A theoretical development on the entropy of interval-valued fuzzy sets based on the intuitionistic distance and its relationship with similarity measure

Abstract

In this contribution, we generalize recent results on the entropy of interval-valued fuzzy sets (IVFSs) based on the intuitionistic distance and its relationship with similarity measure. The intuitionistic distance provides us with a new axiomatic definition of entropy of IVFSs. Based on the set of new axioms, we also prove some theorems that entropy and similarity measure for IVFSs can be transformed by each other in a general way. The contribution ends by deriving a formula that creates a variety of new entropies by the use of given entropies of IVFSs.

Introduction

Among the extensions of the classic fuzzy sets, interval-valued fuzzy sets (IVFSs) [22], [27] and intuitionistic fuzzy sets (IFSs) [1], [31] are the most popular sets treating imprecision and uncertainty. In the setting of fuzzy set theory, it is pointed out that IVFSs and IFSs are equipollent generalizations of fuzzy sets [4], [6], [25].

In fuzzy set theory, entropy and similarity measure have drawn the attention of many researchers who studied these two concepts from different points of view. The notion of entropy of IFSs which is first introduced by Burillo and Bustince [3] allows us to measure the degree of intuitionism of an IFS. A non-probabilistic-type entropy measure with a geometric interpretation of IFSs was then proposed by Szimidt and Kacprzyk [21]. Making use of exploiting the concept of probability, Hung and Yang [9] gave their axiomatic definitions of entropy for IVFSs and IFSs. Furthermore, the other scholars introduced various entropy formulas for IVFSs [29], [30], [31] and IFSs [23], [24]. The notion of similarity measure of IFSs indicates the similarity degree of IFSs and was widely applied in many fields such as decision making [5], [19], [20] and pattern recognition [11], [17]. Szimidt and Kacprzyk [20] introduced a family of similarity measures using a distance measure. Hong and Kim [7], Hung and Yang [10] and Xu [26] proposed independently various similarity measures based on different distance measures for IFSs.

In view of the relationship between the entropy and the similarity measure for IFSs, Zeng and Guo [28] showed that a number of similarity measures and entropies for IVFSs can be deduced by normalized distances of IVFSs on the basis of their axiomatic definitions. Several researchers showed that similarity measures and entropies for IVFSs can be transformed by each other [29], [31]. For more study of the entropy and the similarity measure, the interested reader is referred to [8], [13], [14], [15], [16], [21], [26].

Consistently with a new axiomatic definition of entropy for IVFSs [13], [15], [31] which is briefly reviewed in Section 2.1, we propose here a class of entropies of IVFSs that generalizes the entropy of IVFSs induced by the intuitionistic distance. In order to investigate the relationship between the entropy and the similarity measure for IVFSs, we give some theorems which demonstrate that the entropy and the similarity measure for IVFSs can be transformed by each other. On the basis of the axiomatic definitions, we put forward some formulas to calculate the entropy and the similarity measure for IVFSs in a general way. Finally, we give a theorem that allows us to create a variety of entropies by the use of given entropies of IVFSs.

The structure of this contribution is as follows: Section 2 reviews some definitions and axioms used in the next analysis throughout this paper. Section 3 is devoted to the main results on the transformation of the information measures by each other. This paper is concluded in Section 4.