Improved cosine and cotangent function-based similarity measures for q-rung orthopair fuzzy sets and TOPSIS method

Muhammad Jabir Khan,Poom Kumam,Nasser Aedh Alreshidi,Wiyada Kumam

doi:10.1007/s40747-021-00425-7

Abstract

Despite the importance of cosine and cotangent function- based similarity measures, the literature has not provided a satisfactory formulation for the case of q-rung orthopair fuzzy set (qROFS). This paper criticizes the existing attempts in terms of respect of the basic axioms of a similarity measure and strict inclusion relation. In addition, the maximum operator-based similarity measures are criticized. Then, new improved, axiomatically supported cosine and cotangent function-based similarity measures for qROFSs are proposed. Additional properties of the new similarity measures are discussed to guarantee their good performance. Two algorithmic procedures of TOPSIS method that based on fixed and relative ideal solutions are discussed. The numerical examples are provided to support the findings

Highlights

Fuzziness, as developed in [1], is a kind of uncertainty which appears often in human decision-making problems
We propose modified similarity measures based on cosine and cotangent functions for q-rung orthopair fuzzy set (qROFS)
Equality holds for equal qROFSs and inequality for distinct qROFSs

Summary

Introduction

As developed in [1], is a kind of uncertainty which appears often in human decision-making problems. The fuzzy set theory deals with daily life uncertainties successfully. The membership degree is assigned to each element in a fuzzy set. In real-life situations, the nonmembership degrees should be considered in many cases as well, and it is not necessary that the non-membership degree be equal to the one minus the membership degree. Atanassov [2] introduced the concept of intuitionistic fuzzy set (IFS) that considers both membership and nonmembership degrees. The non-membership degree is not always obtained from a membership degree, which leads to the concept of hesitancy degree. The sum of its membership (ξ ), non-membership (ν), and hesitancy (π ) degrees should be equal to one for IFSs, that is ξ + ν + π = 1

Methods

Results

Conclusion