On stability in possibilistic linear equality systems with Lipschitzian fuzzy numbers

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1 On stability in possibilistic linear equality systems with Lipschitzian fuzzy numbers Robert Fullér Abstract Linear equality systems with fuzzy parameters and crisp variables defined by the extension principle are called possibilistic linear equality systems. The study focuses on the problem of stability (with respect to perturbations of fuzzy parameters) of the solution in these systems. 1 Introduction Modelling real world problems mathematically we often have to find a solution to a linear system a i1 x a in x n = b i, i = 1,..., m, (1) where a ij and b i are real numbers. It is known that the system of equations (1) generally belongs to the class of ill-posed problems, so a small perturbation of the parameters ã ij, b i may cause a large deviation in the solution. System (1) with fuzzy parameters ã ij, b i and crisp variables x j is considered in a lot of paper [2,4,7]. Moreover, in [4] has been shown a stability property (with respect to changes of centres of fuzzy parameters) of the solution to the system (1) with symmetrical triangular fuzzy numbers. Our results are connected with those presented ones in [4] and generalize and extend them. Namely, we shall prove that the system (1) with Lipschitzian fuzzy numbers always belongs to the class of well-posed problems, so a small perturbation of the fuzzy parameters may cause only a small deviation in the solution. 2 Preliminaries A fuzzy number is a fuzzy set on the real axis, i.e. mapping ã: R I = [0, 1] associating with each real number t its grade of membership ã(t). To distinguish a fuzzy number from a crisp (non-fuzzy) one, the former will always be denoted with a tilde. By F we denote the set of all fuzzy numbers ã with the membership function having the following properties: (i) ã is upper semicontinuous; (ii) ã(t) = 0, outside of some interval [c, d]; (iii) there are real numbers a and b, c a b d such that ã is strictly increasing on the interval [c, a], strictly decreasing on [b, d] and ã(t) = 1 for each x [a, b]. If ã, b F and λ R then ã + b, ã b, λã are defined by the Zadeh s extension principle [9] in the usual way. An α-level set of a fuzzy number ã is a non-fuzzy set denoted by [ã] α and is defined by { {t R ã(t) α} if α > 0 [ã] α = cl(suppã) if α = 0 The final version of this paper appeared in R. Fullér, On stability in possibilistic linear equality systems with Lipschitzian fuzzy numbers, Fuzzy Sets and Systems, 34(1990) doi: / (90)90219-V 1

2 where cl(suppã) denotes the closure of the support of ã. Let ã F be a fuzzy number. Then from (i)-(iii) it follows that the α-level set [ã] α is a nonempty compact interval [a 1 (α), a 2 (α)] where a 1 (α) = min[ã] α, a 2 (α) = max[ã] α for each α I. Let ã and b F and [ã] α = [a 1 (α), a 2 (α)], [ b] α = [b 1 (α), b 2 (α)]. It is well known that [ã + b] α = [a 1 (α) + b 1 (α), a 2 (α) + b 2 (α)]. (2) [ã b] α = [a 1 (α) b 2 (α), a 2 (α) b 1 (α)]. (3) The truth value of the assertion ã is equal to b, which we write ã = b is Poss(ã = b) defined as Poss(ã = b) = sup ã(t) b(t). t R It is easily verified that Poss(ã = b) = (ã b)(0) (4) We define a metric D [3] in F by the equation D(ã, b) = sup max { a i(α) b i (α) } (5) α I i=1,2 The following lemma can be proved directly by using the definition (5). Lemma 2.1 Let ã, b, c and d F and λ R. Then D(λã, λ b) = λ D(ã, b), D(ã + c, b + d) D(ã, b) + D( c, d), D(ã c, b d) D(ã, b) + D( c, d). Let L > 0 be a real number. By F(L) we denote the set of all fuzzy numbers ã F with membership function satisfying the Lipschitz condition with constant L, i.e. ã(t) ã(t ) L t t, t, t R (6) Let ã F(L) and [ã] α = [a 1 (α), a 2 (α)]. Then it is easily verified that ã(a 1 (α)) = α, α I, (7) a 1 (ã(t)) = t if a 1 (0) t a 1 (1). (8) In the following lemma we see that the linear combination of Lipschitzian fuzzy numbers is a Lipschitzian one, too. Lemma 2.2 Let L > 0, λ 0, µ 0 be real numbers and let ã, b F(L) be fuzzy numbers. Then (i) λã F(L/ λ ), (ii) λã + µ b F(L/( λ + µ ). 2

3 Proof. (i) From the equation (λã)(u) = ã(u/λ) for each u R and (6) we have for all t, t R. Thus (i) holds. (λã)(t) (λã)(t ) = ã(t/λ) ã(t /λ) L λ t t (ii) With the notations c := λã, d := µ b we need show that for all u, v R. ( c + d)(u) ( c + d)(v) L u v λ + µ Let u, v R be arbitrarily fixed. We assume without loss of generality that u < v. From the definition of addition on fuzzy numbers (2) it follows that there are real numbers u 1, u 2, v 1, v 2 with the properties Since from (8) and (i) we have u = u 1 + u 2, v = v 1 + v 2, u 1 v 1, u 2 v 2 (9) c(u 1 ) = d(u 2 ) = ( c + d)(u) (10) c(v 1 ) = d(v 2 ) = ( c + d)(v) (11) c(u 1 ) c(v 1 ) L λ u 1 v 1, c(u 2 ) c(v 2 ) L µ u 2 v 2, hence by (9)-(11) we get ( c + d)(v) ( c + d)(u) min{ L λ u 1 v 1, L µ u 2 v 2 } L λ + µ ( v 1 u 1 + v 2 u 2 ) = L v u λ + µ Which proves the lemma. The following lemma that will be used in the sequel shows one of the basic properties of Lipschitzian fuzzy numbers. Lemma 2.3 Let ã, b F(L) and [ã] α = [a 1 (α), a 2 (α)], [ b] α = [b 1 (α), b 2 (α)]. Suppose that D(ã, b) δ. Then sup ã(t) b(t) Lδ. t R Proof. Let t R be arbitrarily fixed. It will be sufficient to show that ã(t) b(t) Lδ (12) If t / suppã supp b then we obtain (12) trivially. Suppose that t suppã supp b. With no loss of generality we will assume 0 b(t) < ã(t). Then either 1. t (b 1 (0), b 1 (1)), 2. t b 1 (0), 3. t (b 2 (1), b 2 (0)), 3

4 4. t b 2 (0), must occur. In the case of 1) consider (6) with t and t = a 1 ( b(t)). Then from (7) and t = b 1 ( b(t)) and (5) we have ã(t) b(t) = ã(t) ã(a 1 ( b(t))) L t (a 1 ( b(t)) = L b 1 ( b(t)) a 1 ( b(t)) LD(ã, b) Lδ. In the case of 2) we have b(t) = 0; therefore from δ D(ã, b) a 1 (0) b 1 (0) and t > a 1 (0) we get ã(t) b(t) = ã(t) 0 = ã(t) ã(a 1 (0)) L t a 1 (0) = L(t a 1 (0)) L(b 1 (0) a 1 (0)) Lδ. In the cases of 3) and 4) the proofs are carried out in a similar manner. 3 Stability in possibilistic linear equality systems Generalizing the system (1) consider the following possibilistic linear equality system ã i1 x ã in x n = b i, i = 1,..., m (13) where ã ij, b i are fuzzy numbers. We denote by µ i (x) the degree of satisfaction of the i-th equation at the point x R n in (13), i.e. µ i (x) = Poss(ã i1 x ã in x n = b i ). (14) Following Bellman and Zadeh [1] the solution (or the fuzzy set of feasible solutions) of the system (13) can be viewed as the intersection of the µ i s such that A measure of consistency [2] for the system (13) can be µ(x) = min i=1,m µ i(x) (15) µ = sup µ(x) (16) x R In many important cases the fuzzy parameters ã ij, b i of the system (13) are not known exactly [8] and we have to work with their approximations ã δ ij, b δ i such that max (ã ij, ã δ ij) δ, max D( b i, b δ i ) δ, (17) i,j i where δ 0 is a real number. Then we get the following system with perturbed fuzzy parameters ã δ i1x ã δ inx n = b δ i, i = 1,..., m (18) In a similar manner we can define the solution and the measure of consistency of the perturbed system (18) µ δ (x) = min i=1,m µδ i (x), µ (δ) = sup µ δ (x), x R n where µ δ i (x) denotes the degree of satisfaction of the i-th equation at the point x R n in (18). In the following theorem we establish a stability property (with respect to perturbations (17)) of the solution to the system (13). 4

5 Theorem 3.1 Let L > 0 and ã ij, ã δ ij, b i, b δ i F(L). If (17) holds, then µ µ δ C = sup x R n µ(x) µ δ (x) Lδ, (19) where µ(x) and µ δ (x) are the solutions to the systems (13) and (18) respectively. Proof. It is sufficient to show that for each x R n and i = 1,..., m. µ i (x) µ δ i (x) Lδ Let x R n and i {1,..., m} be arbitrarily fixed. From (4) it follows that Applying Lemma 1. we have ( n ( n ) µ i (x) = ã ij x j b i )(0), µ δ i (x) = ã δ ijx j b δ i (0). ( n D ã ij x j b i, n ) ã δ ijx j b δ i n x j D(ã ij, ã δ ij) + D( b i, b δ i ) δ( x 1 + 1), where x 1 = x x n. By Lemma 2. we have n ã ij x j b i, Finally, applying Lemma 3 we get n ( ) ã δ ijx j b L δ i F x ( µ i (x) µ δ n ( n ) i (x) = ã ij x j b i )(0) ã δ ijx j b δ i (0) sup t R Which proves the theorem. ( n Remark 3.1 From (19) it follows that ( n ã ij x j b i )(t) L x δ( x 1 + 1) = Lδ. µ µ (δ) Lδ, ã δ ijx j b δ i (t)) where µ, µ (δ) are the measures of consistency for the systems (13) and (18) respectively. Remark 3.2 It is easily checked that in the general case ã ij, b i F the solution to the possibilistic linear equality system (13) may be unstable (in metric C) under small variations in the membership function of fuzzy parameters (in metric D). 5

6 Remark 3.3 Let X be the set of points x R n for which µ(x) attains its maximum, if it exist. If x X, then x is called a maximizing (or best) solution of the system (13). When the problem is to find a maximizing solution to a possibilistic linear equality system (13), then according to Negoita [6], we are led to solve the following optimization problem max x n+1 ; x n+1 µ i (x 1,..., x n ), i = 1,..., m, x j R, j = 1,..., n. (20) Finding the solutions of problem (20) generally requires use of nonlinear programming techniques, and could be tricky [10]. However, if the fuzzy numbers in (13) are of trapezoidal form, then the problem (20) turns into quadratically constrained programming problem [5]. 4 Concluding remarks In this paper we have shown that the solution and the measure of consistency of the system (13) have a stability property with respect to changes of the fuzzy parameters. Nevertheless,the behavior of the maximizing solution towards a small perturbations of the fuzzy parameters has not been described yet, i.e. supposing that X, what can be said about the distance ρ(x (δ), X ) as δ 0, where x (δ) is a maximizing solution of perturbed problem (18) and ρ is a metric in R n. 5 Aknowledgement The author would like to express his gratitude to Dr S.A.Orlovsky for his helpful discussion. References [1] R.E.Bellman and L.A.Zadeh, Decision-making in a fuzzy environment, Management Sciences 17(1970), B141-B154. [2] D.Dubois and H.Prade, System of linear fuzzy constraints, Fuzzy Sets and Systems, 3(1980), [3] R.Goetschel and W.Voxman, Topological properties of fuzzy numbers, Fuzzy Sets and Systems, 10(1983), [4] M.Kovács, Fuzzification of ill-posed linear systems,in: D. Greenspan and P.Rózsa, Eds., Colloquia mathematica Societitas János Bolyai 50, Numerical Methods, North-Holland, Amsterdam, 1988, [5] M.Kovács, F.P.Vasiljev and R.Fullér, On stability in fuzzified linear equality systems, Proceedings of the Moscow State University, Ser. 15, 1(1989), 5-9 (in Russian), translation in Moscow Univ. Comput. Math. Cybernet., 1(1989), 4-9. [6] C.V.Negoita, Fuzzinies in management, ORSA/TIMS, Miami (Nov.1976). [7] H.Tanaka, Fuzzy data analysis by possibilistic linear modells, Fuzzy Sets and Systems, 24(1987),

7 [8] H.Tanaka,H.Ichihashi and K.Asai, A value of information in FLP problems via sensitivity analysis, Fuzzy Sets and Systems, 18(1986), [9] L.A.Zadeh, The concept of a linguistic variable and its application to approximate reasoning - 1, Inform. Sci. 8(1975), [10] H.J.Zimmermann, Fuzzy set theory and mathematical programming, in: A.Jones et al. (eds.), Fuzzy Sets Theory and Applications, 1986, D.Reidel Publishing Company, Dordrecht, Extensions For more results on stability properties of possibilistic linear equality and inequality systems see: M.Kovács, F.P.Vasiljev and R.Fullér, On stability in fuzzified linear equality systems, Proceedings of the Moscow State University, Ser. 15, 1(1989), 5-9 (in Russian), translation in Moscow Univ. Comput. Math. Cybernet., 1(1989), 4-9 [MR 91c:94039] [Zbl ]; R.Fullér, Well-posed fuzzy extensions of ill-posed linear equality systems, Fuzzy Systems and Mathematics, 5(1991) Follow ups The results of this paper have been mentioned later in the following works in journals A34-c11 Vroman A, Deschrijver G, Kerre EE, Using Parametric Functions to Solve Systems of Linear Fuzzy Equations with a Symmetric Matrix, INTERNATIONAL JOURNAL OF COMPUTA- TIONAL INTELLIGENCE SYSTEMS, 1(2008), Number 3, pp A34-c10 Vroman A, Deschrijver G, Kerre EE Solving systems of linear fuzzy equations by parametric functions - An improved algorithm FUZZY SETS AND SYSTEMS, 158 (14): JUL Consequently, the exact solution does not exist and therefore the search for an alternative solution has a solid ground. There are already some alternative approaches known in literature. Fuller [A34] considers a system of linear fuzzy equations with Lipschitzian fuzzy numbers. He assigns a degree of satisfaction to each equation in the system and then calculates a measure of consistency for the whole system. Abramovich et al. [1] try to minimize the deviation of the left-hand side from the right-hand side of the system with LR-type fuzzy numbers. Both methods try to approximate the exact solution, i.e. they try to minimize the error when one reenters the solution into the system. (page 1516) A34-c9 Rybkin VA, Yazenin AV On the problem of stability in possibilistic optimization, NTERNA- TIONAL JOURNAL OF GENERAL SYSTEMS, 30 (1):

8 A34-c8 E.B. Ammar and M.A.El-Hady Kassem, On stability analysis of multicriteria LP problems with fuzzy parameters, FUZZY SETS AND SYSTEMS, 82(1996) Fullér in [A34] introduced the stability of the FLP problems with fuzzy parameters. In the present paper we investigate the stability of the solution in fuzzy... (page 331) A34-c7 S.Jenei, Continuity in approximate reasoning, Annales Univ. Sci. Budapest, Sect. Comp., 15(1995) A34-c6 M.Kovács, Stable embeddings of linear equality and inequality systems into fuzzified systems, FUZZY SETS AND SYSTEMS, 45(1992) The obtained stability results are the generalizations of the stability properties of the fuzzified linear systems proved in [A35, A34, 3,5]. (page 311) in proceedings and in edited volumes A34-c5 D.Dubois, E.Kerre, R.Mesiar and H.Prade, Fuzzy interval analysis, in: Didier Dubois and Henri Prade eds., Fundamentals of Fuzzy Sets, The Handbooks of Fuzzy Sets Series, Volume 7, Kluwer Academic Publishers, [ISBN X], 2000, Another type of fuzzy intervals is considered hy Fullér (l990): Lipschitzian fuzzy intervals M such that there is positive constant k such that M(a) M(a ) k a a. He proves that the class of Lipschitzian fuzzy intervals is closed under fuzzy addition and scalar multiplication. (page 507) A34-c4 V.A.Rybkin and A.V.Yazenin, Regularization and stability of possibilistic linear programming problems, in: Proceedings of the Sixth European Congress on Intelligent Techniques and Soft Computing (EUFIT 98), Aachen, September 7-10, 1998, Verlag Mainz, Aachen, Vol. I, in books A34-c3 R. Lowen, Fuzzy Set Theory, Kluwer Academic Publishers, [ISBN: ], A34-c2 Y.J.Lai and C.L.Hwang, Fuzzy Multiple Objective Decision Making, Lecture Notes in Economics and Mathematical Systems, No. 404, Springer Verlag, [ISBN: ], Berlin A34-c1 Y.J.Lai and C.L.Hwang, Fuzzy Mathematical Programming, Methods and Applications, Lecture Notes in Economics and Mathematical Systems, No. 394, Springer Verlag, [ISBN X], Berlin

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