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 à and the function f(a,x), we consider the fuzzy extension induced by f. If K is a convex subset of R, the problem we consider is "maximizing" f¨(Ã,x) on 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 it exists the maximum for f¨ in K, when f(a,x) is convex-concave .