Distances and similarities in intuitionistic fuzzy sets

This book presents the state-of-the-art in theory and practice regarding similarity and distance measures for intuitionistic fuzzy sets. Quantifying similarity and distances is crucial for many applications, e.g. data mining, machine learning, decision making, and control. The work provides readers with a comprehensive set of theoretical concepts and practical tools for both defining and determining similarity between intuitionistic fuzzy sets. It describes an automatic algorithm for deriving intuitionistic fuzzy sets from data, which can aid in the analysis of information in large databases. The book also discusses other important applications, e.g. the use of similarity measures to evaluate the extent of agreement between experts in the context of decision making

Symmetry Between True, False, and Uncertain: An Explanation

In intuitionistic fuzzy sets, there is a natural symmetry between degrees of truth and falsity. As a result, for such sets, natural similarity measures are symmetric relative to an exchange of true and false values. It has been recently shown that among such measures, the most intuitively reasonable are the ones which are also symmetric relative to an arbitrary permutation of degrees of truth, falsity, and uncertainty. This intuitive reasonableness leads to a conjecture that such permutations are not simply mathematical constructions, that these permutations also have some intuitive sense. In this paper, we show that each such permutation can indeed be represented as a composition of intuitively reasonable operations on truth values. Need for intuitionistic fuzzy logic: a brief reminder. In the traditional fuzzy logic (see, e.g., [3, 4]), the degree of truth of each property P on an object x is characterized by a number µP (x) from the interval [0, 1]. It is usuall

New transitivity of Atanassov’s intuitionistic fuzzy sets in a decision making model

Atanassov’s intuitionistic fuzzy sets and especially his intuitionistic fuzzy relations are tools that make it possible to model effectively imperfect information that we meet in many real-life situations. In this paper, we discuss the new concepts of the transitivity problem of Atanassov’s intuitionistic fuzzy relations in an epistemic aspect. The transitivity property reflects the consistency of a preference relation. Therefore, transitivity is important from the point of view of real problems appearing, e.g., in group decision making in preference procedures. We propose a new type of optimistic and pessimistic transitivity among the alternatives (options) considered and their use in the procedure of ranking the alternatives in a group decision-making problem

Similarity measures for Atanassov’s intuitionistic fuzzy sets : some dilemmas and challenges

We discuss some aspects of similarity measures in the context of Atanassov’s intuitionistic fuzzy sets (IFSs, for short). IFSs, proposed in 1983, are a relatively new tool for the modeling and simulation and, because of their construction, present us with new challenges as far the similarity measures are concerned. Specifically, we claim that the distances alone are not a proper measure of similarity for the IFSs. We stress the role of a lack of knowledge concerning elements (options, decisions, etc.) and point out the role of the opposing (complementing) elements. We also pay attention to the fact that it is not justified to talk about similarity when one has not enough knowledge about the compared objects/elements. Some novel measures of similarity are presented

A novel approach to type-reduction and design of interval type-2 fuzzy logic systems

Fuzzy logic systems, unlike black-box models, are known as transparent artificial intelligence systems that have explainable rules of reasoning. Type 2 fuzzy systems extend the field of application to tasks that require the introduction of uncertainty in the rules, e.g. for handling corrupted data. Most practical implementations use interval type-2 sets and process interval membership grades. The key role in the design of type-2 interval fuzzy logic systems is played by the type-2 inference defuzzification method. In type-2 systems this generally takes place in two steps: type-reduction first, then standard defuzzification. The only precise type-reduction method is the iterative method known as Karnik-Mendel (KM) algorithm with its enhancement modifications. The known non-iterative methods deliver only an approximation of the boundaries of a type-reduced set and, in special cases, they diminish the profits that result from the use of type-2 fuzzy logic systems. In this paper, we propose a novel type-reduction method based on a smooth approximation of maximum/minimum, and we call this method a smooth type-reduction. Replacing the iterative KM algorithm by the smooth type-reduction, we obtain a structure of an adaptive interval type-2 fuzzy logic which is non-iterative and as close to an approximation of the KM algorithm as we like

Tailoring of the Magnetic Properties of Co, Fe and Ni Nanocrystallites

score: 0collation: 671-67