Counterintuitive Test Problems for Distance-Based Similarity Measures Between Intuitionistic Fuzzy Sets

Hui-Chin Tang,Shen-Tai Yang

doi:10.3390/math7050437

Abstract

This paper analyzes the counterintuitive behaviors of adopted twelve distance-based similarity measures between intuitionistic fuzzy sets. Among these distance-based similarity measures, the largest number of components of the distance in the similarity measure is four. We propose six general counterintuitive test problems to analyze their counterintuitive behaviors. The results indicate that all the distance-based similarity measures have some counterintuitive test problems. Furthermore, for the largest number of components of the distance-based similarity measure, four types of counterintuitive examples exist. Therefore, the counterintuitive behaviors are inevitable for the distance-based similarity measures between intuitionistic fuzzy sets.

Highlights

Fuzzy sets (FSs) theory, proposed by Zadeh [1], has successfully been applied in various fields.As a generalization of FSs, intuitionistic fuzzy sets (IFSs) proposed by Atanassov [2] is characterized by a membership function and a non-membership function
Much literature has been written on the counterintuitive examples for the similarity measures between two IFSs
This implies that A(a1, a2 ), B(a1, a2 ± α), C(c1, a2 ± α ∓ β) is a counterintuitive example for the similarity measure SLS3 (A, B)

Summary

Introduction

Fuzzy sets (FSs) theory, proposed by Zadeh [1], has successfully been applied in various fields. Adopted the normalized Hamming distance and the normalized Euclidean distance to present the similarity measures for IFSs. Szmidt and Kacprzyk [33] used hesitancy along with the membership and non-membership values of the elements in the sets being compared. This paper proposes six general counterintuitive test problems to analyze the counterintuitive behaviors of the distance-based similarity measures. We further present four types of counterintuitive test problems for the third Liang and Shi’s [30] similarity measure. To the best of our knowledge, the general test problems have not been yet presented to analyze the counterintuitive behaviors of the distance-based similarity measures.

IFSs and Similarity Measures

General Counterintuitive Test Problems

Conclusions and Future Research