Scores for Hesitant Fuzzy Sets: Aggregation Functions and Generalized Integrals

There are several extensions of fuzzy sets. Hesitant fuzzy sets are one of them. They are defined in terms of a set of membership degrees. For a typical hesitant fuzzy set, this set of membership degrees has a finite number of values. One of the motivations to introduce score functions was to rank alternatives. In this case, as the membership degrees are a set, the comparison of membership values does not lead, in general, to a total order. Score functions can be seen as functions that transform the set of membership degrees into a single membership value. In this way, we can construct the total order. In this article, we propose a general framework to define score functions for hesitant fuzzy sets based on fuzzy integrals. This framework permits to see most relevant indices as particular cases. Moreover, previous approaches focused on typical hesitant fuzzy sets. Our approach is more general in the sense that we can process both typical hesitant fuzzy sets and the nontypical ones (with membership values that are not finite). We also frame the problem into a more general setting. That is, the problem of hesitant fuzzy set transformation. Score functions can be seen as functions that transform a hesitant fuzzy set into a standard fuzzy set. Similarly, we can consider its transformation to interval-valued fuzzy sets and type-2 fuzzy sets. Aggregation functions can also be used for the same purpose.