Information measures for hesitant fuzzy sets and interval-valued hesitant fuzzy sets

The main purpose of this paper is to investigate the relationship between the entropy, the similarity measure and the distance measure for hesitant fuzzy sets (HFSs) and interval-valued hesitant fuzzy sets (IVHFSs). The primary goal of the study is to suggest the systematic transformation of the entropy into the similarity measure for HFSs and vice versa. Achieving this goal is important to the task of introducing new formulas for the entropy and the similarity measure of HFSs. With results having been obtained for HFSs, similar results are also obtainable for IVHFSs. This paper also discusses the need for proposing a new entropy for HFSs and subsequently a new similarity measure for HFSs. Finally, two clustering algorithms are developed under a hesitant fuzzy environment in which indices of similarity measures of HFSs and IVHFSs are applied in data analysis and classification. Moreover, two practical examples are examined to compare the proposed methods with the existing ones. (C) 2013 Elsevier Inc. All rights reserved.