Sensitivity analysis for linear optimization problem with fuzzy data in the objective function

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1 Sensitivity analysis for linear optimization problem with fuzzy data in the objetive funtion Stephan Dempe, Tatiana Starostina May 5, 2004 Abstrat Linear programming problems with fuzzy oeffiients in the objetive funtion are onsidered. Emphasis is on the dependene of the optimal solution from linear perturbations of the membership funtions of the objetive funtion oeffiients as well as on the omputation of a robust solution of the fuzzy linear problem if the membership funtions are not surely nown. Keywords: Fuzzy linear programming, sensitivity analysis, robust optimization 1 Fuzzy linear optimization We onsider fuzzy linear optimization problems given by F (x) = n j x j max The oeffiients = ( 1, 2,..., n ) of the objetive funtion are fuzzy numbers of the type L L [4]: (1) j = ( j ; j ; α j ; β j ) L L, j = 1, 2,..., n, (2) where j, j - are the left and right borders of the fuzzy number j orresponding to the maximal reliability level (λ = 1) and α j and β j are non-negative real numbers (see Fig. 1). A fuzzy number j, (j = 1, 2,..., n) is defined as a fuzzy set in the spae of real numbers with the following membership funtion [4]: ( 1 ) if z, z L if z, (z) = µ ( α ) (3) z L if z, β Tehnial University Bergaademie Freiberg, Germany, dempe@math.tufreiberg.de Taganrog State University of Radio Engineering, Taganrog, Russia 1

2 Figure 1: A membership funtion where L is a shape funtion, whih satisfies to following onditions: - L is a ontinuous non-inreasing funtion on [0, ) with L(0) = 1; - L is stritly dereasing on that part of [0, ) on whih it is positive. Problem (1) an be assoiated with a set of the following problems (4), whih depend on a parameter λ (0, 1) [2]: F (x) = n λ j x j max (4) where the oeffiients λ j in the objetive funtion represent intervals orresponding to the λ-level of the fuzzy number j, j = 1, 2,..., n: λ j := {y : µ j (y) λ} = [ j L 1 (λ)α j, j + L 1 (λ)β j ]. Define the abbreviations λ j := j L 1 (λ)α j, λ j := j + L 1 (λ)β j. Similar as in [2] we use the following definition of an optimal solution of problem (4). Definition 1 A point x 0 with Ax = b is alled an t 0, t 1 -optimal solution of the problem (4) iff there is no x 0 with Ax = b satisfying the following inequalities ( λ j + t 0( λ j λ j ))x j ( λ j + t 1( λ j λ j ))x j with at least one strit inequality. ( λ j + t 0( λ j λ j ))x j ( λ j + t 1( λ j λ j ))x j Using this definition it is easy to see that x is an t 0, t 1 -optimal solution iff it is a Pareto-optimal solution of the following biriterial optimization problem, where 2

3 θ = L 1 (λ) is used [2]: f 1 (x) = n f 2 (x) = n ( p 1 1 (j) + p 1 2(j)θ ) x j max ( p 2 1 (j) + p 2 2(j)θ ) x j max x j 0, j = 1, 2,..., n, where p 1 1(j) = j + t 0 ( j j ), p 1 2(j) = t 0 (α j + β j ) α j, p 2 1(j) = j + t 1 ( j j ), p 2 2(j) = t 1 (α j + β j ) α j, (5) To ompute t 0, t 1 -optimal solutions for all λ-levels of the problem (1) we have to ompute Pareto-optimal solutions of problem (5) for all θ [θ, θ] with θ = L 1 (1), θ = L 1 (0). The seletion of different values for t 0, t 1 orresponds to different preferene relations between intervals [3]. For simpliity of our investigations we shall be limited only to the onsideration of the following preferene relation between intervals a = [a, a] and b = [b, b]: a b a b a b, (6) a < b a b a b, (7) In aordane with [3] this ase orresponds to the problem (5) with t 0 = 0 and t 1 = 1. Substituting t 0 = 0 and t 1 = 1 in (5) we obtain the following model: f 1 (x) = n ( j α j θ ) x j max f 2 (x) = n ( j + β j θ) x j max Summing up we have to ompute the sets of Pareto-optimal solutions of problem (8) for all θ [θ, θ]. Definition 2 A point x 0 with Ax = b is a Pareto-optimal solution [5] of problem (8) iff there does not exist x 0 with Ax = b satisfying with at least one strit inequality. f 1 (x) f 1 (x ), f 2 (x) f 2 (x ) (8) 3

4 Figure 2: First objetive funtion for θ [0, 1] Figure 3: Seond objetive funtion for θ [0, 1] We denote the set of Pareto-optimal solutions of problem (8) for fixed θ by Ψ(θ). Sine the objetive funtions in model (8) depend on the parameter θ, eah objetive funtion in (8) is one element in a set of funtions whih are loated between two borders, aording to θ = 0 and θ = 1. This is shown in Fig. 2 and 3. Coming ba to problem (1) and using the above approah for treating fuzziness we see that it is neessary to find all the feasible points whih are Pareto-optimal for problem (8) for at least one value of θ [0, 1]. Definition 3 A point x whih is Pareto-optimal for problem (8) for at least one value of θ [0, 1] is alled essential. The solution of a fuzzy optimization problem is again a fuzzy set in the set of feasable solutions. To define a membership funtion for this solution, we ompute the set of all θ for whih one essential point is Pareto-optimal for 4

5 problem (8) [2]. Then, µ F S (x) = { λ [0, 1] : x is a Pareto-optimal solution of the problem (8) for θ = L 1 (λ) } = l = (L(θ 2i 1 ) L(θ 2i )) (9) i=1 where {θ i } 2l i=1 is suh that x is Pareto-optimal for problem (8) for all θ [θ 2i 1, θ 2i ], i = 1,..., l. Here, Q means the geometri measure of the set Q. 2 Computing Pareto-optimal points In the following we will assume for simpliity that the set M := {x 0 : Ax = b} is not empty and bounded. Then, M = onv {x 1, x 2,..., x r }. For the motivation, we will restrit us to the omputation of the set of all verties of M belonging to the set of Pareto-optimal solutions Ψ(θ) for problem (8) for fixed θ. Clearly, this an be done sine the set of Pareto-optimal solutions itself an easily be determined if we now the subset of verties in this set. Let Ψ v (θ) denote the subset of verties of M in Ψ(θ). Then, Ψ v (θ) = {x i1,..., x ip } {x 1,..., x r }. (10) We sort these Pareto-optimal points suh that f 1 (x i1 ) > f 1 (x i2 ) >... > f 1 (x ip ), f 2 (x i1 ) < f 2 (x i2 ) <... < f 2 (x ip ). (11) This means that the solution x i1 is optimal for the first objetive funtion without onsideration of the seond one and x ip is an optimal solution for the seond objetive funtion if the first funtion is not taen into aount. For simpliity assume that the optimization problems of maximizing the funtion f 1 (x) respetively the funtion f 2 (x) on the set M are non-degenerate, i.e. that they have unique optimal solutions. If this would not be the ase then we have to selet ertain optimal solutions out of the sets of optimal ones. Then, to ompute Ψ v (θ) we an proeed as follows. We solve linear optimization problems f 2 (x) max f 1 (x) z (12) x M with parameter z on the right-hand side of the first onstraint. It follows from the Charnes-Cooper observation [7] that we obtain Ψ(θ) as Ψ(θ) = Φ(z), z [f 1(x ip ),f 1(x i1 )] where Φ(z) is the set of optimal solutions of problem (12). Let ϕ(z) denote the optimal value funtion of the parametri optimization problem (12). The 5

6 funtion ϕ(z) is pieewise linear and onave [6]. The ins of this funtion orrespond to verties of M, hene, to ompute Ψ v (θ) we have to find the ins of the funtion ϕ(z). Theorem 4 Let the funtion ϕ(z) be (affine) linear on both the intervals [z 0, z 1 ] and [z 1, z 2 ] with different slopes. Then, there exists a vertex x of the set M with x Φ(z 1 ). Vie versa, if Φ(z) = 1 for all z and if there is a vertex x of the set M in Φ(z 1 ) then the funtion ϕ(z) has a in at z = z 1. Proof: If the onstraint f 1 (x) = z is not ative, the optimal solution of the problem (12) is equal to x ip independent of z, i.e. ϕ(z) is onstant. Hene, assume that f 1 (x) = z is ative. For right-hand side perturbed linear programming problems there exists a pieewise (affine-) linear solution funtion x(z) (whih an be an arbitrary seletion funtion of Φ( ) in ase of nonuniquely optimal solutions) [6]. To verify this onsider the optimality onditions of problem (12) and use that the harateristi index set is pieewise onstant [6]. This means that ϕ(z) = f 2 (x(z)) is linear whenever x(z) is linear and that this funtion is not differentiable whenever x(z) is not. If the funtion x(z) does not go through a vertex of M, i.e. if no vertex of M belongs to Φ(z 1 ), the funtions x(z) and ϕ(z) are loally (affine-) linear. The assumption that no vertex of M belongs to Φ(z 1 ) equivalently means that the harateristi index set is onstant for z near z 1. Now let the harateristi index set hange at the point z 1. Sine the onstraint f 1 (x) = z is ative this means that the harateristi index set orresponding to the inequalities x 0 hanges. Due to the assumption Φ(z 1 ) = 1 this means that a unique vertex x of M is the optimal solution of (12) at z = z 1. Hene, the funtion x(z) is not differentiable at the point z = z 1 and so is the funtion ϕ(z). q.e.d. 3 Sensitivity analysis Now we assume that the fuzzy numbers j are perturbed by δ j, j = 1,..., n. Applying these perturbations to the problem (8) with the two objetive funtions f 1 (x) and f 2 (x), we obtain the following model: f 1 (x) = n [( j α j θ ) + δ j ] xj max f 2 (x) = n [( j + β j θ) + δ j ] x j max (13) Here we are interested to determine how muh the fuzzy numbers j, j = 1,..., n an be perturbed suh that one Pareto-optimal solution of the initial fuzzy linear optimization problem remains optimal. Problem (13) is alled a hanged problem and the set R(x, θ) := {δ : x Ψ δ (θ)} 6

7 Figure 4: Perturbation of a membership funtion is alled region of stability of the feasible point x, where Ψ δ (θ) denotes the set of Pareto-optimal solutions of problem (13). The orresponding perturbation of the membership funtion of one fuzzy number is illustrated in Fig. 4. Theorem 5 For fixed θ and eah feasible point x the set l R(x) is a onvex polyhedron. Proof: Let N M (x) denote the normal one to the feasible set M of problem (13) at a feasible point x: N M (x) = {z : u, v 0 with z = A u Ev, x v = 0}. Then, x is Pareto-optimal for problem (13) iff there exists w (0, 1) suh that x is an optimal solution of the problem wf 1 (x) + (1 w)f 2 (x) max (14) x j 0, j = 1, 2,..., n, f.e.g. [7]. Now, x is an optimal solution of problem (14) if and only if w f 1 (x) + (1 w) f 2 (x) N M (x) by linear programming where f i (x) is onstant sine f i (x) is a linear funtion of x. Evaluating this ondition we get w( αθ + δ) + (1 w)( + βθ + δ) N M (x). (15) Here, = ( 1,..., n ), and, α, β are aordingly determined vetors. Now, putting the equations (15) and w (0, 1) and the definition of N M (x) together we see that the losure of the set of all solutions (w, δ, u, v) of the resulting system is equal to the set of solutions of a linear system of equations and inequalities. Hene, this set is a onvex polyhedron and this is also true for the projetion of this set onto the δ-spae. q.e.d. To illustrate this theorem let us onsider in the following example the speial ase that δ i = 0 for all i > 1. F (x) = 1 x x 2 max x 1 + 2x 2 6 x 1 + x 2 2 (16) 2x 1 + x 2 6 x 1, x 2 0, 7

8 Figure 5: Set of solutions for the perturbed biriterial problem where 1 = (2; 5; 1; 2) and 2 = (8; 9; 2; 5) (see (2)). If we hange this problem by perturbing the membership funtion of the first fuzzy number 1 by δ 1 we obtain the following problem: f 1 (x) = [(2 θ) + δ 1 ] x 1 + (8 2θ) x 2 max f 2 (x) = [(5 + 2θ) + δ 1 ] x 1 + (9 + 5θ) x 2 max x 1 + 2x 2 6 x 1 + x 2 2 2x 1 + x 2 6 x 1, x 2 0. (17) Consider the biriterial problem (17) with θ = 0. Then the set of Pareto-optimal solutions for δ = 0 is Ψ 0 (θ = 0) = onv {x 2, x 3 } = onv {(2/3, 8/3), (2, 2) } (see Fig. 5). Now we onsider a hanged problem (17) with the parameter δ. We intend to find all suh δ 1 for whih x 2 is no longer Pareto-optimal, i.e. we need to find {δ 1 : x 2 Ψ δ1 (θ = 0)}. Now we transform the problem (17) into the problem (14). w((2 + δ 1 )x 1 + 8x 2 ) + (1 w)((5 + δ 1 )x 1 + 9x 2 ) max x 1 + 2x 2 6 x 1 + x 2 2 2x 1 + x 2 6 x 1, x 2 0. (18) Applying the optimality onditions (15) to problem (18) we obtain w (2 + δ 1 ) (1 w) (5 + δ 1 ) + u 1 u 2 = 0 w 8 (1 w) 9 + 2u 1 + u 2 = 0 u 1, u 2 0, w [0, 1]. (19) Note that the Lagrange multipliers to the onstraints x i 0 and to the last inequality 2x 1 + x 2 6 are zero sine these onditions are not ative. After the 8

9 transformation of equations (19) the following equalities are obtained: δ 1 = 3w 5 + u 1 u 2 w + 2u 1 + u 2 = 9 u 1, u 2 0, w [0, 1]. (20) This implies that the following inequality gives an upper bound for δ 1 suh that the Pareto-optimal point x 2 remains Pareto-optimal, i.e. it is no longer Pareto-optimal for the problem (17) if δ 1 is larger than the right-hand side of the following inequality: δ 1 d, (21) where d is the optimal objetive funtion value of the problem 3w 5 + u 1 u 2 max w + 2u 1 + u 2 = 9 u 1, u 2 0, w [0, 1]. The optimal funtion value of the last problem is equal to two. Hene, R(x 2 ) = (, 2] whih an easily be verified in figure 5. It should be remared that the region of stability is losed here sine the problem (17) is nondegenerate. Remar 6 By the same way as in the proof of Theorem 5 we get the onditions for the omputation of the bounds θ i for evaluating the membership funtion (9) of the solution of problem (1). For this, set δ = 0 in equation (15). If this ondition is satisfied then the feasible solution x is Pareto-optimal for problem (8) and, hene, belongs to the solution of problem (1) with positive membership funtion value. The θ i are the bounds of θ for whih x enters the set of Paretooptimal solutions respetively leaves this set. Note that the equation (15) is no longer linear if θ is not onstant whih results in a nononvex region of stability. The latter result is refleted also by the investigations in the paper [2]. 4 Robust solution Let us assume now that the oeffiients in the objetive funtion in problem (1) are unnown. This is another situation than that in the previous setion where we analysed the dependeny of one Pareto-optimal solution and hene of one feasible solution with positive value of the membership funtion on the fuzzy objetive funtion oeffiients. In [1] linear optimization problems with unown oeffiients in the onstraints have been onsidered. To treat the unertainty resulting from the unnown oeffiients the authors developed a robust ounterpart of the linear programming problem demanding that a robust feasible solution has to satisfy all the onstraints resulting from all possible realizations of the oeffiients. And a robust optimal solution is a best robust feasible solution with respet to the original objetive funtion. Here the situation is slightly different in that we assume that the membership funtions of the fuzzy oeffiients in the objetive funtion are unertain. Again 9

10 we solve the fuzzy optimization problem (1) using the union of the sets of Pareto-optimal solutions of the problem (8) for θ [0, 1]. Then, to adopt the approah in [1] we move the objetive funtions of this problem for fixed θ into the onstraint set. This results in z 1 max z 2 max f 1 (x) = n ( j α j θ ) x j z 1 f 2 (x) = n ( j + β j θ) x j z (22) 2 Now the oneption in [1] was to all a point x a robust feasible point if it satisfies all the onstraints for all the possible realizations. This means for our problem that we get z 1 max z 2 max f 1 (x) := min n f 2 (x) := min n ( j α j θ ) x j z 1 ( j + β j θ) x j z 2 (23) where the minimum in the first two onstraints is to be taen with respet to all possible funtions obtained for all possible realizations of the fuzzy oeffiients in the objetive funtion of (1). This problem is equivalent to f 1 (x) := min n ( j α j θ ) x j max f 2 (x) := min n ( j + β j θ) x j max (24) Hene, if we also onstrut a robust ounterpart of our problem (1) we should demand feasibility of a solution with respet to the onstraints of the original problem but all a feasible solution robust optimal if it is nearly optimal with respet to all possible realizations of the fuzzy objetive funtion. A fuzzy oeffiient j in the objetive funtion of problem (1) is haraterized by ( j, j, α j, β j ). If the fuzzy number j is unertain this means that its membership funtion is unertain and this is refleted by ( j, j, α j, β j ) being taen as one element of a ertain set. Let P be the set of all possible vetors of tuples ( j, j, α j, β j ) resulting from all possible realizations of the fuzzy oeffiients in (1). Then, the minimum in the objetive funtions in problem (24) is 10

11 to be taen w.r.t. p = ( ( j, j, α j, β j ) ) n P. We assume in the following that i=1 the set P is a ompat polyhedron. A robust solution for problem (1) is a feasible solution of this problem, whih is essential for problem (24). Its membership funtion value an be determined aordingly to (9) with (8) being replaed by (24). To solve the resulting problem (24) we use a similar approah as in Setion 2. For that, let the ompat polyhedron P be given as onvex hull of its verties P = onv {( j, j, αj, β j )n i=1, = 1,..., K}. Then, f 1 (x) := min p P ( j α j θ ) x j = min =1,...,K ( j αj θ ) x j and f 2 (x) := min p P ( j + β j θ) x j == min =1,...,K ( j + βj θ ) x j. To ompute the bounds for the variations of both objetive funtions f 1 (x) and f 2 (x) we again solve the two problems of maximizing only one of both funtions on the feasible set M. These problems an be transformed into ξ max ( j αj θ) x j ξ, = 1,..., K (25) x j 0, j = 1, 2,..., n and ξ max ( j + βj θ) x j ξ, = 1,..., K (26) Let x 1 be an optimal solution of (25) and x p denote an optimal solution of (26), then the first objetive f 1 (x) varies between f 1 (x 1 ) and f 1 (x p ), while f 2 (x) runs from f 2 (x p ) to f 2 (x 1 ). Now the problem aording to (12) reads as ξ max ( j + βj θ) x j ξ, = 1,..., K f 1 (x) z (27) Similarly to Setion 2, if we solve this problem for z [ f 1 (x p ), f 1 (x 1 )], then we trae the set of Pareto-optimal solutions of the problem (24). Note, that using 11

12 the regions of stability n R(, θ) := x : ( j αj θ ) x j ( s j αjθ s ) x j s = 1,..., K we an further transform problem (27) into a sequene of linear optimization problems. 5 Computation of the membership funtion To ompute values of the membership funtion of a robust solution of the problem (1) we go along the lines of (9), i.e. we ompute for eah feasible point x M the set of all θ for whih this point is Pareto-optimal with respet to the problem (24). Problem (24) again is a onvex biriterial optimization problem. Hene, for eah Pareto-optimal solution x of this problem there is w [0, 1] suh that x is an optimal solution of the problem w f 1 (x) + (1 w) f 2 (x) max (28) Vie versa, if x is an optimal solution of (28) for some w (0, 1) (and this is also true for w {0, 1} if the solution is unique) then it is also Pareto-optimal for problem (24). Let (w f 1 + (1 w) f ) 2 (x) denote the superdifferential (in the sense of onvex analysis [8]) of the funtion (w f 1 + (1 w) f ) 2 (x). Theorem 7 ([8]) Let all optimal solutions of the problems (25) and (26) be uniquely determined. Then, a feasible solution x M is Pareto-optimal to (24) if and only if there is w [0, 1] with ( ( w f 1 + (1 w) f 2 ) ) ) (x) N M (x). By onvex analysis, (w f 1 + (1 w) f ) 2 (x) = w f 1 (x) + (1 w) f 2 (x). To ompute f 1 (x) remember that f 1 (x) = min =1,...,K ( j αj θ ) x j and let K(x, θ) := : n ( j αj θ ) x j = f 1 (x). 12

13 Then, f 1 (x) = onv { α θ : K(x, θ)}. Hene, to ompute the region of stability of some Pareto-optimal solution x for problem (24) we have to solve a system of nonlinear equations resulting from insertion of the last equation into the result of Theorem 7. Anowledgements: Part of the wor was done during the seond author s stay at the Tehnial University Bergaademie Freiberg. The finanial support from the Deutsher Aademisher Austaush Dienst e.v. is gratefully anowledged. Referenes [1] Ben-Tal A., Nemirovsi A. Robust solution of unertain linear programs. Operations Researh Letters 25, 1999, [2] Chanas S., Kuhta D. Linear programming problem with fuzzy oeffiients in the objetive funtion. In: M. Delgado, J. Kaprzy, J.-L. Verdegay, M.A. Vila, Fuzzy optimization. Physia Verlag, 1994, pp [3] Chanas S., Kuhta D. Multiobjetive programming in optimization of interval objetive funtions - A generalized approah. European Journal of Operations Researh 94, 1996, [4] Dubois D., Prade H. Operations on fuzzy numbers. Int. J. of Systems Siene 6, 1978, [5] Liu B. Theory and pratie of unertain programming. Heidelberg; New Yor: Physia-Verlag, [6] Nožiča F., Guddat J., Hollatz H, Ban B.: Theorie der linearen parametrishen Optimierung. Berlin: Aademie Verlag, [7] Zlobe S.: Stable Parametri Programming. Kluwer Aademi Publishers, Dordreht, [8] Roafellar R.T.: Convex analysis. Prineton: Prineton University Press,