Evaluation and ranking of fuzzy sets under equivalence fuzzy relations as [formula omitted]certainty and [formula omitted]possibility

Abstract

In this paper, we deal with a finite collection of fuzzy sets that are used to represent a given linguistic variable in real-world problems. These fuzzy sets are usually represented as fuzzy singletons, fuzzy numbers, or fuzzy G-numbers. In many cases, there are significant interactions between elements of a given set, so it is useful to evaluate and rank elements while taking the degree of interactions into consideration. To find the degree of interaction, we develop a fuzzy equivalence relation based on a similarity function and a triangular norm operator. The compositions of the (max -(T-norm)) and (min -(S-norm)) cause the membership function to change from the minimum values (certainty version) to the maximum values (possibility version). These concepts are also useful for improving fuzzy data mining, rough fuzzy set theory, etc. Furthermore, two indices, α and β, are introduced to determine the degree of certainty (α-certainty) and degree of possibility (β-possibility). Examples are provided to compare the proposed method with the existing method. After comparing the experimental results, it can be concluded that the proposed approach is satisfactory for many situations. Finally, we apply the proposed methodology to evaluate the results of a hospital efficiency study conducted for the Office of Inspector General(OIG)-Eastern Virginia Department of Health and Human Services.