Entropy for Hesitant Fuzzy Sets Based on Hausdorff Metric with Construction of Hesitant Fuzzy TOPSIS

In this paper, new entropy measures for hesitant fuzzy sets (HFSs) are proposed. Measuring uncertainty for an HFS is computed by an amount of distinction between an HFS and its complement. Hausdorff metric is used to calculate a distance between an HFS and its complement which assists us to construct novel entropy of HFSs. An axiomatic definition of entropy measure for HFSs is also given in this paper. The proposed entropy is proved to satisfy all axioms. Furthermore, more generalizations of the proposed entropy allow us to construct different entropy measures of HFSs which reflect that the closer of an HFS to its complement shows less distinction between them and produces the larger entropy measure of the HFS, and also the more distinction between them gives smaller amount of uncertainty. Several examples are presented to compare our entropy measures with existing methods. Finally, the TOPSIS method is used to construct hesitant fuzzy TOPSIS based on the proposed measures to solve multicriteria decision-making problems. We finally use expository examples to manifest simplicity, practicability and application effectiveness of our proposed entropies compared to existing methods.