Interpretation Of Intuitionistic Fuzzy Sets Research Articles

Generalised operations on hesitant fuzzy values in the framework of Dempster–Shafer theory

The hesitant fuzzy sets theory (HFS) is probably the latest generalisation of fuzzy sets theory and seems to be especially useful in the solution of multiple criteria group decision making (MCGDM) problems, where it enables us to avoid some specific problems concerned with the aggregation of expert’s opinions. Currently there are only few different definitions and generalisations of HFS proposed in the literature. The key issue of hesitant fuzzy sets theory is the formulation of operation laws on the hesitant fuzzy elements (HFE), as they make it possible to use HFS for the solution of real-world problems. This paper presents a critical analysis of conventional operations on HFE and their applicability to the solution of multiple criteria decision making (MCDM) problems. It is shown that the known approaches to the definitions of HFS and corresponding operation laws have some important limitations and drawbacks. Therefore, a new generalised definition of HFS and operation laws based on the interpretation of intuitionistic fuzzy sets in the framework of the Dempster–Shafer theory of evidence (DST) are proposed and analysed. With the use of corresponding theorems it is proved that the proposed approach is free of limitations and drawbacks of known methods.The corresponding methods for aggregation of local criteria presented by HFEs in the framework of DST are proposed and analysed. The proposed approach allows us to solve MCDM and MCGDM problems without intermediate defuzzification when not only criteria, but their weights are HFEs. The advantages of the proposed approach are illustrated with numerical examples and the case study.

English

As Similarity measure for fuzzy sets is one of the important research topics of fuzzy set theory, there are several methods to measure similarity between two fuzzy sets (FS), Intuitionistic fuzzy sets (IFS) and Intuitionistic fuzzy multi sets (IFMS). In this paper, the Normalized Hamming Similarity measure of Intuitionistic Fuzzy Multi sets (IFMS) is introduced. This new measure for IFMS is based on the geometrical interpretation of IFS which involves both similarity and dissimilarity. Using this measure, the application of medical diagnosis and pattern recognition are shown.

New Entropy, Similarity Measure of Intuitionistic Fuzzy Sets and their Applications in Group Decision Making

Abstract In this paper, we propose a new entropy measure with geometrical interpretation of intuitionistic fuzzy sets. Compared with the entropy measure provided by Szmidt and Kacprzyk, the new entropy formula in this paper can measure both fuzziness and intuitionism for intuitionistic fuzzy sets. According to the relationship between entropy and similarity measure, we construct a new similarity measure for intuitionistic fuzzy sets. Then we present two methods, based on entropy and similarity measure, to determine weights of experts for multi-attribute group decision making with intuitionistic fuzzy information.

The operations on intuitionistic fuzzy values in the framework of Dempster–Shafer theory

This paper presents a critical analysis of conventional operations on intuitionistic fuzzy values (IFVs) and their applicability to the solution of multiple criteria decision making (MCDM) problems in the intuitionistic fuzzy setting. Two sets of operations on IFVs based on the interpretation of intuitionistic fuzzy sets in the framework of the Dempster–Shafer theory of evidence (DST) are proposed and analysed. This interpretation makes it possible to represent mathematical operations on IFVs as operations on belief intervals. The corresponding methods for aggregation of local criteria presented by IFVs in the framework of DST are proposed and analysed. The proposed approach allows us to solve MCDM problems without intermediate defuzzification when not only criteria, but their weights are IFVs. The advantages of the proposed approach are illustrated by numerical examples.

Consistency of 2D and 3D distances of intuitionistic fuzzy sets

For intuitionistic fuzzy sets, it is argued that three dimensional distance functions are not necessary because two dimensional distance functions already provide a simple and concise expression of the distance between two intuitionistic fuzzy sets. However, we show that a three dimensional interpretation of intuitionistic fuzzy sets could give different comparison results to the ones obtained with their two dimensional counterparts. In addition to the existing distances, we define a three dimensional Hausdorff distance and compare its consistency with its two dimensional counterpart, which shows the usefulness of the three dimensional functions to model and provide the expression of the distance between two intuitionistic fuzzy sets.

Fuzzy-logic-based network for complex systems risk assessment: Application to ship performance analysis

In this paper, a new interpretation of intuitionistic fuzzy sets in the advanced framework of the Dempster–Shafer theory of evidence is extended to monitor safety-critical systems’ performance. Not only is the proposed approach more effective, but it also takes into account the fuzzy rules that deal with imperfect knowledge/information and, therefore, is different from the classical Takagi–Sugeno fuzzy system, which assumes that the rule (the knowledge) is perfect. We provide an analytical solution to the practical and important problem of the conceptual probabilistic approach for formal ship safety assessment using the fuzzy set theory that involves uncertainties associated with the reliability input data. Thus, the overall safety of the ship engine is investigated as an object of risk analysis using the fuzzy mapping structure, which considers uncertainty and partial truth in the input–output mapping. The proposed method integrates direct evidence of the frame of discernment and is demonstrated through references to examples where fuzzy set models are informative. These simple applications illustrate how to assess the conflict of sensor information fusion for a sufficient cooling power system of vessels under extreme operation conditions. It was found that propulsion engine safety systems are not only a function of many environmental and operation profiles but are also dynamic and complex.

Objective weights with intuitionistic fuzzy entropy measures and computational experiment analysis

It is important to properly assess the weights of attributes when solving multi-attribute decision problems because variations in the weights often influence the rankings of the alternatives. In this paper, we propose an alternative objective weighting method to generate objective weights based on intuitionistic fuzzy (IF) entropy measures, which depend on the nature of a decision matrix in an intuitionistic fuzzy environment. Instead of the traditional fuzzy entropy measure, which is characterized by using the discriminating power to calculate the attribute weights, the proposed approach adopts the IF entropy measure, which emphasizes the credibility of the data. The IF entropy measure applied here is derived from a geometric interpretation of intuitionistic fuzzy sets and incorporates the concept of a ratio of distance measures. We implement four distance measures in the proposed approach and compare them in a computational experiment. The experimental results indicate that different IF entropy measures used in weighting methods can generate distinct objective attribute weights. In particular, when the number of attributes increases, the discrepancy between the IF entropy measures increases.

An interpretation of intuitionistic fuzzy sets in terms of evidence theory: Decision making aspect

This paper presents a new interpretation of intuitionistic fuzzy sets in the framework of the Dempster–Shafer theory of evidence ( DST). This interpretation makes it possible to represent all mathematical operations on intuitionistic fuzzy values as the operations on belief intervals. Such approach allows us to use directly the Dempster’s rule of combination to aggregate local criteria presented by intuitionistic fuzzy values in the decision making problem. The usefulness of the developed method is illustrated with the known example of multiple criteria decision making problem. The proposed approach and a new method for interval comparison based on DST, allow us to solve multiple criteria decision making problem without intermediate defuzzification when not only criteria, but their weights are intuitionistic fuzzy values.

Intuitionistic fuzzy sets: Spherical representation and distances

Most existing distances between intuitionistic fuzzy sets are defined in linear plane representations in 2D or 3D space. Here, we define a new interpretation of intuitionistic fuzzy sets as a restr...

Fuzzy entropy on intuitionistic fuzzy sets: Research Articles

In this article we exploit the concept of probability for defining the fuzzy entropy of intuitionistic fuzzy sets (IFSs). We then propose two families of entropy measures for IFSs and also construct the axiom definition and properties. Two definitions of entropy for IFSs proposed by Burillo and Bustince in 1996 and Szmidt and Kacprzyk in 2001 are used. The first one allows us to measure the degree of intuitionism of an IFS, whereas the second one is a nonprobabilistic-type entropy measure with a geometric interpretation of IFSs used in comparison with our proposed entropy of IFSs in the numerical comparisons. The results show that the proposed entropy measures seem to be more reliable for presenting the degree of fuzziness of an IFS. © 2006 Wiley Periodicals, Inc. Int J Int Syst 21: 443–451, 2006.

On entropy for intuitionistic fuzzy sets applying the Euclidean distance

Recently, Szmidt and Kacprzyk[Fuzzy Sets and Systems 118(2001) 467-477] proposed a non-probabilistic-type entropy measure for intuitionistic fuzzy sets. Tt is a result of a geometric interpretation of intuitionistic fuzzy sets and uses a ratio of distances between them. They showed that the proposed measure can be defined in terms of the ratio of intuitionistic fuzzy cardinalities: of <TEX>$F\bigcapF^c and F\bigcupF^c$</TEX>, while applying the Hamming distances. In this note, while applying the Euclidean distances, it is also shown that the proposed measure can be defined in terms of the ratio of some function of intuitionistic fuzzy cardinalities: of <TEX>$F\bigcapF^c and F\bigcupF^c$</TEX>.

Entropy for intuitionistic fuzzy sets

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 .