Lucian C. Coroianu

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.

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.

Nearest piecewise linear approximation of fuzzy numbers

Abstract The problem of the nearest approximation of fuzzy numbers by piecewise linear 1-knot fuzzy numbers is discussed. By using 1-knot fuzzy numbers one may obtain approximations which are simple enough and flexible to reconstruct the input fuzzy concepts under study. They might be also perceived as a generalization of the trapezoidal approximations. Moreover, these approximations possess some desirable properties. Apart from theoretical considerations approximation algorithms that can be applied in practice are also given.

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.

Lipschitz functions and fuzzy number approximations

We prove that some important properties of convex subsets of vector topological spaces remain valid in the space of fuzzy numbers endowed with the Euclidean distance. We use these results to obtain a characterization of fuzzy number-valued Lipschitz functions. This fact helps us to find the best Lipschitz constant of the trapezoidal approximation operator preserving the value and ambiguity introduced in a recent paper. Finally, applications in finding within a reasonable error the trapezoidal approximation of a fuzzy number preserving the value and ambiguity in the case when the direct formula is not applicable and an estimation for the defect of additivity of the trapezoidal approximation preserving the value and ambiguity are given.

Approximation and Shape Preserving Properties of the Bernstein Operator of Max-Product Kind

Starting from the study of the Shepard nonlinear operator of max-prod type by Bede et al. (2006, 2008), in the book by Gal (2008), Open Problem 5.5.4, pages 324–326, the Bernstein max-prod-type operator is introduced and the question of the approximation order by this operator is raised. In recent paper, Bede and Gal by using a very complicated method to this open question an answer is given by obtaining an upper estimate of the approximation error of the form 𝐶𝜔1√(𝑓;1/𝑛) (with an unexplicit absolute constant 𝐶g0) and the question of improving the order of approximation 𝜔1√(𝑓;1/𝑛) is raised. The first aim of this note is to obtain this order of approximation but by a simpler method, which in addition presents, at least, two advantages: it produces an explicit constant in front of 𝜔1√(𝑓;1/𝑛) and it can easily be extended to other max-prod operators of Bernstein type. However, for subclasses of functions 𝑓 including, for example, that of concave functions, we find the order of approximation 𝜔1(𝑓;1/𝑛), which for many functions 𝑓 is essentially better than the order of approximation obtained by the linear Bernstein operators. Finally, some shape-preserving properties are obtained.

Best Lipschitz constant of the trapezoidal approximation operator preserving the expected interval

In this paper, we prove new distance properties between a fuzzy number and its trapezoidal approximation preserving the expected interval. Then, we find the best Lipschitz constant of the trapezoidal approximation operator preserving the expected interval. Finally, we use this result in finding within a reasonable error the trapezoidal approximation of a fuzzy number preserving the expected interval in the case when the direct formula leads to complex calculations.

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.

Existence, uniqueness and continuity of trapezoidal approximations of fuzzy numbers under a general condition

Abstract The main aim of this paper is to characterize the set of real parameters associated to a fuzzy number, represented in a general form which include the most important characteristics, with the following property: for any given fuzzy number there exists at least a trapezoidal fuzzy number which preserves a fixed parameter. The uniqueness of the nearest trapezoidal fuzzy number with this property is proved, the average Euclidean distance being considered. As an important property, each resulting trapezoidal approximation operator is continuous. The main results are illustrated by examples.