An order-oriented approach to scoring hesitant fuzzy elements
arXiv:2602.16827v1 Announce Type: new 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
arXiv:2602.16827v1 Announce Type: new 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.
Executive Summary
This article proposes a novel order-oriented approach to scoring hesitant fuzzy elements, addressing the limitations of traditional methods. By explicitly defining scores with respect to a given order, the framework enables more flexible and coherent scoring mechanisms. The authors examine various classical orders and demonstrate that scores defined with respect to the symmetric order meet key normative criteria. They introduce dominance functions for ranking hesitant fuzzy elements and provide examples, showcasing their applicability in group decision-making. This contribution is significant, as it provides a unified framework for scoring hesitant fuzzy elements and facilitates the comparison of fuzzy sets.
Key Points
- ▸ Order-oriented approach to scoring hesitant fuzzy elements
- ▸ Explicit definition of scores with respect to a given order
- ▸ Symmetric order satisfies key normative criteria for scoring functions
- ▸ Introduction of dominance functions for ranking hesitant fuzzy elements
- ▸ Examples of dominance functions for finite sets: discrete and relative dominance functions
Merits
Strength in theoretical foundation
The order-oriented approach provides a unified framework for scoring hesitant fuzzy elements, grounded in order theory. This theoretical foundation enhances the coherence and flexibility of the framework.
Practical applicability
The dominance functions introduced in the article can be employed to construct fuzzy preference relations and support group decision-making, making the framework applicable in real-world scenarios.
Demerits
Complexity of the order-oriented approach
The order-oriented approach may be more complex to apply in practice, especially for those without a strong background in order theory. This could be a barrier to adoption, particularly in fields where hesitant fuzzy elements are not widely used.
Limited empirical evaluation
While the article provides theoretical foundations and examples, it would be beneficial to include empirical evaluations to further demonstrate the effectiveness and practicality of the order-oriented approach and dominance functions.
Expert Commentary
The article's contribution to the field of fuzzy set theory and decision-making is significant, as it addresses a long-standing limitation of traditional scoring approaches. The order-oriented approach and dominance functions introduced in the article offer a more flexible and coherent framework for scoring and comparing hesitant fuzzy elements. However, the complexity of the approach and the need for empirical evaluations to further demonstrate its effectiveness are important considerations. The article's findings have the potential to influence the development of new decision-making tools and methods, and its implications for policy decisions are also noteworthy.
Recommendations
- ✓ Further research is needed to investigate the empirical applicability and effectiveness of the order-oriented approach and dominance functions in real-world scenarios.
- ✓ Exploring the potential extensions and generalizations of the framework to other types of fuzzy sets and applications is a promising area of future research.
