Efficient solution of interval optimization problem

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1 Math Meth Oper Res (212) 76: DOI 1.17/s ORIGINAL ARTICLE Efficient solution of interal optimization problem A. K. Bhurjee G. Panda Receied: 6 June 212 / Accepted: 4 August 212 / Published online: 18 August 212 Springer-Verlag 212 Abstract In this paper the interal alued function is defined in the parametric form and its properties are studied. A methodology is deeloped to study the existence of the solution of a general interal optimization problem, which is expressed in terms of the interal alued functions. The methodology is applied to the interal alued conex quadratic programming problem. Keywords Interal optimization Interal alued function Interal matrix Efficient solution 1 Introduction Interal optimization problem (IOP) has been studied by many researchers in seeral directions by Ishibuchi and Tanaka (199), Tong and Shaocheng (1994), Lein (1999), Hu and Wang (26a,b), Liu and Wang (27), Jiang and Liu (28), Li and Tian (28), Wu (28, 29), Jayswal et al.(211), Hladík (211), Jeyakumar and Li (211), Hladik (212). In a linear/nonlinear interal optimization problem, uncertainty may appear in the form of closed interals either in the objectie function or in at least one of the constraints or in both. The interal optimization problem, addressed by Ishibuchi and Tanaka (199) has linear interal alued objectie function and free from interal uncertainty in the set of constraints. The maximization and minimization A. K. Bhurjee Indian Institute of Technology Kharagpur, Kharagpur 72132, West Bengal, India ajaybhurji@gmail.com G. Panda (B) Department of Mathematics, Faculty of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur 72132, West Bengal, India geetanjali@maths.iitkgp.ernet.in

2 274 A. K. Bhurjee, G. Panda problems are soled with respect to different partial orderings. Liu and Wang (27) and Li and Tian (28) consider interal quadratic programming problem and determine the bounds for a pair of two leel mathematical programming problems. Suppose a(x) and b(y) are the minimum alue of lower and upper leel problems of the interal optimization problem. Then x lies in the maximum feasible region and y lies in the minimum feasible region. They are not optimal feasible solutions of the original problem. They proide optimal alue range. Since the interal objectie function is a set alued mapping, so the interal optimization problem possesses a set of efficient(pareto-optimal) solutions and the corresponding objectie alues are compromising optimal alues. Here we hae deeloped a methodology to find the optimal alue as an interal, [α(x ), β(x )] corresponding to the decision ariable x in place of two decision ariables x and y. Also, we hae justified that the solution, x, is both feasible and efficient(pareto- optimal). We focus on a general interal optimization problem, where the interal alued functions may be linear or nonlinear, and the interal uncertainty appears in both objectie function and constraints in any form. The interal optimization problem is conerted to a general optimization problem in the parametric form. Finally we establish a relationship between the solution of the original problem and the transformed problem in Sect. 3. This is a new deelopment, which is different from our predecessors. Section 4 describes an application of the proposed method to interal quadratic programming problem. Conexity plays an important role to proe the existence of the solution of a general optimization problem. Hence there is a need to study the conex property of the interal optimization problems. Since the set of interals is not totally ordered, conexity has to be studied with respect to a partial order. In this contest we introduce conex interal programming problem and study the positie definiteness of the interal Hessian matrix of an interal alued conex function using the concepts deeloped by Rohn (1994). We define two partial orderings and w in the set of interals through parameters and see that all other partial orderings, used so far in the interal optimization problems, are particular cases of our order relation. Throughout the paper the following notations are used. Bold capital letters denote closed interals, small letters denote real numbers. I (R) = The set of all closed interals in R. (I (R)) k = The product space I (R) I (R) I (R) k times C k = k dimensional column ector whose elements are interals. That is, Ck (I (R)) k, C k = (C 1, C 2,...,C k ) T, C j =[c L j, cr j ], j = 1,...,k. (I (R)) p q = The set all of interal matrices A m of order p q. A m = (A ij ) p q, A ij =[aij L, a ij R ], i = 1, 2,...,p; j = 1, 2,...,q. 2 Interal alued function in the parametric form Let +,,,/ be a binary operation on the set of real numbers. The binary operation between two interals A =[a L, a R ] and B =[b L, b R ] in I (R), denoted

3 Efficient solution of interal optimization problem 275 by A B is the set a b : a A, b B. In the case of diision, A/B, itis assumed that / B. An interal can be expressed in terms of a parameter in seeral ways. Any point in A maybeexpressedasa(t), where a(t) = a L + t(a R a L ). Throughout this paper we consider a specific parametric representation of an interal as A =[a L, a R ]=a(t) t [, 1]. The algebraic operations of interals in the classical form are defined in terms of either lower and upper bound or mean and spread of the interals. These interal operations can also be performed with respect to parameters as follows. A B may be restated as the set a(t 1 ) b(t 2 ) t 1, t 2 [, 1]. Hence we hae A B =a(t 1 ) + b(t 2 ) t 1, t 2 [, 1], A B =a(t 1 ) b(t 2 ) t 1, t 2 [, 1], A B =a(t 1 ) b(t 2 ) t 1, t 2 [, 1], ka =ka(t) t [, 1], A B = a(t 1 )/a(t 2 ) t 1, t 2 [, 1], a(t 2 ) =. C k d=the product of a real ector d Rk and an interal ector C k (I (R))k, which is equal to k j=1 C j d j. C k (I (R))k is the set c(t) c(t) = (c 1 (t 1 ), c 2 (t 2 ),..., c k (t k )) T, c j (t j ) = c L j + t j (c R j c L j ), t = (t 1,...,t k ) T, t j 1, j = 1, 2,...,k. (1) A m (I (R)) p q is the set of real matrices A(t) A(t) = (a ij (t ij )) p q is a real matrix of order p q, where a ij (t ij ) = a L ij + t ij(a R ij al ij ), t ij 1, i = 1, 2,...,p, j = 1, 2,...,q I (R) is not a totally order set. Seeral partial order relations exist in I (R) (see Ishibuchi and Tanaka 199; Moore 1966), which is summarized as below. A LR B if a L b L and a R b R, A LR B if A LR B and A = B, A LC B if a L b L and a C b C, A LC B if A LC B and A = B, A RC B if a R b R and a C b C, A RC B if A RC B and A = B, where a C = al + a R. 2. (2) Here we introduce two partial orderings using parameters, which are stronger than the other ones.

4 276 A. K. Bhurjee, G. Panda Definition 1 For A, B I (R), (i). A B if a(t 1 ) b(t 2 ) t 1, t 2 [, 1]. (ii). A w B if a(t) b(t) t [, 1]. (A B if a(t 1 )<b(t 2 ) t 1, t 2 [, 1]; A w B if a(t) <b(t) t [, 1].) Remark 1 In particular cases the aboe partial orderings become the existing partial orderings in I (R). For (t 1, t 2 ) = (, ) and (1, 1), becomes LR ; for (t 1, t 2 ) = (, ) and ( 1 2, 1 2 ), becomes LC; for (t 1, t 2 ) = (1, 1) and ( 1 2, 1 2 ), becomes RC; for t = and t = 1, w becomes LR ; t = and t = 1 2, w becomes LC ; and t = 1 and t = 1 2, becomes RC. w is a particular case of. SoA B always implies A w B. Conerse may not be true. For A =[2, 4], B =[3, 5], a(t) = 2 + 2t, b(t) = 3 + 2t t 3 + 2t for eery t [, 1]. SoA w B is true. But 2 + 2t t 2 for t 1 = 1, t 2 =. So A B is false. The adantage of the use of the parametric form of interal alued function may be marked while applying calculus of interal alued function to study the positie definiteness of the interal Hessian matrix of a conex interal alued function. In the following subsections and Remark 4, we emphasize these adantages. 2.1 Interal alued function Interal alued function is defined in seeral ways by many authors like Hansen (24), Moore (1966), Wu (28) etc. Hansen (24) and Moore (1966) defined an interal function as a function of one or more interal arguments onto an interal. Wu (28) considered the interal alued function, F : R n I (R) as, F(x) =[F L (x), F R (x)], where F L, F R : R n R, F L (x) F R (x) x R n. We represent an interal alued function in a different way. Consider a function f : R n R. Since R n is a ector space oer R, so parameters of f are real numbers. Let c denotes the set of all parameters which are present in f. Without loss of generality we may consider c as an order set with respect to the order maintained on their presence in f (x). For example, if f (x 1, x 2 ) = 3x x 1e 4x 2 then c = (3, 2, 4) T. In general for c = (c 1, c 2, c 3 ) T, we denote f c (x 1, x 2 ) = c 1 x1 2 + c 2 x 1 e c 3x 2. Suppose c 1, c 2, and c 3 lie in the interals C 1, C 2, and C 3, respectiely. C j =[c L j, cr j ], c j(t j ) = c L j + t j (c R j c L j ), t j 1, j = 1, 2, 3. Then c(t) = (c 1 (t 1 ), c 2 (t 2 ), c 3 (t 3 )) T C 3. So f c(t)(x 1, x 2 ) = c 1 (t 1 )x1 2 + c 2(t 2 )x 1 e c 3(t 3 )x 2. Each c j (t j ) is a linear function. For a gien interal ector C 3,theset f c(t)(x 1, x 2 ) f c(t) : R 2 R, c(t) C 3 is an interal, [min t f c(t) (x 1, x 2 ), max t f c(t) (x 1, x 2 )]. In the light of the concept of this example, we redefine an interal alued function as follows. For gien C k (I (R))k, in other words for gien c(t)(see Sect. 2, Expression 1), let f c(t) : R n R. For eery c(t), f c(t) is a function of x. Suppose for eery x, f c(t)

5 Efficient solution of interal optimization problem 277 is continuous in t. Then for a gien interal ector C k, we define an interal alued function F C k : R n I (R) by F C k (x) = f c(t) (x) f c(t) : R n R, c(t) C k. Since for ery fixed x, f c(t) (x) is continuous in t so min c(t) C k f c(t) (x), which is min t [,1] k f c(t) (x) and max c(t) C k f c(t) (x), which is max t [,1] k f c(t) (x), exist. In that case F C k (x) =[min t [,1] k f c(t) (x), max t [,1] k f c(t) (x)]. If f c(t) (x) is linear in t then min c(t) C k f c(t) (x) and max c(t) C k f c(t) (x) exist in the set of ertices of C k. For x, y R n, we define algebraic operations of the interal alued functions as follows. Definition 2 For C k, Dk (I (R))k and +,,,/, (i) F C k (x) F C k (y) = f c(t) (x) f c(t) (y) f c(t) : R n R, c(t) C k, (ii) F C k (x) F D k (x) = f c(t )(x) f d(t )(x) f c(t ), f d(t ) : R n R, c(t ) C k, d(t ) D k. 2.2 Calculus of interal alued function Interal alued functions are defined in seeral ways. Accordingly calculus for the interal alued functions are deeloped. Neumaier (199), Stahl (1995) discussed calculus for the interal alued function F : (I (R)) n I (R). Marko (1979) considered the interal alued function F : R I (R) and defined limit, continuity, differentiability etc. In these classical forms an interal alued function is expressed as [F L (x), F R (x)]. So the existence of the deriatie of an interal alued function F(x) depends upon the existence of the deriatie of the boundary functions F L (x) and F R (x). Here we deelop the calculus for the interal alued function, F C k : R n I (R) so that the existence of the deriatie of F C k (x) depends upon the existence of the deriatie of f c(t) (x) for eery alue of t. The distance between two interals A and B is d(a, B) = max max t min t a(t ) b(t ), max min a(t ) b(t ). t t d is a metric. For an interal alued function F C k : R n I (R), suppose lim x x f c(t) (x) exists for eery c(t) C k (in other words for eery t [, 1]k ) and equal to a(t) say. Let a(t ) = min t [,1] k lim x x f c(t) (x) and a(t ) = max t [,1] k lim x x f c(t) (x) for some t, t [, 1] k. Then max min f c(t t t )(x) a(t ) = f c(t ) (x) a(t ) f c(t ) (x) a(t ), and max min f c(t t t )(x) a(t ) = f c(t ) (x) a(t ) f c(t ) (x) a(t ).

6 278 A. K. Bhurjee, G. Panda Since lim x x f c(t) (x) exists for eery t, so from aboe relations we conclude that for ɛ>, δ>such that d(f C k, A) = max max t min t f c(t )(x) a(t ), max min f c(t t t )(x) a(t ) <ɛ, wheneer x x <δ,where A =[a(t ), a(t )]. Hence we can write lim x x F C k (x) = A. So limit of an interal alued function F C k exists at a point if limit of f c(t) (x) exists at that point for eery t and can be ealuated following the aboe discussion. Similarly F C k is continuous at x if f c(t) (x) is continuous at x for eery t. We write [ lim F x x C k (x) = F C k (x ) = min lim f t c(t)(x), max lim x x t x x f c(t)(x) For each t [, 1] k, suppose f c(t) is differentiable with respect to x. Hence lim h f c(t) (x+h) f c(t) (x) h exists. Using Definition(2-i), we hae, F C k (x + h) F C k (x) = f c(t) (x + h) f c(t) (x) t [, 1] k, c(t) C k. F C k lim (x + h) F C k (x) = α(x, t) t [, 1] k,α(x, t) h h = lim f c(t) (x + h) f c(t) (x) h h [ ] = α(x, t), max α(x, t) k k. min t [,1] t [,1] The interal alued function, F C k : R n I (R) is differentiable at x = x if f c(t) is differentiable at x = x for eery t [, 1] k. The partial deriaties of F C k at x may be calculated as follows. F C k (x ) fc(t) (x ) = for eery t [, 1] k, c(t) C k x i x i If f c(t) (x ) is continuous in t then F C k (x ) ] f c(t) (x ) x i x i = [ ]. min f c(t)(x ) t [,1] k x i, max t [,1] k. The gradient of an interal alued function, F C k : R n I (R) at x = x ( is an interal ector, F C k (x FC k (x ) ) = x 1, F C k (x ) x 2,..., F C k (x )) T x n. Similarly the second order partial deriatie of F C k at x = x, can be calculated as 2 F C k (x ) 2 f c(t) (x ) = c(t) C k x i x j x i x j

7 Efficient solution of interal optimization problem 279 If f c(t) (x ) is continuous in t then 2 F C k (x ) x i x j = [ min t [,1] k 2 f c(t) (x ) x i x j, max t [,1] k 2 f c(t) (x ) x i x j ].Let 2 F C k (x ) denotes the interal Hessian matrix of F C k at x = x, which is an interal square matrix with components 2 F C k (x ) x i x j. Hence 2 F C k (x ) = 2 f c(t) (x ) c(t) C k (3) 2.3 Interal alued conex function From the construction of the interal alued function in the parametric form, it is clear that the conexity of F C k depends upon the conexity of f c(t) and a partial ordering. Here we define conex interal function with respect to and w (Definition 1) as follows. Definition 3 Suppose D R n is a conex set and for gien C k (I (R))k, F C k : R n I (R). Forx 1, x 2 D, λ 1. (i) F C k is said to be conex with respect to if F C k (λx 1 + (1 λ)x 2 ) λf C k (x 1 ) (1 λ)f C k (x 2 ). (ii) F C k is said to be conex with respect to w if F C k (λx 1 + (1 λ)x 2 ) w λf C k (x 1 ) (1 λ)f C k (x 2 ). Remark 2 From Definitions 1 and 3, one may obsere that F C k is conex with respect to means f c(t )(λx 1 + (1 λ)x 2 ) λf c(t )(x 1 ) + (1 λ) f c(t )(x 2 ), (4) for all t, t [, 1] k ; t may or may not be equal to t and F C k is conex with respect to w means f c(t) (λx 1 + (1 λ)x 2 ) λf c(t) (x 1 ) + (1 λ) f c(t) (x 2 ), (5) for all t [, 1] k ; t is same in both sides. So we can conclude that F C k is conex with respect to w if and only if f c(t) (x) is a conex function on D for eery t. Similar result does not hold for conexity with respect to. To justify this, consider the interal alued function, F C 2 (x) =f c(t) (x) = (1 + t 1 )x + (2 + t 2 )x 2 t 1, t 2 [, 1]. The interal alued function F C 2 (x) is conex with respect w, since f c(t) (x) is conex for eery alue of t [, 1] 2. To test the conexity with respect to, consider

8 28 A. K. Bhurjee, G. Panda Inequality (4). At (x 1, x 2 ) = (1, 2), λ = 1/2, this inequality becomes t t t t 2. In particular for t 1 = 1, t 1 =, t 2 = 1, t 2 = the aboe inequality is not true. Hence F C 2 (x) is not conex with respect to. Remark 3 Our definition of conexity with respect to w implies LR conexity if t is and 1; LC conexity if t is and 2 1 ; RC-conexity if t is 1 and 2 1, which are defined by Wu (28) Relation between positie definite interal matrix and interal conex function Rohn (1994) has introduced positie definite interal matrix and studied its properties. Here we quote some properties of positie definite matrices by Rohn (1994). Let A m I (R) n n, A m m = ( al ij +a ij R 2 ) n n, A s m = ( a ij R al ij 2 ) n n, where (ij)th component of A m is an interal [aij L, a ij R]. A m maybeexpressedasa m =[A m m As m, Am m + As m ]. Ā A m means, Ā = (α ij ) n n, where α ij [aij L, a ij R]. A m is said to be symmetric if both A m m and As m are symmetric. With each interal matrix A m we shall associate the symmetric interal matrix A m =[A m m A s m, A m m + A s m ], where A m m and A s m are gien by A m m = Am m +AmT m 2 and A m m = As m +AsT m 2. Let Y =z R n z j =1, for j = 1, 2,...n, and T z be the n n diagonal matrix. Now for each z Y, define the matrix A z by A z = A m m T z A s m T z. Then for each i, j we hae (A z ) ij = (A m m ) ij z i (A s m ) ijz j.so (A z ) ij = (A m A s m ) ij, if z i z j = 1; (A m m + As m ) ij, if z i z j = 1. Definition 4 A m is said to be positie (semi) definite if eery A(t) A m is positie (semi) definite. (A(t) is defined in Expression 2, Sect. 2). Proposition 1 (Rohn 1994) The following statements are equialent : (i) A m is positie (semi) definite, (ii) A m is positie (semi) definite, (iii) A z is positie (semi) definite for each z Y. We proe the following result which relates an interal alued conex function with positie definite interal matrix. Theorem 1 For gien C k (I (R))n,letF C k be a twice differentiable interal alued function on an open conex set D of R n. Then F C k is conex with respect to w if and only if its interal Hessian matrix is positie definite.

9 Efficient solution of interal optimization problem 281 Proof From Remark 2 it follows that F C k is conex on D with respect to w if and only if for all (x 1, x 2 ) D, λ [, 1], F C k (λx 1 + (1 λ)x 2 ) w λf C k (x 1 ) (1 λ)f C k (x 2 ) f c(t) (λx 1 + (1 λ)x 2 ) λf c(t) (x 1 ) + (1 λ) f c(t) (x 2 ), c(t) C k f c(t) is conex on D c(t) C k 2 f c(t) (x) is positie definite c(t) C k 2 F C k (x) is a positie definite interal matrix.(from (3) and Definition 4) Example 1 Consider the interal alued function F C 3 : R+ 2 I (R) as the set f c(t) (x 1, x 2 ) f c(t) (x 1, x 2 ) = (2t 1 + 4)x (t 2 + 1)x 1 x 2 + (3t 3 + 2)x2 2, t 1, t 2, t 3 [, 1]. It can be erified from Definition 3 that F C 3 is a conex interal alued function with respect to [ w in R 2. One ] may erify [ this ] through the [ aboe ] theorem also. A m = 2 [4,6] [1,2] F C 3 (x) =, A [1,2] [2,5] m 5 3 m = , and A s 1 1 m = Forz = (1, 1) [ ] 41 and z = ( 1, 1), A z = is a positie definite matrix. Further for z = (1, 1) 12 [ ] 42 and z = ( 1, 1), A z =, which is also a positie definite matrix. So for eery 22 z Y, A z is positie definite. Hence F C 3 is conex function with respect to w in R+ 2. Alter: For this example we may argue the positie definiteness of 2 F C 3 (x) from [ ] αβ different angle. Here Ā = γδ 2 F C 3 (x), where 4 α 6, 1 β 2, 1 γ 2 and 2 α 5. Here α, δ > and min(αδ βγ) >. This implies eery Ā is a positie definite matrix. But this process will be cumbersome to test the conexity of the interal alued function in higher dimension, in which case one may follow Proposition 1. Remark 4 In the classical form, the deriatie of an interal alued function, F(x) = [F L (x), F R (x)] depends upon the deriatie of F L (x) and F R (x). So F(x) is the interal ector ([ F L (x), F R (x)]) and 2 F(x) is the interal matrix ([ 2 F L (x), 2 F R (x)]). In classical form, F is a conex function with respect to LC iff F L and F C are conex functions; F is a conex function with respect to RC iff F R and F C are conex functions; F is a conex function with respect to LR iff F L and F R are conex functions (Wu 28). This means, conexity of F depends upon the positie definiteness of the matrices, 2 F L, 2 F R and 2 F C in three different cases. In particular, positie definiteness of 2 F L and 2 F R does not gie guarantee that the interal matrix ([ 2 F L, 2 F R ]) is positie definite because an interal matrix A m is positie definite if eery real matrix A A m is positie definite(see Definition 4). So it is not true that F is conex with respect to LR if and only if ([ 2 F L, 2 F R ]) is positie definite. Similar arguments follow for other two partial order relations.

10 282 A. K. Bhurjee, G. Panda This difficulty can be resoled if we consider the interal alued function and its calculus in the parametric form. In the parametric form, the existence of the deriatie of F C k (x) depends upon the existence of the deriatie of f c(t) (x) for eery alue of t.so 2 F C k (x) = 2 f c(t) (x) f c(t) (x) F C k (x). The interal alued function F C k (x) is a conex function if and only if f c(t) (x) is a conex function in x for eery t. This means for eery t, 2 f c(t) (x) is a positie definite matrix. This forces 2 F C k (x) to be a positie definite interal Hessian matrix. 3 Existence of solution of interal alued optimization problem In this section we consider interal optimization problem as, (IOP) min F C k (x) subject to G m j jd (x) ( or w )B j, j = 1, 2,...,p, where B j I (R), the interal alued functions F C k, G jd m j : R n I (R) are the sets, F C k (x) = f c(t) (x) f c(t) : R n R, c(t) C k, and G jd m j (x) = g jd(t j )(x) g jd(t j )(x) : R n R, d(t j ) Dm j. Following the partial orderings as discussed in Sect. 2, the feasible region of (IOP) can be expressed as the set, Ϝ =x R n : G m j jd (x) ( or w ) B j, j = 1, 2,...,p x R n : g jd(t j )(x) b j (t j ), j if G jd m j (x) B j, = x R n : g jd(tj )(x) b j (t j ), j if G m j jd (x) w B j, x R n : g jd(tj )(x) b j, j if G m j jd (x) ( or w ) Bˆ j, x R n : max t j [,1] m j g jd(tj )(x) b j (), j if G m j jd (x) B j, x R n : max t = j [,1] m j g jd(tj )(x) b j (1), min t j [,1] m j g jd(tj )(x) b j () if G m j jd (x) w B j, x R n : max t j [,1] m j g jd(tj )(x) b j, j if G m j jd (x) (or w ) Bˆ j, (6) where d(t j ), d(t j ) Dm j, b j (t j ), b j(t j ) B j, b j Bˆ j =[b j, b j ], t j [, 1] m j, t j, t j [, 1]. Since min x Ϝ F C k (x) = min x Ϝ f c(t) (x) : c(t) C k, t [, 1]k,so(IOP) can be treated as a multi-objectie problem in t for eery x Ϝ oer a continuous domain, which is a rectangular parallelepiped C k. This means, for eery c(t), the optimization problem (IOP) t min x Ϝ f c(t)(x)

11 Efficient solution of interal optimization problem 283 has a minimum solution. So we define the solution of (IOP) ( which is parallel to the concept of efficient solution in case of multi objectie programming problem ), as follows. Definition 5 x Ϝ is called an efficient solution of (IOP) if there is no x Ϝ with f c(t) (x) f c(t) (x ) t [, 1] k and F C k (x) = F C k (x ). Definition 6 x Ϝ is called a properly efficient solution of (IOP),ifx Ϝ is an efficient solution and there is a real number μ> so that for some t [, 1] k and eery x Ϝ with f c(t) (x) < f c(t) (x ), at least one t [, 1] k, t = t exists with f c(t )(x) > f c(t )(x ) and f c(t) (x ) f c(t) (x) f c(t )(x) f c(t )(x ) μ. Consider the following optimization problem with respect to a weight function w : [, 1] k R + as, (IOP I ) min x Ϝ... w(t) f c(t) (x) dt 1 dt 2...dt k, where w(t) = w(t 1, t 2,...,t k ). Here t 1, t 2,...,t k are mutually independent and each t i aries from to 1. So... w(t) f c(t)(x) dt 1 dt 2...dt k is a function of x only, say h(x). (IOP I ) becomes min x Ϝ h(x), which is a general non linear programming problem and free from interal uncertainty. This can be soled by non linear programming technique. The following theorem establishes the relationship between the solution of the transformed problem (IOP I ) and the original problem (IOP). Theorem 2 If x Ϝ is an optimal solution of (IOP I ), then x is a properly efficient solution of (IOP). Proof Let x Ϝ be an optimal solution of (IOP I ). Assume that x is not properly efficient solution of (IOP). Soforsomet [, 1] k and some x Ϝ with f c(t) (x) < f c(t) (x ). Select a weight function w : [, 1] k R +, which is continuous. Than we choose μ = max w(t ) w(t), t = t, t, t [, 1] k,w(t) >, satisfying f c(t) (x ) f c(t) (x) f c(t )(x) f c(t )(x ) >μ for all t [, 1] k with f c(t )(x) > f c(t )(x ). f c(t) (x ) f c(t) (x) >μ(f c(t )(x) f c(t )(x )) > w(t ) w(t) So w(t) f c(t) (x ) w(t) f c(t) (x) >w(t ) f c(t )(x) w(t ) f c(t )(x ). ( ) f c(t )(x) f c(t )(x ).

12 284 A. K. Bhurjee, G. Panda... w(t) f (c(t), x ) dt 1 dt 2...dt k... w(t) f c(t) (x) dt 1 dt 2...dt k... > 1 w(t ) f (c(t ), x) dt 1 dt 2...dt k Hence... w(t ) f c(t )(x ) dt 1 dt 2...dt k.... w(t) f (c(t), x ) dt 1 dt 2...dt k >... w(t) f c(t) (x) dt 1 dt 2...dt k. This contradicts to the assumption that x Ϝ is an optimal solution of (IOP I ). 4 Interal alued conex programming problem An interal alued optimization problem (IOP) is said to be an interal alued conex programming problem if F C k, G m j D are conex functions with respect to or w. Theorem 3 If (IOP) is an interal alued conex programming problem then (IOP) I is a conex programming problem. Proof Suppose (IOP) is conex then its feasible set Ϝ is conex set and its objectie function F C k (x) is conex with respect to or w.forx 1, x 2 Ϝ and λ 1, F C k (λx 1 + (1 λ)x 2 ) (or w )λf C k (x 1 ) (1 λ)f C k (x 2 ). In case of both the partial orderings or w, the aboe relation implies that for eery t [, 1] k, f c(t) (λx 1 + (1 λ)x 2 ) λ f (c(t) (x 1 ) + (1 λ) f c(t) (x 2 ). Multiplying by w(t) and integrating with respect to t 1, t 2,...,t k,wehae... λ w(t) f c(t) (λx 1 + (1 λ)x 2 ) dt... w(t) f c(t) (x 1 ) dt + (1 λ)... w(t) f c(t) (x 2 ) dt, where dt = dt 1 dt 2...dt k. This implies that objectie function of (IOP I ) is a conex function and its feasible set is equal to Ϝ, which is conex set, so (IOP I ) is conex programming problem.

13 Efficient solution of interal optimization problem Interal alued conex quadratic programming problem A general interal quadratic programming (IOP) is of the following form, (IOP) min C n x x T Q m x subject to A m x (or w )B p, x, x Rn, where C n (I (R))n, B p (I (R)) p and A m = (A ij ) p n be interal alued matrix with A ij =[aij L, a ij R], and Q m = (Q ij ) n n is a symmetric positie definite interal matrix. Any element in the interal ectors C n, Bp are the real ectors of order n, p, respectiely, which may be expressed in the parametric form as described in (1)asc(t) and b(t) say. Similarly any element in the interal matrix A m and Q m are the real matrices of order p q and n n, respectiely. These may be expressed in the parametric form as described in (2) asa(t) = (a ij (t ij )) p q and Q(t) = (q ij (t ij )) n n. From (6), the feasible set for this problem can be calculated as x R n A(1)x b() if A m x B, Ϝ = x R n A(1)x b(1), A()x b() if A m x w B, x R n A(1)x b if A m x or w ˆB, ˆB =[b j, b j ], b j R. In all three cases Ϝ is a conex set. The Hessian interal matrix of the interal alued function C n x x T Q m x can be calculated using (3), which becomes Q m. The interal matrix Q m is positie definite. So by Theorem 1, C n x x T Q m x is an interal alued conex function with respect to w. Hence the interal quadratic programming problem (IOP) is an interal conex programming problem. The corresponding optimization problem with respect to a weight function is, (7) (IOP I ) min x Ϝ w(t) c(t ) T x x T Q(t )x dt dt, where c(t ) C n, Q(t ) Q m (see (1) and (2)), dt = dt 1 dt 2...dt n, dt = dt ij, i, j = 1, 2,...,n, w :[, 1]n2 +n R +, t = (t, t ) T. By Theorem 3, (IOP I ) is a conex quadratic programming problem. Optimal solution of this problem can be obtained by soling the following KKT optimality conditions and it s solution is an efficient solution of (IOP) by Theorem 2. Denote h(x) = w(t) c(t ) T x x T Q(t )x dt dt. If Ϝ =x R n A(1)x b() when A m x B, then the Lagrange function is L(x,λ,μ) = h(x) + λ T (A(1)x b()) μ T x, λ R m,μ R n,λ. The KKT

14 286 A. K. Bhurjee, G. Panda optimality conditions are x L(x,λ,μ) =, λ T (A(1)x b()) =, x Ϝ, which are equialent to w(t) c(t ) + Q(t )x dt dt + λ T A(1) = μ, (8) λ T (A(1)x b()) =, μ T x =,λ, x Ϝ. (9) IfϜ takes different form in (7), then KKT optimality conditions can be determined in a similar way. This methodology is explained in the following numerical example in Li and Tian (28). Example 2 Consider the interal optimization problem, (IOP) min [ 1, 6]x 1 +[2, 3]x 2 +[4, 1]x1 2 +[ 1, 1]x 1x 2 +[1, 2]x2 2 s.t. [1, 2]x 1 + 3x 2 [1, 1], [ 2, 8]x 1 +[4, 6]x 2 [4, 6], x 1, x 2. According to Li and Tian (28), the lower leel problem has optimal alue 6.25 corresponding to the solution (1.25, ) and the upper leel problem has optimal alue.9 corresponding to the solution (.3, ). By the methodology described in this paper, we get an efficient solution. Here t = (t 1, t 2, t 3, t 4, t 5 ) T, t j [, 1]. f c(t) (x) = ( 1 + 4t 1 )x 1 + (2 + t 2 )x 2 + (4 + 6t 3 )x ( 1 + 2t 4)x 1 x 2 + (1 + 1t 5 )x 2 2 and Ϝ =(x 1, x 2 ) : 2x 1 + 3x 2 1, 8x 1 +6x 2 4, x 1, x 2. For some w :[, 1] 5 R, the corresponding (IOP I ) becomes: min Ϝ... w(t) ( 1 + 4t 1 )x 1 + (2 + t 2 )x 2 + (4 + 6t 3 )x ( 1 + 2t 4 )x 1 x 2 + (1 + 1t 5 )x 2 2 dt, where dt = dt 1 dt 2 dt 3 dt 4 dt 5.TheKKT conditions (8) and (9) for(iop I ) are as follows w(t) ( 1 + 4t 1 ) + 2(4 + 6t 3 )x 1 + ( 1 + 2t 4 )x 2 dt + 2λ 1 + 8λ 2 = μ 1, w(t) (2 + t 2 ) + ( 1 + 2t 4 )x 1 + 2(1 + 1t 5 )x 2 dt + 3λ 1 + 6λ 2 = μ 2, λ 1 (2x 1 + 3x 2 1) =, λ 2 (8x 1 + 6x 2 4) =, μ 1 x 1 =, μ 2 x 2 =, λ 1,λ 2, x 1, x 2 Ϝ,

15 Efficient solution of interal optimization problem 287 For a particular weight function w(t) = t 1 + t 3, the aboe system can be simplified as x λ 1 + 8λ 2 = μ 1, 2 + 3x 2 + 3λ 1 + 6λ 2 = μ 2, λ 1 (2x 1 + 3x 2 1) =, λ 2 (8x 1 + 6x 2 4) =, μ 1 x 1 =, μ 2 x 2 =, λ 1,λ 2, x 1, x 2 Ϝ. This is a system of linear equations with complementary slackness conditions, which can be soled by Wolfe s method or Lamke s algorithm. Here we use Lingo software and obtain an efficient solution as x = (x1, x 2 ) = (.5, ). The corresponding compromising optimal alue of (IOP)is[ 4,.5]. 5 Conclusion In this paper we hae introduced interal alued function, studied its conex property and use this property to justify the existence of the solution of conex interal alued programming problem. The interal alued problem is conerted to a general optimization problem, which is free from interal uncertainty. It is mathematically proed that the solution of this transformed problem is an efficient solution of the original problem. The methodology of this paper can be applicable to any type of (conex/ non conex/linear/nonlinear) optimization problems with inexact data as interals. Duality theory plays an important role for the existence of solution of a general optimization problem. In the light of this methodology we may deelop the duality theory for a general interal optimization problem. Acknowledgments The authors thank two anonymous referees whose justified critical remarks on the original ersion led to an essential reworking of the paper. References Hansen WGE (24) Global optimization using interal analysis. Marcel Dekker Inc, New York Hladík M (211) Optimal alue bounds in nonlinear programming with interal data. TOP 19(1):93 16 Hladik M (212) Interal linear programming: a surey. Noa Science Publishers, New York Hu B, Wang S (26) A noel approch in uncertain programming. part i: new arithemetic and order relation ofr interal numbers. J Ind Manag Optim 2(4): Hu B, Wang S (26) A noel approch in uncertain programming. Part II: a class of constrained nonlinear programming problems with interal objectie functions. J Ind Manag Optim 2(4): Ishibuchi H, Tanaka H (199) Multiobjectie programming in optimization of the interal objectie function. Eur J Oper Res 48(2): Jayswal A, Stancu-Minasian I, Ahmad I (211) On sufficiency and duality for a class of interal-alued programming problems. Appl Math Comput 218(8): Jeyakumar V, Li GY (211) Robust duality for fractional programming problems with constraint-wise data uncertainty. Eur J Oper Res 151(2): Jiang C, Han X, Liu GR (28) A nonlinear interal number programming method for uncertain optimization problems. Eur J Oper Res 188(1):1 13 Lein VI (1999) Nonlinear optimization under interal uncertainty. Cybern Syst Anal 35(2): Li W, Tian X (28) Numerical solution method for general interal quadratic programming. Appl Math Comput 22(2):

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