Stability in multiobjective possibilistic linear programs

Stability in multiobjective possibilistic linear programs Robert Fullér rfuller@abo.fi Mario Fedrizzi fedrizzi@cs.unitn.it Abstract This paper continues the authors research in stability analysis in possibilistic programming in that it extends the results in [7] to possibilistic linear programs with multiple objective functions. Namely, we show that multiobjective possibilistic linear programs with continuous fuzzy number coefficients are well-posed, i.e. small changes in the membership function of the coefficients may cause only a small deviation in the possibility distribution of the objective function. Keywords: Multiobjective possibilistic linear programs (MPLP), possibility theory, fuzzy number, stability 1 Introduction Stability and sensitivity analysis becomes more and more attractive also in the area of multiple objective mathematical programming (for excellent surveys see e.g. Gal [10] and Rios Insua [18]). Publications on this topic usually investigate the impact of parameter changes (in the righthand side or/and the objective functions or/and the A-matrix or/and the domination structure) on the solution in various models of vectormaximization problems, e.g. linear or non-linear, deterministic or stochastic, static or dynamic (see e.g. [5, 17, 19]). There are only few paper dealing with stability or sensitivity analysis in fuzzy mathematical programming: Sensitivity analysis in LP problems with crisp coefficients and soft constraints was first considered in [11], where a functional relationship between changes of parameters of the righthand side and those of the optimal value of the primal objective function was derived for almost all conceivable cases. The stability of the fuzzy solution in LP problems with fuzzy coefficients of symmetric triangular form was shown in [9]. In this paper, generalizing the authors earlier result [7], we show that the possibility distribution of the objectives of an MPLP with continuous fuzzy number coefficients is stable under small changes in the membership function of the parameters. In this section we recall Buckley s [2, 3, 4] and The final version of this paper appeared in: R. Fullér and M. Fedrizzi, Stability in multiobjective possibilistic linear programs, European Journal of Operational Research, 74(1994) 179-187. 1

Luhandjula s [16] solution concept for MPLP and set up the notation needed for our main result in the next section. A fuzzy quantity ã: IR [0, 1] is a fuzzy set of the real line IR. The support of a fuzzy quantity ã (denoted by suppã) is the crisp set given by {t ã(t) > 0}. A fuzzy number ã is a fuzzy quantity with a normalized, upper semicontinuous, fuzzy convex and bounded supported membership function. A symmetric triangular fuzzy number ã denoted by (a, α) is defined as a t 1 if a t α, ã(t) = α where a IR is the centre and 2α > 0 is the spread of ã. The fuzzy numbers will represent the continuous possibility distributions for fuzzy parameters. A multiobjective possibilistic linear programming is maximize Z = ( c 11 x 1 + + c 1n x n,..., c k1 x 1 + + c kn x n ) (1) subject to ã i1 x 1 + ã in x n b i, i = 1,..., m, x 0, where ã ij, b i, and c lj are fuzzy numbers, x = (x 1,..., x n ) is a vector of (non-fuzzy) decision variables, the operations addition and multiplication by a real number of fuzzy numbers are defined by Zadeh s extension principle [20], and denotes <,, =, or > for each i, i = 1,..., m. Even though may vary from row to row in the constraints, we will rewrite the MPLP (1) as maximize Z = ( c 1 x,..., c k x) (2) subject to Ãx b, x 0, where ã = [ã ij ] is an m n matrix of fuzzy numbers and b = ( b 1,..., b m ) is a vector of fuzzy numbers. The fuzzy numbers are the possibility distributions associated with the fuzzy variables and hence place a restriction on the possible values the variable may assume [20, 21]. For example, Poss[ã ij = t] = ã ij (t). We will assume that all fuzzy numbers ã ij, b i, c j are non-interactive [20]. Following Buckley [2-4], we define Poss[Z = z], the possibility distribution of the objective function Z. We first specify the possibility that x satisfies the i-th constraints. Let π(a i, b i ) = min{ã i1 (a i1 ),..., ã in (a in ), b i (b i }, where a i = (a i1,..., a in ), which is the joint distribution of ã ij, j = 1,..., n, and b i. Then Poss[x F i ] = sup a i,b i { π(a i, b i ) a i1 x 1 +... + a in x n b i }, which is the possibility that x is feasible with respect to the i-th constraint. Therefore, for x 0, Poss[x F] = min 1 i m Poss[x F i], which is the possibility that x is feasible. We next construct Poss[Z = z x] which is the conditional possibility that Z equals z given x. The joint distribution of the c lj, j = 1,..., n, is π(c l ) = min{ c l1 (c l1 ),..., c ln (c ln )} 2

where c l = (c l1,..., c ln ), l = 1,..., k. Therefore, Poss[Z = z x] = Poss[ c 1 x = z 1,..., c k x = z k ] = min 1 l k Poss[ c lx = z l ] = min 1 l k sup {π(c l ) c l1 x 1 + + c ln x n = z l }. c l1,...,c lk Finally, applying Bellman and Zadeh s method of fuzzy decision making [1], the possibility distribution of the objective function is defined as follows Poss[Z = z] = sup min{poss[z = z x], Poss[x F]} x 0 Let ã be a fuzzy number. Then for any θ 0 we define ω(ã, θ), the modulus of continuity of ã as [12] ω(ã, θ) = max ã(u) ã(v). u v θ An α-level set of a fuzzy numer ã is a non-fuzzy set denoted by [ã] α and is defined by { {t IR ã(t) α} if α > 0 [ã] α = cl(suppã) if α = 0 where cl(suppã) denotes the closure of the support of ã. Let ã and b be fuzzy numbers and [ã] α = [a 1 (α), a 2 (α)], [ b] α = [b 1 (α), b 2 (α)]. We metricize the set of fuzzy numbers by the metric [13] D(ã, b) = sup max { a i(α) b i (α) }, α [0,1] i=1,2 where d denotes the classical Hausdorff metric in the family of compact subsets of IR 2, i.e. d([ã] α, [ b] α ) = max{ a 1 (α) b 1 (α), a 2 (α) b 2 (α) } The proof of the next two lemmas is very simple. Lemma 1.1 Let n N and let x i, y i [0, 1], i = 1,..., n. Then min i=1,...,n x i min i=1,...,n y i max i=1,...,n x i y i Lemma 1.2 Let X be a set and let f, g be fuzzy sets of X. Then sup x X f(x) sup g(x) sup f(x) g(x) x X x X The following lemma shows that if all the α-level sets of two continuous fuzzy numbers are close to each other, then there can be only a small deviation between their membership grades. Lemma 1.3 [7] Let δ 0 and let ã, b be fuzzy numbers. If D(ã, b) δ, then sup ã(t) b(t) max{ω(ã, δ), ω( b, δ)}. t IR Remark 1.1 It should be noted that if ã or b are discontinuous fuzzy numbers, then there can be a big deviation between their membership grades even if D(ã, b) is arbitrarily small. 3

2 Stability in multiobjective possibilistic linear programs An important question [6, 7, 9, 22] is the influence of the perturbations of the fuzzy parameters to the Possibility distribution of the objective function. We will assume that there is a collection of fuzzy parameters ã δ ij, b δ i, c δ j available with the property max D(ã ij, ã δ ij) δ, i,j max D( b i, b δ i ) δ, max D( c lj, c δ i lj) δ. (3) l,j Then we have to solve the following problem: max/min Z δ = ( c δ 1x,... c δ kx) (4) subject to Ãδ x b δ, x 0. Let us denote by Poss[x F δ i ] the Possibility that x is feasible with respect to the i-th constraint in (4). Then the Possibility distribution of the objective function Z δ in (4) is defined as follows: Poss[Z δ = z] = sup(min{poss[z δ = z x], Poss[x F δ ]}). x 0 The following theorem shows a stability property (with respect to perturbations (3)) of the possibility dostribution of the objective function of MPLP (1) and (4). Theorem 2.1 Let δ 0 be a real number and let ã ij, b i, ã δ ij, c lj, c δ lj fuzzy numbers. If (3) hold, then be continuous sup Poss[Z δ = z] Poss[Z = z] ω(δ) (5) z IR where ω(δ) = max i,j,l {ω(ã ij, δ), ω(ã δ ij, δ), ω( b i, δ), ω(b δ i, δ), ω( c lj, δ), ω( c δ lj, δ)}. i.e. ω(δ) denotes the maximum of modulus of continuity of all fuzzy coefficients at the point δ. Proof. It is sufficient to show that Poss[Z = z] Poss[Z δ = z] ω(δ), for any (z 1,..., z k ) IR k, (6) Applying Lemmas 1.1-1.3 we have... (for full proof see EJOR, 74(1994) 179-187)... which proves the theorem. Remark 2.1 From lim δ 0 ω(δ) = 0 and (5) it follows that sup Poss[Z δ = z] Poss[Z = z] 0 as δ 0 z IR which means the stability of the possibility distribution of the objective function with respect to perturbations (3). 4

Remark 2.2 As an immediate consequence of this theorem we obtain the following result (see [9]: If the fuzzy numbers in (1) and (4) satisfy the Lipschitz condition with constant L > 0, then sup Poss[Z δ = z] Poss[Z = z] Lδ z IR Furthermore, similar estimations can be obtained in the case of symmetrical trapezoidal fuzzy number parameters (see [15]) and in the case of symmetrical triangular fuzzy number parameters (see [8, 14]). Remark 2.3 It is easy to see that in the case of non-continuous fuzzy parameters the possibility distribution of the objective function may be unstable under small changes of the parameters. 3 Illustration As an example, consider the following biobjective possibilistic linear program max/min ( cx, cx) (7) subject to ãx b, x 0. where ã = (1, 1), b = (2, 1) and c = (3, 1) are fuzzy numbers of symmetric triangular form. Here x is one-dimensional (n = 1) and there is only one constraint (m = 1). We find 1 if x 2, Pos[x F] = 3 if x > 2. x + 1 and Pos[Z = (z 1, z 2 ) x] = min{pos[ cx = z 1 ], Pos[ cx = z 2 ]} where for i = 1, 2 and x 0, and Pos[ cx = z i ] = 4 z i x if z i/x [3, 4], z i x 2 if z i/x [2, 3], Pos[Z = (z 1, z 2 ) 0] = Pos[0 c = z] = { 1 if z = 0, 0 otherwise. Both possibilities are nonlinear functions of x, however the calculation of Pos[Z = (z 1, z 2 )] is easily performed and we obtain θ 1 if z M 1, Pos[Z = (z 1, z 2 )] = min{θ 1, θ 2, θ 3 } if z M 2, 5

where and for i = 1, 2 and Figure 1: Pos[ cx = z 2 ] for z 2 = 8. M 1 = {z IR 2 z 1 z 2 min{z 1, z 2 }, z 1 + z 2 12}, M 2 = {z IR 2 z 1 z 2 min{z 1, z 2 }, z 1 + z 2 > 12}, θ i = 24 z i + 7 + z 2 i + 14z i + 1. θ 3 = 4 min{z 1, z 2 } 2 max{z 1, z 2 } z 1 + z 2. Consider now a perturbed biobjective problem with two different objectives (derived from (7) by a simple δ-shifting of the centres of ã and c): max/min ( cx, c δ x) (8) subject to ã δ x b, x 0. where ã = (1 + δ, 1), b = (2, 1), c = (3, 1), c δ = (3 δ, 1) and δ 0 is the error of measurement. Then 1 if x 2 Pos[x F δ ] = 1 + δ, 3 δx if x > 2 x + 1 1 + δ. and Pos[Z δ = (z 1, z 2 ) x] = min{pos[ cx = z 1 ], Pos[ c δ x = z 2 ]} where Pos[ cx = z 1 ] = 4 z 1 x if z 1/x [3, 4], z 1 x 2 if z 1i/x [2, 3], 4 δ z 2 x if z 2/x [3 δ, 4 δ], Pos[ c δ x = z 2 ] = z 2 x 2 + δ if z 2/x [2 δ, 3 δ], 6

x 0, and So, where and M 1 (δ) = M 2 (δ) = Pos[Z δ = (z 1, z 2 ) 0] = Pos[0 c = z] = { 1 if z = 0, 0 otherwise. θ 1 (δ) if z M 1 (δ), Pos[Z δ = (z 1, z 2 )] = min{θ 1 (δ), θ 2 (δ), θ 3 (δ)} if z M 2 (δ), { { z IR 2 z 1 z 2 (1 0.5δ) min{z 1, z 2 }, z 1 + z 2 z IR 2 z 1 z 2 (1 0.5δ) min{z 1, z 2 }, z 1 + z 2 > ( 24 + δ 7 z 1 ) z1 2 + 14z 1 + 1 + 4z 1 δ θ 1 (δ) = z 1 + 7 + z1 2 + 14z 1 + 1 + 4z 1 δ + 2δ 24 δ (δ + z 2 1 + ) (1 δ z 2 ) 2 + 16z 2 θ 2 (δ) = z 2 + 7 + (1 δ z 2 ) 2 + 16z 2 + δ θ 3 (δ) = (4 δ) min{z 1, z 2 } 2 max{z 1, z 2 } z 1 + z 2. It is easy to check that sup Pos[x F] Pos[x F δ ] δ, x 0 sup Pos[Z = z x] Pos[Z δ = z x] δ, x 0, z } 2(6 δ), 1 + δ } 2(6 δ), 1 + δ sup Pos[Z = z] Pos[Z δ = z] δ. z On the other hand, from the definition of metric D the modulus of continuity and Theorem 2.1 it follows that D(ã, ã δ ) = δ, D( c, c δ ) = δ, D( c, c) = 0, D( b, b) = 0, ω(δ) = δ, and, therefore, sup Pos[Z = z] Pos[Z δ = z] δ. z 7

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