Abstract
Fuzzy number ranking is an important and fundamental subject matter within fuzzy set theory, just as the ranking of crisp values is important and fundamental to classical mathematics. For a user of fuzzy set theory, a fuzzy number ranking method that is user-configurable and user-explainable is highly desirable -- being user-configurable is important for ensuring the fitness of the fuzzy ranking method in relation with the assumptions of the application domain of interest, and being user-explainable is important for providing confidence to the user and the stakeholders of the appropriateness of the ranking outcomes in an expert system. In this article, a simple framework for ranking fuzzy numbers is proposed and developed. This simple framework consists of concrete constructs similar to the building blocks in a LEGO set, for the user to configure his/her own fuzzy ranking operator. The proposed framework is applied to examples taken from previous comparative studies from the fuzzy set theory literature, and to multiple-criteria-decision-making and other fuzzy ranking problems. First of all, these demonstrations will illustrate that fuzzy number ranking can indeed be done in a concrete, user-configurable, and user-explainable manner. Thus, the proposed framework represents a viable alternative for the fuzzy ranking literature, which currently consists of various ranking methods that are mathematically sophisticated yet oftentimes highly non-trivial for the user to interpret and to adapt/re-configure for his/her applications. Secondly, we note that the precursor of the proposed framework has been successfully employed to re-derive, instead of postulate, commonly used fuzzy set-theoretic operators, and to derive fuzzy arithmetic operators that do not produce anomalies plaguing other fuzzy arithmetic approaches. Thus, the applicability of such a framework to fuzzy ranking will further demonstrate its potential as a fundamental framework for tackling a wide range of issues in fuzzy set theory, rather than being an isolated approach limited to narrow aspects of fuzzy set theory.
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