An order-oriented approach to scoring hesitant fuzzy elements
This work addresses a foundational issue in fuzzy set theory for researchers and practitioners in decision-making, but it is incremental as it builds on existing scoring approaches with a new theoretical perspective.
The paper tackled the problem of scoring hesitant fuzzy elements by proposing an order-oriented framework that defines scores relative to a given order, proving that classical orders do not induce lattice structures and that scores based on the symmetric order meet key normative criteria like strong monotonicity and the Gärdenfors condition.
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 Gärdenfors 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.