Modified expression to evaluate the correlation coefficient of dual hesitant fuzzy sets and its application to multi-attribute decision making

Abstract

The main objective of this paper is to understand all the existing correlation coefficients (CoCfs) to determine the relation and dependency between two variables of the fuzzy sets and its extensions for solving decision-making (DM) problems. To study the weighted CoCfs between two variables the environment chosen here is dual hesitant fuzzy set (DHFS) which is a generalization of a fuzzy set which considers the hesitant value of both the membership and non-membership elements of a set. Although there exists CoCfs for DHFS but a detailed mathematical analysis suggests that there exists some shortcomings in the existing CoCfs for DHFS. Thus, an attempt has been made to properly understand the root cause of the posed limitation in the weighted CoCfs for DHFS and hence, modified weighted CoCfs for DHFS has been proposed for solving DHFS multi-attribute decision making (MADM) problems i.e., DM problems in which rating value of each alternative over each criterion is represented by a DHFS in the real-life. Also, to validate the proposed expressions of weighted CoCfs for solving DHFS MADM problems, an existing real-life problem is evaluated and a systematic comparison of the solution is presented for clarification.

Modified Expression to Evaluate the Correlation Coefficient of Dual Hesitant Fuzzy Sets and Its Application to Multi-Attribute Decision Making

“…To rank fuzzy sets and its generalizations there exists various ranking measures like distance measures, similarity measures, score function, accuracy function, certainty function, divergence measure, CoCfs etc. Although in literature there exist expressions to evaluate the CoCf between fuzzy sets and many of its extensions as proposed by several researchers like, the CoCf between two fuzzy sets [8], the CoCf between two intuitionistic fuzzy sets [9][10][11][12][13][14][15][16][17], the CoCf between two interval-valued intuitionistic fuzzy sets [18], the CoCf between two Pythagorean fuzzy sets [19,20], the CoCf between two intuitionistic multiplicative sets [21], the CoCf between two hesitant fuzzy sets [22][23][24][25][26][27], the CoCf between two dual hesitant fuzzy sets [28][29][30] etc.. Ye [29] proposed an expressions for evaluating the weighted CoCfs between two DHFSs and solved a real-life problem (finding the best investment company) where the uncertainty is represented as a DHFS. However, after a deep study, it is observed that some mathematical incorrect assumptions are considered in the existing weighted CoCf and hence it is scientifically incorrect to apply existing weighted CoCf in real-life MADM problems for DHFSs in its present form.…”