On the levels of fuzzy mappings and applications to optimization Y. Chalco-Cano Universidad de Tarapac´ a Arica - Chile ychalco@uta.cl H. Rom´ an-Flores Universidad de Tarapac´ a Arica - Chile hroman@uta.cl M.A. Rojas-Medar Universidade de Campinas Campinas - Brazil marko@ime.unicamp.br Abstract Using order relations on the space of fuzzy num- bers, we define the levels of fuzzy number valued mappings and we present some topologicals and geometrics properties. We use this concept to locate points and the set of points that solve a problem of fuzzy optimization. Consequently we obtain a necessary condition, for means of fuzzy integrals, for fuzzy global optimality. Keywords: Fuzzy numbers, Fuzzy functions, Fuzzy optimization. 1 Introduction Since the concept of fuzzy sets arose in 1965, seve- ral authors have discussed various aspects of its theory and applications. The concept of fuzzy number, that initially it was introduced by Dubois and Prade in [1], plays a key role in this theory and it is motive of study of diverse authors. An order, or a fuzzy order relation on the space of fuzzy numbers has been studied by several au- thors, see for example [6], [7]. Consequently, arise the problems of fuzzy optimization that consists in find the minimum of a fuzzy number valued function, where the space of fuzzy numbers is endowed with an order relation. The collection of papers on fuzzy optimization and applications edited by Slowi´ nski [7] gives the main stream of this topic (see also [3]). In [8], [10], among o- thers, different types of convexity and continuity of fuzzy mappings are defined and they are used to study the existence of solution, as well as the study of topological and geometric properties of the minimal points set, for the problem of fuzzy optimization. This way, diverse tools of the classic theory of optimization are extended to the context fuzzy, for the development of this. In this paper, we introduced the concept of levels of a fuzzy function and we study some of their properties. This new tool is a generalization of the concept of levels of a real function, which is enough important in classic optimization theory, as we can see in [11]. Using level of fuzzy function obtain some results concerning to locate the global minimal points set. Consequently, we define the mean value over lev- els sets of a fuzzy function, we study its proper- ties and obtain some results concerning to fuzzy optimization, in particular we obtain a necessary condition for fuzzy global optimality. In each case we give some examples. These results are a gener- alization of some results obtained by Zhen Quan in [12], where it develops an integral theory for global optimization classic. 2 Preliminaries Throughout this paper, X denote a topological space, and we denote by R the set of real numbers and by χ A the characteristic function of the set A. Let ˜ u be a fuzzy set in R. We denote by [˜ u] α the α- level set of ˜ u, which is a subset of R and is defined by [˜ u] α = {x ∈ R / ˜ u(x) ≥ α} for every α ∈ (0, 1] and [˜ u] 0 = {x ∈ R / ˜ u(x) > 0}, respectively. A fuzzy set ˜ u is said to be a fuzzy number if [˜ u] α is a closed interval in R for all α ∈ [0, 1]. Then the EUSFLAT - LFA 2005 1076