Distances and Similarities in Intuitionistic Fuzzy Sets

Dealing with imprecise information is a common task and challenge in everyday life, as uncertainty is inevitably involved in every real world system. Models are constructed to control, predict, and diagnose such systems, and so uncertainty should be properly incorporated into system description.

In the mid-1980s Atanassov introduced the concept of an intuitionistic fuzzy set. Basically, his idea was that unlike the conventional fuzzy sets in which imprecision is just modeled by the membership degree from [0,1], and for which the non-membership degree is just automatically the complementation to 1 of the membership degree, in an intuitionistic fuzzy set both the membership and nonmembership degrees are numbers from [0,1], but their sum is not necessarily 1. Thus, one can express a well known psychological fact that a human being who expresses the degree of membership of an element in a fuzzy set, very often does not express, when asked, the degree of non-membership as the complementation to 1. This idea has led to an interesting theory whose point of departure is such a concept of intuitionistic fuzzy set. In this chapter we give brief introduction to intuitionistic fuzzy sets. After recalling main definitions, concepts, operations and relations over crisp sets, fuzzy sets, and intuitionistic fuzzy sets we discuss interrelationships among the three types of sets. Two geometrical representations of the intuitionistic fuzzy sets, useful in further considerations are discussed. Finally, two approaches of constructing the intuitionistic fuzzy sets from data are presented. First approach is via asking experts. Second one - the automatic, and mathematically justified method to construct the intuitionistic fuzzy sets from data seems to be especially important in the context of analyzing information in big data bases.

In many theoretical and practical issues we face the following problem. Having two sets in the same universe, we want to calculate a difference between them exemplified by a distance. In this Chapter we consider distances between the intuitionistic fuzzy sets in two ways: while using the two term intuitionistic fuzzy set representation (membership values and non-membership values only are taken into account), and the three term intuitionistic fuzzy set representation (membership values, non-membership values, and hesitation margins are taken into account).We discuss norms and metrics for both types of representations. Both types are correct from the mathematical point of view but, in the practical perspective, the three term approach seems to be more justified. We discuss the problem in detail, considering its analytical, and geometrical aspects. We also show some problems with the Hausdorff distance, while the Hamming metric is applied when using the two term intuitionistic fuzzy set representation. We also show that the method of calculating the Hausdorff distances, which is correct for the interval-valued fuzzy sets, does not work for the intuitionistic fuzzy sets. Finally, we show the usefulness of the three term distances in a measure for ranking the intuitionistic fuzzy alternatives.

In this chapter we consider similarity measures between intuitionistic fuzzy sets starting fromreminding the axiomatic relation between distance and similarity measures.We show that this relation is not satisfied for the intuitionistic fuzzy sets. We also consider some similarity measures for the intuitionistic fuzzy sets, known from the literature. We show that neither similarity measures treating an intuitionistic fuzzy set as a simple interval-valued fuzzy set, nor straightforward generalizations of the similarity measures well-known for the classic fuzzy sets work under reasonable circumstances. Next, expanding upon our previous work, we consider a family of similarity measures constructed by taking into account both all the three terms (membership values, non-membership values, and hesitation margins) describing an intuitionistic fuzzy set, and the complements of the elements we compare. That is, we use all kinds and fine shades of information available. We also point out the traps one should be aware of while examining similarity between intuitionistic fuzzy sets. Finally, we consider correlation of the intuitionistic fuzzy sets.

The intuitionistic fuzzy sets are a generalization of fuzzy sets with an additional degree of freedom, as compared to fuzzy sets, which are fully described by the degree of membership. In the definition of an intuitionistic fuzzy set a degree of non-membership is added, and the value of membership plus the value of nonmembership for an element does not necessarily make one. Some psychological experiments demonstrate that in many judgments of human beings such a phenomenon happens. The additional degree of freedom means inherent possibility to model and process more adequately and more human consistently the imprecise information, and makes the intuitionistic fuzzy sets a useful tool in decision making.