Cross Entropy Measures of Bipolar and Interval Bipolar Neutrosophic Sets and Their Application for Multi-Attribute Decision-Making

Surapati Pramanik, Partha Dey, Florentin Smarandache, Jun Ye
2018 Axioms  
Bipolar neutrosophic set is an important extension of bipolar fuzzy set. This set is a 15 hybridization of bipolar fuzzy set and neutrosophic set. Every element of a bipolar neutrosophic 16 set consists of three independent positive membership functions and three independent negative 17 membership functions. In this paper, we develop cross entropy measures of bipolar neutrosophic 18 sets and prove its properties. We also define cross entropy measures of interval bipolar 19 neutrosophic sets and
more » ... utrosophic sets and prove its properties. Thereafter, we develop two novel multi-attribute 20 decision making methods based on the proposed cross entropy measures. In the decision making 21 framework, we calculate the weighted cross entropy measures between each alternative and the 22 ideal alternative to rank the alternatives and choose the best one. We solve two illustrative 23 examples of multi-attribute decision making problems and compare the obtained result with the 24 results of other existing methods to show the applicability and effectiveness of the developed 25 method. In the end, the main conclusion and future scope of research are summarized. 26 Keywords: neutrosophic set; bipolar neutrosophic set; interval bipolar neutrosophic set; 27 multi-attribute decision making; cross entropy measure 28 29 30 1. Introduction 31 According to Shannon and Weaver [1] and Shannon [2], entropy measure is an important 32 decision making apparatus for computing uncertain information. Shannon [2] introduced the 33 concept of cross entropy approach in information theory. In neutrosophic environment [3], Ye [4] 34 proposed single valued neutrosophic cross entropy measures between two single valued 35 neutrosophic sets (SVNSs) [5] by extending the concept of cross entropy and symmetric 36 discrimination information measures between two fuzzy sets [6] due to Shang and Jiang [7]. Şahin 37 [8] proposed two techniques to convert the interval neutrosophic information to single valued 38 neutrosophic information and fuzzy information. In the same study, Şahin [8] defined interval 39 neutrosophic cross entropy measure by utilizing two reduction methods. Tian et al. [9] developed a 40 transformation operator to convert interval neutrosophic numbers to single valued neutrosophic Preprints ( | NOT PEER-REVIEWED | Posted: numbers and defined cross entropy measures for two SVNSs. In the same study, Tian et al. [9] 42 developed a multi-criteria decision making (MCDM) approach based on cross entropy and TOPSIS 43 [10] where the weight of the criterion is incomplete. Ye [11] pointed out that entropy measure of 44 SVNSs defined by Ye [4] has some drawbacks in some situations. Therefore, Ye [11] proposed an 45 improved cross entropy measures of SVNSs in order to overcome the drawbacks discussed in the 46 paper [4] and extended the concept to interval neutrosophic sets (INSs) environment [12] and 47 developed MCDM models using cross entropy measures of SVNSs and INSs. 48 Bipolar neutrosophic sets (BNSs) was developed by Deli et al. [13] by hybridizing the concepts 49 of bipolar fuzzy sets [14, 15] and neutrosophic sets [3]. A BNS has two fully independent parts, 50 which are positive membership degree T + → [0, 1], I + → [0, 1], F + → [0, 1], and negative 51 membership degree T -→ [-1, 0], I -→ [-1, 0], F -→ [-1, 0] where the positive membership degrees 52 T + , I + , F + represent truth membership degree, indeterminacy membership degree and false 53 membership degree respectively of an element and the negative membership degrees T -, I -, F -54 represent truth membership degree, indeterminacy membership degree and false membership 55 degree respectively of an element to some implicit counter property corresponding to a BNS. Deli et 56 al. [13] defined some operations namely score, accuracy, and certainty functions to compare BNSs 57 and provided some operators in order to aggregate BNSs. Deli and Subas [16] defined correlation 58 coefficient similarity measure for dealing with MCDM problems under single valued neutrosophic 59 setting. Şahin et al. [17] proposed Jaccard vector similarity measure for MCDM problems with single 60 valued neutrosophic information. Uluçay et al. [18] introduced Dice similarity measure, weighted 61 Dice similarity measure, hybrid vector similarity measure, weighted hybrid vector similarity 62 measure for BNSs and established a MCDM method by using the proposed similarity measures. Dey 63 et al. [19] investigated TOPSIS method for solving multi-attribute decision making (MADM) 64 problems with bipolar neutrosophic information where the weights of the attributes are completely 65 unknown to the decision maker. Pramanik et al. [20] defined projection, bidirectional projection and 66 hybrid projection measures for BNSs and proved their basic properties. In the same study, Pramanik 67 et al. [20], developed three new MADM methods based on the proposed projection, bidirectional 68 projection and hybrid projection measures with bipoar neutrosophic information. Wang et al. [21] 69 defined Frank operations of bipolar neutrosophic numbers (BNNs) and proposed Frank bipolar 70 neutrosophic Choquet Bonferroni mean operators by combining Choquet integral operators and 71 Bonferroni mean operators based on Frank operations of BNNs. In the same study, Wang et al. [21] 72 established MCDM method based on Frank Choquet Bonferroni operators of BNNs in bipolar 73 neutrosophic environment. Wu [22] defined several cross entropy measures between two 74 multivalued neutrosophic sets and employed the proposed method for selecting the middle-level 75 managers. In 2015, Ezhilmaran and Shankar [23] defined bipolar intuitionistic fuzzy sets, bipolar 76 intuitionistic fuzzy relations, and bipolar intuitionistic fuzzy graphs. The isomorphism of these 77 graphs and several properties of the graphs were also discussed in the same paer [23]. Mahmood et 78 al. [24] and Deli et al. [25] introduced the hybridized structure called interval bipolar neutrosophic 79 sets (IBNSs) by combining BNSs and INSs and defined some operations and operators for IBNSs. 80 In this paper, we define a cross entropy and weighted cross entropy measures of BNSs and 81 prove some of their properties and extend the concept to a cross entropy measure of IBNSs. Based on 82 the proposed cross entropy measures, we develop two new MADM methods to rank the alternatives Preprints ( | NOT PEER-REVIEWED | Posted: 3 of 23 3 and find the best alternative. Furthermore, two illustrative numerical examples are solved and 84 comparison analysis is given. 85 The rest of the paper is organized as follows. In section 2, we present some concepts regarding 86 SVNSs, INSs, BNSs, IBNSs. Section 3 proposes cross entropy and weighted cross entropy measures 87 of BNSs and investigates their properties. In the next section, we extend cross entropy measures of 88 BNSs to the cross entropy measures of IBNSs and discuss its basic properties. Two novel MADM 89 methods based on the proposed cross entropy measures under bipolar and interval bipolar 90 neutrosophic setting are devoted in section 5. In section 6, two numerical examples are solved and 91 comparison with other existing methods is given. At the end of the article, conclusions and scope of 92 future work are provided. 93 2. Preliminary 94
doi:10.3390/axioms7020021 fatcat:2d62nhhsgndhnivxusahn25moi