQ (e) (x, q), TΨQ (e) (y, q)
n
o
= min min TΓQ (e) (x, q), TΨQ (e) (x, q) , min TΓQ (e) (y, q), TΨQ (e) (y, q)
n
o
= min TΛQ (e) (x, q), TΛQ (e) (y, q) ,
−1
also,
n
−1
−1
n
o
IΛQ (e) (x.y −1 , q) = max IΓQ (e) (x.y −1 , q), IΨQ (e) (x.y −1 , q)
n
o
≤ max max IΓQ (e) (x, q), IΓQ (e) (y, q) , max IΨQ (e) (x, q), IΨQ (e) (y, q)
n
o
= max max IΓQ (e) (x, q), IΨQ (e) (x, q) , max IΓQ (e) (y, q), IΨQ (e) (y, q)
n
o
= max IΛQ (e) (x, q), IΛQ (e) (y, q)
n
o
similarly, we can show FΛQ (e) (x.y −1 , q) ≤ max FΛQ (e) (x, q), FΛQ (e) (y, q) . This completes the proof.
Remark 3.7. For two Q-NSFs (ΓQ , A) and (ΨQ , B) over (F, +, .), (ΓQ , A) ∪ (ΨQ , B) is not generally a QNSF.
For example, let F = (Q, +, .), E = 2Z. Consider two Q-NSFs (ΓQ , E) and (ΨQ , E) over F as follows: for
x ∈ Q, q ∈ Q and m ∈ Z
(
0.50 if x = 4tm, ∃t ∈ Z,
TΓQ (4m) (x, q) =
0
otherwise,
(
0
if x = 4tm, ∃t ∈ Z,
IΓQ (4m) (x, q) =
0.25 otherwise,
(
0.40 if x = 4tm, ∃t ∈ Z,
FΓQ (4m) (x, q) =
0.10 otherwise,
and
TΨQ (4m) (x, q) =
(
0.70
0
if x = 6tm, ∃t ∈ Z,
otherwise,
IΨQ (4m) (x, q) =
(
0
0.50
if x = 6tm, ∃t ∈ Z,
otherwise,
FΨQ (4m) (x, q) =
(
0.20
0.40
if x = 6tm, ∃t ∈ Z,
otherwise.
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Neutrosophic Sets and Systems, Vol. 32 2020
86
Let (ΓQ , A) ∪ (ΨQ , B) = (ΛQ , E). For m = 2, x = 8, y = 12 we have
n
o
TΛQ (8) (8 − 12, q) = TΛQ (8) (−4, q) = max TΓQ (8) (−4, q), TΨQ (8) (−4, q) = max{0, 0} = 0
and
n
o
min TΛQ (8) (8, q),TΛQ (8) (12, q)
n
o
= min max TΓQ (8) (8, q), TΨQ (8) (8, q) , max TΓQ (8) (12, q), TΨQ (8) (12, q)
o
n
= min max 0.50, 0 , max 0, 0.7
n
o
= min 0.50, 0.70 = 0.50.
n
o
Hence, TΛQ (8) (8 − 12, q) < min TΛQ (8) (8, q), TΛQ (8) (12, q) . Thus, the union is not a Q-NSF.
4
Q-Neutrosophic Soft Homomorphism
In this section, we define the Q-neutrosophic soft function, then define the image and pre-image of a QNSS under a Q-neutrosophic soft function. In continuation, we introduce the notion of Q-neutrosophic soft
homomorphism along with some of it’s properties.
Definition 4.1. Let g : X × Q → Y × Q and h : A → B be two functions where A and B are parameter sets.
Then, the pair (g, h) is called a Q-neutrosophic soft function from X × Q to Y × Q.
Definition 4.2. Let (ΓQ , A) and (ΨQ , B) be two Q-NSSs defined over X × Q and Y × Q, respectively, and
(g, h) be a Q-neutrosophic soft function from X × Q to Y × Q. Then,
1. The image of (ΓQ , A) under (g, h), denoted by (g, h)(ΓQ , A), is a Q-NSS over Y × Q and is defined by:
nD
Eo
(g, h)(ΓQ , A) = g(ΓQ ), h(A) =
b, g(ΓQ )(b) : b ∈ h(A) ,
where for all b ∈ h(A), y ∈ Y and q ∈ Q,
(
maxg(x,q)=(y,q) maxh(a)=b [TΓQ (a) (x, q)] if (x, q) ∈ g −1 (y, q),
Tg(ΓQ )(b) (y, q) =
0
otherwise,
(
ming(x,q)=(y,q) minh(a)=b [IΓQ (a) (x, q)] if (x, q) ∈ g −1 (y, q),
Ig(ΓQ )(b) (y, q) =
1
otherwise,
(
ming(x,q)=(y,q) minh(a)=b [FΓQ (a) (x, q)] if (x, q) ∈ g −1 (y, q),
Fg(ΓQ )(b) (y, q) =
1
otherwise,
2. The preimage of (ΨQ , B) under (g, h), denoted by (g, h)−1 (ΨQ , B), is a Q-NSS over X and is defined
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Neutrosophic Sets and Systems, Vol. 32 2020
by:
nD
Eo
a, g −1 (ΨQ )(a) : a ∈ h−1 (B) ,
(g, h)−1 (ΨQ , B) = g −1 (ΨQ ), h−1 (B) =
where for all a ∈ h−1 (B), x ∈ X and q ∈ Q,
Tg−1 (ΨQ )(a) (x, q) = TΨQ [h(a)] (g(x, q)),
Ig−1 (ΨQ )(a) (x, q) = IΨQ [h(a)] (g(x, q)),
Fg−1 (ΨQ )(a) (x, q) = FΨQ [h(a)] (g(x, q)).
If g and h are injective (surjective), then (g, h) is injective (surjective).
Definition 4.3. Let (g, h) be a Q-neutrosophic soft function from X × Q to Y × Q. If g is a homomorphism
from X × Q to Y × Q, then (g, h) is said to be a Q-neutrosophic soft homomorphism. If g is an isomorphism
from X × Q to Y × Q and h is a one-to-one mapping from A to B, then (g, h) is said to be a Q-neutrosophic
soft isomorphism.
Example 4.4. Let A = N (the set of natural numbers) be the parametric set and F = (Z5 , +, .) be a field.
Define a Q-NSS (ΓQ , A) as follows, for any a ∈ A, q ∈ Q and x ∈ Z5 ,
(
0
if x ∈ {1̄, 3̄}
,
TΓQ (a) (x, q) = 1
if x ∈ {0̄, 2̄, 4̄}
3a
(
1 − a1 if x ∈ {1̄, 3̄}
IΓQ (a) (x, q) =
,
0
if x ∈ {0̄, 2̄, 4̄}
(
3
if x ∈ {1̄, 3̄}
FΓQ (a) (x, q) = a+1
.
0
if x ∈ {0̄, 2̄, 4̄}
Now, let g : Z5 × Q → Z5 × Q and h : N → N be given by g(x, q) = 3x + 1 and h(a) = a2 . Then for
b ∈ N2 , y ∈ 3Z5 + 1 , the image of (ΓQ , A) under (g, h) as follows :
Tg(ΓQ )(b) (y, q) =
(
0
Ig(ΓQ )(b) (y, q) =
(
1−
Fg(ΓQ )(b) (y, q) =
if y ∈ {0̄, 2̄, 4̄}
,
if y ∈ {1̄, 3̄}
1
√
3 b
0
(
√1
b
if y ∈ {0̄, 2̄, 4̄}
if y ∈ {1̄, 3̄}
1√
1+ b
if y ∈ {0̄, 2̄, 4̄}
0
if y ∈ {1̄, 3̄}
,
.
Theorem 4.5. Let (ΓQ , A) be a Q-NSF over F1 and (g, h) : F1 × Q → F2 × Q be a Q-neutrosophic soft
homomorphism. Then, (g, h)(ΓQ , A) is a Q-NSF over F2 .
Proof. Let b ∈ h(A) and y1 , y2 ∈ F2 . For g −1 (y1 , q) = φ or g −1 (y2 , q) = φ, the proof is straight forward.
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So, assume there exists x1 , x2 ∈ F1 such that g(x1 , q) = (y1 , q) and g(x2 , q) = (y2 , q). Then,
h
i
Tg(ΓQ )(b) (y1 − y2 , q) =
max
max TΓQ (a) (x, q)
g(x,q)=(y1 −y2 ,q) h(a)=b
h
i
≥ max TΓQ (a) (x1 − x2 , q)
h(a)=b
oi
h
n
≥ max min TΓQ (a) (x1 , q), TΓQ (a) (−x2 , q)
h(a)=b
h
n
oi
≥ max min TΓQ (a) (x1 , q), TΓQ (a) (x2 , q)
h(a)=b
h
h
io
i
n
= min max TΓQ (a) (x1 , q) , max TΓQ (a) (x2 , q)
h(a)=b
h(a)=b
h
i
max −1 max TΓQ (a) (x, q)
g(x,q)=(y1 .y2 ,q) h(a)=b
i
h
,
q)
≥ max TΓQ (a) (x1 .x−1
2
h(a)=b
h
n
oi
≥ max min TΓQ (a) (x1 , q), TΓQ (a) (x2−1 , q)
h(a)=b
h
n
oi
≥ max min TΓQ (a) (x1 , q), TΓQ (a) (x2 , q)
h(a)=b
h
h
i
io
n
= min max TΓQ (a) (x1 , q) , max TΓQ (a) (x2 , q)
Tg(ΓQ )(b) (y1 .y2−1 , q) =
h(a)=b
h(a)=b
Since, the inequality is satisfied for each x1 , x2 ∈ F1 , satisfying g(x1 , q) = (y1 , q) and g(x2 , q) = (y2 , q).
Then,
io
i
n
h
h
Tg(ΓQ )(b) (y1 − y2 , q) ≥ min
max
max TΓQ (a) (x2 , q)
max
max TΓQ (a) (x1 , q) ,
g(x2 ,q)=(y1 ,q) h(a)=b
g(x1 ,q)=(y1 ,q) h(a)=b
o
n
= min Tg(ΓQ )(b) (y1 , q), Tg(ΓQ )(b) (y2 , q) .
Tg(ΓQ )(b) (y1 .y2−1 , q)
n
h
i
h
io
≥ min
max
max TΓQ (a) (x1 , q) ,
max
max TΓQ (a) (x2 , q)
g(x1 ,q)=(y1 ,q) h(a)=b
g(x2 ,q)=(y1 ,q) h(a)=b
o
n
= min Tg(ΓQ )(b) (y1 , q), Tg(ΓQ )(b) (y2 , q) .
Similarly, we show that
n
o
Ig(ΓQ )(b) (y1 − y2 , q) ≤ max Ig(ΓQ )(b) (y1 , q), Ig(ΓQ )(b) (y2 , q) ,
n
o
Ig(ΓQ )(b) (y1 .y2−1 , q) ≤ max Ig(ΓQ )(b) (y1 , q), Ig(ΓQ )(b) (y2 , q) ,
n
o
Fg(ΓQ )(b) (y1 − y2 , q) ≤ max Fg(ΓQ )(b) (y1 , q), Fg(ΓQ )(b) (y2 , q) ,
n
o
Fg(ΓQ )(b) (y1 .y2−1 , q) ≤ max Fg(ΓQ )(b) (y1 , q), Fg(ΓQ )(b) (y2 , q) .
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Theorem 4.6. Let (ΨQ , B) be a Q-NSF over F2 and (g, h) be a Q-neutrosophic soft homomorphism from
F1 × Q to F2 × Q. Then, (g, h)−1 (ΨQ , B) is a Q-NSF over over F1 .
Proof. For a ∈ h−1 (B) and x1 , x2 ∈ F1 , we have
Tg−1 (ΨQ )(a) (x1 − x2 , q) = TΨQ [h(a)] (g(x1 − x2 , q))
= TΨQ [h(a)] (g(x1 , q) − g(x2 , q))
n
o
≥ min TΨQ [h(a)] (g(x1 , q)), TΨQ [h(a)] (−g(x2 , q))
n
o
≥ min TΨQ [h(a)] (g(x1 , q)), TΨQ [h(a)] (g(x2 , q))
n
o
= min Tg−1 (ΨQ )(a) (x1 , q), Tg−1 (ΨQ )(a) (x2 , q)
and
−1
Tg−1 (ΨQ )(a) (x1 .x−1
2 , q) = TΨQ [h(a)] (g(x1 .x2 , q))
Similarly, we can obtain
= TΨQ [h(a)] (g(x1 , q).g(x−1
2 , q))
n
o
−1
≥ min TΨQ [h(a)] (g(x1 , q)), TΨQ [h(a)] (g(x2 , q) )
n
o
≥ min TΨQ [h(a)] (g(x1 , q)), TΨQ [h(a)] (g(x2 , q))
n
o
= min Tg−1 (ΨQ )(a) (x1 , q), Tg−1 (ΨQ )(a) (x2 , q)
n
o
Ig−1 (ΨQ )(a) (x1 − x2 , q) ≤ max Ig−1 (ΨQ )(a) (x1 , q), Ig−1 (ΨQ )(a) (x2 , q) ,
n
o
−1
−1
Ig−1 (ΨQ )(a) (x1 .x−1
,
q)
≤
max
I
(x
,
q),
I
(x
,
q)
,
g (ΨQ )(a) 1
g (ΨQ )(a) 2
2
n
o
Fg−1 (ΨQ )(a) (x1 − x2 , q) ≤ max Fg−1 (ΨQ )(a) (x1 , q), Fg−1 (ΨQ )(a) (x2 , q) ,
n
o
−1
−1
Fg−1 (ΨQ )(a) (x1 .x−1
,
q)
≤
max
F
(x
,
q),
F
(x
,
q)
.
1
2
g (ΨQ )(a)
g (ΨQ )(a)
2
Thus, the theorem is proved.
5
Cartesian Product of Q-Neutrosophic Soft Fields
In this section, we define the Cartesian product of Q-NSFs and prove that it is also a Q-NSF.
Definition 5.1. Let (ΓQ , A) and (ΨQ , B) be two Q-NSFs over (F1 , +, .) and (F2 , +, .), respectively. Then, their
Cartesian product (ΛQ , A × B) = (ΓQ , A) × (ΨQ , B), where ΛQ (a, b) = ΓQ (a) × ΨQ (b) for (a, b) ∈ A × B.
Analytically, for x ∈ F1 , y ∈ F2 and q ∈ Q
nD
Eo
, where
(x, y), q , TΛQ (a,b) (x, y), q , IΛQ (a,b) (x, y), q , FΛQ (a,b) (x, y), q
ΛQ (a, b) =
Majdoleen Abu Qamar, Abd Ghafur Ahmad and Nasruddin Hassan, On Q-Neutrosophic Soft Fields.
90
Neutrosophic Sets and Systems, Vol. 32 2020
TΛQ (a,b) (x, y), q = min TΓQ (a) x, q , TΨQ (b) y, q ,
IΛQ (a,b) (x, y), q = max IΓQ (a) x, q , IΨQ (b) y, q ,
FΛQ (a,b) (x, y), q = max FΓQ (a) x, q , FΨQ (b) y, q .
Theorem 5.2. Let (ΓQ , A) and (ΨQ , B) be two Q-NSFs over (F1 , +, .) and (F2 , +, .), respectively. Then, their
Cartesian product (ΓQ , A) × (ΨQ , B) is a Q-NSF over (F1 × F2 ).
Proof. Let (ΛQ, A × B) = (ΓQ , A) × (ΨQ , B), where ΛQ (a, b) = ΓQ (a) × ΨQ (b) for (a, b) ∈ A × B. Then,
for (x1 , y1 ), q , (x2 , y2 ), q ∈ (F1 × F2 ) × Q we have,
TΛQ (a,b) (x1 , y1 ) − (x2 , y2 ), q
= TΛQ (a,b) (x1 − x2 , y1 − y2 ), q
= min TΓQ (a) (x1 − x2 ), q , TΨQ (b) (y1 − y2 ), q
n
o
≥ min min TΓQ (a) x1 , q , TΓQ (a) − x2 , q , min TΨQ (b) y1 , q , TΨQ (b) − y2 , q
n
o
≥ min min TΓQ (a) x1 , q , TΓQ (a) x2 , q , min TΨQ (b) y1 , q , TΨQ (b) y2 , q
n
o
= min min TΓQ (a) x1 , q , TΨQ (b) y1 , q , min TΓQ (a) x2 , q , TΨQ (b) y2 , q
n
o
= min TΛQ (a,b) (x1 , y1 ), q , TΛQ (a,b) (x2 , y2 ), q
also,
IΛQ (a,b)
(x1 , y1 ) − (x2 , y2 ), q
= IΛQ (a,b) (x1 − x2 , y1 − y2 ), q
= max IΓQ (a) (x1 − x2 ), q , IΨQ (b) (y1 − y2 ), q
n
o
≤ max max IΓQ (a) x1 , q , IΓQ (a) − x2 , q , max IΨQ (b) y1 , q , IΨQ (b) − y2 , q
n
o
≤ max max IΓQ (a) x1 , q , IΓQ (a) x2 , q , max IΨQ (b) y1 , q , IΨQ (b) y2 , q
n
o
= max max IΓQ (a) x1 , q , IΨQ (b) y1 , q , max IΓQ (a) x2 , q , IΨQ (b) y2 , q
n
o
= max IΛQ (a,b) (x1 , y1 ), q , IΛQ (a,b) (x2 , y2 ), q ,
similarly, FΛQ (a,b)
TΛQ (a,b)
(x1 , y1 ) − (x2 , y2 ), q
(x1 , y1 ).(x2 , y2 )−1 , q
n
≤ max FΛQ (a,b) (x1 , y1 ), q , FΛQ (a,b) (x2 , y2 ), q
o
. Next,
−1
= TΛQ (a,b) (x1 .x−1
2 , y1 .y2 ), q
−1
= min TΓQ (a) (x1 .x−1
2 ), q , TΨQ (b) (y1 .y2 ), q
n
o
−1
−1
≥ min min TΓQ (a) x1 , q , TΓQ (a) x2 , q , min TΨQ (b) y1 , q , TΨQ (b) y2 , q
Majdoleen Abu Qamar, Abd Ghafur Ahmad and Nasruddin Hassan, On Q-Neutrosophic Soft Fields.
91
Neutrosophic Sets and Systems, Vol. 32 2020
n
o
min TΓQ (a) x1 , q , TΓQ (a) x2 , q , min TΨQ (b) y1 , q , TΨQ (b) y2 , q
n
o
= min min TΓQ (a) x1 , q , TΨQ (b) y1 , q , min TΓQ (a) x2 , q , TΨQ (b) y2 , q
n
o
= min TΛQ (a,b) (x1 , y1 ), q , TΛQ (a,b) (x2 , y2 ), q ,
≥ min
IΛQ (a,b)
(x1 , y1 ).(x2 , y2 )−1 , q
−1
= IΛQ (a,b) (x1 .x−1
2 , y1 .y2 ), q
−1
= max IΓQ (a) (x1 .x−1
2 ), q , IΨQ (b) (y1 .y2 ), q
n
o
−1
≤ max max IΓQ (a) x1 , q , IΓQ (a) x−1
y
y
,
q
,
max
I
,
q
,
I
,
q
ΨQ (b) 1
ΨQ (b) 2
2
n
o
≤ max max IΓQ (a) x1 , q , IΓQ (a) x2 , q , max IΨQ (b) y1 , q , IΨQ (b) y2 , q
n
o
= max max IΓQ (a) x1 , q , IΨQ (b) y1 , q , max IΓQ (a) x2 , q , IΨQ (b) y2 , q
n
o
= max IΛQ (a,b) (x1 , y1 ), q , IΛQ (a,b) (x2 , y2 ), q ,
n
o
similarly, FΛQ (a,b) (x1 , y1 ), q . (x2 , y2 )−1 , q
≤ max FΛQ (a,b) (x1 , y1 ), q , FΛQ (a,b) (x2 , y2 ), q . This
completes the proof.
6
Conclusions
In this study, we have introduced the concept of Q-neutrosophic soft fields. We have investigated some of
its structural characteristics. Also, we have discussed the concepts of homomorphic image and pre-image of
Q-neutrosophic soft fields. Moreover, we have defined the Cartesian product of Q-neutrosophic soft fields and
discussed some related properties. The proposed notion enriches knowledge on neutrosophic sets in the branch
of algebra. Also, it illuminates the way for more further deep discussion in algebra under neutrosophic and
Q-neutrosophic soft environment for example, by establishing the notions of n-valued neutrosophic soft fields
Q-neutrosophic soft modules and more.
Acknowledgments: We are indebted to Universiti Kebangsaan Malaysia for providing support and facilities for this research. Also, we are indebted to Zerqa University, since this paper is an extended paper of a
short paper published in the 6th International Arab Conference on Mathematics and Computations (IACMC
2019), Special Session of Neutrosophic Set and Logic, Organized by F. Smarandache and S. Alkhazaleh Zerqa
University, Jordan.
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Received: Oct 30, 2019. Accepted: Mar 19, 2020
Majdoleen Abu Qamar, Abd Ghafur Ahmad and Nasruddin Hassan, On Q-Neutrosophic Soft Fields.
Neutrosophic Sets and Systems, Vol. 32, 2020
University of New Mexico
Neutrosophic projective G-submodules
Binu R.1,∗ and P. Isaac2
1
Department of Mathematics,Rajagiri School of Engineering and Technology, India; 1984binur@gmail.com
2
Department of Mathematics,Bharata Mata College, Thrikkakara, India; pibmct@gmail.com
∗
Correspondence: 1984binur@gmail.com; Tel.: ( 91-9946167116)
Abstract. A significant area of module theory is the concept of free modules, projective modules and injective
modules. The goal of this study is to characterize the projective G-modules under a single-valued neutrosophic
set. So we define neutrosophic G-submodule as a generic version of projective G-submodule. It also describes
and derives fundamental algebraic properties including quotient space and direct sum of neutrosophic projective
G-submodules
Keywords: Neutrosophic set; Neutrosophic G-module;Direct sum; Projective G-module; Neutrosophic projective G-module
—————————————————————————————————————————-
1. Introduction
The projective G-module in the abstract algebra plays a pivotal role to analyze the algebraic
structure G-module and its characteristics. Cartan and Eilemberge [16] introduced the concept
of projective modules that offer significant ideas through the theoretical approach to module
theory. The algebraic structure G-module widely used to study the representation of finite
groups developed by Frobenius G and Burnside [11] in the 19th century. Several researchers
have studied the algebraic structure in pure mathematics associated with uncertainty. Since
Zadeh [35] introduced fuzzy sets, fuzzification of algebraic structures was an important milestone in classical algebraic studies. The notion of a fuzzy submodule was introduced by Negoita
and Ralescu [25] and further developed by Mashinchi and Zahedi [24]. This basic notion has
been generalized in several ways after Zadeh’s implementation of fuzzy sets [4, 5]. In 1986
Atanassov [6] put forward intuitionistic fuzzy set theory in which each element coincides with
membership grades and non-membership grades. Biswas [9] applied the idea of the intuitionistic fuzzy set to the algebraic structure group and K. Hur et.al. [21] additionally studied it. In
Binu R & P.Isaac, Neutrosophic projective G-submodules
Neutrosophic Sets and Systems, Vol. 32, 2020
95
2011 P. Isaac, P.P.John [22] studied about algebraic nature of intuitionistic fuzzy submodule
of a classical module.
The theory of neutrosophy first appeared in philosophy [30] and then evolved neutrosophic
set as a mathematical tool. In 1995, Smarandache [31] outlined the neutrosophicset as a
combination of tri valued logic with non-standard analysis in which three different types of
membership values represent each element of a set.The main objective of the neutrosophic set
is to narrow the gap between the vague, ambiguous and imprecise real-world situations. Neutrosophic set theory gives a thorough scientific and mathematical model knowledge in which
speculative and uncertain hypothetical phenomena can be managed by hierarchal membership
of the components “ truth / indeterminacy / falsehood ” [2,3,32]. Neutrosophic set generalizes
a classical set, fuzzy set, interval-valued fuzzy set and intuitionistic fuzzy set that can be used
to make a mathematical model for the real problems of science and engineering. From a scientific and engineering perspective, Wang et.al. [20] specified the definition of a neutrosophic set,
which is called a single-valued neutrosophic set. Several scientists dealt with the neutrosophic
set notion as a new evolving instrument for uncertain information processing and a general
framework for uncertainty analysis in data set [1, 7, 17, 28].
The consolidation of the neutrosophic set hypothesis with algebraic structures is a growing
trend in mathematical research. Among the various branches of applied and pure mathematics,
abstract algebra was one of the first few topics where the research was carried out using
the neutrosophic set concept. W. B. Vasantha Kandasamy and Florentin Smarandache [23]
initially presented basic algebraic neutrosophic structures and their application to advanced
neutrosophic models. Vidan Cetkin [12, 13] consolidated the neutrosophic set theory and
algebraic structures, creating neutrosophic subgroups and neutrosophic submodules. F. Sherry
[18, 19] introduced the concept of fuzzy G-modules in which the concept of fuzzy sets was
combined with G-module and the theory of group representation. One of the key developments
in the neutrosophic set theory is the hybridization of the neutrosophic set with the algebraic
structure G-module. The above fact leads to inspiration for conducting an exploratory study
in the field of abstract algebra, especially in the theory of G-modules in conjunction with
neutrosophic set. In this paper we described neutrosophic projective G-submodule as the
general case of projective G-module and derived its algebraic properties.
The reminder of this work is structured as follows. Section 2 briefs about necessary preliminary definitions and results which are basic for a better and clear cognizance of next
sections. Section 3 defines neutrosophic projective G-modules, algebraic extension of projective G-submodules and derive the theorems related to quotient space and direct sum of
neutrosophic G-submodules. A comprehensive overview, relevance and future study of this
work is defined at the end of the paper in Section 4.
Binu R & Paul Isaac,Neutrosophic projective G-submodules
Neutrosophic Sets and Systems, Vol. 32, 2020
96
2. Preliminaries
In this section, we recall some of the preliminary definitions and results which are essential
for a better and clear comprehension of the upcoming sections.
Definition 2.1. [14] Let (G, ∗) be a group. A vector space M over the field K is called a
G-module, denoted as GM , if for every g ∈ G and m ∈ M ; ∃ a product (called the action of
G on M ), g · m ∈ M satisfies the following axioms
(1) 1G · m = m; ∀m ∈ M (1G being the identity element of G)
(2) (g ∗ h) · m = g · (h · m); ∀m ∈ M and g, h ∈ G
(3) g · (k1 m1 + k2 m2 ) = k1 (g · m1 ) + k2 (g · m2 ); ∀ k1 , k2 ∈ K; m1 , m2 ∈ M ”.
Example 2.1. [18] Let G = {1, −1, i, −i} and M = Cn ; (n ≥ 1). Then M is a vector space
over C and under the usual addition and multiplication of complex numbers we can show that
M is a G-module.
Definition 2.2. [15] Let M be a G-module. A vector subspace N of M is a G-submodule if
N is also a G-module under the same action of G.
Definition 2.3. [15] Let M and M ∗ be G-modules. A mapping f : M → M ∗ is called a
G module homomorphism (HomG (M, M ∗ )) if ∀ k1 , k2 ∈ K, m1 , m2 ∈ M, g ∈ G satisfies the
following conditions
(1) f (k1 m1 + k2 m2 ) = k1 f (m1 ) + k2 f (m2 )
(2) f (gm) = gf (m)
Definition 2.4. [10, 29] A G-module M is projective if for any G-module M ∗ and any Gsubmodule N ∗ of M ∗ , every homomorphism ϕ : M → M ∗ /N ∗ can be lifted to a homomorphism
ψ : M → M ∗ or π ◦ ψ = ϕ where π : M ∗ → M ∗ /N ∗ .
Remark 2.1. A G-module M is projective if and only if M is M ∗ projective for every Gmodule M ∗
Theorem 2.2. [29] Let M and M ∗ be G-modules such that M is M ∗ projective. Let N ∗ be
any G-submodule of M ∗ . Then M is N ∗ projective and M is M ∗ /N ∗ projective.
Proposition 2.1. [29] Let M and Mi be G-modules.Then M is ⊕ni=1 Mi -projective if and
only if M is Mi -projective ∀ i
Definition 2.5.
[32, 34] A neutrosophic set P of the universal set X is defined as P =
{(η, tP (η), iP (η), fP (η)) : η ∈ X} where tP , iP , fP : X → (− 0, 1+ ). The three components
tP , iP and fP represent membership value (Percentage of truth), indeterminacy (Percentage
Binu R & Paul Isaac,Neutrosophic projective G-submodules
97
Neutrosophic Sets and Systems, Vol. 32, 2020
of indeterminacy) and non membership value (Percentage of falsity) respectively. These components are functions of non standard unit interval (− 0, 1+ ) [27].
Remark 2.3. [20, 32]
(1) If tP , iP , fP : X → [0, 1], then P is known as single valued neutrosophic set(SVNS).
(2) In this paper, we discuss about the algebraic structure R-module with underlying set
as SVNS. For simplicity SVNS will be called neutrosophic set.
(3) U X denotes the set of all neutrosophic subset of X or neutrosophic power set of X.
Definition 2.6. [26, 32] Let P, Q ∈ U X . Then P is contained in Q, denoted as P ⊆ Q if
and only if P (η) 6 Q(η) ∀η ∈ X, this means that tP (η) ≤ tQ (η), iP (η) ≤ iQ (η), fP (η) ≥
fQ (η), ∀ η ∈ X.
Definition 2.7. [26, 33]For any neutrosophic subset P = {(η, tP (η), iP (η), fP (η)) : η ∈ X},
the support P ∗ of the neutrosophic set P can be defined as P ∗ = {η ∈ X, tP (η) > 0, iP (η) >
0, fP (η) < 1}.
Definition 2.8. [8] Let (G, ∗) be a group and M be a G module over a field K. A neutrosphic
G-submodule is a neutrosophic set P = {(η, tP (η), iP (η), fP (η)) : η ∈ M } in GM such that the
following conditions are satisfied;
(1) tP (̺η + τ θ) ≥ tP (η) ∧ tP (θ)
iP (̺η + τ θ) ≥ iP (η) ∧ iP (θ)
fP (̺η + τ θ) ≤ fP (η) ∨ fP (θ),
∀ η, θ ∈ M, ̺, τ ∈ K
(2) tP (ξη) ≥ tP (η)
iP (ξη) ≥ iP (η)
fP (ξη) ≤ fP (η) ∀ ξ ∈ G, η ∈ M
Remark 2.4. We denote neutrosophic G-submodules using single valued neutrosophic set by
U (GM ).
Example 2.2. Consider the example 2.1 for G-module M . Define a neutrosophic set
P = {η, tP (η), iP (η), fP (η) : η ∈ M }
of M where
tP (η) =
1
0.5
if η = 0
if η 6= 0
, iP (η) =
1
0.5
if η = 0
if η 6= 0
, fP (η) =
Then P is a neutrosophic G-submodule of M .
Binu R & Paul Isaac,Neutrosophic projective G-submodules
0
0.25
if η = 0
if η 6= 0
Neutrosophic Sets and Systems, Vol. 32, 2020
98
Definition 2.9. [8] Let P = {(x, tP (x), iP (x), fP (x)) : x ∈ X} ∈ U ((GM ) ). The support
P ∗ of the neutrosophic G-submodule P can be defined as P ∗ = {x ∈ X, tP (x) > 0, iP (x) >
0, fP (x) < 1, ∀x ∈ GM }.
Proposition 2.2. If P ∈ U (GM ), then the support P ∗ ∈ GM .
Definition 2.10. [8] Let P ∈ U (GM ) and N be a G-submodule of M . Then the restriction
of P to N is denoted by P |N and it is a neutrosophic set of N defined as follows P |N (η) =
(η, tP |N (η), iP |N (η), fP |N (η)) where
tP |N (η) = tP (η), iP |N (η) = iP (η), fP |N (η) = fP (η), ∀η ∈ N .
Proposition 2.3. [8] Let P ∈ U (GM ) and N ⊆ M then P |N ∈ U (GN ).
Definition 2.11. [8] Let M ∈ GM and N be a G-submodule of M . Then the neutrosophic
set PN of M/N defined as PN (η + N ) = {η + N, tPN (η + N ), iPN (η + N ), fPN (η + N )},where
tPN (η + N ) = ∨tP (η + n) : n ∈ N
iPN (η + N ) = ∨iP (η + n) : n ∈ N
fPN (η + N ) = ∧fP (η + n) : n ∈ N , ∀η ∈ M
Proposition 2.4. [8] Let M ∈ GM . Let N be a G-submodule of M . Then PN ∈ U (GM/N ).
Proposition 2.5. [8] Let P ∈ U (GM ) and Q ∈ U (GM ∗ ) where M and M ∗ are G-modules
over the field K. Let r ∈ [0, 1], the neutrosophic set Qr = {η, tQr (η), iQr (η), fQr (η) : η ∈ M ∗ }
defined by tQr (η) = tQ (η) ∧ r, iQr (η) = iQ (η) ∧ r, fQr (η) = fQ (η) ∨ (1 − r) ∀ η ∈ M ⋆ be a
neutrosophic G-submodule.
Definition 2.12. [8] Let M and M ∗ be G-modules over K and a mapping Υ : M → M ∗ is
a G-module homomorphism. Also P ∈ U (GM ) and Q ∈ U (GM ∗ ). A homomorphism Υ of M
on to M ∗ is called weak neutrosophic G-submodule homomorphism of P into Q if Υ(P ) ⊆ Q.
If Υ is a weak neutrosophic G-module homomorphism of P into Q, then P is weakly
homomorphic to Q and we write P ∼ Q.
A homomorphism Υ of M on to M ∗ is called a neutrosophic G-module homomorphism
of P onto Q if Υ(P ) = Q and we represent it as P ≈ Q.
3. Neutrosophic Projective G module
In this section we discuss the generalized notion of projective G-modules, called neutrosophic projective G-modules, and study several characteristics of projective G-modules in the
neutrosophic domain.
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Definition 3.1. Let M and M ∗ be G-modules. Let P = {η, tP (η), iP (η), fP (η) : η ∈ M } be
neutrosophic G submodule of M and Q = {η, tQ (η), iQ (η), fQ (η) : η ∈ M ∗ } be neutrosophic
G-submodule of M ∗ . Then P is said to be Q projective, if the following conditions are satisfied;
(1) M is M ∗ projective
(2) tP (η) ≤ tQ (ψ(η))
(3) iP (η) ≤ iQ (ψ(η))
(4) fP (η) ≥ fQ (ψ(η)), ∀ ψ ∈ Hom(M, M ∗ ), η ∈ M
Theorem 3.1. Let P and Q be neutrosophic G-submodules of finite dimensional G-modules
of M and M ∗ respectively and M is M ∗ projective. Let {β1 , β2 , ..., βn } be a basis for M ∗ . If
(1) tP (η) ≤ min{tQ (βj ); j = 1, 2, ..., n}
(2) iP (η) ≤ min{iQ (βj ); j = 1, 2, ..., n}
(3) fP (η) ≥ max{fQ (βj ); j = 1, 2, ..., n}, ∀ η ∈ M
Then P is Q-projective.
Proof. Let Q = {η, tB (η), iB (η), fB (η) : η ∈ M ∗ } be a neutrosophic G submodule of M ∗ . Then
∀ η1 , η2 ∈ M ∗ ; ̺, τ ∈ K;
(1) tQ (̺η1 + τ η2 ) ≥ tQ (η1 ) ∧ tQ (η2 )
(2) iQ (̺η1 + τ η2 ) ≥ iQ (η1 ) ∧ iQ (η2 )
(3) fQ (̺η1 + τ η2 ) ≤ fQ (η1 ) ∨ fQ (η2 )
(4) tQ (ξη) ≥ tP (η), iQ (ξη) ≥ iQ (η), fQ (ξη) ≤ fQ (η) ∀ η ∈ M ∗ , ξ ∈ G
Also P is a neutrosophic G-submodule of M and M is M ∗ projective G-module and ψ ∈
Hom(M, M ∗ ) be any G-module homomorphism. For any η ∈ M, ψ(η) ∈ M ∗ .
∴ ψ(η) = α1 β1 + α2 β2 + ... + αn βn , αi ∈ K, βi ∈ M ∗ , i = 1, 2, ..., n
tQ (ψ(η)) = tQ (α1 β1 + α2 β2 + ... + αn βn )
≥ tQ (β1 ) ∧ tQ (β2 ) ∧ ...
∧tQ (βn )
= mini{tQ (β1 ), tQ (β2 ), ...,
tQ (βn )}
≥ tP (η)
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Similarly iQ (ψ(η)) ≥ iP (η)
fQ (ψ(η)) = fQ (α1 β1 + α2 β2 + ... + αn βn )
≤ fQ (β1 ) ∧ tQ (β2 ) ∧ ... ∧ tQ (βn )
= max{fQ (β1 ), fQ (β2 ),
... , fQ (βn )}
≤ fP (η)
∴ P is Q projective.
Theorem 3.2. Let P ∈ U (GM ), Q ∈ U (GM ∗ ) and P is Q projective. If N ∗ is a G-submodule
of M ∗ and C ∈ U (GN ∗ ), then P is C-Projective if Q|N ∗ ⊆ C
Proof. Given P is Q projective, then
(1) M is M ∗ projective
(2) tP (η) ≤ tQ (ψ(η)),
iP (η) ≤ iQ (ψ(η))
fP (η) ≥ fQ (ψ(η))
∀ ψ ∈ HomG (M, M ∗ ), η ∈ M . Since N ∗ is a G-submodule of M ∗ , by a theorem 2.2, M is N ∗
projective. Let ϕ ∈ HomG (M, N ∗ ) and θ : N ∗ → M ∗ be the inclusion homomorphism. Then
θ◦ϕ=ψ
∴ from the condition 2
tP (η) ≤ tQ (ψ(η)) = tQ (θ ◦ ϕ)(η)
= tQ (θ(ϕ(η))) = tQ (ϕ(η)).
Similarly iP (η) ≤ iQ (ϕ(η)) and fP (η) ≥ fQ (ϕ(η) ∀ η ∈ M, ϕ ∈ HomG (M, N ∗ ).
Given C ∈ U (GN ∗ ), ϕ(η) ∈ N ∗ and Q|N ∗ ⊆ C
tQ|N ∗ (ϕ(η)) = tQ (ϕ(η)) ≤ tC (ϕ(η)
⇒ tP (η) ≤ tC (ϕ(η)). Similarly, iP (η) ≤ iC (ϕ(η)) and fP (η) ≥ fC (ϕ(η)). Hence P is CProjective.
Theorem 3.3. Let M and M ∗ be G-modules where P and Q are neutrosophic G-submodules
of M and M ∗ respectively. Let r ∈ [0, 1], the neutrosophic set Qr = {η, tQr (η), iQr (η), fQr (η) :
η ∈ M ∗ } defined by tQr (η) = tQ (η) ∧ r, iQr (η) = iQ (η) ∧ r, fQr (η) = fQ (η) ∨ (1 − r) ∀ η ∈ M ⋆
be a neutrosophic G- submodule. If P is Qr projective, then P is Q projective.
Proof. Consider P as Qr projective where r ∈ [0, 1]. Then
(1) M is M ∗ projective
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101
(2) tP (η) ≤ tQr (ψ(η)),
iP (η) ≤ iQr (ψ(η)),
fP (η) ≥ fQr (ψ(η)),
ψ ∈ HomG (M, M ∗ ) and η ∈ M
Since Qr ⊆ Q, ⇒ tQr (ψ(η)) ≤ tQ (ψ(η),
iQr (ψ(η)) ≤ iQ (ψ(η)) and
fQr (ψ(η)) ≥ fQ (ψ(η)), ∀ ψ(η) ∈ M ∗ .
⇒ tP (η) ≤ tQ (ψ(η)),
iP (η) ≤ iQ (ψ(η)) and
fP (η) ≥ fQ (ψ(η)) ∀ η ∈ M.
∴ P is Q projective.
Proposition 3.1. Let M = ⊕ni=1 Mi be a G-module where Mi′ s are G-submodules of M . If
Pi ∈ U (GMi ) (1 ≤ i ≤ n), then the neutrosophic set P of M defined by tP (η) = ∧{tpi (ηi ) :
i = 1, 2, ..., n} ,iP (η) = ∧{ipi (ηi ) : i = 1, 2, ..., n} and fP (η) = ∨{fPi (ηi ) : i = 1, 2, ..., n} where
P
η = i=n
i=1 (ηi ), ηi ∈ Mi , is a neutrosophic G-submodule of M .
Pi=n
P
Proof. Let η, ν ∈ M where η = i=n
i=1 νi . Each ηi , νi ∈ Mi and ̺, b ∈ K.
i=1 ηi and ν =
Pi=n
Then by definition, ̺η + τ ν = i=1 [̺ηi + τ νi ] where ̺ηi + τ νi ∈ Mi (1 ≤ i ≤ n). Now
tP (̺η + τ ν) = ∧ tPi (̺ηi + τ νi )
≥ ∧ {tPi (ηi ), tPi (νi )}
= {∧ tPi (ηi )} ∧ {∧ tPi (νi )}
= tP (η) ∧ tP (ν)
Similarly iP (̺η + τ ν) ≥ iP (η) ∧ iP (ν)
Now consider
fP (̺η + τ ν) = ∨ fPi (̺ηi + τ νi )
≤ ∨ {fPi (ηi ), fPi (νi )}
= {∨ fAi (ηi )} ∨ {∨ fPi (νi )}
= fP (η) ∨ fP (ν)
Now, for g ∈ G, η ∈ M
tP (gη) = ∧ tPi (gηi )
≥ ∧ {tPi (ηi )}
= tP (η)
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Similarly iP (gη) ≥ iP (η), fP (gη) ≤ fP (η) ∴ P ∈ U (GM ).
Definition 3.2. Let M = ⊕ni=1 Mi be a G module where Mi′ s are G-submodules of M . If Pi ∈
U (GMi ) (1 ≤ i ≤ n) and P ∈ U (GM =⊕ni=1 Mi ) with tP (0) = tPi (0), iP (0) = iPi (0) and fP (0) =
fPi (0) ∀ i Then P is called the direct sum of Pi and it is denoted as P = ⊕ni=1 Pi .
Theorem 3.4. Let M = ⊕ni=1 Mi be G module where Mi′ s are G submodules of M . Let
P ∈ U (GM ) and Qi ∈ U (GMi ) such that Q = ⊕ni=1 Qi . Then P is Q projective if and only if
P is Qi projective ∀i.
Proof. Assume that P is Q-projective, then
(1) M is M projective
(2) tP (η) ≤ tQ (ψ(η)),
iP (η) ≤ iQ (ψ(η)
fP (η) ≥ fQ (ψ(η)
ψ ∈ HomG (M, M ); η ∈ M
To prove that P is Qi projective where i = 1, 2, ..., n, it is enough to prove the following
conditions.
(1) M is Mi -projective
(2) tP (η) ≤ tQi (ϕ(η)),
iP (η) ≤ iQi (ϕ(η))
fP (η) ≥ fQi (ϕ(η))
where ∀ ϕ ∈ HomG (M, Mi ), η ∈ M .
Here M is M = ⊕ni=1 Mi -projective and by the the proposition 2.2, M is Mi projective ∀ i =
1, 2, ..., n. Let ϕ ∈ HomG (M, Mi ) and θ : Mi → M ∈ HomG (Mi , M ) (inclusion) such that
ψ = θ ◦ ϕ. Then ∀ ϕ ∈ HomG (M, Mi )
tP (η) ≤ tQ (ψ(η)
= tQ ((θ ◦ ϕ)(η))
= tQ (θ(ϕ(η)))
= tQ (ϕ(η))
Similarly iP (η) ≤ iQ (ϕ(η)) and
fP (η) ≥ fQ (ψ(η))
= fQ ((θ ◦ ϕ)(η))
= fQ (θ(ϕ(η)))
= fQ (ϕ(η))
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Now ϕ(η) ∈ Mi ⊆ M and η ∈ M and consider
ϕ(η) = 0 + 0 + ... + ϕ(η) + ... + 0
Then
tQ (ϕ(η)) = tQ (0 + 0 + ... + ϕ(η) + ... + 0)
= tQ1 (0) ∧ tQ2 (0) ∧ ... ∧ tQi ϕ(η) ∧ ... ∧ tQn (0)
= tQi (ϕ(η))
Similarly iQ (ϕ(η)) = iQi (ϕ(η)) and
fQ (ϕ(η)) = fQi (ϕ(η)) ∀ i
⇒ tP (η) ≤ tQ (ϕ(η)) = tQi (ϕ(η)).
Also iP (η) ≤ iQ (ϕ(η)) = iQi (ϕ(η) and
fP (η) ≥ fQ (ϕ(η)) = fQi (ϕ(η), ∀ η ∈ M, ϕ ∈ HomG (M, Mi ).
Then P is Qi projective.
Conversely Assume that P is Qi projective where i = 1, 2, ..., n.Then
(1) M is Mi -projective
(2) tP (m) ≤ tQi (ϕi (m)),
iP (m) ≤ iQi (ϕi (m) and
fP (m) ≥ fQi (ϕi (m),
ϕi ∈ HomG (M, Mi ); m ∈ M
To prove P is Q projective, it is enough to prove the following conditions
(1) M is M projective
(2) tP (η) ≤ tQ (ψ(η)),
iP (η) ≤ iQ (ψ(η))
fP (η) ≥ fQ (ψ(η)), ψ ∈ HomG (M, M ); η ∈ M
1. :- Since P is Qi projective and proposition 2.1, M is M -Projective where M = ⊕ni=1 Mi .
2. :- Let ψ ∈ HomG (M, M ) where M = ⊕ni=1 Mi such that ∀ η ∈ M,
ψ(η) ∈ M, i.e. ψ(η) = η1 + η2 + ... + ηn , ∀ ηi ∈ Mi , 1 ≤ i ≤ n and πi : M → Mi be the
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projection map where i = 1, 2, ..., n such that πi (ψ(η)) = ηi , ∀ i., then
ψ(η) = η1 + η2 + ... + ηn ,
∀ ηi ∈ Mi , 1 ≤ i ≤ n
= π1 (ψ(η)) + π2 (ψ(η)) + ...
... + πn (ψ(η))
= (π1 ◦ ψ)(η) + (π2 ◦ ψ)(η) + ... +
(πn ◦ ψ)(η)
= ϕ1 (η) + ϕ2 (η) + ... + ϕn (η)
Also
tQ (ψ(η)) = tQ (ϕ1 (η)) + tQ (ϕ2 (η)) + ... +
tQ (ϕn (η))
= ∧{tQi (ϕi (η)) : 0 ≤ i ≤ n}
[by the proposition 3.1]
≥ tP (η)
Similarly iQ (ψ(η)) ≥ iP (η) and
fQ (ψ(η)) = fQ (ϕ1 (η)) + fQ (ϕ2 (η)) +
... + fQ (ϕn (η))
≤ ∨{fQi (ϕi (η)) : 0 ≤ i ≤ n}
≤ fP (m)
∴ A is Q projective.
4. Conclusion
The study of G-module in a neutrosophic set domain using a single-valued neutrosophic set
provides a new step in the algebra sector and helps to analyze group action in application level
on a vector space. Projective G-modules expand the free G-modules class by maintaining a
portion of the free module’s primary properties. Neutrosophic projective G-module is one of
the most generalizations of classical projective G-module. This paper has developed, the notion
of projectivity of neutrosophic G-modules and its quotient and direct sum properties of M
projectivity. This analysis leads to the extension of the quasi projective module, neutrosophic
injective & projective modules and its features in neutrosophic domain.
Binu R & Paul Isaac,Neutrosophic projective G-submodules
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Conflicts of Interest: Declare conflicts of interest or state ”The authors declare no conflict
of interest.”
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Received: Nov 21, 2019. Accepted: Mar 20, 2020
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Neutrosophic Sets and Systems, Vol. 32, 2020
University of New Mexico
Quadripartitioned Single valued Neutrosophic Dombi Weighted Aggregation
Operators for Multiple Attribute Decision Making
M.Mohanasundari1 , K.Mohana2
1
Department of Mathematics, Bannari Amman Institute of Technology; mohanadhianesh@gmail.com
2
Department of mathematics, Nirmala college for women; riyaraju1116@gmail.com
∗
Correspondence: mohanadhianesh@gmail.com
Abstract. In this paper we have introduced the concept of score and accuracy function of the Quadripartitioned Single valued Neutrosophic Numbers (QSVNN) and also defined ranking methods between two QSVNNs
which is based on its score function. Dombi operators are used in solving many Multicriteria Attribute Group
decision making (MAGDM) problems because of its very good flexibility with a general parameter.Here Dombi
T-norm and T-conorm operations of two QSVNNs are defined. Based on this Dombi operations, we introduced
two Dombi weighted aggregation operators QSVNDWAA and QSVNDWGA under Quadripartitioned Single
valued Neutrosophic environment and also studied its properties. Finally, we discussed about Multicriteria
Attribute Decision making method (MADM) using QSVNDWAA or QSVNDWGA operator and also an illustrative example is given for the proposed method which gives a detailed results to select the best alternative
based upon the ranking orders.
Keywords: Quadripartitioned single valued neutrosophic sets, Score and Accuracy functions, Dombi Weighted
Aggregation Operators .
—————————————————————————————————————————-
1. Introduction
Fuzzy sets which allows the elements to have a degrees of membership in the set and it was
introduced by Zadeh [31] in 1965. The degrees of membership lies in the real unit interval
[0, 1]. Intuitionstic fuzzy set (IF S) allows both membership and non membership to the elements and this was introduced by Atnassov [1] in 1983. By introducing one more component
in IFS set neutrosophic set was introduced by Smarandache [19] in 1998. Neutrosophic set
has three components truth membership function, indeterminacy membership function and
falsity membership function respectively. This neutrosophic set helps to handle the indeterminate and inconsistent information effectively. Later Wang [21] (2010) introduced the concept
of Single valued Neutrosophic set (SVNS) which is a generalization of classic set, fuzzy set,
M.Mohanasundari1 , K.Mohana2 , Quadripartitioned Single valued Neutrosophic Dombi Weighted Aggregation Operators for Multiple Attribute Decision Making
Neutrosophic Sets and Systems, Vol. 32, 2020
108
interval valued fuzzy set and intuitionstic fuzzy set.
In 1982 Pawlak [17] defined the standard version of rough set theory which is given in terms
of a pair of sets that is lower and upper approximation sets. It provided a new approach
to vagueness which is defined by a boundary region of a set. Later Yang(2017) [23] defined
a new hybrid model of single valued neutrosophic rough set model and it has many applications in medical diagnosis, decision making problems, image processing etc., Neutrosophic
set helps to solve many real life world problems [2–6] because of its uncertainty analysis in
data sets. K.Mohana, M.Mohanasundari [15] studied On Some Similarity Measures of Single Valued Neutrosophic Rough Sets and applied the concept in Medical Diagnosis problem.
When indeterminacy component in neutrosophic set is divided into two parts namely ’Contradiction’ ( both true and false ) ’Unknown’ ( neither true nor false) we get four components
that is T,C,U,F which define a new set called ’Quadripartitioned Single valued neutrosophic
set’ (QSVNS)introduced by Rajashi Chatterjee., et al. [18] And this is completely based on
Belnap’s four valued logic and Smarandache’s ’Four Numerical valued neutrosophic logic’.
By combining the concept of rough set and QSVNS a new hybrid model of ’Quadripartitioned Single valued neutrosophic Rough set’ (QSVNRS) was introduced by K.Mohana and
M.Mohanasundari. [16]
Many mathematical operations like average, aggregate, sum, count, max, min are performed
with the help of aggregation operations.Multicriteria Attribute decision making (MADM) is
an approach which is used to select a best one when several alternatives are included under consideration of many attributes. So many researchers [8, 11, 24–27, 29] pay attention to
solve the Multicriteria Attribute decision making problems using the concept of various correlation coefficients of the different sets like fuzzy set, IFS, SVNS, QSVNS. And also many
researchers [12–14, 20, 22, 28, 30, 33] used aggregation operators as one of the tool to solve a
Multicriteria decision making problem and also studied its properties. Dombi Bonferroni mean
operators were introduced by Dombi [10] in 1982 which is used in many Multicriteria Attribute
Group decision making (MAGDM) problems because of its very good flexibility with a general
parameter. J.Chen and J.Ye [7] studied Some Single-valued Neutrosophic Dombi Weighted
Aggregation Operators for Multiple Attribute Decision-Making problem.
In this paper Section 2 deals about the basic definitions of Quadripartitioned Single valued
neutrosophic sets, Score and accuracy function of single valued neutrosophic number, Dombi
T norm and T conorm operations of two single valued neutrosophic numbers(SVNN) and its
properties. We have defined Score and accuracy function of quadripartitioned single valued
M.Mohanasundari1 , K.Mohana2 , Quadripartitioned Single valued Neutrosophic Dombi
Weighted Aggregation Operators for Multiple Attribute Decision Making
Neutrosophic Sets and Systems, Vol. 32, 2020
109
neutrosophic number, Dombi T norm and T conorm operations of two quadripartitioned single
valued neutrosophic numbers(QSVNN) in Section 3. Based on the operations of Dombi T norm
and T conorm on two QSVNNs we have defined two aggregation operators QSVNDWAA and
QSVNDWGA and also studied its properties. Section 4 deals about Multicriteria Attribute
Decision making (MADM) method using the above proposed operators QSVNDWAA and
QSVNDWGA. Finally an illustrative example is given in the method which we have discussed
in Section 4.
2. Preliminaries
2.1 Quadripartitioned single valued neutrosophic sets
Definition 2.1. [19]
Neutrosophic set is defined over the non-standard unit interval ]− 0, 1+ [ whereas single valued
neutrosophic set is defined over standard unit interval [0, 1].It means a single valued neutrosophic set A is defined by
A = {hx, TA (x), IA (x), FA (x)i : x ∈ X}
where TA (x), IA (x), FA (x) : X → [0, 1] such that 0 ≤ TA (x) + IA (x) + FA (x) ≤ 3
Definition 2.2. [18]
Let X be a non-empty set. A quadripartitioned single valued neutrosophic set (QSVNS)
A over X characterizes each element in X by a truth-membership function TA (x), a contradiction membership function CA (x), an ignorance
membership function UA (x) and a
falsity membership function FA (x) such that for each x ∈ X , TA , CA , UA , FA ∈ [0, 1]
and 0 ≤ TA (x) + CA (x) + UA (x) + FA (x) ≤ 4 when X is discrete, A is represented as
P
A = ni=1 hTA (xi ), CA (xi ), UA (xi ), FA (xi )i /xi , xi ∈ X.
Definition 2.3. [18]
The complement of a QSVNS A is denoted by AC and is defined as,
P
AC = ni=1 hFA (xi ), UA (xi ), CA (xi ), TA (xi )i /xi , xi ∈ X i.e., TAC (xi ) = FA (xi ),
CAC (xi ) = UA (xi ), UAC (xi ) = CA (xi ), FAC (xi ) = TA (xi ), xi ∈ X
Definition 2.4. [18]
Consider two QSVNS A and B, over X. A is said to be contained in B, denoted by A ⊆ B iff
TA (x) ≤ TB (x), CA (x) ≤ CB (x), UA (x) ≥ UB (x), and FA (x) ≥ FB (x)
Definition 2.5. [18]
The union of two QSVNS A and B is denoted by A ∪ B and is defined as,
M.Mohanasundari1 , K.Mohana2 , Quadripartitioned Single valued Neutrosophic Dombi
Weighted Aggregation Operators for Multiple Attribute Decision Making
Neutrosophic Sets and Systems, Vol. 32, 2020
A∪B =
xi , xi ∈ X
Pn
i=1 hTA (xi )
110
∨ TB (xi ), CA (xi ) ∨ CB (xi ), UA (xi ) ∧ UB (xi ), FA (xi ) ∧ FB (xi )i /
Definition 2.6. [18]
The intersection of two QSVNS A and B is denoted by A ∩ B and is defined as,
A∩B =
xi , xi ∈ X
Pn
i=1 hTA (xi )
∧ TB (xi ), CA (xi ) ∧ CB (xi ), UA (xi ) ∨ UB (xi ), FA (xi ) ∨ FB (xi )i /
Definition 2.7. [21]
Let X be a universal set. A SV N S N in X is described by a truth-membership function tN (x) ,
an indeterminacy-membership function uN (x), and a falsity-membership function vN (x). Then
a SV N S N can be denoted as the following form:
N = {hx, tN (x), uN (x), vN (x)i |x ∈ X}
where the functions tN (x), uN (x), vN (x) ∈ [0, 1] satisfy the condition 0 ≤ tN (x) + uN (x) +
vN (x) ≤ 3 for x ∈ X. For convenient expression, a basic element hx, tN (x), uN (x), vN (x)i in
N is denoted by s = ht, u, v, i which is called a SVNN. For any SVNN s = ht, u, v, i, its score
and accuracy functions can be introduced, respectively as follows:
E(s) = (2 + t − u − v)/3,
H(s) = t − v,
E(s) ∈ [0, 1],
H(s) ∈ [−1, 1]
According to the two functions E(s) and H(s), the comparison and ranking of two SVNNs are
introduced by the following definition.
Definition 2.8. [32] Let s1 = ht1 , u1 , v1 i and s2 = ht2 , u2 , v2 i be two SVNNs. Then the
ranking method for s1 and s2 is defined as follows:
(1) If E(s1 ) > E(s2 ) then s1 ≻ s2 ,
(2) If E(s1 ) = E(s2 ) and H(s1 ) > H(s2 ) then s1 ≻ s2 ,
(3) If E(s1 ) = E(s2 ) and H(s1 ) = H(s2 ) then s1 = s2 .
Definition 2.9. [10] Let p and q be any two real numbers. Then, the Dombi T-norm and
T-conorm between p and q are defined as follows:
OD (p, q) =
c (p, q)
OD
M.Mohanasundari1 ,
K.Mohana2 ,
1+
n
=1−
1+
1−p
p
n
1
ρ
ρ o1/ρ ,
+ 1−q
q
p
1−p
1
ρ
ρ o1/ρ ,
q
+ 1−q
Quadripartitioned Single valued Neutrosophic Dombi
Weighted Aggregation Operators for Multiple Attribute Decision Making
Neutrosophic Sets and Systems, Vol. 32, 2020
111
where ρ ≥ 1 and (p, q) ∈ [0, 1] × [0, 1].
According to the Dombi T-norm and T-conorm, we define the Dombi operations of SVNNs.
Definition 2.10. [7] Let s1 = ht1 , u1 , v1 i and s2 = ht2 , u2 , v2 i be two SVNNs, ρ ≥ 1, and
λ > 0. Then, the Dombi T-norm and T-conorm operations of SVNNs are defined below:
(1) *
s1
⊕
1−
(2) s*1
1+
1+
n
n
1−t1
t1
*
t1
1−t1
(4)
=
=
1
1
, n 1−v ρ 1 1−v ρ o1/ρ
ρ
ρ o1/ρ ,
o
n
t2
1−u1 ρ
1−u2 ρ 1/ρ
1
+ 1−t
+
+ v 2
1+
1+
u
u
v
2
1
⊗
2
1
2
+
s2
1
ρ
ρ o1/ρ , 1
1−t
+ t 2
(3) λs1 = 1 −
*
sλ1
s2
2
−
1+
n
u1
1−u1
1
ρ
ρ o1/ρ , 1
u2
+ 1−u
−
1+
2
n
1
1
1
ρ o1/ρ ,
ρ o1/ρ ,
ρ o1/ρ
n
n
n
t1
1−u
1−v
1+ λ 1−t
1+ λ u 1
1+ λ v 1
1
1
1
n
ρ o1/ρ , 1
1−t
1+ λ t 1
1
−
1
1
n
ρ o1/ρ , 1
u1
1+ λ 1−u
1
−
v1
1−v1
+
1
n
ρ o1/ρ
v1
1+ λ 1−v
1
1
ρ
ρ o1/ρ
v2
+ 1−v
2
+=
+
[7] Let sj = htj , uj , vj i (j = 1, 2, ..., n) be a collection of SVNNs and
P
w = (w1 , w2 , ..., wn ) be the weight vector for sj with wj ∈ [0, 1] and nj=1 wj = 1. Then, the
Definition 2.11.
SVNDWAA and SVNDWGA operators are defined respectively as follows:
n
L
SVNDWAA (s1 , s2 , ..., sn ) =
w j sj
j=1
n
N
SVNDWGA (s1 , s2 , ..., sn ) =
j=1
w
sj j
3. Quadripartitioned single valued Neutrosophhic Dombi Operations
Definition 3.1. For an QSVNNs q = ht, c, u, f i its score and accuracy functions are defined
by,
E(q) = (3 + t − c − u − f )/4,
H(q) = t − f,
E(q) ∈ [0, 1],
H(q) ∈ [−1, 1]
(1)
(2)
The following definition defined the comparison and ranking of any two QSVNNs based on
the two functions E(s) and H(s).
Definition 3.2. Let q1 = ht1 , c1 , u1 , f1 i and q2 = ht2 , c2 , u2 , f2 i be two QSVNNs. Then the
ranking method for q1 and q2 is defined as follows:
M.Mohanasundari1 , K.Mohana2 , Quadripartitioned Single valued Neutrosophic Dombi
Weighted Aggregation Operators for Multiple Attribute Decision Making
Neutrosophic Sets and Systems, Vol. 32, 2020
112
(1) If E(q1 ) > E(q2 ) then q1 ≻ q2 ,
(2) If E(q1 ) = E(q2 ) and H(q1 ) > H(q2 ) then q1 ≻ q2 ,
(3) If E(q1 ) = E(q2 ) and H(q1 ) = H(q2 ) then q1 = q2 .
Definition 3.3. The Dombi T-norm and T-conorm operations of any two QSVNNs q1 =
ht1 , c1 , u1 , f1 i and q2 = ht2 , c2 , u2 , f2 i are defined as follows:
*
(1) q1 ⊕ q2 =
1−
1+
1+
(2) q1 ⊗ q2 =
*
1+
n
n
n
t1
1−t1
1−u1
u1
1−t1
t1
(4) q1λ =
1+
2
n
c1
1−c1
1
ρ
ρ o1/ρ ,
c2
+ 1−c
2
1
1
n
ρ
ρ o1/ρ ,
o
1−u
1−f1 ρ
1−f2 ρ 1/ρ
1+
+ u 2
+
f
f
2
1
2
2
1
1
1+
(3) λq1 = 1 −
*
−
+
1
1
,
n
ρ
ρ o1/ρ ,
o
1−t
1−c1 ρ
1−c2 ρ 1/ρ
1+
+ t 2
+
c
c
1−
*
1
ρ
ρ o1/ρ , 1
t2
+ 1−t
n
u1
1−u1
ρ
ρ o1/ρ , 1
u2
+ 1−u
1
n
ρ o1/ρ , 1
t1
1+ λ 1−t
2
−
1
2
1
−
1+
n
f1
1−f1
ρ
ρ o1/ρ
f2
+ 1−f
2
1
1
1
n
n
n
ρ o1/ρ ,
ρ o1/ρ ,
ρ o1/ρ
c1
1−u
1−f
1+ λ 1−c
1+ λ u 1
1+ λ f 1
1
1
1
1
1
+
ρ o1/ρ ,
ρ o1/ρ , 1
n
n
1−t
1−c
1+ λ t 1
1+ λ c 1
−
1
1
ρ o1/ρ , 1
n
u1
1+ λ 1−u
1
1
−
1
ρ o1/ρ
n
f1
1+ λ 1−f
1
+
+
4. Dombi Weighted Aggregation Operators of QSVNNs
In this section we introduce two Dombi weighted aggregation operators QSVNDWAA and
QSVNDWGA which is based on the Dombi operations of QSVNNs in Definition 3.3 and also
studied its properties.
Definition 4.1. A collection of QSVNNs is denoted by qj = htj , cj , uj , fj i (j = 1, 2, ..., n) and
P
w = (w1 , w2 , ..., wn ) be the weight vector for qj with wj ∈ [0, 1] and nj=1 wj = 1. Then the
QSVNDWAA and QSVNDWGA operators are defined as follows.
n
L
w j qj
QSVNDWAA (q1 , q2 , ..., qn ) =
j=1
QSVNDWGA (q1 , q2 , ..., qn ) =
n
N
j=1
w
qj j
Theorem 3.1 A collection of QSVNNs is denoted by qj = htj , cj , uj , fj i (j = 1, 2, ..., n)
P
and w = (w1 , w2 , ..., wn ) be the weight vector for qj with wj ∈ [0, 1] and nj=1 wj = 1. Then
the aggregated value of the QSVNDWAA operator is still a QSVNN and is calculated by the
M.Mohanasundari1 , K.Mohana2 , Quadripartitioned Single valued Neutrosophic Dombi
Weighted Aggregation Operators for Multiple Attribute Decision Making
Neutrosophic Sets and Systems, Vol. 32, 2020
113
following formula,
QSV N DW AA(q1 , q2 , ..., qn ) =
*
1
1−
1+
nP
n
j=1 wj
1
1+
nP
n
j=1 wj
1−uj
uj
tj
1−tj
ρ o1/ρ ,
ρ o1/ρ , 1 −
1+
nP
1
1+
1
n
j=1 wj
nP
1−fj
fj
n
j=1 wj
ρ o1/ρ
cj
1−cj
+
ρ o1/ρ ,
(3)
We can prove this theorem using mathematical induction.
Proof: When n = 2 by using the Dombi operations of QSVNNs in Definition (3.3) we can have
the following result
QSV N DW AA(q1 , q2 ) = q1 ⊕ q2
*
1
1
= 1−
n
n
ρ
ρ o1/ρ , 1 −
ρ
ρ o1/ρ ,
t1
t2
c1
c2
1 + w1 1−t1 + w2 1−t2
1 + w1 1−c1 + w2 1−c2
1
1
=
*
ρ
ρ o1/ρ ,
n
n
ρ
ρ o1/ρ
1−f1
1−f2
1−u2
1
+
w
1 + w1 1−u
1
+
w
+
w
2
1
2
u1
u2
f1
f2
1
1−
1+
nP
2
j=1 wj
tj
1−tj
1+
ρ o1/ρ , 1 −
1
1+
1
nP
2
j=1 wj
1−uj
uj
nP
2
j=1 wj
ρ o1/ρ ,
1+
cj
1−cj
nP
ρ o1/ρ ,
1
2
j=1 wj
1−fj
fj
ρ o1/ρ
+
+
when n = k, Equation (1) becomes,
QSV N DW AA(q1 , q2 , ..., qk ) =
*
1
1−
1+
nP
k
j=1 wj
1
1+
nP
k
j=1 wj
1−uj
uj
tj
1−tj
ρ o1/ρ , 1 −
ρ o1/ρ ,
1+
nP
1
1+
nP
k
j=1 wj
1
k
j=1 wj
1−fj
fj
cj
1−cj
ρ o1/ρ
M.Mohanasundari1 , K.Mohana2 , Quadripartitioned Single valued Neutrosophic Dombi
Weighted Aggregation Operators for Multiple Attribute Decision Making
+
ρ o1/ρ ,
Neutrosophic Sets and Systems, Vol. 32, 2020
114
When n = k + 1 we have the following result
QSV N DW AA(q1 , q2 , ..., qk , qk+1 ) =
*
=
*
1+
1+
nP
1−
k
j=1 wj
1
k+1
j=1 wj
1+
1−uj
uj
tj
1−tj
nP
1+
1
nP
1
1−
ρ o1/ρ ,
k
j=1 wj
1+
ρ o1/ρ , 1 −
1+
1
nP
k+1
j=1 wj
nP
1−uj
uj
tj
1−tj
1
k
j=1 wj
nP
ρ o1/ρ , 1 −
1−fj
fj
1
k+1
j=1 wj
ρ o1/ρ ,
1+
ρ o1/ρ
cj
1−cj
nP
1
1+
+
nP
k
j=1 wj
cj
1−cj
ρ o1/ρ ,
⊕ wk+1 qk+1
ρ o1/ρ ,
1
k+1
j=1 wj
1−fj
fj
ρ o1/ρ
+
Hence we proved that Theorem 3.1 is true for n = k + 1 . Thus Equation (1) is true for all n.
The operator QSVNDWAA satisfies the following properties.
(1) Reducibility : If w = (1/n, 1/n, ..., 1/n), then it is obvious that there exits,
QSV N DW AA(q1 , q2 , ..., qn ) =
*
1−
1+
1+
nP
nP
1
n
1
j=1 n
1
n
1
j=1 n
tj
1−tj
1−uj
uj
ρ o1/ρ , 1 −
ρ o1/ρ ,
1+
1+
nP
nP
1
n
1
j=1 n
1
n
1
j=1 n
1−fj
fj
cj
1−cj
ρ o1/ρ
ρ o1/ρ ,
+
(2) Idempotency : Let all the QSVNNs be denoted by qj = htj , cj , uj , fj i = q(j = 1, 2, ..., n).
Then QSVNDWAA (q1 , q2 , ..., qn ) = q .
′
′
′
(3) Commutativity: Let any QSVNS (q1 , q2 , ..., qn ) be any permutation of (q1 , q2 , ..., qn ). Then
′
′
′
there is QSVNDWAA (q1 , q2 , ..., qn ) = QSVNDWAA (q1 , q2 , ..., qn ).
(4) Boundedness: Let qmin = min(s1 , s2 , ..., sn ) and qmax = max(s1 , s2 , ..., sn ). Then qmin ≤
QSV N DW AA(q1 , q2 , ..., qn ) ≤ qmax
Proof: (1) Given qj = htj , cj , uj , fj i = q(j = 1, 2, ..., n) Property (1) is trivially true based on
equation (3)
M.Mohanasundari1 , K.Mohana2 , Quadripartitioned Single valued Neutrosophic Dombi
Weighted Aggregation Operators for Multiple Attribute Decision Making
Neutrosophic Sets and Systems, Vol. 32, 2020
115
(2) The following result is derived from the equation (3) and we get,
QSV N DW AA(q1 , q2 , ..., qn ) =
*
1
1−
1+
nP
n
j=1 wj
1
nP
1+
=
*
1
1−
1+
n
t
1−t
=
*
ρ o1/ρ , 1 −
1−
n
j=1 wj
1
n
1+
c
1−c
1−uj
uj
ρ o1/ρ ,
ρ o1/ρ , 1 −
tj
1−tj
ρ o1/ρ ,
nP
1+
1
1+
1−u ρ 1/ρ
u
1
nP
1+
n
j=1 wj
1
1−fj
fj
n
1−f
f
n
j=1 wj
,
1+
+
1
1
1
1
t ,1 − 1 + c ,1 −
1−u , 1 −
1 + 1−t
1+ u
1 + 1−f
1−c
f
1
cj
1−cj
ρ o1/ρ
ρ o1/ρ
+
ρ o1/ρ ,
+
= ht, c, u, f i = q
QSVNDWAA (q1 , q2 , ..., qn ) = q holds.
(3) This property is obvious.
(4) Consider qmin = min(q1 , q2 , ..., qn ) = ht− , c− , u− , f − i and qmax = max(q1 , q2 , ..., qn ) =
ht+ , c+ , u+ , f + i Then,
t− = min(tj ), c− = min(cj ), u− = max(uj ), f − = max(fj )
j
j
j
j
t+ = max(tj ), c+ = max(cj ), u+ = min(uj ), f + = min(fj )
j
j
j
j
Therefore we get the following inequalities.
1+
nP
n
1
t−
1−t−
ρ o1/ρ
1+
nP
n
1
c−
1−c−
ρ o1/ρ
1−
1−
j=1 wj
j=1
wj
1+
nP
n
1
1−u+
u+
ρ o1/ρ
1+
nP
n
1
1−f +
f+
ρ o1/ρ
j=1 wj
j=1
wj
≤1−
1+
≤1−
1+
≤
1+
≤
1+
Pn
j=1
Pn
j=1
Pn
wj
wj
j=1
Pn
wj
1
tj
1−tj
ρ 1/ρ
1
cj
1−cj
ρ 1/ρ
j=1 wj
1
1−uj
uj
ρ 1/ρ
1
1−fj
fj
ρ 1/ρ
nP
≤1−
1+
1
nP
1
1+
1+
1+
nP
n
≤
≤
≤1−
j=1 wj
n
j=1
wj
n
j=1
nP
n
j=1
wj
1
t+
1−t+
ρ o1/ρ
1
c+
1−c+
ρ o1/ρ
wj
1−u−
u−
ρ o1/ρ
1−f −
f−
ρ o1/ρ
Hence qmin ≤ QSV N DW AA(q1 , q2 , ..., qn ) ≤ qmax holds.
Theorem 3.2 A collection of QSVNNs is denoted by qj = htj , cj , uj , fj i (j = 1, 2, ..., n)
Pn
wj = 1. Then
and w = (w1 , w2 , ..., wn ) be the weight vector for qj with wj ∈ [0, 1] and j=1
the aggregated value of the QSVNDWGA operator is still a QSVNN and is calculated by the
following formula:
M.Mohanasundari1 , K.Mohana2 , Quadripartitioned Single valued Neutrosophic Dombi
Weighted Aggregation Operators for Multiple Attribute Decision Making
Neutrosophic Sets and Systems, Vol. 32, 2020
QSV N DW GA(q1 , q2 , ..., qn ) =
*
116
1
nP
1+
1−
1+
nP
n
j=1 wj
1
n
j=1 wj
1−tj
tj
uj
1−uj
ρ o1/ρ ,
1
nP
1+
ρ o1/ρ , 1 −
1+
n
j=1 wj
nP
1−cj
cj
1
n
j=1 wj
ρ o1/ρ ,
fj
1−fj
ρ o1/ρ
The proof is similar to the proof of Theorem (3.1).
This QSVNDWGA operator also satisfies the following properties.
(1) Reducibility : If w = (1/n, 1/n, ..., 1/n), then it is obvious that there exits,
QSV N DW GA(q1 , q2 , ..., qn ) =
*
1−
1+
nP
1+
nP
1
n
1
j=1 n
1
n
1
j=1 n
ρ o1/ρ ,
1+
ρ o1/ρ , 1 −
1+
1−tj
tj
uj
1−uj
nP
1
n
1
j=1 n
nP
1−cj
cj
1
n
1
j=1 n
ρ o1/ρ ,
fj
1−fj
ρ o1/ρ
+
(2) Idempotency : Let all the QSVNNs be denoted by qj = htj , cj , uj , fj i = q(j = 1, 2, ..., n).
Then QSVNDWGA (q1 , q2 , ..., qn ) = q .
′
′
′
(3) Commutativity: Let any QSVNS (q1 , q2 , ..., qn ) be any permutation of (q1 , q2 , ..., qn ). Then
′
′
′
there is QSVNDWGA (q1 , q2 , ..., qn ) = QSVNDWGA (q1 , q2 , ..., qn ).
(4) Boundedness: Let qmin = min(q1 , q2 , ..., qn ) and qmax = max(q1 , q2 , ..., qn ).
Then
qmin ≤ QSV N DW GA(q1 , q2 , ..., qn ) ≤ qmax
To prove the above properties it is similar to the operator properties of QSVNDWAA. Hence
it is not repeated here.
5. MADM method using QSVNDWAA operator or QSVNDWGA operator
This section deals about the MADM method to handle the MADM problems effectively
with QSVNN information by using the QSVNDWAA operator or QSVNDWGA operator.
Let A = {A1 , A2 , ..., Am } and C = {C1 , C2 , ..., Cn } be a discrete set of alternatives and attributes respectively. The weight vector of the above attributes is given by w = {w1 , w2 , ..., wn }
P
such that wj ∈ [0, 1] and nj=1 wj = 1.
M.Mohanasundari1 , K.Mohana2 , Quadripartitioned Single valued Neutrosophic Dombi
Weighted Aggregation Operators for Multiple Attribute Decision Making
+
(4)
Neutrosophic Sets and Systems, Vol. 32, 2020
117
To make a better decision to choose the alternative Ai (i = 1, 2, ..., m), a decision maker
needs to analyse the attributes Cj (j = 1, 2, ..., n) by the QSVNN qij = htij , cij , uij , fij i (i =
1, 2, ..., m; j = 1, 2, ..., n) then we get a QSVNN decision matrix D = (dij )m×n
The following decision steps are needed to handle the MADM problems under QSVNN information by using the operator QSVNDWAA or QSVNDWGA.
Step 1 : Collect the QSVNN qi (i = 1, 2, ..., m) for the given alternative Ai (i = 1, 2, ..., m)
by using the operator QSVNDWAA
qi = QSV N DW AA(qi1 , qi2 , ..., qin )
*
1
1
= 1−
nP
nP
ρ o1/ρ , 1 −
ρ o1/ρ ,
tij
cij
n
n
1+
1+
j=1 wj 1−tij
j=1 wj 1−cij
1
1+
nP
n
j=1 wj
or by using QSVNDWGA operator
1−uij
uij
ρ o1/ρ ,
1
1+
nP
n
j=1 wj
qi = QSV N DW GA(qi1 , qi2 , ..., qin )
*
1
1
=
ρ o1/ρ ,
o ,
nP
nP
1−tij
1−cij ρ 1/ρ
n
n
w
w
1+
1
+
j=1 j
j=1 j
tij
cij
1
1−
1+
nP
n
j=1 wj
uij
1−uij
ρ o1/ρ , 1 −
1+
nP
where w = (w1 , w2 , ..., wn ) is the weight vector such that wj ∈ [0, 1] and
1−fij
fij
1
n
j=1 wj
Pn
ρ o1/ρ
+
j=1 wj
fij
1−fij
=1
ρ o1/ρ
Step 2: Score values E(qi ) can be calculated by using Equation (1) with the collective
QSVNN qi (i = 1, 2, ..., m)
Step 3: Select the best one according to rank given to the alternatives.
6. Illustrative Example
This section illustrates an example for a MADM problem about investment alternatives
under a QSVNN environment. An investment company chooses three possible alternatives for
M.Mohanasundari1 , K.Mohana2 , Quadripartitioned Single valued Neutrosophic Dombi
Weighted Aggregation Operators for Multiple Attribute Decision Making
+
Neutrosophic Sets and Systems, Vol. 32, 2020
118
investing their money by considering the four attributes. Let A1 , A2 , A3 be three alternatives
which represent food, car and computer company respectively. Let C1 , C2 , C3 , C4 be the four
attributes which denotes i) Knowledge (or) Expertise ii) Start up costs iii) Market or Demand
iv) Competition respectively. Here the alternatives under given attributes are expressed by
the form of QSVNNs. When three alternatives under four attributes are evaluated we get
a quadripartitioned single valued neutrosophic decision matrix D = (qij )m×n where qij =
htij , cij , uij , fij i (i = 1, 2, 3; j = 1, 2, 3, 4) which is given below.
h0.5, 0.6, 0.2, 0.1i h0.4, 0.2, 0.3, 0.1i h0.4, 0.2, 0.3, 0.1i h0.6, 0.7, 0.1, 0.5i
D = h0.5, 0.1, 0.8, 0.7i h0.2, 0.1, 0.8, 0.7i h0.5, 0.4, 0.7, 0.3i h0.5, 0.4, 0.7, 0.3i
h0.1, 0.2, 0.5, 0.7i h0.1, 0.5, 0.3, 0.4i h0.3, 0.2, 0.7, 0.8i h0.9, 0.8, 0.4, 0.1i
The weight vector for the above four attributes is given as w = (0.35, 0.25, 0.25, 0.15). Hence
the proposed operator of QSVNDWAA (or) QSVNDWGA are used here to solve MADM problem under QSVNN information.
The following steps are needed to solve MADM problem when we use the operator QSVNDWAA. Step 1 : By using Equation(1) for ρ = 1 derive the collective QSVNNs of qi for the
alternative Ai (i = 1, 2, 3) which is given below.
q1 = h0.4760, 0.6667, 0.2034, 0.1136i ,
q2 = h0.4483, 0.25, 0.7568, 0.4565i ,
q3 = h0.6038, 0.5, 0.4414, 0.3404i
Step 2 : Score values E(qi ) can be calculated by using Equation (1) of the collective QSVNN
qi (i = 1, 2, 3) for the alternatives Ai (i = 1, 2, 3) gives the following results.
E(q1 ) = 0.6231, E(q2 ) = 0.4962, E(q3 ) = 0.5805
Step 3: The ranking order is given according to the obtained score values
q1 > q3 > q2 and the best one is q1
The same MADM problem can also be solved by using the another proposed operator that
is QSVNDWGA. The following steps are needed to solve the MADM problem.
Step 1 : By using Equation (4) for ρ = 1 derive the collective QSVNNs of qi for the alternative Ai (i = 1, 2, 3) which is given below.
q1 = h0.4545, 0.3033, 0.2416, 0.1965i ,
q2 = h0.3636, 0.1429, 0.7692, 0.6111i ,
q3 = h0.1429, 0.2712, 0.5328, 0.6667i
M.Mohanasundari1 , K.Mohana2 , Quadripartitioned Single valued Neutrosophic Dombi
Weighted Aggregation Operators for Multiple Attribute Decision Making
119
Neutrosophic Sets and Systems, Vol. 32, 2020
Step 2 : Score values E(qi ) can be calculated by using Equation (1) of the collective QSVNN
qi (i = 1, 2, 3) for the alternatives Ai (i = 1, 2, 3) gives the following results.
E(q1 ) = 0.6783, E(q2 ) = 0.4601, E(q3 ) = 0.4181
Step 3: The ranking order is given according to the obtained score values
q1 > q2 > q3 and the best one is q1
The following Table 1 and 2 shows the ranking results for the parameters of ρ ∈ [1, 10] of
the quadripartitioned single valued neutrosophic Dombi weighted arithmetic average (QSVNDWAA) operator and quadripartitioned single valued neutrosophic Dombi weighted geometric
average (QSVNDWGA) operator respectively.
We can observe the following results from Tables 1 and 2.
1) Different aggregation operators that is QSVNDWAA and QSVNDWGA shows different
ranking orders. But the ranking orders due to different operational parameters are same according to the one operator. This results that the operational parameter ρ is not sensitive
in this decision making problem since we get the same ranking orders corresponding to the
QSVNDWAA and QSVNDWGA operator.
Table 1. Ranking results of the operator QSVNDWAA for different operational parameters.
ρ
E(q1 ), E(q2 ), E(q3 )
Ranking Order
1
0.6231,0.4962,0.5805
q1 > q3 > q2
2
0.6596,0.5044,0.6323
q1 > q3 > q2
3
0.6601,0.5089,0.6468
q1 > q3 > q2
4
0.6619,0.5118,0.6535
q1 > q3 > q2
5
0.6637,0.5139,0.6577
q1 > q3 > q2
6
0.6652,0.5154,0.6605
q1 > q3 > q2
7
0.6665,0.5166,0.6626
q1 > q3 > q2
8
0.6675,0.5181,0.6641
q1 > q3 > q2
9
0.6683,0.5184,0.6654
q1 > q3 > q2
10
0.6689,0.5191,0.6664
q1 > q3 > q2
M.Mohanasundari1 , K.Mohana2 , Quadripartitioned Single valued Neutrosophic Dombi
Weighted Aggregation Operators for Multiple Attribute Decision Making
Neutrosophic Sets and Systems, Vol. 32, 2020
120
Table 2. Ranking results of the operator QSVNDWGA for different operational parameters.
ρ
E(q1 ), E(q2 ), E(q3 )
Ranking Order
1
0.6783,0.4601,0.4181
q1 > q2 > q3
2
0.6619,0.4424,0.3994
q1 > q2 > q3
3
0.6475,0.4308,0.3877
q1 > q2 > q3
4
0.6380,0.4236,0.3799
q1 > q2 > q3
5
0.6315,0.4196,0.3745
q1 > q2 > q3
6
0.6269,0.4158,0.3706
q1 > q2 > q3
7
0.6236,0.4136,0.3678
q1 > q2 > q3
8
0.6204,0.4119,0.3656
q1 > q2 > q3
9
0.6185,0.4105,0.3638
q1 > q2 > q3
10
0.6167,0.4094,0.3624
q1 > q2 > q3
1) The ranking orders according to the operators QSVNDWAA and QSVNDWGA are different
2) Ranking orders are not affected by different operational parameters of ρ ∈ [0, 1] in both the
operators which shows that ρ is not sensitive in this decision making problem.
3) These aggregation methods of the operators QSVNDWAA and QSVNDWGA provides new
method to solve MADM problems under an QSVNN environment.
7. Conclusion
In this paper we have studied the Dombi operations of QSVNN based on the Dombi T-norm
and T-conorm operations and also we have proposed the two weighted aggregation operators
QSVNDWAA , QSVNDWGA and investigate their properties. Multiple Attribute Decision
making is one of the effective approach which helps us to the problems involving a selection
from a finite number of alternatives are included under finite number of attributes. To solve
these type of MADM problems ranking orders are used to select the best one among the given
alternatives. This paper also deals about MADM method by using the proposed QSVNDWAA
and QSVNDWGA operator under a QSVNN environment. Using these aggregation operators
we calculate the score function of the alternatives with respect to the given attributes and this
score function helps us to rank the alternatives and choose the best one. Finally we illustrated
an example of a MADM problem for the proposed aggregation operators.
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Received: Oct 21, 2019. Accepted: Mar 20, 2020
M.Mohanasundari1 , K.Mohana2 , Quadripartitioned Single valued Neutrosophic Dombi
Weighted Aggregation Operators for Multiple Attribute Decision Making
Neutrosophic Sets and Systems, Vol. 32, 2020
University of New Mexico
Polarity of generalized neutrosophic subalgebras in
BCK/BCI-algebras
Rajab Ali Borzooei1 , Florentin Smarandache2,∗ and Young Bae Jun1,3
1
First Department of Mathematics, Shahid Beheshti University, Tehran 1983963113, Iran
E-mail: borzooei@sbu.ac.ir
2
Mathematics & Science Department, University of New Mexico, 705 Gurley Ave., Gallup, NM 87301, USA.
E-mail: fsmarandache@gmail.com
3
Department of Mathematics Education, Gyeongsang National University, Jinju 52828, Korea.
E-mail: skywine@gmail.com
∗
Correspondence: F. Smarandache (fsmarandache@gmail.com)
Abstract: k-polar generalized neutrosophic set is introduced, and it is applied to BCK/BCI-algebras. The notions
of k-polar generalized subalgebra, k-polar generalized (∈, ∈ ∨q)-neutrosophic subalgebra and k-polar generalized
(q, ∈ ∨q)-neutrosophic subalgebra are defined, and several properties are investigated. Characterizations of k-polar
generalized neutrosophic subalgebra and k-polar generalized (∈, ∈ ∨q)-neutrosophic subalgebra are discussed, and
the necessity and possibility operator of k-polar generalized neutrosophic subalgebra are are considered. We show
that the generaliged neutrosophic q-sets and the generaliged neutrosophic ∈ ∨q-sets subalgebras by using the k-polar
generalized (∈, ∈ ∨q)-neutrosophic subalgebra and the k-polar generalized (q, ∈ ∨q)-neutrosophic subalgebra. A
k-polar generalized (∈, ∈ ∨q)-neutrosophic subalgebra is established by using the generaliged neutrosophic ∈ ∨qsets, conditions for a k-polar generalized neutrosophic set to be a k-polar generalized neutrosophic subalgebra and a
k-polar generalized (q, ∈ ∨q)-neutrosophic subalgebra are provided.
Keywords: k-polar generalized neutrosophic subalgebra, k-polar generalized (∈, ∈ ∨q)-neutrosophic subalgebra,
k-polar generalized (q, ∈ ∨q)-neutrosophic subalgebra.
1
Introduction
In the fuzzy set which is introduced by Zadeh [35], the membership degree is expressed by only one function so
called the truth function. As a generalization of fuzzy set, intuitionistic fuzzy set is introduced by Atanassove
by using membership function and nonmembership function. The membership (resp. nonmembership) function represents truth (resp. false) part. Smarandache introduced a new notion so called neutrosophic set by
using three functions, i.e., membership function (t), nonmembership function (f) and neutalitic/indeterministic
membership function (i) which are independent components. Neutrosophic set is applied to BCK/BCIalgebras which are discussed in the papers [13, 19, 20, 21, 22, 26, 27, 30]. Indeterministic membership function is leaning to one side, membership function or nonmembership function, in the application of neutrosophic
set to algebraic structures. In order to divide the role of the indeterministic membership function, Song et al.
R.A. Borzooei, F. Smarandache, Y.B. Jun, Polarity of generalized neutrosophic subalgebras in
BCK/BCI-algebras.
Neutrosophic Sets and Systems, Vol. 32, 202 0
124
[31] introduced the generalized neutralrosophic set, and discussed its application in BCK/BCI-algebras. Borzooei et al. [8] introduced the notion of a commutative generalized neutrosophic ideal in a BCK-algebra, and
investigated related properties. They considered characterizations of a commutative generalized neutrosophic
ideal. Using a collection of commutative ideals in BCK-algebras, they established a commutative generalized
neutrosophic ideal. They also introduced the notion of equivalence relations on the family of all commutative
generalized neutrosophic ideals in BCK-algebras, and investigated related properties. Zhang [36] introduced
the notion of bipolar fuzzy sets as an extension of fuzzy sets, and it is applied in several (algebraic) structures
such as (ordered) semigroups (see [12, 7, 10, 28]), (hyper) BCK/BCI-algebras (see [6, 14, 15, 23, 16, 17])
and finite state machines (see [18, 32, 33, 34]). The bipolar fuzzy set is an extension of fuzzy sets whose
membership degree range is [−1, 1]. So, it is possible for a bipolar fuzzy set to deal with positive information
and negative information at the same time. Chen et al. [9] raised a question: “How to generalize bipolar
fuzzy sets to multipolar fuzzy sets and how to generalize results on bipolar fuzzy sets to the case of multipolar
fuzzy sets?” To solve their question, they tried to fold the negative part into positive part, that is, they used
positive part instead of negative part in bipolar fuzzy set. And then they introduced introduced an m-polar
fuzzy set which is an extension of bipolar fuzzy sets. It is applied to BCK/BCI-algebra, graph theory and
decision-making problems etc. (see [4, 2, 1, 3, 29, 5, 25]).
In this paper, we introduce k-polar generalized neutrosophic set and apply it to BCK/BCI-algebras to study.
We define k-polar generalized neutrosophic subalgebra, k-polar generalized (∈, ∈ ∨q)-neutrosophic subalgebra and k-polar generalized (q, ∈ ∨q)-neutrosophic subalgebra and study various properties. We discuss characterization of k-polar generalized neutrosophic subalgebra and k-polar generalized (∈, ∈ ∨q)-neutrosophic
subalgebra. We show that the necessity and possibility operator of k-polar generalized neutrosophic subalgebra
are also a k-polar generalized neutrosophic subalgebra. Using the k-polar generalized (∈, ∈ ∨q)-neutrosophic
subalgebra, we show that the generaliged neutrosophic q-sets and the generaliged neutrosophic ∈ ∨q-sets subalgebras. Using the k-polar generalized (q, ∈ ∨q)-neutrosophic subalgebra, we show that the generaliged
neutrosophic q-sets and the generaliged neutrosophic ∈ ∨q-sets are subalgebras. Using the generaliged neutrosophic ∈ ∨q-sets, we establish a k-polar generalized (∈, ∈ ∨q)-neutrosophic subalgebra. We provide
conditions for a k-polar generalized neutrosophic set to be a k-polar generalized neutrosophic subalgebra and
a k-polar generalized (q, ∈ ∨q)-neutrosophic subalgebra.
2
Preliminaries
If a set X has a special element 0 and a binary operation ∗ satisfying the conditions:
(I) (∀u, v, w ∈ X) (((u ∗ v) ∗ (u ∗ w)) ∗ (w ∗ v) = 0),
(II) (∀u, v ∈ X) ((u ∗ (u ∗ v)) ∗ v = 0),
(III) (∀u ∈ X) (u ∗ u = 0),
(IV) (∀u, v ∈ X) (u ∗ v = 0, v ∗ u = 0 ⇒ u = v),
then we say that X is a BCI-algebra. If a BCI-algebra X satisfies the following identity:
(V) (∀u ∈ X) (0 ∗ u = 0),
then X is called a BCK-algebra.
R.A. Borzooei, F. Smarandache, Y.B. Jun, Polarity of generalized neutrosophic subalgebras in
BCK/BCI-algebras.
Neutrosophic Sets and Systems, Vol. 32, 202 0
125
Any BCK/BCI-algebra X satisfies the following conditions:
(∀u ∈ X) (u ∗ 0 = u) ,
(∀u, v, w ∈ X) (u ≤ v ⇒ u ∗ w ≤ v ∗ w, w ∗ v ≤ w ∗ u) ,
(∀u, v, w ∈ X) ((u ∗ v) ∗ w = (u ∗ w) ∗ v)
(2.1)
(2.2)
(2.3)
where u ≤ v if and only if u ∗ v = 0. A subset S of a BCK/BCI-algebra X is called a subalgebra of X if
u ∗ v ∈ S for all u, v ∈ S.
See the books [11] and [24] for more information on BCK/BCI-algeebras.
A fuzzy set µ in a BCK/BCI-algebra X is called a fuzzy subalgebra of X if µ(u ∗ v) ≥ min{µ(u), µ(v)}
for all u, v ∈ X.
For any family {ai | i ∈ Λ} of real numbers, we define
_
max{ai | i ∈ Λ} if Λ is finite,
{ai | i ∈ Λ} :=
sup{ai | i ∈ Λ} otherwise.
^
min{ai | i ∈ Λ} if Λ is finite,
{ai | i ∈ Λ} :=
inf{ai | i ∈ Λ} otherwise.
W
V
If Λ = {1, 2}, we will also use a1 ∨ a2 and a1 ∧ a2 instead of {ai | i ∈ Λ} and {ai | i ∈ Λ}, respectively.
3 k-polar generalized neutrosophic subalgebras
A k-polar generalized neutrosophic set over a universe X is a structure of the form:
z
Lb :=
| z ∈ X, ℓbIT (z) + ℓbIF (z) ≤ 1̂
(ℓbT (z),ℓbIT (z),ℓbIF (z),ℓbF (z))
(3.1)
where ℓbT , ℓbIT , ℓbIF and ℓbF are mappings from X into [0, 1]k . The membership values of every element z ∈ X
in ℓbT , ℓbIT , ℓbIF and ℓbF are denoted by
b
b
b
b
ℓT (z) = (π1 ◦ ℓT )(z), (π2 ◦ ℓT )(z), · · · , (πk ◦ ℓT )(z) ,
ℓbIT (z) = (π1 ◦ ℓbIT )(z), (π2 ◦ ℓbIT )(z), · · · , (πk ◦ ℓbIT )(z) ,
(3.2)
ℓbIF (z) = (π1 ◦ ℓbIF )(z), (π2 ◦ ℓbIF )(z), · · · , (πk ◦ ℓbIF )(z) ,
b
b
b
b
ℓF (z) = (π1 ◦ ℓF )(z), (π2 ◦ ℓF )(z), · · · , (πk ◦ ℓF )(z) ,
respectively, and satisfies the following condition
(πi ◦ ℓbIT )(z) + (πi ◦ ℓbIF )(z) ≤ 1
for all i = 1, 2, · · · , k.
We shall use the ordered quadruple Lb := ℓbT , ℓbIT , ℓbIF , ℓbF for the k-polar generalized neutrosophic set in
(3.1).
R.A. Borzooei, F. Smarandache, Y.B. Jun, Polarity of generalized neutrosophic subalgebras in
BCK/BCI-algebras.
Neutrosophic Sets and Systems, Vol. 32, 202 0
126
Note that for every k-polar generalized neutrosophic set Lb := ℓbT , ℓbIT , ℓbIF , ℓbF over X, we have
b
b
b
b
(∀z ∈ X) 0̂ ≤ ℓT (z) + ℓIT (z) + ℓIF (z) + ℓF (z) ≤ 3̂ ,
that is, 0 ≤ (πi ◦ ℓbT )(z) + (πi ◦ ℓbIT )(z) + (πi ◦ ℓbIF )(z) + (πi ◦ ℓbF )(z) ≤ 3 for all z ∈ X and i = 1, 2, · · · , k.
Unless otherwise stated in this section, X will represent a BCK/BCI-algebra.
Definition 3.1. A k-polar generalized neutrosophic set Lb := ℓbT , ℓbIT , ℓbIF , ℓbF over X is called a k-polar
generalized neutrosophic subalgebra of X if it satisfies:
ℓbT (z ∗ y) ≥ ℓbT (z) ∧ ℓbT (y)
b
ℓIT (z ∗ y) ≥ ℓbIT (z) ∧ ℓbIT (y)
,
(3.3)
(∀z, y ∈ X)
ℓbIF (z ∗ y) ≤ ℓbIF (z) ∨ ℓbIF (y)
ℓbF (z ∗ y) ≤ ℓbF (z) ∨ ℓbF (y)
that is,
for i = 1, 2, · · · , k.
(πi ◦ ℓbT )(z ∗ y) ≥ (πi ◦ ℓbT )(z) ∧ (πi ◦ ℓbT )(y)
(π ◦ ℓb )(z ∗ y) ≥ (π ◦ ℓb )(z) ∧ (π ◦ ℓb )(y)
i
IT
i
IT
i
IT
(πi ◦ ℓbIF )(z ∗ y) ≤ (πi ◦ ℓbIF )(z) ∨ (πi ◦ ℓbIF )(y)
(πi ◦ ℓbF )(z ∗ y) ≤ (πi ◦ ℓbF )(z) ∨ (πi ◦ ℓbF )(y)
(3.4)
Example 3.2. Consider a BCK-algebra X = {0, α, β, γ} with the binary operation “∗” which is given below.
∗
0
α
β
γ
0
0
α
β
γ
α
0
0
β
γ
β
0
α
0
γ
γ
0
α
β
0
b
b
b
b
b
Let L := ℓT , ℓIT , ℓIF , ℓF be a 4-polar neutrosophic set over X in which ℓbT , ℓbIT , ℓbIF and ℓbF are defined as
follows:
(0.6, 0.7, 0.8, 0.9) if z = 0,
(0.4, 0.4, 0.8, 0.5) if z = α,
ℓbT : X → [0, 1]4 , z 7→
(0.5, 0.6, 0.7, 0.3) if z = β,
(0.3, 0.5, 0.4, 0.7) if z = γ,
R.A. Borzooei, F. Smarandache, Y.B. Jun, Polarity of generalized neutrosophic subalgebras in
BCK/BCI-algebras.
Neutrosophic Sets and Systems, Vol. 32, 202 0
ℓbIT
ℓbIF
127
(0.7, 0.6, 0.8, 0.9)
(0.6, 0.4, 0.7, 0.5)
4
: X → [0, 1] , z 7→
(0.5, 0.5, 0.4, 0.8)
(0.2, 0.6, 0.5, 0.7)
(0.2, 0.3, 0.4, 0.5)
(0.4, 0.7, 0.5, 0.8)
: X → [0, 1]4 , z 7→
(0.5, 0.5, 0.8, 0.6)
(0.7, 0.3, 0.6, 0.7)
if
if
if
if
z
z
z
z
= 0,
= α,
= β,
= γ,
if
if
if
if
z
z
z
z
= 0,
= α,
= β,
= γ,
(0.4, 0.4, 0.3, 0.2) if z = 0,
(0.8, 0.7, 0.5, 0.3) if z = α,
ℓbF : X → [0, 1]4 , z 7→
(0.6, 0.5, 0.6, 0.6) if z = β,
(0.4, 0.6, 0.8, 0.4) if z = γ,
It is routine to verify that Lb := ℓbT , ℓbIT , ℓbIF , ℓbF is a 4-polar generalized neutrosophic subalgebra of X.
If we take z = y in (3.3) and use (III), then we have the following lemma.
Lemma 3.3. Let Lb := ℓbT , ℓbIT , ℓbIF , ℓbF be a k-polar generalized neutrosophic subalgebra of a BCK/BCIalgebr X. Then
!
ℓbT (0) ≥ ℓbT (z), ℓbIT (0) ≥ ℓbIT (z)
.
(3.5)
(∀z, y ∈ X)
ℓbIF (0) ≤ ℓbIF (z), ℓbF (0) ≤ ℓbF (z)
b
b
b
b
b
Proposition 3.4. Let L := ℓT , ℓIT , ℓIF , ℓF be a k-polar generalized neutrosophic set over X. If there exists
a sequence {zn } in X such that lim ℓbT (zn ) = 1̂ = lim ℓbIT (zn ) and lim ℓbIF (zn ) = 0̂ = lim ℓbF (zn ), then
n→∞
n→∞
n→∞
n→∞
ℓbT (0) = 1̂ = ℓbIT (0) and ℓbIF (0) = 0̂ = ℓbF (0).
Proof. Using Lemma 3.3, we have
1̂ = lim ℓbT (zn ) ≤ ℓbT (0) ≤ 1̂ = lim ℓbIT (zn ) ≤ ℓbIT (0) ≤ 1̂,
n→∞
n→∞
0̂ = lim ℓbIF (zn ) ≥ ℓbIF (0) ≥ 0̂ = lim ℓbF (zn ) ≥ ℓbF (0) ≥ 0̂.
n→∞
n→∞
This completes the proof.
Proposition 3.5. Let Lb := ℓbT , ℓbIT , ℓbIF , ℓbF be a k-polar generalized neutrosophic subalgebra of X such that
(∀z, y ∈ X)
ℓbT (z ∗ y) ≥ ℓbT (y), ℓbIT (z ∗ y) ≥ ℓbIT (y)
ℓbIF (z ∗ y) ≤ ℓbIF (y), ℓbF (z ∗ y) ≤ ℓbF (y)
Then Lb is constant on X, that is, ℓbT , ℓbIT , ℓbIF and ℓbF are constants on X.
!
.
R.A. Borzooei, F. Smarandache, Y.B. Jun, Polarity of generalized neutrosophic subalgebras in
BCK/BCI-algebras.
(3.6)
Neutrosophic Sets and Systems, Vol. 32, 202 0
128
Proof. Since z ∗ 0 = z for all z ∈ X, it follows from the condition (3.6) that
ℓbT (z) = ℓbT (z ∗ 0) ≥ ℓbT (0), ℓbIT (z) = ℓbIT (z ∗ 0) ≥ ℓbIT (0),
ℓbIF (z) = ℓbIF (z ∗ 0) ≤ ℓbIF (0), ℓbF (z) = ℓbF (z ∗ 0) ≤ ℓbF (0)
(3.7)
(3.8)
for all z ∈ X. Combining (3.5) and (3.7) induces ℓbT (z) = ℓbT (0), ℓbIT (z) = ℓbIT (0), ℓbIF (z) = ℓbIF (0) and
ℓbF (z) = ℓbF (0) for all z ∈ X. Therefore ℓbT , ℓbIT , ℓbIF and ℓbF are constants on X, that is, Lb is constant on
X.
Given a k-polar generalized neutrosophic set Lb := ℓbT , ℓbIT , ℓbIF , ℓbF over a universe X, consider the
following cut sets.
U (ℓbT , n̂T ) := {z ∈ X | ℓbT (z) ≥ n̂T },
U (ℓbIT , n̂IT ) := {z ∈ X | ℓbIT (z) ≥ n̂IT },
L(ℓbIF , n̂IF ) := {z ∈ X | ℓbIF (z) ≤ n̂IF },
L(ℓbF , n̂F ) := {z ∈ X | ℓbF (z) ≤ n̂F }
for n̂T , n̂IT , n̂IF , n̂F ∈ [0, 1]k , that is,
U (ℓbT , n̂T ) := {z ∈ X | (πi ◦ ℓbT )(z) ≥ n̂iT for all i = 1, 2, · · · , k},
U (ℓbIT , n̂IT ) := {z ∈ X | (πi ◦ ℓbIT )(z) ≥ n̂i for all i = 1, 2, · · · , k},
IT
L(ℓbIF , n̂IF ) := {z ∈ X | (πi ◦ ℓbIF )(z) ≤ n̂iIF for all i = 1, 2, · · · , k},
L(ℓbF , n̂F ) := {z ∈ X | (πi ◦ ℓbF )(z) ≤ n̂iF for all i = 1, 2, · · · , k}
where n̂T = (n1T , n2T , · · · , nkT ), n̂IT = (n1IT , n2IT , · · · , nkIT ), n̂IF = (n1IF , n2IF , · · · , nkIF ) and n̂F = (n1F ,
T
T
n2F , · · · , nkF ). It is clear that U (ℓbT , n̂T ) = ki=1 U (ℓbT , n̂T )i , U (ℓbIT , n̂IT ) = ki=1 U (ℓbIT , n̂IT )i , L(ℓbIF , n̂IF ) =
Tk
Tk
i
i
b
b
b
i=1 L(ℓF , n̂F ) , where
i=1 L(ℓIF , n̂IF ) and L(ℓF , n̂F ) =
U (ℓbT , n̂T )i := {z ∈ X | (πi ◦ ℓbT )(z) ≥ n̂iT },
U (ℓbIT , n̂IT )i := {z ∈ X | (πi ◦ ℓbIT )(z) ≥ n̂iIT },
L(ℓbIF , n̂IF )i := {z ∈ X | (πi ◦ ℓbIF )(z) ≤ n̂i },
IF
L(ℓbF , n̂F )i := {z ∈ X | (πi ◦ ℓbF )(z) ≤ n̂iF }
for i = 1, 2, · · · , k.
We handle the characterization of k-polar generalized neutrosophic subalgebra.
Theorem 3.6. Let Lb := ℓbT , ℓbIT , ℓbIF , ℓbF be a k-polar generalized neutrosophic set over X. Then Lb is a kpolar generalized neutrosophic subalgebra of X if and only if the cut sets U (ℓbT , n̂T ), U (ℓbIT , n̂IT ), L(ℓbIF , n̂IF )
and L(ℓbF , n̂F ) are subalgebras of X for all n̂T , n̂IT , n̂IF , n̂F ∈ [0, 1]k .
Proof. Assume that Lb is a k-polar generalized neutrosophic subalgebra of X. Let z, y ∈ X. If z, y ∈
U (ℓbT , n̂T ) for all n̂T ∈ [0, 1]k , then (πi ◦ ℓbT )(z) ≥ niT and (πi ◦ ℓbT )(y) ≥ niT for i = 1, 2, · · · , k. It folR.A. Borzooei, F. Smarandache, Y.B. Jun, Polarity of generalized neutrosophic subalgebras in
BCK/BCI-algebras.
Neutrosophic Sets and Systems, Vol. 32, 202 0
lows that
129
(πi ◦ ℓbT )(z ∗ y) ≥ (πi ◦ ℓbT )(z) ∧ (πi ◦ ℓbT )(y) ≥ niT
i = 1, 2, · · · , k. Hence z ∗ y ∈ U (ℓbT , n̂T ), and so U (ℓbT , n̂T ) is a subalgebra of X. If z, y ∈ L(ℓbF , n̂F ) for all
n̂F ∈ [0, 1]k , then (πi ◦ ℓbF )(z) ≤ niF and (πi ◦ ℓbF )(y) ≤ niF for i = 1, 2, · · · , k. Hence
(πi ◦ ℓbF )(z ∗ y) ≤ (πi ◦ ℓbF )(z) ∨ (πi ◦ ℓbF )(y) ≤ niF
i = 1, 2, · · · , k, and so z ∗ y ∈ L(ℓbF , n̂F ). Therefore L(ℓbF , n̂F ) is a subalgebra of X. Similarly, we can verify
that U (ℓbIT , n̂IT ) and L(ℓbIF , n̂IF ) are subalgebras of X.
Conversely, suppose that the cut sets U (ℓbT , n̂T ), U (ℓbIT , n̂IT ), L(ℓbIF , n̂IF ) and L(ℓbF , n̂F ) are subalgebras
of X for all n̂T , n̂IT , n̂IF , n̂F ∈ [0, 1]k . If there exists α, β ∈ X such that ℓbIT (α ∗ β) < ℓbIT (α) ∧ ℓbIT (β), that
is,
(πi ◦ ℓbIT )(α ∗ β) < (πi ◦ ℓbIT )(α) ∧ (πi ◦ ℓbIT )(β)
for i = 1, 2, · · · , k, then α, β ∈ U (ℓbIT , n̂IT )i and α∗β ∈
/ U (ℓbIT , n̂IT )i where n̂iIT = (πi ◦ ℓbIT )(α)∧(πi ◦ ℓbIT )(β)
for for i = 1, 2, · · · , k. This is a contradiction, and so
ℓbIT (z ∗ y) ≥ ℓbIT (z) ∧ ℓbIT (y)
for all z, y ∈ X. By the similarly way, we know that ℓbT (z ∗ y) ≥ ℓbT (z) ∧ ℓbT (y) for all z, y ∈ X. Now, suppose
that ℓbF (α ∗ β) > ℓbF (α) ∨ ℓbF (β) for some α, β ∈ X. Then
(πi ◦ ℓbF )(α ∗ β) > (πi ◦ ℓbF )(α) ∨ (πi ◦ ℓbF )(β)
for i = 1, 2, · · · , k. If we take niF = (πi ◦ ℓbF )(α) ∨ (πi ◦ ℓbF )(β) for i = 1, 2, · · · , k, then α, β ∈ L(ℓbF , n̂F )i
but α ∗ β ∈
/ L(ℓbF , n̂F )i , a contradiction. Hence
ℓbF (z ∗ y) ≤ ℓbF (z) ∨ ℓbF (y)
for all z, y ∈ X. Similarly, we can check that ℓbIF (z ∗ y) ≤ ℓbIF (z) ∨ ℓbIF (y) for all z, y ∈ X. Therefore Lb is a
k-polar generalized neutrosophic subalgebra of X.
Theorem 3.7. Let Lb :=
ℓbT , ℓbIT , ℓbIF , ℓbF
be a k-polar generalized neutrosophic set over X. Then Lb is a
k-polar generalized neutrosophic subalgebra of X if and only if the fuzzy sets πi ◦ ℓbT , πi ◦ ℓbIT , πi ◦ ℓbcF and
πi ◦ ℓbcIF are fuzzy subalgebras of X where (πi ◦ ℓbcF )(z) = 1 − (πi ◦ ℓbF )(z) and (πi ◦ ℓbcIF )(z) = 1 − (πi ◦ ℓbIF )(z)
for all z ∈ X and i = 1, 2, · · · , k.
Proof. Suppose that Lb is a k-polar generalized neutrosophic subalgebra of X. For any i = 1, 2, · · · , k, it is
clear that πi ◦ ℓbT and πi ◦ ℓbIT are fuzzy subalgebras of X. For any z, y ∈ X, we get
(πi ◦ ℓbcF )(z ∗ y) = 1 − (πi ◦ ℓbF )(z ∗ y) = 1 − (πi ◦ ℓbF )(z) ∨ (πi ◦ ℓbF )(y)
= (1 − (πi ◦ ℓbF )(z)) ∧ (1 − (πi ◦ ℓbF )(y))
= (πi ◦ ℓbcF )(z) ∧ (πi ◦ ℓbcF )(y)
R.A. Borzooei, F. Smarandache, Y.B. Jun, Polarity of generalized neutrosophic subalgebras in
BCK/BCI-algebras.
Neutrosophic Sets and Systems, Vol. 32, 202 0
130
and
(πi ◦ ℓbcIF )(z ∗ y) = 1 − (πi ◦ ℓbIF )(z ∗ y) = 1 − (πi ◦ ℓbIF )(z) ∨ (πi ◦ ℓbIF )(y)
= (1 − (πi ◦ ℓbIF )(z)) ∧ (1 − (πi ◦ ℓbIF )(y))
= (πi ◦ ℓbcIF )(z) ∧ (πi ◦ ℓbcIF )(y).
Hence πi ◦ ℓbcF and πi ◦ ℓbcIF are fuzzy subalgebras of X .
Conversely, suppose that the fuzzy sets πi ◦ ℓbT , πi ◦ ℓbIT , πi ◦ ℓbcF and πi ◦ ℓbcIF are fuzzy subalgebras of X
for i = 1, 2, · · · , k and let z, y ∈ X. Then
(πi ◦ ℓbT )(z ∗ y) ≥ (πi ◦ ℓbT )(z) ∧ (πi ◦ ℓbT )(y),
(πi ◦ ℓbIT )(z ∗ y) ≥ (πi ◦ ℓbIT )(z) ∧ (πi ◦ ℓbIT )(y)
for all i = 1, 2, · · · , k. Also we have
1 − (πi ◦ ℓbF )(z ∗ y) = (πi ◦ ℓbcF )(z ∗ y) ≥ (πi ◦ ℓbcF )(z) ∧ (πi ◦ ℓbcF )(y)
= (1 − (πi ◦ ℓbF )(z)) ∧ (1 − (πi ◦ ℓbF )(y))
and
= 1 − ((πi ◦ ℓbF )(z) ∨ (πi ◦ ℓbF )(y))
1 − (πi ◦ ℓbIF )(z ∗ y) = (πi ◦ ℓbcIF )(z ∗ y) ≥ (πi ◦ ℓbcIF )(z) ∧ (πi ◦ ℓbcIF )(y)
= (1 − (πi ◦ ℓbIF )(z)) ∧ (1 − (πi ◦ ℓbIF )(y))
= 1 − ((πi ◦ ℓbIF )(z) ∨ (πi ◦ ℓbIF )(y))
which imply that (πi ◦ ℓbF )(z ∗ y) ≤ (πi ◦ ℓbF )(z) ∨ (πi ◦ ℓbF )(y) and
(πi ◦ ℓbIF )(z ∗ y) ≤ (πi ◦ ℓbIF )(z) ∨ (πi ◦ ℓbIF )(y)
for all i = 1, 2, · · · , k. Hence Lb is a k-polar generalized neutrosophic subalgebra of X .
Theorem 3.8. If Lb := ℓbT , ℓbIT , ℓbIF , ℓbF is a k-polar generalized neutrosophic subalgebra of X, then so are
c
c
c
c b b
b
b
b
b
b
b
b
b
✷L := ℓT , ℓIT , ℓIT , ℓT and ✸L := ℓIF , ℓF , ℓF , ℓIF .
Proof. Note that (πi ◦ℓbIT )(z)+(πi ◦ℓbcIT )(z) = (πi ◦ℓbIT )(z)+1−(πi ◦ℓbIT )(z) = 1 and (πi ◦ℓbF )(z)+(πi ◦ℓbcF )(z) =
b
bc
b
bc
(πi ◦ ℓbF)(z) + 1 − (πi ◦ℓbF )(z) = 1, that
is, ℓIT (z) + ℓIT (z) = 1̂ and ℓF (z) + ℓF (z) = 1̂ for all z ∈ X. Hence
✷Lb := ℓbT , ℓbIT , ℓbcIT , ℓbcT and ✸Lb := ℓbcIF , ℓbcF , ℓbF , ℓbIF are k-polar generalized neutrosophic sets over X. For
any z, y ∈ X, we get
(πi ◦ ℓbcIT )(z ∗ y) = 1 − (πi ◦ ℓbIT )(z ∗ y) ≤ 1 − ((πi ◦ ℓbIT )(z) ∧ (πi ◦ ℓbIT )(y))
= (1 − (πi ◦ ℓbIT )(z)) ∨ (1 − (πi ◦ ℓbIT )(y))
= (πi ◦ ℓbcIT )(z) ∨ (πi ◦ ℓbcIT )(y),
R.A. Borzooei, F. Smarandache, Y.B. Jun, Polarity of generalized neutrosophic subalgebras in
BCK/BCI-algebras.
Neutrosophic Sets and Systems, Vol. 32, 202 0
131
(πi ◦ ℓbcT )(z ∗ y) = 1 − (πi ◦ ℓbT )(z ∗ y) ≤ 1 − ((πi ◦ ℓbT )(z) ∧ (πi ◦ ℓbT )(y))
= (1 − (πi ◦ ℓbT )(z)) ∨ (1 − (πi ◦ ℓbT )(y))
= (πi ◦ ℓbcT )(z) ∨ (πi ◦ ℓbcT )(y),
(πi ◦ ℓbcIF )(z ∗ y) = 1 − (πi ◦ ℓbIF )(z ∗ y) ≥ 1 − ((πi ◦ ℓbIF )(z) ∨ (πi ◦ ℓbIF )(y))
= (1 − (πi ◦ ℓbIF )(z)) ∧ (1 − (πi ◦ ℓbIF )(y))
= (πi ◦ ℓbcIF )(z) ∧ (πi ◦ ℓbcIF )(y),
and
(πi ◦ ℓbcF )(z ∗ y) = 1 − (πi ◦ ℓbF )(z ∗ y) ≥ 1 − ((πi ◦ ℓbF )(z) ∨ (πi ◦ ℓbF )(y))
= (1 − (πi ◦ ℓbF )(z)) ∧ (1 − (πi ◦ ℓbF )(y))
= (πi ◦ ℓbcF )(z) ∧ (πi ◦ ℓbcF )(y).
Therefore ✷Lb := ℓbT , ℓbIT , ℓbcIT , ℓbcT and ✸Lb := ℓbcIF , ℓbcF , ℓbF , ℓbIF are kpolar generalized neutrosophic subalgebras of X.
Theorem 3.9. Let Λ1 ×Λ2 ×· · ·×Λk ⊆ [0, 1]k , that is, Λi ⊆ [0, 1] for i = 1, 2, · · · , k. Let Si := {Sti | ti ∈ Λi }
be a family of subalgebras of X for i = 1, 2, · · · , k such that
[
Si ,
(3.9)
X=
ti ∈Λi
(∀si , ti ∈ Λi ) (si > ti ⇒ Ssi ⊂ Sti )
(3.10)
b
b
b
b
b
for i = 1, 2, · · · , k. Let L := ℓT , ℓIT , ℓIF , ℓF be a k-polar generalized neutrosophic set over X defined by
(∀z ∈ X)
W
(πi ◦ ℓbT )(z) = {qi ∈ Λi | z ∈ Sqi } = (πi ◦ ℓbIT )(z),
V
(πi ◦ ℓbIF )(z) = {ri ∈ Λi | z ∈ Sr } = (πi ◦ ℓbF )(z)
i
!
(3.11)
for i = 1, 2, · · · , k. Then Lb := ℓbT , ℓbIT , ℓbIF , ℓbF is a k-polar generalized neutrosophic subalgebra of X.
Proof. For any i = 1, 2, · · · , k, we consider the following two cases.
_
_
ti = {qi ∈ Λi | qi < ti } and ti 6= {qi ∈ Λi | qi < ti }.
The first case implies that
z ∈ U (ℓbT , ti ) ⇔ (∀qi < ti )(z ∈ Sqi ) ⇔ z ∈
\
Sqi ,
qi <ti
z ∈ U (ℓbIT , ti ) ⇔ (∀qi < ti )(z ∈ Sqi ) ⇔ z ∈
\
Sq i .
qi <ti
R.A. Borzooei, F. Smarandache, Y.B. Jun, Polarity of generalized neutrosophic subalgebras in
BCK/BCI-algebras.
Neutrosophic Sets and Systems, Vol. 32, 202 0
132
T
Sqi = U (ℓbIT , ti ), and so U (ℓbT , ti ) and U (ℓbIT , ti ) are subalgebras of X for all i =
Hence U (ℓbT , ti ) =
qi <ti
T
T
U (ℓbIT , ti ) are subalgebras of X. For
U (ℓbT , ti ) and U (ℓbIT , t̂) =
1, 2, . . . , k. Hence U (ℓbT , t̂) =
i=1,2,...,k
i=1,2,...,k
S
S
b
b
Sq i ,
Sqi = U (ℓIT , ti ) for all i = 1, 2, . . . , k. If z ∈
the second case, we will show that U (ℓT , ti ) =
qi ≥ti
qi ≥ti
then z ∈ Sqi for some qi ≥ ti . Hence (πi ◦ ℓbIT )(z) = (πi ◦ ℓbT )(z) ≥ qi ≥ ti , and so z ∈ U (ℓbT , ti ) and
S
W
z ∈ U (ℓbIT , ti ). If z ∈
/
Sqi , then z ∈
/ Sqi for all qi ≥ ti . The condition ti 6= {qi ∈ Λi | qi < ti } induces
qi ≥ti
(ti − εi , ti ) ∩ Λi = ∅ for some εi > 0. Hence z ∈
/ Sqi for all qi > ti − εi , which means that if z ∈ Sqi then
b
b
qi ≤ ti − εi . Hence (πi ◦ ℓIT )(z) = (πi ◦ ℓT )(z) ≤ ti − εi < ti and so z ∈
/ U (ℓbIT , ti ) = U (ℓbT , ti ). Therefore
S
S
Sqi . Consequently, U (ℓbT , ti ) = U (ℓbIT , ti ) =
Sqi which is a subalgebra of X,
U (ℓbT , ti ) = U (ℓbIT , ti ) ⊆
qi ≥ti
qi ≥ti
T
T
U (ℓbIT , ti ) are subalgebras of X. Now, we
U (ℓbT , ti ) and U (ℓbIT , t̂) =
and therefore U (ℓbT , t̂) =
i=1,2,...,k
i=1,2,...,k
consider the following two cases.
^
^
si = {ri ∈ Λi | ri > si } and si 6= {ri ∈ Λi | ri > si }.
For the first case, we get
z ∈ L(ℓbIF , si ) ⇔ (∀si < ri )(z ∈ Sri ) ⇔ z ∈
z ∈ L(ℓbF , si ) ⇔ (∀si < ri )(z ∈ Sri ) ⇔ z ∈
It follows that L(ℓbIF , si ) = L(ℓbF , si ) =
T
\
Sri ,
ri >si
\
Sri .
ri >si
Sri , which is a subalgebra of X. The second case induces
S
(si , si + εi ) ∩ Λi = ∅ for some εi > 0. If z ∈
Sri , then z ∈ Sri for some ri ≤ si , and thus (πi ◦ ℓbIF )(z) =
ri ≤si
S
(πi ◦ ℓbF )(z) ≤ ri ≤ si , i.e., z ∈ L(ℓbIF , si ) and z ∈ L(ℓbF , si ). Hene
Sri ⊆ L(ℓbIF , si ) = L(ℓbF , si ).
ri ≤si
S
If z ∈
/
/ Sri for all ri ≤ si + εi , that is, if
/ Sri for all ri ≤ si which implies that z ∈
Sri , then z ∈
ri >si
ri ≤si
z ∈ Sri then ri ≥ si + εi . Thus (πi ◦ ℓbIF )(z) = (πi ◦ ℓbF )(z) ≥ si + εi ≥ si and so z ∈
/ L(ℓbIF , si ) =
S
L(ℓbF , si ). This shows that L(ℓbIF , si ) = L(ℓbF , si ) =
Sri , which is a subalgebra of X. Therefore L(ℓbF , ŝ) =
ri ≤si
T
T
L(ℓbIF , si ) are subalgebras of X. Using Theorem 3.6, we know that
L(ℓbF , si ) and U (ℓbIF , ŝ) =
i=1,2,...,k
i=1,2,...,k
Lb := ℓbT , ℓbIT , ℓbIF , ℓbF is a k-polar generalized neutrosophic subalgebra of X.
4
k-polar generalized (∈, ∈ ∨q)-neutrosophic subalgebras
Let n̂T = (n1T , n2T , · · · , nkT ), n̂IT = (n1IT , n2IT , · · · , nkIT ), n̂IF = (n1IF, n2IF , · · · , nkIF )and n̂F = (n1F , n2F ,
· · · , nkF ) in [0, 1]k . Given a k-polar generalized neutrosophic set Lb := ℓbT , ℓbIT , ℓbIF , ℓbF over a universe X,
R.A. Borzooei, F. Smarandache, Y.B. Jun, Polarity of generalized neutrosophic subalgebras in
BCK/BCI-algebras.
Neutrosophic Sets and Systems, Vol. 32, 202 0
133
we consider the following sets.
Tq (ℓbT , n̂T ) := {z ∈ X | ℓbT (z) + n̂T > 1̂},
ITq (ℓbIT , n̂IT ) := {z ∈ X | ℓbIT (z) + n̂IT > 1̂},
IFq (ℓbIF , n̂IF ) := {z ∈ X | ℓbIF (z) + n̂IF < 1̂},
Fq (ℓbF , n̂F ) := {z ∈ X | ℓbF (z) + n̂F < 1̂},
which are called generaliged neutrosophic q-sets, and
T ∈∨q (ℓbT , n̂T ) := {z ∈ X | ℓbT (z) ≥ n̂T or ℓbT (z) + n̂T > 1̂},
IT ∈∨q (ℓbIT , n̂IT ) := {z ∈ X | ℓbIT (z) ≥ n̂IT or ℓbIT (z) + n̂IT > 1̂},
IF ∈∨q (ℓbIF , n̂IF ) := {z ∈ X | ℓbIF (z) ≤ n̂IF or ℓbIF (z) + n̂IF < 1̂},
F ∈∨q (ℓbF , n̂F ) := {z ∈ X | ℓbF (z) ≤ n̂F or ℓbF (z) + n̂F < 1̂}
which are called generaliged neutrosophic ∈ ∨q-sets. Then
Tq (ℓbT , n̂T ) =
and
i=1
IFq (ℓbIF , n̂IF ) =
T ∈∨q (ℓbT , n̂T ) =
where
k
\
k
\
i=1
IF ∈∨q (ℓbIF , n̂IF ) =
Tq (ℓbT , n̂T )i , ITq (ℓbIT , n̂IT ) =
k
\
i=1
k
\
i=1
IFq (ℓbIF , n̂IF )i , Fq (ℓbF , n̂F ) =
T ∈∨q (ℓbT , n̂T )i , IT ∈∨q (ℓbIT , n̂IT ) =
k
\
i=1
ITq (ℓbIT , n̂IT )i ,
k
\
i=1
k
\
i=1
Fq (ℓbF , n̂F )i
IT ∈∨q (ℓbIT , n̂IT )i ,
IF ∈∨q (ℓbIF , n̂IF )i , F ∈∨q (ℓbF , n̂F ) =
k
\
i=1
F ∈∨q (ℓbF , n̂F )i
Tq (ℓbT , n̂T )i = {z ∈ X | (πi ◦ ℓbT )(z) + niT > 1},
ITq (ℓbIT , n̂IT )i = {z ∈ X | (πi ◦ ℓbIT )(z) + ni > 1},
IT
IFq (ℓbIF , n̂IF ) = {z ∈ X | (πi ◦ ℓbIF )(z) + niIF < 1},
Fq (ℓbF , n̂F )i = {z ∈ X | (πi ◦ ℓbF )(z) + ni < 1}
i
F
R.A. Borzooei, F. Smarandache, Y.B. Jun, Polarity of generalized neutrosophic subalgebras in
BCK/BCI-algebras.
Neutrosophic Sets and Systems, Vol. 32, 202 0
134
and
T ∈∨q (ℓbT , n̂T )i = {z ∈ X | (πi ◦ ℓbT )(z) ≥ niT or (πi ◦ ℓbT )(z) + niT > 1},
IT ∈∨q (ℓbIT , n̂IT )i = {z ∈ X | (πi ◦ ℓbIT )(z) ≥ niIT or (πi ◦ ℓbIT )(z) + niIT > 1},
IF ∈∨q (ℓbIF , n̂IF )i = {z ∈ X | (πi ◦ ℓbIF )(z) ≤ ni or (πi ◦ ℓbIF )(z) + ni < 1},
IF
IF
F ∈∨q (ℓbF , n̂F ) = {z ∈ X | (πi ◦ ℓbF )(z) ≤
i
niF
or (πi ◦ ℓbF )(z) +
niF
< 1}.
It is clear that T ∈∨q (ℓbT , n̂T ) = U (ℓbT , n̂T ) ∪ Tq (ℓbT , n̂T ), IT ∈∨q (ℓbIT , n̂IT ) = U (ℓbIT , n̂IT ) ∪ ITq (ℓbIT , n̂IT ),
IF ∈∨q (ℓbIF , n̂IF ) = L(ℓbIF , n̂IF ) ∪ IFq (ℓbIF , n̂IF ), and F ∈∨q (ℓbF , n̂F ) = L(ℓbF , n̂F ) ∪ Fq (ℓbF , n̂F ).
By routine calculations, we have the following properties.
b
b
b
b
b
Proposition 4.1. Given a k-polar generalized neutrosophic set L := ℓT , ℓIT , ℓIF , ℓF over a universe X, we
have
1. If n̂T , n̂IT ∈ [0, 0.5]k , then T ∈∨q (ℓbT , n̂T ) = U (ℓbT , n̂T ) and IT ∈∨q (ℓbIT , n̂IT ) = U (ℓbIT , n̂IT ).
2. If n̂F , n̂IF ∈ [0.5, 1]k , then IF ∈∨q (ℓbIF , n̂IF ) = L(ℓbIF , n̂IF ) and F ∈∨q (ℓbF , n̂F ) = L(ℓbF , n̂F ).
3. If n̂T , n̂IT ∈ (0.5, 1]k , then T ∈∨q (ℓbT , n̂T ) = Tq (ℓbT , n̂T ) and IT ∈∨q (ℓbIT , n̂IT ) = ITq (ℓbIT , n̂IT ).
4. If n̂F , n̂IF ∈ [0, 0.5)k , then IF ∈∨q (ℓbIF , n̂IF ) = IFq (ℓbIF , n̂IF ) and F ∈∨q (ℓbF , n̂F ) = Fq (ℓbF , n̂F ).
Unless otherwise stated in this section, X will represent a BCK/BCI-algebra.
b
b
b
b
b
Definition 4.2. Let L := ℓT , ℓIT , ℓIF , ℓF be a k-polar generalized neutrosophic set over X. Then Lb is called
a k-polar generalized (∈, ∈ ∨q)-neutrosophic subalgebra of X if it satisfies:
z ∈ U (ℓbT , n̂T ), y ∈ U (ℓbT , n̂T ) ⇒ z ∗ y ∈ T ∈∨q (ℓbT , n̂T ),
z ∈ U (ℓbIT , n̂IT ), y ∈ U (ℓbIT , n̂IT ) ⇒ z ∗ y ∈ IT ∈∨q (ℓbIT , n̂IT ),
z ∈ L(ℓbIF , n̂IF ), y ∈ L(ℓbIF , n̂IF ) ⇒ z ∗ y ∈ IF ∈∨q (ℓbIF , n̂IF ),
(4.1)
z ∈ L(ℓbF , n̂F ), y ∈ L(ℓbF , n̂F ) ⇒ z ∗ y ∈ F ∈∨q (ℓbF , n̂F )
for all z, y ∈ X, n̂T , n̂IT ∈ (0, 1]k and n̂F , n̂IF ∈ [0, 1)k .
Example 4.3. Consider a BCI-algebra X = {0, 1, 2, α, β} with the binary operation “∗” which is given below.
∗
0
1
2
α
β
0
0
1
2
α
β
1
0
0
2
α
α
2
0
1
0
α
β
α
α
β
α
0
1
β
α
α
α
0
0
R.A. Borzooei, F. Smarandache, Y.B. Jun, Polarity of generalized neutrosophic subalgebras in
BCK/BCI-algebras.
Neutrosophic Sets and Systems, Vol. 32, 202 0
135
Let Lb := ℓbT , ℓbIT , ℓbIF , ℓbF be a 3-polar neutrosophic set over X in which ℓbT , ℓbIT , ℓbIF and ℓbF are defined as
follows:
(0.6, 0.5, 0.5) if z = 0,
(0.7, 0.7, 0.2) if z = 1,
(0.7, 0.8, 0.5) if z = 2,
ℓbT : X → [0, 1]3 , z 7→
(0.3, 0.4, 0.5) if z = α,
(0.3, 0.4, 0.2) if z = β,
ℓbIT
ℓbIF
(0.6, 0.5, 0.6)
(0.4, 0.3, 0.7)
3
(0.6, 0.8, 0.4)
: X → [0, 1] , z 7→
(0.7, 0.4, 0.1)
(0.4, 0.3, 0.1)
(0.3, 0.1, 0.5)
(0.8, 0.3, 0.7)
3
(0.3, 0.8, 0.5)
: X → [0, 1] , z 7→
(0.7, 0.9, 0.6)
(0.8, 0.9, 0.7)
if
if
if
if
if
z
z
z
z
z
= 0,
= 1,
= 2,
= α,
= β,
if
if
if
if
if
z
z
z
z
z
= 0,
= 1,
= 2,
= α,
= β,
(0.2, 0.2, 0.5) if z = 0,
(0.3, 0.9, 0.8) if z = 1,
(0.5, 0.2, 0.4) if z = 2,
ℓbF : X → [0, 1]3 , z 7→
(0.6, 0.4, 0.6) if z = α,
(0.6, 0.9, 0.8) if z = β,
It is routine to verify that Lb := ℓbT , ℓbIT , ℓbIF , ℓbF is 3-polar generalized (∈, ∈ ∨q)-neutrosophic subalgebra.
b
b
b
b
ℓT , ℓIT , ℓIF , ℓF is a k-polar generalized neutrosophic subalgebra of X, then the
generaliged neutrosophic q-sets Tq (ℓbT , n̂T ), ITq (ℓbIT , n̂IT ), IFq (ℓbIF , n̂IF ) and Fq (ℓbF , n̂F ) are subalgebras of
X for all n̂T , n̂IT ∈ (0, 1]k and n̂F , n̂IF ∈ [0, 1)k .
Theorem 4.4. If Lb :=
Proof. Let z, y ∈ Tq (ℓbT , n̂T ). Then ℓbT (z) + n̂T > 1̂ and ℓbT (y) + n̂T > 1̂, that is, (πi ◦ ℓbT )(z) + niT > 1 and
(πi ◦ ℓbT )(y) + niT > 1 for i = 1, 2, · · · , k. It follows that
(πi ◦ ℓbT )(z ∗ y) + niT ≥ ((πi ◦ ℓbT )(z) ∧ (πi ◦ ℓbT )(y)) + niT
= ((πi ◦ ℓbT )(z) + nT )i ∧ ((πi ◦ ℓbT )(y) + nT )i > 1
for i = 1, 2, · · · , k. Hence ℓbT (z ∗ y) + n̂T > 1̂, that is, z ∗ y ∈ Tq (ℓbT , n̂T ). Therefore Tq (ℓbT , n̂T ) is a subalgebra
of X. Let z, y ∈ IFq (ℓbIF , n̂IF ). Then (πi ◦ ℓbIF )(z) + niIF < 1 and (πi ◦ ℓbIF )(y) + niIF < 1 for i = 1, 2, · · · , k.
R.A. Borzooei, F. Smarandache, Y.B. Jun, Polarity of generalized neutrosophic subalgebras in
BCK/BCI-algebras.
136
Neutrosophic Sets and Systems, Vol. 32, 202 0
Hence
(πi ◦ ℓbIF )(z ∗ y) + niIF ≤ ((πi ◦ ℓbIF )(z) ∨ (πi ◦ ℓbIF )(y)) + niIF
= ((πi ◦ ℓbIF )(z) + nIF )i ∨ ((πi ◦ ℓbIF )(y) + nIF )i < 1
for i = 1, 2, · · · , k and so ℓbIF (z ∗ y) + n̂IF < 1̂. Thus z ∗ y ∈ IFq (ℓbIF , n̂IF ) and IFq (ℓbIF , n̂IF ) is a subalgebra
of X. By the similar way, we can verify that ITq (ℓbIT , n̂IT ) and Fq (ℓbF , n̂F ) are subalgebras of X.
We handle characterizations of a k-polar generalized (∈, ∈ ∨q)-neutrosophic subalgebra.
b
b
b
b
b
Theorem 4.5. Let L := ℓT , ℓIT , ℓIF , ℓF be a k-polar generalized neutrosophic set over X. Then Lb is a
k-polar generalized (∈, ∈ ∨q)-neutrosophic subalgebra of X if and only if it satisfies:
V
c
ℓbT (z ∗ y) ≥ {ℓbT (z), ℓbT (y), 0.5}
V b
b
c
(z
∗
y)
≥
{ℓIT (z), ℓbIT (y), 0.5}
ℓ
IT
(∀z, y ∈ X)
,
(4.2)
W
c
ℓbIF (z ∗ y) ≤ {ℓbIF (z), ℓbIF (y), 0.5}
W
c
ℓbF (z ∗ y) ≤ {ℓbF (z), ℓbF (y), 0.5}
that is,
V
(πi ◦ ℓbT )(z ∗ y) ≥ {(πi ◦ ℓbT )(z), (πi ◦ ℓbT )(y), 0.5},
V
(πi ◦ ℓbIT )(z ∗ y) ≥ {(πi ◦ ℓbIT )(z), (πi ◦ ℓbIT )(y), 0.5},
W
(πi ◦ ℓbIF )(z ∗ y) ≤ {(πi ◦ ℓbIF )(z), (πi ◦ ℓbIF )(y), 0.5},
W
(πi ◦ ℓbF )(z ∗ y) ≤ {(πi ◦ ℓbF )(z), (πi ◦ ℓbF )(y), 0.5}
(4.3)
for all z, y ∈ X and i = 1, 2, · · · , k.
Proof. Suppose that Lb := ℓbT , ℓbIT , ℓbIF , ℓbF is a k-polar generalized (∈, ∈ ∨q)-neutrosophic subalgebra of X
and let z, y ∈ X. For any i = 1, 2, . . . , k, assume that (πi ◦ ℓbIT )(z) ∧ (πi ◦ ℓbIT )(y) < 0.5. Then
(πi ◦ ℓbIT )(z ∗ y) ≥ (πi ◦ ℓbIT )(z) ∧ (πi ◦ ℓbIT )(y)
because if (πi ◦ ℓbIT )(z ∗ y) < (πi ◦ ℓbIT )(z) ∧ (πi ◦ ℓbIT )(y), then there exists niIT ∈ (0, 0.5) such that
(πi ◦ ℓbIT )(z ∗ y) < niIT ≤ (πi ◦ ℓbIT )(z) ∧ (πi ◦ ℓbIT )(y).
/ U (ℓbIT , nIT )i . Also (πi ◦ ℓbIT )(z ∗y)+niIT < 1,
It follows that z ∈ U (ℓbIT , nIT )i and y ∈ U (ℓbIT , nIT )i but z ∗y ∈
i.e., z ∗ y ∈
/ ITq (ℓbIT , n̂IT ). Hence z ∗ y ∈
/ IT ∈∨q (ℓbIT , n̂IT ) which is a contradiction. Therefore
^
(πi ◦ ℓbIT )(z ∗ y) ≥ {(πi ◦ ℓbIT )(z), (πi ◦ ℓbIT )(y), 0.5}
for all z, y ∈ X with (πi ◦ ℓbIT )(z) ∧ (πi ◦ ℓbIT )(y) < 0.5. Now suppose that (πi ◦ ℓbIT )(z) ∧ (πi ◦ ℓbIT )(y) ≥ 0.5.
Then z ∈ U (ℓbIT , 0.5)i and y ∈ U (ℓbIT , 0.5)i , and so z ∗ y ∈ IT ∈∨q (ℓbIT , 0.5)i = U (ℓbIT , 0.5)i ∪ ITq (ℓbIT , 0.5)i .
R.A. Borzooei, F. Smarandache, Y.B. Jun, Polarity of generalized neutrosophic subalgebras in
BCK/BCI-algebras.
Neutrosophic Sets and Systems, Vol. 32, 202 0
137
Hence z ∗ y ∈ U (ℓbIT , 0.5)i . Otherwise, (πi ◦ ℓbIT )(z ∗ y) + 0.5 < 0.5 + 0.5 = 1, a contradiction. Consequently,
^
(πi ◦ ℓbIT )(z ∗ y) ≥ {(πi ◦ ℓbIT )(z), (πi ◦ ℓbIT )(y), 0.5}
for all z, y ∈ X. Similarly, we know that
(πi ◦ ℓbT )(z ∗ y) ≥
^
{(πi ◦ ℓbT )(z), (πi ◦ ℓbT )(y), 0.5}
c If ℓbF (z ∗ y) > ℓbF (z) ∨ ℓbF (y) := n̂F , then z, y ∈ L(ℓbF , n̂F ),
for all z, y ∈ X. Suppose that ℓbF (z) ∨ ℓbF (y) > 0.5.
z∗y ∈
/ L(ℓbF , n̂F ) and ℓbF (z ∗ y) + n̂F > 2n̂F > 1, i.e., z ∗ y ∈
/ Fq (ℓbF , n̂F ). This is a contradiction, and so
W
c whenever ℓbF (z) ∨ ℓbF (y) > 0.5.
c Now assume that ℓbF (z) ∨ ℓbF (y) ≤ 0.5.
c Then
ℓbF (z ∗ y) ≤ {ℓbF (z), ℓbF (y), 0.5}
c and thus z ∗ y ∈ F ∈∨q (ℓbF , 0.5)
c = L(ℓbF , 0.5)
c ∪ Fq (ℓbF , 0.5).
c If z ∗ y ∈
c that is,
/ L(ℓbF , 0.5),
z, y ∈ L(ℓbF , 0.5)
c then ℓbF (z ∗ y) + 0.5
c > 0.5
c + 0.5
c = 1̂, i.e., z ∗ y ∈
c This is a contradiction. Hence
ℓbF (z ∗ y) > 0.5,
/ Fq (ℓbF , 0.5).
W
c Therefore ℓbF (z ∗ y) ≤
c and so ℓbF (z ∗ y) ≤ {ℓbF (z), ℓbF (y), 0.5}
c whenever ℓbF (z) ∨ ℓbF (y) ≤ 0.5.
ℓbF (z ∗ y) ≤ 0.5
W b
W
c for
c for all z, y ∈ X. By the similar way, we have ℓbIF (z ∗ y) ≤ {ℓbIF (z), ℓbIF (y), 0.5}
{ℓF (z), ℓbF (y), 0.5}
all z, y ∈ X.
Conversely, let Lb := ℓbT , ℓbIT , ℓbIF , ℓbF be a k-polar generalized neutrosophic set over X which satisfies
the condition (4.2). Let z, y ∈ X and n̂T = (n1T , n2T , · · · , nkT ) ∈ [0, 1]k . If z, y ∈ U (ℓbT , n̂T ), then ℓbT (z) ≥ n̂T
c Otherwise, we get
and ℓbT (y) ≥ n̂T . If ℓbT (z ∗ y) < n̂T , then ℓbT (z) ∧ ℓbT (y) ≥ 0.5.
ℓbT (z ∗ y) ≥
which is a contradiction. Hence
^
c = ℓbT (z) ∧ ℓbT (y) ≥ n̂T ,
{ℓbT (z), ℓbT (y), 0.5}
ℓbT (z ∗ y) + n̂T > 2ℓbT (z ∗ y) ≥ 2
^
c = 1̂
{ℓbT (z), ℓbT (y), 0.5}
and so z ∗ y ∈ Tq (ℓbT , n̂T ) ⊆ T ∈∨q (ℓbT , n̂T ). Similarly, if z, y ∈ U (ℓbIT , n̂IT ), then z ∗ y ∈ IT ∈∨q (ℓbIT , n̂IT ) for
n̂IT = (n1IT , n2IT , · · · , nkIT ) ∈ [0, 1]k . Now, let z, y ∈ L(ℓbIF , n̂IF ) for n̂IF = (n1IF , n2IF , · · · , nkIF ) ∈ [0, 1]k .
c because if not, then
Then ℓbIF (z) ≤ n̂IF and ℓbIF (y) ≤ n̂IF . If ℓbIF (z ∗ y) > n̂IF , then ℓbIF (z) ∨ ℓbIF (z) ≤ 0.5
W
c ≤ ℓbIF (z) ∨ ℓbIF (y) ≤ n̂IF , which is a contradiction. Thus
ℓbIF (z ∗ y) ≤ {ℓbIF (z), ℓbIF (y), 0.5}
_
c = 1̂
ℓbIF (z ∗ y) + n̂IF < 2ℓbIF (z ∗ y) ≤ 2 {ℓbIF (z), ℓbIF (y), 0.5}
and so z ∗ y ∈ IFq (ℓbIF , n̂IF ) ⊆ IF ∈∨q (ℓbIF , n̂IF ). Similarly, we know that if z, y ∈ L(ℓbF , n̂F ), then z ∗ y ∈
Fq (ℓbF , n̂F ) ⊆ F ∈∨q (ℓbF , n̂F ) for n̂F = (n1F , n2F , · · · , nkF ) ∈ [0, 1]k . Therefore Lb is a k-polar generalized (∈,
∈ ∨q)-neutrosophic subalgebra of X.
Using the k-polar generalized (∈, ∈ ∨q)-neutrosophic subalgebra, we show that the generaliged neutrosophic q-sets subalgebras.
Theorem 4.6. If Lb := ℓbT , ℓbIT , ℓbIF , ℓbF is a k-polar generalized (∈, ∈ ∨q)-neutrosophic subalgebra of X,
then the generaliged neutrosophic q-sets Tq (ℓbT , n̂T ), ITq (ℓbIT , n̂IT ), IFq (ℓbIF , n̂IF ) and Fq (ℓbF , n̂F ) are subalgebras of X for all n̂T , n̂IT ∈ (0.5, 1]k and n̂F , n̂IF ∈ [0, 0.5)k .
R.A. Borzooei, F. Smarandache, Y.B. Jun, Polarity of generalized neutrosophic subalgebras in
BCK/BCI-algebras.
Neutrosophic Sets and Systems, Vol. 32, 202 0
138
Proof. Suppose that Lb := ℓbT , ℓbIT , ℓbIF , ℓbF is a k-polar generalized (∈, ∈ ∨q)-neutrosophic subalgebra of
X. Let z, y ∈ X. If z, y ∈ ITq (ℓbIT , n̂IT ) for n̂IT ∈ (0.5, 1]k , then ℓbIT (z) + n̂IT > 1̂ and ℓbIT (y) + n̂IT > 1̂. It
follows from Theorem 4.5 that
^
c + n̂IT
ℓbIT (z ∗ y) + n̂IT ≥ {ℓbIT (z), ℓbIT (y), 0.5}
^
c + n̂IT }
= {ℓbIT (z) + n̂IT , ℓbIT (y) + n̂IT , 0.5
> 1̂,
i.e., z ∗ y ∈ ITq (ℓbIT , n̂IT ). Thus ITq (ℓbIT , n̂IT ) is a subalgebra of X. Suppose that z, y ∈ Fq (ℓbF , n̂F ) for
n̂F ∈ [0, 0.5)k . Then (πi ◦ ℓbF )(z) + niF < 1 and (πi ◦ ℓbF )(z) + niF < 1. Using Theorem 4.5, we have
_
(πi ◦ ℓbF )(z ∗ y) + niF ≤ {(πi ◦ ℓbF )(z), (πi ◦ ℓbF )(y), 0.5} + niF
_
= {(πi ◦ ℓbF )(z) + niF , (πi ◦ ℓbF )(y) + niF , 0.5 + niF }
<1
T
and thus z ∗ y ∈ Fq (ℓbF , n̂F )i for all i = 1, 2, · · · , k. Hence z ∗ y ∈ ki=1 Fq (ℓbF , n̂F )i = Fq (ℓbF , n̂F ), and
therefore Fq (ℓbF , n̂F ) is a subalgebra of X. Similarly, we can induce that Tq (ℓbT , n̂T ) and IFq (ℓbIF , n̂IF ) are
subalgebras of X for n̂IT ∈ (0.5, 1]k and n̂F ∈ [0, 0.5)k .
Using the generaliged neutrosophic ∈ ∨q-sets, we establish a k-polar generalized (∈, ∈ ∨q)-neutrosophic
subalgebra.
Theorem 4.7. Given a k-polar generalized neutrosophic set Lb := ℓbT , ℓbIT , ℓbIF , ℓbF over X, if the generaliged
neutrosophic ∈ ∨q-sets T ∈∨q (ℓbT , n̂T ), IT ∈∨q (ℓbIT , n̂IT ), IF ∈∨q (ℓbIF , n̂IF ) and F ∈∨q (ℓbF , n̂F ) are subalgebras of
X for all n̂T , n̂IT ∈ (0, 1]k and n̂F , n̂IF ∈ [0, 1)k , then Lb is a k-polar generalized (∈, ∈ ∨q)-neutrosophic
subalgebra of X.
Proof. Assume that there exist α, β ∈ X such that
^
(πi ◦ ℓbT )(α ∗ β) < {(πi ◦ ℓbT )(α), (πi ◦ ℓbT )(β), 0.5}
for i = 1, 2, · · · , k. Then there exists niT ∈ (0, 0.5] such that
^
(πi ◦ ℓbT )(α ∗ β) < niT ≤ {(πi ◦ ℓbT )(α), (πi ◦ ℓbT )(β), 0.5}.
T
Hence α, β ∈ U (ℓbT , n̂T )i , and so α, β ∈ ki=1 U (ℓbT , n̂T )i = U (ℓbT , n̂T ) ⊆ T ∈∨q (ℓbT , n̂T ). Since T ∈∨q (ℓbT , n̂T ) is
T
a subalgebra of X, it follows that α ∗ β ∈ T ∈∨q (ℓbT , n̂T ) = ki=1 T ∈∨q (ℓbT , n̂T )i . Thus (πi ◦ ℓbT )(α ∗ β) ≥ niT
or (πi ◦ ℓbT )(α ∗ β) + niT > 1 for i = 1, 2, · · · , k. This is a contradiction, and thus (πi ◦ ℓbT )(z ∗ y) ≥
V
{(πi ◦ ℓbT )(z), (πi ◦ ℓbT )(y), 0.5} for all z, y ∈ X and i = 1, 2, · · · , k. Now, if there exist α, β ∈ X such that
_
(πi ◦ ℓbIF )(α ∗ β) > {(πi ◦ ℓbIF )(α), (πi ◦ ℓbIF )(β), 0.5}
R.A. Borzooei, F. Smarandache, Y.B. Jun, Polarity of generalized neutrosophic subalgebras in
BCK/BCI-algebras.
Neutrosophic Sets and Systems, Vol. 32, 202 0
139
for i = 1, 2, · · · , k, then
(πi ◦ ℓbIF )(α ∗ β) > niIF ≥
_
{(πi ◦ ℓbIF )(α), (πi ◦ ℓbIF )(β), 0.5}
(4.4)
T
for some niIF ∈ [0.5, 1). Hence α, β ∈ L(ℓbIF , n̂IF )i , and so α, β ∈ ki=1 L(ℓbIF , n̂IF )i = L(ℓbIF , n̂IF ) ⊆
IF ∈∨q (ℓbIF , n̂IF ). This implies that α ∗ β ∈ IF ∈∨q (ℓbIF , n̂IF ), and (4.4) induces α ∗ β ∈
/ L(ℓbIF , n̂IF )i and
T
(πi ◦ ℓbIF )(α ∗ β) + niIF > 2niIF > 1 for i = 1, 2, · · · , k. Thus α ∗ β ∈
/ ki=1 L(ℓbIF , n̂IF )i = L(ℓbIF , n̂IF )
Tk
and α ∗ β ∈
/ i=1 IFq (ℓbIF , n̂IF )i = IFq (ℓbIF , n̂IF ). Hence α ∗ β ∈
/ IF ∈∨q (ℓbIF , n̂IF ) which is a contradiction.
Therefore
_
(πi ◦ ℓbIF )(z ∗ y) ≤ {(πi ◦ ℓbIF )(z), (πi ◦ ℓbIF )(y), 0.5}
W
c for all z, y ∈ X. Similarly,
for for all z, y ∈ X and i = 1, 2, · · · , k, i.e., ℓbIF (z ∗ y) ≤ {ℓbIF (z), ℓbIF (y), 0.5}
V
W
b
b
b
b
we show that (πi ◦ ℓIT )(z ∗ y) ≥ {(πi ◦ ℓIT )(z), (πi ◦ ℓIT )(y), 0.5} and (πi ◦ ℓF )(z ∗ y) ≤ {(πi ◦ ℓbF )(z), (πi ◦
ℓbF )(y), 0.5} for all z, y ∈ X and i = 1, 2, · · · , k. Using Theorem 4.5, we conclude that Lb is a k-polar
generalized (∈, ∈ ∨q)-neutrosophic subalgebra of X.
Using the k-polar generalized (∈, ∈ ∨q)-neutrosophic subalgebra, we show that the generaliged neutrosophic ∈ ∨q-sets subalgebras.
Theorem 4.8. If Lb := ℓbT , ℓbIT , ℓbIF , ℓbF is a k-polar generalized (∈, ∈ ∨q)-neutrosophic subalgebra of X,
then the generaliged neutrosophic ∈ ∨q-sets T ∈∨q (ℓbT , n̂T ), IT ∈∨q (ℓbIT , n̂IT ), IF ∈∨q (ℓbIF , n̂IF ) and F ∈∨q (ℓbF , n̂F )
are subalgebras of X for all n̂T , n̂IT ∈ (0, 0.5]k and n̂F , n̂IF ∈ [0.5, 1)k .
Proof. Let z, y ∈ IT ∈∨q (ℓbIT , n̂IT ). Then
and
z ∈ U ((ℓbIT , n̂IT )i or z ∈ ITq ((ℓbIT , n̂IT )i
y ∈ U ((ℓbIT , n̂IT )i or y ∈ ITq ((ℓbIT , n̂IT )i
for i = 1, 2, · · · , k. Thus we get the following four cases:
(i) z ∈ U ((ℓbIT , n̂IT )i and y ∈ U ((ℓbIT , n̂IT )i ,
(ii) z ∈ U ((ℓbIT , n̂IT )i and y ∈ ITq ((ℓbIT , n̂IT )i ,
(iii) z ∈ ITq ((ℓbIT , n̂IT )i and y ∈ U ((ℓbIT , n̂IT )i ,
(iv) z ∈ ITq ((ℓbIT , n̂IT )i and y ∈ ITq ((ℓbIT , n̂IT )i .
For the first case, we have z ∗ y ∈ IT ∈∨q ((ℓbIT , n̂IT )i for i = 1, 2, · · · , k and so
z∗y ∈
k
\
i=1
IT ∈∨q ((ℓbIT , n̂IT )i = IT ∈∨q (ℓbIT , n̂IT ).
R.A. Borzooei, F. Smarandache, Y.B. Jun, Polarity of generalized neutrosophic subalgebras in
BCK/BCI-algebras.
Neutrosophic Sets and Systems, Vol. 32, 202 0
140
In the the case (ii) (resp., (iii)), y ∈ ITq ((ℓbIT , n̂IT )i (resp., z ∈ ITq ((ℓbIT , n̂IT )i ) induce ℓbIT (y) > 1−niIT ≥ niIT
(resp., ℓbIT (z) > 1 − niIT ≥ niIT ), that is, y ∈ U ((ℓbIT , n̂IT )i (resp., z ∈ U ((ℓbIT , n̂IT )i ). Thus z ∗ y ∈
IT ∈∨q ((ℓbIT , n̂IT )i for i = 1, 2, · · · , k which implies that
z∗y ∈
k
\
i=1
IT ∈∨q ((ℓbIT , n̂IT )i = IT ∈∨q (ℓbIT , n̂IT ).
The last case induces ℓbIT (z) > 1 − niIT ≥ niIT and ℓbIT (y) > 1 − niIT ≥ niIT , i.e., z, y ∈ U ((ℓbIT , n̂IT )i for
i = 1, 2, · · · , k. It follows that
z∗y ∈
k
\
i=1
IT ∈∨q ((ℓbIT , n̂IT )i = IT ∈∨q (ℓbIT , n̂IT ).
Therefore IT ∈∨q (ℓbIT , n̂IT ) is a subalgebra of X for all n̂IT ∈ (0, 0.5]k . Similarly, we can show that the set
T ∈∨q (ℓbT , n̂T ) is a subalgebra of X for all n̂T ∈ (0, 0.5]k . Let z, y ∈ F ∈∨q (ℓbF , n̂F ). Then
ℓbF (z) ≤ n̂F or ℓbF (z) + n̂F < 1̂
and
ℓbF (y) ≤ n̂F or ℓbF (y) + n̂F < 1̂.
If ℓbF (z) ≤ n̂F and ℓbF (y) ≤ n̂F , then
ℓbF (z ∗ y) ≤
_
c ≤ n̂F ∨ 0.5
c = n̂F
{ℓbF (z), ℓbF (y), 0.5}
by Theorem 4.5, and so z ∗ y ∈ L(ℓbF , n̂F ) ⊆ F ∈∨q (ℓbF , n̂F ). If ℓbF (z) ≤ n̂F or ℓbF (y) + n̂F < 1̂, then
_
_
c ≤ {n̂F , 1̂ − n̂F , 0.5}
c = n̂F
ℓbF (z ∗ y) ≤ {ℓbF (z), ℓbF (y), 0.5}
by Theorem 4.5. Hence z ∗ y ∈ L(ℓbF , n̂F ) ⊆ F ∈∨q (ℓbF , n̂F ). Similarly, if ℓbF (z) + n̂F < 1̂ and ℓbF (y) ≤ n̂F ,
then z ∗ y ∈ F ∈∨q (ℓbF , n̂F ). If ℓbF (z) + n̂F < 1̂ and ℓbF (y) + n̂F < 1̂, then
_
c ≤ (1̂ − n̂F ) ∨ 0.5
c = 0.5
c < n̂F
ℓbF (z ∗ y) ≤ {ℓbF (z), ℓbF (y), 0.5}
by Theorem 4.5. Thus z ∗ y ∈ L(ℓbF , n̂F ) ⊆ F ∈∨q (ℓbF , n̂F ). Consequencly, F ∈∨q (ℓbF , n̂F ) is a subalgebra of
X for all n̂F ∈ [0.5, 1)k . By the similar way, we can verify that IF ∈∨q (ℓbIF , n̂IF ) is a subalgebra of X for all
n̂IF ∈ [0.5, 1)k .
R.A. Borzooei, F. Smarandache, Y.B. Jun, Polarity of generalized neutrosophic subalgebras in
BCK/BCI-algebras.
Neutrosophic Sets and Systems, Vol. 32, 202 0
141
5 k-polar generalized (q, ∈ ∨q)-neutrosophic subalgebras
Definition 5.1. Let Lb := ℓbT , ℓbIT , ℓbIF , ℓbF be a k-polar generalized neutrosophic set over X. Then Lb is called
a k-polar generalized (q, ∈ ∨q)-neutrosophic subalgebra of X if it satisfies:
z ∈ Tq (ℓbT , n̂T ), y ∈ Tq (ℓbT , n̂T ) ⇒ z ∗ y ∈ T ∈∨q (ℓbT , n̂T ),
z ∈ ITq (ℓbIT , n̂IT ), y ∈ ITq (ℓbIT , n̂IT ) ⇒ z ∗ y ∈ IT ∈∨q (ℓbIT , n̂IT ),
z ∈ IFq (ℓbIF , n̂IF ), y ∈ IFq (ℓbIF , n̂IF ) ⇒ z ∗ y ∈ IF ∈∨q (ℓbIF , n̂IF ),
(5.1)
z ∈ Fq (ℓbF , n̂F ), y ∈ Fq (ℓbF , n̂F ) ⇒ z ∗ y ∈ F ∈∨q (ℓbF , n̂F )
for all z, y ∈ X, n̂T , n̂IT ∈ (0, 1]k and n̂F , n̂IF ∈ [0, 1)k .
Example 5.2. Let X = {0, 1, 2, α, β} be the BCI-algebra which is given in Example 4.3. Let Lb := ℓbT , ℓbIT ,
ℓbIF , ℓbF be a 3-polar generalized neutrosophic set over X in which ℓbT , ℓbIT , ℓbIF and ℓbF are defined as follows:
(0.6, 0.7, 0.8) if z = 0,
(0.7, 0.0, 0.0) if z = 1,
3
b
(0.0, 0.0, 0.9) if z = 2,
ℓT : X → [0, 1] , z 7→
(0.0, 0.0, 0.0) if z = α,
(0.0, 0.0, 0.0) if z = β,
ℓbIT
ℓbIF
(0.6, 0.7, 0.8)
(0.7, 0.0, 0.0)
3
(0.5, 0.8, 0.9)
: X → [0, 1] , z 7→
(0.0, 0.0, 0.7)
(0.0, 0.0, 0.0)
(0.2, 0.3, 0.1)
(1.0, 1.0, 0.2)
(0.3, 0.4, 1.0)
: X → [0, 1]3 , z 7→
(0.4, 1.0, 1.0)
(1.0, 1.0, 1.0)
if
if
if
if
if
z
z
z
z
z
= 0,
= 1,
= 2,
= α,
= β,
if
if
if
if
if
z
z
z
z
z
= 0,
= 1,
= 2,
= α,
= β,
(0.2, 0.4, 0.4) if z = 0,
(0.4, 1.0, 1.0) if z = 1,
(1.0, 0.2, 0.1) if z = 2,
ℓbF : X → [0, 1]3 , z 7→
(1.0, 0.3, 1.0) if z = α,
(1.0, 1.0, 1.0) if z = β,
It is routine to verify that Lb := ℓbT , ℓbIT , ℓbIF , ℓbF is a 3-polar generalized (q, ∈ ∨q)-neutrosophic subalgebra
of X.
R.A. Borzooei, F. Smarandache, Y.B. Jun, Polarity of generalized neutrosophic subalgebras in
BCK/BCI-algebras.
142
Neutrosophic Sets and Systems, Vol. 32, 202 0
Using the k-polar generalized (q, ∈ ∨q)-neutrosophic subalgebra, we show that the generaliged neutrosophic q-sets and the generaliged neutrosophic ∈ ∨q-sets are subalgebras.
b
b
b
b
b
Theorem 5.3. If L := ℓT , ℓIT , ℓIF , ℓF is a k-polar generalized (q, ∈ ∨q)-neutrosophic subalgebra of X,
then the generaliged neutrosophic q-sets Tq (ℓbT , n̂T ), ITq (ℓbIT , n̂IT ), IFq (ℓbIF , n̂IF ) and Fq (ℓbF , n̂F ) are subalgebras of X for all n̂T , n̂IT ∈ (0.5, 1]k and n̂F , n̂IF ∈ [0, 0.5)k .
Proof. Let z, y ∈ Tq (ℓbT , n̂T ). Then z ∗ y ∈ T ∈∨q (ℓbT , n̂T ), and so z ∗ y ∈ U (ℓbT , n̂T ) or z ∗ y ∈ Tq (ℓbT , n̂T ).
If z ∗ y ∈ U (ℓbT , n̂T ), then (πi ◦ ℓbT )(z ∗ y) ≥ niT > 1 − niT since niT > 0.5 for all i = 1, 2, · · · , k. Hence
z ∗y ∈ Tq (ℓbT , n̂T ), and so Tq (ℓbT , n̂T ) is a subalgebra of X. By the similar way, we can verify that ITq (ℓbIT , n̂IT )
is a subalgebra of X. Let z, y ∈ Fq (ℓbF , n̂F ). Then z ∗ y ∈ F ∈∨q (ℓbF , n̂F ), and so z ∗ y ∈ L(ℓbF , n̂F ) of
z ∗ y ∈ Fq (ℓbF , n̂F ). If z ∗ y ∈ L(ℓbF , n̂F ), then (πi ◦ ℓbF )(z ∗ y) ≤ niF < 1 − niF since niF < 0.5 for all
i = 1, 2, · · · , k. Thus z ∗ y ∈ Fq (ℓbF , n̂F ), and hence Fq (ℓbF , n̂F ) is a subalgebra of X. Similarly, the set
IFq (ℓbIF , n̂IF ) is a subalgebra of X.
Theorem 5.4. If Lb := ℓbT , ℓbIT , ℓbIF , ℓbF is a k-polar generalized (q, ∈ ∨q)-neutrosophic subalgebra of X,
then the generaliged neutrosophic ∈ ∨q-sets T ∈∨q (ℓbT , n̂T ), IT ∈∨q (ℓbIT , n̂IT ), IF ∈∨q (ℓbIF , n̂IF ) and F ∈∨q (ℓbF , n̂F )
are subalgebras of X for all n̂T , n̂IT ∈ (0.5, 1]k and n̂F , n̂IF ∈ [0, 0.5)k .
Proof. Let z, y ∈ T ∈∨q (ℓbT , n̂T ) for n̂T ∈ (0.5, 1]k . If z, y ∈ Tq (ℓbT , n̂T ), then obviously z ∗ y ∈ T ∈∨q (ℓbT , n̂T ).
If z ∈ U (ℓbT , n̂T ) and y ∈ Tq (ℓbT , n̂T ), then ℓbT (z) + n̂T ≥ 2n̂T > 1̂, i.e., z ∈ Tq (ℓbT , n̂T ). It follows that
z ∗ y ∈ T ∈∨q (ℓbT , n̂T ). We can prove z ∗ y ∈ T ∈∨q (ℓbT , n̂T ) whenever y ∈ U (ℓbT , n̂T ) and z ∈ Tq (ℓbT , n̂T )
in the same way. If z, y ∈ U (ℓbT , n̂T ), then ℓbT (z) + n̂T ≥ 2n̂T > 1̂ and ℓbT (y) + n̂T ≥ 2n̂T > 1̂ and so
z, y ∈ Tq (ℓbT , n̂T ). Thus z ∗ y ∈ T ∈∨q (ℓbT , n̂T ). Therefore T ∈∨q (ℓbT , n̂T ) is a subalgebra of X for n̂T ∈ (0.5, 1]k .
Now, let z, y ∈ F ∈∨q (ℓbF , n̂F ) for n̂F ∈ [0, 0.5)k . If z, y ∈ Fq (ℓbF , n̂F ), then obviously z ∗ y ∈ F ∈∨q (ℓbF , n̂F ).
If z ∈ L(ℓbF , n̂F ) and y ∈ Fq (ℓbF , n̂F ), then ℓbF (z) + n̂F ≤ 2n̂F < 1̂, i.e., z ∈ Fq (ℓbF , n̂F ). Hence z ∗ y ∈
F ∈∨q (ℓbF , n̂F ). Similarly, we can prove that if y ∈ L(ℓbF , n̂F ) and z ∈ Fq (ℓbF , n̂F ), then z ∗ y ∈ F ∈∨q (ℓbF , n̂F ). If
z, y ∈ L(ℓbF , n̂F ), then ℓbF (z) + n̂F ≤ 2n̂F < 1̂ and ℓbF (y) + n̂F ≤ 2n̂F < 1̂, that is, z, y ∈ Fq (ℓbF , n̂F ). Hence
z ∗ y ∈ F ∈∨q (ℓbF , n̂F ). Therefore F ∈∨q (ℓbF , n̂F ) is a subalgebra of X for all n̂F ∈ [0, 0.5)k . In the same way, we
can show that IT ∈∨q (ℓbIT , n̂IT ) is a subalgebra of X for n̂IT ∈ (0.5, 1]k and IF ∈∨q (ℓbIF , n̂IF ) is a subalgebra of
X for all n̂IF ∈ [0, 0.5)k .
We provide conditions for a k-polar generalized neutrosophic set to be a k-polar generalized (q, ∈ ∨q)neutrosophic subalgebra.
b
b
b
b
b
Theorem 5.5. For a subalgebra S of X, let L := ℓT , ℓIT , ℓIF , ℓF be a k-polar generalized neutrosophic set
over X such that
c ℓbIT (z) ≥ 0.5,
c ℓbIF (z) ≤ 0.5,
c ℓbF (z) ≤ 0.5),
c
(∀z ∈ S)(ℓbT (z) ≥ 0.5,
(∀z ∈ X \ S)(ℓbT (z) = b
0 = ℓbIT (z), ℓbIF (z) = b
1 = ℓbF (z)).
Then Lb is a k-polar generalized (q, ∈ ∨q)-neutrosophic subalgebra of X.
R.A. Borzooei, F. Smarandache, Y.B. Jun, Polarity of generalized neutrosophic subalgebras in
BCK/BCI-algebras.
(5.2)
(5.3)
Neutrosophic Sets and Systems, Vol. 32, 202 0
143
T
Proof. Let z, y ∈ Tq (ℓbT , n̂T ) = ki=1 Tq (ℓbT , n̂T )i . Then (πi ◦ ℓbT )(z) + niT > 1 and (πi ◦ ℓbT )(y) + niT > 1 for all
i = 1, 2, · · · , k. If z ∗ y ∈
/ S, then z ∈ X \ S or y ∈ X \ S since S is a subalgebra of X. Hence (πi ◦ ℓbT )(z) = 0
or (πi ◦ ℓbT )(y) = 0, which imply that niT > 1, a contradiction. Thus z ∗ y ∈ S and so (πi ◦ ℓbT )(z ∗ y) ≥ 0.5
by (5.2). If niT > 0.5, then (πi ◦ ℓbT )(z ∗ y) + niT > 1, ie., z ∗ y ∈ Tq (ℓbT , n̂T )i for all i = 1, 2, · · · , k. Hence
T
z ∗ y ∈ ki=1 Tq (ℓbT , n̂T )i = Tq (ℓbT , n̂T ). Similarly, if z, y ∈ ITq (ℓbIT , n̂IT ), then z ∗ y ∈ ITq (ℓbIT , n̂IT ). Let
T
z, y ∈ IFq (ℓbIF , n̂IF ) = ki=1 IFq (ℓbIF , n̂IF )i . Then (πi ◦ ℓbIF )(z) + niIF < 1 and (πi ◦ ℓbIF )(y) + niIF < 1
for all i = 1, 2, · · · , k, which implies that z ∗ y ∈ S. If niIF ≥ 0.5, then (πi ◦ ℓbIF )(z ∗ y) ≤ 0.5 ≤ niIF
T
for all i = 1, 2, · · · , k which shows that z ∗ y ∈ ki=1 L(ℓbIF , n̂IF )i = L(ℓbIF , n̂IF ). If niIF < 0.5, then
T
(πi ◦ ℓbIF )(z ∗ y) + niIF < 1 for all i = 1, 2, · · · , k and so z ∗ y ∈ ki=1 IFq (ℓbIF , n̂IF )i = IFq (ℓbIF , n̂IF ).
Similarly way is to show that if z, y ∈ Fq (ℓbF , n̂F ), then z ∗ y ∈ F ∈∨q (ℓbF , n̂F ). Therefore Lb is a k-polar
generalized (q, ∈ ∨q)-neutrosophic subalgebra of X.
Combining Theorems 5.3 and 5.5, we have the following corollary.
b
b
b
b
b
Corollary 5.6. If a k-polar generalized neutrosophic set L := ℓT , ℓIT , ℓIF , ℓF satisfies two conditions
(5.2) and (5.3) for a subalgebra S of X, then the generaliged neutrosophic q-sets Tq (ℓbT , n̂T ), ITq (ℓbIT , n̂IT ),
IFq (ℓbIF , n̂IF ) and Fq (ℓbF , n̂F ) are subalgebras of X for all n̂T , n̂IT ∈ (0.5, 1]k and n̂F , n̂IF ∈ [0, 0.5)k .
6
Conclusions
We have introduced k-polar generalized neutrosophic set and have applied it to BCK/BCI-algebras. We have
defined k-polar generalized neutrosophic subalgebra, k-polar generalized (∈, ∈ ∨q)-neutrosophic subalgebra and k-polar generalized (q, ∈ ∨q)-neutrosophic subalgebra and have studid various properties. We have
discussed characterization of k-polar generalized neutrosophic subalgebra and k-polar generalized (∈, ∈ ∨q)neutrosophic subalgebra. We have shown that the necessity and possibility operator of k-polar generalized
neutrosophic subalgebra are also a k-polar generalized neutrosophic subalgebra. Using the k-polar generalized (∈, ∈ ∨q)-neutrosophic subalgebra, we have shown that the generaliged neutrosophic q-sets and the
generaliged neutrosophic ∈ ∨q-sets subalgebras. Using the k-polar generalized (q, ∈ ∨q)-neutrosophic subalgebra, we have shown that the generaliged neutrosophic q-sets and the generaliged neutrosophic ∈ ∨q-sets
are subalgebras. Using the generaliged neutrosophic ∈ ∨q-sets, we have established a k-polar generalized (∈,
∈ ∨q)-neutrosophic subalgebra. We have provided conditions for a k-polar generalized neutrosophic set to be
a k-polar generalized neutrosophic subalgebra and a k-polar generalized (q, ∈ ∨q)-neutrosophic subalgebra.
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Received: Oct 21, 2019. Accepted: Mar 20, 2020
R.A. Borzooei, F. Smarandache, Y.B. Jun, Polarity of generalized neutrosophic subalgebras in
BCK/BCI-algebras.
Neutrosophic Sets and Systems, Vol. 32, 2020
University of New Mexico
Neutrosophic N-Soft Sets with TOPSIS method for Multiple
Attribute Decision Making
Muhammad Riaz 1, Khalid Naeem2, Iqra Zareef 3, Deeba Afzal4
1
2
3
4
Department of Mathematics, University of the Punjab Lahore, Pakistan. E-mail: mriaz.math@pu.edu.pk
Department of Mathematics & Statistics, The University of Lahore, Pakistan. E-mail: khalidnaeem333@gmail.com
Department of Mathematics, University of the Punjab Lahore, Pakistan. E-mail: iqramaan90@yahoo.com
Department of Mathematics & Statistics, The University of Lahore, Pakistan. Pakistan. E-mail: deebafzal@gmail.com
Abstract: The objective of this article is to introduce a new hybrid model of neutrosophic N-soft set
which is combination of neutrosophic set and N-soft set. We introduce some basic operations on
neutrosophic N-soft sets along with their fundamental properties. For multi-attribute decisionmaking (MADM) problems with neutrosophic N-soft sets, we propose an extended TOPSIS
(technique based on order preference by similarity to ideal solution) method. In this method, we first
propose a weighted decision matrix based comparison method to identify the positive and the
negative ideal solutions. Afterwards, we define a separation measurement of these solutions. Finally,
we calculate relative closeness to identify the optimal alternative. At length, a numerical example is
rendered to illustrate the developed scheme in medical diagnosis via hypothetical case study.
Keywords: Neutosophic N-soft set, operations on neutosophic N-soft sets, MADM, TOPSIS, medical
diagnosis.
1. Introduction
In contemporary decision-making science, multi-attribute decision-making (MADM) phenomenon
plays a significant role in solving many real world problems. To deal with uncertainties, researchers
have introduced different theories including, Fuzzy set (FS) [54] that comprises a mapping
communicating the degree of association and intuitionistic fuzzy set (IFS) [10, 11] that comprises a
pair of mappings communicating the degree of association and the degree of non-association of
members of the universe to the unit closed interval with the restriction that sum of degree of
association and degree of non-association should not exceed one. Smarandache [46, 47] introduced
neutrosophic sets as an extension of IFSs. A neutrosophic object comprises three degrees, namely,
degree of association, indeterminacy, and the degree of non-association to each alternative.
Smarandache's Neutrosophic Set [50] is a generalization of Intuitionistic Fuzzy Set, Inconsistent
Intuitionistic Fuzzy Set (Picture Fuzzy Set, Ternary Fuzzy Set), Pythagorean Fuzzy Set (Atanassov’s
Intuitionistic Fuzzy Set of second type), q-Rung Orthopair Fuzzy Set, Spherical Fuzzy Set, and nHyper-Spherical Fuzzy Set; while Neutrosophication is a generalization of Regret Theory, Grey
System Theory, and Three-Ways Decision. In 1999, Molodtsov [32] presented the notion of soft set as
an important mathematical tool to deal with uncertainties. In 2007, Aktas and Cagman [6] extended
the idea of soft sets to soft groups. In 2010, Feng et al. [18, 19] presented several results on soft sets,
fuzzy soft sets and rough sets. In 2009 and 2011, Ali et al. [7, 8] introduced various properties of soft
sets, fuzzy soft sets and rough sets. In 2011, Cagman et al. [12], and Shabir and Naz [51] independently
presented soft topological spaces. Arockiarani et al. [9], in 2013, introduced the notion of fuzzy
M. Riaz, K. Naeem, I. Zareef and D. Afzal, Neutrosophic N-Soft Sets with TOPSIS method
Neutrosophic Sets and Systems, Vol. 32, 2020
147
neutrosophic soft toplogical spaces. In 2016, Davvaz and Sadrabadi [16] presented an interesting
application of IFSs in medicine. Nabeeh et al. [33, 34] worked on neutrosophic multi-criteria decision
making approach for IoT-based enterprises and for personnel selection used the neutrosophicTOPSIS approach in 2019. Chang et al. [35] worked towards a reuse strategic decision pattern
framework-from theories to practices. Garg and Arora [20]-[23] introduced generalized intuitionistic
fuzzy soft power aggregation operator, Dual hesitant fuzzy soft aggregation operators, a novel scaled
prioritized intuitionistic fuzzy soft interaction averaging aggregation operators and their application
to multi criteria decision-making. Peng and Dai [36] presented some approaches to single-valued
neutrosophic MADM based on MABAC, TOPSIS and new similarity measure with score function.
Hashmi et al. [24] introduced m-polar neutrosophic topology with applications to multi-criteria
decision-making in medical diagnosis and clustering analysis. In 2019, Naeem et al. [29] presented
pythagorean fuzzy soft MCGDM methods based on TOPSIS, VIKOR and aggregation operators. In
2019, Naeem et al. [30] established pythagorean m-polar fuzzy sets and TOPSIS method for the
selection of advertisement mode. In 2019, Riaz et al. [37] introduced N-soft topology and its
applications to multi-criteria group decision making (MCGDM). Riaz and Hashmi [38] introduced
the concept of cubic m-polar fuzzy set and presented multi-attribute group decision making
(MAGDM) method for agribusiness in the environment of various cubic m-polar fuzzy averaging
aggregation operators. Riaz and Hashmi [39] introduced the notion of linear Diophantine fuzzy Set
(LDFS) and its applications towards multi-attribute decision making problems. Riaz and Hashmi [40]
introduced soft rough Pythagorean m-polar fuzzy sets and Pythagorean m-polar fuzzy soft rough sets
with application to decision-making. Riaz and Tehrim [41, 42, 43] substantiated the idea of bipolar
fuzzy soft topology, cubic bipolar fuzzy set and cubic bipolar fuzzy ordered weighted geometric
aggregation operators and their application using internal and external cubic bipolar fuzzy data. Riaz
and Tahrim [44] introduced the concept of bipolar fuzzy soft mappings with application to bipolar
disorders.
Smarandache [48] introduced a unifying field in logics: Neutrosophic Logic. Neutrosophy,
Neutrosophic Set, Neutrosophic Probability and Statistics. Smarandache [49] introduced
Neutrosophic Overset, Neutrosophic Underset, and Neutrosophic Offset. Similarly for Neutrosophic
Over-/Under-/Off- Logic, Probability, and Statistics.
Soft sets provide binary evaluation of the objects and other mathematical models like fuzzy sets,
intuitionistic fuzzy sets and neutrosophic sets associate values in the interval [0,1]. These models fail
to deal with the situation when modeling on real world problems associate non-binary evaluations.
Non-binary evaluations are also expected in rating or ranking positions. The ranking can be
expressed in multinary values in the form of number of stars, dots, grades or any generalized
notation. Motivated by these concerns, in 2017, Fatimah et al. [17] floated the idea of N-soft set as an
extended model of soft set, in order to describe the importance of grades in real life. In 2018 and 2019,
Akram et al. [1]-[3] introduced group decision-making methods based on hesitant N-soft sets and
intuitionistic fuzzy N-soft rough set.
The technique for the order of preference by similarity to ideal solution (TOPSIS) was initially
developed by Hwang and Yoon [26] in 1981. The core idea in the TOPSIS method is that selected
alternative should have least geometric distance from positive ideal solution and maximum
geometric distance from negative ideal solution. Positive ideal solution represents the condition for
M. Riaz, K. Naeem, I. Zareef and D. Afzal, Neutrosophic N-Soft Sets with TOPSIS method
Neutrosophic Sets and Systems, Vol. 32, 2020
148
best solution whereas negative ideal solution represents the condition for the worst. In 2000, Chen
[13] extended the TOPSIS method to fuzzy environment and solved a decision making problem based
on fuzzy information. Later, in 2008, Chen and Tsao [14] developed interval-valued fuzzy TOPSIS
method. TOPSIS method in intuitionistic fuzzy framework was proposed by Li and Nan [31] in 2011.
Joshi and Kumar [28] discussed TOPSIS method based on intuitionistic fuzzy entropy and distance
measure for multi-criteria decision making. Recently, in 2016 Dey et al. [15] employed TOPSIS method
for solving decision making problem under bipolar neutrosophic environment. In 2013, Xu and
Zhang [53] developed a novel approach based on maximizing deviation and TOPSIS method for the
explanation of multi-attribute decision making problems. In 2014, Zhang and Xu [55] presented an
extension of TOPSIS in multiple criteria decision making with the help of Pythagorean fuzzy sets.
Chen and Tsao [14] proposed interval-valued fuzzy TOPSIS method and its experimental analysis in
2016. In 2018, Akram and Arshad [4] presented a novel trapezoidal bipolar fuzzy TOPSIS method for
group decision-making. In 2019, Akram and Adeel [5] presented TOPSIS approach for MAGDM
based on interval-valued hesitant fuzzy N-soft environment. In 2019, Tehrim and Riaz [45] presented
a novel extension of TOPSIS method with bipolar neutrosophic soft topology and its applications to
multi-criteria group decision making (MCGDM). Riaz et al. [56]-[57] introduced novel concepts of
soft rough topology with applications to MAGDM.
The goal of this paper is to present a new hybrid model "neutrosophic N-soft set" and their
applications to the decision making (DM). Neutrosophic N-soft set is the generalization of N-soft set,
fuzzy N-soft set and intuitionistic fuzzy N-soft.
The comparison analysis of the proposed model with some existing models is given in Table 1.
Sets
Fuzzy set [54]
Intutionistic
fuzzy set [10]
Neutrosophic
set [46]
Non Binary
Truth
Falsity
Evaluation
Membership
Membership
×
×
×
×
×
Parametrization
×
×
×
×
×
×
N-soft Set [17]
Fuzzy N-soft
×
Set[1]
×
×
N-soft Set [3]
Neutrosophic
×
×
Soft Set [12]
Intutionistic
Indeterminacy
×
×
×
N-soft Set
(Proposed)
Table 1: Comparison with other existing theories
The rest of paper is organized as follows. In Section 2, we recall some fundamental concepts of N-
soft set, fuzzy neutrosophic set and fuzzy neutrosophic soft set. In Section 3, we propose our new
hybrid model fuzzy neutrosophic N-soft set along with their examples. We also present some basic
operations on fuzzy neutrosophic N-soft set with illustrations. We also investigate fundamental
M. Riaz, K. Naeem, I. Zareef and D. Afzal, Neutrosophic N-Soft Sets with TOPSIS method
Neutrosophic Sets and Systems, Vol. 32, 2020
149
properties of the proposed model by using defined operations. In Section 4, we construct relations
by using fuzzy neutrosophic N-soft set and define composition of fuzzy neutrosophic N-soft sets
using relations. We also define some new choice functions and score functions in connection with
fuzzy neutrosophic N-soft sets. In Section 5, we proposed DM method for medical diagnosis by the
model. In Section 6, we give a numerical example of this diagnosis method via conjectural case study.
In Section7, we conclude with some future directions and give suggestions for future work.
2. Preliminaries
In this segment, we review some essential definitions and a few aftereffects of N-soft and
neutrosophic sets that would be accommodating in the following segments.
Definition 2.1
[54] A
fuzzy set 𝜗 in 𝕏 is assessed up by a mapping with 𝕏 as domain and
membership degree in [0,1]. The accumulation of all
signified by 𝜗(𝕏).
fuzzy sets (FSs) in the universal set 𝕏 is
Definition 2.2 [46, 47] A neutrosophic set (NS) ℙ over the universe of discourse 𝕏 is defined as
−
+
where 𝕋ℙ , 𝕀ℙ , 𝔽ℙ : 𝕏 →] 0, 1 [ and
ℙ = {〈𝜑, (𝕋ℙ (𝜑), 𝕀ℙ (𝜑), 𝔽ℙ (𝜑))〉: 𝜑 ∈ 𝕏}
−
0 ≤ 𝕋ℙ (𝜑) + 𝕀ℙ (𝜑) + 𝔽ℙ (𝜑) ≤ 3+ .
The mapping 𝕋ℙ stands for degree of membership, 𝕀ℙ is the degree of indeterminacy and 𝔽ℙ is the
degree of falsity of points of the given set. From philosophical perspective, the neutrosophic set takes
the entries from some subset of ]− 0, 1+ [. But it many actual applications, it is inconvenient to utilize
neutrosophic set with entries from such subsets. Therefore, we consider the neutrosophic set which
takes the entries from some subset of [0,1].
Definition 2.3 [9] Let 𝕏 be a space of objects (points). A fuzzy neutrosophic set (FNS) ℙ in 𝕏 is
dispirit by a truth-membership function 𝕋𝑃 , an indeterminacy membership-function 𝕀𝑃 and a
falsity-membership function 𝔽𝑃 . In mathematical form, this collection is expressed as
ℙ = {〈𝜑, (𝕋ℙ (𝜑), 𝕀ℙ (𝜑), 𝔽ℙ (𝜑))〉: 𝜑 ∈ 𝕏, 𝕋ℙ , 𝕀ℙ , 𝔽ℙ ∈ [0,1]}
with the constraint that sum of 𝕋ℙ (𝜑), 𝕀ℙ (𝜑) and 𝔽ℙ (𝜑) should fall in [0,3] i.e.
0 ≤ 𝕋ℙ (𝜑) + 𝕀ℙ (𝜑) + 𝔽ℙ (𝜑) ≤ 3
Definition 2.4 [32] Let 𝕏 be the set of points and 𝐸 be the set of attributes with ℒ in 𝐸. Assume that
P(𝕏) denotes collection of subsets of 𝕏. The pair (𝜁, ℒ) is said to be a soft set (SS) over 𝕏, where 𝜁
is a function given by
𝜁: ℒ → P(𝕏)
Thus, an SS is expressed in mathematical form as
(𝜁, ℒ) = {(𝜉, 𝜁(𝜉)): 𝜉 ∈ ℒ}.
Definition 2.5 [9] Let 𝕏 be the initial universal set and 𝐸 be the set of parameters. We consider the
̂(𝕏) signifies the set of all NSs of 𝕏. The accretion Ωℒ is called the
non-empty set ℒ ⊆ 𝐸 . Let P
P(𝕏). We can write it as
neutrosophic soft set (NSS) over 𝕏, where Ωℒ is a function given by Ωℒ : ℒ → ̂
Ωℒ = {(𝜉, {〈𝜑, 𝕋ℒ(𝜉) (𝜑), 𝕀ℒ(𝜉) (𝜑), 𝔽ℒ(𝜉) (𝜑)〉: 𝜑 ∈ 𝕏}): 𝜉 ∈ 𝐸}
Notice that if Ωℒ (𝜉) = {〈𝜑, 0,1,1〉: 𝜑 ∈ 𝕏}, then NS-element (𝜉, Ωℒ (𝜉)) does not seem to appear in the
NSS Ωℒ . The set of all NSSs over 𝕏 is symbolized by NS(𝕏𝐸 ).
Definition 2.6 [17] Let 𝕏 be a set of points and 𝐸 be a set of attributes with ℒ in 𝐸. Let 𝒢 =
{0,1,2, ⋯ , 𝑁 − 1} be the set of ordered grades where 𝑁 ∈ {2,3, ⋯ }. The N-soft set (NSS) on 𝕏 is
denoted by (𝜁, ℒ, 𝑁) where 𝜁: ℒ → 2𝕏×𝒢 is a map characterized by
𝜁(𝜉) = (𝜑, 𝓇ℒ(𝜉) )
M. Riaz, K. Naeem, I. Zareef and D. Afzal, Neutrosophic N-Soft Sets with TOPSIS method
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∀𝜑 ∈ 𝕏, 𝜉 ∈ ℒ, 𝓇ℒ(𝜉) ∈ 𝒢.
Definition 2.7 [17] A weak complement of N-soft set (𝜁, ℒ, 𝑁) is another N-soft set (𝜁 ∁ , ℒ, 𝑁) gratifying
𝜁(𝜉)∁ ⊓ 𝜁(𝜉) = 𝜙, ∀𝜉 ∈ 𝕏.
Definition 2.8 [17] A top weak complement of N-soft set (𝜁, ℒ, 𝑁) is an N-soft set (𝜁 ⋆ , ℒ, 𝑁), where
𝜁(𝜉) = (𝜑, 𝑁 − 1), 𝑖𝑓𝓇ℒ(𝜉) (𝜑) < 𝑁 − 1
(𝜁 ⋆ , ℒ, 𝑁) = {
𝜁(𝜉) = (𝜑, 0),
𝑖𝑓𝓇ℒ(𝜉) (𝜑) = 𝑁 − 1
Definition 2.9 [17] A bottom weak complement of N-soft set (𝜁, ℒ, 𝑁) is one more N-soft set (𝜁⋆ , ℒ, 𝑁),
where
(𝜁⋆ , ℒ, 𝑁) = {
3 Neutrosophic N-soft Set
𝜁(𝜉) = (𝜑, 0),
𝜁(𝜉) = (𝜑, 𝑁 − 1),
𝑖𝑓𝓇ℒ(𝜉) (𝜑) > 0,
𝑖𝑓𝓇ℒ(𝜉) (𝜑) = 0.
In this section, we propose a novel structure neutrosophic N-soft set (NNSS), which is blend of NS and
NSS. We present some definitions and operations on NNSS too. Some properties of NNSS associated
with these operations also have been set up.
Definition 3.1 Let 𝕏 be the initial universe set, 𝐸 the set of attributes and 𝒢 the aggregate of
̂(𝕏 × 𝒢) be the collection of all NSSs of
ordered grades. We consider non-empty subset ℒ of 𝐸. Let P
𝕏 × 𝒢. A neutrosophic N-soft set (NNSS) is signified by (𝜆, Ω, 𝑁), where Ω = (𝜁, ℒ, 𝑁) is an NSS. If
there is no ambiguity, we can abbreviate it as 𝜆ℒ represented by the mapping
̂(𝕏 × 𝒢)
𝜆ℒ : ℒ → P
Mathematically,
𝜆ℒ = {(𝜉, Γℒ (𝜉)): Γℒ (𝜉) = {(〈𝜑, 𝕋ℒ(𝜉) (𝜑), 𝕀ℒ(𝜉) (𝜑), 𝔽ℒ(𝜉) (𝜑)〉, 𝓇ℒ(𝜉) (𝜑)), 𝓇ℒ ∈ 𝒢,
𝜑 ∈ 𝕏, 𝕋ℒ , 𝕀ℒ , 𝔽ℒ ∈ [0,1]}, 𝜉 ∈ 𝐸}
In short form, we may write
𝜆ℒ = {(𝜉, Γℒ (𝜉)): 𝜉 ∈ 𝐸}
where
Γℒ (𝜉) = {(〈𝜑, 𝕋ℒ(𝜉) (𝜑), 𝕀ℒ(𝜉) (𝜑), 𝔽ℒ(𝜉) (𝜑)〉, 𝓇ℒ(𝜉) (𝜑)): 𝓇ℒ ∈ 𝒢, 𝜑 ∈ 𝕏, 𝕋ℒ , 𝕀ℒ , 𝔽ℒ ∈ [0,1]}
The accretion of all NNSSs is denoted by NNS(𝕏).
Our proposed structure is more generalized then other existing models. The existing models are
special cases of our proposed model, as shown in Table 2
Neutrosophic N-soft Set (Proposed)
Intutionistic N-soft Set [3]
(𝜉, (〈𝜑, 𝕋ℒ(𝜉) (𝜑), 𝕀ℒ(𝜉) (𝜑), 𝔽ℒ(𝜉) (𝜑)〉, 𝓇ℒ(𝜉) (𝜑))
(𝜉, (〈𝜑, 𝕋ℒ(𝜉) (𝜑),0, 𝔽ℒ(𝜉) (𝜑)〉, 𝓇ℒ(𝜉) (𝜑))
(𝜉, (〈𝜑, 𝕋ℒ(𝜉) (𝜑),0,0〉, 𝓇ℒ(𝜉) (𝜑))
Fuzzy N-soft Set [1]
(𝜉, 𝓇ℒ(𝜉) (𝜑))
N-soft Set [17]
Table 2: Comparison with N-soft set and it's other existing generalization
Example 3.2 Let 𝕏 = {𝜑1 , 𝜑2 } and 𝐸 = {𝜉1 , 𝜉2 , 𝜉3 }. Consider 𝐸 ⊇ ℒ = {𝜉1 , 𝜉2 }. Define N8SS as 𝜆ℒ =
{(𝜉𝑖 , Γℒ (𝜉𝑖 )): 𝜉𝑖 ∈ ℒ, 𝑖 = 1,2}, where 8SS is given in Table 3 below:
(𝜁, ℒ, 8)
𝜉1
𝜉2
𝜑2
4
5
𝜑1
6
3
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Table 3: Tabular representation of 8SS
Now, we define N8SS as
Γℒ (𝜉1 ) = {(〈𝜑1 , 0.8,0.5,0.1〉,6), (〈𝜑2 , 0.6,0.2,0.9〉,4)}
Γℒ (𝜉2 ) = {(〈𝜑1 , 0.5,0.7,0.3〉,3), (〈𝜑2 , 0.7,0.4,0.8〉,5)}
The tabular representation of N8SS is given in Table 4.
𝜆ℒ
𝜉1
𝜑1
𝜑2
𝜉2
(〈0.8,0.5,0.1〉,6)
(〈0.5,0.7,0.3〉,3)
(〈0.6,0.2,0.9〉,4)
(〈0.7,0.4,0.8〉,5)
Table 4: Tabular representation of N8SS
Remarks:
1.
2.
Every N2SS (𝜆, Ω, 2) is generally equal to NSS.
Any arbitrary NNSS over the universe 𝕏 can also be thought of as N(𝑁 + 1)-soft set. For example
an N8SS can also be treated as an N9SS for the grade 8 is never used as can be seen in Table 4.
This observation may be extended on the parallel track.
Now, we head towards presenting some arithmetical notions related to NNSS.
Definition 3.3 Let 𝜆ℒ , 𝜆ℳ ∈NNS(𝕏). 𝜆ℒ is said to be NNS- subset of 𝜆ℳ , if
ℒ ⊑ ℳ,
𝕋ℒ(𝜉) (𝜑) ≤ 𝕋ℳ(𝜉) (𝜑),
𝕀ℒ(𝜉) (𝜑) ≥ 𝕀ℳ(𝜉) (𝜑),
𝔽ℒ(𝜉) (𝜑) ≥ 𝔽ℳ(𝜉) (𝜑),
𝓇ℒ(𝜉) (𝜑) ≤ 𝓇ℳ(𝜉) (𝜑)
∀𝜉 ∈ 𝐸, 𝜑 ∈ 𝒳, 𝓇ℒ ∈ 𝒢. We demonstrate it by 𝜆ℒ ⊑ 𝜆ℳ . 𝜆ℳ is said to be NNS- superset of 𝜆ℒ .
Example 3.4 Let 𝕏 = {𝜑1 , 𝜑2 } and 𝐸 = {𝜉1 , 𝜉2 , 𝜉3 }. Consider 𝐸 ⊇ ℒ = {𝜉1 , 𝜉2 }. Consider N8SS 𝜆ℒ as
given in Example 3.2. Let ℳ = 𝐸. Define N8SS 𝜆ℳ as
𝜆ℳ = {(𝜉𝑖 , Γℳ (𝜉𝑖 )): 𝜉𝑖 ∈ ℳ, 𝑖 = 1,2,3}
where 8SS is given in Table 5 below.
(𝜁, ℳ, 8)
𝜑1
𝜑2
Now, we define N8SS
𝜉1
𝜉2
𝜉3
7
4
6
7
3
5
Table 5: Tabular representation of 8SS
Γℳ (𝜉1 ) = {(〈𝜑1 , 0.9,0.4,0.0〉,7), (〈𝜑2 , 0.7,0.1,0.8〉,5)}
Γℳ (𝜉2 ) = {(〈𝜑1 , 0.6,0.5,0.2〉,4), (〈𝜑2 , 0.9,0.3,0.8〉,7)}
having tabular form
Γℳ (𝜉3 ) = {(〈𝜑1 , 0.8,0.5,0.1〉,6), (〈𝜑3 , 0.5,0.7,0.3〉,3)}
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𝜆ℳ
𝜑1
𝜑2
152
𝜉1
𝜉2
(〈0.9,0.4,0.0〉,7)
𝜉3
(〈0.6,0.5,0.2〉,4)
(〈0.7,0.1,0.8〉,5)
(〈0.8,0.5,0.1〉,6)
(〈0.9,0.3,0.8〉,7)
(〈0.5,0.7,0.3〉,3)
Table 6: Tabular representation of N8SS
It can be seen from Table 4 and Table 6 that 𝜆ℒ ⊑ 𝜆ℳ .
Definition 3.5 Let 𝜆ℒ , 𝜆ℳ ∈NNS(𝕏). Then 𝜆ℒ and 𝜆ℳ are said to be NNS- equal, if
ℒ = ℳ,
𝕋ℒ(𝜉) (𝜑) = 𝕋ℳ(𝜉) (𝜑),
𝕀ℒ(𝜉) (𝜑) = 𝕀ℳ(𝜉) (𝜑),
𝔽ℒ(𝜉) (𝜑) = 𝔽ℳ(𝜉) (𝜑),
𝓇ℒ(𝜉) (𝜑) = 𝓇ℳ(𝜉) (𝜑)
∀𝜉 ∈ 𝐸, 𝜑 ∈ 𝒳, 𝓇ℒ ∈ 𝒢. We demonstrate it by 𝜆ℒ = 𝜆ℳ .
Definition 3.6 Let 𝜆ℒ ∈NNS(𝕏). If 𝕋ℒ(𝜉) (𝜑) = 0, 𝕀ℒ(𝜉) (𝜑) = 1, 𝔽ℒ(𝜉) (𝜑) = 1 and 𝓇ℒ(𝜉) (𝜑) = 0, ∀𝜉 ∈
𝐸, 𝜑 ∈ 𝒳, 𝓇ℒ ∈ 𝒢; then 𝜆ℒ is called null NNSS and symbolized by 𝜆ℒ𝜙 .
Example 3.7 Let 𝕏 = {𝜑1 , 𝜑2 } and 𝐸 = {𝜉1 , 𝜉2 , 𝜉3 }. Consider 𝐸 ⊇ ℒ = {𝜉1 , 𝜉2 }. Define null N8SS as
𝜆ℒ𝜙 = {(𝜉𝑖 , Γℒ𝜙 (𝜉𝑖 )): 𝜉𝑖 ∈ ℒ, 𝑖 = 1,2} where
Γℒ𝜙 (𝜉1 ) = {(〈𝜑1 , 0,1,1〉,0), (〈𝜑2 , 0,1,1〉,0)}
The tabular form given in Table 7
Γℒ𝜙 (𝜉2 ) = {(〈𝜑1 , 0,1,1〉,0), (〈𝜑2 , 0,1,1〉,0)}
𝜆ℒ 𝜙
𝜉1
𝜑1
𝜑2
(〈0,1,1〉,0)
(〈0,1,1〉,0)
𝜉2
(〈0,1,1〉,0)
(〈0,1,1〉,0)
Table 7: Tabular representation of null N8SS
Definition 3.8 Let 𝜆ℒ ∈NNS(𝕏). If 𝕋ℒ(𝜉) (𝜑) = 1, 𝕀ℒ(𝜉) (𝜑) = 0, 𝔽ℒ(𝜉) (𝜑) = 0 and 𝓇ℒ(𝜉) (𝜑) = 𝑁 − 1,
∀𝜉 ∈ 𝐸, 𝜑 ∈ 𝕏, 𝓇ℒ ∈ 𝒢, then 𝜆ℒ is called absolute NNSS and symbolized by 𝜆ℒ̂ .
Example 3.9 Let 𝕏 = {𝜑1 , 𝜑2 } and 𝐸 = {𝜉1 , 𝜉2 , 𝜉3 }. Consider 𝐸 ⊇ ℒ = {𝜉1 , 𝜉2 }. Define absolute N8SS
as 𝜆ℒ̂ = {(𝜉𝑖 , Γℒ̂ (𝜉𝑖 )): 𝜉𝑖 ∈ ℒ, 𝑖 = 1,2} where
Γℒ̂ (𝜉1 ) = {(〈𝜑1 , 1,0,0〉,7), (〈𝜑2 , 1,0,0〉,7)}
Γℒ̂ (𝜉2 ) = {(〈𝜑1 , 1,0,0〉,7), (〈𝜑2 , 1,0,0〉,7)}
having tabular representation that is given in Table 8:
𝜆ℒ̂
𝜑1
𝜑2
𝜉1
(〈1,0,0〉,7)
(〈1,0,0〉,7)
𝜉2
(〈1,0,0〉,7)
(〈1,0,0〉,7)
Table 8: Tabular representation of absolute N8SS
Proposition 3.10 Let 𝜆𝒦 , 𝜆ℒ , 𝜆ℳ ∈NNS(𝕏). Then,
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1. 𝜆ℒ ⊑ 𝜆ℒ̂ .
2. 𝜆ℒ𝜙 ⊑ 𝜆ℒ .
3. 𝜆ℒ ⊑ 𝜆ℒ .
4. 𝜆𝒦 ⊑ 𝜆ℒ and 𝜆ℒ ⊑ 𝜆ℳ ⇒ 𝜆𝒦 ⊑ 𝜆ℳ .
Proof. The proof follows directly from definitions of related terms.
Proposition 3.11 Let 𝜆𝒦 , 𝜆ℒ , 𝜆ℳ ∈NNS(𝕏). Then,
1. 𝜆𝒦 = 𝜆ℒ and 𝜆ℒ = 𝜆ℳ ⇒ 𝜆𝒦 = 𝜆ℳ .
2. 𝜆ℒ ⊑ 𝜆ℳ and 𝜆ℳ ⊑ 𝜆ℒ ⇒ 𝜆ℒ = 𝜆ℳ .
Proof. Straight forward.
Definition 3.12 Let 𝜆ℒ ∈NNS(𝕏). Then weak complement of NNSS 𝜆ℒ is symbolized by 𝜆ℒ∁ and defined
as
𝜆ℒ∁ = {(𝜉, Γℒ∁ ): 𝜉 ∈ 𝐸}
where
∁
(𝜑)): 𝜑 ∈ 𝑋}
Γℒ∁ = {(〈𝜑, 𝔽ℒ(𝜉) (𝜑),1 − 𝕀ℒ(𝜉) (𝜑), 𝕋ℒ(𝜉) (𝜑)〉, 𝓇ℒ(𝜉)
∁
(𝜑) denotes weak complement defined in Definition 2.7.
Here 𝓇ℒ(𝜉)
Example 3.13 Let 𝕏 = {𝜑1 , 𝜑2 } and 𝐸 = {𝜉1 , 𝜉2 , 𝜉3 }. Consider 𝐸 ⊇ ℒ = {𝜉1 , 𝜉2 }. Define complement
of N8SS 𝜆ℒ given in Example 3.2 as 𝜆ℒ∁ = {(𝜉𝑖 , Γℒ∁ (𝜉𝑖 )): 𝜉𝑖 ∈ ℒ, 𝑖 = 1,2} i.e.
Γℒ∁ (𝜉1 ) = {(〈𝜑1 , 0.1,0.5,0.8〉,5), (〈𝜑2 , 0.9,0.8,0.6〉,7)}
Γℒ∁ (𝜉2 ) = {(〈𝜑1 , 0.3,0.3,0.5〉,4), (〈𝜑2 , 0.8,0.6,0.7〉,2)}
The tabular form is given in Table 9.
𝜆∁ℒ
𝜑1
𝜑2
𝜉1
(〈0.8,0.5,0.1〉,5)
(〈0.6,0.2,0.9〉,7)
𝜉2
(〈0.5,0.7,0.3〉,4)
(〈0.7,0.4,0.8〉,2)
Table 9: Tabular representation of weak complement of N8SS
Proposition 3.14 Let 𝜆ℒ ∈NNS(𝕏), then
1. (𝜆ℒ∁ )∁ ≠ 𝜆ℒ .
2. 𝜆ℒ∁ 𝜙 ≠ 𝜆ℒ̂ .
3. 𝜆ℒ∁̂ ≠ 𝜆ℒ𝜙 .
Proof. Straight forward.
Definition 3.15 Let 𝜆ℒ ∈NNS(𝕏). Then top weak complement of NNSS 𝜆ℒ is symbolized by 𝜆ℒ⋆ and
defined as
Where,
𝜆ℒ⋆ = {(𝜉, Γℒ⋆ ): 𝜉 ∈ 𝐸}
⋆
Γℒ⋆ = {(〈𝜑, 𝔽ℒ(𝜉) (𝜑),1 − 𝕀ℒ(𝜉) (𝜑), 𝕋ℒ(𝜉) (𝜑)〉, 𝓇ℒ(𝜉)
(𝜑)): 𝜑 ∈ 𝑋}
⋆
where, 𝓇ℒ(𝜉)
(𝜑) denotes top weak complement defined in Definition 2.8.
Example 3.16 Let 𝕏 = {𝜑1 , 𝜑2 } and 𝐸 = {𝜉1 , 𝜉2 , 𝜉3 }. Consider 𝐸 ⊇ ℒ = {𝜉1 , 𝜉2 }. Define complement
of N8SS 𝜆ℒ given in Example 3.2 as 𝜆ℒ⋆ = {(𝜉𝑖 , Γℒ⋆ (𝜉𝑖 )): 𝜉𝑖 ∈ ℒ, 𝑖 = 1,2} i.e.
Γℒ⋆ (𝜉1 ) = {(〈𝜑1 , 0.1,0.5,0.8〉,7), (〈𝜑2 , 0.9,0.8,0.6〉,7)}
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Γℒ⋆ (𝜉2 ) = {(〈𝜑1 , 0.3,0.3,0.5〉,7), (〈𝜑2 , 0.8,0.6,0.7〉,7)}
In tabular form given in Table 10.
𝜆ℒ⋆
𝜉1
𝜑1
(〈0.8,0.5,0.1〉,7)
𝜑2
(〈0.6,0.2,0.9〉,7)
𝜉2
(〈0.5,0.7,0.3〉,7)
(〈0.7,0.4,0.8〉,7)
Table 10: Tabular representation of top weak complement of N8SS
Proposition 3.17 Let 𝜆ℒ ∈NNS(𝕏). Then,
1. (𝜆ℒ⋆ )⋆ ≠ 𝜆ℒ .
2. 𝜆ℒ⋆ 𝜙 = 𝜆ℒ̂ .
3. 𝜆ℒ⋆̂ = 𝜆ℒ𝜙 .
Proof. The proof follows quickly from definitions of relevant terms.
Definition 3.18 Let 𝜆ℒ ∈NNS(𝕏). Then bottom weak complement of NNSS 𝜆ℒ is symbolized by 𝜆ℒ⋆ and
defined as follows
𝜆ℒ⋆ = {(𝜉, Γℒ⋆ ): 𝜉 ∈ 𝐸}
where
Γℒ⋆ = {(〈𝜑, 𝔽ℒ(𝜉) (𝜑),1 − 𝕀ℒ(𝜉) (𝜑), 𝕋ℒ(𝜉) (𝜑)〉, 𝓇ℒ(𝜉)⋆ (𝜑)): 𝜑 ∈ 𝑋}
Here 𝓇ℒ(𝜉)⋆ (𝜑) denotes top weak complement defined in Definition 2.9.
Example 3.19
Let 𝕏 = {𝜑1 , 𝜑2 } and 𝐸 = {𝜉1 , 𝜉2 , 𝜉3 } . Consider 𝐸 ⊇ ℒ = {𝜉1 , 𝜉2 } . Bottom weak
complement of N8SS 𝜆ℒ defined in Example 3.2 as 𝜆ℒ⋆ = {(𝜉𝑖 , Γℒ⋆ (𝜉𝑖 )): 𝜉𝑖 ∈ ℒ, 𝑖 = 1,2} where
Γℒ⋆ (𝜉1 ) = {(〈𝜑1 , 0.1,0.5,0.8〉,7), (〈𝜑2 , 0.9,0.8,0.6〉,7)}
Γℒ⋆ (𝜉2 ) = {(〈𝜑1 , 0.3,0.3,0.5〉,7), (〈𝜑2 , 0.8,0.6,0.7〉,7)}
In tabular form the bottom weak complement of N8SS is given in Table 11.
𝜆ℒ ⋆
𝜑1
𝜑2
𝜉1
(〈0.8,0.5,0.1〉,0)
(〈0.6,0.2,0.9〉,0)
𝜉2
(〈0.5,0.7,0.3〉,0)
(〈0.7,0.4,0.8〉,0)
Table 11: Tabular representation of bottom weak complement of N8SS
Proposition 3.20 Let 𝜆ℒ ∈NNS(𝕏). Then,
1. (𝜆ℒ⋆ )⋆ ≠ 𝜆ℒ .
2. (𝜆ℒ𝜙 )⋆ = 𝜆ℒ̂ .
3. 𝜆ℒ̂⋆ = 𝜆ℒ𝜙 .
Proof. Straight forward.
Definition 3.21 Let 𝜆ℒ , 𝜆ℳ ∈NNS(𝕏). Then difference of 𝜆ℒ and 𝜆ℳ is symbolized by 𝜆ℒ \𝜆ℳ and
is defined as
𝜆ℒ \𝜆ℳ = {(𝜉, {(〈𝜑, 𝕋ℒ(𝜉)\ℳ(𝜉) (𝜑), 𝕀ℒ(𝜉)\ℳ(𝜉) (𝜑), 𝔽ℒ(𝜉)\ℳ(𝜉) (𝜑)〉, 𝓇ℒ(𝜉)\ℳ(𝜉) (𝜑)):
𝜑 ∈ 𝕏}): 𝜉 ∈ 𝐸}
where 𝕋ℒ(𝜉)\ℳ(𝜉) (𝜑), 𝕀ℒ(𝜉)\ℳ(𝜉) (𝜑) and 𝔽ℒ(𝜉)\ℳ(𝜉) (𝜑) are defined as
𝕋ℒ(𝜉)\ℳ(𝜉) (𝜑) = min{𝕋ℒ(𝜉) (𝜑), 𝔽ℳ(𝜉) (𝜑)}
𝕀ℒ(𝜉)\ℳ(𝜉) (𝜑) = max{𝕀ℒ(𝜉) (𝜑),1 − 𝕀ℳ(𝜉) (𝜑)}
𝔽ℒ(𝜉)\ℳ(𝜉) (𝜑) = max{𝔽ℒ(𝜉) (𝜑), 𝕋ℳ(𝜉) (𝜑)}
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Definition 3.22
and is defined as
155
𝓇 (𝜑) − 𝓇ℳ(𝜉) (𝜑), 𝑖𝑓𝓇ℒ(𝜉) (𝜑) > 𝓇ℳ(𝜉) (𝜑),
𝓇ℒ(𝜉)\ℳ(𝜉) (𝜑) = { ℒ(𝜉)
0,
𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
Let 𝜆ℒ , 𝜆ℳ ∈NNS(𝕏). Then addition of 𝜆ℒ and 𝜆ℳ is symbolized by 𝜆ℒ ⊕ 𝜆ℳ
𝜆ℒ ⊕ 𝜆ℳ = {(𝜉, {(〈𝜑, 𝕋ℒ(𝜉)⊕ℳ(𝜉) (𝜑), 𝕀ℒ(𝜉)⊕ℳ(𝜉) (𝜑), 𝔽ℒ(𝜉)⊕ℳ(𝜉) (𝜑)〉, 𝓇ℒ(𝜉)⊕ℳ(𝜉) (𝜑)): 𝜑 ∈ 𝕏}): 𝜉 ∈ 𝐸}
where 𝕋ℒ(𝜉)⊕ℳ(𝜉) (𝜑), 𝕀ℒ(𝜉)⊕ℳ(𝜉) (𝜑) and 𝔽ℒ(𝜉)⊕ℳ(𝜉) (𝜑) are given as
𝕋ℒ(𝜉)⊕ℳ(𝜉) (𝜑) = min{𝕋ℒ(𝜉) (𝜑) + 𝕋ℳ(𝜉) (𝜑),1}
𝕀ℒ(𝜉)⊕ℳ(𝜉) (𝜑) = min{𝕀ℒ(𝜉) (𝜑) + 𝕀ℳ(𝜉) (𝜑),1}
𝔽ℒ(𝜉)⊕ℳ(𝜉) (𝜑) = min{𝔽ℒ(𝜉) (𝜑) + 𝔽ℳ(𝜉) (𝜑),1}
𝓇ℒ(𝜉) (𝜑) + 𝓇ℳ(𝜉) (𝜑), 𝑖𝑓0 ≤ 𝓇ℒ(𝜉) (𝜑) + 𝓇ℳ(𝜉) (𝜑) < 𝑁 − 1,
𝓇ℒ(𝜉)⊕ℳ(𝜉) (𝜑) = {
𝑁 − 1,
𝑖𝑓𝓇ℒ(𝜉) (𝜑) + 𝓇ℳ(𝜉) (𝜑) ≥ 𝑁 − 1
Let 𝜆ℒ , 𝜆ℳ ∈ NNS (𝕏) be expressed as 𝜆ℒ = (𝜆1 , Ω1 , 𝑁) and 𝜆ℳ = (𝜆2 , Ω2 , 𝑁1 )
Definition 3.23
where Ω1 = (𝜁1 , ℒ, 𝑁2 ) and Ω2 = (𝜁2 , ℳ, 𝑁1 ) are NSSs. Then their restricted union is symbolized by
(𝜆1 , Ω1 , 𝑁2 ) ⊔ℜ (𝜆2 , Ω2 , 𝑁1 ) and defined as (𝑤, Ω1 ⊔ℜ Ω2 , max(𝑁1 , 𝑁2 )) where Ω1 ⊔ℜ Ω2 = (𝑊, ℒ ⊓
ℳ, max(𝑁1 , 𝑁2 )) i.e.
(𝜆1 , Ω1 , 𝑁2 ) ⊔ℜ (𝜆2 , Ω2 , 𝑁1 )
= {(𝜉, {(〈𝜑, 𝕋ℒ(𝜉) (𝜑) ∨ 𝕋ℳ(𝜉) (𝜑), 𝕀ℒ(𝜉) (𝜑) ∧ 𝕀ℳ(𝜉) (𝜑), 𝔽ℒ(𝜉) (𝜑) ∧ 𝔽ℳ(𝜉) (𝜑)〉, 𝓇ℒ(𝜉) (𝜑)
∨ 𝓇ℳ(𝜉) (𝜑)): 𝜑 ∈ 𝕏}): 𝜉 ∈ ℒ ⊓ ℳ}
Example 3.24 Consider again 𝜆ℒ , 𝜆ℳ as given in Examples 3.2 and 3.4 respectively. The restricted
union 𝜆ℒ ⊔ℜ 𝜆ℳ is given in Table 12.
𝜆ℒ ⊔ℜ 𝜆ℳ
𝜑1
𝜉1
𝜉2
(〈0.9,0.4,0.0〉,7)
𝜑2
(〈0.6,0.5,0.2〉,4)
(〈0.7,0.1,0.8〉,5)
(〈0.9,0.3,0.8〉,7)
Table 12: Tabular representation of restricted union of two N8SSs
Let 𝜆ℒ , 𝜆ℳ ∈ NNS (𝕏) be expressed as 𝜆ℒ = (𝜆1 , Ω1 , 𝑁) and 𝜆ℳ = (𝜆2 , Ω2 , 𝑁1 )
Definition 3.25
where Ω1 = (𝜁1 , ℒ, 𝑁2 ) and Ω2 = (𝜁2 , ℳ, 𝑁1 ) are NSSs. Then their extended union is symbolized by
(𝜆1 , Ω1 , 𝑁2 ) ⊔ℰ (𝜆2 , Ω2 , 𝑁1 ) and defined as (𝑤, Ω1 ⊔ℰ Ω2 , max(𝑁1 , 𝑁2 )) where Ω1 ⊔ℰ Ω2 = (𝑊, ℒ ⊔
ℳ, max(𝑁1 , 𝑁2 )) i.e.
(𝜆1 , Ω1 , 𝑁2 ) ⊔ℰ (𝜆2 , Ω2 , 𝑁1 ) = {(𝜉, {(〈𝜑, 𝕋ℒ(𝜉) (𝜑) ∨ 𝕋ℳ(𝜉) (𝜑), 𝕀ℒ(𝜉) (𝜑) ∧ 𝕀ℳ(𝜉) (𝜑), 𝔽ℒ(𝜉) (𝜑) ∧ 𝔽ℳ(𝜉) (𝜑)〉,
𝓇ℒ(𝜉) (𝜑) ∨ 𝓇ℳ(𝜉) (𝜑)): 𝜑 ∈ 𝕏}): 𝜉 ∈ ℒ ⊔ ℳ}
Example 3.26 Consider again 𝜆ℒ , 𝜆ℳ as given in Examples 3.2 and 3.4 respectively. The extended
union 𝜆ℒ ⊔ℰ 𝜆ℳ is given in Table 13.
𝜆ℒ ⊔ℰ 𝜆ℳ
𝜑1
𝜑2
𝜉1
(〈0.9,0.4,0.0〉,7)
(〈0.7,0.1,0.8〉,5)
𝜉2
(〈0.6,0.5,0.2〉,4)
(〈0.9,0.3,0.8〉,7)
𝜉3
(〈0.8,0.5,0.1〉,6)
(〈0.5,0.7,0.3〉,3)
Table 13: Tabular representation of extended union of two N8SSs
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Theorem 3.27 Let 𝜆ℒ , 𝜆ℳ ∈NNS(𝕏). Then their extended-union 𝜆ℒ ⊔ℰ 𝜆ℳ is the smallest NNSS containing
both 𝜆ℒ and 𝜆ℳ .
Proof. Straight forward.
Let 𝜆ℒ , 𝜆ℳ ∈ NNS (𝕏) be expressed as 𝜆ℒ = (𝜆1 , Ω1 , 𝑁) and 𝜆ℳ = (𝜆2 , Ω2 , 𝑁1 )
Definition 3.28
where Ω1 = (𝜁1 , ℒ, 𝑁2 ) and Ω2 = (𝜁2 , ℳ, 𝑁1 ) are NSSs. Then their restricted intersection is symbolized
by (𝜆1 , Ω1 , 𝑁2 ) ⊓ℜ (𝜆2 , Ω2 , 𝑁1 ) and is defined as (𝑦, Ω1 ⊔ℜ Ω2 , min(𝑁1 , 𝑁2 )) where Ω1 ⊓ℜ Ω2 = (𝑌, ℒ ⊓
ℳ, min(𝑁1 , 𝑁2 )) i.e.
(𝜆1 , Ω1 , 𝑁2 ) ⊓ℜ (𝜆2 , Ω2 , 𝑁1 ) = {(𝜉, {(〈𝜑, 𝕋ℒ(𝜉) (𝜑) ∧ 𝕋ℳ(𝜉) (𝜑), 𝕀ℒ(𝜉) (𝜑) ∨ 𝕀ℳ(𝜉) (𝜑), 𝔽ℒ(𝜉) (𝜑) ∨ 𝔽ℳ(𝜉) (𝜑)〉,
𝓇ℒ(𝜉) (𝜑) ∧ 𝓇ℳ(𝜉) (𝜑)): 𝜑 ∈ 𝕏}): 𝜉 ∈ ℒ ⊓ ℳ}
Consider again 𝜆ℒ , 𝜆ℳ as given in Examples 3.2, 3.4 respectively. The restricted
Example 3.29
intersection 𝜆ℒ ⊓ℜ 𝜆ℳ is given in Table 14.
𝜆ℒ ⊓ ℜ 𝜆 ℳ
𝜉1
𝜑1
𝜉2
(〈0.8,0.5,0.1〉,6)
𝜑2
(〈0.5,0.7,0.3〉,3)
(〈0.6,0.2,0.9〉,4)
(〈0.7,0.4,0.8〉,5)
Table 14: Tabular representation of restricted intersection of two N8SSs
Theorem 3.30 Let 𝜆ℒ , 𝜆ℳ ∈ NNS(𝕏). Then their restricted-intersection 𝜆ℒ ⊓ℜ 𝜆ℳ is the largest NNSS
contained in both 𝜆ℒ and 𝜆ℳ .
Proof. Straight forward.
Let 𝜆ℒ , 𝜆ℳ ∈ NNS (𝕏) be expressed as 𝜆ℒ = (𝜆1 , Ω1 , 𝑁) and 𝜆ℳ = (𝜆2 , Ω2 , 𝑁1 )
Definition 3.31
where Ω1 = (𝜁1 , ℒ, 𝑁2 ) and Ω2 = (𝜁2 , ℳ, 𝑁1 ) are NSSs. Then their restricted intersection is symbolized
by (𝜆1 , Ω1 , 𝑁2 ) ⊓ℰ (𝜆2 , Ω2 , 𝑁1 ) and defined as (𝑦, Ω1 ⊓ℰ Ω2 , min(𝑁1 , 𝑁2 )) , where Ω1 ⊓ℰ Ω2 = (𝑌, ℒ ⊔
ℳ, min(𝑁1 , 𝑁2 )) i.e.
(𝜆1 , Ω1 , 𝑁2 ) ⊓ℰ (𝜆2 , Ω2 , 𝑁1 ) = {(𝜉, {(〈𝜑, 𝕋ℒ(𝜉) (𝜑) ∧ 𝕋ℳ(𝜉) (𝜑), 𝕀ℒ(𝜉) (𝜑) ∨ 𝕀ℳ(𝜉) (𝜑), 𝔽ℒ(𝜉) (𝜑) ∨ 𝔽ℳ(𝜉) (𝜑)〉,
Example 3.32
𝓇ℒ(𝜉) (𝜑) ∧ 𝓇ℳ(𝜉) (𝜑)): 𝜑 ∈ 𝕏}): 𝜉 ∈ ℒ ⊔ ℳ}
Consider again 𝜆ℒ , 𝜆ℳ as given in Examples 3.2, 3.4 respectively. The extended
intersection 𝜆ℒ ⊓ℰ 𝜆ℳ is given in Table 15.
𝜆ℒ ⊓ ℰ 𝜆ℳ
𝜑1
𝜑2
𝜉1
(〈0.8,0.5,0.1〉,6)
(〈0.6,0.2,0.9〉,4)
𝜉2
(〈0.5,0.7,0.3〉,3)
(〈0.7,0.4,0.8〉,5)
𝜉3
(〈0.8,0.5,0.1〉,6)
(〈0.5,0.7,0.3〉,3)
Table 15: Tabular representation of extended intersection of two N8SSs
For any two NNSS 𝜆ℒ and 𝜆ℳ over same set of points 𝕏 and using the operations defined above,
we conclude the following proposition:
Proposition 3.33 Let 𝜆ℒ and 𝜆ℳ be two NNSS
(1) 𝜆ℒ ⊔ℰ 𝜆ℒ = 𝜆ℒ
(2) 𝜆ℒ ⊔ℰ 𝜆ℳ = 𝜆ℳ ⊔ℰ 𝜆ℒ
(3) 𝜆ℒ ⊓ℛ 𝜆ℒ = 𝜆ℒ
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(4) 𝜆ℒ ⊓ℛ 𝜆ℳ = 𝜆ℳ ⊓ℛ 𝜆ℒ
(5) 𝜆ℒ ⊔ℰ 𝜆ℒ𝜙 = 𝜆ℒ
(6) 𝜆ℒ ⊓ℛ 𝜆ℒ𝜙 = 𝜆ℒ𝜙
For any three NNSS 𝜆ℒ , 𝜆ℳ and 𝜆𝒩 over same set of points 𝕏 and using the operations defined
above, we conclude the following proposition:
Proposition 3.34 Let 𝜆ℒ , 𝜆ℳ and 𝜆𝒩 be three NNSS
(1) 𝜆ℒ ⊔ℰ (𝜆ℳ ⊔ℰ 𝜆𝒩 ) = (𝜆ℒ ⊔ℰ 𝜆ℳ ) ⊔ℰ 𝜆𝒩
(2) 𝜆ℒ ⊓ℛ (𝜆ℳ ⊓ℛ 𝜆𝒩 ) = (𝜆ℒ ⊓ℛ 𝜆ℳ ) ⊓ℛ 𝜆𝒩
(3) 𝜆ℒ ⊔ℰ (𝜆ℳ ⊓ℛ 𝜆𝒩 ) = (𝜆ℒ ⊔ℰ 𝜆ℳ ) ⊓ℛ (𝜆ℒ ⊔ℰ 𝜆𝒩 )
(4) 𝜆ℒ ⊓ℛ (𝜆ℳ ⊔ℰ 𝜆𝒩 ) = (𝜆ℒ ⊓ℛ 𝜆ℳ ) ⊔ℰ (𝜆ℒ ⊓ℛ 𝜆𝒩 )
Definition 3.35
Let 𝜆ℒ , 𝜆ℳ ∈ NNS (𝕏) be expressed as 𝜆ℒ = (𝜆1 , Ω1 , 𝑁) and 𝜆ℳ = (𝜆2 , Ω2 , 𝑁1 )
where Ω1 = (𝜁1 , ℒ, 𝑁2 ) and Ω2 = (𝜁2 , ℳ, 𝑁1 ) are NSSs. Then AND Operation symbolized by
(𝜆1 , Ω1 , 𝑁2 ) ∧ (𝜆2 , Ω2 , 𝑁1 ) or shortly 𝜆ℒ ∧ 𝜆ℳ and is defined as (𝜆1 , Ω1 , 𝑁2 ) ∧ (𝜆2 , Ω2 , 𝑁1 ) = (𝜆𝒦 , ℒ ×
ℳ, min(𝑁1 , 𝑁2 )), where degree of membership , indeterminacy and non-membership are given as
follows:
𝕋𝒦(𝜉𝑖,𝜉𝑗 ) (𝜑) = min{𝕋ℒ(𝜉𝑖) (𝜑), 𝕋ℳ(𝜉𝑗 ) (𝜑)},
𝕀𝒦(𝜉𝑖,𝜉𝑗 ) (𝜑) =
{𝕀ℒ(𝜉 ) (𝜑)+𝕀ℳ(𝜉 ) (𝜑)}
𝑖
𝑗
2
,
𝔽𝒦(𝜉𝑖 ,𝜉𝑗 ) (𝜑) = max{𝔽ℒ(𝜉) (𝜑), 𝔽ℳ(𝜉) (𝜑)},
for all 𝜑 ∈ 𝕏.
Definition 3.36
𝓇𝒦(𝜉𝑖 ,𝜉𝑗 ) (𝜑) = max{𝓇ℒ(𝜉𝑖 ) (𝜑), 𝓇ℳ(𝜉𝑗 ) (𝜑)}, ∀𝜉𝑖 ∈ ℒ, 𝜉𝑗 ∈ ℳ
Let 𝜆ℒ , 𝜆ℳ ∈ NNS (𝕏) be two NNS be expressed as 𝜆ℒ = (𝜆1 , Ω1 , 𝑁) and 𝜆ℳ =
(𝜆2 , Ω2 , 𝑁1 ) where Ω1 = (𝜁1 , ℒ, 𝑁2 ) and Ω2 = (𝜁2 , ℳ, 𝑁1 ) are NSSs. Then OR operation is symbolized
by (𝜆1 , Ω1 , 𝑁2 ) ∨ (𝜆2 , Ω2 , 𝑁1 ) or shortly 𝜆ℒ ∨ 𝜆ℳ and is defined as (𝜆1 , Ω1 , 𝑁2 ) ∨ (𝜆2 , Ω2 , 𝑁1 ) = (𝜆𝒦 , ℒ ×
ℳ, min(𝑁1 , 𝑁2 )), where degree of membership ,indeterminacy and non-membership are given as
follows:
𝕋ℋ(𝜉𝑖 ,𝜉𝑗 ) (𝜑) = max{𝕋ℒ(𝜉𝑖) (𝜑), 𝕋ℳ(𝜉𝑗 ) (𝜑)},
𝕀ℋ(𝜉𝑖 ,𝜉𝑗 ) (𝜑) =
{𝕀ℒ(𝜉 ) (𝜑)+𝕀ℳ(𝜉 ) (𝜑)}
𝑖
𝑗
2
,
𝔽ℋ(𝜉𝑖 ,𝜉𝑗 ) (𝜑) = min{𝔽ℒ(𝜉) (𝜑), 𝔽ℳ(𝜉) (𝜑)},
for all 𝜑 ∈ 𝕏.
𝓇ℋ(𝜉𝑖,𝜉𝑗 ) (𝜑) = min{𝓇ℒ(𝜉𝑖 ) (𝜑), 𝓇ℳ(𝜉𝑗 ) (𝜑)}, ∀𝜉𝑖 ∈ ℒ, 𝜉𝑗 ∈ ℳ
̂ 𝜆ℒ and is defined by
Definition 3.37 The Truth-favorite of an NNSS 𝜆ℒ is denoted by 𝜆ℳ =△
𝕋ℒ(𝜉) (𝜑) = min{𝕋ℒ(𝜉) (𝜑) + 𝕀ℒ(𝜉) (𝜑),1}
𝕀ℒ(𝜉) (𝜑) = 0
𝔽ℒ(𝜉) (𝜑) = 𝔽ℳ(𝜉) (𝜑)
𝓇ℒ(𝜉) (𝜑) = 𝓇ℳ(𝜉) (𝜑)
for all 𝜑 ∈ 𝕏, 𝜉 ∈ ℒ.
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̂ 𝜆ℒ and is defined by
Definition 3.38 The Falsity-favorite of an NNSS 𝜆ℒ is denoted by 𝜆ℳ =▽
𝕋ℒ(𝜉) (𝜑) = 𝕋ℳ(𝜉) (𝜑)
𝕀ℒ(𝜉) (𝜑) = 0
𝔽ℒ(𝜉) (𝜑) = min{𝔽ℒ(𝜉) (𝜑) + 𝕀ℒ(𝜉) (𝜑),1}
𝓇ℒ(𝜉) (𝜑) = 𝓇ℳ(𝜉) (𝜑)
for all 𝜑 ∈ 𝕏, 𝜉 ∈ ℒ.
Proposition 3.39 Let 𝜆ℒ be an NNSS, then
̂△
̂ 𝜆ℒ =△
̂ 𝜆ℒ .
1. △
̂▽
̂ 𝜆ℒ =▽
̂ 𝜆ℒ .
2. ▽
Proof. Follows immediately from definitions.
Definition 3.40 Let 𝜆ℒ ∈NNS(𝕏). Then scalar multiplication of 𝜆ℒ with 𝛼 is symbolized by 𝜆ℒ ⊗ 𝛼
and is defined as
𝜆ℒ ⊗ 𝛼 = {(𝜉, {(〈𝜑, 𝕋ℒ(𝜉) (𝜑) ⊗ 𝛼, 𝕀ℒ(𝜉) (𝜑) ⊗ 𝛼, 𝔽ℒ(𝜉) (𝜑) ⊗ 𝛼〉, 𝓇ℒ(𝜉) (𝜑) ⊗ 𝛼): 𝜑 ∈ 𝕏}):
𝜉 ∈ 𝐸}
where 𝕋ℒ(𝜉)⊗𝛼 (𝜑), 𝕀ℒ(𝜉)⊗𝛼 (𝜑)𝔽ℒ(𝜉)⊗𝛼 (𝜑) and 𝓇ℒ(𝜉) (𝜑) ⊗ 𝛼 are defined by
𝕋ℒ(𝜉) (𝜑) ⊗ 𝛼 = min{𝕋ℒ(𝜉) (𝜑) × 𝛼, 1}
𝕀ℒ(𝜉) (𝜑) ⊗ 𝛼 = min{𝕀ℒ(𝜉) (𝜑) × 𝛼, 1}
𝔽ℒ(𝜉) (𝜑) ⊗ 𝛼 = min{𝔽ℒ(𝜉) (𝜑) × 𝛼, 1}
𝓇ℒ(𝜉) (𝜑) ⊗ 𝛼 = (
𝓇ℒ(𝜉) (𝜑) × 𝛼,
𝑁 − 1,
𝑖𝑓0 ≤ 𝓇ℒ(𝜉) (𝜑) × 𝛼 < 𝑁 − 1,
𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
Definition 3.41 Let 𝜆ℒ ∈NNS(𝕏). Then scalar division of 𝜆ℒ by 𝛼 is symbolized by 𝜆ℒ /̃𝛼 and is
defined as
𝜆ℒ /̃𝛼 = {(𝜉, {(〈𝜑, 𝕋ℒ(𝜉) (𝜑)/̃𝛼, 𝕀ℒ(𝜉) (𝜑)/̃𝛼, 𝔽ℒ(𝜉) (𝜑)/̃𝛼〉, 𝓇ℒ(𝜉) (𝜑)/̃𝛼): 𝜑 ∈ 𝕏}): 𝜉 ∈ 𝐸}
where 𝕋ℒ(𝜉)/̃𝛼 (𝜑), 𝕀ℒ(𝜉)/̃𝛼 (𝜑)𝔽ℒ(𝜉)/̃𝛼 (𝜑) and 𝓇ℒ(𝜉) (𝜑)/̃𝛼 are defined by
𝕋ℒ(𝜉) (𝜑)/̃𝛼 = min{𝕋ℒ(𝜉) (𝜑)/𝛼, 1}
𝕀ℒ(𝜉) (𝜑)/̃𝛼 = min{𝕀ℒ(𝜉) (𝜑)/𝛼, 1}
𝔽ℒ(𝜉) (𝜑)/̃𝛼 = min{𝔽ℒ(𝜉) (𝜑)/𝛼, 1}
𝓇ℒ(𝜉) (𝜑)/̃𝛼 = (
𝓇ℒ(𝜉) (𝜑)/𝛼, 𝑖𝑓0 ≤ 𝓇ℒ(𝜉) (𝜑)/𝛼 < 𝑁 − 1,
𝑁 − 1,
𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
𝓇 (𝜑)/𝛼, 𝑖𝑓0 ≤ 𝓇ℒ(𝜉) (𝜑)/𝛼 < 𝑁 − 1,
𝓇ℒ(𝜉) (𝜑)/̃𝛼 = { ℒ(𝜉)
𝑁 − 1,
𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
4 Relations On Neutrosophic N-Soft Sets
Let 𝜆ℒ and 𝜆ℳ be two NNSSs defined over the universe (𝕏, ℒ) and (𝕏, ℳ)
̆ is defined as ℜ
̆ (𝜉𝑖 , 𝜉𝑗 ) = 𝜆ℒ (𝜉𝑖 ) ⊓ℛ 𝜆ℳ (𝜉𝑗 ), ∀𝜉𝑖 ∈ ℒ and
respectively. Neutrosophic N-soft relation ℜ
Definition 4.1
∀𝜉𝑗 ∈ ℳ, where
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is an NNSS over (𝕏, 𝒩), where 𝒩 ⊑ ℒ × ℳ.
̆ : 𝒩 → P(𝕏)
ℜ
̆ 2 is defined by
̆ 1 and ℜ
Definition 4.2 The composition ⋄ of two neutrosophic N-soft relations ℜ
̆ 2 (𝑚, 𝑛)
̆ 1 (𝑙, 𝑚) ⊓ ℜ
̆1 ⋄ ℜ
̆ 2 )(𝑙, 𝑛) = ℜ
(ℜ
̆ 1 is neutrosophic N-soft relations from 𝜆ℒ to 𝜆ℳ over the universe (𝕏, ℒ) and (𝕏, ℳ)
where ℜ
̆ 2 is neutrosophic N-soft relations from 𝜆ℳ to 𝜆𝒩 over the universe (𝕏, ℳ) and
respectively and ℜ
(𝕏, 𝒩) respectively.
̆ 2 is
̆ 1 is neutrosophic N-soft relation over the universe (𝕏, ℒ) and ℜ
Let ℜ
Definition 4.3
̆ 1 and ℜ
̆2
neutrosophic N-soft relation over the universe (𝕏, ℳ). The union and intersection of ℜ
defined as below
̆ 2 )(𝑙, 𝑚) = max{ℜ
̆1 ⊔ ℜ
̆ 1 (𝑙, 𝑚), ℜ
̆ 2 (𝑙, 𝑚)}
(ℜ
̆ 2 )(𝑙, 𝑚) = min{ℜ
̆1 ⊓ ℜ
̆ 1 (𝑙, 𝑚), ℜ
̆ 2 (𝑙, 𝑚)}
(ℜ
̆ 2 : ℒ × ℳ → ℙ(𝕏).
̆ 1 : ℒ × ℳ → ℙ(𝕏) and ℜ
where ℜ
̆ for 𝜆ℒ to 𝜆ℳ . Then max-minDefinition 4.4 Let 𝜆ℒ in (𝕏, ℒ) be a neutrosophic N-soft set. Let ℜ
max composition of neutrosophic N-soft set with 𝜆ℒ is another neutrosophic N-soft set 𝜆ℳ of (𝕏, ℳ)
̆ ⋄ 𝜆ℒ . The membership function, indeterminate function, non-membership
which is denoted by ℜ
function and grading function of 𝜆ℳ are defined, respectively, as
𝕋ℜ̆⋄𝜆ℒ (𝑚) = max{min(𝕋ℒ (𝑙), 𝕋ℒ (𝑙, 𝑚))},
𝑙
𝕀ℜ̆⋄𝜆ℒ (𝑚) = min{max(𝕀ℒ (𝑙), 𝕀ℒ (𝑙, 𝑚))},
𝑙
𝔽ℜ̆⋄𝜆ℒ (𝑚) = min{max(𝔽ℒ (𝑙), 𝔽ℒ (𝑙, 𝑚))},
𝑙
𝓇ℜ̆⋄𝜆ℒ (𝑚) = max{min(𝓇ℒ (𝑙), 𝓇ℒ (𝑙, 𝑚))},
𝑙
Definition 4.5
∀𝑙 ∈ ℒ, 𝑚 ∈ ℳ, 𝓇ℒ ∈ 𝒢.
Let 𝜆ℒ be a neutrosophic N-soft set. Then the choice function of 𝜆ℒ is defined as
𝐶(𝜆ℒ ) = 𝓇ℒ + 𝕋ℒ − 𝕀ℒ − 𝔽ℒ
Definition 4.6 Let 𝜆ℒ and 𝜆ℳ be two neutrosophic N-soft sets. Then the score function of 𝜆ℒ and
𝜆ℳ is defined as
𝒮𝐿𝑀 = 𝐶(𝜆ℒ ) − 𝐶(𝜆ℳ )
Definition 4.7 Let 𝜆ℒ be a neutrosophic N-soft set. We define score function for 𝜆ℒ as
𝒮𝐿 = 𝓇𝑖 + 𝕋𝑖 − 𝕀𝑖 𝔽𝑖
5 Application of Neutrosophic N-Soft Set to Medical Diagnosis
In this Section, we discuss the execution of N-soft set and neutrosophic set in medical diagnosis . In
some previous studies of the neutrosophic set and neutrosophic soft set, there are many examples of
medical diagnosis but all of them have lack of parameterized evaluation characterization. First we
propose Algorithm 1 as given below.
Algorithm 1
Step 1: Input a set 𝔓 of patients, a set 𝒮 of symptoms as parameter set and a set 𝔇 of diseases .
Step 2: Construct a relation 𝔏(𝔓 ↪ 𝒮) between the patients and symptoms.
Step 3: Construct a relation a relation 𝔐(𝒮 ↪ 𝔇) between the symptoms and the diseases.
Step 4: Compute the composition relation 𝔑(𝔓 ↪ 𝔇) the relation of patients and diseases by using
Definition 4.4.
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Step 5: Obtain the choice function of 𝔑 by using Definition 4.5.
Step 6: Choose the highest choice value of patient corresponding to disease gives the higher
possibility of the patient affected with the respective disease.
Flow chart portrayal of Algorithm 1 is given in Figure 1:
Figure 1: Flow chart representation of Algorithm 1
Now we demonstrate how neutrosophic N-soft set (NNSS) can be efficiently employed in multicriteria group decision making (MCGDM). First of all, we propose an extension of TOPSIS to NNSS.
In this study, we choose TOPSIS because our goal is to solve a medical diagnosis decision making
problem. Since medical diagnosis involves similarities (in symptoms) and TOPSIS method is most
appropriate method for handling such problems. A detailed study of TOPSIS may be found in [26].
The procedural steps of Neutrosophic N-soft set TOPSIS Method to examine critical situation of each
patient is given in Algorithm 2.
Algorithm 2 (Neutrosophic N-soft set TOPSIS Method)
Step 1: Constructing weighed parameter matrix ℋ by using ranking values obtained in Step 4 of
Algorithm 1 composition relation 𝔑(𝔓 ↪ 𝔇) and relates it with linguistic ratings from Table 26.
𝓇11
𝓇21
⋮
ℋ = 𝓇𝑖1
⋮
𝓇𝑚1
[
𝓇12
𝓇12
⋮
𝓇𝑖2
⋮
𝓇𝑚2
⋯
⋯
⋯
⋯
𝓇1𝑛
𝓇2𝑛
⋮
𝓇𝑖𝑛 = [𝓇𝑖𝑗 ]𝑚×𝑛
⋮
𝓇𝑚𝑛
]
Step 2: Creating normalized decision matrix ℬ. Throughout from now, we shall use
𝐿𝑛 = {1,2,3, ⋯ , 𝑛} ∀𝑛 ∈ 𝑁
𝑏𝑖𝑗 =
𝓇𝑖𝑗
2
√ ∑𝑚
𝑘=1 𝓇𝑘𝑗
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𝑏11
𝑏21
⋮
ℬ = 𝑏𝑖1
⋮
𝑏𝑚1
[
𝑏12
𝑏12
⋮
𝑏𝑖2
⋮
𝑏𝑚2
⋯
⋯
⋯
⋯
𝑏1𝑛
𝑏2𝑛
⋮
𝑏𝑖𝑛 = [𝑏𝑖𝑗 ]𝑚×𝑛
⋮
𝑏𝑚𝑛
]
Step 3: Creating weighted vector 𝐖 = {𝐖1 , 𝐖2 , 𝐖3 , ⋯ , 𝐖𝑛 } by using the expression
𝐰𝑗
𝐖𝑗 = ∑𝑚
𝑘=1 𝐰𝑘
Step 4: Constructing weighted decision matrix
𝜇11 𝜇12
𝜇21 𝜇12
⋮
⋮
𝜇 = 𝜇𝑖1 𝜇𝑖2
⋮
⋮
𝜇𝑚1 𝜇𝑚2
[
𝜇.
⋯
⋯
⋯
⋯
where 𝜇𝑖𝑗 = 𝐖𝑗 𝑏𝑖𝑗
, 𝐰𝑘 =
1
𝑚
∑𝑚
𝑖=1 𝑏𝑖𝑗
(2)
𝜇1𝑛
𝜇2𝑛
⋮
𝜇𝑖𝑛 = [𝜇𝑖𝑗 ]𝑚×𝑛
⋮
𝜇𝑚𝑛
]
(3)
Step 5: Finding positive ideal solution (PIS) and negative ideal solution (NIS) by using the Equations
𝑃𝐼𝑆 = {𝜇1+ , 𝜇2+ , 𝜇3+ , ⋯ , 𝜇𝑗+ ⋯ , 𝜇𝑛+ } = {max(𝜇𝑖𝑗 ): 𝑖 ∈ 𝐿𝑛 }
(4)
𝑁𝐼𝑆 = {𝜇1− , 𝜇2− , 𝜇3− , ⋯ , 𝜇𝑗− ⋯ , 𝜇𝑛− } = {min(𝜇𝑖𝑗 ): 𝑖 ∈ 𝐿𝑛 }
(5)
𝒮𝑖+ = √∑𝑛𝑗=1 (𝜇𝑖𝑗 − 𝜇𝑗+ )2 , ∀𝑖 ∈ 𝐿𝑚
(6)
𝒮𝑖− = √∑𝑛𝑗=1 (𝜇𝑖𝑗 − 𝜇𝑗− )2 , ∀𝑖 ∈ 𝐿𝑚
(7)
Step 6: Calculate separation measurements of PIS
equations
and
(𝒮𝑖+ )
and NIS
(𝒮𝑖− )
for each parameter by using the
Step 7: Calculating of relative closeness of alternative to the ideal solution by using the equation
𝒞𝑖+ =
Step 8: Ranking the preference order.
𝒮𝑖−
−
𝒮𝑖 +𝒮𝑖+
, 0 ≤ 𝒞𝑖+ ≤ 1, ∀𝑖 ∈ 𝐿𝑚
Flow chart portrayal of neutrosophic N-soft set TOPSIS method is shown in Figure 2.
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Figure 2: Flow chart of neutrosophic N-soft set TOPSIS method
5.1 Numerical Example
Now we employ the above Algorithm 1 to find the decision factor about the following top four
deadliest diseases in the world. Due to the following risk factors, these diseases progress slowly. Here
is some detail about these diseases:
𝐃𝟏 : Coronary artery disease (CAD)
CAD occurs when the vessels that transfer blood towards heart become narrowed. CAD leads to
heart failure, arrhythmias and chest pain. Risk factors for CAD are
High blood
High cholesterol
Smoking
Diabetes
Obesity
pressure
Family history
of CAD
Table 16: Risk factors for CAD
𝐃𝟐 : Stroke
This fatal disease occurs when some artery is in brain blocked or leaks. The risk factors for Stroke are:
High blood
Being female
Smoking
pressure
Family history Being American Being African
of stroke
Table 17: Risk factors for Stroke
𝐃𝟑 : Lower respiratory infections (LRI)
This disease occurs due to tuberculosis, pneumonia, influenza, flu, or bronchitis. Risk factors for LRI
contain
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Poor air quality
Asthma
Smoking
Weak immune
HIV
Crowded child-care
system
settings
Table 18: Risk factors for LRI
𝐃𝟒 : Chronic obstructive pulmonary disease (COPD)
This disease is a long-term, progressive lung disease that makes breathing difficult. Risk factors for
COPD are
Family history
Lungs irritation
History of respiratory infections
Smoking
Table 19: Risk factors for COPD
𝐃𝟓 : Trachea, bronchus and lungs cancers
Respiratory cancers incorporate diseases of the bronchus, larynx, lungs and trachea. The risk factors
for Trachea, bronchus and lungs cancers involve
Use of coal for
Tobacco
cooking
usage
Family history of
Smoking
Poor air quality
Diesel fumes
disease
Table 20: Risk factors for Trachea, bronchus and lungs cancers
Core in certain sense is the most basic part occurring in the considered knowledge. Core can be
translated as the arrangement of most trademark some portion of knowledge, which cannot be
abstained from when decreasing the data. The core risk factor of all diseases discussed above is
"smoking". For computational purpose, let's decide the grading values depending upon the degree
of membership function as in Table 21:
Degree of membership
Grading values
function
𝕋=0
0
0.2 < 𝕋 ≤ 0.4
2
0.6 < 𝕋 ≤ 0.8
4
0 < 𝕋 ≤ 0.2
1
0.4 < 𝕋 ≤ 0.6
3
0.8 < 𝕋 ≤ 1.0
5
Table 21: Ranking scale
Table 22 yields relation between symptoms and patients:
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Headache(𝔰1 )
𝔏
𝔭1
(〈0.7,0.2,0.5〉,4)
𝔭3
(〈0.6,0.6,0.4〉,3)
𝔭2
Shortness of breath(𝔰2 )
(〈0.9,0.3,0.1〉,5)
𝔭4
(〈0.6,0.3,0.4〉,3)
(〈0.4,0.6,0.5〉,2)
(〈0.5,0.5,0.8〉,3)
(〈0.2,0.4,0.8〉,1)
(〈0.7,0.4,0.3〉,4)
(〈0.2,0.5,0.8〉,1)
Angina(𝔰3 )
(〈0.8,0.5,0.2〉,4)
(〈0.3,0.1,0.7〉,2)
Table 22: Relation between symptoms and patients
(〈0.7,0.1,0.3〉,4)
The relation between the symptoms and the diseases is given in Table 23:
𝔐
Headache(𝔰1 )
Shortness of
breath(𝔰2 )
Angina(𝔰3 )
𝔇1
𝔇2
𝔇3
𝔇4
(〈0.8,0.4,0.2〉,4)
(〈0.9,0.2,0.1〉,5)
(〈0.6,0.3,0.4〉,3)
(〈0.7,0.5,0.3〉,4)
(〈0.5,0.7,0.5〉,3)
(〈0.4,0.6,0.6〉,2)
(〈0.3,0.5,0.7〉,2)
(〈0.9,0.1,0.1〉,5)
(〈0.1,0.8,0.9〉,1)
(〈0.2,0.9,0.8〉,1)
(〈0.5,0.7,0.5〉,3)
Table 23: Relation between the symptoms and the diseases
(〈0.3,0.7,0.6〉,2)
The composition relation of patients and diseases in Table 24:
𝔑
𝔇1
𝔇2
𝔇3
𝔇4
𝔭1
(〈0.7,0.4,0.5〉,4)
(〈0.7,0.2,0.5〉,4)
(〈0.6,0.3,0.5〉,3)
(〈0.7,0.5,0.5〉,4)
𝔭3
(〈0.6,0.6,0.4〉,3)
(〈0.6,0.6,0.4〉,3)
(〈0.6,0.6,0.4〉,3)
(〈0.60.4,0.4〉,3)
𝔭2
𝔭4
(〈0.8,0.4,0.2〉,4)
(〈0.5,0.5,0.5〉,3)
(〈0.9,0.3,0.1〉,5)
(〈0.4,0.5,0.6〉,2)
(〈0.6,0.3,0.4〉,3)
(〈0.7,0.5,0.2〉,4)
(〈0.3,0.5,0.7〉,2)
(〈0.7,0.5,0.3〉,4)
𝔇4
Table 24: Composition relation of patients and diseases
Table 25 gives choice values of the relation 𝔑:
𝔑
𝔇1
𝔇2
𝔇3
𝔭1
3.8
4
2.8
3.7
𝔭2
4.2
5.5
2.9
4
𝔭3
2.6
2.6
2.6
2.8
𝔭4
2.5
1.3
1.1
3.9
Table 25: Choice values of relation 𝔑
From Table 25, we conclude that the patients 𝔭1 and 𝔭2 are likely to be suffering from 𝔇2 whereas
𝔭3 and 𝔭4 are suffering from 𝔇4 .
In order to examine the intensity level of the disease of the patients, we use neutrosophic N-soft
TOPSIS method which is demonstrated in Algorithm 2. First, we decide the grading values as a
function of linguistic terms as Table 26:
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Linguistic Terms
Grading Values
Undetermined (U)
0
Very Stable (VS)
1
Stable (S)
2
Grave (G)
3
Critical (C)
4
Very Critical (VC)
5
Table 26: Linguistic terms for evaluation of parameters
Now we construct weighted parameter matrix by using Step 9 and Table 26 as
4
4
ℋ= 3
3
[
4
5
3
2
3
3
3
2
𝐶
𝐶
= 𝐺
𝐺
] [
𝐶
𝑉𝐶
𝐺
𝑆
Creating normalized decision matrix ℬ by using Equation 1
0.57
0.57
ℬ = 0.43
0.43
[
0.54
0.68
0.41
0.27
Now by using Equation 2 construct weight vector
𝐺
𝐺
𝐺
𝑆
0.54 0.53
0.54 0.53
0.54 0.40
0.36 0.53
]
]
𝐖 = {𝐖1 , 𝐖2 , 𝐖3 , 𝐖4 } = {0.58,0.14,0.14,0.14}
By using Equation 3 the weighted decision matrix 𝜇 is
0.33
0.33
𝜇 = 0.25
0.25
[
0.07
0.09
0.06
0.04
0.07
0.07
0.07
0.05
0.07
0.07
0.06
0.07
]
The positive ideal solution (PIS) and negative ideal solution (NIS) by using the Equations 4 and 5 as
𝑃𝐼𝑆 = {0.33,0.09,0.07,0.07}
𝑁𝐼𝑆 = {0.25,0.04,0.05,0.06}
The separation measurements of PIS and NIS for each parameter by using the Equations 6 and 7 are
𝒮1+ = 0.11
𝒮2+ = 0.06
𝒮3+ = 0.02
𝒮4+ = 0.01
𝒮1− = 0.11
𝒮2− = 0.06
𝒮3− = 0.03
𝒮4− = 0.02
The relative closeness of alternatives to the ideal solution by using Equation 8 are
𝒞1+ = 0.5
𝒞2+ = 0.5
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𝒞3+ = 0.6
Ranking the preference order is
𝒞4+ = 0.7
𝒞4+ ≥ 𝒞3+ ≥ 𝒞2+ ≥ 𝒞1+
which indicates that condition of patient 𝔭4 is most critical. The pictorial representation of the
rankings of the patients is demonstrated with the assistance of a chart as given in Figure 3.
Figure 3: Ranking of patients w.r.t. intensity level of disease
5. Conclusion
The purpose of this work is to lay the foundation of theory of neutrosophic N-soft set as a hybrid
model of neutrosophic sets and N-soft sets. We established some basic operations on neutrosophic
N-soft sets along with their fundamental properties. We introduced the notions of NNS-subset, nullNNS, absolute-NNS, complements of NNS, truth-favorite, falsity-favorite, relations on NNS,
composition of NNSS and score function of NNS. We explained these concepts with the help of
illustrations. We presented a novel application of multi-attribute decision-making (MADM) based on
neutrosophic N-soft set by using Algorithm 1. We proposed neutrosophic N-soft sets TOPSIS method
as demonstrated in Algorithm 2 for MADM in medical diagnosis. We defined separation
measurements of positive ideal solution and negative ideal solution to compute a relative closeness
to identify the optimal alternative. Lastly, a numerical example is given to illustrate the developed
method for medical diagnosis.
This may be the starting point for neutrosophic N-soft set mathematical concepts and information
structures that are based on neutrosophic set and N-soft set theoretic operations. We have studied a
few concepts only, it will be necessary to carry out more theoretical research to recognize a general
framework for the practical applications. The proposed model of neutrosophic N-soft set can be
elaborated with new research topics such as image processing, expert systems, soft computing
techniques, fusion rules, cognitive maps, graph theory and decision-making of real world problems.
M. Riaz, K. Naeem, I. Zareef and D. Afzal, Neutrosophic N-Soft Sets with TOPSIS method
Neutrosophic Sets and Systems, Vol. 32, 2020
167
We hope that this study will prove a ground-breaking and will open new doors for the vibrant
researchers in this field.
Acknowledgement
The authors are highly thankful to the Editor-in-chief and the referees for their valuable comments
and suggestions for improving the quality of our paper.
Conflicts of Interest
The authors declare that they have no conflict of interest.
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Received: Oct 21, 2019. Accepted: Mar 20, 2020
M. Riaz, K. Naeem, I. Zareef and D. Afzal, Neutrosophic N-Soft Sets with TOPSIS method
Neutrosophic Sets and Systems, Vol. 32, 2020
University of New Mexico
A Novel of neutrosophic τ -Structure
Ring ExtB and ExtV Spaces
R. Narmada Devi
Department of Mathematics, Vel Tech Rangarajan Dr. Sagunthala R & D Institute of Science and Technology, Chennai, India.
E-mail: narmadadevi23@gmail.com
Abstract: In this paper, the concepts of a neutrosophic τ -structure ring spaces, neutrosophic τ -structure ring Gδ T1/2
spaces and neutrosophic τ -structure ring exterior B spaces and neutrosophic τ -structure ring exterior V spaces are
introduced. Some interesting functions that preserve neutrosophic τ -structure ring exterior B spaces and neutrosophic
τ -structure ring exterior V spaces in the context of image and preimage are derived with the necessary examples.
Keywords: neutrosophic τ -structure ring space, neutrosophic τ -structure ring Gδ T1/2 space, neutrosophic τ -structure
ring ExtB space and neutrosophic τ -structure ring ExtV space.
1
Introduction
The concept of fuzzy sets was introduced by Zadeh [16]. consequent to the introduction of fuzzy sets, fuzzy
logic has been applied in many real life situations to handle uncertainty. Chang [7] introduced the concept
of fuzzy topological spaces. There are several kinds of fuzzy set extensions such as intuitionistic fuzzy set,
interval-valued fuzzy sets, etc. After the introduction of intuitionistic fuzzy sets and its topological spaces
by Atanassov [6] and Coker [8], the concept of imprecise data called neutrosophic sets was introduced by
Smarandache [9]. The concept of neutrosophic topological space was introduced by Salama [15]. Later
R.Narmada Devi [10,11,12,13,14] introduced the concepts of intuitionistic fuzzy Gδ sets, intuitionistic fuzzy
exterior spaces and neutrosophic complex topological spaces. Moreover, the neutrosophic theory plays a viral role in all fields of branches like medial diagnosis [1,2,5], multiple criteria group decision making [3,4],
etc. In this paper, the concepts of neutrosophic τ -structure ring spaces, neutrosophic Gδ rings, neutrosophic
first category rings, neutrosophic τ -structure ring Gδ T1/2 spaces and neutrosophic τ -structure ring exterior B
spaces and neutrosophic τ -structure ring exterior V spaces are introduced. Further, neutrosophic τ -structure
ring continuous (resp. open, hardly open) functions and somewhat neutrosophic τ -structure ring continuous
functions are presented. Some interesting properties among of functions along with the spaces are discussed
and necessary examples are provided.
2
Preliminiaries
We need the following basic definitions for our study.
R. Narmada Devi and A New Novel of neutrosophic τ -Structure Ring ExtB and ExtV Spaces
Neutrosophic Sets and Systems, Vol. 32 2020
172
Definition 2.1. [9] Let X be a nonempty set. A neutrosophic set A in X is defined as an object of the form A =
{hx, TA (x), IA (x), FA (x)i : x ∈ X} such that TA , IA , FA : X → [0, 1]. and 0 ≤ TA (x) + IA (x) + FA (x) ≤ 3.
Definition 2.2. [9] Let A = hx, TA (x), IA (x), FA (x)i and B = hx, TB (x), IB (x), FB (x)i be any two neutrosophic sets in X. Then
(i) A ∪ B = hx, TA∪B (x), IA∪B (x), FA∪B (x)i where TA∪B (x) = TA (x) ∨ TB (x), IA∪B (x) = IA (x) ∨ IB (x)
and FA∪B (x) = FA (x) ∧ FB (x).
(ii) A ∩ B = hx, TA∩B (x), IA∩B (x), FA∩B (x)i where TA∩B (x) = TA (x) ∧ TB (x), IA∩B (x) = IA (x) ∧ IB (x)
and FA∩B (x) = FA (x) ∨ FB (x).
(iii) A ⊆ B if TA (x) ≤ TB (x), IA (x) ≤ IB (x) and FA (x) ≥ FB (x), for all x ∈ X.
(iv) the complement of A is defined as C(A) = hx, TC(A) (x), IC(A) (x), FC(A) (x)i where TC(A) (x) = 1 −
TA (x), IC(A) (x) = 1 − IA (x) andFC(A) (x) = 1 − FA (x).
(v) 0N = {hx, 0, 0, 1i : x ∈ X} and 1N = {hx, 1, 1, 0i : x ∈ X}
Definition 2.3. [10,11] Let (X, T ) be an intuitionistic fuzzy topological space. Let A = hx, µA , γA i be an
intuitionistic fuzzy setTon an intuitionistic fuzzy topological space (X, T ). Then A is said be an intuitionistic
fuzzy Gδ set if A = ∞
i=1 Ai , where Ai = hx, µAi , γAi i is an intuitionistic fuzzy open set in an intuitionistic
fuzzy topological space (X, T ). The complement of an intuitionistic fuzzy Gδ set is said to be an intuitionistic
fuzzy Fσ set.
Definition 2.4. [12,13] Let A = hµA , γA i be an intuitionistic fuzzy set on an intuitionistic fuzzy topological
space (X, τ ). An intuitionistic fuzzy exterior of A is defines as follows: if IF Ext(A) = IF int(A)
Definition 2.5. [12,13] Let R be a ring. An intuitionistic fuzzy set A = hx, µA , γA i in R is called an intuitionistic fuzzy ring on R if it satisfies the following conditions on the membership and nonmembership
values:
(i) µA (x + y) ≥ µA (x) ∧ µA (y),
(ii) µA (xy) ≥ µA (x) ∧ µA (y),
(iii) γA (x + y) ≤ γA (x) ∨ γA (y),
(iv) γA (xy) ≤ µA (x) ∨ γA (y),
for all x, y ∈ R.
3
Properties of neutrosophic τ -Structure Ring Exterior B Spaces
Definition 3.1. Let R be a ring. A neutrosophic set A = hx, TA (x), IA (x), FA (x)i in R is called a neutrosophic
ring on R if it satisfies the following conditions:
(i) TA (x + y) ≥ TA (x) ∧ TA (y) and TA (xy) ≥ TA (x) ∧ TA (y)
(ii) IA (x + y) ≥ IA (x) ∧ IA (y) and IA (xy) ≥ IA (x) ∧ IA (y)
R. Narmada Devi and A New Novel of neutrosophic τ -Structure Ring ExtB and ExtV Spaces
Neutrosophic Sets and Systems, Vol. 32 2020
173
(iii) FA (x + y) ≤ FA (x) ∨ FA (y) and FA (xy) ≤ FA (x) ∨ FA (y), for all x, y ∈ R.
Definition 3.2. Let R be a ring. A family S of a neutrosophic rings in R is said to be neutrosophic τ -structure
ring on R if it satisfies the following conditions:
(i) 0N , 1N ∈ S .
(ii) G1 ∩ G2 ∈ S for any G1 , G2 ∈ S .
(iii) ∪Gi ∈ S for arbitrary family {Gi | i ∈ I} ⊆ S .
The ordered pair (R, S ) is called a neutrosophic τ -structure ring space. Every member of S is called a
neutrosophic τ -open ring in (R, S ). The complement C(A) of a neutrosophic τ -open ring A is a neutrosophic
τ -closed ring in (R, S ).
Example 3.1. Let R = {0, 1} be a set of integers module 2 with two binary operations ’+’ and ’.’ are specified
by the following tables:
+ 0 1
· 0 1
0 0 1 and 0 0 0
1 1 0
1 0 1
Then (R, +, ·) is a ring. Define neutrosophic rings B and D on R as follows: TB (0) = 0.5, TB (1) =
0.7, IB (0) = 0.5, IB (1) = 0.7, FB (0) = 0.3, FB (1) = 0.2, TD (0) = 0.3, TD (1) = 0.4, ID (0) = 0.3, ID (1) =
0.4, FD (0) = 0.5, FD (1) = 0.6. Then S = {0N , B, D, 1N } is a neutrosophic τ -structure ring on R. Thus the
pair (R, S ) is a neutrosophic τ - structure ring space.
Notation 3.1. Let (R, S ) be any neutrosophic τ -structure ring space. Then N O(R) ( resp. N C(R) ) denotes
the family of all neutrosophic τ -open( resp. closed ) rings of (R, S ).
Definition 3.3. Let (R, S ) be any neutrosophic τ -structure ring space. Let A be a neutrosophic ring in R.
Then the neutrosophic ring interior and neutrosophic ring closure A are defined and denoted as N FR int(A) =
∪{B | B ∈ N O(R) and B ⊆ A} and N FR cl(A) = ∩{B | B ∈ N C(R) and A ⊆ B respectively.
Remark 3.1. Let (R, S ) be any neutrosophic τ -structure ring space. Let A be any neutrosophic ring in R.
Then the following statements hold:
(i) N FR cl(A) = A if and only if A is a neutrosophic τ -closed ring.
(ii) N FR int(A) = A if and only if A is a neutrosophic τ -open ring.
(iii) N FR int(A) ⊆ A ⊆ N FR cl(A).
(iv) N FR int(1N ) = 1N and N FR int(0N ) = 0N .
(v) N FR cl(1N ) = 1N and N FR cl(0N ) = 0N .
(vi) N FR cl(C(A)) = C(N FR int(A)) and N FR int(C(A)) = C(N FR cl(A)).
∞
(vii) ∪∞
i=1 N FR cl(Ai ) ⊆ N FR cl(∪i=1 Ai ).
(viii) ∩ni=1 N FR cl(Ai ) = N FR cl(∪ni=1 Ai ).
R. Narmada Devi and A New Novel of neutrosophic τ -Structure Ring ExtB and ExtV Spaces
174
Neutrosophic Sets and Systems, Vol. 32 2020
∞
(ix) ∩∞
i=1 N FR cl(Ai ) ⊆ N FR cl(∪i=1 Ai ).
∞
(x) ∪∞
i=1 N FR int(Ai ) ⊆ N FR int(∪i=1 Ai ).
Definition 3.4. Let (R, S ) be any neutrosophic τ -structure ring space. Let A be a neutrosophic ring in R.
Then N FR int(C(A)) is called a neutrosophic ring exterior of A and is denoted by N FR Ext(A).
Proposition 3.1. Let (R, S ) be any neutrosophic τ -structure ring space. Let A and B be any two neutrosophic
rings in R. Then the following statements hold:
(i) N FR Ext(A) ⊆ C(A).
(ii) N FR Ext(A) = C(N FR cl(A)).
(iii) N FR Ext(N FR Ext(A)) = N FR int(N FR cl(A)).
(iv) If A ⊆ B then N FR Ext(A) ⊇ N FR Ext(B).
(v) N FR Ext(1N ) = 0N and N FR Ext(0N ) = 1N .
(vi) N FR Ext(A ∪ B) = N FR Ext(A) ∩ N FR Ext(B).
Definition 3.5. Let (R, S ) be a neutrosophic τ -structure ring space.
T∞ Let A be any neutrosophic ring in R.
Then A is said be to a neutrosophic Gδ ring in (R, S ) if A = i=1 Ai , where Ai = hx, TAi , IAi , FAi i is a
neutrosophic τ -open ring in (R, S ). The complement of a neutrosophic Gδ ring is a neutrosophic Fσ ring in
(R, S ).
Definition 3.6. Let (R, S ) be a neutrosophic τ -structure ring space. Let A be any neutrosophic ring in R.
Then A is said be to a
(i) neutrosophic dense ring if there exists no neutrosophic τ -closed ring B in (R, S ) such that A ⊂ B ⊂
1N .
(ii) neutrosophic nowhere dense ring if there exists no neutrosophic τ -open ring B in (R, S ) such that
B ⊂ N FR cl(A). That is, N FR int(N FR cl(A)) = 0N .
Definition 3.7. Let (R, S ) be any neutrosophic τ -structure ring space. Let A be any neutrosophic fuzzy
ring in R. Then A is said be to a neutrosophic first category ring in (R, S ) if A = ∪∞
i=1 Ai where Ai ’s
are neutrosophic nowhere dense rings in (R, S ). The complement of a neutrosophic first category ring is a
neutrosophic residual ring in (R, S ).
Proposition 3.2. Let (R, S ) be any neutrosophic τ -structure ring space. If A is a neutrosophic Gδ ring and
the neutrosophic ring exterior of C(A) is a neutrosophic dense ring in (R, S ), then C(A) is a neutrosophic
first category ring in (R, S ).
Proof:
A being a neutrosophic Gδ ring in (R, S ), A = ∩∞
i=1 Ai where Ai ’s are neutrosophic τ -open rings. Since
the neutrosophic ring exterior of C(A) is a neutrosophic dense ring in (R, S ), N FR cl(N FR Ext(C(A))) =
1N . Because N FR Ext(C(A)) ⊆ A ⊆ N FR cl(A), one has N FR Ext(C(A)) ⊆ N FR cl(A).
This implies that N FR cl(N FR Ext(C(A))) ⊆ N FR cl(A), that is, 1N ⊆ N FR cl(A). Therefore, N FR cl(A) =
∞
∞
1N . That is, N FR cl(A) = N FR cl(∩∞
i=1 Ai ) = 1N . However, IFR cl(∩i=1 Ai ) ⊆ ∩i=1 N FR cl(Ai ). Hence,
R. Narmada Devi and A New Novel of neutrosophic τ -Structure Ring ExtB and ExtV Spaces
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∞
1N ⊆ ∩ ∞
i=1 N FR cl(Ai ). That is, ∩i=1 N FR cl(Ai ) = 1N . This implies that N FR cl(Ai ) = 1N , for each Ai ∈
S . Hence N FR cl(N FR int(Ai )) = 1N . Now, N FR int(N FR cl(C(Ai ))) = N FR int(C(N FR int(Ai ))) =
C(N FR cl(N FR int(Ai ))) = 0N . Therefore, C(Ai ) is a neutrosophic nowhere dense ring in (R, S ). Now,
∞
∞
C(A) = C(∩∞
i=1 Ai ) = ∪i=1 C(Ai ). Hence, C(A) = ∪i=1 C(Ai ) where C(Ai )’s are neutrosophic nowhere
dense rings in (R, S ). Consequently, C(A) is a neutrosophic first category ring in (R, S ).
Proposition 3.3. If A is a neutrosophic first category ring in a neutrosophic τ -structure ring space (R, S )
such that B ⊆ C(A) where B is non-zero neutrosophic Gδ ring and the neutrosophic ring exterior of C(B) is
a neutrosophic dense ring in (R, S ), then A is a neutrosophic nowhere dense ring in (R, S ).
Proof:
Let A be a neutrosophic first category ring in (R, S ). Then A = ∪∞
i=1 Ai where Ai ’s are neutrosophic
nowhere dense rings in (R, S ). Now C(N FR cl(Ai )) is a neutrosophic τ -open ring in (R, S ). Let B =
∞
∩∞
i=1 C(N FR cl(Ai )). Then B is non-zero neutrosophic Gδ ring in (R, S ). Now, B = ∩i=1 C(N FR cl(Ai )) =
∞
C(∪∞
i=1 N FR cl(Ai )) ⊆ C(∪i=1 Ai ) = C(A). Hence B ⊆ C(A). Then A ⊆ C(B). Now,
N FR int(N FR cl((A)) ⊆ N FR int(N FR cl((C(B)))
= N FR int(C(N FR int(B)))
= C(N FR cl(N FR int(B)))
= C(N FR cl(N FR Ext(C(B)))
Since N FR Ext(C(B)) is a neutrosophic dense ring in (R, S ), N FR cl(Ext(C(B)))
= 1N . Therefore, N FR int(N FR cl(A)) ⊆ 0N . Then, N FR int(N FR cl(A)) = 0N . Hence A is a neutrosophic
nowhere dense ring in (R, S ).
Definition 3.8. Let (R, S ) be a neutrosophic τ -structure ring space. Let A be any neutrosophic ring in R. Then
A is said to be a neutrosophic τ -regular closed ring in (R, S ) if N FR cl(N FR int(A)) = A. The complement
of a neutrosophic τ -regular closed ring in (R.S ) is a neutrosophic τ -regular open ring in (R.S ).
Remark 3.2. Every neutrosophic τ -regular closed ring is a neutrosophic τ -closed ring.
Definition 3.9. Let (R, S ) be a neutrosophic τ -structure ring space. Then (R, S ) is called a neutrosophic
τ -structure ring Gδ T1/2 space if every non-zero neutrosophic Gδ ring in (R, S ) is a neutrosophic τ -open ring
in (R, S ).
Proposition 3.4. If the neutrosophic τ -structure ring space (R, S ) is a neutrosophic τ -structure ring Gδ T1/2
space and if A is a neutrosophic first category ring in (R, S ), then A is not a neutrosophic dense ring in
(R, S ).
Proof:
Assume the contrary. Suppose that A is a neutrosophic first category ring in (R, S ) such that A is a
neutrosophic dense ring in (R, S ), that is, N FR cl(A) = 1N . Then, A = ∪∞
i=1 Ai where Ai ’s are neutrosophic
nowhere dense rings in (R, S ). Now, C(N FR cl(Ai )) is a neutrosophic τ -open ring in (R, S ). Let B =
∞
∩∞
i=1 C(N FR cl(Ai )). Then, B is non-zero neutrosophic Gδ ring in (R, S ). Now, B = ∩i=1 C(N FR cl(Ai )) =
∞
C(∪∞
i=1 N FR cl(Ai )) ⊆ C(∪i=1 Ai ) = C(A). Hence B ⊆ C(A). Then, N FR int(B) ⊆ N FR int(C(A)) ⊆
C(N FR cl(A)) = 0N . That is, N FR int(B) = 0N . Since (R, S ) is a neutrosophic τ -structure ring Gδ T1/2
space, B = N FR int(B), which implies that B = 0N . This is a contradiction. Hence A is not a neutrosophic
dense ring in (R, S ).
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Proposition 3.5. If (R, S ) is a neutrosophic τ -structure ring Gδ T1/2 space, then N FR Ext(∪∞
i=1 C(Ai )) =
∩∞
A
.
i=1 i
Proof:
Let (R, S ) be a neutrosophic τ -structure ring Gδ T1/2 space. Assume that Ai ’s are neutrosophic τ -regular
closed rings in (R, S ). Then, the Ai ’s are neutrosophic τ -closed rings in (R, S ), which implies that C(Ai )’s
are neutrosophic τ -open rings in (R, S ). Let B = ∩∞
i=1 Ai . Then B is a non-zero neutrosophic Gδ ring in
(R, S ). Since (R, S ) is a neutrosophic τ -ring Gδ T1/2 space, B = N FR int(B) is a neutrosophic τ -open ring,
∞
∞
∞
which implies that N FR int(∩∞
i=1 Ai ) = ∩i=1 Ai . Now, N FR Ext(∪i=1 C(Ai )) = N FR int(C(∪i=1 C(Ai ))) =
∞
∞
N FR int(∩i=1 Ai ) = ∩i=1 Ai . Hence the proof.
Definition 3.10. Let (R, S ) be a neutrosophic τ -structure ring space. Then (R, S ) is called a neutrosophic
τ -structure ring exterior B ( in short, ExtB ) space if N FR Ext(∩∞
i=1 C(Ai )) = 0N where Ai ’s are neutrosophic
nowhere dense rings in (R, S ).
Example 3.2. Let R = {0, 1} be a set of integers of module 2
following tables:
+ 0 1
· 0
0 0 1 and 0 0
1 1 0
1 0
with two binary operations provided by the
1
0
1
Then (R, +, ·) is a ring. Define neutrosophic rings A, B, M, D, E, F and G on R as follows: TA (0) =
0.5, TA (1) = 0.7, IA (0) = 0.5, IA (1) = 0.7, FA (0) = 0.3, FA (1) = 0.3, TB (0) = 0.5, TB (1) = 0.7, IB (0) =
0.5, IB (1) = 0.7, FB (0) = 0.3, FB (1) = 0.2, TM (0) = 0.3, TM (1) = 0.4, IM (0) = 0.3, IM (1) = 0.4, FM (0) =
0.5, FM (1) = 0.6, TD (0) = 0.4, TD (1) = 0.5, ID (0) = 0.4, ID (1) = 0.5, FD (0) = 0.3, FD (1) = 0.5, TE (0) =
0.3, TE (1) = 0.2, IE (0) = 0.3, IE (1) = 0.2, FE (0) = 0.5, FE (1) = 0.7, TF (0) = 0.3, TF (1) = 0.2, IF (0) =
0.3, IF (1) = 0.2, FF (0) = 0.5, FF (1) = 0.8, TG (0) = 0.3, TG (1) = 0.2, IG (0) = 0.3, IG (1) = 0.2, FG (0) =
0.6, FG (1) = 0.7, TH (0) = 0.3, TH (1) = 0.2, IH (0) = 0.3, IH (1) = 0.2, FH (0) = 0.6, FH (1) = 0.8. Then
S = {0N , A, B, M, D, 1N } is a neutrosophic τ -structure ring on R. Thus the pair (R, S ) is a neutrosophic
τ -structure ring space. Let {E, F, G, H} be neutrosophic nowhere dense rings in (R, S ).
Then N FR Ext(∩{C(E), C(F ), C(G), C(H)}) = N FR Ext(C(E)) = N FR int(E) = 0N . Therefore,
(R, S ) is a neutrosophic τ -structure ring ExtB space.
Proposition 3.6. Let (R, S ) be a neutrosophic τ -structure ring space. Then the following statements are
equivalent:
(i) (R, S ) is a neutrosophic τ -structure ring ExtB space.
(ii) N FR int(A) = 0N , for every neutrosophic first category ring A in (R, S ).
(iii) N FR cl(A) = 1N , for every neutrosophic residual ring A in (R, S ).
Proof:
(i)⇒(ii)
Let A be any neutrosophic first category ring in (R, S ). Then A = ∪∞
i=1 Ai where Ai ’s are neutro∞
sophic nowhere dense rings in (R, S ). Now, N FR int(A) = N FR int(∪i=1 Ai ) = N FR int(C(∩∞
i=1 C(Ai ))) =
∞
N FR Ext(∩∞
Ext(∩
C(A
)).
Since
(R,
S
)
is
a
neutrosophic
τ
-structure
ring
ExtB
space,
N
F
R
i
i=1 C(Ai )) =
i=1
0N . Therefore, N FR int(A) = 0N . Hence (i) ⇒ (ii).
(ii)⇒(iii)
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177
Let A be any neutrosophic residual ring in (R, S ). Then C(A) is a neutrosophic first category ring in
(R, S ). By (ii), N FR int(C(A)) = 0N . That is, N FR int(C(A)) = 0N = C(N FR cl(A)). Therefore,
N FR cl(A) = 1N . Hence (ii) ⇒ (iii).
(iii)⇒(i)
Let A be any neutrosophic first category ring in (R, S ). Then A = ∪∞
i=1 Ai where Ai ’s are neutrosophic
nowhere dense rings in (R, S ). Since A is a neutrosophic first category ring, C(A) is a neutrosophic residual
∞
ring in (R, S ). Then by (iii), N FR cl(C(A)) = 1N . Now, N FR Ext(∩∞
i=1 C(Ai )) = N FR int(C(∩i=1 C(Ai ))) =
∞
N FR int(∪∞
i=1 Ai ) = N FR int(A) = C(N FR cl(C(A))) = 0N . Hence, N FR Ext(∩i=1 C(Ai )) = 0N where Ai ’s
are neutrosophic nowhere dense rings in (R, S ). Therefore, (R, S ) is a neutrosophic τ -structure ring ExtB
space.
Proposition 3.7. If A is a neutrosophic first category ring in a neutrosophic τ -structure ring space (R, S )
such that B ⊆ C(A) where B is non-zero neutrosophic Gδ ring and the neutrosophic ring exterior of C(B) is
a neutrosophic dense ring in (R, S ), then (R, S ) is a neutrosophic τ -structure ring ExtB space.
Proof:
Let A be any neutrosophic first category ring in (R, S ) such that B ⊆ C(A) where B is non-zero neutrosophic Gδ ring and the neutrosophic ring exterior of C(B) is aneutrosophic dense ring in (R, S ). Then
by Proposition 3.3., A is a neutrosophic nowhere dense ring (R, S ), that is, N FR int(N FR cl(A)) = 0N .
Then, N FR int(A) ⊆ N FR int(N FR cl(A)) implies that N FR int(A) = 0N . By Proposition 3.6., (R, S ) is a
neutrosophic τ -structure ring ExtB space.
Proposition 3.8. If (R, S ) is a neutrosophic τ -structure ring ExtB space and if ∪∞
i=1 Ai = 1N where Ai ’s are
∞
neutrosophic τ -regular closed rings in (R, S ), then N FR cl(∪i=1 N FR Ext(C(Ai ))) = 1N .
Proof:
Let (R, S ) be any neutrosophic τ -structure ring ExtB space. Assume that Ai ’s are neutrosophic τ regular closed rings in (R, S ). Suppose that N FR int(Ai ) = 0N , for each i ∈ J. Since Ai is a neutrosophic
τ - regular closed ring in (R, S ), Ai is a neutrosophic τ -closed ring in (R, S ). Also, N FR int(Ai ) = 0N
implies that N FR int(N FR cl(Ai )) = 0N . Therefore, Ai ’s are neutrosophic nowhere dense rings in (R, S ).
∞
∞
∞
Since ∪∞
i=1 Ai = 1N , N FR Ext(∩i=1 C(Ai )) = N FR Ext(C(∪i=1 Ai )) = N FR int(∪i=1 Ai ) = N FR int(1N ) =
1N . Hence, N FR Ext(∩∞
i=1 C(Ai )) = 1N . Since (R, S ) is a neutrosophic τ -structure ring ExtB space,
∞
N FR Ext(∩i=1 C(Ai )) = 0N , which is a contradiction. Hence N FR int(Ai ) 6= 0N , for atleast one i ∈
J. Therefore, ∪∞
i=1 N FR int(Ai ) 6= 0N . Since Ai is a neutrosophic τ -regular closed rings in (R, S ) and
∞
∪i=1 N FR cl(Ai ) ⊆ N FR cl(∪∞
i=1 Ai ),
∞
⇒ ∪∞
i=1 N FR cl(N FR int(Ai )) ⊆ N FR cl(∪i=1 N FR int(Ai ))
∞
⇒ ∪∞
i=1 Ai ⊆ N FR cl(∪i=1 N FR int(Ai ))
∞
⇒ ∪∞
i=1 Ai ⊆ N FR cl(∪i=1 N FR Ext(C(Ai )))
⇒ 1N ⊆ N FR cl(∪∞
i=1 N FR Ext(C(Ai ))).
∞
But 1N ⊇ N FR cl(∪∞
i=1 N FR Ext(C(Ai ))). Hence, N FR cl(∪i=1 N FR Ext(C(Ai ))) = 1N .
4
On neutrosophic τ -Structure Ring Exterior V Spaces
Definition 4.1. Let (R, S ) be any neutrosophic τ -structure ring space. Then (R, S ) is called a neutrosophic
τ -structure ring exterior V ( in short, ExtV )space if N FR cl(∩ni=1 Ai ) = 1N where Ai ’s are neutrosophic Gδ
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rings and the neutrosophic ring exterior of C(Ai )’s are neutrosophic dense rings in (R, S ).
Example 4.1. Let R = {0, 1, 2} be a set of integers of module 3
follows:
+ 0 1 2
· 0 1
0 0 1 2
0 0 0
and
1 1 2 0
1 0 1
2 2 0 1
2 0 2
together with two binary operations as
2
0
2
1
Then (R, +, ·) is a ring. Define neutrosophic rings A, B and D on R as follows: TA (0) = 1, TA (1) =
0.2, TA (2) = 0.9, IA (0) = 1, IA (1) = 0.2, IA (2) = 0.9, FA (0) = 0, FA (1) = 0.8, FA (2) = 0.1, TB (0) =
0.3, TB (1) = 1, TB (2) = 0.2, IB (0) = 0.3, IB (1) = 1, IB (2) = 0.2, FB (0) = 0.7, FB (1) = 0, FB (2) =
0.8, TD (0) = 0.7, TD (1) = 0.4, TD (2) = 1, ID (0) = 0.7, ID (1) = 0.4, ID (2) = 1, FD (0) = 0.3, FD (1) =
0.6, FD (2) = 0.
Then S = {0N , A, B, D, A∩B, A∪B, A∩D, A∪D, B∩D, B∪D, D∩(A∪B), A∪(B∩D), B∪(A∩D), 1N }
is a neutrosophic τ -structure ring on R. Thus the pair (R, S ) is a neutrosophic τ -structure ring space.
Now, A ∩ D = ∩{B ∪ (A ∩ D), D ∩ (A ∪ B), D, A} and D ∩ (A ∪ B) = ∩{A ∪ B, D ∩ (A ∪ B), A ∪ D} are
neutrosophic Gδ rings in (R, S ). Also, the neutrosophic ring exterior of C(A ∩ D) and C(D ∩ (A ∪ B)) are
neutrosophic dense rings in (R, S ). Now, N FR cl(∩{A∩D, D∩(A∪B)}) = N FR cl(A∩D) = 1N .Therefore,
(R, S ) is a neutrosophic τ -structure ring ExtV space.
Proposition 4.1. Let (R, S ) be a neutrosophic structure ring space. Then (R, S ) is a neutrosophic τ -structure
ring ExtV space iff N FR int(∪ni=1 C(Ai )) = 0N where Ai ’s are neutrosophic Gδ rings and the neutrosophic
ring exterior of C(Ai )’s are neutrosophic dense rings in (R, S ).
Proof:
Let (R, S ) be a neutrosophic ring ExtV space. Assume that Ai ’s are neutrosophic Gδ rings and the
neutrosophic ring exterior of C(Ai )’s are neutrosophic dense rings in (R, S ). Since (R, S ) is a neutrosophic
τ -structure ring ExtV space, N FR cl(∩ni=1 Ai ) = 1N . Now, N FR int(∪ni=1 C(Ai )) = N FR int(C(∩ni=1 Ai )) =
C(N FR cl(∩ni=1 Ai )) = 0N . Therefore, N FR int(∪ni=1 C(Ai )) = 0N where Ai ’s are neutrosophic Gδ rings and
the neutrosophic ring exterior of C(Ai )’s are neutrosophic dense rings in (R, S ).
Conversely, let N FR int(∪ni=1 C(Ai )) = 0N where Ai ’s are neutrosophic Gδ rings and the neutrosophic ring
exterior of C(Ai )’s are neutrosophic dense rings in (R, S ). Now, N FR cl(∩ni=1 Ai ) = N FR cl(C(∪ni=1 Ai )) =
C(N FR int(∪ni=1 Ai )) = 1N . Therefore, (R, S ) is a neutrosophic τ -structure ring ExtV space.
Proposition 4.2. Let (R, S ) be a neutrosophic τ -structure ring space. If every neutrosophic first category ring
in (R, S ) is formed from the neutrosophic Gδ rings and the neutrosophic ring exterior of its complements are
neutrosophic dense rings in a neutrosophic τ -structure ring ExtV space (R, S ), then (R, S ) is a neutrosophic
τ -structure ring ExtB space.
Proof:
Assume that Ai ’s are neutrosophic Gδ rings in (R, S ) and the neutrosophic ring exterior of C(Ai )’s are
neutrosophic dense rings in (R, S ), for i = 1, ..., n. Since (R, S ) is a neutrosophic τ -structure ring ExtV
space and by Proposition 4.1., N FR int(∪ni=1 C(Ai )) = 0N . But ∪ni=1 N FR int(C(Ai )) ⊆ N FR int(∪ni=1 C(Ai )),
which implies that ∪ni=1 N FR int(C(Ai )) = 0N . Then N FR int(C(Ai )) = 0∼ . Since Ai ’s are neutrosophic Gδ
rings in (R, S ) and the neutrosophic ring exterior of C(Ai )’s are neutrosophic dense rings in (R, S ), for
i = 1, ..., n. By Proposition 3.2., C(Ai )’s are neutrosophic first category rings in (R, S ), for i = 1, ..., n.
Therefore, N FR int(C(Ai )) = 0N , for every C(Ai ) is a neutrosophic first category rings in (R, S ). By
Proposition 3.6., (R, S ) is a neutrosophic τ -structure ring ExtB space.
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179
Definition 4.2. Let (R1 , S1 ) and (R2 , S2 ) be any two neutrosophic τ -structure ring spaces. Let f : (R1 , S1 ) →
(R2 , S2 ) be any function. Then f is said to be a
(i) neutrosophic τ -structure ring continuous function if f −1 (A) is a neutrosophic τ -open ring in (R1 , S1 ),
for every neutrosophic τ -open ring A in (R2 , S2 ).
(ii) somewhat neutrosophic τ -structure ring continuous function if A ∈ S2 and f −1 (A) 6= 0∼ implies that
there exists a neutrosophic τ -open ring B in (R1 , S1 ) such that B 6= 0N and B ⊆ f −1 (A).
(iii) neutrosophic τ -structure ring hardly open function if for each neutrosophic dense ring A in (R2 , S2 )
such that A ⊆ B ⊂ 1N for some neutrosophic τ -open ring B in (R2 , S2 ), f −1 (A) is a neutrosophic
dense ring in (R1 , S1 ).
(iv) neutrosophic τ -structure ring open function if f (A) is a neutrosophic τ -open ring in (R2 , S2 ), for every
neutrosophic τ -open ring A in (R1 , S1 ).
Proposition 4.3. Let (R1 , S1 ) and (R2 , S2 ) be any two neutrosophic τ -structure ring spaces. Let f : (R1 , S1 ) →
(R2 , S2 ) be any function. Then the following statements are equivalent:
(i) f is a neutrosophic τ -structure ring continuous function.
(ii) f −1 (B) is a neutrosophic τ -closed ring in (R1 , S1 ), for every neutrosophic τ -closed ring B in (R2 , S2 ).
(iii) N FR cl(f −1 (A)) ⊆ f −1 (N FR cl(A)), for each neutrosophic ring A in (R2 , S2 ).
(iv) f −1 (N FR int(A)) ⊆ N FR int(f −1 (A)), for each neutrosophic ring A in (R2 , S2 ).
Remark 4.1. Let (R1 , S1 ) and (R2 , S2 ) be any two neutrosophic τ -structure ring spaces. If f : (R1 , S1 ) →
(R2 , S2 ) is a neutrosophic τ -structure ring continuous function, then f −1 (N FR Ext(C(A)) ⊆ N FR Ext(C(f −1 (A))),
for each neutrosophic ring A in (R2 , S2 ).
Proof:
The proof follows from the Definition 3.4 and Proposition 4.3..
Proposition 4.4. If a function f : (R1 , S1 ) → (R2 , S2 ) from a neutrosophic τ -structure ring space (R1 , S1 )
into another neutrosophic τ -structure ring space (R2 , S2 ) is neutrosophic τ -structure ring continuous, 1-1 and
if A is a neutrosophic dense ring in (R1 , S1 ), then f (A) is a neutrosophic dense ring in (R2 , S2 ).
Proof:
Suppose that f (A) is not a neutrosophic dense ring in (R2 , S2 ). Then there exists a neutrosophic τ -closed
ring in (R2 , S2 ) such that f (A) ⊂ D ⊂ 1N . Then, f −1 (f (A)) ⊂ f −1 (D) ⊂ f −1 (1N ). Since f is 1-1,
f −1 (f (A)) = A. Hence A ⊂ f −1 (D) ⊂ 1N . Since f is a neutrosophic τ -structure ring continuous function
and D is a neutrosophic τ -closed ring in (R2 , S2 ), f −1 (D) is a neutrosophic τ -closed ring in (R1 , S1 ). Then
N FR cl(A) 6= 1N , which is a contradiction. Therefore f (A) is a neutrosophic dense ring in (R2 , S2 ).
Remark 4.2. Let (R1 , S1 ) and (R2 , S2 ) be any two neutrosophic τ -structure ring spaces. Then
(i) the neutrosophic τ -structure ring continuous image of a neutrosophic τ -structure ring ExtV space
(R1 , S1 ) may fail to be a neutrosophic τ -structure ring ExtV space (R2 , S2 ).
(ii) the neutrosophic τ -structure ring open image of a neutrosophic τ -structure ring ExtV space (R1 , S1 )
may fail to be a neutrosophic τ -structure ring ExtV space (R2 , S2 ).
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Proof: It is clear from the following Examples.
Example 4.2. Let R = {0, 1, 2} be a set of integers of module 3
follows:
+ 0 1 2
· 0 1
0 0 1 2
0 0 0
and
1 1 2 0
1 0 1
2 2 0 1
2 0 2
together with two binary operations as
2
0
2
1
Then (R, +, ·) is a ring. Define neutrosophic rings A, B, V, D, E, and F on R as follows: TA (0) =
1, TA (1) = 0.2, TA (2) = 0.9, IA (0) = 1, IA (1) = 0.2, IA (2) = 0.9, FA (0) = 0, FA (1) = 0.8, FA (2) =
0.1, TB (0) = 0.3, TB (1) = 1, TB (2) = 0.2, IB (0) = 0.3, IB (1) = 1, IB (2) = 0.2, FB (0) = 0.7, FB (1) =
0, FB (2) = 0.8, TV (0) = 0.7, TV (1) = 0.4, TV (2) = 1, IV (0) = 0.7, IV (1) = 0.4, IV (2) = 1FV (0) =
0.3, FV (1) = 0.6, FV (2) = 0, TD (0) = 0.9, TD (1) = 1, TD (2) = 0.2, ID (0) = 0.9, ID (1) = 1, ID (2) =
0.2, FD (0) = 0.1, FD (1) = 0, FD (2) = 0.8, TE (0) = 0.2, TE (1) = 0.2, TE (2) = 1, IE (0) = 0.2, IE (1) =
0.2, IE (2) = 1, FE (0) = 0.8, FE (1) = 0.8, FE (2) = 0, TF (0) = 1, TF (1) = 0.7, TF (2) = 0.4, IF (0) =
1, IF (1) = 0.7, IF (2) = 0.4, FF (0) = 0, FF (1) = 0.3, FF (2) = 0.6.
Then S1 = {0N , A, B, V, A∩B, A∪B, A∩V, A∪V, B∩V, B∪V, V ∩(A∪B), A∪(B∩V ), B∪(A∩V ), 1N }
and S2 = {0N , D, E, F, D ∩E, D ∪E, D ∩F, D∪F, E ∩F, E ∪F, F ∩(D ∪E), D∪(E ∩F ), E ∪(D ∩F ), 1N }
are two neutrosophic τ -structure rings on R. Thus the pair (R, S1 ) and (R, S2 ) are neutrosophic τ -structure
ring spaces. Now, A∩V = ∩{B∪(A∩V ), V ∩(A∪B), V, A} and V ∩(A∪B) = ∩{A∪B, V ∩(A∪B), A∪V }
are neutrosophic Gδ rings in (R, S1 ). Also, the neutrosophic ring exterior of C(A ∩ V ) and C(V ∩ (A ∪ B))
are neutrosophic dense rings in (R, S1 ). Now, N FR cl(∩{A ∩ V, V ∩ (A ∪ B)}) = N FR cl(A ∩ V ) = 1N .
Therefore, (R, S1 ) is a neutrosophic τ -structure ring ExtV space. Define a function f : (R, S1 ) → (R, S2 )
by f (0) = 1, f (1) = 2 and f (2) = 0. Clearly, f is a neutrosophic τ -structure ring continuous function.
Also, f (A) = D, f (B) = E and f (V ) = F . Now, D = ∩{D, D ∪ E, D ∪ (E ∩ F )}, D ∩ F = ∩{F, D ∪
F, D ∩ F, F ∩ (D ∪ E)} and E = ∩{E, E ∪ F, E ∪ (D ∩ F )} are neutrosophic Gδ rings in (R, S2 ). Also,
the neutrosophic ring exterior of C(D), C(F ) and C(D ∩ F ) are neutrosophic Gδ rings in (R, S2 ). But,
N FR cl(∩{D, E, D ∩ F }) = C(E ∩ F ) 6= 1N . Therefore, (R, S2 ) is not a neutrosophic τ -structure ring ExtV
space. Therefore the neutrosophic τ -structure ring continuous image of a neutrosophic τ -structure ring ExtV
space (R1 , S1 ) may fail to be a neutrosophic τ -structure ring ExtV space (R2 , S2 ).
Example 4.3. Let R = {0, 1, 2} be a set of integers of module 3
follows:
+ 0 1 2
. 0 1
0 0 1 2
0 0 0
and
1 1 2 0
1 0 1
2 2 0 1
2 0 2
together with two binary operations as
2
0
2
1
Then (R, +, ·) is a ring. Define neutrosophic rings A, B, V and D on R as follows: TA (0) = 1, TA (1) =
0.2, TA (2) = 0.9, IA (0) = 1, IA (1) = 0.2, IA (2) = 0.9, FA (0) = 0, FA (1) = 0.8, FA (2) = 0.1, TB (0) =
0.3, TB (1) = 1, TB (2) = 0.2, IB (0) = 0.3, IB (1) = 1, IB (2) = 0.2, FB (0) = 0.7, FB (1) = 0, FB (2) =
0.8, TV (0) = 0.7, TV (1) = 0.4, TV (2) = 1, IV (0) = 0.7, IV (1) = 0.4, IV (2) = 1, FV (0) = 0.3, FV (1) =
0.6, FV (2) = 0, TD (0) = 0.5, TD (1) = 0.6, TD (2) = 0.4, ID (0) = 0.5, ID (1) = 0.6, ID (2) = 0.4, FD (0) =
0.5, FD (1) = 0.4, FD (2) = 0.6.
Then S1 = {0N , A, B, V, A∩B, A∪B, A∩V, A∪V, B∩V, B∪V, V ∩(A∪B), A∪(B∩V ), B∪(A∩V ), 1N }
and S2 = {0N , A, B, V, D, A ∪ B, A ∪ V, A ∪ D, B ∪ V, B ∪ D, V ∪ D, A ∩ B, A ∩ V, A ∩ D, B ∩ V, B ∩
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D, V ∩ D, D ∪ (A ∩ V ), V ∩ (A ∪ B), A ∪ (B ∩ V ), B ∪ (A ∩ V ), 1N } are two neutrosophic τ -structure
rings on R. Thus the pair (R, S1 ) and (R, S2 ) are neutrosophic τ -structure ring spaces. Now, A ∩ V =
∩{B ∪ (A ∩ V ), V ∩ (A ∪ B), V, A} and V ∩ (A ∪ B) = ∩{A ∪ B, V ∩ (A ∪ B), A ∪ V } are neutrosophic
Gδ rings in (R, S1 ). Also, the neutrosophic ring exterior of C(A ∩ V ) and C(V ∩ (A ∪ B)) are neutrosophic
dense rings in (R, S1 ). Now, N FR cl(∩{A ∩ V, V ∩ (A ∪ B)}) = N FR cl(A ∩ V ) = 1V . Therefore, (R, S1 )
is a neutrosophic ring ExtV space. Define a function f : (R, S1 ) → (R, S2 ) by f (0) = 0, f (1) = 1 and
f (2) = 2. Clearly, f is a neutrosophic τ -structure ring open function. Also, f (A) = A, f (B) = B, f (V ) = V
and f (D) = D. Now, A = ∩{A, A ∪ B, A ∪ V, A ∪ (B ∩ V )}, D ∪ (A ∩ V ) = ∩{V, V ∪ D, A ∩ V, D ∪ (A ∩
V ), V ∩ (A ∪ B)} and B = ∩{B, B ∪ V, B ∪ D, B ∪ (A ∩ V )} are neutrosophic Gδ rings in (R, S2 ). Also,
the neutrosophic ring exterior of C(A), C(B) and C(D ∪ (A ∩ V )) are neutrosophic Gδ rings in (R, S2 ). But,
N FR cl(∩{A, B, D ∪ (A ∩ V )}) = C(B ∩ V ) 6= 1N . Therefore, (R, S2 ) is not a neutrosophic τ -ring ExtV
space. Therefore the neutrosophic τ -structure ring open image of a neutrosophic τ -structure ring ExtV space
(R1 , S1 ) may fail to be a neutrosophic τ -structure ring ExtV space (R2 , S2 ).
Proposition 4.5. Let (R1 , S1 ) and (R2 , S2 ) be any two neutrosophic τ -structure ring spaces. If f : (R1 , S1 ) →
(R2 , S2 ) is onto function, then the following statements are equivalent:
(i) f is a neutrosophic τ -structure ring hardly open function.
(ii) N FR int(f (A)) 6= 0N , for all neutrosophic ring A in (R1 , S1 ) with N FR int(A) 6= 0N and there exists a
neutrosophic τ -closed ring B 6= 0N in (R2 , S2 ) such that B ⊆ f (A).
(iii) N FR int(f (A)) 6= 0N , for all neutrosophic ring A in (R1 , S1 ) with N FR int(A) 6= 0N and there exists a
neutrosophic τ -closed ring B 6= 0N in (R2 , S2 ) such that f −1 (B) ⊆ A.
Proof:
(i)⇒(ii)
Assume that (i) is true. Let A be any neutrosophic ring A in (R1 , S1 ) with N FR int(A) 6= 0N and
B 6= 0N be a neutrosophic τ -closed ring in (R2 , S2 ) such that B ⊆ f (A). Suppose that N FR int(A) =
0N . This implies that N FR cl(C(f (A))) = 1N . Thus, C(f (A)) is a neutrosophic dense ring in (R2 , S2 )
and C(f (A)) ⊆ C(B). By assumption, f −1 (C(f (A))) is a neutrosophic dense ring in (R1 , S1 ). That is,
N FR cl(f −1 (C(f (A)))) = 1N . Now, N FR int(A) = N FR int(f −1 (f (A))) = C(N FR cl(C(f −1 (f (A))))) =
C(N FR cl(f −1 (C(f (A))))) = 0N . This is a contradiction. Hence (i)⇒(ii).
(ii)⇒(iii)
Assume that (ii) is true. Since f is onto function and by assumption, B ⊆ f (A). This implies that
f −1 (B) ⊆ f −1 (f (A)), that is, f −1 (B) ⊆ A. Hence (ii)⇒(iii).
(iii)⇒(i)
Let V ⊆ C(D) where C is a neutrosophic dense ring and D is non-zero neutrosophic τ -open ring in
(R2 , S2 ). Let A = f −1 (C(V )) and B = C(D). Now, f −1 (B) = f −1 (C(D)) ⊆ f −1 (C(V )) = A.
Consider, N FR int(f (A)) = N FR int(f (f −1 (C(V ))) = N FR int(C(V )) = C(N FR int(V )) = 0N .
Therefore, N FR int(A) = 0N , which implies that N FR int(f −1 (C(V ))) = N FR int(C(f −1 (V ))) = 0N .
Therefore, C(N FR cl(f −1 (V ))) = 0N . Thus, N FR cl(f −1 (V )) = 1N . Therefore, f −1 (V ) is a neutrosophic
dense ring in (R1 , S1 ). This implies that f is a neutrosophic τ -structure ring hardly open function. Hence
(iii)⇒(i). This completes the proof.
Proposition 4.6. If a function f : (R1 , S1 ) → (R2 , S2 ) from a neutrosophic τ -structure ring space (R1 , S1 )
onto another neutrosophic τ -structure ring space (R2 , S2 ) is neutrosophic τ -structure ring continuous, 1-1 and
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neutrosophic τ -structure ring hardly open function and if (R1 , S1 ) is a neutrosophic τ -structure ring ExtV
space, then (R2 , S2 ) is a neutrosophic τ -structure ring ExtV space.
Proof:
Let (R1 , S1 ) be a neutrosophic τ -structure ring ExtV space. Assume that Ai ’s (i = 1, ..., n) are neutrosophic Gδ rings in (R2 , S2 ) and the neutrosophic ring exterior of C(Ai )’s are neutrosophic dense ring in
(R2 , S2 ). Then N FR cl(N FR Ext(C(Ai ))) = 1N and Ai = ∩∞
j=1 Bij where Bij ’s are neutrosophic τ -open
rings in (R2 , S2 ). Hence
∞
−1
f −1 (Ai ) = f −1 (∩∞
(Bij )
(4.1)
j=1 Bij ) = ∩j=1 f
Since f is a neutrosophic τ -structure ring continuous function and Bij ’s are neutrosophic τ -open rings
−1
in (R2 , S2 ), f −1 (Bij )’s are neutrosophic τ -open rings in (R1 , S1 ). Hence f −1 (Ai ) = ∩∞
(Bij ) is an
j=1 f
neutrosophic Gδ rings in (R1 , S1 ). Since f is a neutrosophic τ -structure ring hardly open function and
N FR Ext(C(Ai )) is a neutrosophic dense ring in (R2 , S2 ), f −1 (N FR Ext(C(Ai ))) is a neutrosophic dense
ring in (R1 , S1 ). Now,
f −1 (N FR Ext(C(Ai ))) = f −1 (N FR int(Ai ))
⊆ N FR int(f −1 (Ai ))
= N FR Ext(C(f −1 (Ai ))).
Therefore 1N = N FR cl(f −1 (N FR Ext(C(Ai )))) ⊆ N FR cl(N FR Ext(C(f −1 (Ai )))), which implies that
1N = N FR cl(N FR Ext(C(f −1 (Ai )))). Hence N FR Ext(C(f −1 (Ai ))) is a neutrosophic dense ring in (R1 , S1 ).
Since (R1 , S1 ) is a neutrosophic τ -strucuture ring ExtV space, N FR cl(∩ni=1 f −1 (Ai )) = 1N where f −1 (Ai )’s
are neutrosophic Gδ rings in (R1 , S1 ) and the neutrosophic ring exterior of C(f −1 (Ai ))’s are neutrosophic
dense ring in (R1 , S1 ). Thus, N FR cl(∩ni=1 f −1 (Ai )) = 1N = N FR cl(f −1 (∩ni=1 Ai )). Therefore, f −1 (∩ni=1 Ai )
is a neutrosophic dense rings in (R1 , S1 ). Since f is a neutrosophic τ -structure ring continuous, 1-1 and by
Proposition 3.4., f (f −1 (∩ni=1 Ai )) is a neutrosophic dense ring in (R2 , S2 ). Hence N FR cl(f (f −1 (∩ni=1 Ai ))) =
1N . Since f is 1-1, f (f −1 (∩ni=1 Ai )) = ∩ni=1 Ai . Then, N FR cl(∩ni=1 Ai ) = 1N . Therefore, (R2 , S2 ) is a
neutrosophic τ -structure ring ExtV space.
Conversely, let (R2 , S2 ) be a neutrosophic τ -structure ring ExtV space. Assume that Ai ’s (i = 1, ..., n)
are neutrosophic Gδ rings in (R2 , S2 ) and the neutrosophic ring exterior of C(Ai )’s are neutrosophic dense
rings in (R2 , S2 ).
Then N FR cl(N FR Ext(C(Ai ))) = 1N and Ai = ∩∞
j=1 Bij where Bij ’s are neutrosophic τ -open rings in
(R2 , S2 ). Hence
−1
∞
f −1 (Ai ) = f −1 (∩∞
(Bij )
(4.2)
j=1 Bij ) = ∩j=1 f
Since f is a neutrosophic τ -structure ring continuous function and Bij ’s are neutrosophic τ -open rings
−1
(Bij ) is a
in (R2 , S2 ), f −1 (Bij )’s are neutrosophic τ -open rings in (R1 , S1 ). Hence f −1 (Ai ) = ∩∞
j=1 f
neutrosophic Gδ rings in (R1 , S1 ). Since f is a neutrosophic τ -structure ring hardly open function and
N FR Ext(C(Ai )) is a neutrosophic dense ring in (R2 , S2 ), f −1 (N FR Ext(C(Ai ))) is a neutrosophic dense
ring in (R1 , S1 ). By Remark 4.2., f −1 (N FR Ext(C(Ai ))) ⊆ N FR Ext(C(f −1 (Ai ))).
Thus, N FR cl(f −1 (N FR Ext(C(Ai )))) = 1N ⊆ N FR cl(N FR Ext(C(f −1 (Ai )))). Hence, N FR Ext(C(f −1 (Ai )))
is a neutrosophic dense ring in (R1 , S1 ). Suppose that N FR cl(∩ni=1 f −1 (Ai )) 6= 1N . This implies that
N FR cl(∩ni=1 f −1 (Ai )) 6= 0N
⇒ N FR int(∪ni=1 C(f −1 (Ai ))) 6= 0N
⇒ N FR int(∪ni=1 f −1 (C(Ai ))) 6= 0N .
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Then, there is a non-zero neutrosophic τ -open ring Ei in (R1 , S1 ) such that Ei ⊆ ∪ni=1 f −1 (C(Ai )). Now,
f (Ei ) ⊆ f (∪ni=1 f −1 (C(Ai )))
⊆ ∪ni=1 f (f −1 (C(Ai )))
⊆ ∪ni=1 C(Ai )
= C(∩ni=1 Ai ).
Then, N FR int(f (Ei )) ⊆ N FR int(C(∩ni=1 Ai )) = C(N FR cl(∩ni=1 Ai )).
(4.3)
Since (R2 , S2 ) is a neutrosophic τ -structure ring ExtV space, N FR cl(∩ni=1 Ai ) = 1N . Hence from
(4.3), N FR int(f (Ei )) ⊆ 0N . This implies that N FR int(f (Ei )) = 0N , which is a contradiction. Hence
N FR cl(∩ni=1 f −1 (Ai )) = 1N . Therefore, (R1 , S1 ) is a neutrosophic τ -structure ring ExtV space.
Proposition 4.7. Let (R1 , S1 ) and (R2 , S2 ) be any two neutrosophic τ -structure ring spaces. Let f : (R1 , S1 ) →
(R2 , S2 ) be any bijective function. Then the following statements are equivalent:
(i) f is somewhat neutrosophic τ -structure ring continuous function.
(ii) If A is a neutrosophic τ -closed ring in (R2 , S2 ) such that f −1 (A) 6= 1N , then there exists a neutrosophic
τ -closed ring 0N 6= E 6= 1N in (R1 , S1 ) such that f −1 (A) ⊂ E.
(iii) If A is a neutrosophic dense ring in (R1 , S1 ), then f (A) is a neutrosophic dense ring in (R2 , S2 ).
Proof:
(i)⇒(ii)
Assume that (i) is true. Let A be a neutrosophic τ -closed ring in (R2 , S2 ) such that f −1 (A) 6= 1N .
Then C(A) is a neutrosophic τ -open ring in (R2 , S2 ) such that C(f −1 (A)) = f −1 (C(A)) 6= 0N . Since f
is somewhat neutrosophic τ -structure ring continuous, there exists a neutrosophic τ -open ring E in (R1 , S1 )
such that E ⊆ f −1 (C(A)). Then there exists a neutrosophic τ -closed ring C(E) 6= 0N in (R1 , S1 ) such that
C(E) ⊂ f −1 (A). Hence (i)⇒(ii).
(ii)⇒(iii)
Assume that (ii) is true. Let A be a neutrosophic dense ring in (R1 , S1 ) such that f (A) is a neutrosophic
dense ring in (R2 , S2 ). Then, there exists a neutrosophic τ -closed ring C in (R2 , S2 ) such that
f (A) ⊂ E ⊂ 1N .
This implies that f −1 (E) 6= 1N . Then by (ii), there exists a neutrosophic τ -closed ring 0N 6= D 6= 1N such
that A ⊂ f −1 (E) ⊂ D ⊂ 1N . This is a contradiction. Hence (ii)⇒(iii).
(iii)⇒(ii)
Assume that (iii) is true. Suppose (ii) is not true. Then there exists a neutrosophic τ -closed ring A in
(R2 , S2 ) such that f −1 (A) 6= 1N . But there is no neutrosophic τ -closed ring 0N 6= E 6= 1N in (R1 , S1 ) such
that f −1 (A) ⊆ E. This implies that f −1 (A) is a neutrosophic dense ring in (R1 , S1 ). But from hypothesis
f (f −1 (A)) = A must be neutrosophic dense ring in (R2 , S2 ), which is a contradiction. Hence (iii)⇒(ii).
(ii)⇒(i)
Let A be a neutrosophic τ -open ring in (R2 , S2 ) and f −1 (A) 6= 0N . Then, f −1 (C(A)) = C(f −1 (A)) =
0N . Then by (ii), there exists a neutrosophic τ -closed ring 0N 6= B 6= 1N such that f −1 (C(A)) ⊂ B. This
implies that C(B) ⊂ f −1 (A) and C(B) 6= 0N is a neutrosophic τ -open ring in (R1 , S1 ). Hence (ii)⇒(i).
Hence the proof.
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Proposition 4.8. If a function f : (R1 , S1 ) → (R2 , S2 ) from a neutrosophic τ -structure ring space (R1 , S1 )
onto another neutrosophic τ -structure ring space (R2 , S2 ) is somewhat neutrosophic τ -structure ring continuous, 1-1 and neutrosophic τ -structure ring open function and if (R1 , S1 ) is a neutrosophic τ -structure ring
ExtV space, then (R2 , S2 ) is a neutrosophic τ -structure ring ExtV space.
Proof:
Let (R1 , S1 ) be a neutrosophic τ -structure ring ExtV space. Assume that Ai ’s (i = 1, ..., n) are neutrosophic Gδ rings in (R1 , S1 ) and the neutrosophic ring exterior of C(Ai )’s are neutrosophic dense rings in
(R1 , S1 ). Then, N FR cl(N FR Ext(C(Ai ))) = 1N and Ai = ∩∞
j=1 Bij where Bij ’s are neutrosophic τ -open
rings in (R1 , S1 ). Since f is a neutrosophic τ -structure ring open function, f (Bij )’s are neutrosophic τ -open
rings in (R2 , S2 ). Now, ∩∞
j=1 f (Bij ) is a neutrosophic Gδ rings in (R2 , S2 ). Since f is 1-1,
−
∞
∞
f −1 (∩∞
j=1 f (Bij )) = ∩j=1 f (f (Bij )) = ∩j=1 Bij = Ai
(4.4)
∞
Since f is onto, f (Ai ) = f (f −1 (∩∞
j=1 f (Bij ))) = ∩j=1 f (Bij )
(4.5)
Therefore, f (Ai ) is a neutrosophic Gδ rings in (R2 , S2 ). Since f is somewhat neutrosophic τ -structure
ring continuous function, N FR Ext(C(Ai ‘)) is a neutrosophic dense ring in (R1 , S1 ) and by Proposition 4.7.,
f (N FR Ext(C(Ai ))) is a neutrosophic dense ring in (R2 , S2 ), which implies that N FR Ext((f (Ai ))). Now
n
we claim that N FR cl(∩∞
i=1 f (Ai )) = 1N . Suppose that N FR cl(∩i=1 f (Ai )) 6= 1N . This implies that
C(N FR cl(∩ni=1 f (Ai ))) 6= 0N
⇒ N FR int(∪ni=1 C(f (Ai ))) 6= 0N
⇒ N FR int(∪ni=1 f (C(Ai ))) 6= 0N .
Therefore there is an non-zero neutrosophic τ -open ring Ei in (R2 , S2 ) such that Ei ⊆ ∪ni=1 f (C(Ai )).
Then f −1 (Ei ) ⊆ f −1 (∪ni=1 f (C(Ai ))). Since f is somewhat neutrosophic τ -structure ring continuous function
and Ei ∈ S2 , N FR int(f −1 (Ei )) 6= 0N implies that N FR int(f −1 (∪ni=1 f (C(Ai )))) 6= 0N .
Then N FR int(∪ni=1 f −1 (f (C(Ai )))) 6= 0N . Since f is a bijective function, N FR int(∩ni=1 C(Ai )) 6= 0N , which
implies that C(N FR cl(∩ni=1 Ai )) 6= 0N . That is, N FR cl(∩ni=1 Ai ) 6= 1N . This is a contradiction. Hence
(R2 , S2 ) is a neutrosophic τ -structure ring ExtV space.
Conversely, let (R2 , S2 ) be a neutrosophic τ -structure ring ExtV space. Assume that Ai ’s (i = 1, ..., n) are
neutrosophic Gδ rings in (R1 , S1 ) and the neutrosophic ring exterior of C(Ai )’s are neutrosophic dense ring
in (R1 , S1 ). Then N FR cl(N FR Ext(C(Ai ))) = 1N and Ai = ∩∞
j=1 Bij where Bij ’s are neutrosophic τ -open
rings in (R1 , S1 ). Since f is somewhat neutrosophic τ -structure ring continuous function, N FR Ext(C(Ai ))’s
are neutrosophic dense rings in (R1 , S1 ) and By Proposition 4.7., f (N FR Ext(C(Ai ))) is a neutrosophic
dense ring in (R2 , S2 ). That is, N FR cl(N FR Ext(C(Ai ))) = 1N . Since f is a neutrosophic τ -structure ring
open function and Bij ’s are neutrosophic τ -open rings in (R1 , S1 ), f (Bij )’s are neutrosophic τ -open rings in
(R2 , S2 ). Hence ∩∞
j=1 f (Bij ) is a neutrosophic Gδ ring in (R2 , S2 ). Since f is 1-1,
f −1 (∩ni=1 f (Bij )) = ∩ni=1 (f −1 (f (Bij )) = ∩ni=1 Bij .
(4.6)
∞
f (Ai ) = f (f −1 (∩∞
j=1 f (Bij ))) = ∩j=1 f (Bij ).
(4.7)
Since f is onto,
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Hence f (Ai ) is a neutrosophic Gδ ring in (R2 , S2 ). Now,
N FR cl(N FR Ext(C(f (Ai ))) = N FR cl(N FR Ext(f (C(Ai )))
= N FR cl(N FR int(f (Ai ))
⊇ N FR cl(f (N FR int(Ai ))
⊇ f (N FR cl(N FR int(Ai )))
= f (1N ) = 1N .
This implies that N FR Ext(C(f (Ai )) is a neutrosophic dense ring in (R2 , S2 ). Hence the neutrosophic
ring exterior of C(f (Ai )) is a neutrosophic dense ring in (R2 , S2 ). Since (R2 , S2 ) is a neutrosophic τ structure ring ExtV space, N FR cl(∩ni=1 f (Ai )) = 1N . Now we claim that N FR cl(∩ni=1 f (Ai )) = 1N where
Ai ’s (i = 1, ..., n) are neutrosophic Gδ rings in (R1 , S1 ) and the neutrosophic ring exterior of C(Ai )’s are
neutrosophic dense rings in (R1 , S1 ). Suppose that N FR cl(∩ni=1 Ai ) 6= 1N . This implies that
C(N FR cl(∩ni=1 Ai )) 6= 0N
⇒ N FR int(C(∩ni=1 Ai )) 6= 0N
⇒ N FR int(∪ni=1 C(Ai )) 6= 0N .
Then there is a non-zero neutrosophic τ -open ring Ei in (R1 , S1 ) such that Ei ⊆ ∪ni=1 C(Ai ). Now,
f (Ei ) ⊆ f (∪ni=1 C(Ai ))
⊆ ∪ni=1 f (C(Ai ))
⊆ ∪ni=1 C(f (Ai ))
= C(∩ni=1 f (Ai )).
Then, N FR int(f (Ei )) ⊆ N FR int(C(∩ni=1 f (Ai ))) ⊆ C(N FR cl(∩ni=1 f (Ai )))
(4.8)
Since (R2 , S2 ) is a neutrosophic τ -structure ring ExtV space, N FR cl(∩ni=1 f (Ai )) = 1N . Hence from
(4.8), N FR int(f (Ei )) ⊆ 0N , which implies that N FR int(f (Ei )) = 0N , which is a contradiction. Hence
N FR cl(∩ni=1 Ai ) = 1N . Therefore (R1 , S1 ) is a neutrosophic τ -structure ring ExtV space.
5
Conclusion
A neutrosophic set model provides a mechanism for solving the modeling problems which involve indeterminacy, and inconsistent information in which human knowledge is necessary and human evaluation is needed.
It deals more flexibility and compatibility to the system as compared to the classical theory, fuzzy theory
and intuitionistic fuzzy models. In this paper, a new idea of a neutrosophic τ -structure ring spaces, neutrosophic τ -structure ring Gδ T1/2 spaces and neutrosophic τ -structure ring exterior B spaces and neutrosophic
τ -structure ring exterior V spaces have been introduced. Further, neutrosophic τ -structure ring continuous
(resp. open,hardly open)functions, somewhat neutrosophic τ -structure ring continuous functions are studied.
Their characterization are derived and illustrated with examples.
R. Narmada Devi and A New Novel of neutrosophic τ -Structure Ring ExtB and ExtV Spaces
186
Neutrosophic Sets and Systems, Vol. 32 2020
References
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Received:Ocotober 11, 2019 / Accepted: December 5,2019
R. Narmada Devi and A New Novel of neutrosophic τ -Structure Ring ExtB and ExtV Spaces
Neutrosophic Sets and Systems, Vol. 32, 2020
University of New Mexico
An MCDM Method under Neutrosophic Cubic Fuzzy Sets with Geometric
Bonferroni Mean Operator
D. Ajay
1
1,∗
, Said Broumi 2 , J. Aldring
3
Assistant Professor, Department of Mathematics, Sacred Heart College, Tamilnadu, India ;
dajaypravin@gmail.com
2
Laboratory of Information Processing, Faculty of Science Ben MSik, University Hassan II, Casablanca,
Morocco ; broumisaid78@gmail.com
3
Research Scholar, Department of Mathematics, Sacred Heart College, Tamilnadu, India ;
jaldring24@gmail.com
∗
Correspondence: dajaypravin@gmail.com
Abstract. Neutrosophic cubic fuzzy sets (N CF Ss) involve interval valued and single valued neutrosophic sets,
and are used to describe uncertainty or fuzziness in a more efficient way. Aggregation of neutrosopic cubic fuzzy
information is crucial and necessary in a decision making theory. In order to get a better solution to decision
making problems under neutrosophic cubic fuzzy environment, this paper introduces an aggregating operator to
neutrosophic cubic fuzzy sets with the help of Bonferroni mean and geometric mean, and proposes neutrosophic
cubic fuzzy geometric Bonferroni mean operator (N CF GBM u,v ) with its properties. Then, an efficient decision
making technique is introduced based on weighted operator W N CF GBMwu,v . An application of the established
method is also examined for a real life problem.
Keywords: Neutrosophic Sets; Cubic Fuzzy Sets; Bonferroni Geometric Mean; Aggregation Operators; MCDM
—————————————————————————————————————————-
1. Introduction
Fuzzy set [1] deals with fuzziness in terms of degree of truthness or membership within the
range of interval [0, 1]. The traditional fuzzy sets are not efficient when the decision makers
face more complex problems and it is difficult to quantify their truth values. Y.B.Jun et al. [2]
introduced the notion of cubic sets which represents the degree of belongingness or certainty
by interval valued fuzzy sets and single valued fuzzy sets simultaneously. Therefore, cubic sets
are made up of two parts, where the first one is the interval valued fuzzy sets which represents
belongingness in a particular range of interval, and the second one is exact belongingness or
fuzzy sets.
D. Ajay, Said Broumi, J. Aldring ; An MCDM Method under Neutrosophic Cubic Fuzzy Sets with Geometric
Bonferroni Mean Operator
Neutrosophic Sets and Systems, Vol. 32, 2020
188
Smarandache [3] introduced the philosophical idea of neutrosophic sets (NS) which is formulated from the general concept of fuzzy sets and many real life applications are avaliable under
NS. Ajay, D., et al. used neutrosophic theory in fuzzy SAW method [4] and Abdel-Basset.M
et al. utilized neutrosophic sets to asseses the uncertainty of linear time-cost tradeoffs [5] and
also they applied to resource levelling problem in construction projects [6]. Further, biploar
neutrosophic sets have been used in medical diagnosis [7] and decision making suituations [8].
Moreover, Y.B.Jun et al. [9] and M.Ali et al. [10] effectively utilized cubic fuzzy sets to the
neutrosophic sets and introduced the concept of neutrosophic cubic fuzzy sets (NCFSs) with
some basic operations. Therefore the hybrid form of neutrosophic cubic fuzzy set may be
more adequate to address problems of more complexity using interval valued and exact valued neutrosophic information and it has been broadly used in the fields of MCDM [12–19].
Neutrosophic cubic fuzzy sets contain more information than general form of NS and therefore
NCFSs provide better and efficient solution in MCDM.
Aggregating the fuzzy information plays an important role in decision theory and in particular decision making in real life problems. Variety of aggregating operators exist, but very few
aggregating operators are available under neutrosophic cubic fuzzy numbers such as Heronian
mean operators [21], Einstein Hybrid Geometric Aggregation Operators [22, 23], Dombi Aggregation Operators [24], weighted arithmetic averaging (NCNWAA) operator and weighted
geometric averaging (NCNWGA) operator [25]. Still the Bonferroni geometric mean aggregating operator has not been studied in NCF environment. So the main purposes of this study
are: (1) to establish a neutrosophic cubic fuzzy Bonferroni weighted geometric mean operator
W N CF BW GMwu,v .(2) to develop an MCDM method using W N CF BW GMwu,v operator to
rank the alternatives under NCFS environment.
The content of the paper is organized as follows. Section 2 and 3 briefly introduce the basic
concepts and operations of neutrosophic cubic fuzzy sets. The concepts of Bonferroni mean and
geometric Bonferroni mean are explained in section 4. The neutrosophic cubic fuzzy geometric
Bonferroni mean N CF GBM u,v and weighted neutrosophic cubic fuzzy geometric Bonferroni
mean W N CF GBMwu,v operators are established and examined with their properties in section
5. An MCDM method based on W N CF GBMwu,v is presented in section 6. Finally conclusions
and scope for future research are given in section 7.
2. Neutrosophic Cubic Fuzzy Set
Definition 2.1. [9] Let X be a non empty universal set or universe of discourse. A neutrosophic cubic fuzzy set S̃ in X is constructed in the following form:
S˜ = {x, hT (x), I(x), F (x)i ; hTλ (x), Iλ (x), Fλ (x)i |x ∈ X}
D. Ajay, Said Broumi, J. Aldring ; An MCDM Method under Neutrosophic Cubic Fuzzy Sets
with Geometric Bonferroni Mean Operator
Neutrosophic Sets and Systems, Vol. 32, 2020
189
where T (x), I(x), F (x) are interval valued neutrosophic sets; T (x) = [T − (x), T + (x)] ⊆ [0, 1] is
the degree of truth interval values; I(x) = [I − (x), I + (x)] ⊆ [0, 1] is the degree of indeterminacy
interval values; F (x) = [F − (x), F + (x)] ⊆ [0, 1] is the degree of falsity interval values; and
hTλ (x), Iλ (x), Fλ (x)i ∈ [0, 1] are truth, indeterminacy, and falsity degrees of membership values
respectively. For convenience, a neutrosophic cubic fuzzy element in a neutrosophic cubic fuzzy
set (NCFSs) S̃ is simply denoted by S̃ = {hT, I, F i ; hTλ , Iλ , Fλ i}, where hT, I, F i ⊆ [0, 1] and
hTλ , Iλ , Fλ i ∈ [0, 1], satisfying the conditions that 0 ≤ hT + , I + , F + i ≤ 3 and 0 ≤ hTλ , Iλ , Fλ i ≤
3.
Definition 2.2. [10] Let S̃ be a neutrosophic cubic fuzzy set in X given by
S˜ =
T − (x), T + (x) , I − (x), I + (x) , F − (x), F + (x) ; hTλ (x), Iλ (x), Fλ (x)i |x ∈ X
S̃ is said to be internal NCFSs if T − (x) ≤ Tλ (x) ≤ T + (x), I − (x) ≤ Iλ (x) ≤ I + (x), F − (x) ≤
/ [T − (x), T + (x)] , Iλ (x) ∈
/
Fλ (x) ≤ F + (x)∀x; S̃ is said to be external NCFSs if Tλ (x) ∈
/ [F − (x), F + (x)] ∀x.
[I − (x), I + (x)] , Fλ (x) ∈
Definition 2.3. Let S̃ be a neutrosophic cubic fuzzy set in X. Then the support of neutrosophic cubic fuzzy set S̃ ∗ is defined by
S˜∗ = T − (x), T + (x) ⊃ [0, 0], I − (x), I + (x) ⊃ [0, 0], F − (x), F + (x) ⊂ [1, 1];
hTλ (x) > 0, Iλ (x) > 0, Fλ (x) < 1i |x ∈ X}
Definition 2.4. [25] Let S̃ be a non empty neutrosophic cubic fuzzy number given by
S˜ = {x, hT (x), I(x), F (x)i ; hTλ (x), Iλ (x), Fλ (x)i |x ∈ X}
= T − (x), T + (x) , I − (x), I + (x) , F − (x), F + (x) ; hTλ (x), Iλ (x), Fλ (x)i |x ∈ X ,
then its score, accuracy and certainty functions can be defined respectively, as follows:
[4+T − (x)−I − (x)−F − (x)+T + (x)−I + (x)−F + (x)]
˜ =
s(S)
6
+
[2+Tλ (x)−Iλ (x)−Fλ (x)]
3
2
−
−
+
+
˜ = [(T (x) − F (x) + T (x) − F (x)) /2 + Tλ (x) − Fλ (x)] ,
a(S)
2
−
+
˜ = [(T (x) + T (x)) /2 + Tλ (x)] ;
c(S)
2
˜ a(S),
˜ ∈ [0, 1]
˜ c(S)
s(S),
,
(1)
(2)
(3)
3. Operations on NCFNs
− + − + − +
T , T , Ii , Ii , Fi , Fi ; hTλi , Iλi , Fλi i |x ∈ X (i = 1, 2, 3, · · · n) and
Let Ai (x) =
i ih i
i h
i
o
nh
−
+
Aj (y) = Tj , Tj , Ij− , Ij+ , Fj− , Fj+ ; hTλj , Iλj , Fλj i |y ∈ Y (j = 1, 2, 3, · · · n) be two collections of NCFNs. Then the following operations are defined [25]:
D. Ajay, Said Broumi, J. Aldring ; An MCDM Method under Neutrosophic Cubic Fuzzy Sets
with Geometric Bonferroni Mean Operator
Neutrosophic Sets and Systems, Vol. 32, 2020
190
(1) Union
nh
i h
i
Ai (x) ∪ Aj (y) = min(Ti− , Tj− ), max(Ti+ , Tj+ ) , max(Ii− , Ij− ), min(Ii+ , Ij+ ) ,
h
i
o
max(Fi− , Fj− ), min(Fi+ , Fj+ ) ; hmax(Tλi , Tλj ), min(Iλi , Iλj ), min(Fλi , Fλj )i
(2) Intersection
nh
i h
i
Ai (x) ∩ Aj (y) = max(Ti− , Tj− ), min(Ti+ , Tj+ ) , min(Ii− , Ij− ), max(Ii+ , Ij+ ) ,
h
i
o
min(Fi− , Fj− ), max(Fi+ , Fj+ ) ; hmin(Tλi , Tλj ), max(Iλi , Iλj ), max(Fλi , Fλj )i
(3) Complement
Aci (x) =
Fi− , Fi+ , 1 − Ii− , 1 − Ii+ , Ti− , Ti+ ; hFλi , 1 − Iλi , Tλi i |x ∈ X
i
i
h
h
(4) Ai (x) ⊆ Aj (y) if and only if Ti− , Ti+ ⊆ Tj− , Tj+ , Ii− , Ii+ ⊇ Ij− , Ij+ , Fi− , Fi+ ⊇
h
i
Fj− , Fj+ and Tλi ≤ Tλi , Iλi ≥ Iλj , Fλi ≥ Fλj ∀x ∈ X, y ∈ Y.
(5) Ai (x) = Aj (y) if and only if Ai (x) ⊆ Aj (y) and Ai (x) ⊇ Aj (y) i.e. Ti− , Ti+ =
i
i
i
h
h
h
Tj− , Tj+ , Ii− , Ii+ = Ij− , Ij+ , Fi− , Fi+ = Fj− , Fj+ ; hTλi , Iλi , Fλi i = hTλj , Iλj , Fλj i
(6) For ω > 0
ωAi =
1 − 1 − Ti−
ω
, 1 − 1 − Ti+
h1 − (1 − Tλi )ω , (Iλi )ω , (Fλi )ω i}
ω − ω
ω − ω
ω
, Ii+
, Fi+
, Ii
, Fi
;
(7) For ω > 0
ω
ω
ω
− ω
(Ai )ω =
,
, 1 − 1 − Ii− , 1 − 1 − Ii+
, Ti+
Ti
ω
ω
; h(Tλi )ω , 1 − (1 − Iλi )ω , 1 − (1 − Fλi )ω i
1 − 1 − Fi− , 1 − 1 − Fi+
(8) Algebraic Sum
nh
i
i h
Ai (x) ⊕ Aj (y) = Ti− + Tj− − Ti− Tj− , Ti+ + Tj+ − Ti+ Tj+ , Ii− Ij− , Ii+ Ij+ ,
h
i
o
Fi− Fj− , Fi+ Fj+ ; hTλi + Tλj − Tλi Tλj , Iλi Iλj , Fλi Fλj i
(9) Algebraic Product
i
i h
nh
Ai (x) ⊗ Aj (y) = Ti− Tj− , Ti+ Tj+ , Ii− + Ij− − Ii− Ij− , Ii+ + Ij+ − Ii+ Ij+ ,
h
i
o
Fi− + Fj− − Fi− Fj− , Fi+ + Fj+ − Fi+ Fj+ ; hTλi Tλj , Iλi + Iλj − Iλi Iλj , Fλi + Fλj − Fλi Fλj i
D. Ajay, Said Broumi, J. Aldring ; An MCDM Method under Neutrosophic Cubic Fuzzy Sets
with Geometric Bonferroni Mean Operator
Neutrosophic Sets and Systems, Vol. 32, 2020
191
4. Geometric Bonferroni Mean
Bonferroni proposed the concept of Bonferroni mean (BM) which is defined as follows:
Definition 4.1. [11] Let si (i = 1, 2, . . . , n) be n number of positive crisp data. For any
u, v ≥ 0,
B u,v (s1 , s2 , . . . , sn ) =
1
u+v
n
X
1
sui svj
n(n − 1)
(4)
i,j=1,
i6=j
We call Eq.(4) as the Bonferroni mean (BM) operator. Especially, if v=0, Eq.(4) reduces to
the generalized mean operator given by
1
u+0
n
n
X
1 X
1
B u,0 (s1 , s2 , . . . , sn ) =
sui
s0j
n
(n − 1)
i=1
n
=
1X u
si
n
i=1
j=1,
i6=j
(5)
!1
u
If u = 1 and v = 0, the above equation produces the very known arithmetic mean (AM):
n
B 1,0 (s1 , s2 , . . . , sn ) =
1X u
si
n
(6)
i=1
With the usual notion of geometric mean and the BM , the geometric Bonferroni mean
operator is formulated.
Definition 4.2. Let u, v > 0, and si (i = 1, 2, . . . , n) be a collection of non negative crisp
numbers. If
GB u,v (s1 , s2 , . . . , sn ) =
n
Y
1
1
(usi + vsj ) n(n−1)
(u + v)
(7)
i,j=1,
i6=j
then GB u,v is called the geometric Bonferroni mean (GBM).
Obviously, the GBM statisfies the following properties:
(1) GB u,v (0, 0, . . . , 0) = 0
(2) GB u,v (s1 , s2 , . . . , sn ) = s if si = s, for all i = 1, 2, . . . , n.
(3) GB u,v (s1 , s2 , . . . , sn ) ≥ GB u,v (t1 , t2 , . . . , tn ) if si ≥ ti ∀i that is, GB u,v is monotonic.
(4) Min(si ) ≤ GB u,v ≤ Max(si ).
Furthermore, if v = 0, Eq.(7) generates the geometric mean:
GB
u,0
n
n
Y
1
1
1 Y
n(n−1)
=
(si ) n
(s1 , s2 , . . . , sn ) =
(usi )
u
i,j=1,
i6=j
(8)
i=1
D. Ajay, Said Broumi, J. Aldring ; An MCDM Method under Neutrosophic Cubic Fuzzy Sets
with Geometric Bonferroni Mean Operator
Neutrosophic Sets and Systems, Vol. 32, 2020
192
5. Neutrosophic Cubic Fuzzy Geometric Bonferroni Mean
Definition 5.1. Let S̃i = Ti− , Ti+ , Ii− , Ii+ , Fi− , Fi+ ; hTλi , Iλi , Fλi i
neutrosophic cubic fuzzy numbers (N CF N ). For any u, v > 0,
N CF GBM u,v S̃1 , S̃2 , ..., S̃n =
be a collection of
n
1
1 O ˜
˜
n(n−1)
⊕
vS
)
(uS
i
j
u+v
i,j=1,
i6=j
is called the neutrosophic cubic fuzzy geometric bonferroni mean operator.
Theorem 5.2. Let u, v > 0 and S̃i =
Ti− , Ti+ , Ii− , Ii+ , Fi− , Fi+ ; hTλi , Iλi , Fλi i be a col-
lection of neutrosophic cubic fuzzy numbers (N CF N ). Then the aggregated value is calculated
using the operator N CF GBM u,v
N CF GBM u,v S̃1 , S̃2 , ..., S̃n =
n
1
1 O ˜
˜
n(n−1)
(uS
⊕
vS
)
i
j
u+v
i,j=1,
i6=j
1
u+v
n
1
Y
n(n−1)
1 − (1 − Ti− )u (1 − Tj− )v
,
= 1 − 1 −
i,j=1
i6=j
n
1
Y
n(n−1)
1 − (1 − Ti+ )u (1 − Tj+ )v
1 − 1 −
i,j=1,
i6=j
n
1
Y
n(n−1)
1 − (Ii− )u (Ij− )v
1 −
i,j=1
i6=j
,
n
1
Y
n(n−1)
, 1 −
1 − (Ii+ )u (Ij+ )v
1
u+v
i,j=1
i6=j
1
u+v
n
1
Y
− u
− v n(n−1)
1
−
1
−
(F
)
(F
)
i
j
i,j=1
i6=j
*
1
u+v
1
u+v
n
1
Y
n(n−1)
1 − (Fi+ )u (Fj+ )v
, 1 −
i,j=1
i6=j
1
u+v
n
Y
1
1 − 1 −
(1 − (1 − Tλi )u (1 − Tλj )v ) n(n−1)
n
Y
1
(1 − (Iλi )u (Iλj )v ) n(n−1)
1
−
i,j=1
i6=j
1
u+v
n
Y
1
, 1 −
(1 − (Fλi )u (Fλj )v ) n(n−1)
i,j=1
i6=j
,
1
u+v
(9)
;
,
i,j=1
i6=j
1
u+v
+
.
D. Ajay, Said Broumi, J. Aldring ; An MCDM Method under Neutrosophic Cubic Fuzzy Sets
with Geometric Bonferroni Mean Operator
Neutrosophic Sets and Systems, Vol. 32, 2020
193
Proof. . Using the operational laws on N CF N described in section (3), we have
uS˜i =
1 − 1 − Ti−
u
, 1 − 1 − Ti+
h1 − (1 − Tλi )u , (Iλi )u , (Fλi )u i}
vS˜j =
nh
u − u
u − u
u
, Ii
, Ii+
, Fi
, Fi+
;
v i h v v i h v v i
v
;
, Fj− , Fj+
, Ij− , Ij+
1 − 1 − Tj− , 1 − 1 − Tj+
h1 − (1 − Tλj )v , (Iλj )v , (Fλj )v i}
i h
i
nh
uS˜i ⊕ vS˜j = 1 − (1 − Ti− )u (1 − Tj− )v , 1 − (1 − Ti+ )u (1 − Tj+ )v , (Ii− )u (Ij− )v , (Ii+ )u (Ij+ )v ,
h
i
o
(Fi− )u (Fj− )v , (Fi+ )u (Fj+ )v ; h1 − (1 − Tλi )u (1 − Tλj )v , (Iλi )u (Iλj )v , (Fλi )u (Fλj )v i .
Next, we have the following equation which has been derived by Xu and Yager [28].
n
O
uS˜i ⊕ vS˜j
i,j=1,
i6=j
1
n(n−1)
n
n
Y
1
1
Y
n(n−1)
n(n−1)
=
1 − (1 − Ti− )u (1 − Tj− )v
,
1 − (1 − Tj+ )u (1 − Tj+ )v
,
i,j=1,
i,j=1,
i6=j
i6=j
*
n
n
1
1
Y
Y
+ u + v n(n−1)
− u − v n(n−1)
1
−
,
1
−
1
−
(I
)
(I
)
)
(I
)
1
−
(I
j
i
j
i
,
i,j=1,
i6=j
i,j=1,
i6=j
(10)
n
n
1
1
Y
Y
− u
− v n(n−1)
+ u
+ v n(n−1)
1
−
(F
1
−
(F
)
(F
)
,
1
−
)
(F
)
1
−
i
j
i
j
;
i,j=1,
i6=j
n
Y
i,j=1,
i6=j
1
(1 − (1 − Tλi )u (1 − Tλj )v ) n(n−1) , 1 −
n
Y
v
u
)
) (Iλj
1 − (Iλi
i,j=1,
i6=j
i,j=1,
i6=j
1−
n
Y
i,j=1,
i6=j
u
v
1 − (Fλi
) (Fλj
)
1
n(n−1)
+
1
n(n−1)
,
.
Using NCF operational laws, Eq.(10) yields neutrosophic cubic fuzzy geometric bonferroni
mean operator N CF GBM u,v (S˜1 , S̃2 , · · · , S̃n ) given by Eq.(9). In addition, it satisfies the
D. Ajay, Said Broumi, J. Aldring ; An MCDM Method under Neutrosophic Cubic Fuzzy Sets
with Geometric Bonferroni Mean Operator
Neutrosophic Sets and Systems, Vol. 32, 2020
194
following conditions
n
1
Y
− u
− v n(n−1)
1 − (1 − Ti ) (1 − Tj )
1 − 1 −
1
u+v
,
i,j=1
i6=j
1
u+v
n
1
Y
n(n−1)
1 − (1 − Ti+ )u (1 − Tj+ )v
1 − 1 −
i,j=1,
i6=j
n
1
Y
n(n−1)
1 − (Ii− )u (Ij− )v
1 −
1
u+v
n
1
Y
n(n−1)
, 1 −
1 − (Ii+ )u (Ij+ )v
i,j=1
i6=j
⊆ [0, , 1],
1
u+v
i,j=1
i6=j
1
u+v
n
1
Y
− u
− v n(n−1)
1 − (Fi ) (Fj )
1 −
n
1
Y
+ u
+ v n(n−1)
1 − (Fi ) (Fj )
, 1 −
i,j=1
i6=j
i,j=1
i6=j
⊆ [0, 1],
1
u+v
⊆ [0, 1];
1
u+v
n
Y
1
(1 − (1 − Tλi )u (1 − Tλj )v ) n(n−1)
0 ≤ 1 − 1 −
≤ 1,
i,j=1
i6=j
0 ≤ 1 −
n
Y
i,j=1
i6=j
1
u+v
1
(1 − (Iλi )u (Iλj )v ) n(n−1)
n
Y
1
(1 − (Fλi )u (Fλj )v ) n(n−1)
0 ≤ 1 −
≤ 1,
1
u+v
≤1
i,j=1
i6=j
which completes the proof of the theorem.
We discuss some of the important properties of the N CF GBM u,v :
(1) Idempotency: Suppose the colletive data of neutrosophic cubic fuzzy numbers
S˜i = T − , T + , I − , I + , F − , F + ; hTλi , Iλi , Fλi i (i = 1, 2, 3, · · · n) are equal, for
i
i
i
i
i
i
D. Ajay, Said Broumi, J. Aldring ; An MCDM Method under Neutrosophic Cubic Fuzzy Sets
with Geometric Bonferroni Mean Operator
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any u, v > 0, the aggregate operator be
˜ S̃, ..., S̃)
N CF GBM u,v (S˜1 , S̃2 , ..., S̃n ) = N CF GBM u,v (S,
n
1
1 O ˜
n(n−1)
˜
uS ⊕ vS
=
u+v
i,j=1,
i6=j
(11)
n
1
1 O
n(n−1)
(u
+
v)
S̃
=
u+v
i,j=1,
i6=j
n(n−1)
1
n(n−1)
(u + v) S̃
= S̃
=
u+v
(2) Commuatativity: Let S̃i (i = 1, 2, 3, · · · n) be a collection of neutrosophic cubic numbers. For any u, v > 0,
N CF GBM u,v S̃1 , S̃2 , ..., S̃n = N CF GBM u,v Ṡ˜1 , Ṡ˜2 , . . . , Ṡ˜n
Let Ṡ˜1 , Ṡ˜2 , . . . , Ṡ˜n be any permuation of S̃1 , S̃2 , ..., S̃n . Then
N CF GBM
u,v
S̃1 , S̃2 , ..., S̃n
(12)
n
1
1 O ˜
˜j ) n(n−1)
(uS
⊕
vS
=
i
u+v
i,j=1,
i6=j
=
n
1
1 O ˜
˜ ) n(n−1)
(u
Ṡ
⊕
v
Ṡ
i
j
u+v
i,j=1,
i6=j
= N CF GBM u,v Ṡ˜1 , Ṡ˜2 , . . . , Ṡ˜n
(3) Monotonicity: Let S̃i (i = 1, 2, 3, · · · n) and S̃j (j = 1, 2, 3, · · · n) be two collections
of neutrosophic cubic numbers. For any u, v > 0, if [Ti− , Ti+ ], ⊆ [Tj− , Tj+ ], [Ii− , Ii+ ] ⊇
[Ij− , Ij+ ], Fi− , Fi+ ⊇ [Fj− , Fj+ ]; Tλi ≤ Tλj , Iλi ≥ Iλj , Fλi ≥ Fλj (∀i, j = 1, 2, 3, . . . n),
Then
(4) Boundedness:
(13)
N CF GBM u,v S̃i ≤ N CF GBM u,v S̃j
− + − + − +
Ti , Ti , Ii , Ii , Fi , Fi ; hTλi , Iλi , Fλi i (i =
Let S̃i =
1, 2, 3, · · · n) be a collection of neutrosophic cubic fuzzy numbers, and let
S˜i− = inf Ti− , Ti+ , sup Ii− , Ii+ , sup Fi− , Fi+ ; min (Tλi ) , max (Iλi ) , max (Fλi ) ,
S̃i+ = sup Ti− , Ti+ , inf Ii− , Ii+ , inf Fi− , Fi+ ; max (Tλi ) , min (Iλi ) , min (Fλi ) .
For any u, v > 0,
S̃i− ≤ N CF GBM u,v S̃i (i = 1, 2, 3, . . . n) ≤ S̃i+
(14)
D. Ajay, Said Broumi, J. Aldring ; An MCDM Method under Neutrosophic Cubic Fuzzy Sets
with Geometric Bonferroni Mean Operator
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196
Thus the boundedness is easily obtained.
If parameters u and v are modified in N CF GBM u,v , then a special case can be obtained as
follows:
If v → 0, then by equation (9), we have
n
1
1
1 O ˜
n(n−1)
˜
N CF GBM u,v S̃1 , S̃2 , ..., S̃n =
uS
⊕
vS
i
j
= u
u+v
= 1 −
i,j=1,
i6=j
1−
1−
1−
1−
1 − (1 − Ti− )
i=1
*
n
Y
1−
n
Y
n
Y
1 − (Ii− )
i=1
n
Y
n
1 − (Fi− )u
i=1
1
(1 − (1 − Tλi )u ) n
i=1
n
1
u
!1
u
1
u
!1
1
n
,1 −
, 1−
n
Y
1 − (Ii+ )
i=1
u
!1
1 − (1 − Ti+ )
i=1
u
!1
1−
n
Y
, 1−
n
Y
1
u
n
1 − (Fi+ )u
i=1
u
, 1−
n
Y
1
(1 − (Iλi )u ) n
i=1
!1
1
u
n
n
O
(uS˜i )
1
n
i=1
!
!1
u
,
u
,
1
n
!1
!1
u
u
;
, 1−
n
Y
! 1 +
u
1
(1 − (Fλi )u ) n
i=1
which we call the generalized neutrosophic cubic fuzzy geometric mean (N CF BGM u,v ).
5.1. Weighted Neutrosophic Cubic Fuzzy Bonferroni Geometric Mean
Generally weighted aggregating operator plays a significant role in decision-making processes to aggregate information. Therefore we propose a weighted aggregate operator based
on neutrosophic cubic fuzzy bonferroni geometric mean (W N CF GBMwu,v ).
Definition 5.3. Let S̃i =
Ti− , Ti+ , Ii− , Ii+ , Fi− , Fi+ ; hTλi , Iλi , Fλi i
be a collection of
neutrosophic cubic numbers (NCN), and w = (W1 , W2 , . . . , Wn )T the wieght vector of S̃i =
P
S˜1 , S̃2 , . . . , S̃n , where wi indicates the importance degree of S̃i such that wi > 0 and ni=1 wi =
1 (i = 1, 2, 3, . . . , n). For any u, v > 0,
W N CF GBMwu,v S̃1 , S̃2 , ..., S̃n =
1
u+v
n
1
O
˜i )wi ⊕ v(S˜j )wj n(n−1)
u(S
(15)
i,j=1,
i6=j
is called the weighted neutrosophic cubic fuzzy geometric bonferroni mean operator.
Theorem 5.4. Let u, v > 0 and S̃i (i = 1, 2, 3, . . . , n) be a collection of neutrosophic cubic
fuzzy numbers (NCFN), whose weight vector is wi = (W1 , W2 , . . . , Wn )T , which satisfies that
D. Ajay, Said Broumi, J. Aldring ; An MCDM Method under Neutrosophic Cubic Fuzzy Sets
with Geometric Bonferroni Mean Operator
.
Neutrosophic Sets and Systems, Vol. 32, 2020
wi > 0, and
197
Pn
i=1 wi
= 1 (i = 1, 2, 3, . . . , n). Then the aggregated value using the operator is
n
1
1 O ˜ wi
u,v ˜
wj n(n−1)
˜
W N CF GBMw (S1 , S̃2 , ..., S̃n ) =
u(Si ) ⊕ v(Sj )
u+v
i,j=1,
i6=j
1
u+v
n
1
Y
n(n−1)
1 − (1 − (Ti− )wi )u (1 − (Tj− )wj )v
,
= 1 − 1 −
i,j=1
i6=j
1
u+v
,
n
1
Y
n(n−1)
1 − (1 − (Ti+ )wi )u (1 − (Tj+ )wj )v
1 − 1 −
i,j=1,
i6=j
1
u+v
n
1
Y
− wj v n(n−1)
− wi u
1
−
)
)
(1
−
(1
−
I
)
1
−
(1
−
(1
−
I
)
j
i
,
i,j=1
i6=j
1
u+v
,
n
1
Y
+ wi u
+ wj v n(n−1)
1 − (1 − (1 − Ii ) ) (1 − (1 − Ij ) )
1 −
i,j=1
i6=j
1
u+v
n
1
Y
n(n−1)
1 − (1 − (1 − Fi− )wi )u (1 − (1 − Fj− )wj )v
1 −
(16)
,
i,j=1
i6=j
1
u+v
;
n
1
Y
+ wj v n(n−1)
+ wi u
)
)
(1
−
(1
−
F
)
1
−
(1
−
(1
−
F
)
1
−
j
i
i,j=1
i6=j
*
n
Y
1
(1 − (1 − (Tλi )wi )u (1 − (Tλj )wj )v ) n(n−1)
1 − 1 −
1
u+v
,
i,j=1
i6=j
n
Y
1
wj v n(n−1)
wi u
1
−
)
)
)
(1
−
(1
−
I
)
(1
−
(1
−
(1
−
I
)
λj
λi
1
u+v
,
i,j=1
i6=j
n
Y
1
(1 − (1 − (1 − Fλi )wi )u (1 − (1 − Fλj )wj )v ) n(n−1)
1
−
i,j=1
i6=j
1
u+v
+
.
D. Ajay, Said Broumi, J. Aldring ; An MCDM Method under Neutrosophic Cubic Fuzzy Sets
with Geometric Bonferroni Mean Operator
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Proof. The proof is identical with the proof of theorem (5.2) and therefore is omitted.
6. An application of weighted neutrosophic cubic fuzzy geometric bonferroni mean
operator to MCDM problems
In this section, we propose an algorithm for MCDM method based on neutrosophic cubic
fuzzy geometric Bonferroni mean operators and illustrate it with a numerical example.
Algorithm. Let Ãi = {γ̃1 , γ̃2 , . . . , γ̃n } and C̃j = {η̃1 , η̃2 , . . . , η̃m } be collections of n alternatives and m attributes respectively. According to the appropriate weight of attributes
P
(ω
bj )T = {ω̃1 , ω̃2 , . . . ω̃m } is determined, which satisfies the condition that ω̃j > 0 and ω
bj = 1.
Then the following steps are used in process of MCDM method.
Step 1. Construct neutrosophic cubic fuzzy decision matrix D = [Nij ]n×m .
Step 2. The decision matrix is aggregated using N CF GBM u,v or W N CF GBMwu,v to
m attributes.
Step 3. Utilize the score formula (Eq.1) to calculate the values of s(A˜i )
Step 4. The n alternatives are ranked according to their score values
6.1. Numerical Example and Investigation
An illustrative example on the selection problem of investment alternatives is adapted
(Ref. [25, 26]) to validate the proposed MCDM method with NCF data. A company wants a
sum of money to be invested in an industry. Then the committee suggests the following four
feasible alternatives: (a) γ̃1 is a textile company; (b) γ̃2 is an automobile company; (c) γ̃3 is a
computer company; (d) γ̃4 is a software company. Suppose that three attributes namely, (1) η̃1
is the risk; (2) η̃2 is the growth; (3) η̃3 is the environmental impact; are taken into the evaluation requirements of the alternatives. The weight vectors of the three attributes η̃j (j = 1, 2, 3)
are (b
ωj )T = (0.32, 0.38, 0.3) respectively. Then the experts or decision makers are asked to
evaluate each alternative on attributes by the form of NCFNs. Thus, the assessment data can
be represented by neutrosophic cubic decision matrix D = [Sij ]m×n .
step 1. Neutrosophic cubic fuzzy decision matrix D = [Sij ]4×3
D=
!
[0.5, 0.6],
[0.1, 0.3],
[0.2, 0.4];
h0.6, 0.2, 0.3i
[0.6, 0.8],
[0.1, 0.2],
[0.2, 0.3];
h0.7, 0.1, 0.2i
[0.4, 0.6],
[0.2, 0.3],
[0.1, 0.3];
h0.6, 0.2, 0.2i
!
!
[0.7, 0.8],
[0.1, 0.2],
[0.1, 0.2];
h0.8, 0.1, 0.2i
!
,
,
,
,
!
[0.5, 0.6],
[0.1, 0.3],
[0.2, 0.4];
h0.6, 0.2, 0.3i
[0.6, 0.7],
[0.1, 0.2],
[0.2, 0.3];
h0.6, 0.1, 0.2i
[0.5, 0.6],
[0.2, 0.3],
[0.3, 0.4];
h0.6, 0.3, 0.4i
!
!
[0.6, 0.7],
[0.1, 0.2],
[0.1, 0.3];
h0.7, 0.1, 0.2i
!
,
,
,
,
!
[0.2, 0.4],
[0.7, 0.8],
[0.8, 0.9];
h0.3, 0.8, 0.9i
[0.7, 0.8];
h0.3, 0.7, 0.8i
!
[0.3, 0.4],
[0.6, 0.7],
[0.8, 0.9]; h0.3, 0.6, 0.9i
!
[0.3, 0.5],
[0.7, 0.8],
[0.6, 0.7]; h0.4, 0.8, 0.7i
!
[0.3, 0.4],
[0.6, 0.7],
D. Ajay, Said Broumi, J. Aldring ; An MCDM Method under Neutrosophic Cubic Fuzzy Sets
with Geometric Bonferroni Mean Operator
Neutrosophic Sets and Systems, Vol. 32, 2020
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step 2. The decision matrix is aggregated by W N CF GBMwu,v (S˜i1 , S̃i2 , S̃i3 )(i = 1, 2, . . . n)
operators (Using Eq.16) to the three (η̃j , j = 1, 2, 3) attributes.
(1,1)
If we take the parameter values u = v = 1, then using Ãi = W N CF GBMw
, we get
the following values
A˜1 = {[0.7345, 0.8126] , [0.0881, 0.1861] , [0.1453, 0.2523] ; h0.7951, 0.1453, 0.2093i} ,
A˜2 = {[0.7951, 0.8635] , [0.0790, 0.1307] , [0.1453, 0.2093] ; h0.8124, 0.0790, 0.1642i} ,
A˜3 = {[0.7378, 0.8287] , [0.1307, 0.1861] , [0.1195, 0.1876] ; h0.8126, 0.1674, 0.1703i} ,
A˜4 = {[0.8124, 0.8635] , [0.0790, 0.1307] , [0.0881, 0.1674] ; h0.8491, 0.0881, 0.1453i} .
step 3. Utilizing Eq.(1), the score values s(A˜i ) are found
s(A˜1 ) = 0.8130, s(A˜2 ) = 0.8527, s(A˜3 ) = 0.8244, s(A˜4 ) = 0.8702.
step 4. Since the values s(A˜4 ) > s(A˜2 ) > s(A˜3 ) > s(A˜1 ), the rank of alternatives are in
the order of γ̃4 > γ̃2 > γ̃3 > γ̃1 .
From the results, we could see that the ranking order and the best choice of alternatives
are the same as the results in [25, 26].
(2,2)
If the parameters u = v = 2 , then using Ãi = W N CF GBMw
, we get the following
aggregate values
A˜1 = {[0.7306, 0.8111] , [0.0950, 0.1940] , [0.1542, 0.2619] ; h0.7916, 0.1542, 0.2204i} ,
A˜2 = {[0.7916, 0.8563] , [0.0847, 0.1376] , [0.1542, 0.2204] ; h0.8055, 0.0847, 0.1757i} ,
A˜3 = {[0.7371, 0.8283] , [0.1376, 0.1940] , [0.1354, 0.1945] ; h0.8111, 0.1797, 0.1841i} ,
A˜4 = {[0.8055, 0.8563] , [0.0847, 0.1376] , [0.0950, 0.1797] ; h0.8395, 0.0950, 0.1542i} .
Then we calculate the score of the alternatives s(A˜1 ) = 0.8059, s(A˜2 ) = 0.8451, s(A˜3 ) =
0.8165, s(A˜4 ) = 0.8621.
Since s(A˜4 ) > s(A˜2 ) > s(A˜3 ) > s(A˜1 ), the order of the rank is γ̃4 > γ̃2 > γ̃3 > γ̃1 .
As the values of parameters u and v change according to the subjective preference of the
decision maker, we can find that the ranking order of the alternatives are the same, which
indicates that the proposed method can obtain the most optimistic results than the existing
MCDM methods based on GBM [29]. For a detailed comparision, we represent the scores of
each alternatives in Fig.1 by changing the values of parameters u, v between 0 and 10.
D. Ajay, Said Broumi, J. Aldring ; An MCDM Method under Neutrosophic Cubic Fuzzy Sets
with Geometric Bonferroni Mean Operator
200
Neutrosophic Sets and Systems, Vol. 32, 2020
(a) Scores of alternative γ̃1
(b) Scores of alternative γ̃2
(c) Scores of alternative γ̃3
(d) Scores of alternative γ̃4
Figure 1. Scores of alternative γ̃i obtained by W N CF GBMwu,v
7. Conclusions
In this paper, we have applied geometric Bonferroni mean to neutrosophic cubic fuzzy
sets. A new aggregating operator N CF GBM u,v has been established and its properties are
discussed. The MCDM method is developed based on the weighted operator W N CF GBMwu,v
and is verified with a numerical example where four alternatives are ranked under three criteria.
The graphical representation of the results depicted above shows that the ranking of the
alternatives remains unaffected when the parameters are changed due to subjective preferences.
This proves that the method is objective and moreover the result obtained, when compared
with the results of existing techniques, shows that the proposed method is more effective
in dealing with neutrosophic fuzzy information. In future, N CF GBM u,v operator could be
applied to various other MCDM methods.
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[23] Khan, M., Gulistan, M., Yaqoob, N., Khan, M., Smarandache, F. (2019). Neutrosophic Cubic Einstein
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Received: Oct 08, 2019. Accepted: Mar 15, 2020
Received: Oct 21, 2019. Accepted: Mar 20, 2020
D. Ajay, Said Broumi, J. Aldring ; An MCDM Method under Neutrosophic Cubic Fuzzy Sets
with Geometric Bonferroni Mean Operator
Neutrosophic Sets and Systems, Vol. 32, 2020
University of New Mexico
Single valued neutrosophic mappings defined by single
valued neutrosophic relations with applications
Abdelkrim Latreche1 , Omar Barkat2 , Soheyb Milles2,∗ and Farhan Ismail3
1
Department of Technology, Faculty of Technology, University of Skikda, Algeria; a.latreche@univ-skikda.dz
2
Laboratory of Pure and Applied Mathematics, University of Msila, Algeria; omar.bark@gmail.com
3
Faculty of Technology, Sakarya University, Turkey; farhanismail@subu.edu.tr
∗
Correspondence: soheyb.milles@univ-msila.dz; Tel.: +213664081002
Abstract: In this paper, we introduce the notion of single valued neutrosophic mapping defined by single valued
neutrosophic relation which is considered as a generalization of fuzzy mapping defined by fuzzy relation and several
properties related to this notion are studied. Moreover, we generalize the notion of fuzzy topology on fuzzy sets
introduced by Kandil et al. to the setting of single valued neutrosophic sets. As applications, we establish the property
of continuity in single valued neutrosophic topological space and investigate relationships among various types of
single valued neutrosophic continuous mapping.
Keywords: Single valued neutrosophic set; Binary relation; Mapping; Topology; Continuous mapping.
1
Introduction
It is a well-known fact by now that mappings in crisp set theory are among the oldest acquaintances of modern
mathematics and, play an important role in many mathematical branches (both pure and applied), as well as
in topology and its analysis approaches. The uses of mappings appear also in formal logic [13], category
theory [35], graph theory [11], group theory [6] and in computer science [31]. In general, it was and still more
common.
In fuzzy setting, the concept of fuzzy mapping has received far attention. It has appeared in many papers,
for instance, S. Heilpern [12] introduced this concept and proved a fixed point theorem for fuzzy contraction
mappings. In [17], S. Lou and L. Cheng proved that fuzzy controllers can be regarded as a fuzzy mapping
from the set of linguistic variables describing the observed object to that of linguistic variables describing the
controlled objects. Thereafter, Lim et al. [18] investigated the equivalence relations and mappings for fuzzy
sets and relationship between them. Ismail and Massa’deh [9] defined L-fuzzy mappings and studied their
operations, also they developed many properties of classical mappings into L-fuzzy case. For the study of
fuzzy continuous mappings in fuzzy topological space, an extended approaches are proposed, R.N. Bhaumik
and M.N Mukherjee [5] investigated some properties of fuzzy completely continuous mapping. Mukherjee and
B. Ghosh [27] pay attention to the introduction and studying of the concepts of certain classes of mappings
between fuzzy topological spaces. Each of these mappings presents a stronger form of the fuzzy continuous
mappings. In this regard, we find that other authors also contributed a lot to this field, like M. K. Single and A.
R. Single [36], B. Ahmed [1] and M. K. Mishra et al. [26].
A. Latreche, O. Barkat, S. Milles, F. Ismail. Single valued neutrosophic mappings defined by single valued
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In [3], Attanassov introduced the concept of intuitionistic fuzzy set which is an extension of fuzzy set, characterized by a membership (truth-membership) function and a non-membership (falsity-membership) function
for the elements of a universe X. Moreover, there is a restriction that the sum of both values is less and equal
to one. Recently, F. Smarandache [32] generalized the Atanassov’s intuitionistic fuzzy sets and other types of
sets to the notion of neutrosophic sets. He introduced this concept to deal with imprecise and indeterminate
data. Neutrosophic sets are characterized by truth membership function (T ), indeterminacy membership function (I) and falsity membership function (F ). Many researchers have studied and applied in different fields the
neutrosophic sets and its various extensions such as decision making problems (e.g. [39, 41]), image processing (e.g. [8, 44]), educational problem (e.g. [25]), conflict resolution (e.g. [28]), social problems (e.g. [29, 24]),
medical diagnosis (e.g. [22, 40, 42]), supply chain management (e.g. [20]), construction projects (e.g. [21])
and to address the conditions of uncertainty and inconsistency (e.g. [23]) and others. In particular, to exercise
neutrosophic sets in real life applications suitably, Wang et al. [37] introduced the concept of single valued
neutrosophic set as a subclass of a neutrosophic set, and investigated some of its properties. Very recently,
Kim et al. [15] studied a single valued neutrosophic (relation/ transitive closure/ equivalence relation class/
partition). The studies, whether theoretical or applied on single valued neutrosophic set have been progressing
rapidly. For instance, [2, 7, 14] and more others.
Motivated by recent developments relating to this framework, in this paper, we introduce the notion of
single valued neutrosophic mapping defined by single valued neutrosophic relation as a generalization of fuzzy
mappings introduced by Ismail and Massa’deh [9] and many properties related to this notion are studied. Also,
we generalize the notion of fuzzy topology on fuzzy sets introduced by A. Kandil et al. [16] to the setting of
single valued neutrosophic sets to establish the continuity property of single valued neutrosophic mapping. To
that end, we investigate relation among various types of single valued neutrosophic continuous mappings.
The contents of the paper are organized as follows. In Section 2, we recall the necessary basic concepts and
properties of single valued neutrosophic sets, single valued neutrosophic relations and some related notions that
will be needed throughout this paper. In Section 3, the notion of single valued neutrosophic mapping defined
by single valued neutrosophic relation is introduced and some properties related to this notion are studied.
In Section 4, we establish as an application the single valued neutrosophic continuous mapping in single
valued neutrosophic topological space and relationships between various types of single valued neutrosophic
continuous mapping are explained. Finally, we present some conclusions and discuss future research in Section
5.
2
Preliminaries
This section contains the basic definitions and properties of single valued neutrosophic sets and some related
notions that will be needed throughout this paper.
2.1
Single valued neutrosophic sets
The notion of fuzzy sets was first introduced by Zadeh [43].
Definition 2.1. [43] Let X be a nonempty set. A fuzzy set A = {hx, µA (x)i | x ∈ X} is characterized by a
membership function µA : X → [0, 1], where µA (x) is interpreted as the degree of membership of the element
x in the fuzzy subset A for any x ∈ X.
A. Latreche, O. Barkat, S. Milles, F. Ismail. Single valued neutrosophic mappings defined by single valued
neutrosophic relations with applications
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205
In 1983, Atanassov [3] proposed a generalization of Zadeh membership degree and introduced the notion
of the intuitionistic fuzzy set.
Definition 2.2. [3] Let X be a nonempty set. An intuitionistic fuzzy set (IFS, for short) A on X is an object
of the form A = {hx, µA (x), νA (x)i | x ∈ X} characterized by a membership function µA : X → [0, 1] and a
non-membership function νA : X → [0, 1] which satisfy the condition:
0 ≤ µA (x) + νA (x) ≤ 1, for any x ∈ X.
In 1998, Smarandache [32] defined the concept of a neutrosophic set as a generalization of Atanassov’s
intuitionistic fuzzy set. Also, he introduced neutrosophic logic, neutrosophic set and its applications in [33, 34].
In particular, Wang et al. [37] introduced the notion of a single valued neutrosophic set.
Definition 2.3. [33] Let X be a nonempty set. A neutrosophic set (NS, for short) A on X is an object of the
form A = {hx, µA (x), σA (x), νA (x)i | x ∈ X} characterized by a membership function µA : X →]− 0, 1+ [
and an indeterminacy function σA : X →]− 0, 1+ [ and a non-membership function νA : X →]− 0, 1+ [ which
satisfy the condition:
−
0 ≤ µA (x) + σA (x) + νA (x) ≤ 3+ , for any x ∈ X.
Certainly, intuitionistic fuzzy sets are neutrosophic sets by setting σA (x) = 1 − µA (x) − νA (x).
Next, we show the notion of single valued neutrosophic set as an instance of neutrosophic set which can be
used in real scientific and engineering applications.
Definition 2.4. [37] Let X be a nonempty set. A single valued neutrosophic set (SVNS, for short) A on X is
an object of the form A = {hx, µA (x), σA (x), νA (x)i | x ∈ X} characterized by a truth-membership function
µA : X → [0, 1], an indeterminacy-membership function σA : X → [0, 1] and a falsity-membership function
νA : X → [0, 1].
The class of single valued neutrosophic sets on X is denoted by SV N (X).
For any two SVNSs A and B on a set X, several operations are defined (see, e.g., [37, 38]). Here we will
present only those which are related to the present paper.
(i) A ⊆ B if µA (x) ≤ µB (x) and σA (x) ≤ σB (x) and νA (x) ≥ νB (x), for all x ∈ X,
(ii) A = B if µA (x) = µB (x) and σA (x) = σB (x) and νA (x) = νB (x), for all x ∈ X,
(iii) A ∩ B = {hx, µA (x) ∧ µB (x), σA (x) ∧ σB (x), νA (x) ∨ νB (x)i | x ∈ X},
(iv) A ∪ B = {hx, µA (x) ∨ µB (x), σA (x) ∨ σB (x), νA (x) ∧ νB (x)i | x ∈ X},
(v) A = {hx, 1 − νA (x), 1 − σA (x), 1 − µA (x)i | x ∈ X},
(vi) [A] = {hx, µA (x), σA (x), 1 − µA (x)i | x ∈ X},
(vii) hAi = {hx, 1 − νA (x), σA (x), νA (x)i | x ∈ X}.
In the sequel, we need the following definition of level sets (which is also often called (α, β, γ)-cuts) of a
single valued neutrosophic set.
A. Latreche, O. Barkat, S. Milles, F. Ismail. Single valued neutrosophic mappings defined by single valued
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Definition 2.5. [2] Let A be a single valued neutrosophic set on a set X. The (α, β, γ)-cut of A is a crisp
subset
Aα,β,γ = {x ∈ X | µA (x) ≥ α and σA (x) ≥ β and νA (x) ≤ γ},
where α, β, γ ∈]0, 1].
Definition 2.6. [2] Let A be a single valued neutrosophic set on a set X. The support of A is the crisp subset
on X given by
Supp(A) = {x ∈ X | µA (x) 6= 0 and σA (x) 6= 0 and νA (x) 6= 0}.
2.2
Single valued neutrosophic relations
Kim et al. [15] introduced the concept of single valued neutrosophic relation as a natural generalization of
fuzzy and intuitionistic fuzzy relation.
Definition 2.7. [15] A single valued neutrosophic binary relation (A single valued neutrosophic relation, for
short) from a universe X to a universe Y is a single valued neutrosophic subset in X × Y , i.e., is an expression
R given by
R = {h(x, y), µR (x, y), σR (x, y), νR (x, y)i | (x, y) ∈ X × Y } ,
where µR : X × Y → [0, 1], and σR : X × Y → [0, 1], and νA : X × Y → [0, 1].
For any (x, y) ∈ X × Y . The value µR (x, y) is called the degree of a membership of (x, y) in R, σR (x, y) is
called the degree of indeterminacy of (x, y) in R and νR (x, y) is called the degree of non-membership of (x, y)
in R.
Example 2.8. Let X = {a, b, c, d, e}. Then the single valued neutrosophic relation R defined on X by
R = {h(x, y), µR (x, y), σR (x, y), νR (x, y)i | x, y ∈ X},
where µR , σR and νR are given by the following tables:
µR (., .)
a
b
c
d
e
a
0.35
0
0.20
0
0.25
b
c
d
0
0
0.35
0.40
0
0.35
0
0.65
0
0
0
1
0.35
0
0
σR (., .)
a
b
c
d
e
a
0.5
0.60
0
0.33
0.20
b
c
d
e
0.5 0.42 0.2
0
0.12 0.40 0.80 0.10
1
0.02 0.75 0.15
1
0.88
0
0.10
0.55
1
0.55 0.30
e
0.30
0.45
0.70
0
0.60
A. Latreche, O. Barkat, S. Milles, F. Ismail. Single valued neutrosophic mappings defined by single valued
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νR (., .)
a
b
c
d
e
a
b
c
d
0
1
0.40 0.25
0.30 0.35 0.20 0.35
0.80
1
0
0.85
1
1
1
0
0.70 0.55
1
0.90
207
e
0.25
0.10
0.15
1
0.30
Next, the following definitions is needed to recall.
Definition 2.9. [30] Let R and P be two single valued neutrosophic relations from a universe X to a universe
Y.
(i) The transpose (inverse) Rt of R is the single valued neutrosophic relation from the universe Y to the
universe X defined by
Rt = {h(x, y), µRt (x, y), σRt (x, y), νRt (x, y)i | (x, y) ∈ X × Y },
where
for any (x, y) ∈ X × Y.
µRt (x, y) = µR (y, x)
and
σRt (x, y) = σR (y, x)
and
νRt (x, y) = νR (y, x) ,
(ii) R is said to be contained in P or we say that P contains R, denoted by R ⊆ P , if for all (x, y) ∈ X × Y
it holds that µR (x, y) ≤ µP (x, y), σR (x, y) ≤ σP (x, y) and νR (x, y) ≥ νP (x, y).
(iii) The intersection (resp. the union) of two single valued neutrosophic relations R and P from a universe
X to a universe Y is a single valued neutrosophic relation defined as
R ∩ P = {h(x, y), min(µR (x, y), µP (x, y)), min(σR (x, y), σP (x, y)), max(νR (x, y), νP (x, y))i | (x, y)
∈X ×Y}
and
R ∪P = {h(x, y), max(µR (x, y), max(σR (x, y), σP (x, y)), min(νR (x, y), νP (x, y))i | (x, y) ∈ X ×Y } .
Definition 2.10. [30, 38] Let R be a single valued neutrosophic relation from a universe X into itself.
(i) Reflexivity: µR (x, x) = σR (x, x) = 1 and νR (x, x) = 0, for any x ∈ X.
(ii) Symmetry: for any x, y ∈ X then
µR (x, y) = µR (y, x)
σR (x, y) = σR (y, x) ,
νR (x, y) = νR (y, x)
A. Latreche, O. Barkat, S. Milles, F. Ismail. Single valued neutrosophic mappings defined by single valued
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(iii) Antisymmetry: for any x, y ∈ X, x 6= y then
µR (x, y) 6= µR (y, x)
σR (x, y) 6= σR (y, x) ,
νR (x, y) 6= νR (y, x)
(iv) Transitivity: R ◦ R ⊂ R i.e., R2 ⊂ R.
Single valued neutrosophic mappings defined by single valued neutrosophic relations
3
In this section, we generalize the notion of fuzzy mapping defined by fuzzy relation introduced by Ismail and
Massa’deh [9] to the setting of single valued neutrosophic sets. Also, the main properties related to single
valued neutrosophic mapping are studied.
Definition 3.1. Let A be a single valued neutrosophic set on X and B be a single valued neutrosophic set on
Y , let f : Supp A → Supp B be an ordinary mapping and R be a single valued neutrosophic relation on
X × Y . Then fR is called a single valued neutrosophic mapping if for all (x, y) ∈ Supp A × Supp B the
following condition is satisfied:
min(µA (x), µB (f (x)) , if y = f (x)
µR (x, y) =
0 , Otherwise ,
and
σR (x, y) =
min(σA (x), σB (f (x)) , if y = f (x)
0 , Otherwise ,
νR (x, y) =
max(νA (x), νB (f (x)) , if y = f (x)
1 , Otherwise ,
and
Example 3.2. Let X = {α, β, γ}, Y = {a, b, c}, A ∈ SV N S(X) and B ∈ SV N S(Y ) given by
A = {hα, 0.5, 0.2, 0.8i, hβ, 0.1, 0.7, 0.3i, hγ, 0, 0.9, 1i}
B = {ha, 0, 1, 0.3i, hb, 0.1, 0.5, 0.2i, hc, 0.7, 0.2, 0.4i}.
We will construct the single valued neutrosophic mapping fR by :
(i) an ordinary mapping f : {α, β} → {b, c} such that f (α) = b and f (β) = c,
(ii) a single valued neutrosophic relation R defined by :
µR (α, f (α)) = µR (α, b) = µA (α) ∧ µB (b) = 0.1
µR (β, f (β)) = µR (β, c) = µA (β) ∧ µB (c) = 0.1
µR (α, a) = µR (α, c) = µR (β, a) = µR (β, b) = µR (γ, a) = µR (γ, b) = µR (γ, c) = 0
In similar way, it holds that
A. Latreche, O. Barkat, S. Milles, F. Ismail. Single valued neutrosophic mappings defined by single valued
neutrosophic relations with applications
Neutrosophic Sets and Systems, Vol. 32, 202 0
209
σR (α, f (α)) = σR (α, b) = σA (α) ∧ σB (b) = 0.2
σR (β, f (β)) = σR (β, c) = σA (β) ∧ σB (c) = 0.2
σR (α, a) = σR (α, c) = σR (β, a) = σR (β, b) = σR (γ, a) = σR (γ, b) = σR (γ, c) = 0
and
νR (α, f (α)) = νR (α, b) = νA (α) ∨ νB (b) = 0.8
νR (β, f (β)) = νR (β, c) = νA (β) ∨ νB (c) = 0.4
νR (α, a) = νR (α, c) = νRI (β, a) = νRI (β, b) = σR (γ, a) = σR (γ, b) = σR (γ, c) = 1.
Hence, µR (x, y) = {h(α, f (α)), 0.1, 0.2, 0.8i, h(β, f (β)), 0.1, 0.2, 0.4i, h(α, a), 0, 0, 1i,
h(α, c), 0, 0, 1i, h(β, a), 0, 0, 1i, h(β, b), 0, 0, 1i, h(γ, a), 0, 0, 1i, h(γ, b), 0, 0, 1i, h(γ, c), 0, 0, 1i}.
Thus, fR is a single valued neutrosophic mapping.
Example 3.3. Let X = Q , Y = R , A ∈ SV N S(X) and B ∈ SV N S(Y ) given by:
µA (x) = 0.3 , σA (x) = 0.25 and νA (x) = 0.5 , for any x ∈ Q.
µB (x) = σB (x) = νB (x) = 0.5 , for any x ∈ R.
We will construct the single valued neutrosophic mapping fR by :
(i) an ordinary mapping f : Q → R such that f (x) = x2 ,
(ii) a single valued neutrosophic relation R defined by :
µR (x, f (x)) = µR (x, x2 ) = µA (x) ∧ µB (x2 ) = 0.3
σR (x, f (x)) = σR (x, x2 ) = σA (x) ∧ µB (x2 ) = 0.25
νR (x, f (x)) = νR (x, x2 ) = νA (x) ∨ νB (x2 ) = 0.5
Thus, fR is a single valued neutrosophic mapping.
Remark 3.4. From the above definition, we can construct the single valued neutrosophic mapping by this
method
(i) We determine the Supp A and Supp B.
(ii) We determine the ordinary mapping from Supp A to Supp B.
(iii) We determine the single valued neutrosophic relation by its membership function, indeterminacy function and non-membership function.
(iv) Finally, we conclude the construction of the single valued neutrosophic mapping.
Definition 3.5. Let fR , gS be two single valued neutrosophic mappings, then fR and gS are equal if and only
if f = g and R = S i.e., (µR (x, f (x)) = µS (x, g(x)), σR (x, f (x)) = σS (x, g(x)), and νR (x, f (x)) =
νS (x, g(x))).
Definition 3.6. Let A be a single valued neutrosophic set on X, let f : Supp A → Supp A be an ordinary
mapping such that f (x) = x and R be a single valued neutrosophic relation on X × X. Then fR is called a
single valued neutrosophic identity mapping if for all x, y ∈ Supp A the following conditions are satisfied:
A. Latreche, O. Barkat, S. Milles, F. Ismail. Single valued neutrosophic mappings defined by single valued
neutrosophic relations with applications
Neutrosophic Sets and Systems, Vol. 32, 202 0
210
µR (x, y) =
µA (x) , if x = y
0 , Otherwise ,
σR (x, y) =
σA (x) , if x = y
0 , Otherwise ,
νR (x, y) =
νA (x) , if x = y
1 , Otherwise ,
and
and
Definition 3.7. Let A, B and C are a single valued neutrosophic sets on X, Y and Z respectively, let f :
Supp A → Supp B and g : Supp B → Supp C are an ordinary mappings and R, S are a single valued
neutrosophic relations on X×Y and Y ×Z respectively. Then (g◦f )T is called the composition of single valued
neutrosophic mappings fR and gR such that g ◦ f : Supp A → Supp C and the single valued neutrosophic
relation T is defined by
µT (x, z) = supy (min(µR (x, y), µS (y, z)))
and
σT (x, z) = supy (min(σR (x, y), σS (y, z)))
and
νT (x, z) = infy (max(νR (x, y), νS (y, z))) ,
for any (x, z) ∈ Supp A × Supp C.
Example 3.8. Let X = N, Y = R and Z = R, and let A ∈ SV N S(X), B ∈ SV N S(Y ) and C ∈ SV N S(Z),
defined as follows :
n
1
and νA (n) = 2+2n
, for any n ∈ N.
1+n
0.25 , if x ∈ [−1, 1]
0.5 , if x ∈ [−1, 1]
and νB (x) =
µB (x) = σB (x) =
0 , Otherwise ,
1 , Otherwise ,
|cos(x)|
|sin(x)|
µC (x) = σC (x) = 3 and νC (x) = 3 , for any x ∈ R.
We define a single valued neutrosophic mappings fR : A → B and gS : B → C by :
µA (n) = σA (n) =
(i) an ordinary mappings f : Supp A −→ Supp B, defined for any n ∈ Supp A by :
1 , if n is an even number,
f (n) =
−1 , if n is an odd number ,
and g : Supp B −→ Supp C defined by g(x) = 2x, for any x ∈ [−1, 1].
(ii) a single valued neutrosophic relations R and S defined by :
1
µR (n, f (n)) = σR (n, f (n)) = ∧{µA (n), µB (f (n))} = ∧{ 1+n
, 0.25},
n
, 0.5} and
νR (n, f (n)) = ∨{νA (n), νB (f (n))} = ∨{ 2+2n
∧{0.25, |cos(2x)|
} , x ∈ [−1, 1],
3
µS (x, g(x)) = σS (x, g(x)) = ∧{µB (x), µC (g(x))} =
0 , otherwise ,
∨{0.5, |sin(2x)|
} , x ∈ [−1, 1],
3
and νS (x, g(x)) = ∨{νB (x), νC (g(x))} =
1 , otherwise.
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Then, the composition gS ◦ fR = (g ◦ f )T is defined by :
(i) an ordinary mapping f : Supp A −→ Supp C, defined for any n ∈ Supp A by :
2 , if n is an even number,
(g ◦ f )(n) =
−2 , if n is an odd number ,
(ii) a single valued neutrosophic relation T defined by :
(
1
, 0.25,
∧{ 1+n
µT (n, (g ◦ f )(n)) = σT (n, (g ◦ f )(n)) =
1
∧{ 1+n , 0.25,
|cos(2)|
},
3
|cos(−2)|
}
3
if n is an even number
, if n is an odd number
| cos(2) |
1
}
, 0.25,
3
1+n
1
, 0.25},
= ∧{
1+n
= ∧{
νT (n, (g ◦ f )(n)) =
(
n
, 0.25,
∨{ 2+2n
n
∨{ 2+2n , 0.25,
|sin(2)|
},
3
|sin(−2)|
}
3
if n is an even number
, if n is an odd number
| sin(2) |
n
}
, 0.25,
3
2 + 2n
2
= ∨{
, 0.25}.
2 + 2n
= ∨{
Remark 3.9. The single valued neutrosophic identity mapping IdR is neutral for the composition of single
valued neutrosophic mappings.
In the sequel, we need to introduce the notion of the direct image and the inverse image of a single valued
neutrosophic set by a single valued neutrosophic mapping.
Definition 3.10. Let fR : A → B be a single valued neutrosophic mapping from a single valued neutrosophic
set A to another single valued neutrosophic set B and C ⊆ A. The direct image of C by fR is defined by
fR (C) = {hy, µfR (C) (y), σfR (C) (y), νfR (C) (y)i | y ∈ Y }, where
µB (y) , if y ∈ f (supp(C))
µfR (C) (y) =
0 , Otherwise ,
and
σfR (C) (y) =
σB (y) , if y ∈ f (supp(C))
0 , Otherwise ,
νfR (C) (y) =
νB (y) , if y ∈ f (supp(C))
1 , Otherwise.
and
Similarly, if C ′ ⊆ B. The inverse image of C ′ by f is defined by
fR−1 (C ′ ) = {hx, µf −1 (C ′ ) (x), σf −1 (C ′ ) (x), νf −1 (C ′ ) (x)i | x ∈ X},
R
R
R
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where
µf −1 (C ′ ) (x) =
µA (x) , if x ∈ f −1 (supp(C ′ ))
0 , Otherwise ,
σf −1 (C ′ ) (x) =
σA (x) , if x ∈ f −1 (supp(C ′ ))
0 , Otherwise ,
νf −1 (C ′ ) (x) =
νA (x) , if x ∈ f −1 (supp(C ′ ))
1 , Otherwise.
R
and
R
and
R
Example 3.11. Let X
= [0, +∞[, Y = R and A ∈ SV N S(X)defined for any x ∈ X by :
cos(x) , if x ∈ [0, π2 ]
0.9 , if x ∈ [0, π2 ]
νA (x) =
µA (x) = σA (x) =
0 , Otherwise ,
1 , Otherwise.
Also, let B ∈ SV NS(Y ) given by :
y , if y ∈ [0, 1]
0.2 , if y ∈ [0, 1]
νB (y) =
µB (y) = σB (y) =
0 , Otherwise ,
1 , Otherwise.
We define the single valued neutrosophic mapping fR : A → B by:
(i) an ordinary mapping f : Supp A −→ Supp B, defined for any x ∈ [0, π2 ] by
f (x) = x4 .
(ii) a single valued neutrosophic relation R defined by µR (x, f (x)) = σR (x, f (x)) = µA (x) ∧ µB (f (x)) =
cos(x) ∧ 14 x and νR (x, f (x)) = νA (x) ∨ νB (f (x)) = 0.9
Now, if we take C an
⊆ A given by :
SVNS on X, where C
1
−x + 1 , if x ∈ [0, 2 ]
0.99 , if y ∈ [0, 12 ]
µC (x) = σC (x) =
νC (x) =
0 , Otherwise ,
1 , Otherwise ,
Then, thedirect image of C by fR is defined by :
µB (y) , if y ∈ f (supp(C))
y , if y ∈ [0, 18 ]
µfR (C) (y) =
=
0 , Otherwise ,
,
0 , Otherwise
1
σB (y) , if y ∈ f (supp(C))
y , if y ∈ [0, 8 ]
σfR (C) (y) =
=
0 , Otherwise ,
0 , Otherwise ,
and
0.2 , if y ∈ [0, 18 ]
νB (y) , if y ∈ f (supp(C))
νfR (C) (y) =
=
1 , Otherwise.
0 , Otherwise ,
Moreover, it is easy to show that fR (C) ⊆ B.
Next, if we take
by :
C ′ an SVNS on Y , where C ′ ⊆ B given
1
sin(y) , if y ∈ [0, 3 ]
0.4 , if y ∈ [0, 13 ]
µC ′ (y) = σC ′ (y) =
νC ′ (y) =
0 , Otherwise ,
1 , Otherwise ,
′
is defined by :
Then, the inverse
image of C by f −1
cos(x) , if x ∈ [0, 43 ]
µA (x) , if x ∈ f (supp(C ′ ))
µf −1 (C ′ ) (x) =
=
R
0 , Otherwise ,
0 , Otherwise ,
−1
′
cos(x) , if x ∈ [0, 43 ]
σA (x) , if x ∈ f (supp(C ))
σf −1 (C ′ ) (x) =
=
R
0 , Otherwise ,
,
0 , Otherwise
4
−1
′
0.9 , if x ∈ [0, 3 ]
νA (x) , if x ∈ f (supp(C ))
and νf −1 (C ′ ) (x) =
=
R
1 , Otherwise.
1 , Otherwise ,
−1
′
Moreover, it is easy to show that fR (C ) ⊆ A.
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Now, we introduce the product of single valued neutrosophic sets and single valued neutrosophic projection
mappings.
Definition 3.12. Let A be a single valued neutrosophic set on X and B be a single valued neutrosophic set on
Y . The product of A and B, denoted by A × B is a single valued neutrosophic set on X × Y defined by :
µX×Y (x, y) = min{µA (x), µB (y)}, σX×Y (x, y) = min{σA (x), σB (y)}, νX×Y (x, y) = max{νA (x), νB (y)}.
Also, we introduce the first single valued neutrosophic projection mapping (P1 )R : A × B −→ A by:
(i) an ordinary mapping P1 : Supp(A×B) −→ Supp(A) such that P1 (x, y) = x for any (x, y) ∈ Supp(A×
B),
(i) a single valued neutrosophic relation R defined by :
µR ((x, y), P1 (x, y)) = min{µA×B (x, y), µA (P1 (x, y))}}
= min{µA (x), µB (y), µA (x)}}
= min{µA (x), µB (y)}
and
σR ((x, y), P1 (x, y)) = min{σA×B (x, y), σA (P1 (x, y))}}
= min{σA (x), σB (y), σA (x)}}
= min{σA (x), σB (y)}
and
νR ((x, y), P1 (x, y)) = max{νA×B (x, y), νA (P1 (x, y))}}
= max{νA (x), νB (y), νA (x)}}
= max{νA (x), νB (y)}
The second single valued neutrosophic projection mapping is defined analogously.
Continuity property in single valued neutrosophic topological space
4
The aim of the present section, is to introduce and study the notion of single valued neutrosophic continuous
mapping in single valued neutrosophic topological spaces. The basic properties, and relationships with some
types of continuity are also obtained.
4.1
Single valued neutrosophic topology
In this subsection, we generalize the notion of fuzzy topology on fuzzy sets introduced by Kandil et al. [16] to
the setting of single valued neutrosophic sets to establish the continuity property of single valued neutrosophic
mapping.
Definition 4.1. Let A be a single valued neutrosophic set on the set X and OA = {U is an SVNS on X :
U ⊆ A}. We define a single valued neutrosophic topology on single valued neutrosophic set A by the family
T ⊆ OA which satisfies the following conditions :
(i) A, 0∼ ∈ T ;
(ii) if U1 , U2 ∈ T , then U1 ∩ U2 ∈ T ;
A. Latreche, O. Barkat, S. Milles, F. Ismail. Single valued neutrosophic mappings defined by single valued
neutrosophic relations with applications
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(iii) if Ui ∈ T for all i ∈ I, then ∪I Ui ∈ T .
T is called a single valued neutrosophic topology of A and the pair (A, T ) is a single valued neutrosophic
topological space (SVN-TOP, for short). Every element of T is called a single valued neutrosophic open set
(SVNOS, for short).
Example
4.2. Let X = P(R2 ) and α ∈]0, 1[,
A be a single valued neutrosophicset on X given by :
0 , if θ = ∅
1 , if θ = ∅
1 , if θ = ∅
α
α , 0 < |θ| < ∞ ,
1
−
α
, 0 < |θ| < ∞ ,
,
0
<
|θ|
<
∞
,
(θ)
=
µA (θ) =
σA (θ) =
ν
A
2
0 , Otherwise ,
0.5 , Otherwise.
0 , Otherwise ,
Then, the family T = {A, 0∼ , U } where:
α
α
, |θ| < ∞ ,
, |θ| < ∞ ,
3
4
σU (θ) =
µU (θ) =
0 , Otherwise ,
0 , Otherwise ,
νU (θ) =
is a single valued neutrosophic topology on A.
1 , |θ| < ∞ ,
0.8 , Otherwise ,
Inspired by the notion of interior (resp. closure) on intuitionistic fuzzy topological space on a set introduced by Atanassov [4], we generalize these notions in single valued neutrosophic topology on a single valued
neutrosophic set.
Definition 4.3. Let (A, T ) be a single valued neutrosophic topological space, for every single valued neutrosophic subset G of X we define the interior and closure of G by:
int(G) = {hx, max µU (x), max σU (x), min νU (x)i | x ∈ U ⊆ G} and
x∈X
x∈X
x∈X
cl(G) = {hx, min µK (x), min σK (x), max νK (x)i | x ∈ A and G ⊆ K}
x∈X
x∈X
x∈X
Example 4.4. Let X = {a, b, c} and A, B, C, D ∈ SV N S(X) such that
A = {< a, 0.5, 0.7, 0.1 >, < b, 0.7, 0.9, 0.2 >, < c, 0.6, 0.8, 0 >}
B = {< a, 0.5, 0.6, 0.2 >, < b, 0.5, 0.6, 0.4 >, < c, 0.4, 0.5, 0.4 >}
C = {< a, 0.4, 0.5, 0.5 >, < b, 0.6, 0.7, 0.3 >, < c, 0.2, 0.3, 0.3 >}
D = {< a, 0.5, 0.6, 0.2 >, < b, 0.6, 0.7, 0.3 >, < c, 0.4, 0.5, 0.3 >}
E = {< a, 0.4, 0.5, 0.5 >, < b, 0.5, 0.6, 0.4 >, < c, 0.2, 0.3, 0.4 >}
Then the family T = {A, 0∼ , B, C, D, E} is an SVN-TOP of A.
Now, we suppose that G ∈ SV N S(X) given by G = {< a, 0.41, 0.5, 0.6), < b, 0.3, 0.2, 0.6 >, <
c, 0.2, 0.3, 0.7 >}. Then, int(G) = 0∼ and cl(G) = E ∩ 1∼ = E.
Definition 4.5. Let (A, T ) be a single valued neutrosophic topological space and U ∈ SV N S(A, T ). Then U
is called :
1. a single valued neutrosophic semiopen set (SVNSOS) if U ⊆ cl(int(U ));
2. a single valued neutrosophic α-open set (SVNαOS) if U ⊆ int(cl(int(U )));
3. a single valued neutrosophic preopen set (SVNPOS) if U ⊆ int(cl(U ));
4. a single valued neutrosophic regular open set (SVNROS) if U = int(cl(U )).
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4.2
215
Single valued neutrosophic continuous mappings
In this subsection, we will study some interesting properties of single valued neutrosophic continuous mappings in single valued neutrosophic topological space and relations between various types of single valued
neutrosophic continuous mapping. First, we introduce the notion of single valued neutrosophic continuous
mapping.
Definition 4.6. Let (A, T ) (B, L) be two single valued neutrosophic topological spaces. The mapping fR :
(A, T ) → (B, L) is a single valued neutrosophic continuous if and only if the inverse of each L-open single
valued neutrosophic set is T -open single valued neutrosophic set.
Example 4.7. Let (A, T ) and (B, T ′ ) be two single valued neutrosophic topological spaces, where
µA (x) = 0.8,σA (x) = 0.88 and νA (x) = 0.1, for
any x ∈ R+ and
0.5 , if y ≥ 0
0.88 , if y ≥ 0
0.1 , if y ≥ 0
σB (y) =
νB (y) =
µB (y) =
0.8 , Otherwise ,
0 , Otherwise ,
0.3 , Otherwise ,
We suppose that T = {A, 0∼ , U1 }, where
√
√
√
0.88 , if x ∈ [0, 2]
0.8 , if x ∈ [0, 2]
0.1 , if x ∈ [0, 2]
σU1 (x) =
νU1 (x) =
µU1 (x) =
0 , Otherwise ,
0 , Otherwise ,
1 , Otherwise ,
′
′
Also, we suppose
that T = {B, 0∼ , U1 }, where
0.5 , if y ∈ [0, 2]
0.8 , if y ∈ [0, 2]
0.2 , if y ∈ [0, 2]
σU1′ (y) =
νU1′ (y) =
µU1′ (y) =
0 , Otherwise ,
0 , Otherwise ,
0.4 , Otherwise.
Then, the single valued neutrosophic mapping fR : A → B define by :
(i) an ordinary mapping f : R+ −→ R+ such that f (x) = x2 , for any x ∈ R+ ,
(ii) a single valued neutrosophic relation R defined by :
µR (x, f (x)) = 0.5 σR (x, f (x)) = 0.88 and νR (x, f (x)) = 0.1.
is a single valued neutrosophic continuous mapping. Indeed, it is easy to show that fR−1 (B) = A and
fR−1 (0∼ ) = 0∼ and we have,
µA (x) , if x ∈ f −1 (supp(U1′ ))
µf −1 (U1′ ) (x) =
R
0 , Otherwise ,
√
0.8 , if x ∈ [0, 2]
=
0 , Otherwise ,
= µU1 (x),
σA (x) , if x ∈ f −1 (supp(U1′ ))
0 , Otherwise ,
√
0.88 , if x ∈ [0, 2]
=
0 , Otherwise ,
= σU1 (x),
σf −1 (U1′ ) (x) =
R
A. Latreche, O. Barkat, S. Milles, F. Ismail. Single valued neutrosophic mappings defined by single valued
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and
νA (x) , if x ∈ f −1 (supp(U1′ ))
1 , Otherwise ,
√
νA (x) , if x ∈ [0, 2]
=
1 , Otherwise ,
√
0.1 , if x ∈ [0, 2]
=
1 , Otherwise ,
= νU1 (x).
νf −1 (U1′ ) (x) =
R
Hence, fR−1 (U1′ ) = U1 ∈ T . Thus, fR is a single valued neutrosophic continuous mapping.
Remark 4.8. Let (A, T ) be a single valued neutrosophic topological space. Then the single valued neutrosophic identity mapping IdR : (A, T ) → (A, T ) is a single valued neutrosophic continuous mapping.
Next, we provide the relationships between various types of single valued neutrosophic continuous map¨
ping. First, we generalize the notions of precontinuous mapping, α-continuous mapping introduced by Guray
et al. [10] to the setting of single valued neutrosophic sets.
Definition 4.9. Let fR : (A, T ) → (B, T ′ ) be a single valued neutrosophic mapping. Then fR is called :
1. a single valued neutrosophic precontinuous mapping if fR−1 (U ′ ) is a SVNPOS on A for every SVNOS
U ′ on B;
2. a single valued neutrosophic α-continuous mapping if fR−1 (U ′ ) is a SVNαOS on A for every SVNOS U ′
on B.
The following proposition shows the relationship between single valued neutrosophic continuous mapping
and single valued neutrosophic α-continuous mapping.
Proposition 4.10. Let fR : (A, T ) → (B, T ′ ) be a single valued neutrosophic mapping. If fR is a single
valued neutrosophic continuous mapping, then fR is a single valued neutrosophic α-continuous mapping.
Proof. Let U ′ be a SVNOS in B and we need to show that fR−1 (U ′ ) is an SVNαOS in A. The fact that fR is a
single valued neutrosophic continuous mapping implies that fR−1 (U ′ ) is a SVNOS in A. From Definition 3.10,
it follows that
µA (x) , if x ∈ f −1 (supp(U ′ ))
σA (x) , if x ∈ f −1 (supp(U ′ ))
µf −1 (U ′ ) (x) =
σf −1 (U ′ ) (x) =
R
R
0 , Otherwise ,
0 , Otherwise ,
−1
′
νA (x) , if x ∈ f (supp(U ))
and νf −1 (U ′ ) (x) =
R
1 , Otherwise.
−1
′
We conclude that, fR (U ) is a SVNαOS in A. Hence, fR is a single valued neutrosophic α-continuous
mapping.
Remark 4.11. The converse of the above implication is not necessarily holds. Indeed, let us consider the single
valued neutrosophic mapping fR given in Example 4.7 and T be a SVN-topology given by T = {0∼ , A, U1 },
where: µA (x)
= 1, σA (x) = 0.99, νA (x) = 0.001and
1 , if x ∈ [0, 1]
0.99 , if x ∈ [0, 1]
0.001 , if x ∈ [0, 1]
σU1 (x) =
νU1 (x) =
µU1 (x) =
0 , Otherwise,
0 , Otherwise,
1 , Otherwise.
A. Latreche, O. Barkat, S. Milles, F. Ismail. Single valued neutrosophic mappings defined by single valued
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Hence, int(fR−1 (U1′ )) = U1 , cl(U1 ) = 1∼ and int(1∼ ) = A. Thus, fR−1 (U1′ ) ⊆ int(cl(int(fR−1 (U1′ ))).
We conclude that fR−1 (U1′ ) is an SVNαS but not SVNOS and fR is a single valued neutrosophic α-continuous
mapping but not a single valued neutrosophic continuous mapping.
The following proposition shows the relationship between single valued neutrosophic α-continuous mapping and single valued neutrosophic pre-continuous mapping.
Proposition 4.12. Let fR : (A, T ) → (B, T ′ ) be a single valued neutrosophic mapping. If fR is a single
valued neutrosophic α-continuous mapping, then fR is a single valued neutrosophic pre-continuous mapping.
Proof. Let U ′ be an SVNOS in B and we need to show that fR−1 (U ′ ) is a SVNPOS in A. The fact that
fR is a single valued neutrosophic α-continuous mapping implies that fR−1 (U ′ ) is a SVNαOS in A. From
Definition 3.10, it
follows that
µA (x) , if x ∈ f −1 (supp(U ′ ))
σA (x) , if x ∈ f −1 (supp(U ′ ))
µf −1 (U ′ ) (x) =
σf −1 (U ′ ) (x) =
R
R
0 , Otherwise ,
0 , Otherwise ,
−1
′
νA (x) , if x ∈ f (supp(U ))
and νf −1 (U ′ ) (x) =
R
1 , Otherwise.
−1
′
We conclude that, fR (U ) is an SVNPOS in A. Hence, fR is a single valued neutrosophic pre-continuous
mapping.
Remark 4.13. The converse of the above implication is not necessarily holds. Indeed, let (A, T ) and (B, T ′ )
be two single valued neutrosophic topological spaces, where µA (x) = 1, σA (x) = 1 and νA (x) = 0.005, for
any x ∈ R+ and
0.7 , if y ≥ 0
0.9 , if y ≥ 0
0.01 , if y ≥ 0
σB (y) =
νB (y) =
µB (y) =
0 , Otherwise ,
0.8 , Otherwise ,
0.03 , Otherwise ,
We suppose that T = {A, 0∼ , U1 }, where
µU1 (x) = 0 σU1 (x) = 1 and νU1 (x) = 1.
Also, we suppose
that T ′ = {B, 0∼ , U1′ }, where
0.7 , if y ∈ [0, 4]
0.5 , if y ∈ [0, 4]
0.12 , if y ∈ [0, 4]
σU1′ (y) =
νU1′ (y) =
µU1′ (y) =
0 , Otherwise ,
0 , Otherwise ,
0.32 , Otherwise.
Then, the single valued neutrosophic mapping fR : A → B define by :
√
(i) an ordinary mapping f : R+ −→ R+ such that f (x) = x , for any x ∈ R+ ,
(ii) a single valued neutrosophic relation R defined by :
µR (x, f (x)) = 0.7 σR (x, f (x)) = 0.9 and νR (x, f (x)) = 0.01.
µA (x) , if x ∈ f −1 (supp(U1′ ))
0 , Otherwise ,
=
1 , if x ∈ [0, 16]
0 , Otherwise ,
σf −1 (U1′ ) (x) =
σA (x) , if x ∈ f −1 (supp(U1′ ))
0 , Otherwise ,
µf −1 (U1′ ) (x) =
R
R
=
1 , if x ∈ [0, 16]
0 , Otherwise ,
A. Latreche, O. Barkat, S. Milles, F. Ismail. Single valued neutrosophic mappings defined by single valued
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νf −1 (U1′ ) (x) =
R
=
=
νA (x) , if x ∈ f −1 (supp(U1′ ))
1 , Otherwise ,
0.01 , if x ∈ [0, 16]
1 , Otherwise ,
νA (x) , if x ∈ [0, 16]
1 , Otherwise ,
Hence, cl(fR−1 (U1′ )) = 0∼ = 1∼ and int(1∼ ) = A. Thus, fR−1 (U1′ ) ⊆ int(cl(fR−1 (U1′ ))). We conclude
that fR−1 (U1′ ) is an SVNPOS and fR is a single valued neutrosophic pre-continuous but not a single valued
neutrosophic continuous.
5
Conclusion
In this work, we have generalized the notion of fuzzy mapping defined by fuzzy relation introduced by Ismail
and Massa’deh to the setting of single valued neutrosophic sets. Also, the main properties related to the single
valued neutrosophic mapping have been studied. Next, as an application we have established the single valued
neutrosophic continuous mapping in the single valued neutrosophic topological spaces. Future work will be
directed to study the notion of the single valued neutrosophic mapping for other types of topologies based on
the single valued neutrosophic sets.
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neutrosophic relations with applications
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Received: Oct 07, 2019. Accepted: Mar 22, 2020
A. Latreche, O. Barkat, S. Milles, F. Ismail. Single valued neutrosophic mappings defined by single valued
neutrosophic relations with applications
Neutrosophic Sets and Systems, Vol. 32, 2020
University of New Mexico
A Study on Bipolar Single-Valued Neutrosophic Graphs With Novel Application
M. Aslam Malik1 , Hossein Rashmanlou2,∗ , Muhammad Shoaib1 , R. A. Borzooei4 and Morteza Taheri2 and
Said Broumi
1
5
Department of Mathematics, University of the Punjab, New Campus, Lahore, Pakistan;
aslam.math@pu.edu.pk; muhammadshoaibe14@gmail.com
2
Department of Mathematics, University of the Mazandaran, Babolsar, Iran; M.taheri@stu.umz.ac.ir
4
5
Department of Mathematics, Shahid Beheshti University of the Tehran, Iran; Borzoeei@sbu.ac.ir
Laboratory of Information Processing, Faculty of Science Ben MSik, University Hassan II, B.P 7955, Sidi
Othman, Casablanca, Morocco; broumisaid78@gmail.com
∗
Correspondence: h.rashmanlou@stu.umz.ac.ir; Tel.: (989384012972)
Abstract. Unipolar is less fundamental than bipolar cognition based on truth, and composure is a restraint
for truth-based worlds. Bipolarity is the most powerful phenomenon that survives when truth disappeared in
a black hole due to Hawking radiation or particular / anti-particular emission. The purpose of this research
study is to define few four operations, including residue product, rejection, maximal product and symmetric
difference of bipolar single-valued neutrosophic graph (BSVNG) and to explore some of their related properties
with examples. Bipolar single-valued neutrosophic graph (BSVNG) is the generalization of the single-valued
neutrosophic graph (SVNG), intuitionistic fuzzy graph, bipolar intuitionistic fuzzy graph, bipolar fuzzy graph
and fuzzy graph. BSVNG plays a significant role in the study of neural networks, daily energy issues, energy
systems, and coding. Moreover, we will determine related properties like the degree of a vertex in a BSVNG or
total degree of a vertex in a BSVNG. We provide examples of the vertex degree in BSVNG and the total vertex
degree in BSVNG. In order to make this useful, we develop an algorithm for our useful method in steps.
Keywords: keyword 1; symmetric difference, residue product, maximal product, rejection of BSVNG, Application, algorithm.
—————————————————————————————————————————-
1. Introduction
In 1965, Zadeh [36] put forward the idea of the one-degree fuzzy set concept that determined the true membership function. Since Zadeh’s pioneering work, the fuzzy set theory has
been used in various disciplines such as management sciences, engineering, mathematics, social
sciences, statistics, signal processing, artificial intelligence, automata theory, medical and life
sciences. In the 20th century, Smarandache [31] includes the concept where uncertainty occurs
M. Aslam Malik, Hossein Rashmanlou, Muhammad Shoaib, R. A. Borzooei and Morteza Taheri, A Study
on Bipolar Single-Valued Neutrosophic Graphs With Novel Application
Neutrosophic Sets and Systems, Vol. 32, 2020
222
in the form of Neutrosophic set and extend the intuitionistic fuzzy set. There is also a nonmembership degree that Atanassove [1] defines in an intuitionistic fuzzy set with two degrees
in a set. Abdel-Basset et al. [2–6] studied many concepts on neutrosophic sets. Broumi et
al. [7,9–13,28,29] investigated the extension of the fuzzy graph in the form of the single-valued
neutrosophic graphs, shortest path problem using bellman algorithm under neutrosophic environment, shortest path problem in fuzzy, intuitionistic fuzzy and neutrosophic environment,
single valued neutrosophic coloring, and operations of single valued neutrosophic coloring.
A bipolar fuzzy theory has more scope when we compare to simply a fuzzy theory as compatibility and flexibility. Overall its model is better than the fuzzy model. Borzooei and
Rashmanlou [8, 25–27] studied very well on vague graphs and bipolar fuzzy graph. Rashmanlou studied about interval-valued fuzzy graph [22–24]. The neutrosophic set has much scope
in neutrosophy and the neutrosophy theory is widely used in graph theory. In this extension,
Wang et el. [35] described subclass of a Neutrosophic set known as a single-valued neutrosophic
set. In the fields of bio and physics, SVNG has numerous applications. In these days, its purpose evaluates incomplete and uncertainty information. BSVNG has numerous applications in
the fields of geometry and operational research. It has been a useful scope in various fields of
computer science.Later, Deli et al. [14] described the idea of the bipolar neutrosophic set as the
extension of the Neutrosophic set. He also described the concept of the bipolar fuzzy graph
with some related properties. One problem of an Fuzzy graph, Intuitionistic fuzzy graph,
bipolar fuzzy graph and intuitionistic bipolar fuzzy graph found when uncertainty occurs in
the relationship between two vertices. Need for the neutrosophic graph is necessary because
these are not suitable properly. Many researchers [32, 33] was famous due to their research
work application approach to real-world problems.
The idea of the fuzzy graph is presented by Rosenfeld [30] and [34]. Malik and Hassan [16] both
described the classification of the BSVNG together. Later Malik and Naz [21] presented the
operations on the SVNG. Gomathi and Keerthika [15] studied neutrosophic labeling graph.
Kousik Das et al. [17] defined generalized neutrosophic competition graphs. Mordeson and
Peng [18] given some operations on Fuzzy Graphs. Gani et al. [19, 20] defined order, size, and
irregular fuzzy graphs. The various application of graph theory in the fields of information
technology, operational research, image segmentation, social science, capturing the image, algebra. It is also applicable to bioscience, chemistry, and computer science. The fuzzy is very
useful to deduce the unsolved problems in various fields like networking, clustering with a great
role in the algorithm. The use of fuzzy graph by which a great extent in a few years and has
a scope from 19th century [19, 20]. Neutrosophy is the type of philosophy which studies the
nature and scope of neutralities. We will discuss some new properties on a BSVNG. Bipolar
fuzzy set has many applications in image processing. It gives more advantages in real problems
M. Aslam Malik, Hossein Rashmanlou, Muhammad Shoaib, R. A. Borzooei and Morteza Taheri, A Study on
Bipolar Single-Valued Neutrosophic Graphs With Novel Application
Neutrosophic Sets and Systems, Vol. 32, 2020
223
Figure 1. BSVNG
to make it in an easier form. BSVNG is the extension of an Fuzzy graph, Intuitionistic fuzzy
graph, interval-valued intuitionistic fuzzy graph and SVNG. Bipolar fuzzy graphs are very useful in the fields of signal processing, computer science, and database theory. The operations
we will establish are the symmetric difference and residue product in this paper. Peng [18]
defined Some operations which are the join of two graphs, cartesian product of two graphs
and the union of two graphs. Also, we discuss examples of these operations. We will find the
degree and total degree of BSVNG. In the end, we will make an application on BSVNG with
algorithm.
2. Operations on BSVNGs
In this section, we define four operations, including residue product, rejection, maximal
product and symmetric difference of bipolar single-valued neutrosophic graph (BSVNG) and
to explore some of their related properties with examples.
Definition 2.1. [13] A bipolar single valued neutrosophic graph is such a pair G = (X, Y )
which is of crisp graph G=(V,E) is defined as(i) αM : V → [0, 1], βM : V → [0, 1], γM : V →
[0, 1], δM : V → [−1, 0], ηM : V → [−1, 0], θM : V → [−1, 0]. (ii)
αN (mn) ≤ min{αM (m), αM (n)}, βN (mn) ≥ max{βM (m), βM (n)}
γN (mn) ≥ max{γM (m), γM (n)}, δN (mn) ≥ max{δM (m), δM (n)}
ηN (mn) ≤ min{ηM (m), ηM (n)}, θN (mn) ≤ min{θM (m), θM (n)}.
and 0≤ αN (mn)+βN (mn) + γN (mn) ≤ 3 and −3 ≤ δN (mn)+ηN (mn) + θN (mn) ≤ 0.
Example 2.2. In Figure 1, we see a graph with eight vertices {a,b,c,d,e,f,g,h} and eight edges
{ab, bc, cd ,ef, fg, gh ,bf, cg} that is a bipolar single valued neutrosophic graph. It is easy to
see that all conditions of Definition 2.1 is true for this example.
M. Aslam Malik, Hossein Rashmanlou, Muhammad Shoaib, R. A. Borzooei and Morteza Taheri, A Study on
Bipolar Single-Valued Neutrosophic Graphs With Novel Application
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Definition 2.3. The height of a bipolar single valued neutrosophic set (BSVNs) (in universe
discourse Y)
Q = (αQ (y), βQ (y), γQ , δQ (y), ηQ (y), θQ (y)) is defined by:
h(Q) = (h1 (Q), h2 (Q), h3 (Q), h4 (Q), h5 (Q), h6 (Q))
= (Supy∈Y αQ (y), Infy∈Y βQ (y), Infy∈Y βQ (y), Supy∈Y δQ (y), Infy∈Y ηQ (y), Infy∈Y θQ (y))
Example 2.4. Take Q = {(a, 0.5, 0.4, 0.5, −0.2, −0.4, −0.5), (b, 0.5, 0.6, 0.4, −0.4, −0.3, −0.6),
(c, 0.4, 0.6, 0.4, −0.4, −0.5, −0.3)} be BSVNs then height is defined as h(Q) = (0.5, 0.4, 0.4,
0.4, 0.3, 0.3).
Definition 2.5. let G1 = (M1 , N1 ) and G2 = (M2 , N2 ) are two bipolar single valued neutrosophic fuzzy graphs defined on G1 = (V1 , E1 ) and G2 = (V2 , E2 ) respectively. The symmetric
difference of G1 and G2 is represented by G1 ⊕ G2 = (M1 ⊕ M2 , N1 ⊕ N2 ). Symmetric difference
of G1 and G2 is defined as the following conditions:
(i)
(αM1 ⊕ αM2 )((m1 , m2 )) = min{αM1 (m1 ), αM2 (m2 )}, (βM1 ⊕ βM2 )((m1 , m2 ))
= max{βM1 (m1 ), βM2 (m2 )}
(γM1 ⊕ γM2 )((m1 , m2 )) = max{γM1 (m1 ), γM2 (m2 )}, (δM1 ⊕ δM2 )((m1 , m2 ))
= max{δM1 (m1 ), δM2 (m2 )}
(ηM1 ⊕ ηM2 )((m1 , m2 )) = min{ηM1 (m1 ), ηM2 (m2 )}, (θM1 ⊕ θM2 )((m1 , m2 ))
= min{θM1 (m1 ), θM2 (m2 )}
∀(m1 , m2 ) ∈ (V1 × V2 )
(ii)
(αN1 ⊕ αN2 )((m, m2 )(m, n2 )) = min{αM1 (m), αN2 (m2 n2 )}, (βN1 ⊕ βN2 )((m, m2 )(m, n2 ))
= max{βM1 (m), βN2 (m2 n2 )}
(γN1 ⊕ γN2 )((m, m2 )(m, n2 )) = max{γM1 (m), γN2 (m2 n2 )}, (δN1 ⊕ δN2 )((m, m2 )(m, n2 ))
= max{δM1 (m), δN2 (m2 n2 )}
(ηN1 ⊕ ηN2 )((m, m2 )(m, n2 )) = min{ηM1 (m), ηN2 (m2 n2 )}, (θN1 ⊕ θN2 )((m, m2 )(m, n2 ))
= min{θM1 (m), θN2 (m2 n2 )}
∀ m ∈ V1 and m2 n2 ∈ E2
M. Aslam Malik, Hossein Rashmanlou, Muhammad Shoaib, R. A. Borzooei and Morteza Taheri, A Study on
Bipolar Single-Valued Neutrosophic Graphs With Novel Application
Neutrosophic Sets and Systems, Vol. 32, 2020
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(iii)
(αN1 ⊕ αN2 )((m1 , m)(n1 , m)) = min{αN1 (m1 n1 ), αM2 (m)}, (βN1 ⊕ βN2 )((m1 , m)(n1 , m))
= max{βN1 (m1 n1 ), βM2 (m)}
(γN1 ⊕ γN2 )((m1 , m)(n1 , m)) = max{γN1 (m1 n1 ), γM2 (m)}, (δN1 ⊕ δN2 )((m1 , m)(n1 , m))
= max{δN1 (m1 n1 ), δM2 (m)}
(ηN1 ⊕ ηN2 )((m1 , m)(n1 , m)) = min{ηN1 (m1 n1 ), ηM2 (m)}, (θN1 ⊕ θN2 )((m1 , m)(n1 , m))
= min{θN1 (m1 n1 ), θM2 (m)}
∀ z ∈ V2 and m1 n1 ∈ E1
(i∨)
(αN1 ⊕ αN2 )((m1 , m2 )(n1 , n2 )) = min{αM1 (m1 ), αM1 (n1 ), αN2 (m2 n2 )}
f or all m1 n1 ̸∈ E1 and m2 n2 ∈ E2
or
= min{αM2 (m2 ), αM2 (n2 ), αN1 (m1 n1 )}f or all m1 n1 ∈ E1 and m2 n2 ̸∈ E2
(βN1 ⊕ βN2 )((m1 , m2 )(n1 , n2 )) = max{βM1 (m1 ), βM1 (n1 ), βN2 (m2 n2 )}
f orall m1 n1 ̸∈ E1 and m2 n2 ∈ E2
or
= max{βM2 (m2 ), βM2 (n2 ), βN1 (m1 n1 )} f orall m1 n1 ∈ E1 and m2 n2 ̸∈ E2
(γN1 ⊕ γN2 )((m1 , m2 )(n1 , n2 )) = max{γM1 (m1 ), γM1 (n1 ), FN2 (m2 n2 )}
f orall m1 n1 ̸∈ E1 and m2 n2 ∈ E2
or
= max{γM2 (m2 ), γM2 (n2 ), γN2 (m1 n1 )} f orall m1 n1 ∈ E1 and m2 n2 ̸∈ E2
(δN1 ⊕ δN2 )((m1 , m2 )(n1 , n2 )) = max{δM1 (m1 ), δM1 (n1 ), δN2 (m2 n2 )}
f orall m1 n1 ̸∈ E1 and m2 n2 ∈ E2
or
= max{δM2 (m2 ), δM2 (n2 ), δN1 (m1 n1 )} f orall m1 n1 ∈ E1 and m2 n2 ̸∈ E2
M. Aslam Malik, Hossein Rashmanlou, Muhammad Shoaib, R. A. Borzooei and Morteza Taheri,
A Study on Bipolar Single-Valued Neutrosophic Graphs With Novel Application
Neutrosophic Sets and Systems, Vol. 32, 2020
226
Figure 2. G1
Figure 3. G2
(ηN1 ⊕ ηN2 )((m1 , m2 )(n1 , n2 )) = min{ηM1 (m1 ), ηM1 (n1 ), ηN2 (m2 n2 )}
f orall m1 n1 ̸∈ E1 and m2 n2 ∈ E2
or
= min{ηM2 (m2 ), ηM2 (n2 ), ηN1 (m1 n1 )} f orall m1 n1 ∈ E1 and m2 n2 ̸∈ E2
(θN1 ⊕ θN2 )((m1 , m2 )(n1 , n2 )) = min{θM1 (m1 ), θM1 (n1 ), FN2 (m2 n2 )}
f orall m1 n1 ̸∈ E1 and m2 n2 ∈ E2
or
= min{θM2 (m2 ), θM2 (n2 ), θN2 (m1 n1 )} f orall m1 n1 ∈ E1 and m2 n2 ̸∈ E2
Example 2.6. Let G1 = (M1 , N1 ) and G2 = (M2 , N2 ) be two BSVNGs on V1 = {a, b} and V2 =
{c, d} respectively which shown in Figure 2 and Figure 3. Also symmetric difference shown in
Figure 4.
Proposition 2.7. Let G1 = (M1 , N1 ) and G2 = (M2 , N2 ) be two BSVNGs of graph G1 =
(V1 , E1 ) and G2 = (V2 , E2 ), respectively. Then the symmetric difference G1 ⊕G2 of G1 = (V1 , E1 )
and G2 = (V2 , E2 ) is again a BSVNG.
Proof. Let G1 = (M1 , N1 ) and G2 = (M2 , N2 ) be two BSVNGs of graph G1 = (V1 , E1 ) and
G2 = (V2 , E2 ), respectively. Then the symmetric difference G1 ⊕ G2 of G1 = (V1 , E1 ) and
G2 = (V2 , E2 ) can be proved. Let (m1 , m2 )(n1 , n2 ) ∈ E1 × E2
(i) If m1 = n1 = m
(αN1 ⊕ αN2 )((m, m2 )(m, n2 )) = min{αM1 (m), αN2 (m2 n2 )}
≤ min{αM1 (m), min{αM2 (m2 ), αM2 (n2 )}}
= min{min{{αM1 (m), αM2 (m2 )}, min{{αM1 (m), αM2 (n2 )}}
= min{(αM1 ⊕ αM2 )(m, m2 ), (αM1 ⊕ αM2 )(m, n2 )}
M. Aslam Malik, Hossein Rashmanlou, Muhammad Shoaib, R. A. Borzooei and Morteza Taheri,
A Study on Bipolar Single-Valued Neutrosophic Graphs With Novel Application
Neutrosophic Sets and Systems, Vol. 32, 2020
227
Figure 4. G1 ⊕ G2
(βN1 ⊕ βN2 )((m, m2 )(m, n2 )) = max{βM1 (m), βN2 (m2 n2 )}
≥ max{βM1 (m), max{βM2 (m2 ), βM2 (n2 )}}
= max{max{{βM1 (m), βM2 (m2 )}, max{{βM1 (m), βM2 (n2 )}}
= max{(βM1 ⊕ βM2 )(m, m2 ), (βM1 ⊕ βM2 )(m, n2 )}
(γN1 ⊕ γN2 )((m, m2 )(m, n2 )) = max{γM1 (m), γN2 (m2 n2 )}
≥ max{γM1 (m), max{γM2 (m2 ), γM2 (n2 )}}
= max{max{{γM1 (m), γM2 (m2 )}, max{{γM1 (m), γM2 (n2 )}}
= max{(γM1 ⊕ γM2 )(m, m2 ), (γM1 ⊕ γM2 )(m, n2 )}
(δN1 ⊕ δN2 )((m, m2 )(m, n2 )) = max{δM1 (m), δN2 (m2 n2 )}
≥ max{δM1 (m), max{δM2 (m2 ), δM2 (n2 )}}
= max{max{{δM1 (m), δM2 (m2 )}, min{{δM1 (m), δM2 (n2 )}}
= max{(δM1 ⊕ δM2 )(m, m2 ), (δM1 ⊕ δM2 )(m, n2 )}
M. Aslam Malik, Hossein Rashmanlou, Muhammad Shoaib, R. A. Borzooei and Morteza Taheri,
A Study on Bipolar Single-Valued Neutrosophic Graphs With Novel Application
Neutrosophic Sets and Systems, Vol. 32, 2020
228
(ηN1 ⊕ ηN2 )((m, m2 )(m, n2 )) = min{ηM1 (m), ηN2 (m2 n2 )}
≤ min{ηM1 (m), min{ηM2 (m2 ), ηM2 (n2 )}}
= min{min{{ηM1 (m), ηM2 (m2 )}, min{{ηM1 (m), ηM2 (n2 )}}
= min{(ηM1 ⊕ ηM2 )(m, m2 ), (ηM1 ⊕ ηM2 )(m, n2 )}
(θN1 ⊕ θN2 )((m, m2 )(m, n2 )) = min{θM1 (m), θN2 (m2 n2 )}
≤ min{θM1 (m), min{θM2 (m2 ), θM2 (n2 )}}
= min{min{{θM1 (m), θM2 (m2 )}, min{{θM1 (m), θM2 (n2 )}}
= min{(θM1 ⊕ θM2 )(m, m2 ), (θM1 ⊕ θM2 )(m, n2 )}
(ii) if m2 = n2 = m
(αN1 ⊕ αN2 )((m1 , m)(n1 , m)) = min{αN1 (m1 n1 ), αM2 (m)}
≤ min{min{αN1 (m1 n1 ), αM2 (m)}
= min{min{{αM1 (m1 ), αM2 (m)}, min{{αM1 (n1 ), αM2 (m)}}
= min{(αM1 ⊕ αM2 )(m1 , m), (αM1 ⊕ αM2 )(n1 , m)}
(βN1 ⊕ βN2 )((m1 , m)(n1 , m)) = max{βN1 (m1 n1 ), βM2 (m)}
≥ max{max{βN1 (m1 n1 ), βM2 (m)}
= max{max{{βM1 (m1 ), βM2 (m)}, max{{βM1 (n1 ), βM2 (m)}}
= max{(βM1 ⊕ βM2 )(m1 , m), (βM1 ⊕ βM2 )(n1 , m)}
(γN1 ⊕ γN2 )((m1 , m)(n1 , m)) = max{γN1 (m1 n1 ), γM2 (m)}
≥ max{max{γN1 (m1 n1 ), γM2 (m)}
= max{max{{γM1 (m1 ), γM2 (m)}, max{{γM1 (n1 ), γM2 (m)}}
= max{(γM1 ⊕ γM2 )(m1 , m), (γM1 ⊕ γM2 )(n1 , m)}
M. Aslam Malik, Hossein Rashmanlou, Muhammad Shoaib, R. A. Borzooei and Morteza Taheri,
A Study on Bipolar Single-Valued Neutrosophic Graphs With Novel Application
Neutrosophic Sets and Systems, Vol. 32, 2020
229
(δN1 ⊕ δN2 )((m1 , m)(n1 , m)) = max{δN1 (m1 n1 ), δM2 (m)}
≥ max{max{δN1 (m1 n1 ), δM2 (m)}
= max{max{{δM1 (m1 ), δM2 (m)}, max{{δM1 (n1 ), δM2 (m)}}
= max{(δM1 ⊕ δM2 )(m1 , m), (δM1 ⊕ δM2 )(n1 , m)}
(ηN1 ⊕ ηN2 )((m1 , m)(n1 , m)) = min{ηN1 (m1 n1 ), ηM2 (m)}
≤ min{min{ηN1 (m1 n1 ), ηM2 (m)}
= min{min{{ηM1 (m1 ), ηM2 (m)}, min{{ηM1 (n1 ), ηM2 (m)}}
= min{(ηM1 ⊕ ηM2 )(m1 , m), (ηM1 ⊕ ηM2 )(n1 , m)}
(θN1 ⊕ θN2 )((m1 , m)(n1 , m)) = min{θN1 (m1 n1 ), θM2 (m)}
≤ min{min{θN1 (m1 n1 ), θM2 (m)}
= min{min{{θM1 (m1 ), θM2 (m)}, min{{θM1 (n1 ), θM2 (m)}}
= min{(θM1 ⊕ θM2 )(m1 , m), (θM1 ⊕ θM2 )(n1 , m)}
(iii) If m1 n1 ̸∈ E1 and m2 n2 ∈ E2
(αN1 ⊕ αN2 )((m1 , m2 )(n1 , n2 )) = min{αM1 (m1 ), αM1 (n1 ), αN2 (m2 n2 )}
≤ min{αM1 (m1 ), αM1 (n1 ), min{αM2 (m2 )αM2 (n2 )}}
= min{min{αM1 (m1 ), αM2 (m2 )}, {αM1 (m1 ), αM2 (n2 )}
= min{(αM1 ⊕ αM2 )(m1 , m2 ), (αM1 ⊕ αM2 )(n1 , n2 )}
(βN1 ⊕ βN2 )((m1 , m2 )(n1 , n2 )) = max{βM1 (m1 ), βM1 (n1 ), βN2 (m2 n2 )}
≥ max{βM1 (m1 ), βM1 (n1 ), max{βM2 (m2 )βM2 (n2 )}}
= max{max{βM1 (m1 ), βM2 (m2 )}, {βM1 (m1 ), βM2 (n2 )}
= max{(βM1 ⊕ βM2 )(m1 , m2 ), (βM1 ⊕ βM2 )(n1 , n2 )}
(γN1 ⊕ γN2 )((m1 , m2 )(n1 , n2 )) = max{γM1 (m1 ), γM1 (n1 ), γN2 (m2 n2 )}
≥ max{γM1 (m1 ), γM1 (n1 ), max{γM2 (m2 )γM2 (n2 )}}
= max{max{γM1 (m1 ), γM2 (m2 )}, {γM1 (m1 ), γM2 (n2 )}
= max{(γM1 ⊕ γM2 )(m1 , m2 ), (γM1 ⊕ γM2 )(n1 , n2 )}
M. Aslam Malik, Hossein Rashmanlou, Muhammad Shoaib, R. A. Borzooei and Morteza Taheri,
A Study on Bipolar Single-Valued Neutrosophic Graphs With Novel Application
Neutrosophic Sets and Systems, Vol. 32, 2020
230
(δN1 ⊕ δN2 )((m1 , m2 )(n1 , n2 )) = max{δM1 (m1 ), δM1 (n1 ), δN2 (m2 n2 )}
≥ max{δM1 (m1 ), δM1 (n1 ), max{δM2 (m2 )δM2 (n2 )}}
= max{max{δM1 (m1 ), δM2 (m2 )}, {δM1 (m1 ), δM2 (n2 )}
= max{(δM1 ⊕ δM2 )(m1 , m2 ), (δM1 ⊕ δM2 )(n1 , n2 )}
(ηN1 ⊕ ηN2 )((m1 , m2 )(n1 , n2 )) = min{ηM1 (m1 ), ηM1 (n1 ), ηN2 (m2 n2 )}
≤ min{ηM1 (m1 ), ηM1 (n1 ), min{ηM2 (m2 )ηM2 (n2 )}}
= min{min{ηM1 (m1 ), ηM2 (m2 )}, {ηM1 (m1 ), ηM2 (n2 )}
= min{(ηM1 ⊕ ηM2 )(m1 , m2 ), (ηM1 ⊕ ηM2 )(n1 , n2 )}
(θN1 ⊕ θN2 )((m1 , m2 )(n1 , n2 )) = min{θM1 (m1 ), θM1 (n1 ), θN2 (m2 n2 )}
≤ min{θM1 (m1 ), θM1 (n1 ), min{θM2 (m2 )θM2 (n2 )}}
= min{min{θM1 (m1 ), θM2 (m2 )}, {θM1 (m1 ), θM2 (n2 )}
= min{(θM1 ⊕ θM2 )(m1 , m2 ), (θM1 ⊕ θM2 )(n1 , n2 )}
(i∨) If m1 n1 ∈ E1 and m2 n2 ̸∈ E2
(αN1 ⊕ αN2 )((m1 , m2 )(n1 , n2 )) = min{αM2 (m2 ), αM2 (n2 ), αN1 (m1 n1 )}
≤ min{αM2 (m2 ), αM2 (n2 ), min{αM1 (m1 )αM1 (n1 )}}
= min{min{αM2 (m2 ), αM1 (m1 )}, {αM2 (m2 ), αM1 (n1 )}
= min{(αM1 ⊕ αM2 )(m1 , m2 ), (αM1 ⊕ αM2 )(n1 , n2 )}
(βN1 ⊕ βN2 )((m1 , m2 )(n1 , n2 )) = max{βM2 (m2 ), βM2 (n2 ), βN1 (m1 n1 )}
≥ max{βM2 (m2 ), βM2 (n2 ), max{βM1 (m1 )βM1 (n1 )}}
= max{max{βM2 (m2 ), βM1 (m1 )}, {βM2 (m2 ), βM1 (n1 )}
= max{(βM1 ⊕ βM2 )(m1 , m2 ), (βM1 ⊕ βM2 )(n1 , n2 )}
(γN1 ⊕ γN2 )((m1 , m2 )(n1 , n2 )) = max{γM2 (m2 ), γM2 (n2 ), γN1 (m1 n1 )}
≥ max{γM2 (m2 ), γM2 (n2 ), max{γM1 (m1 )γM1 (n1 )}}
= max{max{γM2 (m2 ), γM1 (m1 )}, {γM2 (m2 ), γM1 (n1 )}
= max{(γM1 ⊕ γM2 )(m1 , m2 ), (γM1 ⊕ γM2 )(n1 , n2 )}
M. Aslam Malik, Hossein Rashmanlou, Muhammad Shoaib, R. A. Borzooei and Morteza Taheri,
A Study on Bipolar Single-Valued Neutrosophic Graphs With Novel Application
Neutrosophic Sets and Systems, Vol. 32, 2020
231
(δN1 ⊕ δN2 )((m1 , m2 )(n1 , n2 )) = max{δM2 (m2 ), δM2 (n2 ), δN1 (m1 n1 )}
≥ max{δM2 (m2 ), δM2 (n2 ), max{δM1 (m1 )δM1 (n1 )}}
= max{max{δM2 (m2 ), δM1 (m1 )}, {δM2 (m2 ), δM1 (n1 )}
= max{(δM1 ⊕ δM2 )(m1 , m2 ), (δM1 ⊕ δM2 )(n1 , n2 )}
(ηN1 ⊕ ηN2 )((m1 , m2 )(n1 , n2 )) = min{ηM2 (m2 ), ηM2 (n2 ), ηN1 (m1 n1 )}
≤ min{ηM2 (m2 ), ηM2 (n2 ), min{ηM1 (m1 )ηM1 (n1 )}}
= min{min{ηM2 (m2 ), ηM1 (m1 )}, {ηM2 (m2 ), ηM1 (n1 )}
= min{(ηM1 ⊕ ηM2 )(m1 , m2 ), (ηM1 ⊕ ηM2 )(n1 , n2 )}
(θN1 ⊕ θN2 )((m1 , m2 )(n1 , n2 )) = min{θM2 (m2 ), θM2 (n2 ), θN1 (m1 n1 )}
≤ min{θM2 (m2 ), θM2 (n2 ), min{θM1 (m1 )θM1 (n1 )}}
= min{min{θM2 (m2 ), θM1 (m1 )}, {θM2 (m2 ), θM1 (n1 )}
= min{(θM1 ⊕ θM2 )(m1 , m2 ), (θM1 ⊕ θM2 )(n1 , n2 )}
. Hence G1 ⊕ G2 is a BSVNG.
Definition 2.8. Let G1 = (M1 , N1 ) and G2 = (M2 , Y2 ) be two BSVNGs. ∀(m1 , m2 ) ∈ V1 × V2
(dα )G1 ⊕G2 (m1 , m2 ) =
∑
(αN1 ⊕ αN2 )((m1 , m2 )(n1 , n2 ))
(m1 ,m2 )(n1 ,n2 )∈E1 ×E2 .
=
∑
min{αM1 (m1 ), αN2 (m2 n2 )}
m1 =n1 ,m2 n2 ∈E2
+
∑
min{αN1 (m1 n1 , αM2 (m2 )}
m1 n1 ∈E1 ,m2 =n2
+
∑
min{αM1 (m1 ), αM1 (n1 ), αN2 (m2 n2 )}
m1 n1 ̸∈E1 and m2 n2 ∈E2
+
∑
min{αN1 (m1 n1 ), αM2 (m2 ), αM2 (n2 )}
m1 n1 ∈E1 and m2 n2 ̸∈E2
M. Aslam Malik, Hossein Rashmanlou, Muhammad Shoaib, R. A. Borzooei and Morteza Taheri,
A Study on Bipolar Single-Valued Neutrosophic Graphs With Novel Application
Neutrosophic Sets and Systems, Vol. 32, 2020
(dβ )G1 ⊕G2 (m1 , m2 ) =
232
∑
(βN1 ⊕ βN2 )((m1 , m2 )(n1 , n2 ))
(m1 ,m2 )(n1 ,n2 )∈E1 ×E2 .
=
∑
max{βM1 (m1 ), βN2 (m2 n2 )}
m1 =n1 ,m2 n2 ∈E2
+
∑
max{βN1 (m1 n1 , βM2 (m2 )}
m1 n1 ∈E1 ,m2 =n2
+
∑
max{βM1 (m1 ), βM1 (n1 ), βN2 (m2 n2 )}
m1 n1 ̸∈E1 and m2 n2 ∈E2
+
∑
max{βN1 (m1 n1 ), βM2 (m2 ), βM2 (n2 )}
m1 n1 ∈E1 and m2 n2 ̸∈E2
(dγ )G1 ⊕G2 (m1 , m2 ) =
∑
(γN1 ⊕ βN2 )((m1 , m2 )(n1 , n2 ))
(m1 ,m2 )(n1 ,n2 )∈E1 ×E2 .
=
∑
max{γM1 (m1 ), γN2 (m2 n2 )}
m1 =n1 ,m2 n2 ∈E2
+
∑
max{γN1 (m1 n1 , γM2 (m2 )}
m1 n1 ∈E1 ,m2 =n2
+
∑
max{γM1 (m1 ), γM1 (n1 ), γN2 (m2 n2 )}
m1 n1 ̸∈E1 and m2 n2 ∈E2
+
∑
max{γN1 (m1 n1 ), γM2 (m2 ), γM2 (n2 )}
m1 n1 ∈E1 and m2 n2 ̸∈E2
(dδ )G1 ⊕G2 (m1 , m2 ) =
∑
(δN1 ⊕ δN2 )((m1 , m2 )(n1 , n2 ))
(m1 ,m2 )(n1 ,n2 )∈E1 ×E2 .
=
∑
max{δM1 (m1 ), δN2 (m2 n2 )}
m1 =n1 ,m2 n2 ∈E2
+
∑
max{δN1 (m1 n1 , δM2 (m2 )}
m1 n1 ∈E1 ,m2 =n2
+
∑
max{δM1 (m1 ), δM1 (n1 ), δN2 (m2 n2 )}
m1 n1 ̸∈E1 and m2 n2 ∈E2
+
∑
max{δN1 (m1 n1 ), δM2 (m2 ), δM2 (n2 )}
m1 n1 ∈E1 and m2 n2 ̸∈E2
M. Aslam Malik, Hossein Rashmanlou, Muhammad Shoaib, R. A. Borzooei and Morteza Taheri,
A Study on Bipolar Single-Valued Neutrosophic Graphs With Novel Application
Neutrosophic Sets and Systems, Vol. 32, 2020
(dη )G1 ⊕G2 (m1 , m2 ) =
233
∑
(ηN1 ⊕ ηN2 )((m1 , m2 )(n1 , n2 ))
(m1 ,m2 )(n1 ,n2 )∈E1 ×E2 .
=
∑
min{ηM1 (m1 ), ηN2 (m2 n2 )}
m1 =n1 ,m2 n2 ∈E2
+
∑
min{ηN1 (m1 n1 , ηM2 (m2 )}
m1 n1 ∈E1 ,m2 =n2
+
∑
min{ηM1 (m1 ), ηM1 (n1 ), ηN2 (m2 n2 )}
m1 n1 ̸∈E1 and m2 n2 ∈E2
+
∑
min{ηN1 (m1 n1 ), ηM2 (m2 ), ηM2 (n2 )}
m1 n1 ∈E1 and m2 n2 ̸∈E2
(dθ )G1 ⊕G2 (m1 , m2 ) =
∑
(θN1 ⊕ ηN2 )((m1 , m2 )(n1 , n2 ))
(m1 ,m2 )(n1 ,n2 )∈E1 ×E2 .
=
∑
min{θM1 (m1 ), θN2 (m2 n2 )}
m1 =n1 ,m2 n2 ∈E2
+
∑
min{θN1 (m1 n1 , θM2 (m2 )}
m1 n1 ∈E1 ,m2 =n2
+
∑
min{θM1 (m1 ), θM1 (n1 ), θN2 (m2 n2 )}
m1 n1 ̸∈E1 and m2 n2 ∈E2
+
∑
min{θN1 (m1 n1 ), θM2 (m2 ), θM2 (n2 )}
m1 n1 ∈E1 and m2 n2 ̸∈E2
Theorem 2.9. Let G1 = (M1 , N1 ) and G2 = (M2 , Y2 ) be two BSVNGs. If αM1 ≥ αN2 , βM1 ≤
βN2 , γM1 ≤ γN2 and αM2 ≥ αN1 , βM2 ≤ βN1 , γM2 ≤ γN1 . Also if δM1 ≤ δN2 , ηM1 ≥ ηN2 , θM1 ≥
θN2 and δM2 ≤ δN1 , ηM2 ≥ ηN1 , θM2 ≥ θN1 . Then for every ∀(m1 , m2 ) ∈ V1 × V2
(d)G1 ⊕G2 (m1 , m2 ) =q(d)G1 (m1 )+ s(d)G2 (m2 ) where s=| V1 | -(d)G1 (m1 ) and q=| V2 | -(d)G2 (m2 )
.
M. Aslam Malik, Hossein Rashmanlou, Muhammad Shoaib, R. A. Borzooei and Morteza Taheri,
A Study on Bipolar Single-Valued Neutrosophic Graphs With Novel Application
Neutrosophic Sets and Systems, Vol. 32, 2020
234
Proof.
∑
(dα )G1 ⊕G2 (m1 , m2 ) =
(αN1 ⊕ αN2 )((m1 , m2 )(n1 , n2 ))
(m1 ,m2 )(n1 ,n2 )∈E1 ×E2 .
=
∑
min{αM1 (m1 ), αN2 (m2 n2 )}
m1 =n1 ,m2 n2 ∈E2
+
∑
min{αN1 (m1 n1 ), αM2 (m2 )}
m1 n1 ∈E1 ,m2 =n2
∑
+
min{αM1 (m1 ), αM1 (n1 ), αN2 (m2 n2 )}
m1 n1 ̸∈E1 and m2 n2 ∈E2
∑
+
min{αN1 (m1 n1 ), αM2 (m2 ), αM2 (n2 )}
m1 n1 ∈E1 and m2 n2 ̸∈E2
=
∑
αN2 (m2 n2 ) +
m2 n2 ∈E2
+
∑
αN1 (m1 n1 )
m1 n1 ∈E1
∑
∑
αN2 (m2 n2 )} +
αN1 (m1 n1 )
m1 n1 ∈E1 and m2 n2 ̸∈E2
m1 n1 ̸∈E1 and m2 n2 ∈E2
= q(dα )G1 (m1 ) + s(dα )G2 (m2 )
∑
(dθ )G1 ⊕G2 (m1 , m2 ) =
(θN1 ⊕ θN2 )((m1 , m2 )(n1 , n2 ))
(m1 ,m2 )(n1 ,n2 )∈E1 ×E2 .
=
∑
min{θM1 (m1 ), θN2 (m2 n2 )}
m1 =n1 ,m2 n2 ∈E2
+
∑
min{θN1 (m1 n1 ), θM2 (m2 )}
m1 n1 ∈E1 ,m2 =n2
∑
+
min{θM1 (m1 ), θM1 (n1 ), θN2 (m2 n2 )}
m1 n1 ̸∈E1 and m2 n2 ∈E2
∑
+
min{θN1 (m1 n1 ), θM2 (m2 ), θM2 (n2 )}
m1 n1 ∈E1 and m2 n2 ̸∈E2
=
∑
m2 n2 ∈E2
+
θN2 (m2 n2 ) +
∑
∑
θN1 (m1 n1 )
m1 n1 ∈E1
θN2 (m2 n2 )} +
m1 n1 ̸∈E1 and m2 n2 ∈E2
∑
θN1 (m1 n1 )
m1 n1 ∈E1 and m2 n2 ̸∈E2
= q(dθ )G1 (m1 ) + s(dθ )G2 (m2 )
In a similar way others four will proved obviously.
We conclude that (d)G1 ⊕G2 (m1 , m2 ) =q(d)G1 (m1 ) + s(d)G2 (m2 ) where s=| V1 | -(d)G1 (m1 ) and
q=| V2 | -(d)G2 (m2 ) .
M. Aslam Malik, Hossein Rashmanlou, Muhammad Shoaib, R. A. Borzooei and Morteza Taheri,
A Study on Bipolar Single-Valued Neutrosophic Graphs With Novel Application
Neutrosophic Sets and Systems, Vol. 32, 2020
235
Definition 2.10. Let G1 = (M1 , N1 ) and G2 = (M2 , Y2 ) be two BSVNGs. ∀(m1 , m2 ) ∈ V1 ×V2
∑
(tdα )G1 ⊕G2 (m1 , m2 ) =
(αN1 ⊕ αN2 )((m1 , m2 )(n1 , n2 )) + (αM1 ⊕ αM2 (m1 , m2 )
(m1 ,m2 )(n1 ,n2 )∈E1 ×E2 .
=
∑
min{αM1 (m1 ), αN2 (m2 n2 )}
m1 =n1 ,m2 n2 ∈E2
+
∑
min{αN1 (m1 n1 , αM2 (m2 )}
m1 n1 ∈E1 ,m2 =n2
+
∑
min{αM1 (m1 ), αM1 (n1 ), αN2 (m2 n2 )}
m1 n1 ̸∈E1 and m2 n2 ∈E2
+
∑
min{αN1 (m1 n1 ), αM2 (m2 ), αM2 (n2 )}
m1 n1 ∈E1 and m2 n2 ̸∈E2
+ min{αM1 (m1 ), αM2 (m2 )}
(tdβ )G1 ⊕G2 (m1 , m2 ) =
∑
(βN1 ⊕ βN2 )((m1 , m2 )(n1 , n2 )) + (βM1 ⊕ βM2 (m1 , m2 )
(m1 ,m2 )(n1 ,n2 )∈E1 ×E2 .
=
∑
max{βM1 (m1 ), βN2 (m2 n2 )}
m1 =n1 ,m2 n2 ∈E2
+
∑
max{βN1 (m1 n1 , βM2 (m2 )}
m1 n1 ∈E1 ,m2 =n2
+
∑
max{βM1 (m1 ), βM1 (n1 ), βN2 (m2 n2 )}
m1 n1 ̸∈E1 and m2 n2 ∈E2
+
∑
max{βN1 (m1 n1 ), βM2 (m2 ), βM2 (n2 )}
m1 n1 ∈E1 and m2 n2 ̸∈E2
+ max{βM1 (m1 ), βM2 (m2 )}
(tdγ )G1 ⊕G2 (m1 , m2 ) =
∑
(γN1 ⊕ γN2 )((m1 , m2 )(n1 , n2 )) + (γM1 ⊕ γM2 (m1 , m2 )
(m1 ,m2 )(n1 ,n2 )∈E1 ×E2 .
=
∑
max{γM1 (m1 ), γN2 (m2 n2 )}
m1 =n1 ,m2 n2 ∈E2
+
∑
max{γN1 (m1 n1 , γM2 (m2 )}
m1 n1 ∈E1 ,m2 =n2
+
∑
max{γM1 (m1 ), γM1 (n1 ), γN2 (m2 n2 )}
m1 n1 ̸∈E1 and m2 n2 ∈E2
+
∑
max{γN1 (m1 n1 ), γM2 (m2 ), γM2 (n2 )}
m1 n1 ∈E1 and m2 n2 ̸∈E2
+ max{γM1 (m1 ), γM2 (m2 )}
M. Aslam Malik, Hossein Rashmanlou, Muhammad Shoaib, R. A. Borzooei and Morteza Taheri,
A Study on Bipolar Single-Valued Neutrosophic Graphs With Novel Application
Neutrosophic Sets and Systems, Vol. 32, 2020
(tdδ )G1 ⊕G2 (m1 , m2 ) =
∑
236
(δN1 ⊕ δN2 )((m1 , m2 )(n1 , n2 )) + (δM1 ⊕ δM2 (m1 , m2 )
(m1 ,m2 )(n1 ,n2 )∈E1 ×E2 .
=
∑
max{δM1 (m1 ), δN2 (m2 n2 )}
m1 =n1 ,m2 n2 ∈E2
+
∑
max{δN1 (m1 n1 , δM2 (m2 )}
m1 n1 ∈E1 ,m2 =n2
+
∑
max{δM1 (m1 ), δM1 (n1 ), δN2 (m2 n2 )}
m1 n1 ̸∈E1 and m2 n2 ∈E2
+
∑
max{δN1 (m1 n1 ), δM2 (m2 ), δM2 (n2 )}
m1 n1 ∈E1 and m2 n2 ̸∈E2
+ min{δM1 (m1 ), δM2 (m2 )}
(tdη )G1 ⊕G2 (m1 , m2 ) =
∑
(ηN1 ⊕ ηN2 )((m1 , m2 )(n1 , n2 )) + (ηM1 ⊕ ηM2 (m1 , m2 )
(m1 ,m2 )(n1 ,n2 )∈E1 ×E2 .
=
∑
min{ηM1 (m1 ), ηN2 (m2 n2 )}
m1 =n1 ,m2 n2 ∈E2
+
∑
min{ηN1 (m1 n1 , ηM2 (m2 )}
m1 n1 ∈E1 ,m2 =n2
+
∑
min{ηM1 (m1 ), ηM1 (n1 ), ηN2 (m2 n2 )}
m1 n1 ̸∈E1 and m2 n2 ∈E2
+
∑
min{ηN1 (m1 n1 ), ηM2 (m2 ), ηM2 (n2 )}
m1 n1 ∈E1 and m2 n2 ̸∈E2
+ max{ηM1 (m1 ), ηM2 (m2 )}
(tdθ )G1 ⊕G2 (m1 , m2 ) =
∑
(θN1 ⊕ θN2 )((m1 , m2 )(n1 , n2 )) + (θM1 ⊕ θM2 (m1 , m2 )
(m1 ,m2 )(n1 ,n2 )∈E1 ×E2 .
=
∑
min{θM1 (m1 ), θN2 (m2 n2 )}
m1 =n1 ,m2 n2 ∈E2
+
∑
min{θN1 (m1 n1 , θM2 (m2 )}
m1 n1 ∈E1 ,m2 =n2
+
∑
min{θM1 (m1 ), θM1 (n1 ), θN2 (m2 n2 )}
m1 n1 ̸∈E1 and m2 n2 ∈E2
+
∑
min{θN1 (m1 n1 ), θM2 (m2 ), θM2 (n2 )}
m1 n1 ∈E1 and m2 n2 ̸∈E2
+ max{θM1 (m1 ), θM2 (m2 )}
Theorem 2.11. Let G1 = (M1 , N1 ) and G2 = (M2 , Y2 ) be two BSVNGs. If
(i)
M. Aslam Malik, Hossein Rashmanlou, Muhammad Shoaib, R. A. Borzooei and Morteza Taheri,
A Study on Bipolar Single-Valued Neutrosophic Graphs With Novel Application
Neutrosophic Sets and Systems, Vol. 32, 2020
237
αM1 ≥ αN2 and αM2 ≥ αN1 then ∀(m1 , m2 ) ∈ V1 × V2
(tdα )G1 ⊕G2 (m1 , m2 ) = q(tdα )G1 (m1 ) + s(tdα )G2 (m2 )
− (q − 1)TG1 (m1 ) − max{TG1 (m1 ), TG1 (m1 )}
and
δM1 ≤ δN2 and δM2 ≤ δN1 then ∀(m1 , m2 ) ∈ V1 × V2
(tdδ )G1 ⊕G2 (m1 ), m2 ) = q(tdδ )G1 (m1 ) + s(tdδ )G2 (m2 )
− (q − 1)TG1 (m1 ) − min{TG1 (m1 ), TG1 (m1 )}
(ii) βM1 ≤ βN2 and βM2 ≤ βN1 then ∀(m1 , m2 ) ∈ V1 × V2
(tdβ )G1 ⊕G2 (m1 , m2 ) = q(tdβ )G1 (m1 ) + s(tdβ )G2 (m2 )
− (q − 1)IG1 (m1 ) − min{IG1 (m1 ), IG1 (m1 }
and
ηM1 ≥ ηN2 and ηM2 ≥ ηN1 then ∀(m1 , m2 ) ∈ V1 × V2
(tdη )G1 ⊕G2 (m1 , m2 ) = q(tdη )G1 (m1 ) + s(tdη )G2 (m2 )
− (q − 1)IG1 (m1 ) − max{IG1 (m1 ), IG1 (m1 )}
(iii) γM1 ≤ γN2 and γM2 ≥ γN1 then ∀(m1 , m2 ) ∈ V1 × V2
(tdγ )G1 ⊕G2 (m1 , m2 ) = q(tdγ )G1 (m1 ) + s(tdγ )G2 (m2 )
− (q − 1)FG1 (m1 ) − min{FG1 (m1 ), FG1 (m1 )}
and
θM1 ≥ θN2 and θM2 ≤ θN1 then ∀(m1 , m2 ) ∈ V1 × V2
(tdθ )G1 ⊕G2 (m1 , m2 ) = q(tdθ )G1 (m1 ) + s(tdθ )G2 (m2 )
− (q − 1)FG1 (m1 ) − max{FG1 (m1 ), FG1 (m1 )}
∀(m1 , m2 ) ∈ V1 × V2 ,s=| V1 | -(d)G1 (m1 ) and q=| V2 | -(d)G2 (m2 ) .
M. Aslam Malik, Hossein Rashmanlou, Muhammad Shoaib, R. A. Borzooei and Morteza Taheri,
A Study on Bipolar Single-Valued Neutrosophic Graphs With Novel Application
Neutrosophic Sets and Systems, Vol. 32, 2020
238
Proof. ∀(m1 , m2 ) ∈ V1 × V2
∑
(tdα )G1 ⊕G2 (m1 , m2 ) =
(αN1 ⊕ αN2 )((m1 , m2 )(n1 , n2 )) + (αM1 ⊕ αM2 )(m1 , m2 )
(m1 ,m2 )(n1 ,n2 )∈E1 ×E2 .
=
∑
min{αM1 (m1 ), αN2 (m2 n2 )}
m1 =n1 ,m2 n2 ∈E2
+
∑
min{αN1 (m1 n1 ), αM2 (m2 )}
m1 n1 ∈E1 ,m2 =n2
∑
+
min{αM1 (m1 ), αM1 (n1 ), αN2 (m2 n2 )}
m1 n1 ̸∈E1 and m2 n2 ∈E2
∑
+
min{αN1 (m1 n1 ), αM2 (m2 ), αM2 (n2 )}
m1 n1 ∈E1 and m2 n2 ̸∈E2
+ max{αM1 (m1 ), αM2 (m2 )}
∑
∑
αN2 (m2 n2 ) +
=
m2 n2 ∈E2
+
αN1 (m1 n1 )
m1 n1 ∈E1
∑
+ max{αM1 (m1 ), αM2 (m2 )}
∑
∑
αN2 (m2 n2 ) +
=
+
αN1 (m1 n1 ) +
m1 n1 ∈E1
∑
αN1 (m1 n1 )
m1 n1 ∈E1 and m2 n2 ̸∈E2
m1 n1 ̸∈E1 and m2 n2 ∈E2
m2 n2 ∈E2
∑
αN2 (m2 n2 )} +
∑
αN2 (m2 n2 )}
m1 n1 ̸∈E1 and m2 n2 ∈E2
αN1 (m1 n1 ) + αM1 (m1 ) + αM2 (m2 ) − max{αM1 (m1 ), αM2 (m2 )}
m1 n1 ∈E1 and m2 n2 ̸∈E2
= q(tdα )G1 (m1 ) + s(tdα )G2 (m2 )
− (q − 1)TG1 (m1 ) − max{TG1 (m1 ), TG1 (m1 )}
∑
(tdδ )G1 ⊕G2 (m1 , m2 ) =
(δN1 ⊕ δN2 )((m1 , m2 )(n1 , n2 )) + (δM1 ⊕ δM2 )(m1 , m2 )
(m1 ,m2 )(n1 ,n2 )∈E1 ×E2 .
=
∑
max{δM1 (m1 ), δN2 (m2 n2 )}
m1 =n1 ,m2 n2 ∈E2
+
∑
max{δN1 (m1 n1 ), δM2 (m2 )}
m1 n1 ∈E1 ,m2 =n2
∑
+
max{δM1 (m1 ), δM1 (n1 ), δN2 (m2 n2 )}
m1 n1 ̸∈E1 and m2 n2 ∈E2
∑
+
max{δN1 (m1 n1 ), δM2 (m2 ), δM2 (n2 )}
m1 n1 ∈E1 and m2 n2 ̸∈E2
+ min{δM1 (m1 ), δM2 (m2 )}
∑
∑
δN2 (m2 n2 ) +
=
m2 n2 ∈E2
δN1 (m1 n1 )
m1 n1 ∈E1
M. Aslam Malik, Hossein Rashmanlou, Muhammad Shoaib, R. A. Borzooei and Morteza Taheri,
A Study on Bipolar Single-Valued Neutrosophic Graphs With Novel Application
Neutrosophic Sets and Systems, Vol. 32, 2020
∑
+
∑
δN1 (m1 n1 ) +
m1 n1 ∈E1
∑
δN1 (m1 n1 )
m1 n1 ∈E1 and m2 n2 ̸∈E2
+ min{δM1 (m1 ), δM2 (m2 )}
∑
∑
δN2 (m2 n2 ) +
=
+
∑
δN2 (m2 n2 )} +
m1 n1 ̸∈E1 and m2 n2 ∈E2
m2 n2 ∈E2
239
δN2 (m2 n2 )}
m1 n1 ̸∈E1 and m2 n2 ∈E2
δN1 (m1 n1 ) + δM1 (m1 ) + δM2 (m2 ) − min{δM1 (m1 ), δM2 (m2 )}
m1 n1 ∈E1 and m2 n2 ̸∈E2
= q(tdδ )G1 (m1 ) + s(tdδ )G2 (m2 )
− (q − 1)TG1 (m1 ) − min{TG1 (m1 ), TG1 (m1 )}
In a similar way others four will proved obviously.
where s=| V1 | -(d)G1 (m1 ) and q=| V2 | -(d)G2 (m2 )
Example 2.12. In Example 2.6 we have to find the degree and total degree of vertices of
G1 ⊕ G2 by using Figure 2, Figure 3, and Figure 4.
(dα )G1 ⊕G2 (a, c) = q(dα )G1 (a) + s(dα )G2 (c)
where s=| V1 | -(d)G1 (a) and q=| V2 | -(d)G2 (e)
s =| V1 | −(d)G1 (a) = 2 − 1 = 1, q =| V2 | −(d)G2 (e) = 2 − 1 = 1
(dα )G1 ⊕G2 (a, c) = q(dα )G1 (a) + s(dα )G2 (c) = 1(0.4) + 1(0.5) = 0.4 + 0.5 = 0.9
(dβ )G1 ⊕G2 (a, c) = q(dβ )G1 (a) + s(dβ )G2 (c) = 1(0.2) + 1(0.4) = 0.2 + 0.4 = 0.6
(dγ )G1 ⊕G2 (a, c) = 0.7, (dδ )G1 ⊕G2 (a, c) = −1.1
(dη )G1 ⊕G2 (a, c) = −0.5, (dθ )G1 ⊕G2 (a, c) = −0.7
So (d)G1 ⊕G2 (a, e) = (0.9, 0.6, −1.1, −0.5, −0.7)
By applying this technique we can find degree of all vertices in a similar way. Now we will
find total degree of vertices. For this select vertex (a,e)
(tdα )G1 ⊕G2 (a, c) = q(tdα )G1 (a) + s(tdα )G2 (c)
− (s − 1)αG2 (c) − (q − 1)αG1 (a) − max{αG1 (a), αG2 (c)}
= 1(0.7 + 0.4) + 1(0.6 + 0.5) − (1 − 1)(0.6) − (1 − 1)(0.7)
− max{0.6, 0.7} = 1(1.1) + 1.1 − 0.7 = 1.5
M. Aslam Malik, Hossein Rashmanlou, Muhammad Shoaib, R. A. Borzooei and Morteza Taheri,
A Study on Bipolar Single-Valued Neutrosophic Graphs With Novel Application
Neutrosophic Sets and Systems, Vol. 32, 2020
240
(tdδ )G1 ⊕G2 (a, c) = q(tdδ )G1 (a) + s(tdδ )G2 (c)
− (s − 1)δG2 (c) − (q − 1)δG1 (a) − min{δG1 (a), δG2 (c)}
= 1(−0.6 − 0.2) + 1(−0.5 − 0.3) − (1 − 1)(−0.5) − (1 − 1)(−0.6)
− min{−0.5, −0.6} = (−0.8 − 0.8 + 0.6 = −1.0
(tdβ )G1 ⊕G2 (a, c) = 1.0, (tdγ )G1 ⊕G2 (a, c) = 1.3
(tdη )G1 ⊕G2 (a, c) = −1.1, (tdθ )G1 ⊕G2 (a, c) = −1.7
(td)G1 ⊕G2 (a, c) = (1.5, 1.0, 1.3, −1.0, −1.1, −1.7)
By applying this technique we can find total degree of all vertices in a similar way.
Definition 2.13. let G1 = (M1 , N1 ) and G2 = (M2 , N2 ) are two bipolar single valued neutrosophic fuzzy graphs defined on G1 = (V1 , E1 ) and G2 = (V2 , E2 ) respectively. The Residue
product of G1 and G2 is represented by G1 • G2 = (M1 • M2 , N1 • N2 ). Residue product of
G1 and G2 is defined as the following conditions: (i)
(αM1 • αM2 )((m1 , m2 )) = max{αM1 (m1 ), αM2 (m2 )}, (βM1 • βM2 )((m1 , m2 ))
= min{βM1 (m1 ), βM2 (m2 )}
(γM1 • γM2 )((m1 , m2 )) = min{γM1 (m1 ), γM2 (m2 )}, (δM1 • δM2 )((m1 , m2 ))
= min{δM1 (m1 ), δM2 (m2 )}
(ηM1 • ηM2 )((m1 , m2 )) = max{ηM1 (m1 ), ηM2 (m2 )}, (θM1 • θM2 )((m1 , m2 ))
= max{θM1 (m1 ), θM2 (m2 )}
∀(m1 , m2 ) ∈ (V1 × V2 )
(ii)
(αN1 • αN2 )((m1 , m2 )(n1 , n2 )) = αN1 (m1 n1 ), (βN1 • βN2 )((m1 , m2 )(n1 , n2 )) = βN1 (m1 n1 )
(γN1 • γN2 )((m1 , m2 )(n1 , n2 )) = γN1 (m1 n1 ), (δN1 • δN2 )((m1 , m2 )(n1 , n2 )) = δN1 (m1 n1 )
(ηN1 • ηN2 )((m1 , m2 )(n1 , n2 )) = ηN1 (m1 n1 ), (θN1 • θN2 )((m1 , m2 )(n1 , n2 )) = θN1 (m1 n1 )
∀m1 n1 ∈ E1 , m2 ̸= n2 .
M. Aslam Malik, Hossein Rashmanlou, Muhammad Shoaib, R. A. Borzooei and Morteza Taheri,
A Study on Bipolar Single-Valued Neutrosophic Graphs With Novel Application
Neutrosophic Sets and Systems, Vol. 32, 2020
241
Figure 5. G1
Figure 6. G2
Figure 7. G1 • G2
Example 2.14. Let G1 = (M1 , N1 ) and G2 = (M2 , N2 ) be two BSVNGs on V1 =
{a, b, c, d} and V2 = {e, f } respectively which shown in Figure 5 and Figure 6. Also Residue
product is shown in Figure 7.
Proposition 2.15. Let G1 = (M1 , N1 ) and G2 = (M2 , N2 ) be two BSVNGs of graph G1 =
(V1 , E1 ) and G2 = (V2 , E2 ), respectively. Then the Residue product G1 • G2 of G1 = (V1 , E1 )
and G2 = (V2 , E2 ) is a BSVNG.
M. Aslam Malik, Hossein Rashmanlou, Muhammad Shoaib, R. A. Borzooei and Morteza Taheri,
A Study on Bipolar Single-Valued Neutrosophic Graphs With Novel Application
Neutrosophic Sets and Systems, Vol. 32, 2020
242
Proof. Let G1 = (M1 , N1 ) and G2 = (M2 , N2 ) be two BSVNGs of graph G1 = (V1 , E1 ) and
G2 = (V2 , E2 ), respectively. Let (m1 , m2 )(n1 , n2 ) ∈ E1 × E2 If m1 n1 ∈ E1 and m2 ̸= n2 then
(αN1 • αN2 )((m1 , m2 )(n1 , n2 )) = αN1 (m1 n1 )
≤ min{αM1 (m1 ), αM1 (n1 )}
≤ max{min{αM1 (m1 ), αM1 (n1 )}, min{αM2 (m2 ), αM2 (n2 )}}
= min{max{αM1 (m1 ), αM1 (n1 )}, max{αM2 (m2 ), αM2 (n2 )}}
= min{(αM1 • αM2 )(m1 , m2 ), (αM1 • αM2 )(n1 , n2 )}
(βN1 • βN2 )((m1 , m2 )(n1 , n2 )) = βN1 (m1 n1 )
≥ max{βM1 (m1 ), βM1 (n1 )}
≥ min{max{βM1 (m1 ), βM1 (n1 )}, max{βM2 (m2 ), βM2 (n2 )}}
= max{min{βM1 (m1 ), βM1 (n1 )}, min{βM2 (m2 ), βM2 (n2 )}}
= max{(βM1 • βM2 )(m1 , m2 ), (βM1 • βM2 )(n1 , n2 )}
(γN1 • γN2 )((m1 , m2 )(n1 , n2 )) = γN1 (m1 n1 )
≥ max{γM1 (m1 ), γM1 (n1 )}
≥ min{max{γM1 (m1 ), γM1 (n1 )}, max{γM2 (m2 ), γM2 (n2 )}}
= max{min{γM1 (m1 ), γM1 (n1 )}, min{γM2 (m2 ), γM2 (n2 )}}
= max{(γM1 • γM2 )(m1 , m2 ), (γM1 • γM2 )(n1 , n2 )}
(δN1 • δN2 )((m1 , m2 )(n1 , n2 )) = δN1 (m1 n1 )
≥ max{δM1 (m1 ), δM1 (n1 )}
≥ min{max{δM1 (m1 ), δM1 (n1 )}, max{δM2 (m2 ), δM2 (n2 )}}
= max{min{δM1 (m1 ), δM1 (n1 )}, min{δM2 (m2 ), δM2 (n2 )}}
= max{(δM1 • δM2 )(m1 , m2 ), (δM1 • δM2 )(n1 , n2 )}
(ηN1 • ηN2 )((m1 , m2 )(n1 , n2 )) = ηN1 (m1 n1 )
≤ min{ηM1 (m1 ), ηM1 (n1 )}
≤ max{min{ηM1 (m1 ), ηM1 (n1 )}, min{ηM2 (m2 ), ηM2 (n2 )}}
= min{max{ηM1 (m1 ), ηM1 (n1 )}, max{ηM2 (m2 ), ηM2 (n2 )}}
= min{(ηM1 • ηM2 )(m1 , m2 ), (ηM1 • ηM2 )(n1 , n2 )}
M. Aslam Malik, Hossein Rashmanlou, Muhammad Shoaib, R. A. Borzooei and Morteza Taheri,
A Study on Bipolar Single-Valued Neutrosophic Graphs With Novel Application
Neutrosophic Sets and Systems, Vol. 32, 2020
243
(θN1 • θN2 )((m1 , m2 )(n1 , n2 )) = θN1 (m1 n1 )
≤ min{θM1 (m1 ), θM1 (n1 )}
≤ max{min{θM1 (m1 ), θM1 (n1 )}, min{θM2 (m2 ), θM2 (n2 )}}
= min{max{θM1 (m1 ), θM1 (n1 )}, max{θM2 (m2 ), θM2 (n2 )}}
= min{(θM1 • θM2 )(m1 , m2 ), (θM1 • θM2 )(n1 , n2 )}
Definition 2.16. Let G1 = (M1 , N1 ) and G2 = (M2 , N2 ) be two BSVNGs.For any
vertex(m1 , m2 ) ∈ V1 × V2
(dα )G1 •G2 (m1 , m2 ) =
∑
(αN1 • αN2 )((m1 , m2 )(n1 , n2 ))
(m1 ,m2 )(n1 ,n2 )∈E1 ×E2 .
=
∑
αN1 (m1 n1 ) = (dα )G1 (m1 )
m1 n1 ∈E1 ,m2 ̸=n2
(dβ )G1 •G2 (m1 , m2 ) =
∑
(βN1 • βN2 )((m1 , m2 )(n1 , n2 ))
(m1 ,m2 )(n1 ,n2 )∈E1 ×E2 .
=
∑
βN1 (m1 n1 ) = (dβ )G1 (m1 )
m1 n1 ∈E1 ,m2 ̸=n2
(dγ )G1 •G2 (m1 , m2 ) =
∑
(γN1 • γN2 )((m1 , m2 )(n1 , n2 ))
(m1 ,m2 )(n1 ,n2 )∈E1 ×E2 .
=
∑
γN1 (m1 n1 ) = (dγ )G1 (m1 )
m1 n1 ∈E1 ,m2 ̸=n2
(dδ )G1 •G2 (m1 , m2 ) =
∑
(δN1 • δN2 )((m1 , m2 )(n1 , n2 ))
(m1 ,m2 )(n1 ,n2 )∈E1 ×E2 .
=
∑
δN1 (m1 n1 ) = (dδ )G1 (m1 )
m1 n1 ∈E1 ,m2 ̸=n2
(dη )G1 •G2 (m1 , m2 ) =
∑
(ηN1 • ηN2 )((m1 , m2 )(n1 , n2 ))
(m1 ,m2 )(n1 ,n2 )∈E1 ×E2 .
=
∑
ηN1 (m1 n1 ) = (dη )G1 (m1 )
m1 n1 ∈E1 ,m2 ̸=n2
(dθ )G1 •G2 (m1 , m2 ) =
∑
(θN1 • θN2 )((m1 , m2 )(n1 , n2 ))
(m1 ,m2 )(n1 ,n2 )∈E1 ×E2 .
=
∑
θN1 (m1 n1 ) = (dθ )G1 (m1 )
m1 n1 ∈E1 ,m2 ̸=n2
M. Aslam Malik, Hossein Rashmanlou, Muhammad Shoaib, R. A. Borzooei and Morteza Taheri,
A Study on Bipolar Single-Valued Neutrosophic Graphs With Novel Application
Neutrosophic Sets and Systems, Vol. 32, 2020
244
Definition 2.17. Let G1 = (M1 , N1 ) and G2 = (M2 , N2 ) be two BSVNGs.
For any
vertex(m1 , m2 ) ∈ V1 × V2
(tdα )G1 •G2 (m1 , m2 ) =
∑
(αN1 • αN2 )((m1 , m2 )(n1 , n2 )) + (αM1 • αM2 )(m1 , m2 ))
(m1 ,m2 )(n1 ,n2 )∈E1 ×E2 .
=
∑
αN1 (m1 n1 ) + min{αM1 (m1 ), αM2 (m2 )}
m1 n1 ∈E1 ,m2 ̸=n2
=
∑
αN1 (m1 n1 ) + αM1 (m1 ) + αM2 (m2 ) − max{αM1 (m1 ), αM2 (m2 )}
m1 n1 ∈E1 ,m2 ̸=n2
= (tdα )G1 (m1 ) + αM2 (m2 ) − max{αM1 (m1 ), αM2 (m2 )}
(tdβ )G1 •G2 (m1 , m2 ) =
∑
(βN1 • βN2 )((m1 , m2 )(n1 , n2 )) + (βM1 • βM2 (m1 , m2 ))
(m1 ,m2 )(n1 ,n2 )∈E1 ×E2 .
=
∑
βN1 (m1 n1 ) + max{βM1 (m1 ), βM2 (m2 )}
m1 n1 ∈E1 ,m2 ̸=n2
=
∑
βN1 (m1 n1 ) + βM1 (m1 ) + βM2 (m2 ) − min{βM1 (M1 ), βM2 (m2 )}
m1 n1 ∈E1 ,m2 ̸=n2
= (tdβ )G1 (m1 ) + βM2 (m2 ) − min{βM1 (m1 ), βM2 (m2 )}
(tdγ )G1 •G2 (m1 , m2 ) =
∑
(γN1 • γN2 )((m1 , m2 )(n1 , n2 )) + (γM1 • γM2 (m1 , m2 ))
(m1 ,m2 )(n1 ,n2 )∈E1 ×E2 .
=
∑
γN1 (m1 n1 ) + max{γM1 (m1 ), γM2 (m2 )}
m1 n1 ∈E1 ,m2 ̸=n2
=
∑
γN1 (m1 n1 ) + γM1 (m1 ) + γM2 (m2 ) − min{γM1 (m1 ), γM2 (m2 )}
m1 n1 ∈E1 ,m2 ̸=n2
= (tdγ )G1 (m1 ) + γM2 (m2 ) − min{γM1 (m1 ), γM2 (m2 )}
(tdδ )G1 •G2 (m1 , m2 ) =
∑
(δN1 • δN2 )((m1 , m2 )(n1 , n2 )) + (δM1 • δM2 )(m1 , m2 ))
(m1 ,m2 )(n1 ,n2 )∈E1 ×E2 .
=
∑
δN1 (m1 n1 ) + max{δM1 (m1 ), δM2 (m2 )}
m1 n1 ∈E1 ,m2 ̸=n2
=
∑
δN1 (m1 n1 ) + δM1 (m1 ) + δM2 (m2 ) − min{δM1 (m1 ), δM2 (m2 )}
m1 n1 ∈E1 ,m2 ̸=n2
= (tdδ )G1 (m1 ) + δM2 (m2 ) − min{δM1 (m1 ), δM2 (m2 )}
M. Aslam Malik, Hossein Rashmanlou, Muhammad Shoaib, R. A. Borzooei and Morteza Taheri,
A Study on Bipolar Single-Valued Neutrosophic Graphs With Novel Application
Neutrosophic Sets and Systems, Vol. 32, 2020
(tdη )G1 •G2 (m1 , m2 ) =
∑
245
(ηN1 • ηN2 )((m1 , m2 )(n1 , n2 )) + (ηM1 • ηM2 (m1 , m2 ))
(m1 ,m2 )(n1 ,n2 )∈E1 ×E2 .
=
∑
ηN1 (m1 n1 ) + min{ηM1 (m1 ), ηM2 (m2 )}
m1 n1 ∈E1 ,m2 ̸=n2
=
∑
−
(m1 ), ηM2 (m2 )}
ηN1 (m1 n1 ) + ηM1 (m1 ) + ηM2 (m2 ) − max{IM
1
m1 n1 ∈E1 ,m2 ̸=n2
= (tdη )G1 (m1 ) + ηM2 (m2 ) − max{ηM1 (m1 ), ηM2 (m2 )}
(tdθ )G1 •G2 (m1 , m2 ) =
∑
(θN1 • θN2 )((m1 , m2 )(n1 , n2 )) + (θM1 • θM2 (m1 , m2 ))
(m1 ,m2 )(n1 ,n2 )∈E1 ×E2 .
=
∑
θN1 (m1 n1 ) + min{θM1 (m1 ), θM2 (m2 )}
m1 n1 ∈E1 ,m2 ̸=n2
=
∑
θN1 (m1 n1 ) + θM1 (m1 ) + θM2 (m2 ) − max{θM1 (m1 ), θM2 (m2 )}
m1 n1 ∈E1 ,m2 ̸=n2
= (tdθ )G1 (m1 ) + θM2 (m2 ) − max{θM1 (m1 ), θM2 (m2 )}
Example 2.18. In Example 2.14 we have to find the degree and total degree of vertices of
G1 • G2 by using Figure 5, Figure 6, and Figure 7.
(dβ )G1 •G2 (a, f ) = (dβ )G1 (a) = 0.5 + 0.4 = 0.9
(dη )G1 •G2 (a, f ) = (dη )G1 (a) = −0.4 − 0.5 = −0.9
(dα )G1 •G2 (a, f ) = 0.5, (dγ )G1 •G2 (a, f ) = 0.9
(dδ )G1 •G2 (a, f ) = −0.2, (dθ )G1 •G2 (a, f ) = −1.0
(d)G1 •G2 (a, f ) = (0.5, 0.9, 0.9, −0.2, −0.9, −1.0)
By applying same method we can find degree of all vertices. Now we are to find total degree
of vertices. For this select vertices (a,f)
(tdβ )G1 •G2 (a, f ) = (tdβ )G1 (a) + βM2 (f ) − min{βM1 (a), βM2 (f )}
= (0.5 + 0.4 + 0.4) + 0.8 − min(0.3, 0.8)
= 1.3 + 0.8 − 0.3 = 1.8
(tdη )G1 •G2 (a, f ) = (tdη )G1 (a) + ηM2 (f ) − max{ηM1 (a), ηM2 (f )}
= (−0.4 − 0.3 − 0.5) + (−0.2) − max(−0.3, −0.2)
= −1.2 − 0.2 + 0.2 = −1.2
(tdγ )G1 •G2 (a, f ) = 1.1, (tdδ )G1 •G2 (a, f ) = −0.4
M. Aslam Malik, Hossein Rashmanlou, Muhammad Shoaib, R. A. Borzooei and Morteza Taheri,
A Study on Bipolar Single-Valued Neutrosophic Graphs With Novel Application
Neutrosophic Sets and Systems, Vol. 32, 2020
246
(tdθ )G1 •G2 (a, f ) = −1.4, (tdα )G1 •G2 (a, f ) = 0.8
So (td)G1 •G2 (a, f ) = (0.8, 1.8, 1.1 − 0.4, −1.2, −1.4)
by applying similar method we can find total degree of all others vertices in a similar way.
Definition 2.19. let G1 = (M1 , N1 ) and G2 = (M2 , N2 ) are bipolar single valued neutrosophic
fuzzy graphs defined on G1 = (V1 , E1 ) and G2 = (V1 , E2 ) respectively. The maximal product
of G1 and G2 is represented by G1 ∗ G2 = (M1 ∗ M2 , N1 ⊕ N2 ). The Maximal product of G1 and G2
is defined as the following conditions (i)
(αM1 ∗ αM2 )((m1 , m2 )) = max{αM1 (m1 ), αM2 (m2 )}, (βM1 ∗ βM2 )((m1 , m2 ))
= min{βM1 (m1 ), βM2 (m2 )}
(γM1 ∗ γM2 )((m1 , m2 )) = min{γM1 (m1 ), γM2 (m2 )}, (δM1 ∗ δM2 )((m1 , m2 ))
= min{δM1 (m1 ), δM2 (m2 )}
(ηM1 ∗ ηM2 )((m1 , m2 )) = max{ηM1 (m1 ), ηM2 (m2 )}, (θM1 ∗ θM2 )((m1 , m2 ))
= max{θM1 (m1 ), θM2 (m2 )}
∀ (m1 , m2 ) ∈ (V1 × V2 )
(ii)
(αM1 ∗ αM2 )((m, m2 )(m, n2 )) = max{αM1 (m), αN2 (m2 n2 )}, (βM1 ∗ βM2 )((m, m2 )(m, n2 ))
= min{βM1 (m), βN2 (m2 n2 )}
(γM1 ∗ γM2 )((m, m2 )(m, n2 )) = min{γMm1 (m), γN2 (m2 n2 )}, (δM1 ∗ δM2 )((m, m2 )(m, n2 ))
= min{δM1 (m), δN2 (m2 n2 )}
(ηM1 ∗ ηM2 )((m, m2 )(m, n2 )) = max{ηM1 (m), ηN2 (m2 n2 )}, (θM1 ∗ θM2 )((m, m2 )(m, n2 ))
= max{θMm1 (m), θN2 (m2 n2 )}
∀ m ∈ V1 and m2 n2 ∈ E2
(iii)
(αM1 ∗ αM2 )((m1 , m)(n1 , m)) = max{αN1 (m1 n1 ), αM2 (m)}, (βM1 ∗ βM2 )((m1 , m)(n1 , m))
= min{βN1 (m1 n1 ), βM2 (m)}
(γM1 ∗ γM2 )((m1 , m)(n1 , m)) = min{γN1 (m1 n1 ), γM2 (m)}, (δM1 ∗ δM2 )((m1 , m)(n1 , m))
= min{δN1 (m1 n1 ), δM2 (m)}
M. Aslam Malik, Hossein Rashmanlou, Muhammad Shoaib, R. A. Borzooei and Morteza Taheri,
A Study on Bipolar Single-Valued Neutrosophic Graphs With Novel Application
Neutrosophic Sets and Systems, Vol. 32, 2020
247
Figure 8. G1
Figure 9. G2
(ηM1 ∗ ηM2 )((m1 , m)(n1 , m)) = max{ηN1 (m1 n1 ), ηM2 (m)}, (θM1 ∗ θM2 )((m1 , m)(n1 , m))
= max{θN1 (m1 n1 ), θM2 (m)}
∀ m ∈ V2 and m1 n1 ∈ E1
Example 2.20. Let G1 = (M1 , N1 ) and G2 = (M2 , N2 ) be two BSVNGs on V1 =
{a, b} and V2 = {c, d, e} respectively which shown in Figure 8 and Figure 9. Also maximal
product is shown in Figure 10.
Proposition 2.21. Let G1 = (M1 , N1 ) and G2 = (M2 , N2 ) be two BSVNGs of graph G1 =
(V1 , E1 ) and G2 = (V2 , E2 ), respectively. Then then maximal product G1 ∗ G2 of G1 = (V1 , E1 )
and G2 = (V2 , E2 ) is a BSVNG.
Proof. Let G1 = (M1 , N1 ) and G2 = (M2 , N2 ) be two BSVNGs of graph G1 = (V1 , E1 ) and
G2 = (V2 , E2 ), respectively. Then the Maximal product G1 ∗ G2 of G1 = (V1 , E1 ) and G2 =
(V2 , E2 ) can be proved. Let (m1 , m2 )(n1 , n2 ) ∈ E1 × E2
M. Aslam Malik, Hossein Rashmanlou, Muhammad Shoaib, R. A. Borzooei and Morteza Taheri,
A Study on Bipolar Single-Valued Neutrosophic Graphs With Novel Application
Neutrosophic Sets and Systems, Vol. 32, 2020
248
Figure 10. G1 ∗ G2
(i) If m1 = n1 = m
(αN1 ∗ αN2 )((m, m2 )(m, n2 )) = max{αM1 (m), αN2 (m2 n2 )}
≤ max{αM1 (m), min{αM2 (m2 ), αM2 (n2 )}}
= min{max{{αM1 (m), αM2 (m2 )}, max{{αM1 (m), αM2 (n2 )}}
= min{(αM1 ∗ αM2 )(m, m2 ), (αM1 ∗ αM2 )(m, n2 )}
(βN1 ∗ βN2 )((m, m2 )(m, n2 )) = min{βM1 (m), βN2 (m2 n2 )}
≥ min{βM1 (m), max{βM2 (m2 ), βM2 (n2 )}}
= max{min{{βM1 (m), βM2 (m2 )}, min{{βM1 (m), βM2 (n2 )}}
= max{(βM1 ∗ βM2 )(m, m2 ), (βM1 ∗ βM2 )(m, n2 )}
(γN1 ∗ γN2 )((m, m2 )(m, n2 )) = min{γM1 (m), γN2 (m2 n2 )}
≥ min{γM1 (m), max{γM2 (m2 ), γM2 (n2 )}}
= max{min{{γM1 (m), γM2 (m2 )}, min{{γM1 (m), γM2 (n2 )}}
= max{(γM1 ∗ γM2 )(m, m2 ), (γM1 ∗ γM2 )(m, n2 )}
M. Aslam Malik, Hossein Rashmanlou, Muhammad Shoaib, R. A. Borzooei and Morteza Taheri,
A Study on Bipolar Single-Valued Neutrosophic Graphs With Novel Application
Neutrosophic Sets and Systems, Vol. 32, 2020
249
(δN1 ∗ δN2 )((m, m2 )(m, n2 )) = min{δM1 (m), δN2 (m2 n2 )}
≥ min{δM1 (m), max{δM2 (m2 ), δM2 (n2 )}}
= max{min{{δM1 (m), δM2 (m2 )}, min{{δM1 (m), δM2 (n2 )}}
= max{(δM1 ∗ δM2 )(m, m2 ), (δM1 ∗ δM2 )(m, n2 )}
(ηN1 ∗ ηN2 )((m, m2 )(m, n2 )) = max{ηM1 (m), ηN2 (m2 n2 )}
≤ max{ηM1 (m), min{ηM2 (m2 ), ηM2 (n2 )}}
= min{max{{ηM1 (m), ηM2 (m2 )}, max{{ηM1 (m), ηM2 (n2 )}}
= min{(ηM1 ∗ ηM2 )(m, m2 ), (ηM1 ∗ ηM2 )(m, n2 )}
(θN1 ∗ θN2 )((m, m2 )(m, n2 )) = max{θM1 (m), θN2 (m2 n2 )}
≤ max{θM1 (m), min{θM2 (m2 ), θM2 (n2 )}}
= min{max{{θM1 (m), θM2 (m2 )}, max{{θM1 (m), θM2 (n2 )}}
= min{(θM1 ∗ θM2 )(m, m2 ), (θM1 ∗ θM2 )(m, n2 )}
(ii) If m2 = n2 = m
(αN1 ∗ αN2 )((m1 , m)(n1 , m)) = max{αN1 (m1 n1 ), αM2 (m)}
≤ max{min{αN1 (m1 n1 ), αM2 (m)}
= min{max{{αN1 (m1 ), αM2 (m)}, max{{αM1 (n1 ), αM2 (m)}}
= min{(αM1 ∗ αM2 )(m1 , m), (αM1 ∗ αM2 )(n1 , m)}
(βN1 ∗ βN2 )((m1 , m)(n1 , m)) = min{βN1 (m1 n1 ), βM2 (m)}
≥ min{max{βN1 (m1 n1 ), βM2 (m)}
= max{min{{βN1 (m1 ), βM2 (m)}, min{{βM1 (n1 ), βM2 (m)}}
= max{(βM1 ∗ βM2 )(m1 , m), (βM1 ∗ βM2 )(n1 , m)}
M. Aslam Malik, Hossein Rashmanlou, Muhammad Shoaib, R. A. Borzooei and Morteza Taheri,
A Study on Bipolar Single-Valued Neutrosophic Graphs With Novel Application
Neutrosophic Sets and Systems, Vol. 32, 2020
250
(γN1 ∗ γN2 )((m1 , m)(n1 , m)) = min{γN1 (m1 n1 ), γM2 (m)}
≥ min{max{γN1 (m1 n1 ), γM2 (m)}
= max{min{{γN1 (m1 ), γM2 (m)}, min{{γM1 (n1 ), γM2 (m)}}
= max{(γM1 ∗ γM2 )(m1 , m), (γM1 ∗ γM2 )(n1 , m)}
(δN1 ∗ δN2 )((m1 , m)(n1 , m)) = min{δN1 (m1 n1 ), δM2 (m)}
≥ min{max{δN1 (m1 n1 ), δM2 (m)}
= max{min{{δN1 (m1 ), δM2 (m)}, min{{δM1 (n1 ), δM2 (m)}}
= max{(δM1 ∗ δM2 )(m1 , m), (δM1 ∗ δM2 )(n1 , m)}
(ηN1 ∗ ηN2 )((m1 , m)(n1 , m)) = max{ηN1 (m1 n1 ), ηM2 (m)}
≤ max{min{ηN1 (m1 n1 ), ηM2 (m)}
= min{max{{ηN1 (m1 ), ηM2 (m)}, max{{ηM1 (n1 ), ηM2 (m)}}
= min{(ηM1 ∗ ηM2 )(m1 , m), (ηM1 ∗ ηM2 )(n1 , m)}
(θN1 ∗ θN2 )((m1 , m)(n1 , m)) = max{θN1 (m1 n1 ), θM2 (m)}
≤ max{min{θN1 (m1 n1 ), θM2 (m)}
= min{max{{θN1 (m1 ), θM2 (m)}, max{{θM1 (n1 ), θM2 (m)}}
= min{(θM1 ∗ θM2 )(m1 , m), (θM1 ∗ θM2 )(n1 , m)}
Definition 2.22. Let G1 = (M1 , N1 ) and G2 = (M2 , N2 ) be two BSVNGs. ∀(m1 , m2 ) ∈ V1 ×V2
∑
(dα )G1 ∗G2 (m1 , m2 ) =
(αN1 ∗ αN2 )((m1 , m2 )(n1 , n2 ))
(m1 ,m2 )(n1 ,n2 )∈E1 ×E2 .
=
∑
max{αM1 (m1 ), αN2 (m2 n2 )}
m1 =n1 ,m2 n2 ∈E2
+
∑
max{αN1 (m1 n1 ), αM2 (m2 )}
m1 n1 ∈E1 ,m2 =n2
M. Aslam Malik, Hossein Rashmanlou, Muhammad Shoaib, R. A. Borzooei and Morteza Taheri,
A Study on Bipolar Single-Valued Neutrosophic Graphs With Novel Application
Neutrosophic Sets and Systems, Vol. 32, 2020
(dβ )G1 ∗G2 (m1 , m2 ) =
251
∑
(βN1 ∗ βN2 )((m1 , m2 )(n1 , n2 ))
(m1 ,m2 )(n1 ,n2 )∈E1 ×E2 .
=
∑
min{βM1 (m1 ), βN2 (m2 n2 )}
m1 =n1 ,m2 n2 ∈E2
+
∑
min{βN1 (m1 n1 ), βM2 (m2 )}
m1 n1 ∈E1 ,m2 =n2
(dγ )G1 ∗G2 (m1 , m2 ) =
∑
(γN1 ∗ γN2 )((m1 , m2 )(n1 , n2 ))
(m1 ,m2 )(n1 ,n2 )∈E1 ×E2 .
=
∑
min{γM1 (m1 ), γN2 (m2 n2 )}
m1 =n1 ,m2 n2 ∈E2
+
∑
min{γN1 (m1 n1 ), γM2 (m2 )}
m1 n1 ∈E1 ,m2 =n2
(dδ )G1 ∗G2 (m1 , m2 ) =
∑
(δN1 ∗ δN2 )((m1 , m2 )(n1 , n2 ))
(m1 ,m2 )(n1 ,n2 )∈E1 ×E2 .
=
∑
min{δM1 (m1 ), δN2 (m2 n2 )}
m1 =n1 ,m2 n2 ∈E2
+
∑
min{δN1 (m1 n1 ), δM2 (m2 )}
m1 n1 ∈E1 ,m2 =n2
(dη )G1 ∗G2 (m1 , m2 ) =
∑
(ηN1 ∗ ηN2 )((m1 , m2 )(n1 , n2 ))
(m1 ,m2 )(n1 ,n2 )∈E1 ×E2 .
=
∑
max{ηM1 (m1 ), ηN2 (m2 n2 )}
m1 =n1 ,m2 n2 ∈E2
+
∑
max{ηN1 (m1 n1 ), ηM2 (m2 )}
m1 n1 ∈E1 ,m2 =n2
(dθ )G1 ∗G2 (m1 , m2 ) =
∑
(θN1 ∗ θN2 )((m1 , m2 )(n1 , n2 ))
(m1 ,m2 )(n1 ,n2 )∈E1 ×E2 .
=
∑
max{θM1 (m1 ), θN2 (m2 n2 )}
m1 =n1 ,m2 n2 ∈E2
+
∑
max{θN1 (m1 n1 ), θM2 (m2 )}
m1 n1 ∈E1 ,m2 =n2
Theorem 2.23. Let G1 = (M1 , N1 ) and G2 = (M2 , N2 ) are two BSVNGs. If αM1 ≥ αN2 , βM1 ≤
βN2 , γM1 ≤ γN2 and αM2 ≥ αN1 , βM2 ≤ βN1 , γM2 ≤ γN1 . Also If δM1 ≤ δN2 , ηM1 ≥ ηN2 , θM1 ≥
θN2 and δM2 ≤ δN1 , ηM2 ≥ ηN1 , θM2 ≥ θN1 Then for every ∀(m1 , m2 ) ∈ V1 × V2
(dα )G1 ∗G2 (m1 , m2 ) =(d)G2 (m2 )αM1 (m1 ) + (d)G1 (m1 )αM2 (m2 )
(dβ )G1 ∗G2 (m1 , m2 )=(d)G2 (m2 )βM1 (m1 ) + (d)G1 (m1 )βM2 (m2 )
(dγ )G1 ∗G2 (m1 , m2 )=(d)G2 (m2 )γM1 (m1 ) + (d)G1 (m1 )γM2 (m2 )
(dδ )G1 ∗G2 (m1 , m2 ) =(d)G2 (m2 )δM1 (m1 ) + (d)G1 (m1 )δM2 (m2 )
M. Aslam Malik, Hossein Rashmanlou, Muhammad Shoaib, R. A. Borzooei and Morteza Taheri,
A Study on Bipolar Single-Valued Neutrosophic Graphs With Novel Application
Neutrosophic Sets and Systems, Vol. 32, 2020
252
(dη )G1 ∗G2 (m1 , m2 )=(d)G2 (m2 )ηM1 (m1 ) + (d)G1 (m1 )ηM2 (m2 )
(dθ )G1 ∗G2 (m1 , m2 )=(d)G2 (m2 )θM1 (m1 ) + (d)G1 (m1 )θM2 (m2 )
Proof.
∑
(dα )G1 ∗G2 (m1 , m2 ) =
(αN1 ∗ αN2 )((m1 , m2 )(n1 , n2 ))
(m1 ,m2 )(n1 ,n2 )∈E1 ×E2 .
∑
=
max{αM1 (m1 ), αN2 (m2 n2 )}
m1 =n1 ,m2 n2 ∈E2
∑
+
max{αN1 (m1 n1 ), αM2 (m2 )}
m1 n1 ∈E1 ,m2 =n2
∑
=
∑
αN2 (m2 n2 ) +
αN1 (m1 n1 )
m1 n1 ∈E1 ,m2 =n2
m2 n2 ∈E2 ,m1 =n1
= (d)G2 (m2 )αM1 (m1 ) + (d)G1 (m1 )αM2 (m2 )
∑
(dδ )G1 ∗G2 (m1 , m2 ) =
(δN1 ∗ δN2 )((m1 , m2 )(n1 , n2 ))
(m1 ,m2 )(n1 ,n2 )∈E1 ×E2 .
∑
=
min{δM1 (m1 ), δN2 (m2 n2 )}
m1 =n1 ,m2 n2 ∈E2
∑
+
min{δN1 (m1 n1 ), δM2 (m2 )}
m1 n1 ∈E1 ,m2 =n2
∑
=
∑
δN2 (m2 n2 ) +
m2 n2 ∈E2 ,m1 =n1
δN1 (m1 n1 )
m1 n1 ∈E1 ,m2 =n2
= (d)G2 (m2 )δM1 (m1 ) + (d)G1 (m1 )δM2 (m2 )
In a similar way others four will proved obviously.
Definition 2.24. Let G1 = (M1 , N1 ) and G2 = (M2 , N2 ) be two BSVNGs. ∀(m1 , m2 ) ∈ V1 ×V2
(tdα )G1 ∗G2 (m1 , m2 ) =
∑
(αN1 ∗ αN2 )((m1 , m2 )(n1 , n2 )) + (αM1 ∗ αM2 (m1 , m2 )
(m1 ,m2 )(n1 ,n2 )∈E1 ×E2 .
=
∑
max{αM1 (m1 ), αN2 (m2 n2 )}
m1 =n1 ,m2 n2 ∈E2
+
∑
max{αN1 (m1 n1 ), αM2 (m2 )}
m1 n1 ∈E1 ,m2 =n2
+ max{αM1 (m1 ), αM2 (m2 )}
M. Aslam Malik, Hossein Rashmanlou, Muhammad Shoaib, R. A. Borzooei and Morteza Taheri,
A Study on Bipolar Single-Valued Neutrosophic Graphs With Novel Application
Neutrosophic Sets and Systems, Vol. 32, 2020
∑
(tdβ )G1 ∗G2 (m1 , m2 ) =
253
(βN1 ∗ βN2 )((m1 , m2 )(n1 , n2 )) + (βM1 ∗ βM2 (m1 , m2 )
(m1 ,m2 )(n1 ,n2 )∈E1 ×E2 .
∑
=
min{βM1 (m1 ), βN2 (m2 n2 )}
m1 =n1 ,m2 n2 ∈E2
∑
+
min{βN1 (m1 n1 ), βM2 (m2 )}
m1 n1 ∈E1 ,m2 =n2
+ min{βM1 (m1 ), βM2 (m2 )}
(tdγ )G1 ∗G2 (m1 , m2 ) =
∑
(γN1 ∗ γN2 )((m1 , m2 )(n1 , n2 )) + (γM1 ∗ γM2 (m1 , m2 )
(m1 ,m2 )(n1 ,n2 )∈E1 ×E2 .
=
∑
min{γM1 (m1 ), γN2 (m2 n2 )}
m1 =n1 ,m2 n2 ∈E2
+
∑
min{γN1 (m1 n1 , γM2 (m2 )}
m1 n1 ∈E1 ,m2 =n2
+ max{γM1 (m1 ), γM2 (m2 )}
∑
(tdδ )G1 ∗G2 (m1 , m2 ) =
(δN1 ∗ δN2 )((m1 , m2 )(n1 , n2 )) + (δM1 ∗ δM2 (m1 , m2 )
(m1 ,m2 )(n1 ,n2 )∈E1 ×E2 .
=
∑
min{δM1 (m1 ), δN2 (m2 n2 )}
m1 =n1 ,m2 n2 ∈E2
+
∑
min{δN1 (m1 n1 ), δM2 (m2 )}
m1 n1 ∈E1 ,m2 =n2
+ min{δM1 (m1 ), δM2 (m2 )}
(tdη )G1 ∗G2 (m1 , m2 ) =
∑
(ηN1 ∗ ηN2 )((m1 , m2 )(n1 , n2 )) + (ηM1 ∗ ηM2 (m1 , m2 )
(m1 ,m2 )(n1 ,n2 )∈E1 ×E2 .
=
∑
max{ηM1 (m1 ), ηN2 (m2 n2 )}
m1 =n1 ,m2 n2 ∈E2
+
∑
max{ηN1 (m1 n1 ), ηM2 (m2 )}
m1 n1 ∈E1 ,m2 =n2
+ max{ηM1 (m1 ), ηM2 (m2 )}
(tdθ )G1 ∗G2 (m1 , m2 ) =
∑
(θN1 ∗ θN2 )((m1 , m2 )(n1 , n2 )) + (θM1 ∗ θM2 (m1 , m2 )
(m1 ,m2 )(n1 ,n2 )∈E1 ×E2 .
=
∑
max{θM1 (m1 ), θN2 (m2 n2 )}
m1 =n1 ,m2 n2 ∈E2
+
∑
max{θN1 (m1 n1 , θM2 (m2 )}
m1 n1 ∈E1 ,m2 =n2
+ max{θM1 (m1 ), θM2 (m2 )}
M. Aslam Malik, Hossein Rashmanlou, Muhammad Shoaib, R. A. Borzooei and Morteza Taheri,
A Study on Bipolar Single-Valued Neutrosophic Graphs With Novel Application
Neutrosophic Sets and Systems, Vol. 32, 2020
254
Theorem 2.25. Let G1 = (M1 , N1 ) and G2 = (M2 , N2 ) be two BSVNGs. If αM1 ≥ αN2 , βM1 ≤
βN2 , γM1 ≤ γN2 and αM2 ≥ αN1 , βM2 ≤ βN1 , γM2 ≤ γN1 . Also If δM1 ≤ δN2 , ηM1 ≥ ηN2 , θM1 ≥
θN2 and δM2 ≤ δN1 , ηM2 ≥ ηN1 , θM2 ≥ θN1 Then for every ∀(m1 , m2 ) ∈ V1 × V2
(dα )G1 ∗G2 (m1 , m2 ) =(d)G2 (m2 )αM1 (m1 ) + (d)G1 (m1 )αM2 (m2 )
(dβ )G1 ∗G2 (m1 , m2 )=(d)G2 (m2 )βM1 (m1 ) + (d)G1 (m1 )βM2 (m2 )
(dγ )G1 ∗G2 (m1 , m2 )=(d)G2 (m2 )γM1 (m1 ) + (d)G1 (m1 )γM2 (m2 )
(dδ )G1 ∗G2 (m1 , m2 ) =(d)G2 (m2 )δM1 (m1 ) + (d)G1 (m1 )δM2 (m2 )
(dη )G1 ∗G2 (m1 , m2 )=(d)G2 (m2 )ηM1 (m1 ) + (d)G1 (m1 )ηM2 (m2 )
(dθ )G1 ∗G2 (m1 , m2 )=(d)G2 (m2 )θM1 (m1 ) + (d)G1 (m1 )θM2 (m2 )
Proof.
(tdα )G1 ∗G2 (m1 , m2 ) =
∑
(αN1 ∗ αN2 )((m1 , m2 )(n1 , n2 )) + (αM1 ∗ αM2 )(m1 , m2 )
(m1 ,m2 )(n1 ,n2 )∈E1 ×E2 .
=
∑
max{αM1 (m1 ), αN2 (m2 n2 )}
m1 =n1 ,m2 n2 ∈E2
+
∑
max{αN1 (m1 n1 ), αM2 (m2 )}
m1 n1 ∈E1 ,m2 =n2
+ max{αM1 (m1 ), αM2 (m2 )}
∑
αN2 (m2 n2 ) +
=
∑
αN1 (m1 n1 )
m1 n1 ∈E1 ,m2 =n2
m2 n2 ∈E2 ,m1 =n1
+ max{αM1 (m1 ), αM2 (m2 )}
= (d)G2 (m2 )αM1 (m1 ) + (d)G1 (m1 )αM2 (m2 ) + max{αM1 (m1 ), αM2 (m2 )}
(tdδ )G1 ∗G2 (m1 , m2 ) =
∑
(δN1 ∗ δN2 )((m1 , m2 )(n1 , n2 )) + (δM1 ∗ δM2 )(m1 , m2 )
(m1 ,m2 )(n1 ,n2 )∈E1 ×E2 .
=
∑
min{δM1 (m1 ), δN2 (m2 n2 )}
m1 =n1 ,m2 n2 ∈E2
+
∑
min{δN1 (m1 n1 ), δM2 (m2 )}
m1 n1 ∈E1 ,m2 =n2
+ min{δM1 (m1 ), δM2 (m2 )}
∑
δN2 (m2 n2 ) +
=
m2 n2 ∈E2 ,m1 =n1
∑
δN1 (m1 n1 )
m1 n1 ∈E1 ,m2 =n2
+ min{δM1 (m1 ), δM2 (m2 )}
= (d)G2 (m2 )δM1 (m1 ) + (d)G1 (m1 )δM2 (m2 ) + min{δM1 (m1 ), δM2 (m2 )}
In a similar way others four will proved obviously.
M. Aslam Malik, Hossein Rashmanlou, Muhammad Shoaib, R. A. Borzooei and Morteza Taheri,
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255
Example 2.26. In Example 2.20 we have to find the degree and total degree of vertices of
G1 ∗ G2 by using Figure 8, Figure 9, and Figure 10. Select the vertex (e,a).
(dα )G1 ∗G2 (a, c) = (d)G2 (c)αM1 (a) + (d)G1 (a)αM2 (c)
= 2(0.4) + 1(0.5) = 0.8 + 0.5 = 1.3
(tdδ )G1 ∗G2 (a, c) = (d)G2 (c)δM1 (a) + (d)G1 (a)δM2 (c)
= 2(−0.5) + 1(−0.4) = −1.0 − 0.4 = −1.4
, (dβ )G1 ∗G2 (a, c) = 1.0 , (dγ )G1 ∗G2 (e, a) = 1.1 , (tdη )G1 ∗G2 (a, c) = −1.3 , (tdθ )G1 ∗G2 (a, c) = −1.2.
By applying the same method we can find the degree of all vertices.now we are find the total
degree of vertices in maximal product. For this select the same vertex (e,a).
(tdα )G1 ∗G2 (a, c) = (d)G2 (c)αM1 (a) + (d)G1 (a)αM2 (c) + max{αM1 (a), αM2 (c)}
= 2(0.4) + 1(0.5) + max(0.4, 0.5) = 0.8 + 0.5 + 0.5 = 1.8
(tdθ )G1 ∗G2 (a, c) = (d)G2 (c)θM1 (a) + (d)G1 (a)θM2 (c) + min{θM1 (a), θM2 (c)}
= 2(−0.3) + 1(−0.6) + min(−0.3, −0.6) = −0.6 − 0.6 − 0.6 = −1.8
(tdβ )G1 ∗G2 (a, c) = 1.3 ,(tdγ )G1 ∗G2 (a, c) = 1.4, (tdδ )G1 ∗G2 (a, c) = −1.8 ,(tdη )G1 ∗G2 (a, c) = −1.8. By
applying same method or technique we can find all other vertices total degree.
Definition 2.27. Let G1 = (M1 , N1 ) and G2 = (M2 , N2 ) are two bipolar single valued neutrosophic fuzzy graphs defined on G1 = (V1 , E1 ) and G2 = (V2 , E2 ) respectively. The rejection of
G1 and G2 is represented by G1 |G2 = (M1 |M2 , N1 |N2 ). Rejection of G1 and G2 is defined as the
following conditions:
(i)
(αM1 |αM2 )((m1 , m2 )) = min{αM1 (m1 ), αM2 (m2 )}, (βM1 |βM2 )((m1 , m2 )) = max{βM1 (m1 ), βM2 (m2 )}
(γM1 |γM2 )((m1 , m2 )) = max{γM1 (m1 ), γM2 (m2 )}, (δM1 |δM2 )((m1 , m2 )) = max{δM1 (m1 ), δM2 (m2 )}
(ηM1 |ηM2 )((m1 , m2 )) = min{ηM1 (m1 ), ηM2 (m2 )}, (θM1 |θM2 )((m1 , m2 )) = min{θM1 (m1 ), θM2 (m2 )}
∀ (m1 , m2 ) ∈ (V1 × V2 ).
(ii)
(αN1 |αN2 )((m, m2 )(m, n2 )) = min{αM1 (m), αM2 (m2 ), αM2 (n2 )}, (βN1 |βN2 )((m, m2 )(m, n2 ))
= max{βM1 (m), βM2 (m2 ), βM2 (n2 )}
M. Aslam Malik, Hossein Rashmanlou, Muhammad Shoaib, R. A. Borzooei and Morteza Taheri,
A Study on Bipolar Single-Valued Neutrosophic Graphs With Novel Application
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(γN1 |γN2 )((m, m2 )(m, n2 )) = max{γM1 (m), γM2 (m2 ), γM2 (n2 )}, (δN1 |δN2 )((m, m2 )(m, n2 ))
= max{δM1 (m), δM2 (m2 ), δM2 (n2 )}
(ηN1 |ηN2 )((m, m2 )(m, n2 )) = min{ηM1 (m), ηM2 (m2 ), ηM2 (n2 )}, (θN1 |θN2 )((m, m2 )(m, n2 ))
= min{θM1 (m), θM2 (m2 ), θM2 (n2 )}
∀ m ∈ V2 and m2 n2 ̸∈ E2 .
(iii)
(αN1 |αN2 )((m, m2 )(m, n2 )) = min{αM1 (m), αM2 (m2 ), αM2 (n2 )}, (βN1 |βN2 )((m, m2 )(m, n2 ))
= max{βM1 (m), βM2 (m2 ), βM2 (n2 )}
(γN1 |γN2 )((m, m2 )(m, n2 )) = max{γM1 (m), γM2 (m2 ), γM2 (n2 )}, (δN1 |δN2 )((m, m2 )(m, n2 ))
= max{δM1 (m), δM2 (m2 ), δM2 (n2 )}
(ηN1 |ηN2 )((m, m2 )(m, n2 )) = min{ηM1 (m), ηM2 (m2 ), ηM2 (n2 )}, (θN1 |θN2 )((m, m2 )(m, n2 ))
= min{θM1 (m), θM2 (m2 ), θM2 (n2 )}
∀ z ∈ V2 and m1 n1 ̸∈ E1 .
(i∨) (αN1 |αN2 )((m1 , m2 )(n1 , n2 )) = min{αM1 (m1 ), αM1 (n1 ), αM2 (m2 ), αM2 (n2 )},
(βN1 |βN2 )((m1 , m2 )(n1 , n2 ))
max{βM1 (m1 ), βM1 (n1 ), βM2 (m2 ), αN2 (n2 )},
=
(γN1 |γN2 )((m1 , m2 )(n1 , n2 ))
=
max{γM1 (m1 ), γM1 (n1 ), γM2 (m2 ), αM2 (n2 )},
(δN1 |δN2 )((m1 , m2 )(n1 , n2 )) = max{δM1 (m1 ), δM1 (n1 ), δM2 (m2 ), δM2 (n2 )}
,
(ηN1 |ηN2 )((m1 , m2 )(n1 , n2 )) = min{ηM1 (m1 ), ηM1 (n1 ), ηM2 (m2 ), δN2 (n2 )}
,
(θN1 |θN2 )((m1 , m2 )(n1 , n2 )) = min{θM1 (m1 ), θM1 (n1 ), θM2 (m2 ), δM2 (n2 )}
∀ m1 n1 ̸∈ E1 and m2 n2 ̸∈ E2 .
Example 2.28. Let G1 = (M1 , N1 ) and G2 = (M2 , N2 ) be two BSVNGs on V1 =
{a, b, c, d} and V2 = {e, f }, respectively which shown in Figure 11 and Figure 12. Also rejection
shown in Figure 13.
Proposition 2.29. Let G1 = (M1 , N1 ) and G2 = (M2 , N2 ) be two BSVNGs of graph G1 =
(V1 , E1 ) and G2 = (V2 , E2 ), respectively. Then the rejection G1 |G2 of G1 = (V1 , E1 ) and
G2 = (V2 , E2 ) is a BSVNG.
M. Aslam Malik, Hossein Rashmanlou, Muhammad Shoaib, R. A. Borzooei and Morteza Taheri, A Study
on Bipolar Single-Valued Neutrosophic Graphs With Novel Application
Neutrosophic Sets and Systems, Vol. 32, 2020
257
Figure 11. G1
Figure 12. G2
Figure 13. G1 | G2
Proof. Suppose that G1 = (M1 , N1 ) and G2 = (M2 , N2 ) be two BSVNGs of graph G1 = (V1 , E1 )
and G2 = (V2 , E2 ) respectively. Then for (m1 , m2 )(n1 , n2 ) ∈ E1 × E2 .
M. Aslam Malik, Hossein Rashmanlou, Muhammad Shoaib, R. A. Borzooei and Morteza Taheri,
A Study on Bipolar Single-Valued Neutrosophic Graphs With Novel Application
Neutrosophic Sets and Systems, Vol. 32, 2020
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(i) If m1 = n1 , m2 n2 ̸∈ E2
(βN1 |βN2 )((m1 , m2 )(n1 , n2 )) = max{βM1 (m1 ), βM2 (m2 ), βM2 (n2 )}
= max{max{βM1 (m1 ), βM2 (m2 )}, max{βM1 (n1 ), βM2 (n2 )}}
= max{(βM1 |βM2 )(m1 , m2 ), (βM1 |βM2 )(n1 , n2 )}
(ηN1 |ηN2 )((m1 , m2 )(n1 , n2 )) = min{ηM1 (m1 ), ηM2 (m2 ), ηM2 (n2 )}
= min{min{ηM1 (m1 ), ηM2 (m2 )}, min{ηM1 (n1 ), ηM2 (n2 )}}
= min{(ηM1 |ηM2 )(m1 , m2 ), (ηM1 |ηM2 )(n1 , n2 )}
In a similar way others four will proved obviously.
(ii) If m2 = n2 , m1 n1 ̸∈ E1
(αN1 |αN2 )((m1 , m2 )(n1 , n2 )) = min{αM1 (m1 ), αM1 (n1 ), αM2 (m2 )}
= min{min{αM1 (m1 ), αM2 (m2 )}, min{αM1 (n1 ), αM2 (n2 )}}
= min{(αM1 |αM2 )(m1 , m2 ), (αM1 |αM2 )(n1 , n2 )}
(δN1 |δN2 )((m1 , m2 )(n1 , n2 )) = max{δM1 (m1 ), δM1 (n1 ), δM2 (m2 )}
= max{max{δM1 (m1 ), δM2 (m2 )}, max{δM1 (n1 ), δM2 (n2 )}}
= max{(δM1 |δM2 )(m1 , m2 ), (δM1 |δM2 )(n1 , n2 )}
In a similar way others four will proved obviously.
(iii) If m1 n1 ̸∈ E1 and m2 n2 ̸∈ E2
(γN1 |γN2 )((m1 , m2 )(n1 , n2 )) = max{γM1 (m1 ), γM1 (n1 ), γM2 (m2 ), γM2 (n2 )}
= max{max{γM1 (m1 ), γM2 (m2 )}, max{γM1 (n1 ), γM2 (n2 )}}
= max{(γM1 |γM2 )(m1 , m2 ), (γM1 |γM2 )(n1 , n2 )}.
(θN1 |θN2 )((m1 , m2 )(n1 , n2 )) = min{θM1 (m1 ), θM1 (n1 ), θM2 (m2 ), θM2 (n2 )}
= min{min{θM1 (m1 ), θM2 (m2 )}, min{θM1 (n1 ), θM2 (n2 )}}
= min{(θM1 |θM2 )(m1 , m2 ), (θM1 |θM2 )(n1 , n2 )}.
In a similar way others four will proved obviously.
Hence all properties are satisfied truly, so in all cases N1 |N2 is a BSVNG on M1 |M2 . Therefore
we can say G1 |G2 = (M1 |M2 , N1 |N2 ) is a BSVNG.
M. Aslam Malik, Hossein Rashmanlou, Muhammad Shoaib, R. A. Borzooei and Morteza Taheri,
A Study on Bipolar Single-Valued Neutrosophic Graphs With Novel Application
Neutrosophic Sets and Systems, Vol. 32, 2020
259
Definition 2.30. Let G1 = (M1 , N1 ) and G2 = (M2 , Y2 ) be two BSVNGs. ∀(m1 , m2 ) ∈ V1 ×V2
∑
(dα )G1 |G2 (m1 , m2 ) =
(αN1 |αN2 )((m1 , m2 )(n1 , n2 ))
(m1 ,m2 )(n1 ,n2 )∈E1 ×E2 .
∑
=
min{αM1 (m1 ), αM2 (m2 ), αM2 (n2 )}
m1 =n1 ,m2 n2 ̸∈E2
∑
+
min{αM1 (m1 ), αM1 (n1 ), αM2 (m2 )}
m2 =n2 ,m1 n1 ̸∈E1
∑
+
min{αM1 (m1 ), αM1 (n1 ), αM2 (m2 ), αM2 (n2 )}
m1 n1 ̸∈E1 and m2 n2 ̸∈E2
∑
(dβ )G1 |G2 (m1 , m2 ) =
(βN1 |βN2 )((m1 , m2 )(n1 , n2 ))
(m1 ,m2 )(n1 ,n2 )∈E1 ×E2 .
∑
=
max{βM1 (m1 ), βM2 (m2 ), βM2 (n2 )}
m1 =n1 ,m2 n2 ̸∈E2
∑
+
max{βM1 (m1 ), βM1 (n1 ), βM2 (m2 )}
m2 =n2 ,m1 n1 ̸∈E1
∑
+
max{βM1 (m1 ), βM1 (n1 ), βM2 (m2 ), βM2 (n2 )}
m1 n1 ̸∈E1 and m2 n2 ̸∈E2
(dγ )G1 |G2 (m1 , m2 ) =
∑
(γN1 |γN2 )((m1 , m2 )(n1 , n2 ))
(m1 ,m2 )(n1 ,n2 )∈E1 ×E2 .
=
∑
max{γM1 (m1 ), γM2 (m2 ), γM2 (n2 )}
m1 =n1 ,m2 n2 ̸∈E2
+
∑
max{γM1 (m1 ), γM1 (n1 ), γM2 (m2 )}
m2 =n2 ,m1 n1 ̸∈E1
+
∑
max{γM1 (m1 ), γM1 (n1 ), γM2 (m2 ), γM2 (n2 )}
m1 n1 ̸∈E1 and m2 n2 ̸∈E2
(dδ )G1 |G2 (m1 , m2 ) =
∑
(δN1 |δN2 )((m1 , m2 )(n1 , n2 ))
(m1 ,m2 )(n1 ,n2 )∈E1 ×E2 .
=
∑
max{δM1 (m1 ), δM2 (m2 ), δM2 (n2 )}
m1 =n1 ,m2 n2 ̸∈E2
+
∑
max{δM1 (m1 ), δM1 (n1 ), δM2 (m2 )}
m2 =n2 ,m1 n1 ̸∈E1
+
∑
max{δM1 (m1 ), δM1 (n1 ), δM2 (m2 ), δM2 (n2 )}
m1 n1 ̸∈E1 and m2 n2 ̸∈E2
M. Aslam Malik, Hossein Rashmanlou, Muhammad Shoaib, R. A. Borzooei and Morteza Taheri,
A Study on Bipolar Single-Valued Neutrosophic Graphs With Novel Application
Neutrosophic Sets and Systems, Vol. 32, 2020
∑
(dη )G1 |G2 (m1 , m2 ) =
260
(ηN1 |ηN2 )((m1 , m2 )(n1 , n2 ))
(m1 ,m2 )(n1 ,n2 )∈E1 ×E2 .
∑
=
min{ηM1 (m1 ), ηM2 (m2 ), ηM2 (n2 )}
m1 =n1 ,m2 n2 ̸∈E2
∑
+
min{ηM1 (m1 ), ηM1 (n1 ), ηM2 (m2 )}
m2 =n2 ,m1 n1 ̸∈E1
∑
+
min{ηM1 (m1 ), ηM1 (n1 ), ηM2 (m2 ), ηM2 (n2 )}
m1 n1 ̸∈E1 and m2 n2 ̸∈E2
∑
(dθ )G1 |G2 (m1 , m2 ) =
(θN1 |θN2 )((m1 , m2 )(n1 , n2 ))
(m1 ,m2 )(n1 ,n2 )∈E1 ×E2 .
∑
=
min{θM1 (m1 ), θM2 (m2 ), θM2 (n2 )}
m1 =n1 ,m2 n2 ̸∈E2
∑
+
min{θM1 (m1 ), θM1 (n1 ), θM2 (m2 )}
m2 =n2 ,m1 n1 ̸∈E1
∑
+
min{θM1 (m1 ), θM1 (n1 ), θM2 (m2 ), θM2 (n2 )}
m1 n1 ̸∈E1 and m2 n2 ̸∈E2
Definition 2.31. Let G1 = (M1 , N1 ) and G2 = (M2 , Y2 ) be two BSVNGs. ∀(m1 , m2 ) ∈ V1 ×V2
(tdα )G1 |G2 (m1 , m2 ) =
∑
(αN1 |αN2 )((m1 , m2 )(n1 , n2 )) + (αM1 |αM2 )(m1 , m2 )
(m1 ,m2 )(n1 ,n2 )∈E1 ×E2 .
=
∑
min{αM1 (m1 ), αM2 (m2 ), αM2 (n2 )}
m1 =n1 ,m2 n2 ̸∈E2
+
∑
min{αM1 (m1 ), αM1 (n1 ), αM2 (m2 )}
m2 =n2 ,m1 n1 ̸∈E1
+
∑
min{αM1 (m1 ), αM1 (n1 ), αM2 (m2 ), αM2 (n2 )}
m1 n1 ̸∈E1 and m2 n2 ̸∈E2
(tdβ )G1 |G2 (m1 , m2 ) =
∑
(βN1 |βN2 )((m1 , m2 )(n1 , n2 )) + (βM1 |βM2 )(m1 , m2 )
(m1 ,m2 )(n1 ,n2 )∈E1 ×E2 .
=
∑
max{βM1 (m1 ), βM2 (m2 ), βM2 (n2 )}
m1 =n1 ,m2 n2 ̸∈E2
+
∑
max{βM1 (m1 ), βM1 (n1 ), βM2 (m2 )}
m2 =n2 ,m1 n1 ̸∈E1
+
∑
max{βM1 (m1 ), βM1 (n1 ), βM2 (m2 ), βM2 (n2 )}
m1 n1 ̸∈E1 and m2 n2 ̸∈E2
M. Aslam Malik, Hossein Rashmanlou, Muhammad Shoaib, R. A. Borzooei and Morteza Taheri,
A Study on Bipolar Single-Valued Neutrosophic Graphs With Novel Application
Neutrosophic Sets and Systems, Vol. 32, 2020
(tdγ )G1 |G2 (m1 , m2 ) =
∑
261
(γN1 |γN2 )((m1 , m2 )(n1 , n2 )) + (γM1 |γM2 )(m1 , m2 )
(m1 ,m2 )(n1 ,n2 )∈E1 ×E2 .
=
∑
max{γM1 (m1 ), γM2 (m2 ), γM2 (n2 )}
m1 =n1 ,m2 n2 ̸∈E2
+
∑
max{γM1 (m1 ), γM1 (n1 ), γM2 (m2 )}
m2 =n2 ,m1 n1 ̸∈E1
+
∑
max{γM1 (m1 ), γM1 (n1 ), γM2 (m2 ), γM2 (n2 )}
m1 n1 ̸∈E1 and m2 n2 ̸∈E2
∑
(tdδ )G1 |G2 (m1 , m2 ) =
(δN1 |δN2 )((m1 , m2 )(n1 , n2 )) + (δM1 |δM2 )(m1 , m2 )
(m1 ,m2 )(n1 ,n2 )∈E1 ×E2 .
=
∑
max{δM1 (m1 ), δM2 (m2 ), δM2 (n2 )}
m1 =n1 ,m2 n2 ̸∈E2
+
∑
max{δM1 (m1 ), δM1 (n1 ), δM2 (m2 )}
m2 =n2 ,m1 n1 ̸∈E1
+
∑
max{δM1 (m1 ), δM1 (n1 ), δM2 (m2 ), δM2 (n2 )}
m1 n1 ̸∈E1 and m2 n2 ̸∈E2
(tdη )G1 |G2 (m1 , m2 ) =
∑
(ηN1 |ηN2 )((m1 , m2 )(n1 , n2 )) + (ηM1 |ηM2 )(m1 , m2 )
(m1 ,m2 )(n1 ,n2 )∈E1 ×E2 .
=
∑
min{ηM1 (m1 ), ηM2 (m2 ), ηM2 (n2 )}
m1 =n1 ,m2 n2 ̸∈E2
+
∑
min{ηM1 (m1 ), ηM1 (n1 ), ηM2 (m2 )}
m2 =n2 ,m1 n1 ̸∈E1
+
∑
min{ηM1 (m1 ), ηM1 (n1 ), ηM2 (m2 ), ηM2 (n2 )}
m1 n1 ̸∈E1 and m2 n2 ̸∈E2
(tdθ )G1 |G2 (m1 , m2 ) =
∑
(θN1 |θN2 )((m1 , m2 )(n1 , n2 )) + (θM1 |θM2 )(m1 , m2 )
(m1 ,m2 )(n1 ,n2 )∈E1 ×E2 .
=
∑
min{θM1 (m1 ), θM2 (m2 ), θM2 (n2 )}
m1 =n1 ,m2 n2 ̸∈E2
+
∑
min{θM1 (m1 ), θM1 (n1 ), θM2 (m2 )}
m2 =n2 ,m1 n1 ̸∈E1
+
∑
min{θM1 (m1 ), θM1 (n1 ), θM2 (m2 ), θM2 (n2 )}
m1 n1 ̸∈E1 and m2 n2 ̸∈E2
Example 2.32. Let G1 = (M1 , N1 ) and G2 = (M2 , N2 ) be two BSVNGs as in Example 2.28.
Their rejection is also shown in Figure 13. We will find the vertex degree in rejection. Consider
M. Aslam Malik, Hossein Rashmanlou, Muhammad Shoaib, R. A. Borzooei and Morteza Taheri,
A Study on Bipolar Single-Valued Neutrosophic Graphs With Novel Application
Neutrosophic Sets and Systems, Vol. 32, 2020
262
the vertex (d,a) here:
(dγ )G1 |G2 (e, a) = max{γM2 (e), γM1 (a), γM1 (d)} + max{γM2 (a), γM1 (a), γM1 (c)}
= max{0.2, 0.2, 0.3} + max{0.2, 0.2, 0.3}
= 0.3 + 0.3
= 0.6
(dθ )G1 |G2 (e, a) = min{θM2 (e), θM1 (a), θM1 (d)} + min{θM2 (a), θM1 (a), θM1 (c)}
= min{−0.3, −0.4, −0.6} + min{−0.3, −0.4, −0.2}
= −0.6 − 0.4
= −1.0
(dα )G1 |G2 (e, a) = 0.6, (dβ )G1 |G2 (e, a) = 0.8
(dδ )G1 |G2 (e, a) = −0.5, (dη )G1 |G2 (e, a) = −0.5
In a similar way, we can find degree of all vertices of a graph in rejection. Now we will find
out the total vertex degree of graph in rejection. Consider the same vertex (d,a) here:
(tdγ )G1 |G2 (e, a) = max{γM2 (e), γM1 (a), γM1 (d)} + max{γM2 (a), γM1 (a), γM1 (c)} + min{γM2 (e), γM1 (a)}
= max{0.2, 0.2, 0.3} + max{0.2, 0.2, 0.3} + min{0.2, 0.2}
= 0.3 + 0.3 + 0.2
= 0.8
(tdθ )G1 |G2 (e, a) = min{θM2 (e), θM1 (a), θM1 (d)} + min{θM2 (a), θM1 (a), θM1 (c) + min{θM2 (e), θM1 (a)}}
= min{−0.3, −0.4, −0.6} + min{−0.3, −0.4, −0.2} + min{−0.3, −0.4}
= −0.6 − 0.4 − 0.4
= −1.4
(tdα )G1 |G2 (e, a) = 0.9, (tdβ )G1 |G2 (e, a) = 1.1
(tdδ )G1 |G2 (e, a) = −0.8, (tdη )G1 |G2 (e, a) = −0.7
In a similar way we can find total vertex degree in rejection.
M. Aslam Malik, Hossein Rashmanlou, Muhammad Shoaib, R. A. Borzooei and Morteza Taheri,
A Study on Bipolar Single-Valued Neutrosophic Graphs With Novel Application
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263
3. Application of bipolar single valued neutrosophic graph (BSVNG)
3.1. Educational Designation participation
Let {Bilal, Asif, Shoaib, Ijaz } be the set of four applicants for designations {Head of
department(HOD),Director of Department(DOD),Assistant director of department(ADOD)}.
For this purpose p=4 (say) be number of applicants and d=3 be number of designations. Consider bipolar single valued-neutrosophic diagraph which is shown in figure ?? representing the
competition between applicants for designation in organization. α(y) is the positive degree
of membership for every applicants denote the percentage of ability toward the purpose of
organization , β(y) and γ(y) are indeterminacy and false in percentage. δ(y) is the is the
negative degree of membership for every applicants denote the percentage of non ability toward the purpose of organization, η(y) and θ(y) are represents the indeterminacy and false
in percentage. α(y) of every directed edge between both designations and applicants denote
the eligibility or positive response from designation in organization , β(y) and γ(y) are indeterminacy and false in this percentage. δ(y) of every directed edge between both designations
and applicants denote the non-eligibility or negative response from designation in organization
, η(y) and θ(y) are indeterminacy and false in this percentage. Edge membership degree of
Table 1
y∈ Y
N(y)
Bilal
{(ADOD,0.5,0.3,0.4,−0.4,−0.5,−0.8),(HOD,0.6,0.4,0.2,−0.4,−0.6,−0.5)}
Asif
{(ADOD,0.8,0.6,0.5,−0.1,−0.4,−0.5),(HOD,0.5,0.6,0.6,−0.3,−0.4,−0.7),(DOD,0.4,0.6,0.4,−0.2,−0.3,−0.5)}
Shoaib
{(DOD,0.5,0.4,0.5,−0.5,−0.4,−0.4)})
Ijaz
{(HOD,0.7,0.5,0.6,−0.3,−0.5,−0.4),(DOD,0.7,0.4,0.5,−0.4,−0.3,−0.2)})
graph is also determined by the following
N (Bilal) ∩ N (Asif ) = {(ADOD, 0.5, 0.6, 0.5, −0.1, −0.5, −0.8), (HOD, 0.5, 0.6, 0.6,
− 0.3, −0.6, −0.7)}
N (Bilal) ∩ N (Shoaib) = ø
N (Bilal) ∩ N (Ijaz) = {(HOD, 0.6, 0.5, 0.6, −0.3, −0.6, −0.5)}
N (Asif ) ∩ N (Shoaib) = {(DOD, 0.4, 0.6, 0.5, −0.2, −0.4, −0.5)}
N (Asif ) ∩ N (Ijaz) = {(HOD, 0.5, 0.6, 0.6, −0.3, −0.5, −0.7), (DOD, 0.4, 0.6, 0.5, −0.2,
− 0.3, −0.5)}
N (Shoaib) ∩ N (Ijaz) = {(DOD, 0.5, 0.4, 0.5, −0.4, −0.4, −0.4)}
There is no edge between Shoaib and Bilal because there is no common designation.
M. Aslam Malik, Hossein Rashmanlou, Muhammad Shoaib, R. A. Borzooei and Morteza Taheri,
A Study on Bipolar Single-Valued Neutrosophic Graphs With Novel Application
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264
Figure 14. Bipolar single valued neutrosophic digraph
(Bilal, Asif ) = (0.4, 0.7, 0.8, −0.2, −0.7, −0.8)(0.5, 0.6, 0.5, 0.3, 0.6, 0.7)
= (0.20, 0.42, 0.40, −0.06, −0.42, −0.56)
(Bilal, Shoaib) = ø
(Bilal, Ijaz) = (0.4, 0.6, 0.8, −0.1, −0.7, −0.8)(0.6, 0.5, 0.6, 0.3, 0.6, 0.5)
= (0.24, 0.30, 0.48, −0.03, −0.42, −0.40)
(Asif, Shoaib) = (0.3, 0.7, 0.8, −0.2, −0.8, −0.8)(0.4, 0.6, 0.5, 0.2, 0.4, 0.5)
= (0.12, 0.42, 0.40, −0.04, −0.32, −0.40)
(Asif, Ijaz) = (0.5, 0.7, 0.8, −0.1, −0.6, −0.8)(0.5, 0.6, 0.5, 0.3, 0.3, 0.5)
= (0.25, 0.42, 0.40, −0.03, −0.18, −0.40)
(Shoaib, Ijaz) = (0.3, 0.6, 0.8, −0.1, −0.8, −0.5)(0.5, 0.4, 0.5, 0.4, 0.4, 0.4)
= (0.15, 0.24, 0.40, −0.04, −0.32, −0.20)
Bipolar single-valued neurotrophic graph for competition of all participant is shown in figure 15. Competition between two individually applicants and when applicant competing for
designation is also given in graph 15.
M. Aslam Malik, Hossein Rashmanlou, Muhammad Shoaib, R. A. Borzooei and Morteza Taheri,
A Study on Bipolar Single-Valued Neutrosophic Graphs With Novel Application
Neutrosophic Sets and Systems, Vol. 32, 2020
265
Figure 15. Bipolar single valued neutrosophic competition graph
0.20 + 0.24 0.20 + 0.24 0.42 + 0.30 0.40 + 0.48 −0.06 − 0.03
,
,
,
,
2
2
2
2
2
−0.42 − 0.42 −0.56 − 0.40
) = (0.22, 0.36, 0.44, −0.045, −0.42, −0.48)
,
,
2
2
R(Bilal, HOD) = (
Similarly we will find others R(applicant,Designation).
S(Bilal, HOD) = 1 + 0.22 − 0.045 − (0.36 + 0.44 − 0.42 − 0.48) = 1.275
S(Asif, HOD) = 1 + 0.225 − 0.045 − (0.42 + 0.40 − 0.30 − 0.48) = 1.14
S(Ijaz, HOD) = 1 + 0.245 − 0.03 − (0.36 + 0.44 − 0.225 − 0.29) = 0.93
S(Bilal, ADOD) = 1 + 0.20 − 0.06 − (0.42 + 0.40 − 0.42 − 0.56) = 1.30
S(Asif, ADOD) = 1 + 0.20 − 0.06 − (0.42 + 0.40 − 0.42 − 0.56) = 1.30
S(Asif, DOD) = 1 + 0.185 − 0.035 − (0.42 + 0.40 − 0.25 − 0.40) = 0.98
S(Shoaib, DOD) = 1 + 0.135 − 0.04 − (0.33 + 0.40 − 0.32 − 0.30) = 0.985
S(Ijaz, DOD) = 1 + 0.20 − 0.035 − (0.33 + 0.40 − 0.25 − 0.30) = 0.985
Black solid lines show comparison between two applicants and dot line means applicant compete for designation. From above table, applicants compete other if it has a more strength.
For example, in HOD designation Bilal has more strength from all. Its eligibility is strong
than other. In ADOD designation Asif and Bilal are in equal position. In DOD designation
Shoaib and Ijaz compete the others but equally compete to each other. [H]In this algorithm
these are the steps
Step 1: Start. Step 2: Input α(y), β(y) and γ(y) membership values for set p applicants.
M. Aslam Malik, Hossein Rashmanlou, Muhammad Shoaib, R. A. Borzooei and Morteza Taheri,
A Study on Bipolar Single-Valued Neutrosophic Graphs With Novel Application
Neutrosophic Sets and Systems, Vol. 32, 2020
266
Table 2
(Applicant,designation)
in competition
R(applicant,Designation)
S(applicant,Designation)
(Bilal,HOD)
Asif, Ijaz
(0.22,0.36,0.44,-0.045,-0.42,-0.48)
1.275
(Asif,HOD)
Bilal,Ijaz
(0.225,0.42,0.40,-0.045,-0.30,-0.48)
1.14
(Ijaz,HOD)
Bilal,Asif
(0.245,0.36,0.44,-0.03,-0.225,-0.29)
0.93
(Bilal,ADOD)
Asif
(0.20,0.42,0.40,-0.06,-0.42,-0.56)
1.30
(Asif,ADOD)
Bilal
(0.20,0.42,0.40,-0.06,-0.42,-0.56)
1.30
(Asif,DOD)
Shoaib,Ijaz
(0.185,0.42,0.40,-0.035,-0.25,-0.40)
0.98
(Shoaib,DOD)
Asif,Ijaz
(0.135,0.33,0.40,-0.04,-0.32,-0.30)
0.985
(Ijaz,DOD)
Asif,Shoaib
(0.20,0.33,0.40,-0.035,-0.25,-0.30)
0.985
Step3: For any two vertices xi and xj taking α(xi xj ), β(xi xj ) and γ(xi xj ) are positive but
δ(xi xj ), η(xi xj )
and
θ(xi xj )
are
negative.
Then
(xi , α(xi xj ), β(xi xj ), γ(xi xj ), δ(xi xj ), η(xi xj ), θ(xi xj ))
Step4: To obtain bipolar single valued neutrosohic out-neighbourhoods N (xi ) Repeat step 3
for all vertices xi and xj .
Step5: Find out N (xi ) ∩ N (xj ). Step6: Calculate height h(N (xi ) ∩ N (xj )). Step7: Draw
all edge where N (xi ) ∩ N (xj ) is non empty. Step8: Give a membership value to every edge
xi xj by using the following conditions
α(xi xj = (min{xi ∩ xj })[N (xi ∩ N (xj )], β(xi xj = (max{xi ∩ xj })[N (xi ∩ N (xj )]
γ(xi xj = (max{xi ∩ xj })[N (xi ∩ N (xj )], δ(xi xj = (max{xi ∩ xj })[N (xi ∩ N (xj )]
η(xi xj = (min{xi ∩ xj })[N (xi ∩ N (xj )], θ(xi xj = (min{xi ∩ xj })[N (xi ∩ N (xj )]
Step9: If x, z1 , z2 , z3 , ..., zp are applicants for designations d, then strength of applicants competition is R(x,d)=(α(x, d), β(x, d), γ(x, d), δ(x, d), η(x, d), θ(x, d)) of every applicants x and
designation
d
is
given
by
the
following
α(xz1 )+...α(xzp ) β(xz1 )+...β(xzp ) γ(xz1 )+...γ(xzp ) δ(xz1 )+...δ(xzp ) η(xz1 )+...η(xzp ) θ(xz1 )+...θ(xzp )
,
,
,
,
)
R(x,d)=(
,
p
p
p
p
p
p
Step10:Find out S(x, d) = 1+α(x, d)+δ(x, d)−(β(x, d)+γ(x, d)+η(x, d)+θ(x, d)). Step11:
End
4. Conclusion
There are more advantages of a bipolar fuzzy set than fuzzy set in real life phenomenon. A
BSVNG has many applications in the field of economics, medical science as well as in scientific
engineering. The flexibility and compatibility of BSVNG are higher than SVNG. We presented
the new properties on a bipolar single-valued neutrosophic graph known as Residue product,
maximal product, Symmetric difference and Rejection of a graph. These all graph products
are suggestive of some aspects of network design. They can be applicable for the configuration
processing of space structures. The repeated application of these operations in constructing
M. Aslam Malik, Hossein Rashmanlou, Muhammad Shoaib, R. A. Borzooei and Morteza Taheri,
A Study on Bipolar Single-Valued Neutrosophic Graphs With Novel Application
Neutrosophic Sets and Systems, Vol. 32, 2020
267
a network generates graphs that display fractal properties. We also discussed the idea with
examples to find the degree and total degree of vertices of some graphs. We have established
some related theorems of these graphs. We have also proved the theorems which are related to
these properties. In the future, our goal is to extend this work on the (1) complex neutrosophic
graphs and some (2) bipolar complex neutrosophic graph.
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Received: Oct 18, 2019. Accepted: Mar 21, 2020
M. Aslam Malik, Hossein Rashmanlou, Muhammad Shoaib, R. A. Borzooei and Morteza Taheri,
A Study on Bipolar Single-Valued Neutrosophic Graphs With Novel Application
Neutrosophic Sets and Systems, Vol. 32, 2020
University of New Mexico
Neutrosophic Geometric Programming (NGP) with
(Max, Product) Operator; An Innovative Model
Huda E. Khalid
Telafer University, Head of the Scientific Affairs and Cultural Relations Department, Telafer, Iraq.
E-mail: hodaesmail@yahoo.com
Abstract. In this paper, a neutrosophic optimization model has been first constructed
for the neutrosophic geometric programming subject to (max-product) neutrosophic relation
constraints. For finding the maximum solution, two new operations (i.e. ⋈, Θ) between a ij and
bi have been defined, which have a key role in the structure of the maximum solution. Also,
two new theorems and some propositions are introduced that discussed the cases of the
incompatibility in the relational equations Aox = b, with some properties of the operation Θ.
Numerical examples have been solved to illustrate new concepts.
Keyword:
Neutrosophic
Geometric
Programming
(NGP);
(max-product)
Operator;
Neutrosophic Relation Constraints; Maximum Solution; Incompatible Problem; Pre-Maximum
Solution; Relational Neutrosophic Geometric Programming (RNGP).
1. Introduction
The first scientist who put forward the fuzzy relational equations was Elie Sanchez, a
famous fuzzy biology mathematician in 1976 [2], while the theoretical concept of the
neutrosophic logic has been put by the popular polymath Florentin Smarandache at 1995 [11].
B. Y. Cao constructed the mathematical models of fuzzy relation geometric programming
(FRGP) at 2005 [1], his works include the structuring of the maximum and minimum solution
of the (FRGP) depending upon the original model for the maximum solution and the minimum
solution for the fuzzy relation equations that was put by Elie Sanchez. At 2015, Huda E. Khalid
introduced an original structure of the maximum solution for the fuzzy neutrosophic relation
geometric programming (FNRGP) [6], Also at 2016, she put a novel algorithm for finding the
minimum solution for the same (FNRGP) problems [7]. As of 2016 so far Huda E. Khalid et al
[3-10] introducing a big qualitative shift in the concept of neutrosophic geometric
programming (NGP) by establishing new concepts for the notion of (over, off, under) in the
same (NGP), as well as she introduced and for the first time, a new type of the neutrosophic
geometric programming using (over, off, under) neutrosophic less than or equal which
contained a new version of the convex condition, furthermore, new decomposition theorems
of neutrosophic sets were presented, and new representations for the neutrosophic sets using
(α, β, γ)-cuts, with strong (α, β, γ)-cuts had been defined.
In this article, section 2 contains the preliminaries which are necessary for the sake of
this paper, while in section 3, a max- product neutrosophic relation geometric programming
model has been proposed with an innovative investigation of the maximum solution for this
Huda E. Khalid, “Neutrosophic Geometric Programming (NGP) with (max-product) Operator, An
Innovative Model”
Neutrosophic Sets and Systems, Vol. 32, 2020
model and two new theorems with some propositions, section 4 presents numerical examples
to illustrate the proposed method. The final section was dedicated to the conclusion.
2. Basic Concepts
Without loss of generality, the elements of b must be rearranged in decreasing or increasing
order and the elements of the matrix A are correspondingly rearranged.
2.1 Definition [7]
In this definition, the author proposed the following axioms:
a- decreasing partial order
1-The greatest element in [0,1) ∪ I is equal to I, max(I, x) = I
∀ x ∈ [0,1)
2- The fuzzy values in a decreasing order will be rearranged as follows: 1 > x1 > x2 > x3 >
⋯ > xn ≥ 0
3- One is the greatest element in [0,1] ∪ I, max(I, 1) = 1
b- Increasing partial order
1- the smallest element in (0,1] ∪ I is I , min(I, x) = I
∀ x ∈ (0,1]
2- The fuzzy values in increasing order will be rearranged as follows: 0 < x1 < x2 < x3 <
⋯ < xn ≤ 1
3- Zero is the smallest element in [0,1] ∪ I, min(I, 0) = 0
2.2 Definition [7]
If there exists a solution to 𝐴𝑜𝑥 = 𝑏 it's called compatible. Suppose 𝑋(𝐴, 𝑏) = {(𝑥1 , 𝑥2 , … , 𝑥𝑛 )𝑇 ∈
[0,1]𝑛 ∪ 𝐼, 𝐼 𝑛 = 𝐼, 𝑛 > 0 |𝐴𝑜𝑥 = 𝑏, 𝑥𝑗 ∈ [0,1] ∪ 𝐼 } is a solution set of 𝐴𝑜𝑥 = 𝑏 we define 𝑥 1 ≤
𝑥 2 ⟺ 𝑥𝑗1 ≤ 𝑥𝑗2 (1 ≤ 𝑗 ≤ 𝑛), ∀ 𝑥 1 , 𝑥 2 ∈ X(A, b). Where " ≤ " is a partial order relation on X(A, b).
2.3 Corollary [1]
If X(A, b) ≠ ∅. Then 𝑥̂ ∈ 𝑋(𝐴, 𝑏).
Similar to fuzzy relation equations, the above corollary works on neutrosophic relation
equations.
Huda E. Khalid, “Neutrosophic Geometric Programming (NGP) with (max-product) Operator, An
Innovative Model”
Neutrosophic Sets and Systems, Vol. 32, 2020
2.4 Basic Notes [3, 10]
1. A component I to the zero power is undefined value, (i.e. 𝐼 0 is undefined), since. 𝐼 0 =
𝐼
𝐼1+(−1) = 𝐼1 ∗ 𝐼 −1 = , which is an impossible case (avoid to divide by 𝐼).
𝐼
2. The value of 𝐼 to the negative power is undefined (i.e. 𝐼 −𝑛 , 𝑛 > 0 is undefined).
3. The Innovative Structure of the Maximum Solution.
We call
𝛾
𝛾
𝛾
min 𝑓(𝑥) = (𝑐1 . 𝑥1 1 ) ∨ (𝑐2 . 𝑥22 ) ∨ … ∨ (𝑐𝑛 . 𝑥𝑛𝑛 )
}
𝑠. 𝑡.
𝐴𝑜𝑥 = 𝑏
𝑥𝑗 ∈ [0,1] ∪ 𝐼,
1≤𝑗≤𝑛
(1)
A ( ∨, . ) (max- product) neutrosophic geometric programming, where 𝐴 = (𝑎𝑖𝑗 ), 1 ≤
𝑖 ≤ 𝑚 , 1 ≤ 𝑗 ≤ 𝑛, is (𝑚 × 𝑛) dimensional neutrosophic matrix, 𝑥 = (𝑥1 , 𝑥2 , … , 𝑥𝑛 )𝑇 an n-
dimensional variable vector, 𝑏 = (𝑏1 , 𝑏2 , … , 𝑏𝑚 )𝑇 (𝑏𝑖 ∈ [0,1] ∪ 𝐼) an m- dimensional constant
vector, 𝑐 = (𝑐1 , 𝑐2 , … , 𝑐𝑛 )𝑇 (𝑐𝑗 ≥ 0) an n- dimensional constant vector, 𝛾𝑗 is an arbitrary real
number, and the composition operator ‘’𝑜’’ is ( ∨, . ) , i.e. ⋁𝑛𝑗=1(𝑎𝑖𝑗 . 𝑥𝑗 ) = 𝑏𝑖 .
Note that the program (1) is undefined and has no minimal solution in the case of 𝛾𝑗 < 0 with
some 𝑥𝑗 ′𝑠 taking indeterminacy value. Therefore, if 𝛾𝑗 < 0 with indeterminacy value in some
𝑥𝑗 ′𝑠, then the greatest solution 𝑥̂𝑗 is an optimal solution for problem (1), the author introduced
theorem 3.4 to treat this issue.
3.1 The Shape of the Maximum Solution 𝐱̂.
Since 1976, the biological mathematician Elie Sanchez put the formula of the maximum
solution in both composite fuzzy relation equations of type (⋁, ⋀) operator and (⋁, . ) operator
[2], these definitions won’t be adequate with neutrosophic relation equations especially
neutrosophic geometric programming type, therefore and for the importance of relational
neutrosophic geometric programming (RNGP) in real-world problems, the author established
a new structure for the maximum solution of (RNGP) with the (⋁, ⋀)operator in ref. [6], while
this article was dedicated to set up the maximum solution of (RNGP) with the (⋁, . ) operator.
Every mathematician who works with neutrosophic theory know that the generality
which characterizes the neutrosophic theory are determined in many ways of which,
Huda E. Khalid, “Neutrosophic Geometric Programming (NGP) with (max-product) Operator, An
Innovative Model”
Neutrosophic Sets and Systems, Vol. 32, 2020
max(𝐼, 𝑥) = min(𝐼, 𝑥) = 𝐼 ∀ 𝑥 ∈ (0,1)
This property gives some vague and difficulty for determining the maximum solution of the
relation equations 𝐴𝑜𝑥 = 𝑏, the author still searches about the answer of the following question.
How will be the shape of the greatest solution 𝑥̂ ?
Actually, any single solution (the same solution that suggested by Elie Sanchez 1976) would
not be accepted and won’t be appropriate for the program (1), unless there are two integrated
pre-maximum solutions gathered to get the final shape of 𝑥̂, as follow:
1.
The first integrated pre-maximum solution named 𝑥̂𝑣1 which supports the fuzzy part
of the problem, this solution has an adjoint matrix named 𝐴𝑣1 , this adjoint matrix is
2.
derived from the matrix 𝐴.
The second integrated pre-maximum solution named 𝑥̂𝑣2 which supports the
neutrosophic part of the problem, this solution has an adjoint matrix named 𝐴𝑣2 , which
is derived from the matrix 𝐴 too.
The following definition describes the mathematical formula of 𝑥̂𝑣1 and 𝑥̂𝑣2 .
3.2 Definition
𝑏𝑖
,
𝑎𝑖𝑗
𝑎𝑖𝑗 ⋈ 𝑏𝑖 = { 1,
1,
𝑎𝑖𝑗 Θ𝑏𝑖 =
𝑛𝐼
,
𝑎𝑖𝑗
𝑖𝑓 𝑎𝑖𝑗 > 𝑏𝑖 , 𝑎𝑖𝑗 ∈ [0,1], 𝑏𝑖 ∈ [0,1]
𝑖𝑓 𝑎𝑖𝑗 ≤ 𝑏𝑖 , 𝑎𝑖𝑗 ∈ [0,1], 𝑏𝑖 ∈ [0,1]
𝑖𝑓
𝑎𝑖𝑗 ∈ [0,1], 𝑏𝑖 = 𝑛𝐼, 𝑛 ∈ (0,1]
𝑖𝑓 𝑎𝑖𝑗 > 𝑛 , 𝑎𝑖𝑗 ∈ [0,1], 𝑏𝑖 = 𝑛𝐼, 𝑛 ∈ (0,1]
1,
𝑖𝑓 𝑎𝑖𝑗 ≤ 𝑛 , 𝑎𝑖𝑗 ∈ [0,1], 𝑏𝑖 = 𝑛𝐼, 𝑛 ∈ (0,1]
𝑛𝑜𝑡 𝑐𝑜𝑚𝑝. 𝑖𝑓
𝑎𝑖𝑗 = 𝑚𝐼 , 𝑚 ∈ (0,1] , 𝑏𝑖 ∈ [0,1] ∪ 𝐼
1
𝑖𝑓
𝑎𝑖𝑗 , 𝑏𝑖𝑗 ∈ [0,1]
{
Where ⋈ is an operator defined at [0,1], while the operator Θ is defined at [0,1] ∪ 𝐼.
Let 𝑥̂𝑗 = ⋀𝑚
𝑖=1(𝑎𝑖𝑗 ⋈ 𝑏𝑖 ),
(1 ≤ 𝑗 ≤ 𝑛) ,
be the components of the pre-maximum solution 𝑥̂𝑣1 , (i.e. 𝑥̂𝑣1 = (𝑥̂1 , 𝑥̂2 , … , 𝑥̂𝑛 )).
Let 𝑥̂𝑗 = ⋀𝑚
𝑖=1(𝑎𝑖𝑗 Θ𝑏𝑖 ),
(1 ≤ 𝑗 ≤ 𝑛) ,
Huda E. Khalid, “Neutrosophic Geometric Programming (NGP) with (max-product) Operator, An
Innovative Model”
(2)
(3)
(4)
(5)
Neutrosophic Sets and Systems, Vol. 32, 2020
be the components of the pre maximum solution 𝑥̂𝑣2 , (i.e. 𝑥̂𝑣2 = (𝑥̂1 , 𝑥̂2 , … , 𝑥̂𝑛 )).
Now the following question will be raised,
Which one 𝑥̂𝑣1 or 𝑥̂𝑣2 should be the exact maximum solution?
Neither 𝑥̂𝑣1 nor 𝑥̂𝑣2 will be the exact solution! the exact solution is the integration between
them. Before solving 𝐴𝑜𝑥̂ = 𝑏, we first define the matrices 𝐴𝑣1 , 𝐴𝑣2 .
Let 𝐴𝑣1 be a matrix has the same dimension and the same rows elements of 𝐴 except for those
rows of the indexes 𝑖 = 𝑖𝑜 corresponding to those indexes of 𝑏𝑖𝑜 = 𝑛𝐼, those special rows of
𝐴𝑣1 will be zeros.
Let 𝐴𝑣2 be a matrix has the same dimension and the same rows elements of 𝐴 except for those
rows of the indexes 𝑖 = 𝑖𝑜 corresponding to those indexes of 𝑏𝑖𝑜 ∈ [0,1], those special rows of
𝐴𝑣2 will be zeros.
Consequently,
𝐴𝑜𝑥̂ = 𝑏 = (𝐴𝑣1 𝑜𝑥̂𝑣1 ) + (𝐴𝑣2 𝑜𝑥̂𝑣2 )
(6)
The formula (6) is the greatest solution in 𝑋(𝐴, 𝑏).
The maximum value of the objective function 𝑓(𝑥̂) = 𝑓(𝑥̂𝑣1 ) ∨ 𝑓(𝑥̂𝑣2 ).
3.3 Theorem
If 𝑎𝑖𝑗 = 𝑚𝐼, 𝑚 ∈ (0,1], 𝑏𝑖 ∈ [0,1] ∪ 𝐼 then 𝐴𝑜𝑥 = 𝑏, is not compatible.
Proof
Let 𝑎𝑖𝑗 = 𝑚𝐼 , 𝑏𝑖 ∈ [0,1] ∪ 𝐼 , the essential question in this case is
What is the value of 𝑥𝑗 ∈ [0,1] ∪ 𝐼 satisfying
⋁1≤𝑗≤𝑛(𝑎𝑖𝑗 . 𝑥𝑗 ) = 𝑏𝑖 ?
(7)
It is well known that the equation (7) can be written as an upper-bound constraint and a lowerbound constraint, that is,
⋁1≤𝑗≤𝑛(𝑎𝑖𝑗 . 𝑥𝑗 ) ≤ 𝑏𝑖
⋁1≤𝑗≤𝑛(𝑎𝑖𝑗 . 𝑥𝑗 ) ≥ 𝑏𝑖
Huda E. Khalid, “Neutrosophic Geometric Programming (NGP) with (max-product) Operator, An
Innovative Model”
(8)
(9)
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First,
The inequality (8) can be written in 𝑛 constraints:
𝑎𝑖𝑗 . 𝑥𝑗 ≤ 𝑏𝑖 , 𝑖. 𝑒. 𝑥𝑗 ≤
𝑏𝑖
,1 ≤ 𝑗 ≤ 𝑛 .
𝑎𝑖𝑗
𝑏
Hence 𝑥𝑗 ≤ ∧ ( 𝑖 ), where the notation ‘’ ∧’’ denotes the minimum operator.
𝑎𝑖𝑗
𝑏
So, we have 𝑥𝑗 ∈ [0, ∧ ( 𝑖 )] ∪ 𝐼, but 𝑎𝑖𝑗 = 𝑚𝐼, this is a contradict for the fact that the variables
𝑎𝑖𝑗
of the system 𝐴𝑜𝑥 = 𝑏 are being in the interval [0,1] ∪ 𝐼.
Second,
The inequality (9) can be written in 𝑛 constraints:
(𝑎𝑖𝑗 . 𝑥𝑗 ) ≥ 𝑏𝑖 , 𝑖. 𝑒. 𝑥𝑗 ≥
𝑏𝑖
𝑎𝑖𝑗
,1 ≤ 𝑗 ≤ 𝑛 .
𝑏
Hence, 𝑥𝑗 ≥∨ ( 𝑖 ), where the notation ‘’ ∨’’ denotes the maximum operator.
𝑎𝑖𝑗
Thus, we have 𝑥𝑗 ∈ [∨ (
𝑏𝑖
𝑎𝑖𝑗
) , 1] ∪ 𝐼, but 𝑎𝑖𝑗 = 𝑚𝐼, in this proof we faced the division on the
indeterminate component (𝐼) which is prohibited behavior. Consequently the variable 𝑥𝑗 will
either belong to the interval [0,∧ (𝑏𝑖 /𝐼)] ∪ 𝐼 or belong to the interval[∨ (𝑏𝑖 /𝐼),1] ∪ 𝐼, this implies
that the system of the relation equation 𝐴𝑜𝑥 = 𝑏 will be not compatible.
Therefore, the system of the relative equations 𝐴𝑜𝑥 = 𝑏 is incompatible at 𝑎𝑖𝑗 = 𝑚𝐼, 𝑚 ∈ (0,1].
So, the restriction of 𝐴𝑜𝑥 = 𝑏 for being compatible is that all elements of the matrix 𝐴 (𝑖. 𝑒. 𝑎𝑖𝑗 )
are belonging to the interval [0,1].
3.4 Theorem
If 𝛾𝑗 < 0 (1 ≤ 𝑗 ≤ 𝑛), then the greatest solution to the problem (1) is an optimal solution.
Proof
Since 𝛾𝑗 < 0 (1 ≤ 𝑗 ≤ 𝑛), with 𝑥𝑗 ∈ [0,1] ∪ 𝐼, then
𝛾𝑗
𝑑(𝑥𝑗 )
𝑑𝑥𝑗
𝛾𝑗 −1
= 𝛾𝑗 𝑥𝑗
< 0 for each 𝑥𝑗 ∈ [0,1] ∪ 𝐼, this
𝛾𝑗
means that 𝑥𝑗 𝛾𝑗 is monotone decreasing function of 𝑥𝑗 . It is clear that 𝑐𝑗 𝑥𝑗 is also a monotone
decreasing function about
𝛾𝑗
𝑐𝑗 . 𝑥̂𝑗
𝛾𝑗
𝑥𝑗 . Therefore, ∀ 𝑥 ∈ 𝑋(𝐴, 𝑏), when 𝑥 ≤ 𝑥̂, then 𝑐𝑗 . 𝑥𝑗 ≥
(1 ≤ 𝑗 ≤ 𝑛), such that 𝑓(𝑥) ≥ 𝑓(𝑥̂), so 𝑥̂ is an optimal solution to the problem (1).
Huda E. Khalid, “Neutrosophic Geometric Programming (NGP) with (max-product) Operator, An
Innovative Model”
Neutrosophic Sets and Systems, Vol. 32, 2020
It remains to study the case that if 𝛾𝑗 < 0 with the component 𝑥̂𝑗 in 𝑥̂𝑣2 equal to 𝐼, we know that
𝐼 𝑛 is undefined for 𝑛 ≤ 0, in this case, the component 𝑥𝑗 = 𝐼 that has a power 𝛾𝑗 < 0 will be
replaced by that corresponding 𝑥𝑗 in the 𝑥̂𝑣1 .
3.5 Proposition
Let 𝑎 ∈ (0,1), 𝑏 = 𝑚𝐼 & 𝑐 = 𝑛𝐼, 𝑛, 𝑚 ∈ (0,1], 𝑖𝑓 𝑚 ≥ 𝑛 , then 𝑎 Θ𝑏 ≥ 𝑎 Θ 𝑐.
Proof
1) Let 𝑎 > 𝑚 ⟹ 𝑎 > 𝑛,
But we have 𝑚 ≥ 𝑛 ⟹ 𝑏 ≥ 𝑐 ⟹
𝑏
𝑎
≥
𝑐
𝑎
⟹ 𝑎 Θ𝑏 ≥ 𝑎 Θ 𝑐.
2) Let 𝑎 ≤ 𝑚 ⟹ 𝑎 Θ 𝑏 = 1, since 𝑚 ≥ 𝑛 ⟹ 𝑎 Θ 𝑐 ≤ 1
Hence, 𝑎 Θ 𝑐 ≤ 𝑎 Θ 𝑏.
3.6 Corollary
Let 𝑎 ∈ (0,1), 𝑏 = 𝑚𝐼, 𝑐 = 𝑛𝐼, 𝑚, 𝑛 ∈ (0,1], if 𝑚 ≥ 𝑛 then 𝑎 Θ (𝑏⋁𝑐) ≥ 𝑎 Θ 𝑐
Proof
Since 𝑚 ≥ 𝑛 ⟹ 𝑏 ≥ 𝑐 ⟹ 𝑏⋁𝑐 = 𝑏, from proposition 2.5, we have
𝑎Θ𝑏 ≥𝑎Θ𝑐
3.7 Proposition
(𝑟𝑒𝑝𝑙𝑎𝑐𝑖𝑛𝑔 𝑏⋁𝑐 𝑖𝑛𝑠𝑡𝑒𝑎𝑑 𝑜𝑓 𝑏) ⟹ 𝑎 Θ (𝑏⋁𝑐) ≥ 𝑎 Θ 𝑐.
Let 𝑎 ∈ (0,1), 𝑏 = 𝑚𝐼, 𝑚 ∈ (0,1], then 𝑎. (𝑎 Θ 𝑏) = 𝑎⋀𝑏.
Proof
1) Let 𝑎 > 𝑚 ⟹
𝑏 = 𝑎. (𝑎 Θ 𝑏)
𝑚𝐼
𝑎
=
𝑏
𝑎
= 𝑎 Θ 𝑏 [multiply both sides by 𝑎] ⟹
2) Let 𝑎 ≤ 𝑚 ⟹ 𝑎 Θ 𝑏 = 1 [multiply both sides by 𝑎] ⟹
𝑎 = 𝑎. (𝑎 Θ 𝑏)
From (10) & (11) we have 𝑎. (𝑎 Θ 𝑏) = 𝑎⋀𝑏.
Huda E. Khalid, “Neutrosophic Geometric Programming (NGP) with (max-product) Operator, An
Innovative Model”
(10)
(11)
Neutrosophic Sets and Systems, Vol. 32, 2020
3.8 Proposition
Let 𝑎 ∈ (0,1), 𝑏 = 𝑚𝐼, 𝑚 ∈ (0,1], then 𝑎. (𝑎 Θ 𝑏) = {
Proof
𝑏
1
𝑎 > 𝑎𝑚
.
𝑎 ≤ 𝑎𝑚
1) Let 𝑎 > 𝑎𝑚 , from definition (3.2) we have 𝑎 Θ (𝑎. 𝑚) =
𝑎.𝑚𝐼
𝑎
= 𝑚𝐼 = 𝑏.
2) Let 𝑎 ≤ 𝑎𝑚 , again from definition (3.2) we have 𝑎 Θ (𝑎. 𝑏) = 1.
𝑏
Hence, 𝑎 Θ (𝑎. 𝑏) = {
1
𝑎 > 𝑎𝑚
𝑎 ≤ 𝑎𝑚
4 Numerical examples
In the upcoming examples, the (max- product) neutrosophic geometric problem is considered.
4.1 Example
Let min 𝑓(𝑥) =
s. t. 𝐴𝑜𝑥 = 𝑏
𝑥𝑗 ∈ [0,1]⋃𝐼
(0.3. 𝑥12 ) ∨
1
3
1
4
(1.8𝐼 . 𝑥2 ) ∨ (𝐼 . 𝑥3 )
(1 ≤ 𝑗 ≤ 𝑛)
.6
1 1
Where 𝑏 = (1, 3 𝐼, 5 𝐼)𝑇 , 𝐴 = (. 5
.3
1 .2
. 2 . 1)
. 5 . 1 3×3
Using the formula (2), we can find the components of 𝑥̂𝑣1 as follows
3
𝑥̂1 = ⋀(𝑎𝑖1 ⋈ 𝑏𝑖 ) = (𝑎11 ⋈ 𝑏1 ) ∧ (𝑎21 ⋈ 𝑏2 ) ∧ (𝑎31 ⋈ 𝑏3 )
𝑖=1
3
= (0.6 ⋈ 1) ∧ (0.5 ⋈
1
𝐼) ∧ (0.3 ⋈ 0.2𝐼) = 1 ∧ 1 ∧ 1 = 1
3
𝑥̂2 = ⋀(𝑎𝑖2 ⋈ 𝑏𝑖 ) = (𝑎12 ⋈ 𝑏1 ) ∧ (𝑎22 ⋈ 𝑏2 ) ∧ (𝑎32 ⋈ 𝑏3 )
𝑖=1
= (1 ⋈ 1) ∧ (0.2 ⋈
1
𝐼) ∧ (0.5 ⋈ 0.2𝐼) = 1 ∧ 1 ∧ 1 = 1
3
Huda E. Khalid, “Neutrosophic Geometric Programming (NGP) with (max-product) Operator, An
Innovative Model”
Neutrosophic Sets and Systems, Vol. 32, 2020
3
𝑥̂3 = ⋀(𝑎𝑖3 ⋈ 𝑏𝑖 ) = (𝑎13 ⋈ 𝑏1 ) ∧ (𝑎23 ⋈ 𝑏2 ) ∧ (𝑎33 ⋈ 𝑏3 )
𝑖=1
= (0.2 ⋈ 1) ∧ (0.1 ⋈
1
𝐼) ∧ (0.1 ⋈ 0.2𝐼) = 1 ∧ 1 ∧ 1 = 1
3
∴ 𝑥̂𝑣1 = (𝑥̂1 , 𝑥̂2 , 𝑥̂3 )𝑇 = (1,1,1)𝑇
Using the formula (3), we can find the components of 𝑥̂𝑣2 as follows
3
𝑥̂1 = ⋀(𝑎𝑖1 Θ 𝑏𝑖 ) = (𝑎11 Θ 𝑏1 ) ∧ (𝑎21 Θ 𝑏2 ) ∧ (𝑎31 Θ 𝑏3 )
𝑖=1
3
= (0.6 Θ 1) ∧ (0.5 Θ
1⁄
1
0.2
2
𝐼) ∧ (0.3 Θ 0.2𝐼) = 1 ∧ 3 𝐼 ∧
𝐼= 𝐼
0.5
0.3
3
3
𝑥̂2 = ⋀(𝑎𝑖2 Θ 𝑏𝑖 ) = (𝑎12 Θ 𝑏1 ) ∧ (𝑎22 Θ 𝑏2 ) ∧ (𝑎32 Θ 𝑏3 )
𝑖=1
= (1 Θ 1) ∧ (0.2 Θ
2
2
1
𝐼) ∧ (0.5 Θ 0.2𝐼) = 1 ∧ 1 ∧ 𝐼 = 𝐼
5
5
3
3
𝑥̂3 = ⋀(𝑎𝑖3 Θ𝑏𝑖 ) = (𝑎13 Θ 𝑏1 ) ∧ (𝑎23 Θ 𝑏2 ) ∧ (𝑎33 Θ 𝑏3 )
𝑖=1
= (0.2 Θ 1) ∧ (0.1 Θ
1
𝐼) ∧ (0.1 Θ 0.2𝐼) = 1 ∧ 1 ∧ 1 = 1
3
𝑇
2 2
∴ 𝑥̂𝑣2 = (𝑥̂1 , 𝑥̂2 , 𝑥̂3 )𝑇 = ( 𝐼, 𝐼, 1)
3 5
.6 1 .2
0 0
In this example, 𝐴𝑣1 = ( 0 0 0 ) , 𝐴𝑣2 = (. 5 . 2
0 0 0
.3 .5
.6
𝐴𝑜𝑥̂ = (𝐴𝑣1 𝑜𝑥̂𝑣1 ) + (𝐴𝑣2 𝑜𝑥̂𝑣2 ) = ( 0
0
1
1
𝐼
= 3 =𝑏
1
[5 𝐼 ]
0
. 1),
.1
0
1
1 .2
0 0 ) 𝑜 [1] + (. 5
.3
0 0
1
2
𝐼
0 0
3
. 2 . 1) 𝑜 2
𝐼
.5 .1
5
[1]
Since 𝐴𝑜𝑥̂ = 𝑏, then there is a solution in 𝑋(𝐴, 𝑏) and 𝑥̂ is the greatest solution
to 𝐴𝑜𝑥 = 𝑏. The value of 𝑓(𝑥̂) is calculated as follow,
Huda E. Khalid, “Neutrosophic Geometric Programming (NGP) with (max-product) Operator, An
Innovative Model”
Neutrosophic Sets and Systems, Vol. 32, 2020
𝑓(𝑥̂) = 𝑓(𝑥̂𝑣1 ) ∨ 𝑓(𝑥̂𝑣2 )
1
1
2
2
1
3
𝑓(𝑥̂) = 〈(0.3 . (1)2 ) ∨ (1.8𝐼 . (1)3 ) ∨ (𝐼 . (1)4 )〉 ∨ 〈(0.3. ( 𝐼)2 ) ∨ (1.8𝐼 . ( 𝐼) ) ∨
3
5
1
(𝐼 . (1)4 )〉 = 〈(0.3 ) ∨ (1.8𝐼) ∨ (𝐼 )〉 ∨ 〈(0.133𝐼) ∨ (1.33𝐼) ∨ (𝐼)〉 = 1.8𝐼
Do not forget that the indeterminate component 𝐼 to the power 𝑛 where 𝑛 > 0
is equal to 𝐼 (i.e. 𝐼 𝑛 = 𝐼 𝑓𝑜𝑟 𝑛 > 0).
4.2 Example
0.1
1
Let 𝐴 = ( 𝐼
0.9
0.5 0.2𝐼
0.4
1
0 ) , 𝑏 = (0.3𝐼 ),
0.7
0.6
It easy to see that some components of the matrix 𝐴 are of the form
𝑎𝑖𝑗 = 𝑚𝐼, 𝑚 ∈ (0,1], while 𝑏𝑖 ∈ [0,1] ∪ 𝐼, in this case, and by theorem (3.2), the
system of the relation equation 𝐴𝑜𝑥 = 𝑏 is incompatible.
4.3 Example
−
2
1
1
Let min 𝑓(𝑥) = (0.2𝐼. 𝑥1 3 ) ∨ (1.3. 𝑥23 ) ∨ (𝐼 . 𝑥32 ) ∨ (0.35. 𝑥4−2 )
s. t. 𝐴𝑜𝑥 = 𝑏
𝑥𝑗 ∈ [0,1]⋃𝐼
(1 ≤ 𝑗 ≤ 𝑛)
.2 .3
Where 𝑏 = (0.3, 0.7𝐼, 0.5, 0.2𝐼) , 𝐴 = (. 3 . 2
1 0
0 .5
𝑇
.4
.9
.1
1
.6
. 8)
1
0 4×4
Using the formula (2), the components of 𝑥̂𝑣1 are
4
𝑥̂1 = ⋀(𝑎𝑖1 ⋈ 𝑏𝑖 ) = 0.5
𝑖=1
4
𝑥̂2 = ⋀(𝑎𝑖2 ⋈ 𝑏𝑖 ) = 1
𝑖=1
Huda E. Khalid, “Neutrosophic Geometric Programming (NGP) with (max-product) Operator, An
Innovative Model”
Neutrosophic Sets and Systems, Vol. 32, 2020
4
𝑥̂3 = ⋀(𝑎𝑖3 ⋈ 𝑏𝑖 ) =
𝑖=1
4
𝑥̂4 = ⋀(𝑎𝑖4 ⋈ 𝑏𝑖 ) =
𝑖=1
∴ 𝑥̂𝑣1 = (𝑥̂1 , 𝑥̂2 , 𝑥̂3 , 𝑥̂4
3
4
1
2
)𝑇
𝑇
3
= (0.5,1, , 0.5)
4
Using the formula (3), the components of 𝑥̂𝑣2 are
4
𝑥̂1 = ⋀(𝑎𝑖1 Θ 𝑏𝑖 ) = 1
𝑖=1
4
2
𝑥̂2 = ⋀(𝑎𝑖2 Θ 𝑏𝑖 ) = 𝐼
5
𝑖=1
4
𝑥̂3 = ⋀(𝑎𝑖3 Θ𝑏𝑖 ) = 0.2𝐼
𝑖=1
4
𝑥̂4 = ⋀(𝑎𝑖4 Θ𝑏𝑖 ) = 0.875𝐼
𝑖=1
𝑇
2
∴ 𝑥̂𝑣2 = (𝑥̂1 , 𝑥̂2 , 𝑥̂3 , 𝑥̂4 )𝑇 = ( 𝐼, 1,0.2𝐼, 0.875𝐼)
5
.2 .3 .4 .6
0
0 0 ) , 𝐴 = (. 3
In this example, 𝐴𝑣1 = ( 0 0
𝑣2
1 0 .1 1
0
0 0
0 0
0
𝐴𝑜𝑥̂ = (𝐴𝑣1 𝑜𝑥̂𝑣1 ) + (𝐴𝑣2 𝑜𝑥̂𝑣2 )
0.5
0 0
.2 .3 .4 .6
1
0 0 ) 𝑜 3 + (. 3 . 2
= (0 0
0 0
1 0 .1 1
4
0 .5
0 0
0 0
[0.5]
0.3
0.7𝐼
=[
]=𝑏
0.5
0.2𝐼
0
0
.2 .9
0
0
.5 1
0
.9
0
1
0
. 8),
0
0
2
0
𝐼
. 8) 𝑜 5
1
0
0.2𝐼
0
[0.875𝐼 ]
Huda E. Khalid, “Neutrosophic Geometric Programming (NGP) with (max-product) Operator, An
Innovative Model”
Neutrosophic Sets and Systems, Vol. 32, 2020
Since 𝐴𝑜𝑥̂ = 𝑏, then there is a solution in 𝑋(𝐴, 𝑏) and 𝑥̂ is the greatest solution
to 𝐴𝑜𝑥 = 𝑏. The value of 𝑓(𝑥̂) is calculated as follow,
𝑓(𝑥̂) = 𝑓(𝑥̂𝑣1 ) ∨ 𝑓(𝑥̂𝑣2 )
1
3
1
1
3 2
𝑓(𝑥̂) = 〈(0.2𝐼 . ( )−2 ) ∨ (1.3. (1)3 ) ∨ (𝐼 . ( ) ) ∨ (0.35. (0.5)−2 )〉 ∨
2
4
3
1
1
〈(0.2𝐼 . (1)−2 ) ∨ (1.3. (0.4𝐼)3 ) ∨ (𝐼 . (0.2𝐼)2 ) ∨ (0.35. (0.5)−2 )〉 = 〈(0.57𝐼 ) ∨
(1.3) ∨ (0.87𝐼 ) ∨ (0.5𝐼 )〉 ∨ 〈(0.2𝐼) ∨ (0.96𝐼) ∨ (0.45𝐼) ∨ (0.5𝐼)〉 = 1.3
5 Conclusion
It is important to know that the fuzzy geometric programming problems (FGPP) have
wide applications in the business management, communication system, civil engineering,
mechanical engineering, structural design and optimization, chemical engineering, optimal
control, decision making, and electrical engineering, unfortunately, the fuzzy logic lacks to
cover the indeterminate solution of any real-world problems, this pushed the author to
construct a new branch of the neutrosophic geometric programming (NGP) problems subject
to neutrosophic relation equations (NRE) and made a series of articles in an attempt to cover
the theoretical sides of (NGP) problems. This paper contains a new (NGP) model subject to
(NRE) with setting up a definition for the maximum solution of this program as well as some
new theorems dealt with the consistency of the problem and some propositions of the new
operation Θ. The future prospects are to make a deep study for the above-mentioned
applications from the point of view of relational neutrosophic geometric programming (RNGP)
problems.
Reference
[1] B. Y. Cao "Optimal models and methods with fuzzy quantities". Springer-Verlag, Berlin (2010).
[2] E. Sanchez, Resolution of Composite Fuzzy Relation Equations, Information and Control 30,38-48
(1976).
[3] F. Smarandache & Huda E. Khalid "Neutrosophic Precalculus and Neutrosophic Calculus". Second
enlarged edition, Pons asbl 5, Quai du Batelage, Brussels, Belgium, European Union, 2018.
[4] F. Smarandache, H. E. Khalid & A. K. Essa, “Neutrosophic Logic: the Revolutionary Logic in Science
and Philosophy”, Proceedings of the National Symposium, EuropaNova, Brussels, 2018.
[5] F. Smarandache, H. E. Khalid, A. K. Essa, M. Ali, “The Concept of Neutrosophic Less Than or Equal
To: A New Insight in Unconstrained Geometric Programming”, Critical Review, Volume XII, 2016, pp.
72-80.
[6] H. E. Khalid, “An Original Notion to Find Maximal Solution in the Fuzzy Neutrosophic Relation
Equations (FNRE) with Geometric Programming (GP) ”, Neutrosophic Sets and Systems, vol. 7, 2015, pp.
3-7.
[7] H. E. Khalid, “The Novel Attempt for Finding Minimum Solution in Fuzzy Neutrosophic Relational
Geometric Programming (FNRGP) with (max, min) Composition”, Neutrosophic Sets and Systems, vol.
11, 2016, pp. 107-111.
[8]. H. E. Khalid, F. Smarandache, & A. K. Essa, (2018). The Basic Notions for (over, off, under)
Neutrosophic Geometric Programming Problems. Neutrosophic Sets and Systems, 22, 50-62.
Huda E. Khalid, “Neutrosophic Geometric Programming (NGP) with (max-product) Operator, An
Innovative Model”
Neutrosophic Sets and Systems, Vol. 32, 2020
[9] H. E. Khalid, (2020). Geometric Programming Dealt with a Neutrosophic Relational Equations Under
the (𝑚𝑎𝑥 − 𝑚𝑖𝑛) Operation, book chapter ‘’Neutrosophic Sets in Decision Analysis and Operations
Research’’, IGI Global Publishing House.
[10] H. E. Khalid, F. Smarandache, & A. K. Essa, (2016). A Neutrosophic Binomial Factorial Theorem with
their Refrains. Neutrosophic Sets and Systems, 14, 50-62.
[11] V. Kandasamy, F. Smarandache, “Fuzzy Relational Maps and Neutrosophic Relational Maps”,
American Research Press, Rehoboth,2004.
Received: 20 Feb, 2020. Accepted: 21 Mar, 2020
Huda E. Khalid, “Neutrosophic Geometric Programming (NGP) with (max-product) Operator, An
Innovative Model”
Neutrosophic Sets and Systems, Vol.32, 2020
University of New Mexico
Neutrosophic Soft Sets Applied on Incomplete Data
Abhijit Saha 1, Said Broumi 2, and Florentin Smarandache 3
Faculty of Mathematics, Techno College of Engineering Agartala, , Tripura, India, Pin-799004; Email:
abhijit84.math@gmail.com
2 Faculty of Science, University of Hassan II, B.P 7955, Sidi Othman, Casablanca, Morocco; Email: broumisaid78@gmail.com
3 Faculty of Mathematics ,University of New Mexico, Gallup, New Mexico 87301, USA; Email: fsmarandache@gmail.com
1
* Correspondence: abhijit84.math@gmail.com
Abstract: A neutrosophic set is a part of neutrosophy that studies the origin, nature and scope of neutralities as well as their interactions with different ideational spectra. In this present paper first we have
introduced the concept of a neutrosophic soft set having incomplete data with suitable examples. Then
we have tried to explain the consistent and inconsistent association between the parameters. We have
introduced few new definitions, namely- consistent association number between the parameters, consistent association degree, inconsistent association number between the parameters and inconsistent association degree to measure these associations. Lastly we have presented a data filling algorithm. An illustrative example is employed to show the feasibility and validity of our algorithm in practical situation.
Keywords: Soft set, neutrosophic set, neutrosophic soft set, data filling.
1. Introduction
In 1999, Molodstov [01] initiated the concept of soft set theory as a new mathematical tool for modelling uncertainty, vague concepts and not clearly defined objects. Although various traditional tools,
including but not limited to rough set theory [02], fuzzy set theory [03], intuitionistic fuzzy set theory
[04] etc. have been used by many researchers to extract useful information hidden in the uncertain data, but there are immanent complications connected with each of these theories. Additionally, all these
approaches lack in parameterizations of the tools and hence they couldn’t be applied effectively in real
life problems, especially in areas like environmental, economic and social problems. Soft set theory is
standing uniquely in the sense that it is free from the above mentioned impediments and obliges approximate illustration of an object from the beginning, which makes this theory a natural mathematical formalism for approximate reasoning.
The Theory of soft set has excellent potential for application in various directions some of which are
reported by Molodtsov in his pioneer work. Later on Maji et al. [05] introduced some new annotations
on soft sets such as subset, complement, union and intersection of soft sets and discussed in detail its
applications in decision making problems. Ali et al. [06] defined some new operations on soft sets and
shown that De Morgan's laws holds in soft set theory with respect to these newly defined operations.
Atkas and Cagman [07] compared soft sets with fuzzy sets and rough sets to show that every fuzzy set
and every rough set may be considered as a soft set. Jun [08] connected soft sets to the theory of
BCK/BCI-algebra and introduced the concept of soft BCK/BCI-algebras. Feng et al. [09] characterized
soft semi rings and a few related notions to establish a relation between soft sets and semi rings. In
2001, Maji et al. [10] defined the concept of fuzzy soft set by combining of fuzzy sets and soft sets . Roy
and Maji [11] proposed a fuzzy soft set based decision making method. Xiao et al. [12] presented a
combined forecasting method based on fuzzy soft set. Feng et al. [13] discussed the validity of the
A. Saha, S. Broumi and F. Smarandache; Neutrosophic soft sets applied on incomplete data
Neutrosophic Sets and Systems, Vol. 32, 2020
283
Roy-Maji method and presented an adjustable decision-making method based on fuzzy soft set. Yang
et al. [14] initiated the idea of interval valued fuzzy soft set (IVFS-set) and analyzed a decision making method using the IVFS-sets. The notion of intuitionistic fuzzy set (IFS) was initiated by Atanassov
as a significant generalization of fuzzy set. Intuitionistic fuzzy sets are very useful in situations when
description of a problem by a linguistic variable, given in terms of a membership function only, seems
too complicated. Recently intuitionistic fuzzy sets have been applied to many fields such as logic programming, medical diagnosis, decision making problems etc. Smarandache [15] introduced the concept of neutrosophic set which is a mathematical tool for handling problems involving imprecise, indeterminacy and inconsistent data. Thao and Smaran [16] proposed the concept of divergence measure on neutrosophic sets with an application to medical problem. Song et al. [17] applied neutrosophic
sets to ideals in BCK/BCI algebras. Some recent applications of neutrosophic sets can be found in [18],
[19], [20], [21], [22], [23] and [24]. Maji [25] introduced the concept of neutrosophic soft set and established some operations on these sets. Mukherjee et al [26] introduced the concept of interval valued
neutrosophic soft sets and studied their basic properties. In 2013, Broumi and Smarandache [27, 28]
combined the intuitionistic neutrosophic and soft set which lead to a new mathematical model called
“intuitionistic neutrosophic soft set”. They studied the notions of intuitionistic neutrosophic soft set
union, intuitionistic neutrosophic soft set intersection, complement of intuitionistic neutrosophic soft
set and several other properties of intuitionistic neutrosophic soft set along with examples and proofs
of certain results. Also, in [29] S. Broumi presented the concept of “generalized neutrosophic soft set”
by combining the generalized neutrosophic sets and soft set models, studied some properties on it,
and presented an application of generalized neutrosophic soft set in decision making problem. Recently, Deli [30] introduced the concept of interval valued neutrosophic soft set as a combination of interval neutrosophic set and soft set. In 2014, S. Broumi et al. [31] initiated the concept of relations on interval valued neutrosophic soft sets.
The soft sets mentioned above are based on complete information. However, incomplete information widely exists in various real life problems. Soft sets under incomplete information become incomplete soft sets. H. Qin et al [32] studied the data filling approach of incomplete soft sets. Y. Zou et
al [33] investigated data analysis approaches of soft sets under incomplete information. In this paper
first we have introduced the concept of a neutrosophic soft set with incomplete data supported by examples. Then we have introduced few new definitions to measure the consistent and inconsistent association between the parameters. Lastly we have presented a data filling algorithm supported by an
illustrative example to show the feasibility and validity of our algorithm.
2. Preliminaries:
2.1 Definition: [03] Let U be a non empty set. Then a fuzzy set
τ
x, μ
τ
x :x U
τ
on U is a set having the form
where the function μ τ :U [0, 1] is called the membership function and
μ τ x represents the degree of membership of each element x U .
2.2 Definition: [04] Let U be a non empty set. Then an intuitionistic fuzzy set (IFS for short)
object
having
the
form
τ
x, μ
τ
x , γτ x
: xU
where
the
τ
is an
functions
μ τ :U [0, 1] and γ τ :U [0, 1] are called membership function and non-membership function
respectively.
μ τ x and γ τ x represent the degree of membership and the degree of non-membership
respectively of each element x U and 0 μ τ x + γ τ x 1 for each x U. We denote the class of
all intuitionistic fuzzy sets on U by IFSU.
A. Saha, S. Broumi and F. Smarandache; Neutrosophic soft sets applied on incomplete data
Neutrosophic Sets and Systems, Vol. 32, 2020
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2.3 Definition: [01] Let U be a universe set and E be a set of parameters. Let P U denotes the
power set of U and AE. Then the pair F, A is called a soft set over
U , where
F is a mapping
given by F: A P U .
In other words, the soft set is not a kind of set, but a parameterized family of subsets of
eA, F e U may be considered as the set of e-approximate elements of the soft set F, A .
2.4 Definition: [10] Let U be a universe set, E be a set of parameters and
U . For
A E . Then the pair
F, A is called a fuzzy soft set over U , where F is a mapping given by F: A FSU .
2.5 Definition: [34] Let U be a universe set, E be a set of parameters and
A E . Then the pair
F, A is called an intuitionistic fuzzy soft set over U , where F is a mapping given by F: A IFSU .
For e A , F e is an intuitionistic fuzzy subset of U and is called the intuitionistic fuzzy value
set of the parameter ‘e’.
Let us denote μ F e x by the membership degree that object ‘x’ holds parameter ‘e’ and γ F e x
by the membership degree that object ‘x’ doesn’t hold parameter ‘e’ , where eA and x U . Then
F e can be written as an intuitionistic fuzzy set such that F e = x, μ F e x , γ F e x : x U .
2.6 Definition: [15] A neutrosophic set A on the universe of discourse U is defined as
A x, A x , A x , A x x U , where
A , A , A U 0,1 are functions such that the
condition: x U , 0 A x A x A x 3 is satisfied.
Here A x , A x , A x represent the truth-membership, indeterminacy-membership and
falsity-membership respectively of the element x U .
Smarandache [15] applied neutrosophic sets in many directions after giving examples of
neutrosophic sets. Then he introduced the neutrosophic set operations namely-complement, union,
intersection, difference, Cartesian product etc.
2.7 Definition: [21] Let U be an initial universe, E be a set of parameters and A E . Let NP U
f , A is termed to be the neutrosophic
A NP U .
denotes the set of all neutrosophic sets of U . Then the pair
soft set over U , where f is a mapping given by f
2.8 Example: Let us consider a neutrosophic soft set
f , A which describes the “attractiveness of the
house”. Suppose U = {u1 , u2 , u3 , u4 , u5 , u6 } be the set of six houses under consideration and
E = {e1 (beautiful), e2 (expensive), e3 (cheap), e4 (good location), e5 (wooden)}be the set of parameters. Then
a neutrosophic soft set
f , A over U can be given by:
U
e1
e2
e3
e4
e5
u1
(0.8,0.5,0.2)
(0.3,0.4,0.6)
(0.1,0.6,0.4)
(0.7,0.3,0.6)
(0.3,0.4,0.6)
u2
(0.4,0.1,0.7)
(0.8,0.2,0.4)
(0.4,0.1,0.7)
(0.2,0.4,0.4)
(0.1,0.1,0.3)
u3
(0.2,0.6,0.4)
(0.5,0.5,0.5)
(0.8,0.1,0.7)
(0.5,0.3,0.5)
(0.5,0.5,0.5)
u4
(0.3,0.4,0.4)
(0.1,0.3,0.3)
(0.3,0.4,0.4)
(0.6,0.6,0.6)
(0.1,0.1,0.5)
A. Saha, S. Broumi and F. Smarandache; Neutrosophic soft sets applied on incomplete data
Neutrosophic Sets and Systems, Vol. 32, 2020
285
u5
(0.1,0.1,0.7)
(0.2,0.6,0.7)
(0.4,0.2,0.1)
(0.8,0.6,0.1)
(0.6,0.7,0.7)
u6
(0.5,0.3,0.9)
(0.3,0.6,0.6)
(0.1,0.5,0.5)
(0.3,0.6,0.5)
(0.4,0.4,0.4)
3. Neutrosophic soft sets with incomplete (missing) data:
Suppose that ( f , E ) is a neutrosophic soft set over U, such that $ xi Î U and e j Î E so that none
of mf (e ) ( xi ), g f (e ) ( xi ) and d f (e ) ( xi ) is known. In this case, in the tabular representation of the
j
j
j
neutrosophic soft set ( f , E ) , we write
(mf (e ) (x ), g f (e ) (x ), d f (e ) (x ))= * . Here we say that the data
i
j
j
i
j
i
for f (e j ) is missing and the neutrosophic soft set ( f , E ) over U has incomplete data.
3.1 Example: Suppose Tech Mahindra is recruiting some new Graduate Trainee for the session 20192020 and suppose that eight candidates have applied for the job. Assume that U = {u1 , u2 , u3 ,......, u8 }
be
the
set
of
candidates
and
E = {e1 (communication skill), e2 (domain knowledge), e3 (experienced), e4 (young),
e5 (highest academic degree), e6 (professional attitute)}be the set of parameters. Then a neutrosophic soft
set over U having missing data can be given by Table-1.
Table-1
U
e1
e2
e3
e4
e5
e6
u1
(0.8,0.5,0.2)
(0.3,0.4,0.6)
(0.1,0.6,0.4)
(0.7,0.3,0.6)
(0.3,0.4,0.6)
(0.2,0.5,0.5)
u2
(0.4,0.1,0.7)
(0.8,0.2,0.4)
(0.4,0.1,0.7)
(0.2,0.4,0.4)
*
(0.6,0.6,0.4)
u3
(0.2,0.6,0.4)
(0.5,0.5,0.5)
*
(0.5,0.5,0.5)
(0.5,0.5,0.5)
(0.3,0.4,0.6)
u4
(0.3,0.4,0.4)
(0.1,0.3,0.3)
(0.3,0.4,0.4)
(0.6,0.6,0.6)
(0.1,0.1,0.5)
(0.3,0.4,0.4)
u5
(0.1,0.1,0.7)
*
(0.4,0.2,0.1)
(0.8,0.6,0.1)
(0.6,0.7,0.7)
(0.3,0.4,0.3)
u6
(0.5,0.3,0.9)
(0.3,0.6,0.6)
(0.1,0.5,0.5)
(0.3,0.6,0.6)
(0.4,0.4,0.4)
(0.3,0.6,0.6)
u7
(0.2,0.4,0.6)
(0.4,0.4,0.5)
(0.5,0.5,0.6)
*
(0.7,0.5,0.8)
(0.4,0.4,0.5)
u8
(0.2,0.3,0.1)
(0.6,0.6,0.1)
(0.8,0.3,0.8)
(0.4,0.3,0.4)
(0.5,0.6,0.3)
(0.9,0.3,0.3)
In case of soft set theory, there always exist some obvious or hidden associations between
parameters. Let us focus on this to find the associations between the parameters of a neutrosophic soft
set.
In example 2.8, one can easily find that if a house is expensive, the house is not cheap and vice
versa. Thus there is an inconsistent association between the parameters ‘expensive’ and ‘cheap’.
Generally, if a house is beautiful or situated in a good location, the house is expensive. Thus there is a
consistent association between the parameters ‘beautiful’ and ‘expensive’ or the parameters ‘good
location’ and ‘expensive’.
In example 3.1, we find that if a candidate is experienced or have highest academic degree, he/she
is not young. Thus there is an inconsistent association between parameters ‘experienced’ and ‘young’
or between ‘highest academic degree’ and ‘young’.
The above two examples reveal the interior relations of parameters. In a neutrosophic soft set,
these associations between parameters will be very useful for filling incomplete data. If it is found that
A. Saha, S. Broumi and F. Smarandache; Neutrosophic soft sets applied on incomplete data
Neutrosophic Sets and Systems, Vol. 32, 2020
286
the parameters ei and e j are associated and the data for f (ei ) is missing, then we can fill the
( )
missing data according to the corresponding data in f e j . To measure these associations, let us
define the notion of association degree and some relevant concepts.
For the rest of the paper we shall assume that U be the universe set and E be the set of parameters.
Let U ij denotes the set of objects that have specified values in the form of an ordered triplet (a, b, c)
where a, b, c[0, 1] on both parameters ei and e j such that
ìï
æ
ö ü
ï
U ij = ïí x Î U : mf (e ) (x), g f (e ) (x), d f (e ) (x) ¹ *, çççmf e (x), g f e (x), d f e (x)÷÷÷¹ * ïý
i
i
i
( j)
( j ) ø ïþï
ïîï
è ( j)
(
)
In other words U ij is the collection of those objects that have known data both on ei and e j .
3.2 Definition: Let ei , e j Î E . Then the consistent association number between the parameters ei and
ej
is
denoted
by
CANij
and
is
defined
ïì
ïü
CANij = ïí x Î Uij : mf e (x) = m
x), g f (e ) (x) = g
x), d f (e ) (x) = d
x)ïý
(
(
(
(
)
f (e j )
f (e j )
f (e j )
ïï
ïï
i
i
i
î
þ
as:
where
.
denotes the cardinality of a set.
3.3 Definition: Let ei , e j Î E . Then the consistent association degree between the parameters ei and
e j is denoted by CADij and is defined as: CADij =
CANij
U ij
where . denotes the cardinality of a set.
It can be easily verified that the value of CADij lies in [0, 1]. Actually CADij measures the extent to
which the value of parameter ei keeps consistent with that of parameter e j over
Uij . Next we define
inconsistent association number and inconsistent association degree as follows:
3.4 Definition: Let ei , e j Î E . Then the inconsistent association number between the parameters ei
and e j is denoted by ICANij and is defined as
ïì
ïü
ïý
ICANij = ïí x Î Uij : mf e (x) ¹ m
x
x
x
or g f e (x) ¹ g
or d f e (x) ¹ d
(
)
(
)
(
)
( i)
( i)
( i)
f (e j )
f (e j )
f (e j )
ïï
ïï
î
þ
where . denotes the cardinality of a set.
3.5 Definition: Let ei , e j Î E . Then the inconsistent association degree between the parameters ei
and e j is denoted by ICADij and is defined as: ICADij =
ICANij
U ij
where . denotes the cardinality of
a set.
It can be easily verified that the value of ICADij lies in [0, 1]. Actually ICADij measures the extent
to which the parameters ei and e j is inconsistent.
3.6 Definition: Let ei , e j Î E . Then the association degree between the parameters ei and e j
{
}
denoted by ADij and is defined by ADij = max CADij , ICADij .
A. Saha, S. Broumi and F. Smarandache; Neutrosophic soft sets applied on incomplete data
is
Neutrosophic Sets and Systems, Vol. 32, 2020
287
If CADij > ICADij , then ADij = CADij , which means that most of the objects over
Uij
have
consistent values on parameters ei and e j . If CADij < ICADij , then ADij = ICADij , which means that
Uij
ei and e j . Again if
CADij = ICADij , then it means that there is the lowest association degree between the parameters ei
and e j .
most of the objects over
have inconsistent values on parameters
3.7 Theorem: For parameters ei and e j , ADij ³ 0.5 for all i, j.
Proof: Follows from the fact that CADij + ICADij = 1 .
3.8 Definition: If ei Î E , then the maximal association degree of parameter ei is denoted by MADi
and is defined by MADi = max ADij .
j
4. DATA Filling Algorithm for a neutrosophic soft set:
Step-1: Input the neutrosophic soft set ( f , E ) which has incomplete data.
Step-2: Find all parameters ei for which data is missing.
Step-3: Compute ADij for j=1,2,3….,m (where ‘m’ is the number of parameters in E).
Step-4: Compute MADi .
Step-5:
Find out all parameters e j which have the maximal association degree MADi with the
parameter ei .
ei and e j ’s (j=1,2,3,….)
Step-6: In case of consistent association between the parameter
æ
ö
(mf (e ) (x), g f (e ) (x), d f (e ) (x)) = çççèmax mf (e ) (x), max g f (e ) (x), max d f (e ) (x)ø÷÷÷ .
i
i
i
association
between
the
æ
(mf (e ) (x), g f (e ) (x), d f (e ) (x)) = çççè1i
i
j
j
i
j
j
ei
parameter
max m
j
f (e j )
(x),1 -
max g
j
ej
and
f (e j )
In case of inconsistent
j
j
(x),1 -
max d
j
f (e j )
’s
(j=1,2,3,….)
ö
ø
(x)÷÷÷.
Step-7: If all the missing data are filled then stop else go to step-2.
An Illustrative example: Consider the neutrosophic soft set given in example 3.1.
Step-1:
U
e1
e2
e3
e4
e5
e6
u1
(0.8,0.5,0.2)
(0.3,0.4,0.6)
(0.1,0.6,0.4)
(0.7,0.3,0.6)
(0.3,0.4,0.6)
(0.2,0.5,0.5)
u2
(0.4,0.1,0.7)
(0.8,0.2,0.4)
(0.4,0.1,0.7)
(0.2,0.4,0.4)
*
(0.6,0.6,0.4)
u3
(0.2,0.6,0.4)
(0.5,0.5,0.5)
*
(0.5,0.5,0.5)
(0.5,0.5,0.5)
(0.3,0.4,0.6)
u4
(0.3,0.4,0.4)
(0.1,0.3,0.3)
(0.3,0.4,0.4)
(0.6,0.6,0.6)
(0.1,0.1,0.5)
(0.3,0.4,0.4)
u5
(0.1,0.1,0.7)
*
(0.4,0.2,0.1)
(0.8,0.6,0.1)
(0.6,0.7,0.7)
(0.3,0.4,0.3)
u6
(0.5,0.3,0.9)
(0.3,0.6,0.6)
(0.1,0.5,0.5)
(0.3,0.6,0.6)
(0.4,0.4,0.4)
(0.3,0.6,0.6)
u7
(0.2,0.4,0.6)
(0.4,0.4,0.5)
(0.5,0.5,0.6)
*
(0.7,0.5,0.8)
(0.4,0.4,0.5)
u8
(0.2,0.3,0.1)
(0.6,0.6,0.1)
(0.8,0.3,0.8)
(0.4,0.3,0.4)
(0.5,0.6,0.3)
(0.9,0.3,0.3)
A. Saha, S. Broumi and F. Smarandache; Neutrosophic soft sets applied on incomplete data
Neutrosophic Sets and Systems, Vol. 32, 2020
288
Step-2: Clearly there are missing data in f (e2 ), f (e3 ), f (e4 ), f (e5 ). We shall fill these missing data.
Step-3:
(a) For the parameter e2 .
\ U 21 = {u1, u2 , u3 , u4 , u6 , u7 , u8 },U 23 = {u1, u2 , u4 , u6 , u7 , u8 },U 24 = {u1, u2 , u3 , u4 , u6 , u8 },
U 25 = {u1, u3 , u4 , u6 , u7 , u8 },U 26 = {u1, u2 , u3 , u4 , u6 , u7 , u8 }.
Now CAN 21 = { } = 0 and so CAD21 = 0 . Again ICAN21 = {u1, u2 , u3 , u4 , u6 , u7 , u8 } = 7 and so
ICAN21 7
ICAD21 =
= = 1 . Hence AD21 = max {CAD21, ICAD21}= max{0,1} = 1.
7
U
21
CAN23 = { } = 0 and
ICAD23 =
ICAN23
U 23
CAN 24 =
ICAD24 =
=
so
CAD23 = 0 .
U 24
=
ICAN23 = {u1, u2 , u4 , u6 , u7 , u8 } = 6
and
so
6
= 1 . Hence AD23 = max {CAD23 , ICAD23 }= max{0,1} = 1.
6
{u3 , u6 } = 2 and so CAD24 =
ICAN 24
Again
2
= 0.33 . Again ICAN24 = {u1, u2 , u4 , u8 } = 4 and so
6
4
= 0.66 . Hence AD24 = max {CAD24 , ICAD24 }= max{0.33,0.66} = 0.66.
6
2
CAN25 = {u3 , u1} = 2 and so CAD25 = = 0.33 . Again ICAN25 = {u4 , u6 , u7 , u8 } = 4 and so
6
ICAN24 4
ICAD25 =
= = 0.66 . Hence AD25 = max {CAD25 , ICAD25 }= max{0.33, 0.66} = 0.66.
U
6
24
1
CAN26 = {u4 } = 1 and so CAD26 = = 0.14 . Again ICAN26 = {u1, u2 , u3 , u6 , u7 , u8 } = 6 and so
7
ICAN 26 6
ICAD26 =
= = 0.85 . Hence AD26 = max {CAD26 , ICAD26 }= max{0.14, 0.85} = 0.85.
7
U
26
Thus MAD2 = max AD2 j = max {AD21, AD23 , AD24 , AD25 , AD26 }= max{1,1, 0.66, 0.66, 0.85} = 1. .
j
(b) For the parameter e3 .
\ U31 = {u1, u2 , u4 , u6 , u7 , u8 },U32 = {u1, u2 , u4 , u6 , u7 , u8 },U34 = {u1, u2 , u4 , u5 , u6 , u8 },
U35 = {u1, u4 , u5 , u6 , u7 , u8 },U36 = {u1, u2 , u4 , u5 , u6 , u7 , u8 }.
2
Now CAN31 = {u2 , u4 } = 2 and so CAD31 = = 0.33 . Again ICAN31 = {u1, u6 , u7 , u8 } = 4 and so
6
ICAN31 4
ICAD31 =
= = 0.66 . Hence AD31 = max {CAD31, ICAD31}= max{0.33, 0.66} = 0.66.
U
6
31
CAN32 = { } = 0 and so CAD32 = 0 . Again ICAN32 = {u1, u2 , u4 , u6 , u7 , u8 } = 6
ICAN32 6
ICAD32 =
= = 1 . Hence AD32 = max {CAD32 , ICAD32 }= max{0,1} = 1.
6
U 32
CAN34 = { } = 0 and so CAD34 = 0 . Again ICAN34 = {u1, u2 , u4 , u5 , u6 , u8 } = 6
ICAN34 4
ICAD34 =
= = 0.66 . Hence AD34 = max {CAD34 , ICAD34 }= max{0, 0.66} = 0.66.
6
U 34
A. Saha, S. Broumi and F. Smarandache; Neutrosophic soft sets applied on incomplete data
and
so
and
so
Neutrosophic Sets and Systems, Vol. 32, 2020
289
CAN35 = { } = 0 and so CAD35 = 0 . Again ICAN35 = {u1, u4 , u5 , u6 , u7 , u8 } = 6
ICAN35 6
ICAD35 =
= = 1 . Hence AD35 = max {CAD35 , ICAD35 }= max{0,1} = 1.
6
U 35
and
so
1
CAN36 = {u4 } = 1 and so CAD36 = = 0.14 . Again ICAN36 = {u1, u2 , u5 , u6 , u7 , u8 } = 6 and so
7
ICAN36 6
ICAD36 =
= = 0.85 . Hence AD36 = max {CAD36 , ICAD36 }= max{0.14, 0.85} = 0.85.
7
U
36
Thus MAD3 = max AD3 j = max {AD31, AD32 , AD34 , AD35 , AD36 }= max{0.66,1, 0.66,1, 0.85} = 1.
j
(c) For the parameter e4 .
\ U 41 = {u1, u2 , u3 , u4 , u5 , u6 , u8 },U 42 = {u1, u2 , u3 , u4 , u6 , u8 },U 43 = {u1, u2 , u4 , u5 , u6 , u8 },
U 45 = {u1, u3 , u4 , u5 , u6 , u8 },U 46 = {u1, u2 , u3 , u4 , u5 , u6 , u8 }.
Now CAN 41 = { } = 0 and so CAD41 = 0 . Again ICAN41 = {u1, u2 , u3 , u4 , u5 , u6 , u8 } = 7 and so
ICAN41 7
ICAD41 =
= = 1 . Hence AD41 = max {CAD41, ICAD41}= max{0,1} = 1.
7
U
41
2
CAN42 = {u3 , u6 } = 2 and so CAD42 = = 0.33 . Again ICAN42 = {u1, u2 , u4 , u8 } = 4 and so
6
ICAN 42 4
ICAD42 =
= = 0.66 . Hence AD42 = max {CAD42 , ICAD42 }= max{0.33,0.66} = 0.66.
U
6
42
CAN43 = { } = 0
ICAD43 =
ICAN43
U 43
=
and
so
CAD43 = 0 .
Again
ICAN43 = {u1, u2 , u4 , u5 , u6 , u8 } = 6
and
so
6
= 1 . Hence AD43 = max {CAD43 , ICAD43 }= max{0,1} = 1.
6
1
CAN45 = {u3 } = 1 and so CAD45 = = 0.16 . Again ICAN45 = {u1, u4 , u5 , u6 , u8 } = 5 and so
6
ICAN45 5
ICAD45 =
= = 0.83 . Hence AD45 = max {CAD45 , ICAD35 }= max{0.16, 0.83} = 0.83.
6
U
45
1
CAN46 = {u6 } = 1 and so CAD46 = = 0.14 . Again ICAN46 = {u1, u2 , u3 , u4 , u5 , u8 } = 6 and so
7
ICAN 46 6
ICAD46 =
= = 0.85 . Hence AD46 = max {CAD46 , ICAD46 }= max{0.14, 0.85} = 0.85.
7
U
46
Thus MAD4 = max AD4 j = max {AD41, AD42 , AD43 , AD45 , AD46 }= max{1, 0.66,1, 0.83, 0.85} = 1.
j
(d) For the parameter e5 .
\ U51 = {u1, u3 , u4 , u5 , u6 , u7 , u8 },U52 = {u1, u3 , u4 , u6 , u7 , u8 },U53 = {u1, u4 , u5 , u6 , u7 , u8 },
U54 = {u1, u3 , u4 , u5 , u6 , u8 },U56 = {u1, u3 , u4 , u5 , u6 , u7 , u8 }.
Now CAN 51 = { } = 0 and so CAD51 = 0 . Again ICAN51 = {u1, u3 , u4 , u5 , u6 , u7 , u8 } = 7 and so
ICAN51 7
ICAD51 =
= = 1 . Hence AD51 = max {CAD51, ICAD51}= max{0,1} = 1.
7
U
51
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290
2
CAN52 = {u1 , u3 } = 2 and so CAD52 = = 0.33 . Again ICAN52 = {u4 , u6 , u7 , u8 } = 4 and so
6
ICAN52 4
ICAD52 =
= = 0.66 . Hence AD52 = max {CAD52 , ICAD52 }= max{0.33, 0.66} = 0.66.
6
U
52
CAN53 = { } = 0 and so CAD53 = 0 . Again ICAN53 = {u1, u4 , u5 , u6 , u7 , u8 } = 6 and
ICAN53 6
ICAD53 =
= = 1 . Hence AD53 = max {CAD53 , ICAD53 }= max{0,1} = 1.
U 53
6
so
1
CAN54 = {u3 } = 1 and so CAD54 = = 0.16 . Again ICAN54 = {u1, u4 , u5 , u6 , u8 } = 5 and so
6
ICAN54 5
ICAD54 =
= = 0.83 . Hence AD54 = max {CAD54 , ICAD54 }= max{0.16, 0.83} = 0.83.
6
U
54
CAN56 = { } = 0 and so CAD56 = 0 . Again ICAN56 = {u1, u3 , u4 , u5 , u6 , u7 , u8 } = 7 and so
ICAN56 7
ICAD56 =
= = 1 . Hence AD56 = max {CAD56 , ICAD56 }= max{0,1} = 1.
7
U 56
Thus MAD5 = max AD5 j = max {AD51, AD52 , AD53 , AD54 , AD56 }= max{1, 0.66,1, 0.83,1} = 1.
j
The association degree table for the neutrosophic soft set ( f , E ) is given below:
e1
e2
e3
e4
e5
e6
e2
1
_
1
0.66
0.66
0.85
e3
0.66
1
_
0.66
1
0.85
e4
1
0.66
1
_
0.83
0.85
e5
1
0.66
1
0.83
_
1
Step-4: From step-3, we have, MAD2 = 1, MAD3 = 1, MAD4 = 1, MAD5 = 1 .
e1 and e3 have the maximal association degree AD21
respectively with the parameter e2 .
Step-5: The parameters
and AD23
The parameters e2 and e5 have the maximal association degree AD32 and AD35 respectively with
the parameter e3 .
The parameters e1 and e3 have the maximal association degree AD41 and AD43 respectively with
the parameter e4 .
The parameters e1, e3 and e6 have the maximal association degree AD51 , AD53 and AD56 respectively
with the parameter e5 .
Step-6: There is a consistent association between the parameters e2 and e1 , e2 and e3 , e5 and e1 ,
e3 and e5 ; while there is an inconsistent association between the parameters e4 and e1 , e4 and
e3 .So we have,
A. Saha, S. Broumi and F. Smarandache; Neutrosophic soft sets applied on incomplete data
Neutrosophic Sets and Systems, Vol. 32, 2020
291
(mf (e ) (u ), g f (e ) (u ), d f (e ) (u ))
= (max (mf e (u ), mf e (u )), max (g f e (u ), g f e (u )), max (d f e (u ), d f e (u )))
( )
( )
( )
( )
( )
( )
2
5
5
2
5
1
5
2
5
3
1
5
3
5
1
5
5
3
= (max (0.1, 0.4), max (0.1, 0.2), max (0.7, 0.1)) = (0.4, 0.2, 0.7),
(mf (e ) (u ), g f (e ) (u ), d f (e ) (u ))
= (max (mf (e ) (u ), mf (e ) (u )), max (g f (e ) (u ), g f (e ) (u )), max (d f (e ) (u ), d f (e ) (u )))
3
3
3
3
3
2
3
3
3
5
2
3
5
3
2
3
3
5
= (max (0.5, 0.5), max (0.5, 0.5), max (0.5, 0.5)) = (0.5, 0.5, 0.5),
(mf (e ) (u ), g f (e ) (u ), d f (e ) (u ))
= (1- max (mf e (u ), mf e (u )), 1 - max (g f e (u ), g f e (u )),1 - max (d f e (u ), d f e (u )))
( )
( )
( )
( )
( )
( )
4
7
7
4
4
7
1
7
7
3
1
7
3
7
7
1
3
7
= (max (0.2, 0.5), max (0.4, 0.5), max (0.6, 0.6)) = (0.5, 0.5, 0.6),
(mf (e ) (u ), g f (e ) (u ), d f (e ) (u ))
= (max (mf e (u ), mf e (u )), max (g f e (u ), g f e (u )), max (d f e (u ), d f e (u )))
( )
( )
( )
( )
( )
( )
5
2
5
1
2
2
5
3
2
2
1
2
3
2
1
2
3
2
= (max (0.4, 0.4), max (0.1, 0.1), max (0.7, 0.7)) = (0.4, 0.1, 0.7).
Thus we have the following table which gives the tabular representation of the filled neutrosophic soft
set:
U
e1
e2
e3
e4
e5
e6
u1
(0.8,0.5,0.2)
(0.3,0.4,0.6)
(0.1,0.6,0.4)
(0.7,0.3,0.6)
(0.3,0.4,0.6)
(0.2,0.5,0.5)
u2
(0.4,0.1,0.7)
(0.8,0.2,0.4)
(0.4,0.1,0.7)
(0.2,0.4,0.4)
(0.4,0.1,0.7)
(0.6,0.6,0.4)
u3
(0.2,0.6,0.4)
(0.5,0.5,0.5)
(0.5,0.5,0.5)
(0.5,0.5,0.5)
(0.5,0.5,0.5)
(0.3,0.4,0.6)
u4
(0.3,0.4,0.4)
(0.1,0.3,0.3)
(0.3,0.4,0.4)
(0.6,0.6,0.6)
(0.1,0.1,0.5)
(0.3,0.4,0.4)
u5
(0.1,0.1,0.7)
(0.4,0.2,0.7)
(0.4,0.2,0.1)
(0.8,0.6,0.1)
(0.6,0.7,0.7)
(0.3,0.4,0.3)
u6
(0.5,0.3,0.9)
(0.3,0.6,0.6)
(0.1,0.5,0.5)
(0.3,0.6,0.6)
(0.4,0.4,0.4)
(0.3,0.6,0.6)
u7
(0.2,0.4,0.6)
(0.4,0.4,0.5)
(0.5,0.5,0.6)
(0.5,0.5,0.6)
(0.7,0.5,0.8)
(0.4,0.4,0.5)
u8
(0.2,0.3,0.1)
(0.6,0.6,0.1)
(0.8,0.3,0.8)
(0.4,0.3,0.4)
(0.5,0.6,0.3)
(0.9,0.3,0.3)
Conclusion: Incomplete information or missing data in a neutrosophic soft set restricts the usage of
the neutrosophic soft set. To make the neutrosophic soft set (with missing / incomplete data) more
useful, in this paper, we have proposed a data filling approach, where missing data is filled in terms of
the association degree between the parameters. We have validated the proposed algorithm by an example and drawn the conclusion that relation between parameters can be applied to fill the missing
data.
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A. Saha, S. Broumi and F. Smarandache; Neutrosophic soft sets applied on incomplete data
Neutrosophic Sets and Systems, Vol. 32, 2020
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Received: Nov 15, 2019. Accepted: Mar 25, 2020
A. Saha, S. Broumi and F. Smarandache; Neutrosophic soft sets applied on incomplete data
Neutrosophic Sets and Systems, Vol. 32, 2020
University of New Mexico
Aggregate Operators of Neutrosophic Hypersoft Set
Muhammad Saqlain 1, Sana Moin2, Muhammad Naveed Jafar 3, Muhammad Saeed3 and Florentin
Smarandache4
1
2
3
Lahore Garrison University, DHA Phase-VI, Sector C, Lahore, 54000, Pakistan. E-mail: msaqlain@lgu.edu.pk
Lahore Garrison University, DHA Phase-VI, Sector C, Lahore, 54000, Pakistan. E-mail: moinsana64@gmail.com
Lahore Garrison University, DHA Phase-VI, Sector C, Lahore, 54000, Pakistan. E-mail: naveedjafar@lgu.edu.pk
3
University of Management and Technology, C-II Johar Town, Lahore, 54000, Pakistan. E-mail: Muhammad.saeed@umt.edu.pk
4
Department of Mathematics, University of New Mexico, Gallup, NM 87301, USA. E-mail: smarand@unm.edu
Abstract: Multi-criteria decision making (MCDM) is concerned about organizing and taking care of
choice and planning issues including multi-criteria. When attributes are more than one, and further
bifurcated, neutrosophic softset environment cannot be used to tackle such type of issues. Therefore,
there was a dire need to define a new approach to solve such type of problems, So, for this purpose
a new environment namely, Neutrosophic Hypersoft set (NHSS) is defined. This paper includes
basics operator’s like union, intersection, complement, subset, null set, equal set etc., of Neutrosophic
Hypersoft set (NHSS). The validity and the implementation are presented along with suitable
examples. For more precision and accuracy, in future, proposed operations will play a vital role is
decision-makings like personal selection, management problems and many others.
Keywords: MCDM, Uncertainty, Soft set, Neutrosophic soft set, Hyper soft set.
1. Introduction
The idea of fuzzy sets was presented by Lotfi A. Zadeh in 1965 [1]. From that point the fuzzy
sets and fuzzy logic have been connected in numerous genuine issues in questionable and uncertain
conditions. The conventional fuzzy sets are based on the membership value or the level of
membership value. A few times it might be hard to allot the membership values for fuzzy sets.
Therefore, the idea of interval valued fuzzy sets was proposed [2] to catch the uncertainty for
membership values. In some genuine issues like real life problems, master framework, conviction
framework, data combination, etc., we should consider membership just as the non- membership
values for appropriate depiction of an object in questionable and uncertain condition. Neither the
fuzzy sets nor the interval valued fuzzy sets is convenient for such a circumstance. Intuitionistic fuzzy
sets proposed by Atanassov [3] is convenient for such a circumstance. The intuitionistic fuzzy sets
can just deal with the inadequate data considering both the membership and non-membership
values. It doesn't deal with the vague and conflicting data which exists in conviction framework.
Smarandache [4] presented the idea of Neutrosophic set which is a scientific apparatus for taking
care of issues including uncertain, indeterminacy and conflicting information. Neutrosophic set
indicate truth membership value (T), indeterminacy membership value (I) and falsity membership
value (F). This idea is significant in numerous application regions since indeterminacy is evaluated
exceptionally and the truth membership values, indeterminacy membership values and falsity
membership values are independent.
The idea of soft sets was first defined by Molodtsov [5] as a totally new numerical device for
taking care of issues with uncertain conditions. He defines a soft set as a parameterized family of
Muhammad Saqlain and Sana Moin, Aggregate Operators of Neutrosophic Hypersoft Set
Neutrosophic Sets and Systems, Vol. 32, 2020
295
subsets of universal set. Soft sets are useful in various regions including artificial insight, game
hypothesis and basic decision-making problems [6] and it serves to define various functions for
various parameters and utilize values against defined parameters. These functions help us to oversee
various issues and choices throughout everyday life.
In the previous couple of years, the essentials of soft set theory have been considered by different
researchers. Maji et al. [7] gives a hypothetical study of soft sets which covers subset and super set of
a soft set, equality of soft sets and operations on soft sets, for Example, union, intersection, AND and
OR-Operations between different sets. Ali at el. [8] presented new operations in soft set theory which
includes restricted union, intersection and difference. Cagman and Enginoglu [9, 10] present soft
matrix theory which substantiated itself a very significant measurement in taking care of issues while
making various choices. Singh and Onyeozili [11] come up with the research that operations on soft
set is equivalent to the corresponding soft matrices. From Molodsov [9, 6, 5, 12] up to present,
numerous handy applications identified with soft set theory have been presented and connected in
numerous fields of sciences and data innovation.
Maji [13] come up with Neutrosophic soft set portrayed by truth, indeterminacy, and falsity
membership values which are autonomous in nature. Neutrosophic soft set can deal with inadequate,
uncertain, and inconsistence data, while intuitionistic fuzzy soft set and fuzzy soft set can just deal
with partial data.
Smarandache [14] presented a new technique to deal with uncertainty. He generalized the soft
to hyper soft set by converting the function into multi-decision function. Smarandache, [15, 16, 17, 18,
19, 20] also discuss the various extension of neutrosophic sets in TOPSIS and MCDM. Saqlain et.al.
[21] proposed a new algorithm along with a new decision-making environment. Many other novel
approaches are also used by many researches [22-39] in decision makings.
1.1 Contribution
Since uncertainty is human sense which for the most part surrounds a man while taking any
significant choice. Let’s say if we get a chance to pick one best competitor out of numerous applicants,
we originally set a few characteristics and choices that what we need in our chose up-and-comer.
based on these objectives we choose the best one. To make our decision easy we use different
techniques. The purpose of this paper is to overcome the uncertainty problem in more precise way
by combing Neutrosophic set with Hypersoft set. This combination will produce a new mathematical
tool “Neutrosophic Hypersoft Set” and will play a vital role in future decision-making research.
2.Preliminaries
Definition 2.1: Soft Set
Let ξ be the universal set and € be the set of attributes with respect to ξ. Let P(ξ) be the power set of
ξ and Ą ⊆ € . A pair (₣, Ą) is called a soft set over ξ and its mapping is given as
It is also defined as:
₣: Ą → 𝑃(𝜉)
(₣, Ą) = {₣(𝑒) ∈ 𝑃(𝜉): 𝑒 ∈ € , ₣(𝑒) = ∅ 𝑖𝑓 𝑒 ≠ Ą}
Definition 2.2: Neutrosophic Soft Set
Let ξ be the universal set and € be the set of attributes with respect to ξ. Let P(ξ) be the set of
Neutrosophic values of ξ and Ą ⊆ € . A pair (₣, Ą) is called a Neutrosophic soft set over ξ and its
mapping is given as
₣: Ą → 𝑃(𝜉)
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Definition 2.3: Hyper Soft Set:
Let ξ be the universal set and 𝑃(ξ ) be the power set of ξ. Consider 𝑙1 , 𝑙 2 , 𝑙 3 … 𝑙 𝑛 for 𝑛 ≥ 1, be 𝑛 well-
defined attributes, whose corresponding attributive values are respectively the set 𝐿1 , 𝐿2 , 𝐿3 … 𝐿𝑛 with
𝐿𝑖 ∩ 𝐿𝑗 = ∅, for 𝑖 ≠ 𝑗 and 𝑖, 𝑗𝜖{1,2,3 … 𝑛} , then the pair (₣, 𝐿1 × 𝐿2 × 𝐿3 … 𝐿𝑛 ) is said to be Hypersoft
set over ξ where
₣: 𝐿1 × 𝐿2 × 𝐿3 … 𝐿𝑛 → 𝑃(𝜉)
3. Calculations
Definition 3.1: Neutrosophic Hypersoft Set (NHSS)
Let ξ be the universal set and 𝑃(ξ ) be the power set of ξ. Consider 𝑙1 , 𝑙 2 , 𝑙 3 … 𝑙 𝑛 for 𝑛 ≥ 1, be 𝑛 well-
defined attributes, whose corresponding attributive values are respectively the set 𝐿1 , 𝐿2 , 𝐿3 … 𝐿𝑛 with
𝐿𝑖 ∩ 𝐿𝑗 = ∅, for 𝑖 ≠ 𝑗 and 𝑖, 𝑗𝜖{1,2,3 … 𝑛} and their relation 𝐿1 × 𝐿2 × 𝐿3 … 𝐿𝑛 = $, then the pair (₣, $)
is said to be Neutrosophic Hypersoft set (NHSS) over ξ where
₣: 𝐿1 × 𝐿2 × 𝐿3 … 𝐿𝑛 → 𝑃(𝜉) and
₣(𝐿1 × 𝐿2 × 𝐿3 … 𝐿𝑛 ) = {< 𝑥, 𝑇(₣($)), 𝐼(₣($)), 𝐹(₣($)) >, 𝑥 ∈ 𝜉 } where T is the membership value of
truthiness, I is the membership value of indeterminacy and F is the membership value of falsity such
that 𝑇, 𝐼, 𝐹: 𝜉 → [0,1] also 0 ≤ 𝑇(₣($)) + 𝐼(₣($)) + 𝐹(₣($)) ≤ 3.
Example 3.1:
Let ξ be the set of decision makers to decide best mobile phone given as
ξ = {𝑚1 , 𝑚2 , 𝑚3 , 𝑚4 , 𝑚5 }
also consider the set of attributes as
𝑠1 = 𝑀𝑜𝑏𝑖𝑙𝑒 𝑡𝑦𝑝𝑒, 𝑠 2 = 𝑅𝐴𝑀, 𝑠 3 = 𝑆𝑖𝑚 𝐶𝑎𝑟𝑑, 𝑠 4 = 𝑅𝑒𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛, 𝑠 5 = 𝐶𝑎𝑚𝑒𝑟𝑎, 𝑠 6 = 𝐵𝑎𝑡𝑡𝑒𝑟𝑦 𝑃𝑜𝑤𝑒𝑟
And their respective attributes are given as
𝑆 1 = 𝑀𝑜𝑏𝑖𝑙𝑒 𝑡𝑦𝑝𝑒 = {𝐼𝑝ℎ𝑜𝑛𝑒, 𝑆𝑎𝑚𝑠𝑢𝑛𝑔, 𝑂𝑝𝑝𝑜, 𝑙𝑒𝑛𝑜𝑣𝑜}
𝑆 2 = 𝑅𝐴𝑀 = {8 𝐺𝐵, 4 𝐺𝐵, 6 𝐺𝐵, 2 𝐺𝐵 }
𝑆 3 = 𝑆𝑖𝑚 𝐶𝑎𝑟𝑑 = {𝑆𝑖𝑛𝑔𝑙𝑒, 𝐷𝑢𝑎𝑙}
𝑆 4 = 𝑅𝑒𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛 = {1440 × 3040 𝑝𝑖𝑥𝑒𝑙𝑠, 1080 × 780 𝑝𝑖𝑥𝑒𝑙𝑠, 2600 × 4010 𝑝𝑖𝑥𝑒𝑙𝑠}
𝑆 5 = 𝐶𝑎𝑚𝑒𝑟𝑎 = {12 𝑀𝑃, 10𝑀𝑃, 15𝑀𝑃}
𝑆 6 = 𝐵𝑎𝑡𝑡𝑒𝑟𝑦 𝑃𝑜𝑤𝑒𝑟 = {4100 𝑚𝐴ℎ, 1000 𝑚𝐴ℎ, 2050 𝑚𝐴ℎ}
Let the function be ₣: 𝑆 1 × 𝑆 2 × 𝑆 3 × 𝑆 4 × 𝑆 5 × 𝑆 6 → 𝑃(𝜉)
Below are the tables of their Neutrosophic values
Table 1: Decision maker Neutrosophic values for mobile type
1
𝑆 (𝑀𝑜𝑏𝑖𝑙𝑒 𝑡𝑦𝑝𝑒)
Iphone
Samsung
Oppo
Lenovo
𝑚1
(0.3, 0.6, 0.7)
(0.7, 0.5, 0.6)
(0.5, 0.2, 0.1)
(0.5, 0.3, 0.2)
𝑚2
(0.7, 0.6, 0.4)
(0.3, 0.2, 0.1)
(0.9, 0.5, 0.3)
(0.5, 0.2, 0.1)
𝑚3
(0.4, 0.5, 0.7)
(0.3, 0.6, 0.2)
(0.9, 0.4, 0.1)
(0.8, 0.5, 0.2)
𝑚4
(0.6, 0.5, 0.3)
(0.8, 0.1, 0.2)
(0.9, 0.3, 0.1)
(0.6, 0.4, 0.3)
𝑚5
(0.5, 0.3, 0.8)
(0.5, 0.4, 0.5)
(0.6, 0.1, 0.2)
(0.7, 0.4, 0.2)
Table 2: Decision maker Neutrosophic values for RAM
𝑆 2 (𝑅𝐴𝑀)
8 GB
4 GB
6 GB
2 GB
𝑚1
(0.3, 0.4, 0.7)
(0.4, 0.2, 0.5)
(0.7, 0.2, 0.3)
(0.8, 0.2, 0.1)
𝑚2
(0.4, 0.5, 0.7)
(0.3, 0.6, 0.2)
(0.9, 0.4, 0.1)
(0.8, 0.5, 0.2)
𝑚3
(0.5, 0.6, 0.8)
(0.4, 0.7, 0.3)
(0.8, 0.3, 0.2)
(0.9 0.4, 0.1)
𝑚4
(0.5, 0.3, 0.8)
(0.5, 0.4, 0.5)
(0.6, 0.1, 0.2)
(0.7, 0.4, 0.2)
Muhammad Saqlain and Sana Moin, Aggregate Operators of Neutrosophic Hypersoft Set
𝑚5
(0.3, 0.6, 0.7)
(0.7, 0.5, 0.6)
(0.5, 0.2, 0.1)
(0.5, 0.3, 0.2)
Neutrosophic Sets and Systems, Vol. 32, 2020
297
Table 3: Decision maker Neutrosophic values for sim card
3
𝑆 (𝑆𝑖𝑚 𝐶𝑎𝑟𝑑)
Single
Dual
𝑚1
(0.6, 0.4, 0.3)
(0.8, 0.2, 0.1)
𝑚2
(0.6, 0.5, 0.3)
(0.4, 0.8, 0.7)
𝑚3
(0.5, 0.4, 0.3)
(0.7, 0.3, 0.2)
𝑚4
(0.7, 0.8, 0.3)
(0.3, 0.6, 0.4)
𝑚5
(0.9, 0.2, 0.1)
(0.8, 0.4, 0.2)
Table 4: Decision maker Neutrosophic values for resolution
4
𝑆 (𝑅𝑒𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛)
1440 × 3040
1080 × 780
2600 × 4010
𝑚1
(0.7, 0.8, 0.3)
(0.3, 0.6, 0.4)
(0.5, 0.2, 0.1)
𝑚2
(0.7, 0.5, 0.3)
(0.7, 0.3, 0.2)
(0.6, 0.3, 0.4)
𝑚3
(0.6, 0.4, 0.3)
(0.8, 0.3, 0.1)
(0.5, 0.7, 0.2)
𝑚4
(0.5, 0.6, 0.9)
(0.6, 0.4, 0.7)
(0.9, 0.3, 0.1)
𝑚5
(0.4, 0.5, 0.3)
(0.3, 0.5, 0.8)
(0.7, 0.4, 0.3)
Table 5: Decision maker Neutrosophic values for camera
5
𝑆 (𝐶𝑎𝑚𝑒𝑟𝑎)
12 MP
10 MP
15 MP
𝑚1
(0.6, 0.4, 0.3)
(0.8, 0.3, 0.1)
(0.5, 0.7, 0.2)
𝑚2
𝑚3
(0.7, 0.8, 0.3)
(0.3, 0.6, 0.4)
(0.5, 0.2, 0.1)
(0.6, 0.4, 0.3)
(0.8, 0.2, 0.1)
(0.8, 0.5, 0.2)
𝑚4
(0.4, 0.5, 0.3)
(0.3, 0.5, 0.8)
(0.7, 0.4, 0.3)
𝑚5
(0.9, 0.2, 0.1)
(0.8, 0.4, 0.2)
(0.7, 0.4, 0.2)
Table 6: Decision maker Neutrosophic values for battery power
6
𝑆 (𝐵𝑎𝑡𝑡𝑒𝑟𝑦 𝑃𝑜𝑤𝑒𝑟)
4100 mAh
1000 mAh
2050 mAh
𝑚1
(0.7, 0.8, 0.3)
(0.3, 0.6, 0.4)
(0.5, 0.2, 0.1)
𝑚2
(0.7, 0.6, 0.4)
(0.3, 0.2, 0.1)
(0.9, 0.5, 0.3)
𝑚3
(0.4, 0.5, 0.7)
(0.3, 0.6, 0.2)
(0.9, 0.4, 0.1)
𝑚4
(0.9, 0.2, 0.1)
(0.8, 0.4, 0.2)
(0.7, 0.4, 0.2)
𝑚5
(0.5, 0.3, 0.8)
(0.5, 0.4, 0.5)
(0.6, 0.1, 0.2)
Neutrosophic Hypersoft set is define as,
₣: (𝑆 1 × 𝑆 2 × 𝑆 3 × 𝑆 4 × 𝑆 5 × 𝑆 6 ) → 𝑃(𝜉)
Let’s assume ₣($) = ₣(𝑠𝑎𝑚𝑠𝑢𝑛𝑔, 6 𝐺𝐵, 𝐷𝑢𝑎𝑙) = {𝑚1 , 𝑚4 }
Then Neutrosophic Hypersoft set of above assumed relation is
₣($) = ₣(𝑠𝑎𝑚𝑠𝑢𝑛𝑔, 6 𝐺𝐵, 𝐷𝑢𝑎𝑙 ) = {
< 𝑚1 , (𝑠𝑎𝑚𝑠𝑢𝑛𝑔{0.7, 0.5, 0.6}, 6 𝐺𝐵{0.7, 0.2, 0.3}, 𝐷𝑢𝑎𝑙{0.8,0.2,0.1}) >
< 𝑚4 (𝑠𝑎𝑚𝑠𝑢𝑛𝑔{0.8,0.1,0.2}, 6 𝐺𝐵{0.6, 0.1, 0.2}, 𝐷𝑢𝑎𝑙{0.3, 0.6,0.4}) >}
Its tabular form is given as
Table 7: Tabular Representation of Neutrosophic Hypersoft Set
₣($) = ₣(𝒔𝒂𝒎𝒔𝒖𝒏𝒈, 𝟔 𝑮𝑩, 𝑫𝒖𝒂𝒍 )
Samsung
6 GB
Dual
𝒎𝟏
(0.7,0.5, 0.6)
(0.7, 0.2, 0.3)
(0.8, 0.2, 0.1)
𝒎𝟒
(0.8, 0.1, 0.2)
(0.6, 0.1, 0.2)
(0.3, 0.6, 0.4)
Definition 3.2: Neutrosophic Hypersoft Subset
Let ₣($1 ) and ₣($2 ) be two Neutrosophic Hypersoft set over ξ. Consider 𝑙1 , 𝑙 2 , 𝑙 3 … 𝑙 𝑛 for 𝑛 ≥ 1, be
𝑛 well-defined attributes, whose corresponding attributive values are respectively the set
𝐿1 , 𝐿2 , 𝐿3 … 𝐿𝑛 with 𝐿𝑖 ∩ 𝐿𝑗 = ∅, for 𝑖 ≠ 𝑗 and 𝑖, 𝑗𝜖{1,2,3 … 𝑛} and their relation 𝐿1 × 𝐿2 × 𝐿3 … 𝐿𝑛 = $
then ₣($1 ) is the Neutrosophic Hypersoft subset of ₣($2 ) if
𝑇(₣($1 )) ≤ 𝑇(₣($2 ))
𝐼(₣($1 )) ≤ 𝐼(₣($2 ))
Muhammad Saqlain and Sana Moin, Aggregate Operators of Neutrosophic Hypersoft Set
Neutrosophic Sets and Systems, Vol. 32, 2020
298
𝐹(₣($1 )) ≥ 𝐹(₣($2 ))
Numerical Example of Subset
Consider the two NHSS ₣($1 ) and NHSS ₣($2 ) over the same universe ξ = {𝑚1 , 𝑚2 , 𝑚3 , 𝑚4 , 𝑚5 }.
The
NHSS
₣($) = ₣(𝑠𝑎𝑚𝑠𝑢𝑛𝑔, 6 𝐺𝐵, 𝐷𝑢𝑎𝑙 ) = {𝑚1 , 𝑚4 }
is
the
subset
of
NHSS
₣($2 ) =
₣(𝑆𝑎𝑚𝑠𝑢𝑛𝑔, 6𝐺𝐵) = {𝑚1 } if 𝑇(₣($1 )) ≤ 𝑇(₣($2 )) , 𝐼(₣($1 )) ≤ 𝐼(₣($2 )) , 𝐹(₣($1 )) ≥ 𝐹(₣($2 )) . Its
tabular form is given below
Table 8: Tabular Representation of NHSS ₣($1 )
𝟏)
𝒎𝟏
(0.7,0.5, 0.6)
(0.7, 0.2, 0.3)
(0.8, 0.2, 0.1)
₣($ = ₣(𝒔𝒂𝒎𝒔𝒖𝒏𝒈, 𝟔 𝑮𝑩, 𝑫𝒖𝒂𝒍 )
Samsung
6 GB
Dual
𝒎𝟒
(0.8, 0.1, 0.2)
(0.6, 0.1, 0.2)
(0.3, 0.6, 0.4)
Table 9: Tabular Representation of NHSS ₣($2 )
₣($𝟐 ) = ₣(𝒔𝒂𝒎𝒔𝒖𝒏𝒈, 𝟔 𝑮𝑩)
Samsung
6 GB
𝒎𝟏
(0.9, 0.6, 0.3)
(0.8, 0.4, 0.1)
This can also be written as
={
₣($1 ) ⊂ ₣($2 ) = ₣(𝑠𝑎𝑚𝑠𝑢𝑛𝑔, 6 𝐺𝐵, 𝐷𝑢𝑎𝑙 ) ⊂ ₣(𝑠𝑎𝑚𝑠𝑢𝑛𝑔, 6 𝐺𝐵)
< 𝑚1 , (𝑠𝑎𝑚𝑠𝑢𝑛𝑔{0.7, 0.5, 0.6}, 6 𝐺𝐵{0.7, 0.2, 0.3}, 𝐷𝑢𝑎𝑙{0.8,0.2,0.1}) >,
}
< 𝑚4 (𝑠𝑎𝑚𝑠𝑢𝑛𝑔{0.8,0.1,0.2}, 6 𝐺𝐵{0.6, 0.1, 0.2}, 𝐷𝑢𝑎𝑙{0.3, 0.6,0.4}) >
⊂ {< 𝑚1 , (𝑠𝑎𝑚𝑠𝑢𝑛𝑔{0.9, 0.6, 0.3}, 6 𝐺𝐵{0.8, 0.4, 0.1})>}
Here we can see that membership value of Samsung for 𝑚1 in both sets is (0.7, 0.5, 0.6) and
(0.9, 0.6, 0.3) which satisfy the Definition of Neutrosophic Hypersoft subset as 0.7 < 0.9, 0.5 < 0.6,
and 0.6 > 0.3. This shows that (0.7, 0.5, 0.6) ⊂ (0.9, 0.6, 0.3) and same was the case with the rest of
the attributes of NHSS ₣($1 ) and NHSS ₣($2 ).
Definition 3.3: Neutrosophic Equal Hypersoft Set
Let ₣($1 ) and ₣($2 ) be two Neutrosophic Hypersoft set over ξ. Consider 𝑙1 , 𝑙 2 , 𝑙 3 … 𝑙 𝑛 for 𝑛 ≥ 1, be
𝑛 well-defined attributes, whose corresponding attributive values are respectively the set
𝐿1 , 𝐿2 , 𝐿3 … 𝐿𝑛 with 𝐿𝑖 ∩ 𝐿𝑗 = ∅, for 𝑖 ≠ 𝑗 and 𝑖, 𝑗𝜖{1,2,3 … 𝑛} and their relation 𝐿1 × 𝐿2 × 𝐿3 … 𝐿𝑛 = $
then ₣($1 ) is the Neutrosophic equal Hypersoft subset of ₣($2 ) if
𝑇(₣($1 )) = 𝑇(₣($2 ))
𝐼(₣($1 )) = 𝐼(₣($2 ))
𝐹(₣($1 )) = 𝐹(₣($2 ))
Numerical Example of Equal Neutrosophic Hypersoft Set
Consider the two NHSS ₣($1 ) and NHSS ₣($2 ) over the same universe ξ = {𝑚1 , 𝑚2 , 𝑚3 , 𝑚4 , 𝑚5 }.
The NHSS ₣($1 ) = ₣(𝑠𝑎𝑚𝑠𝑢𝑛𝑔, 6 𝐺𝐵, 𝐷𝑢𝑎𝑙 ) = {𝑚1 , 𝑚4 } is the equal to NHSS ₣($2 ) =
₣(𝑠𝑎𝑚𝑠𝑢𝑛𝑔, 6 𝐺𝐵) = {𝑚1 }
if
𝑇(₣($1 )) = 𝑇(₣($2 )) ,
𝐹(₣($2 )). Its tabular form is given below
𝐼(₣($1 )) = 𝐼(₣($2 )) ,
Table 10: Tabular Representation of NHSS ₣($1 )
Muhammad Saqlain and Sana Moin, Aggregate Operators of Neutrosophic Hypersoft Set
𝐹(₣($1 )) =
Neutrosophic Sets and Systems, Vol. 32, 2020
₣($1 )
= ₣(𝑠𝑎𝑚𝑠𝑢𝑛𝑔, 6 𝐺𝐵, 𝐷𝑢𝑎𝑙 )
Samsung
6 GB
Dual
𝟐)
299
𝑚1
𝑚4
(0.7,0.5, 0.6)
(0.7, 0.2, 0.3)
(0.8, 0.2, 0.1)
(0.8, 0.1, 0.2)
(0.6, 0.1, 0.2)
(0.3, 0.6, 0.4)
Table 11: Tabular Representation of NHSS ₣($2 )
𝒎𝟏
(0.7,0.5, 0.6)
(0.7, 0.2, 0.3)
₣($ = ₣(𝒔𝒂𝒎𝒔𝒖𝒏𝒈, 𝟔 𝑮𝑩)
Samsung
6 GB
This can also be written as
(₣($1 ) = ₣($2 )) = (₣(𝑠𝑎𝑚𝑠𝑢𝑛𝑔, 6 𝐺𝐵, 𝐷𝑢𝑎𝑙 ) = ₣(𝑠𝑎𝑚𝑠𝑢𝑛𝑔, 6 𝐺𝐵))
= (({< 𝑚1 , (𝑠𝑎𝑚𝑠𝑢𝑛𝑔{0.7, 0.5, 0.6}, 6 𝐺𝐵{0.7, 0.2, 0.3}, 𝐷𝑢𝑎𝑙{0.8,0.2,0.1}) >,
< 𝑚4 (𝑠𝑎𝑚𝑠𝑢𝑛𝑔{0.8,0.1,0.2}, 6 𝐺𝐵{0.6, 0.1, 0.2}, 𝐷𝑢𝑎𝑙{0.3, 0.6,0.4}) >}
= {< 𝑚1 , (𝑠𝑎𝑚𝑠𝑢𝑛𝑔{0.7, 0.5, 0.6}, 6 𝐺𝐵{0.7, 0.2, 0.3}) >}))
Here we can see that membership value of Samsung for 𝑚1 in both sets is (0.7, 0.5, 0.6) and
(0.7, 0.5, 0.6) which satisfy the Definition of Neutrosophic Equal Hypersoft set as 0.7 = 0.7, 0.5 = 0.5
and 0.6 = 0.6. This shows that (0.7, 0.5, 0.6) = (0.7, 0.5, 0.6) and same was the case with the rest of
the attributes of NHSS ₣($1 ) and NHSS ₣($2 ).
Definition 3.4: Null Neutrosophic Hypersoft Set
Let ₣($1 ) be the Neutrosophic Hypersoft set over ξ. Consider 𝑙1 , 𝑙 2 , 𝑙 3 … 𝑙 𝑛 for 𝑛 ≥ 1, be 𝑛 welldefined attributes, whose corresponding attributive values are respectively the set 𝐿1 , 𝐿2 , 𝐿3 … 𝐿𝑛 with
𝐿𝑖 ∩ 𝐿𝑗 = ∅, for 𝑖 ≠ 𝑗 and 𝑖, 𝑗𝜖{1,2,3 … 𝑛} and their relation 𝐿1 × 𝐿2 × 𝐿3 … 𝐿𝑛 = $ then ₣($1 ) is Null
Neutrosophic Hypersoft set if
𝑇(₣($1 )) = 0
𝐼(₣($1 )) = 0
𝐹(₣($1 )) = 0
Numerical Example of Null Neutrosophic Hypersoft Set
Consider the NHSS ₣($1 ) over the universe
ξ = {𝑚1 , 𝑚2 , 𝑚3 , 𝑚4 , 𝑚5 } . The NHSS ₣($1 ) =
₣(𝑠𝑎𝑚𝑠𝑢𝑛𝑔, 6 𝐺𝐵, 𝐷𝑢𝑎𝑙 ) = {𝑚1 , 𝑚4 } is said to be null NHSS if its Neutrosophic values are 0. Its
tabular form is given below
Table 12: Tabular Representation of NHSS ₣($1 )
1)
₣($
= ₣(𝑠𝑎𝑚𝑠𝑢𝑛𝑔, 6 𝐺𝐵, 𝐷𝑢𝑎𝑙 )
Samsung
6 GB
Dual
(0, 0, 0)
(0, 0, 0)
(0, 0, 0)
𝑚1
(0, 0, 0)
(0, 0, 0)
(0, 0, 0)
𝑚4
This can also be written as
₣($1 ) = ₣(𝑠𝑎𝑚𝑠𝑢𝑛𝑔, 6 𝐺𝐵, 𝐷𝑢𝑎𝑙 )
= {< 𝑚1 , (𝑠𝑎𝑚𝑠𝑢𝑛𝑔{0, 0, 0}, 6 𝐺𝐵{0, 0, 0}, 𝐷𝑢𝑎𝑙{0,0,0}) >,
< 𝑚4 (𝑠𝑎𝑚𝑠𝑢𝑛𝑔{0,0,0}, 6 𝐺𝐵{0, 0, 0}, 𝐷𝑢𝑎𝑙{0, 0,0}) >}
Definition 3.5: Compliment of Neutrosophic Hypersoft Set
Muhammad Saqlain and Sana Moin, Aggregate Operators of Neutrosophic Hypersoft Set
Neutrosophic Sets and Systems, Vol. 32, 2020
300
Let ₣($1 ) be the Neutrosophic Hypersoft set over ξ. Consider 𝑙1 , 𝑙 2 , 𝑙 3 … 𝑙 𝑛 for 𝑛 ≥ 1, be 𝑛 well-
defined attributes, whose corresponding attributive values are respectively the set 𝐿1 , 𝐿2 , 𝐿3 … 𝐿𝑛 with
𝐿𝑖 ∩ 𝐿𝑗 = ∅, for 𝑖 ≠ 𝑗 and 𝑖, 𝑗𝜖{1,2,3 … 𝑛} and their relation 𝐿1 × 𝐿2 × 𝐿3 … 𝐿𝑛 = $ then ₣𝑐 ($1 ) is the
Compliment of Neutrosophic Hypersoft set of ₣($1 ) if
₣𝑐 ($1 ): (⇁ 𝐿1 ×⇁ 𝐿2 ×⇁ 𝐿3 … ⇁ 𝐿𝑛 ) → 𝑃(𝜉)
Such that
𝑇 𝐶 (₣($1 )) = 𝐹(₣($1 ))
𝐼 𝐶 (₣($1 )) = 𝐼(₣($1 ))
𝐹 𝐶 (₣($1 )) = 𝑇(₣($1 ))
Numerical Example of Compliment of NHSS
Consider the NHSS ₣($1 ) over the universe ξ = {𝑚1 , 𝑚2 , 𝑚3 , 𝑚4 , 𝑚5 }. The compliment of NHSS
₣($1 ) = ₣(𝑠𝑎𝑚𝑠𝑢𝑛𝑔, 6 𝐺𝐵, 𝐷𝑢𝑎𝑙 ) = {𝑚1 , 𝑚4 }
is given as 𝑇 𝐶 (₣($1 )) = 𝐹(₣($1 )) , 𝐼 𝐶 (₣($1 )) =
𝐼(₣($1 )), 𝐹 𝐶 (₣($1 )) = 𝑇(₣($1 )).Its tabular form is given below
Table 13: Tabular Representation of NHSS ₣($1 )
₣𝐶 ($1 ) = ₣(𝑁𝑜𝑡 𝑠𝑎𝑚𝑠𝑢𝑛𝑔, 𝑁𝑜𝑡 6 𝐺𝐵, 𝑁𝑜𝑡 𝐷𝑢𝑎𝑙 )
Not Samsung
Not 6 GB
Not Dual
𝑚1
(0.6, 0.5, 0.7)
(0.3, 0.2, 0.7)
(0.1, 0.2, 0.8)
𝑚4
(0.2, 0.1, 0.8)
(0.2, 0.1, 0.6)
(0.4, 0.6, 0.3)
This can also be written as
₣𝑐 ($1 ) = ₣( 𝑛𝑜𝑡 𝑠𝑎𝑚𝑠𝑢𝑛𝑔, 𝑛𝑜𝑡 6 𝐺𝐵, 𝑛𝑜𝑡 𝐷𝑢𝑎𝑙 )
= {< 𝑚1 , (𝑛𝑜𝑡 𝑠𝑎𝑚𝑠𝑢𝑛𝑔{0.6, 0.5, 0.7}, 𝑛𝑜𝑡 6 𝐺𝐵{0.3, 0.2, 0.7}, 𝑛𝑜𝑡 𝐷𝑢𝑎𝑙{0.1,0.2,0.8}) >,
< 𝑚4 (𝑛𝑜𝑡 𝑠𝑎𝑚𝑠𝑢𝑛𝑔{0.2,0.1,0.8}, 𝑛𝑜𝑡 6 𝐺𝐵{0.2, 0.1, 0.6}, 𝑛𝑜𝑡 𝐷𝑢𝑎𝑙{0.4, 0.6,0.3}) >}
Here we can see that membership value of Samsung for 𝑚1 in ₣($1 ) is (0.7, 0.5, 0.6) and its
compliment is (0.6, 0.5, 0.7) which satisfy the Definition of compliment of Neutrosophic Hypersoft
set. This shows that (0.6, 0.5, 0.7) is the compliment of (0.7, 0.5, 0.6) and same was the case with the
rest of the attributes of NHSS ₣($1 ).
Definition 3.6: Union of Two Neutrosophic Hypersoft Set
Let ₣($1 ) and ₣($2 ) be two Neutrosophic Hypersoft set over ξ. Consider 𝑙1 , 𝑙 2 , 𝑙 3 … 𝑙 𝑛 for 𝑛 ≥ 1, be
𝑛 well-defined attributes, whose corresponding attributive values are respectively the set
𝐿1 , 𝐿2 , 𝐿3 … 𝐿𝑛 with 𝐿𝑖 ∩ 𝐿𝑗 = ∅, for 𝑖 ≠ 𝑗 and 𝑖, 𝑗𝜖{1,2,3 … 𝑛} and their relation 𝐿1 × 𝐿2 × 𝐿3 … 𝐿𝑛 = $
then ₣($1 ) ∪ ₣($2 ) is given as
𝑇(₣($1 ) ∪ ₣($2 )) = {
𝑇(₣($1 ))
𝑇(₣($2 ))
max (𝑇(₣($1 )), 𝑇(₣($2 )))
𝐼(₣($1 ) ∪ ₣($2 )) =
{
𝐼(₣($1 ))
𝐼(₣($2 ))
(𝐼(₣($1 ))+𝐼(₣($2 )))
2
𝑖𝑓 𝑥 ∈ $1
𝑖𝑓 𝑥 ∈ $2
𝑖𝑓 𝑥 ∈ $1 ∩ $2
𝑖𝑓 𝑥 ∈ $1
𝑖𝑓 𝑥 ∈ $2
𝑖𝑓 𝑥 ∈ $1 ∩ $2
Muhammad Saqlain and Sana Moin, Aggregate Operators of Neutrosophic Hypersoft Set
Neutrosophic Sets and Systems, Vol. 32, 2020
𝐹(₣($1 ) ∪ ₣($2 )) = {
Numerical Example of Union
301
𝐹(₣($1 ))
𝑖𝑓 𝑥 ∈ $1
𝐹(₣($2 ))
𝑖𝑓 𝑥 ∈ $2
min (𝐹(₣($1 )), 𝐹(₣($2 )))
𝑖𝑓 𝑥 ∈ $1 ∩ $2
Consider the two NHSS ₣($1 ) and NHSS ₣($2 ) over the same universe ξ = {𝑚1 , 𝑚2 , 𝑚3 , 𝑚4 , 𝑚5 }.
Tabular representation of NHSS ₣($1 ) = ₣(𝑠𝑎𝑚𝑠𝑢𝑛𝑔, 6 𝐺𝐵, 𝐷𝑢𝑎𝑙 ) = {𝑚1 , 𝑚4 } and NHSS ₣($2 ) =
₣(𝑠𝑎𝑚𝑠𝑢𝑛𝑔, 6 𝐺𝐵) = {𝑚1 } is given below,
Table 14: Tabular Representation of NHSS ₣($1 )
𝟏)
𝒎𝟏
(0.7,0.5, 0.6)
(0.7, 0.2, 0.3)
(0.8, 0.2, 0.1)
₣($ = ₣(𝒔𝒂𝒎𝒔𝒖𝒏𝒈, 𝟔 𝑮𝑩, 𝑫𝒖𝒂𝒍 )
Samsung
6 GB
Dual
𝒎𝟒
(0.8, 0.1, 0.2)
(0.6, 0.1, 0.2)
(0.3, 0.6, 0.4)
Table 15: Tabular Representation of NHSS ₣($2 )
₣($𝟐 ) = ₣(𝒔𝒂𝒎𝒔𝒖𝒏𝒈, 𝟔 𝑮𝑩)
Samsung
6 GB
𝒎𝟏
(0.9, 0.5, 0.3)
(0.8, 0.4, 0.1)
Then the union of above NHSS is given as
Samsung
6 GB
Dual
₣($
Table 16: Union of NHSS ₣($1 ) and NHSS ₣($2 )
𝟏)
𝒎𝟏
(0.9, 0.5, 0.3)
(0.8, 0.3, 0.1)
(0.8, 0.1, 0.0)
∪ ₣($𝟐 )
𝒎𝟒
(0.8, 0.1, 0.2)
(0.6, 0.1, 0.2)
(0.3, 0.6, 0.4)
This can also be written as
₣($1 ) ∪ ₣($2 ) = ₣(𝑠𝑎𝑚𝑠𝑢𝑛𝑔, 6 𝐺𝐵, 𝐷𝑢𝑎𝑙 ) ∪ ₣(𝑠𝑎𝑚𝑠𝑢𝑛𝑔, 6 𝐺𝐵)
= {< 𝑚1 , (𝑠𝑎𝑚𝑠𝑢𝑛𝑔{0.9, 0.5, 0.3}, 6 𝐺𝐵{0.8, 0.3, 0.1}, 𝐷𝑢𝑎𝑙{0.8,0.1,0.0}) >,
< 𝑚4 (𝑠𝑎𝑚𝑠𝑢𝑛𝑔{0.8,0.1,0.2}, 6 𝐺𝐵{0.6, 0.1, 0.2}, 𝐷𝑢𝑎𝑙{0.3, 0.6,0.4}) >}
Definition 3.7: Intersection of Two Neutrosophic Hypersoft Set
Let ₣($1 ) and ₣($2 ) be two Neutrosophic Hypersoft set over ξ. Consider 𝑙1 , 𝑙 2 , 𝑙 3 … 𝑙 𝑛 for 𝑛 ≥ 1, be
𝑛 well-defined attributes, whose corresponding attributive values are respectively the set
𝐿1 , 𝐿2 , 𝐿3 … 𝐿𝑛 with 𝐿𝑖 ∩ 𝐿𝑗 = ∅, for 𝑖 ≠ 𝑗 and 𝑖, 𝑗𝜖{1,2,3 … 𝑛} and their relation 𝐿1 × 𝐿2 × 𝐿3 … 𝐿𝑛 = $
then ₣($1 ) ∩ ₣($2 ) is given as
𝑇(₣($1 ) ∩ ₣($2 )) = {
Numerical Example of Intersection
𝑇(₣($2 ))
min (𝑇(₣($1 )), 𝑇(₣($2 )))
𝐼(₣($1 ) ∩ ₣($2 )) =
𝐹(₣($1 ) ∩ ₣($2 )) = {
𝑇(₣($1 ))
{
𝐼(₣($1 ))
𝐼(₣($2 ))
(𝐼(₣($1 ))+𝐼(₣($2 )))
2
𝐹(₣($1 ))
𝐹(₣($2 ))
max (𝐹(₣($1 )), 𝐹(₣($2 )))
𝑖𝑓 𝑥 ∈ $1
𝑖𝑓 𝑥 ∈ $2
𝑖𝑓 𝑥 ∈ $1 ∩ $2
𝑖𝑓 𝑥 ∈ $1
𝑖𝑓 𝑥 ∈ $2
𝑖𝑓 𝑥 ∈ $1 ∩ $2
𝑖𝑓 𝑥 ∈ $1
𝑖𝑓 𝑥 ∈ $2
𝑖𝑓 𝑥 ∈ $1 ∩ $2
Muhammad Saqlain and Sana Moin, Aggregate Operators of Neutrosophic Hypersoft Set
Neutrosophic Sets and Systems, Vol. 32, 2020
302
Consider the two NHSS ₣($1 ) and NHSS ₣($2 ) over the same universe ξ = {𝑚1 , 𝑚2 , 𝑚3 , 𝑚4 , 𝑚5 }.
Tabular representation of NHSS ₣($1 ) = ₣(𝑠𝑎𝑚𝑠𝑢𝑛𝑔, 6 𝐺𝐵, 𝐷𝑢𝑎𝑙 ) = {𝑚1 , 𝑚4 } and NHSS ₣($2 ) =
₣(𝑠𝑎𝑚𝑠𝑢𝑛𝑔, 6 𝐺𝐵) = {𝑚1 } is given below
Table 17: Tabular Representation of NHSS ₣($1 )
₣($𝟏 )
= ₣(𝒔𝒂𝒎𝒔𝒖𝒏𝒈, 𝟔 𝑮𝑩, 𝑫𝒖𝒂𝒍 )
Samsung
6 GB
Dual
𝒎𝟏
(0.7,0.5, 0.6)
(0.7, 0.2, 0.3)
(0.8, 0.2, 0.1)
𝒎𝟒
(0.8, 0.1, 0.2)
(0.6, 0.1, 0.2)
(0.3, 0.6, 0.4)
Table 18: Tabular Representation of NHSS ₣($2 )
₣($𝟐 ) = ₣(𝒔𝒂𝒎𝒔𝒖𝒏𝒈, 𝟔 𝑮𝑩)
Samsung
6 GB
𝒎𝟏
(0.9, 0.5, 0.3)
(0.8, 0.4, 0.1)
Then the intersection of above NHSS is given as
Table 19: Intersection of NHSS ₣($1 ) and NHSS ₣($2 )
₣($𝟏 ) ∩ ₣($𝟐 )
Samsung
6 GB
Dual
𝒎𝟏
(0.7, 0.5, 0.6)
(0.7, 0.3, 0.3)
(0.0, 0.1, 0.1)
This can also be written as
₣($1 ) ∩ ₣($2 ) = ₣(𝑠𝑎𝑚𝑠𝑢𝑛𝑔, 6 𝐺𝐵, 𝐷𝑢𝑎𝑙 ) ∩ ₣(𝑠𝑎𝑚𝑠𝑢𝑛𝑔, 6 𝐺𝐵)
= {< 𝑚1 , (𝑠𝑎𝑚𝑠𝑢𝑛𝑔{0.7, 0.5, 0.6}, 6 𝐺𝐵{0.7, 0.3, 0.3}, 𝐷𝑢𝑎𝑙{0.0,0.1,0.1}) >}
Definition 3.8: AND Operation on Two Neutrosophic Hypersoft Set
Let ₣($1 ) and ₣($2 ) be two Neutrosophic Hypersoft set over ξ. Consider 𝑙1 , 𝑙 2 , 𝑙 3 … 𝑙 𝑛 for 𝑛 ≥ 1, be
𝑛 well-defined attributes, whose corresponding attributive values are respectively the set
𝐿1 , 𝐿2 , 𝐿3 … 𝐿𝑛 with 𝐿𝑖 ∩ 𝐿𝑗 = ∅, for 𝑖 ≠ 𝑗 and 𝑖, 𝑗𝜖{1,2,3 … 𝑛} and their relation 𝐿1 × 𝐿2 × 𝐿3 … 𝐿𝑛 = $
then ₣($1 ) ∧ ₣($2 ) = ₣($1 × $2 ) is given as
𝑇($1 × $2 ) = 𝑚𝑖𝑛 (𝑇(₣($1 )), 𝑇(₣($2 )))
1
𝐼($ × $
2)
=
(𝐼(₣($1 )), 𝐼(₣($2 )))
2
𝐹($1 × $2 ) = 𝑚𝑎𝑥 (𝐹(₣($1 )), 𝐹(₣($2 )))
Numerical Example of AND-Operation
Consider the two NHSS ₣($1 ) and NHSS ₣($2 ) over the same universe ξ = {𝑚1 , 𝑚2 , 𝑚3 , 𝑚4 , 𝑚5 }.
Tabular representation of NHSS ₣($1 ) = ₣(𝑠𝑎𝑚𝑠𝑢𝑛𝑔, 6 𝐺𝐵, 𝐷𝑢𝑎𝑙 ) = {𝑚1 , 𝑚4 } and NHSS ₣($2 ) =
₣(𝑠𝑎𝑚𝑠𝑢𝑛𝑔, 6 𝐺𝐵, ) = {𝑚1 } is given below
Table 20: Tabular representation of NHSS ₣($1 )
₣($1 )
= ₣(𝑠𝑎𝑚𝑠𝑢𝑛𝑔, 6 𝐺𝐵, 𝐷𝑢𝑎𝑙 )
Samsung
6 GB
Dual
𝑚1
(0.7,0.5, 0.6)
(0.7, 0.2, 0.3)
(0.8, 0.2, 0.1)
Muhammad Saqlain and Sana Moin, Aggregate Operators of Neutrosophic Hypersoft Set
𝑚4
(0.8, 0.1, 0.2)
(0.6, 0.1, 0.2)
(0.3, 0.6, 0.4)
Neutrosophic Sets and Systems, Vol. 32, 2020
303
Table 21: Tabular representation of NHSS ₣($2 )
₣($𝟐 ) = ₣(𝒔𝒂𝒎𝒔𝒖𝒏𝒈, 𝟔 𝑮𝑩)
Samsung
6 GB
𝒎𝟏
(0.9, 0.5, 0.3)
(0.8, 0.4, 0.1)
Then the AND Operation of above NHSS is given as
Table 22: AND of NHSS ₣($1 ) and NHSS ₣($2 )
₣($𝟏 ) ∧ ₣($𝟐 )
𝑆𝑎𝑚𝑠𝑢𝑛𝑔 × 𝑆𝑎𝑚𝑠𝑢𝑛𝑔
𝑆𝑎𝑚𝑠𝑢𝑛𝑔 × 6 𝐺𝐵
6 𝐺𝐵 × 𝑆𝑎𝑚𝑠𝑢𝑛𝑔
6 𝐺𝐵 × 6 𝐺𝐵
𝐷𝑢𝑎𝑙 × 𝑆𝑎𝑚𝑠𝑢𝑛𝑔
𝐷𝑢𝑎𝑙 × 6 𝐺𝐵
𝒎𝟏
(0.7,0.5,0.6)
(0.7, 0.45,0.6)
(0.7, 0.35,0.3)
(0.7,0.3, 0.3)
(0.8,0.35,0.3)
(0.8, 0.3, 0.1)
𝒎𝟒
(0.0,0.1,0.2)
(0.0,0.1,0.2)
(0.0,0.1,0.2)
(0.0,0,1,0.2)
(0.0,0.6,0.4)
(0.0,0.6,0.4)
Definition 3.9: OR Operation on Two Neutrosophic Hypersoft Set
Let ₣($1 ) and ₣($2 ) be two Neutrosophic Hypersoft set over ξ. Consider 𝑙1 , 𝑙 2 , 𝑙 3 … 𝑙 𝑛 for 𝑛 ≥ 1, be
𝑛 well-defined attributes, whose corresponding attributive values are respectively the set
𝐿1 , 𝐿2 , 𝐿3 … 𝐿𝑛 with 𝐿𝑖 ∩ 𝐿𝑗 = ∅, for 𝑖 ≠ 𝑗 and 𝑖, 𝑗𝜖{1,2,3 … 𝑛} and their relation 𝐿1 × 𝐿2 × 𝐿3 … 𝐿𝑛 = $
then ₣($1 ) ∨ ₣($2 ) = ₣($1 × $2 ) is given as
𝑇($1 × $2 ) = 𝑚𝑎𝑥 (𝑇(₣($1 )), 𝑇(₣($2 )))
𝐼($1 × $2 ) =
(𝐼(₣($1 )), 𝐼(₣($2 )))
2
𝐹($1 × $2 ) = 𝑚𝑖𝑛 (𝐹(₣($1 )), 𝐹(₣($2 )))
Numerical Example of OR-Operation
Consider the two NHSS ₣($1 ) and NHSS ₣($2 ) over the same universe ξ = {𝑚1 , 𝑚2 , 𝑚3 , 𝑚4 , 𝑚5 }.
Tabular representation of NHSS ₣($1 ) = ₣(𝑠𝑎𝑚𝑠𝑢𝑛𝑔, 6 𝐺𝐵, 𝐷𝑢𝑎𝑙 ) = {𝑚1 , 𝑚4 } and NHSS ₣($2 ) =
₣(𝑠𝑎𝑚𝑠𝑢𝑛𝑔, 6 𝐺𝐵, ) = {𝑚1 } is given below
Table 23: Tabular representation of NHSS ₣($1 )
₣($𝟏 )
= ₣(𝒔𝒂𝒎𝒔𝒖𝒏𝒈, 𝟔 𝑮𝑩, 𝑫𝒖𝒂𝒍 )
Samsung
6 GB
Dual
𝒎𝟏
(0.7,0.5, 0.6)
(0.7, 0.2, 0.3)
(0.8, 0.2, 0.1)
₣($𝟐 ) = ₣(𝒔𝒂𝒎𝒔𝒖𝒏𝒈, 𝟔 𝑮𝑩)
Samsung
6 GB
𝒎𝟒
(0.8, 0.1, 0.2)
(0.6, 0.1, 0.2)
(0.3, 0.6, 0.4)
𝒎𝟏
(0.9, 0.5, 0.3)
(0.8, 0.4, 0.1)
Table 24: Tabular representation of NHSS ₣($2 )
Then the OR Operation of above NHSS is given as
Muhammad Saqlain and Sana Moin, Aggregate Operators of Neutrosophic Hypersoft Set
Neutrosophic Sets and Systems, Vol. 32, 2020
304
Table 25: OR of NHSS ₣($1 ) and NHSS ₣($2 )
4. Result Discussion
₣($𝟏 ) ∨ ₣($𝟐 )
𝑆𝑎𝑚𝑠𝑢𝑛𝑔 × 𝑆𝑎𝑚𝑠𝑢𝑛𝑔
𝑆𝑎𝑚𝑠𝑢𝑛𝑔 × 6 𝐺𝐵
6 𝐺𝐵 × 𝑆𝑎𝑚𝑠𝑢𝑛𝑔
6 𝐺𝐵 × 6 𝐺𝐵
𝐷𝑢𝑎𝑙 × 𝑆𝑎𝑚𝑠𝑢𝑛𝑔
𝐷𝑢𝑎𝑙 × 6 𝐺𝐵
𝒎𝟏
(0.9,0.5,0.3)
(0.8, 0.45,0.1)
(0.9, 0.35,0.3)
(0.8,0.3, 0.1)
(0.9,0.35,0.1)
(0.8, 0.3, 0.1)
𝒎𝟒
(0.8,0.1,0.0)
(0.8,0.1,0.0)
(0.6,0.1,0.0)
(0.6,0,1,0.0)
(0.3,0.6,0.0)
(0.3,0.6,0.0)
Decision-making is a complex issue due to vague, imprecise and indeterminate environment
specially, when attributes are more than one, and further bifurcated. Neutrosophic softset
environment cannot be used to tackle such type of issues. Therefore, there was a dire need to define
a new approach to solve such type of problems, So, for this purpose neutrosophic hypersoft set
environment is defined along with necessary operations and elaborated with examples.
5. Conclusions
In this paper, operations of Neutrosophic Hypersoft set like union, intersection, compliment, AND
OR operations are presented. The validity and implementation of the proposed operations and
definitions are verified by presenting suitable example. Neutrosophic hypersoft set NHSS will be a
new tool in decision-making problems for suitable selection. In future, many decision-makings like
personal selection, office management, industrial equipment and many other problems can be solved
with the proposed operations [23]. Properties of Union and Intersection operations, cardinality and
functions on NHSS are to be defined in future.
Acknowledgement
The authors are highly thankful to the Editor-in-chief and the referees for their valuable comments
and suggestions for improving the quality of our paper.
Conflicts of Interest
The authors declare that they have no conflict of interest.
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Received: Nov 13, 2019. Accepted: Mar 16, 2020
Muhammad Saqlain and Sana Moin, Aggregate Operators of Neutrosophic Hypersoft Set
Neutrosophic Sets and Systems, Vol. 32, 2020
University of New Mexico
A New Approach of Neutrosophic Soft Set with Generalized
Fuzzy TOPSIS in Application of Smart Phone Selection
Muhammad Saqlain 1, Muhammad Naveed Jafar 2 and Muhammad Riaz 2
1
2
Lahore Garrison University, DHA Phase-VI, Sector C, Lahore, 54000, Pakistan. E-mail: msaqlain@lgu.edu.pk
Lahore Garrison University, DHA Phase-VI, Sector C, Lahore, 54000, Pakistan. E-mail: naveedjafar@lgu.edu.pk
2 Department of Mathematics, University of Punjab, Lahore, 54000, Pakistan. E-mail: mriaz.math@pu.edu.pk
Abstract: With the invention of new technologies, the competition elevates in market. Therefore, it
creates more difficulties for consumer to select the right smart phone. In this paper, a new approach
is proposed to select smart phone, in which environment of decision-making is MCDM. Firstly, an
algorithm is proposed in which problem is formulated in the form of neutrosophic soft set and then
solved with generalized fuzzy TOPSIS (GFT). Secondly, rankings are compared with [10]. Finally, it
is concluded that proposed approach is applicable in decision-making where uncertainty and
imprecise information-based environment is confronted. In future, this evolutionary algorithm can
be used along with other methodologies to solve MCDM problems.
Keywords: Accuracy Function, MCDM, TOPSIS, Mobile Phone, Soft set, Neutrosophic Numbers
NNs, Neutrosophic Soft set, Linguistic Variable.
________________________________________________________________________________________
1. Introduction
Mobile / cell phones are widely used for making call, SMS, MMS, email or to access internet. The first
portable cell phone was manifest by Martin in 1973 [8], using a handset weighing 4.4 IBS. In the
advance world, smart-phone have currently overtaken the usage of earlier telecommunication
system. There may be an outstanding doubt and complications concerning the reputation of cellular
technologies by decision makers, provider, trader, and clients alike. To help this selection process
amongst different available options for technology evaluation, multi-standards decision-making
approach appears to be suitable. Due to brutal market competition by inventions of different models
with innovative designs and characteristics have made the buying decision making more complex
[10]. It is typically tough for a decision-maker to assign a particular performance rating to another for
the attributes into consideration. The advantage of employing a fuzzy approach is to assign the
relative importance of attributes victimization fuzzy ranges rather than a particular number for textile
the $64000 world during a fuzzy atmosphere. MCDM approach [9] with cluster deciding is employed
to judge smartphones as another per client preferences [6]. TOPSIS methodology is especially
appropriate for finding the cluster call –making drawback beneath fuzzy atmosphere. TOPSIS
methodology [22] is predicated on the idea that the chosen various ought to have the shortest distance
from the positive ideal solution. In decision making problems TOPSIS method have been studied by
many researchers: Adeel et al. [3-5, 7 ,11, 13, 18, 21, 24]. This technique of MCDM is used by Saqlain
et. al. [16] to predict CWC 2019. Maji [12] introduced the idea of Neutrosophic soft set. Riaz and
Muhammad Saqlain, Naveed Jafar and Muhammad Riaz, A New Approach of Neutrosophic Soft Set with Generalized Fuzzy
TOPSIS in Application of Smart Phone Selection.
Neutrosophic Sets and Systems, Vol. 32, 2020
308
Naeem [14, 15] presented some essential ideas of soft sets together with soft sigma algebra.
Neutrosophic set could be a terribly powerful tool to agitate incomplete and indeterminate data
planned by F. Smarandache [20] and has attracted the eye of the many students [1], which might offer
the credibleness of the given linguistic analysis worth and linguistic set can offer qualitative analysis
values. At the primary, soft set theory was planned by a Russian scientist [2] that was used as a
standard mathematical mean to come back across the difficulty of hesitant and uncertainty [19]. He
additionally argues that however, the same theory of sentimental set is free from the parameterization
inadequacy syndrome of fuzzy set theory [23], rough set theory, and applied mathematics.
Nowadays, researchers are focusing to present new theories to deal with uncertainty, imprecision
and vagueness [25-35], along with suitable examples to elaborate their theories. Neutrosophic soft
sets along with TOPSIS technique is widely used in decision making problems, every day many
researchers are working in this era [36-45] to discuss the validity of Neutrosophy in decision
problems.
1.1 Novelties
It is a very complicated decision to select the utmost suitable phone. In this condition Neutrosophic
soft-set-environment is considered and simplified with Generalized TOPSIS. An algorithm is
proposed to tackle uncertain, vague and imprecise environment in selection problems.
1.2 Contribution
Cell phone selection is a challenging problem in current generation. To solve this complexity, a few
methods regarding the usage of fuzzy ideas has been proposed. For the few kinds of uncertainty
within the selection method fuzzy linguistic method is used. The objective of the study is to
investigate the uncertainty in selection criteria of cell phone with respect to the consumer’s choice
under Neutrosophic softset environment by applying Generalized fuzzy TOPSIS.
2.Preliminaries
Definition 2.1: Neutrosophic Set [2]
Let U be a universe of discourse then the neutrosophic set A is an object having the form
A = {< x: TA (𝑥), IA (𝑥), FA (𝑥), >; x ∈ U}
where the functions T, I, F : U→ [0,1] define respectively the degree of membership, the degree of
indeterminacy, and the degree of non-membership of the element x ∈ X to the set A with the
condition. ≤TA (𝑥) + IA (𝑥) + FA (𝑥) ≤ 3.
Definition 2.2: Soft Set [2]
Let ℧ be a universe of discourse, Ρ(℧)the power set of ℧, and A set of parameters. Then, the pair (Ϝ,
℧), where
is called a softset over ℧.
Definition 2.3: Neutrosophic Soft Set [12]
Ϝ ∶ Α ⟶ Ρ(℧)
Let ℧ be an initial universal set and E be a set of parameters. Assume, Α ⊂ E. Let Ρ(℧)denotes the
set of all neutrosophic sets over ℧, where F is a mapping given by
Definition 2.4: Accuracy Function [17]
Ϝ ∶ Α ⟶ Ρ(℧)
Muhammad Saqlain, Naveed Jafar and Muhammad Riaz, A New Approach of Neutrosophic Soft Set with Generalized Fuzzy
TOPSIS in Application of Smart Phone Selection.
Neutrosophic Sets and Systems, Vol. 32, 2020
309
Accuracy function is used to convert neutrosophic number NFN into fuzzy number
(Deneutrosophication using 𝑨𝐹 ).
A(F) = { 𝑥 =
[𝑇𝑥 +𝐼𝑥 +𝐹𝑥 ]
3
}
𝑨𝐹 represents the De-Neutrosophication of neutrosophic number into Fuzzy Number.
3. Calculations
In this section an algorithm is proposed to solve MCDM problem under neutrosophic environment.
3.1 Algorithm
Cell phone selection is a challenging problem in current generation. To solve this complexity, a few
methods regarding the usage of neutrosophic fuzzy TOPSIS ideas have been proposed. For the few
kinds of uncertainty within the selection method fuzzy linguistic method is used. The objective of the
study is to investigate the uncertainty in selection criteria of cell phone.
To solve this problem following algorithm is applied as in sequence.
Step 1:
defining a problem
Step 2:
Consideration of problem as MCDM (alternatives and attributes)
Step 3:
Assigning linguistic variables to alternatives and criteria’s / attributes
Step 4:
Substitution of NNs to linguistic variables
Step 5:
Conversion of NNs to fuzzy numbers by using accuracy function [?] defined as,
A(F) = { 𝑥 =
[𝑇𝑥 +𝐼𝑥 +𝐹𝑥 ]
3
}
𝑊ℎ𝑒𝑟𝑒 𝑇𝑥 , 𝐼𝑥 , 𝐹𝑥 𝜖 𝑁𝑁𝑠 𝑎𝑠𝑠𝑖𝑔𝑛𝑒𝑑 𝑏𝑦 𝑑𝑒𝑐𝑖𝑠𝑖𝑜𝑛 𝑚𝑎𝑘𝑒𝑟𝑠 𝑡𝑜 𝑒𝑎𝑐ℎ 𝑐𝑟𝑖𝑡𝑒𝑟𝑖𝑎 𝑖𝑛𝑑𝑖𝑣𝑖𝑑𝑢𝑎𝑙𝑙𝑦
Step 6:
Apply TOPSIS technique
Step 7:
Arrange by ascending order and rank accordingly.
Step 8:
Discussion
Defining
Assigning
Formulation as MCDM
Computation
of Relative
Ranking of
Linguistic
Apply Algorithm
Assigning
of Fuzzy TOPSIS
Neutrosophic
Conversion of NNs to Fuzzy No’s
Figure 1: Algorithm used in mobile selection, under neutrosophic softset environment
Muhammad Saqlain, Naveed Jafar and Muhammad Riaz, A New Approach of Neutrosophic Soft Set with Generalized Fuzzy
TOPSIS in Application of Smart Phone Selection.
Neutrosophic Sets and Systems, Vol. 32, 2020
310
3.2: Case Study
To discuss the;
Validity
Applicability
of the proposed algorithm, mobile selection is considered as a MCDM problem.
3.2.1
Problem Formulation
The mobile phone has been identified for choosing criterion and after that the criterion is depending
upon the public choice. The result gets from criterion, some mobile phone has been selected according
to their criterion. With invention of new technologies, the competition is raised upon in market it
makes more difficult for consumer to select the right phone. In fast growing market, we think that
the result got from fuzzy idea has been improved, so we applied Neutrosophic set to get more
accuracy in result. The aim of the study is to explore the accuracy in the selection of criteria of mobile
phone.
3.2.2
Parameters
Selection is a complex issue, to resolve this problem criteria and alternative plays an important role.
Following criteria and alternatives are considered in this problem formulation.
Criteria’s
Ƈ𝟏
Ram
Ƈ2
Rom
Ƈ3
Processor
Ƈ4
Ƈ5
Display
Size
Camera
Ƈ6
Model
Ƈ7
Price
Mobiles as Alternatives
Ṃ𝟏
SAMSUNG
3.2.3
Ṃ2
NOKIA
Ṃ3
Ṃ4
HTC
HUAWEI
Ṃ5
Q-MOBILE
Ṃ6
RIVO
Assumptions
The decision makers {Ɗ1 , Ɗ2 , Ɗ3 , Ɗ4 } will assign linguistic values from Table .1 according to his own
interest, knowledge and experience, to the above-mentioned criteria and alternatives and shown in
Table.2.
Table 1: Linguistic variables, codes and neutrosophic numbers obtained by expert opinion
Sr # No
Linguistic variable
Code
Neutrosophic Number
1
2
3
4
5
Very Low
ṼḸ
Ḹ
Ș
Ḫ
ṼḪ
(0.1, 0.3,0.7)
(0.3,0.5,0.6)
(0.5,0.5,0.5)
(0.7,0.3,0.4)
(1.0,0.1,0.2)
Low
Satisfactory
High
Very High
3.3 Application of Proposed Algorithm
Step 1: Problem consideration 3.2.
Step 2: Formulation and assumptions 3.2.1 and 3.2.2.
Muhammad Saqlain, Naveed Jafar and Muhammad Riaz, A New Approach of Neutrosophic Soft Set with Generalized Fuzzy
TOPSIS in Application of Smart Phone Selection.
Neutrosophic Sets and Systems, Vol. 32, 2020
311
Step 3: Assigning linguistic variables to each alternatives and criteria’s / attributes.
Table 2: Each decision maker, will assign linguistic values to each attribute, from Table .1
Strategies
Ƈ𝟏 = RAM
Ṃ𝟏
Ṃ2
Ṃ3
Ṃ4
Ƈ𝟐 = ROM
Ƈ𝟑 = PROCESSOR
Ƈ𝟒 = CAMERA
Ɗ𝟐
Ș
Ɗ𝟑
Ɗ𝟒
Ḹ
Ḫ
ṼḪ
Ḫ
Ș
ṼḸ
ṼḸ
Ș
Ḫ
ṼḪ
Ṃ5
ṼḪ
ṼḸ
Ṃ𝟏
Ḹ
Ș
Ṃ6
Ṃ2
Ṃ3
ṼḸ
Ș
Ḹ
Ș
Ḹ
Ḫ
Ḫ
Ḫ
Ș
Ḫ
Ḹ
ṼḸ
Ș
Ḫ
ṼḸ
ṼḪ
Ș
Ḫ
Ṃ3
ṼḪ
Ṃ5
Ḫ
ṼḪ
ṼḸ
Ḫ
Ș
Ḫ
Ḫ
ṼḪ
ṼḸ
ṼḸ
Ș
Ḹ
Ḹ
Ṃ2
Ṃ3
Ș
Ḫ
Ṃ4
ṼḪ
Ṃ6
ṼḪ
Ṃ5
Ș
ṼḸ
ṼḸ
Ș
Ṃ𝟏
ṼḪ
ṼḸ
ṼḪ
ṼḪ
Ḫ
Ḫ
Ḹ
Ṃ5
Ș
Ș
Ḹ
Ṃ2
Ṃ4
ṼḪ
ṼḪ
ṼḪ
Ṃ3
Ș
Ḫ
Ṃ6
Ṃ𝟏
Ș
Ḫ
ṼḸ
Ș
Ḹ
ṼḸ
Ḹ
ṼḪ
ṼḪ
Ṃ4
ṼḪ
Ș
Ḫ
Ḹ
Ṃ2
ṼḸ
ṼḪ
Ṃ6
Ṃ𝟏
Ș
ṼḪ
ṼḪ
Ṃ5
Ḫ
Ḫ
Ṃ4
Ṃ6
Ƈ𝟓 = DISPLAY SIZE
Ɗ𝟏
Ḫ
ṼḸ
ṼḪ
Ḹ
Ș
Ḹ
ṼḪ
Ḹ
Ḹ
Ḫ
Ḫ
Ḫ
Ḫ
Ș
ṼḪ
ṼḪ
Ḫ
ṼḪ
ṼḸ
Ș
Ḹ
ṼḪ
Ḹ
ṼḪ
Ḹ
Ḫ
Ḫ
ṼḸ
ṼḸ
Step 4: Substitution of Neutrosophic Numbers (NNs) to each linguistic variable.
Muhammad Saqlain, Naveed Jafar and Muhammad Riaz, A New Approach of Neutrosophic Soft Set with Generalized Fuzzy
TOPSIS in Application of Smart Phone Selection.
Neutrosophic Sets and Systems, Vol. 32, 2020
312
Table3: Assign neutrosophic number to each linguistic value from table 1.
Ƈ𝟏
(0.1, 0.3,0.7)
(0.3,0.5,0.6)
(0.5,0.5,0.5)
(0.7,0.3,0.4)
(1,0.1,0.2)
(0.5,0.5,0.5)
Ṃ𝟏
Ṃ2
Ṃ3
Ṃ4
Ṃ5
Ṃ6
Ƈ𝟐
(1,0.1,0.2)
(0.5,0.5,0.5)
(0.1, 0.3,0.7)
(1,0.1,0.2)
(0.3,0.5,0.6)
(0.1, 0.3,0.7)
Ƈ𝟑
(0.7,0.3,0.4)
(0.1, 0.3,0.7)
(0.3,0.5,0.6)
(0.5,0.5,0.5)
(0.7,0.3,0.4)
(1,0.1,0.2)
Ƈ𝟒
(0.7,0.3,0.4)
(1,0.1,0.2)
(1,0.1,0.2)
(0.3,0.5,0.6)
(0.5,0.5,0.5)
(0.7,0.3,0.4)
Ƈ𝟓
(0.5,0.5,0.5)
(0.7,0.3,0.4)
(0.7,0.3,0.4)
(1,0.1,0.2)
(0.1, 0.3,0.7)
(0.1, 0.3,0.7)
Ƈ𝟔
(0.1, 0.3,0.7)
(0.3,0.5,0.6)
(0.5,0.5,0.5)
(0.7,0.3,0.4)
(1,0.1,0.2)
(0.5,0.5,0.5)
Ƈ𝟕
(0.7,0.3,0.4)
(0.1, 0.3,0.7)
(1,0.1,0.2)
(0.1, 0.3,0.7)
(0.5,0.5,0.5)
(0.7,0.3,0.4)
Step 5: Conversion of fuzzy neutrosophic numbers NNs of step 4, into fuzzy numbers by using
accuracy function.
A(F) = { 𝑥 =
[𝑇𝑥 +𝐼𝑥 +𝐹𝑥 ]
3
}
Table: 4 After applied accuracy function the obtain result converted into fuzzy value
Ƈ1
0.367
Ƈ2
0.433
Ƈ3
0.467
Ƈ4
0.467
0.5
0.367
Ṃ4
0.467
Ṃ6
Ṃ𝟏
0.433
Ƈ5
0.5
0.467
Ƈ6
0.367
Ƈ7
0.467
0.467
0.433
0.467
0.5
0.433
0.433
0.5
0.467
0.433
0.467
0.367
0.433
0.467
0.467
0.5
0.367
0.433
0.5
0.5
0.367
0.433
0.467
0.367
0.5
0.467
0.467
Ṃ2
Ṃ3
Ṃ5
0.5
0.367
0.467
0.367
Step 6: Now we apply algorithm of TOPSIS to obtain relative closeness.
Table 5: Normalized decision matrices
Ƈ1
Ƈ2
Ƈ3
Ƈ4
Ƈ5
Ƈ6
Ƈ7
Ṃ𝟏
0.327
0.410
0.422
0.413
0.468
0.327
0.437
Ṃ2
0.416
0.474
0.332
0.383
0.437
0.416
0.343
Ṃ3
0.446
0.348
0.422
0.383
0.437
0.446
0.405
Ṃ4
0.416
0.410
0.452
0.413
0.405
0.416
0.343
Ṃ5
0.386
0.443
0.422
0.442
0.343
0.386
0.468
Ṃ6
0.446
0.348
0.391
0.413
0.343
0.446
0.437
Step 6.1: Calculation of weighted normalized matrix
Table6: Weighted normalized decision matrices
weight
0.2
0.3
0.17
0.02
0.25
0.05
0.01
Ƈ1
Ƈ2
Ƈ3
Ƈ4
Ƈ5
Ƈ6
Ƈ7
Ṃ𝟏
0.0654
0.123
0.07174
0.00826
0.117
0.01635
0.00437
Ṃ2
0.0832
0.1422
0.05644
0.00766
0.10925
0.0208
0.00343
Ṃ3
0.0892
0.1044
0.07174
0.00766
0.10925
0.0223
0.00405
Ṃ4
0.0832
0.123
0.07684
0.00826
0.1015
0.0208
0.00343
Ṃ5
0.0772
0.1329
0.07174
0.00884
0.08575
0.0193
0.00468
Ṃ6
0.0892
0.1044
0.06647
0.00826
0.08575
0.0223
0.00437
Step 6.2: Calculation of the ideal best and ideal worst value,
Muhammad Saqlain, Naveed Jafar and Muhammad Riaz, A New Approach of Neutrosophic Soft Set with Generalized Fuzzy
TOPSIS in Application of Smart Phone Selection.
Neutrosophic Sets and Systems, Vol. 32, 2020
v j
v j
313
=Indicates the ideal (best)
= Indicates the ideal (worst)
Table 7: Ideal worst and Ideal best values
v j
v j
Ƈ1
0.0654
0.0832
0.0892
0.0832
0.0772
0.0892
0.0892
0.0654
Ṃ𝟏
Ṃ2
Ṃ3
Ṃ4
Ṃ5
Ṃ6
Ƈ2
0.123
0.1422
0.1044
0.123
0.1329
0.1044
0.1422
0.1044
Ƈ3
0.07174
0.05644
0.07174
0.07684
0.07174
0.06647
0.07684
0.05644
Ƈ4
0.00826
0.00766
0.00766
0.00826
0.00884
0.00826
0.0084
0.00766
Ƈ5
0.117
0.10925
0.10925
0.1015
0.08575
0.08575
0.117
0.08575
Ƈ6
0.01635
0.0208
0.0223
0.0208
0.0193
0.0223
0.0223
0.01635
Ƈ7
0.00437
0.00343
0.00405
0.00343
0.00468
0.00437
0.00343
0.00437
Step 6.3: Calculation of rank.
pi
sij_
sij sij_
Table 8: Calculation of rank by relative closeness
s
Ṃ𝟏
Ṃ2
Ṃ3
Ṃ4
Ṃ5
Ṃ6
j
0.0316
0.0245
0.0400
0.0249
0.0671
0.0500
s j
0.0400
0.0843
0.0374
0.0374
0.0346
0.0271
sij sij_
0.0716
0.1088
0.0774
0.0623
0.1017
0.0771
p
0.5587
0. 3402
0.4832
0.6003
0.7748
0.3515
Rank
3
6
4
2
1
5
Step 7: Calculation of rank and discussion.
4.
Result Discussion
Firstly, the generalized neutrosophic TOPSIS approach is used to simplify mobile selection MCDM
problem. In this calculation, the ranking of each mobile with respect to each criterion is represented
below in Table 8 and Figure 2. To test the validity and the implementation of the technique proposed
by Saqlain et. al. [17], in neutrosophic soft set environment and multi-criteria decision making, mobile
selection problem is considered. Result shows that generalized neutrosophic TOPSIS along with
proposed algorithm can be used to find best alternative.
Secondly, results are compared with [10], in which fuzzy multi-criteria group decision making
approach was used by considering same alternative and attributes. Graphical and tabular
comparison is presented in Table 8 and Figure 2, which shows that under Generalized TOPSIS and
Fuzzy TOPSIS 𝑀5 and 𝑀5 are best alternative whereas, 𝑀2 and 𝑀3 is the worst selection
respectively.
If we compare the results of Generalized fuzzy TOPSIS and Fuzzy TOPSIS 𝑀1 , 𝑀4 , 𝑀5 has same raking
whereas, 𝑀2 , 𝑀3 , 𝑀6 .
Muhammad Saqlain, Naveed Jafar and Muhammad Riaz, A New Approach of Neutrosophic Soft Set with Generalized Fuzzy
TOPSIS in Application of Smart Phone Selection.
Neutrosophic Sets and Systems, Vol. 32, 2020
314
Result Comparison of Generalized
Fuzzy TOPSIS and Fuzzy TOPSIS
7
6
5
4
3
2
1
0
M1
M2
M3
M4
G.F. TOPSIS
M5
M6
F.TOPSIS
Figure 2: Ranking comparison of alternatives
Table 9: Ranking comparison of alternatives using G.F. TOPSIS and F. TOPSIS
Generalized Fuzzy
Fuzzy TOPSIS
TOPSIS-Result
Ranking
Strategy
Ranking
3
3
6
5
4
6
2
2
1
1
5
4
5. Conclusions
In MCDM problems, TOPSIS is widely used to find the best alternative, whereas, due to the vague
and imprecise information in fuzzy environment, ranking of alternatives may not be accurate. Thus,
neutrosophic soft set environment plays a vital role in selection problem. In this article, firstly, an
algorithm is proposed based on accuracy function under neutrosophic soft set environment and to
check the validity of the proposed technique in this environment, mobile selection problem is
considered. Secondly, results are compared with same problem under FMCGDM [10] environment.
However, the article may open a new avenue of research in competitive Neutrosophic decisionmaking arena. Thus, this proposed technique can be used in decision-makings such as supplier
selection, personal selection in academia and many other areas of management system.
Conflicts of Interest
The authors declare no conflict of interest.
Muhammad Saqlain, Naveed Jafar and Muhammad Riaz, A New Approach of Neutrosophic Soft Set with Generalized Fuzzy
TOPSIS in Application of Smart Phone Selection.
Neutrosophic Sets and Systems, Vol. 32, 2020
315
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Punjab University Journal of Mathematics, 51(8): 111-126.
41. Saqlain M, Sana M, Jafar N, Saeed. M, Smarandache, F., Aggregate Operators of Neutrosophic
Hypersoft Sets, Neutrosophic Sets and Systems (NSS), 32: (2020).
42. İ. Deli and S. Broumi, Neutrosophic Soft Matrices and NSM-decision Making, Journal of Intelligent
and Fuzzy Systems, 28 (5) (2015) 2233–2241.
43. I. Deli and N. Çağman, Intuitionistic fuzzy parameterized soft set theory and its decision making,
Applied Soft Computing 28 (2015) 109–113.
44. İ. Deli and S. Broumi, Neutrosophic soft relations and some properties, Annals of Fuzzy Mathematics
and Informatics 9(1) ( 2015) 169–182.
45. İ. Deli, npn-Soft Sets Theory and Applications, Annals of Fuzzy Mathematics and Informatics, 10/6
(2015) 847–862.
Received: 28 Oct, 2019 Accepted: 20 Mar, 2020
Muhammad Saqlain, Naveed Jafar and Muhammad Riaz, A New Approach of Neutrosophic Soft Set with Generalized Fuzzy
TOPSIS in Application of Smart Phone Selection.
Neutrosophic Sets and Systems, Vol. 32, 2020
University of New Mexico
Single and Multi-valued Neutrosophic Hypersoft set and
Tangent Similarity Measure of Single valued Neutrosophic
Hypersoft Sets
Muhammad Saqlain 1, Naveed Jafar2 , Sana Moin2 , Muhammad Saeed3 and Said Broumi4
1
2
2
3
Lahore Garrison University, DHA Phase-VI, Sector C, Lahore, 54000, Pakistan. E-mail: msaqlain@lgu.edu.pk
Lahore Garrison University, DHA Phase-VI, Sector C, Lahore, 54000, Pakistan. E-mail: naveedjafar@lgu.edu.pk
Lahore Garrison University, DHA Phase-VI, Sector C, Lahore, 54000, Pakistan. E-mail: moinsana64@gmail.com
University of Management and Technology, Township, Lahore, 54000, Pakistan. E-mail: Muhammad.saeed@umt.edu.pk
4
Laboratory of Information Processing, Faculty of Science Ben M’Sik, University Hassan II, B.P 7955, Sidi Othman,
Casablanca, Morocco. E-mail: broumisaid78@gmail.com
Abstract: In this paper, we present a single-valued Neutrosophic Hypersoft set, multi-valued
Neutrosophic Hypersoft set and tangent similarity measure for single-valued neutrosophic hypersoft
sets and its properties. Then we use this technique in an application namely selection of cricket
players for different types of matches (ODI, T20, and test) based on Neutrosophic Hypersoft set in
decision making of single-valued neutrosophic hypersoft sets. This technique will help us to decide
the best option for the players.
Keywords: Neutrosophic hypersoft set (NHSS), single-valued neutrosophic hypersoft set (SVNHSS),
multi-valued Neutrosophic Hypersoft set (MVNHSS), tangent similarity measure (TSM), multiple
attribute decision making, cricket player
1. Introduction
As the analysis of classical sets, fuzzy set [1] and intuitionistic fuzzy set [2], the neutrosophic set was
introduced by Smarandache [3, 4] to capture the insufficient, indicate, uncertain and conflicting
information. The neutrosophic set has three free parts, which are truth, indeterminacy and falsity
membership degree; subsequently, it is applied in a wide range, for example, basic decision-making
problems [5-20].
By accomplishing that the neutrosophic sets are difficult to be applied in some genuine issues
on account of truth, indeterminacy and falsity membership degree, Wang, Smarandache, Zhang, and
Sunderraman [21] presented the idea of a single-valued neutrosophic set. The single-valued
neutrosophic set can freely express truth-membership degree, indeterminacy-membership degree,
and falsity-membership degree and manages inadequate, uncertain and conflicting data. All the
aspects of the elements depicted by the single-valued neutrosophic set are entirely appropriate for
human intuition because of the flaw of information that human gets or sees from the surrounding.
The single-valued neutrosophic set has been growing quickly because of its wide scope of
hypothetical distinction and application zones, as discussed in [22-30].
The idea of similarity is significant in examining approximately every logical field. Literature
audit indicates that numerous strategies have been proposed for estimating the degree of similarity
Muhammad Saqlain and Sana Moin, Single and Multi-valued Neutrosophic Hypersoft set and Tangent Similarity Measure
of Single valued Neutrosophic Hypersoft Sets
Neutrosophic Sets and Systems, Vol. 32, 2020
318
between fuzzy sets has been examined by Chen [32], Chen, et al., [33], Hyung et al. [34], Pappis and
Karacapilidis [35] and Wang [36]. It is also a powerful instrument in building multi-criteria decisionmaking
techniques
in
numerous
regions,
for
example,
therapeutic
diagnosis,
design
acknowledgment, grouping investigation, decision making, etc. But these strategies are not fit for
managing the similarity measures including indeterminacy. In the literature, few investigations have
studied to similarity measures for neutrosophic sets and single-valued neutrosophic sets [37-46].
Ye [47] present the distance-based similarity measure of single-valued neutrosophic sets
and applied it to the group decision-making problems with single-valued neutrosophic data. Broumi
and Smarandache [48] invent another similarity measure known as cosine similarity measure of
interval-valued neutrosophic sets. Ye [49] further considered and found that there exist a few flaws
in existing cosine similarity measure characterized in vector space [50] in certain circumstances. He
[49] referenced that they may deliver an unreasonable outcome in some real cases. To conquer these
problems, Ye [49] proposed improved cosine similarity measure dependent on cosine function,
including single-valued neutrosophic cosine similarity measures and interval neutrosophic cosine
similarity measures.
Working on the similarity measures Pramanik and Mondal [51] also present a cotangent
similarity measure of rough neutrosophic sets and their application to the medical field. Pramanik
and Mondal [52] also give tangent similarity measures between intuitionistic fuzzy sets and some of
its properties and applications.
Smarandache [53] presented a new technique to deal with uncertainty. He generalized the
soft set to hypersoft set by converting the function into a multi-decision function. In the same way,
we convert hypersoft set to neutrosophic Hypersoft set to overcome the uncertainty problems. [54]
introduced the TOPSIS by using accuracy function in his work and an application of MCDM is
proposed. Application of fuzzy numbers in mobile selection in metros like Lahore is proposed by
[55]. In medical the application of fuzzy numbers is proposed by Naveed et.al [56]. TOPSIS technique
of MCDM can also be used for the prediction of games, and it’s applied in FIFA 2018 by [57].
prediction of games is a very complex topic and this game is also predicted by [58]. Many researches
presented theories along with application in neutrosophic environment [59-66].
1.1 Novelties
In this paper, we have continued the idea of intuitionistic tangent similarity measure to neutrosophic
class. We have characterized another similarity measure known as Tangent similarity measure for
neutrosophic Hypersoft set and its properties with the application.
2.Preliminaries
Definition 2.1: Neutrosophic Soft Set
Let Ů be the universal set and the set for respective attributes is given by Ë. Let P(Ů) be the set of
Neutrosophic values of Ů and Ǻ ⊆ Ë. A pair (₣, Ǻ) is called a Neutrosophic soft set over Ů and its
mapping is given as
Definition 2.2: Hyper Soft Set
₣: Ǻ → 𝑃(Ů)
Muhammad Saqlain and Sana Moin, Single and Multi-valued Neutrosophic Hypersoft set and Tangent Similarity Measure
of Single valued Neutrosophic Hypersoft Sets
Neutrosophic Sets and Systems, Vol. 32, 2020
319
Let Ů be the universal set and 𝑃(Ů) be the power set of Ů. Consider 𝑝1 , 𝑝2 , 𝑝3 … 𝑝𝑛 for 𝑛 ≥ 1, be 𝑛
well-defined attributes, whose corresponding attributive values are respectively the set
𝑃1 , 𝑃2 , 𝑃3 … 𝑃𝑛 with 𝑃𝑖 ∩ 𝑃𝑗 = ∅, for 𝑖 ≠ 𝑗 and 𝑖, 𝑗𝜖{1,2,3 … 𝑛}, then the pair (₣, 𝑃1 × 𝑃2 × 𝑃3 … 𝑃𝑛 )
is said to be Hypersoft set over Ů where
₣: 𝑃1 × 𝑃2 × 𝑃3 … 𝑃𝑛 → 𝑃(Ů)
Definition 2.3: Neutrosophic Hypersoft Set
Let Ů be the universal set and 𝑃(Ů ) be the power set of Ů. Consider 𝑝1 , 𝑝2 , 𝑝3 … 𝑝𝑛 for 𝑛 ≥ 1, be 𝑛
well-defined attributes, whose corresponding attributive values are respectively the set
𝑃1 , 𝑃2 , 𝑃3 … 𝑃𝑛 with 𝑃𝑖 ∩ 𝑃𝑗 = ∅ , for 𝑖 ≠ 𝑗 and 𝑖, 𝑗𝜖{1,2,3 … 𝑛} and their relation 𝑃1 × 𝑃2 ×
𝑃3 … 𝑃𝑛 = ß, then the pair (₣, ß) is said to be Neutrosophic Hypersoft set (NHSS) over Ů where
₣: 𝑃1 × 𝑃2 × 𝑃3 … 𝑃𝑛 → 𝑃(Ů) and
₣(𝑃1 × 𝑃2 × 𝑃3 … 𝑃𝑛 ) = {< 𝑥, 𝑇(₣(ß)), 𝐼(₣(ß)), 𝐹(₣(ß)) >, 𝑥 ∈ Ů } where T is the membership value
of truthiness, I is the membership value of indeterminacy and F is the membership value of falsity
such that 𝑇, 𝐼, 𝐹: Ů → [0,1] also 0 ≤ 𝑇(₣(ß)) + 𝐼(₣(ß)) + 𝐹(₣(ß)) ≤ 3.
3. Calculations
Definition 3.1: Single valued Neutrosophic Hypersoft Set
Let Ů be the universal set and 𝑃(Ů ) be the power set of Ů. Consider 𝑝1 , 𝑝2 , 𝑝3 … 𝑝𝑛 for 𝑛 ≥ 1, be 𝑛
well-defined attributes, whose corresponding attributive values are respectively the set
𝑃1 , 𝑃2 , 𝑃3 … 𝑃𝑛 with 𝑃𝑖 ∩ 𝑃𝑗 = ∅ , for 𝑖 ≠ 𝑗 and 𝑖, 𝑗𝜖{1,2,3 … 𝑛} and their relation 𝑃1 × 𝑃2 ×
𝑃3 … 𝑃𝑛 = ß, then the pair (₣, ß) is said to be Single valued Neutrosophic Hypersoft set (SVNHSS)
over Ů where
₣: 𝑃1 × 𝑃2 × 𝑃3 … 𝑃𝑛 → 𝑃(Ů) and this mapping to 𝑃(Ů) is single-valued.
₣(𝑃1 × 𝑃2 × 𝑃3 … 𝑃𝑛 ) = {< 𝑥, 𝑇(₣(ß)), 𝐼(₣(ß)), 𝐹(₣(ß)) >, 𝑥 ∈ Ů } where T is the membership value
of truthiness, I is the membership value of indeterminacy and F is the membership value of falsity
such that 𝑇, 𝐼, 𝐹: Ů → [0,1] also 0 ≤ 𝑇(₣(ß)) + 𝐼(₣(ß)) + 𝐹(₣(ß)) ≤ 3.
Example 3.1:
Let ξ be the set of doctors under consideration given as
also consider the set of attributes as
ξ = {𝑑1 , 𝑑 2 , 𝑑 3 , 𝑑 4 , 𝑑 5 }
𝑙1 = 𝑄𝑢𝑎𝑙𝑖𝑓𝑖𝑐𝑎𝑡𝑖𝑜𝑛, 𝑙 2 = 𝐸𝑥𝑝𝑒𝑟𝑖𝑒𝑛𝑐𝑒, 𝑙 3 = 𝐺𝑒𝑛𝑑𝑒𝑟, 𝑙 4 = 𝑆𝑘𝑖𝑙𝑙𝑠
And their respective attributes are given as
𝐿1 = 𝑄𝑢𝑎𝑙𝑖𝑓𝑖𝑐𝑎𝑡𝑖𝑜𝑛
= {𝑀𝐵𝐵𝑆, 𝑀𝑆 𝑑𝑖𝑝𝑙𝑜𝑚𝑎, 𝐷𝑖𝑝𝑙𝑜𝑚𝑎 𝑜𝑓 𝑛𝑎𝑡𝑖𝑜𝑛𝑎𝑙 𝑏𝑜𝑎𝑟𝑑(𝐷𝑁𝐵), 𝐷𝑖𝑝𝑙𝑜𝑚𝑎 𝑖𝑛 𝑐𝑙𝑖𝑛𝑖𝑐𝑎𝑙 𝑟𝑒𝑠𝑒𝑎𝑟𝑐ℎ(𝐷𝐶𝑅)}
𝐿2 = 𝐸𝑥𝑝𝑒𝑟𝑖𝑒𝑛𝑐𝑒 = {5𝑦𝑟, 8𝑦𝑟, 10𝑦𝑟, 15𝑦𝑟}
𝐿3 = 𝐺𝑒𝑛𝑑𝑒𝑟 = {𝑀𝑎𝑙𝑒, 𝐹𝑒𝑚𝑎𝑙𝑒}
𝐿4 = 𝑆𝑘𝑖𝑙𝑙𝑠 = {𝐶𝑜𝑚𝑝𝑎𝑠𝑠𝑖𝑜𝑛𝑎𝑡𝑒, 𝑃𝑟𝑜𝑏𝑙𝑒𝑚 𝑠𝑜𝑙𝑣𝑖𝑛𝑔, 𝐶𝑜𝑚𝑚𝑢𝑛𝑖𝑐𝑎𝑡𝑖𝑣𝑒, 𝑙𝑒𝑎𝑑𝑒𝑟𝑠ℎ𝑖𝑝}
Let the function be ₣: 𝐿1 × 𝐿2 × 𝐿3 × 𝐿4 → 𝑃(𝜉)
Below are the tables of their Neutrosophic values from different decision makers
Muhammad Saqlain and Sana Moin, Single and Multi-valued Neutrosophic Hypersoft set and Tangent Similarity Measure
of Single valued Neutrosophic Hypersoft Sets
Neutrosophic Sets and Systems, Vol. 32, 2020
320
Table 1: Decision maker Neutrosophic values for Qualification
1
𝐿 (𝑄𝑢𝑎𝑙𝑖𝑓𝑖𝑐𝑎𝑡𝑖𝑜𝑛)
𝑑1
𝑑2
𝑑3
𝑑4
𝑑5
MBBS
(0.4, 0.5, 0.8)
(0.7, 0.6, 0.4)
(0.4, 0.5, 0.7)
(0.5, 0.3, 0.7)
(0.5, 0.3, 0.8)
MS diploma
(0.5, 0.3, 0.6)
(0.3, 0.2, 0.1)
(0.3, 0.6, 0.2)
(0.7, 0.3, 0.6)
(0.5, 0.4, 0.5)
DNB
(0.8, 0.2, 0.4)
(0.9, 0.5, 0.3)
(0.9, 0.4, 0.1)
(0.6, 0.3, 0.2)
(0.6, 0.1, 0.2)
DCR
(0.9, 0.3, 0.1)
(0.5, 0.2, 0.1)
(0.8, 0.5, 0.2)
(0.8, 0.2, 0.1)
(0.7, 0.4, 0.2)
Table 2: Decision maker Neutrosophic values for Experience
𝐿2 (𝐸𝑥𝑝𝑒𝑟𝑖𝑒𝑛𝑐𝑒)
𝑑1
𝑑2
𝑑3
𝑑4
𝑑5
5 yr.
(0.3, 0.4, 0.7)
(0.6, 0.5, 0.3)
(0.5, 0.6, 0.8)
(0.6, 0.4, 0.8)
(0.3, 0.6, 0.7)
8 yr.
(0.4, 0.2, 0.5)
(0.8, 0.1, 0.2)
(0.4, 0.7, 0.3)
(0.4, 0.8, 0.7)
(0.7, 0.5, 0.6)
10 yr.
(0.7, 0.2, 0.3)
(0.9, 0.3, 0.1)
(0.8, 0.3, 0.2)
(0.5, 0.4, 0.3)
(0.5, 0.2, 0.1)
15 yr.
(0.8, 0.2, 0.1)
(0.6, 0.4, 0.3)
(0.9 0.4, 0.1)
(0.6, 0.2, 0.3)
(0.5, 0.3, 0.2)
Table 3: Decision maker Neutrosophic values for Gender
𝐿3 (𝐺𝑒𝑛𝑑𝑒𝑟)
𝑑1
𝑑2
𝑑3
𝑑4
𝑑5
Male
(0.5, 0.6, 0.9)
(0.7, 0.8, 0.3)
(0.6, 0.4, 0.3)
(0.8, 0.5, 0.4)
(0.9, 0.2, 0.1)
Female
(0.6, 0.4, 0.7)
(0.3, 0.6, 0.4)
(0.8, 0.2, 0.1)
(0.4, 0.5, 0.6)
(0.8, 0.4, 0.2)
Table 4: Decision maker Neutrosophic values for Skills
𝐿4 (𝑆𝑘𝑖𝑙𝑙𝑠)
𝑑1
𝑑2
𝑑3
𝑑4
𝑑5
Compassionate
(0.6, 0.4, 0.5)
(0.7, 0.5, 0.3)
(0.6, 0.4, 0.3)
(0.6, 0.2, 0.1)
(0.4, 0.5, 0.3)
Problem solving
(0.8, 0.2, 0.4)
(0.7, 0.3, 0.2)
(0.8, 0.3, 0.1)
(0.3, 0.4, 0.5)
(0.3, 0.5, 0.8)
Communicative
(0.5, 0.3, 0.4)
(0.6, 0.3, 0.4)
(0.5, 0.7, 0.2)
(0.8, 0.4, 0.1)
(0.7, 0.4, 0.3)
Leadership
(0.4, 0.9, 0.6)
(0.8, 0.4, 0.2)
(0.2, 0.6, 0.5)
(0.7, 0.5, 0.2)
(0.6, 0.4, 0.7)
1
2
3
4
Single valued neutrosophic hypersoft set is define as ₣: (𝐿 × 𝐿 × 𝐿 × 𝐿 ) → 𝑃(𝜉)
Let’s assume ₣(£) = ₣(𝐷𝑁𝐵, 10 𝑦𝑟, 𝑚𝑎𝑙𝑒, 𝑐𝑜𝑚𝑝𝑎𝑠𝑠𝑖𝑜𝑛𝑎𝑡𝑒) = {𝑑1 }
Then the single-valued neutrosophic hypersoft set of above-assumed relation is
₣(£) = ₣(𝐷𝑁𝐵, 10 𝑦𝑟, 𝑚𝑎𝑙𝑒, 𝑐𝑜𝑚𝑝𝑎𝑠𝑠𝑖𝑜𝑛𝑎𝑡𝑒) = {
≪ 𝑑1 , (𝐷𝑁𝐵{0.8, 0.2, 0.4}, 10 𝑦𝑟{0.7, 0.2, 0.3}, 𝑚𝑎𝑙𝑒{0.5, 0.6, 0.9}, 𝑐𝑜𝑚𝑝𝑎𝑠𝑠𝑖𝑜𝑛𝑎𝑡𝑒{0.6, 0.4, 0.5}) ≫}
Its tabular form is given as
Table 5: Tabular Representation of Single Valued Neutrosophic Hypersoft Set
𝒅𝟏
₣(£) = ₣(𝑫𝑵𝑩, 𝟏𝟎 𝒚𝒓, 𝒎𝒂𝒍𝒆, 𝒄𝒐𝒎𝒑𝒂𝒔𝒔𝒊𝒐𝒏𝒂𝒕𝒆)
DNB
(0.8, 0.2, 0.4)
10 yr.
(0.7, 0.2, 0.3)
Male
(0.5, 0.6, 0.9)
Compassionate
(0.6, 0.4, 0.5)
Muhammad Saqlain and Sana Moin, Single and Multi-valued Neutrosophic Hypersoft set and Tangent Similarity Measure
of Single valued Neutrosophic Hypersoft Sets
Neutrosophic Sets and Systems, Vol. 32, 2020
321
Definition 3.2: Multi-valued Neutrosophic Hypersoft Set
Let Ů be the universal set and 𝑃(Ů ) be the power set of Ů. Consider 𝑝1 , 𝑝2 , 𝑝3 … 𝑝𝑛 for 𝑛 ≥ 1, be 𝑛
well-defined attributes, whose corresponding attributive values are respectively the set
𝑃1 , 𝑃2 , 𝑃3 … 𝑃𝑛 with 𝑃𝑖 ∩ 𝑃𝑗 = ∅ , for 𝑖 ≠ 𝑗 and 𝑖, 𝑗𝜖{1,2,3 … 𝑛} and their relation 𝑃1 × 𝑃2 ×
𝑃3 … 𝑃𝑛 = ß, then the pair (₣, ß) is said to be Single valued Neutrosophic Hypersoft set (SVNHSS)
over Ů where
₣: 𝑃1 × 𝑃2 × 𝑃3 … 𝑃𝑛 → 𝑃(Ů) and this mapping to 𝑃(Ů) is multi-valued.
₣(𝑃1 × 𝑃2 × 𝑃3 … 𝑃𝑛 ) = {< 𝑥, 𝑇(₣(ß)), 𝐼(₣(ß)), 𝐹(₣(ß)) >, 𝑥 ∈ Ů } where T is the membership value
of truthiness, I is the membership value of indeterminacy and F is the membership value of falsity
such that 𝑇, 𝐼, 𝐹: Ů → [0,1] also 0 ≤ 𝑇(₣(ß)) + 𝐼(₣(ß)) + 𝐹(₣(ß)) ≤ 3.
Example 3.2:
Let ξ be the set of doctors under consideration given as ξ = {𝑑1 , 𝑑 2 , 𝑑 3 , 𝑑 4 , 𝑑 5 }
also consider the set of attributes as
𝑙1 = 𝑄𝑢𝑎𝑙𝑖𝑓𝑖𝑐𝑎𝑡𝑖𝑜𝑛, 𝑙 2 = 𝐸𝑥𝑝𝑒𝑟𝑖𝑒𝑛𝑐𝑒, 𝑙 3 = 𝐺𝑒𝑛𝑑𝑒𝑟, 𝑙 4 = 𝑆𝑘𝑖𝑙𝑙𝑠
And their respective attributes are given as
𝐿1 = 𝑄𝑢𝑎𝑙𝑖𝑓𝑖𝑐𝑎𝑡𝑖𝑜𝑛
= {𝑀𝐵𝐵𝑆, 𝑀𝑆 𝑑𝑖𝑝𝑙𝑜𝑚𝑎, 𝐷𝑖𝑝𝑙𝑜𝑚𝑎 𝑜𝑓 𝑛𝑎𝑡𝑖𝑜𝑛𝑎𝑙 𝑏𝑜𝑎𝑟𝑑(𝐷𝑁𝐵), 𝐷𝑖𝑝𝑙𝑜𝑚𝑎 𝑖𝑛 𝑐𝑙𝑖𝑛𝑖𝑐𝑎𝑙 𝑟𝑒𝑠𝑒𝑎𝑟𝑐ℎ(𝐷𝐶𝑅)}
𝐿2 = 𝐸𝑥𝑝𝑒𝑟𝑖𝑒𝑛𝑐𝑒 = {5𝑦𝑟, 8𝑦𝑟, 10𝑦𝑟, 15𝑦𝑟}
𝐿3 = 𝐺𝑒𝑛𝑑𝑒𝑟 = {𝑀𝑎𝑙𝑒, 𝐹𝑒𝑚𝑎𝑙𝑒}
𝐿4 = 𝑆𝑘𝑖𝑙𝑙𝑠 = {𝐶𝑜𝑚𝑝𝑎𝑠𝑠𝑖𝑜𝑛𝑎𝑡𝑒, 𝑃𝑟𝑜𝑏𝑙𝑒𝑚 𝑠𝑜𝑙𝑣𝑖𝑛𝑔, 𝐶𝑜𝑚𝑚𝑢𝑛𝑖𝑐𝑎𝑡𝑖𝑣𝑒, 𝑙𝑒𝑎𝑑𝑒𝑟𝑠ℎ𝑖𝑝}
Let the function be ₣: 𝐿1 × 𝐿2 × 𝐿3 × 𝐿4 → 𝑃(𝜉)
Below are the tables of their Neutrosophic values from different decision makers
Table 6: Decision maker Neutrosophic values for Qualification
𝐿1 (𝑄𝑢𝑎𝑙𝑖𝑓𝑖𝑐𝑎𝑡𝑖𝑜𝑛)
𝑑1
𝑑2
𝑑3
𝑑4
𝑑5
MBBS
(0.4, 0.5, 0.8)
(0.7, 0.6, 0.4)
(0.4, 0.5, 0.7)
(0.5, 0.3, 0.7)
(0.5, 0.3, 0.8)
MS diploma
(0.5, 0.3, 0.6)
(0.3, 0.2, 0.1)
(0.3, 0.6, 0.2)
(0.7, 0.3, 0.6)
(0.5, 0.4, 0.5)
DNB
(0.8, 0.2, 0.4)
(0.9, 0.5, 0.3)
(0.9, 0.4, 0.1)
(0.6, 0.3, 0.2)
(0.6, 0.1, 0.2)
DCR
(0.9, 0.3, 0.1)
(0.5, 0.2, 0.1)
(0.8, 0.5, 0.2)
(0.8, 0.2, 0.1)
(0.7, 0.4, 0.2)
Table 7: Decision maker Neutrosophic values for Experience
𝐿2 (𝐸𝑥𝑝𝑒𝑟𝑖𝑒𝑛𝑐𝑒)
𝑑1
𝑑2
𝑑3
𝑑4
𝑑5
5 yr.
(0.3, 0.4, 0.7)
(0.6, 0.5, 0.3)
(0.5, 0.6, 0.8)
(0.6, 0.4, 0.8)
(0.3, 0.6, 0.7)
8 yr.
(0.4, 0.2, 0.5)
(0.8, 0.1, 0.2)
(0.4, 0.7, 0.3)
(0.4, 0.8, 0.7)
(0.7, 0.5, 0.6)
10 yr.
(0.7, 0.2, 0.3)
(0.9, 0.3, 0.1)
(0.8, 0.3, 0.2)
(0.5, 0.4, 0.3)
(0.5, 0.2, 0.1)
15 yr.
(0.8, 0.2, 0.1)
(0.6, 0.4, 0.3)
(0.9 0.4, 0.1)
(0.6, 0.2, 0.3)
(0.5, 0.3, 0.2)
Table 8: Decision maker Neutrosophic values for Gender
𝐿3 (𝐺𝑒𝑛𝑑𝑒𝑟)
𝑑1
𝑑2
𝑑3
𝑑4
𝑑5
Male
(0.5, 0.6, 0.9)
(0.7, 0.8, 0.3)
(0.6, 0.4, 0.3)
(0.8, 0.5, 0.4)
(0.9, 0.2, 0.1)
Female
(0.6, 0.4, 0.7)
(0.3, 0.6, 0.4)
(0.8, 0.2, 0.1)
(0.4, 0.5, 0.6)
(0.8, 0.4, 0.2)
Muhammad Saqlain and Sana Moin, Single and Multi-valued Neutrosophic Hypersoft set and Tangent Similarity Measure
of Single valued Neutrosophic Hypersoft Sets
Neutrosophic Sets and Systems, Vol. 32, 2020
322
Table 9: Decision maker Neutrosophic values for Skills
𝐿4 (𝑆𝑘𝑖𝑙𝑙𝑠)
𝑑1
𝑑2
𝑑3
𝑑4
𝑑5
Compassionate
(0.6, 0.4, 0.5)
(0.7, 0.5, 0.3)
(0.6, 0.4, 0.3)
(0.6, 0.2, 0.1)
(0.4, 0.5, 0.3)
Problem solving
(0.8, 0.2, 0.4)
(0.7, 0.3, 0.2)
(0.8, 0.3, 0.1)
(0.3, 0.4, 0.5)
(0.3, 0.5, 0.8)
Communicative
(0.5, 0.3, 0.4)
(0.6, 0.3, 0.4)
(0.5, 0.7, 0.2)
(0.8, 0.4, 0.1)
(0.7, 0.4, 0.3)
Leadership
(0.4, 0.9, 0.6)
(0.8, 0.4, 0.2)
(0.2, 0.6, 0.5)
(0.7, 0.5, 0.2)
(0.6, 0.4, 0.7)
Multi-valued neutrosophic hyper soft set is define as
₣: (𝐿1 × 𝐿2 × 𝐿3 × 𝐿4 ) → 𝑃(𝜉)
Let’s assume ₣(£) = ₣(𝐷𝑁𝐵, 10 𝑦𝑟, 𝑚𝑎𝑙𝑒, 𝑐𝑜𝑚𝑝𝑎𝑠𝑠𝑖𝑜𝑛𝑎𝑡𝑒) = {𝑑1 , 𝑑 4 }
Then multi-valued neutrosophic hyper soft set of above assumed relation is
₣(£) = ₣(𝐷𝑁𝐵, 10 𝑦𝑟, 𝑚𝑎𝑙𝑒, 𝑐𝑜𝑚𝑝𝑎𝑠𝑠𝑖𝑜𝑛𝑎𝑡𝑒) = {
≪ 𝑑1 , (𝐷𝑁𝐵{0.8, 0.2, 0.4}, 10 𝑦𝑟{0.7, 0.2, 0.3}, 𝑚𝑎𝑙𝑒{0.5, 0.6, 0.9}, 𝑐𝑜𝑚𝑝𝑎𝑠𝑠𝑖𝑜𝑛𝑎𝑡𝑒{0.6, 0.4, 0.5}) ≫,
≪ 𝑑 4 (𝐷𝑁𝐵{0.6, 0.3, 0.2}, 10 𝑦𝑟{0.5, 0.4, 0.3}, 𝑚𝑎𝑙𝑒{0.8, 0.5, 0.4}, 𝑐𝑜𝑚𝑝𝑎𝑠𝑠𝑖𝑜𝑛𝑎𝑡𝑒{0.6, 0.2, 0.1}) ≫}
Its tabular form is given as
Table 10: Tabular Representation of Multi-valued Neutrosophic Hypersoft Set
₣(£)
𝑑1
= ₣(𝐷𝑁𝐵, 10 𝑦𝑟, 𝑚𝑎𝑙𝑒, 𝑐𝑜𝑚𝑝𝑎𝑠𝑠𝑖𝑜𝑛𝑎𝑡𝑒)
DNB
(0.8, 0.2, 0.4)
10 yr.
(0.7, 0.2, 0.3)
(0.5, 0.4, 0.3)
Male
(0.5, 0.6, 0.9)
(0.8, 0.5, 0.4)
Compassionate
(0.6, 0.4, 0.5)
(0.6, 0.2, 0.1)
(0.6, 0.3, 0.2)
𝑑4
3.3: Tangent similarity measures for single valued neutrosophic hypersoft set
Let Ŕ =< 𝑥, 𝑇 Ŕ (₣(ß)), 𝐼 Ŕ (₣(ß)), 𝐹 Ŕ (₣(ß)) > and Ś =< 𝑥, 𝑇 Ś (₣(ß)), 𝐼 Ś (₣(ß)), 𝐹 Ś (₣(ß)) >
be two
single valued neutrosophic hypersoft set(SVNHSS) for ₣(ß). Tangent similarity measure for these
sets to measure the similarity between them is presented as
1
𝑇𝑆𝑉𝑁𝐻𝑆𝑆 (Ŕ, Ś) =< 𝑥, ∑𝑛𝑖=1 [1 − tan (
𝑛
₣(ß)
𝜋(|𝑇 Ŕ (₣(ß)𝑖 )−𝑇 Ś (₣(ß)𝑖 )|+|𝐼 Ŕ (₣(ß)𝑖 )−𝐼 Ś (₣(ß)𝑖 )|+|𝐹Ŕ (₣(ß)𝑖 )−𝐹Ś (₣(ß)𝑖 )|)
12
)] > , 𝑥 ∈
3.3.1: Proposition
Tangent similarity measure between two single valued Neutrosophic hypersoft set 𝑇𝑆𝑉𝑁𝐻𝑆𝑆 (Ŕ, Ś)
satisfies the following properties.
1.
2.
3.
4.
0 ≤ 𝑇𝑆𝑉𝑁𝐻𝑆𝑆 (Ŕ, Ś) ≤ 1
𝑇𝑆𝑉𝑁𝐻𝑆𝑆 (Ŕ, Ś) = 1 𝑖𝑓 𝑎𝑛𝑑 𝑜𝑛𝑙𝑦 𝑖𝑓 Ŕ = Ś
𝑇𝑆𝑉𝑁𝐻𝑆𝑆 (Ŕ, Ś) = 𝑇𝑆𝑉𝑁𝐻𝑆𝑆 (Ś, Ŕ)
If Ő is a SVNHSS and Ŕ ⊂ Ś ⊂ Ő then 𝑇𝑆𝑉𝑁𝐻𝑆𝑆 (Ŕ, Ő) ≤ 𝑇𝑆𝑉𝑁𝐻𝑆𝑆 (Ŕ, Ś) and 𝑇𝑆𝑉𝑁𝐻𝑆𝑆 (Ŕ, Ő) ≤
𝑇𝑆𝑉𝑁𝐻𝑆𝑆 (Ś, Ő).
It is easy to see that the define similarity measure satisfies the above properties easily so the proofs
are left for the reader.
Muhammad Saqlain and Sana Moin, Single and Multi-valued Neutrosophic Hypersoft set and Tangent Similarity Measure
of Single valued Neutrosophic Hypersoft Sets
Neutrosophic Sets and Systems, Vol. 32, 2020
323
3.4: Decision making using single-valued neutrosophic hypersoft set based on the tangent
similarity measure
Let 𝐿1 , 𝐿2 , 𝐿3 … 𝐿𝑛 be the distinct set of participants, 𝑀1 , 𝑀2 , 𝑀3 … 𝑀𝑛
1
2
3
by the set of norms for
𝑛
participants and 𝑁 , 𝑁 , 𝑁 … 𝑁 be the set of options for each participant. By using a decision-
making technique, the decision-makers add ranking of options concerning each participant. This
ranking gives the effectiveness of participants L against the norms of participants M then theses
values associated with the options for multiple attribute decision making. Algorithm of this
procedure are given below
3.4.1: Algorithm
Step 1: Determine the association between participants and the norms.
The association between participants and the norms is given by the below decision matrix in terms
of single-valued Neutrosophic hyper soft sets.
Table 21: Association between participants and the norms in term of SVNHSS
𝑳𝟏
𝑳𝟐
…
𝑳
𝒎
𝑴𝟏
〈𝑇11 , 𝐼11 , 𝐹11 〉
〈𝑇21 , 𝐼21 , 𝐹21 〉
…
〈𝑇𝑚1 , 𝐼𝑚1 , 𝐹𝑚1 〉
𝑴𝟐
…
〈𝑇12 , 𝐼12 , 𝐹12 〉
…
…
…
〈𝑇22 , 𝐼22 , 𝐹22 〉
〈𝑇𝑚2 , 𝐼𝑚2 , 𝐹𝑚2 〉
Step 2: Determine the association between norms and options.
…
…
𝑴𝒏
〈𝑇1𝑛 , 𝐼1𝑛 , 𝐹1𝑛 〉
〈𝑇2𝑛 , 𝐼2𝑛 , 𝐹2𝑛 〉
…
〈𝑇𝑚𝑛 , 𝐼𝑚𝑛 , 𝐹𝑚𝑛 〉
The association between the norms and the options is given by the below decision matrix in terms of
single-valued Neutrosophic hypersoft sets.
Table 22: Association between the norms and the options in term of SVNHSS
𝑀1
𝑀
2
…
𝑀𝑛
𝑵𝟏
〈𝑇11 , 𝐼11 , 𝐹11 〉
〈𝑇21 , 𝐼21 , 𝐹21 〉
…
〈𝑇𝑛1 , 𝐼𝑛1 , 𝐹𝑛1 〉
𝑵𝟐
…
〈𝑇12 , 𝐼12 , 𝐹12 〉
…
…
…
〈𝑇22 , 𝐼22 , 𝐹22 〉
〈𝑇𝑛2 , 𝐼𝑛2 , 𝐹𝑛2 〉
…
…
Step 3: Determine the association between participants and options.
𝑵𝒌
〈𝑇1𝑘 , 𝐼1𝑘 , 𝐹1𝑘 〉
〈𝑇2𝑘 , 𝐼2𝑘 , 𝐹2𝑘 〉
…
〈𝑇𝑛𝑘 , 𝐼𝑛𝑘 , 𝐹𝑛𝑘 〉
The association between participants and the options is determined with the help of tangent
similarity measures for single-valued neutrosophic hypersoft numbers.
Step 4: Decision of best option
The best option is decided by arranging the results in the descending orders and choosing the highest
value as the highest value represents the best option for the participants.
Muhammad Saqlain and Sana Moin, Single and Multi-valued Neutrosophic Hypersoft set and Tangent Similarity Measure
of Single valued Neutrosophic Hypersoft Sets
Neutrosophic Sets and Systems, Vol. 32, 2020
324
Figure 1: Algorithm design for the proposed technique
4. Example
We have seen a large number of the matches that a team loses because of improper selection of
players. we can't choose which player is perfect for which sort of matches like the test, ODI and T20
due to the presence of the huge amount of uncertainties and a large volume of information about the
players. With such a piece of vast information, we are unable to focus on every aspect because we
may have the cases in which we have the same truth membership, indeterminate membership, and
falsity membership values.
To overcome this issue, let us consider an illustrative example by using proposed method for the
selection of the players in any type of match which is significant for cricket board as cricket board is
the administering body for cricket in the state and the selection of cricket crew is likewise a key duty
of cricket board. For this purpose, let us consider two sets, μ, and η. μ be the set of players and η be
the set of type of matches played by players i.e.
μ = { 𝑃1 , 𝑃2 , 𝑃3 , 𝑃4 , 𝑃5 , 𝑃6 , 𝑃7 , 𝑃8 , 𝑃9 , 𝑃10 , 𝑃11 , 𝑃12 , 𝑃13 } and
η = { Test match, ODI match, T20 match}.
ζ be the set of attributes corresponding to μ and η.
𝜁1 = 𝑃𝑙𝑎𝑦𝑒𝑟𝑠 𝑆𝑡𝑟𝑖𝑘𝑒 𝑅𝑎𝑡𝑒, 𝜁 2 = 𝑃𝑙𝑎𝑦𝑒𝑟𝑠 𝐴𝑣𝑒𝑟𝑎𝑔𝑒, 𝜁 3 = 𝑃𝑙𝑎𝑦𝑒𝑟𝑠 𝐸𝑐𝑜𝑛𝑜𝑚𝑦, 𝜁 4 = 𝑃𝑙𝑎𝑦𝑒𝑟𝑠 𝑎𝑡𝑡𝑖𝑡𝑢𝑑𝑒,
𝜁 5 = 𝑃𝑙𝑎𝑦𝑒𝑟𝑠 𝐹𝑖𝑡𝑛𝑒𝑠𝑠 𝑡𝑒𝑠𝑡
Muhammad Saqlain and Sana Moin, Single and Multi-valued Neutrosophic Hypersoft set and Tangent Similarity Measure
of Single valued Neutrosophic Hypersoft Sets
Neutrosophic Sets and Systems, Vol. 32, 2020
325
And respective attributes for the above-mentioned attributes are given as
ς1 = 𝑃𝑙𝑎𝑦𝑒𝑟𝑠 𝑆𝑡𝑟𝑖𝑘𝑒 𝑅𝑎𝑡𝑒(𝑃𝑆𝑅) = {𝑏𝑒𝑙𝑜𝑤 40 , 40 − 60, 60 − 80, 80 − 100, 100 − 150, 150 𝑎𝑏𝑜𝑣𝑒}
ς2 = 𝑃𝑙𝑎𝑦𝑒𝑟𝑠 𝐴𝑣𝑒𝑟𝑎𝑔𝑒(𝑃𝐴𝑣) = {𝑏𝑒𝑙𝑜𝑤 30, 30 − 50, 50 − 70, 70 𝑎𝑏𝑜𝑣𝑒}
ς3 = 𝑃𝑙𝑎𝑦𝑒𝑟𝑠 𝐸𝑐𝑜𝑛𝑜𝑚𝑦(𝑃𝐸) = {𝑏𝑒𝑙𝑜𝑤 3, 3 − 7, 7 − 13, 𝑎𝑏𝑜𝑣𝑒 13}
ς4 = 𝑃𝑙𝑎𝑦𝑒𝑟𝑠 𝑎𝑡𝑡𝑖𝑡𝑢𝑑𝑒(𝑃𝐴) = {𝑐𝑜𝑝𝑒𝑟𝑎𝑡𝑖𝑣𝑒, 𝑟𝑢𝑑𝑒, 𝑒𝑚𝑜𝑡𝑖𝑜𝑛𝑎𝑙, 𝑚𝑜𝑜𝑑𝑦}
ς5 = 𝑃𝑙𝑎𝑦𝑒𝑟𝑠 𝐹𝑖𝑡𝑛𝑒𝑠𝑠 𝑡𝑒𝑠𝑡(𝑃𝐹𝑇) = {𝑝𝑎𝑠𝑠𝑒𝑑, 𝑛𝑜𝑡 𝑝𝑎𝑠𝑠𝑒𝑑}
Then Neutrosophic Hypersoft set is given as
₣: (ς1 × ς2 × ς3 × ς4 × ς5 ) → 𝑃(μ)
₣: (ς1 × ς2 × ς3 × ς4 × ς5 ) → 𝑃(η)
And
Let’s assume ₣(𝛼) = ₣(100 − 150, 30 − 50, 𝑎𝑏𝑜𝑣𝑒 13, 𝑐𝑜𝑜𝑝𝑒𝑟𝑎𝑡𝑖𝑣𝑒, 𝑝𝑎𝑠𝑠𝑒𝑑) = {𝑃1 , 𝑃3 , 𝑃6 , 𝑃8 , 𝑃9 }
and
₣(𝛽) = ₣(100 − 150, 30 − 50, 𝑎𝑏𝑜𝑣𝑒 13, 𝑐𝑜𝑜𝑝𝑒𝑟𝑎𝑡𝑖𝑣𝑒, 𝑝𝑎𝑠𝑠𝑒𝑑) = {Test match, ODI match, T20 match}
Now using the proposed tangent similarity measures for single-valued neutrosophic hypersoft sets,
we will decide which player is best for which type of match. For this purpose first we will provide
ranking between {100 − 150, 30 − 50, 𝑎𝑏𝑜𝑣𝑒 13, 𝑐𝑜𝑜𝑝𝑒𝑟𝑎𝑡𝑖𝑣𝑒, 𝑝𝑎𝑠𝑠𝑒𝑑} and {𝑃1 , 𝑃3 , 𝑃6 , 𝑃8 , 𝑃9 } in
terms of the single-valued neutrosophic hypersoft sets. In the 2nd step we will provide ranking
{100 − 150, 30 − 50, 𝑎𝑏𝑜𝑣𝑒 13, 𝑐𝑜𝑜𝑝𝑒𝑟𝑎𝑡𝑖𝑣𝑒, 𝑝𝑎𝑠𝑠𝑒𝑑}
between
{Test match, ODI match, T20 match} . In the 3
1
3
6
8
9
rd
and
step, we will find a correlation between
{𝑃 , 𝑃 , 𝑃 , 𝑃 , 𝑃 } and{Test match, ODI match, T20 match} using 𝑇𝑆𝑉𝑁𝐻𝑆𝑆 . In the last step, we will
decide by arranging the results in the descending order and selecting the highest value.
Step 1: Determine the association between {𝐏 𝟏 , 𝐏 𝟑 , 𝐏 𝟔 , 𝐏 𝟖 , 𝐏 𝟗 } and {𝟏𝟎𝟎 − 𝟏𝟓𝟎, 𝟑𝟎 −
𝟓𝟎, 𝐚𝐛𝐨𝐯𝐞 𝟏𝟑, 𝐜𝐨𝐩𝐞𝐫𝐚𝐭𝐢𝐯𝐞, 𝐩𝐚𝐬𝐬𝐞𝐝}.
The
1
association
3
6
8
{100 − 150, 30 − 50, 𝑎𝑏𝑜𝑣𝑒 13, 𝑐𝑜𝑜𝑝𝑒𝑟𝑎𝑡𝑖𝑣𝑒, 𝑝𝑎𝑠𝑠𝑒𝑑}
between
9
and
{𝑃 , 𝑃 , 𝑃 , 𝑃 , 𝑃 } is given by the below decision matrix in terms of single-valued Neutrosophic
hypersoft sets.
Table 13: Association between {𝑃1 , 𝑃 3 , 𝑃 6 , 𝑃 8 , 𝑃 9 } and {100 − 150, 30 − 50, 𝑎𝑏𝑜𝑣𝑒 13, 𝑐𝑜𝑝𝑒𝑟𝑎𝑡𝑖𝑣𝑒, 𝑝𝑎𝑠𝑠𝑒𝑑} in
term of SVNHSS
𝑃1
𝑃3
𝑃6
𝑃8
𝑃
9
Step 2:
𝟏𝟎𝟎 − 𝟏𝟓𝟎(𝑷𝑺𝑹)
𝟑𝟎 − 𝟓𝟎(𝑷𝑨𝒗)
(0.7,0.3,0,2)
(0.4, 0.5, 0.7)
𝑨𝒃𝒐𝒗𝒆 𝟏𝟑(𝑷𝑬)
(0.5, 0.3, 0.8)
𝑪𝒐𝒐𝒑𝒆𝒓𝒂𝒕𝒊𝒗𝒆 (𝑷𝑨)
(0.7, 0.6, 0.4)
𝑷𝒂𝒔𝒔𝒆𝒅 (𝑷𝑭𝑻)
(0.5,0.4,0.7)
(0.3, 0.6, 0.2)
(0.5, 0.4, 0.5)
(0.3, 0.2, 0.1)
(0.7, 0.3, 0.6)
(0.8,0.2,0.1)
(0.9, 0.4, 0.1)
(0.6, 0.1, 0.2)
(0.9, 0.5, 0.3)
(0.6, 0.3, 0.2)
(0.9,0.1,0.3)
(0.8, 0.5, 0.2)
(0.7, 0.4, 0.2)
(0.5, 0.2, 0.1)
(0.8, 0.2, 0.1)
(0.6,0.3,0.3)
(0.5, 0.4, 0.3)
(0.9, 0.2, 0.1)
(0.4, 0.5, 0.7)
Determine the association between {𝐓𝐞𝐬𝐭 𝐦𝐚𝐭𝐜𝐡, 𝐎𝐃𝐈 𝐦𝐚𝐭𝐜𝐡, 𝐓𝟐𝟎 𝐦𝐚𝐭𝐜𝐡} and {𝟏𝟎𝟎 −
𝟏𝟓𝟎, 𝟑𝟎 − 𝟓𝟎, 𝒂𝒃𝒐𝒗𝒆 𝟏𝟑, 𝒄𝒐𝒑𝒆𝒓𝒂𝒕𝒊𝒗𝒆, 𝒑𝒂𝒔𝒔𝒆𝒅}.
The
(0.8, 0.3, 0.2)
(0.5, 0.3, 0.7)
association
between
{100 − 150, 30 − 50, 𝑎𝑏𝑜𝑣𝑒 13, 𝑐𝑜𝑜𝑝𝑒𝑟𝑎𝑡𝑖𝑣𝑒, 𝑝𝑎𝑠𝑠𝑒𝑑}
and
{Test match, ODI match, T20 match} is given by the below decision matrix in terms of single-valued
Neutrosophic hypersoft sets.
Muhammad Saqlain and Sana Moin, Single and Multi-valued Neutrosophic Hypersoft set and Tangent Similarity Measure
of Single valued Neutrosophic Hypersoft Sets
Neutrosophic Sets and Systems, Vol. 32, 2020
326
Table 14: Association between {100 − 150, 30 − 50, 𝑎𝑏𝑜𝑣𝑒 13, 𝑐𝑜𝑝𝑒𝑟𝑎𝑡𝑖𝑣𝑒, 𝑝𝑎𝑠𝑠𝑒𝑑} and
{𝑇𝑒𝑠𝑡 𝑚𝑎𝑡𝑐ℎ, 𝑂𝐷𝐼 𝑚𝑎𝑡𝑐ℎ, 𝑇20 𝑚𝑎𝑡𝑐ℎ} in term of SVNHSS
𝑶𝑫𝑰 𝒎𝒂𝒕𝒄𝒉
𝑻𝟐𝟎 𝒎𝒂𝒕𝒄𝒉
100 − 150(𝑃𝑆𝑅)
(0.7, 0.5, 0.3)
(0.6, 0.4, 0.3)
(0.4, 0.5, 0.3)
(0.7, 0.3, 0.2)
(0.8, 0.3, 0.1)
(0.3, 0.5, 0.8)
𝐴𝑏𝑜𝑣𝑒 13(𝑃𝐸)
(0.6, 0.3, 0.4)
(0.5, 0.7, 0.2)
(0.7, 0.4, 0.3)
(0.5, 0.4, 0.5)
(0.9, 0.2, 0.1)
(0.5, 0.2, 0.1)
(0.6, 0.4, 0.7)
(0.3, 0.6, 0.4)
(0.8, 0.2, 0.1)
30 − 50(𝑃𝐴𝑣)
𝐶𝑜𝑜𝑝𝑒𝑟𝑎𝑡𝑖𝑣𝑒 (𝑃𝐴)
𝑃𝑎𝑠𝑠𝑒𝑑 (𝑃𝐹𝑇)
Step 3:
𝑻𝒆𝒔𝒕 𝒎𝒂𝒕𝒄𝒉
Determine the association between
{𝑷𝟏 , 𝑷𝟑 , 𝑷𝟔 , 𝑷𝟖 , 𝑷𝟗 }.
The association between {𝑃1 , 𝑃3 , 𝑃6 , 𝑃8 , 𝑃9 }
{𝐓𝐞𝐬𝐭 𝐦𝐚𝐭𝐜𝐡, 𝐎𝐃𝐈 𝐦𝐚𝐭𝐜𝐡, 𝐓𝟐𝟎 𝐦𝐚𝐭𝐜𝐡}
and
and{Test match, ODI match, T20 match} is determined
with the help of tangent similarity measures for single-valued neutrosophic hypersoft numbers.
Table 14: Association between {𝑃1 , 𝑃 3 , 𝑃 6 , 𝑃 8 , 𝑃 9 } and {𝑇𝑒𝑠𝑡 𝑚𝑎𝑡𝑐ℎ, 𝑂𝐷𝐼 𝑚𝑎𝑡𝑐ℎ, 𝑇20 𝑚𝑎𝑡𝑐ℎ} using tangent
similarity measure for SVNHSS
𝑻𝒆𝒔𝒕 𝒎𝒂𝒕𝒄𝒉
𝑶𝑫𝑰 𝒎𝒂𝒕𝒄𝒉
𝑻𝟐𝟎 𝒎𝒂𝒕𝒄𝒉
𝑃1
0.8728
0.7752
0.8137
0.8513
0.8143
0.8627
𝑃
6
0.8786
0.8519
0.7798
0.8463
0.8402
0.8875
𝑃9
0.8729
0.8997
0.8289
𝑃3
𝑃
8
Step 4: Decision of best option
The best option is decided by choosing the highest value as the highest value represents the best
match type for the players. The table shows that player 𝑃1 should be selected for a test match, player
𝑃3 should be selected for the T20 match, player 𝑃6 should be selected for a test match, player 𝑃8
should be selected for T20 match and player 𝑃9 should be selected for ODI match.
5.
Conclusions
Decision-making is a complex issue due to vague, imprecise and indeterminate environment
specially, when attributes are more than one, and further bifurcated. Neutrosophic softset
environment cannot be used to tackle such type of issues. Therefore, there was a dire need to define
a new approach to solve such type of problems.
In this paper, we have proposed a single-valued Neutrosophic hypersoft set and multi-valued
neutrosophic hypersoft set, then using a single-valued Neutrosophic hypersoft set we present a
tangent similarity measure and some of its properties. We have also presented an application namely
selection of cricket team players for any type of match based on multi-attribute decision making using
tangent similarity measure. The concept of this paper is to make our decision more precise.
Muhammad Saqlain and Sana Moin, Single and Multi-valued Neutrosophic Hypersoft set and Tangent Similarity Measure
of Single valued Neutrosophic Hypersoft Sets
Neutrosophic Sets and Systems, Vol. 32, 2020
327
Acknowledgement
The authors are highly thankful to the Editor-in-chief and the referees for their valuable comments
and suggestions for improving the quality of our paper.
Conflicts of Interest
The authors declare that they have no conflict of interest.
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Received: 15 Oct, 2019. Accepted 17 Mar, 2020
Muhammad Saqlain and Sana Moin, Single and Multi-valued Neutrosophic Hypersoft set and Tangent Similarity Measure
of Single valued Neutrosophic Hypersoft Sets
Neutrosophic Sets and Systems, Vol. 32, 2020
University of New Mexico
On Optimizing Neutrosophic Complex Programming Using
Lexicographic Order
Hamiden Abd El- Wahed Khalifa1, Pavan Kumar2,* and Florentin Smarandache3
1 Operations Research Department, Faculty of Graduate Studies for Statistical Research, Cairo University, Giza, Egypt,
Email: hamidenkhalifa@cu.edu.eg
1 Present Address: Mathematics Department, College of Science and Arts, Al- Badaya, Qassim University, Qassim, Saudi
Arabia, Email: Ha.Ahmed@qu.edu.sa
2 Mathematics Department, Koneru Lakshmaiah Education Foundation, Vaddeswaram, AP, INDIA-522502, Email:
pavankmaths@kluniversity.in
3 Mathematics Department, University of New Mexico, 705 Gurley Ave., Gallup, NM 87301, USA, Email:
fsmarandache@gmail.com
Abstract: Neutrosophic sets are a generalization of the crisp set, fuzzy set, and intuitionistic fuzzy
set for representing the uncertainty, inconsistency, and incomplete knowledge about the real world
problems. This paper aims to characterize the solution of complex programming (CP) problem with
imprecise data instead of its prices information. The neutrosophic complex programming (NCP)
problem is considered by incorporating single valued trapezoidal neutrosophic numbers in all the
parameters of objective function and constraints. The score function corresponding to the
neutrosophic number is used to transform the problem into the corresponding crisp CP. Here,
Lexicographic order is applied for the comparison between any two complex numbers. The
comparison is developed between the real and imaginary parts separately. Through this manner,
the CP problem is divided into two real sub-problems. In the last, a numerical example is solved for
the illustration that shows the applicability of the proposed approach. The advantage of this
approach is more flexible and makes a real-world situation more realistic.
Keywords: Complex programming; Neutrosophic numbers; Score function; Lexicographic order;
Lingo software; Kuhn- Tucker conditions; Neutrosophic optimal solution
1. Introduction
In many earlier works in complex programming, the researchers considered the real part only
of the complex objective function as the objective function. The constraints of the problem are
considered as a cone in complex space
ℂ𝑛 . Since the concept of complex fuzzy numbers was first
introduced [17], many researchers studied the problems of the concept of fuzzy complex numbers.
This branch subject will be widely applied in fuzzy system theory, especially in fuzzy mathematical
programming, and in complex programming too.
Complex programming problem was studied first by Levinson who studied the linear
programming (LP) in complex space [39]. The duality theorem has extended to the quadratic complex
programming by an adaption of the technique, which introduced by Dorn [27, 22]. The linear
fractional programming in complex space has proposed [45]. Linear and nonlinear complex
programming problems were treated by numerous authors [24, 33- 37, 41]. In applications, many
Hamiden Abd El- Wahed Khalifa, Pavan Kumar and Florentin Smarandache, On Optimizing Neutrosophic Complex
Programming Using Lexicographic Order
Neutrosophic Sets and Systems, Vol. 32, 2020
331
practical problems related to complex variables, for instance, electrical engineering, filter theory,
statistical signal processing, etc., were studied.
Some more general minimax fractional programming problem with complex variables was
proposed with the establishment of the necessary and sufficient optimality conditions [36, 37]. A
certain kind of linear programming with fuzzy complex numbers in the objective function coefficients
also considered as complex fuzzy numbers [52]. The hyper complex neutrosophic similarity measure
was proposed by numerous authors [29]. Also, they discussed its application in multicriteria decision
making problem. There was proposed an interval neutrosophic multiple attribute decision-making
method with credibility information [50]. Later, the multiple attribute group decision making based
on interval neutrosophic uncertain linguistic variables was studied [51].
An extended TOPSIS for multi-attribute decision making problems with neutrosophic cubic
information was proposed [42]. A single valued neutrosophic hesitant fuzzy computational
algorithm was developed for multiple objective nonlinear optimization problem [9]. A computational
algorithm was developed for the neutrosophic optimization model with an application to determine
the optimal shale gas water management under uncertainty [10]. The interval complex neutrosophic
set was studied by the formulation and applications in decision-making [11]. A group decisionmaking method was proposed under hesitant interval neutrosophic uncertain linguistic environment
[40]. The neutrosophic complex topological spaces was studied, and introduced the concept of
neutrosophic complex αѱ connectedness in neutrosophic complex topological spaces [30].
A computational algorithm based on the single-valued neutrosophic hesitant fuzzy was developed
for multiple objective nonlinear optimization problems [9]. A neutrosophic optimization model was
formulated and presented a computational algorithm for optimal shale gas water management under
uncertainty [10]. A multiple objective programming approach was proposed to solve integer valued
neutrosophic shortest path problems [32]. Some linguistic approaches were developed to study the
interval complex neutrosophic sets in decision making applications [39].
Neutrosophic sets were studied to search some applications in the area of transportations and
logistics. A multi-objective transportation model was studied under neutrosophic environment [43].
The multi-criteria decision making based on generalized prioritized aggregation operators was
presented under simplified neutrosophic uncertain linguistic environment [46]. Some dynamic
interval valued neutrosophic set were proposed by modeling decision making in dynamic
environments [48]. A hybrid plithogenic decision-making approach was proposed with quality
function deployment for selecting supply chain sustainability metrics [1]. Some applications of
neutrosophic theory were studied to solve transition difficulties of IT-based enterprises [2].
Based on plithogenic sets, a novel model for the evaluation of hospital medical care systems was
presented [3]. Some decision making applications of soft computing and IoT were proposed for a
novel intelligent medical decision support model [4]. A novel neutrosophic approach was applied to
evaluate the green supply chain management practices [5]. Numerous researchers studied the under
type-2 neutrosophic numbers. An application of under type-2 neutrosophic number was presented
for developing supplier selection with group decision making by using TOPSIS [6]. An application
of hybrid neutrosophic multiple criteria group decision making approach for project selection was
presented [7]. The Resource levelling problem was studied in construction projects under
neutrosophic environment [8].
Hamiden Abd El- Wahed Khalifa, Pavan Kumar and Florentin Smarandache, On Optimizing Neutrosophic Complex
Programming Using Lexicographic Order
Neutrosophic Sets and Systems, Vol. 32, 2020
332
The N-valued interval neutrosophic sets with their applications in the field of medical diagnosis was
presented [16]. Based on the pentagonal neutrosophic numbers, the de-neutrosophication technique
was proposed with some applications in determining the minimal spanning tree [18]. The pentagonal
fuzzy numbers were studied with their different representations, properties, ranking, defuzzification.
The concept of pentagonal fuzzy neutrosophic numbers was proposed with some applications in
game and transportation models [19- 20]. Various forms of linear as well as non-linear form of
trapezoidal neutrosophic numbers, de-neutrosophication techniques were studied. Their application
were also presented in time cost optimization technique and sequencing problems [21]. The
parametric divergence measure of neutrosophic sets was studied with its application in decisionmaking situations [25]. A technique for reducing dimensionality of data in decision-making utilizing
neutrosophic soft matrices was proposed [26].
In this paper, we aim to characterize the solution of complex programming (NCP) neutrosophic
numbers. The score function corresponding to the neutrosophic number is used to convert the
problem into the corresponding crisp CP, and hence lexicographic order used for comparing between
any two complex numbers. The comparison developed between the real and imaginary parts
separately. Through this manner, the CP problem is divided into two real sub-problems.
The outlay of the paper is organized as follows: In section 2; some preliminaries are presented. In
section 3, a NCP problem is formulated. Section 4 characterizes a solution to the NCP problem to
obtain neutrosophic optimal solution.
In section 5, two numerical examples are given for
illustration. Finally some concluding remarks are reported in section6.
2. Preliminaries
In order to discuss our problem conveniently, basic concepts and results related to fuzzy
numbers, trapezoidal fuzzy numbers, intuitionistic trapezoidal fuzzy numbers, neutrosophic set, and
complex mathematical programming are recalled.
Definition 1. (Trapezoidal fuzzy numbers, Kaur and Kumar [31]). A fuzzy number
̃ = (r, s, t, u) is a trapezoidal fuzzy numbers where r, s, t, u ∈ ℝ and its membership
A
function is defined as:
x−r
,
s−r
r ≤ x ≤ s,
1, s ≤ x ≤ t,
μà (x) = u−x
, t ≤ x ≤ u,
u−t
{ 0, otherwise,
̃ is said to be an intuitionistic
Definition 2 (Intuitionistic fuzzy set, Atanassov, [12]). A fuzzy set A
̃ IN of a non empty set
fuzzy set A
X if
IN
̃ = {〈x, μ IN , ρ IN 〉 : x ∈ X}, where μ IN , and ρ IN are
A
̃
̃
̃
̃
B
membership and nonmembership functions such that
ρÃIN ≤ 1, for all x ∈ X.
B
A
B
μ ̃ IN , ρÃIN : X → [0, 1] and 0 ≤ μÃIN +
A
Hamiden Abd El- Wahed Khalifa, Pavan Kumar and Florentin Smarandache, On Optimizing Neutrosophic Complex
Programming Using Lexicographic Order
Neutrosophic Sets and Systems, Vol. 32, 2020
333
̃ IN of
Definition 3 (Intuitionistic fuzzy number, Atanassov, [13]). An intuitionistic fuzzy set A
called an Intuitionistic fuzzy number if the following conditions hold:
There exists c ∈ ℝ: μÃIN (c) = 1, and ρÃIN (c) = 0.
1.
ℝ is
μÃIN : ℝ → [0, 1] is continuous function such that
2.
0 ≤ μÃIN + ρB̃IN ≤ 1, for all x ∈ X.
̃ IN are:
3. The membership and non-membership functions of B
0,
h(x),
1,
μB̃IN (x) =
l(x),
{ 0,
Where f, g, h, l: ℝ
−∞< 𝑥 <𝑟
r≤x≤s
x=s
s≤x≤t
t ≤ x < ∞,
0,
f(x),
1,
ρB̃IN (x) =
g(x),
{ 0,
→ [0, 1] , h
−∞< 𝑥 <𝑎
a≤x≤s
x=s
s≤x≤b
b ≤ x < ∞,
g are strictly increasing functions, l and f are strictly
decreasing functions with the conditions0 ≤ f(x) + f(x) ≤ 1, and0 ≤ l(x) + g(x) ≤ 1.
and
Definition 4 (Trapezoidal intuitionistic fuzzy number, Jianqiang and Zhong, [28]).
̃ IN
A trapezoidal intuitionistic fuzzy number is denoted byB
= (r, s, t, u), (a, s, t, b), where a ≤ r ≤
s ≤ t ≤ u ≤ b with membership and nonmembership functions are defined as:
μB̃INT (x) =
ρB̃INT (x) =
x−r
s−r
1,
u−x
,
,
u−t
{0,
s−x
s−a
0,
,
x−t
b−t
{1,
r ≤ 𝑥 < 𝑠,
s ≤ x ≤ t,
t ≤ x ≤ u,
otherwise,
,
a ≤ 𝑥 < 𝑠,
s ≤ x ≤ t,
t ≤ x ≤ b,
otherwise,
̅ N of non-empty set
Definition 5 (Neutrosophic set, Smarandache, [44]). A neutrosophic set B
defined as:
N
X is
̅ = {〈x, I N (x), J N (x), V N (x)〉 : x ∈ X, I N (x), J N (x), V N (x) ∈ ]0 , 1+ [ }, where
B
−
̅
̅
̅
̅
̅
̅
B
B
B
B
B
B
I ̅ N (x), J ̅ N (x), and V ̅ N (x) are truth membership function, an indeterminacy- membership
B
B
B
function, and a falsity- membership function and there is no restriction on the sum of
Hamiden Abd El- Wahed Khalifa, Pavan Kumar and Florentin Smarandache, On Optimizing Neutrosophic Complex
Programming Using Lexicographic Order
Neutrosophic Sets and Systems, Vol. 32, 2020
334
I ̅ N (x), J ̅ N (x), and V ̅ N (x) , so 0− ≤ I ̅ N (x) + J ̅ N (x) + V ̅ N (x) ≤ 3+ , and ]0− , 1+[ is a
nonstandard unit interval.
B
B
B
B
B
B
Definition 6 (Single-valued neutrosophic set, Wang et al., [49]). A Single-valued neutrosophic set
SVN
̅
B
of a non empty set
̅ SVN = {〈x, I N (x), J N (x), V N (x)〉 : x ∈ X}, where
X is defined as: B
̅
̅
̅
B
B
B
I ̅ N (x), J ̅ N (x), and V ̅ N (x) ∈ [0, 1] for each x ∈ X and 0 ≤ IB̅N (x) + JB̅N (x) + V ̅ N (x) ≤ 3.
B
B
B
B
τb̃ , φb̃ , ωb̃ ∈
Definition 7 (Single-valued neutrosophic number, Thamariselvi and Santhi, [47]). Let
[0, 1] and r, s, t, u ∈ ℝ such thatr ≤ s ≤ t ≤ u. Then a single valued trapezoidal neutrosophic
̃N
number, b
= 〈(r, s, t, u): τb̃ , φb̃ , ωb̃ 〉 is a special neutrosophic set onℝ, whose truth-membership,
indeterminacy-membership, and falsity- membership functions are
x−r
μb̃ N (x) =
τb̃N ( ) ,
r≤𝑥<𝑠
s−r
s≤x≤t
τb̃ ,
τb̃N (
{ 0,
u−x
u−t
),
t≤x≤u
s−x+φ ̃ N (x−r)
b
ρb̃ N (x) =
s−r
{
σb̃ N (x) =
φb̃N ,
x−t+φ ̃ N (u−x)
b
u−t
1,
,
s−x+ω ̃ N (x−r)
b
s−r
ωb̃N ,
x−t+ω ̃ N (u−x)
b
otherwise,
,
,
r≤𝑥<𝑠
s≤x≤t
t≤x≤u
otherwise,
r≤𝑥<𝑠
s≤x≤t
, t≤x≤u
{ 1,
otherwise.
Where τb̃ , φb̃ , and ωb̃ denote the maximum truth, minimum-indeterminacy, and minimum falsity
membership degrees, respectively. A single-valued trapezoidal neutrosophic number
N
u−t
̃ = 〈(r, s, t, u): τ N , φ N , ω N 〉 may express in ill- defined quantity about b , which is
b
̃
̃
̃
b
approximately equal to
̃
Definition 8. Let b
N
b
[s, t].
b
̃ N = 〈(r′ , s′ , t′ , u′ ) : τ N , φ N , ω N 〉 be
= 〈(r, s, t, u): τ ̃ N , φ̃ N , ω̃ N 〉 , and d
̃
̃
̃
b
b
b
two single-valued trapezoidal neutrosophic numbers and
̃ N are
and d
1.
2.
d
d
d
N
̃ ,
v ≠ 0. The arithematic operations on b
b̃N ⊕ d̃N = 〈(r + r ′ , s + s ′ , t + t ′ , u + u′ ); τb̃N ∧ τd̃N , φb̃N ∨ φd̃N , ωb̃N ∨ ωd̃N 〉 ,
b̃N ⊖ d̃N = 〈(r − u′ , s − t ′ , t − s ′ , u′ − r); τb̃N ∧ τd̃N , φb̃N ∨ φd̃N , ωb̃N ∨ ωd̃N 〉,
Hamiden Abd El- Wahed Khalifa, Pavan Kumar and Florentin Smarandache, On Optimizing Neutrosophic Complex
Programming Using Lexicographic Order
Neutrosophic Sets and Systems, Vol. 32, 2020
3.
4.
5.
b̃N ⊗ d̃N =
335
〈(rr ′ , ss ′ , tt ′ , uu′ ); τb̃N ∧ τd̃N , φb̃N ∨ φd̃N , ωb̃N ∨ ωd̃N 〉, u, u′ > 0
〈(ru′ , st ′ , st ′ , ru′ ); τb̃N ∧ τd̃N , φb̃N ∨ φd̃N , ωb̃N ∨ ωd̃N 〉, u < 0, u′ > 0
{〈(uu′ , ss ′ , tt ′ , rr ′ ); τb̃N ∧ τd̃N , φb̃N ∨ φd̃N , ωb̃N ∨ ωd̃N 〉, u < 0, u′ < 0,
〈(r/u′ , s/t ′ , t/s ′ , u/r ′ ); τb̃N ∧ τd̃N , φb̃N ∨ φd̃N , ωb̃N ∨ ωd̃N 〉, u, u′ > 0,
b̃N ⊘ d̃N = 〈(u/u′ , t/t ′ , s/s ′ , r/r ′ ); τ ∧ τ , φ ∨ φ , ω ∨ ω 〉, u < 0, u′ > 0,
̃N
̃N
̃N
̃N
̃N
̃N
d
b
d
b
d
b
{〈(u/r ′ , t/s ′ , s/t ′ , r/u′ ); τb̃N ∧ τd̃N , φb̃N ∨ φd̃N , ωb̃N ∨ ωd̃N 〉, u < 0, u′ < 0,
kr, ks, kt, k ; τ τd N , φ τd N , ω τd N , k 0,
kd N f x
ku, kt, ks, k r ; τ τdN , φ τdN , ωτdN , k 0,
Definition 9 (Score function of single-valued trapezoidal neutrosophic number, Thamaraiselvi and
Santhi [47]). A two single-valued trapezoidal neutrosophic numbers
b, and
d can be compared
based on the score function as
Score function
1
SC b N r s t u [μ b N (1 ρ b N x 1 σ b N x .
16
Definition 10. (Thamaraiselvi and Santhi, [47]). The order relations between b N and d N based on
SC b N are defined as:
1.
If SC b N SC d N then b N
2.
If SC b N SC d N then
3.
If SC b N SC d N then b N
dN
b N d N , and
dN
3. Problem definition and solution concepts
Consider the following single -valued trapezoidal neutrosophic (NCP) problem
min FN x v N x i w N x
(NCP)
Subject to
(1)
where
n
n
n
n
j1
j1
j1
j1
v N x c j N x j , w N x d j N x j , p r N x x jT a rjN x j , q r N x x jT erj N x j
convex functions on X
N
, cj
N
, d j N , a rj N , erj N lr N l1N , l2 N , , lm N
T
are
, hr N h1N , h2 N , , hm N
T
are single-valued trapezoidal neutrosophic numbers.
Hamiden Abd El- Wahed Khalifa, Pavan Kumar and Florentin Smarandache, On Optimizing Neutrosophic Complex
Programming Using Lexicographic Order
Neutrosophic Sets and Systems, Vol. 32, 2020
336
Definition 11. Lexicographic order of two complex numbers
defined as
z1 a ib, and z2 c i d
is
z 1 z 2 a c and b d.
Definition 12. A neutrosophic feasible point x is called single-valued trapezoidal neutrosophic
optimal solution to NCP problem if:
v N x v N x ,and w N x w N x for each x X N .
According to the score function in Definition 9, the NCP problem is converted into the following crisp
CP problem as
Subject to
4.
(2)
Characterization of neutrosophic optimal solution for NCP problem
To characterize the neutrosophic optimal solution of NCP problem, let us divide the CP problem
into the following two subprobems
Pv
Min
v x
Subject to
(3)
M i n w x
Pw
Subject to
(4)
x X x R n : f r x pr x i qr x lr i hr , r 1, 2,, m .
vx
Definition 13. x X is said to be an optimal solution for CP problem if and only if v x
w x for each
and w x
Let us denote
x X.
S v and Sw be the set of solution for Pv and Pw respectively, i.e.,
(5)
Sw x * X : v x * v x ; for all x X .
(6)
Hamiden Abd El- Wahed Khalifa, Pavan Kumar and Florentin Smarandache, On Optimizing Neutrosophic Complex
Programming Using Lexicographic Order
Neutrosophic Sets and Systems, Vol. 32, 2020
Lemma 1. For
337
Sv Sw , the solution of
the CP problem is embedded into
Sv Sw .
Proof. Assume that xˆ be a solution of CP problem this leads to v xˆ v x ; x X i. e.,
xˆ Sv ) . Similarly, w xˆ w x ; x X
(i. e.,
xˆ Sw ) Then, xˆ Sv Sw .
Lemma 2. If S v and Sw are open, Sv Sw , and v, w are strictly convex functions on X then
x Sv is a solution of a conjugate function F x v x i w x .
Proof. Since
x Sv ,
then v x v x ; x X . Also,
v x v x* ; x* Sv X
But
(7)
x* S w which means that w x* v x ; x Sv X and –i w x * i w x
i. e.,
–i w x i w x *
(8)
From (7) and (8), we get
v x – i w x v x* i w x * ; x * Sw , i. e.,
x Sv is a solution of a conjugate function F x v x i w x . Now we will prove that there
is no
xˆ X and xˆ Sv such that:
F xˆ v xˆ i w xˆ F x v x i w x .
(9)
There are two cases:
Case 1: Assume that x X
´
xˆ Sv , x Sw and v xˆ i w xˆ v x i w x i.e.,
w x w xˆ . From the strictly convexity of the function w x and
Sw is open, then
w τ xˆ 1 τ x w xˆ 1 τ w x , 0 1 , this leads to
w τ xˆ 1 τ x w xˆ 1 τ w xˆ i. e.,
For certain
τ such that τ xˆ 1 τ x Sw ,we have
Hamiden Abd El- Wahed Khalifa, Pavan Kumar and Florentin Smarandache, On Optimizing Neutrosophic Complex
Programming Using Lexicographic Order
Neutrosophic Sets and Systems, Vol. 32, 2020
w τ xˆ 1 τ x w xˆ
338
.Which
contradicts
to
xˆ Sw
i.e.,
there
is
no
xˆ X, xˆ Sv , xˆ Sw such that:
F xˆ v xˆ i w xˆ F x v x i w x .
Case 2: Assume that
xˆ X , xˆ S v , xˆ S w and v xˆ i w xˆ v x i w x i.e.,
v xˆ v x , and w x w xˆ . Since the function v x is strictly convex and
S v is open,
then
v τ x 1 τ xˆ v x 1 τ v xˆ , 0 1 This leads to
v τ x 1 τ xˆ v x 1 τ v xˆ , i.e., for certain , we have
x 1 xˆ S v , such that x 1 xˆ Sv
,we have
v τ x 1 τ xˆ v x . Contradicts that x S v .
Thus, there is no
xˆ X such that:
v xˆ i w xˆ v x i w x
5.
Numerical examples
Example1. (Illustration of Lemma1)
Consider the following complex problem
min cosx i sinx
Subject to
(10)
x X x R : 0 x
Problem (10) is divided into the following two problems as:
Pv
mincosx
Subject to
Subject to
(11)
(12)
Hamiden Abd El- Wahed Khalifa, Pavan Kumar and Florentin Smarandache, On Optimizing Neutrosophic Complex
Programming Using Lexicographic Order
Neutrosophic Sets and Systems, Vol. 32, 2020
339
x Sv
The optimal solution of problem (11) is
optimal
solution of
problem (10) is
problem (12) is x 0, ,
i. e.,
, i. e.,
Sv . Also,
Sw 0, Thus, the optimal
the
solution of
x Sv Sw .
Example2. (Illustration of Lemma2)
Consider the following NCP problem:
Min F N x c1N x1 c 2 N x 2 i d1N x1 d 2 N x 2
Subject to
(13)
Where,
Using the score function of the single- valued trapezoidal neutrosophic number introduced in
definition9, problem (13) becomes:
Min F x 3x1 x 2 i 5x1 11x 2
Subject to
(14)
x12 x 22 i x1 x 2 5 i.
According to the Lexicographic order, the problem is divided into the following two subproblems as:
Pv
Min v x 3x1 x 2
Subject to
(15)
Hamiden Abd El- Wahed Khalifa, Pavan Kumar and Florentin Smarandache, On Optimizing Neutrosophic Complex
Programming Using Lexicographic Order
Neutrosophic Sets and Systems, Vol. 32, 2020
340
x12 x 22 5, x1 x 2 1, and
Pw
Min w x 5x1 11x 2
Subject to
(16)
x12 x 22 5, x1 x 2 1.
By applying the Kuhn- Tucker conditions [14, 22], the optimal solutions of problems (15), (16) and
problem (9) are illustrated in the following tables.
𝑆𝑣
Table 1. The set of solution of (Pv )
Optimum value
{(−2, −1)}
𝑆𝑤
{(−2, 1)}
Pv = −7
𝑁
𝑃̃𝑣 = 〈−34, −23, −17, −10; 0.3, 0.6, .06〉
Table 2. The set of solution of (Pw )
Optimum value
Pw = −21
𝑁
𝑃̃𝑤 = 〈−60, −44, −34, −24; 0.6, 0.3, 0.4〉
Therefore, Sv ∩ Sw = ∅ and the solution of problem 𝑆𝑣 is not a solution of the conjugate function
v(x) − i w(x) , because of v(x), and w(x) are not strictly convex functions.
6.
Concluding Remarks
In this paper, the solution of complex programming (NCP) with single valued trapezoidal
neutrosophic numbers in all the parameters of objective function and constraints has characterized.
Based on the score function definition, the NCP has converted into the corresponding crisp CP
problem and hence Lexicographic order has used for comparing between any two complex numbers.
The comparison has developed between the real and imaginary parts separately. Through this
manner, the CP problem has divided into two real sub-problems. The main contribution of this
approach is more flexible and makes a situation realistic to real world application. The obtained
results are more significant to enhance the applicability of single-valued trapezoidal neutrosophic
number in various new fields of decision-making situations. The future research scope is to apply the
proposed approach to more complex and new applications. Another possibility is to work on the
interval type of complex neutrosophic sets for the applications in forecasting filed.
Acknowledgments
Hamiden Abd El- Wahed Khalifa, Pavan Kumar and Florentin Smarandache, On Optimizing Neutrosophic Complex
Programming Using Lexicographic Order
Neutrosophic Sets and Systems, Vol. 32, 2020
341
The authors gratefully thanks the anonymous referees for their valuable suggestions and helpful
comments, which reduced the length of the paper and led to an improved version of paper.
Conflicts and Interest
The authors declare no conflict of interest.
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Received: Sep 10, 2019. Accepted: Mar 12, 2020
Hamiden Abd El- Wahed Khalifa, Pavan Kumar and Florentin Smarandache, On Optimizing Neutrosophic Complex
Programming Using Lexicographic Order
Neutrosophic Sets and Systems, Vol. 32, 2020
University of New Mexico
A New Model for the Selection of Information Technology Project
in a Neutrosophic Environment
Maikel Leyva-Vázquez1, Miguel A. Quiroz-Martínez2, Yoenia Portilla-Castell3, Jesús R. HechavarríaHernández4, and Erick González-Caballero5
Universidad Politécnica Salesiana, Carrera de Ingeniería en Sistema/Instituto Superior Bolivariano de Tecnología, Guayaquil, Guayas, Ecuador,
E-mail: mleyvaz@ups.edu.ec
2 Universidad Politécnica Salesiana, Carrera de Ingeniería en Sistema/Instituto Superior Bolivariano de Tecnología, Guayaquil, Guayas, Ecuador,
E-mail: mquiroz@ups.edu.ec
3 Higher Technological Institute of Administrative and Commercial Professional Training, Ecuador, E-mail: yoenia.portilla@formacion.edu.ec
4 Universidad Católica de Santiago de Guayaquil, Guayaquil, Guayas, Ecuador, E-mail: jesusr2h@ups.edu.ec
5 Universidad Tecnológica de La Habana, La Habana, Cuba. E-mail: erickgc@yandex.com
1
Abstract. Usually, companies confront the difficulty to make the best decision about the way to invest
their recourses in different project alternatives. The company acquires competitive advantages when
their software development projects are well evaluated and correctly selected. Selecting projects in the
Information Technology field presents challenges in many senses; e.g., the difficulty that entails assessing intangible benefits, projects are interdependent and companies impose self-constraints. In addition, the framework to make the decision is generally uncertain with many unknown factors. This paper aims to propose a model that integrates methods, techniques and tools such as the Balanced Scorecard Model, neutrosophic Analytic Hierarchy Process and zero-one linear programming. The proposed
model is designed to select the best portfolio of Information Technology projects, it overcomes the obstacles mentioned above and can be coherently incorporated in the strategic plan process of any company. In addition, it eases the course of experts’ decision making, because it is based on Neutrosophy
and hence incorporates the indeterminacy term.
Keywords: Information Technology Project, Balanced Scorecard Model, Neutrosophic Analytic Hierarchy
Process, zero-one linear programming.
1.
Introduction
According to the guide to the project management body of knowledge (PMBOK) [1], “project
management is the application of knowledge, skills, tools and techniques to projects activities to meet
project requirements”. The guide to the PMBOK also makes reference to the multiple project management. Some authors acknowledge that sometimes exist missing or vaguely defined processes in
any commercial corporations; some of them are the coordination in a multi-project environment and
the strategic processes [2].
Later on, Project Management Institute published in detail additional standards for the Programs
and Portfolio management [1, 3, 4]. A Program is defined as a related group of projects, which are coordinately managed to obtain benefits and controls, under the constraint that these benefits and controls would not be available, in the case they were managed individually.
On the other hand, a Project Portfolio is a group of projects performed during a certain time span
and which share common resources. Some kinds of relationships that can exist among the projects are
complementariness, incompatibility and synergies, which are derived from the division of costs and
benefits obtained from the performance of more than one project simultaneously [5]. See schematized
Maikel Leyva-Vázquez, Miguel A. Quiroz-Martínez, Jesús R. Hechavarría-Hernández, and Erick González-Caballero. A new
model for the selection of information technology project in a neutrosophic environment
Neutrosophic Sets and Systems, Vol. 32, 2020
345
of an example in Fig. 1.
The foundations of project portfolio management have been developing since the seventies. Its
roots can be found in the theory of Harry Markowitz, which deserved the Nobel Prize in Economic
Sciences. He shared this award with Merton H. Miller and William F. Sharpe, for their work in the
field of financial economics theory. Its basic contribution is the "portfolio choice theory". He proposed
a model for the choice of a portfolio of securities in conditions of uncertainty in which it reduced it to a
two-dimensional dilemma: the expected income and the variance.
Nevertheless, some authors point out that significant differences exist between the theory of project portfolio management and Markowitz’s theory [6, 7].
Four of the six responsibilities in project portfolios management, which were emphasized by Kendall and Rollins, are the following, [8]:
To determine a suitable combination of projects such that the company’s goal could be
achieved.
To attain an adequate balance in the portfolio, where the combination of projects has an
adequate balance between risks and rewards, research and development and so on.
To assess the possible existence of new opportunities for the present portfolio, taking into
account the company’s capacity for execution.
To provide information and recommendations for decision makers at every level.
Figure 1: Scheme of a possible Portfolio-Program-Project relationship
The project portfolio management is inherently strategic, it is more related to efficacy (to perform
the adequate project) than the efficiency (to execute the project correctly). It should avail a framework
of work for assessing decisions about to invest, maintain and remove [9].
According to the reports of A. T. Kearney, which is an American global management consulting
firm that focuses on strategic and operational CEO-agenda issues, the plan in investment projects have
barely changed in enterprises since the 1920s, see [10]. The forthcoming necessities of the company are
not forecasted, instead, decision makers assign the budget that they consider sufficient to carry out
each project individually, no doubt this is a drawback, see [11, 12]. The second drawback is when decision makers do not identify potential synergies that could exist among the projects and therefore,
unexpected increases in project costs could arise.
Kaplan and Norton introduced a framework of work to measure the effectiveness of a company;
they called it Balanced Scorecard (BSC). This model integrates four perspectives, namely, financial,
customer, business process and learning and growth [13]. Additionally, this is a way to display the
strategies inside the company. Particularly, BSC is useful to select measures that guarantee the balance
in project portfolios of Information Technologies [6].
The relationship existing between strategy and Project Management is a subject that has considerMaikel Leyva-Vázquez, Miguel A. Quiroz-Martínez, Jesús R. Hechavarría-Hernández, and Erick González-Caballero. A new
model for the selection of information technology project in a neutrosophic environment
Neutrosophic Sets and Systems, Vol. 32, 2020
346
ably evolved during the years pass. One example is project portfolio management, consisting of a
close relationship that connects strategy with Project Management by selecting and prioritizing those
projects which satisfy strategic objectives. Both selection and prioritization are based on criteria that
could perfectly coincide with indicators proper of the Balanced Scorecard model designed for this
company [5, 8].
The economic importance of Information Technology projects is evident. Frequently, Information
Technology projects represent a significant portion of the set of projects inside a company [2]. In the
present-day, the hardware is considered as a commodity, whereas software provides the major part of
a computational system [14].
Information Technology (IT) management is a subject that has quickly grown since the very near
past. Pells in [15] presented the factors which have repercussions on the growth of the IT projects
management, they are the following:
The massive investment in IT all over the world.
The natural orientation of the project management toward the IT industry.
The fast change of technologies.
Failures in IT projects.
The arrival of the Information Era.
IT embraces every industry, company and project.
When these factors are taken into consideration as a whole, they conduce to other important trends
and developments in the fields of project management, project portfolio management and complex
project management.
In this present research, the authors used a balanced scorecard model as a tool to determine the
coherence of the project with company’s strategy, particularly considering their perspectives. Moreover, the criteria to determine the project feasibility have been included. The proposed model is based
on the balanced scorecard model, neutrosophic analytic hierarchy process and zero-one linear programming.
The analytic hierarchy process (AHP) was created by Aczél et al. [16]. It is a well-known multicriteria decision-making technique founded on mathematics and cognitive psychology. This technique has been widely applied to make decisions in complex situations.
Buckley in [17, 18] designed a fuzzy hierarchical analysis, where the crisp decision ratio of the classical AHP is substituted by a fuzzy ratio represented by a trapezoidal membership function. This approach introduces uncertainty and imprecision from the fuzzy viewpoint.
Abdel-Basset et al. in [19] designed a neutrosophic AHP-SWOT model, based on neutrosophic sets,
where a neutrosophic set is a part of neutrosophy that studies the origin, nature and scope of neutralities, as well as their interactions with different ideational spectra [20]. The neutrosophy included for
the first time the notion of indeterminacy in the fuzzy set theory, which is also part of real-world situations. Neutrosophic AHP permits that experts could express their criteria more realistically, by indicating the truthfulness, falseness and indeterminacy of the decision ratio.
This paper aims to present a new mathematical model to select the best information technology
projects. In the first step, a balanced scorecard model is applied to establish the criteria selection. The
second stage consists in applying a neutrosophic AHP technique, where crisp weights of project importance are output. During this step neutrosophic triangular numbers and the operations among
them are used for calculating. These weights of each project's importance are inputs to the third stage.
The third stage consists of a zero-one linear programming model for selecting the best projects that
satisfy the feasible constraints.
Hybridizing different Multicriteria Decision-Making (MCDM) methods for creating new project
selection models have become recurrent in the literature that is why the model proposed in this paper
can also be of interest to researches and decision makers. In [21] the state of the art in project selection
problem is studied for 60 papers published in the period from 1980 to 2017 and it is concluded that the
most popular techniques to perform hybridizations are the Order of Preference by Similarity to Ideal
Solution (TOPSIS) and the analytic hierarchy process / analytic network process followed by the VIMaikel Leyva-Vázquez, Miguel A. Quiroz-Martínez, Jesús R. Hechavarría-Hernández, and Erick González-Caballero. A new
model for the selection of information technology project in a neutrosophic environment
Neutrosophic Sets and Systems, Vol. 32, 2020
347
KOR method. For example, in [22] the AHP technique is hybridized with PROMETHEE with the goal
of urban renewal project selection. Papers in [23-30] introduce the hybridization of methods and techniques of MCDM within the framework of neutrosophy, obtaining more complete models than those
based on fuzzy logic theory because uncertainty in decision-making also incorporates indeterminacy.
In addition that the hybridization of MCDM methods seems to be an inexhaustible source of creating new models for project selection, the model proposed in this paper differs from the rest of the similar ones. This is specifically designed to select information technology projects, which is why the Balanced Scorecard is included to guide the managers on which aspects to test in decision-making. BSC is
so far infrequent in the published papers on hybridization. The AHP technique avoids bias in decision
making due to the use of the consistency index. zero-one linear programming is the tool used to make
the final decision, while neutrosophy is used to model the indeterminacy that decision makers might
have. Another advantage of the model is that it allows decision makers to rate based on linguistic
terms. To the best of the authors’ knowledge, this seems to be the first model for selecting information
technology projects by using the hybridization of Balanced Scorecard, neutrosophic AHP and zeroone linear programming, where a scale of linguistic terms serves to evaluate.
This paper is distributed as follows; section 2 contains the main theories used as the basis of this
document. The proposed mathematical model is developed in section 3. In section 4 the application of
the model is illustrated with an example. Section 5 states the conclusions.
2 Preliminaries
This section exposes the theories used to design the model. It is started with part of the theory of
the project portfolio. Further, the authors summarize the AHP technique and neutrosophic set theory.
Finally, the main concepts of zero-one linear programming are written.
2.1 Approaches to Portfolio IT Project
An important part of IT projects is related to software development. The difference of software development projects with respect to other engineerings, e.g., electronic engineering, is that the former
one imposes additional challenges to project management, mainly due to the particular characteristics
of software [30] and these characteristics are the following:
The software is an intangible product.
The standard software processes do not exist.
The uniqueness of the large scale projects of software developments.
When a computer product will be developed, or an information system, or any other modifications,
in that case, the elaboration of an innovative project is needed for planning and executing the introduction of this product inside the company. Technological innovation projects are elaborated to introduce scientific results obtained from scientific creation. This is related to applied researches, technological developments; and the commercialization of novel technologies, products, systems and processes. This is the final stage in the cycle of science-technology-production [31].
Literature had paid attention to project selection, see [2, 21-34], especially for research and development projects (R&D), see [35, 36]. One main difference exists between IT and (R&D) projects, it is
that projects interdependence in the former has elevated importance [1, 3, 4]. Moreover, two IT projects can share identical code sections or hardware.
The project selection process in general, including IT projects, is a very complex process that is influenced by several factors. One key aspect of IT control is the prioritization of investments. Projects
have to be assessed as an investment viewpoint, by having as a goal to analyze the project capacity for
maximizing the company’s value [32].
One of the criteria to approve the start of one project would be to determine its possibility of success and impact; evidently, most companies cannot start simultaneously every project. The project assessment consists of gathering pertinent information in the end to facilitate the project selection process and to determine the value of every project [8, 37]. The closing phases assessment allows us to
build a base of knowledge that shall be communicated during the organization’s continuous learning
Maikel Leyva-Vázquez, Miguel A. Quiroz-Martínez, Jesús R. Hechavarría-Hernández, and Erick González-Caballero. A new
model for the selection of information technology project in a neutrosophic environment
Neutrosophic Sets and Systems, Vol. 32, 2020
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[6].
One of the goals in portfolio management is to maximize the portfolio value, by carefully assessing
those projects and programs which could be included in the portfolio and also to opportunely exclude
those of them which do not fulfill the portfolio strategic objectives [38]. IT portfolio management is basically a selection process to locate resources to develop/maintain those projects that better satisfy strategic objectives [39].
There exist a number of difficulties in evaluating projects. Rebaza points out, referring to computer
projects that in most cases the projects are evaluated according to cost-benefit criteria [40]. The task of
evaluating projects is not simple and involves many difficulties, some of them are methodological.
These difficulties include the following:
Lack of information availability,
Lack of qualified staff for evaluation,
Lack of evaluation processes in the company.
Use of limited criteria for evaluation.
Project selection methods are used to determine which project the organization will select. Generally, these methods are divided into four major categories according to Bonham, see [5]:
A. Mathematical programming—Integer programming, linear programming, nonlinear programming, goal programming and dynamic programming
B. Economic models—IRR, NPV, PB period, ROI, cost-benefit analysis, option pricing theory, the
average rate of return and profitability index;
C. Decision analysis—Multiattribute utility theory, decision trees, risk analysis, analytic hierarchy
process, unweighted 0–1 factor model, unweighted (1 – n) factor scoring model and weighted
factor scoring model;
D. Interactive comparative models—Delphi, Q-sort, behavioral decision aids and decentralized
hierarchical modeling.
A relatively recent trend in the information technology area is value-based software engineering
(VBSE) [41]. VBSE is considered as part of the life cycle of software engineering management activities
such as the development of the Business Case, project evaluation, project planning etc, which have so
far been considered peripheral. The VBSE aims to guide proposals and solutions based on the maximization of the value provided.
Any decision to construct (or re-engineering) a software system should be guided by its “value” ([42]).
Thus, a system brings more “value” to their users if it provides greater benefits, either in terms of return on investment (ROI), social benefits, reduced management costs, strategic advantages, or any
other aspect. As can be assumed, the quantification of all these types of benefits is complex [42].
Sometimes intangible benefits, such as learning and opportunity for growth, are the fundamental
sources of value. As a result, other indicators to be taken into consideration for investment have
emerged. An example of this is the social return on investment [42], which seeks to capture social values by translating social goals into financial and non-financial measures. Kendal and Rolling ([8])
claim that the more projects that are initiated with insufficient resources, the fewer projects that are
completed and the longer each project takes to complete. Surveys indicate that companies with the
highest number of project selection criteria are associated with better performance ([6]).
Bonham [5] proposes a model for project selection based on three phases, viz., strategic analysis,
individual project analysis (maximization) and portfolio selection (balance). He also noted the importance of analyzing the interdependence between projects.
Bergman and Mark ([2]) present a way to issue the problem of project selection using the requirement analysis to better inform each project option. As a project option develops through the selection
process, its specification of requirements is detailed and refined. Project requirements provide a better
technical, economic and organizational understanding of each project.
Value Measuring Methodology (VMM) ([4]) is a methodology for evaluating and selecting initiatives that offer the greatest benefits. Moreover, Rapid Economic Justification ([39]) is a framework deMaikel Leyva-Vázquez, Miguel A. Quiroz-Martínez, Jesús R. Hechavarría-Hernández, and Erick González-Caballero. A new
model for the selection of information technology project in a neutrosophic environment
349
Neutrosophic Sets and Systems, Vol. 32, 2020
veloped by Microsoft to decide the value of investments in information technology.
Wibowo notes that existing approaches present the following limitations, see [43]:
The inability to deal with the subjectivity and the imprecision of the evaluation processes
and the selection of information systems projects.
Failure to properly manage the multidimensional nature of the problem.
It is very cognitively demanding for the decision-maker.
The model proposed in this paper overcomes all the difficulties specified above, as can be further
seen.
2.2 AHP Technique
AHP consists first in designing a hierarchical structure, where the upper elements are more generic
than those situated below. The layer on top contains a single leaf, representing the decision goal, the
second layer that connected with the goal emerges as a set of leaves representing the criteria and the
followed third layer is containing subcriteria and so on. The last bottom layer of this tree contains
leaves representing the alternatives. See, Fig. 2.
Consequently, square matrices represent the expert or experts’ decision, containing the pair-wise
comparison of criteria, subcriteria or alternatives assessment. Aczél et al. in [16] proposed the scale
that they considered is the better to evaluate decisions, as can be seen in Tab. 1.
Goal
Criterion 1
Criterion 2
Subcriterion 1
Subcriterion 2
Alternative 1
Alternative 2
Criterion k
Subcriterion m
Alternative n
Figure 2: Scheme of a generic tree representing an Analytic Hierarchy Process
Maikel Leyva-Vázquez, Miguel A. Quiroz-Martínez, Jesús R. Hechavarría-Hernández, and Erick González-Caballero. A new
model for the selection of information technology project in a neutrosophic environment
Neutrosophic Sets and Systems, Vol. 32, 2020
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Table 1: Intensity of importance according to the classical AHP
The intensity of importance on an absolute scale
1
Definition
Explanation
Equal importance
3
Moderate importance of one over another
5
Essential or strong importance
7
Very strong importance
9
Extreme importance
2, 4, 6, 8
Two
activities
contribute
equally to the objective
Experience and judgment
moderately favor one activity
over another
Experience and judgment
strongly favor one activity
over another
Activity is strongly favored
and its dominance demonstrated in practice
The evidence favoring one activity over another is of the
highest possible order of affirmation
When comprise is needed
Intermediate values between the two
adjacent judgments.
If activity i has one of the above numbers assigned to it when compared
with activity j, i.e., number 𝑎 ∈ {1,2, ⋯ , 9}, then j has the reciprocal value when compared with i, i.e., value 1/𝑎.
Reciprocals
On the other hand, Aczél et al. established that the Consistency Index (CI) should depend on max,
the maximum eigenvalue of the matrix. They defined the equation CI =
λmax −n
n−1
, where n is the order of
the matrix. Additionally, they defined the Consistency Ratio (CR) with equation CR = CI/RI, where the
Random Index or RI is given in Tab. 2.
Table 2: RI associated with every order.
Order (n)
RI
1
0
2
0
3
0.52
4
0.89
5
1.11
6
1.25
7
1.35
8
1.40
9
1.45
10
1.49
Each RI value is an average random consistency index computed for n 10 for very large samples.
Randomly generated reciprocal matrices were created using the scale 1/9, 1/8, …,1/2, …, 8, 9 and the
average of their eigenvalues were calculated. This average is used to form the RI.
If CR10% it is considered that experts’ evaluation is consistent enough and hence, proceed to use
AHP.
AHP aims to score criteria, subcriteria and alternatives and to rank every alternative according to
these scores.
AHP can also be used in group assessment. In such a case, the final value is calculated by the
weighted geometric mean, which satisfies the inverse requirements [44], see Eq. 1 and 2. The weights
are utilized to measure the importance of each expert’s criteria, where some factors are taken into consideration like expert’s authority, knowledge, effort, among others
w
x̅ = (∏ni=1 xi i )
1⁄
∑n
i=1 wi
(1)
If ∑ni=1 wi = 1, i.e., when expert’s weights sum one, Eq. 1 transforms in Eq. 2,
n
wi
x̅ = ∏ xi
i=1
(2)
2.3 Neutrosophic sets
Maikel Leyva-Vázquez, Miguel A. Quiroz-Martínez, Jesús R. Hechavarría-Hernández, and Erick González-Caballero. A new
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Neutrosophic sets extend classical sets, fuzzy sets and intuitionistic fuzzy sets.. Fuzzy set models
are based on the degree of membership of an element to a set. It has been applied in many areas of
knowledge, including decision making.
Fuzzy set theory was introduced by Lotfi A. Zadeh for the first time at 1965. A fuzzy set consists of
the following manners [45, 46]:
Given a Universe of Discourse U containing a set of objects and A being its subset, a membership
function is a function TA : U[0, 1], defined for every 𝑥U, where TA (𝑥) is the degree of truth for which
𝑥 belongs to A.
The intuitionistic fuzzy set theory was introduced by Krassimir T. Atanassov at 1986. An intuitionistic fuzzy set is defined by two membership functions, TA meaning that 𝑥 belongs to U and FA meaning that 𝑥 does not belong to A. They must satisfy the restriction TA (𝑥) + FA (𝑥) 1, [47].
On the other hand, Neutrosophic set includes a third membership function I A, meaning indeterminacy. Thus, a neutrosophic set is a triple of membership functions, T A, IA and FA with no restriction.
The inclusion of indeterminacy is a contribution made by Florentin Smarandache [20], which agreed
that neutrality and ignorance are also part of the uncertainty. Moreover, he accepts the possibility that
truthfulness, indeterminacy and falseness can be simultaneously maximal. Also, he uses the idea of
non-standard analysis of Abraham Robinson and he utilizes hyperreal numbers in calculations.
Let us define formally the concept of neutrosophic set.
Definition 2.3.1([20]): The neutrosophic set N is characterized by three membership functions,
which are the truth-membership function TA, indeterminacy-membership function IA and falsitymembership function FA, where U is the Universe of Discourse and xU ,
TA (𝑥), IA (𝑥), FA (𝑥) ] −0, 1+ [ and −0 𝑖𝑛𝑓 TA (𝑥) + 𝑖𝑛𝑓 IA (𝑥) + 𝑖𝑛𝑓 FA (𝑥) 𝑠𝑢𝑝 TA (𝑥) + 𝑠𝑢𝑝 IA (𝑥) +
𝑠𝑢𝑝 FA (𝑥)3+ .
See that according to the definition, TA (𝑥), IA (𝑥) and FA (𝑥) are real standard or non-standard subsets of ]-0, 1+[ and hence, TA (𝑥), IA (𝑥) and FA (𝑥) can be subintervals of [0, 1].-0 and 1+ belong to the set
of hyperreal numbers.
Definition 2.3.2([20]): The Single Valued Neutrosophic Set (SVN) N over U is A = {<
𝑥, TA (𝑥), IA (𝑥), FA (𝑥) > : 𝑥U}, where TA:U[0, 1], IA:U[0, 1] and FA:U[0, 1]. 0 TA (𝑥) + IA (𝑥) +
FA (𝑥) 3.
The Single Valued Neutrosophic (SVN) number is symbolized by
N = (t, i, f ), such that 0 t, i, f 1 and 0 t + i + f 3.
Definition 3.2.3 ([19, 48]): The single valued triangular neutrosophic number,
ã = 〈(a1 , a 2 . a 3 ); αã , βã , γã 〉, is a neutrosophic set on ℝ, whose truth, indeterminacy and falsity
membership functions are defined as follows:
αã( 𝑥−a1 ),
a2 −a1
αã,
Tã (𝑥) = α a3−𝑥
ã(
),
{ 0,
a3 −a2
a1 ≤𝑥≤a2
𝑥=a2
a2 <𝑥≤a3
(3)
otherwise
(a 2 − 𝑥 + βã (𝑥 − a1 ))
,
a1 ≤ 𝑥 ≤ a 2
a 2 − a1
βã ,
𝑥 = a2
Iã (𝑥) =
(𝑥 − a 2 + βã (a 3 − 𝑥))
, a2 < 𝑥 ≤ a3
a3 − a2
{ 1,
otherwise
(4)
Maikel Leyva-Vázquez, Miguel A. Quiroz-Martínez, Jesús R. Hechavarría-Hernández, and Erick González-Caballero. A new
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(a 2 − 𝑥 + γã (𝑥 − a1 ))
,
a1 ≤ 𝑥 ≤ a 2
a 2 − a1
γã ,
𝑥 = a2
Fã (𝑥) =
(𝑥 − a 2 + γã (a 3 − 𝑥))
, a2 < 𝑥 ≤ a3
a3 − a2
{ 1,
otherwise
352
(5)
Where αã , βã , γã ∈ [0, 1], a1 , a 2 , a 3 ∈ ℝ and a1 ≤ a 2 ≤ a 3.
Definition 2.3.4 ([19, 48]): Given ã = 〈(a1 , a 2 , a 3 ); αã , βã , γã 〉 and b̃ = 〈(b1 , b2 , b3 ); αb̃ , βb̃ , γb̃ 〉 two single-valued triangular neutrosophic numbers and any non-null number in the real line. Then, the following operations are defined:
Addition: ã + b̃ = 〈(a1 + b1 , a 2 + b2 , a 3 + b3 ); αã ∧ αb̃ , βã ∨ βb̃ , γã ∨ γb̃ 〉
Subtraction: ã − b̃ = 〈(a1 − b3 , a 2 − b2 , a 3 − b1 ); αã ∧ αb̃ , βã ∨ βb̃ , γã ∨ γb̃ 〉
Inversion: ã−1 = 〈(a 3 −1 , a 2 −1 , a1 −1 ); αã , βã , γã 〉, where a1 , a 2 , a 3 ≠ 0.
Multiplication by a scalar number:
〈(λa1 , λa 2 , λa 3 ); αã , βã , γã 〉,
λ>0
λã = {
〈(λa 3 , λa 2 , λa1 ); αã , βã , γã 〉,
λ<0
5. Division of two triangular neutrosophic numbers:
a1 a 2 a 3
〈( , , ) ; αã ∧ αb̃ , βã ∨ βb̃ , γã ∨ γb̃ 〉 , a 3 > 0 𝑎𝑛𝑑 b3 > 0
b3 b2 b1
a 3 a 2 a1
ã
= 〈( , , ) ; αã ∧ αb̃ , βã ∨ βb̃ , γã ∨ γb̃ 〉 , a 3 < 0 𝑎𝑛𝑑 b3 > 0
b3 b2 b1
b̃
a 3 a 2 a1
〈( , , ) ; αã ∧ αb̃ , βã ∨ βb̃ , γã ∨ γb̃ 〉 , a 3 < 0 𝑎𝑛𝑑 b3 < 0
{ b1 b2 b3
6. Multiplication of two triangular neutrosophic numbers:
〈(a1 b1 , a 2 b2 , a 3 b3 ); αã ∧ αb̃ , βã ∨ βb̃ , γã ∨ γb̃ 〉,
a 3 > 0 𝑎𝑛𝑑 b3 > 0
a 3 < 0 𝑎𝑛𝑑 b3 > 0
ãb̃ = {〈(a1 b3 , a 2 b2 , a 3 b1 ); αã ∧ αb̃ , βã ∨ βb̃ , γã ∨ γb̃ 〉,
〈(a 3 b3 , a 2 b2 , a1 b1 ); αã ∧ αb̃ , βã ∨ βb̃ , γã ∨ γb̃ 〉,
a3 < 0 𝑎𝑛𝑑 b3 < 0
Where ∧ is a t-norm and ∨ is a t-conorm.
1.
2.
3.
4.
2.4 Zero-one linear programming
A zero-one linear programming theory solves problems like the following:
Max(Min) f(𝒙) = c1 𝑥1 + c2 𝑥2 + ⋯ + cI 𝑥I
(6)
Subject to: 𝑥𝑖 B
Where, 𝒙 = (𝑥1 , 𝑥2 , … , 𝑥𝐼 )𝑇 , 𝑥𝑖 {0, 1} and ci ∈ ℝ, i = 1, 2, …, I; B is the feasible set of solutions. B can
be defined with equalities like A𝑥 = b, inequalities like A𝑥 ≤ b or A𝑥 b, a combination of them, or
simply an empty set. Where A is an mxI matrix and b is an m-column vector.
This theory solves decision problems, where only two alternatives exist, 1 represents to make the
decision and 0 to not make the decision.
Zero-one linear programming problems are part of the Integer programming problems, when xi ∈
ℤ. Despite their seeming simplicity, these problems are NP-complete [49, 50], thus, a good universal
algorithm cannot be found to solve them during a rational time of execution. This subject is out of the
scope of this paper.
To solve the zero-one linear programming problem let us consider the following equivalent problem:
Max f(𝒙) = c1 𝑥1 + c2 𝑥2 + ⋯ + cI 𝑥I
Subject to: 𝑥𝑖 B
Where, 𝒙 = (𝑥1 , 𝑥2 , … , 𝑥𝐼 )𝑇 , xi ∈ ℤ, xi ≤ 1 and ci ∈ ℝ, i = 1, 2, …, I.
3 Neutrosophic model for IT project assessment
The model consists of three main processes, criteria selection, assessment and project portfolio selection. These processes are integrated by means of a Balanced Scorecard Model (BSC), a Neutrosophic
Analytic Hierarchy Process (NAHP) and zero-one linear programming, see Fig. 3.
Maikel Leyva-Vázquez, Miguel A. Quiroz-Martínez, Jesús R. Hechavarría-Hernández, and Erick González-Caballero. A new
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Criteria selection
Assessment
• Balanced
Scorecard (BSC)
Project portfolio
selection
• Neutrosophic
Analytic Hierarchy
Process (NAHP)
• Zero- one linear
Programming
Figure 3: General structure of the model
The first step is to identify a potential group of projects. Next, a criteria selection is made. Some
possible criteria are schematized in Fig. 4. This step is based on the BSC, which is an unusual tool for
use in project selection. This tool could be incorporated because the proposed model is designed to
solve the specific problem of information technology project selection. Fig. 4 can serve as a guide for
decision makers on which aspects are the most important for evaluating information technology projects. The second stage of the model is to apply the NAHP. The proposed linguistic scale is based on
triangular neutrosophic numbers summarized in Tab. 3, according to the scale defined in [19].
The hybridization of AHP with neutrosophic set theory was used in [19]. This is a more flexible
approach to a model of uncertainty in decision making. The indeterminacy is an essential component
to be assumed in real-world organizational decisions.
The neutrosophic pair-wise comparison matrix is defined in Eq. 7.
1̃ ã12 ⋯ ã1n
̃
A= [
⋮
⋱
⋮ ]
(7)
ã n1 ã n2 ⋯ 1̃
̃ satisfies the condition ã ji = ã−1
A
ij , according to the inversion operator defined in Def. 4.
Abdel-Basset et al. in [19] defined two indices to convert a neutrosophic triangular number in a
crisp number. Eqs. 8 and 9 indicate the score and the accuracy respectively as follow:
1
(8)
S(ã) = [a1 + a 2 + a 3 ](2 + αã −βã − γã )
8
1
(9)
A(ã) = [a1 + a 2 + a 3 ](2 + αã −βã + γã )
8
Criteria
Balanced
Scorecard
model
Feasibility
Resource
availability
Financial
perspective
Customer
perspective
Technical
feasibility
Cost
New
costumers
Social
feasibility
Profit
Costumer's
satisfaction
Economical
feasibility
NPV
Consolidation of
customer´s
portfolio
Business
process
perspective
Improvement
in control
process
Compliment
with
government
regulations
Learning and
growth
perspective
Professional
growth
New
knowledge
Patents
Project time
span
Figure 4: Example of possible project selection criteria
Maikel Leyva-Vázquez, Miguel A. Quiroz-Martínez, Jesús R. Hechavarría-Hernández, and Erick González-Caballero. A new
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Table 3: Aczél et al.’s scale translated to a neutrosophic triangular scale.
Original scale
1
3
5
7
9
2, 4, 6, 8
Definition
Equally influential
Slightly influential
Strongly influential
Very strongly influential
Absolutely influential
Sporadic values between two close
scales
Neutrosophic Triangular Scale
1̃ = 〈(1, 1, 1); 0.50, 0.50, 0.50〉
3̃ = 〈(2, 3, 4); 0.30, 0.75, 0.70〉
5̃ = 〈(4, 5, 6); 0.80, 0.15, 0.20〉
7̃ = 〈(6, 7, 8); 0.90, 0.10, 0.10〉
9̃ = 〈(9, 9, 9); 1.00, 1.00, 1.00〉
2̃ = 〈(1, 2, 3); 0.40, 0.65, 0.60〉
4̃ = 〈(3, 4, 5); 0.60, 0.35, 0.40〉
6̃ = 〈(5, 6, 7); 0.70, 0.25, 0.30〉
8̃ = 〈(7, 8, 9); 0.85, 0.10, 0.15〉
Suppose that the criteria in Fig. 4 and the neutrosophic triangular scale in Table 3 are given, then
the steps to apply the NAHP are as follow:
1. To design an AHP tree. This contains the selected criteria, subcriteria and alternatives from the
first stage.
2. To create the matrices per level from the AHP tree, according to experts’ criteria expressed in neutrosophic triangular scales and respecting the matrix scheme in Eq. 7.
3. To evaluate the consistency of these matrices. Abdel-Basset et al. make reference to Buckley, who
̃ = [ã ij ]
demonstrated that if the crisp matrix A = [a ij ] is consistent, then the neutrosophic matrix A
is consistent.
4. To follow the other steps of a classical AHP. Here, operations among neutrosophic triangular
numbers substitute equivalent operations among crisp numbers in classical AHP.
5. The results obtained from step 4 are the project weights expressed in form of neutrosophic triangular numbers. Now, Eq. 8 is applied to convert, w1, w2, …,wn to crisp weights.
6. If more than one expert make the assessment, then w1, w2, …,wn are replaced by w
̅ 1, w
̅ 2, ⋯ , w
̅ n,
which are their corresponding weighted geometric mean values, see Eq.1. and Eq. 2.
The obtained weights are not necessarily expressed in normal form, accordingly, there exists the
̅ 1′ , w
̅ 2′ , ⋯ , w
̅ n′ , such that ∑ni=1 wi′ = 1
choice to calculate equivalent normalized weights w1′ , w2′ , ⋯ , wn′ or w
n
′
or ∑i=1 w
̅ i = 1. The precedent algorithm can be seen in the form of a flow chart in Fig. 5.
Maikel Leyva-Vázquez, Miguel A. Quiroz-Martínez, Jesús R. Hechavarría-Hernández, and Erick González-Caballero. A new
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355
Figure 5: Flow chart of the NAHP algorithm.
̃ is converted in A and later they continue applying
Let us remark that in Abdel-Basset’s method, A
classical AHP to A. In contrast, in the proposed model, data is converted to numeric value only in the
last step. This way seems to be more acceptable because imprecision is kept throughout all the calculations.
The third stage consists of the application of a zero-one linear programming problem defined as
follows:
Max f(𝒙) = 𝑤1 𝑥1 + 𝑤2 𝑥2 + ⋯ + 𝑤𝑛 𝑥𝑛
(10)
Subject to: 𝑥𝑖 B
See that the problem defined in Eq. 10 is a particular case of that appeared in Eq. 6.
1 , if Project i is selected
Where, xi = {
and wi are the weights per project obtained from stage 2.
0
, otherwise
The purpose of this stage is to select the best projects, which optimally satisfy the constraints imposed by B, considering the weights obtained from NAHP.
4 Application of the model to an example
This section contains an example to illustrate the application of the model to a particular case of
project selection. The authors simplified this example significantly for the sake of facilitating readers’
comprehension.
Once the BSC model and the first stage are concluded, suppose that two project assessment criteria
have been chosen; they are financial perspectives and internal processes, see Fig. 6.
Maikel Leyva-Vázquez, Miguel A. Quiroz-Martínez, Jesús R. Hechavarría-Hernández, and Erick González-Caballero. A new
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To apply the AHP technique in the second stage, the elements of the problem were hierarchically
structured. The goal appears on top of the tree, criteria to evaluate the goal were situated in the intermediate level and alternatives to reach that goal are on the bottom. Where, the goal is to assess IT projects, the intermediate level contains three criteria, viz., cost, project time span and profit and the bottom contain the three potential projects, called Project 1, Project 2 and Project 3. The tree is depicted in
Fig. 7.
The expert expresses its criteria by means of the linguistic terms summarized in Tab. 3. The criteria
defined in the intermediate level are pair-wise linguistically compared to determine their relative importance to achieve the objective.
Later, neutrosophic evaluations in the third column of Tab. 3 substitute their equivalent linguistic
terms. Experts’ evaluations can be seen in Tab. 4.
Criteria
Internal process
perspective
Financial
perspective
Project time span
Cost
Profit
Figure 6: Selected criteria for the example
To assess
projects
Cost
Project time
span
Profit
Project 1
Project 2
Project 3
Figure 7: AHP tree of the example
Table 4: Reciprocal matrix corresponding to the second level
Cost
Project time span
Profit
Cost
1̃
2̃−1
5̃
Project Time span
2̃
1̃
4̃
Profit
5̃−1
4̃−1
1̃
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See that evaluations contain the uncertainty and imprecision proper of neutrosophic set theory and
hence the results are more realistic than those obtained from the classical Aczél et. al.’s AHP technique,
now experts can include the indeterminacy term. Also, let us observe that the inverse of the singlevalued triangular neutrosophic numbers can be calculated by using the inversion operator defined in
Def. 4.
In this example, Cost is assessed with a value between equally and slightly more influential than
Project time span, Profit is strongly more influential than Cost and Profit is evaluated between slightly
and strongly more influential than Project time span. When the last three criteria comparisons are analyzed, let us note a certain degree of inconsistency, where it is expected that Profit is at least strongly
more influential than the Project time span.
To measure the neutrosophic reciprocal matrix consistency, it is sufficient to calculate the CI of the
crisp matrix, where ã ij is substituted by a ij , according to the theorem proved in [9], which says that
given a fuzzy reciprocal matrix of fuzzy numbers 𝑎̅𝑖𝑗 = (𝛼𝑖𝑗 /𝛽𝑖𝑗 /𝛾𝑖𝑗 /𝛿𝑖𝑗 ), when choosing 𝑎𝑖𝑗 ∈ [𝛽𝑖𝑗 , 𝛾𝑖𝑗 ],
if the matrix (𝑎𝑖𝑗 )𝑖𝑗 is consistent then (𝑎̅𝑖𝑗 )𝑖𝑗 is also consistent.
Now on, the eig function coded in Octave 4.2.1 shall be used for estimating max, in this case, CI =
9.0404%<10%, i.e., the matrix is consistent.
The values per row are summed and the weights are calculated. The results were summarized in
Tab. 5.
Table 5: Sum per row and neutrosophic triangular weights in the second level criteria
Row sum
Weight
Cost
<(2.17, 3.20, 4.25); 0.40, 0.65, 0.60>
<(0.12, 0.21, 0.36); 0.40, 0.65, 0.60>
Project
time <(1.53, 1.75, 2.33); 0.40, 0.65, 0.60>
<(0.08 , 0.12, 0.12); 0.40, 0.65, 0.60>
span
Profit
<(8.00, 10.0, 12.0); 0.50, 0.50, 0.50>
<(0.43, 0.67, 1.03); 0.40, 0.65, 0.60>
Total
<(11.70, 14.95, 18.58); 0.40, 0.65, 0.60> <(0.63, 1.00, 1.59); 0.40, 0.65, 0.60>
Tabs. 6, 7 and 8 contain reciprocal matrices for the third level and their weights. Where, Tab. 6 is related to the Cost, Tab. 7 with Project time span and Tab. 8 with Profit. The CIs of these matrices are,
5.1558%, 0.53269% and 0.53269%, respectively.
Table 6: Reciprocal matrix of the third level related to Cost and their weights.
Project
1
Project
2
Project
3
Project 1
1̃
Project 2
2̃
Project3
5̃
Weight
<(0.31, 0.50, 0.79); 0.40, 0.65, 0.60>
5̃−1
5̃−1
1̃
<(0.07, 0.09, 0.12); 0.40, 0.65, 0.60>
2̃−1
1̃
5̃
<(0.27, 0.41, 0.63); 0.40, 0.65, 0.60>
Table 7: Reciprocal matrix of the third level related to Project time span and their weights.
Project
1
Project
2
Project
3
Project 1
1̃
Project 2
5̃−1
Project3
2̃−1
Weight
<(0.09, 0.13, 0.23); 0.40, 0.65, 0.60>
2̃
2̃−1
1̃
<(0.14, 0.26, 0.51); 0.40, 0.65, 0.60>
5̃
1̃
2̃
<(0.35, 0.61, 1.02); 0.40, 0.65, 0.60>
Table 8 Reciprocal matrix of the third level related to Profit and their weights.
Project
1
Project
2
Project
3
Project 1
1̃
Project 2
5̃
Project3
2̃
Weight
<( 0.35, 0.61,1.02); 0.40, 0.65, 0.60>
2̃−1
2̃
1̃
<(0.14, 0.26, 0.51); 0.40, 0.65, 0.60>
5̃−1
1̃
2̃−1
<(0.09, 0.13, 0.23); 0.40, 0.65, 0.60>
Maikel Leyva-Vázquez, Miguel A. Quiroz-Martínez, Jesús R. Hechavarría-Hernández, and Erick González-Caballero. A new
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358
Table 9: Global weight matrix
Project 1
Project 2
Project 3
Criterion
Weight
Costs
<(0.31, 0.50, 0.79);
0.40, 0.65, 0.60>
<(0.27, 0.41, 0.63);
0.40, 0.65, 0.60>
<(0.07, 0.09, 0.12);
0.40, 0.65, 0.60>
<(0.12, 0.21, 0.36);
0.40, 0.65, 0.60>
Project time span
<(0.09, 0.13, 0.23);
0.40, 0.65, 0.60>
<(0.35, 0.61, 1.02);
0.40, 0.65, 0.60>
<(0.14, 0.26, 0.51);
0.40, 0.65, 0.60>
<(0.08, 0.12, 0.12);
0.40, 0.65, 0.60>
Profits
<(0.35, 0.61, 1.02);
0.40, 0.65, 0.60>
<(0.09, 0.13, 0.23);
0.40, 0.65, 0.60>
<(0.14, 0.26, 0.51);
0.40, 0.65, 0.60>
<(0.43, 0.67, 1.03);
0.40, 0.65, 0.60>
Global Weight
<(0.19, 0.53, 1.36);
0.40, 0.65, 0.60>
<(0.10, 0.25, 0.59);
0.40, 0.65, 0.60>
<(0.08, 0.22, 0.63);
0.40, 0.65, 0.60>
Tab. 9 contains the global weight matrix, which is calculated similarly to the crisp case, where the
algebra of crisp values is substituted by its equivalent neutrosophic one.
Now, let us calculate crisp global weights of projects applying Eq. 8 to elements in Tab. 9 and normalizing, they are 0.52658 for Project 1, 0.23797 for Project 2 and 0.23545 for Project 3.
Evidently, according to the obtained weights, the projects can be ranked in the following order,
Project 1 ≻ Project 2 ≻ Project 3.
Additionally, in the third stage, if the decision-makers have to make the choice about what projects
should be carried out, which satisfies some constraints, the precedent weights can be used as inputs in
the optimization problem.
Suppose the manager counts on a total budget of $9000. In case of approval, $3000 must be spent in
Project 1, $3500 in Project 2 and $5000 in Project 3. As well, the total possible number of man-hour is
1100 and it is known that Project 1 needs 1000, Project 2 needs 200 and Project 3 needs 700.
Then, none, one, two or all of the three projects can be selected, always that they satisfy the restrictions imposed on the problem. Our goal is to optimize this selection, i.e., the project or projects
which can be simultaneously carried out have to be selected and then to maximize the benefits.
Formally, let us define three variables xi, i = 1, 2, 3 as follows:
1 , if Project i is selected
xi = {
0
, otherwise
Let us divide the data by their upper bounds for calculating with dimensionless magnitudes.
Hence, the mathematical problem is the following:
Max f(𝒙) = 𝑤1 𝑥1 + 𝑤2 𝑥2 + 𝑤3 𝑥3
Subject to:
(3000/9000)𝑥1 + (3500/9000)𝑥2 + (5000/9000)𝑥3 ≤ 1 (Budget constraint)
(1000/1100)𝑥1 + (200/1100)𝑥2 + (700/1100)𝑥3 ≤ 1 (Man-hour constraint)
w1 = 0.52658, w2 = 0.23797 and w3 = 0.23545 are the previously calculated project weights.
This is a problem of zero-one linear programming. The best solution is x = (1, 0, 0), i.e., the best option is to only select Project 1.
Conclusion
To select appropriately an information technology project is generally a complex task and at the
same time an unavoidable one because this kind of project is essential for many companies. One of the
difficulties arisen by decision makers is the environmental uncertainty and limitations of the existent
assessment systems. In this paper, the neutrosophy theory was chosen, which allows us to deal with
uncertainty and imprecision for IT project selection. Analytic hierarchy process is the technique for
making complex decisions. Then, the proposed model is based on a neutrosophic analytic hierarchy
process. This technique was complemented with a balanced scorecard model for determining the IT
selection criteria and zero-one linear programming to make the best feasible choice of projects. Finally,
an example was used for illustrating the advantages that were obtained from integrating these four
tools. It is necessary to emphasize that this model is unique to the set of information technology project selection models, as it was reviewed by the authors in the literature on that subject and it is particularly adjusted for solving the problem of IT project selection.
Maikel Leyva-Vázquez, Miguel A. Quiroz-Martínez, Jesús R. Hechavarría-Hernández, and Erick González-Caballero. A new
model for the selection of information technology project in a neutrosophic environment
Neutrosophic Sets and Systems, Vol. 32, 2020
345
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Received: Oct 2, 2019. Accepted: Mar 17, 2020
Maikel Leyva-Vázquez, Miguel A. Quiroz-Martínez, Jesús R. Hechavarría-Hernández, and Erick González-Caballero. A new
model for the selection of information technology project in a neutrosophic environment
Neutrosophic Sets and Systems, Vol. 32, 2020
University of New Mexico
Analyzing Age Group and Time of the Day Using Interval Valued
Neutrosophic Sets
S. Broumi1, M.Lathamaheswari2, A. Bakali3, M. Talea1, F. Smarandache4, D. Nagarajan2, Kavikumar5 and
Guennoun Asmae1
of Information Processing, Faculty of Science Ben M’Sik, University Hassan II, B.P 7955, Sidi Othman, Casablanca, Morocco,
E-mail: broumisaid78@gmail.com, E-mail: taleamohamed@yahoo.fr
2Department of Mathematics, Hindustan Institute of Technology & Science, Chennai-603 103, India, E-mail: mlatham@hindustanuniv.ac.in,
E-mail: dnagarajan@hindustanuniv.ac.in
3Ecole Royale Navale, Boulevard Sour Jdid, B. P 16303 Casablanca, Morocco, E-mail: assiabakali@yahoo.fr,
4Department of Mathematics, University of New Mexico, 705 Gurley Avenue, Gallup, NM 87301, USA, E-mail: fsmarandache@gmail.com
5Department of Mathematics and Statistics, Faculty of Applied Science and Technology, Universiti Tun Hussein Onn, Malaysia,
E-mail:kavi@uthm.edu.my
1Laboratory
Abstract: Human psychological behavior is always uncertain in nature with the truth,
indeterminacy and falsity of the information and hence neutrosophic logic is able to deal with this
kind of real world problems as it resembles human’s attitude very closely. In this paper, age group
analysis and time (day or night) analysis have been carried out using interval valued neutrosophic
sets. Further, the impact of the present work is presented.
Keywords: Neutrosophic Logic; Human Psychological Behavior; Age Group; Day; Interval Valued
Neutrosophic Set.
1. Introduction
Uncertainty saturates our daily lives and period the entire range from index fluctuations of
stock market to prediction of weather and car parking in a congested area to traffic control
management. Hence almost all the area contains ambiguity or impression. For various real world
problems, intelligent models with many types of mathematical designs of different logics have been
modeled by the researchers. In the area of computational intelligence, fuzzy logic is one of the
superior logic that provides appropriate representation of real world information and permits
reasoning that are almost accurate in nature [1].
Generally the inputs conquered by the fuzzy logic are determinate and complete. Humans
can able to take knowledgeable decisions in those situations, however it is difficult to express in
proper terms. But fuzzy models need complete information. Due to basic non-linearity, huge
erratic substantial disturbances, time varying nature, difficulties to find precise and predictable
measurements, incompleteness and indeterminacy may arise in the data. All these problems can be
dealt by neutrosophic logic proposed by Smarandache in the year 1999 [2-10]. Also this logic can
able to represent mathematical structure of uncertainty, ambiguity, vagueness, imprecision,
inconsistency, incompleteness and contradiction.
Also it is efficient in characterizing various attributes of data such as incompleteness and
inaccuracy and hence gives proper estimation about the authenticity of the information. This
approach proposes extending the proficiencies of representation of fuzzy logic and system of
Said Broumi, M.Lathamaheswari, Assia Bakali, Mohamed Talea, Florentin Smarandache, D. Nagarajan, J. Kavikumar
Analyzing Age Group and Time of the day using Interval Valued Neutrosophic Sets
Neutrosophic Sets and Systems, Vol. 32, 2020
362
reasoning by introducing neutrosophic representation of the information and system of
neutrosophic reasoning. Neutrosophic logic can exhibit various logical behaviors according to the
nature of the problem to be solved and hence it influences its chance to be utilized and experimented
for real world performance and simulations in human psychology [15].
Due to computational complexity of the neutrosophic sets, single valued neutrosophic sets
have been introduced. It can deal with only exact numerical value of the three components truth,
indeterminacy and falsity. While the data in the form of interval, then single valued neutrosophic
sets unable to scope up and hence interval valued neutrosophic sets have been introduced. As it has
lower and upper membership functions it can deal more uncertainty with less computational
complexity than other types [25]. Neutrosophic set has been used in several areas like traffic control
management, solving minimum spanning tree problem, analyzing failure modes and effect analysis,
blockchain technology, resource leveling problem, medical diagnostic system, evaluating time-cost
tradeoffs, analysis of criminal behavior, petal analysis, decision making problem etc. [26-40].
The major advantage of neutrosophic set and its types namely single valued neutrosophic sets
and interval valued neutrosophic sets overrule other sets namely conventional set, fuzzy set, type-2
fuzzy, intuitionistic fuzzy and type-2 intuinistic fuzzy by their capability of dealing with
indeterminacy which is missing with other types of sets. Since there is a possibility of having interval
number than the exact number we consider interval valued neutrosophic set in this study of
analyzing age group and time. Prediction of future trend is one of the interesting areas in the
research field. Hence, in this paper, age group analysis and time (day or night) analysis have been
done using interval valued neutrosophic sets. The remaining part of the paper is organized as
follows. In section 2, review of literature is given. In section 3, preliminaries are given for better
understanding of the paper. In section 4, age group and day and night time have been analyzed
using the concept of interval valued neutrosophic sets. In section 5, impact of the present work is
given. In section 6, concluded the present work with the future direction.
2. Review of Literature
The author in, [1] analyzed uncertainty exists in the project schedule using fuzzy logic. And the
authors of, [2] analyzed power flow using fuzzy logic. [3] Examined specific seasonal prediction
spatially under fuzzy environment for the group of long-term daily rainfall and temperature data
spatiotemporally. [4] examined about the prediction of temperature flow of the atmosphere based on
fuzzy knowledge–rule base for interior cities in India. [5] proposed a novel approach for
intuitionistic fuzzy sets and its applications in the prediction area.
[6] proposed single-valued neutrosophic minimum spanning tree and its aggregation method.
[7] proposed a new approach for the advisory of weather using fuzzy logic. [8] Proposed a method
for prediction of weather under fuzzy neural network environment and Hierarchy particle swarm
optimization algorithm. [9] Proposed various types of neutrosophic graphs and algebraic model and
applied in the field of technology. [10] proposed single valued neutrosophic graphs (SVNGs).
Said Broumi, M.Lathamaheswari, Assia Bakali, Mohamed Talea, Florentin Smarandache, D. Nagarajan, J. Kavikumar
Analyzing Age Group and Time of the day using Interval Valued Neutrosophic Sets
Neutrosophic Sets and Systems, Vol. 32, 2020
363
[11] examined bipolar single valued neutrosophic graphs. [12] Proposed interval valued
neutrosophic graphs. [13] proposed isolated SVNGs. [14] provided an introduction to the theory
bipolar SVNG. [15] proposed the degree, size and order of SVNGs. 16] applied Dijkstra algorithm to
solve shortest path problem under IVN environment. [17] solved minimum spanning tree problem
under trapezoidal fuzzy neutrosophic environment.
[18] applied minimum spanning tree algorithm for shortest path (SP) problem using bipolar
neutrosophic numbers. [19] proposed a novel matrix algorithm for solving MST for undirected
interval value NG. [20] solved a spanning tree problem with neutrosophic edge weights. [21]
proposed a new algorithm to solve MST problem with undirected NGs. [22] analyzed the role of
SVNSs and rough sets with imperfect and incomplete information systems.
[23] Studied about neutrosophic set and its development . [24] studied about the prediction of
long-term weather elements using adaptive neuro-fuzzy system using GIS approach in Jordan. [25]
have done overview of neutrosophic sets. [26] proposed a methodology of traffic control
management using triangular interval type-2 fuzzy sets and interval neutrosophic sets. [27] Solved
MST problem using single valued trapezoidal neutrosophic numbers.
[28] estimated risk priority number in design failure modes and effect analysis using factor
analysis. [29] have done edge detection on DICOM image using type-2 fuzzy logic. [30] made a
review on the applications of type-2 fuzzy in the field of biomedicine. [31] have done image
extraction on DICOM image usingtype-2 fuzzy. [32] made a review on application of type-2 fuzzy in
control system. [33] proposed single and interval valued neutrosophic graphs using blockchain
technology. [34] introduced interval valued neutrosophic graphs using Dombi triangular norms. [35]
solved resource leveling problem under neutrosophic environment.
[36] introduced cosine similarity measures of bipolar neutrosophic sets and applied in
diagnosis of disorder diseases. [37] introduced a methodology for petal analysis using neutrosophic
cognitive maps. [38] analyzed criminal behavior using neutrosophic model. [39] presented
assessments of linear time-cost tradeoffs using neutrosophic sets. [40] solved sustainable supply
chain risk management problem using plithogenic TOPSIS-CRITIC methodology. In view of the
literature, prediction of age group and day or night time under interval neutrosophic set are yet to be
studied and which is the reason of the present study.
3. Preliminaries
In this section, preliminaries of the proposed concept are given
3. 1. Neutrosophic Set (NS) [25]
Consider the space X consists of universal elements characterized by e . The NS A is a
phenomenon which has the structure A TA e , I A e , FA e / e X where the three grades of
Said Broumi, M.Lathamaheswari, Assia Bakali, Mohamed Talea, Florentin Smarandache, D. Nagarajan, J. Kavikumar
Analyzing Age Group and Time of the day using Interval Valued Neutrosophic Sets
Neutrosophic Sets and Systems, Vol. 32, 2020
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memberships are from X to ]−0,1+[of the element e X to the set A, with the criterion:
0 TA e I A e FA e 3
(1)
The functions TA (e) , I A (e) and FA (e) are the truth, indeterminate and falsity grades lies in real
standard/non-standard subsets of ] −0, 1+ [.
Since there is a complication of applying NSs to real issues, Samarandache and Wang et al. [11-12]
proposed the notion of SVNS, which is a specimen of NS and it is useful for realistic applications of
all the fields.
3.2. Single Valued Neutrosophic Set (SVNS) [25]
For the space X of objects contains global elements e . A SVNS is represented by degrees of
bership grades mentioned in Def. 2.8. For all e in X, TA (e) , I A (e), FA (e) [0, 1]. A SVNS can be
written as
A
e : TA e , I A e , FA e
/e X
(2)
3.3. Interval Valued Neutrosophic Set [12]
Let X be a space of objects with generic elements in X denoted by e . An interval valued
neutrosophic set (IVNS)
A in
X is characterized
indeterminacy-membership function
by truth-membership function, TA (e) ,
I A (e) and falsity membership function FA (e) . For each
point e in X , TA (e) , I A (e) , FA (e) 0,1 , and an IVNS A is defined by
A
T
L
A
e , TAU e , I AL e , I UA e , FAL e , FAU e
| e X
(3)
Where, TA (e) TAL e , TAU e , I A (e) I AL e , I UA e and FA (e) FAL e , FAU e
Fig 1 shows the Pictorial Representation of the neutrosophic set [5]
TN
IN
FN
Fig.1. Neutrosophic set
4. Proposed Methodology
In this section, age group and time (day or night) have been analyzed using interval valued
neutrosophic set.
4.1 Application of Interval Valued Neutrosophic Set in Age Group Analysis
Said Broumi, M.Lathamaheswari, Assia Bakali, Mohamed Talea, Florentin Smarandache, D. Nagarajan, J. Kavikumar
Analyzing Age Group and Time of the day using Interval Valued Neutrosophic Sets
Neutrosophic Sets and Systems, Vol. 32, 2020
365
As per our convenience, the age group is divided into three groups: young people, middle aged
people and old people. Assume young people are a truth membership function, middle aged people
are indeterminate membership function and old people are a falsity membership function. Here, the
degree of middle aged people may provide either degree of old people or young people or both. Let
us consider the age group is definitely young at and below 18-40, it is definitely old at and beyond
51-100 and in between the age group is middle. i.e., the level of the young age people decreases and
the level of old age people increases. The age group is represented pictorially for young people,
middle aged people and old people as in Fig. 2.
C
Young Age
Middle Age
Old
Age
L
R
Fig.2. The degrees of ‘young age’, ‘middle age’ and ‘old age’ people.
Let A be the different age groups of the people and N be an interval valued neutrosophic set defined
in the set A. Let TN a be the membership degree of the age group ‘young age people’ at a , here
a denotes a numerical value. For example, a 20. Similarly, indeterminate degree of ‘middle age
people’ can be denoted by I N a and the falsity degree of ‘old age people’ denoted by FN a at a .
Consider A
N
TN
18, 40 , 41,50 , 51,100 and
T 1 8 , 40
4 1, 50
N
I ,N
I,N
1 8, 4 0FN ,
1 8 , 4 0
4 1, 5T0N 5, 1, 1 00
4 1, 5 0FN ,
I N,
,
5 1, 1 0F0N ,
51, 1 0 0
.
Case (i). At and below [18, 40], there is no middle age people and old age people but there exist only
young age people. Therefore the following values are obtained.
TNL , TNU 18, 40 1,1 , I NL , I U
N 18, 40 0, 0
and
Said Broumi, M.Lathamaheswari, Assia Bakali, Mohamed Talea, Florentin Smarandache, D. Nagarajan, J. Kavikumar
Analyzing Age Group and Time of the day using Interval Valued Neutrosophic Sets
Neutrosophic Sets and Systems, Vol. 32, 2020
366
FNL , FNU 18, 40 0, 0
i.e., the membership function of the interval valued neutrosophic set is
1,1,0,0,0,0
Case (ii). At age [41, 50] (at the point C)
TNL , TNU 41,50 0, 0 , I NL , I U
N 41,50 1,1 and
FNL , FNU 41,50 0, 0
i.e., the membership function of the interval valued neutrosophic set is
0,0,1,1,0,0
Case (iii). At and above [51,100], there are no young age people and middle age people, but there
exist only old age people.
L U
TNL , TNU 51,100 0, 0 , I NL , I U
N 51,100 0, 0 and FN , FN 51,100 1,1
i.e., the membership function of the interval valued neutrosophic set is
Hence, N
0,0,0,0,1,1
1,1 , 0, 0 , 0, 0 , 0, 0, 1,1, 0, 0 , 0, 0, 0, 0, 1,1
Also, young age people decreases and middle age people increases in between L and C.
i.e., 1,1 TNL , TNU 0, 0 and 0, 0 I NL , I UN 1,1
Further, middle age people decreases and old age people increases in between C and R.
i.e., 1,1 I NL , I UN 0, 0 and 0, 0 FNL , FNU 1,1
4.2 Application of Interval Valued Neutrosophic Set in Day and Night Time Analysis
As per our convenience, time of the day is divided into three groups: day, day or night (or both) and
night. Assume day time is a truth membership function, day or night (or both) is an indeterminate
membership function and night time is a falsity membership function. Here, the degree of day or
night time may provide either degree of day time or night time or both. Let us consider the time of
the day is definitely day time at and below 7 AM to 6 PM, it is definitely night at and beyond 7 PM
and 5 AM and in between time is day or night. i.e., the level of the day time decreases and the level
of night time increases. The time of the day is represented pictorially for day, day or night people
and night as in Fig. 3.
Said Broumi, M.Lathamaheswari, Assia Bakali, Mohamed Talea, Florentin Smarandache, D. Nagarajan, J. Kavikumar
Analyzing Age Group and Time of the day using Interval Valued Neutrosophic Sets
Neutrosophic Sets and Systems, Vol. 32, 2020
367
Fig.3. The degrees of time for ‘day’, ‘day or night’ and ‘night’
Let B be the different times of the day, M an interval valued neutrosophic set defined in the set B. Let
TM b be the membership degree of the time ‘day’ at b , here, b denotes a numerical value.
For example b 8 AM or PM. Similarly, the indeterminate degree of the time I N b and the falsity
degree of the time FM b
can be represented by b .
Consider two cases.
B 7 AM ,6PM , 5 AM ,6 AM , 7 PM ,5 AM and
M TN 7 AM ,6PM , I N 7 AM ,6PM , FN 7 AM ,6PM ,
.
TN 5 AM ,6 AM , I N 5 AM ,6 AM , FN 5 AM ,6 AM ,
TN 7 PM ,5 AM , I N 7 PM ,5 AM , FN 7 PM ,5 AM
Also we can consider, B 7 AM ,6PM , 6PM ,7 PM , 7 PM ,5 AM and
M TN 7 AM ,6PM , I N 7 AM ,6PM , FN 7 AM ,6PM ,
.
TN 6 P M, 7 P M
, 7 P M ,
, NI 6 P M, 7 PM N, F 6 P M
TN 7 PM ,5 AM , I N 7 PM ,5 AM , FN 7 PM ,5 AM
Case (i). At and below [7AM, 6 PM], there is no hesitation of day or night time and no night time but
there exist only day time. Therefore the following values are obtained.
T L , T U 7 AM ,6 PM 1,1
N N
I NL , I U
7 AM ,6 PM 0,0 and
N
Said Broumi, M.Lathamaheswari, Assia Bakali, Mohamed Talea, Florentin Smarandache, D. Nagarajan, J. Kavikumar
Analyzing Age Group and Time of the day using Interval Valued Neutrosophic Sets
Neutrosophic Sets and Systems, Vol. 32, 2020
368
FNL , FNU 7 AM ,6 PM 0,0
i.e., the membership function of the interval valued neutrosophic set is
1,1,0,0,0,0
Case (ii). At [5AM, 6AM] (at the point C) and at [6 PM, 7PM]
TNL , TNU 5 AM ,6 AM 0,0 and
I L , I U 5 AM ,6 AM 1,1 and
N N
TNL , TNU 6 PM ,7 PM 0,0
I NL , I U
6 PM ,7 PM 1,1
N
FNL , FNU 5 AM ,6 AM 0,0 and FNL , FNU 6 PM ,7 PM 0,0
i.e., the membership function of the interval valued neutrosophic set is
0,0,1,1,0,0
Case (iii). At and above [7 PM, 5 PM], there is no day time and no hesitation of day or night time, but
there exist only night time.
TNL , TNU 7 PM ,5 AM 0,0
I NL , I U
7 PM ,5 AM 0,0 and
N
F L , F U 7 PM ,5 AM 1,1
N N
i.e., the membership function of the interval valued neutrosophic set is
Hence, M
0,0,0,0,1,1
1,1 , 0, 0, 0, 0 , 0, 0, 1,1, 0, 0 , 0, 0, 0, 0, 1,1
Also, day time decreases and day or night time increases in between L and C.
i.e., 1,1 TNL , TNU 0, 0 and 0, 0 I NL , I UN 1,1
Further, day or night time decreases and night time increases in between C and R.
i.e., 1,1 I NL , I UN 0, 0 and 0, 0 FNL , FNU 1,1
5. Impacts of the work
i). The proposed approach is the effective one in determining age group forecasting while the data is
in the form of interval data with indeterminate information too.
ii). Time (day or night) analysis under interval neutrosophic environment will be very useful as it is
the major scientific and technical problems.
iii). Analysing any future trend can be done easily by inferring the existing information into the
future using interval neutrosophic sets as it has the capacity of addressing with the set of numbers in
the real unit interval which is not just a determined number, it is efficient to deal with real world
problems with various possible interval values
Said Broumi, M.Lathamaheswari, Assia Bakali, Mohamed Talea, Florentin Smarandache, D. Nagarajan, J. Kavikumar
Analyzing Age Group and Time of the day using Interval Valued Neutrosophic Sets
Neutrosophic Sets and Systems, Vol. 32, 2020
369
iv). The proposed methodology of age group analysis can be used in facial image analysis as age
detection system.
v). The proposed methodology of time analysis can be utilized in time series analysis.
6. Conclusion
Since neutrosophic logic resembles human behavior for predicting age and time (day or night),
it is suitable for this study. According to the knowledge of human, membership values of the truth,
indeterminacy and falsity may be exact numbers or interval numbers. In this paper, analysis of age
group and time(day or night) have been done using interval valued neutrosophic set with the
detailed description and pictorial representation. Also the impact of the present work has been
given. In future, the proposed concept can be done based on the concept of neutrosophic rough and
soft sets.
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Received: Nov 29, 2019. Accepted: Mar 15, 2020
Said Broumi, M.Lathamaheswari, Assia Bakali, Mohamed Talea, Florentin Smarandache, D. Nagarajan, J. Kavikumar
Analyzing Age Group and Time of the day using Interval Valued Neutrosophic Sets
Neutrosophic Sets and Systems, Vol. 32, 2020
University of New Mexico
Air Pollution Model using Neutrosophic Cubic Einstein
Averaging Operators
Majid Khan1, Muhammad Gulistan1, Nasruddin Hassan2,* and Abdul Muhaimin Nasruddin3
Department of Mathematics and Statistics, Hazara University, Mansehra 21130 Pakistan; majid_swati@yahoo.com,
gulistanmath@hu.edu.pk
2 School of Mathematical Sciences, Faculty of Science and Technology, Universiti Kebangsaan Malaysia, Bangi 43600,
Selangor Malaysia; nas@ukm.edu.my
Department of Management and Marketing, Faculty of Economics and Management, Universiti Putra Malaysia, Serdang
43400, Selangor Malaysia; abdulmuhaimin085@gmail.com
1
3
* Correspondence: nas@ukm.edu.my; Tel.: (+60192145750)
Abstract: The neutrosophic cubic averaging and Einstein averaging aggregation operators are
presented and applied to the air pollution model of the city of Peshawar, Pakistan. Neutrosophic
cubic set (NCS) is a more generalized version of the neutrosophic set (NS) and an interval
neutrosophic set (INS). It is in a better position to express consistent, indeterminant and incomplete
information, thus it is able to be applied to aggregate the air pollution model. Aggregation operators
have a key role in science and engineering problems. Firstly, the neutrosophic cubic weighted
averaging (NCWA) operator, neutrosophic cubic ordered weighted averaging (NCOWA) operator,
neutrosophic cubic hybrid aggregation (NCHA) operator, neutrosophic cubic Einstine weighted
averaging (NCEWA) operator, neutrosophic cubic Einstine ordered weighted averaging
(NCEOWA) operator and neutrosophic cubic Einstine hybrid aggregation (NCEHA) operator are
defined. Secondly, these operators are applied to the air pollution model of particulate matter with
the size of less than 10 micron (PM10) in Peshawar. Subsequently, the results are compared with the
World Health Organization (WHO) standards using score/accuracy function. The pollution of PM10
is found to be very much higher than WHO standards. Hence, strong measures are required to
control air pollution.
Keywords: Air pollution; neutrosophic cubic weighted averaging; neutrosophic cubic hybrid
averaging; neutrosophic cubic Einstein weighted averaging; neutrosophic cubic Einstein hybrid
averaging.
1. Introduction
The uncertainty is a complex phenomenon that occurs in the real world. Since uncertainty is
inevitably involved in problems, it occurs in different areas of life such that conventional methods
have failed to cope with such problems. The big task is to deal with uncertain information. Many
models have been introduced to incorporate uncertainty into the description of the system. The fuzzy
set was initiated by Zadeh [1]. Henceforth, it is applied in different fields of sciences like artificial
intelligence, information sciences, medical sciences, decision making theory and much more. Due to
its applicability in sciences and daily life problem, fuzzy set has been extended into interval valued
fuzzy sets (IVFS) [2,3], intuitionistic fuzzy set (IFS) [4], interval valued fuzzy set (IVIFS) [5] and cubic
set [6] among others, besides Q-fuzzy [7-11] and vague soft set [12]. IFS consists of two components,
membership and non-membership whereas the hesitant component is considered under the
M. Khan, M. Gulistan, N. Hassan and A.M. Nasruddin, Air pollution model using neutrosophic cubic Einstein averaging
operators
Neutrosophic Sets and Systems, Vol. 32, 2020
373
condition that sum of these components is one. Smarandache presented the idea of neutrosophic sets
(NS) [13], which provides a more general form to extend the ideas of classic theory and fuzzy set
theory. The NS expresses three components namely truth, indeterminacy and falsity and all these
components are independent, which makes NS more general than IFS such that NS can be seen as a
generalization of IFS [14]. For sciences and engineering problems Wang et al. [15] presented a single
valued neutrosophic set (SVNS), while Wang et al. [16] introduced the interval neutrosophic set (INS).
Jun et al. [17] combined INS and NS to form a neutrosophic cubic set (NCS) which enables us to
choose both interval value and single value membership, indeterminacy and falsehood components,
hence presenting a more general form for uncertain and vague data.
Aggregation operators are an imperative part of decision making. The lack of data or knowledge
makes it difficult for decision maker to give the exact decision. This uncertain situation can be
minimized due to the vague nature of NS and its extensions. Researchers [18-27] introduced different
aggregation operators and multicriteria decision making methods in NS and INS. Khan et al. [28]
presented neutrosophic cubic Einstein geometric aggregation operators. Zhan et al. [29] worked on
multi criteria decision making on neutrosophic cubic sets. Banerjee et al. [30] used grey rational
analysis (GRA) techniques to neutrosophic cubic sets. Lu and Ye [31] defined a cosine measure to
neutrosophic cubic set. Pramanik et al. [32] used similarity measure to neutrosophic cubic set. Shi and
Ji [33] defined Dombi aggregation operators on neutrosophic cubic sets. Ye [34] defined aggregation
operators over the neutrosophic cubic numbers. Alhazaymeh et al. [35] presented a hybrid geometric
aggregation operator with application to multiple attribute decision making method on neutrosophic
cubic sets.
According to WHO, air pollution causes millions of premature deaths every year globally. 90%
of these deaths are caused by air pollution in middle and low income countries, mainly in Africa and
Asia. Indeed it is a great threat to the environment. Inhaling polluted air may cause different types
of diseases like lung cancer, respiratory diseases etc. In the last few years Pakistan witnessed a
significant increased in cancer, asthma and chronic lung disease. The particulate matter (PM) is one
of the major factors that cause such types of diseases. The data extracted from Alam et al. [36] consists
of particulate matter with the size of less than 10 micron (PM10) in Peshawar, Pakistan.
The collection of accurate data has always been a tough job which may cause some uncertain
results. That is why the need was felt to analyze the data using vague set. The neutrosophic cubic set
is one of the better choices to deal with vague and inconsistent data. For this purpose, firstly the
neutrosophic cubic averaging and Einstein averaging operators are defined. Then these operators are
used to analyze the air pollution of PM10 model for the city of Peshawar, Pakistan with WHO
standards. In this paper, the NCWA, NCOWA, NCHA, NCEWA, NCEOWA and NCEHA are
defined. Both algebraic and Einstein operators are applied to an air pollution model [36] and
compared. The goal of this work is to analyze the PM10 in the city of Peshawar and compare it with
WHO standards.
The methodology to measure the aggregate value of neutrosophic cubic values is as follows.
Firstly, the data is extracted from [36] and converted to neutrosophic cubic values so that the
aggregated value can be measured. Secondly, the data is analyzed using the WHO standard. It is to
be noted that the neutrosophic cubic set is the combination of both interval neutrosophic and
M. Khan, M. Gulistan, N. Hassan and A.M. Nasruddin, Air pollution model using neutrosophic cubic Einstein averaging
operators
Neutrosophic Sets and Systems, Vol. 32, 2020
374
neutrosophic set, which enable us to deal with both interval-valued neutrosophic and neutrosophic
set at the same time.
This paper is structured as follows. In section 2, some preliminaries are reviewed. In section 3,
the air pollution data is formed. In section 4, neutrosophic cubic averaging operators are defined and
applied over the data of section 3. In section 5, neutrosophic cubic Einstein averaging operators are
defined and applied over the data of section 3. The analyzing results are concluded by both
numerically and graphically. We hope to expand the study further to numerical analysis [37-42],
construction management [43-45], Q-neutrosophic soft environment [46], geometric programming
[47] and binomial factorial problem [48].
2. Preliminaries
This section consists of some definitions and results which provide the foundation of the work.
Definition 2.1 [13] A structure N =
TN (u ), I N (u ), FN (u ) 0 ,1
TN (u), I N (u), FN (u) | u U
and TN (u ), I N (u ), FN (u )
is neutrosophic set (NS), where
are truth, indeterminancy and falsity
function respectively.
Definition 2.2 [15] A structure N =
(SVNS), where
TN (u), I N (u), FN (u) | u U is single value neutrosophic set
TN (u), I N (u), FN (u) [0,1]
respectively called truth, indeterminancy and falsity
functions.,simply denoted by N = TN , I N , FN .
Definition 2.3 [16] An interval neutrosophic set (INS) in U is a structure
N =
TN (u), I N (u), FN (u) | u U
where
TN (u), I N (u), FN (u ) D[0,1] are respectively called truth, indeterminacy an falsity function in
Simply denoted by N TN , I N , FN
U.
for convenience being actually
L U
L U
L U
N TN = TN , TN , I N = I N , I N , FN = FN , FN .
Definition 2.4 [17] A structure N
u, TN (u), I N (u), FN (u), TN (u), I N (u), FN (u ) | u U
is
L U
L U
L U
neutrosophic cubic set (NCS) in U in which TN = TN , TN , I N = I N , I N , FN = FN , FN
is an
TN , I N , FN is neutrosophic set in U , where
N = TN , I N , FN , TN , I N , FN , [0, 0] TN I N FN [3, 3] and 0 TN I N FN 3
interval neutrosophic set and
such that N
U
denotes the collection of neutrosophic cubic sets in U .
Definition 2.5 [22] The t-operators are basically union and intersection in the fuzzy sets which are
denoted by t-conorm
and t-norm . The role of t-operators is very important in fuzzy
theory and its applications.
M. Khan, M. Gulistan, N. Hassan and A.M. Nasruddin, Air pollution model using neutrosophic cubic Einstein averaging
operators
Neutrosophic Sets and Systems, Vol. 32, 2020
375
Definition 2.6 [22] : 0,1 0,1 0,1 is t-conorm if the following axioms hold.
Axiom 1 1, u = 1 and 0, u = 0
Axiom 2 u , v = v , u for all u and v.
Axiom 3 u , v, w = u , v , w for all u, v and w.
'
'
'
'
Axiom 4 If u u and v v , then u, v u , v
Definition 2.7 [22] : 0,1 0,1 0,1 is t-norm if it the following axioms hold.
Axiom 1 1, u = u and 0, u = 0
Axiom 2 u , v = v, u for all u and v.
Axiom 3 u , v, w = u , v , w for all u, v and w.
' '
'
'
Axiom 4 If u u and v v , then u , v u , v
The t-conorms and t-norms families have a vast range, which correspond to unions and
intersections, among these Einstein sum and Einstein product are good choices since they give the
smooth approximation like algebraic sum and algebraic product, respectively. Einstein sums E
and Einstein products E are the examples of t-conorm and t-norm respectively:
uv
E (u , v ) =
,
1 uv
E (u , v ) =
uv
1 1 u
1 v
Definition 2.8 [28] The sum of two neutrosophic cubic sets (NCS),
A = TA , I A , FA , TA , I A , FA
and B = TB , I B , FB , TB , I B , FB , where
L U
L U
L U
L U
L U
L U
TA = TA , TA , I A = I A , I A , FA = FA , FA and TB = TB , TB , I B = I B , I B , FB = FB , FB
is defined as
A B =
TAL TBL TALTBL , TAU TBU TAU TBU , I AL I BL I AL I BL , IUA IUB IUA IUB , FAL FBL , FAU FBU , TATB , I AI B , FA FB FAFB .
Definition 2.9 [28] The product between two neutrosophic cubic sets (NCS),
A = TA , I A , FA , TA , I A , FA
L U
TA = TA , TA
,IA
L U
= I A , I A
, FA
and B = TB , I B , FB , TB , I B , FB , where
L U
= FA , FA
and TB = TBL , TBU , I B = I BL , I UB , FB = FBL , FBU
is defined as
A B = TALTBL , TAU TBU , I AL I BL , I UA I BU , FAL FBL FAL FBL , FAU FBU FAU FBU , TA TB TATB , I A I B I A I B , FA FB
Definition 2.10 [28] The scalar multiplication on a neutrosophic cubic set (NCS)
A = TA , I A , FA , TA , I A , FA
L U
TA = TA , TA
, IA
and a scalar k where
L U
= I A , I A
, FA
L U
= FA , FA
is defined as
M. Khan, M. Gulistan, N. Hassan and A.M. Nasruddin, Air pollution model using neutrosophic cubic Einstein averaging
operators
Neutrosophic Sets and Systems, Vol. 32, 2020
376
k
k
k
k
k
kA = 1 (1 TAL )k ,1 (1 TAU ) k , 1 (1 I AL ) k ,1 (1 I UA ) k , FAL , FAU , TA , I A ,1 1 FA
Definition 2.11 [18] The Einstein sum between two neutrosophic cubic sets (NCS),
A = TA , I A , FA , TA , I A , FA and
L U
TA = TA , TA
,IA
L U
= I A , I A
, FA
B = TB , I B , FB , TB , I B , FB , where
L U
= FA , FA
and
L U
L U
L U
TB = TB , TB , I B = I B , I B , FB = FB , FB
is defined as
T L T L TU TU I L I L IU IU
FAL FBL
FAU FBU
TATB
I AIB
F FB
,
A E B = A L BL , A U BU , A L BL , A U BU ,
,
,
, A
L
L
1 TA TB 1 TA TB 1 I A I B 1 I A I B 1 (1 FA ) 1 FB 1 (1 FAU ) 1 FBU 1 (1 TA ) 1 TB 1 (1 I A ) 1 I B 1 FA FB
Definition 2.12 [28] The Einstein product between two neutrosophic cubic sets (NCS),
A = TA , I A , FA , TA , I A , FA
L U
TA = TA , TA
, IA
L U
= I A , I A
, FA
and B = TB , I B , FB , TB , I B , FB , where
L U
= FA , FA
and
L U
L U
L U
TB = TB , TB , I B = I B , I B , FB = FB , FB
is defined as
F L F L FU FU T T I I
TALTBL
TAU TBU
I AL I BL
I UA I BU
FA FB
,
, A L BL , A U BU , A B , A B ,
,
,
A E B =
L
L
U
U
L
L
U
U
1 (1 TA ) 1 TB 1 (1 TA ) 1 TB 1 (1 I A ) 1 I B 1 (1 I A ) 1 I B 1 FA FB 1 FA FB 1 TATB 1 I A I B 1 (1 FA ) 1 FB
Definition 2.13 [28] The Einstein scalar multiplication on a neutrosophic cubic set (NCS),
A = TA , I A , FA , TA , I A , FA , and a scalar k where
L U
L U
L U
TA = TA , TA , I A = I A , I A , FA = FA , FA
is defined as
kE A =
k
k
k
k
L k
L k
U k
U k
L k
L k
U k
U k
2 FAL
2 FAU
2 TA
2 IA
(1 FA )k (1 FA )k
(1 TA ) (1 TA ) , (1 TA ) (1 TA ) , (1 I A ) (1 I A ) , (1 I A ) (1 I A ) ,
,
,
,
,
k
k
k
k
k
k
k
k
L k
L k
U k
U k
L k
L k
U k
U k
k
k
(1 TA ) (1 TA ) (1 TA ) (1 TA ) (1 I A ) (1 I A ) (1 I A ) (1 I A ) 2 F L F L 2 F U F U 2 T T 2 I I (1 FA ) (1 FA )
A A A A A
A
A
A
L U
L U
L U
Definition 2.14 [28] Let N = TN , I N , FN , TN , I N , FN , where TN = TN , TN , I N = I N , I N , FN = FN , FN be a
neutrosophic cubic value. The score function is defined as
L
U
U
L
Scr N = TN FN TN FN TN FN
(1)
If the score function of two values are equal, the accuracy function is used to compare the
neutrosophic cubic values.
L U
L U
L U
Definition 2.15 [28] Let N = TN , I N , FN , TN , I N , FN , where TN = TN , TN , I N = I N , I N , FN = FN , FN be a
neutrosophic cubic vlaue. The accuracy function is defined as
M. Khan, M. Gulistan, N. Hassan and A.M. Nasruddin, Air pollution model using neutrosophic cubic Einstein averaging
operators
Neutrosophic Sets and Systems, Vol. 32, 2020
377
1
Acu (u ) =
9
TNL I NL FNL TNU I UN FNU TN I N FN
(2)
The following definition describes the comparison relation between two neutrosophic cubic
values.
Definition 2.16 [28] Let N1 , N 2 be two neutrosophic cubic values, with score functions
Scr N1 , Scr N 2
and accuracy functions Acu N1 , Acu N 2 . Then
1). Scr N1 > Scr N 2 N1 > N 2
2). If Scr N1 Scr N 2 , then
(i). Acu N1 Acu N 2 N1 > N 2 , (ii). Acu N1 Acu N 2 N1 = N 2
3. Model Formulation of Air Pollution
Air pollution is a great threat to the environment. It causes different diseases to the human being.
Inhaling polluted air may cause different types of disease like lung cancer and other respiratory
diseases. According to WHO, air pollution causes 7 million premature deaths globally in 2016.
Ambient air pollution alone caused 4.2 million deaths, while the atmospheric contamination of
households from the kitchen with fuels and contaminating technologies led to an estimated 3.8
million deaths in the same year. More than 90% of deaths related to air pollution occur in middleand low-income countries, mainly in Africa and Asia. In the last few years Pakistan witnessed a
significant increase in cancer, asthma and chronic lung diseases. The particulate matter (PM) cause
such type of diseases. The PM size is categorized as PM25, PM10 and PM2.5. The recommendation
of the WHO for air quality call the countries to reduce their annual air pollution to the annual mean
value of 20ug/m3 for PM10. In this model, the data for PM10 was considered.
The collection of data is a hard task to do since most of the time we are unable to collect the
correct and appropriate data. The problems may arise due to unskilled data collectors,
inappropriate methods of collecting data and others. These obstacles can be minimized by using
neutrosophic cubic sets which provide a vast variety to choose and decide. In this paper, a problem
regarding PM10 in Peshawar, Pakistan is considered and their values aggregated using a
neutrosophic cubic environment. Data is taken from [36] and converted to neutrosophic cubic form.
To consider overall values, data aggregation operators are being proposed so that its value can be
compared with WHO standards. According to WHO recommendation, the neutrosophic cubic
value for PM10 is calculated as
NWHO = [0.15, 0.30],[0.10, 0.30],[0.70, 0.85], 0.20, 0.40, 0.75
(3)
The neutrosophic cubic data for 1st , 5th, 10th, 15th and 20th April 2014 are respectively shown as
follows.
N A = [0.82, 0.92], 0.39, 0.66 , 0.18, 0.38 , 0.88, 0.7, 0.42 , N B = [0.59, 0.78], 0.68, 0.73 , 0.22, 0.41 , 0.68, 0.78, 0.32 ,
NC = [0.86, 0.96], 0.8, 0.85 , 0.24, 0.36 , 0.17, 0.8, 0.4 , N D = [0.8, 0.93], 0.11, 0.41 , 0.5, 0.9 , 0.9, 0.5, 0.4
M. Khan, M. Gulistan, N. Hassan and A.M. Nasruddin, Air pollution model using neutrosophic cubic Einstein averaging
operators
Neutrosophic Sets and Systems, Vol. 32, 2020
378
and N E = [0.61, 0.79],[0.41, 0.53],[0.39, 0.56], 0.45, 0.30, 0.45 .
The accuracy function is used to rank the air pollution in their relevant dates, in which day the air
is most polluted by PM10.
Acu ( N A ) = 0.5944, Acu( N B ) = 0.5766, Acu( NC ) = 0.6044, Acu( N D ) = 0.6055, and Acu ( N E ) = 0.4988.
We observe that
D > Acu NC > Acu N A > Acu N B > Acu N E .
Acu N
RANKING
0.8
0.6
0.4
0.2
0
NA
NB
NC
ND
NE
Figure 1. Pollution Graph in Peshawar City in April 2014
The graphical analysis can be seen in Figure 1. To analyze the overall pollution of PM10, the
aggregation operators are needed. To fulfill this desire, the notion of neutrosophic cubic
aggregation operators and neutrosophic cubic Einstein aggregation operators are proposed.
4. Neutrosophic Cubic Weighted Averaging Aggregation Operator
This section consist of some fundamental definitions of neutrosophic cubic weighted
averaging (NCWA), neutrosophic cubic ordered weighted averaging (NCOWA) and neutrosophic
cubic Einstein hybrid avregaing (NCEHA) aggregation operator, which are defined as follows.
n
Definition 4.1 The neutrosophic cubic weighted averaging is a function, NCWA : R R defined
by
n
NCWAw ( N1 , N 2 , ...., N n ) = wk N k , where
k =1
(4)
n
T
W = ( w1 , w2 , ..., wn ) of Nk (k = 1, 2, 3, ..., n), be the weight such that wk [0,1] and wk = 1.
k =1
Note that in NCWA, the neutrosophic values are weighted first and then aggregated.
n
Definition 4.2 The neutrosophic cubic ordered weighted averaging is a function, NCOWA : R R
defined by
n
NCOWAw ( N1 , N 2 , ..., N n ) = wk S k , where
k =1
(5)
M. Khan, M. Gulistan, N. Hassan and A.M. Nasruddin, Air pollution model using neutrosophic cubic Einstein averaging
operators
Neutrosophic Sets and Systems, Vol. 32, 2020
379
Sk denotes the ordered position of neutrosophic cubic (NC) values whereby the NC values are
T
of N k ( k = 1, 2, 3, ..., n ), be the weight such that
ordered in descending order, W = ( w1 , w2 , ..., wn )
n
wk [0,1] and wk = 1.
k =1
Note that in NCOWA, the neutrosophic cubic values are first ordered and then aggregated. The
basic concept of NCOWA is to rearrange the neutrosophic cubic values in descending order and
then aggregate them.
Theorem 4.3 Let N k = TN , I N , FN , TN , I N , FN , where TN
k
k
k
k
k
k
k
L U
L U
L U
= TN , TN , I N = I N , I N , FN = FN , FN
k k k k k k k k
( k = 1, 2, ..., n) be a collection of neutrosophic cubic values, then the neutrosophic cubic weighted
average operator (NCWA) operator of N k is also a neutrosophic cubic value and
n
n
n
n
n
w
w
w
w
NCWA( N k ) 1 (1 TNL ) k ,1 (1 TNU ) k , 1 (1 I NL ) k ,1 (1 I NU ) k , FNL
k
k
k
k
k =1
k =1
k =1
k =1 k
k =1
wk
n
, FNU
k =1
wk
k
n
, TNk
k =1
wk
where W = (w1 , w2 ,..., wn )T of N k (k = 1, 2, 3,..., n ), be the weight such that wk [0,1] and
n
, IN
k =1
n
w
k =1
wk
k
n
,1 1 FN
k =1
= 1.
k
Proof: By mathematical induction for n = 2,
2
2
2
2
2
w
w
w
w
w1 N1 w2 N 2 1 (1 TNL ) k ,1 (1 TNU ) k , 1 (1 I NL ) k ,1 (1 I NU ) k , FNL
k
k
k
k
k =1
k =1
k =1
k =1 k
k =1
Assume that, the result holds for 𝑛 = 𝑚. That is
m
w N
k =1
k
k
m
m
m
m
m
w
w
w
w
= 1 (1 TNL ) k ,1 (1 TNU ) k , 1 (1 I NL ) k ,1 (1 I NU ) k , FNL
k
k
k
k
k =1
k =1
k =1
k =1 k
k =1
wk
2
, FNU
k =1
k
,F
wk
m
k =1
wk
U
Nk
wk
2
, TNk
k =1
wk
2
, IN
k =1
k
,I
m
, TN
k =1 k
m
wk
Nk
k =1
wk
wk
2
,1 1 FN
k =1
m
k
,1 1 FN
k =1
wk
wk
k
Consider n = m 1 , the following result will be proven.
m
m 1
m 1
m 1
m
m
m
wk
w
w
wk m
wk
m1
L w
U w
L w
U w
L k
U k
1 (1 TNk ) k ,1 (1 TNk ) k , 1 (1 I Nk ) k ,1 (1 I Nk ) k , FNk , FNk , TNk , I Nk ,1 1 FNk
k =1
k =1
k =1
k =1
k =1
k =1
k =1
k =1
k =1wk Nk wk 1Nk 1 = k =1
wk 1
1 (1 T L ) wk 1 ,1 (1 T U ) wk 1 , 1 (1 I L ) wk 1 ,1 (1 I U ) wk 1 , F L wk 1 , F U wk 1 , T wk 1 , I wk 1 ,1 1 F
Nk 1
Nk 1
Nk 1
Nk 1
Nk 1
Nk 1
N k 1
Nk 1
Nk 1
m
m1
= 1 1 TNL
k
k =1
wk
m 1
,1 1 TNU
k =1
k
wk
m1
L
, 1 1 I Nk
k =1
wk
m 1
,1 1 I NU
k =1
k
wk
m1 L
, FNk
k =1
wk
m 1
, FNU
k =1
k
wk
m1
, TNk
k =1
wk
m 1
, IN
k =1
k
wk
m 1
,1 1 FN
k =1
k
wk
Hence proved.
M. Khan, M. Gulistan, N. Hassan and A.M. Nasruddin, Air pollution model using neutrosophic cubic Einstein averaging
operators
k
wk
Neutrosophic Sets and Systems, Vol. 32, 2020
380
Example 4.4 The NCWA operator is applied on the data as stated in section 3 with corresponding
T
weight, w = (0.21, 0.14, 0.25, 0.29, 0.11) . This weight calculated by Xu and Yager [27] is an essential part
of aggregation operators and will be used throughout this paper.
The value of NCWA = 0.7871, 0.9171 , 0.5311, 0.7292 , 0.2912, 0.5075 , 0.5216, 0.6071, 0.3995 .
Theorem 4.5 Let N k = TN , I N , FN , TN , I N , FN , where
k
k
k
k
k
k
L U
L U
L U
TN = TN , TN , I N I N , I N , FN = FN , FN ,
k k k k k k k k
k
(k = 1, 2,..., n) be collection of neutrosophic cubic values with weight W = ( w1 , w2 , ..., wn )T of
n
N k ( k = 1, 2, 3, ..., n ), such that wk [0,1] and wk = 1. The following properties are true.
k =1
1. Idempotence: If for all N k = TNk , I Nk , FNk , TNk , I Nk , FNk , where TN = TNL , TNU , I Nk = I NL , I NU , FN = FNL , FNU
k
k k
k k k k k
( k = 1, 2, ..., n) are equal, i.e. N k = N for all 𝑘, then NCWA w ( N1 , N 2 , ..., N n ) = N
2. Monotonicity: Let Bk = TB , I B , FB , TB , I B , FB , where TB = TBL , TBU , I B = I BL , I BU , FB = FBL , FBU
k
k k k k k k k k
k
k
k
k
k
k
be the collection of neutrosophic cubic values. If S B (u ) S N (u ) and Bk (u ) N k (u ) , where 𝑢 ∈
𝑈, then
NCWA w ( N1 , N 2 , ..., N n ) NCWA w ( B1 , B2 , ..., Bn )
3. Boundary: N
k
k
NCWAw
N1 , N2 , ..., Nn N ,
where
k
k
N = min TNL , min I NL ,1 max FNL , min TN , min I N ,1 max FNL , min TN , min I N ,1 max FN
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
N max TNL , max I NL ,1 min FNL , max TN , max I N ,1 min FNL , max TN , max I N ,1 min FN
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
Proof.
1. Idempotence: Since N k = N so
k NCWA( N )
NCWA N
n
n
n
n
n
n
n
n
n
wk
wk
wk
L wk
U wk
L wk
U wk
L wk
U wk
1 (1 TN ) ,1 (1 TN ) , 1 (1 I N ) ,1 (1 I N ) , FN , FN , TN , I N ,1 1 FN
k =1
k =1
k =1
k =1
k =1
k =1
k =1
k =1
k =1
= TN , I N , FN , TN , I N , FN
2. Monotonicity: Since neutrosophic cubic ordered weighted average operator (NCOWA) is
strictly monotone function, hence the proof is trivial.
M. Khan, M. Gulistan, N. Hassan and A.M. Nasruddin, Air pollution model using neutrosophic cubic Einstein averaging
operators
Neutrosophic Sets and Systems, Vol. 32, 2020
381
Let u = min N and y = max N , then by the idempotent law we have
3. Boundary:
u NCOWA( Nk ) y N NCOWA( Nk ) N
Theorem 4.6 Let N k = TN k , I N k , FN k , TN k , I N k , FN k ,where TN = TNL , TNU , I Nk = I NL , I NU , FN = FNL , FNU ,
k
k k
k k k k k
(k = 1, 2,..., n ) be the collection of neutrosophic cubic values and W = ( w1 , w2 , ..., wn )T is weight of
the NCOWA with wk [0,1] and
n
w
k =1
k
= 1. The following properties will hold, where
N k is
the largest 𝑘𝑡ℎ of ( N1 , N 2 , ..., N n ) .
T
1. If W = (1, 0, ..., 0) , then NCOWA ( N1, N2 , ..., Nn ) = max Nk
T
2. If W = (0, 0, ...,1) , then NCOWA ( N1 , N 2 , ..., N n ) = min N k
3. If wk = 1, wl = 0, and k l , then NCOWA ( N1, N2 , ..., Nn ) = Nk .
Proof: Since in NCOWA, the neutrosophic values are ordered in descending order, hence NCWA
operator aggregates the weighted values. On the other hand NCOWA weights only the ordering
positions.
The idea of neutrosophic cubic hybrid aggregation operators (NCHA) is developed to not only
weigh the values but also weigh their ordering position as well.
n
Definition 4.7 NCHA : is a mapping of n-dimension, which has associated weight
n
T
W = ( w1 , w2 , ..., wn ) , where wk [0,1] and wk = 1, such that
k =1
NCHAw ( N1, N 2 ,..., N n ) = w1N (1) w2 N (2) ... wn N ( n) where
N k is the largest 𝑘𝑡ℎ of the weighted neutrosophic cubic values N k . The N k can be calculated
T
n
by the following formula N k = nwk N k , k = 1, 2, 3, , , n, W = ( w1 , w2 , ..., wn ) , wk [0,1] and wk = 1
k =1
, where n is the balancing coefficient.
Theorem 4.8 Let N k = TN , I Nk , FN , TN , I N , FN
k
k
k
k
k
, where T
Nk
= TNLk , TNUk , I Nk = I NLk , I NUk , FNk = FNLk , FNUk
( k = 1 , 2 , n. . . ,be)a collection of neutrosophic cubic values. Then the aggregated value by NCHA is
also a neutrosophic cubic value and
M. Khan, M. Gulistan, N. Hassan and A.M. Nasruddin, Air pollution model using neutrosophic cubic Einstein averaging
operators
Neutrosophic Sets and Systems, Vol. 32, 2020
382
NCHAw ( N )
k
n
L
= 1 1 T( k )
k =1
T
wk
n
U
,1 1 T( k )
k =1
wk
T
n
L
, 1 1 I ( k )
k =1
T
where the weight W = ( w1 , w2 , ..., wn )
wk
I
n
U
,1 1 I( k )
k =1
I
wk
n L
, F ( k )
k =1
wk
F
n
, F ( kU)
k =1
wk
F
,I
n
, T k
k =1
wk
n
k =1
k
wk
w
n
k
,1 1 F
k
k =1
n
is such that wk [0,1] and wk = 1.
k =1
Proof: The proof is directly concluded by Theorem 4.3.
Theorem 4.9 The NCWA operator is a special case of NCHA operator when all the components of
w are equal, i.e. w1 w2 ... wn .
1 1
1 T
n n
n
Proof. Let W = ( , , ..., ) .
Then NCHA w w( N1 , N 2 , ..., N n )
= w1N (1) w2 N (2) ... wn N ( n)
=
1
n
( N (1) N (2) ... N ( n ) ) =
1
n
( N1 , N 2 , ..., N n )
= w1N1 , w2 N 2 , ..., wn N n = NCWA( N1 , N 2 , ..., N n ) .
Theorem 4.10 The NCOWA is a special case of NCHA when all the components of w are equal, i.e.
w1 w2 ... wn .
1 1
1 T
n n
n
Proof. Let W = ( , , ..., ) .
Then NCHAw w( N1 , N 2 , ..., N n )
= w1N (1) w2 N (2) ... wn N ( n)
= w1 N (1) w2 N (2) ... wn N ( n )
. .n,
= NCOWA (N1 ,N 2 , . N
)
Example 4.11 The NCHA is applied to the data as stated in section 3 with corresponding weight
w = (0.21, 0.14, 0.25, 0.29, 0.11)
T
of Xu and Yager [27].
Solution The weighted values are
N A = 0.8384, 0.9294 , 0.4049, 0.6778 , 0.1652, 0.3620 , 0.8744, 0.6876, 0.4355
NB =
0.4643, 0.6535 , 0.5496, 0.6001 , 0.3465, 0.5357 , 0.7635, 0.8404, 0.3170
M. Khan, M. Gulistan, N. Hassan and A.M. Nasruddin, Air pollution model using neutrosophic cubic Einstein averaging
operators
Neutrosophic Sets and Systems, Vol. 32, 2020
383
NC =
0.9143, 0.9821 , 0.8662, 0.9066 , 0.1679, 0.2788 , 0.1091, 0.7566, 0.4719
ND =
0.9030, 0.9788 , 0.1555, 0.5347 , 0.3660, 0.8583 , 0.8583, 0.3660, 0.5232
0.4042, 0.5761 , 0.2519, 0.3398 , 0.5958, 0.7269 , 0.6446, 0.5157, 0.2802
NE =
Scr N A = 1.6759, Scr N B = 0.6821, Scr NC = 1.0856, Scr N D = 0.9926, Scr N E = 0.0220. ,
Here, Scr N A > Scr NC > Scr N D > Scr N B > Scr N E .
According to their ranking, the values are
0.8384, 0.9294 , 0.4049, 0.6778 , 0.1652, 0.3620 , 0.8744, 0.6876, 0.4355
0.9143, 0.9821 , 0.8662, 0.9066 , 0.1679, 0.2788 , 0.1091, 0.7566, 0.4719
0.9030, 0.9788 , 0.1555, 0.5347 , 0.3660, 0.8583 , 0.8583, 0.3660, 0.5232
0.4643, 0.6535 , 0.5496, 0.6001 , 0.3465, 0.5357 , 0.7635, 0.8404, 0.3170
0.4042, 0.5761 , 0.2519, 0.3398 , 0.5958, 0.7269 , 0.6446, 0.5157, 0.2802
N (1) =
N (2) =
N (3) =
N (4) =
N (5) =
The new associated weight is derived by the normal distribution method [19]. Here the associated
T
weight W = (0.110, 0.237, 0.303, 0.235, 0.115)
is the weighting of the NCHA operator.
NCHAw ( N (1) , N (2) , N (3) , N (4) , N (5) )
,1 1 T , 1 1 I ,1 1 I , F
5
L
1 1 T (i )
i =1
=
wi
T
5
i =1
wi
U
(i )
T
5
i =1
wi
L
(i )
I
5
i =1
wi
U
(i )
I
5
i =1
L
(i )
, I ,1 1 F
w
5
5
i , F U wi , T
( i ) i
F
F i =1
i =1
wi
wi
5
i =1
i
wi
5
i =1
i
0.8165, 0.9367 , 0.5533, 0.6932 , 0.2911, 0.5251 , 0.4966, 0.5893, 0.4322
In order to analyze these results with WHO standard, the scores of NCWA and NCHA operators
are calculated and indicated as follows.
Ac( N NCWA ) = 0.5879 , Ac ( N NCHA ) = 0.5927 and Ac ( NWHO ) = 0.4166
M. Khan, M. Gulistan, N. Hassan and A.M. Nasruddin, Air pollution model using neutrosophic cubic Einstein averaging
operators
Neutrosophic Sets and Systems, Vol. 32, 2020
384
The graphical analysis is illustrated in Figure 2. It is observed that in both scores that Peshawar has
highly polluted air.
Figure 2. Comparison of aggregations with WHO standard
5. Neutrosophic Cubic Einstein Aggregation Operators
This section consist of some fundamental definitions of neutrosophic cubic Einstein weighted
averaging (NCEWA), neutrosophic cubic Einstein ordered weighted averaging (NCEOWA) and
neutrosophic cubic Einstein hybrid avregaing (NCEHA) aggregation operator, which are defined
as follows. These are defined using the Einstein addition, Einstein multiplication and Einstein scalar
multiplication.
Definition 5.1 The neutrosophic cubic Einstein weighted averaging is a function, NCEWA
:R
n
R defined by
n
NCEWAw ( N1 , N 2 , ...., N n ) = wk N k
k =1
(6)
E
n
T
where W = ( w1 , w2 , ..., wn ) is the weight of Nk (k = 1, 2, 3,..., n), wk [0,1] and wk = 1.
k =1
This implies that the neutrosophic cubic values are weighted and then aggregated using Einstein
operations.
Definition 5.2
Order neutrosophic cubic Einstein weighted average operator (NCEOWA) is
n
n
defined as NCEOWA : R R by NCEOWAw ( N1 , N 2 , ..., N n ) =
k =1
wk Bk
E
where, Bk
denotes
T
the ordered position of neutrosophic cubic (NC) values in descending order, W = ( w1 , w2 , ..., wn )
n
is the weight of N k ( k = 1, 2, 3, ..., n ), be such that wk [0,1] and wk = 1.
k =1
Note that, NCEOWA values are ordered and then weighted. Thereafter, the ordering values
are aggregated using Einstein operations. The basic concept of ordered weighted operator is to
rearrange the values in descending order.
RANKING
0.6
0.4
0.2
0
NCWA
NCHA
WHO
M. Khan, M. Gulistan, N. Hassan and A.M. Nasruddin, Air pollution model using neutrosophic cubic Einstein averaging
operators
Neutrosophic Sets and Systems, Vol. 32, 2020
385
N k = T Nk , I Nk , F Nk , TN , I N , FN , where T Nk = TNL , TNU , I Nk = I NL , I NU , F Nk = FNL , FNU ,
k
k
k
k k
k k
k k
Theorem 5.3 Let
( k = 1, 2, ..., n) be the collection of neutrosophic cubic values. Then their NCEWA operator is also a
T
is the weight vector of N k ( k = 1, 2, 3, ..., n ),
neutrosophic cubic value where W = ( w1 , w2 , ..., wn )
n
such that wk [0,1] and wk = 1.
k =1
Proof. By mathematical induction for n = 2, using Einstein’s addition and ascalar multiplication,
we will have the following.
w
w
w1
w1
w1
w1
w1
w1
L 1
L 1
1 TNU 1 TNU 1 I NL 1 I NL
1 I NU 1 I NU
1 TN1 1 TN1
1
1
1
1
1
1
,
,
,
,
w
w
w
w
w1
w1
w
w
L 1
L 1
U 1
U 1
L
L
U 1
U 1
1 TN 1 TN 1 I N 1 I N
1 IN 1 IN
1 TN1 1 TN1
1
1
1
1
1
1
w
w1
w1
w1
w1
w
L 1
U 1
2 TN
2 IN
1 FN 1 FN
2 FN
2 FN
,
1
1
1
1
1
1
,
,
,
w1
w1
w1
w1
w1
w1
w1
w1
w
w
L 1
L 1
2 FNU FNU 2 TN TN
2 IN IN
1 FN 1 FN
2 FN1 FN1
1
1
1
1
1
1
1
1
w1 N1 E w2 N 2 E =
w2
w2
w2
w2
w2
w2
w2
w2
L
L
U
U
L
L
U
U
1 TN
1 TN
1 IN
1 IN
1 IN
1 TN
1 TN 1 I N
2
1
2
2
2
1
2
2
,
,
,
,
w
w
w
w
w
w
w
w2
2
2
2
2
2
1
2
L
L
U
U
L
L
U
U
I
I
I
I
1
T
1
T
1
T
1
T
1
1
1
1
N2
N2
N2
N2
N1
N2
N2
N2
w2
w
w2
w2
w2
L
U 2
2
2
2
2
1
F
F
T
I
F
1 FN
N
N
N
N
N
,
2
1
2
2
2
2
,
,
,
w
w
w
w
w
w
w
w
w
2
2
2
2
2
1
2
2
L
L
U 2
U
2
2
2
2
1
1
F
F
F
F
T
T
I
I
F
F
N2
N2
N1
N2
N2
N2
N2
N2
N2
N2
2
2
2
2
2
2
wk
wk
wk
wk 2
wk
wk
wk
wk
2
L
L
1 I NL
1 TNU 1 TNU 1 I NL
1 I NU 1 I NU
1 TNk 1 TNk
k
k
k
k
k
k
k =1
k =1
k =1
k =1
k2=1
, k2=1
,
, k2=1
, k2=1
2
2
2
2
wk
wk
wk
wk
wk
wk
L
L
U
U
U
1 I L wk 1 I L wk
1
T
1
T
1
T
1
T
I
1
1 I NU
Nk
Nk
Nk
Nk
Nk
Nk
Nk
k
2
k =1
k =1
k =1
k =1
k =1
k =1
k =1
k =1
wk N k E =
2
2
2
2
2
wk
wk
wk
wk
k =1
2 FNL
2 FNU
2 TN
2 I N
1 FN
k
k
k
k
k
k =1
k =1
k =1
k =1
, 2
, 2
, 2
, k2=1
2
2
2
2
2
wk
wk
wk
wk
wk
wk
wk
wk
L
L
U
U
IN
FN
TN
2 FN
2 TN
2 IN
1 FN
2 FNk FNk
k
k
k
k
k
k
k
k =1
k =1
k =1
k =1
k =1
k =1
k =1
k =1
k =1
Assuming that for
w2
w2
wk
wk
1 F
2
1 FN
k =1
2
k =1
wk
k
wk
Nk
n = m the result holds true, that is
m
m
m
m
m
m
wk
w
w
wk
wk m
wk
wk
wk
m
L k
L k
1 TNU 1 TNU 1 I NL 1 I NL
1 I NU 1 I NU
1 TNk 1 TNk
k
k
k
k
k
k
k =1
k =1
k =1
k =1
k =1
k =1
k =1
km=1
,
,
,
m
m
m
m
m
m
m
wk
wk
w
w
w
w
wk
wk
U
U
L k
L k
U k
U k
L
L
1 TNk 1 TNk 1 TNk 1 TNk 1 I Nk 1 I Nk 1 I Nk 1 I Nk
m
k =1
k =1
k =1
k =1
k =1
k =1
k =1
k =1
k =1 wk Nk E =
m
m
m
m
m
m
w
w
wk
wk
wk
wk
L k
U k
2 FN
2 FN
2 TN
2 I N
(1 FN ) (1 FN )
k
k
k
k
k
k
k =1
k =1
k =1
k =1
k =1
k =1
, m
,
,
,
m
m
m
m
m
m
m
m
m
wk
wk
wk
wk
wk
wk
wk
wk
w
wk
L
L
U
U
k
2 FNk FNk 2 FNk FNk 2 TNk TNk 2 I Nk I Nk (1 FNk ) (1 FNk )
k =1
k =1
k =1
k =1
k =1
k =1
k =1
k =1
k =1
k =1
The result is proven for n = m 1 , since
M. Khan, M. Gulistan, N. Hassan and A.M. Nasruddin, Air pollution model using neutrosophic cubic Einstein averaging
operators
Neutrosophic Sets and Systems, Vol. 32, 2020
386
wm 1
wm 1
wm 1
wm 1
1 I L wm1 1 I L wm1 1 I U wm1 1 I U wm1
L
L
U
U
1
T
1
T
1
T
1
T
N
N
N
N
Nm 1
Nm 1
Nm 1
Nm1
m 1
m 1
m 1
m 1
,
,
,
wm 1
wm 1
wm 1
wm 1
wm 1
wm 1
wm 1
wm 1
L
L
L
L
1 I NU
1 TNU
1 TNU
1 I NU
1 TNm1 1 TNm1
1 I Nm1 1 I Nm1
m 1
m 1
m 1
m 1
wm1 N m1 =
wm 1
wm 1
wm 1
wm 1
wm 1
wm 1
L
U
2 FN
2 TN
2 IN
1 FN
1 FN
2 FNm1
m 1
m 1
m 1
m 1
m 1
,
,
,
,
wm 1
wm 1
wm 1
wm 1
wm 1
wm1
L
L
U
U
2 FNm1 FNm1 2 FNm1 FNm1 2 TNm1 TNm1 2 I Nm1 I Nm1 1 FNm1 1 FNm1
m
wk N k
k =1
wm1N m1
E
=
E
m
m
m
m
m
m
wk
w
w
wk
wk m
wk
wk
wk
m
L k
L k
1 TNU 1 TNU 1 I NL 1 I NL
1 I NU 1 I NU
1 TNk 1 TNk
k
k
k
k
k
k
k =1
k =1
k =1
k =1
k =1
k =1
k =1
k =1
,
,
,
m
m
m
m
m
m
m
wk
wk
w
w
w
w
wk
wk
m
L k
L k
U k
U k
L
L
U
U
1 TN 1 TN
1 IN 1 IN
1 IN 1 IN
1 TNk 1 TNk
k
k
k
k
k
k
k =1
k =1
k =1
k =1
k =1
k =1
k =1
k =1
m
m
m
m
m
m
w
w
wk
wk
wk
wk
L k
U k
2 FN
2 FN
2 TN
2 I N
(1 FN ) (1 FN )
k
k
k
k
k
k
k =1
k =1
k =1
k =1
k =1
k =1
,
,
,
,
m
m
m
m
m
m
m
m
m
m
w
w
w
w
w
w
w
w
k
k
k
k
k
k
wk
wk
U k
U k
F
F
T
T
I
I
(1
F
)
(1
F
)
2
2
2
2 FNLk FNLk
Nk
Nk
Nk
Nk
Nk
Nk
Nk
Nk
k =1
k =1
k =1
k =1
k =1
k =1
k =1
k =1
k =1
k =1
wm 1
wm 1
wm 1
wm 1
wm 1
wm 1
wm 1
wm 1
L
L
U
U
L
L
U
U
1 TN
1 TN
1 TN
1 TN
1 IN
1 IN
1 IN
1 I Nm1
m 1
m 1
m 1
m 1
m 1
m 1
m 1
,
,
,
wm 1
wm 1
wm 1
wm 1
wm 1
wm 1
wm 1
wm 1
L
L
U
U
L
L
U
U
1 TN
1 TN
1 TN
1 IN
1 IN
1 IN
1 I Nm1
1 TNm1
m 1
m 1
m 1
m 1
m 1
m 1
w
wm 1
wm 1
wm 1
wm 1
wm 1
U
2 F L m1
2
2
2
I
1
F
1
F
F
T
N
N
N
N
N
N
m 1
m 1
m 1
m 1
m 1
m 1
,
,
,
,
wm 1
wm 1
wm 1
wm 1
wm 1
wm 1
L
L
U
U
FN
FN 2 TN
2 FN
2 IN
1 FN
TN
IN
1 FN
2 FNm1
m 1
m 1
m 1
m 1
m 1
m 1
m 1
m 1
m 1
m1
w N
k =1 k k
E
=
m 1
m 1
m 1
m 1
w
w m 1
w
w
wk m 1
wk m 1
w
wk
m1
L k
L k
U k
U k
L
L
U k
1
T
1
T
1
T
1
T
1
I
1
I
1
I
1 I NU
Nk
Nk
Nk
Nk
Nk
Nk
Nk
k
k =1
k =1
k =1
k =1
k =1
k =1
mk =11
, mk =11
,
,
1
1
1
1
1
1
m
m
m
m
m
m
w
w
w
w
wk
wk
w
w
L k
L k
U k
U k
L
L
U k
U k
1 TNk 1 TNk 1 TNk 1 TNk 1 I Nk 1 I Nk 1 I Nk 1 I Nk
k =1
k =1
k =1
k =1
k =1
k =1
k =1
k =1
m 1
m 1
m 1
m 1
m 1
m 1
wk
wk
wk
wk
w
w
2 FNL
2 FNU
2 TN
2 I N
(1 FN ) k (1 FN ) k
k
k
k
k
k
k
k =1
k =1
k =1
k =1
k =1
, m1
,
,
, k =1
m 1
m 1
m 1
wk m 1
wk m 1
wk m 1
wk m 1
w
w m 1
w
w
wk
wk
m1
L k
L k
U k
U k
2
2
2
2
(1
)
(1
)
T
I
I
F
F
F
F
F
F
T
k =1 Nk k =1 Nk k =1 Nk k =1 Nk k =1 Nk k =1 Nk k =1 Nk k =1 Nk k =1 Nk
Nk
k =1
Hence proved.
Example 5.4 The NCEWA is applied to data in section 3 with corresponding weight of Xu and Yager
T
[27], w = (0.21, 0.14, 0.25, 0.29, 0.11) .
Then NCEWA 0.7848, 0.9163 , 0.5058, 0.6650 , 0.2957, 0.5241 , 0.5652, 0.6187, 0.3990
Note that the NCEWA operator aggregates the weighted value whereas the NCEOWA operator
weight the ordering position and then aggregates the values. The idea of NCEHA is developed to
overcome to not only weight the neutrosophic cubic values but their order positioning as well.
M. Khan, M. Gulistan, N. Hassan and A.M. Nasruddin, Air pollution model using neutrosophic cubic Einstein averaging
operators
Neutrosophic Sets and Systems, Vol. 32, 2020
387
In order to analyze these results with WHO standard, the score of NCEWA and NCEHA operators
are compared in terms of the accuracy functions as follows as illustrated in Figure 3.
Ac ( N NCEWA ) = 0.5861, , Ac ( N NCEHA ) = 0.6099 and Ac( N
WHO ) = 0.4166.
RANKING
0.8
0.6
0.4
0.2
0
NCEWA
NCEHA
WHO
Figure 3. The comparison of Einstein aggregation with WHO
Observe that using both operators NCEWA and NCEHA, Peshawar has highly polluted air as
shown in Figure 3. The overall graphical presentation is illustrated in Figure 4. Hence we conclude
that serious measures by the relevant government agencies are needed to overcome the situation.
0.7
0.6
RANKING
0.5
0.4
0.3
0.2
WHO
0.1
NCWA, NCHA
0
NCWA, NCHA
NCEWA,NCEHA
WHO
Figure 4. Comparison of neutrosophic cubic aggregation operators with WHO
6. Conclusions
In this research, NCWA, NCHA, NCEWA, NCEHA operators are compared with WHO
standards. The aggregation operators are applied to the numerical data of PM10. These aggregation
operators enabled us to analyze the air pollution model in the city of Peshawar, Pakistan. We
computed the accuracy functions of all of these aggregation operators and WHO standard. The
analysis is then presented graphically to illustrate the comparison. It is observed that in the month of
April 2014, the pollution of PM10 is very much higher than WHO standards. Strong measures are
thus required to control air pollution. Our future research will be to apply further the NCWA, NCHA,
M. Khan, M. Gulistan, N. Hassan and A.M. Nasruddin, Air pollution model using neutrosophic cubic Einstein averaging
operators
Neutrosophic Sets and Systems, Vol. 32, 2020
388
NCEWA, NCEHA operators to construction management, geometric programming, binomial
factorial problem, and numerical convergence of polynomial roots [49-50].
Funding: This research received no external funding
Conflicts of Interest: The authors declare no conflict of interest.
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Abdel-Basset M; Manogaran G; Gamal A; Smarandache F. A group decision making framework based on
neutrosophic TOPSIS approach for smart medical device selection, J. Med. Syst., 2019; 43(2), 38.
Abdel-Baset, M., Chang, V., Gamal, A., & Smarandache, F. (2019). An integrated neutrosophic ANP and
VIKOR method for achieving sustainable supplier selection: A case study in the importing field. Computers
in Industry, 106, 94-110.
Abdel-Basset, M., Manogaran, G., Gamal, A., & Smarandache, F. (2019). A group decision-making
framework based on the neutrosophic TOPSIS approach for smart medical device selection. Journal of
medical systems, 43(2), 38.
Abu Qamar M; Hassan N. Characterizations of group theory under Q-neutrosophic soft environment,
Neutrosophic Sets Syst., 2019; 27, pp. 114-130.
Khalid H.E; Smarandache F; Essa A.K. The basic notions for (over, off, under) neutrosophic geometric
programming problem, Neutrosophic Sets Syst., 2018; 22, pp. 50-62.
Khalid H.E; Smarandache F; Essa A.K. A neutrosophic binomial factorial theorem with their refrains,
Neutrosophic Sets Syst., 2016; 14, pp. 7-11.
Jamaludin N; Monsi M; Said Husain S.K; Hassan N. The interval zoro-symmetric single-step procedure
IZSS2-5D for the simultaneous bounding of simple polynomial zeros, AIP Conf. Proc., 2013; 1557, pp. 268271.
Jamaludin N; Monsi M; Hassan N. On the interval zoro symmetric single step procedure IZSS2-5D for
simultaneous bounding of simple polynomial zeros, Int. J. Math. Anal., 2013; 7(59), pp. 2941-2945.
Received: 02 Oct, 2019. Accepted: 18 Mar, 2020.
M. Khan, M. Gulistan, N. Hassan and A.M. Nasruddin, Air pollution model using neutrosophic cubic Einstein averaging
operators
Neutrosophic Sets and Systems, Vol. 32, 2020
University of New Mexico
Neutrosophic 𝛼-Irresolute Multifunction in Neutrosophic
Topological Spaces
T.RajeshKannan1 and S.Chandrasekar2
1,2
PG and Research Department of Mathematics, Arignar Anna Government Arts College, Namakkal(DT),Tamil Nadu,
India, rajeshkannan03@yahoo.co.in, chandrumat@gmail.com
* Correspondence: Author (chandrumat@gmail.com)
Abstract: Aim of this present paper is, we define some new type of irresolute multifunction
between the two spaces. We obtain some characterization and some properties between such as
Lower & Upper 𝛼- irresolute multifunction
.
Keywords: Neutrosophic 𝛼 -irresolute lower; Neutrosophic 𝛼 irresolute upper; Neutrosophic 𝛼 closed sets; Neutrosophic topological spaces
1. Introduction
C.L. Chang [3] was introduced fuzzy topological space by using .Zadeh’s L.A [18] (uncertain)
fuzzy sets. Further Coker [4] was developed the notion of Intuitionistic fuzzy topological spaces by
using Atanassov’s[1] Intuitionistic fuzzy set. Neutrality the degree of indeterminacy, as an
independent concept was introduced by Smarandache [7]. He also defined the Neutrosophic set of
three component Neutrosophic topological spaces (t, f, i) =(Truth, Falsehood, Indeterminacy),The
Neutrosophic crisp set concept converted to Neutrosophic topological spaces by A.A.Salama [13].
I.Arokiarani.[2] et al, introduced Neutrosophic α -closed sets. T Rajesh kannan[10] et.al
introduced and investigated a new class of continuous multivalued function
is called
Neutrosophic α- continuous multivalued function in Neutrosophic topological spaces.
Aim of this present paper is, we define some new type of irresolute multifunction between the
two spaces. we obtain some characterization and some properties between such as Lower &
Upper 𝛼 - irresolute multifunction.
2. PRELIMINARIES
In this section, we introduce the basic definition for Neutrosophic sets and its operations.
Throughout this presentation, (𝑅𝐶 1 ,𝜏𝑅𝐶 1 ) is namely as classical topological spaces on 𝑅𝐶 1
(represent as CTS𝑅𝐶 1 ) , (𝑅𝑁 2 .,𝜏𝑁 𝑁 ) is namely as an Neutrosophic topological spaces on
𝑅 ,
2
𝑅𝑁 2 .(represent as NUTS𝑅𝑁 2 ,),The family of all open set in 𝑅𝐶 1 (𝛼 −Open in𝑅𝐶 1 , semi-openin 𝑅𝐶 1
and pre-open in 𝑅𝐶 1 respectively ) is denoted by O(CTS𝑅𝐶 1 )( 𝛼 O(CTS𝑅𝐶 1 ) , SO(CTS𝑅𝐶 1 ) and
PO(CTS𝑅𝐶 1 ) respectively). The family of all Neutrosophic open set in 𝑅𝑁 2 ,(𝛼 −Open in 𝑅𝑁 2 ,, semiopen in 𝑅𝑁 2 , and pre-open in 𝑅𝑁 2 , respectively ) is denoted by O(NUTS𝑅𝑁 2 ,).( 𝛼 O(NUTS𝑅𝑁 2 ,) ,
SO(NUTS𝑅𝑁 2 ,) and PO(NUTS𝑅𝑁 2 ,) respectively). The family of all closed set in 𝑅𝐶 1 (𝛼 −closed
in𝑅𝐶 1 , semi-closed in 𝑅𝐶 1 and pre-Closed in 𝑅𝐶 1 respectively )
is denoted by
C(CTS𝑅𝐶 1 ).( 𝛼 C(CTS𝑅𝐶 1 ) , SC(CTS𝑅𝐶 1 ) and PS(CTS𝑅𝐶 1 ) respectively). The family of all
Neutrosophic Closed in 𝑅𝑁 2 (𝛼 −closed in 𝑅𝑁 2 , , semi-closed in 𝑅𝑁 2 , and pre-closed in 𝑅𝑁 2 ,
respectively ) is denoted by C(NUTS𝑅𝑁 2 ,).( 𝛼 C(NUTS𝑅𝑁 2 ,) , SC(NUTS𝑅𝑁 2 ,) and PC(NUTS𝑅𝑁 2 ,)
respectively)
T.RajeshKannan, and S.Chandrasekar, Neutrosophic 𝛼-Irresolute Multifunction In Neutrosophic Topological Spaces
Neutrosophic Sets and Systems, Vol. 32, 2020
391
Definition 2.1 [7]
Let 𝑅𝑁 1 be a non-empty fixed set. A Neutrosophic set 𝐴𝑅𝑁1 is an object having the form𝐴𝑅𝑁1 = {<𝜉 ,
𝜇𝐴𝑅𝑁 (𝜉 ),𝜎𝐴𝑅𝑁 (𝜉 ),𝛾𝐴𝑅𝑁 (𝜉 )> : 𝜉
,𝛾𝐴
1
𝑅𝑁 1
1
(𝜉 )):𝑅𝑁 1
𝜇𝑅𝑁 1 (𝜉 ):𝑅𝑁 1 → [0,1 ],𝜎𝑅𝑁 1 (𝜉 )):𝑅𝑁 1 → [0,1],
𝑅𝑁 1 }.Where
∈
1
→ [0,1], are represent Neutrosophic of the degree of membership function, the
degree indeterminacy and the degree of non membership function respectively of each element 𝜉 ∈
𝑅𝐶 1 to the set 𝐴𝑅𝐶 1 with 0 ≤ 𝜇𝐴 (𝜉 )+𝜎𝐴 𝑁 (𝜉 )+𝛾𝐴 𝑁 (𝜉 ) ≤ 1.This is
called standard form
𝑅𝑁 1
𝑅 1
𝑅 1
generalized fuzzy sets. But also Neutrsophic set may be 0 ≤ 𝜇𝐴
Remark 2.2[7]
we denote𝐴𝑅𝑁1 = {<𝜉 , 𝜇𝐴
𝐴𝑅𝑁1 = {<𝜉 , 𝜇𝐴
𝑅𝑁 1
(𝜉 ),𝜎𝐴
𝑅𝑁 1
𝑅𝑁 1
,𝜎𝐴
,𝛾𝐴
𝑅𝑁 1
(𝜉 ),𝛾𝐴
𝑅𝑁 1
𝑅𝑁 1
𝑅𝑁 1
> } for the Neutrosophic set
(𝜉 )+𝜎𝐴
𝑅𝑁 1
(𝜉 )+𝛾𝐴
𝑅𝑁 1
(𝜉 ) ≤ 3
(𝜉 )> : 𝜉 ∈𝑅𝐶 1 }.
Example 2.3 [7]
Each Intuitionistic fuzzy set 𝐴𝑅𝑁1 is a non-empty set in 𝑅𝑁 1 is obviously on Neutrosophic set
having the form 𝐴𝑅𝑁1 = {<𝜉 , 𝜇𝐴
𝑅𝑁 1
(𝜉 ),(1 − (𝜇𝐴
𝑅𝑁 1
+ 𝛾𝐴
𝑅𝑁 1
(𝜉))) ,𝛾𝐴
𝑅𝑁 1
(𝜉 )> : 𝜉 ∈𝑅𝐶 1 }
Definition 2.4 [7]
We must introduce the Neutrosophic set 0𝑁 and 1𝑁 in 𝑅𝑁 1 as follows: :
0𝑁 = {<𝜉, 0, 0, 1>: 𝜉 ∈ 𝑅𝑁 1 } & 1𝑁 = {< 𝜉, 1, 0, 0>: 𝜉∈ 𝑅𝑁 1 }
Definition 2.5 [7]
Let 𝑅𝑁 1 be a non-empty set and Neutrosophic sets 𝐴𝑅𝑁 1 and 𝐵𝑅𝑁1 in the form NS 𝐴𝑅𝑁1 = {<𝜉 ,
𝜇𝐴𝑅𝑁 (𝜉 ),𝜎𝐴𝑅𝑁 (𝜉 ),𝛾𝐴𝑅𝑁 (𝜉 ))> : 𝜉 ∈𝑅𝐶 1 }&𝐵𝑅𝑁1 = {<𝜉 , 𝜇𝐵𝑅𝑁 (𝜉 ),𝜎𝐵𝑅𝑁 (𝜉 ),𝛾𝐵𝑅𝑁 (𝜉 )> : 𝜉 ∈𝑅𝐶 1 } defined as:
1
1
(1)𝐴𝑅𝑁 1 ⊆ 𝐵𝑅𝑁 1 ⇔ 𝜇𝐴
(2)𝐴𝑅𝑁 1 𝐶 = {<𝜉 , 𝛾𝐵
𝑅𝑁 1
1
𝑅𝑁 1
( 𝜉 ) ≤ 𝜇𝐵
(𝜉 ), 𝜎𝐴
(3)𝐴𝑅𝑁1 ∩𝐵𝑅𝑁 1 ={<𝜉 , 𝜇𝐴
(4)𝐴𝑅𝑁 1 ∪𝐵𝑅𝑁1 ={<x, 𝜇𝐴
𝑅𝑁 1
𝑅𝑁 1
𝑅𝑁 1
𝑅𝑁 1
(𝜉 ),𝜇𝐵
(𝜉 ))⋀𝜇𝐵
(𝜉 )∨ 𝜇 𝐵
(5) ∩ 𝐴𝑗𝑅1𝐶 ={<𝜉 , ∧𝑗 𝜇𝐴𝑗
𝑅𝑁 1
(6) ∪ 𝐴𝑗𝑅𝑁1 = {<𝜉 , ∨𝑗 𝜇𝐴𝑗
(𝜉 ), 𝜎𝐴
𝑅𝑁 1
𝑅𝑁 1
𝑅𝑁 1
(𝜉 )>: 𝜉 ∈𝑅𝐶 1 }
(𝜉 ), 𝜎𝐴
(𝜉 ), 𝜎𝐴
(𝜉), ∧𝑗 𝜎𝐴𝑗
𝑅𝑁 1
𝑅𝑁 1
(𝜉 ), ≤ 𝜎𝐵
𝑅𝑁 1
𝑅𝑁 1
(𝜉), ∨𝑗 𝜎𝐴𝑗
𝑅𝑁 1
1
𝑅𝑁 1
(𝜉 ))⋀ 𝜎𝐵
(𝜉 ) ∨ 𝜎𝐵
𝑅𝑁 1
(𝜉), ∨𝑗 𝛾𝐴𝑗
𝑅𝑁 1
1
(𝜉), and 𝛾𝐵
𝑅𝑁 1
(𝜉), 𝛾𝐴
𝑅𝑁 1
(𝜉), ∧𝑗 𝛾𝐴𝑗
(𝜉),𝛾𝐴
𝑅𝑁 1
𝑅𝑁 1
𝑅𝑁 1
1
(𝜉 ) ≥ 𝛾𝐵
(𝜉 )∨ 𝛾𝐵
(𝜉 ) ⋀𝛾𝐵
(𝜉)> : 𝜉 ∈𝑅𝑁 1 }
𝑅𝑁 1
𝑅𝑁 1
𝑅𝑁 1
𝑅𝑁 1
(𝜉 )
(𝜉)>: 𝜉 ∈𝑅𝑁 1
(𝜉 )> : 𝜉 ∈𝑅𝑁 1 }
(𝜉)> : 𝜉 ∈𝑅𝑁 1 } for all
𝜉 ∈𝑅𝐶 1
Proposition 2.6 [9]
For all 𝐴𝑅𝑁1 and𝐵𝑅𝑁 1 are two Neutrosophic sets then the following condition are true:
(1) (𝐴𝑅𝑁1 ∩ 𝐵𝑅𝑁 1 )𝐶 = (𝐴𝑅𝑁 1 )𝐶 ∪(𝐵𝑅𝑁 1 )𝐶
(2) (𝐴𝑅𝑁1 ∪ 𝐵𝑅𝑁 1 )𝐶 = (𝐴𝑅𝑁 1 )𝐶 ∩ (𝐵𝐴 𝑁 )𝐶
𝑅 1
Definition 2.7 [10]
A Neutrosophic topology is a non -empty set 𝑅𝑁 1 is a family
𝑅𝑁 1 satisfying the following axioms:
(i) 0𝑁 , 1𝑁 ∈ 𝜏𝑁 𝑅𝑁
1
(ii) 𝐺𝑅𝑁 1 ∩𝐻𝑅𝑁1 ∈ 𝜏𝑁 𝑅𝑁 for any 𝐺𝑅𝑁 1 ,𝐻𝑅𝑁1 ∈ 𝜏𝑁 𝑅𝑁
1
(iii) ⋃𝑖 𝐺𝑖𝑅𝑁1 ∈ 𝜏𝑁 𝑅𝑁 for every 𝐺𝑖𝑅𝑁 1 ∈ 𝜏𝑁 𝑅𝑁 , I ∈ J
1
1
𝜏𝑁 𝑅𝑁 1 of Neutrosophic subsets in
1
The pair (𝑅𝑁 1 , 𝜏𝑁 𝑅𝑁 ) is called a Neutrosophic topological space.
1
The element Neutrosophic topological spaces of 𝜏𝑁 𝑅𝑁 are called Neutrosophic open sets.
1
A Neutrosophic set 𝐴𝑅𝑁 1 is closed if and only if 𝐴𝑅𝑁 1 𝐶 is Neutrosophic open.
Definition 2.8[10]
Let (𝑅𝑁 1 , 𝜏𝑁 𝑅𝑁 ) be Neutrosophic topological spaces.
1
T.RajeshKannan, and S.Chandrasekar, Neutrosophic 𝛼-Irresolute Multifunction In Neutrosophic Topological Spaces
Neutrosophic Sets and Systems, Vol. 32, 2020
𝐴𝑅𝑁1 = {<𝜉 ,𝜇𝐴
𝑅𝑁 1
(𝜉 ),𝜎𝐴
.1.Neu-Cl(𝐴𝑅𝑁1 ) =∩{ 𝐾 𝐴
2.Neu-Int(𝐴𝑅𝑁 1 ) = ∪{𝐺𝐴
𝑅𝑁 1
𝑅𝑁 1
𝑅𝑁 1
(𝜉 ),𝛾𝐴
:𝐾 𝐴
:𝐺𝐴
(𝜉 ))> : 𝜉 ∈𝑅𝑁 1 } be a Neutrosophic set in 𝑅𝑁 1
is a Neutrosophic closed set in 𝑅𝑁 1 and 𝐴𝑅𝑁1 ⊆ 𝐾 𝐴
𝑅𝑁 1
𝑅𝑁 1
𝑅𝑁 1
392
is a Neutrosophic open set in 𝑅𝑁 1 and 𝐺𝐴
𝑅𝑁 1
𝑅𝑁 1
}
⊆𝐴𝑅𝑁1 }.
3.Neutrosophic Semi-open if 𝐴𝑅𝑁1 ⊆ Neu-Cl(Neu-Int(𝐴𝑅𝑁1 )).
4.The complement of Neutrosophic Semi-open set is called Neutrosophic semi-closed.
5.Neu-sCl(𝐴𝑅𝑁1 ) =∩ { 𝐾 𝐴 𝑁 :𝐾 𝐴 𝑁 is a Neutrosophic Semi closed set in 𝑅𝑁 1 and 𝐴𝑅𝑁1 ⊆ 𝐾 𝐴
6.Neu-sInt(𝐴𝑅𝑁1 ) = ∪{𝐺𝐴
𝑅 1
𝑅 1
:𝐺𝐴
𝑅𝑁 1
is a Neutrosophic Semi open set in 𝑅𝐶 1 and 𝐺𝐴
𝑅𝑁 1
𝑅𝑁 1
𝑅𝑁 1
⊆𝐴𝑅𝑁 1 }.
7.Neutrosophic α-open set if 𝐴𝑅𝑁1 ⊆ Neu-Int(Neu-Cl(Neu-Int(𝐴𝑅𝑁1 ))).
8.The complement of Neutrosophic α-open set is called Neutrosophic α-closed.
9.Neuα- Cl(𝐴𝑅𝑁1 ) =∩{ 𝐾 𝐴 𝑁 :𝐾 𝐴 𝑁 is a Neutrosophic α - closed set in 𝑅𝑁 1 and 𝐴𝑅𝑁1 ⊆𝐾 𝐴
}
11.Neutrosophicpre open set if 𝐴𝑅𝑁1 ⊆ Neu-Int(Neu-Cl𝐴𝑅𝑁1 )).
12.The complement of Neutrosophic Pre-open set is called Neutrosophic pre-closed.
13.Neu- pCl(𝐴𝑅𝑁 1 ) =∩{ 𝐾 𝐴 𝑁 :𝐾 𝐴 𝑁 is a Neutrosophic P- closed set in 𝑅𝑁 1 and 𝐴𝑅𝑁1 ⊆𝐾 𝐴
}
𝑅 1
10.Neu α -Int(𝐴𝑅𝑁1 ) = ∪ {𝐺𝐴
𝑅𝑁 1
𝑅 1
:𝐺𝐴
𝑅𝑁 1
𝑅 1
𝑅 1
14.Neu- pInt(𝐴𝑅𝑁1 ) = ∪{𝐺𝐴
:𝐺𝐴
𝑅𝑁 1
𝑅𝑁 1
is a Neutrosophic α - open set in 𝑅
𝑁
1
and 𝐺𝐴
is a Neutrosophic P - open set in 𝑅
Remark:2.9[11]
Let 𝐴𝑅𝑁1 be an Neutrosophic topological space (𝑅𝑁 1 , 𝜏𝑁 𝑅𝐶 ).Then
𝑅𝑁 1
𝑁
1
}
𝑅𝑁 1
⊆𝐴𝑅𝑁 1 }.
and 𝐺𝐴
𝑅𝑁 1
𝑅𝑁 1
⊆ 𝐴𝑅𝑁1 }.
1
(i) Neu α-Cl(𝐴𝑅𝑁 1 ) = 𝐴𝑅𝑁 1 ∪ Neu-Cl(Neu-Int(Neu-Cl(𝐴𝑅𝑁1 ))).
(ii) Neu α-Int(𝐴𝑅𝑁1 ) = 𝐴𝑅𝑁1 ∩Neu-Int(Neu-Cl(Neu-Int(𝐴𝑅𝑁1 ))).
Definition 2.10[9]
Take 𝜉1 ,𝜉2 ,𝜉3 are belongs to real numbers 0 to 1 such that 0≤𝜉1 +𝜉2 +𝜉3 ≤1 .An Neutrosophic point
℘(𝜉1 , 𝜉2 , 𝜉3 )is Neutrosophic set defined by
℘(𝜉1 ,𝜉2 ,𝜉3 ) = {(𝜉1 , 𝜉2 , 𝜉3 )𝑖𝑓
𝜉=℘
(0,0,1)𝑖𝑓 𝜉 ≠ ℘
Take ℘(𝜉1 , 𝜉2 , 𝜉3 ) =<℘𝜉1 ℘𝜉2 . ℘𝜉3 > Where ℘𝜉1 ℘𝜉2 . ℘𝜉3 are represent Neutrosophic the degree of
membership function, the degree indeterminacy and the degree of non-membership function
respectively of each element 𝜉 ∈ 𝑅𝑁 1 to the set 𝐴𝑅𝑁1
Definition:2.11
A Neutrosophic set 𝐴𝑅𝑁 1 in 𝑅𝑁 1 is said to be quasi-coincident (q-coincident) with a
Neutrosophic set 𝐵𝑅𝑁 1 denoted by 𝐴𝑅𝑁 1 q𝐵𝑅𝑁1 if and only if there exists 𝜉 ∈𝑅𝑁 1 such that 𝐴𝑅𝑁 1 (𝜉 ) +
𝐵𝑅𝑁1 (𝜉 ) >1.
Remark: 2.12
𝐴𝑅 𝑁 1 q 𝐵𝑅 𝑁 1 ⟺ 𝐴𝑅 𝑁 1 ⊈ 𝐵𝑅 𝑁 1 𝐶
Definition 2.13[9]
let𝑅𝑁 1 and 𝑅𝑁 2 be two finite sets. Define 𝜓1 :𝑅𝑁 1 → 𝑅𝑁 2 .
If𝐴𝑅𝑁2 = {<𝜃, 𝜇𝐴 𝑁 (𝜃),𝜎𝐴 𝑁 (𝜃),𝛾𝐴 𝑁 (𝜃))> : 𝜃∈𝑅𝐶 2 }.is an NS in 𝑅𝑁 2 , then the inverse image( pre
𝑅 2
𝑅 2
𝑅 2
image) 𝐴𝑅𝑁 2 under 𝜓1 is an NS defined by 𝜓1 −1 (𝐴𝑅𝑁 2 )=< 𝜉, 𝜓1 −1 𝜇𝐴
where
𝜓1 (𝜇𝐴
𝑅𝑁 2
𝑅𝑁 2
(𝜃)), 𝜓1 (𝜎𝐴
(𝜃)), ={ sup 𝜇𝐴
0, elsewhere
𝜓1 (𝜎𝐴 𝑁 (𝜃))= { sup𝜎𝐴
𝑅 2
𝑅𝑁
𝑅𝑁 2
2
𝑅𝑁 2
(𝜉), 𝜓1 −1 𝜎𝐴
𝑅𝑁 2
(𝜉), 𝜓1 −1 𝛾𝐴
𝑅𝑁 2
𝜇𝑈 (𝜉), 𝜎𝑈 (𝜉), 𝛾𝑈 (𝜉) : 𝜉∈ 𝑅𝑁 1 :> under 𝜓1 is an NS defined
(𝜃)), 𝜓1 𝛾𝐴 𝑁 (𝜃): 𝜃∈ 𝑅𝑁 2 >
𝑅
(𝜉) : 𝜉∈𝑅𝑁 1 >. Also define image NS U=<𝜉,
by 𝜓1 (U)=< 𝜃, 𝜓1 (𝜇𝐴
𝑅𝑁 2
2
(𝜉), if 𝜓1 −1 (𝜃) ≠𝜙, 𝜉∈ 𝜓1 −1 (𝜃)
(𝜉) if 𝜓1 −1 (𝜃) ≠𝜙, 𝜉∈ 𝜓1 −1 (𝜃)
T.RajeshKannan, and S.Chandrasekar, Neutrosophic 𝛼-Irresolute Multifunction In Neutrosophic Topological Spaces
Neutrosophic Sets and Systems, Vol. 32, 2020
0, elsewhere
𝜓1 (𝛾𝐴 𝑁 (𝜃))= { inf (𝛾𝐴
𝑅𝑁 2
𝑅 2
393
(𝜉) if 𝜓1 −1 (𝜃) ≠ 𝜙, 𝜉∈ 𝜓1 −1 (𝜃)
0,
Elsewhere
Definition 2.14[2]
A mapping 𝜓1 :(𝑅𝑁 1 ,𝜏𝑁 𝑅𝑁 ) → (𝑅𝑁 2 ., 𝜏𝑁 𝑅𝑁 ) is called a
1
2
(1) Neutrosophic continuous(Neu-continuous ) if𝜓1 −1 (𝐴𝑅𝑁 2 )
C(NUTS𝑅𝑁 2 )
(2) Neutrosophic α-continuous(Neu α - continuous) if 𝜓1 −1 (𝐴𝑅𝑁 2 )
whenever𝐴𝑅𝑁 2 ∈ C(NUTS𝑅𝑁 2 )
∈ C(CTS𝑅𝐶 1 )whenever 𝐴𝑅𝑁 2 ∈
∈ αC(CTS𝑅𝐶 1 )
(3) Neutrosophic Semi-continuous(Neu Semi - continuous ) if 𝜓1 −1 (𝐴𝑅𝑁 2 )
whenever𝐴𝑅𝑁 2 ∈ C(NUTS𝑅
𝑁
2
)
∈ 𝑠C(CTS𝑅𝐶 1 )
Definition 2.15.
Let (𝑅𝐶 1 ,𝜏𝑁 𝑅𝐶 ) be a topological space in the classical sense and (𝑅𝑁 2 .,𝜏𝑁 𝑅𝑁 be an Neutrosophic
1
2
topological space. 𝛹 : (𝑅𝐶 1 ,𝜏𝑅𝐶 1 ) → (𝑅𝑁 2 .,𝜏𝑁 𝑅𝑁 ) is called a Neutrosophic multifunction if and only if
2
for each 𝜉 ∈ 𝑅𝐶 1 , 𝛹 (𝜉) is a Neutrosophic set in 𝑅𝑁 2 .
Definition 2.16
For a Neutrosophicmultifunction : 𝛹 : (𝑅𝐶 1 ,𝜏𝑅𝐶 1 ) → (𝑅𝑁 2 .,𝜏𝑁 𝑅𝑁 ),the upper inverse𝛹 + (Γ) and lower
2
inverse 𝛹 − (Γ) of a Neutrosophic set 𝛤𝑅𝑁2. in 𝑅𝑁 2 are defined as follows:
𝛹 + ( 𝛤𝑅𝑁 2 . ) ={ 𝜉 ∈ 𝑅𝐶 1 \ 𝛹 (𝜉 ) ≤ 𝛤𝑅𝑁 2. } and
𝛹 − ( 𝛤𝑅𝑁 2 . ) = {𝜉 ∈ 𝑅𝐶 1 \ 𝛹(𝜉 )q 𝛤𝑅𝑁 2. }.
Lemma 2.17.
For a Neutrosophicmultifunction 𝛹: (𝑅𝐶 1 ,𝜏𝑅𝐶 1 ) → (𝑅𝑁 2 .,𝜏𝑁 𝑅𝑁 ),
2
we have 𝛹 − (1- 𝛤𝑅𝑁 2 ) = 𝑅𝐶 1 - 𝛹 + ( 𝛤𝑅𝑁 2 . ), for any Neutrosophic set 𝛤𝑅𝑁 2 . in 𝑅𝑁 2 .
Lemma:2.18
Let 𝛤𝑅𝑁 2 be a subset of Neutrosophic topology 𝜏𝑁 𝑅𝑁 .then
2
1.𝛤𝑅𝑁 2 is𝛼 -closed in𝑅𝑁 2 iffNeu-SInt (Neu-Cl(𝛤𝑅𝑁 2 ) ⊂ 𝛤𝑅𝑁 2
2.Neu- SInt(Neu-Cl(𝛤𝑅𝑁 2 ) = 𝑁𝑒𝑢 − 𝐶𝑙(𝑁𝑒𝑢 − 𝐼𝑛𝑡(𝑁𝑒𝑢 − 𝐶𝑙(𝛤𝑅𝑁2 ))
Lemma:2.19
Let 𝛤𝑅𝑁 2 be a subset of Neutrosophic topology 𝜏𝑁 𝑅𝑁 .then below are equivalent
2
1𝛤𝑅𝑁 2 isNeu𝛼 -open in𝑅𝑁 2
2.𝑈𝑅𝑁 2 ⊂ 𝛤𝑅𝑁2 ⊂ 𝑁𝑒𝑢 − 𝐼𝑛𝑡(𝑁𝑒𝑢 − 𝐶𝑙(𝑈𝑅𝑁2 )) for some 𝑈𝑅𝑁2 of 𝑅𝑁 2 .
3.𝑈𝑅𝑁 2 ⊂ 𝛤𝑅𝑁2 ⊂ 𝑁𝑒𝑢 − 𝑆(𝐶𝑙(𝑈𝑅𝑁2 )) for some 𝑈𝑅𝑁 2 of 𝑅𝑁 2
4.𝛤𝑅𝑁 2 ⊂ 𝑁𝑒𝑢 − 𝑆𝐶𝑙(𝑁𝑒𝑢 − 𝐼𝑛𝑡(𝛤𝑅𝑁 2 ))
Definition 2.19[6]
A Neutrosophicmultifunction :𝛹 : (𝑅𝐶 1 ,𝜏𝑅𝐶 1 ) → (𝑅𝑁 2 ,𝜏𝑁 𝑅𝑁 ) is said to be 1.Neutrosophic upper semi
continuous at a point 𝜉∈𝑅𝐶 1 if for any 𝛤𝑅𝑁 2
𝑈𝑅𝐶 1 ∈ O(CTS𝑅𝐶 1 ) such that 𝛹 (𝑈𝑅𝐶 1 ) ⊂ 𝛤𝑅𝑁 2 .
2
O(NUTS𝑅𝑁 2 ), 𝛤𝑅𝑁 2 . containing𝛹(𝜉) ,there exist 𝜉 ∈
2.Neutrosophic lower semi continuous at a point 𝜉∈𝑅𝐶 1 if for any𝛤𝑅𝑁 2 O(NUTS𝑅𝑁 2 ), with
𝛹 (𝜉)q𝛤𝑅𝑁 2 , there exist x ∈ 𝑈𝑅𝐶1 ∈ O(CTS𝑅𝐶 1 ) such that 𝛹(𝑈𝑅𝐶 1 )q𝛤𝑅𝑁 2
3.Neutrosophic upper semi continuous (Neutrosophic lower semi continuous) if it is Neutrosophic
upper semi continuous (Neutrosophic lower semi continuous) at each point 𝜉∈𝑅𝐶 1 .
4.Neutrosophic upper pre -continuous at a point 𝜉∈𝑅𝐶 1 if for any 𝛤𝑅𝑁 2 O(NUTS𝑅𝑁 2 ), Γ containing
T.RajeshKannan, and S.Chandrasekar, Neutrosophic 𝛼-Irresolute Multifunction In Neutrosophic Topological Spaces
Neutrosophic Sets and Systems, Vol. 32, 2020
394
𝛹(𝜉) ,there exist 𝜉 ∈ 𝑈𝑅𝐶 1 ∈PO(CTS𝑅𝐶 1 ) such that 𝛹 (𝑈𝑅𝐶1 ) ⊂ 𝛤𝑅𝑁2
5.Neutrosophic lower pre- continuous at a point 𝜉∈𝑅𝐶 1 if for any𝛤𝑅𝑁 2 O(NUTS𝑅𝑁 2 ),
with𝛹 (𝜉)q𝛤𝑅𝑁 2 , there exist 𝜉 ∈ 𝑈𝑅𝐶 1 ∈PO(CTS𝑅𝐶 1 ) such that 𝛹(𝑈𝑅𝐶 1 )q𝛤𝑅𝑁 2
6.Neutrosophic upper pre-continuous (Neutrosophic lower pre-continuous) if it is Neutrosophic
upper pre-continuous (Neutrosophic lower pre-continuous) at each point 𝜉∈𝑅𝐶 1 .
7.Neutrosophic upper 𝛼 -continuous at a point 𝜉∈𝑅𝐶 1 if for any 𝛤𝑅𝑁2 O(NUTS𝑅𝑁 2 ), Γ containing
𝛹(𝜉) (that is , F (𝜉) ⊂ Γ), there exist 𝜉 ∈ 𝑈𝑅𝐶 1 ∈ 𝛼 O(CTS𝑅𝐶 1 ) such that 𝛹 (𝑈𝑅𝐶 1 ) ⊂ 𝛤𝑅𝑁 2
8.Neutrosophic lower 𝛼 - continuous at a point 𝜉∈𝑅𝐶 1 if for any𝛤𝑅𝑁 2 O(NUTS𝑅𝑁 2 ),
with𝛹 (𝜉)q𝛤𝑅𝑁 2 , there exist x ∈ 𝑈𝑅𝐶1 ∈ 𝛼 O(CTS𝑅𝐶 1 ) such that 𝛹(𝑈𝑅𝐶 1 )q𝛤𝑅𝑁 2
9.Neutrosophic upper 𝛼 -continuous (Neutrosophic lower 𝛼 -continuous) if it is Neutrosophic
upper 𝛼 -continuous (Neutrosophic lower 𝛼 -continuous) at each point 𝜉∈𝑅𝐶 1 .
10.Neutrosophic upper quasi-continuous at a point 𝜉∈𝑅𝐶 1 if for any 𝛤𝑅𝑁 2
O(NUTS𝑅𝑁 2 ),𝛤𝑅𝑁 2 containing 𝛹 (𝜉) ,there exist 𝜉 ∈ 𝑈𝑅𝐶 1 ∈ 𝑆 O(CTS𝑅𝐶 1 ) such that 𝛹 (𝑈𝑅𝐶 1 )
⊂ 𝛤𝑅𝑁 2
11.Neutrosophic lower quasi semi continuous at a point 𝜉∈𝑅𝐶 1 if for any𝛤𝑅𝑁 2 O(NUTS𝑅𝑁 2 ), with
𝛹 (𝜉)q𝛤𝑅𝑁 2 , there exist 𝜉 ∈ 𝑈𝑅𝐶1 ∈S O(CTS𝑅𝐶 1 ) such that 𝛹(𝑈𝑅𝐶 1 )q𝛤𝑅𝑁 2
12.Neutrosophic upper quasi semi continuous (Neutrosophic lower quasi semi continuous) if it is
Neutrosophic upper quasi semi continuous (Neutrosophic lower quasi semi continuous) at each
point 𝜉∈𝑅𝐶 1 .
III. Lower 𝛼 -Irresolute Neutrosophic Multifunctions
In this section, we introduce the Definition for Neutrosophic Lower 𝛼 - irresolute
multifunction and its properties
Definition 3.1.
An Neutrosophic multifunction 𝛹 : (𝑅𝐶 1 ,𝜏𝑅𝐶 1 ) → (𝑅𝑁 2 .,𝜏𝑁 𝑅𝑁 ) is said to be
2
(1) Neutrosophic lower α-irresolute at a point 𝑥0 ∈ 𝑅𝐶 1 , if for any 𝛤𝑅𝑁2 ∈ 𝛼𝑂(𝑁𝑈𝑇𝑆𝑅𝑁 2 ) such
that 𝛹(𝑥0 )𝑞 𝛤𝑅𝑁 2 there exists 𝑈𝑅𝐶 1 ∈ 𝛼O(CTS𝑅𝐶 1 ) containing 𝑥0 such that 𝛹(𝜉)𝑞𝛤𝑅𝑁 2 , ∀𝜉 ∈ 𝑈𝑅𝐶 1
(2) Neutrosophic lower 𝛼 -irresolute if it is Neutrosophic lower 𝛼 -irresolute at each point of 𝑅𝐶 1 .
Theorem 3.2
Every Neutrosophic
lower𝛼-irresolute multifunction is Neutrosophic lower 𝛼-continuous
multifunction.
Proof:
Letting𝑥0 ∈ 𝑅𝐶 1 , 𝛹:(𝑅𝐶 1 ,𝜏𝑅𝐶 1 ) → (𝑅𝑁 2 .𝜏𝑁 𝑅𝑁 ) and 𝛤𝑅𝑁 2 ∈ 𝑂(𝑁𝑈𝑇𝑆𝑅𝑁 2 )
2
such that 𝛹(𝑥0 )𝑞 𝛤𝑅𝑁 2 . But we know that , Every𝛤𝑅𝑁2 , 𝛤𝑅𝑁 2 ∈ 𝑂(𝑁𝑈𝑇𝑆𝑅𝑁 2 ) is
𝛤𝑅𝑁2 ∈ 𝛼𝑂(𝑁𝑈𝑇𝑆𝑅𝑁 2 ),.Therefore
𝛤𝑅𝑁2 ∈ 𝛼𝑂(𝑁𝑈𝑇𝑆𝑅𝑁 2 ).By our assumption
lower𝛼 −irresolute multifunction, there
exists 𝑈𝑅𝐶 1 ∈ 𝛼O(CTS𝑅
𝐶
1
,Neutrosophic
) containing 𝑥0 such that
𝛹(𝜉)𝑞𝛤𝑅𝑁 2 , ∀𝜉 ∈ 𝑈𝑅𝐶 1 .Hence𝛹 is Neutrosophic lower 𝛼-continuous multifunction at 𝑥0 .
Theorem 3.3
Every Neutrosophic lower𝛼 − irresolute multifunction is Neutrosophic lower Pre continuous
multifunction.
Proof:
Letting 𝑥0 ∈ 𝑅𝐶 1 , 𝛹 : (𝑅𝐶 1 ,𝜏𝑅𝐶 1 ) → (𝑅𝑁 2 .,𝜏𝑁 𝑅𝑁 )
and 𝛤𝑅𝑁2 ∈ 𝑂(𝑁𝑈𝑇𝑆𝑅𝑁 2 ) such that
2
𝛹(𝑥0 )𝑞 𝛤𝑅𝑁 2 . 𝐵ut we know that , Every𝛤𝑅𝑁2 , 𝛤𝑅𝑁2 ∈ 𝑂(𝑁𝑈𝑇𝑆𝑅𝑁 2 )is
𝛤𝑅𝑁2 ∈ 𝛼𝑂(𝑁𝑈𝑇𝑆𝑅𝑁 2 ).Therefore
𝛼-irresolute
multifunction,
𝛤𝑅𝑁2 ∈ 𝛼𝑂(𝑁𝑈𝑇𝑆𝑅𝑁 2 ).By our assumption ,Neutrosophic lower
there
exists
𝑈𝑅𝐶 1 ∈ 𝛼O(CTS𝑅𝐶 1 )
containing 𝑥0
𝛹(𝑥0 )𝑞𝛤𝑅𝑁 2 , ∀𝑥 ∈ 𝑈𝑅𝐶 1 . every𝑈𝑅𝐶 1 , 𝑈𝑅𝐶 1 ∈ 𝛼𝑂(𝑁𝑈𝑇𝑆𝑅𝑁 2 ) is 𝑈𝑅𝐶 1 ∈ 𝑃𝑂(𝐶𝑇𝑇𝑆𝑅𝑁 2 ).
such
that
T.RajeshKannan, and S.Chandrasekar, Neutrosophic 𝛼-Irresolute Multifunction In Neutrosophic Topological Spaces
Neutrosophic Sets and Systems, Vol. 32, 2020
There
395
exists𝑈𝑅𝐶 1 ∈ 𝑃O(CTS𝑅𝐶 1 ) containing 𝑥0 such that
𝛹(𝜉)𝑞𝛤𝑅𝑁 2 , ∀𝜉 ∈ 𝑈𝑅𝐶 1 .Hence𝛹 is
Neutrosophic lower Pre-continuous multifunction at 𝑥0 .
Theorem 3.4
Every Neutrosophic
lower𝛼-irresolute multifunction is Neutrosophic lower
quasi semi
continuous multifunction.
Proof:
Letting 𝑥0 ∈ 𝑅𝐶 1 , 𝛹 : (𝑅𝐶 1 ,𝜏𝑅𝐶 1 ) → (𝑅𝑁 2 .,𝜏𝑁 𝑅𝑁 ) and 𝛤𝑅𝑁 2 ∈ 𝑂(𝑁𝑈𝑇𝑆𝑅𝑁 2 ) such that
2
𝛹(𝑥0 )𝑞 𝛤𝑅𝑁 2 , But we know that , Every𝛤𝑅𝑁2 , 𝛤𝑅𝑁 2 ∈ 𝑂(𝑁𝑈𝑇𝑆𝑅𝑁 2 )is
𝛤𝑅𝑁2 ∈ 𝛼𝑂(𝑁𝑈𝑇𝑆𝑅𝑁 2 ),Therefore𝛤𝑅𝑁 2 ∈ 𝛼𝑂(𝑁𝑈𝑇𝑆𝑅𝑁 2 ).By our assumption ,Neutrosophic
𝛼 − irresolute multifunction, There
exists 𝑈𝑅𝐶1 ∈ 𝛼O(CTS𝑅
𝐶
lower
) containing 𝑥0 such that
𝛹(𝜉)𝑞𝛤𝑅𝑁 2 , ∀𝜉 ∈ 𝑈𝑅𝐶 1 Here every 𝑈𝑅𝐶 1 , 𝑈𝑅𝐶 1 ∈ 𝛼𝑂(𝑁𝑈𝑇𝑆𝑅 2 ) is 𝑈𝑅𝐶 1 ∈ 𝑆𝑂(𝐶𝑇𝑇𝑆𝑅𝑁 2 ). Finally
we get , There exists 𝑈𝑅𝐶 1 ∈ 𝑆O(CTS𝑅𝐶 1 ) containing 𝑥0 such that 𝛹(𝜉)𝑞𝛤𝑅𝑁 2 , ∀𝜉 ∈ 𝑈𝑅𝐶 1 hence 𝛹 is
𝑁
1
Neutrosophic lower quasi semi continuous multifunction at 𝑥0 .
Theorem 3.5
Let 𝛹 ∶ (𝑅𝐶 1 , 𝜏𝑅𝐶1 ) → (𝑅𝑁 2 , 𝜏𝑁 𝑅𝑁 ), be an Neutrosophic multifunction and letting 𝑥0 ∈ 𝑅𝐶 1 .
2
Then the following statements are equivalent:
(a) 𝛹 is Neutrosophic lower α-irresolute at 𝑥0 .
(b) For any 𝛤𝑅𝑁 2 , 𝛤𝑅𝑁 2 ∈ 𝛼𝑂(𝑁𝑈𝑇𝑆𝑅𝑁 2 ) with (𝑥0 )𝑞𝛤𝑅𝑁 2 , ⟹ 𝑥0 ∈ 𝑠𝐶𝑙(𝐼𝑛𝑡(𝛹 − (𝛤𝑅𝑁2 ))).
𝛤𝑅𝑁 2 , 𝛤𝑅𝑁 2 ∈ 𝛼𝑂(𝑁𝑈𝑇𝑆𝑅𝑁 2 ) with
such that 𝛹(𝜉)𝑞𝑉𝑅𝐶 1 , ∀ 𝜉 ∈ 𝑉𝑅𝐶 1
(c) For any 𝑈𝑅𝐶1 , 𝑈𝑅𝐶 1 ∈ 𝑆𝑂(𝐶𝑇𝑆𝑈𝑅𝐶1 ) ,𝑥0 𝜖𝑈𝑅𝐶 1 and for each
𝛹(𝑥0 )𝑞𝛤𝑅𝑁 2 , there exists a 𝑉𝑅𝐶1 ∈ 𝑂(𝐶𝑇𝑆𝑅𝐶 1 ) , 𝑉𝑅𝐶1 ⊂ 𝑈𝑅𝐶1
Proof.
(a) ⇒ (b). Let 𝑥0 ∈ 𝑅𝐶 1 and𝛤𝑅𝑁 2 ∈ 𝛼𝑂(𝑁𝑈𝑇𝑆𝑅𝑁 2 ) such that
𝛹(𝑥0 )𝑞𝛤𝑅𝑁 2 Then by our assumption
(a) , we get there exists 𝑈𝑅𝐶1 ∈ 𝛼O(CTS𝑅 1 ) such that 𝑥0 ∈ 𝑈𝑅𝐶 1 and 𝐹(𝜉)𝑞𝛤𝑅𝑁 2 , ∀ 𝜉 ∈ 𝑈𝑅𝐶 1 .Thus
𝑥0 ∈ 𝑈𝑅𝐶 1 ⊂ 𝛹 − (𝛤𝑅𝑁 2 ) … … (1) Here 𝑈𝑅𝐶 1 ∈ 𝛼O(CTS𝑅𝐶 1 ) .we know that for any set 𝐴𝑅𝐶 1 , 𝐴𝑅𝐶1 ∈
𝐶
𝛼O(CTS𝑅𝐶 1 )⟺ 𝐴𝑅𝐶 1 ⊂ 𝑠𝐶𝑙 (𝐼𝑛𝑡(𝐴𝑅𝐶 1 )). Therefore, 𝑈𝑅𝐶 1 ⊂ 𝑠𝐶𝑙 (𝐼𝑛𝑡(𝑈𝑅𝐶 1 )) … (2). from(1) and(2),
we get 𝑥0 ∈ 𝑠𝐶𝑙 (𝐼𝑛𝑡𝛹 − (𝛤𝑅𝑁 2 )).Hence (b).
(b) ⇒ (c). Let 𝛤𝑅𝑁 2 ∈ 𝛼𝑂(𝑁𝑈𝑇𝑆𝑅𝑁 2 ) such that (𝑥0 )𝑞𝛤𝑅𝑁 2 , then 𝑥0 ∈ 𝑠𝐶𝑙 (𝐼𝑛𝑡𝛹 − (𝛤𝑅𝑁 2 )). Let 𝑈𝑅𝐶 1 ∈
𝑠O(CTS𝑅𝐶 1 ) and 𝑥0 ∈ 𝑈𝑅𝐶1 .Then 𝑈𝑅𝐶 1 ∩ 𝐼𝑛𝑡 (𝛹 − (𝛤𝑅𝑁2 )) ≠ 𝜑 and 𝑈𝑅𝐶1 ∩ 𝐼𝑛𝑡 (𝛹 − (𝛤𝑅𝑁 2 )) is semi-
open in 𝑅𝐶 1 . Put 𝑉𝑅𝐶 1 = 𝐼𝑛𝑡(𝑈𝑅𝐶 1 ∩ 𝐼𝑛𝑡(𝛹 − (𝛤𝑅𝑁 2 )), Then 𝑉𝑅𝐶 1 is an open set of 𝑅𝐶 1 , 𝑉𝑅𝐶 1 ⊂
𝑈𝑅𝐶1 , 𝑉𝑅𝐶 1 ≠ 𝜑and
𝛹(𝑣)𝑞𝛤𝑅𝑁 2 , ∀𝑣 ∈ 𝑉𝑅𝐶 1 . (c) ⇒ (a).Let {𝑈𝜉 } be the system of the 𝑠O(CTS𝑅𝐶 1 )
containing 𝜉.
Let 𝑈𝑅𝐶 1 ∈ 𝑆O(CTS𝑅𝐶 1 ) and 𝑥0 ∈ 𝑈𝑅𝐶 1 and Any 𝛤𝑅𝑁 2 ∈ 𝛼𝑂(𝑁𝑈𝑇𝑆𝑌) such that 𝛹(𝑥0 )𝑞𝛤𝑅𝑁 2 , there
𝛹(𝑣)𝑞𝛤𝑅𝑁 2 ∀𝑣 ∈ 𝐵𝑈 . Let 𝑊𝑅𝐶1 = ∪ 𝐵𝑈 ∶ 𝑈 ∈
𝑥0 ∈ 𝑠𝐶𝑙(𝑊𝑅𝐶1 ) and 𝛹(𝑣)𝑞𝛤𝑅𝑁 2 , ∀𝑣 ∈ 𝑊𝑅𝐶 1 .Put 𝑆 𝑅𝐶 1 =
exists a nonempty open set 𝐵𝑈 ⊂ 𝑈𝑅𝐶 1 Such that
𝐶
{𝑈𝑥0 }, then 𝑊𝑅𝐶 1 ∊ O(CTS𝑅 1 ),and
𝑊𝑅𝐶 1 ∪ {𝑥0 }, then 𝑊𝑅𝐶 1 ⊂ 𝑆 𝑅𝐶 1 ⊂ 𝑠𝐶𝑙(𝑊𝑅𝐶 1 ). Thus 𝑆 𝑅𝐶 1 ∈ 𝛼𝑂(𝐶𝑇𝑆𝑅𝐶 1 ) ), 𝑥0 ∈ 𝑆 𝑅𝐶1
and
𝛹(𝑣)𝑞𝛤𝑅𝑁 2 , ∀𝑣 ∈ 𝑆 𝑅𝐶 1 . Hence 𝛹is Neutrosophic lower α-irresolute at 𝑥0 .
Theorem 3.6
Let𝛹 ∶ (𝑅𝐶 1 , 𝜏𝑅𝐶 1 ) → (𝑅𝑁 2 . , 𝜏𝑁 𝑅𝑁 ), be
2
an Neutrosophic multifunction. Then the following
statements are equivalent:
(a) 𝛹 is Neutrosophic lower α-irresolute.
(b)𝛹 − (𝜆𝑅𝑁 2 ) ∈ 𝛼𝑂(𝐶𝑇𝑆𝑅𝐶 1 ),for every Neutrosophic α-open set 𝜆𝑅𝑁 2 of 𝑅𝑁 2 .
+
(c) 𝛹 (𝛽𝑅𝑁 2 ) ∈ 𝛼𝐶(𝐶𝑇𝑆𝑅𝐶 1 ),for every Neutrosophic α-closed set 𝛽𝑅𝑁 2 of 𝑅𝑁 2 .
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396
(d) 𝑠𝐼𝑛𝑡(𝐶𝑙(𝛹 + (𝛤𝑅𝑁2 ))) ⊂ 𝛹 + (𝑁𝑒𝑢 − 𝛼𝐶𝑙(𝛤𝑅𝑁2 )), for each Neutrosophic set𝛤𝑅𝑁2 of 𝑅𝑁 2 .
(e) 𝛹 (𝑠𝐼𝑛𝑡 (𝐶𝑙(𝑉𝑅𝐶 1 ))) ⊂ 𝑁𝑒𝑢 − 𝛼𝐶𝑙(𝛹(𝑉𝑅𝐶 1 )),for each subset𝑉𝑅𝐶 1 of 𝑅𝐶 1 .
(f) 𝛹 (𝛼𝐶𝑙(𝑉𝑅𝐶 1 )) ⊂ 𝑁𝑒𝑢 − 𝛼𝐶𝑙(𝛹(𝑉𝑅𝐶 1 )),for each subset 𝑉𝑅𝐶 1 of 𝑅𝐶 1 ,
(g) 𝛼𝐶𝑙(𝛹 + (𝛤𝑅𝑁2 )) ⊂ 𝛹 + (𝑁𝑒𝑢 − 𝛼𝐶𝑙(𝛤𝑅𝑁2 )),for each Neutrosophic set 𝛤𝑅𝑁 2 of 𝑅𝑁 2 .
(h) 𝛹 (𝐶𝑙 (𝐼𝑛𝑡 (𝐶𝑙(𝐴𝑅𝐶 1 )))) ⊂ 𝑁𝑒𝑢 − 𝛼𝐶𝑙(𝛹(𝐴𝑅𝐶 1 )),for each subset 𝐴𝑅𝐶1 of 𝑅𝐶 1 .
Proof.
(a)⇒(b). Let 𝜆𝑅𝑁 2 ∈ 𝛼𝑂(𝑁𝑈𝑇𝑆𝑅𝑁 2 )
and
𝑥0 ∈ 𝛹 − (𝜆𝑅𝑁2 )such
that𝛹 (𝑥0 )𝑞𝜆𝑅𝑁 2 ,
𝛹 is
since
Neutrosophic lower α-irresolute, Applying previous theorem, it follows that 𝑥0 ∈
−
−
𝑠𝐶𝑙(𝐼𝑛𝑡(𝛹 − (𝜆𝑅𝑁 2 ))). As 𝑥0 is chosen arbitray in 𝛹 (𝜆𝑅𝑁2 ), we have 𝛹 (𝜆𝑅𝑁 2 ) ⊂ 𝑠𝐶𝑙(𝐼𝑛𝑡𝛹 − (𝜆𝑅𝑁2 ))
−
−
and thus 𝛹 (𝜆𝑅𝑁2 ) ∈ 𝛼𝑂(𝐶𝑇𝑆𝑅𝐶 1 ). Hence 𝛹 (𝜆𝑅𝑁2 ) is an 𝛼 −open in 𝑅𝐶 1 .(b)⇒(a). Let 𝑥0 ∈ 𝑅𝐶 1 and
𝜆𝑅𝑁 2 ∈ 𝛼𝑂(𝑁𝑈𝑇𝑆𝑅𝑁 2 )such that 𝛹(𝑥0 )𝑞𝜆𝑅𝑁 2 , so that𝑥0 ∈ 𝛹 −(𝜆𝑅𝑁2 ). By hypothesis 𝛹− (𝜆𝑅𝑁2 ) ∈
𝛼𝑂(𝐶𝑇𝑆𝑅𝐶 1 ). We have 𝑥0 ∈ 𝛹 − (𝜆𝑅𝑁 2 ) ⊂ 𝑠𝐶𝑙(𝐼𝑛𝑡(𝛹 − (𝜆𝑅𝑁 2 ))) and we get 𝛹 is Neutrosophic lower
α-irresolute at 𝑥0. . As 𝑥0 was arbitrarily chosen, 𝛹 is Neutrosophic lower α-irresolute.
(b)⇔(c). From the definition, both are equivalent.
𝛤𝑅𝑁 2 ∈ (𝑁𝑈𝑇𝑆𝛤𝑅𝑁 2 ). taking closure , Neu- 𝛼𝐶𝑙(𝛤𝑅𝑁2 ) is Neutrosophic α-closed set in
+
𝑅 2 . By our assumption, 𝛹 (𝑁𝑒𝑢 − 𝛼𝐶𝑙(𝛤𝑅𝑁2 )) ∈ 𝛼𝐶(𝐶𝑇𝑆𝑅𝐶 1 ).
(c)⇒ (d).Let
𝑁
We know that sIntCl(𝐴𝑅𝐶 1 ) ⊂ 𝐴𝑅𝐶 1 iff 𝐴𝑅𝐶 1 ∈ 𝛼𝐶(𝐶𝑇𝑆𝑅𝐶 1 ).
+
we obtain 𝛹 (𝑁𝑒𝑢 − 𝛼𝐶𝑙(𝛤𝑅𝑁2 )) ⊃ 𝑠𝐼𝑛𝑡 (𝐶𝑙 (𝛹 + (𝑁𝑒𝑢 − 𝐶𝑙(𝛤𝑅𝑁 2 )))) ⊃ 𝑠𝐼𝑛𝑡 (𝐶𝑙 (𝛹 + (𝛤𝑅𝑁2 ))).
(d) ⇒ (e) Suppose that (d) is satisfied and let 𝑉𝑅𝐶 1 be an arbitrary subset of 𝑅𝐶 1 . Let us Take𝛤𝑅𝑁 2 =
𝛹(𝑉𝑅𝐶 1 ), Then𝑉𝑅𝐶 1 ⊂ 𝛹 + (𝛤𝑅𝑁 2 ). Therefore, by hypothesis, we have
𝑠𝐼𝑛𝑡(𝐶𝑙(𝑉𝑅𝐶1 )) ⊂ 𝑠𝐼𝑛𝑡(𝐶𝑙(𝛹 + (𝛤𝑅𝑁2 ))) ⊂ 𝛹 + (𝑁𝑒𝑢 − 𝛼𝐶𝑙(𝛤𝑅𝑁2 )).
Therefore,𝛹 (𝑠𝐼𝑛𝑡 (𝐶𝑙(𝑉𝑅𝐶 1 ))) ⊂ 𝛹 (𝛹+ (𝑁𝑒𝑢 − 𝛼𝐶𝑙(𝛤𝑅𝑁 2 ))) ⊂ 𝑁𝑒𝑢 − 𝛼𝐶𝑙(𝛤𝑅𝑁 2 ) = 𝑁𝑒𝑢 −
𝛼𝐶𝑙 (𝛹(𝑉𝑅𝐶 1 )).
(e) ⇒(c).Suppose that (e) is true. and let 𝛤𝑅𝑁 2 ∈ 𝛼𝐶(𝑁𝑈𝑇𝑆𝑅𝑁 2 ). Put 𝑉𝑅𝐶 1 = 𝛹 + (𝛤),Then 𝛹 (𝑉𝑅𝐶 1 ) ⊂
𝛤𝑅𝑁 2 . Therefore, by our
hypothesis, we have
𝑁𝑒𝑢 − 𝛼𝐶𝑙(𝛤𝑅𝑁 2 ) = 𝛤𝑅𝑁 2 .And
𝛹 (𝑠𝐼𝑛𝑡 (𝐶𝑙(𝑉𝑅𝐶 1 ))) ⊂ 𝑁𝑒𝑢 − 𝛼𝐶𝑙 (𝛹(𝑉𝑅𝐶 1 )) ⊂
𝛹+ (𝛹(𝑠𝐼𝑛𝑡(𝐶𝑙( 𝑉𝑅𝐶1 )))) ⊂ 𝛹 + (𝛤𝑅𝑁2 ). Since we always have
𝛹+ (𝛹(𝑠𝐼𝑛𝑡(𝐶𝑙( 𝑉𝑅𝐶1 )))) ⊃ 𝑠𝐼𝑛𝑡(𝐶𝑙( 𝑉𝑅𝐶1 )),Then must verify 𝛹+ (𝛤𝑅𝑁2 ) ⊃ 𝑠𝐼𝑛𝑡 (𝐶𝑙 (𝛹 + (𝛤𝑅𝑁2 ))). We
know that sIntCl𝑉𝑅𝐶1 ⊂ 𝑉𝑅𝐶 1 iff 𝑉𝑅𝐶 1 ∈ 𝛼𝐶(𝐶𝑇𝑆𝑅𝐶 1 ),Finally we get 𝐹 + (𝛤𝑅𝑁 2 ) ∈ 𝛼𝐶(𝐶𝑇𝑆𝑅𝐶 1 ).
(c)⇒ (f). Here 𝑉𝑅𝐶 1 ⊂ 𝛹 + (𝛹( 𝑉𝑅𝐶 1 )), we have 𝑉𝑅𝐶1 ⊂ 𝛹 + (𝑁𝑒𝑢 − 𝐶𝑙(𝛹( 𝑉𝑅𝐶 1 ))). NowNeu+
𝛼𝐶𝑙(𝛹( 𝑉𝑅𝐶 1 )) is an Neutrosophic α-closed set in 𝑅𝑁 2 and so by our assumption,𝛹 (𝑁𝑒𝑢 −
𝐶𝑙(𝛹( 𝑉𝑅𝐶 1 ))) ∈ 𝛼𝐶(𝐶𝑇𝑆𝑅𝐶 1 ).Thus𝛼𝐶𝑙( 𝑉𝑅𝐶 1 ) ⊂ 𝛹𝛹 + (𝑁𝑒𝑢 − 𝛼𝐶𝑙(𝛹( 𝑉𝑅𝐶 1 ))).
Consequently, 𝛹 (𝛼𝐶𝑙(𝑉𝑅𝐶 1 )) ⊂ 𝛹 (𝛹+ (𝑁𝑒𝑢 − 𝛼𝐶𝑙 (𝛹(𝑉𝑅𝐶 1 )))) ⊂ 𝑁𝑒𝑢 − 𝛼𝐶𝑙(𝛹( 𝑉𝑅𝐶 1 )).
(f)⇒ (c).Let
𝛤𝑅𝑁 2 ∈ 𝛼𝐶𝑂(𝑁𝑈𝑇𝑆𝑅𝑁 2 ). Replacing 𝑉𝑅𝐶1 by 𝛹+ we get by(f), 𝛹 (𝛼𝐶𝑙(𝛹+ (𝛤𝑅𝑁 2 ))) ⊂
𝑁𝑒𝑢 − 𝛼𝐶𝑙(𝛹(𝛹+ (𝛤𝑅𝑁 2 ))) ⊂ 𝑁𝑒𝑢 − 𝛼𝐶𝑙(𝛤𝑅𝑁 2 ) = 𝛤𝑅𝑁 2 .Consequently, 𝛼𝐶𝑙(𝛹 + (𝛤𝑅𝑁 2 )) ⊂ 𝛹 + (𝛤𝑅𝑁 2 ).
+
But 𝛹 (𝛤𝑅𝑁2 ) ⊂ 𝛼𝐶𝑙(𝛹 + (𝛤𝑅𝑁 2 )) and so, 𝛼𝐶𝑙(𝛹 + (𝛤𝑅𝑁 2 )) = 𝛹 + (𝛤𝑅𝑁2 ).
+
Thus 𝛹 (𝛤𝑅𝑁2 ) ∈ 𝛼𝐶(𝐶𝑇𝑆𝑅𝐶 1 ).
T.RajeshKannan, and S.Chandrasekar, Neutrosophic 𝛼-Irresolute Multifunction In Neutrosophic Topological Spaces
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397
(f) ⇒ (g). Let𝛤𝑅𝑁 2 be any Neutrosophic set of 𝑅𝑁 2 . Replacing 𝑉𝑅𝐶 1 by
𝛹+ (𝛤𝑅𝑁2 ) we get by
(f),𝛹 (𝛼𝐶𝑙 (𝛹+ (𝛤𝑅𝑁 2 ))) ⊂ 𝑁𝐸𝑈 − 𝛼𝐶𝑙(𝛹(𝛹+ (𝛤𝑅𝑁 2 ))) ⊂ 𝑁𝑒𝑢 − 𝛼𝐶𝑙(𝛤𝑅𝑁 2 ).Therefore
𝛼𝐶𝑙(𝛹 + (𝛤𝑅𝑁 2 )) ⊂ 𝛹 + (𝑁𝑒𝑢 − 𝛼𝐶𝑙(𝛤𝑅𝑁2 )).
we
get
(g)⇒ (f). Replacing 𝛤𝑅𝑁 2 by 𝛹 ( 𝑉𝑅𝐶 1 ), where 𝑉𝑅𝐶1 is a subset of 𝑅𝐶 1 , we get by our result
(g),𝛼𝐶𝑙(𝑉𝑅𝐶 1 ) ⊂ 𝛼𝐶𝑙(𝛹 + (𝛹(𝑉𝑅𝐶 1 ))) = 𝛼𝐶𝑙(𝛹 + (𝛤𝑅𝑁2 )) = 𝛹 + (𝛼𝐶𝑙(𝛤𝑅𝑁2 )) = 𝛹 + (𝛼𝐶𝑙(𝛹(𝑉𝑅𝐶1 ))).Thus
𝛹(𝛼𝐶𝑙(𝑉𝑅𝐶 1 )) ⊂ 𝛹(𝛹+ (𝛼𝐶𝑙(𝛹(𝑉𝑅𝐶 1 ))) ⊂ 𝑁𝑒𝑢 𝛼𝐶𝑙(𝛹(𝑉𝑅𝐶 1 )).
(e)⇒ (h).Clearly is true from the above result.
(h)⇒(a). Let 𝜉 ∈ 𝑅𝐶 1 and 𝛤𝑅𝑁 2 ∈ 𝛼𝑂(𝑁𝑈𝑇𝑆𝑅𝑁 2 )such that (𝜉 )𝑞𝛤𝑅𝑁2 . Then 𝜉 ∈ 𝛹 − (𝛤𝑅𝑁2 ).We shall
show that
𝛹− (𝛤𝑅𝑁2 ) ∈ 𝛼𝑂(𝐶𝑇𝑆𝑅𝐶 1 ). By the hypothesis, We have 𝛹 (𝐶𝑙(𝐼𝑛𝑡(𝐶𝑙(𝛹+ (𝛤𝑅𝑁 2 𝑐 ))))) ⊂
𝑁𝑒𝑢 − 𝛼𝐶𝑙(𝛹(𝛹+ (𝛤𝑅𝑁 2 𝑐 ))) ⊂ (𝛤𝑅𝑁 2 𝑐 ), Which
(𝛹 − (𝛤𝑅𝑁 2 ))𝑐 .Therefore,
𝛼𝑂(𝐶𝑇𝑆𝑅𝐶 1 ). Put 𝑈𝑅𝐶 1
implies
𝐶𝑙(𝐼𝑛𝑡(𝐶𝑙(𝛹 + (𝛤𝑅𝑁 2 + ))))) ⊂ 𝛹 + (𝛤𝑅𝑁 2 𝑐 ) ⊂
𝛹− (𝛤𝑅𝑁2 ) ⊂ 𝐼𝑛𝑡(𝐶𝑙(𝐼𝑛𝑡(𝛹 − (𝛤𝑅𝑁2 )))) . Hence 𝛹−(𝛤𝑅𝑁2 ) ∈
= 𝛹 − (𝛤𝑅𝑁2 ). Then 𝜉 ∈ 𝑈𝑅𝐶 1 ∈ 𝛼𝑂(𝐶𝑇𝑆𝑅𝐶 1 ) and 𝛹 (𝑢)𝑞𝛤𝑅𝑁 2 for every 𝑢 ∈
we obtain
𝑈𝑅𝐶1 . Therefore 𝛹 is Neutrosophic lower α-irresolute.
IV. Upper α-Irresolute Neutrosophic Multifunctions
In this section, we introduce the Definition for Neutrosophic upper 𝛼 - irresolute multifunction
and its properties
Definition 4.1.
An Neutrosophicmultifunction 𝛹 : (𝑅𝐶 1 ,𝜏𝑅𝐶 1 ) → (𝑅𝑁 2 .,𝜏𝑁 𝑅𝑁 ), is called
2
𝛤𝑅𝑁 2 ,𝛤𝑅𝑁2 ∈ 𝛼𝑂(𝑁𝑈𝑇𝑆𝑅𝑁 2 . )
such that 𝛹 (𝑥 0 ) ⊂ 𝛤𝑅𝑁 2 there exists 𝑈𝑅𝐶 1 ∈ 𝛼𝑂(𝐶𝑇𝑆𝑅𝐶 1 ) containing 𝑥 0 such that 𝛹 (𝑈𝑅𝐶 1 ) ⊂ 𝛤𝑅𝑁 2 .
(a) Neutrosophic upper α-irresolute at a point 𝑥 0 ∈ 𝑅𝐶 1 , if for any
(b) Neutrosophic upper α-irresolute if it is satisfied that property at each point of 𝑅𝐶 1 .
Theorem 4.2
Every Neutrosophic upper 𝛼-irresolute multifunction is Neutrosophic upper𝛼-continuous
multifunction.
Proof:
Letting𝑥0 ∈ 𝑅𝐶 1 , 𝛹 : (𝑅𝐶 1 ,𝜏𝑅𝐶 1 ) → (𝑅𝑁 2 .,𝜏𝑁 𝑅𝑁 ) and 𝛤𝑅𝑁 2 ∈ 𝑂(𝑁𝑈𝑇𝑆𝑅𝑁 2 ) such that 𝛹(𝑥0 ) ⊂ 𝛤𝑅𝑁 2 ,
2
But we know that , every𝛤𝑅𝑁 2 , 𝛤𝑅𝑁2 ∈ 𝑂(𝑁𝑈𝑇𝑆𝑅𝑁 2 ) is 𝛤𝑅𝑁2 ∈ 𝛼𝑂(𝑁𝑈𝑇𝑆𝑅𝑁 2 ), Therefore 𝛤𝑅𝑁 2 ∈
𝛼𝑂(𝑁𝑈𝑇𝑆𝑅𝑁 2 ),By our assumption ,Neutrosophic lower 𝛼 − irresolute multifunction, There exists
𝑈𝑅𝐶 1 ∈ 𝛼O(CTS𝑅𝐶 1 ) containing 𝑥0 such that
𝛹(𝜉 ) ⊂ 𝛤𝑅𝑁 2 , ∀𝜉 ∈ 𝑈𝑅𝐶 1 Hence 𝛹 is Neutrosophic
lower 𝛼-continuous multifunction at 𝑥0 .
Theorem 4.3
Every Neutrosophic upper 𝛼-irresolute multifunction is Neutrosophic upper Pre-continuous
multifunction.
Proof:
Letting 𝑥0 ∈ 𝑅𝐶 1 , 𝛹 : (𝑅𝐶 1 ,𝜏𝑅𝐶 1 ) → (𝑅𝑁 2 .,𝜏𝑁 𝑅𝑁 ) and 𝛤𝑅𝑁2 ∈ 𝑂(𝑁𝑈𝑇𝑆𝑅𝑁 2 ) such that 𝛹(𝑥0 ) ⊂
2
𝛤𝑅𝑁 2 .But we know that , Every𝛤𝑅𝑁 2 , 𝛤𝑅𝑁2 ∈ 𝑂(𝑁𝑈𝑇𝑆𝑅𝑁 2 )is 𝛤𝑅𝑁 2 ∈ 𝛼𝑂(𝑁𝑈𝑇𝑆𝑅𝑁 2 ).Therefore
𝛤𝑅𝑁2 ∈ 𝛼𝑂(𝑁𝑈𝑇𝑆𝑅𝑁 2 ).By our assumption ,Neutrosophic upper𝛼 −irresolute multifunction, There
𝛹(𝜉) ⊂ 𝛤𝑅𝑁 2 , ∀𝜉 ∈ 𝑈𝑅𝐶 1 , every𝑈𝑅𝐶 1 , 𝑈𝑅𝐶 1 ∈
∈ 𝑃𝑂(𝐶𝑇𝑇𝑆𝑅 2 ).There exists 𝑈𝑅𝐶 1 ∈ 𝑃O(CTS𝑅𝐶 1 ) containing 𝑥0 such that
exists 𝑈𝑅𝐶 1 ∈ 𝛼O(CTS𝑅𝐶 1 ) containing 𝑥0 such that
𝛼𝑂(𝑁𝑈𝑇𝑆𝑅
𝑁
2
) is 𝑈𝑅𝐶 1
𝑁
𝛹(𝜉) ⊂ 𝛤𝑅𝑁 2 , ∀𝜉 ∈ 𝑈𝑅𝐶 1 hence 𝛹 is Neutrosophic upperPre-continuous multifunction at 𝑥0 .
Theorem 4.4
Every Neutrosophic
upper 𝛼-irresolute multifunction is Neutrosophic upper quasi semi
continuous multifunction.
Proof:
T.RajeshKannan, and S.Chandrasekar, Neutrosophic 𝛼-Irresolute Multifunction In Neutrosophic Topological Spaces
Neutrosophic Sets and Systems, Vol. 32, 2020
Letting 𝑥0 ∈ 𝑅𝐶 1 , 𝛹
398
: (𝑅𝐶 1 ,𝜏𝑅𝐶 1 ) → (𝑅𝑁 2 .,𝜏𝑁 𝑅𝑁 )
2
and 𝛤𝑅𝑁2 ∈ 𝑂(𝑁𝑈𝑇𝑆𝑅𝑁 2 ) such that
𝑁
𝑁
𝛹(𝑥0 ) ⊂
𝛤𝑅𝑁 2 ,.But we know that , Every𝛤𝑅𝑁2 , 𝛤𝑅𝑁 2 ∈ 𝑂(𝑁𝑈𝑇𝑆𝑅 2 )is 𝛤𝑅𝑁 2 ∈ 𝛼𝑂(𝑁𝑈𝑇𝑆𝑅 2 ),
Therefore
𝛤𝑅𝑁2 ∈ 𝛼𝑂(𝑁𝑈𝑇𝑆𝑅𝑁 2 ), By our assumption ,Neutrosophic
upper𝛼 -irresolute
multifunction, there
exists 𝑈𝑅𝐶 1 ∈ 𝛼O(CTS𝑅𝐶 1 ) containing 𝑥0 such that
𝛹(𝜉) ⊂ 𝛤𝑅𝑁 2 , ∀𝜉 ∈
) is 𝑈𝑅𝐶 1 ∈ 𝑆𝑂(𝐶𝑇𝑇𝑆𝑅 2 ). Their
exists
𝑈𝑅 𝐶 1 ∈
𝑆O(CTS𝑅 1 ) containing 𝑥0 such that 𝛹(𝜉) ⊂ 𝛤𝑅𝑁 2 , ∀𝜉 ∈ 𝑈𝑅𝐶 1 Hence 𝛹 is Neutrosophic upper
𝑈𝑅𝐶 1 . Every𝑈𝑅𝐶 1 , 𝑈𝑅𝐶 1 ∈ 𝛼𝑂(𝑁𝑈𝑇𝑆𝑅
𝐶
𝑁
𝑁
2
quasi semi continuous multifunction at 𝑥0 .
Theorem 4.5
Let : (𝑅𝐶 1 ,𝜏𝑅𝐶 1 ) → (𝑅𝑁 2 .,𝜏𝑁 𝑅𝑁 ), be an Neutrosophic multifunction and let
2
𝜉 ∈ 𝑅𝐶 1 . Then the
following statements are equivalent:
(a) 𝛹is Neutrosophic Upper α-irresolute at 𝜉 .
(b) For each 𝛤𝑅𝑁 2 ∈ 𝛼𝑂(𝑁𝑈𝑇𝑆𝑅𝑁 2 ) with (𝜉) ⊂ 𝛤𝑅𝑁2 , Implies 𝜉 ∈ 𝑠𝐶𝑙(𝐼𝑛𝑡(𝛹− (𝛤))).
(c) For any 𝜉 , 𝜉 ∈ 𝑈𝑅𝐶 1 ∈ 𝑆𝑂(𝐶𝑇𝑆𝑅𝐶 1 ) and for any
there exists a nonempty open set 𝑉𝑅𝐶 1
𝛤𝑅𝑁 2 ∈ 𝛼𝑂(𝑁𝑈𝑇𝑆𝑅𝑁 2 ) with (𝜉) ⊂ 𝛤𝑅𝑁2 ,
⊂ 𝑈𝑅𝐶 1 such that 𝛹 ( 𝑉𝑅𝐶 1 ) ⊂ 𝛤𝑅𝑁 2 .
Proof.
(a)⇒ (b) Let 𝜉 ∈ 𝑅𝐶 1 and 𝛤𝑅𝑁 2 ∈ 𝛼𝑂(𝑁𝑈𝑇𝑆𝑅𝑁 2 ) Such that 𝛹(𝜉) ⊂ 𝛤𝑅𝑁 2 Then by our assumption
(a), we get there exists 𝑈𝑅𝐶 1 ∈ 𝛼O(CTS𝑅𝐶 1 ) such that 𝜉 ∈ 𝑈𝑅𝐶 1 and 𝐹(𝑈𝑅𝐶 1 ) ⊂ 𝛤𝑅𝑁2 , Thus 𝜉 ∈
𝑈𝑅𝐶 1 ⊂ 𝛹 + (𝛤𝑅𝑁2 ).
𝐶
here 𝑈𝑅𝐶1 ∈ 𝛼O(CTS𝑅𝐶 1 )
𝛼O(CTS𝑅 1 )⟺ 𝐴𝑅𝐶 1 ⊂ 𝑠𝐶𝑙(𝐼𝑛𝑡(𝐴𝑅𝐶 1 )).
+
𝑠𝐶𝑙(𝐼𝑛𝑡𝛹 (𝛤𝑅𝑁 2 )).hence(b).
.We know that for any set 𝐴𝑅𝐶 1 , 𝐴𝑅𝐶 1
Therefore, 𝑈𝑅𝐶 1 ⊂ 𝑠𝐶𝑙(𝐼𝑛𝑡(𝑈𝑅𝐶1 )). Finally we get
(b)⇒ (c). Let 𝛤𝑅𝑁 2 ∈ 𝛼𝑂(𝑁𝑈𝑇𝑆𝑅𝑁 2 ) such that
∈
𝜉 ∈
𝛹 (𝜉) ⊂ 𝛤𝑅𝑁 2 , then 𝜉 ∈ 𝑠𝐶𝑙(𝐼𝑛𝑡𝛹 − (𝛤𝑅𝑁2 )). Let 𝑈𝑅𝐶1 ∈
𝑆O(CTS𝑅𝐶 1 ) and 𝜉 ∈ 𝑈𝑅𝐶 1 .Then𝑈𝑅𝐶 1 ∩ 𝐼𝑛𝑡(𝛹 − (𝛤𝑅𝑁 2 )) ≠ 𝜑 and 𝑈𝑅𝐶1 ∩ 𝐼𝑛𝑡(𝛹 − (𝛤𝑅𝑁2 )) is semiopen in 𝑅𝐶 1 .Put 𝑉𝑅𝐶1 = 𝐼𝑛𝑡(𝑈𝑅𝐶1 ∩ 𝐼𝑛𝑡(𝛹 − (𝛤𝑅𝑁 2 )), Then 𝑉𝑅𝐶 1 is an open set of 𝑅𝐶 1 , 𝑉𝑅𝐶 1 ⊂
𝑈𝑅𝐶 1 , 𝑉𝑅𝐶1 ≠ 𝜑 and 𝛹 (𝑉𝑅𝐶 1 ) ⊂ 𝛤𝑅𝑁 2 ,
(c) ⇒(a).Let {𝑈𝜉 } be the system of the 𝑆O(CTS𝑅𝐶 1 ) containing𝜉 . Let 𝑈𝑅𝐶1 ∈ 𝑆O(CTS𝑅𝐶 1 ) and
𝜉 ∈
𝛤𝑅𝑁 2 ∈ 𝛼𝑂(𝑁𝑈𝑇𝑆𝑅 2 ) such that 𝛹(𝜉) ⊂ 𝛤𝑅𝑁 2 , there exists a nonempty open set
𝐵𝑈 ⊂ 𝑈𝑅𝐶 1 Such that 𝛹 (𝑣) ⊂ 𝛤𝑅𝑁 2 , ∀𝑣 ∈ 𝐵𝑈 . Let 𝑊𝑅𝐶 1 = ∪ 𝐵𝑈 ∶ 𝑈𝑅𝐶 1 ∈ {𝑈𝜉 }, then
𝑊𝑅𝐶1 ∊O(CTS𝑅𝐶 1 ) and
𝜉 ∈ 𝑠𝐶𝑙(𝑊𝑅𝐶 1 ) and 𝛹(𝑣) ⊂ 𝛤𝑅𝑁 2 , ∀𝑣 ∈ 𝑊𝑅𝐶 1 .Put 𝑆𝑅𝐶1 = 𝑊𝑅𝐶1 ∪
𝜉.Then 𝑊𝑅𝐶 1 ⊂ 𝑆𝑅𝐶 1 ⊂ 𝑠𝐶𝑙(𝑊𝑅𝐶1 ). Thus 𝑆𝑅𝐶 1 ∈ 𝛼𝑂(𝐶𝑇𝑆𝑅𝐶 1 ) ), 𝜉 ∈ 𝑆𝑅𝐶 1 and 𝛹 (𝑣) ⊂
𝛤𝑅𝑁 2 , ∀𝑣 ∈ 𝑆. Hence 𝛹 is Neutrosophic Upper α-irresolute at 𝜉 .
𝑈𝑅𝐶 1 and Let
𝑁
Theorem 4.6
For an Neutrosophicmultifunction : (𝑅𝐶 1 ,𝜏𝑅𝐶 1 ) → (𝑅𝑁 2 ,𝜏𝑁 𝑅𝑁 )the following statements are
2
equivalent:
(a) 𝛹 is Neutrosophic upper α-irresolute.
+
(b) 𝛹 (𝛤𝑅𝑁 2 ) ∈ 𝛼𝑂(𝐶𝑇𝑆𝑅𝐶 1 ),for every Neutrosophic α-open set 𝛤𝑅𝑁 2 𝑜𝑓 𝑅𝑁 2
−
(c) 𝛹 (𝜆𝑅𝑁 2. ) ∈ 𝛼𝐶(𝐶𝑇𝑆𝑅𝐶 1 ),for each Neutrosophic α-closed set 𝜆𝑅𝑁 2. 𝑜𝑓 𝑅𝑁 2 .
(d) For each point 𝜉 ∈ 𝑅𝐶 1 and for each α-neighborhood 𝑉𝑅𝑁 2. of 𝛹 (𝜉 ) 𝑖𝑛 𝑅𝑁 2 . 𝐹+ (𝑉𝑅𝑁 2 . ) is an
α-neighborhood of 𝜉.
(e) For each point 𝜉 ∈ 𝑅𝐶 1 and for each α-neighborhood 𝑉𝑅𝑁 2. of
neighborhood 𝑈𝑅𝐶1 of 𝜉 such that 𝛹 (𝑈𝑅𝐶 1 ) ⊂ 𝑉𝑅𝑁 2 . .
𝛹 (𝜉) in 𝑅𝑁 2 ., there is an α-
(f) 𝛼𝐶𝑙(𝛹 − (𝜆𝑅𝑁 2. )) ⊂ 𝛹 − (𝑁𝑒𝑢 − 𝛼𝐶𝑙(𝜆𝑅𝑁2. )) 𝑓𝑜𝑟 each Neutrosophic set 𝜆𝑅𝑁2. 𝑜𝑓 𝑅𝑁 2 .
(g) 𝑠𝐼𝑛𝑡(𝐶𝑙(𝛹 − (𝜆𝑅𝑁2 . ))) ⊂ 𝛹 − (𝑁𝑒𝑢 − 𝛼𝐶𝑙(𝜆𝑅𝑁 2. ))for any Neutrosophic set 𝜆 𝑜𝑓 𝑅𝑁 2 ..
Proof.
T.RajeshKannan, and S.Chandrasekar, Neutrosophic 𝛼-Irresolute Multifunction In Neutrosophic Topological Spaces
Neutrosophic Sets and Systems, Vol. 32, 2020
399
𝛤𝑅𝑁 2 𝜖 𝛼𝑂(𝑁𝑈𝑇𝑆𝑅𝑁 2 . ) and 𝜉 ∈ 𝛹+ (𝛤𝑅𝑁 2 ).Applying previous theorem, we get 𝜉 ∈
+
+
𝑠𝐶𝑙(𝐼𝑛𝑡𝛹+ (𝛤𝑅𝑁 2 )).Therefore, we obtain 𝛹 (𝛤𝑅𝑁2 ) ⊂ 𝑠𝐶𝑙 (𝐼𝑛𝑡𝛹 + (𝛤𝑅𝑁 2 ))., Finally we get 𝛹 (𝛤𝑅𝑁 2 ) ∈
(a)⇒ (b). Let
𝛼𝑂(𝐶𝑇𝑆𝑅𝐶 1 ).
(b)⇒ (a). Let
𝜉 be arbitrarily point in 𝑅𝐶 1 and 𝛤𝑅𝑁 2 𝜖𝛼𝑂(𝑁𝑈𝑇𝑆𝑅𝑁 2 . ) such that 𝛹(𝜉 ) ⊂ 𝛤𝑅𝑁 2 so ∈
+
𝛹 + (𝛤𝑅𝑁 2 ) . By hypothesis 𝛹 (𝛤𝑅𝑁2 ) ∈ 𝛼𝑂(𝐶𝑇𝑆𝑅𝐶 1 ), we get𝜉 ∈ 𝛹+ (𝛤𝑅𝑁 2 ) ⊂ 𝑠𝐶𝑙(𝐼𝑛𝑡(𝛹+ (𝛤𝑅𝑁 2 )))
and hence F is Neutrosophic upper α-irresolute at 𝜉 .As 𝜉 is arbitrarily chosen, 𝛹 is Neutrosophic
upper α-irresolute.
−
𝐶
𝐶
(b)⇒ (c). This implies easily get from that [𝛹 (𝛤𝑅𝑁 2 )] = [𝛹 + (𝛤𝑅𝑁 2 ) ].where 𝛤𝑅𝑁 2 𝜖𝛼𝑂(𝑁𝑈𝑇𝑆𝑌)
−
(c)⇒ (f).Let 𝜆𝑅𝑁2. 𝜖𝛼𝑂(𝑁𝑈𝑇𝑆𝑅𝑁 2 . ) .Then by our assumption (c), 𝛹 (𝑁𝑒𝑢 − 𝛼𝐶𝑙(𝜆𝑅𝑁 2. )) is an α−
closed set in 𝑅𝐶 1 . We have 𝛹 (𝑁𝑒𝑢 − 𝛼𝐶𝑙(𝜆𝑅𝑁 2. )) ⊃ 𝑠𝐼𝑛𝑡(𝐶𝑙(𝛹 − (𝑁𝑒𝑢 − 𝐶𝑙(𝜆𝑅𝑁 2. )))) ⊃
𝑠𝐼𝑛𝑡(𝐶𝑙(𝛹 − (𝜆𝑅𝑁 2. ))) ⊃ 𝛹 − (𝜆) ∪ 𝑠𝐼𝑛𝑡(𝐶𝑙(𝛹 − (𝜆𝑅𝑁2 . ))) ⊃ 𝛼𝐶𝑙(𝛹 − (𝜆𝑅𝑁 2. )).Hence the result.
(f)⇒(g).Let 𝜆𝑅𝑁2. 𝜖𝛼𝑂(𝑁𝑈𝑇𝑆𝑅𝑁 2 . ). we have 𝛼𝐶𝑙(𝛹 − (𝜆𝑅𝑁 2. )) = 𝛹 − (𝜆𝑅𝑁2 . ) ∪ 𝑠𝐼𝑛𝑡(𝐶𝑙(𝛹 − (𝜆𝑅𝑁2 . ))) ⊂
𝛹 − (𝑁𝑒𝑢 − 𝛼𝐶𝑙(𝜆𝑅𝑁 2. )). Hence (g).
𝐶
(g)⇒(c).Let 𝜆𝑅𝑁2. 𝜖 𝛼𝐶(𝑁𝑈𝑇𝑆𝑅𝑁 2 . )Then by (g) we have,
𝑠𝐼𝑛𝑡 (𝐶𝑙 (𝛹 − (𝜆𝑅𝑁 2. 𝐶 ))) ⊂ 𝛹 − (𝜆𝑅𝑁 2. 𝐶 ) ∪ 𝑠𝐼𝑛𝑡 (𝐶𝑙 (𝛹 − (𝜆𝑅𝑁 2. 𝐶 ))) ⊂ 𝛹 − (𝛼𝐶𝑙(𝜆𝑅𝑁 2. 𝑐 ) = 𝛹 − (𝜆𝑅𝑁2. 𝐶 ).
Hence By our result, 𝛹 − (𝜆𝑅𝑁 2. 𝐶 ) ∈ 𝛼𝐶(𝐶𝑇𝑆𝑅𝐶 1 ).
𝜉 ∈ 𝑅𝐶 1 and 𝑉𝑅𝑁2. be an α-neighborhood of 𝛹(𝜉 ) in 𝑅𝑁 2 .Then there is an
𝜆𝑅𝑁2. 𝜖𝛼𝑂(𝑁𝑈𝑇𝑆𝑅𝑁 2 . ) such that 𝛹(𝜉 ) ⊂ 𝜆𝑅𝑁 2 . ⊂ 𝑉𝑅𝑁 2 . . Hence, 𝜉 ∈ 𝛹+ (𝜆𝑅𝑁 2 ) ⊂ 𝛹+ (𝑉𝑅𝑁 2 ). Now
+
+
by hypothesis 𝛹 (𝜆𝑅𝑁2 ) ∈ 𝛼𝑂(𝐶𝑇𝑆𝑅𝐶 1 ), and Thus 𝛹 (𝑉𝑅𝑁 2. ) is an α-neighborhood of 𝜉 .
(d)⇒(e). Let 𝜉 ∈ 𝑅𝐶 1 and 𝑉𝑅𝑁 2 be an α-neighborhood of 𝛹(𝜉 ) in 𝑅𝑁 2 .Put 𝑈𝑅𝐶 1 = 𝛹 + (𝑉𝑅𝑁2. ).Then
𝑈𝑅𝐶 1 is an α-neighborhood of 𝜉 and 𝛹 (𝑈) ⊂ 𝑉𝑅𝑁 2 .
(e)⇒(a).Let𝜉 ∈ 𝑅𝐶 1 and𝑉𝑅𝑁 2 be an Neutrosophic set in 𝑅𝑁 2 such that 𝛹 (𝜉 ) ⊂ 𝑉𝑅𝑁 2 . . 𝑉𝑅𝑁 2 . being an
Neutrosophic α-open set in 𝑅𝑁 2 . , is an α-neighborhood of 𝛹(𝜉 ) and according to the hypothesis
there is an α-neighborhood 𝑈𝑅𝐶 1 of 𝜉 such that 𝛹(𝑈𝑅𝐶 1 ) ⊂ 𝑉𝑅𝑁 2 . Therefore 𝑉𝑅𝑁2 . ∈ 𝛼𝑂(𝐶𝑇𝑆𝑅𝐶 1 )
such that 𝜉 ∈ 𝐴𝑅𝐶 1 ⊂ 𝑈𝑅𝐶 1 and hence
𝛹(𝐴) ⊂ 𝛹(𝑈𝑅𝐶 1 ) ⊂ 𝑉𝑅𝑁 2 . . Hence 𝛹 is Neutrosophic
upper α-irresolute at 𝜉 .
(b)⇒(d).Let
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Neutrosophic Sets and Systems, Vol. 32, 2020
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Journal Of Computer And Mathematical Sciences,vol.9(10),1400-1408 Octobe2018.
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space, The International Journal of Analytical and Experimental
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XI,IssueIX,September 2019,1360-9367
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C. Maheswari , S. Chandrasekar: Neutrosophic gb-closed Sets and Neutrosophic gb-Continuity,
Neutrosophic Sets and Systems, vol. 29, 2019, pp. 89-100, DOI: 10.5281/zenodo.3514409
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Abdel-Basset, M., El-hoseny, M., Gamal, A., & Smarandache, F. (2019). A novel model for evaluation
Hospital medical care systems based on plithogenic sets. Artificial intelligence in medicine, 100, 101710.
Abdel-Basset, M., Manogaran, G., Gamal, A., & Chang, V. (2019). A Novel Intelligent Medical Decision
Support Model Based on Soft Computing and IoT. IEEE Internet of Things Journal.
Abdel-Basset, M., Mohamed, R., Zaied, A. E. N. H., & Smarandache, F. (2019). A hybrid plithogenic
decision-making approach with quality function deployment for selecting supply chain sustainability
metrics. Symmetry, 11(7), 903.
Abdel-Baset, M., Chang, V., & Gamal, A. (2019). Evaluation of the green supply chain management
practices: A novel neutrosophic approach. Computers in Industry, 108, 210-220.
Abdel-Basset, M., Saleh, M., Gamal, A., & Smarandache, F. (2019). An approach of TOPSIS technique for
developing supplier selection with group decision making under type-2 neutrosophic number. Applied
Soft Computing, 77, 438-452.
Received: 12 Sep, 2019. Accepted: 20 Mar 2020.
T.RajeshKannan, and S.Chandrasekar, Neutrosophic 𝛼-Irresolute Multifunction In Neutrosophic Topological Spaces
Neutrosophic Sets and Systems, Vol. 32, 2020
University of New Mexico
Neutrosophic Fuzzy Matrices and Some Algebraic Operations
Rakhal Das1, Florentin Smarandache2 and Binod Chandra Tripathy3,*
1 Department
of Mathematics, Tripura University Agartala -799022; Tripura, INDIA; E mail: rakhaldas95@gmail.com and
rakhal.mathematics@tripurauniv.in;
2 Department of Mathematics, University of New Mexico, 705 Gurley Avenue, Gallup, NM 87301, USA.; E mail :
fsmarandache@gmail.com
3 Department of Mathematics, Tripura University Agartala -799022; Tripura, INDIA; E mail binodtripathy@tripurauniv.in,
tripathybc@yahoo.com and tripathybc@gmail.com
*Correspondence: e-mail tripathybc@yahoo.com; Tel.: (+91 9864087231)
Abstract: In this article, we study neutrosophic fuzzy set and define the subtraction and
multiplication of two rectangular and square neutrosophic fuzzy matrices. Some properties of
subtraction, addition and multiplication of these matrices and commutative property, distributive
property have been examined.
Keywords: Neutrosophic fuzzy matrix, Neutrosophic set. Commutativity, Distributive, Subtraction
of neutrosophic matrices.
1. Introduction
Neutrosophic set was introduced by Florentin Smarandache [1] in 1998, where each element had
three associated defining functions, namely the membership function (T), the non-membership (F)
function and the indeterminacy function (I) defined on the universe of discourse X, the three
functions are completely independent. Relative to the natural problems sometimes one may not be
able to decide. After the development of the Neutrosophic set theory, one can easily take decision
and indeterminacy function of the set is the nondeterministic part of the situation. The applications
of the theory has been found in various field for dealing with indeterminate and inconsistent
information in real world one may refer to [2,3,4]. Neutrosophic set is a part of neutrosophy which
studied the origin, nature and scope of neutralities, as well as their interactions with ideational
spectra. The neutrosophic set generalizes the concept of classical fuzzy set [10, 11], interval valued
fuzzy set, intuitionistic fuzzy set and so on. In the recent years, the concept of neutrosophic set has
been applied successfully by Broumi et al. [12, 13, 14] and Abdel-Basset et al. [15, 16, 17, 18]
The single-valued neutrosophic number which is a generalization of fuzzy numbers and
intuitionistic fuzzy numbers. A single-valued neutrosophic number is simply an ordinary number
whose precise value is somewhat uncertain from a philosophical point of view. There are two special
forms of single-valued neutrosophic numbers such as single-valued trapezoidal neutrosophic
numbers and single-valued triangular neutrosophic numbers.
The neutrosophic interval matrices have been defined by Vasantha Kandasamy and Florentin
Smarandache in their book “Fuzzy interval matrices, Neutrosophic interval matrices, and
their
R. Das, F. Smarandache and B.C. Tripathy, Neutrosophic Fuzzy Matrices and Some Algebraic Operations
Neutrosophic Sets and Systems, Vol. 32, 2020
applications”.
402
A neutrosophic fuzzy matrix [aij]nxm, whose entries are of the form a + Ib
(neutrosophic number), where a, b are the elements of the interval [0,1] and I is an indeterminate
such that In = I, n being a positive integer.
So the difference between the neutrosophic number of the form a + Ib and the single-valued
neutrosophic numbers is that the generalization of fuzzy number and the single-valued
neutrosophic components <T, I, F> is the generalization of fuzzy numbers and intuitionistic fuzzy
numbers. Since fuzzy number lies between 0 to 1 so the component neutrosophic fuzzy number a
and b lies in [0,1]. In the case of single-valued neutrosophic matrix components will be the true value,
indeterminacy and fails value with three components in each element of a matrix [3, 4, 8].
We know the important role of matrices in science and technology. However, the classical
matrix theory sometimes fails to solve the problems involving uncertainties, occurring in an
imprecise environment. Kandasamy and Smarandache [7] introduced fuzzy relational maps and
neutrosophic relational maps. Thomason [8], introduced the fuzzy matrices to represent fuzzy
relation in a system based on fuzzy set theory and discussed about the convergence of powers of
fuzzy matrix. Dhar, Broumi and Smarandache [2] define Square Neutrosophic Fuzzy Matrices
whose entries are of the form a+Ib, where a and b are fuzzy number from [0, 1] gives the definition of
Neutrosophic Fuzzy Matrices multiplication.
In this paper our ambition is to define the subtraction of fuzzy neutrosophic matrices,
rectangular fuzzy neutrosophic matrices and study some algebraic properties. We shall focus on all
types of neutrosophic fuzzy matrices. The paper unfolds as follows. The next section briefly
introduces some definitions related to neutrosophic set, neutrosophic matrices, Fuzzy integral
neutrosophic matices and fuzzy matrix. Section 3 presents a new type of fuzzy neutrosophic
matrices and investigated some properties such as subtraction, commutative property and
distributive property.
2. Materials and Methods (proposed work with more details)
In this section we recall some concepts of neutrosophic set, neutrosophic matrices and fuzzy
neutrosophic matrices proposed by Kandasamy and Smarandache in their monograph [3], and also
the concept of fuzzy matrix (One may refer to [2])
Definition 2.1 (Smarandache [1]). Let U be an universe of discourse then the neutrosophic set A is an
object having the form A = {< x:TA(x), IA(x), FA(x)>, x
U}, where the functions T, I, F : U→ ]−0, 1+[
define respectively the degree of membership (or Truthness), the degree of indeterminacy, and the
degree of non-membership (or Falsehood) of the element x
U to the set A with the condition.
0 ≤ TA(x) + IA(x) + FA(x) ≤ 3+.
−
From philosophical point of view, the neutrosophic set takes the value from real standard or
non-standard subsets of ]−0, 1+[. So instead of ]−0, 1+[ we need to take the interval [0, 1] for technical
applications, because ]−0, 1+[will be difficult to apply in the real applications such as in scientific and
engineering problems.
Definition 2.2 (Dhar et al. [3]). Let Mmxn= {(aij) : aij
K(I)}, where K(I), is a
R. Das, F. Smarandache and B.C. Tripathy, Neutrosophic Fuzzy Matrices and Some Algebraic Operations
Neutrosophic Sets and Systems, Vol. 32, 2020
403
neutrosophic field. We call Mmxn to be the neutrosophic matrix.
Example 2.1: Let R(I) = 〈R ∪ I 〉be the neutrosophic field
M4x3 =
M4x3 denotes the neutrosophic matrix, with entries from real and the indeterminacy.
Definition 2.3 (Kandasamy and Smarandache [5])
Let N = [0, 1]
I where I is the indeterminacy. The m×n matrices Mmxn = {(aij) : aij
[0, 1]
I} is called
the fuzzy integral neutrosophic matrices. Clearly the class of m×n matrices is contained in the class of
fuzzy integral neutrosophic matrices.
The row vector 1×n and column vector m×1 are the fuzzy neutrosophic row matrices and fuzzy
neutrosophic column matrices respectively.
Example 2.2: Let M4x3 =
be a 4 ×3 integral fuzzy neutrosophic matrix
Definition 2.5 (Kandasamy and Smarandache [5]).
Let Ns = [0, 1] ∪ {bI : b
[0, 1]}; we call the set Ns to be the fuzzy neutrosophic set. Let Ns be the fuzzy
neutrosophic set. Mmxn = {(aij): aij Ns i= 1 to m and j = 1 to n} we call the matrices with entries from Ns
to be the fuzzy neutrosophic matrices.
Example 2.3: Let Ns = [0,1] ∪{bI: b [0,1]} be the fuzzy neutrosophic set and
P=
be a 3 ×3 fuzzy neutrosophic matrix.
Definition 2.6 (Thomas [9]). A fuzzy matrix is a matrix which has its elements from the interval [0,
1], called the unit fuzzy interval. Amxn fuzzy matrix for which m = n (i.e. the number of rows is equal
to the number of columns) and whose elements belong to the unit interval [0, 1] is called a fuzzy
square matrix of order n. A fuzzy square matrix of order two is expressed in the following way
A=
,
where the entries x, y, t, z all belongs to the interval [0,1].
Definition 2.7 (Kandasamy and Smarandache [5]). Let A be a neutrosophic fuzzy matrix, whose
entries is of the form a + Ib (neutrosophic number), where a, b are the elements of [0,1] and I is an
indeterminate such that In = I, n being a positive integer.
R. Das, F. Smarandache and B.C. Tripathy, Neutrosophic Fuzzy Matrices and Some Algebraic Operations
Neutrosophic Sets and Systems, Vol. 32, 2020
404
A=
Definition 2.8 Multiplication Operation of two Neutrosophic Fuzzy Matrices
Consider two neutrosophic fuzzy matrices, whose entries are of the form a + Ib (neutrosophic number), where a, b are the elements of [0,1] and I is an indeterminate such that In = I, n being a positive
integer, given by
A=
, B =
The Multiplication Operation of two Neutrosophic Fuzzy Matrices is given by
AB =
D11 = [max{ min(
,
), min(
D21 = [max {min(
,
), min (
D21 = [max {min{(
D22 = [max {min{(
Hence, AB =
,
,
,
)} + I max{ min{(
,
), min(
,
)}]
,
)} + I max {min(
,
), min(
,
)}]
), min (
,
)} + I max {min{(
,
), min (
), min (
,
)} + I max{ min{(
,
), min (
,
)}]
,
)}]
.
3. Results (examples / case studies related to the proposed work)
In this section we define the subtraction and distributive property of neutrosophic fuzzy matrices
along with some properties associated with such matrices.
3.1 Subtraction Operation of two Neutrosophic Fuzzy Matrices
Consider two neutrosophic fuzzy matrices given by
A=
and
B=
.
Addition and multiplication between two neutrosophic fuzzy matrices have been defined in
Smarandache [2]. We would like to define the subtraction of these two matrices as follows.
A- B = C,
where cij are as follows
c11 = min{x1, t1} + I min{y1, z1}
c12 = min{x2, t2} + I min{y2, z2}
c21 = min{x3, t3} + I min{y3, z3}
c21 = min{x4, t4} + I min{y4, z4}
R. Das, F. Smarandache and B.C. Tripathy, Neutrosophic Fuzzy Matrices and Some Algebraic Operations
Neutrosophic Sets and Systems, Vol. 32, 2020
405
c31 = min{x5, t5} + I min{y5, z5}
c32 = min{x6, t6} + I min{y6, z6}
Since min{a, b} = min{b, a} so based on this we have the following properties.
Proposition 3.1. The following properties hold in the case of neutrosophic fuzzy matrix for
subtraction
(i) A-B = B-A
(ii) (A - B) - C = A - (B - C) = (B- C) – A = (C – B) – A.
Proof. Consider three neutrosophic fuzzy matrices A, B and C as follows.
A=
,B=
and C =
–
A–B=
= D (say),
where,
,
D11 = min{
}+Imin{
}=
D12 = min{
,
}+Imin{
,
}=
D21 = min{
,
}+Imin{
,
}=
D22 = min{
,
}+Imin{
,
}=
D31 = min{
,
}+Imin{
,
}=
D32 = min{
,
}+Imin{
,
}=
and B – A =
D=
[
,
= D,
min(a, c) = min(c, a)]
Hence, A – B = B – A.
Now we have,
D – C = (A – B) – C
=
–
= F (say),
R. Das, F. Smarandache and B.C. Tripathy, Neutrosophic Fuzzy Matrices and Some Algebraic Operations
Neutrosophic Sets and Systems, Vol. 32, 2020
406
where,
F11 = min{
,
}+Imin{
,
} = min{
,
,
}+Imin{
,
,
}=
F12 = min{
,
}+Imin{
,
} = min{
,
,
}+Imin{
,
,
}=
F21 = min{
,
}+Imin{
,
} = min{
,
,
}+Imin{
F22 = min{
,
}+Imin{
,
}= min{
F31 = min{
,
}+Imin{
,
} = min{
,
,
}+Imin{
,
,
}=
F32 = min{
,
}+Imin{
,
} = min{
,
,
}+Imin{
,
,
}=
,
(A – B) – C = F =
,
}+Imin{
,
,
,
}=
,
}=
.
Next we have,
–
B–C=
= E (say),
where
E11 = min{
,
}+Imin{
,
}=
E12 = min{
,
}+Imin{
,
}=
E21 = min{
,
}+Imin{
,
}=
E22 = min{
,
}+Imin{
,
}=
E31 = min{
,
}+Imin{
,
}=
E32 = min{
,
}+Imin{
,
}=
.
We have
B–C=E=
–
A – (B – C) =
,
where
min{
,
}+Imin{
,
} = min{
,
,
}+Imin{
,
,
}
R. Das, F. Smarandache and B.C. Tripathy, Neutrosophic Fuzzy Matrices and Some Algebraic Operations
Neutrosophic Sets and Systems, Vol. 32, 2020
407
min{
,
}+Imin{
,
} = min{
,
,
}+Imin{
min{
,
}+Imin{
,
} = min{
,
,
}+Imin{
min{
,
}+Imin{
,
}= min{
,
,
}+Imin{
min{
,
}+Imin{
,
} = min{
,
,
}+Imin{
min{
,
}+Imin{
,
} = min{
,
,
}+Imin{
,
,
,
}
,
}
,
}
,
,
}
,
,
}
,
F=
Therefore, A – (B – C) = F = (A – B) – C.
3.2 Identity element for subtraction
In the group theory under the operation “*” the identity element IN of a set is an element such that IN
* A = A * IN = A.
Specially the identity element of neutrosophic set is IN = {[aij +bijI]mxn: aij = 1 = bij for all i, j}.
Result 3.1. For a neutrosophic fuzzy matrix, IN is the identity matrix for subtraction.
Let A=
, and IN =
be the neutrosophic identity
matrix of order 3x2.
Then we have the following
A – IN =
–
= IN –A = A,
=
where
min{
, 1}+Imin{
,1} =
min{
, 1}+Imin{
,1}=
min{
, 1}+Imin{
,1} =
min{
, 1}+Imin{
,1} =
min{
, 1}+Imin{
,1} =
R. Das, F. Smarandache and B.C. Tripathy, Neutrosophic Fuzzy Matrices and Some Algebraic Operations
Neutrosophic Sets and Systems, Vol. 32, 2020
min{
, 1}+Imin{
408
,1} =
3.3 Identity element for addition
In neutrosophic matrix addition we can define a identity element IN such that IN = {[aij +bijI]mxn: aij = 0
= bij for all i, j}
, and IN =
Let A=
be the neutrosophic identity
matrix of order 3x2.
Then we have the following
–
A – IN =
=
= IN –A = A,
where
max{
max{
max{
max{
max{
max{
, 0}+Imax{
, 0}+Imax{
, 0}+Imax{
, 0}+Imax{
, 0}+Imax{
, 0}+Imax{
,0} =
,0}=
,0} =
,0} =
,0} =
,0} =
.
Result 3.2. The neutrosophic set forms a groupoid, semigroup, monaid and is commutative under
the neutrosophic matrix operation of subtraction. The distributive law also holds for subtraction, i.e.
A(B – C) = AB – AC.
Result 3.3. The neutrosophic set forms a groupoid, semigroup, monaid and commutative under
the operation of addition. The distributive law also holds for addition, i.e.
A(B + C) = AB + AC.
Thus we have, A(B
C) = AB
AC.
4. Applications
The formation of neutrosophic group structure, neutrosophic matrix set and algebraic structure
on this set, the results are applicable
5. Conclusions
In this paper we have established some neutrosophic algebraic property, and subtraction operation
addition and multiplication of these matrices and commutative property, distributive property had
been examine. This result can be applied further application of neutrosophic fuzzy matric theory.
For the development of neutrosophic group and its algebraic property the results of this paper
would be helpful.
Acknowledgments: this paper formatted by 3 authors those who are author of this article.
Conflicts of Interest: Declare no conflicts of interest involved in it.
R. Das, F. Smarandache and B.C. Tripathy, Neutrosophic Fuzzy Matrices and Some Algebraic Operations
Neutrosophic Sets and Systems, Vol. 32, 2020
409
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(2017).58-66
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A novel neutrosophic approach”. Computers in Industry, 108 ,(2019), 210-220.
M. Abdel-Basset, M. Saleh, A. Gamal, and F. Smarandache,. An approach of TOPSIS technique for
developing supplier selection with group decision making under type-2 neutrosophic number. Applied
Soft Computing, 77, (2019) 438-452.
Abdel-Basset, M., El-hoseny, M., Gamal, A., & Smarandache, F. (2019). A novel model for evaluation
Hospital medical care systems based on plithogenic sets. Artificial intelligence in medicine, 100, 101710.S.
Abdel-Basset, M., Mohamed, M., Elhoseny, M., Chiclana, F., & Zaied, A. E. N. H. (2019). Cosine similarity
measures of bipolar neutrosophic set for diagnosis of bipolar disorder diseases. Artificial Intelligence in
Medicine, 101, 101735.
Received: Oct 07, 2019. Accepted: Mar 18, 2020
R. Das, F. Smarandache and B.C. Tripathy, Neutrosophic Fuzzy Matrices and Some Algebraic Operations
Neutrosophic Sets and Systems, Vol. 32, 2020
University of New Mexico
Neutrosophic Bipolar Vague Soft Set and Its Application to
Decision Making Problems
Anjan Mukherjee1 and Rakhal Das2
1 Department
of Mathematics, Tripura University Agartala -799022; Tripura, INDIA; Email: 1mukherjee123anjan@gmail.com
of Mathematics, Tripura University Agartala -799022; Tripura, INDIA; Email 2rakhaldas95@gmail.com.
2 Department
Abstract: In this paper we study the concept of neutrosophic bipolar vague soft sets and some of its
operations. It is the combination of neutrosophic bipolar vague sets and soft sets. Further we
develop a decision making method based on neutrosophic bipolar vague soft set. A numerical
example has been shown. Some new operations on neutrosophic bipolar vague soft set have also
been designed.
Keywords: Neutrosophic set, Neutrosophic bipolar vague set, Soft set, vague set, Neutrosophic
bipolar vague soft set.
1. Introduction
Most real life problems involve data with a high level of uncertainty and imprecision. Traditionally,
classical mathematical theories such as fuzzy mathematics, probability theories and interval
mathematics are used to deal with uncertain and fuzziness. However all these theories have their
difficulties and weakness as pointed out by Molodtsov. This led to the introduction of the theory of
soft sets by Molodtsov [ 18 ] in 1999. Among the significant milestones in the development of the
theory of soft sets and its generalizations in the introduction of the possibility value which indicates
the degree of possibility of belongingness of the elements in the universal set as well as the elements
of each sets which enables the users to know the opinion of the experts in one model without the
need for any operation. However, in order to handle the indeterminate and inconsistent information,
neutrosophic set is defined [23,24 ] as a new mathematical tool for dealing with problems involving
incomplete, indeterminacy and inconsistent knowledge. The theory of vague set was first proposed
by Gau and Buehrer [13 ] as an extension of fuzzy set theory[29] and vague sets are regarded as a
special case of content-dependent fuzzy sets.
In, [23 ] ,Samerandeche talked about neutrosophic set theory, one of the most important new
mathematical tools for handling problems involving imprecise, indeterminacy and inconsistent data.
Neutrosophic vague set was defined by S. Allehezaleh [2 ] in 2015. Lee [17]
introduced
bipolar-valued fuzzy sets and their operations in 2000. It an extension of fuzzy set [ 29 ]. Ali et al.[
1] introduced the notion of bipolar neutrosophic soft set in 2017.
Hassain et al. [16] introduced the
concept of neutrosophic bipolar vague set and its application to neutrosophic bipolar vague graphs.
For real life problems see the following ([3] to [12],[14],[15],[19]to[22],[25]to[28],[30]to[32]).
Anjan Mukherjee and Rakhal Das. Neutrosophic Bipolar Vague Soft Set And Its Application To Decision
Making Problems
Neutrosophic Sets and Systems, Vol. 32, 2020
411
In this paper, we first introduce the concept of neutrosophic bipolar vague soft set and some of its
operations. It is the combination of neutrosophic bipolar vague set and soft set. We develop a
decision making method based on neutrosophic bipolar vague soft set. A numerical example has
been shown. Some new operations on neutrosophic bipolar vague soft set have been designed.
Finally we present an application of this concept in solving a decision making problem.
2. Materials and Methods (proposed work with more details)
In this section we recall some definitions and results for our future work.
Definition2.1:[ 8 ] Let U be an initial universal set and let E be a set of parameters. Let P(U) denote
the power set of all subsets of U and let A⊆E. A collection of pairs (f, A) is called a soft set over U,
where f is a mapping given by f : A → P(U).
Definition2.2:[ 17 ] Let U be the universe. Then a bipolar fuzzy set A on U is defined by
Here
the positive membership function.
the negative membership function.
Definition2.3:[ 17 ] If A and B be two bipolar fuzzy sets then their union, intersection and
complement are defined as follows:
(i)
(ii)
(iii)
(iv)
and
(v)
.
Definition2.4: [13 ] A vague set A in the universe of discourse U is a pair
(tA, fA) where tA, FA:U
, 1] such that tA + fA
for all U. The function
and
are called the
true membership function and the false membership function respectively. The interval
is called the value of u in A and is denoted by
.
Definition2.5:[ 13 ] Let X be a non-empty set. Let A and B be two vague sets in the form
,
(i)
. Then
and
.
Anjan Mukherjee and Rakhal Das. Neutrosophic Bipolar Vague Soft Set And Its Application To Decision Making Problems
Neutrosophic Sets and Systems, Vol. 32, 2020
412
(ii)
(iii)
.
(iv)
Definition2.6: [ 23,24 ] A neutrosophic set A on the universe of discourse
A
x,
A
x , A x , A x
condition: x U ,
Here
U
is defined as
x U , where A , A , A U 0,1 are functions such that the
0 A x A x A x 3 is satisfied.
A x , A x , A x represent the truth-membership, indeterminacy-membership and
falsity-membership respectively of the element
x U
.From philosophical point of view, the
neutrosophic set takes the value from real standard or non-standard subsets of 0,1 . But in real
life application in scientific and engineering problems it is difficult to use neutrosophic set with
value from real standard or non-standard subset of 0,1 . Hence we consider the neutrosophic set
which takes the value from the subset of
0,1 .
Definition2.7: [ 2 ] A neutrosophic vague set ANV on the universe of discourse U written as ANV = {<x;
(x);
(x);
(x)>; xU } whose truth-membership, indeterminacy-membership, and
falsity-membership functions is defined as
(x) = [T-, T+],
(x) = [I-, I+] and
(x) = [F-,
F+], where (1) T+ = 1 − F−, (2) F+ = 1 − T− and (3) −0 ≤ T− + I− + F− ≤ 2+.
Definition2.8:[ 16 ] Let U be the universe of discourse. The neutrosophic bipolar vague set defined
as ANBV where
Here
,
,
, Where
The condition is
,
.
Anjan Mukherjee and Rakhal Das. Neutrosophic Bipolar Vague Soft Set And Its Application To Decision Making Problems
Neutrosophic Sets and Systems, Vol. 32, 2020
413
Also,
,
,
,
,
where
The condition is
Example 2.9: Let
.
be a set of universe then the NBV set ANBV as follows:
ANBV =
,
.
Definition 2.10: [ 16] The compliment
of
={
is as
},
={
={
},
},
={
},
={
},
={
}.
Example 2.11: Considering the example 2.9, we have
=
,
.
Definition 2.12: [ 16 ] Two NBV sets
all
and
of the universe U are said to be equal if for
,
Anjan Mukherjee and Rakhal Das. Neutrosophic Bipolar Vague Soft Set And Its Application To Decision Making Problems
Neutrosophic Sets and Systems, Vol. 32, 2020
414
,
,
and
,
,
Where
.
(say).
Definition 2.13 [16 ] If in the universe U, two NBV sets
and
,
be given as
,
and
,
,
for
Then
Definition 2.14: [16 ] The union and intersection of two NBV sets
(i)
and
are given as
where
, and
.
(ii)
is given by
, and
Anjan Mukherjee and Rakhal Das. Neutrosophic Bipolar Vague Soft Set And Its Application To Decision Making Problems
Neutrosophic Sets and Systems, Vol. 32, 2020
415
.
. Neutrosophic Bipolar Vague Soft Set.
In this section we study the concept of Neutrosophic bipolar vague soft set. It is a combination
of neutrosophic vague set & the soft set. Further we study some of its operation and properties.
Definition 3.1: Let U be a universal set. E be a set of parameters and
the set of all neutrosophic bipolar vague set of U. Then the pair
bipolar vague soft set (NBVS set in short) over U. Here
collection of all neutrosophic bipolar vague soft sets over
Example 3.2: Let
and
over
,
. Let NBVset(U) denotes
is called an neutrosophic
is a mapping
is denoted by
. The
.
. Then neutrosophic bipolar vague soft sets
are as follows:
Anjan Mukherjee and Rakhal Das. Neutrosophic Bipolar Vague Soft Set And Its Application To Decision Making Problems
Neutrosophic Sets and Systems, Vol. 32, 2020
416
.
Definition 3.3: An empty neutrosophic bipolar vague soft set
in
is defined as
.
Definition 3.4: An absolute neutrosophic bipolar vague soft set
in
is defined as
.
Example 3.5: Let
then
,
(a) Absolute neutrosophic bipolar vague soft set
in
is defined as
,
,
,
,
Anjan Mukherjee and Rakhal Das. Neutrosophic Bipolar Vague Soft Set And Its Application To Decision Making Problems
Neutrosophic Sets and Systems, Vol. 32, 2020
417
,
}
Definition 3.6:
. Where i = 1, 2 be two
neutrosophic bipolar vague soft set over U. then
is denoted by
is neutrosophic bipolar vague soft subset of
if
,
,
And
,
,
.
Example 3.7: Consider the example 3.2. In this case
as per our definition 3.6.
Definition 3.8: Let A be a neutrosophic bipolar vague soft set over . Then the complement of a
neutrosophic bipolar vague soft set A is denoted by
is defined as
.
,
,
and
,
,
.
Anjan Mukherjee and Rakhal Das. Neutrosophic Bipolar Vague Soft Set And Its Application To Decision Making Problems
Neutrosophic Sets and Systems, Vol. 32, 2020
Example 3.9: Let
then the neutrosophic bipolar vague soft set A is
Then the complement of A is
is as
Definition 3.10: Let
. Then the union and
intersection of
and
of two neutrosophic bipolar vague soft set are defined as follows:
=
(a)
Where
Anjan Mukherjee and Rakhal Das. Neutrosophic Bipolar Vague Soft Set And Its Application To Decision Making Problems
Neutrosophic Sets and Systems, Vol. 32, 2020
And
(b)
=
Where
And
Anjan Mukherjee and Rakhal Das. Neutrosophic Bipolar Vague Soft Set And Its Application To Decision Making Problems
Neutrosophic Sets and Systems, Vol. 32, 2020
Example 3.11. Consider the example 3.2 then
Similarly
.
Definition 3.12 Let
,
,
}:
,
,
,
be a neutrosophic bipolar vague soft set over
U. then aggregation neutrosophic bipolar vague soft operator denoted by
is defined as
Where
Where
Anjan Mukherjee and Rakhal Das. Neutrosophic Bipolar Vague Soft Set And Its Application To Decision Making Problems
Neutrosophic Sets and Systems, Vol. 32, 2020
is the cardinality of
Where
.
4. Application of neutrosophic bipolar vague soft set.
We develop an algorithm based on neutrosophic bipolar vague soft sets and give numerical example
to show the fossibility and effectiveness of the approaches in definition 3.12.
Algorithm
1.
First we construct the neutrosophic bipolar vague soft set on
2.
Then we compute the neutrosophic bipolar vague soft set aggregation operator.
3.
Average of each intervals and find
4.
Find the optimum value on U.
.
.
Assume that a farm wants to fill a position in the office. There are three candidates for the post. The
selection committee use the neutrosophic bipolar vague soft decision making method. Assume that
the set of candidate
. Where
which may be characterized by a set of parameters
= “experience”,
= “technical knowledge”,
= “age”.
(a) The selection committee construct a neutrosophic bipolar vague soft set A over the set
as
,
Anjan Mukherjee and Rakhal Das. Neutrosophic Bipolar Vague Soft Set And Its Application To Decision Making Problems
Neutrosophic Sets and Systems, Vol. 32, 2020
.
(b) Then we find the neutrosophic vague soft set aggregation operator
For
,
+[1,1]
For
+
,
+[1,1]
For
of A2 as
+
,
+[1,1]
+
(c) We take the average of each interval.
i.e. [1,1]=1,
Anjan Mukherjee and Rakhal Das. Neutrosophic Bipolar Vague Soft Set And Its Application To Decision Making Problems
Neutrosophic Sets and Systems, Vol. 32, 2020
(d) The
(e) Finally the selection committee choose
for the post since
has the maximum
degree 0.1455 among them.
5. Conclusion
In this paper, we introduce the neutrosophic bipolar vague soft set. It is a combination of soft set and
the neutrosophic bipolar vague set. We develop a decision making method based on neutrosophic
bipolar vague soft set. A numerical example has beengiven. Some new operations on neutrosophic
bipolar vague soft set have been designed. For further study, it may be applied to real world
problems with realistic data and extend proposed algorithm to other decision making problem with
vagueness and uncertainty. Here we require less calculations and few steps to get our result.
Acknowledgements
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: Sep 06, 2019. Accepted: Mar 17, 2020
Anjan Mukherjee and Rakhal Das. Neutrosophic Bipolar Vague Soft Set And Its Application To Decision Making Problems
Neutrosophic Sets and Systems, Vol. 32, 2020
University of New Mexico
On Some Types of Neutrosophic Topological Groups
with Respect to Neutrosophic Alpha Open Sets
Qays Hatem Imran1*, Ali Hussein Mahmood Al-Obaidi2, and Florentin Smarandache3
1Department
of Mathematics, College of Education for Pure Science, Al-Muthanna University, Samawah, Iraq.
E-mail: qays.imran@mu.edu.iq
2Department of Mathematics, College of Education for Pure Science, University of Babylon, Hillah, Iraq.
E-mail: aalobaidi@uobabylon.edu.iq
3Department of Mathematics, University of New Mexico 705 Gurley Ave. Gallup, NM 87301, USA.
E-mail: smarand@unm.edu
* Correspondence: qays.imran@mu.edu.iq
Abstract: In this article, we presented eight different types of neutrosophic topological groups, each
of which depends on the conceptions of neutrosophic -open sets and neutrosophic -continuous
functions. Also, we found the relation between these types, and we gave some properties on the
other side.
Keywords: Neutrosophic -open sets, neutrosophic -continuous functions, neutrosophic
topological groups, and neutrosophic topological groups of type (𝑅), 𝑅 = 1,2,3, … ,8.
1. Introduction
Smarandache [1,2] originally handed the theory of “neutrosophic set”. Recently, Abdel-Basset
et al. discussed a novel neutrosophic approach [3-6]. Salama et al. [7] gave the clue of neutrosophic
topological space. Arokiarani et al. [8] added the view of neutrosophic α -open subsets of
neutrosophic topological spaces. Dhavaseelan et al. [9] presented the idea of neutrosophic
𝛼 𝑚 -continuity. Banupriya et al. [10] investigated the notion of neutrosophic αgs continuity and
neutrosophic αgs irresolute maps. Nandhini et al. [11] presented Nαg#ψ-open map, Nαg#ψ-closed
map, and Nαg#ψ-homomorphism in neutrosophic topological spaces. Sumathi et al. [12] submitted
the perception of neutrosophic topological groups. The target of this article is to perform eight
different types of neutrosophic topological groups, each of which depends on the notions of
neutrosophic α-open sets and neutrosophic α-continuous functions and also we found the relation
between these types.
2. Preliminaries
In all this paper, (𝒢, 𝜏) and (ℋ, 𝜎) (or briefly 𝒢 and ℋ ) frequently refer to neutrosophic
topological spaces (or shortly NTSs). Suppose 𝒜 be a neutrosophic open subset (or shortly Ne-OS)
of 𝒢, then its complement 𝒜𝑐 is closed (or shortly Ne-CS). In addition, its interior and closure are
denoted by 𝑁𝑖𝑛𝑡(𝒜)and 𝑁𝑐𝑙(𝒜), correspondingly.
Definition 2.1 [8]: Let 𝓐 be a Ne-OS in NTS 𝓖, then it is said that a neutrosophic 𝛂-open subset (or
briefly Ne-𝛂OS) if 𝓐 ⊆ 𝑵𝒊𝒏𝒕(𝑵𝒄𝒍(𝑵𝒊𝒏𝒕(𝓐))). Then 𝓐𝒄 is the so-called a neutrosophic 𝛂-closed (or
briefly Ne-𝛂CS). The collection of all such these Ne-𝛂OSs (resp. Ne-𝛂CSs) of 𝓖 is denoted by
𝑵𝜶𝑶(𝓖) (resp. 𝑵𝜶𝑪(𝓖)).
Definition 2.2 [8]: Let 𝒜 be a neurrosophic set in NTS 𝒢. Then the union of all such these Ne-αOSs
involved in 𝒜( symbolized by 𝛼𝑁𝑖𝑛𝑡(𝒜)) is said to be the neutrosophic α-interior of 𝒜.
Qays Hatem Imran, Ali Hussein Mahmood Al-Obaidi and Florentin Smarandache, On Some Types of Neuteosophic
Topological Groups with respect to Neutrosophic Alpha Open Sets
Neutrosophic Sets and Systems, Vol. 32, 2020
426
Definition 2.3 [8]: Let 𝒜 be a neurrosophic set in NTS 𝒢. Then the intersection of all such these
Ne-αCSs that contain 𝒜 ( symbolized by 𝛼𝑁𝑐𝑙(𝒜)) is said to be the neutrosophic α-closure of 𝒜.
Proposition 2.4 [13]: Let 𝒜 be a neutrosophic set in NTS 𝒢. Then 𝒜 ∈ 𝑁𝛼𝑂(ℬ) iff there exists a NeαOS ℬ where ℬ ⊆ 𝒜 ⊆ 𝑁𝑖𝑛𝑡(𝑁𝑐𝑙(ℬ)).
Proposition 2.5 [8]: In any NTS, the following claims hold, and not vice versa:
(i) For each, Ne-OS is a Ne-αOS.
(ii) For each, Ne-CS is a Ne-αCS.
Definition 2.6: Let 𝒽: (𝒢, 𝜏) ⟶ (ℋ, 𝜎) be a function, then 𝒽 is called:
(i) a neutrosophic continuous (in short Ne-continuous) iff for each 𝒜 Ne-OS in ℋ, then 𝒽−1 (𝒜) is
a Ne-OS in 𝒢 [14].
(ii) a neutrosophic α-continuous (in short Ne-α-continuous) iff for each 𝒜 Ne-OS in ℋ , then
𝒽−1 (𝒜) is a Ne-αOS in 𝒢 [8].
(iii) a neutrosophic α-irresolute (in short Ne-α-irresolute) iff for each 𝒜 Ne-αOS in ℋ , then
𝒽−1 (𝒜) is a Ne-αOS in 𝒢.
Proposition 2.7 [8]: Every Ne-continuous function is a Ne-α-continuous, but the opposite is not valid
in general.
Proposition 2.8: Every Ne-α-irresolute function is a Ne-α-continuous, but the opposite is not exact in
general.
Proof: Let 𝒽: (𝒢, 𝜏) ⟶ (ℋ, 𝜎) be a Ne-α-irresolute function and let 𝒜 be any Ne-OS in ℋ. From
proposition 2.5, we get 𝒜 is a Ne-αOS in ℋ . Since 𝒽 is a Ne-α-irresolute, then 𝒽−1 (𝒜) is a
Ne-αOS in 𝒢. Therefore 𝒽 is a Ne-α-continuous.
Example 2.9: Let 𝒢 = {𝑝, 𝑞}. Suppose the neutrosophic sets 𝒜, ℬ, 𝒞 and 𝒟 be in 𝒢 as follows:
𝒜 = 〈𝑥, (
𝒞 = 〈𝑥, (
𝑝
,
𝑞
0.5 0.3
𝑝
,
𝑞
0.6 0.3
),(
),(
𝑝
,
𝑞
0.5 0.3
𝑝
,
𝑞
0.6 0.3
),(
),(
𝑝
,
𝑞
0.5 0.7
𝑝
,
𝑞
0.4 0.7
)〉, ℬ = 〈𝑥, (
𝑝
,
𝑞
0.5 0.6
)〉 and 𝒟 = 〈𝑥, (
𝑝
),(
,
𝑞
𝑝
,
𝑞
0.5 0.6
0.6 0.7
),(
𝑝
),(
,
𝑞
𝑝
,
𝑞
0.5 0.4
0.6 0.7
),(
𝑝
)〉,
,
𝑞
0.4 0.3
)〉.
Then the families 𝜏 = {0𝑁 , 𝒜, 1𝑁 } and 𝜎 = {0𝑁 , 𝒟, 1𝑁 } are neutrosophic topologies on 𝒢.
Thus, (𝒢, 𝜏) and (𝒢, 𝜎) are NTSs. Define 𝒽: (𝒢, 𝜏) ⟶ (𝒢, 𝜎) as (𝑝) = 𝑝, 𝒽(𝑞) = 𝑞. Hence 𝒽 is a
Ne-α-continuous function, but not Ne-α-irresolute.
Definition 2.10: A function 𝒽: (𝒢, 𝜏) ⟶ (ℋ, 𝜎) is said to be ℳ-function iff 𝒽−1 (𝑁𝑖𝑛𝑡(𝑁𝑐𝑙(ℬ))) ⊆
𝑁𝑖𝑛𝑡(𝑁𝑐𝑙(𝒽−1 (ℬ) )), for every Ne-αOS ℬ of ℋ.
Theorem 2.11: If 𝒽: (𝒢, 𝜏) ⟶ (ℋ, 𝜎) is a Ne-α-continuous function and ℳ-function, then 𝒽 is a
Ne-α-irresolute.
Proof: Let 𝒜 be any Ne-αOS of ℋ, there exists a Ne-OS ℬ of ℋ where ℬ ⊆ 𝒜 ⊆ 𝑁𝑖𝑛𝑡(𝑁𝑐𝑙(ℬ)).
Since 𝒽 is ℳ-function, we have 𝒽−1 (ℬ) ⊆ 𝒽−1 (𝒜) ⊆ 𝒽−1 (𝑁𝑖𝑛𝑡(𝑁𝑐𝑙(ℬ)) ) ⊆ 𝑁𝑖𝑛𝑡(𝑁𝑐𝑙(𝒽−1 (ℬ) )).
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By proposition 2.4, we have 𝒽−1 (𝒜) is a Ne-αOS. Hence, 𝒽 is a Ne-α-irresolute.
Definition 2.12 [8]: A function 𝒽: (𝒢, 𝜏) ⟶ (ℋ, 𝜎) is called a neutrosophic α -open (resp.
neutrosophic α -closed) iff for each 𝒜 ∈ 𝑁𝛼𝑂(𝒢) (resp. 𝒜 ∈ 𝑁𝛼𝐶(𝒢) ), 𝒽(𝒜) ∈ 𝑁𝛼𝑂(ℋ)
(resp. 𝒽(𝒜) ∈ 𝑁𝛼𝐶(ℋ)).
Definition
2.13
[15]:
A
bijective
function 𝒽: (𝒢, 𝜏) ⟶ (ℋ, 𝜎)
is
called
a
neutrosophic
homeomorphism iff 𝒽 and 𝒽−1 are Ne-continuous.
Definition 2.14 [12]: A neutrosophic topological group (briefly NTG) is a set 𝒢 which carries a
group structure and a neutrosophic topology with the following two postulates:
(i) The operation function 𝜇: 𝒢 × 𝒢 → 𝒢, given as 𝜇(𝑔, ℎ) = 𝑔 ⋅ ℎ is a Ne-continuous.
(ii) The inversion function 𝐼: 𝒢 → 𝒢, given as 𝐼(𝑔) = 𝑔−1 is a Ne-continuous.
Remark 2.15 [12]:
(i) The function 𝛾: 𝒢 × 𝒢 → 𝒢, given as 𝛾(𝑔, ℎ) = 𝑔 ⋅ ℎ is a Ne-continuous iff for each Ne-OS 𝒞 and
𝑔 ⋅ ℎ ∈ 𝒞, there exist Ne-OS 𝒜, ℬ such that 𝑔 ∈ 𝒜, ℎ ∈ ℬ, and 𝒜 ⋅ ℬ ⊆ 𝒞.
(ii) The function 𝑖𝑛𝑣: 𝒢 → 𝒢 is a Ne-continuous iff for each Ne-OS 𝒜 and 𝑔−1 ∈ 𝒜, there exists a
Ne-OS ℬ and 𝑔 ∈ ℬ where ℬ −1 ⊆ 𝒜.
Definition 2.16 [16]: A group 𝒢 is nice iff its operation is nice.
3. Different Types of Neutrosophic Topological Groups
In this section, we introduce eight types of neutrosophic topological groups, each of which
depends on the notions of neutrosophic α-open sets and neutrosophic α-continuous functions.
Definition 3.1: Let 𝒢 be a set that equips with a group structure and a neutrosophic topology. Then
𝒢 is called:
(i) NTG of type (1) iff the operation function 𝜇: 𝒢 × 𝒢 → 𝒢 and the inversion function 𝐼: 𝒢 → 𝒢 are
both Ne-α-continuous.
(ii) NTG of type (2) iff the operation function 𝜇: 𝒢 × 𝒢 → 𝒢 and the inversion function 𝐼: 𝒢 → 𝒢 are
both Ne-α-irresolute.
(iii) NTG of type (3) iff the operation function 𝜇: 𝒢 × 𝒢 → 𝒢 is Ne-α-continuous and the inversion
function 𝐼: 𝒢 → 𝒢 is Ne-continuous.
(iv) NTG of type (4) iff the operation function 𝜇: 𝒢 × 𝒢 → 𝒢 is Ne-α-irresolute and the inversion
function 𝐼: 𝒢 → 𝒢 is Ne-continuous.
(v) NTG of type (5) iff the operation function 𝜇: 𝒢 × 𝒢 → 𝒢 is Ne-α-irresolute and the inversion
function 𝐼: 𝒢 → 𝒢 is Ne-α-continuous.
(vi) NTG of type (6) iff the operation function 𝜇: 𝒢 × 𝒢 → 𝒢 is Ne-α-continuous and the inversion
function 𝐼: 𝒢 → 𝒢 is Ne-α-irresolute.
(vii) NTG of type (7) iff the operation function 𝜇: 𝒢 × 𝒢 → 𝒢 is Ne-continuous, and the inversion
function 𝐼: 𝒢 → 𝒢 is Ne-α-continuous.
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(viii) NTG of type (8) iff the operation function 𝜇: 𝒢 × 𝒢 → 𝒢 is Ne-continuous, and the inversion
function 𝐼: 𝒢 → 𝒢 is Ne-α-irresolute.
Proposition 3.2:
(i) Every NTG is a NTG of type (𝑅), where 𝑅 = 1,3,7.
(ii) Every NTG of type (2) is a NTG of type (5).
(iii) Every NTG of type (2) is a NTG of type (6).
(iv) Every NTG of type (4) is a NTG of type (3).
(v) Every NTG of type (4) is a NTG of type (5).
(vi) Every NTG of type (𝑅) is a NTG of type (1), where 𝑅 = 2,3, … ,8.
Proof:
(i) Let 𝒢 be a NTG, then the operation function 𝜇 and the inversion function 𝐼 are both
Ne-continuous. By proposition 2.7, we have that the operation function 𝜇 and the inversion
function 𝐼 are both Ne-α-continuous. Hence, 𝒢 is a NTG of type (𝑅), where 𝑅 = 1,3,7.
(ii) Let 𝒢 be a NTG of type (2), then the operation function 𝜇 and the inversion function 𝐼 are both
Ne-α-irresolute. By proposition 2.8, we have that the inversion function 𝐼 is a Ne-α-continuous.
Hence, 𝒢 is a NTG of type (5).
(iii) Let 𝒢 be a NTG of type (2), then the operation function 𝜇 and the inversion function 𝐼 are
both Ne- α -irresolute. By proposition 2.8, we have that the operation function 𝜇 is a
Ne-α-continuous. Hence, 𝒢 is a NTG of type (6).
(iv) Let 𝒢 be a NTG of type (4), then the operation function 𝜇 is a Ne-α-irresolute and the
inversion function 𝐼 is a Ne-continuous. By proposition 2.8, we have that the operation function 𝜇
is a Ne-α-continuous. Hence, 𝒢 is a NTG of type (3).
(v) Let 𝒢 be a NTG of type (4), then the operation function 𝜇 is a Ne-α-irresolute and the inversion
function 𝐼 is a Ne-continuous. By proposition 2.7, we have that the inversion function 𝐼 is a
Ne-α-continuous. Hence, 𝒢 is a NTG of type (5).
(vi) Let 𝒢 be a NTG of type (𝑅), where 𝑅 = 2,3, … ,8. By proposition 2.7 and proposition 2.8, we
have that the operation function 𝜇 and the inversion function 𝐼 are both Ne-α-continuous. Hence, 𝒢
is a NTG of type (1).
Proposition 3.3:
(i) A NTG of type (3) with ℳ-function operation 𝜇 is a NTG of type (4).
(ii) A NTG of type (1) with ℳ-function inversion 𝐼 and ℳ-function operation 𝜇 is a NTG of type
(2).
(iii) A NTG of type (1) with ℳ-function operation 𝜇 is a NTG of type (5).
(iv) A NTG of type (1) with ℳ-function inversion 𝐼 is a NTG of type (6).
(v) A NTG of type (5) with ℳ-function inversion 𝐼 is a NTG of type (2).
(vi) A NTG of type (6) with ℳ-function operation 𝜇 is a NTG of type (2).
(vii) A NTG of type (7) with ℳ-function inversion 𝐼 is a NTG of type (8).
Proof:
(i) Let 𝒢 be a NTG of type (3), then the operation function 𝜇 is a Ne-α-continuous and the inversion
function 𝐼 is a Ne-continuous. Since 𝜇 is ℳ-function. So by Theorem 2.11, we get that operation 𝜇
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is a Ne-α-irresolute. Hence, 𝒢 is a NTG of type (4).
(ii) Let 𝒢 be a NTG of type (1), then the operation function 𝜇 and the inversion function 𝐼 are both
Ne-α-continuous. Since 𝜇, 𝐼 are ℳ-function. So by Theorem 2.11, we get that the operation function
𝜇 and the inversion function 𝐼 are both Ne-α-irresolute. Hence, 𝒢 is a NTG of type (2). The proof is
evident for others.
Remark 3.4: The next illustration displays relationship among different kinds of neutrosophic
topological groups mentioned in this section and the neutrosophic topological group:
Definition 3.5: A bijective function 𝒽: (𝒢, 𝜏) ⟶ (ℋ, 𝜎) is said to be:
(i) Neutrosophic α-homeomorphism iff 𝒽 and 𝒽−1 are Ne-α-continuous.
(ii) Neutrosophic α-irresolute – homeomorphism iff 𝒽 and 𝒽−1 are Ne-α-irresolute.
Definition 3.6: Let (𝒢, 𝜏) be a NTS, then 𝒢 is called neutrosophic α -homogeneous (resp.
neutrosophic α -irresolute – homogeneous) iff for any two elements 𝑔, ℎ ∈ 𝒢 , there exists a
neutrosophic α-homeomorphism (resp. neutrosophic α-irresolute – homeomorphism) from 𝒢 onto
𝒢 which transforms 𝑔 into ℎ.
Proposition 3.7: The inversion function 𝐼 in a NTG of type (𝑅) , where 𝑅 = 1,2, … … ,8 is a
neutrosophic α-homeomorphism.
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Proof: Let 𝒢 be a NTG of type (1). Since 𝒢 is a group, 𝐼(𝒢) = 𝒢 −1 = 𝒢 which implies 𝐼 is onto,
also for any 𝑔 ∈ 𝒢 , there exists a unique inverse which is equal to 𝐼(𝑔) which implies, 𝐼 is
one-to-one. Now; we have 𝐼 is a Ne- α -continuous and 𝐼 −1 : 𝒢 → 𝒢 such that 𝐼 −1 (𝑔) = 𝑔 , i.e
𝐼 −1 (𝑔) = 𝐼(𝑔) for each 𝑔 ∈ 𝒢 , so, 𝐼 −1 is a Ne- α -continuous. Thus, 𝐼 is a neutrosophic
α-homeomorphism. In the case of type (𝑅), we have a similar proof, where 𝑅 = 2,3, … ,8.
Corollary 3.8: Let 𝒢 be a NTG of type (1) and 𝒜 ⊆ 𝒢. If 𝒜 ∈ 𝜏, then 𝒜−1 ∈ 𝑁𝛼𝑂(𝒢).
Proof: Since the inversion function 𝐼 is a neutrosophic α-homeomorphism, then 𝐼(𝒜) = 𝒜 −1 is a
Ne-αOS in 𝒢 for each 𝒜 ∈ 𝜏.
Proposition 3.9: The inversion function 𝐼 in a NTG of type (3) [and type (4)] is a neutrosophic
homeomorphism.
Proof: Suppose 𝒢 be a NTG of type (3). Since 𝒢 is a group, 𝐼(𝒢) = 𝒢 −1 = 𝒢 which implies 𝐼 is
onto, also for any 𝑔 ∈ 𝒢, there exists a unique inverse which is equal to 𝐼(𝑔) which implies, 𝐼 is
one-to-one. Now; we have 𝐼 is a Ne-continuous and 𝐼 −1 : 𝒢 → 𝒢 such that 𝐼 −1 (𝑔) = 𝑔, i.e 𝐼 −1 (𝑔) =
𝐼(𝑔) for each 𝑔 ∈ 𝒢, so, 𝐼 −1 is a Ne-continuous. Thus, 𝐼 is a neutrosophic homeomorphism. In the
case of type (4), we have similar proof.
Proposition 3.10: The inversion function 𝐼 in a NTG of type (𝑅), where 𝑅 = 2,6,8 is a neutrosophic
α-irresolute – homeomorphism.
Proof: Suppose 𝒢 be a NTG of type (2). Since 𝒢 is a group, 𝐼(𝒢) = 𝒢 −1 = 𝒢 which implies 𝐼 is
onto, also for any 𝑔 ∈ 𝒢, there exists a unique inverse which is equal to 𝐼(𝑔) which implies, 𝐼 is
one-to-one. Now; we have 𝐼 is a Ne-α-irresolute and 𝐼 −1 : 𝒢 → 𝒢 such that 𝐼 −1 (𝑔) = 𝑔, i.e 𝐼 −1 (𝑔) =
𝐼(𝑔) for each 𝑔 ∈ 𝒢 , so, 𝐼 −1
is a Ne- α -irresolute. Thus, 𝐼 is a neutrosophic α -irresolute –
homeomorphism. In the case of type (6) and type (8), we have a similar proof.
Proposition 3.11: Let 𝒢 be a set which carries a group structure and a neutrosophic topology, let
𝑘1 , 𝑘2 ∈ 𝒢. Then for each 𝑔 ∈ 𝒢 if one of the following functions:
(i) 𝑙𝑘1 (𝑔) = 𝑘1 ⋅ 𝑔
(ii) 𝑟𝑘1 (𝑔) = 𝑔 ⋅ 𝑘1
(iii) 𝒽𝑘1𝑘2 (𝑔) = 𝑘1 ⋅ 𝑔 ⋅ 𝑘2
is a neutrosophic α-homeomorphism (resp. neutrosophic α-irresolute – homeomorphism), then so
the others.
Proof: Since 𝑘1 and 𝑘2 are arbitrary elements in 𝒢, clear that 𝑙𝑘1 and 𝑟𝑘1 come from 𝒽𝑘1 𝑘2 by taking
𝑘2 = 𝑒 or 𝑘1 = 𝑒 respectively. Hence, when 𝒽𝑘1𝑘2 is a neutrosophic α -homeomorphism, both
𝑙𝑘1
is a neutrosophic
then for each 𝑔 ∈ 𝒢 , 𝑙𝑘1 (ℎ) = 𝑙𝑘1 (𝑔 ⋅ 𝑘2 ) = 𝑘1 ⋅ 𝑔 ⋅ 𝑘2 = 𝒽𝑘1𝑘2 (𝑔) , 𝒽𝑘1𝑘2
is a neutrosophic
𝑙𝑘1 and
𝑟𝑘2 are
neutrosophic
α -homeomorphisms.
Now;
when
α-homeomorphism. Since 𝒢 is a group, 𝒢 ⋅ 𝑘 = 𝒢 for each 𝑘 ∈ 𝒢 then 𝒢 ⋅ 𝑘2 = 𝒢. Hence, for each
ℎ ∈ 𝒢 ⋅ 𝑘2 , 𝑙𝑘1 (ℎ) = 𝑘1 ⋅ ℎ, 𝑙𝑘1 is a neutrosophic α-homeomorphism. But ℎ = 𝑔 ⋅ 𝑘2 for some 𝑔 ∈ 𝒢,
α-homeomorphism. Then by the first part of the proof, 𝑟𝑘1 . And we have a similar proof if we are
beginning with 𝑟𝑘1 is a neutrosophic α-homeomorphism. In the case of neutrosophic α-irresolute –
homeomorphism, we have a similar proof.
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Theorem 3.12: Let 𝒢 be a nice NTG of type (𝑅), where 𝑅 = 1,2,3, … ,8 and let 𝑘1 , 𝑘2 ∈ 𝒢. Then for
each 𝑔 ∈ 𝒢 the following functions:
(i) 𝑙𝑘1 (𝑔) = 𝑘1 ⋅ 𝑔
(ii) 𝑟𝑘1 (𝑔) = 𝑔 ⋅ 𝑘1
(iii) 𝒽𝑘1𝑘2 (𝑔) = 𝑘1 ⋅ 𝑔 ⋅ 𝑘2
are neutrosophic α-homeomorphisms.
Proof: Let 𝒢 be a nice NTG of type (1). It is clear that each of the functions 𝑙𝑘1 , 𝑟𝑘1 and 𝒽𝑘1 𝑘2 is a
bijective function. Let 𝒽 be the operation of 𝒢, then 𝒽 is a Ne-α-continuous. Since 𝒢 is a nice, so
𝑙𝑘1 = 𝒽/{𝑘1 } × 𝒢 is a Ne-α-continuous. Similarly, 𝑙𝑘1 −1 (𝑔) = 𝑘1 −1 ⋅ 𝑔, 𝑙𝑘1 −1 is a Ne-α-continuous.
Hence, 𝑙𝑘1 is a neutrosophic α-homeomorphism. Thus, because of the preceding proposition, 𝑟𝑘1
and 𝒽𝑘1 𝑘2 are neutrosophic α -homeomorphisms. The case of type (𝑅) has a similar proof,
where 𝑅 = 2,3, … ,8.
Theorem 3.13: Let 𝒢 be a nice NTG of type (𝑅), where 𝑅 = 2,4,5 and let 𝑘1 , 𝑘2 ∈ 𝒢. Then for each
𝑔 ∈ 𝒢 the following functions:
(i) 𝑙𝑘1 (𝑔) = 𝑘1 ⋅ 𝑔
(ii) 𝑟𝑘1 (𝑔) = 𝑔 ⋅ 𝑘1
(iii) 𝒽𝑘1𝑘2 (𝑔) = 𝑘1 ⋅ 𝑔 ⋅ 𝑘2
are neutrosophic α-irresolute – homeomorphisms.
Proof: Let 𝒢 be a nice NTG of type (2). It is clear that each of the functions 𝑙𝑘1 , 𝑟𝑘1 and 𝒽𝑘1 𝑘2 is a
bijective function. Let 𝒽 be the operation of 𝒢, then 𝒽 is a Ne-α-irresolute. Since 𝒢 is a nice, so
𝑙𝑘1 = 𝒽/{𝑘1 } × 𝒢
is a Ne-α-irresolute. Similarly, 𝑙𝑘1 −1 (𝑔) = 𝑘1 −1 ⋅ 𝑔, 𝑙𝑘1 −1 is a Ne-α-irresolute.
Hence, 𝑙𝑘1 is a neutrosophic α -irresolute – homeomorphism. Thus, given the preceding
proposition, 𝑟𝑘1 and 𝒽𝑘1 𝑘2 are neutrosophic α-irresolute – homeomorphisms. The case of type (𝑅)
has a similar proof, where 𝑅 = 4,5.
Corollary 3.14: Let 𝒜, ℬ and 𝒞 be subsets of a nice NTG 𝒢 of type (1) (resp. of type (4)) such that
𝒜 is a Ne-CS (resp. Ne-αCS), and ℬ is a Ne-OS (resp. Ne-αOS). Then for each 𝑘 ∈ 𝒢, 𝑘 ⋅ 𝒜 and 𝒜 ⋅
𝑘 are Ne-α-CSs also 𝑘 ⋅ ℬ, ℬ ⋅ 𝑘, 𝒞 ⋅ ℬ and ℬ ⋅ 𝒞 are Ne-αOSs.
Proof: Since 𝒜 is a Ne-CS so in view of the theorem 3.12, 𝑙𝑘 (𝒜) = 𝑘 ⋅ 𝒜 and 𝑟𝑘 (𝒜) = 𝒜 ⋅ 𝑘 are
Ne-αCSs.
Similarly, since ℬ is a Ne-OS so in view of the theorem 3.12, 𝑙𝑘 (ℬ) = 𝑘 ⋅ ℬ and 𝑟𝑘 (ℬ) = ℬ ⋅ 𝑘 are
Ne-αOSs. Also, 𝒞 ⋅ ℬ = ⋃𝒸∈𝒞 𝒸 ⋅ ℬ but 𝒸 ⋅ ℬ is a Ne-αOS for each 𝒸 ∈ 𝒞. Hence, 𝒞 ⋅ ℬ is a Ne-αOS.
Similarly, ℬ ⋅ 𝒞 is a Ne-αOS. In the case of type (4), we have a similar proof.
Corollary 3.15: A nice NTG of type (𝑅), where 𝑅 = 1,2,3, … ,8 is neutrosophic α-homogeneous.
Proof: Let 𝒢 be a nice NTG of type (1) and 𝑎, 𝑏 ∈ 𝒢. Then for any fixed element 𝑘 ∈ 𝒢, 𝑟𝑘 is a
neutrosophic α-homeomorphism, therefore, it is true when 𝑘 = 𝑎 −1 ⋅ 𝑏. Thus, 𝑟𝑎−1 𝑏 (𝑔) = 𝑔 ⋅ 𝑎−1 ⋅ 𝑏
is a neutrosophic α-homeomorphism we need because 𝑟𝑎−1 𝑏 (𝑎) = 𝑏. Therefore, 𝒢 is a neutrosophic
α-homogeneous. In the case of type (𝑅), we have a similar proof, where 𝑅 = 2,3, … ,8.
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Corollary 3.16: A nice NTG of type (𝑅) , where 𝑅 = 2,4,5 is neutrosophic α -irresolute –
homogeneous.
Proof: Let 𝒢 be a nice NTG of type (2) and 𝑎, 𝑏 ∈ 𝒢. Then for any fixed element 𝑘 ∈ 𝒢, 𝑟𝑘 is a
neutrosophic α -irresolute – homeomorphism, therefore, it is true when 𝑘 = 𝑎 −1 ⋅ 𝑏 . Thus,
𝑟𝑎−1𝑏 (𝑔) = 𝑔 ⋅ 𝑎 −1 ⋅ 𝑏 is a neutrosophic α-irresolute – homeomorphism. But 𝑟𝑎−1𝑏 (𝑎) = 𝑏, therefore
𝒢 is a neutrosophic α-irresolute – homogeneous. In the case of type (𝑅), we have a similar proof,
where 𝑅 = 4,5.
Definition 3.17: Let 𝒢 be a NTG of type (2), (5), and ℱ be a fundamental system of neutrosophic
α-open nhds of the identity element 𝑒. Then for any fixed element 𝑘 ∈ 𝒢, 𝑟𝑘 is a neutrosophic
α -irresolute – homeomorphism. So ℱ(𝑘) = {𝑟𝑘 (𝒜) = 𝒜 ⋅ 𝑘: 𝒜 ∈ ℱ} is a fundamental system of
neutrosophic α-open nhds of 𝑘.
Proposition 3.18: Let 𝒢 be a NTG of type (2), (5). Any fundamental system ℱ of neutrosophic
α-open nhds of e in 𝒢 has the below postulates:
(i) If 𝒜, ℬ ∈ ℱ, then ∃𝒞 ∈ ℱ such that 𝒞 ⊆ 𝒜⋂ℬ.
(ii) If 𝑔 ∈ 𝒜 ∈ ℱ, then ∃ℬ ∈ ℱ such that ℬ ⋅ 𝑔 ⊆ 𝒜.
(iii) If 𝒜 ∈ ℱ, then ∃ℬ ∈ ℱ such that ℬ −1 ⋅ ℬ ⊆ 𝒜.
(iv) If 𝒜 ∈ ℱ, 𝑘 ∈ 𝒢, then ∃ℬ ∈ ℱ such that 𝑘 −1 ⋅ ℬ ⋅ 𝑘 ⊆ 𝒜.
(v) ∀𝒜 ∈ ℱ, ∃ℬ ∈ ℱ such that ℬ −1 ⊆ 𝒜.
(vi) ∀𝒜 ∈ ℱ, ∃𝒞 ∈ ℱ such that 𝒞 2 ⊆ 𝒜.
Proof:
(i) Let 𝒜, ℬ ∈ ℱ, then 𝒜⋂ℬ ∈ ℱ, so ∃𝒞 ∈ ℱ such that 𝒞 ⊆ 𝒜⋂ℬ.
(ii) Let 𝒜 ∈ ℱ and 𝑔 ∈ 𝒜 implies 𝒜 ⋅ 𝑔−1 ∈ ℱ, then ∃ℬ ∈ ℱ such that ℬ ⊆ 𝒜 ⋅ 𝑔−1 . Thus, ℬ ⋅ 𝑔 ⊆
𝒜.
(iii) The function 𝜇: 𝒢 × 𝒢 → 𝒢, given by 𝜇(𝑔, ℎ) = 𝑔−1 ⋅ ℎ is a Ne-α-irresolute because 𝒢 is a NTG
of type (2), (5). Thus 𝜇 −1 (𝒜) is a neutrosophic α-open nhd in 𝒢 × 𝒢 contains (𝑒, 𝑒) and hence
includes a set of the from 𝒰 × 𝒱, where 𝒰, 𝒱 are neutrosophic α-open and provide 𝑒. But 𝒰⋂𝒱 is
a neutrosophic α-open contains 𝑒, so ∃ℬ ∈ ℱ such that ℬ ⊆ 𝒰⋂𝒱 then ℬ ⊆ 𝒰 and ℬ ⊆ 𝒱. Thus
ℬ × ℬ ⊆ 𝒰 × 𝒱 ⊆ 𝜇 −1 (𝒜), then 𝜇(ℬ × ℬ) ⊆ 𝒜 but 𝜇(ℬ × ℬ) = ℬ −1 ⋅ ℬ ⊆ 𝒜.
(iv) The function 𝒽: 𝒢 → 𝒢 given by 𝒽(𝑔) = 𝑘 −1 ⋅ 𝑔 ⋅ 𝑘 is a Ne- α -irresolute. Since 𝑙𝑘 −1 , 𝑟𝑘 is
Ne- α -irresolute. So 𝑙𝑘 −1 ∘ 𝑟𝑘 is a Ne- α -irresolute from 𝒢 to 𝒢 put 𝒽 = 𝑙𝑘 −1 ∘ 𝑟𝑘 , 𝒽(𝑔) = (𝑙𝑘 −1 ∘
𝑟𝑘 )(𝑔) = 𝑙𝑘 −1 (𝑟𝑘 (𝑔)) = 𝑙𝑘 −1 (𝑔 ⋅ 𝑘) = 𝑘 −1 ⋅ 𝑔 ⋅ 𝑘.
𝑟𝑘
𝒢
𝒽
𝒢
𝒢
𝑙𝑘 −1
So, 𝒽−1 (𝒜) is a neutrosophic α -open nhd and contains 𝑒 , hence ∃ℬ ∈ ℱ, ℬ ⊆ 𝒽−1 (𝒜) then
𝒽(ℬ) ⊆ 𝒜. Thus, 𝒽(ℬ) = 𝑘 −1 ⋅ ℬ ⋅ 𝑘 ⊆ 𝒜.
Qays Hatem Imran, Ali Hussein Mahmood Al-Obaidi and Florentin Smarandache, On Some Types of Neuteosophic
Topological Groups with respect to Neutrosophic Alpha Open Sets
Neutrosophic Sets and Systems, Vol. 32, 2020
433
(v) Since 𝐼 the inverse function in a NTG of type (2) is a Ne-α-irresolute, then 𝐼 −1 (𝒜) is a
neutrosophic α-open contains 𝑒 so ∃ℬ ∈ ℱ such that ℬ ⊆ 𝐼 −1 (𝒜) then 𝐼(ℬ) ⊆ 𝒜 . Thus, 𝐼(ℬ) =
ℬ −1 ⊆ 𝒜.
(vi) Since 𝜇 in a NTG of type (5) is a Ne- α-irresolute. So 𝜇 −1 (𝒜) is a neutrosophic α-open
contains (𝑒, 𝑒) and thus contains a neutrosophic set of the from 𝒰 × 𝒱, where 𝒰, 𝒱 are neutrosophic
α-open and contain 𝑒 then 𝒰⋂𝒱 is a neutrosophic α-open and contain 𝑒 ∃𝒞 ∈ ℱ such that 𝒞 ⊆
𝒰⋂𝒱, then 𝒞 × 𝒞 ⊆ 𝒰 × 𝒱 ⊆ 𝜇 −1 (𝒜). Thus, 𝜇(𝒞 × 𝒞) = 𝒞 ⋅ 𝒞 = 𝒞 2 ⊆ 𝒜.
Definition 3.19: A neutrosophic α-open nhd 𝒞 of 𝑔 is called symmetric if 𝒞 −1 = 𝒞.
Proposition 3.20: Let 𝒢 be a NTG of type (𝑅), where 𝑅 = 1,2, … ,8, and let ℬ be any neutrosophic
α-open nhd of a point 𝑔 ∈ 𝒢. Then ℬ⋃ℬ −1 is symmetric neutrosophic α-open nhd of 𝑔.
Proof: Let ℬ is a neutrosophic α-open nhd of 𝑔, then ℬ⋃ℬ −1 is a neutrosophic α-open nhd of 𝑔;
ℬ⋃ℬ −1 = {𝑏: 𝑏 ∈ ℬ 𝑜𝑟 𝑏 ∈ ℬ −1 } = {𝑏: 𝑏 −1 ∈ ℬ 𝑜𝑟 𝑏 −1 ∈ ℬ −1 }
= {𝑏: 𝑏 −1 ∈ ℬ⋃ℬ −1 } = {𝑏: 𝑏 ∈ (ℬ⋃ℬ −1 )−1 } = (ℬ⋃ℬ −1 )−1 .
That is, ℬ⋃ℬ −1 is symmetric neutrosophic α-open nhd of 𝑔.
Proposition 3.21: Let ℬ be any neutrosophic α-open nhd of 𝑒 in a nice NTG of type (𝑅), where
𝑅 = 1,2, … . . ,8. Then ℬ ⋅ ℬ −1 is symmetric neutrosophic α-open nhd of 𝑒.
Proof: Let ℬ be a neutrosophic α-open nhd of 𝑒 and since 𝒢 is a nice, then ℬ ⋅ ℬ −1 is neutrosophic
α-open nhd of 𝑒;
ℬ ⋅ ℬ −1 = {𝑥 ⋅ 𝑦 −1 : 𝑥, 𝑦 ∈ ℬ} = {(𝑥 −1 )−1 ⋅ 𝑦 −1 : 𝑥, 𝑦 ∈ ℬ} = (ℬ −1 )−1 ⋅ ℬ −1 = (ℬ ⋅ ℬ −1 )−1 .
That is, ℬ ⋅ ℬ −1 is symmetric neutrosophic α-open nhd of 𝑒.
4. Conclusion
In this work, we examined the conceptions of eight different types of neutrosophic topological
groups, each of which, depending on the notions of neutrosophic α-open sets and neutrosophic
α-continuous function. In the future, we plan to rsearch the ideas of neutrosophic topological
subgroups and the neutrosophic topological quotient groups as well as defining the perception of
neutrosophic topological product groups with some results.
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Topological Groups with respect to Neutrosophic Alpha Open Sets
Neutrosophic Sets and Systems, Vol. 32, 2020
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Received: Dec 11, 2019. Accepted: Mar 20, 2020
Qays Hatem Imran, Ali Hussein Mahmood Al-Obaidi and Florentin Smarandache, On Some Types of Neuteosophic
Topological Groups with respect to Neutrosophic Alpha Open Sets
Neutrosophic Sets and Systems, Vol. 32, 2020
University of New Mexico
A Contemporary Approach on Neutrosophic Nano Topological
Spaces
1 D.
Sasikala and 2 K.C. Radhamani
1Department
of Mathematics, PSGR Krishnammal College for Women, Coimbatore, Tamilnadu, India, Email:
dsasikala@psgrkcw.ac.in
2Department of Mathematics, Dr.N.G.P.Arts and Science College, Coimbatore, Tamilnadu, India, Email:
radhamani@drngpasc.ac.in
Abstract: In this article, we implement a new notion of sets namely neutrosophic nano j-closed set,
neutrosophic nano generalized closed set, neutrosophic nano generalized j-closed set and
neutrosophic nano generalized j*-closed set in neutrosophic nano topological spaces. We also provide
some appropriate examples to study the properties of these sets. The existing relations between some
of these sets in neutrosophic nano topological space have been investigated.
Keywords: Neutrosophic nano j-closed set, neutrosophic nano generalized closed set, neutrosophic
nano generalized j-closed set, neutrosophic nano generalized j*-closed set.
I.
Introduction
In recent years, Topology plays a vast role in research area. In particular, the concept of
neutrosophy is a trending tool in topology. We use fuzzy concept where we consider only the
membership value. The intuitionistic fuzzy concept is used where the membership and the nonmembership values are considered. But, more real life problems deal with indeterminacy. The
suitable concept for the situation where the indeterminacy occurs is neutrosophy which is
represented by the degree of membership (truth value), the degree of non-membership (falsity value)
and the degree of indeterminacy.
The fuzzy concept was initially proposed by Zadeh [22] in 1965 and Chang [7] introduced Fuzzy
topological spaces in 1968. Atanasov [6] defined intuitionistic fuzzy set and Coker [8] developed
intuitionistic fuzzy topology. In 2005, Smarandache [17] introduced neutrosophic set and many
researchers used this concept in engineering, medicine and many fields where the situation of
indeterminacy arises. Abdel-Basset et.al, [1 - 5] working with many practical problems by using
neutrosophy concept in the recent days. Salama et.al, [14] introduced the generalization of
neutrosophic sets, neutrosophic closed sets and neutrosophic crisp sets in neutrosophic topological
spaces.
The nano topology which has the maximum of five elements was introduced by Lellis Thivagar
[9]. He applied nano topology for nutrition modelling [11] and medical diagnosis [12]. Zhang et.al
[23], worked on neutrosophic rough sets over two universes. Lellis Thivagar initiated [10]
neutrosophic nano topology and some closed sets on neutrosophic nano topological spaces were
derived by recent researchers.
D. Sasikala and K.C. Radhamani, A Contemporary Approach on Neutrosophic Nano Topological Spaces
Neutrosophic Sets and Systems, Vol. 32, 2020
436
Sasikala and Arockiarani [15] introduced generalized j-closed set. Sasikala and Radhamani [16]
introduced nano j-closed set in nano topological spaces. In this paper, we present a new set called
neutrosophic nano j-closed set and work with some interesting examples. Also we investigate some
of the properties of the introduced sets.
II.
Preliminaries
Definition 2.1[9] Let U be a nonempty finite set of objects called the universe and R be an
equivalence relation on U , called the indiscernibility relation. The pair ( U , R ) is said to be the
approximation space. Let X ⊆ U .
The lower approximation of X with respect to the relation R is the set of all objects,
(i)
X
which can be for certain classified as
and it is denoted by
LR ( X ) . i.e.,
LR ( X ) = { R( x ) : R( x ) ⊆ U } , where R( x ) denotes the equivalence class determined
by x .
x∈ U
The upper approximation of X with respect to the relation R is the set of all objects,
(ii)
which can be possibly classified as X and it is denoted by
U R ( X ) .i.e.,
U R ( X ) = { R( x ) : R( x ) X ≠ φ }
(iii)
x∈ U
The boundary region of X with respect to the relation R is the set of all objects, which
can be classified neither as X nor as not X and it is denoted by BR ( X ) .i.e.,
BR ( X ) = U R ( X ) - LR ( X )
Remark 2.2[9] If ( U , R )
is an approximation space and X ,Y ⊆ U , then
(i) LR ( X ) ⊆ X ⊆ U R ( X )
(ii) LR ( φ ) = U R ( φ ) = φ and LR ( U ) = U R ( U ) = U
(iii) U R ( X Y ) = U R ( X ) U R ( Y )
(iv) U R ( X Y ) ⊆ U R ( X ) U R ( Y )
(v) L R ( X Y ) ⊇ LR ( X ) LR ( Y )
(vi) L R ( X Y ) = LR ( X ) LR ( Y )
(vii) LR ( X ) ⊆ LR ( Y ) and U R ( X ) ⊆ U R ( Y ) whenever X ⊆ Y
(viii) U R ( X C ) = [ LR ( X )] C and LR ( X C ) = [ U R ( X )] C
(ix) U RU R ( X ) = LRU R ( X ) = U R ( X )
(x) LR LR ( X ) = U R LR ( X ) = LR ( X )
Definition 2.3[9] Let
U
be an universe,
R
be an equivalence relation on
U
and
τ R ( X ) = { U ,φ, LR ( X ), U R ( X ), BR ( X )} where X ⊆ U . Then by the properties mentioned in remark
2.2, τ R ( X ) satisfies the following axioms:
(i)
U and φ are in τ R ( X )
(ii)
The union of the elements of any sub collection of τ R ( X ) is in τ R ( X )
(iii)
The intersection of the elements of any finite sub collection of τ R ( X ) is in τ R ( X )
Then
τ R ( X ) forms a topology on U called the nano topology with respect to X . We call
( U , τ R ( X )) as the nano topological space. The elements of τ R ( X ) are called nano open sets. The
complement of nano open sets are called nano closed sets.
D. Sasikala and K.C. Radhamani, A Contemporary Approach on Neutrosophic Nano Topological Spaces
Neutrosophic Sets and Systems, Vol. 32, 2020
437
Definition 2.4[9] Let ( U , τ R ( X )) be a nano topological space. A subset A is called nano generalized
closed (briefly Ng-closed) set if N Cl( A ) ⊆V where A⊆V and V is nano open in U .
Definition 2.5[16] A subset A of a nano topological space ( U , τ R ( X )) is called a nano j-open set if
A⊆ N Int[ N PCl( A )] . The complement of nano j-open set is called a nano j-closed (briefly Nj-closed)
set.
i.e., if A is Nj-closed, then N Cl [ N PInt( A )] ⊆ A .
Definition 2.6[16] A subset A of a nano topological space ( U , τ R ( X )) is called a nano generalized
j-closed (briefly Ngj-closed) set if N JCl( A ) ⊆V where A⊆V and V is nano open in U .
Definition 2.7[17] Let X be an universe of discourse with a general element x , the neutrosophic
set is an object having the form A = { x , μ A ( X ),σ A ( X ),γ A ( X ) , x ∈ X } where μ,σ , and γ each
take the values from 0 to 1 and called as the degree of membership, degree of indeterminacy, and the
degree of non-membership of the
element
0 ≤ μ A( x ) + σ A( x ) + σ A( x ) ≤ 3 .
x ∈ X to the set
A
with the condition
Definition 2.8[10] Let U be a nonempty set and R be an equivalence relation on U . Let F be a
neutrosophic set in U with the membership function μ F , the indeterminacy function σ F , and the
non-membership function
γ F . The neutrosophic nano lower, neutrosophic nano upper
F in the approximation ( U , R ) , denoted by
approximations and neutrosophic nano boundary of
N , N and B N( F ) are respectively defined as follows:
(i)
(ii)
(iii)
N( F ) = { x, μR( A ) ( x ),σ R( A ) ( x ),γR( A ) ( x ) / y ∈ [x]R , x ∈ U }
N ( F ) = { x , μ R( A ) ( x ),σ R( A ) ( x ),γ R( A ) ( x ) / y ∈ [x]R , x ∈ U }
B N( F ) = N - N
Where μ R( A ) ( x ) = ∧ y∈ [ x ] μ A ( y ) , σ R( A ) ( x ) = ∧ y∈ [ x ] σ A ( y ) ,
R
R
γ R( A ) ( x ) = ∨ y∈ [ x ] γ A ( y ) ,
R
μ R( A ) ( x ) = ∨ y∈ [ x ] μ A ( y ) , σ R( A ) ( x ) = ∨ y∈ [ x ] σ A ( y ) , γ R( A ) ( x ) = ∧ y∈ [ x ] γ A ( y )
R
R
R
Definition 2.9[10] Let U be an universe, R be an equivalence relation on U and F be a
neutrosophic set in U . If the collection τ N ( F ) = { 0N ,1N , N( F ), N( F ),B N( F )} forms a topology,
then it is said to be a neutrosophic nano topology. We call ( U , τ N ( F )) as the neutrosophic nano
topological space. The elements of τ N ( F ) are called neutrosophic nano open sets.
Definition 2.10[17] Let U be a nonempty set and the neutrosophic sets A and B are in the form
A = { x : μ A ( x ),σ A ( x ),γ A ( x ) , x ∈ U } , B = { x : μ B ( x ),σ B ( x ),γ B ( x ) , x ∈ U } . Then the following
statements hold:
(i)
(ii)
(iii)
(iv)
0N = { x ,0 ,0 ,1 : x ∈ U } and 1N = { x ,1,1,0 : x ∈ U }
A ⊆ B iff μ A ( x ) ≤ μB ( x ),σ A ( x ) ≤ σ B ( x ) or σ A ( x ) ≥ σ B ( x ),γ A ( x ) ≥ γ B ( x ) for all x ∈ U
A = B iff A ⊆ B and B ⊆ A
AC = { x, γ A ( x ), 1 - σ A ( x ), μ A ( x ) , x ∈ U }
D. Sasikala and K.C. Radhamani, A Contemporary Approach on Neutrosophic Nano Topological Spaces
Neutrosophic Sets and Systems, Vol. 32, 2020
(v)
(vi)
(vii)
438
A B = { x , μ A ( x ) ∧ μB ( x ), σ A ( x ) ∧ σ B ( x ), γ A ( x ) ∨ γ B ( x ) for all x ∈ U }
A B = { x , μ A ( x ) ∨ μB ( x ), σ A ( x ) ∨ σ B ( x ), γ A ( x ) ∧ γ B ( x ) for all x ∈ U }
A - B = {x, μ A ( x ) ∧ γ B ( x ), σ A ( x ) ∧ 1 - σ B ( x ), γ A ( x ) ∨ μB ( x ) for all x ∈ U }
Definition 2.11[10] [ τ N ( F )] C is called the dual neutrosophic nano topology of τ N ( F ) . The
elements of [ τ N ( F )] C are called neutrosophic nano closed (NN closed) sets. Thus, a neutrosophic
set N( G ) of U is neutrosophic nano closed iff U - N ( G ) is neutrosophic nano open in τ N ( F ) .
Definition
2.12[10]
Let
( U , τ N ( A ))
be
a
neutrosophic
nano
topological
space
and
A = { x , μ A ( x ),σ A ( x ),γ A ( x ) : x ∈ U } be a neutrosophic set in X . Then the neutrosophic closure
and neutrosophic interior of A are defined by N Cl( A ) = intersection of all closed sets which
contains A and N Int( A ) = union of all open sets which is contained in A .
A is a neutrosophic open set iff A = N Int( A ) and A is a neutrosophic closed set iff A = N Cl( A )
III. NEUTROSOPHIC NANO j-CLOSED SETS
Definition 3.1 Let ( U , τ N ( A )) be a neutrosophic nano topological space. Then a neutrosophic nano
subset A in ( U , τ N ( A )) is said to be neutrosophic nano j-closed (briefly NNj-closed) set if
N N Cl( N N PInt( A ))⊆ A .
Theorem 3.2 Every neutrosophic nano closed set is a neutrosophic nano j-closed set.
Proof. Let A be a neutrosophic nano closed set. i.e., N N Cl( A ) = A . We know that N N Int( A ) ⊆
N N PInt( A ) ⊆ A which implies N N Cl ( N N PInt( A ))⊆ N N Cl ( A ) = A . Hence every neutrosophic nano
closed set is neutrosophic nano j-closed.
Remark 3.3 The converse part of the above theorem need not be true as seen from the following
example.
Example 3.4 Let ( U , τ N ( A )) be a neutrosophic nano topological space with U = { p1, p2, p3 } , the
universe of discourse and RU = {{ p1, p2 },{ p3 }} , the equivalence relation on U .
Let A = { p1,( 0.5,0.4,0.3 ) , p2,( 0.5,0.6 ,0.4 ) , p3,( 0.2,0.5,0.2 ) } be the neutrosophic nano subset
of U .
Now,
N N LR ( A ) = { p1,( 0.5,0.4 ,0.4 ) , p2,( 0.5,0.4 ,0.4 ) , p3,( 0.2,0.5,0.2 ) }
N N U R ( A ) = { p1,( 0.5,0.6 ,0.3 ) , p2,( 0.5,0.6 ,0.3 ) , p3,( 0.2,0.5,0.2 ) }
,
,
N N BR ( A ) = { p1,( 0.4 ,0.6 ,0.5 ) , p2,( 0.4 ,0.6 ,0.5 ) , p3,( 0.2,0.5,0.2 ) } and the neutrosophic nano
topology formed by the subset A is τ N ( A ) = { 0N ,1N , N N LR ( A ), N N U R ( A ), N N BR ( A )} .
Here the subsets are called neutrosophic nano open sets and the neutrosophic nano closed sets are
0N ,1N ,[ N N LR ( A )] C , [ N NU R ( A )] C and [ N N BR ( A )] C
,
where
[ N N LR ( A )] C = { p1,( 0.4 ,0.6 ,0.5 ) , p2,( 0.4 ,0.6 ,0.5 ) , p3,( 0.2,0.5,0.2 ) } ,
[ N NU R ( A )] C = { p1,( 0.3,0.4,0.5 ) , p2,( 0.3,0.4,0.5 ) , p3,( 0.2,0.5,0.2 ) }
,
and
N N Int( A ) = { p1,( 0.5,0.6 ,0.3 ) , p2,( 0.5,0.6 ,0.3 ) , p3,( 0.2,0.5,0.2 ) }
and
[ N N BR ( A )] C = { p1,( 0.5,0.4,0.4 ) , p2,( 0.5,0.4,0.4 ) , p3,( 0.2,0.5,0.2 ) } .
Now,
N N PInt( A ) = { p1,( 0.5,0.6 ,0.3 ) , p2,( 0.5,0.6 ,0.3 ) , p3,( 0.2,0.5,0.2 ) } .
D. Sasikala and K.C. Radhamani, A Contemporary Approach on Neutrosophic Nano Topological Spaces
Neutrosophic Sets and Systems, Vol. 32, 2020
439
Let us take a closed set in τ N ( A ) and let it be B .
i.e., B = { p1,( 0.5,0.4,0.4 ) , p2,( 0.5,0.4,0.4 ) , p3,( 0.2,0.5,0.2 ) } .
Clearly
Let
N N Cl ( N N PInt( B )) = B C ⊆ B ⇒ B is NNj-closed.
us
take
an
another
NNj-closed
set
C = { p1,( 0.6 ,0.4,0.2 ) , p2,( 0.6 ,0.5,0.3 ) , p3,( 0.3,0.5,0.1 ) } . But C is not a neutrosophic nano
closed set. Hence a NNj-closed set need not be a NN closed set.
Theorem 3.5 The union (intersection) of two NNj-closed (open) sets need not be a NNj-closed (open)
set as seen in the following example.
Example 3.6 Let ( U , τ N ( A )) be a neutrosophic nano topological space with U = { p1, p2, p3 } , the
universe of discourse and RU = {{ p1, p2 },{ p3 }} , the equivalence relation on U .
Let A = { p1,( 0.5,0.4,0.3 ) , p2,( 0.5,0.6 ,0.4 ) , p3,( 0.2,0.5,0.2 ) } be the neutrosophic nano subset
of
U
.
The
sets
{ p1,( 0.4,0.6 ,0.5 ) , p2,( 0.4,0.6 ,0.5 ) , p3,( 0.2,0.5,0.2 ) }
{ p1,( 0.5,0.4,0.4 ) , p2,( 0.5,0.4,0.4 ) , p3,( 0.2,0.5,0.2 ) }
are
NNj-closed
and
sets.
But
{ p1,( 0.5,0.6 ,0.4 ) , p2,( 0.5,0.6 ,0.4 ) , p3,( 0.2,0.5,0.2 ) } which is the intersection of the above
two sets is not a NNj-closed sets.
Theorem 3.7 Every neutrosophic nano j-closed set is a neutrosophic nano pre closed set.
Proof. Let A be a neutrosophic nano j-closed set. i.e., N N Cl( N N PInt( A ))⊆ A . We know that
N N Int( A ) ⊆ N N PInt( A ) which implies N N Cl ( N N Int( A ))⊆ N N Cl ( N N PInt( A ))⊆ A . Therefore A is a
neutrosophic nano pre closed set. Hence every NNj-closed set is NN pre closed.
Remark 3.8 The converse part of the above theorem need not be true as seen from the following
example.
Example
3.9
Let
RU = {{ p1, p3 },{ p2 }}
U = { p1, p2, p3 }
and
let
be
the
the
universe
neutrosophic
with
nano
the
equivalence
subset
A = { p1,( 0.3,0.4 ,0.2 ) , p2,( 0.4 ,0.5,0.1 ) , p3,( 0.5,0.2,0.3 ) }
on
relation
U
.
Here
N N LR ( A ) = { p1,( 0.3,0.2,0.3 ) , p2,( 0.4 ,0.5,0.1 ) , p3,( 0.3,0.2,0.3 ) }
N N U R ( A ) = { p1,( 0.5,0.4 ,0.2 ) , p2,( 0.4 ,0.5,0.1 ) , p3,( 0.5,0.4 ,0.2 ) }
be
,
,
and
N N B R ( A ) = { p1,( 0.3,0.4 ,0.3 ) , p2,( 0.1,0.5,0.4 ) , p3,( 0.3,0.4 ,0.3 ) } . Then the neutrosophic
nano topology formed by A is τ N ( A ) = { 0N ,1N , N N LR ( A ), N N U R ( A ), N N BR ( A )} .
The subsets of τ N ( A ) are called neutrosophic nano open sets and the neutrosophic nano closed sets
are
0N ,1N ,[ N N LR ( A )] C ,[ N NU R ( A )] C and [ N N BR ( A )] C
where
[ N N LR ( A )] C = { p1,( 0.3,0.8 ,0.3 ) , p2,( 0.1,0.5,0.4 ) , p3,( 0.3,0.8 ,0.3 ) }
,
[ N NU R ( A )] C = { p1,( 0.2,0.6 ,0.5 ) , p2,( 0.1,0.5,0.4 ) , p3,( 0.2,0.6 ,0.5 ) }
[ N N BR ( A )] C = { p1,( 0.3,0.6 ,0.3 ) , p2,( 0.4,0.5,0.1 ) , p3,( 0.3,0.6 ,0.3 ) }
,
.
N N Int( A ) = { p1,( 0.3,0.4 ,0.3 ) , p2,( 0.4 ,0.5,0.1 ) , p3,( 0.3,0.4 ,0.3 ) }
and
Then
,
N N PInt( A ) = { p1,( 0.3,0.4 ,0.2 ) , p2,( 0.4 ,0.5,0.1 ) , p3,( 0.4 ,0.4 ,0.3 ) } and Cl ( A ) = 1N .
Clearly the set A itself is a neutrosophic nano pre closed set, but not a neutrosophic nano j-closed
set, since N N Cl ( N N PInt( A )) = 1N , which is not contained in A .
Theorem: 3.10 Every neutrosophic nano regular closed set is a neutrosophic nano j-closed set.
D. Sasikala and K.C. Radhamani, A Contemporary Approach on Neutrosophic Nano Topological Spaces
Neutrosophic Sets and Systems, Vol. 32, 2020
440
Proof. We know that every NN regular closed set is a NN closed set and also every NN closed set is a
NNj-closed set. Hence every NN regular closed set is a NNj-closed set.
Remark 3.11 The converse part of the above theorem need not be true as seen in the following
example.
Example 3.12 Let
relation on
neutrosphic
U = { p1, p2, p3 } be the universe, RU = {{ p1, p2 },{ p3 }} be the equivalence
U , and
A = { p1,( 0.1,0.4 ,0.2 ) , p2,( 0.4 ,0.2,0.3 ) , p3,( 0.5,0.3,0.3 ) }
nano
subset
of
U
.
be the
Then
N N LR ( A ) = { p1,( 0.1,0.2,0.3 ) , p2,( 0.1,0.2,0.3 ) , p3,( 0.5,0.3,0.3 ) }
,
N N U R ( A ) = { p1,( 0.4 ,0.4 ,0.2 ) , p2,( 0.4 ,0.4 ,0.2 ) , p3,( 0.5,0.3,0.3 ) }
,
N N BR ( A ) = { p1,( 0.3,0.4 ,0.2 ) , p2,( 0.3,0.4 ,0.2 ) , p3,( 0.3,0.3,0.5 ) } , and the neutrosophic nano
topology formed by A is τ N ( A ) = { 0N ,1N , N N LR ( A ), N N U R ( A ), N N BR ( A )} .
Here the subsets are called neutrosophic nano open sets and the neutrosophic nano closed sets are
0N ,1N ,[ N N LR ( A )] C ,[ N NU R ( A )] C ,and [ N N BR ( A )] C
where
[ N N LR ( A )] C = { p1,( 0.3,0.8 ,0.1 ) , p2,( 0.3,0.8 ,0.1 ) , p3,( 0.3,0.7 ,0.5 ) }
,
[ N NU R ( A )] C = { p1,( 0.2,0.6 ,0.4 ) , p2,( 0.2,0.6 ,0.4 ) , p3,( 0.3,0.7 ,0.5 ) }
[ N N BR ( A )] C = { p1,( 0.2,0.6 ,0.3 ) , p2,( 0.2,0.6 ,0.3 ) , p3,( 0.5,0.7 ,0.3 ) }
,
.
and
Then
N N Int( A ) = { p1,( 0.1,0.2,0.3 ) , p2,( 0.1,0.2,0.3 ) , p3,( 0.5,0.3,0.3 ) } and Cl ( A ) = 1N .
Let B = { p1,( 0.2,0.2,0.2 ) , p2,( 0.3,0.4,0.2 ) , p3,( 0.5,0.4,0.2 ) } be an another neutrosophic
nano subset on U . Clearly N N Cl( N N PInt( B )) = N N Cl( N N LR ( A )) = [ N N BR A )] C ⊆ B .
But N N Cl ( N N Int( B )) ≠ B . Hence a NNj-closed set need not be a NN regular closed.
Definition 3.13 Let ( U , τ N ( A )) be a neutrosophic nano topological space. Then a neutrosophic nano
subset A in ( U , τ N ( A )) is said to be neutrosophic nano generalized closed (briefly NNg-closed) set
if N N Cl( A ) ⊆ V whenever A⊆V and V is neutrosophic nano open in U .
Theorem 3.14 Every neutrosophic nano closed set is a neutrosophic nano generalized closed set.
Proof. Let A be the neutrosophic nano closed set. Let A⊆V and V is neutrosophic nano open set
in U . Since A is NN closed, N N Cl( A ) ⊆ A . i.e., N N Cl ( A ) ⊆ A ⊆ V . Hence A is NNg-closed set.
Hence every NN closed set is NNg-closed.
Remark 3.15 The converse of the above theorem need not be true as seen in the following example.
Example 3.16 Let ( U , τ N ( A )) be a neutrosophic nano topological space with U = { p1, p2, p3 } , the
universe of discourse and RU = {{ p1, p2 },{ p3 }} , the equivalence relation on U .
Let A = { p1,( 0.5,0.4,0.3 ) , p2,( 0.5,0.6 ,0.4 ) , p3,( 0.2,0.5,0.2 ) } be the neutrosophic nano subset
of U .
Now,
N N LR ( A ) = { p1,( 0.5,0.4 ,0.4 ) , p2,( 0.5,0.4 ,0.4 ) , p3,( 0.2,0.5,0.2 ) }
N N U R ( A ) = { p1,( 0.5,0.6 ,0.3 ) , p2,( 0.5,0.6 ,0.3 ) , p3,( 0.2,0.5,0.2 ) }
,
,
N N BR ( A ) = { p1,( 0.4 ,0.6 ,0.5 ) , p2,( 0.4 ,0.6 ,0.5 ) , p3,( 0.2,0.5,0.2 ) } and the neutrosophic nano
topology formed by the subset A is τ N ( A ) = { 0N ,1N , N N LR ( A ), N N U R ( A ), N N BR ( A )} .
Let V = N N U R ( A ) and B = { p1,( 0.4 ,0.5,0.6 ) , p2,( 0.3,0.3,0.5 ) , p3,( 0.1,0.4 ,0.3 ) } .
D. Sasikala and K.C. Radhamani, A Contemporary Approach on Neutrosophic Nano Topological Spaces
Neutrosophic Sets and Systems, Vol. 32, 2020
441
Clearly B is a NNg-closed set, since Cl( B ) ⊆V whenever B⊆V . But it is not a NN closed set.
Definition 3.17 Let ( U , τ N ( F )) be a neutrosophic nano topological space. Then a neutrosophic nano
subset A in ( U , τ N ( F )) is said to be neutrosophic nano generalized j-closed (briefly NNgj-closed)
set if N N JCl ⊆ V whenever A⊆V and V is neutrosophic nano open in U .
Definition 3.18 Let ( U , τ N ( F )) be a neutrosophic nano topological space. Then a neutrosophic nano
subset A in ( U , τ N ( F )) is said to be neutrosophic nano generalized j*-closed (briefly NNgj*-closed)
set if N N JCl ⊆ V whenever A⊆V and V is neutrosophic nano j-open in U .
Theorem 3.19 If A is a neutrosophic nano gj-closed set in ( U , τ R ( X )) and A ⊆ B ⊆ N N JCl ( A ) , then
B is neutrosophic nano generalized j-closed set in ( U , τ R ( X )) .
Proof. Let B⊆V where V is neutrosophic nano open in U . Then A⊆ B implies A⊆V . Since
A is NNgj-closed, N N JCl ( A ) ⊆ V . Also A ⊆ N N JCl ( B ) implies N N JCl ( B ) ⊆ N N JCl ( A ) . Thus
N N JCl ( B ) ⊆ V and therefore B is NNgj-closed set in U .
Theorem 3.20 Every neutrosophic nano closed set is a neutrosophic nano generalized j-closed.
Proof. Let A be a neutrosophic nano closed set in U . Let A⊆V and V is neutrosophic nano
A
open in U . Since
is neutrosophic nano closed,
N N Cl ( A ) = A ⊆ V . Also
N N JCl ( A ) ⊆ N N Cl( A ) ⊆ V , where V is NN open in U . Therefore A is a neutrosophic nano
generalized j-closed set. Hence every NN closed set is NNgj-closed.
Remark 3.21 The converse part of the above theorem need not be true as seen in the following
example.
Example 3.22 Let
relation on
neutrosphic
U = { p1, p2, p3 } be the universe, RU = {{ p1, p2 },{ p3 }} be the equivalence
U , and
A = { p1,( 0.1,0.4 ,0.2 ) , p2,( 0.4 ,0.2,0.3 ) , p3,( 0.5,0.3,0.3 ) }
nano
subset
of
U
be the
.
Then
N N LR ( A ) = { p1,( 0.1,0.2,0.3 ) , p2,( 0.1,0.2,0.3 ) , p3,( 0.5,0.3,0.3 ) }
,
N N U R ( A ) = { p1,( 0.4 ,0.4 ,0.2 ) , p2,( 0.4 ,0.4 ,0.2 ) , p3,( 0.5,0.3,0.3 ) }
,
N N BR ( A ) = { p1,( 0.3,0.4 ,0.2 ) , p2,( 0.3,0.4 ,0.2 ) , p3,( 0.3,0.3,0.5 ) } , and the neutrosophic nano
topology formed by A is τ N ( A ) = { 0N ,1N , N N LR ( A ), N N U R ( A ), N N BR ( A )} . Let the open set
V = { p1,( 0.4 ,0.4 ,0.2 ) , p2,( 0.4 ,0.4 ,0.2 ) , p3,( 0.5,0.3,0.3 ) } .
Let B = { p1,( 0.2,0.3,0.3 ) , p2,( 0.2,0.3,0.4 ) , p3,( 0.1,0.2,0.3 ) } . Clearly B⊆V .
Also N N JCl ( B ) ⊆ V . Hence B is a NNgj-closed set, but not a NN closed set.
Theorem 3.23 Every neutrosophic nano j-closed set is a neutrosophic nano generalized j-closed set.
Proof. Let A be a NNj-closed set. Let A⊆V and V is neutrosophic nano open in U. Since A is
NNj-closed, N N JCl ( A ) ⊆ A ⊆ V . Therefore A is NNgj-closed. Hence every NNj-closed set is NNgjclosed.
Remark 3.24 The converse of the above theorem need not be true as seen in the following example.
Example 3.25 In example 3.22, B
is a NNgj-closed set. But N N Cl ( N N PInt( B )) is not contained in
V . i.e., B is not a NNj-closed set. Hence every NNgj-closed set need not a NNj-closed set.
Theorem 3.26 Every NNg-closed set is a NNgj-closed set.
Proof. Let A be a NNg-closed set. Then N N Cl( A ) ⊆ V whenever A⊆V and V is neutrosophic nano
open in U . Since N N JCl ( A ) ⊆ N N Cl( A ) ⊆ V , we have N N JCl ( A ) ⊆ V whenever A⊆V and V is
NN open in U . Therefore A is NNgj-closed. Hence every NNg-closed set is a NNgj-closed set.
Theorem 3.27 Every NNj-closed set is a NNgj*-closed set.
D. Sasikala and K.C. Radhamani, A Contemporary Approach on Neutrosophic Nano Topological Spaces
Neutrosophic Sets and Systems, Vol. 32, 2020
442
Proof. Let A be a NNj-closed set. Let
A⊆V and V is neutrosophic nano j-open in U . Since A
is NNj-closed, N N JCl ( A ) = A ⊆ V , V is NNj- open in U . Therefore A is NNgj*-closed. Hence
every NNj-closed set is a NNgj*-closed set.
Theorem 3.28 Every NNgj*-closed set is a NNgj-closed set.
Proof. Let A be a NNgj*-closed set. Let A⊆V and V is neutrosophic nano open in U . Since every
NN open set is NNj-open, V is NNj-open in U . Since A is NNgj*-closed set, we have N N JCl ( A ) ⊆ V
. Therefore
N N JCl ( A ) ⊆ V whenever A⊆V and V is NNj-open in
U . Therefore A is NNgj-
closed. Hence every NNgj*-closed set is a NNgj-closed set.
IV. Conclusion
Neutrosophic nano j-closed set, neutrosophic nano generalized closed set, neutrosophic nano
generalized j-closed set, neutrosophic nano generalized j*-closed set were introduced and some of
their properties were discussed in this paper. The concept can be used for real life decision making
problems where the situations of indeterminacy occurs. The practical problems may be solved by
finding CORE values through the criterion reduction.
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Received: Sep 28, 2019. Accepted: Mar 19, 2020
D. Sasikala and K.C. Radhamani, A Contemporary Approach on Neutrosophic Nano Topological Spaces
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University of New Mexico
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Dr. Mohamed Abdel-Basset
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