Necessary and Sufficient Optimality Conditions for Nonlinear Fuzzy Optimization Problem

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1 Sutra: International Journal of Mathematical Science Education Technomathematics Research Foundation Vol. 4 No. 1, pp. 1-16, 211 Necessary and Sufficient Optimality Conditions f Nonlinear Fuzzy Optimization Problem V. D. Pathak U. M. Pirzada In this paper we derive the necessary and sufficient Kuhn-Tucker like optimality conditions f nonlinear fuzzy optimization problems with fuzzy valued objective function and fuzzy-valued constraints using the concept of convexity and H- diffrentiability of fuzzy-valued functions. Keywds: Fuzzy numbers Hukuhara differentiability Kuhn-Tucker optimality conditions 1 Introduction Classical optimization techniques have been successfully applied f years. In real process optimization, there exist different types of uncertainties in the system. Zimmermann [16] pointed out various kinds of uncertainties that can be categized as stochastic uncertainty and fuzziness. The optimization under a fuzzy environment which involve fuzziness is called fuzzy optimization. Bellman and Zadeh in 197 [1] proposed the concept of fuzzy decision and the decision model under fuzzy environments. After that, various approaches to fuzzy linear and nonlinear optimization, have been developed over the years by researchers. The nondominated solution of a nonlinear optimization problem with fuzzy-valued objective function was proposed by Wu [1]. Using the concept of continuous differentiability of fuzzy-valued functions, he derived the sufficient optimality conditions f obtaining the nondominated solution of fuzzy optimization problem having fuzzy- valued objective function with real constraints. However, the fuzzy optimization problem having fuzzy-valued constraints can not be solved by using the results of Wu [1]. In this article, we establish Kuhn-Tucker like both necessary and sufficient optimality conditions f obtaining the nondominated solution of a nonlinear fuzzy optimization V. D. Pathak Charotar Institute of Computer Applications, Changa-3421, DI. Anand, Gujarat, India. vdpathak@yahoo.com U. M. Pirzada Department of Applied Mathematics, Faculty of Tech. & Engg.,M.S.University of Baroda, Vadodara 391, India. salmapirzada@yahoo.com

2 2 problem with fuzzy-valued objective function and fuzzy-valued constraints. In Section 2, we introduce definition of fuzzy number, basic properites and arithmetics of fuzzy numbers. In Section 3, we consider the differential calculus of fuzzy-valued functions defined on R and R n using hukuhara differentiability of fuzzy-valued functions. In Section 4, we provide nondominated solution of unconstrained fuzzy optimization problems by proving the first and second der optimality conditions. In Section 5, we provide nondominated solution of nonlinear constrained fuzzy optimization problems by proving the Kuhn-Tucker like optimality conditions f the same. And at last we conclude in Section 6. 2 Preliminaries Definition 1 [6] Let R be the set of real numbers and ã : R [, 1] be a fuzzy set. We say that ã is a fuzzy number if it satisfies the following properties: (i) ã is nmal, that is, there exists x R such that ã(x ) = 1; (ii) ã is fuzzy convex, that is, ã(tx + (1 t)y) min{ã(x), ã(y)}, whenever x, y R and t [, 1]; (iii) ã(x) is upper semicontinuous on R, that is, {x/ã(x) α} is a closed subset of R f each α (,1]; (iv) cl{x R/ã(x) > } fms a compact set. The set of all fuzzy numbers on R is denoted by F(R). F all α (,1], α-level set ã α of any ã F(R) is defined as ã α = {x R/ã(x) α}. The -level set ã is defined as the closure of the set {x R/ã(x) > }. By definition of fuzzy numbers, we can prove that, f any ã F(R) and f each α (, 1], ã α is compact convex subset of R, and we write ã α = [ã L α, ã U α ]. ã F(R) can be recovered from its α-cuts by a well-known decomposition theem (ref. [7]), which states that ã = α [,1] α ã α where union on the right-hand side is the standard fuzzy union. Definition 2 [15] Accding to Zadeh s extension principle, we have addition and scalar multiplication in fuzzy number space F(R) by their α-cuts are as follows: where ã, b F(R), λ R and α [, 1]. (ã b) α = [ã L α + b L α, ã U α + b U α] (λ ã) α = [λ ã L α, λ ã U α], Definition 3 [9] Let A, B R n. The Hausdff metric d H is defined by d H (A,B) = max{sup inf x y, sup inf x y }. x A y B y B x A Then the metric d F on F(R) is defined as d F (ã, b) = sup {d H (ã α, b α)}, α 1 f all ã, b F(R). Since ã α and b α are closed bounded intervals in R, d F (ã, b) = sup max{ ã L α b L α, ã U α b U α }. α 1

3 3 We need the following proposition. Proposition 1 [3] F ã F(R), we have (i) ã L α is bounded left continuous nondecreasing function on (,1]; (ii) ã U α is bounded left continuous nonincreasing function on (,1]; (iii) ã L α and ã U α are right continuous at α = ; (iv) ã L α ã U α. Meover, if the pair of functions ã L α and ã U α satisfy the conditions (i)-(iv), then there exists a unique ã F(R) such that ã α = [ã L α, ã U α ], f each α [, 1]. We define here a partial der relation on fuzzy number space. Definition 4 F ã and b in F(R) and ã α = [ã L α, ã U α] and b α = [ b L α, b U α] are two closed intervals in R, f all α [,1], we define (i) ã b if and only if ã L α b L α and ã U α b U α f all α [, 1]; (ii) ã b if and only if < ã L α < b L α < ã L α b L α < ã L α < b L α : ã U α b U : α ã U α < b U : α ã U α < b U α f all α [, 1]. is partial der relation on fuzzy number space. Definition 5 [14] The membership function of a triangular fuzzy number ã is defined by (r a >< L ) if a L r a (a a L ) ζã(r) = (a U r) if a < r a U (a >: U a) otherwise which is denoted by ã = (a L, a, a U ). The α-level set of ã is then ã α = [(1 α)a L + αa,(1 α)a U + αa]. 3 Differential calculus of fuzzy-valued function 3.1 Continuity of fuzzy-valued function Definition 6 [] Let V be a real vect space and F(R) be a fuzzy number space. Then a function f : V F(R) is called fuzzy-valued function defined on V. Cresponding to such a function f and α [, 1], we define two real-valued functions f L α and f U α on V as f L α (x) = ( f(x)) L α and f U α (x) = ( f(x)) U α f all x V. Definition 7 [4] Let f : R n F(R) be a fuzzy-valued function. We say that f is continuous at c R n if f every ǫ >, there exists a δ = δ(c, ǫ) > such that f all x R n with x c < δ. That is, d F ( f(x), f(c)) < ǫ lim x c f(x) = f(c).

4 4 We prove the following proposition. Proposition 2 Let f : R n F(R) be a fuzzy-valued function. If f is continuous at c R n, then functions f L α (x) and f U α (x) are continuous at c, f all α [, 1]. Proof The result follows using the definitions of continuity of fuzzy-valued function f and metric on fuzzy numbers. 3.2 H-differentiability of fuzzy-valued function on R Let ã and b be two fuzzy numbers. If there exists a fuzzy number c such that c b = ã. Then c is called Hukuhara difference of ã and b and is denoted by ã H b. H-differentiability of fuzzy-valued function due to M.L. Puri and D.A. Ralescu [11] is as follows Definition Let X be a subset of R. A fuzzy-valued function f : X F(R) is said to be H-differentiable at x X if there exists a fuzzy number D f(x ) such that the limits (with respect to metric d F ) 1 lim h + h [ f(x + h) H f(x )], and 1 lim h + h [ f(x ) H f(x h)] both exist and are equal to D f(x ). In this case, D f(x ) is called the H-derivative of f at x. If f is H-differentiable at any x X, we call f is H-differentiable over X. Remark 1 Many fuzzy-valued functions are H-differentaible f which Hukuhara differences f(x +h) H f(x ) and f(x ) H f(x h) both exist. The following example illustrates the fact. Example 1 Given in [11], let f : (, 2π) F(R) be defined on level sets by [ f(x)] α = (1 α)(2 + sin(x))[ 1,1], f α [, 1]. At x = π/2, H-difference does not exist. Therefe, function is not H-differentiable at x = π/2. Now we prove following proposition regarding differentiability of f L α and f U α. Proposition 3 Let X be a subset of R. If a fuzzy-valued function f : X F(R) is H-differentiable at x with derivative D f(x ), then f L α (x) and f U α (x) are differentiable at x, f all α [, 1]. Meover, we have (D f) α(x ) = [D( f L α )(x ), D( f U α )(x )]. Proof The result follows from definitions of H-differentiability of fuzzy-valued function and metric on fuzzy number space.

5 5 3.3 H-differentiability of fuzzy-valued function on R n Definition 9 [1] Let f be a fuzzy-valued function defined on an open subset X of R n and let x = (x 1,..., x n) X be fixed. We say that f has the i th partial H-derivative D i f(x ) at x if the fuzzy-valued function g(x i ) = f(x 1,.., x i 1, x i, x i+1,.., x n) is H-differentiable at x i with H-derivative D i f(x ). We also write D i f(x ) as ( f/ x i )(x ). Definition 1 [1] We say that f is H-differentiable at x if one of the partial H- derivatives f/ x 1,..., f/ x n exists at x and the remaining n-1 partial H-derivatives exist on some neighbhoods of x and are continuous at x (in the sense of fuzzyvalued function). The gradient of f at x is denoted by f(x ) = (D 1 f(x ),..., D n f(x )), and it defines a fuzzy-valued function from X to F n (R) = F(R)... F(R) (n times), where each D i f(x ) is a fuzzy number f i = 1,...,n. The α-level set of f(x ) is defined and denoted by ( f(x )) α = ((D 1 f(x ))α,..., (D n f(x ))α), where (D i f(x ))α = [D i fl α (x ), D i fu α (x )], i = 1,...,n. We say that f is H-differentiable on X if it is H-differentiable at every x X. Proposition 4 Let X be an open subset of R n. If a fuzzy-valued function f : X F(R) is H-differentiable on X. Then f L α and f U α are also differentiable on X, f all α [, 1]. Meover, f each x X, (D i f(x))α = [D i fl α (x),d i fu α (x)], i = 1,...,n. Proof The result follows from Propositions 2 and 3. Definition 11 We say that f is continuously H-differentiable at x if all of the partial H-derivatives f/ x i, i = 1,..,n, exist on some neighbhoods of x and are continuous at x (in the sense of fuzzy-valued function). We say that f is continuously H-differentiable on X if it is continuously H-differentiable at every x X. Proposition 5 Let f : X F(R) is continuously H-differentiable on X. Then f L α and f U α are also continuously differentiable on X, f all α [,1]. Proof Followed by Propositions 2 and 4. Now we define twice continuously H-differentiable fuzzy-valued function.

6 6 Definition 12 Let f : X F(R), X R n be a fuzzy-valued function. Suppose now that there is x X such that gradient of f, f, is itself H-differentiable at x, that is, f each i, the function D i f : X F(R) is H-differentiable at x. Denote the H-partial derivative of D i f in the direction of ēj at x by and D 2 ij f 2 f(x ) x i x j, if i j, Dii 2 f 2 f(x ) x 2, if i = j. i Then we say that f is twice H-differentiable at x, with second H-derivative 2 f(x ) which is denoted by 1 2 f(x )... 2 f(x ) x 2 x 1 1 x n 2 f(x ) = 2 f(x ) x n x f(x ) x 2 n where 2 f(x ) x i x j F(R), i,j = 1,...,n. If f is twice H-differentiable at each x in X, we say that f is twice H-differentiable on X, and if f each i, j = 1,...,n, the cross-partial derivative 2 f x i x j is continuous function from X to F(R), we say that f is twice continuously H-differentiable on X. The α-level set of 2 f(x ) is defined and denoted in matrix notation as ( 2 f(x )) α = ((D 2 ij f(x )) α) i,j = 1,...,n and α [, 1], where (D 2 ij f(x )) α denotes α-cut of (D 2 ij f(x )). C A Proposition 6 Let f : X R n F(R) is differentiable with derivative f on X and let each D i f : X F(R), i = 1,...,n, is also differentiable at x with derivative D 2 ij f(x ), i, j = 1,...,n. Then D i fl α and D i fu α are also differentiable at x, f all α [, 1]. Also, we have (D 2 ij f(x )) α = [D 2 ij ( f L α )(x ), D 2 ij ( f U α )(x )], i,j = 1,...,n. Proof Follows by Proposition 5. In der to define the Kuhn-Tucker like optimality conditions f nonlinear fuzzy optimization problems, we need to provide some properties of fuzzy-valued functions. F that first we state here following two Propositions from Real Analysis. Proposition 7 [13] Let φ be a real-valued function of two variables defined on I [a,b], where I is an interval in R. Suppose that the following conditions are satisfied: (i) F every x I, the real-valued function h(y) = φ(x, y) is Riemann integrable on Z b [a,b]. In this case, we write f(x) = φ(x,y)dy; a (ii) Let x int(i), the interi of I. F every ǫ >, there exists a δ > such that φ φ (x,y) x x (x, y) < ǫ f all y [a, b] and all x (x δ,x + δ).

7 7 Then φ x (x, y) is Riemann integrable on [a,b], f (x ) exists, and Z b f (x φ ) = a x (x, y)dy. Proposition [2] Every function monotonic on an interval is Riemann integrable there. Let f : X F(R) be a fuzzy-valued function defined on X subset of R n. Then f each α [,1], fl α and f α U are real-valued functions defined on X. F any fixed x X, we have the cresponding real-valued functions f α L (x ) and f α U (x ) defined on α [,1]. By Proposition 1 and, we can easily say that f α L (x ) and f α U (x ) are riemann integrable. So, we define new functions F L and F U as follows F L (x) = f α L (x)dα and F U (x) = f α U (x)dα (3.1) f every x X. Then we have following useful proposition. Proposition 9 [1] Let f be a fuzzy-valued function defined on an open subset X of R n. If f is continuously H-differentiable on some neighbhood of x. Then the realvalued functions F L and F U defined as above are continuously differentiable at x and F L (x ) = f α L (x )dα and F U (x ) = f α U (x ) dα x i x i x i x i f all i=1,..,n. Proof We need to show that the partial derivatives F L x i and F U x i exist on some neighbhood of x and are continuous at x f all i =1,..,n. Since f is continuously H- differentiable on some neighbhood of x. By Proposition 5, fl α and f α U are also continuously differentiable real-valued functions at x f all α [, 1]. Therefe, Proposition say that F L x i (x ) = f α L (x )dα and F U (x ) = f α U (x ) dα (3.2) x i x i x i f all i=1,..,n. Since fl x i is continuous at x, that is, f every ǫ > there exists a δ > such that x x < δ implies f α L (x) f L α (x ) < ǫ f all α [, 1] x i x i From (3.2), we have, if x x < δ then F L (x ) F L (x) x i x i = [ f α L (x ) f U α (x)] dα x i x i f α L (x ) f U α (x) dα < ǫ, x i x i f all i =1,...,n. Therefe, F L x i is contiuous f all i = 1,...,n. Similarly we can discuss the case of F U x i. Hence complete the proof.

8 4 Neccesary and sufficient optimality conditions f unconstrained fuzzy optimization problem 4.1 Unconstrained Fuzzy Optimization Problem Let T R n be an open subset of R n and f be fuzzy-valued function defined on T. Consider the following nonlinear fuzzy optimization problem (FOP1) Minimize f(x) = f(x 1,.., x n) Subject to x T We define here nondominated solutions of (FOP1). Definition 13 Let T is an open subset of R n. (i) A point x T is a locally nondominated solution of (FOP1) if there exists no x 1 ( x ) N ǫ(x ) T such that f(x 1 ) f(x ), N ǫ(x ) is a ǫ-neighbhood of x. (ii) A point x T is a nondominated solution of (FOP1) if there exists no x 1 ( x ) T such that f(x 1 ) f(x ). (iii) A point x T is a locally weak nondominated solution of (FOP1) if there exists no x 1 ( x ) N ǫ(x ) T such that f(x 1 ) f(x ). (iv) A point x T is a weak nondominated solution of (FOP1) if there exists no x 1 ( x ) T such that f(x 1 ) f(x ). 4.2 Neccessary and sufficient optimality conditions The first and second der neccessary and sufficient optimality conditions f real unconstrained optimization problem, given in [5], are as follows. Theem 1 Let T is an open subset of R n. (i) (FONC) Let f continuously differentiable function on T. If x is a local minimizer of f over T, then f(x ) =. (ii) (SONC) Let f twice continuously differentiable function on T. If x is a local minimizer of f over T, then 2 f(x ) is positive semidefinite. (iii) (SOSC) Let f twice continuously differentiable function on T. Suppose that 1. f(x ) = and 2. 2 f(x ) is positive definite. Then x is a strict local minimizer of f. We prove here necessary and sufficient optimality conditions f obtaining nondominated solution of (FOP1). F that first we prove the following proposition. Proposition 1 If x is a locally nondominated solution of (FOP1), then x is also a local minimum of real-valued functions f L α (x) and f U α (x) f all α [, 1]. Proof We prove this result by contradiction. Assume that x is not a local minimum of f L α f U α f at least one α [, 1]. Without loss of generality, suppose that x is not a local minimum of f L α f α [, 1]. Therefe, there exists x 1 N ǫ(x ) T such that

9 9 f L α (x 1 ) < f L α (x ) (4.1) Since x is a locally nondominated solution of (FOP1), there exists no x N ǫ(x ) T such that < f α L ( x) < f α L (x ) < f α L ( x) f α L (x ) < f α L ( x) < f α L (x ) : f α U ( x) f α U (x : ) f α U ( x) < f α U (x : ) f α U ( x) < f α U (x ) f all α [,1]. This gives contradiction to inequality (4.1). Therefe x is also a minimum of real-valued functions f α L (x) and f α U (x) f all α [, 1]. Now we prove the first-der necessary condition. Theem 2 Suppose x T is a locally nondominated solution of (FOP1). Suppose also that f is conituously H-differentiable function on T. Then f α L (x )dα + f α U (x )dα = Proof The theem can prove easily using Proposition 1 and Theem 1 (i). Next, we prove first-der sufficient condition. Theem 3 Let f is twice-contiuously H-differentiable fuzzy-valued function defined on T R n. If x is a locally nondominated solution of (FOP1) then 2 F(x ) is positive semidefinite matrix. Here 2 F(x ) = 2 fl α (x )dα + 2 fu α (x )dα Proof Since f twice-contiuously H-differentiable fuzzy-valued function on T. By Proposition 2 and 6, f L α and f U α are also twice-contiuously H-differentiable functions on R n. Also, by Proposition 1, we say that f L α and f U α has local minimum at x. Therefe by second der neccessary condition f real unconstrained optimization stated in Theem 1 (ii), 2 fl α (x ) and 2 fu α (x ) are positive semidefinite, f α [, 1]. That is, x T 2 fl α (x ) x and x T 2 fu α (x ) x f all x T, x and α [, 1], where x T is transpose of x. Therefe, x T 2 fl α (x ) x and x T 2 fu α (x ) x Adding these inequalities, we get x T 2 F(x ) x f all x T, x. Therefe, 2 F(x ) is positive semidefinite.

10 1 Now, we prove second-der sufficient condition. Theem 4 Let f is twice contiuously H-differentiable function on T R n. Suppose that 1. F(x ) 2. 2 F(x ) is positive definite. Then, x is locally weak nondominated solution of (FOP1). Proof We prove this result using contradiction. Suppose x T is not a locally weak nondominated solution of (FOP1). Then, there exists x 1 N ǫ(x ) T such that f(x 1 ) f(x ). That is, there exists x 1 N ǫ(x ) T such that < f α L (x 1 ) < f α L (x ) < f α L (x 1 ) f α L (x ) < f α L (x 1 ) < f α L (x ) : f α U (x 1 ) f α U (x : ) f α U (x 1 ) < f α U (x : ) f α U (x 1 ) < f α U (x ) f all α [, 1]. Therefe, we have where F(x) = F(x 1 ) < F(x ), (4.2) f α L (x)dα + f α U (x)dα. As f is twice continuously H-differentaible function, F(x) is also. Using assumption 2 and Rayleigh s inequality (refer [5]), it follows that if d, then By Tayl s theem and assumption 1, < λ min (F(x ) d 2 d T F(x ) d. F(x + d) F(x ) = 1 2 dt 2 F(x ) d + O( d 2 ) Hence, f all d such that d is sufficiently small, λ min( 2 F(x )) d 2 + O( d 2 ). 2 F(x + d) > F(x ) This gives contradiction to inequality (4.2). Hence proved the result. We give one example to illustrate the above results. Example 2 Let f : R 2 F(R) be defined by f(x 1, x 2 ) = (, 2,4) x 2 1 (,2, 4) x2 2 (1,3, 5), where (, 2,4) and (1, 3,5) are triangular fuzzy numbers. By first der neccessary condition: F(x) =. Here, f L α (x 1, x 2 ) = 2αx αx (1 + 2α) and f α U (x 1, x 2 ) = (4 2α)x (4 2α)x (5 2α). Therefe, «fα L 4x1 α (x 1, x 2 ) = 4x 2 α

11 11 and Therefe, «fα U 2(4 2α)x1 (x 1, x 2 ) = 2(4 2α)x 2 «14x1 F(x 1, x 2 ) = 14x 2 This implies, x = (x 1, x 2 ) = (,). Now to verify second der necessary and sufficient conditions, we find 2 F(x): 2 fl α (x) = «4α 4α and Therefe, 2 fu α (x) = 2(4 2α) «2(4 2α) 2 F(x) = «14 14 which is positive definite. Therefe, by neccessary and sufficient conditions, x = (, ) is a nondominated solution of given problem. 5 Neccesary and sufficient optimality conditions f constrained fuzzy optimization problem 5.1 Constrained Fuzzy Optimization Problem Let T R n be an open subset of R n and f, g j, f j = 1,...,m be fuzzy-valued functions defined on T. Consider the following nonlinear fuzzy optimization problem (FOP2) Minimize f(x) = f(x 1,.., x n) Subject to g j (x), j = 1,.., m, where is a fuzzy number defined as (r) = 1 if r = and (r) = if r and its level set is α = {} f α [, 1]. Definition 14 Let x X = {x T : g j (x), j = 1,.., m}. We say that x is a nondominated solution of (FOP2) if there exists no x 1 ( x ) X such that f(x 1 ) f(x ). That is, x is a nondominated solution of (FOP2) if there exists no x 1 ( x ) X such that < f α L (x 1 ) < f α L (x ) < f α L (x 1 ) f α L (x ) < f α L (x 1 ) < f α L (x ) : f α U (x 1 ) f α U (x : ) f α U (x 1 ) < f α U (x : ) f α U (x 1 ) < f α U (x ) f all α [, 1].

12 Necessary and Sufficient Optimality Conditions Let f and g j, j = 1,.., m, be real-valued functions defined on T R n. Then we consider the following optimization problem (P) Minimize f(x) = f(x 1,.., x n) Subject to g j (x), j = 1,.., m. The well-known Kuhn-Tucker optimality conditions f problem (P) by S. Rangarajan in [12] is stated as follows: Theem 5 Let f be a convex, continuously differentiable function mapping T into R, where T R n is open and convex. F j =1,..m, the constraint functions g j : T R are convex, continuously differentiable functions. Suppose there is some x T such that g j (x) <, j =1,..,m. Then x is an optimal solution of propositionblem (P) over the feasible set {x T : g j (x), j = 1,.., m} if and only if there exist multipliers µ j R, j = 1,..,m, such that the Kuhn-Tucker first der conditions hold: (KT-1) f(x ) + P m j=1 µ j g j (x ) = ; (KT-2) µ j g j (x ) = f all j = 1,..,m. First we introduce the concept of convexity f fuzzy-valued functions. Definition 15 Let T be a convex subset of R n and f be a fuzzy-valued function defined on T. We say that f is convex at x if f each λ (,1) and x T. f(λx + (1 λ)x) (λ f(x ) ((1 λ) f(x)) Proposition 11 f : T F(R) is convex at x if and only if f L α and f U α are convex at x, f all α [, 1]. Proof The result can prove easily using the concepts of arithmetic operations and partial der relation of fuzzy numbers. We now present the Kuhn-Tucker like optimality conditions f (FOP2). Theem 6 Let the fuzzy-valued objective function f : T F(R) is convex and continuously H-differentiable, where T R n is open and convex. F j = 1,..,m, the fuzzyvalued constraint functions g j : T F(R) are convex and continuously H-differentiable. Let X = {x T R n : g j (x), j = 1,.., m} be a feasible set of problem (FOP) and let x X. Suppose there is some x T such that g j (x), j =1,..,m. Then x is a nondominated solution of problem (FOP2) over X if and only if there exist multipliers µ j R, j = 1,..,m, such that the Kuhn-Tucker first der conditions hold: (FKT-1) f α L (x ) dα + f mx α U (x ) dα + µ j g j(x U ) = ; j=1 (FKT-2) µ j g j U (x ) = f all j = 1,..,m.

13 13 Proof Necessary. Define a new function, F(x) = f α L (x)dα + f α U (x)dα. (5.1) Since f is convex and continuously H-differentiable function, by Propositions 5 and 11, we say that F(x) is convex and continuously differentiable real-valued function on T. Since x is a nondominated solution of (FOP2). Then there exists no (x 1 x ) X such that < f α L (x 1 ) < f α L (x ) < f α L (x 1 ) f α L (x ) < f α L (x 1 ) < f α L (x ) : f α U (x ) f α U (x : ) f α U (x 1 ) < f α U (x : ) f α U (x 1 ) < f α U (x ) f all α [, 1]. That is, there exists no x 1 ( x ) X such that Therefe, F(x 1 ) < F(x ) F(x ) F(x 1 ) Since g j are convex and continuously H-differentiable functions f j = 1,..,m implies g jα L and gu jα are real-valued convex and continuously differentiable functions f all α [, 1] and j =1,..,m. By definition of partial dering and Proposition 1, we have X = {x T R n : g j (x), j = 1,..., m} = {x T R n : g jα(x) L and g jα(x) U, j = 1,..., m} = {x T R n : g jα(x) U, j = 1,..., m} = {x T R n : g j(x) U, j = 1,..., m} Therefe, x X = {x T R n : g j U (x), j = 1,.., m} and there is some x T such that g j U (x) <, j =1,..,m. So our problem becomes an optimization problem with real objective function F(x) subject to real constraints. Therefe, by Theem 5, there exist multipliers µ j R, j = 1,..,m, such that the following kuhn-tucker first der conditions hold: (KT-1) F(x ) + P m j=1 µ j g U j (x ) = ; (KT-2) µ j g U j (x ) = f all j = 1,..,m. But F(x) = f α L (x)dα + f α U (x)dα. We obtain the kuhn-tucker conditions f problem (FOP2) as follows (FKT-1) f α L (x ) dα + f mx α U (x ) dα + µ j g j(x U ) = ; j=1 (FKT-2) µ j g j U (x ) = f all j = 1,..,m.

14 14 Sufficient.We are going to prove this part by contradiction. Suppose that x not a nondominated solution. Then there exists a x 1 ( x ) X such that f(x 1 ) f(x ). Therefe, we have f all α [, 1]. From (5.1), we obtain f L α (x 1 ) + f U α (x 1 ) < f L α (x ) + f U α (x ) F(x 1 ) < F(x ) (5.2) Since F is convex and continuously differentiable function. Furtherme, x X = {x T R n : g j U (x), j = 1,.., m}, by conditions (FKT-1) and (FKT-2) of this theem, we obtain the following new conditions: (KT-1) F(x ) + P m j=1 µ j g U j (x ) = ; (KT-2) µ j g U j (x ) = f all j = 1,..,m. Using Theem 5, we say that x is an optimal solution of real-objective function F with real constraints g j U (x), f j = 1,.., m. i.e., F(x ) F(x 1 ), which contradicts to (5.2). Hence the proof. We consider here first fuzzy optimization problem having fuzzy-valued objective function and real constraints. Example 3 Minimize f(x1, x 2 ) = (ã x 2 1) ( b x 2 2) subject to g(x 1, x 2 ) = (x 1 2) 2 + (x 2 2) 2 1, where ã = (1, 2,3) and b = (,1,2) are triangular fuzzy numbers defined on R as (r 1), if 1 r 2, r, if r 1, >< >< ã(r) = (3 r), if 2 < r 3, b(r) = 2 r, if 1 < r 2, >: >: otherwise otherwise Using definition 2, we obtain f L α (x 1, x 2 ) = (1 + α)x αx 2 2 and f U α (x 1, x 2 ) = (3 α)x (2 α)x 2 2 f α [, 1]. We obtain «f α L 2x1 (α + 1) (x 1, x 2 ) =, 2x 2 α f α U (x 1, x 2 ) = «2(x1 2) g(x 1, x 2 ) = 2(x 2 2) Therefe, we have «2x1 (3 α) and 2x 2 (2 α) «f α L 3x1 (x 1, x 2 ) dα =, x f U 5x1 α (x 1, x 2 ) dα = 2 3x 2 From Theem 6, we have the following Kuhn-Tucker conditions «.

15 15 (FKT-1) x 1 + 2µ(x 1 2) =, 4x 2 + 2µ(x 2 2) =, (FKT-2) µ((x 1 2) 2 + (x 2 2) 2 1) =. Solving these equations, we get the solution (x 1, x 2 ) = (6/5, 3/2) and µ = 6. By Theem 6, we say that (x 1, x 2 ) = (6/5, 3/2) is nondominated solution f given problem. Also the minimum value of objective function is f min = (1.44, 5.13,.2) and we can find its defuzzified value 5.13 by using center of area method (ref. [7]). Now we solve the same fuzzy optimization problem having fuzzy-valued objective function with fuzzy constraints. Example 4 Minimize f(x1, x 2 ) = (ã x 2 1) ( b x 2 2) subject to g(x 1, x 2 ) = ( b (x 1 2) 2 ) ( b (x 2 2) 2 ) c, where ã = (1, 2,3), b = (,1, 2) and c = (, 2,4) are triangular fuzzy numbers defined on R as (r 1), if 1 r 2, r, if r 1, >< >< ã(r) = (3 r), if 2 < r 3, b(r) = 2 r, if 1 < r 2, >: >: otherwise otherwise r/2, if r 2, >< c(r) = (4 r)/2, if 2 < r 4, >: otherwise Using definition 2, we obtain f L α (x 1, x 2 ) = (1 + α)x αx2 2 and f U α (x 1, x 2 ) = (3 α)x (2 α)x2 2 f α [, 1]. Meover, g U α (x 1, x 2 ) = (2 α)(x 1 2) 2 + (2 α)(x 2 2) 2 (4 2α) f α [, 1]. Therefe, g U (x 1, x 2 ) = (x 1 2) 2 + (x 2 2) 2 2. Now we obtain «f α L 2x1 (α + 1) (x 1, x 2 ) =, 2x 2 α f α U (x 1, x 2 ) = «2(x1 2) g(x 1, x 2 ) = 2(x 2 2) Therefe, we have «2x1 (3 α) and 2x 2 (2 α) «f α L 3x1 (x 1, x 2 ) dα =, x f U 5x1 α (x 1, x 2 ) dα = 2 3x 2 From Theem 6, we have the following Kuhn-Tucker conditions (FKT-1) x 1 + 2µ(x 1 2) =, 4x 2 + 2µ(x 2 2) =, (FKT-2) µ((x 1 2) 2 + (x 2 2) 2 2) =. «.

16 16 Solving these equations, we get the solution (x 1, x 2 ) = (( )/(1+ 41),( )/( )) and µ = By Theem 6, we say that (x 1, x 2) = (( )/(1+ 41),( )/( 1+ 41)) is nondominated solution f given problem. Also the minimum value of objective function is f min = (.453, , 5.794) and we can find its defuzzified value by using center of area method. Remark 2 By comparing the defuzzified value of minimum objective functions in example 3 and 4, we observe that there is significant effect on minimum value of the fuzzyvalued objective function if consider fuzzy optimization problem with fuzzy constraints. Meover, if we consider the fuzzy optimization problem having fuzzy constraints then we can not apply Theem 6.2 from [1] to find the nondominated solution. In that case, our result is quite useful to get the solution. 6 Conclusion Using partial der relation on fuzzy number space, the necessary and sufficient Kuhn- Tucker like optimality condtions f nonlinear fuzzy optimization problem have been derived in this paper. We have used hukuhara differentiability and convexity of fuzzyvalued function f proving the same. We have also provided an example to illustrate the possible applications in this subject. References 1. Bellman,R.E.and Zadeh,L.A., Decision making in a fuzzy environment, Management Science, 17, 197,pp Brian S. Thomson, Judith B. Bruckner, Andrew M. Bruckner, Elementary Real Analysis. Prentice Hall (Pearson) Publisher, Degang,C. and Liguo,Z., Signed Fuzzy valued Measures and Radon-Nikodym Theem of Fuzzy valued Measurable Functions, Southest Asian Bulletin of Mathematics, Diamond P., Kloeden, Meric spaces of fuzzy sets: They and Applications,Wld Scientific, Chong,E. K. P. and Zak,S. H., An Introduction to Optimization, A Wiley-Interscience Publication, Gege,A. A., Fuzzy Ostrowski Type Inequalities, Computational and Applied mathematics, Vol. 22, 23, pp Gege,J.K., Bo Yuan, Fuzzy Sets and Fuzzy Logic: They and Applications, Prentice-Hall of India, Hsien-Chung Wu, Duality They in Fuzzy Optimization Problems, Fuzzy Optimization and Decision Making,3, 24, pp Hsien-Chung Wu, An (α, β)-optimal Solution Concept in Fuzzy Optimization Problems, Optimization,Vol. 53,April 24, pp Hsien-Chung Wu, The Optimality Conditions f Optimization Problems with Fuzzyvalued Objective Functions, Optimization 57, 2, pp Puri M. L., Ralescu D. A., Differentials of fuzzy functions, J. of Math. Analysis and App., 91, 193, pp Rangarajan K. Sundaram, A First Course in Optimization They,Cambridge University Press, Rudin, W.,Principles of Mathematical Analysis (3rd Ed.), New Yk: McGraw-Hill Book Company, Saito, S. and Ishii, H., L-Fuzzy Optimization Problems by Parametric Representation,IEEE, Song S., Wu C., Existence and uniqueness of solutions to cauchy problem of fuzzy differential equations, Fuzzy Sets and Systems, 11, 2, pp Zimmermann, H.J., Fuzzy Set They and Its Applications(2nd Ed.),Kluwer-Nijhoff, Hinghum, 1991.