On the Continuity and Convexity Analysis of the Expected Value Function of a Fuzzy Mapping
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1 Journal of Uncertain Systems Vol.1, No.2, pp , 2007 Online at: On the Continuity Convexity Analysis of the Expected Value Function of a Fuzzy Mapping Cheng Wang a Wansheng Tang a Ruiqing Zhao a,b a.institute of Systems Engineering, Tianjin University, Tianjin , China b.department of Computer Science, University of Wales, Aberystwyth, SY23 3DB, wangch-tju@hotmail.com tang@tju.edu.cn rzz@aber.ac.uk Received 3 November 2006; Accepted 30 January 2007 Abstract In the optimization problem under a fuzzy environment, the objective function might be given with fuzzy coefficients. In this paper, fuzzy variables are used to characterize these coefficients, such a function is referred as a fuzzy mapping. In order to make a decision in the fuzzy sense, we study some properties of fuzzy mapping. Since the convex analysis plays an important role in the studies of optimization problems, this paper discusses the continuity convexity of the expected value function of a fuzzy mapping. We prove that the continuity convexity can be inherited after calculating the expected value on the hypotheses of monotonicity the upper semi-continuous of the membership functions of independent fuzzy variables. An application in a retailer s optimization problem illuminates how to study programming in a fuzzy environment quantitatively qualitatively. c 2007 World Academic Press, UK. All rights reserved. Keywords: Fuzzy variable; Fuzzy mapping; Convexity; Continuity 1 Introduction In many system-analysis areas, a model has to be set up using data that is only approximately known. Fuzzy sets theory makes it possible. Since Zadeh [20 first proposed the concept of fuzzy set, it has been well developed applied in a wide variety of real problems. In order to measure a fuzzy event, Zadeh [21 initialized the concept of possibility measure, which was developed by several researchers such as Nahmias [13, Kaufman Gupta [4, Zimmermann [22, Dubois Prade [2 Liu [6. Since the possibility measure is not self-dual, Liu Liu [9 proposed the concept of credibility measure, which is a self-dual measure, defined the expected value of a fuzzy variable based on the credibility measure. A framework of credibility theory was given by Liu [8. Fuzzy programming offers powerful means of hling optimization problems with fuzzy factors. Many researchers such as Zimmermann [22, Yazenin [18 [19, Sakawa [15 Tanaka et al [17 applied fuzzy sets theory to optimization problems successfully. A detailed survey on fuzzy optimization was made in 1989 by Luhjula [12. Especially, Liu Liu [9 presented a concept of expected value operator of a fuzzy variable provided a spectrum of fuzzy expected value model EVM. In EVM, the decision is to optimize the expected value of the objective function under constraints. The convexity of the fuzzy expected value model was discussed in their paper. Liu [11 introduced the fuzzy programming with recourse problem discussed the convexity of the model. Corresponding author: W. Tang tang@tju.edu.cn.
2 Journal of Uncertain Systems, Vol.1, No.2, pp , In order to model a fuzzy decentralized decision-making problem, Gao Liu [3 introduced the fuzzy expected value multilevel programming chance-constrained multilevel programming. It is well known that the convex analysis [14 is important for quantitative qualitative studies helps to find optimal solutions in operation research. Since the continuity convexity of a function are the basics for convex analysis, it would be significant for us to consider these properties of the expected value function of a fuzzy mapping. Then some results of the convex analysis can be used to study the programming help to find solutions under a fuzzy environment. This paper is organized as follows. In Section 2, preliminaries of some basic notions several results which can be used in this paper are given. In Section 3, we consider the continuity convexity of the expected value function of a fuzzy mapping, several examples are given corresponding with the results. In Section 4, an application is given on the retailer s optimization problem. 2 Preliminaries Let Θ be a nonempty set, PΘ the power set of Θ. A possibility measure [13 is a set function which satisfies the following conditions: Axiom 1. Pos{Θ} 1, Pos{φ} 0; Pos : PΘ [0, 1, Axiom 2. Pos{ i A i } sup i Pos{A i } for any collection {A i } in PΘ; Axiom 3. Let Θ i be nonempty sets on which Pos i { } satisfy the first two axioms, i 1, 2,..., n, respectively, Θ Θ 1 Θ 2... Θ n. Then Pos{A} sup Pos 1 {θ 1 } Pos 2 {θ 2 }... Pos n {θ n } θ 1,θ 2,...,θ n A for each A PΘ. In that case we write Pos Pos 1 Pos 2... Pos n. Then the triplet Θ, PΘ, Pos is called a possibility space [16. Let the kernel of a possibility space Θ, PΘ, Pos, {θ Θ Pos{θ} > 0}, be denoted by Θ +. Definition 1 Liu Liu [9 Let Θ, PΘ, Pos be a possibility space A a set in PΘ. Then the credibility measure of A is defined by where A c is the complement of A. Cr{A} 1 2 Pos{A} + 1 Pos{Ac }, 1 Definition 2 Nahmias [13, Liu Liu [6 A fuzzy variable ξ is defined as a function from the possibility space Θ, PΘ, Pos to the set of real numbers its membership function is derived from the possibility measure by µ ξ x Pos{θ Θ ξθ x}, x R. 2 Let ξθ + denote the collection {x Pos{θ Θ ξθ x} > 0}. A fuzzy variable ξ is said to be bounded if for any α 0, 1, {x R µ ξ x α} is a nonempty bounded subset of R.
3 150 C. Wang et al.: Analysis of the Expected Value Function of a Fuzzy Mapping Definition 3 Nahmias [13 The fuzzy variables ξ 1, ξ 2,..., ξ n are said to be independent if only if Pos{ξ i B i, i 1, 2,..., n} min Pos{ξ i B i } 3 1 i n for any sets B 1, B 2,..., B n of R. Definition 4 Liu Liu [9 Suppose ξ is a fuzzy variable. Then the expected value of ξ is defined by + 0 E[ξ Cr{ξ r}dr Cr{ξ r}dr 4 0 provided that at least one of the two integrals is finite. Definition 5 Liu Liu [10 Let ξ be a fuzzy variable on the possibility space Θ, PΘ, Pos. Then the α-optimistic value, denoted by ξ U α, of ξ is defined as ξ U α sup{r Pos{θ Θ ξθ r} α}, α 0, 1, the α-pessimistic value, denoted by ξ L α, of ξ is defined as ξ L α inf{r Pos{θ Θ ξθ r} α}, α 0, 1. Lemma 1 Liu Liu [10 Let ξ be a bounded fuzzy variable defined on the possibility space Θ, PΘ, Pos. Then we have E[ξ [ ξ U α + ξ L α dα. Definition 6 Liu [7 Let f : R n R be a measurable function, ξ i fuzzy variables defined on the possibility spaces Θ i, PΘ i, Pos i, i 1, 2,..., n, respectively. Then the function of fuzzy variables, denoted by ξ fξ 1, ξ 2,..., ξ n, is a fuzzy variable defined on the product possibility space Θ, PΘ, Pos, where Θ n i1 Θ i, Pos n i1 Pos i for any θ 1, θ 2,..., θ n Θ. ξθ 1, θ 2,..., θ n f ξ 1 θ 1, ξ 2 θ 2,..., ξ n θ n Definition 7 Aubin [1 Suppose X is a topological space f : X R a real-valued function. We say that f is lower semi-continuous at x 0 X if for all λ < fx 0, there exists a neighborhood Bx 0, δ, where δ > 0, such that λ fx, x Bx 0, δ. 5 We shall say that f is lower semi-continuous if it is lower semi-continuous at every point of X. A function f is upper semi-continuous if f is lower semi-continuous, f is continuous if only if f is lower semi-continuous upper semi-continuous. Lemma 2 Aubin [1 A function f from X to R is lower semi-continuous or upper semi-continuous at x 0 X if only if lim inf x x 0 fx fx 0 or lim sup fx fx 0. x x 0
4 Journal of Uncertain Systems, Vol.1, No.2, pp , Theorem 1 Let ξ be a fuzzy variable with an upper semi-continuous membership function, ξ U α ξ L α the α-optimistic value the α-pessimistic value of ξ, respectively. Then we have { } Pos ξ ξ U α α, α 0, 1 6 { } Pos ξ ξ L α α, α 0, 1. 7 Proof. It follows from Definition 5 that Pos{ξ ξ U α + ɛ} < α Pos{ξ ξ U α ɛ} α, ɛ > 0, α 0, 1. Since the membership function of ξ is upper semi-continuous, we can deduce that Pos{ξ ξ U α} lim sup Pos{ξ r} r ξ U α lim sup ɛ 0 + lim ɛ 0 + lim ξ U α ɛ r ξ U α+ɛ ɛ 0 + ξ U α ɛ r Pos{ξ r} sup Pos{ξ r} ξ U α ɛ r ξ U α+ɛ sup Pos{ξ r} lim ɛ 0 + Pos{ξ ξu α ɛ} α. { } The inequality Pos ξ ξ L α α can be proved similarly. sup Pos{ξ r} ξ U α+ɛ<r Remark 1 If the membership function is not upper semi-continuous, the result may not be right. Consider the fuzzy variable ξ with membership function x, x [0, 1 µ ξ x 2 x, x 1, , 2 0, x 1.5. By Definition 5, we have that ξ U 0.5 sup{r Pos{θ Θ ξθ r} 0.5} 1.5, whilst Pos{ξ ξ U 0.5} Pos{ξ 1.5} 0. Definition 8 A function f : R n R is said to be nondecreasing nonincreasing, respectively with respect to x i if only if for each fixed x i R n 1, where x i denotes the vector without x i, fx i, x i is nondecreasing nonincreasing, respectively with respect to x i. If fx is nondecreasing or nonincreasing with respect to x i, then fx is monotonic with respect to x i. Example 1 Consider the function fx, y xy with x [ 1, 1 y [1, 2. For each y 0 [1, 2, fx, y 0 xy 0 is nondecreasing with respect to x, we say fx, y xy is monotonic with respect to x. Since fx 0, y x 0 y is nonincreasing with respect to y when x 0 [ 1, 0 nondecreasing with respect to y when x 0 [0, 1, fx, y is not monotonic with respect to y. The definition of fuzzy mapping can be given as follows. Definition 9 A fuzzy mapping is a mapping from a topological space X to a collection of fuzzy variables. 8
5 152 C. Wang et al.: Analysis of the Expected Value Function of a Fuzzy Mapping 3 The Expected Value Function of a Fuzzy Mapping In practical analysis, we often come across such functions, some of whose parameters may be of fuzziness. Usually the expected values or the values with the highest possibility of the parameters are considered. However, it will be more accurate for us to use fuzzy variables to characterize such parameters. For example, if the coefficient b is fuzzy in the function fx x + b, we may use a fuzzy variable ξ to depict it, the mapping is denoted by fx, ξ x + ξ as a fuzzy mapping. For simple depiction, a function fx, u x + u is introduced by substituting the variable u for the parameter b in the function fx x + b. More generally, let F Θ i, PΘ i, Pos i denote the collection of all bounded fuzzy variables defined on the possibility spaces Θ i, PΘ i, Pos i, i 1, 2,..., n, respectively, ξ i F Θ i, PΘ i, Pos i, ξ ξ 1, ξ 2,..., ξ n a fuzzy vector, X a topological space f : X n i1 ξ i Θ i R a bounded measurable function. Thus fx, ξ fx, ξ 1, ξ 2,..., ξ n is a fuzzy mapping from X to FΘ, PΘ, Pos, where Θ n i1 Θ i Pos n i1 Pos i. Let E[fx, ξ, f U x, ξ; α f L x, ξ; α denote the expected value function, the α-optimistic value function the α-pessimistic value function of fx, ξ, α 0, 1, respectively, i.e., for each x 0 X, E[fx 0, ξ, f U x 0, ξ; α f L x 0, ξ; α are the expected value, the α-optimistic value the α-pessimistic value of the fuzzy variable fx 0, ξ, respectively. In this section, the continuity convexity of the function E[fx, ξ are studied, since they play crucial roles in quantitative qualitative studies of optimization problem under a fuzzy environment. First, some properties of the function of fuzzy variables are discussed. Theorem 2 Let ξ i be independent fuzzy variables defined on the possibility spaces Θ i, PΘ i, Pos i with upper semi-continuous membership functions, i 1, 2,..., n, respectively f : X R n R a measurable function. If fx is monotonic with respect to x i on ξ i Θ + i, i 1, 2,..., n, respectively, then we have i f U ξα f ξ1 V α, ξv 2 α,..., ξv n α, where ξi V α ξu i α, if fx 1, x 2,..., x n is nondecreasing with respect to x i, ξi V α ξl i α, otherwise, α 0, 1; ii f L ξα f ξ1 V α, ξv 2 α,..., ξv n α, where ξi V α ξl i α, if fx 1, x 2,..., x n is nondecreasing with respect to x i, ξ V i α ξu i α, otherwise, α 0, 1, where f U ξα f L ξα denote the α-optimistic value the α-pessimistic value of the fuzzy variable fξ, respectively. Proof. Without loss of generality, suppose f is nondecreasing with the first k variables nonincreasing with the last n k variables. Then we want to prove that for all α 0, 1, f U ξα f ξ1 U α, ξ2 U α,..., ξk U α, ξk+1α, L..., ξn L α 9 f L ξα f ξ1 L α, ξ2 L α,..., ξk L α, ξk+1α, U..., ξn U α. 10 For any α 0, 1, since ξ i are independent with each other, it follows from Definition 3
6 Journal of Uncertain Systems, Vol.1, No.2, pp , Theorem 1 that { } Pos fξ f ξ1 Uα, ξu 2 α,..., ξu k α, ξl k+1 α,..., ξl n α { } Pos ξ 1 ξ1 Uα; ξ 2 ξ2 Uα;... ; ξ k ξk Uα; ξ k+1 ξk+1 L α;... ; ξ n ξn U α } } k i1 {ξ Pos i ξi Uα n ik+1 {ξ Pos i ξi Lα } } k i1 {ξ Pos i ξi Uα n ik+1 {ξ Pos i ξi Lα α. Following Definition 5, we get Suppose f U ξα > f f U ξα f ξ1 U α, ξ2 U α,..., ξk U α, ξk+1α, L... ξn L α. ξ1 Uα, ξu 2 α,..., ξu k α, ξl k+1 α,... ξl n α, which implies { } Pos fξ > f ξ1 U α, ξ2 U α,..., ξk U α, ξk+1α, L..., ξn L α α by Definition 5. So there exists x x 1, x 2,..., x n, such that fx > f ξ1 U α, ξ2 U α,..., ξk U α, ξk+1α, L..., ξn L α Pos {θ Θ ξθ x } α. According to the monotonicity of fx, we can deduce that there must exist some i k x i > ξu i α or some i > k x i < ξl i α. It follows from Definition 5 that if x i > ξu i α then we have Pos {θ Θ ξθ x } Pos {θ i Θ i ξ i θ i x i } < α. If x i < ξl i α, we also have Pos {θ Θ ξθ x } Pos {θ i Θ i ξ i θ i x i } < α. Those are in contradiction with Pos {θ Θ ξθ x } α, such that 9 holds. The assertion that f L ξα f ξ1 Lα, ξl 2 α,..., ξl k α, ξu k+1 α,..., ξu n α can be proved similarly. Since the α-optimistic value the α-pessimistic value of the function of fuzzy variables can be given by Theorem 2, the expected value of such fuzzy variable function can be calculated directly following from Lemma 1. The following examples illustrate the result. Example 2 Consider the function fx 1, x 2, x 3 e x 1 + x 3 2 x 3 with x 1, x 2, x 3 R. It is obviously that f is increasing with respect to x 1, x 2 decreasing with respect to x 3. Let ξ i be defined on the possibility spaces Θ i, PΘ i, Pos i, where Θ i [0, 2, Pos i {θ i } { θi, 0 θ i 1 2 θ i, 1 < θ i 2, i 1, 2, 3, respectively. Let ξ 1 θ 1 θ 1, ξ 2 θ θ 2 ξ 3 θ θ 3, θ i [0, 2. Then ξ 1, ξ 2 ξ 3 are independent bounded fuzzy variables with upper semi-continuous membership functions equal to the triangular fuzzy variables 0, 1, 2, 1, 2, 3 2, 3, 4, respectively. It follows from Theorem 2 that 3 f U ξ 1, ξ 2, ξ 3 α e ξu 1 α + ξ2 α U ξ L 3 α e 2 α + 3 α α
7 154 C. Wang et al.: Analysis of the Expected Value Function of a Fuzzy Mapping 3 f L ξ 1, ξ 2, ξ 3 α e ξl 1 α + ξ2 α L ξ U 3 α e α α 3 4 α, for all α 0, 1. Following Lemma 1, we get E[fξ 1, ξ 2, ξ e2. 2 f U ξ 1, ξ 2, ξ 3 α + f L ξ 1, ξ 2, ξ 3 α dα [ e 2 α + 3 α α + e α α 3 4 α dα Example 3 Let ξ 1, ξ 2 ξ 3 be given as those in Example 2, α 0, 1 the function with x 1, x 2, x 3 R. Then fx 1, x 2, x 3 x 2 e x 1 + x 1 x 3 f U ξ 1, ξ 2, ξ 3 α ξ U 2 αeξu 1 α + ξ U 1 αξu 3 α 3 αe2 α + 2 α4 α, f L ξ 1, ξ 2, ξ 3 α ξ L 2 αeξl 1 α + ξ L 1 αξl 3 α 1 + αeα + α2 + α E[fξ 1, ξ 2, ξ e2. f U ξ 1, ξ 2, ξ 3 α + f L ξ 1, ξ 2, ξ 3 α dα [ 3 αe 2 α + 2 α4 α αe α + α2 + α dα Now for the fuzzy mapping fx, ξ, we have the following results on the hypothesis: P: ξ i are independent bounded fuzzy variables with upper semi-continuous membership functions the function fx, u 1, u 2,..., u n is monotonic with respect to u i on ξ i Θ + i, i 1, 2,..., n, respectively. Theorem 3 Consider the fuzzy mapping fx, ξ, satisfying the hypothesis P. Furthermore, if f x, ξ1 V α, ξv 2 α,..., ξv n α f x, ξ1 V α, ξv 2 α,..., ξv n α are lower semi-continuous for all α 0, 1, where ξi V α ξv i α are given as those in Theorem 2, then E[fx, ξ is lower semicontinuous with respect to x. Proof. For any α 0, 1 fixed x 0 X, it follows from Theorem 2 Lemma 2 that lim inf f U x, ξ; α lim inf f x, ξ V x x 0 x x 1 α, ξ2 V α,..., ξn V α 0 f x 0, ξ1 V α, ξv 2 α,..., ξv n α f U x 0, ξ; α, so f U x, ξ; α is lower semi-continuous at x 0. Furthermore f U x, ξ; α is lower semi-continuous with respect to x, α 0, 1. The assertion that f L x, ξ; α is lower semi-continuous with respect to x for all α 0, 1 can be proved similarly. Since E[fx, ξ 1 [ f U x, ξ; α + f L x, ξ; α dα, it follows that E[fx, ξ is lower semi-continuous when both f U x, ξ; α f L x, ξ; α are lower semi-continuous for all α 0, 1. 11
8 Journal of Uncertain Systems, Vol.1, No.2, pp , Remark 2 Similarly, E[fx, ξ is upper semi-continuous continuous, respectively on the conditions that the functions f x, ξ1 V α, ξv 2 α,..., ξv n α f x, ξ1 V α, ξv 2 α,..., ξv n α are upper semi-continuous continuous, respectively for all α 0, 1. { } { } Furthermore, since Pos ξ ξ U α α Pos ξ ξ L α α, then ξi Uα, ξl i α ξ iθ +, for all α 0, 1. Then we have the following remark. Remark 3 If the function fx, u is lower semi-continuous upper semi-continuous continuous, respectively with respect to x for each u ξ Θ +, then f U x, ξ; α, f L x, ξ; α E[fx, ξ are lower semi-continuous upper semi-continuous continuous, respectively functions with respect to x, α 0, 1. Proposition 1 Let X be a compact subset of a topological space, the fuzzy mapping fx, ξ satisfy the conditions in Theorem 3. Then the following problem has a solution x X, i.e., E[f x, ξ inf E[fx, ξ. x X Example 4 Let ξ 1, ξ 2 ξ 3 be given as those in Example 2, g 1 x g 2 x lower semi-continuous with g 1 x > 0 g 2 x > 0 for all x [0, +. Consider the continuity of the expected value function of the fuzzy mapping Firstly, let fx, ξ 1, ξ 2, ξ 3 max{x, ξ 3 } ξ 1 g 1 x + ξ 2 ξ 3 + g 2 x, fx, u 1, u 2, u 3 max{x, u 3 } u 1 g 1 x + u 2 u 3 + g 2 x, x [0,. where x [0, +, u 1 ξ 1 Θ + 0, 2, u 2 ξ 2 Θ + 1, 3 u 3 ξ 3 Θ + 2, 4. It is obviously that fx, u 1, u 2, u 3 is increasing with respect to u i on ξ i Θ +, i 1, 2, 3, respectively. Since for each u 1 ξ 1Θ +, u 2 ξ 2Θ + u 3 ξ 3Θ +, max{x, u 3 } is continuous with respect to x. Thus the function fx, u 1, u 2, u 3 is lower semi-continuous with respect to x. It follows from Remark 3 that f U x, ξ 1, ξ 2, ξ 3 ; α, f L x, ξ 1, ξ 2, ξ 3 ; α E[fx, ξ 1, ξ 2, ξ 3 are lower semi-continuous with respect to x, α 0, 1. Furthermore, it follows from Theorem 2 Lemma 1 that for all α 0, 1, } f U x, ξ 1, ξ 2, ξ 3 ; α max {x, ξ3 Uα ξ1 Uα g 1x + ξ2 ξ Uα 3 Uα + g 2x max{x, 4 α} 2 αg 1 x + 3 α4 α + g 2 x { 4 α2 αg1 x + 3 αg 2 x + 4 α, x < 4 α x2 αg 1 x + 3 αg 2 x + 4 α, x 4 α, } f L x, ξ 1, ξ 2, ξ 3 ; α max {x, ξ3 Lα ξ1 Lα g 1x + ξ2 ξ Lα 3 Lα + g 2x max{x, 2 + α} α g 1 x α2 + α + g 2 x { 2 + αα g1 x α2 + α + g 2 x, x < 2 + α 1 x α g 1 x α2 + α + g 2 x, x 2 + α E[fx, ξ 1, ξ 2, ξ 3 1 f U x, ξ 1, ξ 2, ξ 3 ; α + f L x, ξ 1, ξ 2, ξ 3 ; α dα g 1x + 2g 2 x + 13, x [0, x3 1 2 x2 + x + 8 g 1 x + 2g 2 x + 13, x [2, xg 1 x + 2g 2 x , x [4,.
9 156 C. Wang et al.: Analysis of the Expected Value Function of a Fuzzy Mapping It is easy to check that f U x, ξ 1, ξ 2, ξ 3 ; α, f L x, ξ 1, ξ 2, ξ 3 ; α E[fx, ξ 1, ξ 2, ξ 3 are lower semicontinuous with respect to x, α 0, 1, since g 1 x g 2 x are lower-semi-continuous with respect to x. Theorem 4 Consider the fuzzy mapping fx, ξ, satisfying the hypothesis P. Furthermore, if f x, ξ1 V α, ξv 2 α,..., ξv n α f x, ξ1 V α, ξv 2 α,..., ξv n α are convex for all α 0, 1, where ξi V α ξv i α are given as those in Theorem 2, then E[fx, ξ is convex with respect to x. Proof. For each fixed x 0 X, it follows from Theorem 2 that f U x 0, ξ; α f x 0, ξ1 V α, ξ2 V α,..., ξn V α f L x 0, ξ; α f x 0, ξ1 V α, ξ2 V α,..., ξn V α for each α 0, 1. So f U x, ξ; α f L x, ξ; α are convex functions with respect to x, α 0, 1. Then for each λ [0, 1 x 1, x 2 X, we have E[fλx λx 2, ξ 1 1 [ f U λx λx 2, ξ; α + f L λx λx 2, ξ; α dα [ λ f U x 1, ξ; α + f L x 1, ξ; α + 1 λ [ f U x 1, ξ; α + f L x 1, ξ; α dα + 1 λ 2 λ 2 0 λe[fx 1, ξ + 1 λe[fx 2, ξ. So the function E[fx, ξ is convex with respect to x. Remark 4 Similarly, E[fx, ξ is concave on the conditions that f f x, ξ1 V α, ξv 2 α,..., ξv n α are concave for all α 0, f U x 2, ξ; α + f L x 2, ξ; α dα [ f U x 2, ξ; α + f L x 2, ξ; α dα x, ξ1 V α, ξv 2 α,..., ξv n α Remark 5 If fx, u is convex concave, respectively for each u ξθ +, then f U x, ξ; α, f L x, ξ; α E[fx, ξ are convex concave, respectively, α 0, 1. Example 5 Let ξ 1, ξ 2 ξ 3 be given as those in Example 2 the fuzzy mapping fx, ξ 1, ξ 2, ξ 3 eξ 1 ξ xξ 3, where x X R +. It is obviously that for all x 0 X, fx 0, ξ 1, ξ 2, ξ 3 FΘ, PΘ, Pos the function fx, u 1, u 2, u 3 is increasing with respect to u 1 u 2 on ξ 1 Θ 1 ξ 2 Θ 2, respectively, decreasing with respect to u 3 on ξ 3 Θ 3. Since for each θ θ 1, θ 2, θ 3 Θ, fx, ξ 1 θ 1, ξ 2 θ 2, ξ 3 θ 3 is convex with respect to x, then E[fx, ξ 1, ξ 2, ξ 3 is convex with respect to x.
10 Journal of Uncertain Systems, Vol.1, No.2, pp , Example 6 The utility function for a risk averse agent [5 can be given as Ux 1 e rx r with x R r > 0, where r is a measure of the agent s degree of risk aversion. Since the degree of one s risk aversion is fuzziness hard to be calculated, it would be more reasonable for us to depict the degree by a positive fuzzy variable ξ r, i.e., Pos{ξ r 0} 0. Let Ux, ξ r 1 e ξrx Ux, u 1 e ux. ξ r u It is easy to check that Ux 0, u is decreasing with respect to u, for each x 0 R, Ux, u 0 a concave function, for each u 0 > 0. By Remark 5 we can deduce that E[Ux, ξ r is a concave function. 4 An Application in the Retailer s Optimization Problem Consider an optimization problem a retailer usually faces. The retailer does such a kind of business that he makes an order of one item product from one supplier sells them to customers. Assume that the retailer is risk-neutral; the ordering quantity does not affect the ordering price per unit; no discount is allowed for the retailer to sell the goods; no reorder is allowed when lack of goods; there is a salvage value for unsold goods. Furthermore, if the retailer chooses to give up the business, he has an outside opportunity utility level W. In the programming, we have the following notations. Notation x ordering quantity made by the retailer units s quantity of the product the retailer can sell units c ordering price per unit $ p retail price per unit $ v salvage value per unit $ W x the profit for the retailer $ We have the relation of the parameters that 0 < v < c < p, 0 s 0 x M, where M can be explained as the quantity of the product provided by the supplier. Suppose M is great enough. Since the ability of the retailer to sell the product is fuzziness hard to be characterized, it would be more reasonable for us to use fuzzy variable ξ s to depict the quantity of the goods he can sell. Assume that ξ s is a bounded fuzzy variable with upper semi-continuous membership function defined on the possibility space Θ, PΘ, Pos. Since the quantity of the goods the retailer can sell must be greater than or equal to 0, we have Pos{ξ s < 0} 0. Thus, when the retailer makes an order x, his profit W x is a fuzzy variable on the possibility space Θ, PΘ, Pos. The profit for the retailer can be calculated as the fuzzy mapping Let the function W x, ξ s p min{x, ξ s } + vx min{x, ξ s } cx p v min{x, ξ s } c vx. 12 W x, u p v min{x, u} c vx. 13 It is obviously that W x, u is increasing with respect to u, for each u 0 0, W x, u 0 is continuous concave with respect to x. It follows from Remark 3 Remark 5 that W U x, ξ s ; α, W L x, ξ s ; α E[W x, ξ s are continuous concave with respect to x.
11 158 C. Wang et al.: Analysis of the Expected Value Function of a Fuzzy Mapping In the programming, suppose the retailer has a constraint that the possibility of the event that the profit is less than W, is less than α 0, where W is his outside opportunity utility level, α 0 his critical level. Then the optimization problem can be depicted as the following programming model, max E[W x, ξ s x s.t. Pos{W x, ξ s < W 14 } < α 0 0 x M. Obviously, Pos{W x, ξ s < W } < α 0 is equal to W L x, ξ s ; α 0 W by Definition 5. Since all the sections of upper semi-continuous function are closed, we deduce that the set of x which satisfies the constraints, denoted by Dα 0, W, is closed. If Dα 0, W is empty, then the retailer would not do the business with a reserve income W. Furthermore, if the set is nonempty, since E[W x, ξ s is continuous concave with respect to x, we can deduce, by Proposition 1, that there exists an x, which maximizes E[W x, ξ s under the constraints. If the fuzzy variable ξ s is given as a triangular fuzzy variable equals s 1, s, s+ 2, where 1 > 0, 2 > 0 s 1, then we have ξ U s α s + 1 α 2 ξ L s α s 1 α 1, α 0, 1. It follows from Theorem 2 Lemma 1 that W U x, ξ s ; α p v min{x, ξs U α} c vx { p cx, x < s + 1 α 2 p vs + 1 α 2 c vx, x s + 1 α 2, 15 E[W x, ξ s W L x, ξ s ; α p v min{x, ξs L α} c vx { p cx, x < s 1 α 1 16 p vs 1 α 1 c vx, x s 1 α 1 p cx, x s p vs p vs + p 2c + v 1 x p vx 2, 1 p vs 2 2s p vs + p 2c + v 2 x p vx 2, p v s s 1 < x s s < x s + 2 c vx, x > s + 2. Obviously that W U x, ξ s ; α, W L x, ξ s ; α E[W x, ξ s are continuous concave functions. Then the program 14 can be solved through classical analysis. Since max x 0 W L x, ξ s ; α 0 p cs 1 α 0 1, if it is less than W, the set of x satisfying the constraints, Dα 0, W, is empty. Thus the retailer would not choose to do the business with a reserve income W. Otherwise, we have [ W Dα 0, W The programming model can be simplified as p c, p vs 1 α 0 1 W c v max E[W x, ξ s. 19 x Dα 0,W
12 Journal of Uncertain Systems, Vol.1, No.2, pp , Specifically, let p 10, c 7, v 6, W 500, α 0 0.4, s 200, , i.e., ξ s 150, 200, 250. Since max x 0 W L x, ξ s ; α > 500, the equations can rewrite as 3x, x 150 E[W x, ξ s x 1 50 x2, 150 < x x, x > 250 Dα 0, W [500/3, 180, respectively. Solving the program 19, we get the optimal ordering quantity x 180 the maximum expected profit under constraints E[W x, ξ s Conclusion This paper studies the continuity convexity of the expected value function of the fuzzy mapping fx, ξ to provide a way for convex analysis of the programming under a fuzzy environment. Firstly, we give the definition of fuzzy mapping as a mapping from a topological space to a collection of fuzzy variables. Secondly, some properties of fuzzy variable function are surveyed. With the assumptions of monotonicity the independence of fuzzy variables with upper semi-continuous membership functions, the critical value of a fuzzy variable function, which is also a fuzzy variable, can be solved by the function of the critical values of fuzzy variables. Then, based on the properties of fuzzy variable function, we study the continuity convexity of the α-optimistic value function, the α-pessimistic value function the expected value function of a fuzzy mapping, respectively. We have the results that the continuity convexity can be inherited after calculating the α- optimistic value, the α-pessimistic value the expected value, respectively, if the function fx, u is monotonic with respect to u the fuzzy variables ξ i are bounded with upper semi-continuous membership function independent with each other. In the retailer s optimization problem within competition, the quantity of the product he can sell is more reasonable to be considered as a endogenous variable than exogenous variable, since the quantity is dependent on the ability of the retailer. Thus we can consider the quantity as a positive fuzzy variable. Then, following from the properties of fuzzy mapping, the optimization problem can be solved classically. This paper also gives a method to calculate the expected value of the function of fuzzy variables. This might be useful for quantitative qualitative studies in the optimization problem under a fuzzy environment. However, the results are given on the fuzzy mapping with restrictions. Considering the limitation, we should do further research for more general fuzzy mappings. Acknowledgments This work was supported by the National Natural Science Foundation of China Grant No , No , Program for New Century Excellent Talents in University. References [1 Aubin, J., Optima Equilibria: An Introduction to Nonlinear Analysis, Springer-Verlag, [2 Dubois, D., Prade, H., Possibility Theory, Plenum, New york, [3 Gao, J., Liu, B., Fuzzy multilevel programming with a hybrid intelligent algorithm, Computer & Mathematics with Applications
13 160 C. Wang et al.: Analysis of the Expected Value Function of a Fuzzy Mapping [4 Kaufman, A., Gupta, M., Introduction to Fuzzy Arithmetic: Theory Applications, Van Nostr Reinhold, New York, [5 Laffont, J., Martimort, D., The Theory of Incentives: The Principal-Agent Model, Princeton University Press, [6 Liu, B., Toward fuzzy optimization without mathematical anbiguity, Fuzzy Optimization Decision Making [7 Liu, B., Uncertainty Theory: an Introduction to its Axiomatic Foundations, Springer-Verlag, Berlin, [8 Liu, B., A survey of credibility theory, Fuzzy Optimization Decision Making [9 Liu, B., Liu, Y., Expected value of fuzzy variable fuzzy expected value models, IEEE Transactions on Fuzzy Systems [10 Liu, Y., Liu, B., Expected value operator of rom fuzzy variable rom fuzzy expected value models, Internatioanl Journal of Uncertainty, Fuzziness & Knowledge-Based Systems [11 Liu, Y., Fuzzy programming with recourse, International Journal of Uncertainty, Fuzziness & Knowledge- Based Systems [12 Luhjula, M., Fuzzy Optimization: An Appraisal, Fuzzy Sets Systems [13 Nahmias, S., Fuzzy variables, Fuzzy Sets Systems [14 Rockafellar, R., Convex analysis, Princeton University Press, [15 Sakawa, M., Fuzzy Sets Interactive Multiobjective Optimization, New York: Plenum, [16 Tanaka, H., Watada, J., Possibilistic linear systems their application to the linear regression model, Fuzzy Sets Systems [17 Tanaka, H., Guo, P. Zimmermann, H., Possibility distributions of fuzzy decision variables obtained from possibilistic linear programming problems, Fuzzy Sets Systems [18 Yazenin, A., Fuzzy stochastic programming, Fuzzy Sets Systems [19 Yazenin, A., On the problem of possibilistic optimization, Fuzzy Sets Systems [20 Zadeh, L., Fuzzy sets, Information Control [21 Zadeh, L., Fuzzy sets as a basis for a theory of possibility, Fuzzy Sets Systems [22 Zimmermann, H., Applications of fuzzy set theory to mathematical programming, Information Sciences
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