Entropy and Cross-entropy for Generalized Hesitant Fuzzy Information and Their Use in Multiple Attribute Decision Making

In this paper, we present the entropy, cross-entropy, and similarity measure for generalized hesitant fuzzy information and discuss their desirable properties. Some measure formulas are developed, and the relationships among them are investigated. We show that the similarity measure and entropy for generalized hesitant fuzzy information can be transformed by each other based on their axiomatic definitions. Then we develop two approaches for solving multiple attribute decision making, in which the attribute values are given in the form of generalized hesitant fuzzy elements (GHFEs). In the first approach, the attribute weight vector is determined by the generalized hesitant fuzzy entropies, and the optimal alternative is obtained by comparing the generalized hesitant fuzzy cross-entropies between alternatives and positive-ideal or negative-ideal solutions; in the second approach, the attribute weight vector is derived from the maximizing deviation method and optimal alternative is obtained by using the technique for order preference by similarly to ideal solution (TOPSIS) method. Finally, an example is provided to illustrate the practicality and effectiveness of the developed approaches. (C) 2016 Wiley Periodicals, Inc.