An order-oriented approach to scoring hesitant fuzzy elements

TL;DR

This paper introduces an order-based framework for scoring hesitant fuzzy elements, providing a more flexible and theoretically grounded approach that improves ranking and decision-making processes.

Contribution

It develops a unified, order-oriented scoring framework for hesitant fuzzy sets, analyzing classical orders, and proposing dominance functions for better ranking and group decision-making.

Findings

Symmetric order scores satisfy key normative criteria.

Classical orders do not induce lattice structures.

Dominance functions enable effective ranking and decision support.

Abstract

Traditional scoring approaches on hesitant fuzzy sets often lack a formal base in order theory. This paper proposes a unified framework, where each score is explicitly defined with respect to a given order. This order-oriented perspective enables more flexible and coherent scoring mechanisms. We examine several classical orders on hesitant fuzzy elements, that is, nonempty subsets in [0,1], and show that, contrary to prior claims, they do not induce lattice structures. In contrast, we prove that the scores defined with respect to the symmetric order satisfy key normative criteria for scoring functions, including strong monotonicity with respect to unions and the G\"ardenfors condition. Following this analysis, we introduce a class of functions, called dominance functions, for ranking hesitant fuzzy elements. They aim to compare hesitant fuzzy elements relative to control sets incorporating minimum acceptability thresholds. Two concrete examples of dominance functions for finite sets are provided: the discrete dominance function and the relative dominance function. We show that these can be employed to construct fuzzy preference relations on typical hesitant fuzzy sets and support group decision-making.