Similarity and entropy measures for hesitant fuzzy sets

Hesitant fuzzy sets (HFSs) are beneficial tools for expressing the hesitancy of decision makers (DMs) to access alternatives in daily life, thereby enabling the membership of an element to a set that is represented by several possible values. This study proposes an interval bound footprint (IBF), which describes the fluctuation range of the values of hesitant fuzzy elements (HFEs) arranged in order. In addition, a few similarity and entropy measures for HFSs are deduced. First, the interval bound footprint, upper bound footprint, and lower bound footprint for HFEs are defined and their corresponding properties are discussed. Subsequently, several similarity and entropy measures for HFSs are presented based on IBF. Lastly, a hesitant fuzzy multi-criteria decision-making method based on the proposed similarity and entropy measures is introduced. We use a numerical example to discuss the differences among the proposed similarity measures and the applicable environment based on the risk preferences of the different DMs.