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

doi:10.1016/j.fss.2010.10.004

Fuzzy Sets and Systems

Volume 165, Issue 1, 16 February 2011, Pages 81-97

https://doi.org/10.1016/j.fss.2010.10.004 Get rights and content

Abstract

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.

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The following approximations preserving operators had been studied: trapezoidal approximation preserving the expected interval proposed by Grzegorzewski and Mrówka [24,25] and improved in many papers [5,21,23,32,34], trapezoidal approximation preserving cores proposed by Grzegorzewski and Stefanini [26] and further studied by Abbasbandy and Hajjari [4], trapezoidal approximation preserving the value and ambiguity proposed by Ban et al. [8], trapezoidal approximation preserving ambiguity proposed by Ban and Coroianu [13], symmetric trapezoidal approximation preserving the x-centroid proposed by Wang and Li [31], and triangular approximation preserving the x-centroid proposed by Li et al. [28]. Some properties of this type of approximations are also studied [9,10,17]. In [14], the authors proved that not all (linear) operators can be preserved by trapezoidal approximations of fuzzy numbers.
Recently, many scholars studied approximations of fuzzy numbers by specific fuzzy numbers under preservation of some operators. In fact, these approximations may not exist for some linear operators. The purpose of this paper is to study necessary and sufficient conditions of linear operators which are preserved by interval, triangular, symmetric triangular, trapezoidal, or symmetric trapezoidal approximations of fuzzy numbers. In addition, an effective method for solving such problems is proposed.
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Trapezoidal approximation of fuzzy numbers was considered by many authors (see [1–12,14–16,22–24,26–29,32–38]).
Fuzzy number approximation by trapezoidal fuzzy numbers which preserves the expected interval is considered. New operators that fulfill additional requirements for the core and support of the fuzzy number are suggested. These supplementary conditions guarantee the proper interpretation of the solution even for very skew input fuzzy numbers. Moreover, the trapezoidal approximation without expected interval invariance is also examined and the relationship between these two approximations is discussed.

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

Abstract

Applied Mathematics and Computation (New York)

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