Approximation of fuzzy numbers by nonlinear Bernstein operators of max-product kind

Abstract

In this paper firstly we extend from [0, 1] to an arbitrary compact interval [a, b], the definition of the nonlinear Bernstein operators of max-product kind, B

A note on fuzzy-transform approximation of fuzzy numbers

“…The approximation of F u (x) will allow the construction of u(x) by iv), iii) and ii) of Proposition (4). As we have seen, the membership function u(x) of a continuous fuzzy number can be obtained from the corresponding midpoint distribution function F u (x) (a monotonic function) by Proposition (4).32…”

“…First of all we have to notice that any approximation operator which preserves the monotonicity and which interpolates the function F u at the endpoints of supp(u) can be used to generate a fuzzy number which approximates u. There are also nonlinear operators with could be used like so called Bernstein operators of max-product kind (see e. g. [4]). The main advantage of the F-transform approach is that the convergence is linear with respect to n, more exactly the uniform convergence rate is of order ω(u, 1/n).…”

“…In this case, u is a nonlinear shaped triangular (unimodal) fuzzy number with support [1,4] and core {c} where c = 2.345, i.e.,…”