Adrian I. Ban

Approximation of fuzzy numbers by trapezoidal fuzzy numbers preserving the expected interval

The problem to find the nearest trapezoidal approximation of a fuzzy number with respect to a well-known metric, which preserves the expected interval of the fuzzy number, is completely solved. The previously proposed approximation operators are improved so as to always obtain a trapezoidal fuzzy number. Properties of this new trapezoidal approximation operator are studied.

Approximations of fuzzy numbers by trapezoidal fuzzy numbers preserving the ambiguity and value

Value and ambiguity are two parameters which were introduced to represent fuzzy numbers. In this paper, we find the nearest trapezoidal approximation and the nearest symmetric trapezoidal approximation to a given fuzzy number, with respect to the average Euclidean distance, preserving the value and ambiguity. To avoid the laborious calculus associated with the Karush-Kuhn-Tucker theorem, the working tool in some recent papers, a less sophisticated method is proposed. Algorithms for computing the approximations, many examples, proofs of continuity and two applications to ranking of fuzzy numbers and estimations of the defect of additivity for approximations are given.

On the nearest parametric approximation of a fuzzy number---Revisited

The problem of finding the nearest parametric approximation of a fuzzy number with respect to the average Euclidean distance is completely solved. Properties of translation invariance, scale invariance, additivity, preservation of expected value, value and ambiguity of this new approximation operator are studied and an algorithm for computing it is provided.

Nearest interval, triangular and trapezoidal approximation of a fuzzy number preserving ambiguity

The ambiguity was introduced to simplify the task of representing and handling of fuzzy numbers. We find the nearest real interval, nearest triangular (symmetric) fuzzy number, nearest trapezoidal (symmetric) fuzzy number of a fuzzy number, with respect to average Euclidean distance, preserving the ambiguity. A simpler and elementary method, to avoid the Karush-Kuhn-Tucker theorem and the laborious calculus associated with it and to prove the continuity is used. We give algorithms for calculus and several examples. The approximations are discussed in relation to data aggregation.

Triangular and parametric approximations of fuzzy numbers---inadvertences and corrections

Recent papers were dedicated to approximate fuzzy numbers by triangular, trapezoidal or parametric fuzzy numbers, with or without additional conditions. Unfortunately, the result of approximation is not always a fuzzy number, sometimes it is not a fuzzy set. We point out the wrongs and inadvertences in some recent papers, then we correct the results.

Trapezoidal approximation and aggregation

Trapezoidal approximation of fuzzy numbers is discussed in relation to data aggregation. An important problem whether it is better to simplify initial data before using an aggregation operator or conversely, to aggregate original fuzzy values and then to simplify the output is addressed.

Developing new similarity measures of generalized intuitionistic fuzzy numbers and generalized interval-valued fuzzy numbers from similarity measures of generalized fuzzy numbers

Abstract The main aim of this paper is to give a novel method to extend a similarity measure of generalized trapezoidal fuzzy numbers (GTFNs) to similarity measures of generalized trapezoidal intuitionistic fuzzy numbers (GTIFNs) and generalized interval-valued trapezoidal fuzzy numbers (GIVTFNs) such that the initial properties to be preserved. The first idea is to consider the similarity measure of GIVTFNs as a combination of similarity measure between the lower GTFNs with the similarity measure between the upper GTFNs. The second idea is to find a bijective correspondence between GTIFNs and GIVTFNs such that the similarity measures can be transferred in an obvious way. The results in the triangular case are immediate consequences.

On the defect of additivity of fuzzy measures

It is well-known that fuzzy measures are non-additive. In this paper we introduce the concept of defect which gives us the degree of non-additivity of fuzzy measures. For large classes of fuzzy measures this defect is calculated and estimated. Three kinds of applications are given: to the approximative calculation of some fuzzy integrals, to the best approximation of a fuzzy measure by classical measures and to the introduction of an invariant at translations distance on the set of fuzzy measures.

Metric properties of the nearest extended parametric fuzzy number and applications

We propose the notion of extended parametric fuzzy number, which generalizes the extended trapezoidal fuzzy number and parametric fuzzy number, discussed in some recent papers. The metric properties of the nearest extended parametric fuzzy number of a fuzzy number, proved in the present article, help us to obtain the property of continuity for the parametric approximation operator and to simplify the solving of the problems of parametric approximations under conditions.

Simplifying the Search for Effective Ranking of Fuzzy Numbers

We prove that a ranking index which generates an order on some subset of fuzzy numbers can be reduced to a ranking index with a simpler form and which generates an equivalent order. This way, the finding of ranking indices generating orders with desirable properties can be essentially simplified such that we need to search only from additive or scale invariant ranking indices. We determine exactly, making an abstraction of equivalent orders, the class of ranking indices on trapezoidal fuzzy numbers which generate orders satisfying all the basic requirements. Finally, ranking indices used to rank trapezoidal fuzzy numbers are extended to ranking indices used to rank arbitrary fuzzy numbers so that the desired properties are preserved.