An order-oriented approach to scoring hesitant fuzzy elements

Technical Deep Dive: An Order-Oriented Approach to Scoring Hesitant Fuzzy Elements

Overview

• This paper proposes a unified framework for scoring hesitant fuzzy sets (HFSs) that is explicitly defined with respect to a given order.
• Traditional scoring approaches often lack a formal base in order theory. The authors address this by examining several classical orders on hesitant fuzzy elements (HFEs) and showing how the choice of order shapes the properties of scoring functions.
• The authors introduce a new class of functions called "dominance functions" that aim to compare HFEs relative to control sets incorporating minimum acceptability thresholds. This provides a more flexible and interpretable approach to ranking and decision-making with HFEs.

Background and Context

• Hesitant fuzzy sets (HFSs) represent membership degrees as sets of possible values rather than single values, allowing them to handle uncertainty and hesitation more effectively than traditional fuzzy sets.
• Comparing and ranking HFEs is complicated by the lack of a total order in the class of all possible membership degrees.
• Scores have emerged as an essential tool for defining systematic ranking methods over HFEs, but their properties have not been fully examined in the context of order theory.

Methodology

1. Review of orders on hesitant fuzzy elements:
   • Examined classical orders like the list order, pessimistic order, and optimistic order, showing that they do not induce lattice structures.
   • Introduced the symmetric order that endows the class of all nonempty subsets of [0,1] with a lattice structure.
2. Defining scores relative to an order:
   • Proposed a new definition of scores that explicitly links them to a given order structure, rather than just the arithmetic mean.
   • Showed that scores related to the symmetric order satisfy key admissible properties like strong monotonicity with respect to unions and the Gärdenfors condition.
3. Introducing dominance functions:
   • Defined a new class of "dominance functions" that compare HFEs relative to control sets encoding minimum acceptability thresholds.
   • Provided two concrete examples of dominance functions - the discrete dominance function (DDF) and the relative dominance function (RDF) - and analyzed their properties.
   • Demonstrated how dominance functions can be used to construct fuzzy preference relations for group decision-making.

Results

• Showed that the symmetric order induces a lattice structure on the class of all nonempty subsets of [0,1], unlike other classical orders on HFEs.
• Proved that scores defined with respect to the symmetric order satisfy important normative properties like strong monotonicity and the Gärdenfors condition.
• Introduced dominance functions as a flexible tool for incorporating baseline requirements into the evaluation of HFEs, going beyond the traditional "mean value" interpretation of scores.
• Demonstrated how dominance functions can be used to construct fuzzy preference relations for group decision-making problems.

Interpretation and Significance

• The order-theoretic approach to scoring HFEs provides a more principled and flexible framework, moving beyond the limitations of previous score-based methods.
• Dominance functions offer a novel way to incorporate context-specific requirements into the evaluation of HFEs, which is crucial for many practical decision-making scenarios.
• The lattice structure induced by the symmetric order and the associated scoring properties suggest it may be a more suitable foundation for developing hesitant fuzzy set theory and applications.

Limitations and Uncertainties

• The paper focuses on theoretical analysis and does not provide extensive empirical validation of the proposed methods.
• The practical implications and comparative performance of the DDF and RDF dominance functions require further investigation in diverse decision-making contexts.
• The generalization of the order-oriented scoring approach and dominance functions to broader classes of hesitant fuzzy sets (beyond THFEs) is an open question.

Future Work

• Empirical studies to assess the performance of the proposed dominance functions in real-world decision-making scenarios.
• Exploration of the applications of the symmetric order and associated scoring properties in hesitant fuzzy set theory and related fields.
• Generalization of the order-oriented scoring framework and dominance functions to more expressive hesitant fuzzy set representations.
• Investigation of the connections between order-based scores and other hesitant fuzzy set operators, such as aggregation functions and negations.