Entropy measures for hesitant fuzzy sets and their application in multi-criteria decision-making

Entropy is used to measure the uncertain degree of fuzzy sets and has been widely used in many fields. This paper introduces an axiomatic definition of entropy measure and a novel entropy formula for hesitant fuzzy elements (HFEs). Afterwards, a general form of entropy measures for HFEs is proposed, from which a family of concrete entropy formulas for HFEs can be derived. Compared with the existing ones, these formulas can measure both fuzziness and hesitation of HFEs, as a result, the uncertain information can be described in a more appropriate manner. The proposed axiomatic definition and entropy formulas are used to define the entropy measure for hesitant fuzzy sets. A multi-criteria decision making model which uses the proposed entropy measures for HFEs to compute the criteria weights and obtain a ranking of alternatives is introduced.