130; Summary of the Special Issue &quot;Neutrosophic Information Theory and Applications&quot; at &quot;Information

Seok-Zun Song

an Open Access Journal by MDPI Special Issue Neutrosophic Information Theory and Applications Special Issue Editors: Prof. Dr. Florentin Smarandache and Prof. Jun Ye an Open Access Journal by MDPI Editor-in-Chief Message from the Editor-in-Chief Prof. Dr. Willy Susilo The concept of Information is to disseminate scientific results achieved via experiments and theoretical results in depth. It is very important to enable researchers and practitioners to learn new technology and findings that enable development in the applied field. Information is an online open access journal of information science and technology, data, knowledge and communication. It publishes reviews, regular research papers and short communications. We invite high quality work, and our review and publication processing is very efficient. Author Benefits Open Access Unlimited and free access for readers No Copyright Constraints Retain copyright of your work and free use of your article Discounts on Article Processing Charges (APC) If you belong to an institute that participates with the MDPI Institutional Open Access Program Thorough Peer-Review Coverage by Leading Indexing Services Compendex/Engineering Village (Elsevier), Scopus (Elsevier) High Visibility Emerging Sources Citation Index ( ESCI-Web of Science) No Space Constraints, No Extra Space or Color Charges No restriction on the length of the papers, number of figures or colors Aims and Scope Information (ISSN 2078-2489) is a scientific, open access journal of information science and technology, data, knowledge, and communication. It publishes reviews, regular research papers and short communications. The scope of Information includes: Information technology (IT) and computing science Communication theory, information theory and communication technology Information security Information society Data management, information systems Data mining, data, knowledge, big data Information processing systems Information and its applications Information in organizations and knowledge management Information processes and systems in nature Editorial Office Information Editorial Office information@mdpi.com MDPI AG St. Alban-Anlage 66 4052 Basel, Switzerland Tel: +41 61 683 77 34 Fax: +41 61 302 89 18 www.mdpi.com mdpi.com/journal/information CONTENTS Florentin Smarandache, Jun Ye Summary of the Special Issue “Neutrosophic Information Theory and Applications” at “Information” Journal Information 2018, 9, 49; doi:10.3390/info9030049 Surapati Pramanik, Shyamal Dalapati, Shariful Alam, Florentin Smarandache, Tapan Kumar Roy NS-Cross Entropy-Based MAGDM under Single-Valued Neutrosophic Set Environment Information 2018, 9(2), 37; doi:10.3390/info9020037 Rui Wang, Yanlai Li Generalized Single-Valued Neutrosophic Hesitant Fuzzy Prioritized Aggregation Operators and Their Applications to Multiple Criteria Decision-Making Information 2018, 9(1), 10; doi:10.3390/info9010010 Harish Garg, Nancy Some New Biparametric Distance Measures on Single-Valued Neutrosophic Sets with Applications to Pattern Recognition and Medical Diagnosis Information 2017, 8(4), 162; doi:10.3390/info8040162 Muhammad Akram, Muzzamal Sitara Certain Concepts in Intuitionistic Neutrosophic Graph Structures Information 2017, 8(4), 154; doi:10.3390/info8040154 Surapati Pramanik, Shyamal Dalapati, Shariful Alam, Tapan Kumar Roy NC-TODIM-Based MAGDM under a Neutrosophic Cubic Set Environment Information 2017, 8(4), 149; doi:10.3390/info8040149 Yu-Han Huang, Gui-Wu Wei, Cun Wei VIKOR Method for Interval Neutrosophic Multiple Attribute Group DecisionMaking Information 2017, 8(4), 144; doi:10.3390/info8040144 Muhammad Akram, Maryam Nasir Certain Competition Graphs Based on Intuitionistic Neutrosophic Environment Information 2017, 8(4), 132; doi:10.3390/info8040132 Seok-Zun Song, Florentin Smarandache, Young Bae Jun Neutrosophic Commutative N -Ideals in BCK-Algebras Information 2017, 8(4), 130; doi:10.3390/info8040130 Young Bae Jun, Florentin Smarandache, Hashem Bordbar Neutrosophic N-Structures Applied to BCK/BCI-Algebras Information 2017, 8(4), 128; doi:10.3390/info8040128 Dong-Sheng Xu, Cun Wei, Gui-Wu Wei TODIM Method for Single-Valued Neutrosophic Multiple Attribute Decision Making Information 2017, 8(4), 125; doi:10.3390/info8040125 Keli Hu, En Fan, Jun Ye, Changxing Fan, Shigen Shen, Yuzhang Gu Neutrosophic Similarity Score Based Weighted Histogram for Robust Mean-Shift Tracking Information 2017, 8(4), 122; doi:10.3390/info8040122 Jun Ye Linguistic Neutrosophic Cubic Numbers and Their Multiple Attribute DecisionMaking Method Information 2017, 8(3), 110; doi:10.3390/info8030110 information Editorial Summary of the Special Issue “Neutrosophic Information Theory and Applications” at “Information” Journal Florentin Smarandache 1, * 1 2 * ID and Jun Ye 2 ID Department of Mathematics and Sciences, University of New Mexico, 705 Gurley Ave., Gallup, NM 87301, USA Department of Electrical and Information Engineering, Shaoxing University, 508 Huancheng West Road, Shaoxing 312000, China; yehjun@aliyun.com Correspondence: smarand@unm.edu Received: 27 February 2018; Accepted: 27 February 2018; Published: 28 February 2018 Abstract: Over a period of seven months (August 2017–February 2018), the Special Issue dedicated to “Neutrosophic Information Theory and Applications” by the “Information” journal (ISSN 2078-2489), located in Basel, Switzerland, was a success. The Guest Editors, Prof. Dr. Florentin Smarandache from the University of New Mexico (USA) and Prof. Dr. Jun Ye from the Shaoxing University (China), were happy to select—helped by a team of neutrosophic reviewers from around the world, and by the “Information” journal editors themselves—and publish twelve important neutrosophic papers, authored by 27 authors and coauthors. There were a variety of neutrosophic topics studied and used by the authors and coauthors in Multi-Criteria (or Multi-Attribute and/or Group) Decision-Making, including Cross Entropy-Based MAGDM, Neutrosophic Hesitant Fuzzy Prioritized Aggregation Operators, Biparametric Distance Measures, Pattern Recognition and Medical Diagnosis, Intuitionistic Neutrosophic Graph, NC-TODIM-Based MAGDM, Neutrosophic Cubic Set, VIKOR Method, Neutrosophic Multiple Attribute Group Decision-Making, Competition Graphs, Intuitionistic Neutrosophic Environment, Neutrosophic Commutative N-Ideals, Neutrosophic N-Structures Applied to BCK/BCI-Algebras, Neutrosophic Similarity Score, Weighted Histogram, Robust Mean-Shift Tracking, and Linguistic Neutrosophic Cubic Numbers. Neutrosophic logic, symbolic logic, set, probability, statistics, etc., are, respectively, generalizations of fuzzy and intuitionistic fuzzy logic and set, classical and imprecise probability, classical statistics, and so on. Neutrosophic logic, symbol logic, and set are gaining significant attention in solving many real-life problems that involve uncertainty, impreciseness, vagueness, incompleteness, inconsistency, and indeterminacy. A number of new neutrosophic theories have been proposed and have been applied in computational intelligence, multiple-attribute decision making, image processing, medical diagnosis, fault diagnosis, optimization design, etc. This Special Issue gathers original research papers that report on the state of the art, as well as on recent advancements in neutrosophic information theory in soft computing, artificial intelligence, big and small data mining, decision-making problems, pattern recognition, information processing, image processing, and many other practical achievements. In the first chapter (NS-Cross Entropy-Based MAGDM under Single-Valued Neutrosophic Set Environment), the authors Surapati Pramanik, Shyamal Dalapati, Shariful Alam, Florentin Smarandache, Tapan Kumar Roy propose a new cross entropy measure under a single-valued neutrosophic set (SVNS) environment, namely NS-cross entropy, and prove its basic properties. Additionally, they define the weighted NS-cross entropy measure, investigate its basic properties, and develop a novel multi-attribute group decision-making (MAGDM) strategy that is free from the drawbacks of asymmetrical behavior and undefined phenomena. It is capable of dealing with an unknown weight of attributes and an unknown weight of decision-makers. Finally, a numerical example of multi-attribute Information 2018, 9, 49; doi:10.3390/info9030049 www.mdpi.com/journal/information Information 2018, 9, 49 2 of 4 group decision-making problem of investment potential is solved to show the feasibility, validity and efficiency of the proposed decision-making strategy. Single-valued neutrosophic hesitant fuzzy set (SVNHFS) is a combination of a single-valued neutrosophic set and a hesitant fuzzy set, and its aggregation tools play an important role in the multiple criteria decision-making (MCDM) process. The second paper (Generalized Single-Valued Neutrosophic Hesitant Fuzzy Prioritized Aggregation Operators and Their Applications to Multiple Criteria Decision-Making) investigates MCDM problems in which the criteria under SVNHF environment are in different priority levels. First, the generalized single-valued neutrosophic hesitant fuzzy prioritized weighted average operator and generalized single-valued neutrosophic hesitant fuzzy prioritized weighted geometric operator are developed based on the prioritized average operator. Second, some desirable properties and special cases of the proposed operators are discussed in detail. Third, an approach combining the proposed operators and the score function of single-valued neutrosophic hesitant fuzzy element is constructed to solve MCDM problems. Finally, the authors Rui Wang, Yanlai Li provide the example of investment selection to illustrate the validity and rationality of the proposed method. Single-valued neutrosophic sets (SVNSs) handling the uncertainties characterized by truth, indeterminacy, and falsity membership degrees are a more flexible way of capturing uncertainty. In the third paper (Some New Biparametric Distance Measures on Single-Valued Neutrosophic Sets with Applications to Pattern Recognition and Medical Diagnosis), the authors Harish, Garg and Nancy propose some new types of distance measures, overcoming the shortcomings of the existing measures, for SVNSs with two parameters along with their proofs. The various desirable relations between the proposed measures are also derived. A comparison between the proposed and existing measures is performed in terms of counter-intuitive cases for showing its validity. The proposed measures are illustrated with case studies of pattern recognition, as well as medical diagnoses, along with the effect of the different parameters on the ordering of the objects. A graph structure is a generalization of simple graphs. Graph structures are very useful tools for the study of different domains of computational intelligence and computer science. In the fourth research paper, Certain Concepts in Intuitionistic Neutrosophic Graph Structures, the authors Muhammad Akram and Muzzamal Sitara introduce certain notions of intuitionistic neutrosophic graph structures, illustrating these notions with several examples. They investigate some related properties of intuitionistic neutrosophic graph structures, and also present an application of intuitionistic neutrosophic graph structures. A neutrosophic cubic set is the hybridization of the concept of a neutrosophic set and an interval neutrosophic set. A neutrosophic cubic set has the capacity to express the hybrid information of both the interval neutrosophic set and the single valued neutrosophic set simultaneously. Since the neutroaophic cubic sets have only recently been defined, not much research on the operations and applications of neutrosophic cubic sets is currently available in the literature. In the fifth paper, NC-TODIM-Based MAGDM under a Neutrosophic Cubic Set Environment, the authors Surapati Pramanik, Shyamal Dalapati, Shariful Alam and Tapan Kumar Roy propose score and accuracy functions for neutrosophic cubic sets and prove their basic properties. They also develop a strategy for ranking of neutrosophic cubic numbers based on the score and accuracy functions. The authors firstly develop a TODIM (Tomada de decisao interativa e multicritévio) in the neutrosophic cubic set (NC) environment, which is called the NC-TODIM. They establish a new NC-TODIM strategy for solving multi-attribute group decision-making (MAGDM) problems in neutrosophic cubic set environments. They illustrate the proposed NC-TODIM strategy for solving a multi-attribute group decision-making problem to show the applicability and effectiveness of the developed strategy. They also conduct sensitivity analysis to show the impact of the ranking order of the alternatives on the different values of the attenuation factor of losses for multi-attribute group decision-making strategies. In the sixth paper, VIKOR Method for Interval Neutrosophic Multiple Attribute Group Decision-Making, the authors Yu-Han Huang, Gui-Wu Wei and Cun Wei extend the VIKOR method to multiple-attribute Information 2018, 9, 49 3 of 4 group decision-making (MAGDM) with interval neutrosophic numbers (INNs). Firstly, the basic concepts of INNs are briefly presented. The method first aggregates all individual decision-makers’ assessment information based on an interval neutrosophic weighted averaging (INWA) operator, and then employs the extended classical VIKOR method to solve MAGDM problems with INNs. The validity and stability of this method are verified by example analysis and sensitivity analysis, and its superiority is illustrated by a comparison with the existing methods. The concept of intuitionistic neutrosophic sets provides an additional possibility for representing imprecise, uncertain, inconsistent and incomplete information that exists in real situations. The seventh research article (Certain Competition Graphs Based on Intuitionistic Neutrosophic Environment) presents the notion of intuitionistic neutrosophic competition graphs. Then, the authors Muhammad Akram and Maryam Nasir discuss p-competition intuitionistic neutrosophic graphs and m-step intuitionistic neutrosophic competition graphs. Further, applications of intuitionistic neutrosophic competition graphs in ecosystem and career competition are described. The notion of a neutrosophic commutative N-ideal in BCK-algebras is introduced in the eighth paper (Neutrosophic Commutative N-Ideals in BCK-Algebras), and several properties are investigated. Relations between a neutrosophic N-ideal and a neutrosophic commutative N-ideal are discussed by the authors Seok-Zun Song, Florentin Smarandache, and Young Bae Jun. Characterizations of a neutrosophic commutative N-ideal are considered. Neutrosophic N-Structures Applied to BCK/BCI-Algebras is the title of the ninth paper. The notions of a neutrosophic N-subalgebra and a (closed) neutrosophic N-ideal in a BCK/BCI-algebra are introduced by authors Young Bae Jun, Florentin Smarandache and Hashem Bordbar, and several related properties are investigated. Characterizations of a neutrosophic N-subalgebra and a neutrosophic N-ideal are considered, and relations between a neutrosophic N-subalgebra and a neutrosophic N-ideal are stated. The conditions for a neutrosophic N-ideal being a closed neutrosophic N-ideal are provided. Recently, TODIM has been used to solve multiple attribute decision making (MADM) problems. Single-valued neutrosophic sets (SVNSs) are useful tools for depicting the uncertainty of the MADM. In the tenth paper, TODIM Method for Single-Valued Neutrosophic Multiple Attribute Decision Making, Dong-Sheng Xu, Cun Wei and Gui-Wu Wei extend the TODIM method to the MADM with the single-valued neutrosophic numbers (SVNNs). Firstly, the definition, comparison, and distance of SVNNs are briefly presented, and the steps of the classical TODIM method for MADM problems are introduced. Then, an extended classical TODIM method is proposed for dealing with MADM problems with SVNNs, its significant characteristic being that it can fully consider the decision makers’ bounded rationality, which is a real factor in decision-making. Furthermore, the authors extend the proposed model to interval neutrosophic sets (INSs). Finally, a numerical example is proposed. Visual object tracking is a critical task in computer vision. Challenging things always exist when an object needs to be tracked. For instance, background clutter is one of the most challenging problems. The mean-shift tracker is quite popular because of its efficiency and performance under a range of conditions. However, the challenge of background clutter also disturbs its performance. In the eleventh article, Neutrosophic Similarity Score Based Weighted Histogram for Robust Mean-Shift Tracking, the authors Keli Hu, En Fan, Jun Ye, Changxing Fan, Shigen Shen and Yuzhang Gu propose a novel weighted histogram based on neutrosophic similarity score to help the mean-shift tracker discriminate the target from the background. The authors utilize the single-valued neutrosophic set (SVNS), which is a subclass of NS, to improve the mean-shift tracker. First, two kinds of criteria are considered—object feature similarity and background feature similarity—and each bin of the weight histogram is represented in the SVNS domain via three membership functions: T(Truth), I(indeterminacy), and F(Falsity). Second, the neutrosophic similarity score function is introduced to fuse those two criteria and to build the final weighted histogram. Finally, a novel neutrosophic weighted mean-shift tracker is proposed. The proposed tracker is compared with several mean-shift-based trackers on a dataset of 61 public sequences. The results reveal that this method outperforms other trackers, especially when confronting background clutter. Information 2018, 9, 49 4 of 4 To describe both certain linguistic neutrosophic information and uncertain linguistic neutrosophic information simultaneously in the real world, Jun Ye proposes in the twelfth paper (Linguistic Neutrosophic Cubic Numbers and Their Multiple Attribute Decision-Making Method) the concept of a linguistic neutrosophic cubic number (LNCN), including an internal LNCN and external LNCN. In LNCN, its uncertain linguistic neutrosophic number consists of the truth, indeterminacy, and falsity uncertain linguistic variables, and its linguistic neutrosophic number consists of the truth, indeterminacy, and falsity linguistic variables to express their hybrid information. Then, the author presents the operational laws of LNCNs and the score, accuracy, and certain functions of LNCN for comparing/ranking LNCNs. Next, the author proposes a LNCN weighted arithmetic averaging (LNCNWAA) operator and a LNCN weighted geometric averaging (LNCNWGA) operator to aggregate linguistic neutrosophic cubic information and discuss their properties. Further, a multiple attribute decision-making method based on the LNCNWAA or LNCNWGA operator is developed under a linguistic neutrosophic cubic environment. Finally, an illustrative example is provided to indicate the application of the developed method. © 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). information Article NS-Cross Entropy-Based MAGDM under Single-Valued Neutrosophic Set Environment Surapati Pramanik 1, * ID , Shyamal Dalapati 2 , Shariful Alam 2 , Florentin Smarandache 3 and Tapan Kumar Roy 2 1 2 3 * ID Department of Mathematics, Nandalal Ghosh B.T. College, Panpur, P.O.-Narayanpur, District–North 24 Parganas, Bhatpara 743126, West Bengal, India Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, P.O.-Botanic Garden, Howrah 711103, West Bengal, India; shyamal.rs2015@math.iiests.ac.in (S.D.); salam@math.iiests.ac.in (S.A.); tkroy@math.iiests.ac.in (T.K.R.) Department of Mathematics & Science, University of New Mexico, 705 Gurley Ave., Gallup, NM 87301, USA; smarand@unm.edu Correspondence: surapati.math@gmail.com; Tel.: +91-9477035544 Received: 29 December 2017; Accepted: 6 February 2018; Published: 9 February 2018 Abstract: A single-valued neutrosophic set has king power to express uncertainty characterized by indeterminacy, inconsistency and incompleteness. Most of the existing single-valued neutrosophic cross entropy bears an asymmetrical behavior and produces an undefined phenomenon in some situations. In order to deal with these disadvantages, we propose a new cross entropy measure under a single-valued neutrosophic set (SVNS) environment, namely NS-cross entropy, and prove its basic properties. Also we define weighted NS-cross entropy measure and investigate its basic properties. We develop a novel multi-attribute group decision-making (MAGDM) strategy that is free from the drawback of asymmetrical behavior and undefined phenomena. It is capable of dealing with an unknown weight of attributes and an unknown weight of decision-makers. Finally, a numerical example of multi-attribute group decision-making problem of investment potential is solved to show the feasibility, validity and efficiency of the proposed decision-making strategy. Keywords: neutrosophic set; single-valued neutrosophic set; NS-cross entropy measure; multi-attribute group decision-making 1. Introduction To tackle the uncertainty and modeling of real and scientific problems, Zadeh [1] first introduced the fuzzy set by defining membership measure in 1965. Bellman and Zadeh [2] contributed important research on fuzzy decision-making using max and min operators. Atanassov [3] established the intuitionistic fuzzy set (IFS) in 1986 by adding non-membership measure as an independent component to the fuzzy set. Theoretical and practical applications of IFSs in multi-criteria decision-making (MCDM) have been reported in the literature [4–12]. Zadeh [13] introduced entropy measure in the fuzzy environment. Burillo and Bustince [14] proposed distance measure between IFSs and offered an axiomatic definition of entropy measure. In the IFS environment, Szmidt and Kacprzyk [15] proposed a new entropy measure based on geometric interpretation of IFS. Wei et al. [16] developed an entropy measure for interval-valued intuitionistic fuzzy set (IVIFS) and presented its applications in pattern recognition and MCDM. Li [17] presented a new multi-attribute decision-making (MADM) strategy combining entropy and Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) in an IVIFS environment. Shang and Jiang [18] introduced the cross entropy in the fuzzy environment. Vlachos and Sergiadis [19] presented intuitionistic fuzzy cross entropy by extending fuzzy cross entropy [18]. Ye [20] defined a new cross entropy under an IVIFS environment and presented an Information 2018, 9, 37; doi:10.3390/info9020037 www.mdpi.com/journal/information Information 2018, 9, 37 2 of 21 optimal decision-making strategy. Xia and Xu [21] put forward a new entropy and a cross entropy and employed them for multi-attribute criteria group decision-making (MAGDM) strategy under an IFS environment. Tong and Yu [22] defined cross entropy under an IVIFS environment and applied it to MADM problems. The study of uncertainty took a new direction after the publication of the neutrosophic set (NS) [23] and single-valued neutrosophic set (SVNS) [24]. SVNS appeals more to researchers for its applicability in decision-making [25–54], conflict resolution [55], educational problems [56,57], image processing [58–60], cluster analysis [61,62], social problems [63,64], etc. The research on SVNS gained momentum after the inception of the international journal “Neutrosophic Sets and Systems”. Combining with the neutrosophic set, a number of hybrid neutrosophic sets such as the neutrosophic soft set [65–72], the neutrosophic soft expert set [73–75], the neutrosophic complex set [76], the rough neutrosophic set [77–86], the rough neutrosophic tri complex set [87], the neutrosophic rough hyper complex set [88], the neutrosophic hesitant fuzzy sets/multi-valued neutrosophic set [89–97], the bipolar neutrosophic set [98–103], the rough bipolar neutrosophic set [104], the neutrosophic cubic set [105–113], and the neutrosophic cubic soft set [114,115] has been reported in the literature. Wang et al. [116] defined the interval neutrosophic set (INS). Different interval neutrosophic hybrid sets and their theoretical development and applications have been reported in the literature, such as the interval-valued neutrosophic soft set [117], the interval neutrosophic complex set [118], the interval neutrosophic rough set [119–121], and the interval neutrosophic hesitant fuzzy set [122]. Other extensions of neutrosophic sets, such as trapezoidal neutrosophic sets [123,124], normal neutrosophic sets [125], single-valued neutrosophic linguistic sets [126], interval neutrosophic linguistic sets [127,128], simplified neutrosophic linguistic sets [129], single-valued neutrosophic trapezoid linguistic sets [130], interval neutrosophic uncertain linguistic sets [131–133], neutrosophic refined sets [134–139], linguistic refined neutrosophic sets [140] bipolar neutrosophic refined sets [141], and dynamic single-valued neutrosophic multi-sets [142] have been proposed to enrich the study of neutrosophics. So the field of neutrosophic study has been steadily developing. Majumdar and Samanta [143] defined an entropy measure and presented an MCDM strategy under SVNS environment. Ye [144] proposed cross entropy measure under the single-valued neutrosophic set environment, which is not symmetric straight forward and bears undefined phenomena. To overcome the asymmetrical behavior of the cross entropy measure, Ye [144] used a symmetric discrimination information measure for single-valued neutrosophic sets. Ye [145] defined cross entropy measures for SVNSs to overcome the drawback of undefined phenomena of the cross entropy measure [144] and proposed a MCDM strategy. The aforementioned applications of cross entropy [144,145] can be effective in dealing with neutrosophic MADM problems. However, they also bear some limitations, which are outlined below: i. ii. iii. The strategies [144,145] are capable of solving neutrosophic MADM problems that require the criterion weights to be completely known. However, it can be difficult and subjective to offer exact criterion weight information due to neutrosophic nature of decision-making situations. The strategies [144,145] have a single decision-making structure, and not enough attention is paid to improving robustness when processing the assessment information. The strategies [144,145] cannot deal with the unknown weight of the decision-makers. Research gap: MAGDM strategy based on cross entropy measure with unknown weight of attributes and unknown weight of decision-makers. This study answers the following research questions: i. Is it possible to define a new cross entropy measure that is free from asymmetrical phenomena and undefined behavior? Information 2018, 9, 37 ii. iii. iv. v. vi. 3 of 21 Is it possible to define a new weighted cross entropy measure that is free from the asymmetrical phenomena and undefined behavior? Is it possible to develop a new MAGDM strategy based on the proposed cross entropy measure in single-valued neutrosophic set environment, which is free from the asymmetrical phenomena and undefined behavior? Is it possible to develop a new MAGDM strategy based on the proposed weighted cross entropy measure in the single-valued neutrosophic set environment that is free from the asymmetrical phenomena and undefined behavior? How do we assign unknown weight of attributes? How do we assign unknown weight of decision-makers? Motivation: The above-mentioned analysis describes the motivation behind proposing a comprehensive NS-cross entropy-based strategy for tackling MAGDM under the neutrosophic environment. This study develops a novel NS-cross entropy-based MAGDM strategy that can deal with multiple decision-makers and unknown weight of attributes and unknown weight of decision-makers and free from the drawbacks that exist in [144,145]. The objectives of the paper are: 1. 2. 3. 4. To define a new cross entropy measure and prove its basic properties, which are free from asymmetrical phenomena and undefined behavior. To define a new weighted cross measure and prove its basic properties, which are free from asymmetrical phenomena and undefined behavior. To develop a new MAGDM strategy based on weighted cross entropy measure under single-valued neutrosophic set environment. To develop a technique to incorporate unknown weight of attributes and unknown weight of decision-makers in the proposed NS-cross entropy-based MAGDM under single-valued neutrosophic environment. To fill the research gap, we propose NS-cross entropy-based MAGDM, which is capable of dealing with multiple decision-makers with unknown weight of the decision-makers and unknown weight of the attributes. The main contributions of this paper are summarized below: 1. 2. 3. 4. We define a new NS-cross entropy measure and prove its basic properties. It is straightforward symmetric and it has no undefined behavior. We define a new weighted NS-cross entropy measure in the single-valued neutrosophic set environment and prove its basic properties. It is straightforward symmetric and it has no undefined behavior. In this paper, we develop a new MAGDM strategy based on weighted NS cross entropy to solve MAGDM problems with unknown weight of the attributes and unknown weight of decision-makers. Techniques to determine unknown weight of attributes and unknown weight of decisions makers are proposed in the study. The rest of the paper is presented as follows: Section 2 describes some concepts of SVNS. In Section 3 we propose a new cross entropy measure between two SVNS and investigate its properties. In Section 4, we develop a novel MAGDM strategy based on the proposed NS-cross entropy with SVNS information. In Section 5 an illustrative example is solved to demonstrate the applicability and efficiency of the developed MAGDM strategy under SVNS environment. In Section 6 we present comparative study and discussion. Section 7 offers conclusions and the future scope of research. Information 2018, 9, 37 4 of 21 2. Preliminaries This section presents a short list of mostly known definitions pertaining to this paper. Definition 1 [23] NS. Let U be a space of points (objects) with a generic element in U denoted by u, i.e., u ∈ U. A neutrosophic set A in U is characterized by truth-membership measure TA (u), indeterminacy-membership measure I A (u) and falsity-membership measure FA (u), where TA (u), I A (u), FA (u) are the measures from U to ]− 0, 1+ [i.e., TA (u), I A (u), FA (u):U →]− 0, 1+ [ NS can be expressed as A = {<u; (TA (u), I A (u), FA (u))>: ∀ u ∈U}. Since TA (u), I A (u), FA (u) are the subsets of ]− 0, 1+ [there the sum (TA (u) + I A (u) + FA (u)) lies between − 0 and 3+ . Example 1. Suppose that U = {u1 , u2 , u3 , . . .} be the universal set. Let R1 be any neutrosophic set in U. Then R1 expressed as R1 = {<u1 ; (0.6, 0.3, 0.4)>: u1 ∈ U}. Definition 2 [24] SVNS. Assume that U be a space of points (objects) with generic elements u ∈ U. A SVNS H in U is characterized by a truth-membership measure TH (u), an indeterminacy-membership measure IH (u), and a falsity-membership measure FH (u), where TH (u), IH (u), FH (u) ∈ [0, 1] for each point u in U. Therefore, a SVNS A can be expressed as H = {u, (TH (u), I H (u), FH (u)) | ∀u ∈ U}, whereas, the sum of TH (u), IH (u) and FH (u) satisfy the condition 0 ≤ TH (u) + IH (u) + FH (u) ≤ 3 and H(u) = <(TH (u), IH (u), FH (u)> call a single-valued neutrosophic number (SVNN). Example 2. Suppose that U = {u1 , u2 , u3 , . . .} be the universal set. A SVNS H in U can be expressed as: H = {u1 , (0.7, 0.3, 0.5)| u1 ∈ U} and SVNN presented H = <0.7, 0.3, 0.5>. Definition 3 [24] Inclusion of SVNSs. The inclusion of any two SVNS sets H1 and H2 in U is denoted by H1 ⊆ H2 and defined as follows: H1 ⊆ H2 , TH1 (u) ≤ TH2 (u), IH1 (u) ≥ IH2 (u), FH1 (u) ≥ FH2 (u) i f f f or all u ∈ U. Example 3. Let H1 and H2 be any two SVNNs in U presented as follows: H1 = <(0.7, 0.3, 0.5)> and H2 = <(0.8, 0.2, 0.4)> for all u ∈ U. Using the property of inclusion of two SVNNs, we conclude that H1 ⊆ H2 . Definition 4 [24] Equality of two SVNSs. The equality of any two SVNS H1 and H2 in U denoted by H1 = H2 and defined as follows: TH1 (u) = TH2 (u), IH1 (u) = IH2 (u) and FH1 (u) = FH2 (u) f or all u ∈ U. Definition 5 Complement of any SVNSs. The complement of any SVNS H in U denoted by H c and defined as follows: H c = {u, 1 − TH , 1 − IH , 1 − FH | u ∈ U }. Example 4. Let H be any SVNN in U presented as follows: H = < (0.7, 0.3, 0.5) >. Then compliment of H is obtained as H c = <(0.3, 0.7, 0.5)>. Definition 6 [24] Union. The union of two single-valued neutrosophic sets H1 and H2 is a neutrosophic set H3 (say) written as H3 = H1 ∪H2 . TH3 (u) = max {TH1 (u), TH2 (u)}, IH J3 (u) = min {IH1 (u), IH2 (u)}, FH3 (u) = min {FH1 (u), FH2 (u)}, ∀ u ∈ U. Information 2018, 9, 37 5 of 21 Example 5. Let H1 and H2 be two SVNSs in U presented as follows: H1 = <(0.6, 0.3, 0.4)> and H2 = <(0.7, 0.3, 0.6)>. Then union of them is presented as: H1 ∪ H2 =< (0.7, 0.3, 0.4) > . Definition 7 [24] Intersection. The intersection of two single-valued neutrosophic sets H1 and H2 denoted by H4 and defined as H4 = H1 ∩ H2 TH4 (u) = min {TH1 (u), TH2 (u)}, IH4 (u) = max{IH1 (u), IH2 (u)} FH4 (u) = max {FH1 (u), FH2 (u)}, ∀ u ∈ U. Example 6. Let H1 and H2 be two SVNSs in U presented as follows: H1 = <(0.6, 0.3, 0.4)> and H2 = <(0.7, 0.3, 0.6)>. Then intersection of H1 and H2 is presented as follows: H1 ∩H2 = <(0.6, 0.3, 0.6)> 3. NS-Cross Entropy Measure In this section, we define a new single-valued neutrosophic cross-entropy measure for measuring the deviation of single-valued neutrosophic variables from an a priori one. Definition 8 NS-cross entropy measure. Let H1 and H2 be any two SVNSs in U = { u1 , u2 , u3 , . . . , un }. Then, the single-valued cross-entropy of H1 and H2 is denoted by CENS (H1 , H2 ) and defined as follows: CENS (H1 , H2 ) =  q    1 2 * ∑ q n i=1 2| IH1 (ui )− IH2 (ui )| q 2 2 1+| IH2 (u)| 1+| IH1 (ui )| + + FH2 (ui )|  q 2| FH1 (ui )−q 2 2 1+| FH1 (ui )| + 1+| FH2 (ui )| 2 | TH1 (ui )− TH2 (ui )| q 2 2 1+| TH1 (ui )| + 1+| TH2 (ui )| r + r 2 (1− I H (ui ))−(1− IH2 (ui )) 1 r 2 1+ (1− I H (ui )) + r + 1 2 1 2 1+ (1− T H (ui )) + 2 1+ (1− I H (ui )) 2 2 (1− F H (ui ))−(1− FH2 (ui )) 1 r 1+ (1− F H (ui )) + 2 (1− T H (ui ))−(1− TH2 (ui )) 1 r 1+ (1− F H 2  + 1 1+ (1− T H (ui )) 2 2  + (1) +   2  (u )) i Example 7. Let H1 and H2 be two SVNSs in U, which are given by H1 = {u, (0.7, 0.3, 0.4)| u ∈ U} and H2 = {u, (0.6, 0.4, 0.2)| u ∈ U}. Using Equation (1), the cross entropy value of H1 and H2 is obtained as CENS (H 1 , H2 ) = 0.707. Theorem 1. Single-valued neutrosophic cross entropy CENS (H 1 , H2 ) for any two SVNSs H1 , H2 , satisfies the following properties: i. ii. iii. iv. CENS (H 1 , H2 ) ≥ 0. CENS (H 1 , H2 ) = 0 if and only if TH1 (ui ) = TH2 (ui ), IH1 (ui ) = IH2 (ui ), FH1 (ui ) = FH2 (ui ), ∀ ui ∈ U. CENS (H 1 , H2 ) = CENS (Hc1 , Hc2 ) CENS (H1 , H2 ) = CENS (H2 , H1 ) Information 2018, 9, 37 6 of 21 Proof. (i) For all values of ui ∈ U, TH1 (ui ) ≥ 0, TH2 (ui ) ≥ 0, TH1 (ui ) − TH2 (ui ) ≥ q q 2 2 ≥ 0, (1 − T H1 (ui )) ≥ 0, (1 − TH2 (ui )) ≥ ≥ 0, 1 + TH1 (ui ) 1 + TH2 (ui ) r r 2 (1 − T H1 (ui )) − (1 − TH2 (ui )) ≥ 0, 1 + (1 − T H1 (ui )) ≥ 0, 1 + (1 − T H2 (ui ))   2 (1− T H (ui ))−(1− TH2 (ui )) 2| TH1 (ui )− TH2 (ui )| 1  ≥ 0. q r Then,  q +r 2 2 2 2 1+| TH1 (ui )| + 1+| TH2 (ui )| 1+ (1− T H (ui )) + 1+ (1− T H (ui )) 2 1   ( 1 − I ( u ))−( 1 − I ( u )) 2 IH2 (ui )| H2 i H1 i  q 2| IH1 (ui )−q  r +r Similarly, 2 2 2 2 1+| IH1 (ui )| + 1+| IH2 (u)| 1+ (1− I H (ui )) + 1+ (1− I H (ui )) 2 1   2 (1− F H (ui ))−(1− FH2 (ui )) 2| FH1 (ui )− FH2 (ui )| 1  ≥ 0. q r +r and  q 2 2 2 2 1+| FH1 (ui )| + 1+| FH2 (ui )| 1+ (1− F H (ui )) + 1+ (1− F H (ui )) (ii)  q  2 | TH1 (ui )− TH2 (ui )| q 2 2 1+| TH1 (ui )| + 1+| TH2 (ui )|  q 2 | IH1 (ui )−qIH2 (ui )| 2 1+| IH1 (ui )| + 1+| IH2 (u)  q | + 2 2 | FH1 (ui )− FH2 (ui )| q 2 2 1+| FH2 (ui )| 1+| FH1 (ui )| + + r + r 2 (1− T H (ui ))−(1− TH2 (ui )) 1 r 2 1+ (1− T H (ui )) + 1+ (1− T H (ui )) 2 1+ (1− I H (ui )) + 1+ (1− I H (ui )) 2 2 1 2 (1− F H (ui ))−(1− FH2 (ui )) 1 r 2 1+ (1− F H (ui )) + 1+ (1− F H (ui ))  ≥ 0. ≥ 0,   = 0, ⇔ TH (ui ) = TH (ui ) , 2 1  = 0 ⇔ IH (ui ) = IH (ui ) , and, 2 1 2 2 1 2 2 1 2 (1− I H (ui ))−(1− IH2 (ui )) 1 r r 0, 2 1 Therefore, CENS (H 1 , H2 ) ≥ 0. Hence complete the proof. 2 0,   = 0, ⇔ F H (ui ) = FH (ui ) 2 1 Therefore, CENS (H 1 , H2 ) = 0, iff TH1 (ui ) = TH2 (ui ), IH1 (ui ) = IH2 (ui ), FH1 (ui ) = FH2 (ui ), ∀ ui ∈ U. Hence complete the proof. (iii) Using Definition 5, we obtain the following expression CENS (Hc1 , Hc2 ) =   r r =    1 2   1 2 i =1 2 (1− T H (ui ))−(1− TH2 (ui )) 1 r 2 1+ (1− T H (ui )) + 2 1+ (1− I H (ui )) 2 1 2 (1− F H (ui ))−(1− FH2 (ui )) 1 r 2 1 1+ (1− F H (ui )) 2 2| TH1 (ui )− TH2 (ui )| q 2 2 1+| TH1 (ui )| + 1+| TH2 (ui )| IH2 (ui )|  q 2| IH1 (ui )−q 2 2 1+| IH1 (ui )| + 1+| IH2 (u)| FH2 (ui )|  q 2| FH1 (ui )−q 2 2 1+| FH1 (ui )| + 1+| FH2 (ui )| + r + r + 2 2 2 +  +  2| FH1 (ui )− FH2 (ui )|  q +q 2 2  1+| FH1 (ui )| + 1+| FH2 (ui )| + r 2 (1− T H (ui ))−(1− TH2 (ui )) 1 r 2 1+ (1− T H (ui )) + 1 2 1+ (1− I H (ui )) + 1 2 (1 − TH2 (ui )) − ( 1 − TH1 (ui )) , = 2 1+ (1− I H (ui )) 1+ (1− F H 2 2  2  + + +2    = CESN (H1 , H2 ) 2  (u )) i Therefore, CENS (H 1 , H2 ) = CENS (Hc1 , Hc2 ). Hence complete the proof. TH2 (ui ) − TH1 (ui ) , = TH1 (ui ) − TH2 (ui ) (iv) Since, IH2 (ui ) − IH1 (ui ) , FH1 (ui ) − FH2 (ui ) 1+ (1− T H (ui )) 2 (1− I H (ui ))−(1− IH2 (ui )) 1 r 1+ (1− F H (ui )) + + 2| IH1 (ui )− IH2 (ui )| + q q 2 2 1+| IH1 (ui )| + 1+| IH2 (u)| 2 (1− F H (ui ))−(1− FH2 (ui )) 1 r 1  2 | TH1 (ui )− TH2 (ui )|  q q 2 2 1+| TH1 (ui )| + 1+| TH2 (ui )| 2 1 1+ (1− I H (ui )) + * ∑ q i =1 1+ (1− T H (ui )) 2 (1− I H (ui ))−(1− IH2 (ui )) 1 r 1+ (1− F H (ui )) + n  * ∑ r n IH1 (ui ) − IH2 (ui ) FH2 (ui ) − FH1 (ui ) , (1 − T H1 (ui )) − (1 − TH2 (ui )) (1 − IH1 (ui )) − (1 − IH2 (ui )) = = = (1 − IH2 (ui )) − (1 − IH1 (ui )) , Information 2018, 9, 37 7 of 21 q 2 (1 − F H1 (ui )) − (1 − FH2 (ui )) = (1 − FH2 (ui )) − (1 − FH1 (ui )) , then, + 1 + TH1 (ui ) q q q q q 2 2 2 2 2 = + 1 + TH1 (ui ) , 1 + IH1 (ui ) + = 1 + TH2 (ui ) 1 + TH2 (ui ) 1 + IH2 (ui ) q q q q q 2 2 2 2 2 1 + IH2 (ui ) 1 + FH1 (ui ) 1 + FH2 (ui ) 1 + FH2 (ui ) + 1 + IH1 (ui ) , = + + r q q q 2 2 2 2 = 1 + (1 − T H1 (ui )) 1 + (1 − TH2 (ui )) 1 + (− TH2 (ui ) ) + q q q q 2 2 2 1 + (1 − TH1 (ui )) , 1 + (1 − IH1 (ui )) + 1 + (1 − IH2 (ui )) = 1 + (1 − IH2 (ui )) r q q q 2 2 2 1 + (1 − IH1 (ui )) , 1 + (1 − F H1 (ui )) + 1 + (1 − FH2 (ui )) = 1 + (1 − FH2 (ui )) q 2 1 + (1 − FH1 (ui )) , ∀ ui ∈ U. Therefore, CENS (H 1 , H2 ) = CENS (H2 , H1 ). Hence complete the proof. 1 + FH1 (ui ) , + 2 + 2 + Definition 9 Weighted NS-cross entropy measure. We consider the weight wi (i = 1, 2, ..., n) for the n element ui (i = 1, 2, .., n) with the conditions wi ∈ [0, 1] and ∑ wi = 1. i =1 Then the weighted cross entropy between SVNSs H1 and H2 can be defined as follows:    n 2 (1− T H (ui ))−(1− TH2 (ui )) 2 | TH1 (ui )− TH2 (ui )| 1  + r q r + ∑ wi  q 2 2 2 2  1+| TH1 (ui )| + 1+| TH2 (ui )| i=1 1+ (1− T H (ui )) + 1+ (1− T H (ui )) 2 1   +   2 (1− I H (ui )) −(1− IH2 (ui )) 2 (1− F H (ui ))−(1− FH2 (ui )) 2 | FH1 (ui )− FH2 (ui )| IH2 (ui )| 1 1     q 2 | IH1 (ui )−q r r q r r q + + + 2 2 2 2 2 2 2 2  1+| IH1 (ui ) | + 1+| IH2 (u)| 1+| FH1 (ui )| + 1+| FH2 (ui )| 1+ (1 − I H (ui )) + 1+ (1− I H (ui )) 1+ (1− F H (ui )) + 1+ (1− F H (ui )) CEw NS (H1 , H2 ) = 1 2 * 1 2 1 (2) 2 Theorem 2. Single-valued neutrosophic weighted NS-cross-entropy (defined in Equation (2)) satisfies the following properties: i. ii. iii. iv. CEw NS (H 1 , w CENS (H 1 , ∀ ui ∈ U. CEw NS (H 1 , CEw NS (H 1 , H2 ) ≥ 0. H2 ) = 0, if and only if TH1 (ui ) = TH2 (ui ) IH1 (ui ) = IH2 (ui ), FH1 (ui ) = FH2 (ui ), c c H2 ) = CEw NS (H1 , H2 ) w H2 ) = CENS ( H 2 , H1 ) Proof. (i). For all values of ui ∈ U, TH1 (ui ) ≥ 0 TH2 (ui ) ≥ 0, TH1 (ui ) − TH2 (ui ) ≥ 0, q q 2 2 ≥ 0, ≥ 0, (1 − T H1 (ui )) ≥ 0, (1 − TH2 (ui )) ≥ 0, 1 + TH1 (ui ) 1 + TH2 (ui ) r r 2 2 (1 − T H1 (ui )) − (1 − TH2 (ui )) ≥ 0, 1 + (1 − T H1 (ui )) ≥ 0, 1 + (1 − T H2 (ui )) ≥ 0, then,   2 ( 1 − T ( u ))−( 1 − T ( u )) H2 i H1 i  q 2 | TH1 (ui )−qTH2 (ui )|  ≥ 0. r +r 2 2 2 2 1+| TH1 (ui )| + 1+| TH2 (ui )| 1+ (1− T H (ui )) + 1+ (1− T H (ui )) 2 1   2 (1− I H (ui ))−(1− IH2 (ui )) 2 | IH1 (ui )− IH2 (ui )| 1 q  r q ≥ Similarly, 0, +r 2 2 2 2 1+| IH1 (ui )| + 1+| IH2 (u)| 1+ (1− I H (ui )) + 1+ (1− I H (ui )) 1   2 2 ( 1 − F ( u ))−( 1 − F ( u )) 2 | FH1 (ui )− FH2 (ui )| H2 i H1 i  ≥ 0. q r +r and  q 2 2 2 2 1+| FH1 (ui )| + 1+| FH2 (ui )| 1+ (1− F H (ui )) + 1+ (1− F H (ui )) n Since wi ∈ [0, 1] and ∑ wi = 1, therefore, i =1 Hence complete the proof. 2 1 CEw NS (H 1 , H2 ) ≥ 0. Information 2018, 9, 37 (ii) Since,  q  8 of 21   q 2 | TH1 (ui )−qTH2 (ui )| 2 2 1+| TH1 (ui )| + 1+| TH2 (ui )| 2 | IH1 (ui )− IH2 (ui )| q 2 1+| IH1 (ui ) | + 1+| IH2 (u) | 2  q 2 | FH1 (ui )−qFH2 (ui )| 2 2 1+| FH1 (ui )| + 1+| FH2 (ui )| and wi + + n r r + r 2 (1− T H (ui ))−(1− TH2 (ui )) 1 r 2 1+ (1− T H (ui )) + 1+ (1− T H (ui )) 2 1 2 (1− I H (ui ))−(1− IH2 (ui )) 1 r 2 2 (1− F H (ui ))−(1− FH2 (ui )) 1 r 2 2 1+ (1− I H (ui )) + 1+ (1− I H (ui )) 2 1 2 1+ (1− F H (ui )) + 1+ (1− F H (ui ))   = 0, ⇔ TH (ui ) = TH (ui ) , 2 1 =  ⇔ IH1 (ui ) = IH2 (ui ) , 0,   = 0, ⇔ F H (ui ) = 1 2 1 2  FH2 (ui ) ∈ [0, 1] , ∑ wi = 1, wi ≥ 0. Therefore, CEw NS (H1 , H2 ) = 0 iff TH1 ( ui ) = TH2 ( ui ), i =1 IH1 (ui ) = IH2 (ui ), FH1 (ui ) = FH2 (ui ), ∀ ui ∈ U. Hence complete the proof. (iii) Using Definition 5, we obtain the following expression   c c CEw NS (H1 , H2 ) =  r   2 1+ (1− I H (ui )) + 1  2   1+ (1− I H (ui )) 1 n * ∑ wi  q i =1 2 (1− T H (ui ))−(1− TH2 (ui )) 1 r 2 1+ (1− T H (ui )) + 1 2 + 2 2 (1− F H (ui )) −(1− FH2 (ui )) 1 r 1+ (1− F H (ui )) + 1 2 q i =1 2 (1− I H (ui ))−(1− IH2 (ui )) 1 r r = 1 2 * ∑ wi  r n 1+ (1− F H (ui )) 2 2  q 2 | FH1 (ui )−qFH2 (ui )| 2 2 1+| FH1 (ui )| + 1+| FH2 (ui )| r + = (1 − TH2 (ui )) − (1 − TH1 (ui )) ,  + r 2 (1− T H (ui ))−(1− TH2 (ui )) 1 r 2 1+ (1− T H (ui )) + 1 2 1+ (1− I H (ui )) + 1 2 2 (1− F H (ui )) −(1− FH2 (ui )) 1 r 2 1+ (1− F H (ui )) + 1 2 1+ (1− I H (ui )) 1+ (1− F H 2 c c w Therefore, CEw NS (H1 , H2 ) = CENS (H1 , H2 ). Hence complete the proof. (iv) Since TH1 (ui ) − TH2 (ui ) = TH2 (ui ) − TH1 (ui ) , FH1 (ui ) − FH2 (ui ) + + 2 1 2 (1− I H (ui ))−(1− IH2 (ui )) 1 r r 2 +  2 | FH1 (ui )− FH2 (ui )|  q +q 2 2  1+| FH (ui )| + 1+| FH (ui )| 2 | TH1 (ui )− TH2 (ui )| q 2 2 1+| TH2 (ui )| + 2 2 | IH1 (ui )− IH2 (ui )| + q q 2 2 1+| IH1 (ui )| + 1+| IH2 (u)| 1+| TH1 (ui )| + 2 | IH1 (ui )− IH2 (ui )| q 2 2 1+| IH1 (ui )| + 1+| IH2 (u)| 1+ (1− T H (ui ))  2 | TH1 (ui )− TH2 (ui )|  q q 2 2 1+| TH1 (ui )| + 1+| TH2 (ui )| FH2 (ui ) − FH1 (ui ) , 1+ (1− T H (ui )) 2 2   + + +   = CEw NS (H1 , H2 ) 2  (u )) i IH1 (ui ) − IH2 (ui ) = IH2 (ui ) − IH1 (ui ) , (1 − T H1 (ui )) − (1 − TH2 (ui )) (1 − IH1 (ui )) − (1 − IH2 (ui )) (1 − IH2 (ui )) − (1 − IH1 (ui )) , q 2 + (1 − F H1 (ui )) − (1 − FH2 (ui )) = (1 − FH2 (ui )) − (1 − FH1 (ui )) , we obtain 1 + TH1 (ui ) q q q q q 2 2 2 2 2 = + 1 + TH1 (ui ) , 1 + IH1 (ui ) = 1 + TH2 (ui ) 1 + TH2 (ui ) + 1 + IH2 (ui ) q q q q q 2 2 2 2 2 + 1 + IH1 (ui ) , = + + 1 + IH2 (ui ) 1 + FH1 (ui ) 1 + FH2 (ui ) 1 + FH2 (ui ) r q q q 2 2 2 2 = + 1 + FH1 (ui ) , 1 + (1 − T H1 (ui )) 1 + (1 − TH2 (ui )) 1 + (− TH2 (ui ) ) + q q q q 2 2 2 2 1 + (1 − TH1 (ui )) , 1 + (1 − IH1 (ui )) + 1 + (1 − IH2 (ui )) = 1 + (1 − IH2 (ui )) + r q q q 2 2 2 2 1 + (1 − IH1 (ui )) , 1 + (1 − F H1 (ui )) + 1 + (1 − FH2 (ui )) = 1 + (1 − FH2 (ui )) + q n 2 1 + (1 − FH1 (ui )) , ∀ ui ∈ U and wi ∈ [0, 1] , ∑ wi = 1. w Therefore, CEw NS (H1 , H2 ) = CENS (H2 , H1 ). Hence complete the proof. i =1 = = Information 2018, 9, 37 9 of 21 4. MAGDM Strategy Using Proposed Ns-Cross Entropy Measure under SVNS Environment In this section, we develop a new MAGDM strategy using the proposed NS-cross entropy measure. Description of the MAGDM Problem Assume that A = { A1 , A2 , A3 , . . . , Am } and G = { G1 , G2 , G3 , . . . , Gn } be the discrete set of alternatives and attributes respectively and W = {w1 , w2 , w3 , . . . , wn } be the weight vector of n attributes Gj (j = 1, 2, 3, . . . , n), where w j ≥ 0 and ∑ w j = 1. Assume that E = E1 , E2 , E3 , . . . , Eρ j =1 be the set of decision-makers who are employed to evaluate the alternatives. The weight vector λ1 , λ2 , λ3 , . . . , λρ (where, λk ≥ 0 and of the decision-makers Ek (k = 1, 2, 3, . . . , ρ) is λ = ρ ∑ λk = 1), which can be determined according to the decision-makers’ expertise, judgment quality k =1 and domain knowledge. Now, we describe the steps of the proposed MAGDM strategy (see Figure 1) using NS-cross entropy measure. MAGDM Strategy Using Ns-Cross Entropy Measure Step 1. Formulate the decision matrices For MAGDM with SVNSs information, the rating values of the alternatives Ai (i = 1, 2, 3, . . . , m) based on the attribute Gj ( j = 1, 2, 3, . . . , n) provided by the k-th decision-maker can be expressed in terms of SVNN as aijk =< Tijk , Iijk , Fijk > (i = 1, 2, 3, . . . , m; j = 1, 2, 3, . . . , n; k = 1, 2, 3, . . . , ρ). We present these rating values of alternatives provided by the decision-makers in matrix form as follows:   A  1  M =  A2   . Am k G1 k a11 k a21 . akm1 G2 k a12 ... akm2 .... ... k a2n . ... Gn k a1n k a22 akmn        (3) Step 2. Formulate priori/ideal decision matrix In the MAGDM, the a priori decision matrix has been used to select the best alternatives among the set of collected feasible alternatives. In the decision-making situation, we use the following decision matrix as a priori decision matrix.   A  1  P =  A2   . Am G1 ∗ a11 ∗ a21 . ∗ am1 G2 ∗ a12 ∗ a22 ... a∗m2 .... ... . ... Gn ∗ a1n ∗ a2n a∗mn        (4) where, aij∗ =< max ( Tijk ), min ( Iijk ), min ( Fijk ) >) corresponding to benefit attributes and aij∗ =< i i i min ( Tijk ), max ( Iijk ), max ( Fijk ) > corresponding to cost attributes, and (i = 1, 2, 3, . . . , m; j = 1, 2, 3, . . . , i i n; k = 1, 2, 3, . . . , ρ). i Step 3. Determinate the weights of decision-makers To find the decision-makers’ weights we introduce a model based on the NS-cross entropy measure. The collective NS-cross entropy measure between Mk and P (Ideal matrix) is defined as follows: 1 m (5) CEcNS ( Mk , P) = ∑ CE NS Mk ( Ai ), P( Ai ) m i =1 Information 2018, 9, 37 10 of 21 n where, CE NS Mk ( Ai ), P( Ai ) = ∑ CENS ( Mk ( Ai ( Gj )), P( Ai ( Gj ))). j =1 Thus, we can introduce the following weight model of the decision-makers: λK = 1 ÷ CEcNS ( Mk , P) ρ ∑ 1÷ k =1 CEcNS ( Mk , P) (6) ρ where, 0 ≤ λK ≤ 1 and ∑ λK = 1 for k = 1, 2, 3, . . . , ρ. k =1 Preparatory Phase Multi attribute group decision making problem Decision making analysis phase Formation of decision matrix provided by decision makers Step-1 Formation of ideal decision matrix Step- 2 Determination of the weights of decision maker Step- 3 Formation of weighted aggregated decision matrix Step-4 Determinationof ofthe theweights weight of Determination of attributes attribute Calculationofweighted Calculation weightedNSNS-cross cross entropy measure Step-5 HH Step- 6 Rank the priority Step-7 Selection Selectionofthe thebest bestalternative alternative Step-8 Figure.1 Decision making procedure of the proposed MAGDM strategy Figure 1. Decision-making procedure of the proposed MAGDM strategy. Information 2018, 9, 37 11 of 21 Step 4. Formulate the weighted aggregated decision matrix For obtaining one group decision, we aggregate all the individual decision matrices (Mk ) to an aggregated decision matrix (M) using single valued neutrosophic weighted averaging (SVNWA) operator ([51]) as follows: ρ 3 2 1 aij = SV NSWAλ ( a1ij , a2ij , a3ij , . . . , aρ ij ) = (λ1 aij ⊕ λ2 aij ⊕ λ3 aij ⊕ . . . ⊕ λρ aij ) = ρ ρ ρ λ < 1 − ∏ (1 − Tijk )λk , ∏ ( Iijk ) k , ∏ ( Fijk ) > (7) k =1 k =1 k =1 λk Therefore, the aggregated decision matrix is defined as follows:  G1 a11 a21 . am1  A  1  M =  A2   . Am G2 a12 a22 ... am2 .... ... . ... Gn a1n a2n amn        (8) where, aij =< Tij , Iij , Fij >, (i = 1, 2, 3, . . . , m; j = 1, 2, 3, . . . , n; k = 1, 2, 3, . . . , ρ). Step 5. Determinate the weight of attributes To find the attributes weight we introduce a model based on the NS-cross entropy measure. The collective NS-cross entropy measure between M (Weighted aggregated decision matrix) and P (Ideal matrix) for each attribute is defined by j CE NS ( M, P) = 1 m CE NS M( Ai ( Gj )), P( Ai ( Gj )) m i∑ =1 (9) where, i = 1, 2, 3, . . . , m; j = 1, 2, 3, . . . , n. Thus, we defined a weight model for attributes as follows: j 1 ÷ CE NS ( M, P) wj = n j ∑ 1 ÷ CE NS ( M, P) (10) J =1 n where, 0 ≤ w j ≤ 1 and ∑ w j = 1 for j = 1, 2, 3, . . . , n. j =1 Step 6. Calculate the weighted NS-cross entropy measure Using Equation (2), we calculate weighted cross entropy value between weighted aggregated matrix and priori matrix. The cross entropy values can be presented in matrix form as follows: NS w MCE     =   CEw NS ( A1 ) CEw NS ( A2 ) ............... ................. CEw NS ( Am )        (11) Step 7. Rank the priority Smaller value of the cross entropy reflects that an alternative is closer to the ideal alternative. Therefore, the preference priority order of all the alternatives can be determined according to the Information 2018, 9, 37 12 of 21 increasing order of the cross entropy values CEw NS (Ai ) (i = 1, 2, 3, . . . , m). Smallest cross entropy value indicates the best alternative and greatest cross entropy value indicates the worst alternative. Step 8. Select the best alternative From the preference rank order (from step 7), we select the best alternative. 5. Illustrative Example In this section, we solve an illustrative example adapted from [12] of MAGDM problems to reflect the feasibility, applicability and efficiency of the proposed strategy under the SVNS environment. Now, we use the example [12] for cultivation and analysis. A venture capital firm intends to make evaluation and selection of five enterprises with the investment potential: (1) (2) (3) (4) (5) Automobile company (A1 ) Military manufacturing enterprise (A2 ) TV media company (A3 ) Food enterprises (A4 ) Computer software company (A5 ) On the basis of four attributes namely: (1) (2) (3) (4) Social and political factor (G1 ) The environmental factor (G2 ) Investment risk factor (G3 ) The enterprise growth factor (G4 ). The investment firm makes a panel of three decision-makers. The steps of decision-making strategy (4.1.1.) to rank alternatives are presented as follows: Step: 1. Formulate the decision matrices We represent the rating values of alternatives Ai (i = 1, 2, 3, 4, 5) with respects to the attributes Gj (j = 1, 2, 3, 4) provided by the decision-makers Ek (k = 1, 2, 3) in matrix form as follows: Decision matrix for E1 decision-maker      1 M =    A1 A2 A3 A4 A5 G1 (0.9, 0.5, 0.4) (0.7, 0.2, 0.3) (0.8, 0.4, 0.4) (0.5, 0.8, 0.7) (0.8, 0.4, 0.3) G2 (0.7, 0.4, 0.4) (0.8, 0.4, 0.3) (0.7, 0.4, 0.2) (0.6, 0.3, 0.4) (0.5, 0.4, 0.5) G3 (0.7, 0.3, 0.4) (0.9, 0.6, 0.5) (0.9, 0.7, 0.6) (0.7, 0.2, 0.5) (0.6, 0.4, 0.4) G4 (0.5, 0.4, 0.9) (0.9, 0.1, 0.3) (0.7, 0.3, 0.3) (0.5, 0.4, 0.7) (0.9, 0.7, 0.5)          (12) Decision matrix for E2 decision-maker      M2=    A1 A2 A3 A4 A5 G1 (0.7, 0.2, 0.3) (0.7, 0.4, 0.4) (0.6, 0.4, 0.4) (0.7, 0.5, 0.3) (0.9, 0.4, 0.3) G2 (0.5, (0.7, (0.5, (0.6, (0.6, 0.4, 0.5) 0.3, 0.4) 0.3, 0.5) 0.3, 0.6) 0 .4, 0.5) (0.9, (0.7, (0.9, (0.7, (0.8, G3 0.4, 0.3, 0.5, 0.4, 0.5, 0.5) 0.4) 0.4) 0.4) 0.6) (0.6, (0.6, (0.6, (0.8, (0.5, G4 0.5, 0.4, 0.5, 0.5, 0.4, 0.3) 0.3) 0.6) 0.4) 0.5)          (13) Information 2018, 9, 37 13 of 21 Decision matrix for E3 decision-maker      3 M =    A1 A2 A3 A4 A5 (0.7, (0.6, (0.8, (0.9, (0.8, G1 0.2, 0.5, 0.3, 0.3, 0.3, 0.5) 0.5) 0.5) 0.4) 0.3) (0.6, (0.9, (0.9, (0.6, (0.6, G2 0.4, 0.3, 0.3, 0.3, 0.4, 0.4) 0.4) 0.4) 0.4) 0.3) (0.7, (0.7, (0.8, (0.5, (0.6, G3 0.4, 0.4, 0.3, 0.2, 0.3, 0.5) 0.3) 0.4) 0.4) 0.4) (0.9, (0.8, (0.7, (0.7, (0.7, G4 0.4, 0.4, 0.3, 0.3, 0.3,  0.3) 0.5) 0.4) 0.5) 0.5)         (14) Step: 2. Formulate priori/ideal decision matrix A priori/ideal decision matrix Please provide a sharper picture      P=    A1 A2 A3 A4 A5 (0.9, (0.7, (0.8, (0.9, (0.9, G1 0.2, 0.2, 0.3, 0.3, 0.3, 0.3) 0.3) 0.4) 0.3) 0.3) (0.7, (0.9, (0.9, (0.6, (0.6, G2 0.4, 0.3, 0.3, 0.3, 0.4, 0.4) 0.3) 0.2) 0.4) 0.3) (0.9, (0.9, (0.9, (0.7, (0.8, G3 0.3, 0.3, 0.3, 0.2, 0.3, 0.4) 0.3) 0.4) 0.4) 0.4) (0.9, (0.9, (0.7, (0.7, (0.9, G4 0.4, 0.1, 0.3, 0.3, 0.3, 0.3) 0.3) 0.3) 0.4) 0.5)          (15) Step: 3. Determine the weight of decision-makers By using Equations (5) and (6), we determine the weights of the three decision-makers as follows: λ1 = (1 ÷ 0.9) (1 ÷ 1.2) (1 ÷ .07) ≈ 0.33, λ2 = ≈ 0.25, λ1 = ≈ 0.42. 3.37 3.37 3.37 Step: 4. Formulate the weighted aggregated decision matrix Using Equation (7) the weighted aggregated decision matrix is presented as follows: Weighted Aggregated decision matrix      M=    A1 A2 A3 A4 A5 G1 (0.8, 0.3, 0.4) (0.7, 0.3, 0 .4) (0.8, 0.4, 0.4) (0.7, 0.5, 0.5) (0.8, 0.4, 0.4) G2 (0.6, 0.4, 0.4) (0.8, 0.3, 0.4) (0.8, 0.3, 0.3) (0.6, 0.3, 0.4) (0.6, 0.4, 0.4) G3 (0.8, 0.4, 0.4) (0.8, 0.4, 0.4) (0.9, 0.5, 0.5) (0.6, 0.2, 0.4) (0.7, 0.4, 0.4) G4 (0.7, 0.4, 0.5) (0.8, 0.2, 0.3) (0.7, 0.3, 0.4) (0.7, 0.4, 0.5) (0.8, 0.5, 0.5)          (16) Step: 5. Determinate the weight of the attributes By using Equations (9) and (10), we determine the weights of the four attribute as follows: w1 = (1 ÷ 0.11) (1 ÷ 0.20) (1 ÷ 0.15) (1 ÷ 0.26) ≈ 0.16, w2 = ≈ 0.37, w3 = ≈ 0.20, w4 = ≈ 0.27. 25 25 25 25 Step: 6. Calculate the weighted SVNS cross entropy matrix Using Equation (2) and weights of attributes, we calculate the weighted NS-cross entropy values between ideal matrix and weighted aggregated decision matrix. Information 2018, 9, 37 14 of 21 NS w MCE     =   0.195 0.198 0.168 0.151 0.184        (17) Step: 7. Rank the priority The cross entropy values of alternatives are arranged in increasing order as follows: 0.151 < 0.168 < 0.184 < 0.195 < 0.198. Alternatives are then preference ranked as follows: A4 > A3 > A5 > A1 > A2 . Step: 8. Select the best alternative From step 7, we identify A4 is the best alternative. Hence, Food enterprises (A4 ) is the best alternative for investment. In Figure 2, we draw a bar diagram to represent the cross entropy values of alternatives which shows that A4 is the best alternative according our proposed strategy. In Figure 3, we represent the relation between cross entropy values and acceptance values of alternatives. The range of acceptance level for five alternatives is taken by five points. The high acceptance level of alternatives indicates the best alternative for acceptance and low acceptance level of alternative indicates the poor acceptance alternative. We see from Figure 3 that alternative A4 has the smallest cross entropy value and the highest acceptance level. Therefore A4 is the best alternative for acceptance. Figure 3 indicates that alternative A2 has highest cross entropy value and lowest acceptance value that means A2 is the worst alternative. Finally, we conclude that the relation between cross entropy values and acceptance value of alternatives is opposite in nature. Figure 2. Bar diagram of alternatives versus weighted NS-cross entropy values of alternatives. Information 2018, 9, 37 15 of 21 Figure 3. Relation between weighted NS-cross entropy values and acceptance level line of alternatives. 6. Comparative Study and Discussion In literature only two MADM strategies [144,145] have been proposed. No MADGM strategy is available. So the proposed MAGDM is novel and non-comparable with the existing cross entropy under SVNS for numerical example. i. ii. iii. iv. The MADM strategies [144,145] are not applicable for MAGDM problems. The proposed MAGDM strategy is free from such drawbacks. Ye [144] proposed cross entropy that does not satisfy the symmetrical property straightforward and is undefined for some situations but the proposed strategy satisfies symmetric property and is free from undefined phenomenon. The strategies [144,145] cannot deal with the unknown weight of the attributes whereas the proposed MADGM strategy can deal with the unknown weight of the attributes The strategies [144,145] are not suitable for dealing with the unknown weight of decision-makers, whereas the essence of the proposed NS-cross entropy-based MAGDM is that it is capable of dealing with the unknown weight of the decision-makers. 7. Conclusions In this paper, we have defined a novel cross entropy measure in SVNS environment. The proposed cross entropy measure in SVNS environment is free from the drawbacks of asymmetrical behavior and undefined phenomena. It is capable of dealing with the unknown weight of attributes and the unknown weight of decision-makers. We have proved the basic properties of the NS-cross entropy measure. We also defined weighted NS-cross entropy measure and proved its basic properties. Based on the weighted NS-cross entropy measure, we have developed a novel MAGDM strategy to solve neutrosophic multi-attribute group decision-making problems. We have at first proposed a novel MAGDM strategy based on NS-cross entropy measure with technique to determine the unknown weight of attributes and the unknown weight of decision-makers. Other existing cross entropy measures [144,145] can deal only with the MADM problem with single decision-maker and known weight of the attributes. So in general, our proposed NS-cross entropy-based MAGDM strategy is not comparable with the existing cross-entropy-based MADM strategies [144,145] under the single-valued neutrosophic environment. Finally, we solve a MAGDM problem to show the feasibility, applicability and efficiency of the proposed MAGDM strategy. The proposed NS-cross entropy-based MAGDM can be applied in teacher selection, pattern recognition, weaver selection, medical treatment selection options, and other practical problems. 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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). information Article Generalized Single-Valued Neutrosophic Hesitant Fuzzy Prioritized Aggregation Operators and Their Applications to Multiple Criteria Decision-Making Rui Wang 1,2, * and Yanlai Li 1,3 1 2 3 * School of Traffic and Logistics, Southwest Jiaotong University, Chengdu 610031, China; lyl_swjtu@163.com Tangshan Graduate School, Southwest Jiaotong University, Tangshan 063000, China National Lab of Railway Transportation, Southwest Jiaotong University, Chengdu 610031, China Correspondence: wryuedi@my.swjtu.edu.cn; Tel.: +86-28-8760-0165 Received: 11 December 2017; Accepted: 2 January 2018; Published: 5 January 2018 Abstract: Single-valued neutrosophic hesitant fuzzy set (SVNHFS) is a combination of single-valued neutrosophic set and hesitant fuzzy set, and its aggregation tools play an important role in the multiple criteria decision-making (MCDM) process. This paper investigates the MCDM problems in which the criteria under SVNHF environment are in different priority levels. First, the generalized single-valued neutrosophic hesitant fuzzy prioritized weighted average operator and generalized single-valued neutrosophic hesitant fuzzy prioritized weighted geometric operator are developed based on the prioritized average operator. Second, some desirable properties and special cases of the proposed operators are discussed in detail. Third, an approach combined with the proposed operators and the score function of single-valued neutrosophic hesitant fuzzy element is constructed to solve MCDM problems. Finally, an example of investment selection is provided to illustrate the validity and rationality of the proposed method. Keywords: multiple criteria decision-making (MCDM); single-valued neutrosophic hesitant fuzzy set (SVNHFS); generalized single-valued neutrosophic hesitant fuzzy prioritized weighted average operator; generalized single-valued neutrosophic hesitant fuzzy prioritized weighted geometric operator 1. Introduction In daily life, MCDM problems happen in many fields; decision makers determine the best one from several alternatives through evaluating them with respect to the corresponding criteria. Due to the high complexity of the social environment, the evaluation information given by decision makers is often uncertain, incomplete, and inconsistent. With the demand for accuracy of decision-making results is getting higher and higher, much research in recent years has focused on the MCDM problems under fuzzy environment [1]. In 1965, Zadeh [2] developed the fuzzy set (FS) theory, which is a powerful tool to express the fuzzy information. However, there are several obvious limitations of FS theory in expressing uncertain information, which are attracting widespread interest in improving FS theory. Atanassov [3] introduced the non-membership function to extend FS theory and proposed the intuitionistic fuzzy set (IFS) theory. IFS can express the membership and non-membership information simultaneously; the property can deal with some applications effectively, which FS cannot. For example, ten decision makers vote for an affair, four present agreement, three suggest different opinions, and the others choose to give up. The example above can be characterized by IFS, i.e., the value of membership is 0.4, and the value of non-membership is 0.3. However, expressing the voting information by FS is impossible. To describe the fuzziness of evaluation information more effective, Atanassov and Gargov [4] utilized the interval number to extend the membership and non-membership functions Information 2018, 9, 10; doi:10.3390/info9010010 www.mdpi.com/journal/information Information 2018, 9, 10 2 of 19 and put forward the interval-valued intuitionistic fuzzy set (IVIFS) theory. Nevertheless, in the real decision-making process, only considering the membership and non-membership information is not comprehensive sometimes. For instance, a decision maker gives her/his evaluation on a viewpoint, she/he may think the positive probability is 0.5, the false probability is 0.6, and the indeterminacy probability is 0.2 [5]. Obviously, IFS and IVIFS theory cannot deal with this situation. Therefore, Smarandache [6] defined the neutrosophic set (NS), which can be regarded as a generalization of FS and IFS [7]. NS consists of three independent membership functions, namely, truth-membership, indeterminacy-membership, and falsity-membership functions. Whereas, NS theory was originally proposed from a philosophical point of view, and it is difficult to apply NS theory in the field of science and engineering. To solve this problem, Wang [8,9] defined the concepts of interval neutrosophic set (INS) and single-valued neutrosophic set (SVNS), which are specific cases of NS. Another drawback of FS is that its membership value is single; while determining the exact value of membership may be difficult for decision makers due to doubt. To deal with this situation, Torra and Narukawa [10] and Torra [11] extended the FS theory to hesitant fuzzy set (HFS) theory through allowing decision makers to give several different values of membership. Furthermore, Chen [12] defined the concept of interval-valued hesitant fuzzy set (IVHFS), in which the possible membership values can be expressed by interval numbers. Considering the complex information given by decision makers, Zhu [13] introduced the non-membership hesitancy function to propose the dual hesitant fuzzy set (DHFS) theory. According to the aforementioned analysis of improved FS theory from two directions, Ye [14] developed the single-valued neutrosophic hesitant fuzzy set (SVNHFS) combined with NS and HFS theory, in addition, Liu and Shi [7] extended the SVNHFS to interval neutrosophic hesitant fuzzy set (INHFS). Consequently, SVNHFS and INHFS not only can characterize the inconsistent and indeterminate information but also allow decision makers to give several possible values of truth-membership, indeterminacy-membership, and falsity-membership functions. Besides the evaluation information, aggregation tools also are important parts of MCDM process. Ye [14] developed the operational laws and cosine measure of single-valued neutrosophic hesitant fuzzy elements (SVNHFEs), and proposed the single-valued neutrosophic hesitant fuzzy weighted average (SVNHFWA) operator and single-valued neutrosophic hesitant fuzzy weighted geometric (SVNHFWG) operator to aggregate SVNHFEs. Şahin and Liu [15] constructed the decision-making approach based on the correlation coefficient and weighted correlation coefficient of SVNHFEs. Biswas et al. [16] put forward several approaches for decision-making under SVNHF environment by using distance measures of SVNHFEs. Liu and Luo [17] proposed the single-valued neutrosophic hesitant fuzzy ordered weighted average (SVNHFOWA) operator and single-valued neutrosophic hesitant fuzzy hybrid weighted average (SVNHFHWA) operator, and applied them into MCDM process. Liu and Zhang [18] developed the single-valued neutrosophic hesitant fuzzy Heronian mean aggregation operators to deal with MCDM problems. Liu and Shi [7] defined the operational laws of INHFSs and proposed interval neutrosophic hesitant fuzzy generalized weighted average (INHFGWA) operator, interval neutrosophic hesitant fuzzy generalized ordered weighted average (INHFGOWA) operator, and interval neutrosophic hesitant fuzzy generalized hybrid weighted average (INHFGHWA) operator. Ye [19] determined the ranking of alternatives combined with the correlation coefficient of INHFSs. The aforementioned decision-making methods are applied to the situation of the aggregated arguments and are in the same priority; whereas, in many real situations, criteria always have different priorities. For example, a mother chooses the dried milk for her baby, the criteria she considers are price and safety. Obviously, a prioritization ordering exists between the criteria, i.e., safety is much more important than price [20]. To deal with this situation, Yager [21] proposed the prioritized average (PA) operator to aggregate the evaluation information concerning the criteria of different priorities. Since the PA operator was presented, many scholars have focused on extending the PA operator into the fuzzy environment. For instance, Yu [20] proposed the intuitionistic fuzzy prioritized weighted average (IFPWA) operator and intuitionistic fuzzy prioritized weighted geometric (IFPWG) operator, and investigated their properties. Yu et al. [22] extended the PA operator into IVIF Information 2018, 9, 10 3 of 19 environment and developed the interval-valued intuitionistic fuzzy prioritized weighted average (IVIFPWA) operator and interval-valued intuitionistic fuzzy prioritized weighted geometric (IVIFPWG) operator. Liu and Wang [23] studied the aggregation operator under IN environment and put forward the interval neutrosophic prioritized ordered weighted average (INPOWA) operator. Furthermore, Wei [24] extended the PA operator into hesitant fuzzy MCDM problems. Jin et al. [25] developed interval-valued hesitant fuzzy Einstein prioritized weighted average (IVHFEPWA) operator and the interval-valued hesitant fuzzy Einstein prioritized weighted geometric (IVHFEPWG) operator through improving the operations of IVHFSs. However, to our best knowledge, little attention has been paid to the prioritized aggregation operators under SVNHF environment. This paper proposes the aggregation operators for SVNHFEs, in which the aggregation arguments have different priority levels, and develops an approach for decision-making. To do this, the rest of this paper is organized as follows. Section 2 briefly introduces some basic concepts of SVNS, HFS, SVNHFS, and the PA operator. Section 3 develops the generalized single-valued neutrosophic hesitant fuzzy prioritized weighted average (GSVNHFPWA) operator and generalized single-valued neutrosophic hesitant fuzzy prioritized weighted geometric (GSVNHFPWG) operator, and investigates some desirable properties and special cases of the proposed operators. Section 4 constructs an approach for decision-making based on the proposed operators. Section 5 provides a numerical example to illustrate the applications and advantages of the proposed method. Section 6 summarizes the conclusions of this research. 2. Preliminaries In this section, we briefly introduce some basic concepts, including the definitions of NS, SVNS, HFS, and SVNHFS. The operations of SVNHFEs and the PA operator are also presented, which are used in the subsequent discussion. 2.1. The Single-Valued Neutrosophic Set Definition 1. Ref. [6] Let X be a universe of discourse, with a generic element in X denoted by x. An NS A in X is characterized by a truth-membership function TA ( x ), an indeterminacy-membership function I A ( x ), and a falsity-membership function FA ( x ). The functions TA ( x ), I A ( x ), and FA ( x ) are real standard or non-standard subsets of ]− 0, 1+ [, i.e., TA ( x ) : X →]− 0, 1+ [ , I A ( x ) : X →]− 0, 1+ [ , and FA ( x ) : X →]− 0, 1+ [ . Thus, the sum of three aforementioned functions satisfies the condition of − 0 ≤ supT ( x ) + supI ( x ) + supF ( x ) ≤ 3+ . A A A NS theory was originally proposed from the angle of philosophy and can be regarded as a generalization of FS, IFS, and IVIFS. However, the NS is not easily used for real scientific and engineering decision-making problems. To solve this limitation, Wang [8] defined the concept of SVNS, which is a special case of NS. Definition 2. Ref. [8] Let X be a universe of discourse, with a generic element in X denoted by x. An SVNS A is given by A = {h x, TA ( x ), I A ( x ), FA ( x )i| x ∈ X }, (1) where TA ( x ) is the truth-membership function, I A ( x ) is the indeterminacy-membership function, and FA ( x ) is the falsity-membership function. For each point x in X, the functions TA ( x ), I A ( x ), and FA ( x ) satisfy the conditions of TA ( x ), I A ( x ), FA ( x ) ∈ [0, 1] and 0 ≤ TA ( x ) + I A ( x ) + FA ( x ) ≤ 3. 2.2. The Hesitant Fuzzy Set During the decision-making process, decision makers sometimes may be confused when determining the exact membership value of an element to the set because of the existing several possible membership values. Considering this situation, Torra and Narukawa [10] defined the concept of HFS. Information 2018, 9, 10 4 of 19 Definition 3. Ref. [10] Let X be a non-empty and finite set, an HFS A on X is defined by a function h A ( x ) that when applied to X returns a finite subset of [0, 1], which can be expressed as A = {h x, h A ( x )i| x ∈ X }, (2) where h A ( x ) is a set of some different values in [0, 1], indicating the possible membership degrees of the element x ∈ X to A. 2.3. The Single-Valued Neutrosophic Hesitant Fuzzy Set Based on the combination of SVNS and HFS, Ye [14] proposed the concept of SVNHFS. Definition 4. Ref. [14] Let X be a non-empty and finite set, an SVNHFS N on X is expressed as N= where et( x ) = nD n γ γ ∈ et( x ) , ei ( x ) = o E x, et( x ), ei ( x ), fe( x ) | x ∈ X , (3) n o o δ δ ∈ ei ( x ) , and fe( x ) = η η ∈ fe( x ) are three sets of some different values in [0, 1], denoting the possible truth-membership hesitant, possible indeterminacy-membership hesitant, and possible falsity-membership hesitant degrees of the element x ∈ X to N. And they satisfy the conditions of γ, δ, η ⊆ [0, 1] and 0 ≤ supγ+ + supδ+ + supη + ≤ 3, where γ+ = ∪γ∈et( x) max{γ}, o n e = et(x),ei(x), ef (x) δ+ = ∪δ∈ei( x) max{δ}, and η + = ∪η∈ef (x) max{η } for x ∈ X. For convenience, we call n o n e = et, ei, fe . is an SVNHFE, denoted by n o n o n n o e1 = et1 , ei1 , fe1 and n e2 = et2 , ei2 , fe2 be three SVNHFEs, λ > 0, e = et, ei, fe , n Definition 5. Ref. [14] Let n then the basic operations of SVNHFEs are defined as o n e1 ⊕ n e2 = et1 ⊕ et2 , ei1 ⊗ ei2 , fe1 ⊗ fe2 = n n o e1 ⊗ n e2 = et1 ⊗ et2 , ei1 ⊕ ei2 , fe1 ⊕ fe2 = n λe n= eλ = n ∪ γ1 ∈et1 ,δ1 ∈ei1 ,η1 ∈ fe1 ,γ2 ∈et2 ,δ2 ∈ei2 ,η2 ∈ fe2 ∪ γ1 ∈et1 ,δ1 ∈ei1 ,η1 ∈ fe1 ,γ2 ∈et2 ,δ2 ∈ei2 ,η2 ∈ fe2 ∪ nn γ∈et,δ∈ei,η ∈ fe ∪ nn γ∈et,δ∈ei,η ∈ fe {{γ1 + γ2 − γ1 γ2 }, {δ1 δ2 }, {η1 η2 }}; (4) {{γ1 γ2 }, {δ1 + δ2 − δ1 δ2 }, {η1 + η2 − η1 η2 }}; (5) o n o n oo 1 − (1 − γ ) λ , δ λ , η λ ; (6) o n o n oo γ λ , 1 − (1 − δ ) λ , 1 − (1 − η ) λ . o n e) of n e is given by e = et, ei, fe be an SVNHFE, then the score function s(n Definition 6. Ref. [18] Let n e) = s(n 1 l 1 q 1 p γi + ∑i=1 (1 − δi ) + ∑i=1 (1 − ηi ) ∑ i = 1 l p q 3, (7) (8) e) is limited to [0, 1]. where l, p, q are the numbers of values in et, ei, fe, respectively. Obviously, the range of s(n o n n o e1 = et1 , ei1 , fe1 and n e2 = et2 , ei2 , fe2 be two SVNHFEs, then the comparison Definition 7. Ref. [18] Let n method of them is expressed by (1) (2) (3) e2 ), then n e1 > n e2 ; e1 ) > s(n If s(n e2 ), then n e1 < n e2 ; e1 ) < s(n If s(n e1 ) = s(n e2 ), then n e1 = n e2 . If s(n Information 2018, 9, 10 5 of 19 2.4. The Prioritized Average Operator Aggregation operators play an important role in group decision-making to fusion the evaluation information. In view of priority relations between the criteria, Yager [21] developed the PA operator to solve this problem. Definition 8. Ref. [21] Let C = {C1 , C2 , . . . , Cn } be a collection of criteria, and priority relations between the criteria exist which can be expressed by the ordering of C1 ≻ C2 ≻ C3 ≻ · · · ≻ Cn . That means criteria Cj has a higher priority level than criteria Ck if j < k. The value Cj ( x ) is the evaluation information of alternative x with respect to criteria Cj . Thus, if n PA Cj ( x ) = ∑ j=1 w j Cj ( x ), (9) j −1 then the function PA is called the prioritized average (PA) operator, where w j = Tj /∑nj=1 Tj ,Tj = ∏k=1 Ck ( x ), T1 = 1. 3. Generalized Single-Valued Neutrosophic Hesitant Fuzzy Prioritized Aggregation Operators The PA operator can effectively solve the decision-making problems that the criteria have different priorities; however, it can only be used in the situation where the aggregated arguments are exact values. Combined with the PA operator and the generalized mean operators [26], we extend the PA operator to deal with the decision-making problems under SVNHF environment. In this section, the GSVNHFPWA operator and GSVNHFPWG operator are proposed, and their properties are presented simultaneously. Besides, several special cases of the GSVNHFPWA operator and GSVNHFPWG operator are also discussed through changing the values of the parameter λ. 3.1. Generalized Single-Valued Neutrosophic Hesitant Fuzzy Prioritized Average Operator n o ej = et j , ei j , fej ( j = 1, 2, . . . , n) be a collection of SVNHFEs, and let GSVNHFPWA : Definition 9. Let n Ωn → Ω , if e1 , n e2 , . . . , n en ) = GSVNHFPWAλ (n T1 ∑nj=1 Tj e1λ n ⊕ T2 ∑nj=1 Tj e2λ n ⊕···⊕ T1 ∑nj=1 Tj j −1 eλn n !1/λ , (10) ek )( j = 2, . . . , n), then the function GSVNHFPWA is called the GSVNHFPWA operator. Where Tj = ∏k=1 s(n ek ) is the score function value of SVNHFE n ek . T1 = 1, and s(n According to the operational laws of SVHFEs in Definition 5, we can obtain the theorem as follows. n o ej = et j , ei j , fej ( j = 1, 2, . . . , n) be a collection of SVNHFEs, then their aggregated value by Theorem 1. Let n using the GSVNHFPWA operator is also an SVNHFE, and e1 , n e2 , . . . , n en ) = GSVNHFPWAλ (n = ∪ e1 ∈et1 ,e γ γ2 ∈et2 ,...,e γn ∈etn ,δe1 ∈ei1 ,δe2 ∈ei2 ,...,δen ∈ein ,e η1 ∈ fe1 ,e η2 ∈ fe2 ,...,e ηn ∈ fen j −1 T1 ∑nj=1 Tj e1λ ⊕ n T2 ∑nj=1 Tj e2λ ⊕ · · · ⊕ n Tn ∑nj=1 Tj eλn n !1/λ  1/λ   1/λ   Tj T          njT n n λ ∑nj=1 Tj  1 − ∏ 1 − γ λ ∑ j =1 j    , 1 − 1 − ∏ 1 − 1 − δj , j    j =1 j =1             1/λ  Tj      n n  λ ∑ j=1 Tj    1 − 1 − η 1− 1 − ∏ j    .  j =1        ek )( j = 2, . . . , n), T1 = 1, and s(n ek ) is the score function value of SVNHFE n ek . where Tj = ∏k=1 s(n (11) Information 2018, 9, 10 6 of 19 Proof. We can use mathematical induction to prove the Theorem 1: (a) For n = 1, since T1 e1 ) = GSVNHFPWAλ (n (b) ∑nj=1 Tj e1λ n !1/λ = T1 λ e n T1 1 1/λ e1 . =n Obviously, Equation (11) holds for n = 1. For n = 2, since e1λ = n e2λ = n Then T1 ∑nj=1 Tj e1λ = n T2 eλ n ∑nj=1 Tj 2 = We have ∪ γ1 ∈et1 ,δ1 ∈ei1 ,η1 ∈ fe1 ∪ γ2 ∈et2 ,δ2 ∈ei2 ,η2 ∈ fe2 1 − 1 − γ1λ ∪ (( ∑n 1 Tj 1 − γ2λ ∑n 2 Tj γ1 ∈et1 ,δ1 ∈ei1 ,η1 ∈ fe1 γ2 ∈et2 ,δ2 ∈ei2 ,η2 ∈ fe2 1 − (1 − δ1 )λ 1 − (1 − δ1 )λ (( T1 T ∑n j =1 j = ( o n o n oo γ2λ , 1 − (1 − δ2 )λ , 1 − (1 − η2 )λ , (( ∪ nn o n o n oo γ1λ , 1 − (1 − δ1 )λ , 1 − (1 − η1 )λ , ∪ γ1 ∈et1 ,δ1 ∈ei1 ,η1 ∈ fe1 ,γ2 ∈et2 ,δ2 ∈ei2 ,η2 ∈ fe2 ( nn T j =1 T 1− j =1 T 1− ! 1 − γ1λ ) ( , ) ( , ∑n 1 Tj j =1 1 − (1 − δ2 ) λ T1 T ∑n j =1 j T2 T ∑n j =1 j 1 − (1 − δ2 )λ 1 − (1 − δ2 )λ ) ( , ) ( , +1− T2 T ∑n j =1 j T2 T ∑n j =1 j 1 − γ2λ !) ( , (( ∑n 2 Tj j =1 − 1− 1 − ( 1 − η1 ) λ 1 − γ1λ T1 T ∑n j =1 j ! T 1 − γ1λ 1− !) ( , 1 − ( 1 − η1 ) λ 1 − ( 1 − η2 ) T T ∪ ! 1 − (1 − δ1 )λ T1 T eλ ⊕ n 2 n eλ = n ∑nj=1 Tj 1 ∑ j=1 Tj 2 γ1 ∈et1 ,δ1 ∈ei1 ,η1 ∈ fe1 ,γ2 ∈et2 ,δ2 ∈ei2 ,η2 ∈ fe2 T1 T ∑n j =1 j ∑n 1 Tj j =1 T1 T ∑n j =1 j ∑n 2 Tj j =1 ! ! ) T1 T ∑n j =1 j T2 T ∑n j =1 j )) , )) . !) , !!1/λ   . T 1− 1 − γ2λ 1 − ( 1 − η2 ) λ T 1 − γ2λ j =1 1 − ( 1 − η1 ) λ ∑n 1 Tj λ ∑n 2 Tj T2 T ∑n j =1 j T2 T ∑n j =1 j j =1 !) . !) . , 1 − ( 1 − η2 ) λ Thus e1 , n e1 ) = GSVNHFPWAλ (n    1− 1− 1 − (1 − δ1 ) λ T1 ∑nj=1 Tj e1λ ⊕ n T2 ∑nj=1 Tj e2λ n !1/λ =  1/λ     T2 T1         n n ∑ j=1 Tj ∑ j=1 Tj  λ λ   1 − γ2 , 1 − 1 − γ1 ∪     γ1 ∈et1 ,δ1 ∈ei1 ,η1 ∈ fe1 ,γ2 ∈et2 ,δ2 ∈ei2 ,η2 ∈ fe2          T1 T ∑n j =1 j ! 1 − (1 − δ2 ) i.e., Equation (11) holds for n = 2. λ T2 T ∑n j =1 j !!1/λ     , 1−   1− 1 − ( 1 − η1 ) λ T1 T ∑n j =1 j ! 1 − ( 1 − η2 ) λ T2 T ∑n j =1 j  Information 2018, 9, 10 (c) 7 of 19 If Equation (11) holds for n = k, we have T1 ∑nj=1 e1 , n e2 , . . . , n ek ) = GSVNHFPWAλ (n = Tj e1λ ⊕ n T2 ∑nj=1 Tj e2λ ⊕ · · · ⊕ n Tk ∑nj=1 Tj eλk n !1/λ  1/λ   1/λ   Tj Tj               ∑n Tj ∑n Tj k k j = 1 j = 1 λ     λ 1 − 1 − γ , , 1 − 1 − 1 − δ 1 − ∏ ∏     j j      j =1 j =1          ∪ e1 ∈et1 ,e γ γ2 ∈et2 ,...,e γk ∈etk ,δe1 ∈ei1 ,δe2 ∈ei2 ,...,δek ∈eik ,e η1 ∈ fe1 ,e η2 ∈ fe2 ,...,e ηk ∈ fek 1/λ  Tj      ∑n Tj j =1 λ   . 1 − 1 − ∏ 1 − 1 − η j     j = 1           k When n = k + 1, combined with the operations of SVNHFE in Definition 5, we have T1 T T T eλ ⊕ n 2 n eλ ⊕ · · · ⊕ n k n eλ ⊕ nk+1 n eλ = n ∑nj=1 Tj 1 ∑ j=1 Tj 2 ∑ j=1 Tj k ∑ j=1 Tj k+1 =       ∪  e1 ∈et1 ,e γ γ2 ∈et2 ,...,e γk ∈etk ,δe1 ∈ei1 ,δe2 ∈ei2 ,...,δek ∈eik ,e η1 ∈ fe1 ,e η2 ∈ fe2 ,...,e ηk ∈ fek           ∪  γk+1 ∈etk+1 ,δk+1 ∈eik+1 ,ηk+1 ∈ fek+1     = k 1 − ∏ 1 − γλj 1 − 1 − γkλ+1 j =1 Tj ∑nj=1 Tj      Tj   Tj         k   λ ∑nj=1 Tj   k λ ∑nj=1 Tj  , ∏ 1 − 1 − δj , ∏ 1 − 1 − ηj ⊕      j = 1 j = 1                Tk+1   Tk+1   Tk+1             n n n ∑ j=1 Tj ∑ j=1 Tj ∑ j=1 Tj , , 1 − (1 − δk+1 )λ 1 − (1 − ηk +1 ) λ                       Tj Tj nTj  k+1 k +1 λ ∑n Tj  k+1 λ ∑n Tj  ∑ j=1 Tj j =1 j =1 1 − ∏ 1 − γλj , ∏ 1 − 1 − δj , ∏ 1 − 1 − ηj .   j =1   j =1  e1 ∈et1 ,e j =1 γ γ2 ∈et2 ,...,e γk+1 ∈etk+1 ,δe1 ∈ei1 ,δe2 ∈ei2 ,...,δek+1 ∈eik+1 ,e η1 ∈ fe1 ,e η2 ∈ fe2 ,...,e ηk+1 ∈ fek+1     ∪ Then e1 , n e2 , . . . , n ek+1 ) = GSVNHFPWAλ (n T1 ∑nj=1 Tj e1λ ⊕ n T2 ∑nj=1 Tj e2λ ⊕ · · · ⊕ n Tk+1 λ e n ∑nj=1 Tj k+1 !1/λ  1/λ   1/λ   Tj Tj               ∑n Tj ∑n Tj k + 1 k + 1 j =1 j =1 λ     , 1 − 1 − ∏ 1 − 1 − δj , = ∪ 1 − ∏ 1 − γλj       e1 ∈et1 ,e j = 1 j = 1 γ γ2 ∈et2 ,...,e γk+1 ∈etk+1 ,δe1 ∈ei1 ,δe2 ∈ei2 ,...,δek+1 ∈eik+1 ,e η1 ∈ fe1 ,e η2 ∈ fe2 ,...,e ηk+1 ∈ fek+1               1/λ  Tj      ∑n Tj j =1 λ   1 − 1 − ∏ 1 − 1 − η j .     j = 1       k +1 i.e., Equation (11) holds for n = k + 1, thus we can confirm Equation (11) holds for all n. The proof of Theorem 1 is completed. Some desirable properties of the GSVNHFPWA operator are presented as below. ej = Theorem 2. (Idempotency) Let n j −1 n o et j , ei j , fej ( j = 1, 2, . . . , n) be a collection of SVNHFEs, where ek . If all e )( j = 2, . . . , n), T1 = 1, and s(n ek ) is the score function value of SVNHFE n Tj = ∏k=1 s(n n o ok n e e e e e e e e e ej = t j , i j , f j ( j = 1, 2, . . . , n) are equal, i.e., n ej = n e = t, i, f , t = γ, i = δ, and f = η, then n o n e1 , n e2 , . . . , n en ) = n e = et, ei, fe . GSVNHFPWAλ (n (12) Information 2018, 9, 10 8 of 19 o n ej = n e = et, ei, fe , by Theorem 1, we have Proof. Since n e1 , n e2 , . . . , n en ) = GSVNHFPWAλ (n T1 ∑nj=1 Tj eλ ⊕ n T2 ∑nj=1 Tj eλ ⊕ · · · ⊕ n Tn ∑nj=1 Tj eλ n !1/λ    1/λ   1/λ   1/λ  T T Tj               ∑n j T ∑n j T n n n ∑nj=1 Tj j j λ λ j = 1 j = 1 λ 1 − ∏ 1 − γ    = ∪ , 1 − 1 − ∏ 1 − (1 − δ ) , 1 − 1 − ∏ 1 − (1 − η )        e e e e e∈t,δ∈i,e j =1 j =1 j =1 γ η ∈ f         1/λ    1/λ    1/λ  T ∑n T ∑n T ∑n     j =1 j           nj=1 j nj=1 j n T T T ∑ ∑ ∑  1 − 1 − γ λ j =1 j  , 1 −  1 − 1 − ( 1 − δ ) λ j =1 j  , 1 −  1 − 1 − ( 1 − η ) λ j =1 j  = ∪    e∈et,δe∈ei,e γ η ∈ fe           = n ∪ e∈et,δe∈ei,e γ η ∈ fe γλ 1/λ o 1/λ 1/λ , 1 − (1 − δ ) λ , 1 − (1 − η ) λ = Then, the proof of Theorem 2 is completed. n o e = et, ei, fe . {{γ}, {δ} , {η }} = n ∪ e∈et,δe∈ei,e γ η ∈ fe n o ej = et j , ei j , fej ( j = 1, 2, . . . , n) be a collection of SVNHFEs, where (Boundedness) Let n Theorem 3. j −1 e. ek )( j = 2, . . . , n), T1 = 1, and s(n ek ) is the score function value of SVNHFE n Tj = ∏k=1 s(n k e− = {{γ− }, {δ+ }, {η + }} and n e+ = {{γ+ }, {δ− }, {η − }}, where γ+ = ∪γ ∈et max γ j , And let n j j δ+ = ∪δ ∈ei max δj , η + = ∪η ∈ fe max η j , γ− = ∪γ ∈et min γ j , δ− = ∪δ ∈ei min δj , j j j j j j j j and η − = ∪η ∈ fe min η j . Then j j e− ≤ GSVNHFPWAλ (n e1 , n e2 , . . . , n en ) ≤ n e+ . n (13) Proof. Since γ− ≤ γ j ≤ γ+ , δ− ≤ δj ≤ δ+ , and η − ≤ η j ≤ η + . First, when λ ∈ (0, ∞), then γλj ≥ γ − λ , 1 − γλj n ∏ j =1 n 1−∏ j =1  n 1 − γλj ∑n 1 − γλj ∑n ∑n ∑n  λ 1 − ∏ 1 − γ j j =1 ≤ 1− γ Tj T j =1 j Similarly, we have  n  λ 1 − ∏ 1 − γ j j =1 Tj T j =1 j And as δ− ≤ δj ≤ δ+ , then 1 − δj ≤ 1 − δ − , 1 − δj T j =1 j 1/λ λ 1 − γλj n n ≤ 1− γ ∑nj=1 Tj − λ ≥ 1−∏ 1− γ  n  ≥ 1 − ∏ j =1  n  ≤ 1 − ∏ j =1 ≤ 1 − δ− Tj T j =1 j λ Tj j =1 λ ∑n Tj T ∑n j =1 j ≤ ∏ 1− γ j =1 Tj T j =1 j     , Tj 1/λ 1 − 1 − δj − λ − λ − λ Tj T ∑n j =1 j , , Tj T j =1 j ∑n , 1/λ Tj n λ ∑ j=1 Tj  1 − γ− = γ− .  1/λ Tj n T λ ∑ j =1 j  1 − γ+ = γ+ .  , 1 − 1 − δj ≥ 1− 1−δ λ − λ ≥ 1 − 1 − δ− Tj T ∑n j =1 j , λ , Information 2018, 9, 10 9 of 19 n ∏ j =1 n Tj 1 − 1 − δj   n  1 − ∏ j =1 n  1 − 1 − ∏ j =1 n ≥ ∏ 1− 1−δ j =1 Tj 1 − ∏ 1 − 1 − δj j =1 λ ∑nj=1 Tj λ ∑nj=1 Tj n − λ , − λ ∑n ≤ 1−∏ 1− 1−δ j =1 Tj T j =1 j ∑n Tj T j =1 j , 1/λ  1/λ Tj Tj n n n λ ∑ j=1 Tj  λ ∑ j=1 Tj   1 − 1 − δj ≤ 1 − ∏ 1 − 1 − δ −   , j =1 1/λ 1/λ  Tj Tj ∑n Tj n λ ∑nj=1 Tj  λ j =1   ≥ 1 − 1 − ∏ 1 − 1 − δ − = δ− . 1 − 1 − δj   j =1 Similarly, we have  n  1 − 1 − ∏ j =1 1/λ 1/λ  Tj Tj n n n λ ∑ j=1 Tj  λ ∑ j=1 Tj   ≤ 1 − 1 − ∏ 1 − 1 − δ + = δ+ . 1 − 1 − δj   j =1 On the other hand,  n  η − ≤ 1 − 1 − ∏ j =1 1/λ Tj λ ∑nj=1 Tj  ≤ η+. 1 − 1 − δj  e1 , n e2 , . . . , n en ) = n e = {{γ}, {δ}, {η }}, then Let GSVNHFPWAλ (n e) = s(n 1 l l ∑ i =1 p q γi + 1p ∑i=1 (1 − δi ) + 1q ∑i=1 (1 − ηi ) 3 ≥ 1 l− l − ∑ i =1 γi− + p− 1 p − ∑ i =1 1 − δi+ + q− 1 q − ∑ i =1 1 − ηi+ 1 − δi− + q+ 1 q + ∑ i =1 1 − ηi− 3 And e) = s(n 1 l l ∑ i =1 p q γi + 1p ∑i=1 (1 − δi ) + 1q ∑i=1 (1 − ηi ) 3 ≤ 1 l+ l + ∑ i =1 γi+ + p+ 1 p + ∑ i =1 3 e) < s ( n e+ ), we have e− ) < s(n If s(n e) = s ( n e− ), i.e., If s(n 1 l l ∑ i =1 p e− ), = s(n e+ ). = s(n e− < GSVNHFPWAλ (n e1 , n e2 , . . . , n en ) < n e+ . n q γi + 1p ∑i=1 (1 − δi ) + 1q ∑i=1 (1 − ηi ) 3 = 1 l− l − ∑ i =1 γi− + p− 1 p − ∑ i =1 1 − δi+ + q− 1 q − ∑ i =1 1 − ηi+ 1 − δi− + q+ 1 q + ∑ i =1 1 − ηi− 3 , , Then e) = s ( n e− ), i.e., If s(n 1 l l ∑ i =1 p e1 , n e2 , . . . , n en ) = n e− . GSVNHFPWAλ (n q γi + 1p ∑i=1 (1 − δi ) + 1q ∑i=1 (1 − ηi ) 3 ≤ 1 l+ l + ∑ i =1 γi+ + p+ 1 p + ∑ i =1 3 Information 2018, 9, 10 10 of 19 Then e1 , n e2 , . . . , n en ) = n e+ . GSVNHFPWAλ (n Based on analysis above, we have e− ≤ GSVNHFPWAλ (n e1 , n e2 , . . . , n en ) ≤ n e+ λ ∈ (0, ∞). n Similarly, we can obtain e− ≤ GSVNHFPWAλ (n e1 , n e2 , . . . , n en ) ≤ n e+ λ ∈ (−∞, 0). n The proof of Theorem 3 is completed. n n o o et j , ei j , fej ( j = 1, 2, . . . , n) and n e∗j = et∗j , ei∗j , fej∗ ( j = 1, 2, . . . , n) j −1 j −1 ek )( j = 2, . . . , n), Tj∗ = ∏k=1 s n e∗k ( j = 2, . . . , n), be two collections of SVNHFEs, where Tj = ∏k=1 s(n ek and n e∗k , respectively. If n ej ≤ n e∗j ek ) and s n e∗k are the score values of SVNHFE n T1 = T1∗ = 1, s(n ( j = 1, 2, . . . , n), then ej = Theorem 4. (Monotonicity) Let n e1 , n e2 , . . . , n en ) ≤ GSVNHFPWAλ (n e1∗ , n e2∗ , . . . , n e∗n ). GSVNHFPWAλ (n (14) Proof. It directly follows from Theorem 3. Special cases of the GSVNHFPWA operator are shown as follows. (1) (2) (3) (4) If λ = 1, then the GSVNHFPWA operator is reduced to the single-valued neutrosophic hesitant fuzzy prioritized weighted average (SVNHFPWA) operator: e1 , n e2 , . . . , n en ) = SVNHFPWA(n T1 ∑nj=1 Tj e1 ⊕ n T2 ∑nj=1 Tj e2 ⊕ · · · ⊕ n Tn ∑nj=1 Tj ! en . n (15) If λ → 0 , then the GSVNHFPWA operator is reduced to the single-valued neutrosophic hesitant fuzzy prioritized weighted geometric (SVNHFPWG) operator: e1 , n e2 , . . . , n en ) = SVNHFPWG(n e1 ) (n T1 T ∑n j =1 j e2 ) ⊗ (n T2 T ∑n j =1 j en ) ⊗ · · · ⊗ (n Tn T ∑n j =1 j ! . (16) If λ = 2, then the GSVNHFPWA operator is reduced to the single-valued neutrosophic hesitant fuzzy prioritized weighted quadratic average (SVNHFPWQA) operator: e1 , n e2 , . . . , n en ) = SVNHFPWQA(n T1 ∑nj=1 Tj e21 n ⊕ T2 ∑nj=1 Tj e22 n ⊕···⊕ Tn ∑nj=1 Tj e2n n !1/2 . (17) If λ = 3, then the GSVNHFPWA operator is reduced to the single-valued neutrosophic hesitant fuzzy prioritized weighted cubic average (SVNHFPWCA) operator: e1 , n e2 , . . . , n en ) = SVNHFPWCA(n T1 ∑nj=1 Tj e31 n ⊕ T2 ∑nj=1 Tj e32 n ⊕···⊕ Tn ∑nj=1 Tj e3n n !1/3 . (18) Information 2018, 9, 10 11 of 19 If λ = 1 and the aggregated arguments are in the same priority level, then the GSVNHFPWA operator is reduced to the single-valued neutrosophic hesitant fuzzy weighted average (SVNHFWA) operator [14]: (5) e1 , n e2 , . . . , n en ) = (w1 n e1 ⊕ w2 n e2 ⊕ · · · ⊕ wn n en ). SVNHFWA(n (19) If λ → 0 and the aggregated arguments are in the same priority level, then the GSVNHFPWA operator is reduced to the single-valued neutrosophic hesitant fuzzy weighted geometric (SVNHFWG) operator [14]: (6) n e1 , n e2 , . . . , n en ) = n e1w1 ⊗ n ew e2w2 ⊗ · · · ⊗ n SVNHFWG(n n . (20) If w = (1/n, 1/n, . . . , 1/n) T , λ = 1, and the aggregated arguments are in the same priority level, then the GSVNHFPWA operator is reduced to the single-valued neutrosophic hesitant fuzzy arithmetic average (SVNHFAA) operator: (7) e1 , n e2 , . . . , n en ) = SVNHFAA(n 1 e ⊕n e2 ⊕ · · · ⊕ n en ). (n n 1 (21) If w = (1/n, 1/n, . . . , 1/n) T , λ → 0 , and the aggregated arguments are in the same priority level, then the GSVNHFPWA operator is reduced to the single-valued neutrosophic hesitant fuzzy geometric average (SVNHFGA) operator: (8) e1 , n e2 , . . . , n en ) = (n e1 ⊗ n e2 ⊗ · · · ⊗ n en )1/n . SVNHFGA(n (22) 3.2. Generalized Single-Valued Neutrosophic Hesitant Fuzzy Prioritized Geometric Operator Based on the GSVNHFPWA operator investigated above, we develop the GSVNHFPWG operator as the following. n o ej = et j , ei j , fej ( j = 1, 2, . . . , n) be a collection of SVNHFEs, and let GSVNHFPWG : Definition 10. Let n Ωn → Ω , if T1 T2 Tn 1 ∑n T ∑n T ∑n T e1 , n e2 , . . . , n en ) = GSVNHFPWGλ (n n1 ) j=1 j ⊗ (λe n2 ) j=1 j ⊗ · · · ⊗ (λe n n ) j =1 j (λe λ ! , (23) j −1 ek )( j = 2, . . . , n), then the function GSVNHFPWG is called the GSVNHFPWG operator. Where Tj = ∏k=1 s(n ek ) is the score function value of SVNHFE n ek . T1 = 1, and s(n Similarly, according to the operations of SVHFEs in Definition 5, the theorem is obtained as below. n o ej = et j , ei j , fej ( j = 1, 2, . . . , n) be a collection of SVNHFEs, then their aggregated value by Theorem 5. Let n using the GSVNHFPWG operator is also an SVNHFE, and   T1 T2 Tn n n n T  1 T T ∑ ∑ ∑ e1 , n e2 , . . . , n en ) =  n1 ) j=1 j ⊗ (λe GSVNHFPWGλ (n n2 ) j=1 j ⊗ · · · ⊗ (λe n n ) j =1 j  (λe  λ = ∪ e1 ∈et1 ,e γ γ2 ∈et2 ,...,e γn ∈etn ,δe1 ∈ei1 ,δe2 ∈ei2 ,...,δen ∈ein ,e η1 ∈ fe1 ,e η2 ∈ fe2 ,...,e ηn ∈ fen j −1         n 1 − 1 − ∏ 1 − 1 − γ j j =1 Tj λ ∑nj=1 Tj  1/λ  T      njT n  1 − ∏ 1 − η λ ∑ j =1 j  . j   j =1    1/λ   1/λ  T       ∑n j T n j  , 1 − ∏ 1 − δjλ j=1  ,    j =1    ek )( j = 2, . . . , n), T1 = 1, and s(n ek ) is the score function value of SVNHFE n ek . where Tj = ∏k=1 s(n (24) Information 2018, 9, 10 12 of 19 Proof. The proof procedure of Theorem 5 is similar to Theorem 1. Some desirable properties of the GSVNHFPWG operator are presented as below. ej = Theorem 6. (Idempotency) Let n j −1 n o et j , ei j , fej ( j = 1, 2, . . . , n) be a collection of SVNHFEs, where ek . If all e )( j = 2, . . . , n), T1 = 1, and s(n ek ) is the score function value of SVNHFE n Tj = ∏k=1 s(n o n n ok e e e e e e e e e ej = t j , i j , f j ( j = 1, 2, . . . , n) are equal, i.e., n ej = n e = t, i, f , t = γ, i = δ, and f = η, then n o n e1 , n e2 , . . . , n en ) = n e = et, ei, fe . GSVNHFPWGλ (n (25) Proof. The proof procedure of Theorem 6 is similar to Theorem 2. Theorem 7. ej (Boundedness) Let n j −1 = n o et j , ei j , fej ( j = 1, 2, . . . , n) be a collection of SVNHFEs, ek )( j = 2, . . . , n), T1 = 1, and s(n ek ) is the score value of SVNHFE n e. where Tj = ∏k=1 s(n k e− = {{γ− }, {δ+ }, {η + }} and n e+ = {{γ+ }, {δ− }, {η − }}, where γ+ = ∪γ ∈et max γ j , And let n j j δ+ = ∪δ ∈ei max δj , η + = ∪η ∈ fe max η j , γ− = ∪γ ∈et min γ j , δ− = ∪δ ∈ei min δj , and j j j j j j j j η − = ∪η ∈ fe min η j . Then j j e− ≤ GSVNHFPWGλ (n e1 , n e2 , . . . , n en ) ≤ n e+ . n (26) Proof. The proof procedure of Theorem 7 is similar to Theorem 3. n n o o et j , ei j , fej ( j = 1, 2, . . . , n) and n e∗j = et∗j , ei∗j , fej∗ ( j = 1, 2, . . . , n) j −1 j −1 ek )( j = 2, . . . , n), Tj∗ = ∏k=1 s n e∗k ( j = 2, . . . , n), be two collections of SVNHFEs, where Tj = ∏k=1 s(n ek ) and s n e∗k are the score function values of SVNHFE n ek and n e∗k , respectively. T1 = T1∗ = 1, s(n ∗ ej ≤ n ej ( j = 1, 2, . . . , n), then If n ej = Theorem 8. (Monotonicity) Let n e1 , n e2 , . . . , n en ) ≤ GSVNHFPWGλ (n e1∗ , n e2∗ , . . . , n e∗n ). GSVNHFPWGλ (n (27) Proof. It directly follows from Theorem 7. Special cases of the GSVNHFPWG operator are shown as follows: (1) (2) (3) If λ = 1, then the GSVNHFPWG operator is reduced to the single-valued neutrosophic hesitant fuzzy prioritized weighted geometric (SVNHFPWG) operator: e1 , n e2 , . . . , n en ) = SVNHFPWG(n e1 ) (n T1 T ∑n j =1 j e2 ) ⊗ (n T2 T ∑n j =1 j en ) ⊗ · · · ⊗ (n Tn T ∑n j =1 j ! . (28) If λ = 1 and the aggregated arguments are in the same priority level, then the GSVNHFPWG operator is reduced to the SVNHFWG operator [14]: e2 )w2 ⊗ · · · ⊗ (n en )wn . e1 , n e2 , . . . , n en ) = (n e1 )w1 ⊗ (n SVNHFWG(n (29) If w = (1/n, 1/n, . . . , 1/n) T , λ = 1, and the aggregated arguments are in the same priority level, then the GSVNHFPWG operator is reduced to the SVNHFGA operator: e1 , n e2 , . . . , n en ) = (n e1 ⊗ n e2 ⊗ · · · ⊗ n en )1/n . SVNHFGA(n (30) Information 2018, 9, 10 13 of 19 4. An Approach for Decision-Making under Single-Valued Neutrosophic Hesitant Fuzzy Environment In this section, we utilize the GSVNHFPWA operator and GSVNHFPWG operator to solve the MCDM problems under SVNHF environment, respectively. For a MCDM problem, let A = { A1 , A2 , . . . Am } be a set of m alternatives to be evaluated, C = {C1 , C2 , . . . , Cn } be a collection of criteria that prioritizations between the criteria expressed by the linear ordering C1 ≻ C2 ≻ · · · ≻ Cn exist, i.e., criteria Cj has a higher priority level than the criteria Ck if j < k. Decision makers evaluates eij m×n (i = 1, 2, . . . , m; j = 1, 2, . . . , n) the alternatives over the criteria by using SVNHFEs, let N = n n o eij = etij , eiij , feij is the evaluation information given by be an SVNHF decision matrix, and n decision maker. Where etij = γij γij ∈ etij represents the possible degrees that the alternative o n Ai satisfies the criteria Cj provided by decision maker, eiij = δij δij ∈ eiij represents the possible indeterminacy degrees that decision maker judges whether the alternative Ai satisfies the criteria Cj , n o and feij = ηij ηij ∈ feij represents the possible degrees that the alternative Ai does not satisfy the criteria Cj provided by decision maker. Based on the assumptions above, we use the GSVNHFPWA operator or GSVNHFPWG operator to construct an approach for decision-making under SVNHF environment. The main steps are presented below. Step 1. Calculate the values of Tij (i = 1, 2, . . . , m; j = 1, 2, . . . , n) by the equations as follows. Tij = j −1 ∏k=1 s(neik )(i = 1, 2, . . . , m; j = 1, 2, . . . , n), Ti1 = 1. (31) Step 2. Utilize the GSVNHFPWA operator: Ti1 ei = GSVNHFPWAλ (n ei1 , n ei2 , . . . , n ein ) = n ∑nj=1 Tij λ ei1 ) ⊕ (n Ti2 ∑nj=1 Tij λ ei2 ) ⊕ · · · ⊕ (n Tin ∑nj=1 Tij ein ) (n λ !1/λ  1/λ  Tij      n n T 1 − ∏ 1 − γij λ ∑ j=1 ij  , = ∪  ei1 ∈eti1 ,e j =1 γ γi2 ∈eti2 ,...,e γin ∈etin ,δei1 ∈eii1 ,δei2 ∈eii2 ,...,δein ∈eiin ,e ηi1 ∈ fei1 ,e ηi2 ∈ fei2 ,...,e ηin ∈ fein           n T 1 − 1 − ∏ 1 − 1 − δij j =1 ij λ ∑nj=1 Tij or the GSVNHFPWG operator: 1/λ   1/λ   Tij        n n λ ∑ j=1 Tij   , 1 − 1 − ∏ 1 − 1 − ηij .   j =1      ei = GSVNHFPWGλ (n ei1 , n ei2 , . . . , n ein ) = n = ∪ (32)  1 (λe ni1 λ Ti1 ∑nj=1 Tij )     ei1 ∈eti1 ,e γ γi2 ∈eti2 ,...,e γin ∈etin ,δei1 ∈eii1 ,δei2 ∈eii2 ,...,δein ∈eiin ,e ηi1 ∈ fei1 ,e ηi2 ∈ fei2 ,...,e ηin ∈ fein    ⊗ (λe ni2  Ti2 ∑nj=1 Tij ) n ⊗ · · · ⊗ (λe nin ) 1 − 1 − ∏ 1 − 1 − γij j =1 m×n Tij λ ∑nj=1 Tij  1/λ   1/λ  Tij Tij          n n λ ∑nj=1 Tij λ ∑nj=1 Tij   1 − ∏ 1 − δij , 1 − ∏ 1 − ηij .    j =1 j =1       eij to aggregate the SVNHF decision matrix N = n each alternative. Tin T ∑n j=1 ij     1/λ     ,   (33) n o ei = eti , eii , fei of into the SVNHFE n Information 2018, 9, 10 14 of 19 n o ei = eti , eii , fei Step 3. Rank all the alternatives by calculating the score function value of the SVNHFE n combined with Definition 6. 1 1 1 ei ) = s(n γ + 3. (34) (1 − δi ) + ∑η ∈ fe (1 − ηi ) i i li ∑γi ∈eti i pi ∑δi ∈eii qi ei ), the higher the ranking of alternative xi will be. Then the bigger the score function value s(n 5. Numerical Example In this section, we apply a numerical example of MCDM problem under SVNHF environment to illustrate the applications and advantages of the proposed method [14]. 5.1. Implementation Suppose that an investment company wants to invest a sum of money in a target company. After a market survey, four alternative companies are identified to be chosen from, namely, a car company (A1 ), a food company (A2 ), a computer company (A3 ), and an arms company (A4 ). To evaluate the investment potential of a company needs to consider many aspects, such as the growth prospects of the company, risk degree of the investment, and the impact of the company on the environment. Therefore, the investment company shall evaluate the four alternative companies above with respect to three criteria, namely, the environmental impact (C1 ), the risk (C2 ), and the growth (C3 ). In the real decision-making process, compared with determining the weights of criteria, identifying the priority level of criteria is more feasible and accurate. Then, according to the weight vector of three criteria w = (0.40, 0.35, 0.25) T [14], we set up the criteria C1 with the first priority level, followed by criteria C2 and C3 . Decision makers from the investment company express the evaluation eij m×n is obtained information combined with SVNHFEs, and the SVNHF decision matrix N = n shown in Table 1 [14]. Table 1. SVNHF decision matrix. Alternatives C1 C2 C3 A1 A2 A3 A4 {{0.2, 0.3}, {0.1, 0.2}, {0.5, 0.6}} {{0.6, 0.7}, {0.1, 0.2}, {0.1, 0.2}} {{0.5, 0.6}, {0.1}, {0.3}} {{0.3, 0.5}, {0.2}, {0.1, 0.2, 0.3}} {{0.3, 0.4, 0.5}, {0.1}, {0.3, 0.4}} {{0.6, 0.7}, {0.1, 0.2}, {0.2, 0.3}} {{0.5, 0.6}, {0.4}, {0.2, 0.3}} {{0.7, 0.8}, {0.1}, {0.1, 0.2}} {{0.5, 0.6}, {0.2, 0.3}, {0.3, 0.4}} {{0.6, 0.7}, {0.1}, {0.3}} {{0.6}, {0.3}, {0.4}} {{0.6, 0.7}, {0.1}, {0.2}} Then, we use the proposed method to determine the ranking result of the four alternative companies, which are presented as follows. Step 1. Calculate the values of Tij (i = 1, 2, 3, 4; j = 1, 2, 3) according to Equation (31) as follows:    Tij =   1.000 1.000 1.000 1.000 0.5167 0.7833 0.7167 0.6667 0.3358 0.5875 0.4539 0.5556    .  Step 2. Utilize the GSVNHFPWA operator (which the parameter λ = 1) to aggregate the SVNHF decision n o ei = eti , eii , fei (i = 1, 2, 3, 4) of eij matrix N = n (i = 1, 2, 3, 4; j = 1, 2, 3) into the SVNHFE n m×n each alternative company. Take the alternative company A1 for instance, we have Information 2018, 9, 10 15 of 19 e1 = GSVNHFPWA1 (n e11 , n e12 , n e13 ) = n = ∪ e11 ∈et11 ,e γ γ12 ∈et12 ,e γ13 ∈et13 ,δe11 ∈ei11 ,δe12 ∈ei12 ,δe13 ∈ei13 ,e η11 ∈ fe11 ,e η12 ∈ fe12 ,e η13 ∈ fe13 = nn T11 e11 )1 (n ∑3j=1 T1j ⊕ T12 e12 )1 (n ∑3j=1 T1j ⊕ T13 e13 )1 (n ∑3j=1 T1j !1/1   1/1   1/1  T1j T1j               ∑3 T ∑3 T 3 3 1 1   j=1 1j  j=1 1j  , 1 − 1 − ∏ 1 − 1 − δ1j ,   1 − ∏ 1 − γ1j      j = 1 j = 1         1/1  T1j    3 1 ∑3j=1 T1j    1 − 1 − ∏ 1 − 1 − η1j     j = 1           1 − (1 − 0.2)0.54 (1 − 0.3)0.28 (1 − 0.5)0.18 , 1 − (1 − 0.2)0.54 (1 − 0.3)0.28 (1 − 0.6)0.18 , 1 − (1 − 0.2)0.54 (1 − 0.4)0.28 (1 − 0.5)0.18 , 1 − (1 − 0.2)0.54 (1 − 0.4)0.28 (1 − 0.6)0.18 , 1 − (1 − 0.2)0.54 (1 − 0.5)0.28 (1 − 0.5)0.18 , 1 − (1 − 0.2)0.54 (1 − 0.5)0.28 (1 − 0.6)0.18 , 1 − (1 − 0.3)0.54 (1 − 0.3)0.28 (1 − 0.5)0.18 , 1 − (1 − 0.2)0.54 (1 − 0.3)0.28 (1 − 0.6)0.18 , 1 − (1 − 0.2)0.54 (1 − 0.4)0.28 (1 − 0.5)0.18 , o 1 − (1 − 0.3)0.54 (1 − 0.4)0.28 (1 − 0.6)0.18 , 1 − (1 − 0.2)0.54 (1 − 0.5)0.28 (1 − 0.5)0.18 , 1 − (1 − 0.2)0.54 (1 − 0.5)0.28 (1 − 0.6)0.18 , n 1 − 1 − (1 − (1 − 0.1))0.54 (1 − (1 − 0.1))0.28 (1 − (1 − 0.2))0.18 , 1 − 1 − (1 − (1 − 0.1))0.54 (1 − (1 − 0.1))0.28 (1 − (1 − 0.3))0.18 , o 1 − 1 − (1 − (1 − 0.2))0.54 (1 − (1 − 0.1))0.28 (1 − (1 − 0.2))0.18 , 1 − 1 − (1 − (1 − 0.2))0.54 (1 − (1 − 0.1))0.28 (1 − (1 − 0.3))0.18 , n 1 − 1 − (1 − (1 − 0.5))0.54 (1 − (1 − 0.3))0.28 (1 − (1 − 0.3))0.18 , 1 − 1 − (1 − (1 − 0.5))0.54 (1 − (1 − 0.3))0.28 (1 − (1 − 0.4))0.18 , 1 − 1 − (1 − (1 − 0.5))0.54 (1 − (1 − 0.4))0.28 (1 − (1 − 0.3))0.18 , 1 − 1 − (1 − (1 − 0.5))0.54 (1 − (1 − 0.4))0.28 (1 − (1 − 0.4))0.18 , 1 − 1 − (1 − (1 − 0.6))0.54 (1 − (1 − 0.3))0.28 (1 − (1 − 0.3))0.18 , 1 − 1 − (1 − (1 − 0.6))0.54 (1 − (1 − 0.3))0.28 (1 − (1 − 0.4))0.18 , o 1 − 1 − (1 − (1 − 0.6))0.54 (1 − (1 − 0.4))0.28 (1 − (1 − 0.3))0.18 , 1 − 1 − (1 − (1 − 0.6))0.54 (1 − (1 − 0.4))0.28 (1 − (1 − 0.4))0.18 . e1 as the following. and obtain the SVNHFE n e1 = {{0.2922, 0.3203, 0.3220, 0.3414, 0.3489, 0.3556, 0.3675, 0.3691, 0.3811, 0.3941, n 0.4004, 0.4242}, {0.1134, 0.1220, 0.1648, 0.1774}, {0.3953, 0.4164, 0.4283, 0.4512, 0.4361, 0.4595, 0.4726, 0.4979}}. Similarly, the SVNHFEs of other alternative companies can be computed as follows: e2 = {{0.6000, 0.6275, 0.6363, 0.6613, 0.6457, 0.6701, 0.6778, 0.7000}, {0.1000, 0.1257, n 0.1340, 0.1684}, {0.1651, 0.1887, 0.2211, 0.2528}}; e3 = {{0.5228, 0.5567, 0.5694, 0.6000}, {0.1989}, {0.2787, 0.3186}}; n e4 = {{0.5280, 0.5608, 0.5821, 0.6111, 0.5943, 0.6225, 0.6408, 0.6657}, {0.1366}, n {0.1189, 0.1464, 0.1625, 0.2000, 0.1950, 0.2400}}. ei by using Equation (34): Step 3. Calculate the score function value of the SVNHFE n e1 ) = 0.5902, s(n e2 ) = 0.7711, s(n e3 ) = 0.6882, s(n e4 ) = 0.7623. s(n Then, we can obtain the ranking order of four alternative companies is A2 ≻ A4 ≻ A3 ≻ A1 , the food company A2 is the best alternative. If we replace the GSVNHFPWA operator in the aforementioned procedures with the GSVNHFPWG operator, the decision-making steps of the proposed method can be described as follows. Step 1′ . See Step 1. Step 2′ . Utilize the GSVNHFPWG operator (which the parameter λ = 1) to aggregate the SVNHF decision n o eij ei = eti , eii , fei (i = 1, 2, 3, 4) of matrix N = n (i = 1, 2, 3, 4; j = 1, 2, 3) into the SVNHFE n m×n each alternative company. Take an alternative company A1 for example, we have Information 2018, 9, 10 16 of 19  T11 T12 T13   ∑3j=1 T1j ∑3j=1 T1j ∑3j=1 T1j    e11 ) e11 , n e12 , n e13 ) = (n e1 = GSVNHFPWG1 (n e12 ) e13 ) n ⊗ (n ⊗ (n  = ∪ e11 ∈et11 ,e γ γ12 ∈et12 ,e γ13 ∈et13 ,δe11 ∈ei11 ,δe12 ∈ei12 ,δe13 ∈ei13 ,e η11 ∈ fe11 ,e η12 ∈ fe12 ,e η13 ∈ fe13 = 1/1   1/1  T1j T1j       3 3 1 ∑3j=1 T1j   1 ∑3j=1 T1j     , 1 − ∏ 1 − δ1j , 1 − 1 − ∏ 1 − 1 − γ1j        j = 1 j = 1                  1/1  T1j       3 1 ∑3j=1 T1j     1 − ∏ 1 − η1j    j =1      1 − 1 − (1 − (1 − 0.2))0.54 (1 − (1 − 0.3))0.28 (1 − (1 − 0.5))0.18 , 1 − 1 − (1 − (1 − 0.2))0.54 (1 − (1 − 0.3))0.28 (1 − (1 − 0.6))0.18 , 1 − 1 − (1 − (1 − 0.2))0.54 (1 − (1 − 0.4))0.28 (1 − (1 − 0.5))0.18 , 1 − 1 − (1 − (1 − 0.2))0.54 (1 − (1 − 0.4))0.28 (1 − (1 − 0.6))0.18 , 1 − 1 − (1 − (1 − 0.2))0.54 (1 − (1 − 0.5))0.28 (1 − (1 − 0.5))0.18 , 1 − 1 − (1 − (1 − 0.2))0.54 (1 − (1 − 0.5))0.28 (1 − (1 − 0.6))0.18 , 1 − 1 − (1 − (1 − 0.3))0.54 (1 − (1 − 0.3))0.28 (1 − (1 − 0.5))0.18 , 1 − 1 − (1 − (1 − 0.3))0.54 (1 − (1 − 0.3))0.28 (1 − (1 − 0.6))0.18 , 1 − 1 − (1 − (1 − 0.3))0.54 (1 − (1 − 0.4))0.28 (1 − (1 − 0.5))0.18 , 1 − 1 − (1 − (1 − 0.3))0.54 (1 − (1 − 0.4))0.28 (1 − (1 − 0.6))0.18 , o 1 − 1 − (1 − (1 − 0.2))0.54 (1 − (1 − 0.5))0.28 (1 − (1 − 0.5))0.18 , 1 − 1 − (1 − (1 − 0.2))0.54 (1 − (1 − 0.5))0.28 (1 − (1 − 0.6))0.18 , n 1 − (1 − 0.1)0.54 (1 − 0.1)0.28 (1 − 0.2)0.18 , 1 − (1 − 0.1)0.54 (1 − 0.1)0.28 (1 − 0.3)0.18 , 1 − (1 − 0.2)0.54 (1 − 0.1)0.28 (1 − 0.2)0.18 , o n 1 − (1 − 0.2)0.54 (1 − 0.1)0.28 (1 − 0.3)0.18 , 1 − (1 − 0.5)0.54 (1 − 0.3)0.28 (1 − 0.3)0.18 , 1 − (1 − 0.5)0.54 (1 − 0.3)0.28 (1 − 0.4)0.18 , nn 1 − (1 − 0.5)0.54 (1 − 0.4)0.28 (1 − 0.3)0.18 , 1 − (1 − 0.5)0.54 (1 − 0.4)0.28 (1 − 0.4)0.18 , 1 − (1 − 0.6)0.54 (1 − 0.3)0.28 (1 − 0.3)0.18 , o 1 − (1 − 0.6)0.54 (1 − 0.3)0.28 (1 − 0.4)0.18 , 1 − (1 − 0.6)0.54 (1 − 0.4)0.28 (1 − 0.3)0.18 , 1 − (1 − 0.6)0.54 (1 − 0.4)0.28 (1 − 0.4)0.18 . e1 as the following: and obtain the SVNHFE n e1 = {{0.2644, 0.2733, 0.2865, 0.2961, 0.3049, 0.3151, 0.3291, 0.3402, 0.3566, 0.3686, n 0.3795, 0.3923}, {0.1190, 0.1401, 0.1733, 0.1931}, {0.4163, 0.4324, 0.4408, 0.4562, 0.4825, 0.4968, 0.5043, 0.5179}}. Similarly, the SVNHFEs of other alternative companies can be computed as follows: e2 = {{0.6000, 0.6234, 0.6314, 0.6559, 0.6403, 0.6652, 0.6738, 0.7000}, {0.1000, 0.1344, n 0.1436, 0.1763}, {0.1866, 0.2217, 0.2260, 0.2594}}; e3 = {{0.5194, 0.5517, 0.5649, 0.6000}, {0.2531}, {0.2917, 0.3222}}; n e4 = {{0.4600, 0.4781, 0.4788, 0.4976, 0.5789, 0.6016, 0.6026, 0.6262}, {0.1465}, n {0.1261, 0.1565, 0.1712, 0.2000, 0.2196, 0.2467}}. ei by using Equation (34): Step 3′ . Calculate the score function value of the SVNHFE n e2 ) = 0.7622, s(n e3 ) = 0.6663, s(n e1 ) = 0.5669, s(n e4 ) = 0.7358. s(n Then, we can obtain the ranking order of four alternative companies is A2 ≻ A4 ≻ A3 ≻ A1 , and the food company A2 is also the best alternative. In real life, decision makers may determine the value of the parameter λ according to the decision-making problem itself or their preference. To analyze the influence of the parameter λ on the final ranking result, we change the parameter λ of the GSVNHFPWA operator and GSVNHFPWG operator in the numerical example above. Different values of the parameter λ are provided, such as 0.001, 0.5, 1, 2, 3, 5, 10, 20, and 50, which is determined by decision makers in decision-making process. Combined with the proposed method, we can obtain the score function values of four alternative companies, then the ranking results are determined as shown in Tables 2 and 3. Tables 2 and 3 show that when the GSVNHFPWA operator is used to aggregate arguments, the best alternative is the food company A2 for 0 < λ ≤ 3, but the best alternative is the arms company A4 for 5 ≤ λ ≤ 50. Besides, when the GSVNHFPWG operator is used to aggregate arguments, the best alternative is always the food company A2 for 0 < λ ≤ 50, however, there are some differences in specific ranking for λ = 50. Thus, the different ranking results indicate that the parameter λ plays a very important Information 2018, 9, 10 17 of 19 role in the aggregation process; decision makers should be cautious to determine the value of λ in real decision-making process. Table 2. Score function values obtained by the GSVNHFPWA operator and the rankings of alternatives for different values of λ. The Value of λ s(e n1 ) s(e n2 ) s(e n3 ) s(e n4 ) λ = 0.001 λ = 0.5 λ=1 λ=2 λ=3 λ=5 λ = 10 λ = 20 λ = 50 0.5834 0.5866 0.5902 0.5984 0.6071 0.6232 0.6500 0.6734 0.6927 0.7702 0.7706 0.7711 0.7721 0.7732 0.7753 0.7810 0.7902 0.8023 0.6856 0.6869 0.6882 0.6910 0.6937 0.6991 0.7109 0.7253 0.7394 0.7571 0.7596 0.7623 0.7676 0.7727 0.7811 0.7954 0.8104 0.8261 Ranking A2 A2 A2 A2 A2 A4 A4 A4 A4 ≻ ≻ ≻ ≻ ≻ ≻ ≻ ≻ ≻ A4 A4 A4 A4 A4 A2 A2 A2 A2 ≻ ≻ ≻ ≻ ≻ ≻ ≻ ≻ ≻ A3 A3 A3 A3 A3 A3 A3 A3 A3 ≻ ≻ ≻ ≻ ≻ ≻ ≻ ≻ ≻ A1 A1 A1 A1 A1 A1 A1 A1 A1 Table 3. Score function values obtained by the GSVNHFPWG operator and the rankings of alternatives for different values of λ. The Value of λ s(e n1 ) s(e n2 ) s(e n3 ) s(e n4 ) λ = 0.01 λ = 0.5 λ=1 λ=2 λ=3 λ=5 λ = 10 λ = 20 λ = 50 0.5735 0.5704 0.5669 0.5592 0.5512 0.5372 0.5166 0.5013 0.5718 0.7667 0.7647 0.7622 0.7569 0.7518 0.7435 0.7317 0.7311 1.0000 0.6766 0.6718 0.6663 0.6553 0.6459 0.6324 0.6132 0.5964 0.8765 0.7454 0.7408 0.7358 0.7251 0.7152 0.6998 0.6806 0.6686 0.8030 Ranking A2 A2 A2 A2 A2 A2 A2 A2 A2 ≻ ≻ ≻ ≻ ≻ ≻ ≻ ≻ ≻ A4 A4 A4 A4 A4 A4 A4 A4 A3 ≻ ≻ ≻ ≻ ≻ ≻ ≻ ≻ ≻ A3 A3 A3 A3 A3 A3 A3 A3 A4 ≻ ≻ ≻ ≻ ≻ ≻ ≻ ≻ ≻ A1 A1 A1 A1 A1 A1 A1 A1 A1 5.2. Comparison and Discussion To further verify the effectiveness of the proposed method, we compare the aforementioned ranking order with the results of other decision-making methods for analyzing the same numerical example as shown in Table 4; these methods include the SVNHFWA operator and SVNHFWG operator [14], correlation coefficient of DHFSs [27], correlation coefficient of SVNEs [28], and correlation coefficient of SVNHFEs [15]. From Table 4, we can see that the ranking order of four alternatives obtained by the SVNHFWA operator is A4 ≻ A2 ≻ A3 ≻ A1 due to the feature of emphasizing group major points; besides, the ranking order of four alternatives in other methods are always A2 ≻ A4 ≻ A3 ≻ A1 , which is consistent with our proposed method. Table 4. Comparison result of different decision-making methods. Decision-Making Method The GSVNHFPWA operator (λ = 1) The GSVNHFPWG operator (λ = 1) The SVNHFWA operator The SVNHFWG operator Correlation coefficient of DHFSs Correlation coefficient of SVNEs Correlation coefficient of SVNHFEs Ranking A2 A2 A4 A2 A2 A2 A2 ≻ ≻ ≻ ≻ ≻ ≻ ≻ A4 A4 A2 A4 A4 A4 A4 ≻ ≻ ≻ ≻ ≻ ≻ ≻ A3 A3 A3 A3 A3 A3 A3 ≻ ≻ ≻ ≻ ≻ ≻ ≻ A1 A1 A1 A1 A1 A1 A1 Information 2018, 9, 10 18 of 19 With regard to the existing five decision-making methods above, the methods based on the correlation coefficient of DHFSs and correlation coefficient of SVNEs are only applicable to the DHF and SVN environment, respectively, while DHFS and SVNS are the specific cases of SVNHFS. On the other hand, the other three methods can only solve the decision-making problems that the criteria are in the same priority level. Therefore, the comparison result indicates that the proposed method, not only can deal with the decision-making problems effectively but, also has several advantages as follows: (1) decision makers evaluate the alternatives by using SVNHFEs, which contains truth-membership, indeterminacy-membership, and falsity-membership degrees, and SVNHFS is also a generalization of HFS, DHFS, and SVNS; thus, SVNHFEs can express more reliable evaluation information of decision makers; (2) the GSVNHFPWA operator and GSVNHFPWG operator can solve the decision-making problems that the criteria are in different priority levels, which is not considered in other decision-making methods under SVNHF environment; and (3) the GSVNHFPWA operator and GSVNHFPWG operator can be reduced to several aggregation operators through adjusting the value of the parameter λ, including the SVNHFWA operator and SVNHFWG operator [14]. Decision makers can determine the exact value of the parameter λ to respond to the possible situations in real life. 6. Conclusions This paper studies the MCDM problems under SVNHF environment, while the criteria are in different priority levels. Motivated by the idea of the PA operator, we develop the GSVNHFPWA operator and GSVNHFPWG operator for aggregating SVNHFEs based on the related researches of SVNS and HFS theory. Some desirable properties of the proposed operators are investigated in detail, such as idempotency, boundedness, and monotonicity. Furthermore, we obtained several special cases that reduced from the proposed operators by changing the value of the parameter λ. Then, an approach for MCDM in which the criteria have different priorities is constructed combined with these operators. Finally, a numerical example is provided to illustrate the applications of the proposed method, and several advantages are reflected by the comparison between the proposed method and several existing decision-making methods. In the future, we shall investigate the SVNHF prioritized aggregation operators according to the different t-norm and t-conorm operational laws, and develop more aggregation operators for SVNHFSs. Acknowledgments: This work was supported by the National Natural Science Foundation of China (No. 71371156), the National Express Academic Scholarship Funding Project of Southwest Jiaotong University (No. A0920502051703-1), and Doctoral Innovation Fund Program of Southwest Jiaotong University. 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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). information Article Some New Biparametric Distance Measures on Single-Valued Neutrosophic Sets with Applications to Pattern Recognition and Medical Diagnosis Harish Garg * ID and Nancy ID School of Mathematics, Thapar Institute of Engineering & Technology, Deemed University Patiala, Punjab 147004, India; nancyverma16@gmail.com * Correspondence: harishg58iitr@gmail.com or harish.garg@thapar.edu; Tel.: +91-86990-31147 Received: 29 November 2017; Accepted: 11 December 2017; Published: 15 December 2017 Abstract: Single-valued neutrosophic sets (SVNSs) handling the uncertainties characterized by truth, indeterminacy, and falsity membership degrees, are a more flexible way to capture uncertainty. In this paper, some new types of distance measures, overcoming the shortcomings of the existing measures, for SVNSs with two parameters are proposed along with their proofs. The various desirable relations between the proposed measures have also been derived. A comparison between the proposed and the existing measures has been performed in terms of counter-intuitive cases for showing its validity. The proposed measures have been illustrated with case studies of pattern recognition as well as medical diagnoses, along with the effect of the different parameters on the ordering of the objects. Keywords: decision-making; single-valued neutrosophic sets; distance measure; pattern recognition; uncertainties 1. Introduction The classical measure theory has been widely used to represent uncertainties in data. However, these measures are valid only for precise data, and hence they may be unable to give accurate judgments for data uncertain and imprecise in nature. To handle this, fuzzy set (FS) theory, developed by Zadeh [1], has received much attention over the last decades because of its capability of handling uncertainties. After this, Atanassov [2] proposed the concept of an intuitionistic fuzzy set (IFS), which extends the theory of FSs with the addition of a degree of non-membership. As IFS theory has widely been used by researchers [3–16] in different disciplines for handling the uncertainties in data, hence its corresponding analysis is more meaningful than FSs’ crisp analysis. Nevertheless, neither the FS nor IFS theory are able to deal with indeterminate and inconsistent information. For instance, we take a person giving their opinion about an object with 0.5 being the possibility that the statement is true, 0.7 being the possibility that the statement is false and 0.2 being the possibility that he or she is not sure. To resolve this, Smarandache [17] introduced a new component called the “indeterminacy-membership function” and added the “truth membership function” and “falsity membership function”, all which are independent components lying in ]0− , 1+ [, and hence the corresponding set is known as a neutrosophic set (NS), which is the generalization of the IFS and FS. However, without specification, NSs are difficult to apply to real-life problems. Thus, a particular case of the NS called a single-valued NS (SVNS) has been proposed by Smarandache [17], Wang et al. [18]. After this pioneering work, researchers have been engaged in extensions and applications to different disciplines. However, the most important task for the decision-maker is to rank the objects so as to obtain the desired object(s). For this, researchers have made efforts to enrich the concept of information measures in neutrosophic environments. Broumi and Smarandache [19] introduced the Hausdorff distance, while Majumdar [20] presented the Hamming and Euclidean Information 2017, 8, 162; doi:10.3390/info8040162 www.mdpi.com/journal/information Information 2017, 8, 162 2 of 20 distance for comparing the SVNSs. Ye [21] presented the concept of correlation for single-valued neutrosophic numbers (SVNNs). Additionally, Ye [22] improved the concept of cosine similarity for SVNSs, which was firstly introduced by Kong et al. [23] in a neutrosophic environment. Nancy and Garg [24] presented an improved score function for ranking the SVNNs and applied them to solve the decision-making problem. Garg and Nancy [25] presented the entropy measure of order α and applied them to solve decision-making problems. Recently, Garg and Nancy [26] presented a technique for order preference by similarity to ideal solution (TOPSIS) method under an interval NS environment to solve decision-making problems. Aside from these, various authors have incorporated the idea of NS theory into the similarity measures [27,28], distance measures [29,30], the cosine similarity measure [19,22,31], and aggregation operators [22,31–40]. Thus, on the basis of the above observations, it has been observed that distance or similarity measures are of key importance in a number of theoretical and applied statistical inference and data processing problems. It has been deduced from studies that similarity, entropy and divergence measures could be induced by the normalized distance measure on the basis of their axiomatic definitions. On the other hand, SVNSs are one of the most successful theories to handle the uncertainties and certainties in the system, but little systematic research has explored these problems. The gap in the research motivates us to develop some families of the distance measures of the SVNS to solve the decision-making problem, for which preferences related to different alternatives are taken in the form of neutrosophic numbers. The main contributions of this work are summarized as follows: (i) to highlight the shortcomings of the various existing distance measures under the single-valued neutrosophic information through illustrative examples; (ii) to overcome the shortcomings of the existing measures, this paper defines some new series of biparametric distance measures between SVNSs, which depend on two parameters, namely, p and t, where p is the L p norm and t identifies the level of uncertainty. The various desirable relations between these have been investigated in detail. Then, we utilized these measures to solve the problem of pattern recognition as well as medical diagnosis and compared their performance with that of some of the existing approaches. The rest of this paper is organized as follows. Section 2 briefly describes the concepts of NSs, SVNSs and their corresponding existing distance measures. Section 3 presents a family of the normalized and weighted normalized distance measures between two SVNSs. Some of their desirable properties have also been investigated in detail, while generalized distance measures have been proposed in Section 4. The defined measures are illustrated, by an example in Section 5, using the field of pattern recognition and medical diagnosis for demonstrating the effectiveness and stability of the proposed measures. Finally, a concrete conclusion has been drawn in Section 6. 2. Preliminaries An overview of NSs and SVNSs is addressed here on the universal set X. 2.1. Basic Definitions Definition 1 ([17,41]). A neutrosophic set (NS) A in X is defined by its truth membership function ( TA ( x )), an indeterminacy-membership function ( I A ( x )) and a falsity membership function ( FA ( x )), where all are subsets of ]0− , 1+ [. There is no restriction on the sum of TA ( x ), I A ( x ) and FA ( x ); thus 0− ≤ sup TA ( x ) + sup I A ( x ) + sup FA ( x ) ≤ 3+ for all x ∈ X. Here, sup represents the supremum of the set. Wang et al. [18], Smarandache [41] defined the SVNS, which is an instance of a NS. Definition 2 ([18,41]). A single-valued neutrosophic set (SVNS) A is defined as A = {h x, TA ( x ), I A ( x ), FA ( x )i | x ∈ X } Information 2017, 8, 162 3 of 20 where TA : X → [0, 1], I A : X → [0, 1] and FA : X → [0, 1] with TA ( x ) + I A ( x ) + FA ( x ) ≤ 3 for all x ∈ X. The values TA ( x ), I A ( x ) and FA ( x ) denote the truth-membership degree, the indeterminacy-membership degree and the falsity-membership degree of x to A, respectively. The pairs of these are called single-valued neutrosophic numbers (SVNNs), which are denoted by α = hµ A , ρ A , νA i, and class of SVNSs is denoted by Φ( X ). Definition 3. Let A = hµ A , ρ A , νA i and B = hµ B , ρ B , νB i be two single-valued neutrosophic sets (SVNSs). Then the following expressions are defined by [18]: (i) (ii) (iii) (iv) (v) A ⊆ B if and only if (iff) µ A ( x ) ≤ µ B ( x ), ρ A ( x ) ≥ ρ B ( x ) and νA ( x ) ≥ νB ( x ) for all x in X; A = B iff A ⊆ B and B ⊆ A; Ac = {hνA ( x ), 1 − ρ A ( x ), µ A ( x ) | x ∈ X i}; A ∩ B = hmin(µ A ( x ), µ B ( x )), max(ρ A ( x ), ρ B ( x )), max(νA ( x ), νB ( x ))i; A ∪ B = hmax(µ A ( x ), µ B ( x )), min(ρ A ( x ), ρ B ( x )), min(νA ( x ), νB ( x ))i. 2.2. Existing Distance Measures Definition 4. A real function d : Φ( X ) × Φ( X ) → [0, 1] is called a distance measure [19], where d satisfies the following axioms for A, B, C ∈ Φ( X ): (P1) (P2) (P3) (P4) 0 ≤ d( A, B) ≤ 1; d( A, B) = 0 iff A = B; d( A, B) = d( B, A); If A ⊆ B ⊆ C, then d( A, C ) ≥ d( A, B) and d( A, C ) ≥ d( B, C ). On the basis of this, several researchers have addressed the various types of distance and similarity measures between two SVNSs A = h xi , µ A ( xi ), ρ A ( xi ), νA ( xi )| xi ∈ X i and B = h xi , µ B ( xi ), ρ B ( xi ), νB ( xi )| xi ∈ X i, i = 1, 2, ..., n, which are given as follows: (i) The extended Hausdorff distance [19]: 1 n D H ( A, B) = n ∑ max i =1 n |µ A ( xi ) − µ B ( xi )|, |ρ A ( xi ) − ρ B ( xi )|, |νA ( xi ) − νB ( xi )| o (1) (ii) The normalized Hamming distance [20]: D NH ( A, B) = 1 3n n ∑ i =1 n |µ A ( xi ) − µ B ( xi )| + |ρ A ( xi ) − ρ B ( xi )| + |νA ( xi ) − νB ( xi )| o (2) (iii) The normalized Euclidean distance [20]: D NE ( A, B) = 1 3n n ∑ i =1 n (µ A ( xi ) − µ B ( xi ))2 + (ρ A ( xi ) − ρ B ( xi ))2 + (νA ( xi ) − νB ( xi ))2 o !1/2 (3) (iv) The cosine similarities [22]: 1 SCS1 ( A, B) = n n " # (4) " # (5) π |µ A ( xi ) − µ B ( xi )| ∨ |ρ A ( xi ) − ρ B ( xi )| ∨ |νA ( xi ) − νB ( xi )| ∑ cos 2 i =1 and 1 SCS2 ( A, B) = n n π |µ A ( xi ) − µ B ( xi )| + |ρ A ( xi ) − ρ B ( xi )| + |νA ( xi ) − νB ( xi )| ∑ cos 6 i =1 and their corresponding distances denoted by DCS1 = 1 − SCS1 and DCS2 = 1 − SCS2 . Information 2017, 8, 162 4 of 20 (v) The tangent similarities [42]: 1 ST1 ( A, B) = 1 − n n " # (6) " # (7) π |µ A ( xi ) − µ B ( xi )| ∨ |ρ A ( xi ) − ρ B ( xi )| ∨ |νA ( xi ) − νB ( xi )| ∑ tan 4 i =1 and 1 ST2 ( A, B) = 1 − n n π |µ A ( xi ) − µ B ( xi )| + |ρ A ( xi ) − ρ B ( xi )| + |νA ( xi ) − νB ( xi )| ∑ tan 12 i =1 and their corresponding distances denoted by DT1 = 1 − ST1 and DT2 = 1 − ST2 . 2.3. Shortcomings of the Existing Measures The above measures have been widely used; however, simultaneously they have some drawbacks, which are illustrated with the numerical example that follows. Example 1. Consider two known patterns A and B, which are represented by SVNSs in a universe X given by A = h x, 0.5, 0.0, 0.0 | x ∈ X i, B = h x, 0.0, 0.5, 0.0 | x ∈ X i. Consider an unknown pattern C ∈ SV NSs( X ), which is recognized where C = h x, 0.0, 0.0, 0.5 | x ∈ X i; then the target of this problem is to classify the pattern C in one of the classes A or B. If we apply the existing measures [19,20,22,42] defined in Equations (1)–(7) above, then we obtain the following: Pair DH DN H D NE DCS1 DCS2 DT1 DT1 (A,C) (B,C) 0.5 0.5 0.3333 0.3333 0.4048 0.4048 0.2929 0.2929 0.1340 0.1340 0.4142 0.4142 0.2679 0.2679 Thus, from this, we conclude that these existing measures are unable to classify the pattern C with A and B. Hence these measures are inconsistent and unable to perform ranking. Example 2. Consider two SVNSs defined on the universal set X given by A = h x, 0.3, 0.2, 0.3 | x ∈ X i and B = h x, 0.4, 0.2, 0.4 | x ∈ X i. If we replace the degree of falsity membership of A (0.3) with 0.4, and that of B (0.4) with 0.3, then we obtain new SVNSs as C = h x, 0.3, 0.2, 0.4 | x ∈ X i and D = h x, 0.4, 0.2, 0.3 | x ∈ X i. Now, by using the distance measures defined in Equations (1)–(7), we obtain their corresponding values as follows: Pair DH DN H D NE DCS1 DCS2 DT1 DT1 (A,B) (C,D) 0.1 0.1 0.066 0.066 0.077 0.077 0.013 0.013 0.006 0.006 0.078 0.078 0.052 0.052 Thus, it has been concluded that by changing the falsity degree of SVNSs and keeping the other degrees unchanged, the values of their corresponding measures remain the same. Thus, there is no effect of the degree of falsity membership on the distance measures. Similarly, we can observe the same for the degree of the truth membership functions. This seems to be worthless to calculate distance using the measures mentioned above. Thus, there is a need to build up a new distance measure that overcomes the shortcomings of the existing measures. 3. Some New Distance Measures between SVNSs In this section, we present the Hamming and the Euclidean distances between SVNSs, which can be used in real scientific and engineering applications. Letting Φ( X ) be the class of SVNSs over the universal set X, then we define the distances for SVNSs, A = hµ A ( xi ), ρ A ( xi ), νA ( xi ) | xi ∈ X i and B = hµ B ( xi ), ρ B ( xi ), νB ( xi ) | xi ∈ X i, by considering the uncertainty parameter t, as follows: Information 2017, 8, 162 (i) 5 of 20 Hamming distance: n d1 ( A, B) =  − t(µ A ( xi ) − µ B ( xi )) + (ρ A ( xi ) − ρ B ( xi )) + (νA ( xi ) − νB ( xi ))  (8)  (9)   1  + − t(ρ A ( xi ) − ρ B ( xi )) − (νA ( xi ) − νB ( xi )) + (µ A ( xi ) − µ B ( xi ))    3(2 + t) i∑ =1 + − t(νA ( xi ) − νB ( xi )) − (ρ A ( xi ) − ρ B ( xi )) + (µ A ( xi ) − µ B ( xi )) (ii) Normalized Hamming distance: n d2 ( A, B) =  − t(µ A ( xi ) − µ B ( xi )) + (ρ A ( xi ) − ρ B ( xi )) + (νA ( xi ) − νB ( xi ))   1  + − t(ρ A ( xi ) − ρ B ( xi )) − (νA ( xi ) − νB ( xi )) + (µ A ( xi ) − µ B ( xi ))    3n(2 + t) i∑ =1 + − t(νA ( xi ) − νB ( xi )) − (ρ A ( xi ) − ρ B ( xi )) + (µ A ( xi ) − µ B ( xi )) (iii) Euclidean distance:   1  d3 ( A, B) =   3(2 + t )2 n    − t(µ A ( xi ) − µ B ( xi )) + (ρ A ( xi ) − ρ B ( xi )) + (νA ( xi ) − νB ( xi )) 2 (10) 1/2 (11)  2  − t(ρ A ( xi ) − ρ B ( xi )) − (νA ( xi ) − νB ( xi )) + (µ A ( xi ) − µ B ( xi ))   2 + − t(νA ( xi ) − νB ( xi )) − (ρ A ( xi ) − ρ B ( xi )) + (µ A ( xi ) − µ B ( xi )) + ∑  i =1 1/2 (iv) Normalized Euclidean distance:   1  d4 ( A, B) =   3n(2 + t)2  2 − t(µ A ( xi ) − µ B ( xi )) + (ρ A ( xi ) − ρ B ( xi )) + (νA ( xi ) − νB ( xi ))   + − t(ρ A ( xi ) − ρ B ( xi )) − (νA ( xi ) − νB ( xi )) + (µ A ( xi ) − µ B ( xi )) ∑  i =1 + − t(νA ( xi ) − νB ( xi )) − (ρ A ( xi ) − ρ B ( xi )) + (µ A ( xi ) − µ B ( xi )) n    2  2 where t ≥ 3 is a parameter. Then, on the basis of the distance properties as defined in Definition 4, we can obtain the following properties: Proposition 1. The above-defined distance d2 ( A, B), between two SVNSs A and B, satisfies the following properties (P1)–(P4): (P1) (P2) (P3) (P4) 0 ≤ d2 ( A, B) ≤ 1, ∀ A, B ∈ Φ( X ); d2 ( A, B) = 0 iff A = B; d2 ( A, B) = d2 ( B, A); If A ⊆ B ⊆ C, then d2 ( A, C ) ≥ d2 ( A, B) and d2 ( A, C ) ≥ d2 ( B, C ). Proof. For two SVNSs A and B, we have (P1) 0 ≤ µ A ( xi ), µ B ( xi ) ≤ 1, 0 ≤ ρ A ( xi ), ρ B ( xi ) ≤ 1 and 0 ≤ νA ( xi ), νB ( xi ) ≤ 1. Thus, | µ A ( xi ) − µ B ( xi ) |≤ 1, | ρ A ( xi ) − ρ B ( xi ) |≤ 1, | νA ( xi ) − νB ( xi ) |≤ 1 and | t(µ A ( xi ) − µ B ( xi )) |≤ t. Therefore, | (tµ A ( xi ) − νA ( xi ) − ρ A ( xi )) − (tµ B ( xi ) − νB ( xi ) − ρ B ( xi )) |≤ (2 + t) | (tρ A ( xi ) + νA ( xi ) − µ A ( xi )) − (tρ B ( xi ) + νB ( xi ) − µ B ( xi )) |≤ (2 + t) | (tνA ( xi ) + ρ A ( xi ) − µ A ( xi )) − (tνB ( xi ) + ρ B ( xi ) − µ B ( xi )) |≤ (2 + t) (P2) Hence, by the definition of d2 , we obtain 0 ≤ d2 ( A, B) ≤ 1. Firstly, we assume that A = B, which implies that µ A ( xi ) = µ B ( xi ), ρ A ( xi ) = ρ B ( xi ), and νA ( xi ) = νB ( xi ) for i = 1, 2, .., n. Thus, by the definition of d2 , we obtain d2 ( A, B) = 0. Conversely, assuming that d2 ( A, B) = 0 for two SVNSs A and B, this implies that Information 2017, 8, 162 6 of 20 | − t(µ A ( xi ) − µ B ( xi )) + (ρ A ( xi ) − ρ B ( xi )) + (νA ( xi ) − νB ( xi ))| + | − t(ρ A ( xi ) − ρ B ( xi )) − (νA ( xi ) − νB ( xi )) + (µ A ( xi ) − µ B ( xi ))| + | − t(νA ( xi ) − νB ( xi )) − (ρ A ( xi ) − ρ B ( xi )) + (µ A ( xi ) − µ B ( xi ))| or    =0   | − t(µ A ( xi ) − µ B ( xi )) + (ρ A ( xi ) − ρ B ( xi )) + (νA ( xi ) − νB ( xi ))| = 0 | − t(ρ A ( xi ) − ρ B ( xi )) − (νA ( xi ) − νB ( xi )) + (µ A ( xi ) − µ B ( xi ))| = 0 | − t(νA ( xi ) − νB ( xi )) − (ρ A ( xi ) − ρ B ( xi )) + (µ A ( xi ) − µ B ( xi ))| = 0 (P3) (P4) After solving, we obtain µ A ( xi ) − µ B ( xi ) = 0, ρ A ( xi ) − ρ B ( xi ) = 0 and νA ( xi ) − νB ( xi ) = 0, which implies µ A ( xi ) = µ B ( xi ), ρ A ( xi ) = ρ B ( xi ) and νA ( xi ) = νB ( xi ). Therefore, A = B. Hence d2 ( A, B) = 0 iff A = B. This is straightforward from the definition of d2 . If A ⊆ B ⊆ C, then µ A ( xi ) ≤ µ B ( xi ) ≤ µC ( xi ), ρ A ( xi ) ≥ ρ B ( xi ) ≥ ρC ( xi ) and νA ( xi ) ≥ νB ( xi ) ≥ νC ( xi ), which implies that µ A ( xi ) − µ B ( xi ) ≥ µ A ( xi ) − µC ( xi ), νA ( xi ) − νB ( xi ) ≤ νA ( xi ) − νC ( xi ), and ρ A ( xi ) − ρ B ( xi ) ≤ ρ A ( xi ) − ρC ( xi ). Therefore, | − t(µ A ( xi ) − µ B ( xi )) + (ρ A ( xi ) − ρ B ( xi )) + (νA ( xi ) − νB ( xi ))| ≤ | − t(µ A ( xi ) − µc ( xi )) + (ρ A ( xi ) − ρC ( xi )) + (νA ( xi ) − νC ( xi ))| | − t(ρ A ( xi ) − ρ B ( xi )) − (νA ( xi ) − νB ( xi )) + (µ A ( xi ) − µ B ( xi ))| ≤ | − t(ρ A ( xi ) − ρC ( xi )) − (νA ( xi ) − νC ( xi )) + (µ A ( xi ) − µC ( xi ))| | − t(νA ( xi ) − νB ( xi )) − (ρ A ( xi ) − ρ B ( xi )) + (µ A ( xi ) − µ B ( xi ))| ≤ | − t(νA ( xi ) − νC ( xi )) − (ρ A ( xi ) − ρC ( xi )) + (µ A ( xi ) − µC ( xi ))| By adding, we obtain d2 ( A, B) ≤ d2 ( A, C ). Similarly, we obtain d2 ( B, C ) ≤ d2 ( A, C ). Proposition 2. Distance d4 as defined in Equation (11) is also a valid measure. Proof. For two SVNSs A and B, we have (P1) 0 ≤ µ A ( xi ), µ B ( xi ) ≤ 1, 0 ≤ ρ A ( xi ), ρ B ( xi ) ≤ 1 and 0 ≤ νA ( xi ), νB ( xi ) ≤ 1. Thus, | µ A ( xi ) − µ B ( xi ) |≤ 1, | ρ A ( xi ) − ρ B ( xi ) |≤ 1, | νA ( xi ) − νB ( xi ) |≤ 1 and | t(µ A ( xi ) − µ B ( xi )) | ≤ t. Therefore, | − t(µ A ( xi ) − µ B ( xi )) + (ρ A ( xi ) − ρ B ( xi )) + (νA ( xi ) − νB ( xi ))|2 ≤ (2 + t)2 | − t(ρ A ( xi ) − ρ B ( xi )) − (νA ( xi ) − νB ( xi )) + (µ A ( xi ) − µ B ( xi ))|2 ≤ (2 + t)2 | − t(νA ( xi ) − νB ( xi )) − (ρ A ( xi ) − ρ B ( xi )) + (µ A ( xi ) − µ B ( xi ))|2 ≤ (2 + t)2 (P2) Hence, by the definition of d4 , we obtain 0 ≤ d4 ( A, B) ≤ 1. Assuming that A = B implies that µ A ( xi ) = µ B ( xi ), ρ A ( xi ) = ρ B ( xi ) and νA ( xi ) = νB ( xi ) for i = 1, 2, . . . , n, and hence using Equation (11), we obtain d4 ( A, B) = 0. Conversely, assuming that d4 ( A, B) = 0 implies | − t(µ A ( xi ) − µ B ( xi )) + (ρ A ( xi ) − ρ B ( xi )) + (νA ( xi ) − νB ( xi ))|2 = 0 | − t(ρ A ( xi ) − ρ B ( xi )) − (νA ( xi ) − νB ( xi )) + (µ A ( xi ) − µ B ( xi ))|2 = 0 | − t(νA ( xi ) − νB ( xi )) − (ρ A ( xi ) − ρ B ( xi )) + (µ A ( xi ) − µ B ( xi ))|2 = 0 Information 2017, 8, 162 (P3) (P4) 7 of 20 After solving these, we obtain µ A ( xi ) − µ B ( xi ) = 0, ρ A ( xi ) − ρ B ( xi ) = 0 and νA ( xi ) − νB ( xi ) = 0; that is, µ A ( xi ) = µ B ( xi ), ρ A ( xi ) = ρ B ( xi ) and νA ( xi ) = νB ( xi ) for t ≥ 3. Hence A = B. Therefore, d4 ( A, B) = 0 iff A = B. This is straightforward from the definition of d4 . If A ⊆ B ⊆ C, then µ A ( xi ) ≤ µ B ( xi ) ≤ µC ( xi ), ρ A ( xi ) ≥ ρ B ( xi ) ≥ ρC ( xi ), and νA ( xi ) ≥ νB ( xi ) ≥ νC ( xi ). Therefore | − t(µ A ( xi ) − µ B ( xi )) + (ρ A ( xi ) − ρ B ( xi )) + (νA ( xi ) − νB ( xi ))|2 ≤ | − t(µ A ( xi ) − µc ( xi )) + (ρ A ( xi ) − ρC ( xi )) + (νA ( xi ) − νC ( xi ))|2 | − t(ρ A ( xi ) − ρ B ( xi )) − (νA ( xi ) − νB ( xi )) + (µ A ( xi ) − µ B ( xi ))|2 ≤ | − t(ρ A ( xi ) − ρC ( xi )) − (νA ( xi ) − νC ( xi )) + (µ A ( xi ) − µC ( xi ))|2 | − t(νA ( xi ) − νB ( xi )) − (ρ A ( xi ) − ρ B ( xi )) + (µ A ( xi ) − µ B ( xi ))|2 ≤ | − t(νA ( xi ) − νC ( xi )) − (ρ A ( xi ) − ρC ( xi )) + (µ A ( xi ) − µC ( xi ))|2 Hence by the definition of d4 , we obtain d4 ( A, B) ≤ d4 ( A, C ). Similarly, we obtain d4 ( B, C ) ≤ d4 ( A, C ). Now, on the basis of these proposed distance measures, we conclude that this successfully overcomes the shortcomings of the existing measures as described above. Example 3. If we apply the proposed distance measures d2 and d4 on the data considered in Example 1 to classify the pattern C, then corresponding to the parameter t = 3, we obtain d2 ( A, C ) = 0.3333, d2 ( B, C ) = 0.1333, d4 ( A, C ) = 0.3464 and d4 ( B, C ) = 0.1633. Thus, the pattern C is classified with the pattern B and hence is able to identify the best pattern. Example 4. If we utilize the proposed distances d2 and d4 for the above-considered Example 2, then their corresponding values are d2 ( A, B) = 0.0267, d2 (C, D ) = 0.0667, d4 ( A, B) = 0.0327 and d4 (C, D ) = 0.6930. Therefore, there is a significant effect of the change in the falsity membership on the measure values and hence consequently on the ranking values. Proposition 3. Measures d1 and d3 satisfy the following properties: (i) (ii) 0 ≤ d1 ≤ n; 0 ≤ d3 ≤ n1/2 . Proof. We can easily obtain that d1 ( A, B) = nd2 ( A, B), and thus by Proposition 1, we obtain 0 ≤ d1 ( A, B) ≤ n. Similarly, we can obtain 0 ≤ d3 ( A, B) ≤ n1/2 . However, in many practical situations, the different sets may have taken different weights, and thus weight ωi (i = 1, 2, . . . , n) of the element xi ∈ X should be taken into account. In the following, we develop a weighted Hamming distance and the normalized weighted Euclidean distance between SVNSs. (i) The normalized weighted Hamming distance: d5 ( A, B) n =  − t(µ A ( xi ) − µ B ( xi )) + (ρ A ( xi ) − ρ B ( xi )) + (νA ( xi ) − νB ( xi ))    1  ωi   + − t(ρ A ( xi ) − ρ B ( xi )) − (νA ( xi ) − νB ( xi )) + (µ A ( xi ) − µ B ( xi ))  3n(2 + t) i∑ =1 + − t(νA ( xi ) − νB ( xi )) − (ρ A ( xi ) − ρ B ( xi )) + (µ A ( xi ) − µ B ( xi )) (12) Information 2017, 8, 162 8 of 20 (ii) The normalized weighted Euclidean distance: = d6 ( A, B)      1  3n(2 + t)2    2  − t(µ A ( xi ) − µ B ( xi )) + (ρ A ( xi ) − ρ B ( xi )) + (νA ( xi ) − νB ( xi ))   + − t(ρ A ( xi ) − ρ B ( xi )) − (νA ( xi ) − νB ( xi )) + (µ A ( xi ) − µ B ( xi )) ∑ ωi   i =1 + − t(νA ( xi ) − νB ( xi )) − (ρ A ( xi ) − ρ B ( xi )) + (µ A ( xi ) − µ B ( xi )) n where t ≥ 3 is a parameter. 1/2      2    2   (13) It is straightforward to check that the normalized weighted distance dk ( A, B)(k = 5, 6) between SVNSs A and B also satisfies the above properties (P1)–(P4). Proposition 4. Distance measures d2 and d5 satisfy the relation d5 ≤ d2 . Proof. Becausenωi ≥ 0, ∑in=1 ωi = 1, then for any two SVNSs A and B, we have d5 ( A, B) = n 1 ω | − t(µ A ( xi ) − µ B ( xi )) + (ρ A ( xi ) − ρ B ( xi )) + (νA ( xi ) − νB ( xi ))| + | − t(ρ A ( xi ) − 3n(2+t) ∑i =1 i ρ B ( xi )) − (νA ( xi ) − µ B ( xi ))| + | − t(νA ( xi ) − νB ( xi )) − (ρ A ( xi ) − ρ B ( xi )) + oνB ( xi )) + (µ A ( xi ) − ≤ 3n(21+t) ∑in=1 | − t(µ A ( xi ) − µ B ( xi )) + (ρ A ( xi ) − ρ B ( xi )) + (νA ( xi ) − (µ A ( xi ) − µ B ( xi ))| νB ( xi ))| + | − t(ρ A ( xi ) − ρ B ( xi )) − (νA ( xi ) − νB ( xi )) + (µ A ( xi ) − µ B ( xi ))| + | − t(νA ( xi ) − νB ( xi )) − (ρ A ( xi ) − ρ B ( xi )) + (µ A ( xi ) − µ B ( xi ))| ; that is, d5 ( A, B) ≤ d2 ( A, B). Proposition 5. Let A and B be two SVNSs in X; then d5 and d6 are the distance measures. n Proof. Because ωi ∈ [0, 1] and ∑ ωi = 1 then we can easily obtain 0 ≤ d5 ( A, B) ≤ d2 ( A, B). i =1 Thus, d5 ( A, B) satisfies (P1). The proofs of (P2)–(P4) are similar to those of Proposition 1. Similar is true for d6 . Proposition 6. The distance measures d4 and d6 satisfy the relation d6 ≤ d4 . Proof. The proof follows from Proposition 4. Proposition 7. The distance measures d2 and d4 satisfy the inequality d4 ≤ √ d2 . Proof. For two SVNSs A and B, we have | − t(µ A ( xi ) − µ B ( xi )) + (ρ A ( xi ) − ρ B ( xi )) + (νA ( xi ) − νB ( xi ))|2 ≤ (2 + t)2 | − t(ρ A ( xi ) − ρ B ( xi )) − (νA ( xi ) − νB ( xi )) + (µ A ( xi ) − µ B ( xi ))|2 ≤ (2 + t)2 | − t(νA ( xi ) − νB ( xi )) − (ρ A ( xi ) − ρ B ( xi )) + (µ A ( xi ) − µ B ( xi ))|2 ≤ (2 + t)2 which implies that −t(µ A ( xi ) − µ B ( xi )) + (ρ A ( xi ) − ρ B ( xi )) + (νA ( xi ) − νB ( xi )) 2+t −t(ρ A ( xi ) − ρ B ( xi )) − (νA ( xi ) − νB ( xi )) + (µ A ( xi ) − µ B ( xi )) 2+t −t(νA ( xi ) − νB ( xi )) − (ρ A ( xi ) − ρ B ( xi )) + (µ A ( xi ) − µ B ( xi )) 2+t 2 2 2 ≤1 ≤1 ≤1 Information 2017, 8, 162 9 of 20 For any a ∈ [0, 1], we have a2 ≤ a. Therefore, −t(µ A ( xi ) − µ B ( xi )) + (ρ A ( xi ) − ρ B ( xi )) + (νA ( xi ) − νB ( xi )) 2 2+t −t(µ A ( xi ) − µ B ( xi )) + (ρ A ( xi ) − ρ B ( xi )) + (νA ( xi ) − νB ( xi )) ≤ 2+t −t(ρ A ( xi ) − ρ B ( xi )) − (νA ( xi ) − νB ( xi )) + (µ A ( xi ) − µ B ( xi )) 2 2+t −t(ρ A ( xi ) − ρ B ( xi )) − (νA ( xi ) − νB ( xi )) + (µ A ( xi ) − µ B ( xi )) ≤ 2+t −t(νA ( xi ) − νB ( xi )) − (ρ A ( xi ) − ρ B ( xi )) + (µ A ( xi ) − µ B ( xi )) 2 2+t −t(νA ( xi ) − νB ( xi )) − (ρ A ( xi ) − ρ B ( xi )) + (µ A ( xi ) − µ B ( xi )) ≤ 2+t and By adding these inequalities and by the definition of d4 , we have d4 ( A, B) =   1    3n(2 + t)2 n    − t(µ A ( xi ) − µ B ( xi )) + (ρ A ( xi ) − ρ B ( xi )) + (νA ( xi ) − νB ( xi )) 2 1/2  2  − t(ρ A ( xi ) − ρ B ( xi )) − (νA ( xi ) − νB ( xi )) + (µ A ( xi ) − µ B ( xi ))   i =1 2 + − t(νA ( xi ) − νB ( xi )) − (ρ A ( xi ) − ρ B ( xi )) + (µ A ( xi ) − µ B ( xi ))  1/2  − t(µ A ( xi ) − µ B ( xi )) + (ρ A ( xi ) − ρ B ( xi )) + (νA ( xi ) − νB ( xi )) n    1  + − t(ρ A ( xi ) − ρ B ( xi )) − (νA ( xi ) − νB ( xi )) + (µ A ( xi ) − µ B ( xi ))  ≤ ∑    3n(2 + t) i =1 + − t(νA ( xi ) − νB ( xi )) − (ρ A ( xi ) − ρ B ( xi )) + (µ A ( xi ) − µ B ( xi )) + ∑  ≤ (d2 ( A, B))1/2 As A and B are arbitrary SVNSs, thus we obtain d4 ≤ Proposition 8. Measures d6 and d5 satisfy the inequality d6 ≤ √ d2 . √ d5 . Proof. The proof follows from Proposition 7. The Hausdroff distance between two non-empty closed and bounded sets is a measure of the resemblance between them. For example, we consider A = [ x1 , x2 ] and B = [y1 , y2 ] in the Euclidean domain R; the Hausdroff distance in the additive set environment is given by the following [8]: H ( A, B) = max | x1 − y1 |, | x2 − y2 | Now, for any two SVNSs A and B over X = { x1 , x2 , . . . , xn }, we propose the following utmost distance measures: • Utmost normalized Hamming distance: d1H ( A, B) =  | − t(µ A ( xi ) − µ B ( xi )) + (ρ A ( xi ) − ρ B ( xi )) + (νA ( xi ) − νB ( xi ))|,  n   1 max  | − t(ρ A ( xi ) − ρ B ( xi )) − (νA ( xi ) − νB ( xi )) + (µ A ( xi ) − µ B ( xi ))|, ∑   3n(2 + t) i=1 i | − t(νA ( xi ) − νB ( xi )) − (ρ A ( xi ) − ρ B ( xi )) + (µ A ( xi ) − µ B ( xi ))| (14) Information 2017, 8, 162 • 10 of 20 Utmost normalized weighted Hamming distance: d2H ( A, B) = • | − t(µ A ( xi ) − µ B ( xi )) + (ρ A ( xi ) − ρ B ( xi )) + (νA ( xi ) − νB ( xi ))|,  n   1 ωi max  | − t(ρ A ( xi ) − ρ B ( xi )) − (νA ( xi ) − νB ( xi )) + (µ A ( xi ) − µ B ( xi ))|, ∑   3n(2 + t) i=1 i | − t(νA ( xi ) − νB ( xi )) − (ρ A ( xi ) − ρ B ( xi )) + (µ A ( xi ) − µ B ( xi ))| (15) Utmost normalized Euclidean distance: = •  d3H ( A, B)     1 1/2 | − t(µ A ( xi ) − µ B ( xi )) + (ρ A ( xi ) − ρ B ( xi )) + (νA ( xi ) − νB ( xi ))|2 ,      | − t(ρ A ( xi ) − ρ B ( xi )) − (νA ( xi ) − νB ( xi )) + (µ A ( xi ) − µ B ( xi ))|2 , max ∑  (16)  3n(2 + t)2 i=1 i    2  | − t(ν ( x ) − ν ( x )) − (ρ ( x ) − ρ ( x )) + (µ ( x ) − µ ( x ))|  n  A i B i A i B i A i B i Utmost normalized weighted Euclidean distance: = d4H ( A, B)      1  3n(2 + t)2    1/2 | − t(µ A ( xi ) − µ B ( xi )) + (ρ A ( xi ) − ρ B ( xi )) + (νA ( xi ) − νB ( xi ))|2 ,      n    | − t(ρ A ( xi ) − ρ B ( xi )) − (νA ( xi ) − νB ( xi )) + (µ A ( xi ) − µ B ( xi ))|2 , ∑ ωi max    i  i =1   | − t(νA ( xi ) − νB ( xi )) − (ρ A ( xi ) − ρ B ( xi )) + (µ A ( xi ) − µ B ( xi ))|2  (17) Proposition 9. The distance d1H ( A, B) defined in Equation (14) for two SVNSs A and B is a valid distance measure. Proof. The above measure satisfies the following properties: (P1) As A and B are SVNSs, so | µ A ( xi ) − µ B ( xi ) |≤ 1, | ρ A ( xi ) − ρ B ( xi ) |≤ 1 and | νA ( xi ) − νB ( xi ) |≤ 1. Thus, | (tµ A ( xi ) − νA ( xi ) − ρ A ( xi )) − (tµ B ( xi ) − νB ( xi ) − ρ B ( xi )) |≤ (2 + t) | (tρ A ( xi ) + νA ( xi ) − µ A ( xi )) − (tρ B ( xi ) + νB ( xi ) − µ B ( xi )) |≤ (2 + t) | (tνA ( xi ) + ρ A ( xi ) − µ A ( xi )) − (tνB ( xi ) + ρ B ( xi ) − µ B ( xi )) |≤ (2 + t) (P2) (P3) (P4) Hence, by the definition of d1H , we obtain 0 ≤ d1H ( A, B) ≤ 1. Similar to the proof of Proposition 1. This is clear from Equation (14). Let A ⊆ B ⊆ C, which implies µ A ( xi ) ≤ µ B ( xi ) ≤ µC ( xi ), ρ A ( xi ) ≥ ρ B ( xi ) ≥ ρC ( xi ) and νA ( xi ) ≥ νB ( xi ) ≥ νC ( xi ). Therefore, | − t(µ A ( xi ) − µ B ( xi )) + (ρ A ( xi ) − ρ B ( xi )) + (νA ( xi ) − νB ( xi ))| ≤ | − t(µ A ( xi ) − µc ( xi )) + (ρ A ( xi ) − ρC ( xi )) + (νA ( xi ) − νC ( xi ))|, | − t(ρ A ( xi ) − ρ B ( xi )) − (νA ( xi ) − νB ( xi )) + (µ A ( xi ) − µ B ( xi ))| ≤ | − t(ρ A ( xi ) − ρC ( xi )) − (νA ( xi ) − νC ( xi )) + (µ A ( xi ) − µC ( xi ))| and | − t(νA ( xi ) − νB ( xi )) − (ρ A ( xi ) − ρ B ( xi )) + (µ A ( xi ) − µ B ( xi ))| ≤ | − t(νA ( xi ) − νC ( xi )) − (ρ A ( xi ) − ρC ( xi )) + (µ A ( xi ) − µC ( xi ))|, which implies that max | − i t(µ A ( xi ) − µ B ( xi )) + (ρ A ( xi ) − ρ B ( xi )) + (νA ( xi ) − νB ( xi ))|, | − t(ρ A ( xi ) − ρ B ( xi )) − (νA ( xi ) − νB ( xi )) + (µ A ( xi ) − µ B ( xi ))|, | − t(νA ( xi ) − νB ( xi )) − (ρ A ( xi ) − ρ B ( xi )) + (µ A ( xi ) − µ B ( xi ))| ≤ max | − t(µ A ( xi ) − µc ( xi )) + (ρ A ( xi ) − ρC ( xi )) + (νA ( xi ) − νC ( xi ))|, | − t(ρ A ( xi ) − ρC ( xi )) − i (νA ( xi ) − νC ( xi )) + (µ A ( xi ) − µC ( xi ))| and | − t(νA ( xi ) − νC ( xi )) − (ρ A ( xi ) − ρC ( xi )) + (µ A ( xi ) − µC ( xi ))| . Hence d1H ( A, B) ≤ d1H ( A, C ). Similarly, we obtain d1H ( B, C ) ≤ d1H ( A, C ). Proposition 10. For A, B ∈ Φ( X ), d2H , d3H and d4H are the distance measures. Information 2017, 8, 162 11 of 20 Proof. The proof follows from the above proposition. Proposition 11. The measures d2H and d1H satisfy the following inequality: d2H ≤ d1H . Proof. Because wi ∈ [0, 1], therefore d2H ( A, B) = ≤ = n 1 wi max | − t(µ A ( xi ) − µ B ( xi )) + (ρ A ( xi ) − ρ B ( xi )) + (νA ( xi ) − νB ( xi ))|, ∑ 3n(2 + t) i=1 i | − t(ρ A ( xi ) − ρ B ( xi )) − (νA ( xi ) − νB ( xi )) + (µ A ( xi ) − µ B ( xi ))|, | − t(νA ( xi ) ! −νB ( xi )) − (ρ A ( xi ) − ρ B ( xi )) + (µ A ( xi ) − µ B ( xi ))| n 1 max | − t(µ A ( xi ) − µ B ( xi )) + (ρ A ( xi ) − ρ B ( xi )) + (νA ( xi ) − νB ( xi ))|, ∑ 3n(2 + t) i=1 i | − t(ρ A ( xi ) − ρ B ( xi )) − (νA ( xi ) − νB ( xi )) + (µ A ( xi ) − µ B ( xi ))|, | − t(νA ( xi ) −νB ( xi )) − (ρ A ( xi ) − ρ B ( xi )) + (µ A ( xi ) − µ B ( xi ))| d1H ( A, B) Hence, d2H ≤ d1H . Proposition 12. The measures d3H and d4H satisfy the inequality d4H ( A, B) ≤ d3H ( A, B). Proof. The proof follows from Proposition 11. Proposition 13. The measures d3H and d1H satisfy the inequality d3H ≤ q d1H . Proof. Because for any a ∈ [0, 1], a2 ≤ a ≤ a1/2 , the remaining proof follows from Proposition 7. Proposition 14. The measures d4H and d2H satisfy the inequality d4H ≤ q d2H . Proof. The proof follows from Proposition 13. Proposition 15. The measures d1H and d2 satisfy the following inequality: d1H ≤ d2 .Proof. For positive numbers a , i = 1, 2, ..., n, we have max{ a } ≤ n a . Thus, for any two ∑ i i i i i =1 SVNSs A and B, we have d1H ( A, B) = 3n(21+t) ∑in=1 maxi | − t(µ A ( xi ) − µ B ( xi )) + (ρ A ( xi ) −ρ B ( xi )) + (νA ( xi ) − νB ( xi ))|, | − t(ρ A ( xi ) − ρ B ( xi )) −(νA (xi ) − νB ( xi )) + (µ A ( xi ) − µ B ( xi ))|, | − t(νA ( xi ) − ∑in=1 | − t(µ A ( xi ) − µ B ( xi )) + (ρ A ( xi ) − ρ B ( xi )) + (νA ( xi ) − νB ( xi )) + | − t(ρ A ( xi ) − ρ B ( xi )) − (νA ( xi ) − νB ( xi )) + (µ A ( xi ) − µ B ( xi ))| + | − t(νA ( xi ) − νB ( xi )) − (ρ A ( xi ) − ρ B ( xi )) + (µ A ( xi ) − µ B ( xi ))| = d2 ( A, B). Hence d1H ≤ d2 . νB ( xi )) − (ρ A ( xi ) − ρ B ( xi )) + (µ A ( xi ) − µ B ( xi ))| ≤ 1 3n(2+t) Proposition 16. The measures d3H and d4 satisfy the following inequality: d3H ≤ d4 Proof. The proof follows from Proposition 15. Proposition 17. The measures d2 , d5 and d1H satisfy the following inequalities: Information 2017, 8, 162 (i) (ii) 12 of 20 d +d H d2 ≥ q5 2 1 ; d2 ≥ d5 · d1H . Proof. Because d2 ≥ d5 and d2 ≥ d1H , by adding these inequalities, we obtain d2 ≥ q other hand, by multiplying these, we obtain d2 ≥ d5 · d1H . d5 + d1H . 2 On the 4. Generalized Distance Measure The above-defined Hamming and Euclidean distance measures are generalized for the two SVNSs A and B on the universal set X as follows: p d ( A, B) = ( n 1 | − t(µ A ( xi ) − µ B ( xi )) + (ρ A ( xi ) − ρ B ( xi )) + (νA ( xi ) − νB ( xi ))| p ∑ 3n(2 + t) p i=1 +| − t(ρ A ( xi ) − ρ B ( xi )) − (νA ( xi ) − νB ( xi )) + (µ A ( xi ) − µ B ( xi ))| p +| − t(νA ( xi ) − νB ( xi )) − (ρ A ( xi ) − ρ B ( xi )) + (µ A ( xi ) − µ B ( xi ))| p (18) )1/p where p ≥ 1 is an L p norm and t ≥ 3 represents the uncertainty index parameters. In particular, if p = 1 and p = 2, then the above measure, given in Equation (18), reduces to measures d2 and d4 defined in Equations (9) and (11), respectively. Proposition 18. The above-defined distance d p ( A, B), between SVNSs A and B, satisfies the following properties (P1)–(P4): (P1) (P2) (P3) (P4) 0 ≤ d p ( A, B) ≤ 1, ∀ A, B ∈ Φ( X ); d p ( A, B) = 0, iff A = B; d p ( A, B) = d p ( B, A); If A ⊆ B ⊆ C, then d p ( A, C ) ≥ d p ( A, B) and d p ( A, C ) ≥ d p ( B, C ). Proof. For p ≥ 1 and t ≥ 3, we have the following: (P1) For SVNSs, | µ A ( xi ) − µ B ( xi ) |≤ 1, | ρ A ( xi ) − ρ B ( xi ) |≤ 1 and | νA ( xi ) − νB ( xi ) |≤ 1. Thus, we obtain −(2 + t) ≤ t(µ A ( xi ) − µ B ( xi )) − (νA ( xi ) − νB ( xi )) − (ρ A ( xi ) − ρ B ( xi )) ≤ (2 + t) −(2 + t) ≤ −t(ρ A ( xi ) − ρ B ( xi )) − (νA ( xi ) − νB ( xi )) + (µ A ( xi ) − µ B ( xi )) ≤ (2 + t) −(2 + t) ≤ −t(νA ( xi ) − νB ( xi )) − (ρ A ( xi ) − ρ B ( xi )) + (µ A ( xi )) − µ B ( xi ) ≤ (2 + t) which implies that 0 ≤ t(µ A ( xi ) − µ B ( xi )) − (νA ( xi ) − νB ( xi )) − (ρ A ( xi ) − ρ B ( xi )) p 0 ≤ − t(νA ( xi ) − νB ( xi )) − (ρ A ( xi ) − ρ B ( xi )) + (µ A ( xi )) − µ B ( xi ) p 0 ≤ − t(ρ A ( xi ) − ρ B ( xi )) − (νA ( xi ) − νB ( xi )) + (µ A ( xi ) − µ B ( xi )) (P2) p ≤ (2 + t ) p ≤ (2 + t ) p ≤ (2 + t ) p Thus, by adding these inequalities, we obtain 0 ≤ d p ( A, B) ≤ 1. Assuming that A = B ⇔ µ A ( x ) = µ B ( xi ), ρ A ( xi ) = ρ B ( xi ), and νA ( x ) = νB ( xi ), thus, d p ( A, B) = 0. Information 2017, 8, 162 13 of 20 Conversely, assuming that d p ( A, B) = 0 implies that | − t(µ A ( xi ) − µ B ( xi )) + (ρ A ( xi ) − ρ B ( xi )) + (νA ( xi ) − νB ( xi ))| = 0 | − t(ρ A ( xi ) − ρ B ( xi )) − (νA ( xi ) − νB ( xi )) + (µ A ( xi ) − µ B ( xi ))| = 0 | − t(νA ( xi ) − νB ( xi )) − (ρ A ( xi ) − ρ B ( xi )) + (µ A ( xi ) − µ B ( xi ))| = 0 and hence, after solving, we obtain µ A ( xi ) = µ B ( xi ), ρ A ( xi ) = ρ B ( xi ) and νA ( xi ) = νB ( xi ). Thus, A = B. This is straightforward. Let A ⊆ B ⊆ C; then µ A ( xi ) ≤ µ B ( xi ) ≤ µC ( xi ), ρ A ( xi ) ≥ ρ B ( xi ) ≥ ρC ( xi ) and νA ( xi ) ≥ νB ( xi ) ≥ νC ( xi ). Thus, µ A ( xi ) − µ B ( xi ) ≥ µ A ( xi ) − µC ( xi ), ρ A ( xi ) − ρ B ( xi ) ≤ ρ A ( xi ) − ρC ( xi ) and νA ( xi ) − νB ( xi ) ≤ νA ( xi ) − νC ( xi ). Hence, we obtain (P3) (P4) | − t(µ A ( xi ) − µ B ( xi )) + (ρ A ( xi ) − ρ B ( xi )) + (νA ( xi ) − νB ( xi ))| p ≤ | − t(µ A ( xi ) − µc ( xi )) + (ρ A ( xi ) − ρC ( xi )) + (νA ( xi ) − νC ( xi ))| p | − t(ρ A ( xi ) − ρ B ( xi )) − (νA ( xi ) − νB ( xi )) + (µ A ( xi ) − µ B ( xi ))| p ≤ | − t(ρ A ( xi ) − ρC ( xi )) − (νA ( xi ) − νC ( xi )) + (µ A ( xi ) − µC ( xi ))| p | − t(νA ( xi ) − νB ( xi )) − (ρ A ( xi ) − ρ B ( xi )) + (µ A ( xi ) − µ B ( xi ))| p and ≤ | − t(νA ( xi ) − νC ( xi )) − (ρ A ( xi ) − ρC ( xi )) + (µ A ( xi ) − µC ( xi ))| p Thus, we obtain d p ( A, B) ≤ d p ( A, C ). Similarly, d p ( B, C ) ≤ d p ( A, C ). If the weight vector ωi , (i = 1, 2, . . . , n) of each element is considered such that ωi ∈ [0, 1] and ∑i ωi = 1, then a generalized parametric distance measure between SVNSs A and B takes the following form: p dw ( A, B) = 1 3n(2 + t) p n ∑ ωi i =1 ( | − t(µ A ( xi ) − µ B ( xi )) + (ρ A ( xi ) − ρ B ( xi )) + (νA ( xi ) − νB ( xi ))| p +| − t(ρ A ( xi ) − ρ B ( xi )) − (νA ( xi ) − νB ( xi )) + (µ A ( xi ) − µ B ( xi ))| p +| − t(νA ( xi ) − νB ( xi )) − (ρ A ( xi ) − ρ B ( xi )) + (µ A ( xi ) − µ B ( xi ))| p )!1/p (19) In particular, if p = 1 and p = 2, Equation (19) reduces to Equations (12) and (13), respectively. Proposition 19. Let ω = (ω1 , ω2 , . . . , ωn ) T be the weight vector of xi , (i = 1, 2, . . . , n) with ωi ≥ 0 n and ∑ ωi = 1; then the generalized parametric distance measure between the SVNSs A and B defined by i =1 Equation (19) satisfies the following: (P1) (P2) (P3) (P4) p 0 ≤ dw ( A, B) ≤ 1, ∀ A, B ∈ Φ( X ); p dw ( A, B) = 0 iff A = B; p p dw ( A, B) = dw ( B, A); p p p p A ⊆ B ⊆ C then dw ( A, C ) ≥ dw ( A, B) and dw ( A, C ) ≥ dw ( B, C ). Proof. The proof follows from Proposition 18. 5. Illustrative Examples In order to illustrate the performance and validity of the above-proposed distance measures, two examples from the fields of pattern recognition and medical diagnosis have been taken into account. Information 2017, 8, 162 14 of 20 5.1. Example 1: Application of Distance Measure in Pattern Recognition Consider three known patterns A1 , A2 and A3 , which are represented by the following SVNSs in a given universe X = { x1 , x2 , x3 , x4 }: A1 = {h x1 , 0.7, 0.0, 0.1i, h x2 , 0.6, 0.1, 0.2i, h x3 , 0.8, 0.7, 0.6i, h x4 , 0.5, 0.2, 0.3i} A2 = {h x1 , 0.4, 0.2, 0.3i, h x2 , 0.7, 0.1, 0.0i, h x3 , 0.1, 0.1, 0.6i, h x4 , 0.5, 0.3, 0.6i} A3 = {h x1 , 0.5, 0.2, 0.2i, h x2 , 0.4, 0.1, 0.2i, h x3 , 0.1, 0.1, 0.4i, h x4 , 0.4, 0.1, 0.2i} Consider an unknown pattern B ∈ SV NS( X ), which will be recognized where B = {h x1 , 0.4, 0.1, 0.4i, h x2 , 0.6, 0.1, 0.1i, h x3 , 0.1, 0.0, 0.4i, h x4 , 0.4, 0.4, 0.7i} Then the target of this problem is to classify the pattern B in one of the classes A1 , A2 or A3 . For this, proposed distance measures, d1 , d2 , d3 , d4 , d1H and d3H , have been computed from B to Ak (k = 1, 2, 3) corresponding to t = 3, and the results are given as follows: d1 ( A1 , B) = 0.5600; d1 ( A2 , B) = 0.2932; d1 ( A3 , B) = 0.4668 d2 ( A1 , B) = 0.1400; d2 ( A2 , B) = 0.0733; d2 ( A3 , B) = 0.1167 d3 ( A1 , B) = 0.3499; d3 ( A2 , B) = 0.1641; d3 ( A3 , B) = 0.3120 d4 ( A1 , B) = 0.1749; H d1 ( A1 , B) = 0.0633; d3H ( A1 , B) = 0.1252; d4 ( A2 , B) = 0.0821; H d1 ( A2 , B) = 0.0300; d3H ( A2 , B) = 0.0560; d4 ( A3 , B) = 0.1560 d1H ( A3 , B) = 0.0567 d3H ( A3 , B) = 0.1180 Thus, from these distance measures, we conclude that the pattern B belongs to the pattern A2 . On the other hand, if we assume that the weights of x1 , x2 , x3 and x4 are 0.3, 0.4, 0.2 and 0.1, respectively, then we utilize the distance measures d5 , d6 , d2H and d4H for obtaining the most suitable pattern as follows: d5 ( A1 , B) = 0.0338; d5 ( A2 , B) = 0.0162; d5 ( A3 , B) = 0.0233 d6 ( A1 , B) = 0.0861; d6 ( A2 , B) = 0.0369; d6 ( A3 , B) = 0.0604 d2H ( A1 , B) = 0.0148; d2H ( A2 , B) = 0.0068; d2H ( A3 , B) = 0.0117 d4H ( A1 , B) = 0.0603; d4H ( A2 , B) = 0.0258; d4H ( A3 , B) = 0.0464 Thus, the ranking order of the three patterns is A2 , A3 and A1 , and hence A2 is the most desirable pattern to be classified with B. Furthermore, it can be easily verified that these results validate the above-proposed propositions on the distance measures. Comparison of Example 1 Results with Existing Measures The above-mentioned measures have been compared with some existing measures under a NS environment for showing the validity of the approach whose results are summarized in Table 1. From these results, it has been shown that the final ordering of the pattern coincides with the proposed measures, and hence it shows the conservative nature of the measures. Information 2017, 8, 162 15 of 20 Table 1. Ordering value of Example 1. Methods Measure Value of B from A1 A2 A3 Ranking Order D H (defined in Equation (1)) [19] Correlation coefficient [19] D NE (defined in Equation (3)) [20] SCS1 (defined in Equation (4)) [22] SCS2 (defined in Equation (5)) [22] ST1 (defined in Equation (6)) [42] ST2 (defined in Equation (7)) [42] 0.3250 0.7883 0.5251 0.8209 0.8949 0.7275 0.9143 A2 A2 A1 A2 A2 A2 A2 0.1250 0.9675 0.7674 0.9785 0.9911 0.9014 0.9673 0.2500 0.8615 0.6098 0.8992 0.9695 0.7976 0.9343 ≻ ≻ ≻ ≻ ≻ ≻ ≻ A3 A3 A3 A3 A3 A3 A3 ≻ ≻ ≻ ≻ ≻ ≻ ≻ A1 A1 A2 A1 A1 A1 A1 5.2. Example 2: Application of Distance Measure in Medical Diagnosis Consider a set of diseases Q = { Q1 (Viral fever), Q2 (Malaria), Q3 (Typhoid), Q4 (Stomach Problem), Q5 (Chest problem)} and a set of symptoms S = {s1 (Temperature), s2 (HeadAche), s3 (Stomach Pain), s4 (Cough), s5 (Chest pain)}. Suppose a patient, with respect to all the symptoms, can be represented by the following SVNS: P(Patient) = {(s1 , 0.8, 0.2, 0.1), (s2 , 0.6, 0.3, 0.1), (s3 , 0.2, 0.1, 0.8), (s4 , 0.6, 0.5, 0.1), (s5 , 0.1, 0.4, 0.6)} and each diseases Qk (k = 1, 2, 3, 4, 5) is as follows: Q1 (Viral fever) = {(s1 , 0.4, 0.6, 0.0), (s2 , 0.3, 0.2, 0.5), (s3 , 0.1, 0.3, 0.7), (s4 , 0.4, 0.3, 0.3), (s5 , 0.1, 0.2, 0.7)} Q2 (Malaria) = {(s1 , 0.7, 0.3, 0.0), (s2 , 0.2, 0.2, 0.6), (s3 , 0.0, 0.1, 0.9), (s4 , 0.7, 0.3, 0.0), (s5 , 0.1, 0.1, 0.8)} Q3 (Typhoid) = {(s1 , 0.3, 0.4, 0.3), (s2 , 0.6, 0.3, 0.1), (s3 , 0.2, 0.1, 0.7), (s4 , 0.2, 0.2, 0.6), (s5 , 0.1, 0.0, 0.9)} Q4 (Stomach problem) = {(s1 , 0.1, 0.2, 0.7), (s2 , 0.2, 0.4, 0.4), (s3 , 0.8, 0.2, 0.0), (s4 , 0.2, 0.1, 0.7), (s5 , 0.2, 0.1, 0.7)} Q5 (Chest problem) = {(s1 , 0.1, 0.1, 0.8), (s2 , 0.0, 0.2, 0.8), (s3 , 0.2, 0.0, 0.8), (s4 , 0.2, 0.0, 0.8), (s5 , 0.8, 0.1, 0.1)} Now, the target is to diagnose the disease of patient P among Q1 , Q2 , Q3 , Q4 and Q5 . For this, proposed distance measures, d1 , d2 , d3 , d4 , d1H and d3H , have been computed from P to Qk (k = 1, 2, . . . , 5) and are given as follows: d1 ( Q1 , P) = 0.6400; d1 ( Q2 , P) = 0.9067; d1 ( Q3 , P) = 0.6333; d1 ( Q4 , P) = 1.4600; d1 ( Q5 , P) = 1.6200 d2 ( Q1 , P) = 0.1280; d2 ( Q2 , P) = 0.1813; d2 ( Q3 , P) = 0.1267; d2 ( Q4 , P) = 0.2920; d2 ( Q5 , P) = 0.3240 d3 ( Q1 , P) = 0.3626; d3 ( Q2 , P) = 0.4977; d3 ( Q3 , P) = 0.4113; d3 ( Q4 , P) = 0.7566; d3 ( Q5 , P) = 0.8533 d4 ( Q4 , P) = 0.3383; d4 ( Q5 , P) = 0.3816 d4 ( Q1 , P) = 0.1622; d1H ( Q1 , P) = 0.0613; d3H ( Q1 , P) = 0.1175; d4 ( Q2 , P) = 0.2226; d1H ( Q2 , P) = 0.0880; d3H ( Q2 , P) = 0.1760; d4 ( Q3 , P) = 0.1840; d1H ( Q3 , P) = 0.0627; d3H ( Q3 , P) = 0.1373; = 0.1320; d1H ( Q5 , P) = 0.1400 d3H ( Q4 , P) = 0.2439; d3H ( Q5 , P) = 0.2661 d1H ( Q4 , P) Thus, from these distance measures, we conclude that the patient P suffers from the disease Q3 . On the other hand, if we assign weights 0.3, 0.2, 0.2, 0.1 and 0.2 corresponding to Qk (k = 1, 2, . . . , 5), respectively, then we utilize the distance measures d5 , d6 , d2H and d4H for obtaining the most suitable pattern as d5 ( Q1 , P) = 0.0284; d6 ( Q1 , P) = 0.0795; d2H ( Q1 , P) = 0.0135; d4H ( Q1 , P) = 0.0572; d5 ( Q2 , P) = 0.0403; d6 ( Q2 , P) = 0.1101; d2H ( Q2 , P) = 0.0200; d4H ( Q2 , P) = 0.0885; d5 ( Q3 , P) = 0.0273; d6 ( Q3 , P) = 0.0862; d2H ( Q3 , P) = 0.0129; d4H ( Q3 , P) = 0.0636; d5 ( Q4 , P) = 0.0625; d5 ( Q5 , P) = 0.0684 d6 ( Q4 , P) = 0.1599; d6 ( Q5 , P) = 0.1781 d2H ( Q4 , P) = 0.0276; d2H ( Q5 , P) = 0.0289 d4H ( Q4 , P) = 0.1139; d4H ( Q5 , P) = 0.1226 Thus, on the basis of the ranking order, we conclude that the patient P suffers from the disease Q3 . Information 2017, 8, 162 16 of 20 Comparison of Example 2 Results with Existing Approaches In order to verify the feasibility of the proposed decision-making approach based on the distance measure, we conducted a comparison analysis based on the same illustrative example. For this, various measures as presented in Equations (1)–(7) were taken, and their corresponding results are summarized in Table 2, which shows that the patient P suffers from the disease Q1 . Table 2. Comparison of diagnosis result using existing measures. Approach D H (defined in Equation (1)) [19] Correlation [19] Distance measure [27] p=1 p=2 p=3 p=5 D NH (defined in Equation (2)) [20] D NH (defined in Equation (3)) [20] SCS1 (defined in Equation (4)) [22] SCS1 (defined in Equation (5)) [22] ST1 (defined in Equation (6)) [42] ST1 (defined in Equation (7)) [42] Ranking Order Q1 ≻ Q3 ≻ Q2 ≻ Q4 ≻ Q5 Q1 ≻ Q2 ≻ Q3 ≻ Q4 ≻ Q5 Q3 Q1 Q1 Q1 ≻ ≻ ≻ ≻ Q1 Q3 Q3 Q3 ≻ ≻ ≻ ≻ Q3 Q2 Q3 Q3 ≻ ≻ ≻ ≻ Q2 Q2 Q2 Q2 ≻ ≻ ≻ ≻ Q2 Q3 Q2 Q2 ≻ ≻ ≻ ≻ Q4 Q4 Q4 Q4 ≻ ≻ ≻ ≻ Q4 Q4 Q4 Q4 ≻ ≻ ≻ ≻ Q5 Q5 Q5 Q5 ≻ ≻ ≻ ≻ Q5 Q5 Q5 Q5 Q3 ≻ Q1 ≻ Q2 ≻ Q4 ≻ Q5 Q1 ≻ Q3 ≻ Q2 ≻ Q4 ≻ Q5 Q1 Q1 Q1 Q1 5.3. Effect of the Parameters p and t on the Ordering However, in order to analyze the effect of the parameters t and p on the measure values, an experiment was performed by taking different values of p (p = 1, 1.5, 2, 3, 5, 10) corresponding to a different value of the uncertainty parameter t (t = 3, 5, 7). On the basis of these different pairs of parameters, distance measures were computed, and their results are summarized in Tables 3 and 4, respectively, for Examples 1 and 2 corresponding to different criterion weights. From these, the following have been computed: For a fixed value of p, it has been observed that the measure values corresponding to each alternative increase with the increase in the value of t. On the other hand, by varying the value of t from 3 to 7, corresponding to a fixed value of p, this implies that values of the distance measures of each diagnosis from the patient P increase. (ii) It has also been observed from this table that when the weight vector has been assigned to each criterion weight, then the measure values are less than that of an equal weighting case. (iii) Finally, it is seen from the table that the measured values corresponding to each alternative Qk (k = 1, 2, 3, 4, 5) are conservative in nature. (i) For each pair, the measure values lie between 0 and 1, and hence, on the basis of this, we conclude that the patient P suffers from the Q1 disease. The ranking order for the decision-maker is shown in the table as (13245), which indicates that the order of the different attributes is of the form Q1 ≻ Q3 ≻ Q2 ≻ Q4 ≻ Q5 . Hence Q1 is the most desirable, while Q5 is the least desirable for different values of t and p. Information 2017, 8, 162 17 of 20 Table 3. Results of classification of given sample using proposed distance measure. When Equal Importance Is given to Each Criteria When Weight Vector (0.3, 0.4, 0.2, 0.1) T Is Taken p t d p ( A1 , B ) d p ( A2 , B ) d p ( A3 , B ) Ranking p d w ( A1 , B ) 1 3 5 7 0.1400 0.1667 0.1815 0.0733 0.0762 0.0778 0.1167 0.1214 0.1241 A2 ≻ A3 ≻ A1 A2 ≻ A3 ≻ A1 A2 ≻ A3 ≻ A1 0.0338 0.0387 0.0414 0.0162 0.0170 0.0175 0.0233 0.0248 0.0256 1.5 3 5 7 0.1598 0.1924 0.2116 0.0783 0.0817 0.0838 0.1374 0.1437 0.1480 0.0620 0.0723 0.0784 0.0277 0.0293 0.0304 0.0426 0.0452 0.0469 2 3 5 7 0.1749 0.2137 0.2374 0.0821 0.0859 0.0885 0.1560 0.1646 0.1705 0.0861 0.1021 0.1120 0.0369 0.0392 0.0408 0.0604 0.0644 0.0671 3 3 5 7 0.1970 0.2469 0.2785 0.0880 0.0929 0.0962 0.1875 0.02012 0.2098 0.1229 0.1497 0.1672 0.0507 0.0543 0.0566 0.0927 0.1000 0.1046 5 3 5 7 0.2240 0.2902 0.3326 0.0967 0.1041 0.1087 0.2314 0.2526 0.2650 0.1680 0.2128 0.2426 0.0689 0.0749 0.0786 0.1469 0.1605 0.1685 10 3 5 7 0.2564 0.3421 0.3942 0.1107 0.1231 0.1304 0.2830 0.3131 0.3301 0.2203 0.2915 0.3356 0.0939 0.1047 0.1109 0.2248 0.2487 0.2622 A2 ≻ A3 ≻ A1 A2 ≻ A3 ≻ A1 A2 ≻ A3 ≻ A1 A2 ≻ A3 ≻ A1 A2 ≻ A3 ≻ A1 A2 ≻ A3 ≻ A1 A2 ≻ A3 ≻ A1 A2 ≻ A3 ≻ A1 A2 ≻ A3 ≻ A1 A2 ≻ A1 ≻ A3 A2 ≻ A3 ≻ A1 A2 ≻ A3 ≻ A1 A2 ≻ A1 ≻ A3 A2 ≻ A3 ≻ A1 A2 ≻ A3 ≻ A1 p p d w ( A2 , B ) d w ( A3 , B ) Ranking A2 ≻ A3 ≻ A1 A2 ≻ A3 ≻ A1 A2 ≻ A3 ≻ A1 A2 ≻ A3 ≻ A1 A2 ≻ A3 ≻ A1 A2 ≻ A3 ≻ A1 A2 ≻ A3 ≻ A1 A2 ≻ A3 ≻ A1 A2 ≻ A3 ≻ A1 A2 ≻ A3 ≻ A1 A2 ≻ A3 ≻ A1 A2 ≻ A3 ≻ A1 A2 ≻ A3 ≻ A1 A2 ≻ A3 ≻ A1 A2 ≻ A3 ≻ A1 A2 ≻ A1 ≻ A3 A2 ≻ A3 ≻ A1 A2 ≻ A3 ≻ A1 Table 4. Diagnosis result on basis of proposed distance measure. When Weight Vector (0.3, 0.2, 0.2, 0.1, 0.2) T is Taken When Equal Importance Is Given to Each Criteria d p (Q 1 , P) d p (Q 2 , P) d p (Q 3 , P) d p (Q 4 , P) d p (Q 5 , P) p d w ( Q1 , P ) d w ( Q2 , P ) p d w ( Q3 , P ) p d w ( Q4 , P ) p d w ( Q5 , P ) p p t 1 3 5 7 0.1280 0.1410 0.1481 0.1813 0.1867 0.1896 0.1267 0.1457 0.1563 0.2920 0.3076 0.3178 0.3240 0.3400 0.3489 0.0284 0.0304 0.0315 0.0403 0.0413 0.0419 0.0273 0.0300 0.0315 0.0625 0.0643 0.0656 0.0684 0.0700 0.070 1.5 3 5 7 0.1465 0.1612 0.1711 0.2023 0.2131 0.2205 0.1600 0.1794 0.1916 0.3175 0.3364 0.3492 0.3574 0.3778 0.3913 0.0553 0.0598 0.0630 0.0768 0.0808 0.0836 0.0579 0.0628 0.0658 0.1154 0.1202 0.1237 0.1282 0.1334 0.1369 2 3 5 7 0.1622 0.1787 0.1895 0.2226 0.2391 0.2501 0.1840 0.2038 0.2168 0.3383 0.3609 0.3760 0.3816 0.4052 0.4211 0.0795 0.0867 0.0914 0.1101 0.1183 0.1238 0.0862 0.0928 0.0972 0.1599 0.1686 0.1744 0.1781 0.1872 0.1933 3 3 5 7 0.1870 0.2061 0.2175 0.2601 0.2876 0.3047 0.2163 0.2376 0.2516 0.3715 0.4004 0.4185 0.4142 0.4421 0.4601 0.1182 0.1297 0.1365 0.1662 0.1842 0.1954 0.1312 0.1409 0.1475 0.2276 0.2436 0.2535 0.2509 0.2666 0.2765 5 3 5 7 0.2185 0.2405 0.2529 0.3187 0.3625 0.3877 0.2531 0.2782 0.2940 0.4170 0.4531 0.4740 0.4504 0.4826 0.5023 0.1675 0.1841 0.1934 0.2471 0.2817 0.3016 0.1892 0.2045 0.2145 0.3127 0.3384 0.3532 0.3354 0.3588 0.3729 10 3 5 7 0.2519 0.2771 0.2912 0.3980 0.4586 0.4624 0.2969 0.3271 0.3451 0.4731 0.5170 0.5420 0.4896 0.5252 0.5466 0.2215 0.2434 0.2556 0.3524 0.4063 0.4363 0.2599 0.2840 0.2981 0.4095 0.4464 0.4675 0.4235 0.4547 0.4730 5.4. Advantages of the Proposed Method According to the above comparison analysis, the proposed method for addressing decision-making problems has the following advantages: (i) The distance measure under the IFS environment can only handle situations in which the degree of membership and non-membership is provided to the decision-maker. This kind of measure is unable to deal with indeterminacy, which commonly occurs in real-life applications. Because SVNSs are a successful tool in handling indeterminacy, the proposed distance measure in the neutrosophic domain can effectively be used in many real applications in decision-making. (ii) The proposed distance measure depends upon two parameters p and t, which help in adjusting the hesitation margin in computing data. The effect of hesitation will be diminished or almost neglected if the value of t is taken very large, and for smaller values of t, the effect of hesitation will rise. Thus, according to requirements, the decision-maker can adjust the parameter to handle Information 2017, 8, 162 18 of 20 incomplete as well as indeterminate information. Therefore, this proposed approach is more suitable for engineering, industrial and scientific applications. (iii) As has been observed from existing studies, various existing measures under NS environments have been proposed by researchers, but there are some situations that cannot be distinguished by these existing measures; hence their corresponding algorithm may give an irrelevant result. The proposed measure has the ability to overcome these flaws; thus it is a more suitable measure to tackle problems. 6. Conclusions SVNSs are applied to problems with imprecise, uncertain, incomplete and inconsistent information existing in the real world. Although several measures already exist to deal with such kinds of information systems, they have several flaws, as described in the manuscript. Here in this article, we overcome these flaws by proposing an alternative way to define new generalized distance measures between the two SVNNs. Further, a family of normalized and weighted normalized Hamming and Euclidean distance measures have been proposed for the SVNSs. Some desirable properties and their relations have been studied in detail. Finally, a decision-making method has been proposed on the basis of these distance measures. To demonstrate the efficiency of the proposed coefficients, numerical examples of pattern recognition as well as medical diagnosis have been taken. A comparative study, as well as the effect of the parameters on the ranking of the alternative, will support the theory and hence demonstrate that the proposed measures are an alternative way to solve the decision-making problems. In the future, we will extend the proposed approach to the soft set environment [43–45], the multiplicative environment [46–48], and other uncertain and fuzzy environments [7,49–53]. Acknowledgments: The authors wish to thank the anonymous reviewers for their valuable suggestions. The second author, Nancy, was supported through the Maulana Azad National Fellowship funded by the University Grant Commission (No. F1-17.1/2017-18/MANF-2017-18-PUN-82613). Author Contributions: H. Garg and Nancy jointly designed the idea of research, planned its development. Nancy reviewed the literature and finding examples. H. Garg wrote the paper. Nancy made a contribution to the case study. H. Garg analyzed the data and checking language. Finally, all the authors have read and approved the final manuscript. Conflicts of Interest: The author declares no conflict of interest. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. Zadeh, L.A. Fuzzy sets. Inf. Control 1965, 8, 338–353. Atanassov, K.T. 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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). information Article Certain Concepts in Intuitionistic Neutrosophic Graph Structures Muhammad Akram * and Muzzamal Sitara Department of Mathematics, University of the Punjab, Quaid-e-Azam Campus, 54590 Lahore, Pakistan; muzzamalsitara@gmail.com * Correspondence: m.akram@pucit.edu.pk Received: 2 November 2017; Accepted: 19 November 2017; Published: 25 November 2017 Abstract: A graph structure is a generalization of simple graphs. Graph structures are very useful tools for the study of different domains of computational intelligence and computer science. In this research paper, we introduce certain notions of intuitionistic neutrosophic graph structures. We illustrate these notions by several examples. We investigate some related properties of intuitionistic neutrosophic graph structures. We also present an application of intuitionistic neutrosophic graph structures. Keywords: graph structure; intuitionistic neutrosophic graph structure; ψ-complement MSC: 03E72; 05C72; 05C78; 05C99 1. Introduction Fuzzy graph models are advantageous mathematical tools for dealing with combinatorial problems of various domains including operations research, optimization, social science, algebra, computer science, environmental science and topology. Fuzzy graphical models are obviously better than graphical models due to natural existence of vagueness and ambiguity. Initially, we needed fuzzy set theory to cope with many complex phenomenons having incomplete information. Fuzzy set theory [1] is a very strong mathematical tool for solving approximate reasoning related problems. These notions describe complex phenomenons very well, which are not properly described using classical mathematics. Atanassov [2] generalized the fuzzy set theory by introducing the notion of intuitionistic fuzzy sets. The intuitionistic fuzzy sets have more describing possibilities as compared to fuzzy sets. An intuitionistic fuzzy set is inventive and more useful due to the existence of non-membership degree. In many situations like information fusion, indeterminacy is explicitly quantified. Smarandache [3] introduced the concept of neutrosophic sets, and he combined the tricomponent logic, non-standard analysis, and philosophy. It is a branch of philosophy which studies the origin, nature and scope of neutralities as well as their interactions with different ideational spectra. Three independent components of neutrosophic set are: truth value, indeterminacy value and falsity value [3]. For convenient use of neutrosophic sets in real-life phenomena, Wang et al. [4] proposed single valued neutrosophic sets, which is a generalization of intuitionistic fuzzy sets [2] and has three independent components having values in a standard unit interval [0, 1]. Ye [5–8] proposed several multi criteria decision-making methods based on neutrosophic sets. Bhowmik and Pal [9,10] introduced the notion of intuitionistic neutrosophic sets. Kauffman [11] introduced fuzzy graphs on the basis of Zadeh’s fuzzy relations [12]. Rosenfeld [13] discussed fuzzy analogue of many graph-theoretic notions. Later on, Bhattacharya [14] gave Information 2017, 8, 154; doi:10.3390/info8040154 www.mdpi.com/journal/information Information 2017, 8, 154 2 of 19 some remarks on fuzzy graphs. The complement of a fuzzy graph was defined by Sunitha and Vijayakumar [15]. Bhutani and Rosenfeld studied the notion of M-strong fuzzy graphs and their properties in [16]. Parvathi et al. defined operations on intuitionistic fuzzy graphs in [17]. Akram and Shahzadi [18] introduced neutrosophic soft graphs with applications. Dinesh and Ramakrishnan [19] introduced the notion of fuzzy graph structures and discussed some related properties. Akram and Akmal [20] introduced the concept of bipolar fuzzy graph structures. Recently, Akram and Sitara [21] introduced the concept of intuitionistic neutrosophic graph structures. Several notions’ graph structures have been studied by the same authors in [22–27]. In this research paper, we introduce certain notions of intuitionistic neutrosophic graph structures and illustrate these notions by examples. We also present an application of intuitionistic neutrosophic graph structures in decision-making. For other notations and applications, readers are referred to [28–45] . 2. Intuitionistic Neutrosophic Graph Structures Sampathkumar [46] introduced the graph structure, which is a generalization of an undirected graph and is quite useful in studying some structures like graphs, signed graphs, labeled graphs and edge colored graphs. Definition 1. [46] A graph structure G = (V, R1 , . . . , Rr ) consists of a non-empty set V together with relations R1 , R2 , . . . , Rr on V, which are mutually disjoint such that each Rh , 1 ≤ h ≤ r is symmetric and irreflexive. One can represent a graph structure G = (V, R1 , ..., Rr ) in the plane, just like a graph where each edge is labeled as Rh , 1 ≤ h ≤ r. Definition 2. [3] An ordered triple < TN , IN , FN > in ]0− , 1+ [ in the universe of discourse V is called neutrosophic set, where TN , IN , FN : V → ]0− , 1+ [, and their sum is without any restriction. Definition 3. [4] An ordered triple < TN , IN , FN > in [0, 1] in a universe of discourse V is called single-valued neutrosophic set, where TN , IN , FN : V → [0, 1], and their sum is restricted between 0 and 3. Definition 4. [47] Let V be a fixed set. A generalized intuitionistic fuzzy set I of V is an object having the form I={(u, µ I (u), νI (u))|u ∈ V }, where the functions µ I (u) :→ [0, 1] and νI (u) :→ [0, 1] define the degree of membership and degree of nonmembership of an element u ∈ V, respectively, such that min{µ I (u), νI (u)} ≤ 0.5, for all u ∈ V. Definition 5. [9,10] An intuitionistic neutrosophic set can be stated as a set having the form I { TI (u), I I (u), FI (u) : u ∈ V }, where = min{ TI (u), I I (u)} ≤ 0.5, min{ FI (u), I I (u)} ≤ 0.5, min{ TI (u), FI (u)} ≤ 0.5, and 0 ≤ TI (u) + I I (u) + FI (u) ≤ 2. Definition 6. Let Ǧ = ( P, P1 , P2 , . . . , Pr ) be a graph structure(GS), and then Ǧi = (O, O1 , O2 and . . . , Or ) is called an intuitionistic neutrosophic graph structure (INGS), if O = < k, T (k), I (k), F (k ) > and Oh = < (k, l ), Th (k, l ), Ih (k, l ), Fh (k, l ) > are intuitionistic neutrosophic sets on P and Ph , respectively, such that 1. 2. Th (k, l ) ≤ T (k ) ∧ T (l ), Th (k, l ) ∧ Ih (k, l ) ≤ 0.5, Ih (k, l ) ≤ I (k ) ∧ I (l ), Fh (k, l ) ≤ F (k) ∨ F (l ); Th (k, l ) ∧ Fh (k, l ) ≤ 0.5, Ih (k, l ) ∧ Fh (k, l ) ≤ 0.5; Information 2017, 8, 154 3. 3 of 19 0 ≤ Th (k, l ) + Ih (k, l ) + Fh (k, l ) ≤ 2, ∀ (k, l ) ∈ Oh , h = 1, 2, . . . , r, where O is an underlying vertex set of Ǧi and Oh (h = 1, 2, . . . , r ) are underlying h-edge sets of Ǧi . Example 1. Consider a GS Ǧ = ( P, P1 , P2 ) such that O, O1 ,O2 are IN subsets of P, P1 , P2 , respectively, where P = { k 1 , k 2 , k 3 , k 4 , k 5 , k 6 , k 7 , k 8 }, P1 = {k1 k2 , k3 k4 , k5 k6 , k3 k7 , k6 k8 }, P2 = {k2 k3 , k4 k5 , k1 k6 , k5 k7 , k2 k8 }. Through direct calculations, it is easy to show that Ǧi = (O, O1 , O2 ) is an INGS of Ǧ as represented in Figure 1. k1 (0.3, 0.4, 0.3) b O 0 b .3) .2 ,0 .4 ,0 .1 (0 k3 O1 0.1 ,0 k8 (0.3, 0.4, 0.3) O 1( ) 0.1 1, ,0 .1, 0.3 . ,0 3) 0. b .3) .3) b .3, O2 (0.1, 0.3, 0.4) .4, .1 (0 ( O1 O 2( 0.3 , .4) .3) ,0 0.4 ) ,0 ,0 b , 0.3 .3 b 0.3 O1 (0.3, 0.1, 0.3) .2, ,0 2 (0 .4 k7 (0.3, 0.4, 0.3) O ,0 b .3) .3 0. ,0 (0 k6 , .2 (0 k2 3, 0.4 ,0 .3, (0 4) O 1 0. 0.1 2 (0 .3 0 .2, (0 ) k5 .4 ,0 .2 ) b (0 O2 k4 (0.2, 0.1, 0.3) Figure 1. An intuitionistic neutrosophic graph structure. Definition 7. Let Ǧi = (O, O1 , O2 , . . . , Or ) be an INGS of Ǧ. If Ȟi = (O′ , O1′ , O2′ , . . . , Or′ ) is an INGS of Ǧ such that T ′ (k ) ≤ T (k), I ′ (k) ≤ I (k), F ′ (k ) ≥ F (k) ∀k ∈ P, Th′ (k, l ) ≤ Th (k, l ), Ih′ (k, l ) ≤ Ih (k, l ), Fh′ (k, l ) ≥ Fh (k, l ), ∀(k, l ) ∈ Ph , h = 1, 2, ..., r. Then, Ȟi is said to be an intuitionistic neutrosophic (IN) subgraph structure of INGS Ǧi . Example 2. Consider an INGS Ȟi = (O′ , O1′ , O2′ ) of GS Ǧ = ( P, P1 , P2 ) as represented in Figure 2. Through routine calculations, it can be easily shown that Ȟi is an IN subgraph structure of INGS Ǧi . Information 2017, 8, 154 4 of 19 k1 (0.2, 0.3, 0.4) b ( O1 O 2( 0.2 .4) , .1, 0.3 0.4 , 0.3 ,0 ) 0.4 ,0 b .3 ,0 .3 ,0 .1 (0 k3 (0 .1, k8 (0.2, 0.3, 0.4) O1 O 1( ) 0.1 1, ,0 .1, 0.4 . ,0 b ) 4) 0. 0.4 .4) 0.3 .3) ,0 b .2, b ,0 2 (0 b .4) 0.1 b ,0 k7 (0.2, 0.4, 0.4) O O2 (0.1, 0.3, 0.4) . (0 k2 0.3 .2, 1, .2, k5 , 1 0. 5) O 1 0. 2 (0 (0 (0 O ) .4 ,0 .3 ,0 O1 (0.2, 0.0, 0.4) .2 (0 k6 .1, , 0.2 ) 0.5 .1 (0 O2 ) b k4 (0.1, 0.1, 0.4) Figure 2. IN subgraph structure. Definition 8. An INGS Ȟi = (O′ , O1′ , O2′ , . . . , Or′ ) is called an IN induced-subgraph structure of Ǧi by Q ⊆ P if T ′ (k) = T (k ), I ′ (k ) = I (k ), F ′ (k) = F (k), ∀k ∈ Q, Th′ (k, l ) = Th (k, l ), Ih′ (k, l ) = Ih (k, l ), Fh′ (k, l ) = Fh (k, l ), ∀k, l ∈ Q, h = 1, 2, . . . , r. Example 3. The INGS in the given Figure 3 is an IN induced-subgraph structure of an INGS in Figure 1. k7 (0.3, 0.4, 0.3) ) .3) ,0 0.4 .1, .3 1 (0 ) O .3) O1 (0.3, 0.1, 0.3) ,0 .3) 1( ,0 ) O .4 ,0 .3 0.3 ,0 ,0 .2 .4, 0.3 (0 O2 b (0 k5 .2 ,0 .4 .3, b k8 (0.3, 0.4, 0.3) ,0 .1 (0 k3 b ,0 0.1 O2 (0.1, 0.3, 0.4) b .4 ,0 .3 b .3, .2 (0 k2 (0 k6 b . ,0 0.1 ) (0 O2 .4 0 3, ) Figure 3. An IN induced-subgraph structure. Definition 9. An INGS Ȟi = (O′ , O1′ , O2′ , . . . , Or′ ) is said to be a IN spanning-subgraph structure of Ǧi if O′ = O and Th′ (k, l ) ≤ Th (k, l ), Ih′ (k, l ) ≤ Ih (k, l ), Fh′ (k, l ) ≥ Fh (k, l ), h = 1, 2, . . . , r. Example 4. An INGS shown in Figure 4 is an IN spanning-subgraph structure of an INGS in Figure 1. Information 2017, 8, 154 5 of 19 k1 (0.3, 0.4, 0.3) b .4) k7 (0.3, 0.4, 0.3) 2 (0 .1, , .1, ( O1 O 2( 0.2 .4) 0.3 ) 0.3 ,0 b .2 ,0 .4 ,0 .1 (0 k3 (0 .1, k8 (0.3, 0.4, 0.3) O1 0.4 , 0.2 ,0 O 1( ) 0.1 1, ,0 .1, 0.5 .4) ,0 b 0.3 . ,0 b 0.3 ) 4) .3) O b ,0 b ,0 b O2 (0.1, 0.2, 0.5) . (0 k2 0.4 0.1 2, .2, k5 , 3 0. 4) O 1 0. 2 (0 .3, (0 O ) .3 ,0 .4 ,0 O1 (0.2, 0.1, 0.4) .3 (0 k6 .1, (0 , 0.1 ) 0.5 0. .1 ) b (0 O2 k4 (0.2, 0.1, 0.3) Figure 4. An IN spanning-subgraph structure. Definition 10. Let Ǧi = (O, O1 , O2 , . . . , Or ) be an INGS. Then, kl ∈ Ph is named as a IN Oh -edge or shortly Oh -edge, if Th (k, l ) > 0 or Ih (k, l ) > 0 or Fh (k, l ) > 0 or all these conditions are satisfied. As a result, support of Oh is: supp(Oh ) = {kl ∈ Oh : Th (k, l ) > 0} ∪ {kl ∈ Oh : Ih (k, l ) > 0} ∪ {kl ∈ Oh : Fh (k, l ) > 0}, h = 1, 2, ..., r. Definition 11. Oh -path in an INGS Ǧi = (O, O1 , O2 , . . . , Or ) is a sequence k1 , k2 , ..., kr of distinct vertices (except kr = k1 ) in P, such that k h−1 k h is an IN Oh -edge ∀h = 2, ..., r. Definition 12. An INGS Ǧi = (O, O1 , O2 , . . . , Or ) is Oh -strong for any h ∈ {1, 2, ..., r } if Th (k, l ) = min{ T (k), T (l )}, Ih (k, l ) = min{ I (k), I (l )}, Fh (k, l ) = max{ F (k ), F (l )}, ∀kl ∈ supp(Oh ). If Ǧi is Oh -strong for all h ∈ {1, 2, . . . , r }, then Ǧi is a strong INGS. Example 5. Consider an INGS Ǧi = (O, O1 , O2 ) as represented in Figure 5. Then, Ǧi is strong INGS, as it is O1 − and O2 − strong. k2 (0.3, 0.3, 0.3) .4) ,0 .3) O b b O1 (0.2, 0.3, 0.5) b , 0. 0.2 k5 ( .4) ,0 0.2 .1, O ) 2 (0 .5 ,0 .2 ,0 .1 (0 O1 4, 0 .3) .5) k4 (0.3, 0.3, 0.4) O2 (0.1, 0.2, 0.4) 4, 0 , 0. 0.2 O1 (0.2, 0.3, 0.4) ,0 0.3 0.3 .2, .3, 1 (0 2 (0 , 0.5) O .2, 0.3 k 3( O2 (0 k1 (0.4, 0.3, 0.4) b O1 (0.3, 0.3, 0.4) b b k6 (0.1, 0.2, 0.4) Figure 5. A strong INGS. Definition 13. An INGS Ǧi = (O, O1 , O2 , . . . , Or ) is a complete INGS, if 1. Ǧi is strong INGS. Information 2017, 8, 154 2. 3. 6 of 19 supp(Oh ) ̸= ∅, for all h = 1, 2, . . . , r. For all k, l ∈ P, kl is a Oh − edge for some h. Example 6. Let Ǧi = (O, O1 , O2 ) be an INGS of GS Ǧ = ( P, P1 , P2 ), such that P = { k 1 , k 2 , k 3 , k 4 , k 5 , k 6 }, P1 = {k1 k6 , k1 k2 , k2 k4 , k2 k5 , k2 k6 , k1 k6 }, P2 = {k2 k6 , k4 k3 , k5 k6 , k1 k4 }, P3 = {k1 k5 , k5 k3 , k2 k3 , k1 k3 , k4 k6 }. By means of direct calculations, it is easy to show that Ǧi is strong INGS. Moreover, supp(O1 ) ̸= ∅, supp(O2 ) ̸= ∅, supp(O3 ) ̸= ∅, and every pair k h k q of vertices of P, is O1 -edge or O2 -edge or an O3 -edge. Hence, Ǧi is a complete INGS, that is, O1 O2 O3 -complete INGS. Definition 14. Let Ǧi = (O, O1 , O2 , . . . , Or ) be an INGS. The truth strength T.POh , falsity strength F.POH , and indeterminacy strength I.POh of an Oh -path, POh = k1 , k2 , . . . , k n is defined as: n ∧ T.POh = i =2 n ∧ I.POh = i =2 n ∨ F.POh = [ TOPh (k i−1 k i )], [ IOP h (k i−1 k i )], i =2 [ FOPh (k i−1 k i )]. Example 7. Consider an INGS Ǧi = (O, O1 , O2 , O3 ) as in Figure 6. We found an O1 -path PO1 = k2 , k1 , k6 . So, T.PO1 = 0.2, I.PO1 = 0.1 and F.PO2 = 0.5. ) O3 (0.2, 0.1, 0.4) O2 (0 .2, 0.3 , 0.4) 0.2 .2, ,0 ) .5) 0.5 ,0 0.2 .3, .5) b k6 (0.3, 0.1, 0.3) (0 O3 O 1( ,0 0.3 .2, O 1 (0 .5) b b .4 ) .4 ,0 .4, 0 ,0 5) b .2, 0 .3 0.3 O1 k3 (0 b ,0 , 0. O1 (0 .2, 0.2 .2, , 0.5) 0.2 ,0 .4) .2, 0.1 0.5) O2 (0.2, 0.1, 0.3) 3 (0 0.3, (0.2, 3 (0 .2 O2 ( (0 .2, O3 (0.2, 0.3, 0.5) k4 1 (0 O k2 (0.3, 0.3, 0.5) O1 (0.2, 0.3, 0.5) O 0.2, 0.3, 0.5) b O k1 (0.2, 0.3, 0.5) k5 (0.2, 0.2, 0.3) Figure 6. A complete INGS. Definition 15. Let Ǧi = (O, O1 , O2 , . . . , Or ) be an INGS. Then, • Oh -strength of connectedness of truth between k and l is defined as: TO∞ (kl ) = TOi (kl ) = ( TOi−1 ◦ TO1 )(kl ) for i ≥ 2 and TO2 (kl ) = ( TO1 ◦ TO1 )(kl ) = h • h h h h h ∨ y h ∨ i ≥1 { TOi h (kl )}, such that ( TO1 h (ky) ∧ TO1 h )(yl ). ∞ ( kl ) = Oh -strength of connectedness of indeterminacy between k and l is defined as: IO h i ( kl ) = ( I i −1 ◦ I 1 )( kl ) for i ≥ 2 and I 2 ( kl ) = ( I 1 ◦ I 1 )( kl ) = that IO O O O O O h h h h h h ∨ 1 ( IO (ky) ∧ y h ∨ { IOi h (kl )}, i ≥1 1 )( yl ). IO i such Information 2017, 8, 154 7 of 19 Oh -strength of connectedness of falsity between k and l is defined as: FO∞ (kl ) = • ∧ FOi (kl ) = ( FOi−1 ◦ FO1 )(kl ) for i ≥ 2 and FQ2 (kl ) = ( FO1 ◦ FO1 )(kl ) = h h h h h h y h ∧ i ≥1 { FOi h (kl )}, such that ( FO1 h (ky) ∨ FO1 h )(yl ). Definition 16. An INGS Ǧi = (O, O1 , O2 , . . . , Or ) is called an Oh -cycle if (supp(O), supp(O1 ), supp(O2 ), . . . , supp(Or )) is an Oh − cycle. Definition 17. An INGS Ǧi = (O, O1 , O2 , . . . , Or ) is an IN fuzzy Oh -cycle (for any h) if 1. 2. Ǧi is an Oh -cycle. There exists no unique Oh -edge kl in Ǧi such that TOh (kl ) = min{ TOh (yz) : yz ∈ Ph = supp(Oh )} or IOh (kl ) = min{ IOh (yz) : yz ∈ Ph = supp(Oh )} or FOh (kl ) = max{ FOh (yz) : yz ∈ Ph = supp(Oh )}. Example 8. Consider an INGS Ǧi = (O, O1 , O2 ) as in Figure 6. Then, Ǧi is an O1 -cycle and IN fuzzy O1 − cycle, since (supp(O), supp(O1 ), supp(O2 )) is an O1 -cycle and no unique O1 -edge kl satisfies the condition: TOh (kl ) = min{ TOh (yz) : yz ∈ Ph = supp(Oh )} or IOh (kl ) = min{ IOh (yz) : yz ∈ Ph = supp(Oh )} or FOh (kl ) = max{ FOh (yz) : yz ∈ Ph = supp(Oh )}. Definition 18. Let Ǧi = (O, O1 , O2 , . . . , Or ) be an INGS and k a vertex in Ǧi . Let (O′ , O1′ , O2′ , . . . , Or′ ) be an IN subgraph structure of Ǧi induced by P \ {k} such that ∀y ̸= k, z ̸= k. TO′ (k) = 0 = IO′ (k) = FO′ (k), TO′ (ky) = 0 = IO′ (ky) = FO′ (ky) ∀ edges ky ∈ Ǧi ; TO′ (y) = TO (y), h h h IO′ (y) = IO (y), FO′ (y) = FO (y), ∀y ̸= k;TO′ (yz) = TOh (yz), IO′ (yz) = IOh (yz), FO′ (yz) = FOh (yz). h h h Then, k is IN fuzzy Oh cut-vertex, for some h, if ∞ ( yz ) > I ∞ ( yz ) TO∞ (yz) > TO∞′ (yz), IO O′ h h h h and FO∞ (yz) > FO∞′ (yz), for some y, z ∈ P \ {k}. h h ∞ ( yz ) > Note that k is an IN fuzzy Oh − T cut-vertex, if TO∞ (yz) > TO∞′ (yz), IN fuzzy Oh − I cut-vertex, if IO h h h ∞ ( yz ) and IN fuzzy O − F cut-vertex, if F ∞ ( yz ) > F ∞ ( yz ). IO ′ h O O′ h h h Example 9. Consider an INGS Ǧi = (O, O1 , O2 ) as represented in Figure 7 and Ǧh′ = (O′ , O1′ , O2′ ) is an IN subgraph structure of an INGS Ǧi , and we found it by deleting the vertex k2 . The vertex k2 is an IN fuzzy O1 -I ∞ ( k k ) = 0 < 0.5 = I ∞ ( k k ), I ∞ ( k k ) = 0.7 = I ∞ ( k k ) and I ∞ ( k k ) = 0.3 < cut-vertex, since IO ′ 2 5 O1 4 3 O1 2 5 O′ 3 5 O′ 4 3 1 1 1 k2 (0.4, 0.7, 0.5) .5, 0.5 O1 ( .7) 0.3, 0.3, 0.4) k1 ( .3, 0.6 , 0.4) 0.3 , 0. 6, 0 .4) b O1 (0.4, 0.4, 0.5) b ,0 O2 (0 O2 (0.1, 0.4, 0.2) b .1, O 2 (0 .2) 0.4, 0 b k 6( O1 (0.3, 0.2, 0.4) ) 0.7 ,0 .5 .5, (0 O1 (0.5, 0.7, 0.5) k4 b 0.4) , 0.5, .3 O 1 (0 O2 (0.2, 0.4, 0.3) b (0 k3 ∞ ( k k ). 0.4 = IO 3 5 1 k5 (0.4, 0.5, 0.6) Figure 7. An INGS Ǧi = (O, O1 , O2 ). .4) 4, 0 , 0. 0.3 Information 2017, 8, 154 8 of 19 Definition 19. Let Ǧi = (O, O1 , O2 , . . . , Or ) be an INGS and kl an Oh − edge. Let (O′ , O1′ , O2′ , . . . , Or′ ) be an IN fuzzy spanning-subgraph structure of Ǧi , such that TO′ (kl ) = 0 = IO′ (kl ) = FO′ (kl ), TO′ (qt) = TOh (qt), IO′ (qt) = IOh (qt), FO′ (qt) = FOh (qt), h h h h h h ∀ edges qt ̸= kl. Then, kl is an IN fuzzy Oh -bridge if ∞ ( yz ) > I ∞ ( yz ) and F ∞ ( yz ) > F ∞ ( yz ), for some y, z ∈ P. TO∞ (yz) > TO∞′ (yz), IO O O′ O′ h h h h h h Note that kl is an IN fuzzy Oh − T bridge if TO∞ (yz) h and IN fuzzy Oh − F bridge if FO∞ (yz) > FO∞′ (yz). h h TO∞′ (yz), h > ∞ ( yz ) > I ∞ ( yz ) IN fuzzy Oh − I bridge if IO O′ h h ′ = (O′′ , O′′ , O′′ ) is IN Example 10. Consider an INGS Ǧi = (O, O1 , O2 ) as shown in Figure 7 and ǦH 2 1 spanning-subgraph structure of an INGS Ǧi found by the deletion of O1 -edge (k2 k5 ). Edge (k2 k5 ) is ∞ ( k k )= 0.3 < 0.4= I ∞ ( k k ), an IN fuzzy O1 -bridge. As TO∞′′ (k2 k5 )= 0.3 < 0.4 = TO∞1 (k2 k5 ), IO ′′ 2 5 O1 2 5 1 1 FO∞′′ (k2 k5 )= 0.4 < 0.5 = FO∞1 (k2 k5 ). 1 Definition 20. An INGS Ǧi = (O, O1 , O2 , . . . , Or ) is an Oh -tree, if (supp(O), supp(O1 ), supp(O2 ), . . . , supp(Or )) is an Oh − tree. Alternatively, Ǧi is an Oh -tree, if there is a subgraph of Ǧi induced by supp(Oh ), which forms a tree. Definition 21. An INGS Ǧi = (O, O1 , O2 , . . . , Or ) is an IN fuzzy Oh -tree if Ǧi has an IN fuzzy spanning-subgraph structure Ȟi = (O′′ , O1′′ , O2′′ , . . . , Or′′ ), such that, for all Oh -edges kl not in Ȟi , ∞ ( kl ), F ( kl ) < F ∞ ( kl ). Ȟi is an Oh′′ -tree, and TOh (kl ) < TO∞′′ (kl ), IOh (kl ) < IO ′′ Oh O′′ h h h In particular, Ǧi is an IN fuzzy Oh -T tree if TOh (kl ) < TO∞′′ (kl ), an IN fuzzy Oh -I tree if h ∞ ( kl ), and an IN fuzzy O -F tree if F ( kl ) > F ∞ ( kl ). IOh (kl ) < IO ′′ Oh h O′′ h h Example 11. Consider an INGS Ǧi = (O, O1 , O2 ) as shown in Figure 8. It is an O2 -tree, not an O1 -tree but it is IN fuzzy O1 -tree because it has an IN fuzzy-spanning subgraph (O′ , O1′ , O2′ ) as an O1′ -tree, which is found by ∞ ( k k ) = 0.3 > 0.1 = the deletion of O1 -edge k2 k5 from Ǧi . Moreover, TO∞′ (k2 k5 ) = 0.3 > 0.2 = TO1 (k2 k5 ), IO ′ 2 5 1 1 IO1 (k2 k5 ) and FO∞′ (k2 k5 ) = 0.4 < 0.5 = FO1 (k2 k5 ). 1 k1 (0.3, 0.6, 0.5) b b , 0.1, 0.5 O2 (0.3, 0.2, 0.3) O1 (0.3, 0.3, 0.4) 0.6 , 0. O 2 (0 .1, 0.4 .5) , 0. 5, 0 .4, 1, 0 1) O 0. 2( .4) 0 b k6 (0.3, 0.4, 0.3) Figure 8. An IN fuzzy O1 -tree. b .4 5, 0 . 4, 0 0. k 5( O 1(0.2 , 0.4) O1 (0 k2 ( 0.4 , 0. 0.6 7, 0 , 0. .5) 2) O1 (0.3, 0.5, 0.5) b k4 ( .2, k3 .5, 0.5 b 2 (0 ) O 4) . ) .4, 0 0.5 .7, (0.2, 0 0 , O2 (0.5 Information 2017, 8, 154 9 of 19 Definition 22. An INGS Ǧi1 = (O1 , O11 , O12 , . . . , O1r ) of graph structure Ǧ1 = ( P1 , P11 , P12 , . . . , P1r ) is said to be isomorphic to an INGS Ǧi2 = (O2 , O21 , O22 , . . . , O2r ) of the graph structure Ǧ2 = ( P2 , P21 , P22 , ..., P2r ), if there is a pair ( g, ψ), where g : P1 → P2 is a bijective mapping and ψ is any permutation on this set {1, 2, . . . , r } such that; TO1 (k ) = TO2 ( g(k )), IO1 (k) = IO2 ( g(k)), FO1 (k) = FO2 ( g(k)), ∀k ∈ P1 , TO1h (kl ) = TO2φ(h) ( g(k) g(l )), IO1h (kl ) = IO2φ(h) ( g(k) g(l ), FQ1h (kl ) = FO2φ(h) ( g(k) g(l )), ∀kl ∈ P1h , h = 1,2,. . . ,r. Example 12. Let Ǧi1 = (O, O1 , O2 ) and Ǧi2 = (O′ , O1′ , O2′ ) be two INGSs as shown in the Figure 9. Ǧi1 and Ǧi2 are isomorphic under ( g, ψ), where g : P → P′ is a bijective mapping and ψ is the permutation on {1, 2}, which is defined as ψ(1) = 2, ψ(2) = 1, and the following conditions hold: TO (k h ) = TO′ ( g(k h )), IO (k h ) = IO′ ( g(k h )), FO (k h ) = FO′ ( g(k h )), ∀k h ∈ P and TOh (k h k q ) = TO′ ( g(k h ) g(k q )), IOh (k h k q ) = IO′ ( g(k h ) g(k q )), FOh (k h k q ) = FO′ ( g(k h ) g(k q )), ψ(h) ψ(h) ψ(h) ∀k h k q ∈ Ph , h = 1, 2. l2 (0.5, 0.5, 0.5) k3 (0.2, 0.7, 0.5) b ) 0.5 .2, 1, 0 1 ( 0. O′ ) , 0.4 , 0.3 0.1 2 5) 2, 0 . ) b l1 (0.3, 0.3, 0.4) O ′( , 0. 0.3 .1, O ) , 0.5 , 0.4 , 0.2 , 0. b O2′ (0.2, 0.2, 0.4) ′ (0.2 O1 , 0.3 (0.2 O1 (0.1 b O2 k4 (0.2, 0.2, 0.5) l4 (0.2, 0.2, 0.5) ) ) b O1 (0.2, 0.2, 0.4) , 0.4 0.5 1 (0 .2, b , 0.3 2, 0 k1 (0.3, 0.4, 0.4) b ′ (0.2 O2 (0. O2 4) b b k2 (0.5, 0.5, 0.5) l3 (0.2, 0.7, 0.5) Figure 9. Two isomorphic INGSs. Definition 23. An INGS Ǧi1 = (O1 , O11 , O12 , . . . , O1r ) of the graph structure Ǧ1 = ( P1 , P11 , P12 , ..., P1r ) is identical with an INGS Ǧi2 = (O2 , O21 , O22 , ..., O2r ) of the graph structure Ǧ2 = ( P2 , P21 , P22 , ..., P2r ) if g : P1 → P2 is a bijective mapping such that TO1 (k ) = TO2 ( g(k )), IO1 (k) = IO2 ( g(k)), FO1 (k) = FO2 ( g(k)), ∀k ∈ P1 , TO1h (kl ) = TO2h ( g(k) g(l )), IO1h (kl ) = IO2h ( g(k) g(l )), FO1h (kl ) = FO2(h) ( g(k) g(l )), Information 2017, 8, 154 10 of 19 ∀kl ∈ P1h , h = 1, 2, . . . , r. Example 13. Let Ǧi1 = (O, O1 , O2 ) and Ǧi2 = (O′ , O1′ , O2′ ) be two INGSs of the GSs Ǧ1 = ( P, P1 , P2 ), Ǧ2 = ( P′ , P1′ , P2′ ), respectively, as they are shown in Figures 10 and 11. SVINGSs Ǧi1 and Ǧi2 are identical under g : P → P′ is defined as : g ( k 1 ) = l2 , g ( k 2 ) = l1 , g ( k 3 ) = l4 , g ( k 4 ) = l3 , g ( k 5 ) = l5 , g ( k 6 ) = l8 , g ( k 7 ) = l7 , g ( k 8 ) = l6 . Moreover, TO (k h ) = TO′ ((k h )), IO (k h ) = IO′ ( g(k h )), FO (k h ) = FO′ ( g(k h )), ∀k h ∈ P and TOh (k h k q ) = TO′ ( g(k h ) g(k q )), IOh (k h k q ) = IO′ ( g(k h ) g(k q )), FOh (k h k q ) = FO′ ( g(k h ) g(k q )), ∀k h k q ∈ Ph , h = 1, 2. h h h .3) 4, 0 0. .5, O1 (0 .4, k ,0 0.b 5 k 2 (0.3, 0.b .5) b .4) , 0.3, 0 O 1 (0.2 .6 (0 k4 .1, 0.2 O 2 (0 k1 (0.2, 0.3, 0.4) , 0.4) O1 (0.3 ) , 0.4, 0 , 0.3, 0.5 .2) O2 (0.3, 0.3, 0.5) O1 (0.3 .6) , 0.2, 0 , 0.3, 0 b .5) O 2 (0.4 O2 (0.1 b ( k8 , 0.4 .4) ,0 .5 ,0 b , 0.3) .2, 0.2 4, 0.5) (0 6 , 0.5, 0 O1 (0.2, 0.3, 0.4) b 0 k 3( .2) O2 k5 (0.5, 0.6, 0.5) b O2 (0.6 , 0.5) (0.5, 0.4 .3) ,0 0.6 k7 (0.5, 0.3, 0.6) Figure 10. An INGS Ǧi1 . l5 (0.5, 0.6, 0.5) b (0.1 (0 .4 b ,0 .5 ,0 .2 ) 2, 0 .2, 0 .4) O ′( 1 1 (0.3 , ′ 0.1 O 2( b 0.3 , 0.2 0.2, , 0. b , 0.4 2, 0 1 (0. O′ 1 (0 . l8 O′ .3, O2 0.6 , 0.2 5, 0 .4) ) ) 0.3 ) , 0.3 l3 ( 0.3 ′ ) , 0.5 0.5) b .6, l1 (0.3, 0.4, 0.5) b b O′ 0.5, ,0 O2 0.6, 0.4 (0 l2 (0.2, 0.3, 0.4) ) O2′ (0.3, 0.3, 0.5) , 0.6 , 0.3 ′ (0.4 ) O2 l6 ( 0.5 2 O1′ (0.2, 0.3, 0.4) ) ,0 ) 0.3 O ′( 0.5) 0.4 l 4( .4, ′ .4, .5, 0 , 0.5 b l7 (0.5, 0.3, 0.6) Figure 11. An INGS Ǧi2 . Definition 24. Let Ǧi = (O, O1 , O2 , ..., Or ) be an INGS and ψ is any permutation on {O1 , O2 , ..., Or } and on set {1, 2, ..., r }, that is, ψ(Oh ) = Oq if and only if ψ(h) = q ∀h. If kl ∈ Oh , for any h and TOψ (kl ) = TO (k ) ∧ TO (l ) − h FOψ (kl ) = FO (k) ∨ FO (l ) − h ∨ Tψ(Oq ) (kl ), IOψ (kl ) = IO (k) ∧ IO (l ) − ∧ Tψ(Oq ) (kl ), h = 1, 2, ..., r, then, kl ∈ q̸=h q̸=h h ∨ Iψ(Oq ) (kl ), q̸=h ψ Ot , where t is chosen such that ψ ψ ψ TOψ (kl ) ≥ TOψ (kl ), IOψ (kl ) ≥ IOψ (kl ), FOψ (kl ) ≥ FOψ (kl ) ∀h. In addition, INGS (O, O1 , O2 , . . . , Or ) is t h t h t h ψc called a ψ-complement of an INGS Ǧi , and it is symbolized as Ǧi . Example 14. Let O = {(k1 , 0.3, 0.4, 0.7), (k2 , 0.5, 0.6, 0.4), (k3 , 0.7, 0.5, 0.3)}, O1 = {(k1 k3 , 0.3, 0.4, 0.3)}, O2 = {(k2 k3 , 0.5, 0.4, 0.3)}, O3 = {(k1 k2 , 0.3, 0.3, 0.4)} be IN subsets of P, P1 , P2 , P3 , respectively. Information 2017, 8, 154 11 of 19 Thus, Ǧi = (O, O1 , O2 , O3 ) is an INGS of GS Ǧ = ( P, P1 , P2 , P3 ). Let ψ(O1 ) = O2 , ψ(O2 ) = O3 , ψ(O3 ) = O1 , where ψ is permutation on {O1 , O2 , O3 }. Now, for k1 k3 , k2 k3 , k1 k2 ∈ O1 , O2 , O3 , respectively: TOψ (k1 k3 ) = 0, IOψ (k1 k3 ) = 0, FOψ (k1 k3 ) = 0.7, TOψ (k1 k3 ) = 0, IOψ (k1 k3 ) = 0, FOψ (k1 k3 ) = 0.7, 1 1 2 1 2 ψ 2 TOψ (k1 k3 ) = 0.3, IOψ (k1 k3 ) = 0.4, FOψ (k1 k3 ) = 0.7. So k1 k3 ∈ O3 , 3 3 3 TOψ (k2 k3 ) = 0.5, IOψ (k2 k3 ) = 0.5, FOψ (k2 k3 ) = 0.4, TOψ (k2 k3 ) = 0, IOψ (k2 k3 ) = 0.1, FOψ (k2 k3 ) = 0.4, 1 1 2 1 2 ψ 2 TOψ (k2 k3 ) = 0, IOψ (k2 k3 ) = 0.1, FOψ (k2 k3 ) = 0.4. So k2 k3 ∈ O1 , 3 3 3 TOψ (k1 k2 ) = 0, IOψ (k1 k2 ) = 0.1, FOψ (k1 k2 ) = 0.7, TOψ (k1 k2 ) = 0.3, IOψ (k1 k2 ) = 0.4, FOψ (k1 k2 ) = 0.7, 1 1 2 1 ψ 2 2 TOψ (k1 k2 ) = 0, IOψ (k1 k2 ) = 0.1, FOψ (k1 k2 ) = 0.7. This shows k1 k2 ∈ O2 . 3 Hence, ψc Ǧi 3 ψ ψ ψ =(O, O1 , O2 , O3 ) is a ψ-complement of an INGS Ǧi as presented in Figure 12. k1 (0.3, 0.4, 0.7) k1 (0.3, 0.4, 0.7) b b 0.3) b O 2( 0.5, 0.4, .7 ,0 .4 0 3, ) . ψ (0 O2 ψ 0.4, k3 (0.7, 0.5, 0.3) 0.3, O3 (0.3, 0.3, 0.4) O1 ( O3 (0.3, 0.4, 0.7) 3 0.3) b b b ψ k2 (0.5, 0.6, 0.4) k2 (0.5, 0.6, 0.4) O1 (0.5, 0.5, 0.4) k3 (0.7, 0.5, 0.3) ψc Figure 12. INGSs Ǧi , Ǧi . Proposition 1. A ψ-complement of an INGS Ǧi = (O, O1 , O2 , . . . , Or ) is a strong INGS. Moreover, ψ if ψ(h) = t, where h, t ∈ {1, 2, ..., r }; then, all Ot -edges in an INGS (O, O1 , O2 , . . . , Or ) become Oh -edges in ψ ψ ψ (O, O1 , O2 , ..., Or ). Proof. By definition of ψ-complement, TOψ (kl ) = TO (k) ∧ TO (l ) − IOψ (kl ) = FOψ (kl ) = h Tψ(Oq ) (kl ), (1) q̸=h h h ∨ ∨ IO (k) ∧ IO (l ) − Iψ(Oq ) (kl ), (2) Fψ(Oq ) (kl ), (3) q̸=h FO (k) ∨ FO (l ) − ∧ q̸=h for h ∈ {1, 2, ..., r }. For Expression 1. ∨ As TO (k) ∧ TO (l ) ≥ 0, Tψ(Oq ) (kl ) ≥ 0 and TOh (kl ) ≤ TO (k) ∧ TO (l ) ∀Oh . ⇒ ∨ q̸=h q̸=h Tψ(Oq ) (kl ) ≤ TO (k ) ∧ TO (l ) ⇒ TO (k ) ∧ TO (l ) − Hence, TOψ (kl ) ≥ 0 ∀h. h Furthermore, TOψ (kl ) gets a maximum value, when h and kl is an Ot -edge, then ∨ q̸=h Tψ(Oq ) (kl ) ≥ 0. ∨ Tψ(Oq ) (kl ) is zero. Clearly, when ψ(Oh ) = Ot q̸=h q̸=h Tψ(Oq ) (kl ) attains zero value. Hence, TOψ (kl ) h ∨ = TO (k) ∧ TO (l ), f or (kl ) ∈ Ot , ψ(Oh ) = Ot . (4) Information 2017, 8, 154 12 of 19 Similarly, for I, the results are: ∨ Since IO (k) ∧ IO (l ) ≥ 0, Iψ(Oq ) (kl ) ≥ 0 and IOh (kl ) ≤ IO (k ) ∧ IO (l ) ∀Oh . ⇒ ∨ q̸=h q̸=h Iψ(Oq ) (kl ) ≤ IO (k) ∧ IO (l ) ⇒ IO (k ) ∧ IO (l ) − Therefore, IOψ (kl ) ≥ 0 ∀ i. h Value of the IOψ (kl ) is maximum when h is an Ot -edge, then ∨ q̸=h ∨ ∨ q̸=h Iψ(Oq ) (kl ) ≥ 0. Iψ(Oq ) (kl ) gets zero value. Clearly, when ψ(Oh ) = Ot and kl q̸=h Iψ(Oq ) (kl ) is zero. Thus, IOψ (kl ) = h IO (k ) ∧ IO (l ), f or (kl ) ∈ Ot , ψ(Oh ) = Ot . (5) On a similar basis for F in ψ-complement, the results are: Since FO (k) ∨ FO (l ) ≥ 0, ⇒ ∧ q̸=h ∧ q̸=h Fψ(Oq ) (kl ) ≥ 0 and FOh (kl ) ≤ FO (k ) ∨ FO (l ) ∀Oh . Fψ(Oq ) (kl ) ≤ FO (k) ∨ FO (l ) ⇒ FO (k) ∨ FO (l ) − Hence, FOψ (kl ) ≥ 0 ∀h. ∧ h Furthermore, FOψ (kl ) is maximum, when h an Ot -edge, then ∧ q̸=h ∧ Fψ(Oq ) (kl ) ≥ 0. q̸=h Fψ(Oq ) (kl ) is zero. Definitely, when ψ(Oh ) = Ot and kl is q̸=h Fψ(Oq ) (kl ) is zero. Hence, FOψ (kl ) = h FO (k) ∨ FO (l ), f or (kl ) ∈ Ot , ψ(Oh ) = Ot . (6) Expressions (4)–(6) give the required proof. Definition 25. Let Ǧi = (O, O1 , O2 , ..., Or ) be an INGS and ψ be any permutation on {1, 2, ..., r }. Then, ψc (i) Ǧi is a self-complementary INGS if Ǧi is isomorphic to Ǧi ; (ii) Ǧi is a strong self-complementary INGS if Ǧi is identical to Ǧi . ψc Definition 26. Let Ǧi = (O, O1 , O2 , . . . , Or ) be an INGS. Then, ψc (i) Ǧi is a totally self-complementary INGS if Ǧi is isomorphic to Ǧi , ∀ permutations ψ on {1, 2, . . . , r }; (ii) Ǧi is a totally-strong self-complementary INGS if Ǧi is identical to Ǧi , ∀ permutations ψ on {1, 2, . . . , r }. ψc Example 15. INGS Ǧi = (O, O1 , O2 , O3 ) in Figure 13 is totally-strong self-complementary INGS. 0.4, 0.5) k1 (0.7, b ) .6) 3, 0.5 .2, 0. 0.4 ,0 O2 (0 .4, 3 (0 O 0.5 ) k3 b .6) b .3, ,0 b b ,0 0.4 0.2 0.2 .4, ( k7 ,0 1( ) 4, 0.5 .4) 0 .3, O ) (0 O1 O (0 3 0.5 .4, 0. O 2( 0 .2, , 0.3 (0 .2, 0.3 ,0 b b .4) k6 (0.4, 0.5, 0.6) k5 (0.2, 0.3, 0.3) k4 (0.4, 0.5, 0.5) k2 (0.4, 0.5, 0.6) Figure 13. Totally-strong self-complementary INGS. Theorem 1. A strong INGS is a totally self-complementary INGS and vice versa. Information 2017, 8, 154 13 of 19 Proof. Consider any strong INGS Ǧi and Permutation ψ on {1,2, . . . , r}. By proposition 1, ψ-complement of an INGS Ǧi = (O, O1 , O2 , . . . , Or ) is a strong INGS. Moreover, if ψ−1 (t) = h, where h, t ∈ {1, 2, ..., r }, ψ ψ ψ ψ then all Ot -edges in an INGS (O, O1 , O2 , ..., Or ) become Oh -edges in (O, O1 , O2 , ..., Or ), this leads TOt (kl ) = TO (k) ∧ TO (l ) = TOψ (kl ), IOt (kl ) = IO (k ) ∧ IO (l ) = IOψ (kl ), h FOt (kl ) = FO (k) ∨ FO (l ) = FOψ (kl ). h h Therefore, under g : P → P (identity mapping), Ǧi and ψ Ǧi are isomorphic, such that TO (k) = TO ( g(k )), IO (k ) = IO ( g(k)), FO (k ) = FO ( g(k )) and TOt (kl ) = TOψ ( g(k) g(l )) = TOψ (kl ), h h IOt (kl ) = IOψ ( g(k) g(l )) = IOψ (kl ) , h h FOt (kl ) = FOψ ( g(k) g(l )) = FOψ (kl ), h h ψ −1 ( t ) ∀kl ∈ Pt , for = h; h,t = 1, 2, . . . , r. For each permutation ψ on {1, 2, ..., r }, this holds. Hence, Ǧi is a totally self-complementary INGS. ψ Conversely, let Ǧi is isomorphic to Ǧi for each permutation ψ on {1, 2, ..., r }. Then, by definitions of ψ-complement of INGS and isomorphism of INGS, we have TOt (kl ) = TOψ ( g(k ) g(l )) = TO ( g(k)) ∧ TO ( g(l )) = TO (k ) ∧ TO (l ), h IOt (kl ) = IOψ ( g(k) g(l )) = IO ( g(k )) ∧ IO ( g(l )) = TO (k) ∧ IO (l ), h FOt (kl ) = FOψ ( g(k) g(l )) = FO ( g(k)) ∨ FO ( g(l )) = FO (k) ∨ FO (l ), h ∀kl ∈ Pt , t = 1,2,...,r. Hence, Ǧi is strong INGS. Remark 1. Each self-complementary INGS is a totally self-complementary INGS. Theorem 2. If Ǧ = ( P, P1 , P2 , . . . , Pr ) is a totally strong self-complementary GS and O = ( TO , IO , FO ) is an IN subset of P, where TO , IO , FO are the constant functions, then any strong INGS of Ǧ with IN vertex set O is necessarily totally-strong self-complementary INGS. Proof. Let u ∈ [0, 1], v ∈ [0, 1] and w ∈ [0, 1] be three constants, and TO (k) = u, IO (k) = v, FO (k) = w ∀k ∈ P. Since Ǧ is a totally strong self-complementary GS, so, for each permutation ψ−1 on {1, 2, . . . , r }, there exists a bijective mapping g : P → P, such that, for each Pt -edge (kl ), (g(k)g(l)) [a Ph -edge in Ǧ ] is a −1 ψ Pt -edge in Ǧ ψ c . Thus, for every Ot -edge (kl ), (g(k)g(l)) [an Oh -edge in Ǧi ] is an Ot -edge in Ǧi Moreover, Ǧi is a strong INGS, so TO (k ) = u = TO ( g(k )), IO (k ) = v = IO ( g(k)), FO (k) = w = FO ( g(k)) ∀k ∈ P and ψ −1 c . Information 2017, 8, 154 14 of 19 TOt (kl ) = TO (k) ∧ TO (l ) = TO ( g(k)) ∧ TO ( g(l )) = TOψ ( g(k) g(l )), h IOt (kl ) = IO (k) ∧ IO (l ) = IO ( g(k)) ∧ IO ( g(l )) = IOψ ( g(k ) g(l )), h FOt (kl ) = FO (k) ∨ IO (l ) = FO ( g(k)) ∨ FO ( g(l )) = FOψ ( g(k) g(l )), h ∀kl ∈ Ph , h = 1, 2, . . . , r. This shows that Ǧi is a strong self-complementary INGS. This exists for each permutation ψ and ψ−1 on set {1, 2, . . . , r }, thus Ǧi is a totally strong self-complementary INGS. Hence, required proof is obtained. Remark 2. Converse of the Theorem 2 may or may not true, as an INGS shown in Figure 2 is totally strong self-complementary INGS, and it is also a strong INGS with a totally strong self-complementary underlying GS but TO , IO , FO are not the constant-valued functions. 3. Application First, we explain the general procedure of this application by the following algorithm. Algorithm: Crucial interdependence relations Step 1. Input vertex set P = { B1 , B2 , . . . , Bn } and IN set O defined on P. Step 2. Input IN set of interdependence relations of any vertex with all other vertices and calculate T, F, and I of every pair of vertices by using, T ( Bi Bj ) ≤ min( T ( Bi ), T ( Bj )), F ( Bi Bj ) ≤ max( F ( Bi ), F ( Bj )), I ( Bi Bj ) ≤ min( I ( Bi ), I ( Bj )). Step 3. Repeat the Step 2 for every vertex in P. Step 4. Define relations P1 , P2 , . . . , Pn on set P such that ( P, P1 , P2 , . . . , Pn ) is a GS. Step 5. Consider an element of that relation, for which its value of T is comparatively high, and its values of F and I are lower than other relations. Step 6. Write down all elements in relations with T, F and I values, corresponding relations O1 , O2 , . . . , On are IN sets on P1 , P2 , P3 , . . . , Pn , respectively, and (O, O1 , O2 , . . . , On ) is an INGS. Human beings, the main creatures in the world, depend on many things for their survival. Interdependence is a very important relationship in the world. It is a natural phenomenon that nobody can be 100% independent, and the whole world is relying on interdependent relationships. Provinces or states of any country, especially of a progressive country, can not be totally independent, more or less they have to depend on each other. They depend on each other for many things, that is, there are many interdependent relationships among provinces or states of a progressive country—for example, education, natural energy resources, agricultural items, industrial products, and water resources, etc. However, all of these interdependent relationships are not of equal importance. Some are very important to run the system of a progressive country. Between any two provinces, all interdependent relationships do not have the same strength. Some interdependent relationships are like the backbone for the country. We can make an INGS of provinces or states of a progressive country, and can highlight those interdependent relationships, due to which the system of the country is running properly. This INGS can guide the government as to which interdependent relationships are very crucial, and they must try to make them strong and overcome the factors destroying or weakening them. We consider a set P of provinces and states of Pakistan: P = {Punjab, Sindh, Khyber Pakhtunkhawa(KPK), Balochistan, Gilgit-Baltistan, Azad Jammu and Kashmir(AJK) }. Let O be the IN set on P, as defined in Table 1. Information 2017, 8, 154 15 of 19 Table 1. IN set O of provinces of Pakistan. Provinces or States T I F Punjab Sindh Khyber Pakhtunkhawa(KPK) Balochistan Gilgit-Baltistan Azad Jammu and Kashmir 0.5 0.5 0.4 0.3 0.3 0.3 0.3 0.4 0.4 0.4 0.4 0.4 0.3 0.4 0.4 0.4 0.4 0.3 In Table 1, symbol T demonstrates the positive role of that province or state for the strength of the Federal Government, and symbol F indicates its negative role, whereas I denotes the percentage of ambiguity of its role for the strength of the Federal Government. Let us use the following alphabets for the provinces’ names: PU = Punjab, SI = Sindh, KPK = Khyber Pakhtunkhwa, BA = Balochistan, GB = Gilgit-Baltistan, AJK = Azad Jammu and Kashmir. For every pair of provinces of Pakistan in set P, different interdependent relationships with their T, I and F values are demonstrated in Tables 2–6. Table 2. IN set of interdependent relations between Punjab and other provinces. Type of Interdependent Relationships (PU, SI) (PU, KPK) (PU, BA) Education Natural energy resources Agricultural items Industrial products Water resources (0.5, 0.1, 0.1) (0.3, 0.2, 0.3) (0.3, 0.2, 0.2) (0.4, 0.2, 0.1) (0.3, 0.1, 0.1) (0.4, 0.3, 0.2) (0.4, 0.2, 0.2) (0.4, 0.2, 0.1) (0.4, 0.1, 0.1) (0.4, 0.3, 0.2) (0.3, 0.2, 0.2) (0.3, 0.2, 0.1) (0.3, 0.2, 0.1) (0.3, 0.1, 0.1) (0.2, 0.2, 0.2) Table 3. IN set of interdependent relationships between Sindh and other provinces. Type of Interdependent Relationships (SI, KPK) (SI, BA) (SI, GB) Education Natural energy resources Agricultural items Industrial products Water resources (0.3, 0.2, 0.1) (0.3, 0.2, 0.3) (0.4, 0.1, 0.1) (0.4, 0.2, 0.1) (0.3, 0.2, 0.2) (0.3, 0.2, 0.3) (0.3, 0.1, 0.0) (0.3, 0.1, 0.2) (0.3, 0.2, 0.2) (0.2, 0.3, 0.2) (0.3, 0.2, 0.4) (0.2, 0.2, 0.4) (0.3, 0.1, 0.1) (0.3, 0.2, 0.2) (0.2, 0.2, 0.3) Table 4. IN set of interdependent relationships between KPK and other provinces. Type of Interdependent Relationships (KPK, BA) (KPK, GB) (KPK, AJK) Education Natural energy resources Agricultural items Industrial products Water resources (0.1, 0.4, 0.3) (0.3, 0.2, 0.1) (0.1, 0.2, 0.4) (0.1, 0.3, 0.4) (0.3, 0.2, 0.2) (0.1, 0.4, 0.3) (0.3, 0.2, 0.2) (0.1, 0.4, 0.4) (0.1, 0.4, 0.3) (0.3, 0.3, 0.2) (0.1, 0.4, 0.4) (0.3, 0.3, 0.2) (0.1, 0.3, 0.3) (0.1, 0.2, 0.2) (0.3, 0.2, 0.2) Information 2017, 8, 154 16 of 19 Table 5. IN set of interdependent relationships between AJK and other provinces. Type of Interdependent Relationships (AJK, PU) (AJK, SI) (AJK, BA) Education Natural energy resources Agricultural items Industrial products Water resources (0.3, 0.1, 0.1) (0.1, 0.2, 0.3) (0.3, 0.2, 0.1) (0.3, 0.2, 0.2) (0.3, 0.2, 0.1) (0.1, 0.4, 0.3) (0.2, 0.4, 0.3) (0.3, 0.3, 0.2) (0.3, 0.2, 0.2) (0.3, 0.3, 0.2) (0.1, 0.3, 0.4) (0.3, 0.3, 0.3) (0.3, 0.2, 0.2) (0.3, 0.2, 0.3) (0.3, 0.0, 0.1) Table 6. IN set of interdependent relationships of Gilgit-Baltistan with other provinces. Type of Interdependent Relationships (GB, PU) (GB, BA) (GB, AJK) Education Natural energy resources Agricultural items Industrial products Water resources (0.3, 0.2, 0.1) (0.1, 0.3, 0.4) (0.3, 0.2, 0.2) (0.3, 0.3, 0.2) (0.2, 0.3, 0.3) (0.1, 0.4, 0.4) (0.3, 0.1, 0.0) (0.1, 0.3, 0.3) (0.2, 0.4, 0.4) (0.2, 0.3, 0.2) (0.2, 0.1, 0.4) (0.2, 0.2, 0.4) (0.1, 0.4, 0.4) (0.1, 0.4, 0.2) (0.3, 0.1, 0.1) Many relations can be defined on the set P, we define following relations on set P as: P1 = Education, P2 = Natural energy resources , P3 = Agricultural items, P4 = Industrial products, P5 = Water resources, such that ( P, P1 , P2 , P3 , P4 , P5 ) is a GS. Any element of a relation demonstrates a particular interdependent relationship between these two provinces. As ( P, P1 , P2 , P3 , P4 , P5 ) is GS; this is why any element can appear in only one relation. Therefore, any element will be considered in that relationship, whose value of T is high, and values of I, F are comparatively low, using the data of above tables. Write down T, I and F values of the elements in relations according to the above data, such that O1 , O2 , O3 , O4 , O5 are IN sets on relations P1 , P2 , P3 , P4 , P5 , respectively. Let P1 = {(Punjab, Sindh), (Gilgit − Baltistan, Punjab), (AzadJammuandKashmir, Punjab)}; P2 = {(Sindh, Balochistan), (Khyber Pakhtunkhawa, Balochistan), (Balochistan, Gilgit-Baltistan), (Khyber Pakhtunkhawa, Gilgit-Baltistan)}; P3 = {(Sindh, Khyber Pakhtunkhwa), (Gilgit-Baltistan, Sindh) }; P4 = {(Punjab, KhyberPakhtunkhwa), (Sindh, AzadJammuandKashmir), (Balochistan, Punjab)}; P5 = {(KheberPakhtunkhwa, AzadJammuandKashmir), (Balochistan, AzadJammuandKashmir), (Gilgit − Baltistan, Azad Jammu and Kashmir)}. Let O1 = {(( PU, SI ), 0.5, 0.1, 0.1), (( GB, PU ), 0.3, 0.2, 0.1), (( AJK, PU ), 0.3, 0.1, 0.1)}, O2 = {((SI, BA), 0.3, 0.1, 0.0), ((KPK, BA), 0.3, 0.2, 0.1), (( BA, GB), 0.3, 0.1, 0.0), ((KPK, GB), 0.3, 0.2, 0.2)}, O3 = {((SI, KPK ), 0.4, 0.1, 0.1), (( GB, SI ), 0.3, 0.1, 0.1), }, O4 = {(( PU, KPK ), 0.4, 0.1, 0.1), ((SI, AJK ), 0.3, 0.2, 0.2), (( BA, PU ), 0.3, 0.1, 0.1)}, O5 = {((KPK, AJK ), 0.3, 0.2, 0.2), (( BA, AJK ), 0.3, 0.0, 0.1), (( GB, AJK ), 0.3, 0.1, 0.1)}. Obviously, (O, O1 , O2 , O3 , O4 , O5 ) is an INGS as shown in Figure 14. Information 2017, 8, 154 17 of 19 Agricultural items (0.4, 0.1, 0.1) Sindh Industrial products (0.3, 0.2, 0.2) Education (0.5, 0.1, 0.1) Ag ric ul tu (0 ral .3, ite 0.1 ms ,0 .1 ) Azad Jammu Kashmir Education (0.3, 0.1, 0.1) Khyber Pakhtun Khwa Punjab Na Na tu ra tu ra (0 l en .3 er , 0 gy .2 re , 0 so .1 u r ) l (0 ener .3, gy 0.1 re , 0 sou .0) rc es Water resources (0.3, 0.2, 0.2) E (0 duc .3, at 0.2 ion ,0 .1 ) Industrial products (0.4, 0.1, 0.1) s rce ou ) es , 0.1 r r 1 ate 0. W 0.3, ( Na ucts ce s Balochistan od l pr stria 1, 0.1) u d . In .3, 0 (0 tur a (0 l ene .3 , 0 rgy .2 , 0 reso ur .2 c ) es Gilgit Baltistan Water resources (0.3, 0.0, 0.1) Natural energy resources (0.3, 0.1, 0.0) Figure 14. INGS identifying crucial interdependence relation between any two provinces. Every edge of this INGS demonstrates the most dominating interdependent relationship between those two provinces—for example, the most dominating interdependent relationship between Punjab and Gilgit-Baltistan is education, and its T, F and I values are 0.3, 0.2 and 0.1, respectively. It shows that education is the strongest connection bond between Punjab and Gilgit-Baltistan; it is 30% stable, 10% unstable, and 20% unpredictable or uncertain. Using INGS, we can also elaborate the strength of any province, e.g., Punjab has the highest vertex degree for interdependent relationship education, and Balochistan has the highest vertex degree for the interdependent relationship natural energy resources. This shows that the strength of Punjab is education, and the strength of Balochistan is the natural energy resources. This INGS can be very helpful for Provincial Governments, and they can easily estimate which kind of interdependent relationships they have with other provinces, and what is the percentage of its stability and instability. It can also guide the Federal Government in regards to, between any two provinces, which relationships are crucial and what is their status. The Federal Government should be conscious of making decisions such that the most crucial interdependent relationships of its provinces are not disturbed and need to overcome the counter forces that are trying to destroy them. 4. Conclusions Graph theory is a useful tool for solving combinatorial problems of different fields, including optimization, algebra, computer science, topology and operations research. An intuitionistic neutrosophic set constitutes a generalization of an intuitionistic fuzzy set. In this research paper, we have introduced the notion of intuitionistic neutrosophic graph structure. 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This article is an open access ⃝ article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). information Article NC-TODIM-Based MAGDM under a Neutrosophic Cubic Set Environment Surapati Pramanik 1, * 1 2 * ID , Shyamal Dalapati 2 , Shariful Alam 2 ID and Tapan Kumar Roy 2 Department of Mathematics, Nandalal Ghosh B.T. College, Panpur, P.O.-Narayanpur, District–North 24 Parganas, West Bengal 743126, India Department of Mathematics, Indian Institute of Engineering Science and Technology, Shibpur, P.O.-Botanic Garden, Howrah, West Bengal 711103, India; shyamal.rs2015@math.iiests.ac.in (S.D.); salam50in@yahoo.co.in (S.A.); roy_t_k@yahoo.co.in (T.K.R.) Correspondence: sura_pati@yahoo.co.in; Tel.: +91-947-703-5544 Received: 19 October 2017; Accepted: 14 November 2017; Published: 18 November 2017 Abstract: A neutrosophic cubic set is the hybridization of the concept of a neutrosophic set and an interval neutrosophic set. A neutrosophic cubic set has the capacity to express the hybrid information of both the interval neutrosophic set and the single valued neutrosophic set simultaneously. As newly defined, little research on the operations and applications of neutrosophic cubic sets has been reported in the current literature. In the present paper, we propose the score and accuracy functions for neutrosophic cubic sets and prove their basic properties. We also develop a strategy for ranking of neutrosophic cubic numbers based on the score and accuracy functions. We firstly develop a TODIM (Tomada de decisao interativa e multicritévio) in the neutrosophic cubic set (NC) environment, which we call the NC-TODIM. We establish a new NC-TODIM strategy for solving multi attribute group decision making (MAGDM) in neutrosophic cubic set environment. We illustrate the proposed NC-TODIM strategy for solving a multi attribute group decision making problem to show the applicability and effectiveness of the developed strategy. We also conduct sensitivity analysis to show the impact of ranking order of the alternatives for different values of the attenuation factor of losses for multi-attribute group decision making strategies. Keywords: neutrosophic cubic set; single valued neutrosophic set; interval neutrosophic set; multi attribute group decision making; TODIM strategy; NC-TODIM 1. Introduction While modelling multi attribute decision making (MADM) and multi attribute group decision making (MAGDM), it is often observed that the parameters of the problem are not precisely known. The parameters often involve uncertainty. To deal with uncertainty, Zadeh [1] left an important mark to represent and compute with imperfect information by introducing the fuzzy set. The fuzzy set fostered a broad research community, and its impact has also been clearly felt at the application level in MADM [2–4] and MAGDM [5–9]. Atanassov [10] incorporated the non-membership function as an independent component and defined the intuitionistic fuzzy set (IFS) at first to express uncertainty in a more meaningful way. IFSs have been applied in many MADM problems [11–13]. Smarandache [14] proposed the notion of the neutrosophic set (NS) by introducing indeterminacy as an independent component. Wang et al. [15] grounded the concept of the single valued neutrosophic set (SVNS), an instance of the neutrosophic set, to deal with incomplete, inconsistent, and indeterminate information in a realistic way. Wang et al. [16] proposed the interval neutrosophic set (INS) as a subclass of neutrosophic sets in which the values of truth, indeterminacy, and falsity membership degrees are interval numbers. Theoretical development Information 2017, 8, 149; doi:10.3390/info8040149 www.mdpi.com/journal/information Information 2017, 8, 149 2 of 21 and applications of SVNSs and INSs are found in [17–37] for MADM or MAGDM. Some studies on MADM in single valued neutrosophic hesitant fuzzy set environments are found in [38–41]. NS and INS are both capable of handling uncertainty and incomplete information. By fusing NS and INS, Ali et al. [42] proposed the neutrosophic cubic set (NCS) and defined external and internal neutrosophic cubic sets, and established some of their properties. In the same study, Ali et al. [42] proposed an adjustable strategy to NCS-based decision making. Jun et al. [43] also defined NCS by combining NS and INS. In decision making process, the advantage of NCSs is that the decision makers can employ the hybrid information comprising of INSs and SVNSs for evaluating and rating of the alternatives with respect to their predefined attributes. However, there are only a few studies in the literature to deal with MADM and MAGDM in the NCS environment. Banerjee et al. [44] established grey relational analysis (GRA) [45–47] based on the new MADM strategy in the NCS environment. In the same study, Banerjee et al. [44] proposed the Hamming distances for weighted grey relational coefficients and ideal grey relational coefficients, and offered the concept of relative closeness coefficients for presenting the ranking order of the alternatives based on the descending order of their relative closeness coefficients. Similarity measure is an important mathematical tool in decision-making problems. Pramanik et al. [48] at first defined similarity measure for NCSs and proved its basic properties. In the same study, Pramanik et al. [48] developed a new MAGDM strategy in the NCS environment. Lu and Ye [49] proposed cosine measures between NCSs and established their basic properties. In the same study, Lu and Ye [49] proposed three new cosine measures-based MADM strategies under a NCS environment. Due to little research on the operations and application of NCSs, Pramanik et al. [50] proposed several operational rules on NCSs, and defined Euclidean distances and arithmetic average operators of NCSs. In the same study, Pramanik et al. [50] also employed the information entropy scheme to calculate the unknown weights of the attributes, and developed a new extended TOPSIS strategy for MADM under the NCS environment. Zhan et al. [51] proposed a new algorithm for multi-criteria decision making (MCDM) in an NCS environment based on a weighted average operator and a weighted geometric operator. Ye [52] established the concept of a linguistic neutrosophic cubic number (LNCN). In the same study, Ye [52] developed a new MADM strategy based on a LNCN weighted arithmetic averaging (LNCNWAA) operator and a LNCN weighted geometric averaging (LNCNWGA) operator under a linguistic NCS environment. In the literature, there are only six strategies [44,48–52] for MADM and MAGDM in NCS environment. However, we say that none of them is generally superior to all others. So, new strategies for MADM and MAGDM should be explored under the NCS environment for the development of neutrosophic studies. TODIM (an acronym in Portuguese for interactive multi-criteria decision making strategy named Tomada de decisao interativa e multicritévio) is an important MADM strategy, since it considers the decision makers’ bounded rationality. Firstly, Gomes and Lima [53] introduced the TODIM strategy based on prospect theory [54]. Krohling and Souza [55] defined the fuzzy TODIM strategy to solve MCDM problems. Several researchers applied the TODIM strategy in various fuzzy MADM or MAGDM problems [56–58]. Fan et al. [59] introduced the extended TODIM strategy to deal with the hybrid MADM problems. Krohling et al. [60] extended the TODIM strategy from fuzzy environment to intuitionistic fuzzy environment to process the intuitionistic fuzzy information. Wang [61] introduced TODIM strategy into multi-valued neutrosophic set environment. Zhang et al. [62] proposed the TODIM strategy for MAGDM problems under a neutrosophic number environment. Ji et al. [63] proposed the TODIM strategy under a multi-valued neutrosophic environment and employed it to solve personal selection problems. In 2017, Xu et al. [64] developed the TODIM strategy in a single valued neutrosophic setting and extended it into interval neutrosophic setting. Neutrosophic TODIM [64] is capable of dealing with only single-valued neutrosophic information or interval neutrosophic information. Information 2017, 8, 149 3 of 21 NCS can be used to express the interval neutrosophic information and neutrosophic information in the process of MADGM. It seems that TODIM in NCSs has an enormous chance of success to deal with group decision making problems. In the NCS environment, the TODIM strategy is yet to appear. Motivated by these, we initiated the study of TODIM in the NCS environment, which we call NC-TODIM. However, NCSs comprise of hybrid information of INSs and SVNSs simultaneously, which are more flexible and elegant for expressing neutrosophic cubic information. To apply NCSs to MADGM problems, we introduce some basic operations of neutrosophic cubic (NC) numbers and the score, and accuracy functions of NC numbers, and the ranking strategy of NC numbers. In this paper we develop a TODIM strategy (for short, NC-TODIM strategy) for MAGDM in the NCS environment. The proposed NC-TODIM strategy was proven to be capable of successfully dealing with MAGDM problems by solving an illustrative example. What is more, a comparative analysis ensured the feasibility of the proposed NC-TODIM strategy. The remainder of the paper is divided into seven sections that are organized as follows: Section 2 presents some basic definitions of NS, SVNS, INS, and NCS. Section 3 presents comparison strategy of two NC-numbers. Section 4 is devoted to present the proposed NC-TODIM strategy. Section 5 presents an illustrative numerical example of MAGDM in the NCS environment. Section 6 is devoted to analyzing the ranking order with different values of attenuation factors of losses. Section 7 presents a comparative analysis between the developed strategy and other existing strategies in the NCS environment. Section 8 presents the conclusion and the future scope of research. 2. Preliminaries In this section, we review some basic definitions which are important to develop the paper. Definition 1. [14] NS. Let U be a space of points (objects) with a generic element in U denoted by u, i.e., u ∈ U. A neutrosophic set R in U is characterized by truth-membership function tR , indeterminacy-membership function iR , and falsity-membership function fR , where tR , iR , fR are the functions from U to ]− 0, 1+ [ i.e., tR , iR , fR : U→]− 0, 1+ [ that means tR (u), iR (u), fR (u) are the real standard or non-standard subset of ]− 0, 1+ [. The neutrosophic set can be expressed as R = {<u; (tR (u), iR (u), fR (u))>: ∀u∈U}. Since tR (u), iR (u), fR (u) are the subset of ]− 0, 1+ [, there the sum (tR (u) + iR (u) + fR (u)) lies between − 0 and 3+ , where − 0 = 0 − ε and 3+ = 3 + ε, ε > 0. Example 1. Suppose that U = {u1 , u2 , u3 , . . .} is the universal set. Let R1 be any neutrosophic set in U. Then R1 expressed as R1 = {<u1 ; (0.6, 0.3, 0.4)>: u1 ∈U}. Definition 2. [15] SVNS. Let U be a space of points (objects) with a generic element in U denoted by u. A single valued neutrosophic set H in U is expressed by H = {<u, (tH (u), iH (u), fH (u))>; u∈U}, where tH (u), iH (u), fH (u)∈[0, 1]. Therefore for each u∈U, tH (u), iH (u), fH (u)∈[0, 1] and 0 ≤ tH (u) + iH (u) + fH (u) ≤ 3. e in G is characterized by Definition 3. [16] INS. Let G be a non-empty set. An interval neutrosophic set G truth-membership function tG e , the indeterminacy membership function iG e and falsity membership function fG e. e (g) ⊆ [0, 1] and G defined as (g), f (g), i For each g∈G, tG e e e G G e = {< g; [t− (g), t+ (g)], [i− (g), i+ (g), ], [f− (g), f+ (g)] : ∀g ∈ G}. G e e e e e e G G G G G G Here, + − − + + − + t− e (g), t e (g), i e (g), i e (g), f e (g), f e (g) : G →] 0, 1 [, G and − G G G G G + + + 0 ≤ sup t+ e (g) + sup i e (g) + sup f e (g) ≤ 3 . G G G Information 2017, 8, 149 4 of 21 In real problems it is difficult to express the truth-memberships function, indeterminacy-membership function and falsity-membership function in the form of + − − + + − + t− e (g), t e (g), i e (g), i e (g), f e (g), f e (g) : G →] 0, 1 [. G Here, G G G G G + − + + − t− e (g), t e (g), i e (g), i e (g), f e (g), f e (g) : G → [0, 1]. G G G G G G e 1 be an INS. Then G e 1 is Example 2. Suppose that G = {g1 , g2 , g3 , . . ., gn } is a non-empty set. Let G expressed as e 1 = {< g ; [0.39, 0.47], [0.17, 0.43], [0.18, 0.36] : g ∈ G}. G 1 1 e e R(g)>: ∀g∈G}, where G Definition 4. [42,43] NCS. A NCS in a non-empty set G is defined as © = {<g; G(g), e and R are the INS and NS in G respectively. NCS can be presented as an order pair © = <G, R>, then we call it as a neutrosophic cubic (NC) number. Example 3. Suppose that G = {g1 , g2 , g3 , . . ., gn } is a non-empty set. Let ©1 be any NC-number. Then ©1 can be expressed as ©1 = {<g1 ; [0.39, 0.47], [0.17, 0.43], [0.18, 0.36], (0.6, 0.3, 0.4)>: g1 ∈G}. Some operations of NC-numbers: i. Union of any two NC-numbers e 1 , R1 > and ©2 =< G e 2 , R2 > be any two NC-numbers in a non-empty set G. Let ©1 =< G Then the union of ©1 and ©2 denoted by ©1 ∪ ©2 and defined as e 1 (g) ∪ G e 1 ( g ) , R 1 ( g ) ∪ R2 ( g ) ∀ g ∈ G > , ©1 ∪ ©2 =< G e 1 (g) ∪ G e 1 (g) = {<g, [max {t− (g), t− (g)},max {t+ (g), t+ (g)}], [max {i− (g), i− (g)}, max {i+ (g), where G e e e e e e e G1 G2 G1 G2 G1 G2 G1 − − + + i+ e (g)}], [min {f e (g), f e (g)}, min {f e (g), f e (g)}]>: g∈G} and R1 (g) ∪ R2 (g) = {<g, max {tR1 (g), tR2 (g)}, G1 G2 G2 G1 G2 max {iR1 (g), iR2 (g)}, min {fR1 (g), fR2 (g)}>:∀g∈U}. Example 4. Let ©1 and ©2 be two NC-numbers in G presented as follows: ©1 =< [0.39, 0.47], [0.17, 0.43], [0.18, 0.36], (0.6, 0.3, 0.4) > and ©2 =< [0.56, 0.70], [0.27, 0.42], [0.15, 0.26], (0.7, 0.3, 0.6) > . Then ©1 ∪ ©2 =< [0.56, 0.70], [0.27, 0.43], [0.15, 0.26], (0.7, 0.3, 0.4) > . ii. Intersection of any two NC-numbers Intersection of two NC-numbers denoted and defined as follows: e 1 (g) ∩ G e 1 ( g ) , R 1 ( g ) ∩ R2 ( g ) ∀ g ∈ G > , ©1 ∩ ©2 =<G e 1 (g) ∩ G e 1 (g) = {<g, [min {t− (g), t− (g)},min {t+ (g), t+ (g)}], [min {i− (g), i− (g)}, min {i+ (g), where G e e e e e e e G1 G2 G1 G2 G1 G2 G1 G2 G1 − − + + i+ e (g)}], [max {f e (g), f e (g)}, max {f e (g), f e (g)}]>: g∈G} and R1 (g) ∩ R2 (g)= {<g, min {tR1 (g), tR2 (g)}, G2 G1 G2 min {iR1 (g),iR2 (g)}, max {fR1 (g), fR2 (g)}>:∀g∈U}. Information 2017, 8, 149 5 of 21 Example 5. Let ©1 and ©2 be any two NC-numbers in G presented as follows: ©1 =< [0.45, 0.57], [0.27, 0.33], [0.18, 0.46], (0.7, 0.3, 0.5) > and ©2 =< [0.67, 0.75], [0.22, 0.44], [0.17, 0.21], (0.8, 0.4, 0.4) > . Then ©1 ∩ ©2 =< [0.45, 0.57], [0.22, 0.33], [0.18, 0.46], (0.7, 0.3, 0.4) > . iii. Compliment of a NC-number e 1 , R1 > be a NCS in G. Then, the compliment of ©1 =< G e 1 , R1 > denoted by Let ©1 =< G c c c e ©1 = {<g, G1 (g), R1 (g)>: ∀g∈G}. e 1 c = {<g, [t− (g), t+ (g)], [i− (g), i+ (g)], [f− (g), f+ (g)]>: ∀g∈G}, where, Here, G e c e c e c e c e c e c G1 G1 G1 G1 G1 G1 − + − + − − + + − t− e c (g) = f e (g), t e c (g) = f e (g), i e c (g) = {1} − i e (g), i e c (g) = {1} − i e (g), f e c (g) = t e (g), G1 G1 G1 G1 G1 G1 G1 G1 G1 + c + c c f+ e c (g) = f e (g) and tR1 (g) = fR 1 (g), iR´ 1 (g) = {1 } − iR1 (g), fR1 (g) = tR1 (g). G1 G1 G1 Example 6. Assume that ©1 be any NC-number in G in the form: ©1 =< [0.45, 0.57], [0.27, 0.33], [0.18, 0.46], (0.7, 0.3, 0.5) > Then compliment of ©1 is obtained as ©c1 =< [0.18, 0.46], [0.73, 0.67], [0.45, 0.57], (0.5, 0.7, 0.7) > . Definition 5. Score function. Let ©1 be a NC-number in a non-empty set G. Then, a score function of ©1 , denoted by Sc (© 1 ) is defined as: 1 + a − 2b − c 1 2 + a1 + a2 − 2b1 − 2b2 − c1 − c2 )+( )] Sc (© 1 ) = [( 2 4 2 (1) where, ©1 = <[a1 , a2 ], [b1 , b2 ], [c1 , c2 ], (a, b, c) > and Sc (© 1 )∈[–1, 1]. Proposition 1. Score function of two NC-numbers lies between −1 to 1. Proof. Using the definition of INS and NS, we have all a1 , a2 , b1 , b2 , c1 , c2 , a, b, and c [0, 1]. Since, 0 ≤ a1 ≤ 1, 0 ≤ a2 ≤ 1 0 ≤ a1 + a2 ≤ 2, (2) ⇒ 2 ≤ 2 + a1 + a2 ≤ 4 0 ≤ b1 ≤ 1 ⇒ 0 ≤ 2b1 ≤ 2 and 0 ≤ b2 ≤ 1 ⇒ 0 ≤ 2b2 ≤ 2 ⇒ − 2 ≤ −2b1 ≤ 0 ⇒ − 2 ≤ −2b2 ≤ 0 (3) ⇒ − 4 ≤ −2b1 − 2b2 ≤ 0 0 ≤ c1 ≤ 1 ⇒ − 1 ≤ − c1 ≤ 0 0 ≤ c2 ≤ 1 ⇒ − 1 ≤ − c2 ≤ 0 ⇒ − 2 ≤ − c1 − c2 ≤ 0 (4) Information 2017, 8, 149 6 of 21 Adding Equations (2)–(4), we obtain ⇒ − 4 ≤ 2 + a1 + a2 − 2b1 − 2b2 − c1 − c2 ≤ 4, ⇒ −1 ≤ 2 + a1 + a2 − 2b1 − 2b2 − c1 − c2 4 ≤ 1 (5) Again, 0 ≤ a ≤ 1⇒ 1 ≤ 1 + a ≤ 2, (6) 0 ≤ b ≤ 1 ⇒ 0 ≤ 2b ≤ 2, 0 ≤ c ≤ 1, ⇒ 0 ≤ 2b + c ≤ 3, (7) ⇒ − 3 ≤ − 2b − c ≤ 0 Adding (6) and (7), we obtain − 2 ≤ 1 + a − 2b − c ≤ 2, ⇒ −1 ≤ 1 + a− 2b − c 2 ≤ 1 (8) Adding (5) and (8) and dividing by 2, we obtain −1≤ 1 + a − 2b − c 1 2 + a1 + a2 − 2b1 − 2b2 − c1 − c2 [( )+( )] ≤ 1 2 4 2 Sc (© 1 ) ∈ [−1, 1]. Hence the proof is complete. Example 7. Let ©1 and ©2 be two NC-numbers in G, presented as follows: ©1 =< [0.39, 0.47], [0.17, 0.43], [0.18, 0.36], (0.6, 0.3, 0.4) > and ©2 =< [0.56, 0.70], [0.27, 0.42], [0.15, 0.26], (0.7, 0.3, 0.6) > . Then, by applying Definition 5, we obtain Sc (© 1 ) = −0.01 and Sc (© 2 ) = 0.07, In this case, we can say that ©2 > ©1 . Definition 6. Accuracy function. Let ©1 be a NC-number in a non-empty set G, an accuracy function of ©1 is defined as: Ac(©1 ) = 1/2 [1/2(a1 + a2 − b2 (1 − a2 ) − b1 (1 − a1 ) − c2 (1 − b1 ) − c1 (1 − b2 ) + a − b(1 − a) − c(1 − b)] (9) Here, Ac (© 1 )∈[–1, 1]. When the value of Ac (© 1 ) increases, we say that the degree of accuracy of the NC-number ©1 increases. Proposition 2. Accuracy function of two NC-numbers lies between −1 to 1. Proof. The values of accuracy function depend upon 1 { (a1 + a2 − b2 (1 − a2 ) − b1 (1 − a1 ) − c2 (1 − b1 ) − c1 (1 − b2 )) and { a − b(1 − a) − c(1 − b)} 2 The values of 1 { (a1 + a2 − b2 (1 − a2 ) − b1 (1 − a1 ) − c2 (1 − b1 ) − c1 (1 − b2 ))} 2 Information 2017, 8, 149 7 of 21 and { a − b(1 − a) − c(1 − b)} lie between −1 to 1 from [37]. Thus, −1 ≤ Ac (© 1 ) ≤ 1. Hence the proof is completed. Example 8. Let ©1 and ©2 be two NC-numbers in G presented as follows: ©1 =< [0.41, 0.52], [0.10, 0.18], [0.06, 0.17], (0.48, 0.11, 0.11) > and ©2 =< [0.40, 0.51], [0.10, 0.20], [0.10, 0.19], (0.50, 0.11, 0.11) > . Then, by applying Definition 6, we obtain Ac (© 1 ) = 0.14 and Ac (© 2 ) = 0.30. In this case, we can say that alternative ©2 is better than ©1 . With respect to the score function Sc and the accuracy function Ac, a strategy for comparing NC-numbers can be defined as follows: 3. Comparison Strategy of Two NC-Numbers Let ©1 and ©2 be any two NC-numbers. Then we define comparison strategy as follows: i. If ii. Sc(©1 ) > Sc(©2 ), then ©1 > ©2 . (10) Sc(©1 ) = Sc(©2 ) and Ac(©1 ) > Ac(©2 ), then ©1 > ©2 . (11) Sc(©1 ) = Sc(©2 ) and Ac(©1 ) = Ac(©2 ), then ©1 = ©2 . (12) If iii. If Example 9. Let ©1 and ©2 be two NC-numbers in G, presented as follows: ©1 =< [0.23, 0.29], [0.37, 0.46], [0.34, 0.42], (0.26, 0.26, 0.26) > and ©2 =< [0.25, 0.31], [0.35, 0.44], [0.35, 0.44], (0.28, 0.28, 0.28) > . Then, applying Definition 5, we obtain Sc (© 1 ) = 0.13 and Sc (© 2 ) = 0.13. Applying Definition 6, we obtain Ac(© 1 ) = −0.20 and Ac (© 2 ) = −0.18. In this case, we say that alternative ©2 > ©1 . (Score values and Accuracy values taking correct up to two decimal places). Definition 7. Let ©1 and ©2 be any two NC-numbers, then the distance between them is defined by ∂(©1 , ©2 ) = 91 [|a1 − d1 | + |a2 − d2 | + |b1 − e1 | + |b2 − e2 | + |c1 − f1 | + |c2 − f2 | + |a − d| + |b − e| + |c − f|] where, ©1 = <[a1 , a2 ], [b1 , b2 ], [c1 , c2 ], (a, b, c)> and ©2 = <[d1 , d2 ], [e1 , e2 ], [f1 , f2 ], (d, e, f)>. Example 10. Let ©1 and ©2 be two NC-numbers in G presented as follows: ©1 =< [0.66, 0.75], [0.25, 0.32], [0.17, 0.34], (0.53, 0.17, 0.22) > (13) ∂ Information 2017, 8, 149 8 of 21 and ©2 =< [0.35, 0.55], [0.12, 0.25], [0.12, 0.20], (0.60, 0.23, 0.43) > Then, applying Definition 7, we obtain ∂ (©1 , ©2 ) = 0.12.   − + − + − + Definition 8. Let ©ij = {< [tij , tij ], [iij , iij ], [fij , fij ], (t, i, f) >} be any neutrosophic cubic value. ©ij used to evaluate i-th alternative with respect to j-th criterion. The normalized form of ©ij is defined as follows:  [ ©ij⊗ = {< [ tij− m 2 2 ( Σ (tij− ) +(tij+ ) ) 2 i=1 iij− m 2 2 1 ( Σ (iij− ) +(iij+ ) ) 2 i=1  [ fij− m 2 2 1 ( Σ (fij− ) +(fij+ ) ) 2 i=1 [ tij m 1 ( Σ (tij )2 +(iij )2 +(fij )2 ) 2 i=1 , 1 , , tij+ m i=1 iij+ m 2 2 1 ( Σ (iij− ) +(iij+ ) ) 2 i=1  , fij+ m ], ], (14) ] i=1 1 ( Σ (tij )2 +(iij )2 +(fij )2 ) 2 i=1 2 1 2 ( Σ (fij− ) +(fij+ ) ) 2 iij m 2 1 2 ( Σ (tij− ) +(tij+ ) ) 2 , fij m 1 ( Σ (tij )2 +(iij )2 +(fij )2 ) 2 ] >}. i=1 A conceptual model of the evolution of the neutrosophic cubic set is shown in Figure 1. Figure 1. Evolution of the neutrosophic cubic set. 4. NC-TODIM Based MAGDM under a NCS Environment Assume that A = {A1 , A2 , . . . , Am } (m ≥ 2) and C = {C1 , C2 , . . . , Cn } (n ≥ 2) are the discrete set of alternatives and attributes respectively. W = {W1 , W2 , . . . , Wn } is the weight vector of attributes n Cj (j = 1, 2, . . . , n), where Wj > 0 and Σ Wj = 1. Let E = {E1 , E2 , . . . , Er } be the set of decision makers j=1 r and γ = {γ1 , γ2 , . . . , γr } be the weight vector of decision makers, where γk > 0 and Σ γk = 1. k=1 NC-TODIM Strategy Now, we describe the NC-TODIM strategy to solve the MAGDM problems with NC-numbers. The NC-TODIM strategy consists of the following steps: Information 2017, 8, 149 9 of 21 Step 1. Formulate the decision matrix e kij , Rk > is the rating value Assume that Mk = (©kij ) be the decision matrix, where ©kij = <G ij m×n provided by the k-th (Ek) decision maker for alternative Ai , with respect to attribute Cj . The matrix form of Mk is presented as:  A  1  Mk = A2   . Am C1 ©k11 ©k21 . ©km1 C2 ©k12 ©k22 ... ©km2 ... ... . ...  Cn ©k1n    ©k2n    ©kmnj (15) Step 2. Normalize the decision matrix The MAGDM problem generally consists of cost criteria and benefit criteria. So, the decision matrix needs to be normalized. For cost criterion Cj , we use the Definition 8 to normalize the decision matrix (Equation (15)) provided by the decision makers. For benefit criterion Cj we don’t need to normalize the decision matrix. When Cj is a cost criterion, the normalized form of decision matrix (see Equation (15)) is presented below. M ⊗k  A  1  = A2   . Am C1 ⊗k ©11 ⊗k ©21 . ⊗k ©m1 C2 ⊗k ©12 ⊗k ©22 ... ⊗k ©m2 ... ... . ...  Cn ⊗k  ©1n  ⊗k  ©2n    ⊗k ©mnj (16) Here ©ij⊗k is the normalized form of the NC-number. Step 3. Determine the relative weight of each criterion The relative weight Wch of each criterion is obtained by the following equation. Wch = WC Wh (17) where, Wh = max {W1 , W2 , . . . , Wn }. Step 4. Calculate score values Using Equation (1), calculate the score value Sc (©ij⊗k ) (i = 1, 2, . . . , m; j = 1, 2, . . . , n) of ©ij⊗k if Cj is a cost criterion. Using Equation (1), calculate the score value Sc ((c)kij ) (i = 1, 2, . . . , m; j = 1, 2, . . . , n) of ©kij if Cj is a benefit criterion. Step 5. Calculate accuracy values Using Equation (9), calculate the accuracy value Ac (©ij⊗k ) (I = 1, 2, . . . , m; j= 1, 2, . . . , n) of ©ij⊗k if Cj is a cost criterion. Using Equation (9), calculate the accuracy value Ac (©kij ) (I = 1, 2, . . . , m; j = 1, 2, . . . , n) of ©kij if Cj is a benefit criterion. Step 6. Formulate the dominance matrix Calculate the dominance of each alternative Ai over each alternative Aj with respect to the criteria C (C1, C2 , . . . , Cn ), of the k-th decision maker Ek by the following Equations (18) and (19). Information 2017, 8, 149 10 of 21 (For cost criteria) Ψkc (Ai , Aj ) = s WCh ( Σ Wch c=1 = 0, = s − α1 ⊗k if ©ic = ©jc⊗k n ( Σ Wch c=1 WCh ⊗k ⊗k ⊗k ∂(©ic , ©jc ) , if ©ic < ©jc⊗k (For benefit criteria) Ψkc (Ai , Aj ) =          ⊗k ⊗k ⊗k ∂(©ic , ©jc ) , if ©ic > ©jc⊗k n s ( = 0, = − α1 WCh n Σ Wch ∂(©kic , ©kjc ) , if ©kic > ©kjc c=1 s if ©kic = ©kjc n Σ Wch ( c=W1 Ch ∂(©kic , , ©kjc ) , if ©kic < ©kjc                        (18) (19) where, parameter α represents the attenuation factor of losses and α must be positive. Step 7. Formulate the individual overall dominance matrix Using Equation (20), calculate the individual total dominance matrix of each alternative Ai over each alternative Aj under the criterion Cj . n φk = (Ai , Aj ) = Σ Ψkc (Ai , Aj ) c=1 (20) Step 8. Aggregate the dominance matrix Using Equation (21), calculate the collective overall dominance of alternative Ai over each alternative Aj . m φ(Ai , Aj ) = Σ γk λk (Ai , Aj ) k=1 (21) Step 9. Calculate global values We present the global value of each alternative as follows: n n Σ φ (Ai , Aj ) − min ( Σ φ (Ai , Aj )) Ωi = 1≤ i ≤ m j=1 j=1 n n (22) max ( Σ φ (Ai , Aj )) − min ( Σ φ (Ai , Aj )) 1≤ i ≤ m j=1 1≤ i ≤ m j=1 Step 10. Rank the priority Sorting the values of Ωi provides the rank of each alternative. A set of alternatives can be preference-ranked according to the descending order of Ωi . The highest global value corresponds to the best alternative. A conceptual model of the NC-TODIM strategy is shown in Figure 2. Information 2017, 8, 149 11 of 21 Figure 2. A flow chart of the proposed neutrosophic cubic set (NC)-TODIM strategy. 5. Illustrative Example In this section, a MAGDM problem is adapted from the study [18] under the NCS environment. An investment company wants to select the best alternative among the set of feasible alternatives. The feasible alternatives are 1. 2. 3. 4. Car company (A1 ) Food company (A2 ) Computer company (A3 ) Arms company (A4 ). The best alternative is selected based on the following criteria: 1. 2. 3. Risk analysis (C1 ) Growth analysis (C2 ) Environmental impact analysis (C3 ). An investment company forms a panel of three decision makers {E1 , E2 , E3 } who evaluate four alternatives in decision making process. The weight vector of attributes and decision makers are considered as W = (0.4, 0.35, 0.25)T γ = (0.32, 0.33, 0.35)T respectively. The proposed strategy is presented using the following steps: Step 1. Formulate the decision matrix Formulate the decision matrices Mk (k = 1, 2, 3) using the rating values of alternatives with respect to three criteria provided by the three decision makers in terms of NC-numbers. Assume e kij , Rk > present the rating value provided by the decision maker Ek for that the NC-numbers ©k = <G ij ij alternative Ai with respect to attribute Cj . Using these rating values ©kij (k = 1, 2, 3; i = 1, 2, 3, 4; j = 1, 2, 3), three decision matrices Mk = (©kij ) (k = 1, 2, 3) are constructed (see Equations (23)–(25)). 4 ×3 Decision matrix for E1  A  1  M = A2  A3 A4 1 < [0.41, < [0.35, < [0.23, < [0.17, 0.52], 0.46], 0.29], 0.23], [0.10, [0.18, [0.36, [0.45, 0.18], 0.27], 0.45], 0.55], C1 [0.06, [0.17, [0.34, [0.42, 0.17], 0.34], 0.42], 0.59], (0.48, (0.43, (0.26, (0.21,  0.11, 0.16, 0.26, 0.32, 0.11) 0.21) 0.26) 0.37) > > > > < [0.40, < [0.22, < [0.34, < [0.22, 0.51], 0.28], 0.45], 0.28], [0.10, [0.40, [0.20, [0.40, 0.20], 0.50], 0.30], 0.50], C2 [0.10, [0.39, [0.19, [0.39, 0.19], 0.48], 0.39], 0.48], (0.50, (0.28, (0.44, (0.28, 0.11, 0.28, 0.16, 0.28, 0.11) 0.28) 0.22) 0.28) > > > > < [0.22, < [0.38, < [0.22, < [0.38, 0.27], 0.49], 0.27], 0.49], [0.41, [0.10, [0.41, [0.10, 0.52], 0.21], 0.52], 0.21], C3 [0.41, [0.10, [0.41, [0.10, 0.52], 0.21], 0.52], 0.21], (0.31, (0.57, (0.31, (0.57, 0.31, 0.12, 0.31, 0.12, 0.31) 0.12) 0.31) 0.12) >   >  > > 0.32, 0.26, 0.11, 0.16, 0.37) 0.26) 0.11) 0.21) > > > > < [0.25, < [0.25, < [0.44, < [0.25, 0.31], 0.31], 0.57], 0.31], [0.35, [0.35, [0.10, [0.35, 0.44], 0.44], 0.17], 0.44], C2 [0.35, [0.35, [0.10, [0.35, 0.44], 0.44], 0.17], 0.44], (0.28, (0.28, (0.51, (0.28, 0.28, 0.28, 0.11, 0.28, 0.28) 0.28) 0.11) 0.28) > > > > < [0.34, < [0.34, < [0.19, < [0.34, 0.43], 0.43], 0.24], 0.43], [0.13, [0.13, [0.53, [0.13, 0.27], 0.27], 0.67], 0.27], C3 [0.13, [0.13, [0.53, [0.13, 0.27], 0.27], 0.67], 0.27], (0.49, (0.49, (0.27, (0.49, 0.11, 0.11, 0.27, 0.11, 0.11) 0.11) 0.27) 0.11) >   >  > > (23) Decision matrix for E2  A  1  M2 = A2  A3 A4 < [0.17, < [0.23, < [0.41, < [0.35, 0.23], 0.29], 0.52], 0.46], [0.46, [0.37, [0.10, [0.20, 0.55], 0.46], 0.18], 0.28], C1 [0.42, [0.34, [0.10, [0.17, 0.59], 0.42], 0.17], 0.34], (0.21, (0.26, (0.48, (0.42,  (24) Information 2017, 8, 149 12 of 21 Decision matrix for E3  A  1  M = A2  A3 A4 3 < [0.22, < [0.22, < [0.38, < [0.38, 0.27], 0.27], 0.49], 0.49], [0.42, [0.42, [0.10, [0.10, 0.52], 0.52], 0.21], 0.21], C1 [0.42, [0.42, [0.10, [0.10, 0.52], 0.52], 0.21], 0.21], (0.28, (0.28, (0.50, (0.50, 0.28, 0.28, 0.11, 0.11, 0.28) 0.28) 0.11) 0.11) > > > > < [0.22, < [0.40, < [0.34, < [0.22, 0.28], 0.51], 0.45], 0.28], [0.40, [0.10, [0.20, [0.40, 0.50], 0.20], 0.30], 0.50], C2 [0.39, [0.10, [0.19, [0.39, 0.48], 0.19], 0.39], 0.48], (0.28, (0.50, (0.44, (0.28, 0.28, 0.11, 0.16, 0.28, 0.28) 0.11) 0.22) 0.28) > > > > < [0.41, < [0.23, < [0.38, < [0.17, 0.52], 0.29], 0.49], 0.23], [0.10, [0.36, [0.10, [0.45, 0.18], 0.45], 0.21], 0.54], C3 [0.10, [0.34, [0.10, [0.42, 0.17], 0.42], 0.21], 0.59], (0.48, (0.26, (0.50, (0.21, 0.11, 0.26, 0.11, 0.32, 0.11) 0.26) 0.11) 0.37)  >   >  > > (25) Step 2. Normalize the decision matrix Since all the criteria are benefit type, we do not need to normalize the decision matrix. Step 3. Determine the relative weight of each criterion Using Equation (17), we obtain the relative weight vector Wch of criteria as follows: Wch = (1, 0.875, 0.625)T . Step 4. Calculate score values The score values of each alternative relative to each criterion obtained by Equation (1) are presented in the Tables 1–3. Table 1. Score values for M1 . A1 A2 A3 A4 C1 C2 C2 0.56 0.40 0.50 −0.03 0.54 0.09 0.38 0.09 0.06 0.54 0.06 0.54 Table 2. Score values for M2 . A1 A2 A3 A4 C1 C2 C2 −0.03 0.13 0.56 0.39 0.13 0.13 0.60 0.13 0.49 0.49 −0.04 0.49 Table 3. Score values for M3 . A1 A2 A3 A4 C1 C2 C2 0.07 0.07 0.51 0.51 0.09 0.52 0.37 0.09 0.56 0.13 0.39 −0.03 Step 5. Calculate accuracy values The accuracy values of each alternative relative to each criterion obtained by Equation (9) are presented in Tables 4–6. Table 4. Accuracy values for M1 . A1 A2 A3 A4 C1 C2 C2 0.14 0.12 −0.20 −0.38 0.30 −0.23 0.09 −0.23 −0.24 0.32 −0.24 0.32 Information 2017, 8, 149 13 of 21 Table 5. Accuracy values for M2 . A1 A2 A3 A4 C1 C2 C2 −0.38 −0.20 0.14 0.12 −0.18 −0.18 0.36 −0.18 0.21 0.21 −0.21 0.21 Table 6. Accuracy values for M3 . A1 A2 A3 A4 C1 C2 C2 −0.24 −0.24 0.26 0.26 −0.23 0.30 0.09 −0.23 0.41 −0.20 0.12 −0.38 Step 6. Formulate the dominance matrix Using Equation (19), we construct dominance matrix for α = 1. The dominance matrices are represented in matrix form (See Equations (26)–(34)). The dominance matrix Ψ11 , the dominance matrix Ψ12 Ψ11  A  1  =A2  A3 A4 Ψ12  A  1  =A2   A3 A4 A1 0 −0.46 −0.74 −0.88  A3 A4 0.30 0.35   −0.58 0.30  0 0.19 −0.47 0 A2 0.18 0 0.23 −0.74 A1 0 −0.82 −0.51 −0.81 A2 0.29 0 0.24 0  A3 A4 0.18 0.28   −0.69 0   0 0.29 −0.65 0 (26) (27) The dominance matrix Ψ13 , the dominance matrix Ψ21 Ψ13 Ψ21  A  1  =  A2   A3 A4  A  1  =  A2   A3 A4 A1 0 0.25 0 0.25 A1 0 0.18 0.35 0.30 A2 −1 0 −1 0 A2 −0.46 0 0.09 0.23 A3 0 0.26 0 0.26 A3 −0.88 −0.75 0 0.19  A4 −1   0   −1 0  A4 −0.74   −0.58  0.04  0 (28) (29) Information 2017, 8, 149 14 of 21 The dominance matrix Ψ22 , the dominance matrix Ψ23  A1 0 0 0.29 0 A  1  Ψ22 =A2   A3 A4  A  1  Ψ23 =A2  A3 A4 A2 0 0 0.29 0 A1 0 0 −1 0  A3 A4 −0.84 0    0.84 0   0 0.29 −0.84 0 A2 A3 0 0.26 0 0.26 −1 0 0 0.26 The dominance matrix Ψ31 , the dominance matrix Ψ32  A  1  Ψ31 =A2   A3 A4   A  1  Ψ32 = A2  A3 0.23 A4 A1 0 0 0.31 0.31  A4 0    0   −1 0 (31)  A4 0.78   0.78  0  0 (32) A2 A3 0 0.78 0 0.78 0.31 0 0.31 0 A1 0 0.29 −0.51 0 A2 −0.83 0 0 −0.83 A3 0.65 0.18 0.23 −0.65 The dominance matrix Ψ33  A  1  Ψ33 =A2   A3 A4 A1 0 0.23 0.59 −1.1 A2 −0.94 0 0.18 0.58 (30) A3 0.59 −0.73 0 −0.94  A4 0    0.29   0  A4 −1.1   0.15   0.23  0 (33) (34) Step 7. Formulate the individual overall dominance matrix The individual overall dominance matrix is calculated by the Equation (20) and the dominance matrices are represented in matrix form (see Equations (35)–(37)). First decision maker’s overall dominance matrix φ1  A  1  φ = A2  A3 A4 1 A1 0 −1 −1.3 −1.5 A2 −0.53 0 0.53 0.74 A3 0.47 −1 0 −0.86  A4 −0.37   0.30   0.52  0 (35) Information 2017, 8, 149 15 of 21 Second decision maker’s overall dominance matrix φ2  A  1  φ2 = A2  A3 A4 A1 0 0.18 −0.36 0.30 A2 −0.46 0 −0.62 0.23 A3 −1.5 −1.3 0 −0.39  A4 −0.74   −0.58  0.67  0 (36) Third decision maker’s overall dominance matrix φ3  A  1  φ =A2  A3 A4 3 A1 0 0.52 −0.05 −0.79 A2 −1.8 0 −0.02 −1.1 A3 −2 −1.3 0 −1.6  A4 1.9    −0.34  0.46  0 (37) Step 8. Aggregate the dominance matrix Using Equation (21), the aggregate dominance matrix φ is constructed (see Equation (38)) as follows:   A1 A2 A3 A4 A 0 −0.94 −1.1 −0.53  1    φ = A2 −0.10 (38) 0 −1.23 −0.22   A3 −0.54 −0.38 0 −0.23 A4 −0.64 −0.55 −0.96 0 Step 9. Calculate global values Using Equation (22), we calculate the values of Ωi (i = 1, 2, 3, 4) and represented in Table 7. Table 7. Global values of alternatives. Ai A1 A2 A3 A4 Ωi 0.49 0.61 1 0 Step 10. Rank the priority Since Ω3 > Ω2 > Ω1 > Ω4 , alternatives are then preference ranked as follows: A3 > A2 > A1 > A4 . Hence A3 is the best alternative. From the illustrative example, we see that the proposed NC-TODIM strategy is more suitable for real scientific and engineering applications because it can handle hybrid information consisting of INS and SVNS information simultaneously to cope with indeterminate and inconsistent information. Thus, NC-TODIM extends the existing decision-making strategies and provides a sophisticated mathematical tool for decision makers. 6. Rank of Alternatives with Different Values of α Table 8 shows that the ranking order of alternatives depends on the values of the attenuation factor, which reflects the importance of the attenuation factor in the NC-TODIM strategy. Information 2017, 8, 149 16 of 21 Table 8. Global values and ranking of alternatives for different values of α. Values of α 0.5 1 1.5 2 3 Global Values of Alternative (Ωi ) Ω1 = 0, Ω2 = 0.89, Ω3 = 1, Ω4 = 0.46 Ω3 > Ω2 > Ω4 > Ω1 Ω1 = 0.49, Ω2 = 0.61, Ω3 = 1, Ω4 = 0 Ω3 > Ω2 > Ω1 > Ω4 Ω1 = 0, Ω2 = 0.72, Ω3 = 1, Ω4 = 0.44 Ω3 > Ω2 > Ω4 > Ω1 Ω1 = 0, Ω2 = 1, Ω3 = 0.81, Ω4 = 0.38 Ω2 > Ω3 >Ω4 > Ω1 Ω1 = 0, Ω2 = 0.56, Ω3 = 1, Ω4 = 0.45 Ω3 > Ω2 > Ω4 > Ω1 Rank Order of Ai A3 > A2 > A4 > A1 A3 > A2 > A1 > A4 A3 > A2 > A4 > A1 A2 > A3 > A4 > A1 A3 > A2 > A4 > A1 Analysis on Influence of the Parameter α to Ranking Order The impact of parameter α on ranking order is examined by comparing the ranking orders taken with varying the different values of α. When α = 0.5, 1, 1.5, 2, 3, ranking order are presented in Table 8. We draw Figures 3 and 4 to compare the ranking order for different values of α. When α = 0.5, α = 1.5 and α = 3, the ranking order is unchanged and A3 is the best alternative, while A1 is the worst alternative. When α = 1, the ranking order is changed and A3 is the best alternative and A4 is the worst alternative. For α = 2, the ranking order is changed and A2 is the best alternative and A1 is the worst alternative. From Table 8, we see that A3 is the best alternative in four cases and A1 is the worst for four cases. We can say that ranking order depends on parameter α. Figure 3. Global values of the alternatives for different values of attenuation factor α = 0.5, 1, 1.5, 2, 3. Information 2017, 8, 149 17 of 21 Figure 4. Ranking of the alternatives for α = 0.5, 1, 1.5, 2, 3. 7. Comparative Analysis and Discussion On comparing with the existing neutrosophic decision making strategies [26–29,33–35,64–69], we see that the decision information used in the proposed NC-TODIM strategy is NC numbers, which comprises of interval neutrosophic information and single-valued neutrosophic information simultaneously; whereas the decision information in the existing literature is either SVNSs or INSs. Since NC numbers comprises of much more information, the NC numbers based on the TODIM strategy proposed in this paper is more elegant, typical and more general in applications, while the existing neutrosophic decision-making strategies cannot deal with the NC number decision-making problem developed in this paper. The first decision making paper in NCS environment was studied by Banerjee et al. [44]. On comparison with existing GRA-based NCS decision making strategies [44], we observe that the proposed NC-TODIM strategy uses the score, and accuracy functions, while the decision making-strategy in [44] uses Hamming distances for weighted grey relational coefficients and standard (ideal) grey relational coefficients, and ranks the alternatives based on the relative closeness coefficients. Hence, the proposed NC-TODIM strategy is relatively simple in the decision making process. On comparing with cosine measures of NCSs [49], we observe that the proposed NC-TODIM involves multiple decision makers, while in [49] only a single decision maker is involved. This shows that [49] cannot deal with group decision making, while the proposed NC-TODIM strategy is more sophisticated as it can deal with single as well as group decision making in the NCS environment. On comparison with extended TOPSIS [50] with neutrosophic cubic information, we observe that nine components are present in NCSs. Therefore, by calculation of a weighted decision matrix, a neutrosophic cubic positive ideal solution (NCPIS), and a neutrosophic cubic negative ideal solution, the distance measures of alternatives from NCPIS and NCNIS (NCNIS,) and entropy weight, and use of an aggregation operator are lengthy, time consuming, and hence expensive. The proposed NC-TODIM strategy is free from different kinds of typical aggregation operators. The calculations required for the proposed strategy are relatively straightforward and time-saving. Therefore, the final ranking obtained by the proposed strategy is more conclusive than those produced by the other strategies, and it is evident that the proposed strategy is accurate and reliable. On comparison with the strategy proposed by Zhan et al. [51], we see that they employ score, accuracy, and certainty functions, and a weighted average operator and weighted geometric operator of NCSs for decision making problem involving only a single decision maker. This reflects that the strategy introduced by Zhan et al. [51] is only applicable for decision making problems involving single decision maker. However, our proposed NC-TODIM strategy is more general as it is capable of dealing with group decision-making problems. Information 2017, 8, 149 18 of 21 A comparative study is conducted with the existing strategy [48] for group decision making under a NCS environment (See Table 9). Since the philosophy of two strategies are different, the obtained results (ranking order) are different. At a glance, it cannot be said which strategy is superior to the other. However, on comparison with similarity measure-based strategies studied in [48], we observed that ideal solutions are needed for ranking of alternatives but in a real world ideal solution, this is an imaginary case, which means that an indeterminacy arises automatically, whereas in our proposed NC-TODIM strategy we can calculate the rank of the alternatives based on global values of alternatives. So, the proposed NC-TODIM strategy is relatively easy to implement and apply for solving MAGDM problems. Table 9. Ranking order of alternatives using three different decision making strategies in the neutrosophic cubic set (NCS) environment. Proposed NC-TODIM Strategy Similarity Measure [48] Ω1 = 0, Ω2 = 0.89, Ω3 = 1, Ω4 = 0.46 Ranking order: A3 > A2 > A4 > A1 ρ1 = 0.20, ρ2 = 0.80, ρ3 = 0.22, ρ4 = 0.19 Ranking order: A2 > A3 > A1 > A4 8. Conclusions NCSs can better describe hybrid information comprising of INSs and NSs. In this study, we proposed a score function and an accuracy function, and established their properties. We developed a NC-TODIM strategy, which is capable for tackling MAGDM problems affected by uncertainty and indeterminacy represented by NC numbers. The standard TODIM, in its original formulation, is only applicable to a crisp environment. Existing neutrosophic TODIM strategies deal with single valued neutrosophic information or interval neutrosophic information. Therefore, proposed NC-TODIM strategy demonstrates the advantages of presenting and manipulating MAGDM problems with NCSs comprising of the hybrid information of INSs and NSs. Furthermore, NC-TODIM strategy that considers the risk preferences of decision makers, is significant to solve MAGDM problems. The proposed NC-TODIM strategy was verified to be applicable, feasible, and effective by solving an illustrative example regarding the selection problem of investment alternatives. In addition, we investigated the influence of attenuation factor of losses α on ranking the order of alternatives. The contribution of this study can be concluded as follows. First, this study utilized NCSs to present the interval neutrosophic information and neutrosophic information in the MAGDM process. Second, the NC-TODIM strategy established in this paper is simpler and easier than the existing strategy proposed by Pramanik et al. [48] for group decision making with neutrosophic cubic information based on similarity measure and demonstrates the main advantage of its simple and easy group decision making process. Third, TODIM strategy was extended to the NCS environment. Fourth, we defined the NC number. Fifth, we defined the score and accuracy functions and proved their basic properties. Sixth, we developed the ranking of NC numbers using score and accuracy functions. Therefore, two functions namely, score function, accuracy function, and proofs of their basic properties, ranking of NC numbers, and NC-TODIM strategy for MAGDM are the main contributions of the paper. Several directions for future research are generated from this study. First, this study employs the NC-TODIM strategy to deal with MAGDM. In addition to MAGDM, MAGDM problems in a variety of other fields can be solved using the NC-TODIM strategy, including logistics center selection, personnel selection, teacher selection, renewable energy selection, medical diagnosis, image processing, fault diagnosis, etc. Second, this study considers the risk preferences of decision makers i.e., the essence of TODIM, while the interrelationship between criteria are ignored. In future research, the NC-TODIM strategy will be improved to address this deficiency. Third, the proposed strategy can only deal with crisp weights of attributes and decision makers, rather than NCS, which reflects its main limitation. This limitation will be effectively addressed in our future research. Fourth, in our illustrative Information 2017, 8, 149 19 of 21 example, three criteria are considered as an example. However, in real world group decision making problems, many other criteria should be included. A comprehensive framework for MAGDM problem comprising of all relevant criteria should be designed based on prior studies and the proposed NC-TODIM strategy in future research. Finally, we conclude that the developed NC-TODIM strategy offers a novel and effective strategy for decision makers under the NCS environment, and will open up a new avenue of research into the neutrosophic hybrid environment. 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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). information Article VIKOR Method for Interval Neutrosophic Multiple Attribute Group Decision-Making Yu-Han Huang 1, *, Gui-Wu Wei 2, * 1 2 3 * ID and Cun Wei 3 College of Mathematics and Software Science, Sichuan Normal University, Chengdu 610068, China School of Business, Sichuan Normal University, Chengdu 610101, China School of Science, Southwest Petroleum University, Chengdu 610500, China; weicun1990@163.com Correspondence: hyh85004267@163.com (Y.-H.H.); weiguiwu1973@sicnu.edu.cn (G.-W.W.) Received: 21 October 2017; Accepted: 8 November 2017; Published: 10 November 2017 Abstract: In this paper, we will extend the VIKOR (VIsekriterijumska optimizacija i KOmpromisno Resenje) method to multiple attribute group decision-making (MAGDM) with interval neutrosophic numbers (INNs). Firstly, the basic concepts of INNs are briefly presented. The method first aggregates all individual decision-makers’ assessment information based on an interval neutrosophic weighted averaging (INWA) operator, and then employs the extended classical VIKOR method to solve MAGDM problems with INNs. The validity and stability of this method are verified by example analysis and sensitivity analysis, and its superiority is illustrated by a comparison with the existing methods. Keywords: MAGDM; INNs; VIKOR method 1. Introduction Multiple attribute group decision-making (MAGDM), which has been increasingly investigated and considered by all kinds of researchers and scholars, is one of the most influential parts of decision theory. It aims to provide a comprehensive solution by evaluating and ranking alternatives based on conflicting attributes with respect to decision-makers’ (DMs) preferences, and has widely been utilized in engineering, economics, and management. Several traditional MAGDM methods have been developed by scholars in literature, such as the TOPSIS (Technique for Order Preference by Similarity to an Ideal Solution) method [1,2], the VIKOR (VIsekriterijumska optimizacija i KOmpromisno Resenje) method [3–5], the PROMETHEE (Preference Ranking Organization Method for Enrichment Evaluations) method [6], the ELECTRE (ELimination Et Choix Traduisant la Realité) method [7], the GRA (Grey Relational Analysis) method [8–10], and the MULTIMOORA (Multiobjective Optimization by Ratio Analysis plus Full Multiplicative Form) method [11,12]. Due to the fuzziness and uncertainty of the alternatives in different attributes, attribute values in MAGDM are not always represented as real numbers, and they can be described as fuzzy numbers in more suitable occasions [13–15]. Since fuzzy set (FS) was first defined by Zadeh [16], is has been used as a better tool to solve MAGDM [17,18]. Smarandache [19,20] proposed a neutrosophic set (NS). Furthermore, the concepts of single-valued neutrosophic sets (SVNSs) [21] and interval neutrosophic sets(INSs) [22] were presented for actual applications. Ye [23] proposed a simplified neutrosophic set (SNS). Broumi and Smarandache [24] defined the correlation coefficient of INS. Zhang et al. [25] gave the correlation coefficient of interval neutrosophic numbers (INNs) in MAGDM. Zhang et al. [26] gave an outranking approach for INN MAGDM. Tian et al. [27] defined a cross-entropy in INN MAGDM. Zhang et al. [28] proposed some INN aggregating. Some other INN operators are proposed in References [29–32]. Ye [33] proposed two similarity measures between INNs. The SVNS and INS have received more and more attention since their appearance [34–42]. Information 2017, 8, 144; doi:10.3390/info8040144 www.mdpi.com/journal/information Information 2017, 8, 144 2 of 10 Opricovic [3] proposed the VIKOR method for a MAGDM problem with conflicting attributes [43–45]. Some scholars proposed fuzzy VIKOR models [46], intuitionistic fuzzy VIKOR models [47–49], the linguistic VIKOR method [50], the interval type-2 fuzzy VIKOR model [51], the hesitant fuzzy linguistic VIKOR method [52], the dual hesitant fuzzy VIKOR method [53], the linguistic intuitionistic fuzzy [54], and the single-valued neutrosophic number (SVNN) VIKOR method [38]. However, there has not yet been an academic investigation of the VIKOR method for MAGDM problems with INNs. Therefore, it is necessary to pay great attention to this novel and worthy research issue. The purpose of our paper is to use the VIKOR idea to solve MAGDM with INNs, to fill this vacancy of knowledge. In Section 2, we give the definition of INNs. We propose the VIKOR method for INN MAGDM. In Section 3, an example is provided, and the comparative analysis is proposed in Section 4. We finish with our conclusions in Section 5. 2. Preliminaries The concepts of SVNSs and INSs are introduced. SVNSs and INSs NSs [19,20] are not easy to apply to real applications. Wang et al. [21] developed SNSs. Furthermore, Wang et al. [22] defined INSs. Definition 1 [21]. Let X be a space of points (objects), a SVNSs A in X is characterized as following: A = {( x, ξ A ( x ), ψ A ( x ), ζ A ( x ))| x ∈ X } (1) where the truth-membership function ξ A ( x ), indeterminacy-membership ψ A ( x ) and falsity-membership function ζ A ( x ), ξ A ( x ) → [0, 1], ψ A ( x ) → [0, 1] and ζ A ( x ) → [0, 1] , with the condition 0 ≤ ξ A ( x ) + ψ A ( x ) + ζ A ( x ) ≤ 3. Definition 2 [22]. Let X be a space of points (objects) with a generic element in fixed set X, denoted by x, where e in X is characterized as follows: an INS A e= A x, ξ Ae ( x ), ψ Ae ( x ), ζ Ae ( x ) | x ∈ X (2) where truth-membership function ξ Ae ( x ), indeterminacy-membership ψ Ae ( x ), and falsity-membership function ζ Ae ( x ) are interval values, ξ Ae ( x ) ⊆ [0, 1], ψ Ae ( x ) ⊆ [0, 1] and ζ Ae ( x ) ⊆ [0, 1], and 0 ≤ sup ξ Ae ( x ) + sup ψ Ae ( x ) + sup ζ Ae ( x ) ≤ 3. i h i h i h i h e = ξ e , ψ e , ζ e = ξ L , ξ R , ψ L , ψ R , ζ L , ζ R , where ξ L , ξ R ⊆ An INN can be expressed as A e e A e e e e A e e A A A A A A A A A h i h i R L R L R R R [0, 1], ψ e , ψ e ⊆ [0, 1], ζ e , ζ e ⊆ [0, 1], and 0 ≤ ξ e + ψ e + ζ e ≤ 3. A A A A e= Definition 3 [45]. Let A e = SF A e= Definition 4 [45]. Let A is defined as: h A A A i h i h i ξ Le , ξ Re , ψ Le , ψ Re , ζ Le , ζ Re be an INN, then a score function, SF, is: A A A A A A 2 + ξ Le − ψ Le − ζ Le + 2 + ξ Re − ψ Re − ζ Re h A A A 6 A A A e ∈ [0, 1] , SF A (3) i h i h i e , ξ Le , ξ Re , ψ Le , ψ Re , ζ Le , ζ Re be an INN, then an accuracy function, AF A A e = AF A A A A A A ξ Le + ξ Re − ζ Le + ζ Re A A A 2 A e ∈ [−1, 1] , AF A (4) Information 2017, 8, 144 3 of 10 i h i h i h i i h i h h e = e = Definition 5 [45]. Let A and B ξ Le , ξ Re , ψ Le , ψ Re , ζ Le , ζ Re ξ Le , ξ Re , ψ Le , ψ Re , ζ Le , ζ Re B B B B A A A A A A B B 2+ξ Le −ψ Le −ζ Le + 2+ξ Re −ψ Re −ζ Re 2+ξ Le −ψ Le −ζ Le + 2+ξ Re −ψ Re −ζ Re B B B B B B A A A A A A e = e = be two INNs, SF A and SF B be 6 6 ξ L +ξ R − ζ Le +ζ Re ξ L +ξ R − ζ Le +ζ Re B B A A e = Ae Ae e = Be Be and AF B be the accuracy the score functions, and AF A 2 2 e < SF B e = SF B e = AF B e < B; e , then A e , then (1) if AF A e , then e if SF A functions, then if SF A e < AF B e < B. e = B; e e , then A e (2) if AF A A e= Definition 6 [22,33]. Let A be two INNs, then: h i i h i h i i h h i h e= ξ Le , ξ Re , ψ Le , ψ Re , ζ Le , ζ Re ξ Le , ξ Re , ψ Le , ψ Re , ζ Le , ζ Re and B A A A A A A B B B B B B L L R R L L R R ψB , ψ A ψB , ζ A ζ!B , ζ A ζ B ; ξ LA + ξ BL − ξ LA ξ BL , ξ RA + ξ BR − ξ RA ξ BR , ψ A L L R R L R ψR , R L L L ξ A ξ B , ξ A ξ B , ψ A + ψB − ψ A ψB , ψ A + ψBR − ψ A B e⊗B e= L ; (2) A ζ A + ζ BL − ζ LA ζ BL , ζ RA + ζ BR − ζ RA ζ BR h i i h i h e = 1 − 1 − ξ L λ , 1 − 1 − ξ R λ , ψ L λ , ψ R λ , ζ L λ , ζ R λ , λ > 0; (3) λ A A A A A A A λ h i i h λ λ i h L λ R λ , 1 − 1 − ζ L λ, 1 − 1 − ζ R λ e = , λ > 0. ξ LA , ξ RA , ψA , ψA (4) A A A e⊕B e= (1) A e and B e and B e be two INNs, then the normalized Hamming distance between A e is Definition 7 [45]. Let A defined as follows: ! L − ξ L + ξ R − ξ R + ψL − I L 1 ξ B B B A A A e B e = d A, (5) R − ψR + ζ L − ζ L + ζ R − ζ R 6 + ψA B B B A A 3. VIKOR Method for INN MAGDM Problems Let φ = {φ1 , φ2 , · · · , φm } be alternatives and ϕ = { ϕ1 , ϕ2 , · · · , ϕn } be attributes. Let τ = n (τ1 , τ2 , · · · , τn ) be the weight of ϕ j , 0 ≤ τj ≤ 1, ∑ τj = 1. Let D = { D1 , D2 , · · · , Dt } be the set j =1 t of DMs, σ = (σ1 , σ2 , · · · , σt ) be the weighting of DMs, with 0 ≤ σk ≤ 1, ∑ σk = 1. Suppose k =1 i i h i h h R(k) L(k) R(k) L(k) (k) L(k) R(k) ek = e , ζ ij , ζ ij , ψij , ψij rij = ξ ij , ξ ij is the INN decision matrix that R m×n m×n h h h i i i R(k) R(k) R(k) L(k) R(k) L(k) R(k) L(k) R(k) ≤ 3, ξ ij , ξ ij ⊆ [0, 1], ψij , ψij ⊆ [0, 1], ζ ij , ζ ij ⊆ [0, 1], 0 ≤ ξ ij + ψij + ζ ij i = 1, 2, · · · , m, j = 1, 2, · · · , n, k = 1, 2, · · · , t. To cope with the MAGDM with INNs, we develop the INN VIKOR model. e k and the interval neutrosophic number weighted averaging Utilize the R Step 1. (INNWA) operator e rij = h i h i h i (t) (1) (2) rij rij , · · · , e rij , e ξ ijL , ξ ijR , ψijL , ψijR , ζ ijL , ζ ijR = INNWAσ e i = 1, 2, · · · , m, j = 1, 2, · · · , n e= e to get R rij m×n (6) . e + and negative ideal solutions R e− . Step 2. Define the positive ideal solutions R h i h i h i ξ jL+ , ξ jR+ , ψjL+ , ψjR+ , ζ jL+ , ζ jR+ i h i h i h e − = ξ L− , ξ R− , ψ L− , ψ R− , ζ L− , ζ R− R j j j j j j e+ = R (7) (8) Information 2017, 8, 144 4 of 10 For the benefit attribute: i h h i i h ξ jL+ , ξ jR+ , ψjL+ , ψjR+ , ζ jL+ , ζ jR+ = maxξ ijL , maxξ ijR , minψijL , minψijR , minζ ijL , minζ ijR i i i i i i i i h i h ξ jL− , ξ jR− , ψjL− , ψjR− , ζ jL− , ζ jR− R R R L L L = minξ ij , minξ ij , maxψij , maxψij , maxζ ij , maxζ ij h i i i i i i i i i h i h ξ jL− , ξ jR− , ψjL− , ψjR− , ζ jL− , ζ jR− R L R L R L = maxξ ij , maxξ ij , minψij , minψij , minζ ij , minζ ij (12) i i i i i (11) i i h i (10) i i For the cost attribute: h i h i i h ξ jL+ , ξ jR+ , ψjL+ , ψjR+ , ζ jL+ , ζ jR+ R R L R L L = minξ ij , minξ ij , maxψij , maxψij , maxζ ij , maxζ ij i (9) Step 3. Compute the Γi and Zi .  h i  i h i h ξ jL+ , ξ jR+ , ψjL+ , ψjR+ , ζ jL+ , ζ jR+ , i h i h i  τj × d h n ξ ijL , ξ ijR , ψijL , ψijR , ζ ijL , ζ ijR  h i h i  i h Γi = ∑ ξ jL+ , ξ jR+ , ψjL+ , ψjR+ , ζ jL+ , ζ jR+ , j =1 d h L− R− i h L− R− i h L− R− i  , ζj , ζj ξ j , ξ j , ψj , ψj i   i h i h  h  R+ L+ R+ L+ L+ R+   , ζ , , ψ , ζ , ξ , ψ ξ   j j i h j j h j  i hj i     τ × d   j   R L R L R L   ξ ij , ξ ij , ψij , ψij , ζ ij , ζ ij i  i h i h  h Zi = max  j    ξ L+ , ξ jR+ , ψjL+ , ψjR+ , ζ jL+ , ζ jR+ ,     d h j i  i h i h     R− L− R− L− L− R−   ξ ,ξ , ζ ,ζ , ψ ,ψ j j j j j (13) (14) j where τj is weight of ϕ j . Step 4. Compute the Θi by the following formula: Θi = θ where (Γi − Γi∗ ) (Zi − Zi∗ ) + 1 − θ ( ) Γi− − Γi∗ Zi− − Zi∗ (15) Γi∗ = min Γi , Γi− = max Γi (16) Zi∗ = min Zi , Zi− = max Γi (17) i i i i where θ depicts the decision-making mechanism coefficient. If θ > 0.5, it is for “the maximum group utility”; If θ < 0.5, it is “the minimum regret”; and it is both if θ = 0.5. Step 5. Rank the alternatives by Θi , Γi and Zi according to the selection rule of the traditional VIKOR method. Information 2017, 8, 144 5 of 10 4. Numerical Example 4.1. Numerical Example In this section, a numerical example is given with INNs. Five possible emerging technology enterprises (ETEs) φi (i = 1, 2, 3, 4, 5) are selected. Four attributes are selected to evaluate the five possible ETEs: 1 ϕ1 is the employment creation; 2 ϕ2 is the development of science and technology; 3 ϕ3 is the technical advancement; 4 ϕ4 is the industrialization infrastructure. The five ETEs are to be evaluated by using INNs under the attributes (τ = (0.2, 0.1, 0.3, 0.4) T ) by the DMs (σ = (0.2, 0.5, 0.3) T ), as listed in Tables 1–3. e1 . Table 1. The decision matrix R φ1 φ2 φ3 φ4 φ5 φ1 φ2 φ3 φ4 φ5 ϕ1 ϕ2 ([0.3, 0.4], [0.6, 0.7], [0.3, 0.5]) ([0.5, 0.7], [0.6, 0.8], [0.2, 0.4]) ([0.4, 0.5], [0.5, 0.6], [0.2, 0.3]) ([0.6, 0.7], [0.2, 0.3], [0.1, 0.2]) ([0.4, 0.5], [0.2, 0.3], [0.2, 0.3]) ([0.4, 0.5], [0.2, 0.3], [0.1, 0.2]) ([0.5, 0.6], [0.3, 0.5], [0.2, 0.3]) ([0.3, 0.4], [0.5, 0.6], [0.1, 0.2]) ([0.4, 0.5], [0.1, 0.2], [0.2, 0.3]) ([0.2, 0.3], [0.6, 0.7], [0.2, 0.3]) ϕ3 ϕ4 ([0.1, 0.2], [0.4, 0.5], [0.1, 0.2]) ([0.5, 0.7], [0.4, 0.6], [0.2, 0.3]) ([0.3, 0.4], [0.1, 0.2], [0.2, 0.3]) ([0.4, 0.5], [0.2, 0.3], [0.1, 0.2]) ([0.5, 0.6], [0.4, 0.5], [0.2, 0.3]) ([0.3, 0.4], [0.5, 0.6], [0.2, 0.3]) ([0.6, 0.7], [0.3, 0.4], [0.2, 0.3]) ([0.4, 0.5], [0.1, 0.2], [0.3, 0.4]) ([0.3, 0.4], [0.4, 0.5], [0.2, 0.3]) ([0.3, 0.4], [0.6, 0.7], [0.3, 0.4]) e2 . Table 2. The decision matrix R φ1 φ2 φ3 φ4 φ5 φ1 φ2 φ3 φ4 φ5 ϕ1 ϕ2 ([0.4, 0.6], [0.5, 0.7], [0.3, 0.4]) ([0.6, 0.9], [0.4, 0.5], [0.3, 0.4]) ([0.8, 0.9], [0.8, 0.9], [0.4, 0.5]) ([0.6, 0.7], [0.3, 0.4], [0.5, 0.6]) ([0.4, 0.5], [0.6, 0.7], [0.6, 0.7]) ([0.6, 0.7], [0.5, 0.6], [0.5, 0.6]) ([0.7, 0.8], [0.6, 0.7], [0.4, 0.5]) ([0.7, 0.8], [0.5, 0.6], [0.5, 0.6]) ([0.8, 0.9], [0.5, 0.6], [0.6, 0.7]) ([0.6, 0.7], [0.3, 0.4], [0.3, 0.4]) ϕ3 ϕ4 ([0.5, 0.6], [0.4, 0.5], [0.3, 0.4]) ([0.7, 0.8], [0.3, 0.4], [0.3, 0.4]) ([0.7, 0.8], [0.1, 0.2], [0.3, 0.4]) ([0.5, 0.6], [0.2, 0.3], [0.4, 0.5]) ([0.9, 1.0], [0.4, 0.5], [0.3, 0.4]) ([0.6, 0.7], [0.4, 0.5], [0.3, 0.4]) ([0.8, 0.9], [0.4, 0.5], [0.3, 0.4]) ([0.8, 0.9], [0.5, 0.6], [0.2, 0.3]) ([0.5, 0.6], [0.7, 0.9], [0.3, 0.4]) ([0.7, 0.8], [0.8, 0.9], [0.1, 0.2]) e3 . Table 3. The decision matrix R φ1 φ2 φ3 φ4 φ5 φ1 φ2 φ3 φ4 φ5 ϕ1 ϕ2 ([0.7, 0.8], [0.4, 0.5], [0.4, 0.5]) ([0.6, 0.7], [0.5, 0.6], [0.4, 0.5]) ([0.7, 0.8], [0.3, 0.4], [0.5, 0.6]) ([0.7, 0.8], [0.4, 0.5], [0.6, 0.7]) ([0.6, 0.7], [0.7, 0.8], [0.2, 0.3]) ([0.7, 0.8], [0.3, 0.4], [0.6, 0.7]) ([0.7, 0.8], [0.6, 0.7], [0.5, 0.6]) ([0.8, 0.9], [0.2, 0.4], [0.6, 0.7]) ([0.6, 0.9], [0.1, 0.2], [0.7, 0.8]) ([0.7, 0.8], [0.3, 0.5], [0.4, 0.5]) ϕ3 ϕ4 ([0.6, 0.7], [0.3, 0.4], [0.4, 0.5]) ([0.8, 0.9], [0.2, 0.3], [0.7, 0.8]) ([0.8, 0.9], [0.2, 0.4], [0.4, 0.5]) ([0.6, 0.7], [0.1, 0.2], [0.5, 0.6]) ([0.7, 0.9], [0.3, 0.4], [0.4 0.5]) ([0.5, 0.6], [0.4, 0.5], [0.4, 0.5]) ([0.6, 0.7], [0.3, 0.4], [0.4, 0.6]) ([0.9, 1.0], [0.1, 0.2], [0.5, 0.6]) ([0.6, 0.7], [0.3, 0.4], [0.4, 0.5]) ([0.8, 0.9], [0.5, 0.6], [0.5, 0.6]) Information 2017, 8, 144 6 of 10 Then, we use the proposed model to select the best ETE. e k (k = 1, 2, 3) and the INNWA operator, in order to obtain matrix R e= e Step 1. Utilize R rij Equation (6) which is listed in Table 4. ϕ1 e Table 4. The decision matrix R. 5×4 by ϕ2 φ1 φ2 φ3 φ4 φ5 ([0.4974, 0.6477], [0.4850, 0.6328], [0.3270, 0.4472]) ([0.5817, 0.8268], [0.4638, 0.5802], [0.3016, 0.4277]) ([0.7186, 0.8301], [0.5426, 0.6507], [0.3723, 0.4768]) ([0.6331, 0.7344], [0.3016, 0.4038], [0.3828, 0.5044]) ([0.4687, 0.5710], [0.5044, 0.6150], [0.3464, 0.4583]) ([0.6021, 0.7058], [0.3571, 0.4625], [0.3828, 0.5044]) ([0.6677, 0.7703], [0.5223, 0.6544], [0.3723, 0.4768]) ([0.6853, 0.7976], [0.3798, 0.5313], [0.3828, 0.5044]) ([0.6933, 0.8620], [0.2236, 0.3464], [0.5044, 0.6150]) ([0.5785, 0.6853], [0.3446, 0.4783], [0.3016, 0.4083]) ϕ3 ϕ4 φ1 φ2 φ3 φ4 φ5 ([0.4740, 0.5785], [0.3669, 0.4676], [0.2625, 0.3723]) ([0.7058, 0.8238], [0.2814, 0.3979], [0.3567, 0.4649]) ([0.6853, 0.7976], [0.1231, 0.2462], [0.3016, 0.4038]) ([0.5150, 0.6163], [0.1625, 0.2656], [0.3241, 0.4397]) ([0.8082, 1.0000], [0.3669, 0.4676], [0.3016, 0.4038]) ([0.5127, 0.6243], [0.4183, 0.5186], [0.3016, 0.4038]) ([0.7172, 0.8268], [0.3464, 0.4472], [0.3016, 0.4265]) ([0.7976, 1.0000], [0.2236, 0.3464], [0.2855, 0.3912]) ([0.4998, 0.6021], [0.4854, 0.6274], [0.3016, 0.4038]) ([0.6853, 0.7976], [0.6559, 0.7579], [0.2019, 0.3194]) e + and R e − by Equations (7) and (8). Step 2. Define the R   ([0.7186, 0.8301], [0.3016, 0.4038], [0.3016, 0.4277]),    0.6933, 0.8620 , 0.2236, 0.3464 , 0.3016, 0.4038 , ([ ] [ ] [ ]) e+ = R  0.8082, 1.0000 , 0.1231, 0.2462 , 0.2625, 0.3723 , ([ ] [ ] [ ])    ([0.7976, 1.1000], [0.2236, 0.3464], [0.2019, 0.3194])   ([0.4687, 0.5710], [0.5426, 0.6507], [0.3828, 0.5044]),    0.5785, 0.6853 , 0.5223, 0.6544 , 0.5044, 0.6150 , ([ ] [ ] [ ]) e− = R  ([0.4740, 0.5785], [0.3669, 0.4676], [0.3567, 0.4649]),    ([0.4998, 0.6021], [0.6559, 0.7579], [0.3016, 0.4265]) Step 3. Compute the Γi and Zi by Equation (14).                   Γ1 = 0.6507, Γ2 = 0.4182, Γ3 = 0.2416, Γ4 = 0.5261, Γ5 = 0.5195 Z1 = 0.2386, Z2 = 0.1515, Z3 = 0.0921, Z4 = 0.2765, Z5 = 0.2252 Step 4. Compute the Θi (let θ = 0.5) by Equation (15). Θ1 = 0.8974, Θ2 = 0.3772, Θ3 = 0.0000, Θ4 = 0.8477, Θ5 = 0.7006 Step 5. The order of ETEs is determined by Θi (i = 1, 2, 3, 4, 5): φ3 ≻ φ2 ≻ φ5 ≻ φ4 ≻ φ1 , and thus the most desirable ETE is φ3 . 4.2. Comparative Analysis In what follows, we compare with the interval neutrosophic number weighted averaging (INNWA) operator and interval neutrosophic number weighted geometric (INNWG) operator [28], INN similarity [33], and INN VIKOR [55]. The results are shown in Table 5. From the above analysis, it can be seen that the five methods have the same best emerging technology enterprise φ3 , and the ranking results of Method 1 and Method 2 are slightly different. The proposed INN VIKOR method can reasonably focus a MAGDM problem with INNs. At the same time, compared with Method 5 based on the INN VIKOR method in Reference [55], our proposed method avoids the interval numbers’ comparison. Information 2017, 8, 144 7 of 10 Table 5. The orders by utilizing five methods. Methods Method 1 with INNWA operator in [28] Method 2 with INNWG operator in [28] Method 3 based on similarity in [33] Method 4 based on similarity in [33] Method 5 based on INN VIKOR in [55] The proposed method Ranking Orders φ3 φ3 φ3 φ3 φ3 φ3 ≻ φ5 ≻ φ2 ≻ φ2 ≻ φ2 ≻ φ2 ≻ φ2 ≻ φ2 ≻ φ5 ≻ φ5 ≻ φ5 ≻ φ5 ≻ φ5 ≻ φ4 ≻ φ4 ≻ φ4 ≻ φ4 ≻ φ4 ≻ φ4 ≻ φ1 ≻ φ1 ≻ φ1 ≻ φ1 ≻ φ1 ≻ φ1 Best Alternatives φ3 φ3 φ3 φ3 φ3 φ3 5. Conclusions The VIKOR method for a MAGDM presents some conflicting attributes. We extended the VIKOR method to MAGDM with INNs. Firstly, the basic concepts of INNs were briefly presented. The method first aggregates all individual decision-makers’ assessment information based on an INNWA operator, and then employs the extended classical VIKOR method for MAGDM problems with INNs. The validity and stability of this method were verified by example analysis and comparative analysis, and its superiority was illustrated by a comparison with the existing methods. 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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). information Article Certain Competition Graphs Based on Intuitionistic Neutrosophic Environment Muhammad Akram * ID and Maryam Nasir Department of Mathematics, University of the Punjab, New Campus, Lahore 54590, Pakistan; maryamnasir912@gmail.com * Correspondence: m.akram@pucit.edu.pk; Tel.: +92-42-99231241 Received: 7 September 2017; Accepted: 19 October 2017; Published: 24 October 2017 Abstract: The concept of intuitionistic neutrosophic sets provides an additional possibility to represent imprecise, uncertain, inconsistent and incomplete information, which exists in real situations. This research article first presents the notion of intuitionistic neutrosophic competition graphs. Then, p-competition intuitionistic neutrosophic graphs and m-step intuitionistic neutrosophic competition graphs are discussed. Further, applications of intuitionistic neutrosophic competition graphs in ecosystem and career competition are described. Keywords: intuitionistic neutrosophic competition graphs; intuitionistic neutrosophic open-neighborhood graphs; p-competition intuitionistic neutrosophic graphs; m-step intuitionistic neutrosophic competition graphs MSC: 03E72; 68R10; 68R05 1. Introduction Euler [1] introduced the concept of graph theory in 1736, which has applications in various fields, including image capturing, data mining, clustering and computer science [2–5]. A graph is also used to develop an interconnection between objects in a known set of objects. Every object can be illustrated by a vertex, and interconnection between them can be illustrated by an edge. The notion of competition graphs was developed by Cohen [6] in 1968, depending on a problem in ecology. The competition graphs have many utilizations in solving daily life problems, including channel assignment, modeling of complex economic, phytogenetic tree reconstruction, coding and energy systems. Fuzzy set theory and intuitionistic fuzzy sets theory are useful models for dealing with uncertainty and incomplete information. However, they may not be sufficient in modeling of indeterminate and inconsistent information encountered in the real world. In order to cope with this issue, neutrosophic set theory was proposed by Smarandache [7] as a generalization of fuzzy sets and intuitionistic fuzzy sets. However, since neutrosophic sets are identified by three functions called truth-membership (t), indeterminacy-membership (i ) and falsity-membership ( f ), whose values are the real standard or non-standard subset of unit interval ]0− , 1+ [. There are some difficulties in modeling of some problems in engineering and sciences. To overcome these difficulties, Smarandache in 1998 [8] and Wang et al. [9] in 2010 defined the concept of single-valued neutrosophic sets and their operations as a generalization of intuitionistic fuzzy sets. Yang et al. [10] introduced the concept of the single-valued neutrosophic relation based on the single-valued neutrosophic set. They also developed kernels and closures of a single-valued neutrosophic set. The concept of the single-valued intuitionistic neutrosophic set was proposed by Bhowmik and Pal [11,12]. Information 2017, 8, 132; doi:10.3390/info8040132 www.mdpi.com/journal/information Information 2017, 8, 132 2 of 26 The valuable contribution of fuzzy graph and generalized structures has been studied by several researchers [13–22]. Smarandache [23] proposed the notion of the neutrosophic graph and separated them into four main categories. Wu [24] discussed fuzzy digraphs. Fuzzy m-competition and p-competition graphs were introduced by Samanta and Pal [25]. Samanta et al. [26] introduced m-step fuzzy competition graphs. Dhavaseelan et al. [27] defined strong neutrosophic graphs. Akram and Shahzadi [28] introduced the notion of a single-valued neutrosophic graph and studied some of its operations. They also discussed the properties of single-valued neutrosophic graphs by level graphs. Akram and Shahzadi [29] introduced the concept of neutrosophic soft graphs with applications. Broumi et al. [30] proposed single-valued neutrosophic graphs and discussed some properties. Ye [31–33] has presented several novel concepts of neutrosophic sets with applications. In this paper, we first introduce the concept of intuitionistic neutrosophic competition graphs. We then discuss m-step intuitionistic neutrosophic competition graphs. Further, we describe applications of intuitionistic neutrosophic competition graphs in ecosystem and career competition. Finally, we present our developed methods by algorithms. Our paper is divided into the following sections: In Section 2, we introduce certain competition graphs using the intuitionistic neutrosophic environment. In Section 3, we present applications of intuitionistic neutrosophic competition graphs in ecosystem and career competition. Finally, Section 4 provides conclusions and future research directions. 2. Intuitionistic Neutrosophic Competition Graphs We have used standard definitions and terminologies in this paper. For other notations, terminologies and applications not mentioned in the paper, the readers are referred to [34–44]. Definition 1. [38] Let X be a fixed set. A generalized intuitionistic fuzzy set I of X is an object having the form I={(u, µ I (u), νI (u))|u ∈ U }, where the functions µ I (u) :→ [0, 1] and νI (u) :→ [0, 1] define the degree of membership and degree of non-membership of an element u ∈ X, respectively, such that: min{µ I (u), νI (u)} ≤ 0.5, for all u ∈ X. This condition is called the generalized intuitionistic condition. Definition 2. [11] An intuitionistic neutrosophic set (IN-set) is defined as Ă = (w, t Ă (w), i Ă (w), f Ă (w)), where: t Ă (w) ∧ f Ă (w) ≤ 0.5, t Ă (w) ∧ i Ă (w) ≤ 0.5, i Ă (w) ∧ f Ă (w) ≤ 0.5, for all, w ∈ X, such that: 0 ≤ t Ă (w) + i Ă (w) + f Ă (w) ≤ 2. Definition 3. [12] An intuitionistic neutrosophic relation (IN-relation) is defined as an intuitionistic neutrosophic subset of X × Y, which has of the form: R = {((w, z), t R (w, z), i R (w, z), f R (w, z)) : w ∈ X, z ∈ Y }, where t R , i R and f R are intuitionistic neutrosophic subsets of X × Y satisfying the conditions: 1. 2. one of these t R (w, z), i R (w, z) and f R (w, z) is greater than or equal to 0.5, 0 ≤ t R (w, z) + i R (w, z) + f R (w, z) ≤ 2. Definition 4. An intuitionistic neutrosophic graph (IN-graph) G = ( X, h, k) (in short G) on X (vertex set) is a triplet such that: Information 2017, 8, 132 1. 2. 3. 3 of 26 tk (w, z) ≤ th (w) ∧ th (z), ik (w, z) ≤ ih (w) ∧ ih (z), f k (w, z) ≤ f h (w) ∨ f h (z), tk (w, z) ∧ ik (w, z) ≤ 0.5, tk (w, z) ∧ f k (w, z) ≤ 0.5, ik (w, z) ∧ f k (w, z) ≤ 0.5, 0 ≤ tk (w, z) + ik (w, z) + f k (w, z) ≤ 2, for all w, z ∈ X, where, th , ih and f h → [0, 1] denote the truth-membership, indeterminacy-membership and falsity-membership of an element w ∈ X and: tk , ik and f k → [0, 1] denote the truth-membership, indeterminacy-membership and falsity-membership of an element (w, z) ∈ E (edge set). We now illustrate this with an example. Example 1. Consider IN-graph G on non-empty set X, as shown in Figure 1. .1 (0 ,0 0 .2 , .4 ) (0.1, 0.2, 0.3) a(0.1, 0.4, 0.5) (0.5, 0.2, 0.2) b(0.6, 0.3, 0.2) (0. 5 ,0 .2 , 0 .1 ) (0.5, 0.2, 0.1) c(0.8, 0.3, 0.4) d(0.7, 0.4, 0.2) Figure 1. Intuitionistic neutrosophic graph (IN-graph). − → Definition 5. Let G be an intuitionistic neutrosophic digraph (IN-digraph), then intuitionistic neutrosophic out-neighborhoods (IN-out-neighborhoods) of a vertex w are an IN-set: + + N+ (w) = ( Xw+ , t+ w , i w , f w ), where, −−−→ −−−→ −−−→ + = { z | k ( w, z ) > 0, k ( w, z ) > 0, k ( w, z ) > 0}, Xw 3 2 1 −−−→ + −−−→ + + + + such that t+ w : Xw → [0, 1] defined by tw ( z ) = k 1 ( w, z ), iw : Xw → [0, 1] defined by iw ( z ) = k 2 ( w, z ) and −−−→ f z+ : Xz+ → [0, 1] defined by f w+ (z) = k3 (w, z). − → Definition 6. Let G be an IN-digraph, then the intuitionistic neutrosophic in-neighborhoods (IN-in-neighborhoods) of a vertex w are an IN-set: − − N− (w) = ( Xw− , t− w , i w , f w ), where, −−−→ −−−→ −−−→ − = { z | k ( z, w ) > 0, k ( z, w ) > 0, k ( z, w ) > 0}, Xw 2 3 1 −−−→ − −−−→ − − − − such that t− w : Xw → [0, 1] defined by tw ( z ) = k 1 ( z, w ), iw : Xw → [0, 1] defined by iw ( z ) = k 2 ( z, w ) and −−−→ − → [0, 1] defined by f − ( z ) = k ( z, w ). f w− : Xw 3 w Information 2017, 8, 132 4 of 26 − → Example 2. Consider G = ( X, h, k) to be an IN-digraph, such that, X = { a, b, c, d, e}, h = {(a, 0.5, 0.3, 0.1), − → → (b, 0.6, 0.4, 0.2), (c, 0.8, 0.3, 0.1), (d, 0.1, 0.9, 0.4), (e, 0.4, 0.3, 0.6)} and k = {( ab, 0.3, 0.3, 0.1), (− ae , 0.3, 0.2, − → − → − → − → 0.4), ( bc , 0.5, 0.2, 0.1), ( ed , 0.1, 0.2, 0.5), ( dc, 0.1, 0.2, 0.3), ( bd, 0.1, 0.3, 0.3)}, as shown in Figure 2. a(0.5, 0.3, 0.1) b(0.6, 0.4, 0.2) e(0.4, 0.3, 0.6) (0.1, 0.2, 0.5) (0.1, 0.3, 0.3) (0.3, 0.2, 0.4) (0.3, 0.3, 0.1) (0.5 , 0 .2 , 0 .1 ) c(0.8, 0.3, 0.1) ,0 (0.1 0 .2, .3) d(0.1, 0.9, 0.4) Figure 2. IN-digraph. Then, N+ ( a) = {(b, 0.3, 0.3, 0.1), (e, 0.3, 0.2, 0.4)}, N+ (c) = ∅, N+ (d) = {(c, 0.1, 0.2, 0.3)}, and N− (b) = {(a, 0.3, 0.3, 0.1)}, N− (c) = {(b, 0.5, 0.2, 0.1), (d, 0.1, 0.2, 0.3)}. Similarly, we can calculate IN-out and in-neighborhoods of the remaining vertices. Definition 7. The height of an IN-set Ă = (w, t Ă , i Ă , f Ă ) is defined as: H ( Ă) = ( sup t Ă (w), sup i Ă (w), inf f Ă (w)) = ( H1 ( Ă), H2 ( Ă), H3 ( Ă)). w∈ X w∈ X w∈ X For example, the height of an IN-set Ă = {( a, 0.5, 0.7, 0.2), (b, 0.1, 0.2, 1), (c, 0.3, 0.5, 0.3)} in X = { a, b, c} is H ( Ă) = (0.5, 0.7, 0.2). − → − → Definition 8. An intuitionistic neutrosophic competition graph (INC-graph) C( G ) of an IN-digraph G = (X, h, k) − → is an undirected IN-graph G = (X, h, k), which has the same intuitionistic neutrosophic set of vertices as in G and − → has an intuitionistic neutrosophic edge between two vertices w, z ∈ X in C( G ) if and only if N+ (w) ∩ N+ (z) is − → a non-empty IN-set in G . The truth-membership, indeterminacy-membership and falsity-membership values of edge − → (w, z) in C( G ) are: tk (w, z) = (th (w) ∧ th (z)) H (N+ (w) ∩ N+ (z)), ik (w, z) = (ih (w) ∧ ih (z)) H (N+ (w) ∩ N+ (z)), f k (w, z) = ( f h (w) ∨ f h (z)) H (N+ (w) ∩ N+ (z)), respectively. − → Example 3. Consider G = ( X, h, k) to be an IN-digraph, such that, X = { a, b, c, d}, h = {( a, 0.1, 0.4, 0.5), − → − → → (b, 0.6, 0.3, 0.2), (c, 0.8, 0.3, 0.4), (d, 0.7, 0.4, 0.2)} and k = {( ab, 0.1, 0.2, 0.4), (− ac , 0.1, 0.2, 0.3), ( bc , 0.5, − → − → 0.2, 0.2), ( bd, 0.5, 0.2, 0.1), ( cd, 0.5, 0.2, 0.1)}, as shown in Figure 3. Information 2017, 8, 132 5 of 26 b(0.6, 0.3, 0.2) 1 .4 ) (0.5, 0.2, 0.2) . (0 . ,0 0 2, (0.1, 0.2, 0.3) (0. 5 , 0. 2, 0 .1) (0.5, 0.2, 0.1) c(0.8, 0.3, 0.4) a(0.1, 0.4, 0.5) d(0.7, 0.4, 0.2) Figure 3. IN-digraph. By direct calculations, we have Tables 1 and 2 representing IN-out and in-neighborhoods, respectively. Table 1. IN-out-neighborhoods. w N+ ( w ) a b c d {(b, 0.1, 0.2, 0.4), (c, 0.1, 0.2, 0.3)} {(d, 0.5, 0.2, 0.1)} {(b, 0.5, 0.2, 0.2), (d, 0.5, 0.2, 0.1)} ∅ Table 2. IN-in-neighborhoods. w N− ( w ) a b c d ∅ {(a, 0.1, 0.2, 0.4), (c, 0.1, 0.2, 0.3)} {(a, 0.1, 0.2, 0.3)} {(b, 0.5, 0.2, 0.1), (c, 0.5, 0.2, 0.1)} The INC-graph of Figure 3 is shown in Figure 4. b(0.6, 0.3, 0.2) (0.3, 0.06, 0.16) (0.01, 0.06, 0.2) a(0.1, 0.4, 0.5) c(0.8, 0.3, 0.4) d(0.7, 0.4, 0.2) Figure 4. Intuitionistic neutrosophic competition graph (INC-graph). − → Therefore, there is an edge between two vertices in INC-graph C( G ), whose truth-membership, indeterminacy-membership and falsity-membership values are given by the above formula. Information 2017, 8, 132 6 of 26 Definition 9. For an IN-graph G = ( X, h, k), where h = ( h1 , h2 , h3 ) and k = (k1 , k2 , k3 ), then an edge (w, z), w, z ∈ X is called independent strong if: 1 [h (w) ∧ h1 (z)] < k1 (w, z), 2 1 1 [h2 (w) ∧ h2 (z)] > k2 (w, z), 2 1 [h3 (w) ∨ h3 (z)] > k3 (w, z). 2 Otherwise, it is called weak. − → − → Theorem 1. Suppose G is an IN-digraph. If N+ (w) ∩ N+ (z) contains only one element of G , then the edge − → (w, z) of C( G ) is independent strong if and only if: |[N+ (w) ∩ N+ (z)]|t > 0.5, |[N+ (w) ∩ N+ (z)]|i < 0.5, |[N+ (w) ∩ N+ (z)]| f < 0.5. − → Proof. Suppose, G is an IN-digraph. Suppose N+ (w) ∩ N+ (z) = ( a, p̆, q, r ), where p̆, q and r are the truth-membership, indeterminacy-membership and falsity-membership values of either the edge (w, a) or the edge (z, a), respectively. Here, |[N+ (w) ∩ N+ (z)]|t = p̆ = H1 (N+ (w) ∩ N+ (z)), |[N+ (w) ∩ N+ (z)]|i = q = H2 (N+ (w) ∩ N+ (z)), |[N+ (w) ∩ N+ (z)]| f = r = H3 (N+ (w) ∩ N+ (z)). Then, k1 (w, z) = p̆ × [ h1 (w) ∧ h1 (z)], k2 (w, z) = q × [ h2 (w) ∧ h2 (z)], k3 (w, z) = r × [ h3 (w) ∨ h3 (z)]. − → Therefore, the edge (w, z) in C( G ) is independent strong if and only if p̆ > 0.5, q < 0.5 and r < 0.5. − → Hence, the edge (w, z) of C( G ) is independent strong if and only if: |[N+ (w) ∩ N+ (z)]|t > 0.5, |[N+ (w) ∩ N+ (z)]|i < 0.5, |[N+ (w) ∩ N+ (z)]| f < 0.5. We illustrate the theorem with an example as shown in Figure 5. Information 2017, 8, 132 7 of 26 a(0.7, 0.5, 0.4) b(0.8, 0.4, 0.5) a(0.7, 0.5, 0.4) (0.6, 0.4, 0.4) b(0.8, 0.4, 0.5) (0 .4 (0.2, 0.3, 0.3) (0.7, 0.3, 0.3) 2, 0.1 5, 0.1 2) (0.2, 0.3, 0.4) c(0.8, 0.5, 0.4) d(0.3, 0.4, 0.5) c(0.8, 0.5, 0.4) d(0.3, 0.4, 0.5) (b) (a) Figure 5. INC-graph. (a) IN-digraph; (b) corresponding INC-graph. − → Theorem 2. If all the edges of an IN-digraph G are independent strong, then: k1 (w, z) > 0.5, (h1 (w) ∧ h1 (z))2 k2 (w, z) < 0.5, (h2 (w) ∧ h2 (z))2 k3 (w, z) < 0.5 (h3 (w) ∨ f 3 (z))2 − → for all edges (w, z) in C( G ). − → Proof. Suppose all the edges of IN-digraph G are independent strong. Then: −−−→ 1 [h1 (w) ∧ h1 (z)] < k1 (w, z), 2 −−−→ 1 [h2 (w) ∧ h2 (z)] > k2 (w, z), 2 −−−→ 1 [h3 (w) ∨ h3 (z)] > k3 (w, z), 2 − → − → for all the edges (w, z) in G . Let the corresponding INC-graph be C( G ). − → Case (1): When N+ (w) ∩ N+ (z) = ∅ for all w, z ∈ X, then there does not exist any edge in C( G ) between w and z. Thus, we have nothing to prove in this case. Case (2): When N+ (w) ∩ N+ (z) 6= ∅, let N+ (w) ∩ N+ (z) = {(a1 , m1 , n1 , p̆1 ), (a2 , m2 , n2 , p̆2 ), . . . , (al , ml , nl , p̆l )}, where mi , ni and p̆i are the truth-membership, indeterminacy-membership and −−−→ −−−→ falsity-membership values of either (w, ai ) or (z, ai ) for i = 1, 2, . . ., l, respectively. Therefore, −−−→ −−−→ mi = [k1 (w, ai ) ∧ k1 (z, ai )], −−−→ −−−→ ni = [k2 (w, ai ) ∧ k2 (z, ai )], −−−→ −−−→ p̆i = [k3 (w, ai ) ∨ k3 (z, ai )], f or i = 1, 2, . . . , l. Suppose, H1 (N+ (w) ∩ N+ (z)) = max{mi , + i = 1, 2, . . . , l } = mmax , + H2 (N (w) ∩ N (z)) = max{ni , i = 1, 2, . . . , l } = nmax , H3 (N+ (w) ∩ N+ (z)) = min{ p̆i , i = 1, 2, . . . , l } = p̆min . Information 2017, 8, 132 8 of 26 −−−→ −−−→ −−−→ −−−→ Obviously, mmax > k1 (w, z) and nmax < k2 (w, z) and p̆min < k3 (w, z) for all edges (w, z) show that: −−−→ k1 (w, z) mmax > > 0.5, h1 ( w ) ∧ h1 ( z ) h1 ( w ) ∧ h1 ( z ) −−−→ k2 (w, z) nmax < < 0.5, h2 ( w ) ∧ h2 ( z ) h2 ( w ) ∧ h2 ( z ) −−−→ k3 (w, z) p̆min < < 0.5, h3 ( w ) ∨ h3 ( z ) h3 ( w ) ∧ h3 ( z ) therefore, k1 (w, z) = (h1 (w) ∧ h1 (z)) H1 (N+ (w) ∩ N+ (z)), k1 (w, z) = [h1 (w) ∧ h1 (z)] × mmax , k1 (w, z) = mmax , (h1 (w) ∧ h1 (z)) k1 (w, z) mmax = > 0.5, 2 (h1 (w) ∧ h1 (z)) (h1 (w) ∧ h1 (z)) k2 (w, z) = (h2 (w) ∧ h2 (z)) H2 (N+ (w) ∩ N+ (z)), k2 (w, z) = [h2 (w) ∧ h2 (z)] × nmax , k2 (w, z) = nmax , (h2 (w) ∧ h2 (z)) nmax k2 (w, z) < 0.5, = 2 (h2 (w) ∧ h2 (z)) (h2 (w) ∧ h2 (z)) and: k3 (w, z) = (h3 (w) ∨ h3 (z)) H3 (N+ (w) ∩ N+ (z)), k3 (w, z) = [h3 (w) ∨ h3 (z)] × p̆min , k3 (w, z) = p̆min , (h3 (w) ∨ h3 (z)) k3 (w, z) p̆min < 0.5. = (h3 (w) ∨ h3 (z)) (h3 (w) ∨ h3 (z))2 Hence, − → C( G ). k1 (w,z) (h1 (w)∧h1 (z))2 > 0.5, k2 (w,z) (h2 (w)∧h2 (z))2 < 0.5, and k3 (w,z) (h3 (w)∨h3 (z))2 < 0.5 for all edges (w, z) in − → − → − → − → Theorem 3. Let C(G1 ) = ( h1 , k1 ) and C(G2 ) = ( h2 , k2 ) be two INC-graph of IN-digraphs G1 = (h1 , l1 ) and − → − → − → − → −→ ∗ −→ is an IN-graph −→ ∗ ∪ G where, G −→ ∗ G2 = (h2 , l2 ), respectively. Then, C(G1 G2 ) = GC(G C(G1 ) C(G2 )∗ 1 ) C(G2 ) − → ∗ − → ∗ − → −→ ∗ E −→ ∗ ), C(G1 ) and C(G2 ) are the crisp competition graphs of G1 and on the crisp graph ( X1 × X2 , EC(G ) C ( G ) 2 1 − → G2 , respectively. D is an IN-graph on ( X1 × X2 , E ) such that: 1. E = {(w1 , w2 )(z1 , z2 ) : z1 ∈ N− (w1 )∗ , z2 ∈ N+ (w2 )∗ } −→ ∗ E −→ ∗ = {( w1 , w2 )( w1 , z2 ) : w1 ∈ X1 , w2 z2 ∈ E −→ ∗ } ∪ {( w1 , w2 )( z1 , w2 ) : w2 ∈ EC(G ) C(G ) C(G ) 1 2 −→ ∗ }. X2 , w1 z1 ∈ EC(G ) 1 2 Information 2017, 8, 132 2. 3. 4. 5. 6. 7. 8. 9 of 26 t h1 h2 = t h1 ( w1 ) ∧ t h2 ( w2 ) , i h1 h2 = i h1 ( w1 ) ∧ i h2 ( w2 ) , f h1 h2 = f h1 ( w1 ) ∨ f h2 ( w2 ) . → ( z2 x2 )}, → ( w2 x 2 ) ∧ t − tk ((w1 , w2 )(w1 , z2 )) = [th1 (w1 ) ∧ th2 (w2 ) ∧ th2 (z2 )] × ∨ x2 {th1 (w1 ) ∧ t− l l 2 2 −→ ∗ E −→ ∗ , (w1 , w2 )(w1 , z2 ) ∈ EC(G x2 ∈ (N+ (w2 ) ∩ N+ (z2 ))∗ . C(G2 ) 1) → ( w2 x 2 ) ∧ i − → ( z2 x2 )}, ik ((w1 , w2 )(w1 , z2 )) = [ih1 (w1 ) ∧ ih2 (w2 ) ∧ ih2 (z2 )] × ∨ x2 {ih1 (w1 ) ∧ i− l l 2 2 −→ ∗ E −→ ∗ , (w1 , w2 )(w1 , z2 ) ∈ EC(G x2 ∈ (N+ (w2 ) ∩ N+ (z2 ))∗ . C(G2 ) 1) → ( z2 x2 )}, → ( w2 x 2 ) ∨ f − f k ((w1 , w2 )(w1 , z2 )) = [ f h1 (w1 ) ∨ f h2 (w2 ) ∨ f h2 (z2 )] × ∨ x2 { f h1 (w1 ) ∨ f − l l 2 2 −→ ∗ E −→ ∗ , (w1 , w2 )(w1 , z2 ) ∈ EC(G x2 ∈ (N+ (w2 ) ∩ N+ (z2 ))∗ . C(G2 ) 1) → ( w1 x 1 ) ∧ t − → ( z1 x1 )}, tk ((w1 , w2 )(z1 , w2 )) = [th1 (w1 ) ∧ th1 (z1 ) ∧ th2 (w2 )] × ∨ x1 {th2 (w2 ) ∧ t− l l 1 1 −→ ∗ E −→ ∗ , (w1 , w2 )(z1 , w2 ) ∈ EC(G x1 ∈ (N+ (w1 ) ∩ N+ (z1 ))∗ . C(G2 ) 1) → ( w1 x 1 ) ∧ i − → ( z1 x1 )}, ik ((w1 , w2 )(z1 , w2 )) = [ih1 (w1 ) ∧ ih1 (z1 ) ∧ ih2 (w2 )] × ∨ x1 {ih2 (w2 ) ∧ i− l l 1 1 −→ ∗ E −→ ∗ , (w1 , w2 )(z1 , w2 ) ∈ EC(G x1 ∈ (N+ (w1 ) ∩ N+ (z1 ))∗ . C(G2 ) 1) → ( w1 x 1 ) ∨ t − → ( z1 x1 )}, f k ((w1 , w2 )(z1 , w2 )) = [ f h1 (w1 ) ∨ f h1 (z1 ) ∨ f h2 (w2 )] × ∨ x1 { f h2 (w2 ) ∨ f − l l 1 1 9. −→ ∗ E −→ ∗ , (w1 , w2 )(z1 , w2 ) ∈ EC(G x1 ∈ (N+ (w1 ) ∩ N+ (z1 ))∗ . C(G2 ) 1) → ( z 1 w1 ) ∧ t h ( z 2 ) ∧ tk ((w1 , w2 )(z1 , z2 )) = [th1 (w1 ) ∧ th1 (z1 ) ∧ th2 (w2 ) ∧ th2 (z2 )] × [th1 (w1 ) ∧ t− 2 l1 → ( w2 z2 )], t− l 10. (w1 , z1 )(w2 , z2 ) ∈ E . → ( z 1 w1 ) ∧ i h ( z 2 ) ∧ ik ((w1 , w2 )(z1 , z2 )) = [ih1 (w1 ) ∧ ih1 (z1 ) ∧ ih2 (w2 ) ∧ ih2 (z2 )] × [ih1 (w1 ) ∧ i− 2 l1 − → i l (w2 z2 )], 11. (w1 , z1 )(w2 , z2 ) ∈ E . → ( z 1 w1 ) ∨ f h ( z 2 ) ∨ f k ((w1 , w2 )(z1 , z2 )) = [ f h1 (w1 ) ∨ f h1 (z1 ) ∨ f h2 (w2 ) ∨ f h2 (z2 )] × [ f h1 (w1 ) ∨ f − 2 l1 − → f l (w2 z2 )], 2 2 2 (w1 , z1 )(w2 , z2 ) ∈ E . Proof. Using similar arguments as in Theorem 2.1. [39], it can be proven. − → − → Example 4. Consider G1 = ( X1 , h1 , l1 ) and G2 = ( X2 , h2 , l2 ) to be two IN-digraphs, respectively, as shown − → − → in Figure 6. The intuitionistic neutrosophic out and in-neighborhoods of G1 and G2 are given in Tables 3 and 4. − → − → The INC-graphs C(G1 ) and C(G2 ) are given in Figure 7. − → Table 3. IN-out and in-neighborhoods of G1 . w ∈ X1 N+ ( w ) N− ( w ) w1 w2 w3 w4 {w2 (0.2, 0.2, 0.3)} ∅ {w2 (0.3, 0.2, 0.1)} {w3 (0.3, 0.1, 0.1)} ∅ {w1 (0.2, 0.2, 0.3), w3 (0.3, 0.1, 0.1)} {w4 (0.3, 0.1, 0.1)} ∅ − → Table 4. IN-out and in-neighborhoods of G2 . w ∈ X2 N+ ( w ) N− ( w ) z1 z2 z3 {z3 (0.3, 0.2, 0.2)} {z3 (0.3, 0.1, 0.1)} ∅ ∅ ∅ {z1 (0.3, 0.2, 0.2), z2 (0.3, 0.1, 0.1)} Information 2017, 8, 132 10 of 26 − → G2 − → G1 z1 (0.4, 0.3, 0.2) (0 .2 3, 0 (0. ,0 0.2 .2 , .3 ) 2) , 0. w1 (0.3, 0.4, 0.5) w2 (0.4, 0.3, 0.1) (0. 3 (0.3, 0.1, 0.1) , 0. 2, 0 .1 ) z2 (0.4, 0.3, 0.5) .3) 2, 0 . 0 , w3 (0.5, 0.2, 0.1) 2 (0. z3 (0.7, 0.2, 0.3) w4 (0.4, 0.3, 0.1) Figure 6. IN-digraphs. → C(G2 ) → C(G1 ) z1 (0.4, 0.3, 0.2) (0.0 6 , 0 .0 4, 0 .1 5 ) (0.12, 0.03, 0.1) w1 (0.3, 0.4, 0.5) w2 (0.4, 0.3, 0.1) z2 (0.4, 0.3, 0.5) w3 (0.5, 0.2, 0.1) z3 (0.7, 0.2, 0.3) w4 (0.4, 0.3, 0.1) − → − → Figure 7. INC-graphs of G1 and G2 . − →∗ − → ∗ ∪ G = (w, k), where w = (tw , iw , f w ) and k = (tk , ik , f k ), We now construct the INC-graph GC(G 1 ) C(G2 ) − →∗ − →∗ from C(G1 ) and C(G2 ) using Theorem 2.14. We obtained two sets of edges by using Condition (1). Information 2017, 8, 132 11 of 26 −→ ∗ E −→ ∗ ={( w1 , z1 )( w1 , z2 ), ( w2 , z1 )( w2 , z2 ), ( w3 , z1 )( w3 , z2 ), EC(G ) C(G ) 1 2 (w4 , z1 )(w4 , z2 ), (w1 , z1 )(w3 , z1 ), (w1 , z2 )(w3 , z2 ), (w1 , z3 )(w3 , z3 )}, E ={(w2 , z1 )(w1 , z3 ), (w2 , z1 )(w3 , z3 ), (w2 , z2 )(w1 , z3 ) (w2 , z2 )(w3 , z3 ), (w3 , z1 )(w4 , z3 ), (w3 , z2 )(w4 , z3 )}. The truth-membership, indeterminacy-membership and falsity-membership of edges can be calculated by using Conditions (3) to (11) as, k((w1 , z1 )(w1 , z2 )) = (th1 (w1 ) ∧ th2 (z1 ) ∧ th2 (z2 ), i h1 ( w1 ) ∧ i h2 ( z 1 ) ∧ i h2 ( z 2 ) , ×(th1 (w1 ) ∧ tl2 (z1 z3 ) ∧ tl2 (z2 z3 ), f h1 (w1 ) ∨ f h2 (z1 ) ∨ f h2 (z2 )) i h 1 ( w 1 ) ∧ i l2 ( z 1 z 3 ) ∧ i l2 ( z 2 z 3 ) , f h 1 ( w1 ) ∨ f l2 ( z 1 z 3 ) ∨ f l2 ( z 2 z 3 ) = (0.3, 0.3, 0.5) × (0.3, 0.1, 0.5) = (0.09, 0.03, 0.25), k((w2 , z1 )(w1 , z3 )) = (th1 (w2 ) ∧ th2 (z1 ) ∧ th1 (w1 ) ∧ th2 (z3 ), i h1 ( w2 ) ∧ i h2 ( z 1 ) ∧ i h1 ( w1 ) ∧ i h2 ( z 3 ) , f h1 (w2 ) ∨ f h2 (z1 ) ∨ f h1 (w1 ) ∨ f h2 (z3 )) ×(th1 (w2 ) ∧ tl1 (w1 w2 ) ∧ tl2 (z3 ) ∧ tl2 (z1 z3 ), i h 1 ( w 2 ) ∧ i l1 ( w 1 w 2 ) ∧ i l2 ( z 3 ) ∧ i l2 ( z 1 z 3 ) , f h1 (w2 ) ∨ f l1 (w1 w2 ) ∨ f l2 (z3 ) ∨ f l2 (z1 z3 )) = (0.3, 0.2, 0.5) × (0.2, 0.2, 0.3) = (0.06, 0.04, 0.15). All the truth-membership, indeterminacy-membership and falsity-membership degrees of adjacent edges of −→ ∗ −→ and G are given in Table 5. GC ( G ) C(G )∗ 1 2 − → ∗ − → ∪ G . Table 5. Adjacent edges of GC(G ) C(G )∗ 1 2 (w, ẃ) (z, ź) k (w, ẃ) (z, ź) (w1 , z1 )(w1 , z2 ) (w2 , z1 )(w2 , z2 ) (w3 , z1 )(w3 , z2 ) (w4 , z1 )(w4 , z2 ) (w1 , z1 )(w3 , z1 ) (w1 , z3 )(w3 , z3 ) (w2 , z1 )(w1 , z3 ) (w2 , z1 )(w3 , z3 ) (w2 , z2 )(w1 , z3 ) (w2 , z2 )(w3 , z3 ) (w3 , z1 )(w4 , z3 ) (w3 , z2 )(w4 , z3 ) (w1 , z2 )(w3 , z2 ) (0.09, 0.03, 0.25) (0.12, 0.03, 0.1) (0.12, 0.02, 0.1) (0.12, 0.03, 0.1) (0.06, 0.04, 0.15) (0.06, 0.04, 0.15) (0.06, 0.04, 0.15) (0.12, 0.04, 0.09) (0.06, 0.02, 0.15) (0.12, 0.02, 0.15) (0.12, 0.02, 0.09) (0.12, 0.02, 0.15) (0.06, 0.04, 0.25) The INC-graph obtained by using this method is given in Figure 8 where solid lines indicate part of INC-graph −→ ∗ −→ ∗ , and the dotted lines indicate the part of G . obtained from GC(G 1 ) C(G2 ) − → − → − → − → The Cartesian product G1 G2 of IN-digraphs G1 and G2 is shown in Figure 9. The IN-out-neighborhoods − → − → − → − → of G1 G2 are calculated in Table 6. The INC-graphs of G1 G2 are shown in Figure 10. Information 2017, 8, 132 12 of 26 w1 z2 (0.09, 0.03, 0.25) (0.3, 0.3, 0.5) (0.06, 0.04, 0.15) (0.0 w2 (0.4, 0.2, 0.2) (0.4, 0.3, 0.2) .0 2 15 , 0. ) .0 (0 6, (0.3, 0.2, 0.5) 2, 0.0 2, 0 .0 4 , 0 .0 , 0 .0 2, 0 .09) 5) (0.06, 0.04, 0.25) (0 .12 ,0 .02 ,0 .15 ) 9) (0.12, 0.02, 0.1) 0.1 (0.4, 0.2, 0.3) (0.12, 0.03, 0.1) (0.1 2 w4 6, 0 (0.3, 0.3, 0.5) (0.4, 0.3, 0.5) (0.4, 0.3, 0.2) (0.1 w3 z3 (0.4, 0.2, 0.5) (0 .12 ,0 .0 2 (0.5, 0.2, 0.3) ,0 .15 ) (0.4, 0.3, 0.5) (0.12, 0.03, 0.1) (0.06, 0.04, 0.15) z1 (0.4, 0.2, 0.3) − → ∗ − → ∗ ∪ G . Figure 8. GC(G 1 ) C(G2 ) z2 z1 0 (0.3, w1 (0.3, 0.3, 0.5) z3 ) .2, 0.5 (0.3, 0.3, 0.5) (0.3, 0.1, 0.5) (0.3, 0.2, 0.5) (0.2, 0.2, 0.5) (0.2, 0.2, 0.3) w2 (0.4, 0.3, 0.2) (0.3, 0.2, 0.2) w3 (0.4, 0.2, 0.2) 0 (0.3, (0.2, 0.2, 0.3) ) .2 , 0 .2 (0.4, 0.3, 0.5) 0.2 (0.3, , 0.2) (0.4, 0.2, 0.5) (0.3, 0.1, 0.1) (0.4, 0.2, 0.3) (0.3, 0.2, 0.5) (0.3, 0.2, 0.3) (0.3, 0.1, 0.1) (0.5, 0.2, 0.3) (0.3, 0.1, 0.5) (0.3, 0.1, 0.2) w4 (0.4, 0.3, 0.2) 0 (0.3, .2 , 0 .2 ) (0.4, 0.3, 0.5) (0.3, 0.1, 0.3) (0.3, 0.1, 0.1) − → − → Figure 9. G1 G2 . (0.4, 0.2, 0.3) Information 2017, 8, 132 13 of 26 − → − → Table 6. IN-out-neighborhoods of G1 G2 . (w, z) N+ (w, z) ( w1 , z 1 ) ( w1 , z 2 ) ( w1 , z 3 ) ( w2 , z 1 ) ( w2 , z 2 ) ( w2 , z 3 ) ( w3 , z 1 ) ( w3 , z 2 ) ( w3 , z 3 ) ( w4 , z 1 ) ( w4 , z 2 ) ( w4 , z 2 ) {((w2 , z1 ), 0.2, 0.2, 0.3), ((w1 , z3 ), 0.3, 0.2, 0.5)} {((w1 , z3 ), 0.3, 0.1, 0.5), ((w2 , z2 ), 0.2, 0.2, 0.5)} {((w2 , z3 ),0.2, 0.2, 0.3)} {((w2 , z3 ),0.3, 0.2, 0.2)} {((w2 , z3 ),0.3, 0.1, 0.1)} ∅ {((w3 , z3 ), 0.3, 0.2, 0.2), ((w2 , z1 ), 0.3, 0.2, 0.2)} {((w2 , z2 ), 0.3, 0.2, 0.5), ((w3 , z3 ), 0.3, 0.1, 0.1)} {((w2 , z3 ),0.3, 0.2, 0.3)} {((w4 , z3 ), 0.3, 0.2, 0.2), ((w3 , z1 ), 0.3, 0.1, 0.2)} {((w4 , z3 ), 0.3, 0.1, 0.1), ((w3 , z2 ), 0.3, 0.1, 0.5)} {((w3 , z3 ),0.3, 0.1, 0.3)} w1 (0.3, 0.3, 0.5) (0.06, 0.04, 0.15) w2 (0.4, 0.3, 0.2) z2 (0.09, 0.03, 0.25) 6 (0.0 (0.4, 0.2, 0.2) (0.4, 0.3, 0.2) 5) 0 .1 02, .0 (0 6, (0.3, 0.2, 0.5) 0 , .0 2 2, 0 .0 4 , 0 .0 9) (0.12, 0.02, 0.1) , 0 .0 2, 0 .0 9 ) (0.12, 0.03, 0.1) 0 .1 5) (0.4, 0.2, 0.3) (0.12, 0.03, 0.1) (0.1 2 w4 , 0. (0.3, 0.3, 0.5) (0.4, 0.3, 0.5) (0.1 w3 z3 (0.06, 0.04, 0.25) (0 .1 2 ,0 .02 ,0 .15 ) (0.4, 0.2, 0.5) (0 .12 ,0 .02 (0.4, 0.3, 0.5) (0.06, 0.04, 0.15) z1 (0.5, 0.2, 0.3) ,0 .1 5 ) (0.4, 0.2, 0.3) − → − → Figure 10. C(G1 G2 ). It can be seen that − → − → −→ ∗ −→ ∪ G from Figures 8 and 10. C(G1 G2 ) ∼ = GC ( G ) C(G )∗ 1 2 Definition 10. The intuitionistic neutrosophic open-neighborhood of a vertex w of an IN-graph G = ( X, h, k) is IN-set N(w) = ( Xw , tw , iw , f w ), where, Xw = {z|k1 (w, z) > 0, k2 (w, z) > 0, k3 (w, z) > 0}, and tw : Xw → [0, 1] defined by tw (z) = k1 (w, z), iw : Xw → [0, 1] defined by iw (z) = k2 (w, z) and f z : Xw → [0, 1] defined by f w (z) = k3 (w, z). For every vertex w ∈ X, the intuitionistic neutrosophic singleton set, Aw = (w, h1′ , h2′ , h3′ ), such that: h1′ : {w} → [0, 1], h2′ : {w} → [0, 1], h3′ : {w} → [0, 1] defined by h1′ (w) = h1 (w), h2′ (w) = h2 (w) and h3′ (w) = h3 (w), respectively. The intuitionistic neutrosophic closed-neighborhood of a vertex w is N[w] = N(w) ∪ Aw . Information 2017, 8, 132 14 of 26 Definition 11. Suppose G = ( X, h, k) is an IN-graph. The single-valued intuitionistic neutrosophic open-neighborhood graph of G is an IN-graph N(G) = ( X, h, k′ ), which has the same intuitionistic neutrosophic set of vertices in G and has an intuitionistic neutrosophic edge between two vertices w, z ∈ X in N(G) if and only if N(w) ∩ N(z) is a non-empty IN-set in G. The truth-membership, indeterminacy-membership and falsity-membership values of the edge (w, z) are given by: k′1 (w, z) = [ h1 (w) ∧ h1 (z)] H1 (N(w) ∩ N(z)), k′2 (w, z) = [h2 (w) ∧ h2 (z)] H2 (N(w) ∩ N(z)), k′3 (w, z) = [h3 (w) ∨ h3 (z)] H3 (N(w) ∩ N(z)), respectively. Definition 12. Suppose G = ( X, h, k) is an IN-graph. The single-valued intuitionistic neutrosophic closed-neighborhood graph of G is an IN-graph N(G) = ( X, h, k′ ), which has the same intuitionistic neutrosophic set of vertices in G and has an intuitionistic neutrosophic edge between two vertices w, z ∈ X in N[G] if and only if N[w] ∩ N[z] is a non-empty IN-set in G. The truth-membership, indeterminacy-membership and falsity-membership values of the edge (w, z) are given by: k′1 (w, z) = [ h1 (w) ∧ h1 (z)] H1 (N[w] ∩ N[z]), k′2 (w, z) = [ h2 (w) ∧ h2 (z)] H2 (N[w] ∩ N[z]), k′3 (w, z) = [ h3 (w) ∨ h3 (z)] H3 (N[w] ∩ N[z]), respectively. Example 5. Consider G = ( X, h, k) to be an IN-graph, such that X = { a, b, c, d}, h = {( a, 0.5, 0.4, 0.3), (b, 0.6, 0.3, 0.1), (c, 0.7, 0.3, 0.1), (d, 0.5, 0.6, 0.3)}, and k = {( ab, 0.3, 0.2, 0.2), ( ad, 0.4, 0.3, 0.2), (bc, 0.5, 0.2, 0.1), (cd, 0.4, 0.2, 0.2)}, as shown in Figure 11. Then, corresponding intuitionistic neutrosophic open and closed-neighborhood graphs are shown in Figure 12. (0.3, 0.2, 0.2) b(0.6, 0.3, 0.1) a(0.5, 0.4, 0.3) (0.4, 0.3, 0.2) (0.5, 0.2, 0.1) c(0.7, 0.3, 0.1) d(0.5, 0.6, 0.3) (0.4, 0.2, 0.2) Figure 11. IN-digraph. Information 2017, 8, 132 a(0.5, 0.4, 0.3) d(0.5, 0.6, 0.3) 15 of 26 (0.2, 0.06, 0.06) (0.2, 0.06, 0.06) b(0.6, 0.3, 0.1) c(0.7, 0.3, 0.1) (0.15, 0.06, 0.06) b(0.6, 0.3, 0.1) a(0.5, 0.4, 0.3) d(0.5, 0.6, 0.3) (0. 2 ,0 .06 ,0 .06 ) (0.3, 0.06, 0.01) (0.2, 0.12, 0.09) (a) c(0.7, 0.3, 0.1) (0.2, 0.06, 0.06) (0.2, 0.06, 0.06) (b) Figure 12. (a) N(G); (b) N[G]. Theorem 4. For each edge of an IN-graph G, there exists an edge in N[G]. Proof. Suppose (w, z) is an edge of an IN-graph G = (V, h, k ). Suppose N[G] = (V, h, k′ ) is the corresponding closed neighborhood of an IN-graph. Suppose w, z ∈ N[w] and w, z ∈ N[z]. Then, w, z ∈ N[w] ∩ N[z]. Hence, H1 (N[w] ∩ N[z]) 6= 0, H2 (N[w] ∩ N[z]) 6= 0, H3 (N[w] ∩ N[z]) 6= 0. Then, k′1 (w, z) = [h1 (w) ∧ h1 (z)] H1 (N[w] ∩ N[z]) 6= 0, k′2 (w, z) = [h2 (w) ∧ h2 (z)] H2 (N[w] ∩ N[z]) 6= 0, k′3 (w, z) = [h3 (w) ∨ h3 (z)] H3 (N[w] ∩ N[z]) 6= 0. Thus, for each edge (w, z) in IN-graph G, there exists an edge (w, z) in N[G]. Definition 13. The support of an IN-set Ă = (w, t Ă , i Ă , f Ă ) in X is the subset Â of X defined by: Â = {w ∈ X : t Ă (w) 6= 0, i Ă (w) 6= 0, f Ă (w) 6= 1} and |supp( Â)| is the number of elements in the set. We now discuss p-competition intuitionistic neutrosophic graphs. − → Suppose p is a positive integer. Then, p-competition IN-graph C p ( G ) of the IN-digraph − → G = ( X, h, k) is an undirected IN-graph G = ( X, h, k), which has the same intuitionistic neutrosophic − → set of vertices as in G and has an intuitionistic neutrosophic edge between two vertices w, z ∈ X in Information 2017, 8, 132 16 of 26 − → − → C p ( G ) if and only if |supp(N+ (w) ∩ N+ (z))| ≥ p. The truth-membership value of edge (w, z) in C p ( G ) (i − p)+1 [h1 (w) ∧ h1 (z)] H1 (N+ (w) ∩ N+ (z)); the indeterminacy-membership value of edge is t(w, z) = i − → (w, z) in C p ( G ) is i (w, z) = (i− pi )+1 [h2 (w) ∧ h2 (z)] H2 (N+ (w) ∩ N+ (z)); and the falsity-membership − → (i − p)+1 value of edge (w, z) in C p ( G ) is f (w, z) = [h3 (w) ∨ h3 (z)] H3 (N+ (w) ∩ N+ (z)) where i + + i = |supp(N (w) ∩ N (z))|. The three-competition IN-graph is illustrated by the following example. − → Example 6. Consider G = ( X, h, k) to be an IN-digraph, such that X = {w1 , w2 , w3 , z1 , z2 , z3 }, h = {(w1 , 0.5, 0.1, 0.2), (w2 , 0.1, 0.6, 0.3), (w3 , 0.1, 0.2, 0.5), (z1 , 0.7, 0.2, 0.1), (z2 , 0.5, 0.2, 0.3), (z3 , 0.3, 0.7, 0.2)} −−−−→ −−−−→ −−−−→ −−−−→ and k = {((w1 , z1 ), 0.4, 0.1, 0.1), ((w1 , z2 ), 0.5, 0.1, 0.3), ((w1 , z3 ), 0.2, 0.1, 0.1), ((w2 , z1 ), 0.1, 0.1, 0.2), −−−−→ −−−−→ −−−−→ −−−−→ ((w2 , z2 ), 0.1, 0.1, 0.2), ((w2 , z3 ), 0.1, 0.5, 0.2), ((w3 , z1 ), 0.1, 0.1, 0.1) ((w3 , z2 ), 0.1, 0.1, 0.2)}, as shown in Figure 13. Then, N+ (w1 ) = {(z1 , 0.4, 0.1, 0.1), (z2 , 0.5, 0.1, 0.3), (z3 , 0.2, 0.1, 0.1)}, N+ (w2 ) = {(z1 , 0.1, 0.1, 0.2), (z2 , 0.1, 0.1, 0.2), (z3 , 0.1, 0.5, 0.2)} and N+ (w3 ) = {(z1 , 0.1, 0.1, 0.1), (z2 , 0.1, 0.1, 0.2)}. Therefore, N+ (w1 ) ∩ N+ (w2 ) = {(z1 , 0.1, 0.1, 0.2), (z2 , 0.1, 0.1, 0.3), (z3 , 0.1, 0.1, 0.2)}, N+ (w1 ) ∩ N+ (w3 ) = {(z1 , 0.1, 0.1, 0.1), (z2 , 0.1, 0.1, 0.3)} and N+ (w2 ) ∩ N+ (w3 ) = {(z1 , 0.1, 0.1, 0.2), (z2 , 0.1, 0.1, 0.2)}. Now, i = |supp(N+ (w1 ) ∩ N+ (w2 ))| = 3. For p = 3, t(w1 , w2 ) = 0.003, i (w1 , w2 ) = 0.003 and f (w1 , w2 ) = 0.02. As shown in Figure 14. w1 (0.5, 0.1, 0.2) z1 (0.7, 0.2, 0.1) w2 (0.1, 0.6, 0.3) z2 (0.5, 0.2, 0.3) z3 (0.3, 0.7, 0.2) w3 (0.1, 0.2, 0.5) Figure 13. IN-digraph. z1 (0.7, 0.2, 0.1) (0.003, 0.003, 0.02) w1 (0.5, 0.1, 0.2) z2 (0.5, 0.2, 0.3) w2 (0.1, 0.6, 0.3) z3 (0.3, 0.7, 0.2) w3 (0.1, 0.2, 0.5) Figure 14. Three-competition IN-graph. Information 2017, 8, 132 17 of 26 We now define another extension of INC-graph known as the m-step INC-graph. − →m P z,w : a directed intuitionistic neutrosophic path of length m from z to w. N+ m ( z ): single-valued intuitionistic neutrosophic m-step out-neighborhood of vertex z. N− m ( z ): single-valued intuitionistic neutrosophic m-step in-neighborhood of vertex z. −→ − → Cm (G): m-step INC-graph of the IN-digraph G . − → − → − → Definition 14. Suppose G = (X, h, k) is an IN-digraph. The m-step IN-digraph of G is denoted by G m = (X, h, k) − → where the intuitionistic neutrosophic set of vertices of G is the same as the intuitionistic neutrosophic set of vertices of − → − → G m and has an edge between z and w in G m if and only if there exists an intuitionistic neutrosophic directed path − →m − → P z,w in G . − → Definition 15. The intuitionistic neutrosophic m-step out-neighborhood of vertex z of an IN-digraph G = (X, h, k) is IN-set: + + + + N+ m ( z ) = ( Xz , t z , i z , f z ), where − → + + Xz+ = {w| there exists a directed intuitionistic neutrosophic path of length m from z to w, P m z,w }, tz : Xz → [0, 1], − −−−− → iz+ : Xz+ → [0, 1] and f z+ : Xz+ → [0, 1] are defined by t+ z = min { t ( w1 , w2 ), ( w1 , w2 ) is an edge of −−−−−→ −−−−−→ − →m − →m + P z,w }, iz = min{i (w1 , w2 ), (w1 , w2 ) is an edge of P z,w } and f z+ = max { f (w1 , w2 ), (w1 , w2 ) is an edge of − →m P z,w }, respectively. − → Definition 16. The intuitionistic neutrosophic m-step in-neighborhood of vertex z of an IN-digraph G = ( X, h, k ) is IN-set: − − − − N− m ( z ) = ( Xz , t z , i z , f z ), where − → − − Xz− = {w| there exists a directed intuitionistic neutrosophic path of length m from w to z, P m w,z }, tz : Xz → [0, 1], − −−−− → iz− : Xz− → [0, 1] and f z− : Xz− → [0, 1] are defined by t− z = min { t ( w1 , w2 ), ( w1 , w2 ) is an edge of −−−−−→ −−−−−→ − →m − →m − P w,z }, iz = min{i (w1 , w2 ), (w1 , w2 ) is an edge of P w,z } and f z− = max { f (w1 , w2 ), (w1 , w2 ) is an edge of − →m P w,z }, respectively. − → − → Definition 17. Suppose G = ( X, h, k ) is an IN-digraph. The m-step INC-graph of IN-digraph G is denoted − → − → by Cm ( G ) = ( X, h, k), which has the same intuitionistic neutrosophic set of vertices as in G and has an − → + edge between two vertices w, z ∈ X in Cm ( G ) if and only if (N+ m ( w ) ∩ Nm ( z )) is a non-empty IN-set in − → − → + G . The truth-membership value of edge (w, z) in Cm ( G ) is t(w, z) = [h1 (w) ∧ h1 (z)] H1 (N+ m ( w ) ∩ Nm ( z )); − → + the indeterminacy-membership value of edge (w, z) in Cm ( G ) is i (w, z) = [ h2 (w) ∧ h2 (z)] H2 (Nm (w) ∩ N+ m ( z )); − → + ( z )). ( w ) ∩ N and the falsity-membership value of edge (w, z) in Cm ( G ) is f (w, z) = [ h3 (w) ∨ h3 (z)] H3 (N+ m m The two-step INC-graph is illustrated by the following example. − → Example 7. Consider G = ( X, h, k) is an IN-digraph, such that, X = {w1 , w2 , z1 , z2 , z3 }, h = {(w1 , 0.3, 0.4, 0.6), (w2 , 0.2, 0.5, 0.3), (z1 , 0.4, 0.2, 0.3), (z2 , 0.7, 0.2, 0.1), (z3 , 0.5, 0.1, 0.2), (z4 , 0.6, 0.3, 0.2)}, −−−−→ −−−−→ −−−−→ −−−−→ and k = {((w1 , z1 ), 0.2, 0.1, 0.2), ((w2 , z4 ), 0.1, 0.2, 0.3), ((z1 , z3 ), 0.3, 0.1, 0.2), ((z1 , z2 ), 0.3, 0.1, 0.2), −−−−→ −−−−→ ((z4 , z2 ), 0.2, 0.1, 0.1), and ((z4 , z3 ), 0.4, 0.1, 0.4)}, as shown in Figure 15. Then, N2+ (w1 ) = {(z2 , 0.2, 0.1, 0.2), (z3 , 0.2, 0.1, 0.2)} and N2+ (w2 ) = {(z2 , 0.1, 0.1, 0.3), (z3 , 0.1, 0.1, 0.4)}. Therefore, N2+ (w1 ) ∩ N2+ (w2 ) = {(z2 , 0.1, 0.1, 0.3), (z3 , 0.1, 0.1, 0.4)}. Thus, t(w1 , w2 ) = 0.02, i (w1 , w2 ) = 0.04 and f (w1 , w2 ) = 0.18. This is shown in Figure 16. Information 2017, 8, 132 18 of 26 w2 (0.2, 0.5, 0.3) w1 (0.3, 0.4, 0.6) (0.2, 0.1, 0.2) (0.1, 0.2, 0.3) z1 (0.4, 0.2, 0.3) z4 (0.6, 0.3, 0.2) ,0 .1 , 0 .2 ) (0 0 .2 , .1 , 0.1 ) .1 , ,0 z2 (0.7, 0.2, 0.1) 0 .2 (0. 4 .1 , ,0 .3 (0 0 .4 ) (0. 3 ) z3 (0.5, 0.1, 0.2) Figure 15. IN-digraph. (0.02, 0.04, 0.18) w1 (0.3, 0.4, 0.6) w2 (0.2, 0.5, 0.3) z1 (0.4, 0.2, 0.3) z4 (0.6, 0.3, 0.2) z2 (0.7, 0.2, 0.1) z3 (0.5, 0.1, 0.2) Figure 16. Two-step INC-graph. − → Definition 18. The intuitionistic neutrosophic m-step out-neighborhood of vertex z of an IN-digraph G = (X, h, k) is IN-set: Nm ( z ) = ( Xz , t z , i z , f z ), where m }, t : X → Xz = {w| there exists a directed intuitionistic neutrosophic path of length m from z to w, Pz,w z z [0, 1], iz : Xz → [0, 1] and f z : Xz → [0, 1] are defined by tz = min{t(w1 , w2 ), (w1 , w2 ) is an edge of m }, i = min {i ( w , w ), ( w , w ) is an edge of Pm } and f = max { f ( w , w ), ( w , w ) is an edge of Pz,w z z 2 2 2 2 1 1 1 1 z,w m }, respectively. Pz,w Information 2017, 8, 132 19 of 26 Definition 19. Suppose G = ( X, h, k ) is an IN-graph. Then, the m-step intuitionistic neutrosophic neighborhood graph (IN-neighborhood-graph) Nm (G) is defined by Nm (G) = ( X, h, κ ), where h = (h1 , h2 , h3 ), κ = (κ1 , κ2 , κ3 ), κ1 : X × X → [0, 1], κ2 : X × X → [0, 1] and κ3 : X × X → [0, 1] are such that: κ1 (w, z) = h1 (w) ∧ h1 (z) H1 (Nm (w) ∩ Nm (z)), κ2 (w, z) = h2 (w) ∧ h2 (z) H2 (Nm (w) ∩ Nm (z)), κ3 (w, z) = h3 (w) ∨ h3 (z) H3 (Nm (w) ∩ Nm (z)), respectively. − → − → Theorem 5. If all the edges of IN-digraph G = ( X, h, k) are independent strong, then all the edges of Cm ( G ) are independent strong. − → − → Proof. Suppose G = ( X, h, k ) is an IN-digraph and Cm ( G ) = ( X, h, k) is the corresponding − → + m-step INC-graph. Since all the edges of G are independent strong, then H1 (N+ m ( w ) ∩ Nm ( z )) > 0.5, + + + + H2 (Nm (w) ∩ Nm (z)) < 0.5 and H3 (Nm (w) ∩ Nm (z)) < 0.5. Then, t(w, z) = (h1 (w) ∧ h1 (z)) H1 (N+ m (w) ∩ t(w,z) + (w) ∩ ( z )) N+ , or t ( w, z ) > 0.5 ( h ( w ) ∧ h ( z )) , or 0.5, i ( w, z ) = ( h ( w ) ∧ h ( z )) H (N > 2 2 2 1 1 m m (h (w)∧h (z)) 1 N+ m ( z )), or f ( w, z ) < 0.5( h3 ( w ) ∨ h3 ( z )), 1 i (w,z) < 0.5 and (h2 (w)∧h2 (z)) f (w,z) or (h (w)∨h (z)) < 0.5. 3 3 N+ m ( z )), or i ( w, z ) < 0.5( h2 ( w ) ∧ h2 ( z )), or f (w, z) = ( h3 (w) ∨ h3 (z)) H3 (N+ m (w) ∩ − → Hence, the edge (w, z) is independent strong in Cm ( G ). Since, (w, z) is taken to be the arbitrary − → − → edge of Cm ( G ), thus all the edges of Cm ( G ) are independent strong. 3. Applications Competition graphs are very important to represent the competition between objects. However, still, these representations are unsuccessful to deal with all the competitions of world; for that purpose, INC-graphs are introduced. Now, we discuss the applications of INC-graphs to study the competition along with algorithms. The INC-graphs have many utilizations in different areas. 3.1. Ecosystem Consider a small ecosystem: human eats trout; bald eagle eats trout and salamander; trout eats phytoplankton, mayfly and dragonfly; salamander eats dragonfly and mayfly; snake eats salamander and frog; frog eats dragonfly and mayfly; mayfly eats phytoplankton; dragonfly eats phytoplankton. These nine species human, bald eagle, salamander, snake, frog, dragonfly, trout, mayfly and phytoplankton are taken as vertices. Let the degree of existence in the ecosystem of human be 60%, the degree of indeterminacy of existence be 30% and the degree of false-existence be 10%, i.e., the truth-membership, indeterminacy-membership and falsity-membership values of the vertex human are (0.6, 0.3, 0.1). Similarly, we assume the truth-membership, indeterminacy-membership and falsity-membership values of other vertices as (0.7, 0.3, 0.2), (0.4, 0.3, 0.5), (0.3, 0.5, 0.1), (0.3, 0.4, 0.5), (0.3, 0.5, 0.2), (0.7, 0.3, 0.2), (0.6, 0.4, 0.2) and (0.3, 0.5, 0.2). Suppose that human likes to eat trout 20%, indeterminate to eat 10% and dislike to eat, say 10%. The likeness, indeterminacy and dislikeness of preys for predators are shown in Table 7. It is clear that if trout is removed from the food cycle, then human must be lifeless, and in such a situation bald eagle, phytoplankton, dragonfly and mayfly grow in an undisciplined manner. Thus, we can evaluate the food cycle with the help of INC-graphs. Information 2017, 8, 132 20 of 26 Table 7. Likeness, indeterminacy and dislikeness of preys and predators. Name of Predator Name of Prey Like to Eat Indeterminate to Eat Dislike to Eat Human Bald eagle Bald eagle Snake Snake Salamander Salamander Frog Trout Trout Trout Dragon f ly May f ly Frog Trout Trout Salamander Salamander Frog Dragon f ly May f ly Dragon f ly Dragon f ly May f ly Phytoplankton Phytoplankton Phytoplankton May f ly 20 20 30 20 30 20 20 30 20 30 20 10 30 10 10 20 20 20 20 20 20 30 40 10 10 10 30 10 10 20 30 10 40 20 40 30 10 10 10 10 20 10 For this food web Figure 17, we have the following Table 8 of IN-out-neighborhoods. Human (0.6, 0.3, 0.1) (0.2, 0.2, 0.1) Snake (0.3, 0.5, 0.1) (0.2, 0.1, 0.1) (0.3, 0.2, 0.3) Bald eagle (0.7, 0.3, 0.2) 2, 0 , 0. (0. 3 Frog (0.3, 0.4, 0.5) .4) Dragonfly (0.6, 0.4, 0.2) (0.3, 0.3, 0.3) Phytoplankton (0.7, 0.3, 0.2) (0.3 . (0 1, 0. 1) (0.1 1, 0. , 0. 1, 0 .1 ) (0.2, 0.2, 0.2) 2, 0 (0.2, 0.1, 0.1) , 0. (0.2, 0.4, 0.1) 2 (0. Trout (0.3, 0.5, 0.2) .4 ) Salamander (0.4, 0.3, 0.5) (0.2, 0.2, 0.2) Mayfly (0.3, 0.5, 0.2) (0.3, 0.3, 0. 2) , 0. 1, 0 .1) Figure 17. IN-food web. Table 8. IN-out-neighborhoods. w∈X N+ ( w ) Human Bald eagle Salamander Snake Frog May f ly Phytoplankton Dragon f ly Trout {( Trout, 0.2, 0.1, 0.1)} {( Trout, 0.2, 0.2, 0.2), (Salamander, 0.3, 0.2, 0.3)} {( Dragon f ly, 0.2, 0.2, 0.2), ( May f ly, 0.2, 0.2, 0.4)} {(Salamander, 0.2, 0.2, 0.1), ( Frog, 0.3, 0.2, 0.4)} {( Dragon f ly, 0.3, 0.3, 0.3), ( May f ly, 0.1, 0.1, 0.1)} {( Phytoplankton, 0.3, 0.3, 0.2)} ∅ {( Phytoplankton, 0.1, 0.1, 0.1)} {( Phytoplankton, 0.2, 0.1, 0.1), ( May f ly, 0.3, 0.1, 0.1), ( Dragon f ly, 0.2, 0.4, 0.1)} Information 2017, 8, 132 21 of 26 Therefore, N+ ( Human ∩ Bald eagle) = {(Trout, 0.2, 0.1, 0.2)}, N+ (Bald eagle ∩ Snake) = {(Salamander, 0.2, 0.2, 0.3)}, N+ (Salamander ∩ Frog) = {(Dragon f ly, 0.2, 0.2, 0.3), ( May f ly, 0.1, 0.1, 0.4)}, N+ (Salamander ∩ Trout) = {(Dragon f ly, 0.2, 0.2, 0.2), ( May f ly, 0.2, 0.1, 0.4)}, N+ (Trout ∩ Frog) = {(Dragon f ly, 0.2, 0.3, 0.3), ( May f ly, 0.1, 0.1, 0.1)}, N+ ( May f ly ∩ Trout) = {(Phytoplankton, 0.2, 0.1, 0.2)}, N+ ( May f ly ∩ Dragon f ly) = {(Phytoplankton, 0.1, 0.1, 0.2)} and N+ (Dragon f ly ∩ Trout) = {(Phytoplankton, 0.1, 0.1, 0.1)}. Now, there is an edge between human and bald eagle; snake and bald eagle; salamander and trout; salamander and frog; trout and frog; trout and dragonfly; trout and mayfly; dragonfly and mayfly in the INC-graph, which highlights the competition between them; and for the other pair of species, there is no edge, which indicates that there is no competition in the INC-graph Figure 18. For example, there is an edge between human and bald eagle indicating a 12% degree of likeness to prey on the same species, a 3% degree of indeterminacy and a 4% degree of non-likeness between them. Human (0.6, 0.3, 0.1) (0.06, 0.06, 0.06) (0.12, 0.03, 0.04) Snake (0.3, 0.5, 0.1) Bald eagle (0.7, 0.3, 0.2) (0.06, 0.06, 0.1) Salamander (0.4, 0.3, 0.5) (0.06, 0.06, 0.15) (0.06, 0.12, 0.05) Trout (0.3, 0.5, 0.2) (0.03, 0.04, 0.02) Phytoplankton (0.7, 0.3, 0.2) Dragonfly (0.6, 0.4, 0.2) Frog (0.3, 0.4, 0.5) (0.03, 0.04, 0.04) (0.06, 0.05, 0.04) Mayfly (0.3, 0.5, 0.2) Figure 18. Corresponding INC-graph We present our method, which is used in our ecosystem application in Algorithm 1. Algorithm 1: Ecosystem. Step 1. Input the truth-membership, indeterminacy-membership and falsity-membership values for set of n species. Step 2. If for any two distinct vertices wi and w j , t(wi w j ) > 0, i (wi w j ) > 0, f (wi w j ) > 0, then (w j , t(wi w j ), i (wi w j ), f (wi w j )) ∈ N+ (wi ). Step 3. Step 4. Step 5. Step 6. Step 7. Step 8. Repeat Step 2 for all vertices wi and w j to calculate IN-out-neighborhoods N+ (wi ). Calculate N+ (wi ) ∩ N+ (w j ) for each pair of distinct vertices wi and w j . Calculate H [N+ (wi ) ∩ N+ (w j )]. If N+ (wi ) ∩ N+ (w j ) 6= ∅, then draw an edge wi w j . Repeat Step 6 for all pairs of distinct vertices. Assign membership values to each edge wi w j using the conditions: t(wi w j ) = (wi ∧ w j ) H1 [N+ (wi ) ∩ N+ (w j )] i (wi w j ) = (wi ∧ w j ) H2 [N+ (wi ) ∩ N+ (w j )] f (wi w j ) = (wi ∨ w j ) H3 [N+ (wi ) ∩ N+ (w j )]. Information 2017, 8, 132 22 of 26 3.2. Career Competition Consider the IN-digraph Figure 19 representing the competition between applicants for a career. Let { Rosaleen, Nazneen, Abner, Amara, Casper } be the set of applicants for the particular careers { Medicine, Pharmacy, Anatomy, Surgery}. The truth-membership value of each applicant represents the degree of loyalty quality; the indeterminacy-value represents the indeterminate state of loyalty; and the false-membership value represents the disloyalty of each applicant towards their careers. Let the degree of truth-membership of Nazneen of her loyalty towards her career be 30%: degree of indeterminacy is 50%, and degree of disloyalty is 10%, i.e., the truth-membership, indeterminacy and falsity-membership values of the vertex Nazneen are (0.3, 0.5, 0.1). The truth-membership value of each directed edge between an applicant and a career represents the eligibility for that career; the indeterminacy-value represents the indeterminate state of that career; and the false-membership value represents non-eligibility for that particular career. 4, 0. 2) 0. 2, Nazneen (0.3, 0.5, 0.1) 2) 0. , .1 (0 . (0 2, 0. (0.2, 0.3, 0.4) Surgery (0.3, 0.4, 0.5) Abner (0.3, 0.5, 0.6) (0. 3 (0.2, 0.4, 0.5) (0.2, , 0. 4, 0 0 .4 , Pharmacy (0.2, 0.5, 0.3) Rosaleen (0.3, 0.5, 0.2) 0 .5 ) .5) 0.2, 0.3) Casper (0.1, 0.6, 0.3) 0 .3, 0.1 ) Anatomy (0.3, 0.6, 0.2) .2 ) , 0.3) (0.1, 1 (0. 5, , 0. ,0 .5 (0.2, 0 (0. 5 Medicine (0.7, 0.3, 0.2) Amara (0.6, 0.5, 0.2) Figure 19. IN-digraph. Thus, in Table 9, N+ ( Nazneen) ∩ N+ ( Rosaleen) = {(Surgery, 0.2, 0.2, 0.4)}, N+ ( Nazneen) ∩ = {( Pharmacy, 0.1, 0.4, 0.3)}, N+ ( Nazneen) ∩ N+ ( Abner ) = {( Pharmacy, 0.1, 0.4, 0.5)}, + N ( Nazneen) ∩ N+ (Casper ) = ∅, N+ ( Rosaleen) ∩ N+ ( Amara) = ∅, N+ ( Rosaleen) ∩ N+ (Casper ) = ∅, N+ ( Rosaleen) ∩ N+ ( Abner ) = ∅, N+ ( Amara) ∩ N+ (Casper ) = {( Medicine, 0.1, 0.2, 0.3)}, N+ ( Amara) ∩ N+ ( Abner ) = {( Medicine, 0.3, 0.3, 0.5), ( Pharmacy, 0.2, 0.4, 0.5)} and N+ (Casper ) ∩ N+ ( Abner ) = {( Medicine, 0.1, 0.2, 0.5), ( Anatomy, 0.1, 0.4, 0.5)}. N+ ( Amara) Table 9. IN-out-neighborhoods. w∈X N+ ( w ) Nazneen Rosaleen Amara Casper Abner {(Surgery, 0.2, 0.2, 0.2), ( Pharmacy, 0.1, 0.4, 0.2)} {(Surgery, 0.2, 0.3, 0.4)} {( Medicine, 0.5, 0.3, 0.1), ( Pharmacy, 0.2, 0.5, 0.3)} {( Medicine, 0.1, 0.2, 0.3), ( Anatomy, 0.1, 0.5, 0.2)} {( Medicine, 0.3, 0.4, 0.5), ( Anatomy, 0.2, 0.4, 0.5), ( Pharmacy, 0.2, 0.4, 0.5)} Information 2017, 8, 132 23 of 26 The INC-graph is shown in Figure 20. The solids lines indicate the strength of competition between two applicants, and dashed lines indicate the applicant competing for the particular career. For example, Nazneen and Rosaleen both are competing for the career, surgery, and the strength of competition between them is (0.06, 0.1, 0.08). In Table 10, W (z, c) represents the competition of applicant z for career c with respect to loyalty quality, indeterminacy and disloyalty to compete with the others. The strength to compete with the other applicants with respect to a particular career is calculated in Table 10. From Table 10, Nazneen and Rosaleen have equal strength to compete with the other for the career, surgery. Abner and Casper have equal strength of competition for the career, anatomy. Amara competes with the others for the career, pharmacy and medicine. (0.0 6 Nazneen (0.3, 0.5, 0.1) . (0 03 ,0 .2 0 0, . 3) , 0 .1 , 0 .0 8) Surgery (0.3, 0.4, 0.5) Pharmacy (0.2, 0.5, 0.3) .0 (0 3, Abner (0.3, 0.5, 0.6) 0 .2 Rosaleen (0.3, 0.5, 0.2) 0, 0.0 6) Medicine (0.7, 0.3, 0.2) 1 .0 (0 ,0 Anatomy (0.3, 0.6, 0.2) 0 .2 ,0 . (0 0 .3 09 ) ,0 0 .2 ,0 .3 0) Casper (0.1, 0.6, 0.3) Amara (0.6, 0.5, 0.2) (0.01, 0.10, 0.09) Figure 20. Corresponding INC-graph. Table 10. Strength of competition of the applicant for a particular career. (Applicant, Career) In Competition W(Applicant, Career) S(Applicant, Career) ( Nazneen, Surgery) ( Rosaleen, Surgery) Rosaleen Nazneen (0.06, 0.1, 0.08) (0.06, 0.1, 0.08) 0.88 0.88 ( Abner, Anatomy) (Casper, Anatomy) Casper Abner (0.01, 0.20, 0.30) (0.01, 0.20, 0.30) 0.51 0.51 ( Nazneen, Pharmacy) ( Abner, Pharmacy) ( Amara, Pharmacy) Abner, Amara Amara, Nazneen Nazneen, Abner (0.03, 0.20, 0.18) (0.06, 0.20, 0.30) (0.06, 0.20, 0.18) 0.65 0.56 0.68 ( Amara, Medicine) (Casper, Medicine) ( Abner, Medicine) Abner, Casper Abner, Amara Casper, Amara (0.05, 0.15, 0.195) (0.01, 0.15, 0.195) (0.05, 0.20, 0.30) 0.705 0.665 0.55 We present our method, which is used in our career competition application in Algorithm 2. Information 2017, 8, 132 24 of 26 Algorithm 2: Career Competition Step 1. Input the truth-membership, indeterminacy-membership and falsity-membership values for set of n applicants. Step 2. If for any two distinct vertices zi and z j , t(zi z j ) > 0, i (zi z j ) > 0, f (zi z j ) > 0, then (z j , t(zi z j ), i (zi z j ), f (zi z j )) ∈ N+ (zi ). Step 3. Step 4. Step 5. Step 6. Step 7. Step 8. Repeat Step 2 for all vertices zi and z j to calculate IN-out-neighborhoods N+ (zi ). Calculate N+ (zi ) ∩ N+ (z j ) for each pair of distinct vertices zi and z j . Calculate H [N+ (zi ) ∩ N+ (z j )]. If N+ (zi ) ∩ N+ (z j ) 6= ∅, then draw an edge zi z j . Repeat Step 6 for all pairs of distinct vertices. Assign membership values to each edge zi z j using the conditions: t(zi z j ) = (zi ∧ z j ) H1 [N+ (zi ) ∩ N+ (z j )] i (zi z j ) = (zi ∧ z j ) H2 [N+ (zi ) ∩ N+ (z j )] f (zi z j ) = (zi ∨ z j ) H3 [N+ (zi ) ∩ N+ (z j )]. Step 9. If z, r1 , r2 , r3 , . . ., rn are the applicants competing for career c, then the strength of competition W (z, c) = (t(z, c), i (z, c), f (z, c)) of each applicant z for the career c is: W (z, c) = (t(zr1 )+t(zr2 )+...+t(zrn ),i (zr1 )+i (zr2 )+...+i (zrn ), f (zr1 )+ f (zr2 )+...+ f (zrn )) . n Step 10. Calculate S(z, c), the strength of competition of each applicant z for career c. S(z, c) = t(z, c) − (i (z, c) + f (z, c)) + 1. 4. Conclusions Graphs serve as mathematical models to analyze many concrete real-world problems successfully. Certain problems in physics, chemistry, communication science, computer technology, sociology and linguistics can be formulated as problems in graph theory. Intuitionistic neutrosophic set theory is a mathematical tool to deal with incomplete and vague information. Intuitionistic neutrosophic set theory deals with the problem of how to understand and manipulate imperfect knowledge. 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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). information Article Neutrosophic Commutative N -Ideals in BCK-Algebras Seok-Zun Song 1, * 1 2 3 * ID , Florentin Smarandache 2 and Young Bae Jun 3 Department of Mathematics, Jeju National University, Jeju 63243, Korea Department of Mathematics & Science, University of New Mexico, 705 Gurley Ave., Gallup, NM 87301, USA; fsmarandache@gmail.com Department of Mathematics Education, Gyeongsang National University, Jinju 52828, Korea; skywine@gmail.com Correspondence: szsong@jejunu.ac.kr Received: 16 September 2017; Accepted: 16 October 2017; Published: 18 October 2017 Abstract: The notion of a neutrosophic commutative N -ideal in BCK-algebras is introduced, and several properties are investigated. Relations between a neutrosophic N -ideal and a neutrosophic commutative N -ideal are discussed. Characterizations of a neutrosophic commutative N -ideal are considered. Keywords: neutrosophic N -structure; neutrosophic N -ideal; neutrosophic commutative N -ideal MSC: 06F35, 03G25, 03B52 1. Introduction As a generalization of fuzzy sets, Atanassov [1] introduced the degree of nonmembership/ falsehood (f) in 1986 and defined the intuitionistic fuzzy set. Smarandache proposed the term “neutrosophic” because “neutrosophic” etymologically comes from “neutrosophy” [French neutre < Latin neuter, neutral, and Greek sophia, skill/wisdom] which means knowledge of neutral thought, and this third/neutral represents the main distinction between “fuzzy”/“intuitionistic fuzzy” logic/set and “neutrosophic” logic/set, i.e., the included middle component (Lupasco–Nicolescu’s logic in philosophy), i.e., the neutral/indeterminate/unknown part (besides the “truth”/“membership” and “falsehood”/“non-membership” components that both appear in fuzzy logic/set). Smarandache introduced the degree of indeterminacy/neutrality (i) as an independent component in 1995 (published in 1998) and defined the neutrosophic set on three components (t, i, f) = (truth, indeterminacy, falsehood). For more details, refer to the site http://fs.gallup.unm.edu/FlorentinSmarandache.htm. Jun et al. [2] introduced a new function which is called negative-valued function, and constructed N -structures. Khan et al. [3] introduced the notion of neutrosophic N -structure and applied it to a semigroup. Jun et al. [4] applied the notion of neutrosophic N -structure to BCK/BCI-algebras. They introduced the notions of a neutrosophic N -subalgebra and a (closed) neutrosophic N -ideal in a BCK/BCI-algebra, and investigated related properties. They also considered characterizations of a neutrosophic N -subalgebra and a neutrosophic N -ideal, and discussed relations between a neutrosophic N -subalgebra and a neutrosophic N -ideal. They provided conditions for a neutrosophic N -ideal to be a closed neutrosophic N -ideal. BCK-algebras entered into mathematics in 1966 through the work of Imai and Iséki [5], and have been applied to many branches of mathematics, such as group theory, functional analysis, probability theory and topology. Such algebras generalize Boolean rings as well as Boolean D-posets (= MV-algebras). Also, Iséki introduced the notion of a BCI-algebra which is a generalization of a BCK-algebra (see [6]). Information 2017, 8, 130; doi:10.3390/info8040130 www.mdpi.com/journal/information Information 2017, 8, 130 2 of 9 In this paper, we introduce the notion of a neutrosophic commutative N -ideal in BCK-algebras, and investigate several properties. We consider relations between a neutrosophic N -ideal and a neutrosophic commutative N -ideal. We discuss characterizations of a neutrosophic commutative N -ideal. 2. Preliminaries By a BCI-algebra, we mean a system X := ( X, ∗, 0) ∈ K (τ ) in which the following axioms hold: (I) (II) (III) (IV) (( x ∗ y) ∗ ( x ∗ z)) ∗ (z ∗ y) = 0, ( x ∗ ( x ∗ y)) ∗ y = 0, x ∗ x = 0, x∗y = y∗x = 0 ⇒ x = y for all x, y, z ∈ X. If a BCI-algebra X satisfies 0 ∗ x = 0 for all x ∈ X, then we say that X is a BCK-algebra. We can define a partial ordering by (∀ x, y ∈ X ) ( x y ⇒ x ∗ y = 0). In a BCK/BCI-algebra X, the following hold: (∀ x ∈ X ) ( x ∗ 0 = x ), (1) (∀ x, y, z ∈ X ) (( x ∗ y) ∗ z = ( x ∗ z) ∗ y). (2) A BCK-algebra X is said to be commutative if it satisfies the following equality: (∀ x, y ∈ X ) ( x ∗ ( x ∗ y) = y ∗ (y ∗ x )) . (3) A subset I of a BCK/BCI-algebra X is called an ideal of X if it satisfies 0 ∈ I, (4) (∀ x, y ∈ X ) ( x ∗ y ∈ I, y ∈ I ⇒ x ∈ I ) . (5) A subset I of a BCK-algebra X is called a commutative ideal of X if it satisfies (4) and (∀ x, y, z ∈ X ) (( x ∗ y) ∗ z ∈ I, z ∈ I ⇒ x ∗ (y ∗ (y ∗ x )) ∈ I ) . (6) Lemma 1. An ideal I is commutative if and only if the following assertion is valid. (∀ x, y ∈ X ) ( x ∗ y ∈ I ⇒ x ∗ (y ∗ (y ∗ x )) ∈ I ) . (7) We refer the reader to the books [7,8] for further information regarding BCK/BCI-algebras. For any family { ai | i ∈ Λ} of real numbers, we define _ { ai | i ∈ Λ } : = ^ { ai | i ∈ Λ } : = ( max{ ai | i ∈ Λ} sup{ ai | i ∈ Λ} if Λ is finite, otherwise. ( min{ ai | i ∈ Λ} inf{ ai | i ∈ Λ} if Λ is finite, otherwise. Denote by F ( X, [−1, 0]) the collection of functions from a set X to [−1, 0]. We say that an element of F ( X, [−1, 0]) is a negative-valued function from X to [−1, 0] (briefly, N -function on X). By an N -structure, we mean an ordered pair ( X, f ) of X and an N -function f on X (see [2]). A neutrosophic N -structure over a nonempty universe of discourse X (see [3]) is defined to be the structure Information 2017, 8, 130 3 of 9 XN : = X ( TN ,IN ,FN ) = n x ( TN ( x ),IN ( x ),FN ( x )) |x∈X o (8) where TN , IN and FN are N -functions on X which are called the negative truth membership function, the negative indeterminacy membership function and the negative falsity membership function, respectively, on X. Note that every neutrosophic N -structure XN over X satisfies the condition: (∀ x ∈ X ) (−3 ≤ TN ( x ) + IN ( x ) + FN ( x ) ≤ 0) . 3. Neutrosophic Commutative N -Ideals In what follows, let X denote a BCK-algebra unless otherwise specified. Definition 1 ([4]). A neutrosophic N -structure XN over X is called a neutrosophic N -ideal of X if the following assertion is valid.  TN (0) ≤ TN ( x ) ≤ W { TN ( x ∗ y), TN (y)} V  (∀ x, y ∈ X )  IN (0) ≥ IN ( x ) ≥ { IN ( x ∗ y), IN (y)} FN (0) ≤ FN ( x ) ≤ W { FN ( x ∗ y), FN (y)}   . (9) Definition 2. A neutrosophic N -structure XN over X is called a neutrosophic commutative N -ideal of X if the following assertions are valid. (∀ x ∈ X ) ( TN (0) ≤ TN ( x ), IN (0) ≥ IN ( x ), FN (0) ≤ FN ( x )) , W   TN ( x ∗ (y ∗ (y ∗ x ))) ≤ { TN (( x ∗ y) ∗ z), TN (z)} V   (∀ x, y, z ∈ X )  IN ( x ∗ (y ∗ (y ∗ x ))) ≥ { IN (( x ∗ y) ∗ z), IN (z)}  . FN ( x ∗ (y ∗ (y ∗ x ))) ≤ W (10) (11) { FN (( x ∗ y) ∗ z), FN (z)} Example 1. Consider a BCK-algebra X = {0, 1, 2, 3, 4} with the Cayley table which is given in Table 1. Table 1. Cayley table for the binary operation “*”. * 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0 0 2 3 4 0 1 0 3 4 0 1 2 0 4 0 1 2 3 0 The neutrosophic N -structure XN = n 0 1 2 3 4 , , , , (−0.8,−0.2,−0.9) (−0.3,−0.9,−0.5) (−0.7,−0.7,−0.4) (−0.3,−0.6,−0.7) (−0.5,−0.3,−0.1) over X is a neutrosophic commutative N -ideal of X. Theorem 1. Every neutrosophic commutative N -ideal is a neutrosophic N -ideal. Proof. Let XN be a neutrosophic commutative N -ideal of X. For every x, z ∈ X, we have o Information 2017, 8, 130 4 of 9 TN ( x ) = TN ( x ∗ (0 ∗ (0 ∗ x ))) ≤ IN ( x ) = IN ( x ∗ (0 ∗ (0 ∗ x ))) ≥ FN ( x ) = FN ( x ∗ (0 ∗ (0 ∗ x ))) ≤ _ ^ { TN (( x ∗ 0) ∗ z), TN (z)} = { IN (( x ∗ 0) ∗ z), IN (z)} = _ _ ^ { FN (( x ∗ 0) ∗ z), FN (z)} = { TN ( x ∗ z), TN (z)}, { IN ( x ∗ z), IN (z)}, _ { FN ( x ∗ z), FN (z)} by putting y = 0 in (11) and using (1). Therefore, XN is a neutrosophic commutative N -ideal of X. The converse of Theorem 1 is not true in general as seen in the following example. Example 2. Consider a BCK-algebra X = {0, 1, 2, 3, 4} with the Cayley table which is given in Table 2. Table 2. Cayley table for the binary operation “∗” * 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0 0 2 3 4 0 1 0 3 4 0 0 0 0 3 0 0 0 0 0 The neutrosophic N -structure XN = n 0 1 2 3 4 , , , , (−0.8,−0.1,−0.7) (−0.7,−0.6,−0.6) (−0.6,−0.2,−0.4) (−0.3,−0.8,−0.4) (−0.3,−0.8,−0.4) o over X is a neutrosophic N -ideal of X. But it is not a neutrosophic commutative N -ideal of X since FN (2 ∗ (3 ∗ W (3 ∗ 2)) = FN (2) = −0.4 −0.7 = { FN ((2 ∗ 3) ∗ 0), FN (0)}. We consider characterizations of a neutrosophic commutative N -ideal. Theorem 2. Let XN be a neutrosophic N -ideal of X. Then, XN is a neutrosophic commutative N -ideal of X if and only if the following assertion is valid.  TN ( x ∗ (y ∗ (y ∗ x ))) ≤ TN ( x ∗ y),    (∀ x, y ∈ X )  IN ( x ∗ (y ∗ (y ∗ x ))) ≥ IN ( x ∗ y),  . (12) FN ( x ∗ (y ∗ (y ∗ x ))) ≤ FN ( x ∗ y) Proof. Assume that XN is a neutrosophic commutative N -ideal of X. The assertion (12) is by taking z = 0 in (11) and using (1) and (10). Conversely, suppose that a neutrosophic N -ideal XN of X satisfies the condition (12). Then,  TN ( x ∗ y) ≤ W { TN (( x ∗ y) ∗ z), TN (z)} V  (∀ x, y ∈ X )  IN ( x ∗ y) ≥ { IN (( x ∗ y) ∗ z), IN (z)} FN ( x ∗ y) ≤ W { FN (( x ∗ y) ∗ z), FN (z)}   . (13) It follows that the condition (11) is induced by (12) and (13). Therefore, XN is a neutrosophic commutative N -ideal of X. Lemma 2 ([4]). For any neutrosophic N -ideal XN of X, we have  W   TN ( x ) ≤ V { TN (y), TN (z)}  (∀ x, y, z ∈ X )  x ∗ y z ⇒ IN ( x ) ≥ { IN (y), IN (z)}   W FN ( x ) ≤ { FN (y), FN (z)}    . (14) Information 2017, 8, 130 5 of 9 Theorem 3. In a commutative BCK-algebra, every neutrosophic N -ideal is a neutrosophic commutative N -ideal. Proof. Let XN be a neutrosophic N -ideal of a commutative BCK-algebra X. For any x, y, z ∈ X, we have (( x ∗ (y ∗ (y ∗ x ))) ∗ (( x ∗ y) ∗ z)) ∗ z = (( x ∗ (y ∗ (y ∗ x ))) ∗ z) ∗ (( x ∗ y) ∗ z) ( x ∗ (y ∗ (y ∗ x ))) ∗ ( x ∗ y) = ( x ∗ ( x ∗ y)) ∗ (y ∗ (y ∗ x )) = 0, that is, ( x ∗ (y ∗ (y ∗ x ))) ∗ (( x ∗ y) ∗ z) z. It follows from Lemma 2 that TN ( x ∗ (y ∗ (y ∗ x ))) ≤ IN ( x ∗ (y ∗ (y ∗ x ))) ≥ FN ( x ∗ (y ∗ (y ∗ x ))) ≤ _ ^ { TN (( x ∗ y) ∗ z), TN (z)}, { IN (( x ∗ y) ∗ z), IN (z)}, _ { FN (( x ∗ y) ∗ z), FN (z)}. Therefore, XN is a neutrosophic commutative N -ideal of X. Let XN be a neutrosophic N -structure over X and let α, β, γ ∈ [−1, 0] be such that −3 ≤ α + β + γ ≤ 0. Consider the following sets. α TN := { x ∈ X | TN ( x ) ≤ α}, β I N : = { x ∈ X | I N ( x ) ≥ β }, γ FN := { x ∈ X | FN ( x ) ≤ γ}. The set XN (α, β, γ) := { x ∈ X | TN ( x ) ≤ α, IN ( x ) ≥ β, FN ( x ) ≤ γ} is called the (α, β, γ)-level set of XN . It is clear that β γ α XN (α, β, γ) = TN ∩ IN ∩ FN . β γ α , I and F are commutative ideals of X for all Theorem 4. If XN is a neutrosophic N -ideal of X, then TN N N α, β, γ ∈ [−1, 0] with −3 ≤ α + β + γ ≤ 0 whenever they are nonempty. β γ α , I and F level commutative ideals of X . We call TN N N N β γ α , I and F are nonempty for all α, β, γ ∈ [−1, 0] with −3 ≤ α + β + γ ≤ 0. Proof. Assume that TN N N α, y ∈ Then, x ∈ TN β IN γ and z ∈ FN for some x, y, z ∈ X. Thus, TN (0) ≤ TN ( x ) ≤ α, IN (0) ≥ IN (y) ≥ β, β γ α ∩ I ∩ F . Let ( x ∗ y ) ∗ z ∈ T α and z ∈ T α . Then, and FN (0) ≤ FN (z) ≤ γ, that is, 0 ∈ TN N N N N TN (( x ∗ y) ∗ z) ≤ α and TN (z) ≤ α, which imply that TN ( x ∗ (y ∗ (y ∗ x ))) ≤ β _ { TN (( x ∗ y) ∗ z), TN (z)} ≤ α, β α . If ( a ∗ b ) ∗ c ∈ I and c ∈ I , then I (( a ∗ b ) ∗ c ) ≥ β and I ( c ) ≥ β. that is, x ∗ (y ∗ (y ∗ x )) ∈ TN N N N N Thus IN ( a ∗ (b ∗ (b ∗ c))) ≥ ^ { IN (( a ∗ b) ∗ c), IN (c)} ≥ β, Information 2017, 8, 130 6 of 9 β γ γ and so a ∗ (b ∗ (b ∗ c)) ∈ IN . Finally, suppose that (u ∗ v) ∗ w ∈ FN and w ∈ FN . FN ((u ∗ v) ∗ w) ≤ γ and FN (w) ≤ γ. Thus, FN (u ∗ (v ∗ (v ∗ w))) ≤ γ _ Then, { FN ((u ∗ v) ∗ w), FN (w)} ≤ γ, β γ α , I and F are commutative ideals of X. that is, u ∗ (v ∗ (v ∗ w)) ∈ FN . Therefore, TN N N Corollary 1. Let XN be a neutrosophic N -structure over X and let α, β, γ ∈ [−1, 0] be such that −3 ≤ α + β + γ ≤ 0. If XN is a neutrosophic commutative N -ideal of X, then the nonempty (α, β, γ)-level set of XN is a commutative ideal of X. Proof. Straightforward. β γ α , I and F are ideals of X Lemma 3 ([4]). Let XN be a neutrosophic N -structure over X and assume that TN N N for all α, β, γ ∈ [−1, 0] with −3 ≤ α + β + γ ≤ 0. Then XN is a neutrosophic N -ideal of X. β γ α , I and F are commutative Theorem 5. Let XN be a neutrosophic N -structure over X and assume that TN N N ideals of X for all α, β, γ ∈ [−1, 0] with −3 ≤ α + β + γ ≤ 0. Then, XN is a neutrosophic commutative N -ideal of X. β γ α , I and F are commutative ideals of X, then they are ideals of X. Hence, X is a Proof. If TN N N N neutrosophic N -ideal of X by Lemma 3. Let x, y ∈ X and α, β, γ ∈ [−1, 0] with −3 ≤ α + β + γ ≤ 0 α ∩ I β ∩ F γ . Since such that TN ( x ∗ y) = α, IN ( x ∗ y) = β and FN ( x ∗ y) = γ. Then, x ∗ y ∈ TN N N β β γ γ α ∩ I ∩ F is a commutative ideal of X, it follows from Lemma 1 that x ∗ ( y ∗ ( y ∗ x )) ∈ T α ∩ I ∩ F . TN N N N N N Hence TN ( x ∗ (y ∗ (y ∗ x ))) ≤ α = TN ( x ∗ y), IN ( x ∗ (y ∗ (y ∗ x ))) ≥ β = IN ( x ∗ y), FN ( x ∗ (y ∗ (y ∗ x ))) ≤ γ = FN ( x ∗ y). Therefore, XN is a neutrosophic commutative N -ideal of X by Theorem 2. Theorem 6. Let f : X → X be an injective mapping. Given a neutrosophic N -structure XN over X, the following are equivalent. (1) XN is a neutrosophic commutative N -ideal of X, satisfying the following condition.  TN ( f ( x )) = TN ( x )    (∀ x ∈ X )  IN ( f ( x )) = IN ( x )  . (15) FN ( f ( x )) = FN ( x ) (2) β γ α , I and F are commutative ideals of X , satisfying the following condition. TN N N N β β γ γ α f ( TN ) = TNα , f ( IN ) = IN , f ( FN ) = FN . (16) α, Proof. Let XN be a neutrosophic commutative N -ideal of X, satisfying the condition (15). Then, TN β γ IN and FN are commutative ideals of XN by Theorem 4. Let α ∈ Im( TN ), β ∈ Im( IN ), γ ∈ Im( FN ) and β γ α ∩ I ∩ F . Then T ( f ( x )) = T ( x ) ≤ α, I ( f ( x )) = I ( x ) ≥ β and F ( f ( x )) = F ( x ) ≤ γ. x ∈ TN N N N N N N N N β γ β β γ γ α ∩ I ∩ F , which shows that f ( T α ) ⊆ T α , f ( I ) ⊆ I and f ( F ) ⊆ F . Let y ∈ X Thus, f ( x ) ∈ TN N N N N N N N N be such that f (y) = x. Then, TN (y) = TN ( f (y)) = TN ( x ) ≤ α, IN (y) = IN ( f (y)) = IN ( x ) ≥ β Information 2017, 8, 130 7 of 9 β γ α ∩ I ∩ F . Thus, x = f ( y ) ∈ and FN (y) = FN ( f (y)) = FN ( x ) ≤ γ, which imply that y ∈ TN N N β β γ β γ γ α ) ∩ f ( I ) ∩ f ( F ), and so T α ⊆ f ( T α ), I ⊆ f ( I ) and F ⊆ f ( F ). Therefore (16) is valid. f ( TN N N N N N N N N β γ α , I and F are commutative ideals of X , satisfying the condition (16). Conversely, assume that TN N N N Then, XN is a neutrosophic commutative N -ideal of X by Theorem 5. Let x, y, z ∈ X be such that TN ( x ) = α, IN (y) = β and FN (z) = γ. Note that α̃ α for all α > α̃, TN ( x ) = α ⇐⇒ x ∈ TN and x ∈ / TN β β̃ IN (y) = β ⇐⇒ y ∈ IN and y ∈ / IN for all β < β̃, γ γ̃ FN (z) = γ ⇐⇒ z ∈ FN and z ∈ / FN for all γ > γ̃. β γ α , f ( y ) ∈ I and f ( z ) ∈ F . Hence, T ( f ( x )) ≤ α, I ( f ( y )) ≥ β It follows from (16) that f ( x ) ∈ TN N N N N α̃ = and FN ( f (z)) ≤ γ. Let α̃ = TN ( f ( x )), β̃ = IN ( f (y)) and γ̃ = FN ( f (z)). If α > α̃, then f ( x ) ∈ TN α̃ α̃ f TN , and thus x ∈ TN since f is one to one. This is a contradiction. Hence, TN ( f ( x )) = α = TN ( x ). β̃ β̃ β̃ If β < β̃, then f (y) ∈ IN = f IN which implies from the injectivity of f that y ∈ IN , a contradiction. γ̃ γ̃ Hence, IN ( f ( x )) = β = IN ( x ). If γ > γ̃, then f (z) ∈ FN = f FN . Since f is one to one, we have γ̃ z ∈ FN which is a contradiction. Thus, FN ( f ( x )) = γ = FN ( x ). This completes the proof. For any elements ωt , ωi , ω f ∈ X, we consider sets: ωt := { x ∈ X | TN ( x ) ≤ TN (ωt )} , XN ω XNi := { x ∈ X | IN ( x ) ≥ IN (ωi )} , n o ωf XN := x ∈ X | FN ( x ) ≤ FN (ω f ) . ωf ω ωt , ωi ∈ XNi and ω f ∈ XN . Obviously, ωt ∈ XN ωt Lemma 4 ([4]). Let ωt , ωi and ω f be any elements of X. If XN is a neutrosophic N -ideal of X, then XN , ω ωf XNi and XN are ideals of X. Theorem 7. Let ωt , ωi and ω f be any elements of X. If XN is a neutrosophic commutative N -ideal of X, ω ωf ωt then XN , XNi and XN are commutative ideals of X. Proof. If XN is a neutrosophic commutative N -ideal of X, then it is a neutrosophic N -ideal of X and ωf ω ω ωi ωt ωt , XNi and XN are ideals of X by Lemma 4. Let x ∗ y ∈ XN ∩ XN ∩ XN f for any x, y ∈ X. Then, so XN TN ( x ∗ y) ≤ TN (ωt ), IN ( x ∗ y) ≥ TN (ωi ) and FN ( x ∗ y) ≤ FN (ω f ). It follows from Theorem 2 that TN ( x ∗ (y ∗ (y ∗ x ))) ≤ TN ( x ∗ y) ≤ TN (ωt ), IN ( x ∗ (y ∗ (y ∗ x ))) ≥ IN ( x ∗ y) ≥ IN (ωi ), FN ( x ∗ (y ∗ (y ∗ x ))) ≤ FN ( x ∗ y) ≤ FN (ω f ). ω ωf ω ωf ωt ωt ∩ XNi ∩ XN , and therefore XN Hence, x ∗ (y ∗ (y ∗ x )) ∈ XN , XNi and XN are commutative ideals of X by Lemma 1. Theorem 8. Any commutative ideal of X can be realized as level commutative ideals of some neutrosophic commutative N -ideal of X. Proof. Let A be a commutative ideal of X and let XN be a neutrosophic N -structure over X in which Information 2017, 8, 130 8 of 9 TN : X → [−1, 0], x 7→ ( α 0 IN : X → [−1, 0], x 7→ ( β if x ∈ A, −1 otherwise, FN : X → [−1, 0], x 7→ ( γ 0 if x ∈ A, otherwise, if x ∈ A, otherwise where α, γ ∈ [−1, 0) and β ∈ (−1, 0]. Division into the following cases will verify that XN is a neutrosophic commutative N -ideal of X. If ( x ∗ y) ∗ z ∈ A and z ∈ A, then x ∗ (y ∗ (y ∗ x ) ∈ A. Thus, TN (( x ∗ y) ∗ z) = TN (z) = TN ( x ∗ (y ∗ (y ∗ x ))) = α, IN (( x ∗ y) ∗ z) = IN (z) = IN ( x ∗ (y ∗ (y ∗ x ))) = β, FN (( x ∗ y) ∗ z) = FN (z) = FN ( x ∗ (y ∗ (y ∗ x ))) = γ, and so (11) is clearly verified. If ( x ∗ y) ∗ z ∈ / A and z ∈ / A, then TN (( x ∗ y) ∗ z) = TN (z) = 0, IN (( x ∗ y) ∗ z) = IN (z) = −1 and FN (( x ∗ y) ∗ z) = FN (z) = 0. Hence TN ( x ∗ (y ∗ (y ∗ x ))) ≤ IN ( x ∗ (y ∗ (y ∗ x ))) ≥ FN ( x ∗ (y ∗ (y ∗ x ))) ≤ _ ^ { TN (( x ∗ y) ∗ z), TN (z)}, { IN (( x ∗ y) ∗ z), IN (z)}, _ { FN (( x ∗ y) ∗ z), FN (z)}. If ( x ∗ y) ∗ z ∈ A and z ∈ / A, then TN (( x ∗ y) ∗ z) = α, TN (z) = 0, IN (( x ∗ y) ∗ z) = β, IN (z) = −1, FN (( x ∗ y) ∗ z) = γ and FN (z) = 0. Therefore, TN ( x ∗ (y ∗ (y ∗ x ))) ≤ IN ( x ∗ (y ∗ (y ∗ x ))) ≥ FN ( x ∗ (y ∗ (y ∗ x ))) ≤ _ ^ { TN (( x ∗ y) ∗ z), TN (z)}, { IN (( x ∗ y) ∗ z), IN (z)}, _ { FN (( x ∗ y) ∗ z), FN (z)}. Similarly, if ( x ∗ y) ∗ z ∈ / A and z ∈ A, then (11) is verified. Therefore, XN is a neutrosophic α = A, I β = A and F γ = A. This completes the proof. commutative N -ideal of X. Obviously, TN N N 4. Conclusions In order to deal with the negative meaning of information, Jun et al. [2] have introduced a new function which is called negative-valued function, and constructed N -structures. The concept of neutrosophic set (NS) has been developed by Smarandache in [9,10] as a more general platform which extends the concepts of the classic set and fuzzy set, intuitionistic fuzzy set and interval valued intuitionistic fuzzy set. In this article, we have introduced the notion of a neutrosophic commutative N -ideal in BCK-algebras, and investigated several properties. We have considered relations between a neutrosophic N -ideal and a neutrosophic commutative N -ideal. We have discussed characterizations of a neutrosophic commutative N -ideal. Acknowledgments: The first author was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (No. 2016R1D1A1B02006812). The authors wish to thank the anonymous reviewers for their valuable suggestions. Author Contributions: Y.B. Jun initiated the main idea of the work and wrote the paper. S.Z. Song and Y.B. Jun conceived and designed the new definitions and results. F. Smarandache and S.Z. Song performed finding examples and checking contents. All authors have read and approved the final manuscript for submission. Information 2017, 8, 130 9 of 9 Conflicts of Interest: The authors declare no conflict of interest. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. Atanassov, K. Intuitionistic fuzzy sets. Fuzzy Sets Syst. 1986, 20, 87–96. Jun, Y.B.; Lee, K.J.; Song, S.Z. N -ideals of BCK/BCI-algebras. J. Chungcheong Math. Soc. 2009, 22, 417–437. Khan, M.; Anis, S.; Smarandache, F.; Jun, Y.B. Neutrosophic N -structures and their applications in semigroups. Ann. Fuzzy Math. Inform. 2017, in press. Jun, Y.B.; Smarandache, F.; Bordbar, H. Neutrosophic N -structures applied to BCK/BCI-algebras. Information 2017, 8, 128. Imai, Y.; Iséki, K. On axiom systems of propositional calculi. Proc. Jpn. Acad. 1966, 42, 19–21. Iséki, K. An algebra related with a propositional calculus. Proc. Jpn. Acad. 1966, 42, 26–29. Huang, Y.S. BCI-Algebra; Science Press: Beijing, China, 2006. Meng, J.; Jun, Y.B. BCK-Algebras; Kyungmoon Sa Co.: Seoul, Korea, 1994. Smarandache, F. A Unifying Field in Logics: Neutrosophic Logic. Neutrosophy, Neutrosophic Set, Neutrosophic Probability; American Reserch Press: Rehoboth, NM, USA, 1999. Smarandache, F. Neutrosophic set-a generalization of the intuitionistic fuzzy set. Int. J. Pure Appl. Math. 2005, 24, 287–297. c 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). information Article Neutrosophic N -Structures Applied to BCK/BCI-Algebras Young Bae Jun 1 , Florentin Smarandache 2 and Hashem Bordbar 3, * 1 2 3 * Department of Mathematics Education, Gyeongsang National University, Jinju 52828, Korea; skywine@gmail.com Mathematics & Science Department, University of New Mexico, 705 Gurley Ave., Gallup, NM 87301, USA; fsmarandache@gmail.com Department of Mathematics, Shiraz University, Shiraz 7616914111, Iran Correspondence: bordbar.amirh@gmail.com Received: 12 September 2017; Accepted: 6 October 2017; Published: 16 October 2017 Abstract: Neutrosophic N -structures with applications in BCK/BCI-algebras is discussed. The notions of a neutrosophic N -subalgebra and a (closed) neutrosophic N -ideal in a BCK/BCI-algebra are introduced, and several related properties are investigated. Characterizations of a neutrosophic N -subalgebra and a neutrosophic N -ideal are considered, and relations between a neutrosophic N -subalgebra and a neutrosophic N -ideal are stated. Conditions for a neutrosophic N -ideal to be a closed neutrosophic N -ideal are provided. Keywords: neutrosophic N -structure; neutrosophic N -subalgebra; (closed) neutrosophic N -ideal MSC: 06F35, 03G25, 03B52 1. Introduction BCK-algebras entered into mathematics in 1966 through the work of Imai and Iséki [1], and they have been applied to many branches of mathematics, such as group theory, functional analysis, probability theory and topology. Such algebras generalize Boolean rings as well as Boolean D-posets (MV-algebras). Additionally, Iséki introduced the notion of a BCI-algebra, which is a generalization of a BCK-algebra (see [2]). A (crisp) set A in a universe X can be defined in the form of its characteristic function µ A : X → {0, 1} yielding the value 1 for elements belonging to the set A and the value 0 for elements excluded from the set A. So far, most of the generalizations of the crisp set have been conducted on the unit interval [0, 1], and they are consistent with the asymmetry observation. In other words, the generalization of the crisp set to fuzzy sets relied on spreading positive information that fit the crisp point {1} into the interval [0, 1]. Because no negative meaning of information is suggested, we now feel a need to deal with negative information. To do so, we also feel a need to supply a mathematical tool. To attain such an object, Jun et al. [3] introduced a new function, called a negative-valued function, and constructed N -structures. Zadeh [4] introduced the degree of membership/truth (t) in 1965 and defined the fuzzy set. As a generalization of fuzzy sets, Atanassov [5] introduced the degree of nonmembership/falsehood (f) in 1986 and defined the intuitionistic fuzzy set. Smarandache introduced the degree of indeterminacy/neutrality (i) as an independent component in 1995 (published in 1998) and defined the neutrosophic set on three components: (t, i, f) = (truth, indeterminacy, falsehood) Information 2017, 8, 128; doi:10.3390/info8040128 www.mdpi.com/journal/information Information 2017, 8, 128 2 of 12 For more details, refer to the following site: http://fs.gallup.unm.edu/FlorentinSmarandache.htm In this paper, we discuss a neutrosophic N -structure with an application to BCK/BCI-algebras. We introduce the notions of a neutrosophic N -subalgebra and a (closed) neutrosophic N -ideal in a BCK/BCI-algebra, and investigate related properties. We consider characterizations of a neutrosophic N -subalgebra and a neutrosophic N -ideal. We discuss relations between a neutrosophic N -subalgebra and a neutrosophic N -ideal. We provide conditions for a neutrosophic N -ideal to be a closed neutrosophic N -ideal. 2. Preliminaries We let K (τ ) be the class of all algebras with type τ = (2, 0). A BCI-algebra refers to a system X := ( X, ∗, θ ) ∈ K (τ ) in which the following axioms hold: (I) (II) (III) (IV) (( x ∗ y) ∗ ( x ∗ z)) ∗ (z ∗ y) = θ, ( x ∗ ( x ∗ y)) ∗ y = θ, x ∗ x = θ, x ∗ y = y ∗ x = θ ⇒ x = y. for all x, y, z ∈ X. If a BCI-algebra X satisfies θ ∗ x = θ for all x ∈ X, then we say that X is a BCK-algebra. We can define a partial ordering by (∀ x, y ∈ X ) ( x y ⇒ x ∗ y = θ ) In a BCK/BCI-algebra X, the following hold: (∀ x ∈ X ) ( x ∗ θ = x ) (1) (∀ x, y, z ∈ X ) (( x ∗ y) ∗ z = ( x ∗ z) ∗ y) (2) A non-empty subset S of a BCK/BCI-algebra X is called a subalgebra of X if x ∗ y ∈ S for all x, y ∈ S. A subset I of a BCK/BCI-algebra X is called an ideal of X if it satisfies the following: (I1) 0 ∈ I, (I2) (∀ x, y ∈ X )( x ∗ y ∈ I, y ∈ I ⇒ x ∈ I ). We refer the reader to the books [6,7] for further information regarding BCK/BCI-algebras. For any family { ai | i ∈ Λ} of real numbers, we define _ { ai | i ∈ Λ } : = ^ { ai | i ∈ Λ } : = ( max{ ai | i ∈ Λ} sup{ ai | i ∈ Λ} if Λ is finite otherwise ( min{ ai | i ∈ Λ} inf{ ai | i ∈ Λ} if Λ is finite otherwise We denote by F ( X, [−1, 0]) the collection of functions from a set X to [−1, 0]. We say that an element of F ( X, [−1, 0]) is a negative-valued function from X to [−1, 0] (briefly, N -function on X). An N -structure refers to an ordered pair ( X, f ) of X and an N -function f on X (see [3]). In what follows, we let X denote the nonempty universe of discourse unless otherwise specified. A neutrosophic N -structure over X (see [8]) is defined to be the structure: XN : = X ( TN ,IN ,FN ) = n x ( TN ( x ),IN ( x ),FN ( x )) |x∈X o (3) Information 2017, 8, 128 3 of 12 where TN , IN and FN are N -functions on X, which are called the negative truth membership function, the negative indeterminacy membership function and the negative falsity membership function, respectively, on X. We note that every neutrosophic N -structure XN over X satisfies the condition: (∀ x ∈ X ) (−3 ≤ TN ( x ) + IN ( x ) + FN ( x ) ≤ 0) 3. Application in BCK/BCI-Algebras In this section, we take a BCK/BCI-algebra X as the universe of discourse unless otherwise specified. Definition 1. A neutrosophic N -structure XN over X is called a neutrosophic N -subalgebra of X if the following condition is valid:  TN ( x ∗ y) ≤ W { TN ( x ), TN (y)} V  (∀ x, y ∈ X )  IN ( x ∗ y) ≥ { IN ( x ), IN (y)} FN ( x ∗ y) ≤ W { FN ( x ), FN (y)}  (4)   Example 1. Consider a BCK-algebra X = {θ, a, b, c} with the following Cayley table. ∗ θ a b c θ θ a b c a θ θ a c b θ θ θ c c θ a b θ The neutrosophic N -structure XN = n θ a b c , , , (−0.7,−0.2,−0.6) (−0.5,−0.3,−0.4) (−0.5,−0.3,−0.4) (−0.3,−0.8,−0.5) over X is a neutrosophic N -subalgebra of X. o Let XN be a neutrosophic N -structure over X and let α, β, γ ∈ [−1, 0] be such that −3 ≤ α + β + γ ≤ 0. Consider the following sets: α TN := { x ∈ X | TN ( x ) ≤ α} β IN := { x ∈ X | IN ( x ) ≥ β} γ FN := { x ∈ X | FN ( x ) ≤ γ} The set XN (α, β, γ) := { x ∈ X | TN ( x ) ≤ α, IN ( x ) ≥ β, FN ( x ) ≤ γ} is called the (α, β, γ)-level set of XN . Note that β γ α XN (α, β, γ) = TN ∩ IN ∩ FN Theorem 1. Let XN be a neutrosophic N -structure over X and let α, β, γ ∈ [−1, 0] be such that −3 ≤ α + β + γ ≤ 0. If XN is a neutrosophic N -subalgebra of X, then the nonempty (α, β, γ)-level set of XN is a subalgebra of X. Information 2017, 8, 128 4 of 12 Proof. Let α, β, γ ∈ [−1, 0] be such that −3 ≤ α + β + γ ≤ 0 and XN (α, β, γ) 6= ∅. If x, y ∈ XN (α, β, γ), then TN ( x ) ≤ α, IN ( x ) ≥ β, FN ( x ) ≤ γ, TN (y) ≤ α, IN (y) ≥ β and FN (y) ≤ γ. It follows from Equation (4) that TN ( x ∗ y) ≤ { TN ( x ), TN (y)} ≤ α, V IN ( x ∗ y) ≥ { IN ( x ), IN (y)} ≥ β, and W FN ( x ∗ y) ≤ { FN ( x ), FN (y)} ≤ γ. W Hence, x ∗ y ∈ XN (α, β, γ), and therefore XN (α, β, γ) is a subalgebra of X. β γ α , I and F are subalgebras of Theorem 2. Let XN be a neutrosophic N -structure over X and assume that TN N N X for all α, β, γ ∈ [−1, 0] with −3 ≤ α + β + γ ≤ 0. Then XN is a neutrosophic N -subalgebra of X. Proof. Assume that there exist a, b ∈ X such that TN ( a ∗ b) > { TN ( a), TN (b)}. Then TN ( a ∗ b) > tα ≥ W tα { TN ( a), TN (b)} for some tα ∈ [−1, 0). Hence a, b ∈ TNtα but a ∗ b ∈ / TN , which is a contradiction. Thus W TN ( x ∗ y) ≤ for all x, y ∈ X. If IN ( a ∗ b) < V W { TN ( x ), TN (y)} { IN ( a), IN (b)} for some a, b ∈ X, then ^ IN ( a ∗ b) < t β < where t β := 12 { IN ( a ∗ b) + { IN ( a), IN (b)}}. contradiction. Therefore tβ tβ Thus a, b ∈ IN and a ∗ b ∈ / IN , which is a V IN ( x ∗ y) ≥ { IN ( a), IN (b)} V { IN ( x ), IN (y)} for all x, y ∈ X. Now, suppose that there exist a, b ∈ X and tγ ∈ [−1, 0) such that FN ( a ∗ b) > tγ ≥ t t _ { FN ( a), FN (b)} Then a, b ∈ FNγ and a ∗ b ∈ / FNγ , which is a contradiction. Hence FN ( x ∗ y) ≤ _ { FN ( x ), FN (y)} for all x, y ∈ X. Therefore XN is a neutrosophic N -subalgebra of X. Because [−1, 0] is a completely distributive lattice with respect to the usual ordering, we have the following theorem. Theorem 3. If { X Ni | i ∈ N} is a family of neutrosophic N -subalgebras of X, then { X Ni | i ∈ N}, ⊆ forms a complete distributive lattice. Proposition 1. If a neutrosophic N -structure XN over X is a neutrosophic N -subalgebra of X, then TN (θ ) ≤ TN ( x ), IN (θ ) ≥ IN ( x ) and FN (θ ) ≤ FN ( x ) for all x ∈ X. Proof. Straightforward. Theorem 4. Let XN be a neutrosophic N -subalgebra of X. If there exists a sequence { an } in X such that lim TN ( an ) = −1, lim IN ( an ) = 0 and lim FN ( an ) = −1, then TN (θ ) = −1, IN (θ ) = 0 and FN (θ ) = −1. n→∞ n→∞ n→∞ Proof. By Proposition 1, we have TN (θ ) ≤ TN ( x ), IN (θ ) ≥ IN ( x ) and FN (θ ) ≤ FN ( x ) for all x ∈ X. Hence TN (θ ) ≤ TN ( an ), IN ( an ) ≤ IN (θ ) and FN (θ ) ≤ FN ( an ) for every positive integer n. It follows that Information 2017, 8, 128 5 of 12 − 1 ≤ TN (θ ) ≤ lim TN ( an ) = −1 n→∞ 0 ≥ IN (θ ) ≥ lim IN ( an ) = 0 n→∞ − 1 ≤ FN (θ ) ≤ lim FN ( an ) = −1 n→∞ Hence TN (θ ) = −1, IN (θ ) = 0 and FN (θ ) = −1. Proposition 2. If every neutrosophic N -subalgebra XN of X satisfies: TN ( x ∗ y) ≤ TN (y), IN ( x ∗ y) ≥ IN (y), FN ( x ∗ y) ≤ FN (y) (5) for all x, y ∈ X, then XN is constant. Proof. Using Equations (1) and (5), we have TN ( x ) = TN ( x ∗ θ ) ≤ TN (θ ), IN ( x ) = IN ( x ∗ θ ) ≥ IN (θ ) and FN ( x ) = FN ( x ∗ θ ) ≤ FN (θ ) for all x ∈ X. It follows from Proposition 1 that TN ( x ) = TN (θ ), IN ( x ) = IN (θ ) and FN ( x ) = FN (θ ) for all x ∈ X. Therefore XN is constant. Definition 2. A neutrosophic N -structure XN over X is called a neutrosophic N -ideal of X if the following assertion is valid: W   TN (θ ) ≤ TN ( x ) ≤ { TN ( x ∗ y), TN (y)} V   (∀ x, y ∈ X )  IN (θ ) ≥ IN ( x ) ≥ { IN ( x ∗ y), IN (y)}  (6) FN (θ ) ≤ FN ( x ) ≤ W { FN ( x ∗ y), FN (y)} Example 2. The neutrosophic N -structure XN over X in Example 1 is a neutrosophic N -ideal of X. Example 3. Consider a BCI-algebra X := Y × Z where (Y, ∗, θ ) is a BCI-algebra and (Z, −, 0) is the adjoint BCI-algebra of the additive group (Z, +, 0) of integers (see [6]). Let XN be a neutrosophic N -structure over X given by n o n o XN = x (α,0,γ) | x ∈ Y × (N ∪ {0}) ∪ x (0,β,0) |x∈ / Y × (N ∪ {0}) where α, γ ∈ [−1, 0) and β ∈ (−1, 0]. Then XN is a neutrosophic N -ideal of X. Proposition 3. Every neutrosophic N -ideal XN of X satisfies the following assertions: ( x, y ∈ X ) ( x y ⇒ TN ( x ) ≤ TN (y), IN ( x ) ≥ IN (y), FN ( x ) ≤ FN (y)) Proof. Let x, y ∈ X be such that x y. Then x ∗ y = θ, and so TN ( x ) ≤ { TN ( x ∗ y), TN (y)} = { TN (θ ), TN (y)} = TN (y) V V IN ( x ) ≥ { IN ( x ∗ y), IN (y)} = { IN (θ ), IN (y)} = IN (y) W W FN ( x ) ≤ { FN ( x ∗ y), FN (y)} = { FN (θ ), FN (y)} = FN (y) W W This completes the proof. Proposition 4. Let XN be a neutrosophic N -ideal of X. Then (1) TN ( x ∗ y) ≤ TN (( x ∗ y) ∗ y) ⇔ TN (( x ∗ z) ∗ (y ∗ z)) ≤ TN (( x ∗ y) ∗ z) (2) IN ( x ∗ y) ≥ IN (( x ∗ y) ∗ y) ⇔ IN (( x ∗ z) ∗ (y ∗ z)) ≥ IN (( x ∗ y) ∗ z) (3) FN ( x ∗ y) ≤ FN (( x ∗ y) ∗ y) ⇔ FN (( x ∗ z) ∗ (y ∗ z)) ≤ FN (( x ∗ y) ∗ z) for all x, y, z ∈ X. (7) Information 2017, 8, 128 6 of 12 Proof. Note that (( x ∗ (y ∗ z)) ∗ z) ∗ z ( x ∗ y) ∗ z (8) for all x, y, z ∈ X. Assume that TN ( x ∗ y) ≤ TN (( x ∗ y) ∗ y), IN ( x ∗ y) ≥ IN (( x ∗ y) ∗ y) and FN ( x ∗ y) ≤ FN (( x ∗ y) ∗ y) for all x, y ∈ X. It follows from Equation (2) and Proposition 3 that TN (( x ∗ z) ∗ (y ∗ z)) = TN (( x ∗ (y ∗ z)) ∗ z) ≤ TN ((( x ∗ (y ∗ z)) ∗ z) ∗ z) ≤ TN (( x ∗ y) ∗ z) IN (( x ∗ z) ∗ (y ∗ z)) = IN (( x ∗ (y ∗ z)) ∗ z) ≥ IN ((( x ∗ (y ∗ z)) ∗ z) ∗ z) ≥ IN (( x ∗ y) ∗ z) and FN (( x ∗ z) ∗ (y ∗ z)) = FN (( x ∗ (y ∗ z)) ∗ z) ≤ FN ((( x ∗ (y ∗ z)) ∗ z) ∗ z) ≤ FN (( x ∗ y) ∗ z) for all x, y ∈ X. Conversely, suppose TN (( x ∗ z) ∗ (y ∗ z)) ≤ TN (( x ∗ y) ∗ z) IN (( x ∗ z) ∗ (y ∗ z)) ≥ IN (( x ∗ y) ∗ z) (9) FN (( x ∗ z) ∗ (y ∗ z)) ≤ FN (( x ∗ y) ∗ z) for all x, y, z ∈ X. If we substitute z for y in Equation (9), then TN ( x ∗ z) = TN (( x ∗ z) ∗ θ ) = TN (( x ∗ z) ∗ (z ∗ z)) ≤ TN (( x ∗ z) ∗ z) IN ( x ∗ z) = IN (( x ∗ z) ∗ θ ) = IN (( x ∗ z) ∗ (z ∗ z)) ≥ IN (( x ∗ z) ∗ z) FN ( x ∗ z) = FN (( x ∗ z) ∗ θ ) = FN (( x ∗ z) ∗ (z ∗ z)) ≤ FN (( x ∗ z) ∗ z) for all x, z ∈ X by using (III) and Equation (1). Theorem 5. Let XN be a neutrosophic N -structure over X and let α, β, γ ∈ [−1, 0] be such that −3 ≤ α + β + γ ≤ 0. If XN is a neutrosophic N -ideal of X, then the nonempty (α, β, γ)-level set of XN is an ideal of X. Proof. Assume that XN (α, β, γ) 6= ∅ for α, β, γ ∈ [−1, 0] with −3 ≤ α + β + γ ≤ 0. Clearly, θ ∈ XN (α, β, γ). Let x, y ∈ X be such that x ∗ y ∈ XN (α, β, γ) and y ∈ XN (α, β, γ). Then TN ( x ∗ y) ≤ α, IN ( x ∗ y) ≥ β, FN ( x ∗ y) ≤ γ, TN (y) ≤ α, IN (y) ≥ β and FN (y) ≤ γ. It follows from Equation (6) that TN ( x ) ≤ IN ( x ) ≥ FN ( x ) ≤ _ ^ { TN ( x ∗ y), TN (y)} ≤ α { IN ( x ∗ y), IN (y)} ≥ β _ { FN ( x ∗ y), FN (y)} ≤ γ so that x ∈ XN (α, β, γ). Therefore XN (α, β, γ) is an ideal of X. Information 2017, 8, 128 7 of 12 β γ α , I and F are ideals of X for Theorem 6. Let XN be a neutrosophic N -structure over X and assume that TN N N all α, β, γ ∈ [−1, 0] with −3 ≤ α + β + γ ≤ 0. Then XN is a neutrosophic N -ideal of X. Proof. If there exist a, b, c ∈ X such that TN (θ ) > TN ( a), IN (θ ) < IN (b) and FN (θ ) > FN (c), respectively, then TN (θ ) > at ≥ TN ( a), IN (θ ) < bi ≤ IN (b) and FN (θ ) > c f ≥ FN (c) for some cf b at at , c f ∈ [−1, 0) and bi ∈ (−1, 0]. Then θ ∈ / TN , θ ∈ / INi and θ ∈ / FN . This is a contradiction. Hence, TN (θ ) ≤ TN ( x ), IN (θ ) ≥ IN ( x ) and FN (θ ) ≤ FN ( x ) for all x ∈ X. Assume that there exist W V at , bt , ai , bi , a f , b f ∈ X such that TN ( at ) > { TN ( at ∗ bt ), TN (bt )}, IN ( ai ) < { IN ( ai ∗ bi ), IN (bi )} and W FN ( a f ) > { FN ( a f ∗ b f ), FN (b f )}. Then there exist st , s f ∈ [−1, 0) and si ∈ (−1, 0] such that TN ( at ) > st ≥ I N ( ai ) < si ≤ _ ^ FN ( a f ) > s f ≥ { TN ( at ∗ bt ), TN (bt )} { IN ( ai ∗ bi ), IN (bi )} _ { FN ( a f ∗ b f ), FN (b f )} s s sf sf st st It follows that at ∗ bt ∈ TN , bt ∈ TN , ai ∗ bi ∈ INi , bi ∈ INi , a f ∗ b f ∈ FN and b f ∈ FN . However, sf s st at ∈ / TN , ai ∈ / INi and a f ∈ / FN . This is a contradiction, and so TN ( x ) ≤ IN ( x ) ≥ FN ( x ) ≤ _ ^ { TN ( x ∗ y), TN (y)} { IN ( x ∗ y), IN (y)} _ { FN ( x ∗ y), FN (y)} for all x, y ∈ X. Therefore XN is a neutrosophic N -ideal of X. Proposition 5. For any neutrosophic N -ideal XN of X, we have  W   TN ( x ) ≤ V { TN (y), TN (z)}  (∀ x, y, z ∈ X )  x ∗ y z ⇒ IN ( x ) ≥ { IN (y), IN (z)}   W FN ( x ) ≤ { FN (y), FN (z)}     Proof. Let x, y, z ∈ X be such that x ∗ y z. Then ( x ∗ y) ∗ z = θ, and so TN ( x ∗ y) ≤ IN ( x ∗ y) ≥ FN ( x ∗ y) ≤ It follows that _ ^ { IN (( x ∗ y) ∗ z), IN (z)} = _ ^ IN ( x ) ≥ FN ( x ) ≤ _ ^ { TN ( x ∗ y), TN (y)} ≤ { IN ( x ∗ y), IN (y)} ≥ _ { FN (θ ), FN (z)} = FN (z) _ ^ { FN ( x ∗ y), FN (y)} ≤ { TN (θ ), TN (z)} = TN (z) { IN (θ ), IN (z)} = IN (z) _ { FN (( x ∗ y) ∗ z), FN (z)} = TN ( x ) ≤ This completes the proof. _ { TN (( x ∗ y) ∗ z), TN (z)} = { TN (y), TN (z)} { IN (y), IN (z)} _ { FN (y), FN (z)} Theorem 7. In a BCK-algebra, every neutrosophic N -ideal is a neutrosophic N -subalgebra. Proof. Let XN be a neutrosophic N -ideal of a BCK-algebra X. For any x, y ∈ X, we have (10) Information 2017, 8, 128 8 of 12 TN ( x ∗ y) ≤ = ≤ IN ( x ∗ y) ≥ = ≥ and FN ( x ∗ y) ≤ = ≤ _ _ _ { TN (( x ∗ y) ∗ x ), TN ( x )} = { TN (θ ∗ y), TN ( x )} = { TN ( x ), TN (y)} ^ ^ ^ _ _ _ _ { IN (θ ∗ y), IN ( x )} = { IN (y), IN ( x )} { FN ( x ), FN (y)} ^ { IN (( x ∗ x ) ∗ y), IN ( x )} { IN (θ ), IN ( x )} { FN (( x ∗ y) ∗ x ), FN ( x )} = { FN (θ ∗ y), FN ( x )} = { TN (( x ∗ x ) ∗ y), TN ( x )} { TN (θ ), TN ( x )} { IN (( x ∗ y) ∗ x ), IN ( x )} = ^ _ _ _ { FN (( x ∗ x ) ∗ y), FN ( x )} { FN (θ ), FN ( x )} Hence XN is a neutrosophic N -subalgebra of a BCK-algebra X. The converse of Theorem 7 may not be true in general, as seen in the following example. Example 4. Consider a BCK-algebra X = {θ, 1, 2, 3, 4} with the following Cayley table. ∗ θ 1 2 3 4 θ θ 1 2 3 4 1 θ θ 1 3 4 2 θ θ θ 3 4 3 θ θ 1 θ 3 4 θ θ θ θ θ Let XN be a neutrosophic N -structure over X, which is given as follows: XN = n 1 θ , , (−0.8,0,−1) (−0.8,−0.2,−0.9) 2 3 4 , , (−0.2,−0.6,−0.5) (−0.7,−0.4,−0.7) (−0.4,−0.8,−0.3) o Then XN is a neutrosophic N -subalgebra of X, but it is not a neutrosophic N -ideal of X as W V TN (2) = −0.2 > −0.7 = { TN (2 ∗ 3), TN (3)}, IN (4) = −0.8 < −0.4 = { IN (4 ∗ 3), IN (3)}, or W FN (4) = −0.3 > −0.7 = { FN (4 ∗ 3), FN (3)}. Theorem 7 is not valid in a BCI-algebra; that is, if X is a BCI-algebra, then there is a neutrosophic N -ideal that is not a neutrosophic N -subalgebra, as seen in the following example. Example 5. Consider the neutrosophic N -ideal XN of X in Example 3. If we take x := (θ, 0) and y := (θ, 1) in Y × (N ∪ {0}), then x ∗ y = (θ, 0) ∗ (θ, 1) = (θ, −1) ∈ / Y × (N ∪ {0}). Hence TN ( x ∗ y) = 0 > α = IN ( x ∗ y) = β < 0 = FN ( x ∗ y) = 0 > γ = _ { TN ( x ), TN (y)} _ { FN ( x ), FN (y)} ^ { IN ( x ), IN (y)} or Therefore XN is not a neutrosophic N -subalgebra of X. Information 2017, 8, 128 9 of 12 For any elements ωt , ωi , ω f ∈ X, we consider sets: ωt XN := { x ∈ X | TN ( x ) ≤ TN (ωt )} ω XNi := { x ∈ X | IN ( x ) ≥ IN (ωi )} n o ωf XN := x ∈ X | FN ( x ) ≤ FN (ω f ) ωf ω ωt Clearly, ωt ∈ XN , ωi ∈ XNi and ω f ∈ XN . ω ωt Theorem 8. Let ωt , ωi and ω f be any elements of X. If XN is a neutrosophic N -ideal of X, then XN , XNi and ωf XN are ideals of X. ωf ω ω ωf ωt ωt Proof. Clearly, θ ∈ XN ∩ XNi ∩ XN and , θ ∈ XNi and θ ∈ XN . Let x, y ∈ X be such that x ∗ y ∈ XN ωf ω ωt ∩ XNi ∩ XN . Then y ∈ XN TN ( x ∗ y) ≤ TN (ωt ), TN (y) ≤ TN (ωt ) I N ( x ∗ y ) ≥ I N ( ωi ), I N ( y ) ≥ I N ( ωi ) FN ( x ∗ y) ≤ FN (ω f ), FN (y) ≤ FN (ω f ) It follows from Equation (6) that TN ( x ) ≤ IN ( x ) ≥ FN ( x ) ≤ ωf ω _ ^ { TN ( x ∗ y), TN (y)} ≤ TN (ωt ) { IN ( x ∗ y), IN (y)} ≥ IN (ωi ) _ { FN ( x ∗ y), FN (y)} ≤ FN (ω f ) ω ωf ωt ωt ∩ XNi ∩ XN , and therefore XN , XNi and XN are ideals of X. Hence x ∈ XN Theorem 9. Let ωt , ωi , ω f ∈ X and let XN be a neutrosophic N -structure over X. Then ωf ω ωt , XNi and XN are ideals of X, then the following assertion is valid: (1) If XN  TN ( x ) ≥ W { TN (y ∗ z), TN (z)} ⇒ TN ( x ) ≥ TN (y) V  (∀ x, y, z ∈ X )  IN ( x ) ≤ { IN (y ∗ z), IN (z)} ⇒ IN ( x ) ≤ IN (y) FN ( x ) ≥ (2) If XN satisfies Equation (11) and W { FN (y ∗ z), FN (z)} ⇒ FN ( x ) ≥ FN (y)    (∀ x ∈ X ) ( TN (θ ) ≤ TN ( x ), IN (θ ) ≥ IN ( x ), FN (θ ) ≤ FN ( x )) ω (11) (12) ωf ωt , XNi and XN are ideals of X for all ωt ∈ Im( TN ), ωi ∈ Im( IN ) and ω f ∈ Im( FN ). then XN ωf ω ωt , XNi and XN are ideals of X for ωt , ωi , ω f ∈ X. Let x, y, z ∈ X be such Proof. (1) Assume that XN W V W that TN ( x ) ≥ { TN (y ∗ z), TN (z)}, IN ( x ) ≤ { IN (y ∗ z), IN (z)} and FN ( x ) ≥ { FN (y ∗ z), FN (z)}. ω ω ω ω f f ωt ωt ∩ XNi ∩ XN and z ∈ XN ∩ XNi ∩ XN , where ωt = ωi = ω f = x. It follows Then y ∗ z ∈ XN ω ωf ωt from (I2) that y ∈ XN ∩ XNi ∩ XN for ωt = ωi = ω f = x. Hence TN (y) ≤ TN (ωt ) = TN ( x ), IN (y) ≥ IN (ωi ) = IN ( x ) and FN (y) ≤ FN (ω f ) = FN ( x ). Information 2017, 8, 128 10 of 12 (2) Let ωt ∈ Im( TN ), ωi ∈ Im( IN ) and ω f ∈ Im( FN ) and suppose that XN satisfies Equations (11) ωf ω ω ωt ωt ∩ XNi ∩ XN by Equation (12). Let x, y ∈ X be such that x ∗ y ∈ XN ∩ XNi ∩ and (12). Clearly, θ ∈ XN ωf ωf ωi ωt XN and y ∈ XN ∩ XN ∩ XN . Then TN ( x ∗ y) ≤ TN (ωt ), TN (y) ≤ TN (ωt ) I N ( x ∗ y ) ≥ I N ( ωi ), I N ( y ) ≥ I N ( ωi ) FN ( x ∗ y) ≤ FN (ω f ), FN (y) ≤ FN (ω f ) which implies that { TN ( x ∗ y), TN (y)} ≤ TN (ωt ), { IN ( x ∗ y), IN (y)} ≥ IN (ωi ), and { FN ( x ∗ y), FN (y)} ≤ FN (ω f ). It follows from Equation (11) that TN (ωt ) ≥ TN ( x ), IN (ωi ) ≤ IN ( x ) and W V ωf ω ωf ω W ωt ωt ∩ XNi ∩ XN , and therefore XN , XNi and XN are ideals of X. FN (ω f ) ≥ FN ( x ). Thus, x ∈ XN Definition 3. A neutrosophic N -ideal XN of X is said to be closed if it is a neutrosophic N -subalgebra of X. Example 6. Consider a BCI-algebra X = {θ, 1, a, b, c} with the following Cayley table. ∗ θ 1 a b c 1 θ θ a b c θ θ 1 a b c a a a θ c b b b b c θ a c c c b a θ Let XN be a neutrosophic N -structure over X which is given as follows: XN = n 1 a θ , , , (−0.9,−0.3,−0.8) (−0.7,−0.4,−0.7) (−0.6,−0.8,−0.3) c b , (−0.2,−0.6,−0.3) (−0.2,−0.8,−0.5) Then XN is a closed neutrosophic N -ideal of X. o Theorem 10. Let X be a BCI-algebra, For any α1 , α2 , γ1 , γ2 ∈ [−1, 0) and β 1 , β 2 ∈ (−1, 0] with α1 < α2 , γ1 < γ2 and β 1 > β 2 , let XN := (T ,IX ,F ) be a neutrosophic N -structure over X given as follows: N N N TN : X → [−1, 0], x 7→ ( α1 α2 if x ∈ X+ otherwise IN : X → [−1, 0], x 7→ ( β1 β2 if x ∈ X+ otherwise FN : X → [−1, 0], x 7→ ( γ1 γ2 if x ∈ X+ otherwise where X+ = { x ∈ X | θ x }. Then XN is a closed neutrosophic N -ideal of X. Proof. Because θ ∈ X+ , we have TN (θ ) = α1 ≤ TN ( x ), IN (θ ) = β 1 ≥ IN ( x ) and FN (θ ) = γ1 ≤ FN ( x ) for all x ∈ X. Let x, y ∈ X. If x ∈ X+ , then TN ( x ) = α1 ≤ IN ( x ) = β 1 ≥ FN ( x ) = γ1 ≤ _ { TN ( x ∗ y), TN (y)} _ { FN ( x ∗ y), FN (y)} ^ { IN ( x ∗ y), IN (y)} Information 2017, 8, 128 11 of 12 Suppose that x ∈ / X+ . If x ∗ y ∈ X+ then y ∈ / X+ , and if y ∈ X+ then x ∗ y ∈ / X+ . In either case, we have TN ( x ) = α2 = IN ( x ) = β 2 = FN ( x ) = γ2 = _ { TN ( x ∗ y), TN (y)} _ { FN ( x ∗ y), FN (y)} ^ { IN ( x ∗ y), IN (y)} For any x, y ∈ X, if any one of x and y does not belong to X+ , then TN ( x ∗ y) ≤ α2 = IN ( x ∗ y) ≥ β 2 = FN ( x ∗ y) ≤ γ2 = If x, y ∈ X+ , then x ∗ y ∈ X+ . Hence TN ( x ∗ y) = α1 = IN ( x ∗ y) = β 1 = FN ( x ∗ y) = γ1 = _ { TN ( x ), TN (y)} _ { FN ( x ), FN (y)} _ { TN ( x ), TN (y)} _ { FN ( x ), FN (y)} ^ ^ { IN ( x ), IN (y)} { IN ( x ), IN (y)} Therefore XN is a closed neutrosophic N -ideal of X. Proposition 6. Every closed neutrosophic N -ideal XN of a BCI-algebra X satisfies the following condition: (∀ x ∈ X ) ( TN (θ ∗ x ) ≤ TN ( x ), IN (θ ∗ x ) ≥ IN ( x ), FN (θ ∗ x ) ≤ FN ( x )) (13) Proof. Straightforward. We provide conditions for a neutrosophic N -ideal to be closed. Theorem 11. Let X be a BCI-algebra. If XN is a neutrosophic N -ideal of X that satisfies the condition of Equation (13), then XN is a neutrosophic N -subalgebra and hence is a closed neutrosophic N -ideal of X. Proof. Note that ( x ∗ y) ∗ x θ ∗ y for all x, y ∈ X. Using Equations (10) and (13), we have TN ( x ∗ y) ≤ IN ( x ∗ y) ≥ FN ( x ∗ y) ≤ _ ^ { TN ( x ), TN (θ ∗ y)} ≤ { IN ( x ), IN (θ ∗ y)} ≥ _ _ ^ { FN ( x ), FN (θ ∗ y)} ≤ { TN ( x ), TN (y)} { IN ( x ), IN (y)} _ { FN ( x ), FN (y)} Hence XN is a neutrosophic N -subalgebra and is therefore a closed neutrosophic N -ideal of X. Author Contributions: In this paper, Y. B. Jun conceived and designed the main idea and wrote the paper, H. Bordbar performed the idea, checking contents and finding examples, F. Smarandache analyzed the data and checking language. Conflicts of Interest: The authors declare no conflict of interest. References 1. 2. 3. 4. Imai,Y.; Iséki, K. On axiom systems of propositional calculi. Proc. Jpn. Acad. 1966, 42, 19–21. Iséki, K. An algebra related with a propositional calculus. Proc. Jpn. Acad. 1966, 42, 26–29. Jun, Y.B.; Lee, K.J.; Song, S.Z. N -ideals of BCK/BCI-algebras. J. Chungcheong Math. Soc. 2009, 22, 417–437. Zadeh, L.A. Fuzzy sets. Inf. Control 1965, 8, 338–353. Information 2017, 8, 128 5. 6. 7. 8. 12 of 12 Atanassov, K. Intuitionistic fuzzy sets. Fuzzy Sets Syst. 1986, 20, 87–96. Huang, Y.S. BCI-Algebra; Science Press: Beijing, China, 2006. Meng, J.; Jun, Y.B. BCK-Algebras; Kyungmoon Sa Co.: Seoul, Korea, 1994. Khan, M.; Amis, S.; Smarandache, F.; Jun, Y.B. Neutrosophic N -structures and their applications in semigroups. Ann. Fuzzy Math. Inform. submitted, 2017. c 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). information Article TODIM Method for Single-Valued Neutrosophic Multiple Attribute Decision Making Dong-Sheng Xu 1 , Cun Wei 1, * and Gui-Wu Wei 2, * 1 2 * ID School of Science, Southwest Petroleum University, Chengdu 610500, China; xudongsheng1976@163.com School of Business, Sichuan Normal University, Chengdu 610101, China Correspondence: weicun1990@163.com (C.W.); weiguiwu1973@sicnu.edu.cn (G.-W.W.) Received: 20 September 2017; Accepted: 11 October 2017; Published: 16 October 2017 Abstract: Recently, the TODIM has been used to solve multiple attribute decision making (MADM) problems. The single-valued neutrosophic sets (SVNSs) are useful tools to depict the uncertainty of the MADM. In this paper, we will extend the TODIM method to the MADM with the single-valued neutrosophic numbers (SVNNs). Firstly, the definition, comparison, and distance of SVNNs are briefly presented, and the steps of the classical TODIM method for MADM problems are introduced. Then, the extended classical TODIM method is proposed to deal with MADM problems with the SVNNs, and its significant characteristic is that it can fully consider the decision makers’ bounded rationality which is a real action in decision making. Furthermore, we extend the proposed model to interval neutrosophic sets (INSs). Finally, a numerical example is proposed. Keywords: multiple attribute decision making (MADM); single-valued neutrosophic numbers; interval neutrosophic numbers; TODIM method; prospect theory 1. Introduction Multiple attribute decision making (MADM) is a hot research area of the decision theory domain, which has had wide applications in many fields, and attracted increasing attention [1,2]. Due to the fuzziness and uncertainty of the alternatives in different attributes, attribute values in decision making problems are not always represented as real numbers, and they can be described as fuzzy numbers in more suitable occasions, such as interval-valued numbers [3,4], triangular fuzzy variables [5–8], linguistic variables [9–13] or uncertain linguistic variables [14–21], intuitionistic fuzzy numbers (IFSs) [22–27] or interval-valued intuitionistic fuzzy numbers (IVIFSs) [28–31], and SVNSs [32] or INSs [33]. Since Fuzzy set (FS), which is a very useful tool to process fuzzy information, was firstly proposed by Zadeh [34], it has been regarded as an useful tool to solve MADM [35,36], fuzzy logic [37], and patterns recognition [38]. Atanassov [22] introduced IFSs with the membership degree and non-membership degree, which were extended to IVIFSs [28]. Smarandache [39,40] proposed a neutrosophic set (NS) with truth-membership function, indeterminacy-membership function, and falsity-membership function. Furthermore, the concepts of a SVNS [32] and an INS [33] were presented for actual applications. Ye [41] proposed a simplified neutrosophic set (SNS), including the SVNS and INS. Recently, SNSs (INSs, and SVNSs) have been utilized to solve many MADM problems [42–67]. In order to depict the increasing complexity in the actual world, the DMs’ risk attitudes should be taken into consideration to deal with MADM [68–70]. Based on the prospect theory, Gomes and Lima [71] established TODIM (an acronym in Portuguese for Interactive Multi-Criteria Decision Making) method to solve the MADM problems with the DMs’ psychological behaviors are considered. Some scholars have paid attention to depict the DMs’ attitudinal characters in the MADM [72–74]. Also, some scholars proposed fuzzy TODIM models [75,76], intuitionistic fuzzy Information 2017, 8, 125; doi:10.3390/info8040125 www.mdpi.com/journal/information Information 2017, 8, 125 2 of 18 TODIM models [77,78], the Pythagorean fuzzy TODIM approach [68], the multi-hesitant fuzzy linguistic TODIM approach [79,80], the interval type-2 fuzzy TODIM model [81], the intuitionistic linguistic TODIM method [82], and the 2-dimension uncertain linguistic TODIM method [83]. However, there is no scholar to investigate the TODIM model with SVNNS. Therefore, it is very necessary to pay abundant attention to this novel and worthy issue. The aim of this paper is to extend the TODIM idea to solve the MADM with the SVNNs, to fill up this vacancy. In Section 2, we give the basic concepts of SVNSs and the classical TODIM method for MADM problems. In Section 3, we propose the TODIM method for SVN MADM problems. In Section 4, we extend the proposed SVN TODIM method to INNs. In Section 5, an illustrative example is pointed out and some comparative analysis is conducted. We give a conclusion in Section 6. 2. Preliminaries Some basic concepts and definitions of NSs and SVNSs are introduced. 2.1. NSs and SVNSs Definition 1 [39,40]. Let X be a space of points (objects) with a generic element in fix set X, denoted by x. NSs A in X is characterized by a truth-membership function TA ( x ), an indeterminacy-membership I A ( x ) and a falsity-membership function FA ( x ), where TA ( x ) : X → ]− 0, 1+ [, I A ( x ) : X → ]− 0, 1+ [ and FA ( x ) : X → ]− 0, 1+ [ and 0− ≤ supTA ( x ) + supI A ( x ) + supFA ( x ) ≤ 3+ . The NSs was difficult to apply to real applications. Wang [32] develop the SNSs. Definition 2 [32]. Let X be a space of points (objects); a SVNSs A in X is characterized as the following: A = {( x, TA ( x ), I A ( x ), FA ( x ))| x ∈ X } (1) where the truth-membership function TA ( x ), indeterminacy-membership I A ( x ) and falsity-membership function FA ( x ), TA ( x ) : X → [0, 1], I A ( x ) : X → [0, 1] and FA ( x ) : X → [0, 1] , with the condition 0 ≤ TA ( x ) + I A ( x ) + FA ( x ) ≤ 3. For convenience, a SVNN can be expressed to be A = ( TA , I A , FA ), TA ∈ [0, 1], I A ∈ [0, 1], FA ∈ [0, 1], and 0 ≤ TA + I A + FA ≤ 3. Definition 3 [50]. Let A = ( TA , I A , FA ) be a SVNN, a score function S( A) is defined: S( A) = (2 + TA − I A − FA ) , S( A) ∈ [0, 1]. 3 (2) Definition 4 [50]. Let A = ( TA , I A , FA ) be a SVNN, an accuracy function H ( A) of a SVNN is defined: H ( A) = TA − FA , H ( A) ∈ [−1, 1]. (3) to evaluate the degree of accuracy of the SVNN A = ( TA , I A , FA ), where H ( A) ∈ [−1, 1] . The larger the value of H ( A) is, the higher the degree of accuracy of the SVNN A. Zhang et al. [50] gave an order relation between two SVNNs, which is defined as follows: Definition 5 [50]. Let A = ( TA , I A , FA ) and B = ( TB , IB , FB ) be two SVNNs, if S( A) < S( B), then A < B; if S( A) = S( B), then (1) (2) if H ( A) = H ( B), then A = B; if H ( A) < H ( B), then A < B. Information 2017, 8, 125 3 of 18 Definition 6 [32]. Let A and B be two SVNNs, the basic operations of SVNNs are: (1) (2) (3) (4) A ⊕ B = ( TA + TB − TA TB , I A IB , FA FB ); A ⊗ B = ( TA TB , I A + IB − I A IB , FA + FB − FA FB ); λA = 1 − (1 − TA )λ , ( I A )λ , ( FA )λ , λ > 0; ( A)λ = ( TA )λ , ( I A )λ , 1 − (1 − FA )λ , λ > 0. Definition 7 [42]. Let A and B be two SVNNs, then the normalized Hamming distance between A and B is: d( A, B) = 1 (| T − TB | + | I A − IB | + | FA − FB |) 3 A (4) 2.2. The TODIM Approach The TODIM approach [71], developed to consider the DM’s psychological behavior, can effectively solve the MADM problems. Based on the prospect theory, this approach depicts the dominance of each alternative over others by constructing a function of multi-attribute values [69]. Let G = { G1 , G2 , · · · , Gn } be the attributes, w = (w1 , w2 , · · · , wn ) be the weight of Gj , 0 ≤ w j ≤ 1, n and ∑ w j = 1. A = { A1 , A2 , · · · , Am } are alternatives. Let A = aij m×n be a decision matrix, j =1 where aij is given for the alternative Ai under the Gj , i = 1, 2, · · · , m, and j = 1, 2, · · · , n. We set w jr = w j /wr ( j, r = 1, 2, · · · , n) are relative weight of Gj to Gr , and wr = max w j | j = 1, 2, · · · , n , and 0 ≤ w jr ≤ 1. Then the traditional TODIM model concludes the following computing steps: Step 1. Normalizing A = aij m×n into B = bij m×n . Step 2. Computing the dominance degree of Ai over every alternative At under attribute Gj : n δ ( Ai , A t ) = ∑ φ j ( A i , A t ), (i, t = 1, 2, · · · , m) (5) j =1 where  s n    w jr bij − btj / ∑ w jr ,    j =1   0, φ j ( Ai , A t ) = v !  u   n u   −1t ∑ w  btj − bij /w jr ,  jr  θ j =1 i f bij − btj > 0 i f bij − btj = 0 (6) i f bij − btj < 0 and the parameter θ shows the attenuation factor of the losses. If bij − btj > 0, then φj ( Ai , At ) represents a gain; if bij − btj < 0, then φj ( Ai , At ) signifies a loss. Step 3. Deriving the overall dominance value of Ai by the Equation (7): m m ∑ δ( Ai , At ) − min ∑ δ( Ai , At ) i t =1 t =1 , i = 1, 2, · · · , m. φ ( Ai ) = m m max ∑ δ( Ai , At ) − min ∑ δ( Ai , At ) i Step 4. t =1 i (7) t =1 Ranking all alternatives and selecting the most desirable alternative in accordance with φ( Ai ). The alternative with minimum value is the worst. Inversely, the maximum value is the best one. Information 2017, 8, 125 4 of 18 3. TODIM Method for SVN MADM Problems Let A = { A1 , A2 , · · · , Am } be alternatives, and G = { G1 , G2 , · · · , Gn } be attributes. Let w = n (w1 , w2 , · · · , wn ) be the weight of attributes, where w j ∈ [0, 1], ∑ w j = 1. Suppose that R = rij m×n = j =1 rij = Tij , Iij , Fij , which is an attribute value, given by an Tij , Iij , Fij m×n be a SVN matrix, where e expert, for the alternative Ai under Gj , Tij ∈ [0, 1], Iij ∈ [0, 1], Fij ∈ [0, 1], 0 ≤ Tij + Iij + Fij ≤ 3, i = 1, 2, · · · , m, j = 1, 2, · · · , n. To solve the MADM problem with single-valued neutrosophic information, we try to present a single-valued neutrosophic TODIM model based on the prospect theory and can depict the DMs’ behaviors under risk. Firstly, we calculate the relative weight of each attribute Gj as: w jr = w j /wr , j, r = 1, 2, · · · , n. (8) where w j is the weight of the attribute of Gj , wr = max w j | j = 1, 2, · · · , n , and 0 ≤ w jr ≤ 1. Based on the Equation (8), we can derive the dominance degree of Ai over each alternative At with respect to the attribute Gj : φ j ( Ai , A t ) = s          v  u   u   1t  −  θ  n w jr d rij , rtj / ∑ w jr , j =1 n 0, ! ∑ w jr d rij , rtj /w jr , j =1 i f rij > rtj i f rij = rtj (9) i f rij < rtj 1 d rij , rtj = Tij − Ttj + Iij − Itj + Fij − Ftj . 3 (10) where the parameter θ shows the attenuation factor of the losses, and d rij , rtj is to measure the distances between the SVNNs rij and rtj by Definition 7. If rij > rtj , then φj ( Ai , At ) represents a gain; if rij < rtj , then φj ( Ai , At ) signifies a loss. For indicating functions φj ( Ai , At ) clearly, a dominance degree matrix φj = φj ( Ai , At ) m×m under Gj is expressed as: φ j = φ j ( Ai , A t ) m×m A1 A2 = . .. Am       A1 o φ j ( A1 , A2 ) 0 φ j ( A2 , A1 ) .. .. . . φ j ( A m , A1 ) φ j ( A m , A2 ) A2 ··· · · · φ j ( A1 , A m ) · · · φ j ( A2 , A m ) .. . ··· ··· 0  Am   , j = 1, 2, · · · , n .   (11) On the basis of Equation (11), the overall dominance degree δ( Ai , At ) of the Ai over each At can be calculated: n δ ( Ai , A t ) = ∑ φ j ( A i , A t ), (i, t = 1, 2, · · · , m). (12) j =1 Thus, the overall dominance degree matrix δ = [δ( Ai , At )]m×m can be derived by Equation (12): δ = [δ( Ai , At )]m×m A1 A2 = . .. Am       A1 o δ ( A2 , A1 ) .. . δ ( A m , A1 ) A2 δ ( A1 , A2 ) 0 .. . δ ( A m , A2 ) ··· Am · · · δ ( A1 , A m ) · · · δ ( A2 , A m ) .. ··· . ··· 0    .    (13) Information 2017, 8, 125 5 of 18 Then, the overall value of each Ai can be calculated Equation (14): m δ A , A − min δ A , A ( ) ( ) ∑ ∑ t t i i i t =1 t =1 , i = 1, 2, · · · , m. δ ( Ai ) = m m max ∑ δ( Ai , At ) − min ∑ δ( Ai , At ) m i i t =1 (14) t =1 Also the greater the overall value δ( Ai ), the better the alternative Ai . In general, single-valued neutrosophic TODIM model includes the computing steps: (Procedure one) Step 1. Identifying the single-valued neutrosophic matrix R = rij MADM, where rij is a SVNN. m×n = Tij , Iij , Fij m×n in the Step 2. Calculating the relative weight of Gj by using Equation (8). Step 3. Calculating the dominance degree φj ( Ai , At ) of Ai over each alternative At under attribute Gj by Equation (9). Step 4. Calculating the overall dominance degree δ( Ai , At ) of Ai over each alternative At by using Equation (12). Deriving the overall value δ( Ai ) of each alternative Ai using Equation (14). Determining the order of the alternatives in accordance with δ( Ai )(i = 1, 2, · · · , m). Step 5. Step 6. 4. TODIM Method for Interval Neutrosophic MADM Problems Furthermore, Wang et al. [33] defined INSs. e in X is Definition 8 [33]. Let X be a space of points (objects) with a generic element in fix set X, an INSs A characterized as follows: e = x, T e ( x ), I e ( x ), F e ( x ) | x ∈ X (15) A A A A where truth-membership function TAe ( x ), indeterminacy-membership I Ae ( x ) and falsity-membership function FAe ( x ) are interval values, TA ( x ) ⊆ [0, 1], I A ( x ) ⊆ [0, 1] and FA ( x ) ⊆ [0, 1], and 0 ≤ sup TAe ( x ) + sup I Ae ( x ) + sup FAe ( x ) ≤ 3. e = = TAe , I Ae , FAe An interval neutrosophic number (INN) can be expressed as A h i h i h i h h h i i i R L R L R L R R R L L L T e , T e , I e , I e , F e , F e , where T e , T e ⊆ [0, 1], I e , I e ⊆ [0, 1], F e , F e ⊆ [0, 1], A A A A A A and 0 ≤ T Re + I Re + F Re ≤ 3. A A A A A A A A A e= Definition 9 [84]. Let A represented as follows: e = S A h i h i h i T Le , T Re , I Le , I Re , F Le , F Re be an INN, a score function S of an INN can be A A A A A A 2 + T Le − I Le − F Le + 2 + T Re − I Re − F Re A A A A 6 A A e ∈ [0, 1]. ,S A (16) h i h i h i e e = be an INN, an accuracy function H A Definition 10 [84]. Let A T Le , T Re , I Le , I Re , F Le , F Re A A A A A A is defined: T Le + T Re − F Le + F Re A A A A e ∈ [−1, 1]. e = (17) ,H A H A 2 Tang [84] defined an order relation between two INNs. Information 2017, 8, 125 6 of 18 i h i h i h i i h i h h e = e = , FeL , FeR Definition 11 [84]. Let A and B T Le , T Re , I Le , I Re , F Le , F Re TeL , TeR , I eL , I R e B B B B B A A A A A A B 2+ T Le − I Le − F Le + 2+ T Re − I Re − F Re 2+ T eL − I eL − FeL + 2+ T eR − I R − FeR e B B B B B B A A A A A A e = e = be two INNs, S A and S B be the 6 T L + T R − F Le + F Re A A e = Ae Ae e = and H B scores, and H A 2 e =S B e<B e <S B e , then e , then A e ; if S A S A (1) (2) T eL + T eR − FeL + FeR B B B 2 B 6 be the accuracy function, then if e =H B e=B e , then A e; if H A e < B. e <H B e ,A e if H A e1 = T L , T R , I L , I R , F L , F R and A e2 = T L , T R , I L , I R , F L , F R Definition 12 [33,61]. Let A 2 2 2 2 2 2 1 1 1 1 1 1 be two INNs, and some basic operations on them are defined as follows: e1 ⊕ A e2 = T L + T L − T L T L , T R + T R − T R T R , I L I L , I R I R , F L F L , F R F R ; (1) A 1 2 1 2 1 2 1 2 1 1 1 1 1 1 1 1 ! L L R R L T1 T2 , T1 T2 , I1 + I1L − I1L I1L , I1R + I1R − I1R I1R , e e (2) A1 ⊗ A2 = ; F1L + F1L − F1L F1L , F1R + F1R − F1R F1R i h h i h i e1 = 1 − 1 − T L λ , 1 − 1 − T R λ , I L λ , I R λ , F L λ , F R λ , λ > 0; (3) λ A 1 1 1 1 1 1 i h i h λ h λ i λ λ λ λ λ e1 = T1L , T1R , 1 − 1 − F1L , 1 − 1 − F1R (4) , I1L , I1R , λ > 0. A e1 = T L , T R , I L , I R , F L , F R and A e2 = T L , T R , I L , I R , F L , F R Definition 13 [84]. Let A 1 1 1 1 1 1 L R2 2 L R2 2 L R2 2 e1 = T , T , I , I , F , F be two INNs, then the normalized Hamming distance between A and 1 1 1 1 1 1 L R L R L R e A2 = T2 , T2 , I2 , I2 , F2 , F2 is defined as follows: e1 , A e2 = 1 T L − T L + T R − T R + I L − I L + I R − I R + F L − F L + F R − F R d A (18) 2 2 2 2 2 2 1 1 1 1 1 1 6 e = e Let A, G and w be presented as in Section 3. Suppose that R rij m×n = i h i h i h TijL , TijR , IijL , IijR , FijL , FijR is the interval neutrosophic decision matrix, where h i h i h i m×n TijL , TijR , IijL , IijR , FijL , FijR is truth-membership function, indeterminacy-membership function and h h h i i i falsity-membership function, TijL , TijR ⊆ [0, 1], IijL , IijR ⊆ [0, 1], FijL , FijR ⊆ [0, 1], 0 ≤ TijR + IijR + FijR ≤ 3, i = 1, 2, · · · , m, j = 1, 2, · · · , n. To cope with the MADM with INNs, we develop interval neutrosophic TODIM model. Firstly, we calculate the relative weight of each attribute Gj as: w jr = w j /wr , j, r = 1, 2, · · · , n (19) where w j is the weight of the attribute of Gj , wr = max w j | j = 1, 2, · · · , n , and 0 ≤ w jr ≤ 1. Based on the Equation (20), we can derive the dominance degree of Ai over each alternative At with respect to the attribute Gj : φ j ( Ai , A t ) =          s v  u   u   1t  −   θ n w jr d e rij , e rtj / ∑ w jr , j =1 n 0, ! rij , e rtj /w jr , ∑ w jr d e j =1 if e rij > e rtj if e rij = e rtj if e rij < e rtj (20) Information 2017, 8, 125 7 of 18 1 L d e rij , e rtj = Tij − TtjL + TijR − TtjR + IijL − ItjL + IijR − ItjR + FijL − FtjL + FijR − FtjR . (21) 6 where the parameter θ shows the attenuation factor of the losses, and d e rij , e rtj is to measure the rij and e rtj by Definition 13. If e rij > e rtj , then φj ( Ai , At ) represents a gain; if distances between the INNs e e rij < e rtj , then φj ( Ai , At ) signifies a loss. For indicating functions φj ( Ai , At ) clearly, a dominance degree matrix φj = φj ( Ai , At ) m×m under Gj is expressed as: φ j = φ j ( Ai , A t ) m×m  A1 A2 = . .. Am      A1 o φ j ( A2 , A1 ) .. . φ j ( A m , A1 ) A2 φ j ( A1 , A2 ) 0 .. . φ j ( A m , A2 ) ··· Am · · · φ j ( A1 , A m ) · · · φ j ( A2 , A m ) .. ··· . ··· 0    , j = 1, 2, · · · , n    (22) On the basis of Equation (22), the overall dominance degree δ( Ai , At ) of the Ai over each At can be calculated: n δ ( Ai , A t ) = ∑ φ j ( A i , A t ), (i, t = 1, 2, · · · , m) (23) j =1 Thus, the overall dominance degree matrix δ = [δ( Ai , At )]m×m can be derived by Equation (23): δ = [δ( Ai , At )]m×m A1 A2 = . .. Am       A1 o δ ( A2 , A1 ) .. . δ ( A m , A1 ) A2 δ ( A1 , A2 ) 0 .. . δ ( A m , A2 ) ··· Am · · · δ ( A1 , A m ) · · · δ ( A2 , A m ) .. . ··· ··· 0       (24) Then, the overall value of each Ai can be calculated Equation (25): m δ ( Ai ) = m ∑ δ( Ai , At ) − min ∑ δ( Ai , At ) i t =1 , i = 1, 2, · · · , m. m m max ∑ δ( Ai , At ) − min ∑ δ( Ai , At ) t =1 i t =1 i t =1 Also the greater the overall value δ( Ai ), the better the alternative Ai . In general, interval neutrosophic TODIM model includes the computing steps: (Procedure two) h i h i h i L , T R , I L , I R , FL , FR e = erij Step 1. Identifying the interval neutrosophic matrix R = T ij ij ij ij ij ij m×n in the MADM, where erij is an INN. (25) m×n Step 2. Calculating the relative weight of Gj by using Equation (19). Step 3. Calculating the dominance degree φj ( Ai , At ) of Ai over each alternative At under attribute Gj by Equation (20). Step 4. Calculating the overall dominance degree δ( Ai , At ) of Ai over each alternative At by using Equation (23). Deriving the overall value δ( Ai ) of each alternative Ai using Equation (25). Determining the order of the alternatives in accordance with δ( Ai )(i = 1, 2, · · · , m). Step 5. Step 6. Information 2017, 8, 125 8 of 18 5. Numerical Example and Comparative Analysis 5.1. Numerical Example 1 In this part, a numerical example is given to show potential evaluation of emerging technology commercialization with SVNNs. Five possible emerging technology enterprises (ETEs) Ai (i = 1, 2, 3, 4, 5) are to be evaluated and selected. Four attributes are selected to evaluate the five possible ETEs: 1 G1 is the employment creation; 2 G2 is the development of science and technology; 3 G3 is the technical advancement; and 4 G4 is the industrialization infrastructure. The five ETEs Ai (i = 1, 2, 3, 4, 5) are to be evaluated by using the SVNNs under the above four attributes (whose weighting vector ω = (0.2, 0.1, 0.3, 0.4) T ), as listed in the following matrix. A1 A2 e = A3 R A4 A5        G1 (0.5, 0.8, 0.1) (0.7, 0.2, 0.1) (0.6, 0.7, 0.2) (0.8, 0.1, 0.3) (0.6, 0.4, 0.4) G2 (0.6, 0.3, 0.3) (0.7, 0.2, 0.2) (0.5, 0.7, 0.3) (0.6, 0.3, 0.4) (0.4, 0.8, 0.1) G3 (0.3, 0.6, 0.1) (0.7, 0.2, 0.4) (0.5, 0.3, 0.1) (0.3, 0.4, 0.2) (0.7, 0.6, 0.1) G4 (0.5, 0.7, 0.2) (0.8, 0.2, 0.1) (0.6, 0.3, 0.2) (0.5, 0.6, 0.1) (0.5, 0.8, 0.2)        Then, we use Procedure One to select the best ETE. Firstly, since w4 = max{w1 , w2 , w3 , w4 }, then G4 is the reference attribute and the reference attribute’s weight is wr = 0.4. Then, we can calculate the relative weights of the attributes Gj ( j = 1, 2, 3, 4) as w1r = 0.50, w2r = 0.25, w3r = 0.75 and w4r = 1.00. Let θ = 2.5, then the dominance degree matrix φj ( Ai , At )( j = 1, 2, 3, 4) with respect to Gj can be calculated: A1 A2 φ1 = A3 A4 A5 A1 A2 φ2 = A3 A4 A5 A1 A2 φ3 = A3 A4 A5 A1 A2 φ4 = A3 A4 A5                             A1 0.0000 0.2309 0.1414 0.2828 0.2309 A2 A3 A4 A5 −0.4619 −0.2828 −0.5657 −0.4619 0.0000 0.2160 0.1633 0.2000 −0.4320 0.0000 −0.4899 −0.3651 −0.3266 0.2449 0.0000 0.2000 −0.4000 0.1826 −0.4000 0.0000 A1 A2 A3 A4 A5 0.0000 −0.4000 0.1291 0.0577 0.1732 0.1000 0.0000 0.1633 0.1155 0.1826 −0.5164 −0.6532 0.0000 −0.5657 −0.4619 −0.2309 −0.4619 0.1414 0.0000 0.1826 −0.6928 −0.7303 0.1155 −0.7303 0.0000 A1 A2 A3 A4 A5 0.0000 −0.4422 −0.2981 −0.2309 −0.2667 0.3317 0.0000 −0.3266 0.2828 0.2646 0.2236 0.2449 0.0000 0.2000 0.2236 0.1732 −0.3771 −0.2667 0.0000 −0.3528 0.2000 −0.3528 −0.2981 0.2646 0.0000 A1 A2 A3 A4 A5 0.0000 −0.3464 −0.2582 −0.1633 0.1155 0.3464 0.0000 0.2309 0.3055 0.3651 0.2582 −0.2309 0.0000 0.2582 0.2828 0.1633 −0.3055 −0.2582 0.0000 0.2000 −0.1155 −0.3651 −0.2828 −0.2000 0.0000                             Information 2017, 8, 125 9 of 18 The overall dominance degree δ( Ai , At ) of the candidate Ai over each candidate At can be derived by Equation (13): A1 A2 δ = A3 A4 A5 A1 0.0000 1.0090 0.1068 0.3884 −0.3774        A2 A3 A4 A5  −1.6505 −0.7100 −0.9022 −0.4399 0.0000 0.2836 0.8671 1.01234    −1.0712 0.0000 −0.5974 −0.3206   −1.4711 −0.1386 0.0000 0.2298  −1.8482 −0.2828 −1.0657 0.0000 Then, we get the overall value δ( Ai )(i = 1, 2, 3, 4, 5) by using Equation (14): δ( A1 ) = 0.0000, δ( A2 ) = 1.0000, δ( A3 ) = 0.2648 δ( A4 ) = 0.3944, δ( A5 ) = 0.0187 Finally, we get order of ETEs by δ( Ai )(i = 1, 2, 3, 4, 5): A2 ≻ A4 ≻ A3 ≻ A5 ≻ A1 , and thus the most desirable ETE is A2 . 5.2. Comparative Analysis 1 In what follows, we compare our proposed method with other existing methods including the SVNWA operator and SVNWG operator proposed by Sahin [85] as follows: Definition 14 [85]. Let A j = Tj , Ij , Fj ( j = 1, 2, · · · , n) be a collection of SVNNs, w = (w1 , w2 , · · · , wn ) T n be the weight of A j ( j = 1, 2, · · · , n), and w j > 0, ∑ w j = 1 . Then j =1 ri = ( Ti , Ii , Fi ) n = SVNWAw (ri1 , ri2 , · · · , rin ) = ⊕ w j rij j =1 n = 1 − ∏ 1 − Tij j =1 w j n , ∏ Iij j =1 ri = ( Ti , Ii , Fi ) w j n , ∏ Fij j =1 n = SVNWGω (ri1 , ri2 , · · · , rin ) = ⊗ rij j =1 = n ∏ Tij j =1 w j n , 1 − ∏ 1 − Iij j =1 w j n w j (26) ! w j , 1 − ∏ 1 − Fij j =1 w j ! (27) e as well as the SVNWA and SVNWG operators, the aggregating values are By utilizing the R, derived in Table 1. Table 1. The aggregating values of the emerging technology enterprises by the SVNWA (SVNWG) operators. A1 A2 A3 A4 A5 SVNWA SVNWG (0.4591, 0.6307, 0.1473) (0.7449, 0.2000, 0.1625) (0.5627, 0.3868, 0.1692) (0.5497, 0.3464, 0.1762) (0.5822, 0.6389, 0.1741) (0.4369, 0.6718, 0.1627) (0.7384, 0.2000, 0.2124) (0.5578, 0.4571, 0.1822) (0.4799, 0.4381, 0.2067) (0.5610, 0.6933, 0.2083) Information 2017, 8, 125 10 of 18 According to the aggregating results in Table 1, the score functions are listed in Table 2. Table 2. The score functions of the emerging technology enterprises. A1 A2 A3 A4 A5 SVNWA SVNWG 0.5604 0.7942 0.6689 0.6757 0.5898 0.5341 0.7753 0.6398 0.6117 0.5531 According to the score functions shown in Table 2, the order of the emerging technology enterprises are in Table 3. Table 3. Order of the emerging technology enterprises. Order SVNWA SVNWG A2 > A4 > A3 > A5 > A1 A2 > A3 > A4 > A5 > A1 From the above analysis, it can be seen that two operators have the same best emerging technology enterprise A2 and two methods’ ranking results are slightly different. However, the SVN TODIM approach can reasonably depict the DMs’ psychological behaviors under risk, and thus, it may deal with the above issue effectively. This verifies the method we proposed is reasonable and effective in this paper. 5.3. Numerical Example 2 If the five possible emerging technology enterprises Ai (i = 1, 2, 3, 4, 5) are to be evaluated by using the INNS under the above four attributes (whose weighting vector ω = (0.2, 0.1, 0.3, 0.4) T ), e then: as listed in the matrix R,  ([0.5, 0.6], [0.8, 0.9], [0.1, 0.2]) ([0.6, 0.7], [0.3, 0.4], [0.3, 0.4])  ([0.7, 0.9], [0.2, 0.3], [0.1, 0.2]) ([0.7, 0.8], [0.1, 0.2], [0.2, 0.3])  e= R  ([0.6, 0.7], [0.7, 0.8], [0.2, 0.3]) ([0.5, 0.6], [0.7, 0.8], [0.3, 0.4])   ([0.8, 0.9], [0.1, 0.2], [0.3, 0.4]) ([0.6, 0.7], [0.3, 0.4], [0.4, 0.5]) ([0.6, 0.7], [0.4, 0.5], [0.4, 0.5]) ([0.4, 0.5], [0.8, 0.9], [0.1, 0.2])  ([0.3, 0.4], [0.6, 0.7], [0.1, 0.2]) ([0.5, 0.6], [0.7, 0.8], [0.1, 0.2]) ([0.7, 0.9], [0.2, 0.3], [0.4, 0.5]) ([0.8, 0.9], [0.2, 0.3], [0.1, 0.2])    ([0.5, 0.6], [0.3, 0.4], [0.1, 0.2]) ([0.6, 0.7], [0.3, 0.4], [0.2, 0.3])   ([0.3, 0.4], [0.4, 0.5], [0.2, 0.3]) ([0.5, 0.6], [0.6, 0.7], [0.1, 0.2])  ([0.7, 0.8], [0.6, 0.7], [0.1, 0.2]) ([0.5, 0.6], [0.8, 0.9], [0.2, 0.3]) Then, we use Procedure Two to select the best ETE. Firstly, since w4 = max{w1 , w2 , w3 , w4 }, then G4 is the reference attribute and the reference attribute’s weight is wr = 0.4. Then, we can calculate the relative weights of the attributes Information 2017, 8, 125 11 of 18 Gj ( j = 1, 2, 3, 4) as: w1r = 0.50, w2r = 0.25, w3r = 0.75 and w4r = 1.00. Let θ = 2.5, then the dominance degree matrix φj ( Ai , At )( j = 1, 2, 3, 4) with respect to Gj can be calculated: A1 A2 φ1 = A3 A4 A5 A1 A2 φ2 = A3 A4 A5 A1 A2 φ3 = A3 A4 A5 A1 A2 φ4 = A3 A4 A5                             A1 0.0000 0.2380 0.1414 0.2828 0.2309 A2 A3 A4 A5 −0.4761 −0.2828 −0.5657 −0.4619 0.0000 0.2236 0.1528 0.2082 −0.4472 0.0000 −0.4899 −0.3651 −0.3055 0.2449 0.0000 0.2000 −0.4163 0.1826 −0.4000 0.0000 A1 A2 A3 A4 A5 0.0000 −0.4619 0.1291 0.0577 0.1732 0.1155 0.0000 0.1732 0.1291 0.1915 −0.5164 −0.6928 0.0000 −0.5657 −0.4619 −0.2309 −0.5164 0.1414 0.0000 0.1826 −0.6928 −0.7659 0.1155 −0.7303 0.0000 A1 A2 A3 A4 A5 0.0000 −0.4522 −0.2981 −0.2309 −0.2667 0.3391 0.0000 0.2550 0.2915 0.2739 0.2236 −0.3399 0.0000 0.2000 0.2236 0.1732 −0.3887 −0.2667 0.0000 −0.3528 0.2000 −0.3651 −0.2981 0.2646 0.0000 A1 A2 A3 A4 A5 0.0000 −0.3266 −0.2828 −0.1155 0.1633 0.3266 0.0000 0.2309 0.3055 0.3651 0.2828 −0.2309 0.0000 0.2582 0.2828 0.1155 −0.3055 −0.2582 0.0000 0.2000 −0.1633 −0.3651 −0.2828 −0.2000 0.0000                             The overall dominance degree δ( Ai , At ) of the candidate Ai over each candidate At can be derived by Equation (24): A1 A2 δ = A3 A4 A5        A1 0.0000 1.0192 0.1314 0.3406 −0.4252 A2 −1.7168 0.0000 −1.0310 −1.5161 −1.9124 A3 −0.7346 0.3727 0.0000 −0.1386 −0.8654 A4 A5  −0.7506 0.0698 0.3513 0.8305    −0.4726 0.0445   0.2000 0.0298  −0.6657 0.0000 Then, we get the overall value δ( Ai )(i = 1, 2, 3, 4, 5) by using Equation (25): δ( A1 ) = 0.1143, δ( A2 ) = 1.0000, δ( A3 ) = 0.3944 δ( A4 ) = 0.4322, δ( A5 ) = 0.0000 Finally, we get order of ETEs by δ( Ai )(i = 1, 2, 3, 4, 5): A2 ≻ A4 ≻ A3 ≻ A1 ≻ A5 , and thus the most desirable ETE is A2 . 5.4. Comparative Analysis 2 In what follows, we compare our proposed method with other existing methods including the INWA operator and INWG operator proposed by Zhang et al. [50] as follows: Information 2017, 8, 125 12 of 18 ej = Definition 15 [50]. Let A h i h i h i TjL , TjR , IjL , IjR , FjL , FjR ( j = 1, 2, · · · , n) be a collection of INNs, n w = (w1 , w2 , · · · , wn ) T be the weight of A j ( j = 1, 2, · · · , n), and w j > 0, ∑ w j = 1 . Then e ri = j =1 TiL , TiR , IiL , IiR , FiL , FiR n = INWAw (eri1 , eri2 , · · · , erin ) = ⊕ w jerij j =1  " # w j w j n n L R , , 1 − ∏ 1 − Tij  1 − ∏ 1 − Tij  j =1 j =1  # " # = " w j n w j n n w j n w j  L R L R , ∏ Fij , ∏ Iij , ∏ Fij ∏ Iij j =1 e ri = j =1 j =1 j =1 TiL , TiR , IiL , IiR , FiL , FiR  w n = INWGw (eri1 , eri2 , · · · , erin ) = ⊗ erij j  " #j="1 # w j w j n w j w j n n n , 1 − ∏ 1 − IijL , ∏ TijR , , 1 − ∏ 1 − IijR  ∏ TijL  j =1 j = 1 j = 1 j =1 " # =  w j w j n n  1 − ∏ 1 − FijL , 1 − ∏ 1 − FijR j =1 (28)      j =1       (29) e and the INWA and INWG operators, the aggregating values By utilizing the decision matrix R, are in Table 4. Table 4. The aggregating values of the emerging technology enterprises by the INWA and INWG operators. INWA A1 ([0.4591, 0.5611], [0.6307, 0.7342], [0.1116, 0.2144]) A2 ([0.7449, 0.8928], [0.1866, 0.2881], [0.1625, 0.2742]) A3 ([0.5627, 0.6634], [0.3868, 0.4925], [0.1692, 0.2734]) A4 ([0.5497, 0.6674], [0.3464, 0.4657], [0.1762, 0.2844]) A5 ([0.5822, 0.6863], [0.6389, 0.7421], [0.1741, 0.2825]) INWG A1 A2 A3 A4 A5 ([0.4369, 0.5395], [0.6718, 0.7805], [0.1223, 0.2227]) ([0.7384, 0.8895], [0.1905, 0.2906], [0.2124, 0.3144]) ([0.5578, 0.6581], [0.4571, 0.5685], [0.1822, 0.2825]) ([0.4799, 0.5851], [0.4381, 0.5440], [0.2067, 0.3077]) ([0.5610, 0.6624], [0.6933, 0.8082], [0.2083, 0.3097]) According to the aggregating values in Table 4, the score functions are in Table 5. Table 5. The score functions of the emerging technology enterprises. A1 A2 A3 A4 A5 INWA INWG 0.5549 0.7877 0.6507 0.6574 0.5718 0.5298 0.7700 0.6209 0.5948 0.5340 Information 2017, 8, 125 13 of 18 According to the score functions shown in Table 5, the order of the emerging technology enterprises are in Table 6. Table 6. Order of the emerging technology enterprises. Ordering INWA INWG A2 > A4 > A3 > A5 > A1 A2 > A3 > A4 > A5 > A1 From the above analysis, it can be seen that two operators have the same best emerging technology enterprise A2 and two methods’ ranking results are slightly different. However, the interval neutrosophic TODIM approach can reasonably depict the DMs’ psychological behaviors under risk, and thus, it may deal with the above issue effectively. This verifies the method we proposed is reasonable and effective. 6. Conclusions In this paper, we will extend the TODIM method to the MADM with the single-valued neutrosophic numbers (SVNNs). Firstly, the definition, comparison and distance of SVNNs are briefly presented, and the steps of the classical TODIM method for MADM problems are introduced. Then, the extended classical TODIM method is proposed to deal with MADM problems with the SVNNs, and its significant characteristic is that it can fully consider the decision makers’ bounded rationality which is a real action in decision making. Furthermore, we extend the proposed model to interval neutrosophic sets (INSs). Finally, a numerical example is proposed to verify the developed approach. In the future, the application of the proposed models and methods of SVNSs and INSs needs to be explored in the decision making [86–99], risk analysis and many other uncertain and fuzzy environment [100–112]. Acknowledgments: The work was supported by the National Natural Science Foundation of China under Grant No. 71571128 and the Humanities and Social Sciences Foundation of Ministry of Education of the People’s Republic of China (17XJA630003) and the Construction Plan of Scientific Research Innovation Team for Colleges and Universities in Sichuan Province (15TD0004). 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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). information Article Neutrosophic Similarity Score Based Weighted Histogram for Robust Mean-Shift Tracking Keli Hu 1, * 1 2 3 * ID , En Fan 1 ID , Jun Ye 2 ID , Changxing Fan 1 , Shigen Shen 1 ID and Yuzhang Gu 3 Department of Computer Science and Engineering, Shaoxing University, Shaoxing 312000, China; fan_en@hotmail.com (E.F.); fcxjszj@usx.edu.cn (C.F.); shigens@126.com (S.S.) Department of Electrical and Information Engineering, Shaoxing University, Shaoxing 312000, China; yehjun@aliyun.com Key Laboratory of Wireless Sensor Network & Communication, Shanghai Institute of Microsystem and Information Technology, Chinese Academy of Sciences, Shanghai 200050, China; gyz@mail.sim.ac.cn Correspondence: ancimoon@gmail.com; Tel.: +86-575-8834-1512 Received: 25 August 2017; Accepted: 30 September 2017; Published: 2 October 2017 Abstract: Visual object tracking is a critical task in computer vision. Challenging things always exist when an object needs to be tracked. For instance, background clutter is one of the most challenging problems. The mean-shift tracker is quite popular because of its efficiency and performance in a range of conditions. However, the challenge of background clutter also disturbs its performance. In this article, we propose a novel weighted histogram based on neutrosophic similarity score to help the mean-shift tracker discriminate the target from the background. Neutrosophic set (NS) is a new branch of philosophy for dealing with incomplete, indeterminate, and inconsistent information. In this paper, we utilize the single valued neutrosophic set (SVNS), which is a subclass of NS to improve the mean-shift tracker. First, two kinds of criteria are considered as the object feature similarity and the background feature similarity, and each bin of the weight histogram is represented in the SVNS domain via three membership functions T(Truth), I(indeterminacy), and F(Falsity). Second, the neutrosophic similarity score function is introduced to fuse those two criteria and to build the final weight histogram. Finally, a novel neutrosophic weighted mean-shift tracker is proposed. The proposed tracker is compared with several mean-shift based trackers on a dataset of 61 public sequences. The results revealed that our method outperforms other trackers, especially when confronting background clutter. Keywords: tracking; mean-shift; neutrosophic set; single valued neutrosophic set; neutrosophic similarity score 1. Introduction Currently, applications in the computer vision field such as surveillance, video indexing, traffic monitoring, and auto-driving have come into our life. However, most of the key algorithms still lack the performance of those applications. One of the most important tasks is visual object tracking, and it is still a challenging problem [1–3]. Challenges like illumination variation, scale variation, motion blur, background clutters, etc. may happen when dealing with the task of visual object tracking [2]. A specific classifier is always considered for tackling such kinds of challenging problems. Boosting [4] and semi-supervised boosting [5] were employed for building a robust classifier; multiple instance learning [6] was introduced into the classifier training procedure due to the interference of the inexact training instance; compressive sensing theory [7] was applied for developing effective and efficient appearance models for robust object tracking, due to factors such as pose variation, illumination change, occlusion, and motion blur. Information 2017, 8, 122; doi:10.3390/info8040122 www.mdpi.com/journal/information Information 2017, 8, 122 2 of 14 The mean-shift procedure was first introduced into visual object tracking by Comaniciu et al. [8,9]. The color histogram was employed as the tracking feature. The location of the target in each frame was decided by minimizing the distance between two probability density functions, which are represented by a target histogram and a target candidate histogram. By utilizing the color histogram feature and the efficient seeking method, such a mean-shift tracker demonstrates high efficiency and good performance, even when confronting motion blur and deformation problems. On the other hand, the color histogram feature cannot help the tracker discriminate the target from the background effectively, especially when background clutter exists. Several new metrics or features were considered to deal with such a problem. For instance, Cross-Bin metric [10], SIFT (Scale-invariant feature transform) [11], and texture feature [12] were introduced into the mean shift based tracker, and the proposed trackers all earn a better performance than the traditional one. Besides, Tomas et al. [13] exploited the background to discriminate the target and proposed the background ratio weighting method. In addition, since estimating an adequate scale is essential for robust tracking, a more robust method for estimating the scale of the searching bounding box was proposed through the forward–backward consistency check. This mean-shift based tracker [13] outperforms several state-of-the-art algorithms. Robert et al. [14] also proposed a scale selecting scheme by utilizing the Lindeberg’s theory [15] based on the local maxima of differential scale-space filters. Although so many kinds of visual trackers have been proposed, the visual tracking is still an open problem, due to the challenging conditions in the real tracking tasks. All in all, the mean-shift tracker demonstrates high efficiency and may earn an even better performance if a more effective method can be found to discriminate the target from the background. Thus, finding a suitable way to represent the information presented by the background, as well as the target, is of high relevance. Neutrosophic set (NS) [16] is a new branch of philosophy to deal with the origin, nature, and scope of neutralities. It has an inherent ability to handle the indeterminate information like the noise included in images [17–21] and video sequences. Until now, NS has been successfully applied in many areas [22]. For the computer vision research fields, the NS theory is widely utilized in image segmentation [17–21], skeleton extraction [23] and object tracking [24], etc. Before calculating the segmentation result for an image, a specific neutrosophic image was usually computed via several criteria in NS domain [17–21]. For object tracking, in order to improve the traditional color based CAMShift tracker, the single valued neutrosophic cross-entropy was employed for fusing color and depth information [24]. In addition, the NS theory is also utilized for improving the clustering algorithms, such as c-means [25]. While several criteria are always proposed to handle a specific image processing problem, an appropriate way for fusing information is needed. Decision-making [26–30] can be regarded as a problem-solving activity terminated by a solution deemed to be satisfactory, and it has been frequently employed for dealing with such an information fusion problem. The similarity measurement [30] using the correlation coefficient under single valued neutrosophic environment was successfully applied into the issue of image thresholding [21]. A single valued neutrosophic set (SVNS) [31] is an instance of a neutrosophic set and provides an additional possibility to represent uncertainty, imprecise, incomplete, and inconsistent information, which exists in the real world. The correlation coefficient of SVNS was proposed by the authors of [30] and was successfully applied for handling the multicriteria decision making problem. For the mean-shift tracker, the color histogram is employed for representing the tracked target. Due to the challenging conditions during the tracking procedure, indeterminate information always exists. For instance, object feature may changes due to object pose or external environment changes between frames. It is difficult to localize the object exactly during the tracking procedure. Thus, there exists indeterminate information when you try to utilize the uncertain bounding box to extract object feature. All in all, how to utilize the information of the object and the corresponding background to help the tracker discriminate the object is also an indeterminate problem. In this work, we propose a novel mean-shift tracker based on the neutrosophic similarity score [21,30] under the SVNS environment. We build a neutrosophic weight histogram, which jointly Information 2017, 8, 122 3 of 14 considered the indeterminate attributes of the object and the background information. First, we propose two criteria of the object feature similarity and the background feature similarity, where each one is represented as its bin of the histogram corresponding to three membership functions for the T(Truth), I(indeterminacy), and F(Falsity) element of the neutrosophic set. Second, the neutrosophic similarity score function is introduced to fuse those two criteria and build the final weighted histogram. Finally, the weight of each bin of the histogram is applied to modify the traditional mean-shift tracker, and a novel neutrosophic weighted mean-shift tracker is proposed. To our own knowledge, it is the first time to introduce the NS theory into the mean-shift procedure. Experiments results revealed that the proposed neutrosophic weighted mean-shift tracker outperforms several kinds of mean-shift based trackers [9,13,14]. The remainder of this paper is organized as follows: in Section 2, the traditional mean-shift procedure for visual object tracking and the definition of the neutrosophic similarity score are first given. Then the details of the method for calculating the neutrosophic weight histogram are presented, and the main steps of the proposed mean-shift tracker are illustrated in the following subsection. Experimental evaluations and discussions are presented in Section 3, and Section 4 has the conclusions. 2. Problem Formulation In this section, we present the algorithmic details of this paper. For the visual tracking problem, the initial location of the target will be given in the first frame, and the location is always represented by a rectangle bounding box [1–3]. Then the critical task for a visual tracker is to calculate the displacement of the bounding box in the following frame corresponding to the previous one. 2.1. Traditional Mean-Shift Tracker The main steps of the traditional mean-shift visual tracker are summarized in this subsection. The kernel-based histogram is employed by the traditional mean-shift tracker. At the beginning, the feature model of the target is calculated by n q̂u = C ∑ k kxi∗ k2 δ[b(xi∗ ) − u] (1) i =1 m where q̂ is the target model, q̂ = {q̂u }u=1...m ; q̂u is the u-th bin of the target model satisfying ∑ q̂u = 1; u =1 xi∗ is the normalized pixel location which located in the initial bounding box; and n is the number of pixels belonging to the target. In order to reduce the interference of the background clutters, the kernel k(x) is utilized. k(x) is an isotropic, convex, and monotonic decreasing kernel. The ( kernel assigns smaller 2 π (1 − x ) i f x < 1 . weights to pixels farther than the center. In this work, k(x) is defined as k ( x ) = 0 else The function b(x): R2 → 1 . . . m associates to the pixel at location x the index b(x) of the histogram bin corresponding to the color of that pixel. Then, C is the normalization constant, which is denoted by C= ∑in=1 1 k kxi∗ k2 (2) The function δ( x ) is the Kronecker delta function. Let y be the center of the target candidate and {xi }i = 1, . . . , nh be the pixel locations in the bounding box of the target candidate. Here, nh is the total number of the pixels falling in the bounding box. Then when using the same kernel profile k(x), the probability of the feature in the target candidate is given by nh y − xi 2 k δ[b(xi − u)] p̂u = Ch ∑ k k h i =1 (3) Information 2017, 8, 122 4 of 14 where h is the bandwidth and Ch is the normalization constant derived by imposing the condition m ∑ p̂u = 1. u =1 The metric based on Bhattacharyya coefficient is proposed to evaluate the similarity between the probability distributions of the target and the candidate target. Let ρ[p(ŷ), q̂] be the similarity probability, then it can be calculated by m ρ[p(ŷ), q̂] = ∑ u =1 q p̂(y)q̂u (4) For the mean-shift tracker, the location ŷ0 in the previous frame is employed as the starting location for searching the new target location in the current frame. The estimate of a new target location is then obtained by maximizing the Bhattacharyya coefficient ρ[p(ŷ), q̂] using a Taylor series expansion, see [8,9] for further details. To reach the maximum of the Bhattacharyya coefficient, the kernel is repeatedly moved from the current location ŷ0 to the new location ŷ1 = n ∑i=h 1 xi wi g n ∑i=h 1 wi g xi 2 k ŷ0 − h k xi 2 k ŷ0 − h k (5) where g(x) is the negative derivative of the kernel k(x), i.e., g( x ) = −k′ ( x ). Furthermore, it is assumed that g(x) exists for all x ∈ [0, ∞) except for a finite set of points. The parameter wi in Equation (5) is denoted by s m wi = ∑ δ [ b ( xi ) − u ] u =1 q̂u p̂u (ŷ0 ) (6) 2.2. Neutrosophic Similarity Score A neutrosophic set with multiple criteria can be expressed as follows: Let A = {A1 , A2 , . . . , Am } be a set of alternatives and C = {C1 , C2 , . . . , Cn } be a set of criteria. Then the character of the alternative Ai (i = 1, 2, . . . , m) can be represented by the following information: Ai = nD Cj , TCj ( Ai ), ICj ( Ai ), FCj ( Ai ) E o Cj ∈ C i = 1 . . . m, j = 1 . . . n (7) where TCj ( Ai ), ICj ( Ai ), FCj ( Ai ) ∈ [0, 1]. Here, TCj ( Ai ) denotes the degree to which the alternative Ai satisfies the criterion Cj ; ICj ( Ai ) indicates the indeterminacy degree to which the alternative Ai satisfies or does not satisfy the criterion Cj ; FCj ( Ai ) indicates the degree to which the alternative Ai does not satisfy the criterion Cj . A method for multicriteria decision-making based on the correlation coefficient under single-valued neutrosophic environment is proposed in [30]. The similarity degree between two elements Ai and Aj is defined as: SCk ( Ai , A j ) = q TCk ( Ai ) TCk ( A j ) + ICk ( Ai ) ICk ( A j ) + FCk ( Ai ) FCk ( A j ) q TCk 2 ( Ai ) + ICk 2 ( Ai ) + FCk 2 ( Ai ) TCk 2 ( A j ) + ICk 2 ( A j ) + FCk 2 ( A j ) (8) Information 2017, 8, 122  Assume the ideal alternative A∗  =  nD  Cj , TCj ( A∗ ), ICj ( A∗ ), FCj ( A∗ ) E 5 of 14  Cj ∈ C  o i = 1 . . . m, j = 1 . . . n. Then the similarity degree between any alternative Ai and the ideal alternative A* can be   calculated by   I ( A ) I ( A∗ ) + F ( A ) F ( A∗ ) TCk ( Ai )TCk ( A∗ ) + Ck Ck i Ck i Ck q (9) 2 2 ∗ 2 2 TCk ( Ai ) + ICk ( Ai ) + FCk ( Ai ) TCk ( A ) + ICk 2 ( A∗ ) + FCk 2 ( A∗ ) ∈   n Suppose wk ∈ [0,1] is the weight of each criterion Ck and ∑ j=1 w j = 1, then the weighted correlation SCk ( Ai , A∗ ) = q coefficient between an alternative Ai and the ideal alternative A* is defined by W ( Ai , A ∗ ) = TCk ( Ai ) TCk ( A∗ ) + ICk ( Ai ) ICk ( A∗ ) + FCk ( Ai ) FCk ( A∗ )   q q w ∑ k  2 TCk ( Ai ) + ICk 2( Ai ) + FCk 2 ( Ai ) TCk 2 ( A∗ ) +ICk 2 ( A∗ )+ FCk 2 ( A∗ ) k =1 n (10) The alternative with high correlation coefficient is considered to be a good choice for the current decision. 2.3. Calculate the Neutrosophic Weight Histogram Employing the information discriminated from the background is one of the most important issues for robustly tacking a visual object. As shown in Figure 1, the smallest region GO inside the red bounding box is the object region and this region corresponds to the location of the object in the corresponding frame. Then GO is decided by the tracker and its accuracy depends on the robustness of the tracker. In this work, the surrounding area of GO is defined as the background region GB . In order to eliminate the indeterminacy of the region GO to some extent, the region far from GO is employed as α GB and GB = βGO − αGO .β Figure 1. Illustration of the object region. To enhance the robustness of the traditional mean-shift tracker, a novel weight histogram wNS is defined in the neutrosophic domain. Each bin of the weighted histogram wNS is expressed in the SVNS domain via three membership functions T(Truth), I(indeterminacy), and F(Falsity). For the proposition of object feature is a discriminative feature, TCO , ICO , and FCO represent the probabilities when a proposition is true, indeterminate and false degrees, respectively. Finding the location of the tracked object in a new frame is the main task for a tracker, and the target model (object feature histogram in the initial frame) is frequently employed as major information to discriminate the object from the background. The region which owns more similarity to the object feature is more likely to be the object region. Using the object feature similarity criterion, we can further give the definitions: TCO (u) = q̂u ˆ ICO (u) = |q̂u − q̂u (t − 1)| (11) (12) Information 2017, 8, 122 6 of 14 FCO (u) = 1 − TCO (u) (13) where q̂u is the u-th bin of the target model corresponding to the object region GO in the first frame of the tracking process and it is calculated by using Equation (1). The indeterminacy degree ICO (u) is defined in Equation (12). Then, q̂u (t − 1) is the u-th bin of the updated object feature histogram in the previous frame. Suppose p̂u (t − 1) is the feature histogram corresponding to the extracted object region at time t−1, then q̂u (t − 1) is calculated by q̂u (t − 1) = (1 − λ)q̂u (t − 2) + λ p̂u (t − 1) (14) where λ is the updating rate for λ ∈ (0,1). As the tracker may drift from the object due to the similar surroundings, using the object features with high similarity to the background will bring risk to the accuracy of the tracker. The background feature similarity criterion is considered in this work. The corresponding three membership functions TCB , ICB and FCB are defined as follows: TCB (u) = q̂u (15)   i f b̂u = 0  0 ICB (u) = (16) 1 i f b̂u > q̂u   b̂ /q̂ else u u FCB (u) = b̂u (17) where b̂u is the u-th bin of the object background feature histogram. This histogram is initialized in the background region GB in the first frame, as shown in Figure 1. For q̂u , Equation (1) is also employed to calculate b̂u , and b̂u , which will be updated when the surroundings of the tracked target change dramatically. By substituting the corresponding T(Truth), I(indeterminacy), and F(Falsity) under the criteria of the object feature similarity and the background feature similarity into Equation (10), the u-th bin of the neutrosophic weight histogram can be calculated by wuNS = wCO SCO (u, A∗ ) + wCB SCB (u, A∗ ) = wCO √ TCO (u) TC ( A∗ )+ ICO (u) IC ( A∗ )+ FCO (u) FC ( A∗ ) k k q k TCO 2 (u)+ ICO 2 (u)+ FCO 2 (u) TC 2 ( A∗ )+ IC 2 ( A∗ )+ FC 2 ( A∗ ) k k k (18) TCB (u) TC ( A∗ )+ ICB (u) IC ( A∗ )+ FCB (u) FC ( A∗ ) k k q k +wCB √ 2 TCB (u)+ ICB 2 (u)+ FCB 2 (u) TC 2 ( A∗ )+ IC 2 ( A∗ )+ FC 2 ( A∗ ) k k k where wCO , wCB ∈ [0,1] are the corresponding weights of criteria and wCO + wCB = 1. The ideal alternative under two criteria is the same as A∗ = h1, 0, 0i. 2.4. Neutrosophic Weighted Mean-Shift Tracker In this work, the neutrosophic weighted histogram is introduced into the traditional mean-shift procedure, and this improved mean-shift tracker is called the neutrosophic weighted mean-shift tracker. The basic flow of the proposed tracker is described below: Initialization Step 1: Read the first frame and select an object on the image plane as the target to be tracked. Step 2: Calculate the object feature histogram q̂ and object background feature histogram b̂ by using Equation (1). Tracking Input: (t + 1)-th video frame Step 3: Employ the location ŷ0 in the previous frame as the starting location for searching the new target location in the current frame. Information 2017, 8, 122 Step 4: 7 of 14 Based on the mean-shift algorithm and neutrosophic weight histogram, derive the new location of the object according to Equation (19) and Equation (5) as follows: m wi = ∑ δ [ b ( xi ) − u ] u =1 Step 5: Step 6: s wuNS q̂u p̂u (ŷ0 ) (19) If kŷ1 − ŷ0 k < ε 0 , stop. Otherwise, set ŷ0 ← ŷ1 and go to Step 4. Derive q̂u (t − 1) according to Equation (14) and then update object background feature h i histogram b̂ ← b̂c when the Bhattacharyya coefficient ρ b̂, b̂c < ε 1 , where b̂c is the corresponding feature histogram in the current background region GB . Output: Tracking location. 3. Experiment Results and Analysis We tested the neutrosophic weighted mean-shift tracker on a challenging benchmark [2]. As mentioned at the outset, background clutter is one of the most challenging problems for the mean-shift tracker. Besides the 50 challenging sequences in this benchmark [2], another 10 sequences with the challenge of background clutter are also selected as testing sequences. The information of those 10 sequences is given in Table 1. The abbreviations of several kinds of challenges included in the testing sequences are shown in the footer of Table 1. Table 1. An overview of another 10 sequences. Sequence Target Challenges Frames Board Bolt2 Box ClifBar Coupon Crowds Car2 Car1 Human3 Car24 board human box book coupon human car car human car SV, MB, FM, OPR, OV, BC DEF, BC IV, SV, OCC, MB, IPR, OPR, OV, BC, LR SV, OCC, MB, FM, IPR, OV, BC OCC, BC IV, DEF, BC IV, SV, MB, FM, BC IV, SV, MB, FM, BC, LR SV, OCC, DEF, OPR, BC IV, SV, BC 698 293 1161 472 327 347 913 1020 1698 3059 Note: IV: Illumination Variation, SV: Scale Variation, OCC: Occlusion, DEF: Deformation, MB: Motion Blur, FM: Fast Motion, IPR: In-Plane Rotation, OPR: Out-of-Plane Rotation, OV: Out-of-View, BC: Background Clutters, and LR: Low Resolution. To gauge the performance of the proposed tracker, we compare our results to another three mean-shift based trackers including ASMS [13], KMS [9] and SMS [14]. Some experimental results have shown that ASMS [13] outperforms several state-of-the-art algorithms. KMS is the traditional mean-shift tracker. Both SMS and ASMS are scale adaptive. All of the tested algorithms employ the color histogram as object feature. 3.1. Setting Parameters For the proposed neutrosophic weighted mean-shift tracker, the parameter α and β relate to the background region GB are set to 1.2 and 1.48 respectively. The parameter λ in Equation (14) decides the updating rate of the object feature histogram. With the assumption that the appearance of the tracked object will not change dramatically, a low updating rate should be given. In this work, λ is set to 0.05. As seen in the Section 2.4, the accuracy of the result of the mean-shift procedure depends on the parameter ε0 to some extent, where ε0 is set to 0.1. A much greater value of ε0 may lead to failure. The parameter ε1 is a threshold for updating the object background feature histogram. During the Information 2017, 8, 122 8 of 14 tracking procedure, the surroundings of the object always change. Hence, it is essential to update the object background feature histogram when the similarity between the current surroundings and the object background feature falls to a specific value. If ε1 is set to 0, the updating process of the background feature will stop. If ε1 is set to 1, the updating frequency will be too high. Thus, a medium value is chosen as ε1 = 0.5. The neutrosophic weight histogram plays an essential role in this proposed mean-shift based tracker. In order to emphasize the background information when constructing the neutrosophic weight histogram, the corresponding parameter wCB should be set to a relatively high value. However, if this value is set too high, the effect of the first neutrosophic criteria will reduce, even to zero. In this work, wCB is set to 0.6, and wCO is set to 0.4. Finally, all the values of these parameters are chosen by hand-tuning, and all of them are constant for all experiments. 3.2. Evaluation Criteria The overlap rate of the bounding box is used as the evaluation criterion, and the overlap rate is defined as area ROITi ∩ ROIGi si = (20) area ROITi ∪ ROIGi where ROITi is the target bounding box in the i-th frame and ROIGi is the corresponding ground truth bounding box. For the video datasets applied in this work, the ground truth bounding boxes of the tracked target are manually labeled for each frame. The success ratio is defined as: R= ∑ N i =1 ui /N, ui = ( 1 0 i f si > r otherwise (21) where N is the number of frames and r is the overlap threshold which decides the corresponding tracking result is correct or not. The success ratio is R ∈ [0,1]. When the overlap ratio si is greater than r on each frame, R achieves the maximum, and then this means the corresponding tracker performs very well in this sequence. On the contrary, R achieves the minimum when si is smaller than r on each frame, and then this means the corresponding tracker performs the worst. Both the one-pass evaluation (OPE) and temporal robustness evaluation (TRE) are employed as the evaluation metric. For the TRE, each testing sequence is partitioned into 20 segments, and each tracker is tested throughout all of the segments. The results for the OPE evaluation metric are derived by testing the tracker with one time initialization from the ground truth position in the first frame of each testing sequence. Finally, we use the area under curve (AUC) of each success plot to rank the tracking algorithms. For each success plot, the tracker with a greater value of AUC ranks better. 3.3. Tracking Results Several screen captures for some of the testing sequences are given in Figures 2–5. Success plots of TRE and OPE for the whole testing sequences are shown in Figures 6a and 7a, and the success plots for those sequences including background clutter challenge are shown in Figures 6b and 7b. In the following section, a more detailed discussion of the tracking results is documented. Information 2017, 8, 122 9 of 14 Figure 2. Screenshots of tracking results of the video sequence used for testing (mountainBike, target is selected in frame #1). Figure 3. Screenshots of tracking results of the video sequence used for testing (Box, target is selected in frame #1). MountainBike sequence: This sequence highlights the challenges of BC, IPR and OPR. As shown in Figure 2, an improper scale of the bounding box is estimated by the SMS tracker, and the SMS tracker has failed in frame #26. The ASMS tracker, as can be seen in frame #32, has drifted from the tracking target because of the similar color of the surroundings, although an appropriate scale is given. During the first half of the tracking process, both of the KMS and our NEUTMS perform well. However, compared to the NEUTMS, the KMS tracker sometimes drifts a little farer from the biker, as seen in frame #38. When the challenge of background clutter appears, the KMS tracker may also drift from the right location of the target, as seen in frame #178. During the whole tracking process, the NEUTMS tracker performs the best result. Information 2017, 8, 122 10 of 14 Figure 4. Screenshots of tracking results of the video sequence used for testing (Football, target is selected in frame #1). Box sequence: The challenges included in this sequence can be found in Table 1. This sequence is more challenging than the MountainBike sequence. As seen in frame #31 in Figure 3, all the trackers except for the SMS tracker can give a right location of the tracked box, and the ASMS performs the best result so far. Due to the black background upon the box, the SMS tracker fails soon. While the box is passing by the circuit board on the table, both the ASMS and the KMS tracker begin to lose the box. By employing the information of the background region, our NEUTMS tracker has successfully overcome the challenges like BC and MB during this sequence. Football sequence: Challenges of BC, OCC, IPR and OPR are presented in this sequence. As shown in Figure 4, the SMS tracker has already failed in frame #10. The ASMS and KMS trackers fail when the tracked player getting close to another player on account of the factor of all the players wear the same helmet. However, the NEUTMS tracker performs well even the tracked player runs through some players with similar feature. Figure 5. Screenshots of tracking results of the video sequence used for testing (Bolt, target is selected in frame #1). Bolt sequence: This sequence presents the challenges of OCC, DEF, IPR and OPR. As shown in Figure 5, all the trackers perform well till frame #117. Compared to the ASMS and SMS trackers, the KMS and NEUTMS trackers cannot calculate a proper size for the bounding box due to the fixed Information 2017, 8, 122 11 of 14 scale. The KMS tracker has begun to drift form the target on the account of the improper size of the bounding box since frame #117. By fusing the information of the feature of the object and background region, the NEUTMS tracker has successfully tracked the target throughout this sequence even with an inappropriate scale. Though a good scale is estimated by the ASMS tracker, it fails when Bolt passes by some other runners, as seen in frame #142 and #160. Success plots of OPE Success plots of OPE - background clutter (31) 0.8 0.7 NEUTMS [0.404] ASMS [0.382] KMS [0.306] SMS [0.220] 0.7 Success rate Success rate 0.6 0.5 0.4 0.3 0.2 0.5 0.4 0.3 0.2 0.1 0.1 0 0 NEUTMS [0.374] ASMS [0.358] KMS [0.284] SMS [0.180] 0.6 0.2 0.4 0.6 0.8 0 0 1 0.2 Overlap threshold 0.4 0.6 0.8 1 Overlap threshold (a) (b) Figure 6. Success plots of one-pass evaluation (OPE): (a) Success plots of OPE over all the testing sequences; (b) Success plots of OPE over all the 31 testing sequences included the challenge of background clutters (BC). The value shown between the brackets is the area under curve (AUC) value corresponds to the tracker. 1) Success plots of TRE - background clutter (31) Success plots of TRE 0.8 0.7 NEUTMS [0.432] ASMS [0.421] KMS [0.354] SMS [0.249] 0.7 Success rate Success rate 0.6 0.5 0.4 0.3 0.2 0.5 0.4 0.3 0.2 0.1 0.1 0 0 NEUTMS [0.395] ASMS [0.389] KMS [0.328] SMS [0.209] 0.6 0.2 0.4 0.6 Overlap threshold (a) 0.8 1 0 0 0.2 0.4 0.6 0.8 1 Overlap threshold (b) Figure 7. Success plots of temporal robustness evaluation (TRE): (a) Success plots of TRE over all the testing sequences; (b) Success plots of TRE over all the 31 testing sequences included the challenge of BC. The value shown between the brackets is the AUC value corresponds to the tracker. We employ all the 61 sequences as the testing sequence dataset. Success plots of OPE and TRE over all the sequences are shown in Figures 6a and 7a respectively, which show our NEUTMS tracker is superior to other trackers. Due to the fact that the focus of our work in this paper to employ both the shiftistracker’s ability of shift tracker’s ability of object and background feature to enhance the mean-shift tracker’s ability of overcoming the problem of similar surroundings, only the success plots for the challenge of BC are given, and then the BC challenge is one of the most challenging problems for the traditional mean-shift tracker [13]. The results Information 2017, 8, 122 12 of 14 of the corresponding success plots are shown in Figures 6b and 7b, which show the robustness of the NEUTMS tracker when handling the challenge of BC. In order to test the performance of the proposed NEUTMS tracker over other kinds of challenges, all the AUC results for each tracker are given in Tables 2 and 3. The best result is highlighted in red italic type and the second result is highlighted in bold type. As seen in Tables 2 and 3, the NEUTMS tracker performs the best result when tackling the challenge of BC, MB, DEF, IPR, OCC or OPR when the OPE evaluation is considered. For TRE, the NEUTMS tracker performs the best result when confronting the same kind of challenge to OPE except for the challenge of MB. The ASMS tracker wins over SV because a robust scale updating scheme is used. The NEUTMS tracker performs the second best result over FM, IV and OV mainly because some inaccurate background information may be brought into the background feature model. The NEUTMS tracker performs the second best result when confronting the challenge of LR on account of less information can be employed for enhancing the tracker. Table 2. AUC results of each tracker on sequences with different challenge for OPE. Challenge BC FM MB DEF IV IPR LR OCC OPR OV SV Total NEUTMS ASMS KMS SMS 0.374 0.358 0.284 0.180 0.409 0.436 0.325 0.255 0.408 0.406 0.322 0.222 0.444 0.399 0.302 0.219 0.306 0.338 0.292 0.193 0.365 0.346 0.277 0.184 0.235 0.271 0.185 0.131 0.413 0.387 0.315 0.251 0.422 0.393 0.315 0.235 0.380 0.413 0.369 0.274 0.340 0.390 0.290 0.242 0.404 0.382 0.306 0.220 Table 3. AUC results of each tracker on sequences with different challenge for TRE. Challenge BC FM MB DEF IV IPR LR OCC OPR OV SV Total NEUTMS ASMS KMS SMS 0.395 0.389 0.328 0.209 0.422 0.442 0.346 0.274 0.418 0.434 0.342 0.243 0.480 0.453 0.371 0.277 0.361 0.392 0.328 0.224 0.402 0.401 0.334 0.220 0.252 0.271 0.237 0.153 0.432 0.416 0.361 0.281 0.442 0.437 0.363 0.268 0.392 0.418 0.357 0.258 0.366 0.387 0.320 0.247 0.432 0.421 0.354 0.249 4. Conclusions In this paper, a neutrosophic weighted mean-shift tracker is proposed. The experimental results have revealed its robustness. While calculating the neutrosophic weighted histogram, two kinds of criteria are considered as the object feature similarity and the background feature similarity, and each bin of the weight histogram is represented in the SVNS domain via three membership functions T, I and F. Both the feature in the object and the background region are fused by introducing the weighted neutrosophic similarity score function. Finally, the neutrosophic weighted histogram is employed to decide the new location of the object. As discussed in this work, we have not considered the scale variation problem. To further improve the performance of our tracker in the future, our primary mission is to introduce a scale updating scheme into this neutrosophic weighted mean-shift tracker. Acknowledgments: This work is supported by National Natural Science Foundation of China under Grant No. 61603258, the public welfare technology application research project of Zhejiang province under Grant No. 2016C31082, and National Natural Science Foundation of China under Grant No. 61703280, 61772018. Author Contributions: Keli Hu conceived and designed the algorithm; Keli Hu, En Fan, Jun Ye and Changxing Fan performed and implemented experiments; Keli Hu and Shigen Shen analyzed the data; Keli Hu wrote the paper; Jun Ye and Yuzhang Gu have fully supervised the work and approved the paper for submission. Conflicts of Interest: The authors declare no conflict of interest. References 1. 2. Yilmaz, A.; Javed, O.; Shah, M. Object tracking: A survey. ACM Comp. Surv. 2006, 38, 13. [CrossRef] Wu, Y.; Lim, J.; Yang, M.H. Online object tracking: A benchmark. 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This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). information Article Linguistic Neutrosophic Cubic Numbers and Their Multiple Attribute Decision-Making Method Jun Ye ID Department of Electrical and Information Engineering, Shaoxing University, 508 Huancheng West Road, Shaoxing 312000, China; yehjun@aliyun.com or yejun@usx.edu.cn; Tel.: +86-575-8832-7323 Received: 25 August 2017; Accepted: 6 September 2017; Published: 8 September 2017 Abstract: To describe both certain linguistic neutrosophic information and uncertain linguistic neutrosophic information simultaneously in the real world, this paper originally proposes the concept of a linguistic neutrosophic cubic number (LNCN), including an internal LNCN and external LNCN. In LNCN, its uncertain linguistic neutrosophic number consists of the truth, indeterminacy, and falsity uncertain linguistic variables, and its linguistic neutrosophic number consists of the truth, indeterminacy, and falsity linguistic variables to express their hybrid information. Then, we present the operational laws of LNCNs and the score, accuracy, and certain functions of LNCN for comparing/ranking LNCNs. Next, we propose a LNCN weighted arithmetic averaging (LNCNWAA) operator and a LNCN weighted geometric averaging (LNCNWGA) operator to aggregate linguistic neutrosophic cubic information and discuss their properties. Further, a multiple attribute decision-making method based on the LNCNWAA or LNCNWGA operator is developed under a linguistic neutrosophic cubic environment. Finally, an illustrative example is provided to indicate the application of the developed method. Keywords: linguistic neutrosophic cubic number; score function; accuracy function; certain function; linguistic neutrosophic cubic number weighted arithmetic averaging (LNCNWAA) operator; linguistic neutrosophic cubic number weighted geometric averaging (LNCNWGA) operator; decision-making 1. Introduction In terms of complex objective aspects of real life, human preference judgments may use linguistic expression, instead of numerical value expression, in order to be more suitable for people’s thinking habits. Hence, Zadeh [1] firstly introduced the concept of a linguistic variable and applied it to fuzzy reasoning. After that, linguistic decision analysis and linguistic aggregation operators have been proposed to solve linguistic decision-making problems [2–5]. Due to the incompleteness and uncertainty of linguistic decision environments, uncertain linguistic variables and their various aggregation operators were developed and applied to uncertain linguistic decision-making problems [6–11]. As to the extension of linguistic variables, the concept of linguistic intuitionistic fuzzy numbers and their linguistic intuitionistic multicriteria group decision-making methods were introduced in the literature [12,13], and then linguistic intuitionistic multicriteria decision-making method was proposed based on the Frank Heronian mean operator [14]. Recently, the concept of a neutrosophic linguistic number, which indicates a changeable uncertain linguistic number corresponding to some specified indeterminate range, and some weighted aggregation operators of neutrosophic linguistic numbers, were presented to solve multiple attribute group decision-making problems with neutrosophic linguistic numbers [15]. Then, the concept of a linguistic neutrosophic number, which is described independently by the truth, indeterminacy, and falsity linguistic variables, and some aggregation operators of linguistic neutrosophic numbers, were proposed to solve multiple attribute group decision-making problems with linguistic neutrosophic numbers [16,17]. Information 2017, 8, 110; doi:10.3390/info8030110 www.mdpi.com/journal/information Information 2017, 8, 110 2 of 11 To express vagueness and uncertainty in real life, the concept of a (fuzzy) cubic set (including the internal cubic set and external cubic set) was introduced based on the hybrid information of both partial certain and partial uncertain values in [18], where the first component is an interval/uncertain value and the second component is an exact/certain value. After that, the concept of a neutrosophic cubic set (including the internal neutrosophic cubic set and external neutrosophic cubic set), where a neutrosophic cubic number (a basic element in a neutrosophic cubic set) is composed of both the interval neutrosophic number and the single-valued neutrosophic number, and the distance measure of neutrosophic cubic sets were proposed and applied to pattern recognition [19,20]. Then, decision-making methods with neutrosophic cubic information were put forward based on grey relational analysis [21] and cosine measures [22], respectively. However, all the existing linguistic variables, including: uncertain linguistic variables, linguistic intuitionistic fuzzy numbers (basic elements in a linguistic intuitionistic fuzzy set), neutrosophic linguistic numbers (basic elements in a neutrosophic linguistic set), and linguistic neutrosophic numbers (basic elements in a linguistic neutrosophic set), cannot express the hybrid information of both uncertain linguistic and certain linguistic neutrosophic numbers simultaneously in linguistic decision-making environments. Furthermore, the cubic set and neutrosophic cubic set cannot also express linguistic arguments and handle linguistic decision-making problems under linguistic environments. Hence, it is necessary to extend neutrosophic cubic sets to linguistic neutrosophic arguments. For this purpose, this study presents a new concept of a linguistic neutrosophic cubic number (LNCN), where the uncertain linguistic neutrosophic number corresponding to its first part is composed of the truth, indeterminacy, and falsity uncertain linguistic variables and the linguistic neutrosophic number corresponding to its second part is composed of the truth, indeterminacy, and falsity linguistic variables. Then, we propose the operational laws of LNCNs and the score, accuracy, and certain functions of LNCN for comparing/ranking LNCNs. Further, we present a LNCN weighted arithmetic averaging (LNCNWAA) operator and a LNCN weighted geometric averaging (LNCNWGA) operator. Moreover, we develop a decision-making method based on the LNCNWAA or LNCNWGA operator and the score, accuracy, and certain functions to solve decision-making problems with the hybrid information of both certain linguistic neutrosophic numbers and uncertain linguistic neutrosophic numbers under linguistic environments. The rest of this paper is structured as follows: Section 2 proposes the concept of LNCN (including the internal LNCN and external LNCN), the operational laws of LNCNs, and the score, accuracy, and certain functions of LNCNs to rank LNCNs. In Section 3, we propose the LNCNWAA and LNCNWGA operators to aggregate LNCNs and discuss their properties. In Section 4, a multiple attribute decision-making method is developed based on the LNCNWAA or LNCNWGA operator under a LNCN environment. In Section 5, an example illustrates the application of the proposed method. Section 6 gives conclusions and future work. 2. Linguistic Neutrosophic Cubic Numbers (LNCNs) and Their Operational Laws This section proposes the concept of LNCN, which include the internal LNCN and external LNCN, and the operational laws of LNCNs. Definition 1. Let a linguistic term set be S = {sj | j ∈ [0, p]}, where p + 1 is an odd number/cardinality. A LNCN h in S is constructed as h = (u, c), where u = h[s Ta , s Tb ], [s Ia , s Ib ], [s Fa , s Fb ]i is an uncertain linguistic neutrosophic number with the truth, indeterminacy, and falsity uncertain linguistic variables [sTa , sTb ], [sIa , sIb ], and [sFa , sFb ] for sTa , sIa , sFa sTb , sIb , sFb ∈ S and Ta ≤ Tb, Ia ≤ Ib, Fa ≤ Fb; c = hs T , s I , s F i is a linguistic neutrosophic number with the truth, indeterminacy, and falsity linguistic variables sT , sI , and sF for sT , sI , sF ∈ S. Definition 2. Let a LNCN be h = (h[s Ta , s Tb ], [s Ia , s Ib ], [s Fa , s Fb ]i, hs T , s I , s F i) for sTa , sIa , sFa sTb , sIb , sFb , sT , sI , sF ∈ S. Then, we call Information 2017, 8, 110 (1) (2) 3 of 11 h an internal LNCN if Ta ≤ T ≤ Tb, Ia ≤ I ≤ Ib, Fa ≤ F ≤ Fb; h an external LNCN if T ∈ / (Ta, Tb), I ∈ / (Ia, Ib), and F ∈ / (Fa, Fb). Based on the operational laws of linguistic intuitionistic fuzzy numbers and linguistic neutrosophic numbers introduced in the existing literature [12–14,16,17], we propose the following operational laws of LNCNs. Definition 3. Let two LNCNs be h1 = (h[s Ta1 , s Tb1 ], [s Ia1 , s Ib1 ], [s Fa1 , s Fb1 ]i, hs T1 , s I1 , s F1 i) and h2 = (h[s Ta2 , s Tb2 ], [s Ia2 , s Ib2 ], [s Fa2 , s Fb2 ]i, hs T2 , s I2 , s F2 i). Then, their operational laws are defined as follows: s Ta1+Ta2− Ta1·Ta2 , s Tb1+Tb2− Tb1·Tb2 , s Ia1· Ia2 , s Ib1· Ib2 , s Fa1· Fa2 , s Fb1· Fb2 ,  p p p p p p h1 ⊕ h2 =   s T1+T2− T1·T2 , s I1· I2 , s F1· F2  p p p      s Ta1·Ta2 , s Tb1·Tb2 , s Ia1+ Ia2− Ia1· Ia2 , s Ib1+ Ib2− Ib1· Ib2 , s Fa1+ Fa2− Fa1· Fa2 , s Fb1+ Fb2− Fb1· Fb2 ,  p p p p p p h1 ⊗ h2 =   s T1·T2 , s I1+ I2− I1· I2 , s F1+ F2− F1· F2 p p p  s Ta1 λ , s Ia1 λ , s Tb1 λ , s Ib1 λ , s p ( Fa1 )λ , s p ( Fb1 )λ  p p p − p (1− p ) p − p (1− p ) p ( p ) p ( p ) λh1 =   s T1 λ , s I1 λ , s F1 λ p − p (1− p( ) p p ) p( g ) λ , s  s Ta1 λ , s Tb1 λ, s Ib1  p− p(1− Ia1 p ) p − p (1− p ) p( p ) p( p ) h1λ =   s T1 λ , s I1 λ , s F1 λ p( p ) p − p (1− p ) p − p (1− p ) λ , s       , λ > 0   ,  λ, s λ p− p(1− Fa1 p− p(1− Fb1 p ) p ) , λ > 0  Then, the above operational results are still LNCNs. Based on the score and accuracy functions of a linguistic neutrosophic number in the literature [16], we present the score, accuracy, and certain functions of LNCN to compare/rank LNCNs. Definition 4. Let a LNCN be h = (h[s Ta , s Tb ], [s Ia , s Ib ], [s Fa , s Fb ]i, hs T , s I , s F i) for sTa , sIa , sFa sTb , sIb , sFb , sT , sI , sF ∈ S. Then, its score, accuracy, and certain functions are defined as follows: S(h) = 1 [(4p + Ta + Tb − Ia − Ib − Fa − Fb) + (2p + T − I − F )], for S(h) ∈ [0, 1] 9p (1) 1 [( Ta + Tb − Fa − Fb) + ( T − F )], for H (h) ∈ [−1, 1] 3p (2) Ta + Tb + T , for C (h) ∈ [0, 1] 3p (3) H (h) = C (h) = Then, we introduce a ranking method based on the values of the score, accuracy, and certain functions. Definition 5. Let two LNCNs be h1 = (h[s Ta1 , s Tb1 ], [s Ia1 , s Ib1 ], [s Fa1 , s Fb1 ]i, hs T1 , s I1 , s F1 i) and h2 = (h[s Ta2 , s Tb2 ], [s Ia2 , s Ib2 ], [s Fa2 , s Fb2 ]i, hs T2 , s I2 , s F2 i). Then, their ranking method based on their score, accuracy, and certain functions are defined as follows: (1) (2) If S(h1 ) > S(h2 ), then h1 ≻ h2 ; If S(h1 ) = S(h2 ) and H(h1 ) > H(h2 ), then h1 ≻ h2 ; Information 2017, 8, 110 (3) (4) 4 of 11 If S(h1 ) = S(h2 ), H(h1 ) = H(h2 ), and C(h1 ) > C(h2 ), then h1 ≻ h2 ; S(h1 ) = S(h2 ), H(h1 ) = H(h2 ), and C(h1 ) = C(h2 ), then h1 ~ h2 Example 1. Let h1 = (<[s4 , s6 ], [s1 , s2 ], [s1 , s3 ]>, <s5 , s1 , s2 >), h2 = (<[s4 , s5 ], [s1 , s2 ], [s1 , s2 ]>, <s4 , s1 , s1 >), and h3 = (<[s6 , s7 ], [s2 , s3 ], [s1 , s3 ]>, <s6 , s2 , s3 >) in the linguistic term set S = {sj | j ∈ [0, 8]} are three LNCNs. Then, we need to compare them. By using Equations (1) to (3), the values of their score, accuracy, and certain functions are as follows: S(h1 ) = [32 + 4 + 6 − (1 + 2 + 1 + 3) + 16 + 5 − (1 + 2)]/72 = 0.7361, S(h2 ) = [32 + 4 + 5 − (1 + 2 + 1 + 2) + 16 + 4− (1 + 1)]/72 = 0.7361, and S(h3 ) = [32 + 6 + 7 − (2 + 3 + 1 + 3) + 16 + 6 − (2 + 3)]/72 = 0.7361; H(h1 ) = [4 + 6 − (1 + 3) + 5 − 2]/24 = 0.375, H(h2 ) = [4 + 5 − (1 + 2) + 4 − 1]/24 = 0.375, and H(h3 ) = [6 + 7 − (1 + 3) + 6 − 3]/24 = 0.5; and C(h1 ) = (4 + 6 + 5)/24 = 0.625 and C(h2 ) = (4 + 5 + 4)/24 = 0.5417. According to the ranking method of Definition 5, their ranking order is h3 ≻ h1 ≻ h2 . 3. Two Weighted Aggregation Operators of LNCNs 3.1. Linguistic Neutrosophic Cubic Number Weighted Arithmetic Averaging (LNCNWAA) Operator D E Definition 6. Let h j = [s Taj , s Tbj ], [s Iaj , s Ibj ], [s Faj , s Fbj ] , s Tj , s I j , s Fj (j =1, 2, . . . , n) be a group of LNCNs, then the LNCNWAA operator can be defined as follows: n LNCNWAA(h1 , h2 , · · · , hn ) = ∑ wj hj (4) j =1 where wj is the weight of hj (j = 1, 2, . . . , n) for wj ∈ [0, 1] and ∑nj=1 w j = 1. According to Definitions 3 and 6, there is the following theorem. E D Theorem 1. Let h j = [s Taj , s Tbj ], [s Iaj , s Ibj ], [s Faj , s Fbj ] , s Tj , s I j , s Fj (j = 1, 2, . . . , n) be a group of LNCNs, then the aggregation result obtained by Equation (4) is still a LNCN, which is calculated by the following aggregation formula: n LNCNWAA(h1 , h2 , · · · , hn ) = ∑ w j h j j =1  *     +   s ,  s n ,s , s n Ibj w j , s n Faj w j , s n Fbj w j  ,  n n   Iaj w Tbj w j Taj w p∏ ( p ) j p∏ ( p ) p∏ ( p ) p∏ ( p ) p − p ∏ (1− p ) j p − p ∏ (1− p )   j =1 j =1 j =1 j =1 j =1  + j =1 =  *     s  , s , s n n n w w w Tj Ij Fj j j j p − p ∏ (1− j =1 p ) p∏ ( p) p∏ ( j =1 j =1 p (5) ) where wj is the weight of hj (j = 1, 2, . . . , n) for wj ∈ [0, 1] and ∑nj=1 w j = 1. In the following, the mathematical induction is used to prove Theorem 1. Proof. (1) Set n = 2, according the operational laws of LNCNs, we have the following results: s p− p(1− Ta1 )w1 , s p− p(1− Tb1 )w1 , s p( Ia1 )w1 , s p( Ib1 )w1 , s p( Fa1 )w1 , s p( Fb1 )w1 ,  p p p p p p w1 h 1 =   s p− p(1− T1 )w1 , s p( I1 )w1 , s p( F1 )w1  p p p     Information 2017, 8, 110 5 of 11  w2 , s w2 , s w2 , s w2 , s w2 , s w2 , s Ta2 Ia2 Fa2 Tb2 Ib2 Fb2  p( p ) p( p ) p − p (1− p ) p − p (1− p ) p( p ) p( p ) w2 h 2 =   s p− p(1− T2 )w2 , s p( I2 )w2 , s p( F2 )w2 p p p Then, there exists the following result: LNCNWAA  (h1 , h2 ) = w1 h1 ⊕ w2 h2  s Ta2 w2 , Ta1 w1 w1 w2 ( p− p(1− p ) )( p− p(1− p ) ) Ta2 Ta1   *  p − p (1− p ) + p − p (1− p ) − + p ,     s  w1 w2 Tb1 Tb2  ( p− p(1− p ) )( p− p(1− p ) ) w1 , Tb2 w2  p− p(1− Tb1 p ) + p − p (1− p ) − p  =   s p( Ia1 )w1 ( Ia2 )w2 , s p( Ib1 )w1 ( Ib2 )w2 , s p( Fa1 )w1 ( Fa2 )w2 , s p( Fb1 )w1 ( Fb2 )w2  p p p p p p p p  *   s w w w w ,s w ,s  p− p(1− T1 p )  w1 + p− p(1− T2 p ) w2 − p( I1 p ) T2 2 1 ( p− p(1− T1 p ) )( p− p(1− p ) ) p p( F1 p ) 1 ( I2 ) 2 p p p p p 1 ( F2 )w2 p p          +    p p p p p p∏ ( p) j =1   w1 w1 Ta2 w2 , s Tb2 w2 ,  * s p− p(1− Ta1 +  p− p(1− Tb1 p ) (1− p ) p ) (1− p )         ,           s 2 Iaj w j , s 2 Ibj w j , s 2 Faj w j , s 2 Fbj w j  =  p∏ ( p ) p∏ ( p ) p∏ ( p ) p∏ ( p )   j = 1 j = 1 j = 1 j = 1  *  +     w w  s p− p(1− T1 ) 1 (1− T2 ) 2 , s 2 I j w j , s 2 Fj w j  p p∏ ( p) p p∏ ( j =1  j=1 p     ,  * + s p− p(1− Tb1 )w1 + p− p(1− Tb2 )w2 −( p− p(1− Tb1 )w1 − p(1− Tb2 )w2 + p(1− Tb1 )w1 (1− Tb2 )w2 )  p p p p p p      ,        = s , s , s , s  2 Iaj w j   2 Faj w j  2 2 Ibj w j Fbj w j  p∏ ( p ) p∏ ( p ) p∏ ( p ) p∏ ( p )  j =1 j =1 j =1 j =1  *   s p− p(1− T1 )w1 + p− p(1− T2 )w2 −( p− p(1− T1 )w1 − p(1− T2 )w2 + p(1− T1 )w1 (1− T2 )w2 ) , s 2 I j w j , s 2  p p∏ ( j =1 Fj p )           +     wj p − p ∏ (1− j =1 p ) p∏ ( p) p∏ ( j =1 j =1 p ) (6) )    * +        , , s s , s , s , s s     2 Iaj w j   2 Faj w j  , 2 2 2 2 Taj w Tbj w j Ibj w j Fbj w j  p − p ∏ (1− p ) j p − p ∏ (1− p ) p∏ ( p ) p∏ ( p ) p∏ ( p ) p∏ ( p )  j =1 j =1 j =1 j =1 j =1 = * + j =1   , s 2 I j w j , s 2 Fj w j  s 2 Tj w j      s p− p(1− Ta1 )w1 + p− p(1− Ta2 )w2 −( p− p(1− Ta1 )w1 − p(1− Ta2 )w2 + p(1− Ta1 )w1 (1− Ta2 )w2 ) , p       .    (2) Set n = k, by Equation (5) we obtain k LNCNWAA(h1 , h2 , · · · , hk ) = ∑ w j h j j =1        + *        , , , s s , s s , s s     k Iaj w j   k Faj w j  , k k k k Tbj w j Ibj w j Fbj w j Taj w  p∏ ( p ) p∏ ( p ) p∏ ( p ) p∏ ( p ) p − p ∏ (1− p ) j p − p ∏ (1− p )  j =1 j =1 j =1 j =1 j =1 = * + j =1   , s k I j w j , s k Fj w j  s k Tj w j p − p ∏ (1− j =1 p ) p∏ ( p) j =1 p∏ ( j =1 p )          (7) Information 2017, 8, 110 6 of 11 (3) Set n = k + 1, based on Equations (6) and (7), we can obtain the following result: k +1 LNCNWAA(h1 , h2 , · · · , hk+1 ) = ∑ w j h j j =1    , s k w Taj wj ( p− p ∏ (1− p ) )( p− p(1− Takp+1 ) k+1 )    k w w j =1  p− p ∏ (1− Taj ) j + p− p(1− Tak+ j ) k+1 −   p p p    j =1 ,  *   s  +  k w k +1   Tbj wj  Tbk + 1 ( p− p ∏ (1− p ) )( p− p(1− p ) )    k j =1 , Tbk + j wk+1 Tbj w j  − p − p ∏ (1− p ) + p − p (1− p ) p  j =1      =        s k Iaj w j Iak+ j wk+1 , s k Ibj w j Ibk+ j wk+1 , s k Faj w j Fak+ j wk+1 , s k Fbj w j Fbk+ j wk+1    p∏ ( p ) ( p ) p∏ ( p ) ( p ) p∏ ( p ) ( p ) p∏ ( p ) ( p )  j =1 j =1 j =1 j =1 +  *   s , s , s  k k k w w w w w Tj wj Ik + j Fj Fk + j Ij ( p− p ∏ (1− p ) ))( p− p(1− Tkp+1 ) k+1 )  p ∏ ( p ) j ( p ) k +1 p ∏ ( p ) j ( p ) k +1 k w w p − p ∏ (1− j =1   j =1 Tk+ j Tj j k +1 − p ) + p − p (1− p ) j =1 j =1 p s , k k k wj w w Taj w j Taj w j Tak+1 wk+1 −( p − p   p− p ∏ (1− Taj ∏ (1− p ) − p(1− Takp+1 ) k+1 + p ∏ (1− p ) (1− Takp+1 ) k+1 ) p ) + p − p (1− p )   j =1 j =1 j =1  *    s k k k w w w Tbj w j Tbj w j Tbj w j  p− p ∏ (1− p ) + p− p(1− Tbkp+1 ) k+1 −( p− p ∏ (1− p ) − p(1− Tbkp+1 ) k+1 + p ∏ (1− p ) (1− Tbkp+1 ) k+1 )  j =1 j= 1  j =1     =       s k Iaj w j Iak+1 wk+1 , s k Ibj w j Ibk+1 wk+1 , s k Faj w j Fak+1 wk+1 , s k Fbj w j Fbk+1 wk+1   p∏ ( p ) ( p ) p∏ ( p ) ( p ) p∏ ( p ) ( p ) p∏ ( p ) ( p )  j =1 j =1 j =1 j =1  *   s , s k Ij  k k k w w w Tj w j Tj w j Tj w j p − p ∏ (1− j =1  p ) + p− p(1− Tkp+1 ) k+1 −( p − p ∏ (1− j =1 p ) − p(1− Tkp+1 )  k +1 + p ∏ (1− j =1 p ) (1− Tkp+1 ) k +1 ) p∏ ( p) j =1     , +                         , w j Ik +1 w ( p ) k +1 ,s k p∏ ( j =1 Fj w j Fk+1 wk+1 p ) ( p )               +         ,s  * s , k k w w +  Taj w Tbj w j   p− p ∏ (1− p ) j (1− Takp+1 ) k+1 p− p ∏ (1− p ) (1− Tbkp+1 ) k+1   j =1 j =1      ,           =   s k Iaj w j Iak+1 wk+1 , s k Ibj w j Ibk+1 wk+1 , s k Faj w j Fak+1 wk+1 , s k Fbj w j Fbk+1 wk+1    p∏ ( p ) ( p ) p∏ ( p ) ( p ) p∏ ( p ) ( p ) p∏ ( p ) ( p )   j =1 j =1 j =1  *  + j =1     , s , s s   k k k I j w j Ik+1 w Fj w j Fk+1 w Tj w j Tk+1 w p − p ∏ (1− j =1  p ) (1− p ) k +1 p∏ ( p) ( j =1  p ) k +1 p∏ ( j =1   p ) ( p ) k +1    * +        s s , , , , s , s , s  s      k +1 k + 1 k + 1 k + 1 k + 1 k + 1 w w w w w w Taj Iaj Faj Tbj Ibj Fbj j j j  p − p ∏ (1− p ) j p − p ∏ (1− p ) p∏ ( p ) j p∏ ( p ) p∏ ( p ) j p∏ ( p )  j =1 j =1 j =1 j =1 j =1 = * + j =1   , s k+1 I j w j , s k+1 Fj w j  s k +1 Tj w j p − p ∏ (1− j =1 p ) p∏ (p) p∏ ( j =1 j =1 p )      .    Based on the above results, Equation (5) can hold for any n. The proof is finished. Clearly, the LNCNWAA operator contains the following properties: (1) Idempotency: Let hj (j = 1, 2, . . . , n) be a group of LNCNs. When hj = h for j = 1, 2, . . . , n, there is LNCNWAA(h1 , h2 , · · · , hn ) = h. (2) Boundedness: Let hj (j = 1, 2, . . . , n) be a group of LNCNs and the minimum and maximum LNCNs be h− = and h+ = min(s Taj ), min(s Tbj ) , max(s Iaj ), max(s Ibj ) , max(s Faj ), max(s Fbj ) , min(s Tj ), max(s I j ), max(s Fj ) j j j j j j j max(s Taj ), max(s Tbj ) , min(s Iaj ), min(s Ibj ) , min(s Faj ), min(s Fbj ) , max(s Tj ), min(s I j ), min(s Fj ) j j j j j j j j j j j respectively. Then, there exists h− ≤ LNCNWAA(h1 , h2 , · · · , hn ) ≤ h+ . (3) Monotonicity: Let hj (j = 1, 2, . . . , n) be a group of LNCNs. When hj ≤ h∗j for j = 1, 2, . . . , n, then there exists LNCNWAA(h1 , h2 , · · · , hn ) ≤ LNCNWAA h1∗ , h2∗ , · · · , h∗n . Information 2017, 8, 110 7 of 11 Proof. (1) For hj = h (j = 1, 2, . . . , n), we have the following result: n LNCNWAA(h1 , h2 , · · · , hn ) = ∑ w j h j j =1  *   +     s ,  s n ,s , s n Ibj w j , s n Faj w j , s n Fbj w j  ,  n n   Tbj w j Taj w Iaj w p∏ ( p ) p − p ∏ (1− p ) j p − p ∏ (1− p ) p∏ ( p ) j p∏ ( p ) p∏ ( p )   j = 1 j = 1 j = 1 j = 1 j = 1 j = 1   + = *      s , s , s n n n w w w Tj Ij Fj j j j p − p ∏ (1− p j =1  * ) p∏ ( p) p∏ ( j =1 j =1 p )       +         s ,s ,s ,s n n n n n n ,  s ,  s  ,    ∑ wj ∑ wj ∑ wj ∑ wj ∑ wj ∑ wj   j =1 j =1 j =1 j =1 Ia j=1 Ta j=1   p( Fa p− p(1− Tb p( Ib p( Fb p ) p ) p ) p ) =  * p − p (1− p )  + p( p )     ,s ,s n n n   s w w w ∑ = p− p(1− Tp ) j=1 j ∑ p( pI ) j=1 j ∑ j F j =1 p( p ) s p− p(1− Ta ) , s p− p(1− Tb ) , s p( Ia ) , s p( Ib ) , s p( Fa ) , s p( Fb ) , s p− p(1− T ) , s p( I ) , s p( F ) p p p p p p p p p = (h[s Ta , s Tb ], [s Ia , s Ib ], [s Fa , s Fb ]i, hs T , s I , s F i) = h. (2) Since the minimum LNCN is h− and the maximum LNCN is h+ , there is h− ≤ hj ≤ h+ . n n n j =1 j =1 Thus, there exists ∑ w j h− ≤ ∑ w j h j ≤ ∑ w j h+ . According to the above property (1), there exists j =1 h− n ≤ ∑ wj hj ≤ h+ . Then, j =1 h− ≤ LNCNWAA(h1 , h2 , · · · , hn ) ≤ h+ can hold. n h∗j (j = 1, 2, . . . , n), there exists ∑ w j h j j =1 LNCNWAA(h1 , h2 , · · · , hn ) ≤ LNCNWAA h1∗ , h2∗ , · · · , h∗n can hold. (3) For h j ≤ ≤ n ∑ w j h∗j . Then, j =1 Hence, we complete the proofs of these properties. Obviously, when wj = 1/n for j = 1, 2, . . . , n, the LNCNWAA operator is reduced to the LNCN arithmetic averaging operator. 3.2. LNCNWGA Operator D E [s Taj , s Tbj ], [s Iaj , s Ibj ], [s Faj , s Fbj ] , s Tj , s I j , s Fj (j =1, 2, . . . , n) be a group of Definition 7. Let h j = LNCNs, then the LNCNWGA operator is defined as follows: n LNCNWGA(h1 , h2 , · · · , hn ) = wj ∏ hj j =1 where wj is the weight of hj (j =1, 2, . . . , n) for wj ∈ [0, 1] and ∑nj=1 w j = 1. According to Definitions 3 and 7, we can introduce the following theorem. (8) Information 2017, 8, 110 8 of 11 D E Theorem 2. Leth j = [s Taj , s Tbj ], [s Iaj , s Ibj ], [s Faj , s Fbj ] , s Tj , s I j , s Fj (j = 1, 2, . . . , n) be a group of LNCNs. Then, the aggregation result of Equation (8) is still a LNCN, which is calculated by the following aggregation equation: n wj LNCNWGA(h1 , h2 , · · · , hn ) = ∏ h j 1   j=  * +  s n ,  s  , , s n Tbj w j , s ,s ,s n n n n  Ibj w j Fbj w j Iaj w Faj w Taj w p − p ∏ (1− p ) j p − p ∏ (1− p ) p − p ∏ (1− p ) j p − p ∏ (1− p ) p∏ ( p ) j p∏ ( p )  j =1 j = 1 j = 1 j = 1 j = 1 j =1  * + =   s n Tj w j , s , s n n w w Ij Fj j j p∏ ( j =1 p ) p − p ∏ (1− p ) p − p ∏ (1− j =1 p j =1         ) (9) where wj is the weight of hj (j = 1, 2, . . . , n) for wj ∈ [0, 1] and ∑nj=1 w j = 1. Obviously, when wj = 1/n for j = 1, 2, ..., n, the LNCNWGA operator is reduced to the LNCN geometric averaging operator. Based on the similar proof manner of Theorem 1, we can prove Theorem 2. Hence, it is omitted here. Obviously, the LNCNWGA operator also contains the following properties: (1) Idempotency: Let hj (j = 1, 2, . . . , n) be a group of LNCNs. When hj = h for j = 1, 2, . . . , n, there exists LNCNWGA(h1 , h2 , · · · , hn ) = h. (2) Boundedness: Let hj (j = 1, 2, . . . , n) be a group of LNCNs and the minimum and maximum LNCNs be h− = and h+ = min(s Taj ), min(s Tbj ) , max(s Iaj ), max(s Ibj ) , max(s Faj ), max(s Fbj ) , min(s Tj ), max(s I j ), max(s Fj ) j j j j j j j max(s Taj ), max(s Tbj ) , min(s Iaj ), min(s Ibj ) , min(s Faj ), min(s Fbj ) , max(s Tj ), min(s I j ), min(s Fj ) j j j j j j j j j j j respectively. Then, there exists h− ≤ LNCNWGA(h1 , h2 , · · · , hn ) ≤ h+ . (3) Monotonicity: Let hj (j = 1, 2, . . . , n) be a group of LNCNs. When hj ≤ h∗j for j = 1, 2, . . . , n, there exists LNCNWGA(h1 , h2 , · · · , hn ) ≤ LNCNWGA h1∗ , h2∗ , · · · , h∗n . Based on the similar proofs of the properties corresponding to the LNCNWAA operator, we can also prove these properties of the LNCNWGA operator. Hence, these proofs are omitted here. 4. Decision-Making Method Based on the LNCNWAA or Linguistic Neutrosophic Cubic Number Weighted Geometric Averaging (LNCNWGA) Operator This section proposes a decision-making method based the LNCNWAA or LNCNWGA operator to solve multiple attribute decision-making problems with LNCN information. If there is a multiple attribute decision-making problem, we consider Q = {Q1 , Q2 , . . . , Qm } as a set of alternatives and R = {R1 , R2 , . . . , Rn } as a set of attributes. The weigh vector of the attributes Rj (j = 1, 2, . . . , n) is specified as w = (w1 , w2 , . . . , wn ). Then, decision-makers are invited to evaluate the alternatives Qi (i = 1, 2, . . . , m) over the attributes Rj (j = 1, 2, . . . , n) by LNCNs from the predefined linguistic term set S = {sj | j ∈ [0, p]}, where p + 1 is an odd number/cardinality. Based on the linguistic term set, the decision-makers can assign the uncertain linguistic arguments corresponding to the truth, indeterminacy, and falsity linguistic terms and the certain linguistic arguments corresponding to the truth, indeterminacy, and falsity linguistic terms in each LNCN as the linguistic evaluation of each attribute Rj (j = 1, 2, . . . , n) on each alternative Qi (i = 1, 2, . . . , m) in the evaluation process. Thus, all the LNCNs can be constructed as a LNCN decision matrix D = (hij )m ×n , D E where hij = [s Taij , s Tbij ], [s Iaij , s Ibij ], [s Faij , s Fbij ] , s Tij , s Iij , s Fij (i = 1, 2, . . . , m; j = 1, 2, . . . , n) is a LNCN. Thus, the decision-making method based on the LNCNWAA or LNCNWGA operator is described by the following decision steps: Step 1 Calculate hi = LNCNWAA(hi1 , hi2 , ..., hin ) or hi = LNCNWGA(hi1 , hi2 , ..., hin ) (i = 1, 2, . . . , m) by using Equation (5) or Equation (9) and obtain the collective overall LNCN hi for Qi (i = 1, 2, . . . , m). Information 2017, 8, 110 9 of 11 Step 2 Calculate the values of S(hi ) (H(hi ) and/or C(hi ) if necessary) (i = 1, 2, . . . , m) for each collective overall LNCN hi (i = 1, 2, . . . , m) by Equation (1) (Equation (2) and/or Equation (3)). Step 3 Rank the alternatives corresponding to the ranking method of Definition 5, and then select the best one. Step 4 End. 5. Illustrative Example This section provides an illustrative example in order to demonstrate the application of the proposed decision-making method under a linguistic neutrosophic cubic environment. A manufacturing company needs to hire a mechanical designer. After all applicants are chosen preliminarily by the human resources department, four potential candidates Q1 , Q2 , Q3 , and Q4 need to be further evaluated according to the three requirements/attributes: (1) R1 is the innovation skill; (2) R2 is the design experience; (3) R3 is the self-confidence. A group of experts is required to conduct the interview and to choose the most suitable candidate. Then, the weigh vector w = (0.45, 0.35, 0.2) is considered as the importance of the three attributes. Herewith, the experts (decision-makers) need to evaluate the four potential candidates/alternatives Qi (i = 1, 2, 3, 4) corresponding to the three attributes Rj (j = 1, 2, 3) by the form of LNCNs based on the given linguistic term set S = {sj | j ∈ [0, p]}, where S = {s0 = extremely poor, s1 = very poor, s2 = poor, s3 = slightly poor, s4 = fair, s5 = slightly good, s6 = good, s7 = very good, s8 = extremely good} for p = 8. Thus, all the LNCNs are given by the experts and constructed as the following LNCN decision matrix D(hij )4×3 : D (hij )4×3  (h[s4 , s6 ], [s1 , s2 ], [s1 , s3 ]i, hs5 , s1 , s2 i)   (h[s3 , s5 ], [s1 , s2 ], [s1 , s2 ]i, hs4 , s1 , s1 i) =  (h[s4 , s7 ], [s1 , s2 ], [s2 , s3 ]i, hs5 , s2 , s3 i) (h[s6 , s7 ], [s2 , s3 ], [s2 , s3 ]i, hs7 , s3 , s3 i) (h[s4 , s6 ], [s1 , s3 ], [s1 , s3 ]i, hs5 , s2 , s2 i) (h[s5 , s7 ], [s1 , s2 ], [s1 , s2 ]i, hs6 , s1 , s2 i) (h[s6 , s7 ], [s1 , s3 ], [s1 , s3 ]i, hs7 , s2 , s2 i) (h[s5 , s7 ], [s1 , s2 ], [s1 , s2 ]i, hs6 , s1 , s2 i) (h[s4 , s7 ], [s1 , s3 ], [s2 , s3 ]i, hs6 , s2 , s3 i) (h[s4 , s6 ], [s2 , s3 ], [s1 , s2 ]i, hs5 , s3 , s1 i) (h[s5 , s7 ], [s1 , s3 ], [s2 , s3 ]i, hs5 , s2 , s3 i) (h[s4 , s6 ], [s1 , s2 ], [s1 , s2 ]i, hs5 , s1 , s1 i)      Thus, the proposed decision-making method can be applied to the decision-making problem with LNCN information. On the one hand, we can use the decision-making method based on the LNCNWAA operator, which is described by the following decision steps: Step 1 By using Equation (5), the collective overall LNCNs of hi for Qi (i = 1, 2, 3, 4) can be given as follows: h1 = (<[s4 , s6.2589 ], [s1 , s2.4997 ], [s1.1487 , s3 ]>, <s5.2337 , s1.4641 , s2.1689 >), h2 = ([s4.0011 , s6.1167 ], [s1.1487 , s2.1689 ], [s1 , s2 ]>, <s5.0371 , s1.2457 , s1.2746 >), h3 = ([s5.0371 , s7 ], [s1 , s2.4997 ], [s1.5692 , s3 ]>, <s5.9577 , s2 , s2.6031 >), and h4 = ([s5.3523 , s6.8513 ], [s1.366 , s2.4003 ], [s1.366 , s2.4003 ]>, <s6.4122 , s1.6395 , s2.0896 >). Step 2 Calculate the score values of S(hi ) (i = 1, 2, 3, 4) by Equation (1): S(h1 ) = 0.7252, S(h2 ) = 0.7544, S(h3 ) = 0.7406, and S(h4 ) = 0.7688. Step 3 The ranking order of the four alternatives is Q4 ≻ Q2 ≻ Q3 ≻ Q1 based on the score values. Thus, the candidate Q4 is the best choice among the four candidates. On the other hand, we can also use the decision-making method based on the LNCNWGA operator, which is described by the following decision procedures: Step 1’ By using Equation (9), the collective overall LNCNs of hi for Qi (i = 1, 2, 3, 4) are given as follows: h1 = (<[s4 , s6.1879 ], [s1 , s2.5725 ], [s1.2125 , s3 ]>, <s5.1857 , s1.569 , s2.2148 >), h2 = (<[s3.7998 , s5.8338 ], [s1.2125 , s2.2148 ], [s1 , s2 ]>, <s4.8203 , s1.4556 , s1.3677 >), h3 = (<[s4.8203 , s7 ], [s1 , s2.5725 ], [s1.6674 , s3 ]>, <s5.6249 , s2 , s2.6705 >), and h4 = (<[s5.1906 , s6.7875 ], [s1.4691 , s2.4726 ], [s1.4691 , s2.4726 ]>, <s6.2007 , s1.9835 , s2.2996 >). Step 2’ By using Equation (1), we calculate the score values of S(hi ) (i = 1, 2, 3, 4) as follows: S(h1 ) = 0.7195, S(h2 ) = 0.7389, S(h3 ) = 0.7296, and S(h4 ) = 0.7502. Information 2017, 8, 110 10 of 11 Step 3’ The ranking order of the four candidates is Q4 ≻ Q2 ≻ Q3 ≻ Q1 . Thus, the candidate Q4 is still the best choice among the four candidates. Obviously, the above two ranking orders based on the LNCNWAA and LNCNWGA operators and the best candidate are identical. Compared with existing currant linguistic neutrosophic decision-making methods [16,17], the decision information in this study is LNCNs, while the decision information used in [16,17] is linguistic neutrosophic numbers. As mentioned above, since LNCN is composed of its uncertain neutrosophic number and its linguistic neutrosophic number, LNCN contains more information than the unique linguistic neutrosophic number in [16,17]. However, existing linguistic neutrosophic decision-making methods in [16,17] cannot handle such a decision-making problem with linguistic neutrosophic cubic information in this paper. Therefore, the decision-making method proposed in this paper can solve decision-making problems with both certain linguistic and uncertain linguistic neutrosophic information. It can also provide a new way for hybrid linguistic decision-making problems under certain and uncertain linguistic environments. Due to no related studies in the existing literature, this is the first study to propose a new concept of LNCN and a new linguistic neutrosophic cubic decision-making method. However, decision-makers can select one of two weighted aggregation operators of LNCNs to solve linguistic neutrosophic cubic decision-making problems according to their preference and actual requirements. 6. Conclusions This paper originally proposed the concept of LNCN, including the internal LNCN and external LNCN, and the operational laws of LNCNs, and introduced the score, accuracy, and certain functions of LNCNs for comparing/ranking LNCNs. Then, we proposed the LNCNWAA and LNCNWGA operators to aggregate LNCNs and discussed their properties. Next, we developed a multiple attribute decision-making method based on the LNCNWAA or LNCNWGA operator for solving multiple attribute decision-making problems with LNCN information. Finally, an example illustrated the application of the developed method under a LNCN environment. The proposed decision-making method can solve decision-making problems with determinate and uncertain linguistic neutrosophic arguments. Obviously, the main advantages of this study are summarized as follows: (1) (2) (3) The LNCN expression is superior to existing linguistic expressions in the certain and uncertain linguistic environment. The developed linguistic neutrosophic cubic decision-making method extends existing ones to deal with linguistic neutrosophic cubic decision-making problems with the hybrid information of both uncertain linguistic neutrosophic arguments and certain linguistic neutrosophic arguments. The developed new method enriches linguistic neutrosophic expressions and linguistic neutrosophic decision-making methods. In the future work, we shall further introduce new aggregation operators of LNCNs and applications in group decision-making, pattern recognition, and medical diagnoses. Acknowledgments: This paper was supported by the National Natural Science Foundation of China (Nos. 71471172, 61703280). Conflicts of Interest: The author declares no conflict of interest. References 1. 2. Zadeh, L.A. The concept of a linguistic variable and its application to approximate reasoning Part I. Inf. Sci. 1975, 8, 199–249. [CrossRef] Herrera, F.; Herrera-Viedma, E.; Verdegay, L. A model of consensus in group decision making under linguistic assessments. Fuzzy Sets Syst. 1996, 79, 73–87. [CrossRef] Information 2017, 8, 110 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 11 of 11 Herrera, F.; Herrera-Viedma, E. 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Neutrosophic cubic sets. New Math. Nat. Comput. 2017, 13, 41–45. [CrossRef] Banerjee, D.; Giri, B.C.; Pramanik, S.; Smarandache, F. GRA for multi attribute decision making in neutrosophic cubic set environment. Neutrosoph. Sets Syst. 2017, 15, 64–73. Lu, Z.K.; Ye, J. Cosine measures of neutrosophic cubic sets for multiple attribute decision-making. Symmetry 2017, 9, 121. [CrossRef] © 2017 by the author. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/). MDPI is a member of Follow Us facebook.com/MDPIOpenAccessPublishing twitter.com/MDPIOpenAccess linkedin.com/company/mdpi plus.google.com/+MdpiOA weibo.com/mdpicn Wechat: MDPI-China medium.com/@MDPIOpenAccess blog.mdpi.com MDPI AG St. Alban-Anlage 66 CH-4052 Basel Switzerland Tel: +41 61 683 77 34 Fax: +41 61 302 89 18 www.mdpi.com mdpi.com/journal/information See www.mdpi.com for a full list of offices and contact information. MDPI AG is a company registered in Basel, Switzerland, No. CH-270.3.014.334-3, whose registered office is at St. Alban-Anlage 66, CH-4052 Basel, Switzerland. Basel, September 2017 Special Issue February 2018 Neutrosophic Information Theory and Applications www.mdpi.com ISSN 2078-2489

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