On the levels of fuzzy mappings and applications to optimization

Transcription

1 On the levels of fuzzy mappings and applications to optimization Y. Chalco-Cano Universidad de Tarapacá Arica - Chile ychalco@uta.cl H. Román-Fles Universidad de Tarapacá Arica - Chile hroman@uta.cl M.A. Rojas-Medar Universidade de Campinas Campinas - Brazil marko@ime.unicamp.br Abstract Using der relations on the space of fuzzy numbers 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 f means of fuzzy integrals f fuzzy global optimality. Keywds: Fuzzy numbers Fuzzy functions Fuzzy optimization. Introduction Since the concept of fuzzy sets arose in 965 several auths have discussed various aspects of its they and applications. The concept of fuzzy number that initially it was introduced by Dubois and Prade in [ plays a key role in this they and it is motive of study of diverse auths. An der a fuzzy der relation on the space of fuzzy numbers has been studied by several auths see f 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 der relation. The collection of papers on fuzzy optimization and applications edited by Slowiński [7 gives the main stream of this topic (see also [3). In [8 [0 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 076 the minimal points set f the problem of fuzzy optimization. This way diverse tools of the classic they of optimization are extended to the context fuzzy f 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 imptant in classic optimization they as we can see in [. Using level of fuzzy function obtain some results concerning to locate the global minimal points set. Consequently we define the mean value over levels sets of a fuzzy function we study its properties and obtain some results concerning to fuzzy optimization in particular we obtain a necessary condition f fuzzy global optimality. In each case we give some examples. These results are a generalization of some results obtained by Zhen Quan in [2 where it develops an integral they f global optimization classic. 2 Preliminaries Throughout this paper 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 ũ be a fuzzy set in R. We denote by [ũ α the α- level set of ũ which is a subset of R and is defined by [ũ α = {x R / ũ(x) α} f every α (0 and [ũ 0 = {x R / ũ(x) > 0} respectively. A fuzzy set ũ is said to be a fuzzy number if [ũ α is a closed interval in R f all α [0. Then the

2 symbol F(R) denotes the set of all fuzzy numbers. We write [ũ α = [ ũ L α ũ U α f all α [0. Let ũ ṽ F(R) then by Zadeh extension principle we define the addition to yield a fuzzy number ũ ṽ by (ũ ṽ)(z) = sup min{ũ(x) ṽ(y)} z=x+y f all z R. Also we define the scalar multiplication to yield a fuzzy number λũ by { ũ( x (λũ)(z) = λ ) if λ 0 0(x) if λ = 0. Then we have the following: If ũ ṽ F(R) ũ ṽ F(R) and λũ F(R). Meover f all α [0 we have [ũ ṽ α = [ũ α + [ṽ α = [ ũ L α + ṽ L α ũ U α + ṽ U α [λũ α = λ [ũ α = [ λũ L α λũ U α. Also the difference of ũ and ṽ is defined as ũ ṽ = ũ ( ṽ). Meover f all α [0 we have [ũ ṽ α = [ũ α [ṽ α = [ ũ L α ṽ U α ũ U α ṽ L α. Definition (i) Let [a b and [c d be closed intervals. We say that [c d [a b if and only if c a and d b. (ii) Let [a b and [c d be closed intervals. We say that [c d [a b if and only if (c < a d b) (c a d < b) (c < a d < b) (iii) Let ũ and ṽ be fuzzy numbers. We say that ũ ṽ (ũ ṽ) if and only if [ũ α [ṽ α ([ũ α [ṽ α respectively) f all α. We can see that is a der relation on F(R). It is well know that there is others preference relation defined in the space F(R) (see [3 [4) f example the relation of inclusions but in this paper we consider the relation defined in the Definition. Remark 2 We also write ũ ṽ if ṽ ũ. Now ũ ṽ not implies ũ ṽ. F example if ũ = χ [ 23 and ṽ = χ [ 2 we have that ũ ṽ and ũ ṽ. Now we consider a distance between two fuzzy numbers ũ and ṽ by D(ũ ṽ) = sup H([ũ α [ṽ α ) α [0 where H is the well-known Hausdff metric defined by H([ũ α [ṽ α ) = max { ũ L α ṽα L ũ U α ṽα U } 3 Levels of a fuzzy function Let f : (F(R) D) be a fuzzy function and let ũ be a fuzzy number. Then we define the level ũ of f by Lũ(f) := {x / f(x) ũ}. Notice that this definition is a generalization of the levels of a real valued function. Now in the classic they of optimization it is imptant the properties topologicals and geometrics of levels of a real valued function. Next we study some properties of this new tool and in the following chapter we present some of their possible applications. Proposition 3 Let f be a fuzzy function. Then (i) Lũ(f) Lṽ(f) f all ũ ṽ. (ii) Suppose {ũ n } is a decreasing sequence which D-converge to ũ as n. Then Lũ(f) = n= Lũn (f) = lim n L ũ n (f) To continue we present some properties topologicals of the levels of f. Theem 4 Let be a topological space and f : (F(R) D) a function continuous. Then Lũ(f) is a closed subset of f each fuzzy number ũ. Proof We consider a fuzzy number ũ arbitrary and let (x n ) Lũ(f) a sequence such que x n x. Then f(x n ) ũ f all n and consequently [f(x n ) α [ũ α = [ ũ L α ũ U α f all n N and α [0. 077

3 Let α [0 be fixed arbitrarily. Let us consider the functions g g 2 : R defined by Since W (x) D then exist y W (x) such that y D. Consequently g (x) = min[f(x) α and g 2 (x) = max[f(x) α. Then g i (y) g i (x) D(f(y) f(x)) i = 2 g (y) ũ L α g 2 (y) > ũ U α g (y) > ũ L α g 2 (y) ũ U α g (y) ũ L α g 2 (y) ũ U α () f all y x. Consequently from the continuity of f we have that g g 2 are continuous. Now also we have that then g (x n ) ũ L α and g 2 (x n ) ũ U α g (x) ũ L α and g 2 (x) ũ U α. Thus [f(x) α [ũ α f all α. Example 5 We consider the fuzzy function f : R F(R) defined by f(x)(r) = r+x x if r [ x 0 x > 0 r x x if r [0 x x > 0 0 other cases i.e. f each x > 0 f(x) is a fuzzy number isosceles triangular with suppt [ x x and f each x 0 f(x) = 0. We can verify that f is D- continuous. Now let ũ be a number fuzzy nonnegative defined by if r [ 0 4 ũ(r) = 2 4r if r [ other cases. Then from the Theem 4 we have that Lũ(f) it is a closed set. Theem 6 Let be a topological space ũ a fuzzy number and f : (F(R) D) a function continuous. Then is a open subset of. D = {x / f(x) ũ} where g g 2 are defined as in the proof of the Theem 4. Now from () and from the continuity of g we have that ũ L α < g (y) < g (x) + ɛ and as ɛ it is arbitrary we obtain that ũ L α < g (x). Similarly ũ L α g (y) < g (x) + ɛ and as ɛ it is arbitrary we obtain that ũ L α g (x). Thus we obtain that g (x) ũ L α g 2 (x) > ũ U α g (x) > ũ L α g 2 (x) ũ U α g (x) ũ L α g 2 (x) ũ U α (2) that it contradicts the statement x D. To continue we present some properties geometrics of the levels of f. F this we will use the following definition: A fuzzy function f : F(R) is said to be quasiconvex if f every λ [0 and x y f(λx + ( λ)y) sup{f(x) f(y)}. Theem 7 A fuzzy function f : F(R) is quasiconvex if and only if Lũ(f) is a convex (crisp) set f each ũ F(R). Proof ) Let f : F(R) be quasiconvex and ũ F(R). If Lũ(f) is a singleton an empty set then it is obvious a convex set. Assume that x y Lũ(f) i.e. f(x) ũ and f(y) ũ. It follows that sup {f(x) f(y)} ũ. Since f is quasiconvex we have f all λ [0 f(λx + ( λ)y) sup {f(x) f(y)} ũ. Proof We suppose that D is not open then exist x D such that V (x) D f all neighbhood V (x) of x. Now from the continuity of f f ɛ > 0 arbitrary exist a neighbhood W (x) of x such that D(f(x) f(y)) < ɛ f all y W (x). 078 Therefe λx + ( λ)y Lũ(f). Thus Lũ(f) is a convex set. ) Let Lũ(f) be a convex set f every fuzzy number ũ. Let x y and ũ = sup {f(x) f(y)}. Then x y Lũ(f) and by the convexity of Lũ(f)

4 it follows that λx + ( λ)y Lũ(f) f all λ [0. Hence f(λx + ( λ)y) ũ = sup {f(x) f(y)}. Thus f is quasiconvex. Remark 8 In the literature other types of convexity fuzzy function are defined they are generalizations of convexity velued-real functions. F example in [9 Yu-Ru Syau defines (Φ Φ 2 )- Convex fuzzy function which generalizes another types of convexity fuzzy (see f example [8). On the other hand in the paper [9 a result similar to the Theem 7 is presented. 4 Optimization fuzzy Suppose is a Hausdff topological space f a fuzzy numbers valued function on and S a subset of. We consider the following problem: } inf f(x) (3) subject to : x S where x 0 is a solution of (3) if f(x 0 ) f(x) f all x S. Assuming that there exist solution f (3) we are interest in localized the set of points global minima f the problem (3). In this section we present a result that it is related to this. Befe we will give some previous definitions. Definition 9 A subset G of a topological space is said to be robust if intg = G. Definition 0 Let be a Hausdff topological space B is a Bel field of i.e. each open set of is in B. A measure µ on ( B) is said to be Q- measure if any nonempty open set A has positive measure i.e. µ(a) > 0 f all nonempty open set A. Theem Let ( B µ) be a Q-measure space and S be a robust subset of and let f : F(R) be continuous. Suppose that there exist solution f (3) and suppose that the intesection of Lũ(f) and S is nonemptyset. If µ (Lũ(f) S) = 0 then ũ is the global minimum value of f over S and Lũ(f) S is the set of global minima. Proof We suppose that ũ is not the global minimum value of f over S whereas ṽ is. Then exist a fuzzy number w such that Consequently w = (ũ ṽ) 2 ṽ w ũ. Now from the continuity of f and taking into account the Theem 6 we have that D = {x / f(x) w} it is a nonempty and open set of. Therefe Lṽ(f) S D S Lũ(f) S. From the Lemma.4 in [2 there is an open set B such that B D S Lũ(f) S. Now as µ is a Q-measure we obtain that 0 < µ(b) µ(lũ(f) S). We have a contradiction with respect to the condition µ(lũ(f) S) = 0. The prove is complete. Collary 2 Under the same conditions of the Theem. If w ũ then µ(l w (f)) > 0 where ũ = min f(x) : x S. Example 3 Let g : R F(R) be a fuzzy number valued mapping defined by [ rx 2 (x ) 2 + r + r x g(x) = 2 (x ) r x 2 (x ) 2 x 2 (x ) 2 r [0 x2 (x ) 2 +. We can see that g is continuous and there exist minimal points f problem 3. Let ũ be the fuzzy number triangular isosceles with suppt [. Then Lũ(f) = {x R / g(x) ũ} = {0 }. Therefe if we take S = ( 2 4 and µ the Lebesgue measure we have that 079 Lũ(f) S and µ(lũ(f) S) = 0.

5 From the Them we have that Lũ(f) S it is global minimal points set and ũ is the global minimun value of f over S. Now if we consider f as in the Example 5 we have that f is continuous. But as we can see in this case doesn t exist a minimum f the problem 3 and therefe the Theem is not applicable. 5 Mean valued condition In this Section we present the concept of mean valued f fuzzy function we given also properties and we present applications to the fuzzy optimization. With this aim we first recall some basic concepts and properties of fuzzy integral. Let be a Banach space let ( Ω µ) be a measure space where Ω denotes the σ-algebra of all Lebesgue-measurable subsets of. A set-valued function f : K(R) is called Bel measurable if its graph i.e. the set {(x t)/ t f(x)} is a Bel subset of R. The Aumman integral of a set-valued function f : K(R) is defined by { } fdµ = gdµ / g S(f) where gdµ is the Bochner-integral and S(f) is the set of all integrable selects of f i.e. S(f) = { g L (; R)/ g(x) f(x) a.e. }. A set-valued function f : K(R) is said to be integrably bounded if there exists a integrable function h : R such that t h(x) f all t and x such that t f(x). If f : K(R) is an integrably bounded then the Aumann integral of f is a nonempty subset of R. Let f : F(R) be a fuzzy function and define f α : K(R) by f α (x) = [f(x) α α [0. Then f is called measurable if f α is measurable f all α [0. Also f is called integrably bounded if f α is an integrably bounded f every α [0. If f is integrably bounded then the fuzzy integral of f denoted by fdµ it is such that [ α fdµ = f α dµ 080 F me details of fuzzy integrals see [2. Proposition 4 If f : F(R) is D- continuous with [f(x) α = [g α (x) h α (x) then g α h α : R are integrable and [ α [ fdµ = f α dµ = g α dµ h α dµ f each α [0. Proposition 5 Let f f 2 : F(R) be two fuzzy function D-continuous. If f (x) f 2 (x) f all x then f dµ f 2dµ. If f : F(R) is D-continuous with [f(x) α = [g α (x) h α (x). Then give A we can define A fdµ by [ α [ fdµ = f α dµ = g α dµ h α dµ. A A A A (4) Next we present the definition of mean value over levels sets of f assumptions that: (i) f : F(R) is continuous; (ii) ( Ω µ) is a Q-measure space; (iii) there exist a minimum valued ũ f the problem 3 i.e. ũ = min f(x) : x S Definition 6 Suppose w ũ. We define M(f w) := f(x)dµ (5) µ(l w (f)) L w (f) to be the mean value of the fuzzy function f over its level set L w (f). Accding to Collary 2 µ(l w (f)) > 0 and taking into account that f is D-continuos the mean valued (5) is well defined. The following are properties of mean valued of f. Proposition 7 F w ũ M(f w) w. Proof By definition f(x) w f all x L w (f). Then from the Proposition 5 we have M(f w) = f(x)dµ µ(l w (f)) L w (f) wdµ µ(l w (f)) L w (f) = µ(l w (f)) µ(l w(f)) w = w.

6 Theem 8 A point x is a global minimun with ũ = f(x) as the cresponding global minimun valued then M(f w) ũ f w ũ. Proof Suppose ũ is the global minimun value of f. Then f(x) ũ f all x S. So f w ũ from the Propositions 5 we have M(f w) = 6 Conclusions µ(l w (f)) µ(l w (f)) L w (f) L w (f) f(x)dµ ũdµ = ũ. In this paper using a der relation on the fuzzy numbers space we introdue the concept levels of fuzzy function. We obtain some imptants topologicals and geometrics properties. We use this new tool to obtain some results f fuzzy optimization basically we find a way to locate points and we obtain a necessary condition of type integral. In this part we are interested in generalizing some results of the classic they of integral global optimization. Again here the der relation not allow to generalize in a natural way necessary and sufficient conditions f means of fuzzy integrals f global optimization (Theem 20). In future wks we will exple this study considering others preference relations. Acknowledgments The auths thank the financial suppt of DIPOG-UTA Fondecyt-Chile through Project and CNPq-Brazil grant No 30354/03-0. References [ D. Dubois and H. Prade (978) Operations on fuzzy numbers Internat. J. of Systems Sci [3 C. Carlsson and R.Fullér (2002) Fuzzy reasoning in decision and optimization (Studies in Fuzziness and Soft Computing 82). Physica-Verlag NY. [4 M.Delgado J.L. Verdegay and M.A. Vila (989) A general model f fuzzy linear programing Fuzzy Sets and Systems [5 H.-C. Wu (2004) Evaluate fuzzy optimization problems based on biobjective programming problems Computers and Mathematics with Applications [6 H.-C. Wu (2004) An (α β)-optimal Solution Concept in Fuzzy Optimization Problems Optimization [7 R. Slowiński (Ed.) (998) Fuzzy Sets in Decision Analysis Operations Research and Statistic Kluwer Academic Publishers Boston. [8 Yu-Ru Syau (200) Generalization of preinvex and B-vex fuzzy mappings Fuzzy Sets and Systems [9 Yu-Ru Syau (2003) (Φ Φ 2 )-Convex fuzzy mappings Fuzzy Sets and Systems [0 M. Amemiya and W. Takahashi (2002) Convexity of fuzzy-valued maps and minimization theems Scientiae Mathematicae Japonicae 56 No [ H. Román-Fles and M. Rojas-Medar (999) Level-Continuity of functions and applications Computers and Mathematics with Applications [2 Chew Soo Hong and Zheng Quan (988) Integral Global Optimization Lectures Notes in Economics and Mathematical Systems 298 Springer-Verlag. [2 P.Diamond P. Kloeden (994) Metric Space of Fuzzy Sets: They and Application Singapure Wld Scientific. 08