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There are several extensions of fuzzy sets in the literature. While the original definition by Zadeh defined membership of an element to a set by means of a value in the [0,1] interval, extensions replace this value in [0,1] by a more general mathematical object. This is the case of hesitant fuzzy sets that replace this value by a subset of [0,1]. More typically, hesitant fuzzy sets are defined in terms of finite subsets of [0,1].

For fuzzy sets, there is a total order for the elements of the reference set based on the membership function. Nevertheless, this is not the case for most of the extensions. In particular, this does not hold for hesitant fuzzy sets.

Because of that, score indices have been introduced for hesitant fuzzy sets to provide a way to order them. In short, score indices permits us to build a total order of the elements of the reference set. This total order is built taking into account the set of membership …