Cross-Entropy of Dual Hesitant Fuzzy Sets for Multiple Attribute Decision-Making

Cross-Entropy of Dual Hesitant Fuzzy Sets for Multiple Attribute Decision-Making

Jun Ye (Department of Electrical and Information Engineering, Shaoxing University, Shaoxing, China)
Copyright: © 2016 |Pages: 11
DOI: 10.4018/IJDSST.2016070102
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Abstract

This paper proposes a cross-entropy measure between dual hesitant fuzzy sets (DHFSs) as an extension of the cross-entropy measures of intuitionistic fuzzy sets. Then the cross-entropy measure between DHFSs is applied to multiple attribute decision making under dual hesitant fuzzy environments. Through the weighted cross-entropy measure between each alternative and the ideal alternative, we can obtain the ranking order of all alternatives and the best one. The decision-making method based on the cross-entropy measure of DHFSs can deal with dual hesitant fuzzy multiple attribute decision making problems and can automatically take into account much more information than existing hesitant (or intuitionistic) fuzzy decision-making methods and the differences of the evaluation data given by different experts or decision makers. Finally, a practical example about investment alternatives is given to demonstrate the application and effectiveness of the developed approach.
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1. Introduction

Cross-entropy measures are an important research topic in the fuzzy set theory and mainly measure the discrimination information. Until now, a lot of research has been done about this issue. Kullback and Leibler (1951) firstly proposed a measure of the ‘‘cross entropy distance’’ between two probability distributions. Later, Lin (1991) introduced a modified cross-entropy measure. Then, Shang and Jiang (1997) presented fuzzy cross-entropy and a symmetric discrimination information measure between two fuzzy sets. As an extension of the De Luca-Termini nonprobabilistic entropy for intuitionistic fuzzy sets (IFSs) (Atanassov, 1986), Vlachos and Sergiadis (2007) introduced the concepts of intuitionistic fuzzy cross entropy and discrimination information for IFSs. Furthermore, Ye (2009) has applied the intuitionistic fuzzy cross entropy to multicriteria fuzzy decision-making problems. Then, Ye (2011) proposed a fuzzy cross entropy measure of the interval-valued intuitionistic fuzzy sets (IVIFSs) (so-called IVIFS cross entropy) by analogy with the intuitionistic fuzzy cross entropy and its optimal decision-making method based on the weights of alternatives in which criterion values for alternatives are IVIFSs and the criteria weights are known information. Recently, Xia and Xu (2012) proposed some cross-entropy and entropy formulas for intuitionistic fuzzy sets and applied them to group decision-making problems. Because the hesitant fuzzy set proposed by Torra and Narukawa (2009) and Torra (2010) can express much more information given by decision makers, Xu and Xia (2011) presented the distance and similarity measures for hesitant fuzzy sets. Xu and Xia (2012) have developed some information measures under hesitant fuzzy environment, including the entropy, cross-entropy, and similarity measures, and have proven several theorems that the entropy, cross-entropy, and similarity measures of hesitant fuzzy elements can be transformed by each other based on their axiomatic definitions, and then have established two hesitant fuzzy multiple attribute decision-making methods, which permit decision-makers to provide several possible values for an alternative under the given attributes with humans’ hesitant thinking.

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