New unified score functions and similarity measures for non-standard fuzzy numbers: an extended TOPSIS method addressing risk attitudes

Abstract

In this study, we propose a new form of score function for both intuitionistic and picture fuzzy sets, which we term the weighted average membership function. By inserting both refusal and neutrality degrees for picture fuzzy sets, we extend the idea of modifying memberships by including hesitancy for intuitionistic fuzzy sets. This new representative membership function can be used to rank and defuzzify fuzzy numbers. Using this idea of updating the membership function allows us to consider all the knowledge that both fuzzy set generalizations can offer, similar to ordinary fuzzy concepts. Moreover, any method of handling uncertainty for standard fuzzy sets can be mimicked. The proposed convex combination type of score function is a generalization that produces special cases for some proper values of the function parameters and collects some linear score functions from the literature under one roof. Also, it provides an extended approach by incorporating decision behaviors into a more general scope. This paper addresses the applications of both assigning a single value to a non-standard fuzzy number and an extended TOPSIS method that considers decision makers' optimism degrees with the aid of a newly defined similarity measure. To demonstrate its effectiveness, illustrative examples and simulation studies are presented.