An Algorithmic Approach for Computing Unions and Intersections Between Fuzzy Multisets

Abstract

Fuzzy multisets represent a particularly challenging generalization of the concept of fuzzy sets. The membership degrees of fuzzy multisets are given by multisets in [0,1] rather than single values. Mathematically, they can be also seen as a generalization of the hesitant fuzzy sets. But in this general setting, the information about repetition is not lost with fuzzy multisets; and so, the opinions given by the experts are more reliably accounted for. The definitions of the complement, union, and intersection operations for these sets and their relation with other extensions of fuzzy sets, however, is not straightforward. Aggregate unions and intersections have been shown to be equivalent to the standard definitions of union and intersection for the typical hesitant fuzzy sets. But computing them is not simple because the definitions of the aggregate operations as multiset unions of sequences based on permutations can potentially result in a huge number of operations. In this paper, we propose a new formulation for the aggregate union and intersection of fuzzy multisets that is computationally less intensive, thereby providing two algorithms amenable to computer-based calculations.