Deviation Degree: A Perspective on Score Functions in Hesitant Fuzzy Sets

Score functions play an important role in ranking hesitant fuzzy elements (HFEs) and hesitant fuzzy sets (HFSs). Currently, various kinds of HFE and HFS score functions have been investigated in the literature. However, the essential characteristic and generation mechanism of these score functions have not been systematically studied. To address these issues, this paper introduces an axiomatic definition of deviation degree measure and proposes a general form of dual HFE and HFS deviation score functions, from which a family of existing HFE and HFS score functions can be derived. Besides, we develop two ranking methods based on a pair of dual deviation score functions for distinguishing HFEs and HFSs that are indiscernible by a single score function. Moreover, the mathematical and behavioral properties of HFS deviation score functions are analyzed for applying them in practice. Finally, the proposed ranking method for HFSs is applied to the multi-criteria decision-making problems with hesitant fuzzy information.