An order-oriented approach to scoring hesitant fuzzy elements

2026-02-18

AI Summary

This study addresses the lack of order-theoretic foundations in conventional scoring methods for hesitant fuzzy sets, which often struggle to balance consistency and flexibility. For the first time, order theory is systematically introduced to construct a unified scoring framework grounded in a given order relation. The proposed approach defines scoring mechanisms from an order-oriented perspective and introduces a novel paradigm of dominance functions based on control sets. Theoretical analysis demonstrates that, under symmetric orders, the proposed scoring satisfies strong monotonicity and the Gärdenfors condition. Moreover, the work clarifies a common misconception regarding the lattice structure of classical orders and constructs two concrete classes of dominance functions. The resulting framework effectively supports the ranking of hesitant fuzzy elements and facilitates group decision-making.

Technology Category

• Reasoning under Uncertainty: Decision/Utility Theory
• Knowledge Representation and Reasoning: Preferences
• Machine Learning: Learning Preferences or Rankings

Application Category

• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• User Modeling, Personalization and Recommendation: Fairness-aware retrieval and ranking
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

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 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.

Problem

• hesitant fuzzy sets
• scoring functions
• order theory
• lattice structures
• fuzzy preference relations

Innovation

• order-oriented scoring
• hesitant fuzzy sets
• symmetric order
• dominance functions
• fuzzy preference relations