Article

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

Computers & Mathematics with Applications (Impact Factor: 2). 03/2011; 61:1379-1401. DOI: 10.1016/j.camwa.2011.01.005

Source: DBLP

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**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.Fuzzy Sets and Systems 01/2014; 257:3–22. · 1.88 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**In this paper firstly we extend to an arbitrary compact interval the definition of the nonlinear Bernstein operators of max-product kind, by proving that their order of uniform approximation is the same as in the particular case of the unit interval. Then, similarly to the particular case of the unit interval, we will prove that these operators preserve the quasi-concavity too. Since these properties will help us to generate in a simple way fuzzy numbers of the same support, it turns out that these results are very suitable in the approximation of fuzzy numbers. Moreover, these operators approximate the (non-degenerate) segment core with a good order of approximation. In addition, in the case when the fuzzy numbers are given in parametric form, the Bernstein max-product operator generates a sequence of fuzzy numbers whose widths, expected intervals, ambiguities and values, approximate with a convergent rate the width, the expected interval, the ambiguity and the value of the approximated fuzzy number. Finally, we obtain a quantitative approximation with respect to a well-known integral type metric for some subclasses of functions, leading to applications to the approximation of some subclasses of fuzzy numbers.Fuzzy Sets and Systems 01/2014; 257:41–66. · 1.88 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**Recently, many scholars investigated interval, triangular, and trapezoidal approximations of fuzzy numbers. These publications can be grouped into two classes: Euclidean distance class and non-Euclidean distance class. Most approximations in Euclidean distance class can be calculated by formulas, but calculating approximations in the other class is more complicated. Furthermore, approximations in Euclidean distance class can be divided into two subclasses. One is to study approximations of fuzzy numbers without constraints, the other one is to study approximations preserving some attributes. In this paper, we use LR-type fuzzy numbers to approximate fuzzy numbers. The proposed approximations will generalize all recent approximations without constraints in Euclidean class. Also, an efficient formula is provided.Fuzzy Sets and Systems 01/2014; 257:23–40. · 1.88 Impact Factor

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