Optimisation of a non linear fuzzy function

Abstract

 We present a first study concerning the optimization of a non linear fuzzy function f depending both on a crisp variable and a fuzzy number: therefore the function value is a fuzzy number. More specifically, given a real fuzzy number Ã∈F and the function f(a,x):R ²→R, we consider the fuzzy extension induced by f, f˜ : F × R → F, f˜(Ã,x) = Y˜. If K is a convex subset of R, the problem we consider is “maximizing”f˜(Ã,x), x¯∈ K. The first problem is the meaning of the word “maximizing”: in fact it is well-known that ranking fuzzy numbers is a complex matter. Following a general method, we introduce a real function (evaluation function) on real fuzzy numbers, in order to get a crisp rating, induced by the order of the real line. In such a way, the optimization problem on fuzzy numbers can be written in terms of an optimization problem for the real-valued function obtained by composition of f with a suitable evaluation function. This approach allows us to state a necessary and sufficient condition in order that ∈K is the maximum for f˜ in K, when f(a,x) is convex-concave (Theorem 4.1).