A user-configurable and explainable framework for ranking fuzzy numbers

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.