Eulalia Szmidt

Distances between intuitionistic fuzzy sets

A geometrical representation of an intuitionistic fuzzy set is a point of departure for our proposal of distances between intuitionistic fuzzy sets. New definitions are introduced and compared with the approach used for fuzzy sets. It is shown that all three parameters describing intuitionistic fuzzy sets should be taken into account while calculating those distances.

Entropy for intuitionistic fuzzy sets

Abstract A non-probabilistic-type entropy measure for intuitionistic fuzzy sets is proposed. It is a result of a geometric interpretation of intuitionistic fuzzy sets and uses a ratio of distances between them proposed in Szmidt and Kacprzyk (to appear). It is also shown that the proposed measure can be defined in terms of the ratio of intuitionistic fuzzy cardinalities: of F ∩ F c and F ∪ F c .

A consensus-reaching process under intuitionistic fuzzy preference relations

We extend the main idea of a fuzzy analysis of consensus—that is based on a concept of a distance from consensus—to a case when individual testimonies are individual intuitionistic fuzzy preference relations, as opposed to fuzzy preference relations commonly used. Intuitionistic fuzzy preference relations, that in addition to a membership degree (from [0, 1]) include a hesitation margin (concerning the membership degree), can better reflect the very imprecision of testimonies (expressing preferences) of the individuals during the consensus‐reaching process. Our new solution, obtained as an interval‐valued measure of a distance from consensus, better reflects both real human perception and a soft nature of consensus.

A Similarity Measure for Intuitionistic Fuzzy Sets and Its Application in Supporting Medical Diagnostic Reasoning

We propose a new similarity measure for intuitionistic fuzzy sets and show its usefulness in medical diagnostic reasoning. We point out advantages of this new concept over the most commonly used similarity measures being just the counterparts of distances. The measure we propose involves both similarity and dissimilarity.

A new concept of a similarity measure for intuitionistic fuzzy sets and its use in group decision making

We propose a new measure of similarity for intuitionistic fuzzy sets, and use it to analyze the extent of agreement in a group of experts. The proposed measure takes into account two kinds of distances – one to an object to be compared, and one to its complement. We infer about the similarity of preferences on the basis of a difference between the two types of distances. We show that infering without taking into account a distance to a complement of an object can be misleading.

Intuitionistic Fuzzy Sets in some Medical Applications

For many real world problems, imperfect, imprecise information is, by nature, part of the problem itself and continuing reasoning without proper modelling tools may led to generating inaccurate inferences.

An intuitionistic fuzzy set based approach to intelligent data analysis: an application to medical diagnosis

We propose a new approach for medical diagnosis by employing intuitionistic fuzzy sets (cf. Atanassov [1]; [2]) which because of additional degree of freedom in comparison with fuzzy sets (Zadeh [14]), can be viewed as their generalization. Employing intuitionistic fuzzy sets, we can simply and adequately express a hesitation concerning the objects considered - both patients and illnesses. Solution is obtained by looking for the smallest distance (cf. Szmidt and Kacprzyk [8], [11]) between symptoms that are characteristic for a patient and symptoms describing illnesses considered. We point out advantages of this new technique over the method proposed by De, Biswas and Roy [4] where intuitionistic fuzzy sets were also applied but the max-min-max composition of intuitionistic fuzzy relations was used instead of taking into account all, unchanged symptom values as proposed in this article.

How to measure the amount of knowledge conveyed by Atanassov's intuitionistic fuzzy sets

We address the problem of how to measure the amount of knowledge conveyed by the Atanassovs intuitionistic fuzzy set (A-IFS for short). The problem is relevant from the point of view of many application areas, notably decision making. An amount of knowledge considered is strongly linked to its related amount of information. Our analysis is concerned with an intrinsic relationship between the positive and negative information and a lack of information expressed by the hesitation margin. Illustrative examples are shown.

Amount of Information and Its Reliability in the Ranking of Atanassov’s Intuitionistic Fuzzy Alternatives

In this paper we discuss the ranking of alternatives represented by elements of Atanassov’s intuitionistic fuzzy sets, to be called A-IFSs, for short. That is, alternatives are elements of the universe of discourse with a degree of membership and a degree of non-membership assigned. First, we show disadvantages of some approaches known from the literature, including a straightforward method based on the calculation of distances from the ideal positive alternative which can be viewed as a counterpart of the approach in the traditional fuzzy setting. Instead, we propose an approach which takes into account not only the amount of information related to an alternative (expressed by a distance from an ideal positive alternative) but also the reliability of information represented by an alternative meant as how sure the information is.

Some Problems with Entropy Measures for the Atanassov Intuitionistic Fuzzy Sets

This paper is a continuation of our previous papers on entropy of the Atanassov intuitionistic fuzzy sets (A-IFSs, for short). We discuss the necessity of taking into account all three functions (membership, non-membership and hesitation margin) describing A-IFSs while considering the entropy.