CHARACTERIZATION OF THE STABILITY SET FOR NON-DIFFERENTIABLE FUZZY PARAMETRIC OPTIMIZATION PROBLEMS

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1 CHARACTERIZATION OF THE STABILITY SET FOR NON-DIFFERENTIABLE FUZZY PARAMETRIC OPTIMIZATION PROBLEMS MOHAMED ABD EL-HADY KASSEM Received 3 November 2003 This note presents the characterization of the stability set of the first kind for multiobjective nonlinear programming MONLP problems with fuzzy parameters either in the constraints or in the objective functions without any differentiability assumptions. These fuzzy parameters are characterized by triangular fuzzy numbers TFNs. The existing results concerning the parametric space in convex programs are reformulated to study for multiobjective nonlinear programs under the concept of α-pareto optimality. 1. Introduction In an earlier work, Mangasarian [6] introduced the Kuhn-Tucker saddle point KTSP necessary and sufficiency optimality theorems for nonlinear programming problems. Tanaka and Asai [10] formulated multiobjective linear programming problems with fuzzy parameters. Sakawa and Yano [9] introduced the concept of α-pareto optimality of fuzzy parametric programs. Orlovski [7] formulated general multiobjective nonlinear programming problems with fuzzy parameters. Kaufmann and Gupta [5] introduced the concept of triangular fuzzy numbers TFNs. Osman and El-Banna [8] introduced the stability of multiobjective nonlinear programming problems with fuzzy parameters. Kassem [2, 3] introduced an algorithm for multiobjective nonlinear programming problems with fuzzy parameters in the constraints and determined the stability set of the first kind for these problems, also introduced a method for decomposing the fuzzy parametric space in multiobjective nonlinear programming problems using the generalized Tchebycheff norm GTN. This note gives the characterization of the stability set of the first kind for convex multiobjective nonlinear programming MONLP problems with fuzzy parameters in the constraints and for convex MONLP problems with fuzzy parameters in the objective functions. These fuzzy parameters are characterized by TFNs and treatment under the concept of α-pareto optimality. In this note, no differentiability assumptions are needed and the KTSP is used in the derivation of the proposed results. Copyright 2005 Hindawi Publishing Corporation International Journal of Mathematics and Mathematical Sciences 2005: DOI: /IJMMS

2 1996 Characterization of the stability set 2. Problem formulation We consider the following convex MONLP problem with fuzzy parameters in the righthand side of constraints: min f 1 x,..., f m x 2.1 x X υ = { x R n : g j x υ j, j = 1,2,...,k }, 2.2 where the functions f i x, i = 1,2,...,m,andg j x, j = 1,2,...,k are assumed to be convex on R n and υ j, j = 1,2,...,k, are any real fuzzy parameters which are characterized by real fuzzy numbers that form a convex continuous fuzzy subset of the real line whose membership functions are µ υj υ j, j = 1,2,...,k. There is an infinite set of fuzzy numbers, but here we will define a special class of fuzzy numbers called TFNs which can be defined by a triplet υ 1,υ 2,υ 3, that is, the membership functions µ υj υ j, j = 1,2,...,k, are functions of υ t, t = 1,2,3. These membership functions µ υj υ j, j = 1,2,...,k,aredefinedby[5] 0, υ j υ 1 j, υ j υ 1 j υ 2 j υ 1, υ 1 j υ j υ 2 j, j µ υj υj = υ 3 j υ j υ 3 j υ 2, υ 2 j υ j υ 3 j, j 0, υ j υ 3 j, which are nondifferentiable on [υ 1 j,υ 3 j ], j = 1,2,...,k. 2.3 Definition 2.1. The α-level set of the fuzzy numbers υ j, j = 1,2,...,k, isdefinedasthe ordinary set L α υ for which the degree of their membership functions exceeds the level α [1]: L α υ = { υ : µ υj υj α, j = 1,2,...,k }. 2.4 For a certain degree of α, the MONLP problem can be written in the following nonfuzzy form [9]: min f 1 x,..., f m x 2.5 Nυ = { x,υ R n+k : g j x υ j, j = 1,2,...,k; µ υj υj α, j = 1,2,...,k }. 2.6 The scalarization form for the problem is minσ m i=1w i f i x 2.7

3 Mohamed Abd El-Hady Kassem 1997 x,υ Nυ, 2.8 where w i 0, i = 1,2,...,m, for at least one i satisfying w i > 0andΣ m i=1w i = 1. Definition 2.2. For a certain degree of α, suppose that the problem is solvable for w,υ = w,υwithanα-optimal point x,υ, then the α-stability set of the first kind of problem corresponding to x,υ denotedbysx,υis defined by { w,υ t Sx,υ = R m+3k : Σ m i=1w i f i x = } min x,υɛnυ Σm i=1w i f i x, 2.9 where R m is the m-dimensional vector space of weights and R 3k is the 3k-dimensional vector space of fuzzy parameters which are characterized by TFNs. Lemma 2.3. For a certain degree of α,iftheproblem isstableforallw,υ t such that Nυ φ, then gx = υ, µ υ υ t = α, w,υ t Sx,υ. Proof. If Sx,υ ={w,υ t }, then the result is clear. Assume that ŵ, υ t Sx, υ, w,υ t ŵ, υ t, then by the assumption and from the KTSP necessary optimality theorem [6], it follows that x,υ andsomeû 0, η 0solveKTSPandû j g j x υ j = 0, η r α µ υr υ t r = 0, that is, for α = α,wehave τ x,w,υ,u,η, υ t τ x,w,υ,û, η, υ t τ x,w,υ,û, η, υ t 2.10 for all x,υ Nυandforallu 0, η 0andû[gx υ] = 0, η[α µ υ υ t ] = 0, where τ x,w,υ,u,η,υ t = Σ m i=1w i f i x+σ k j=1u j [g j x υ j ] Σ k r=1η r [ µ υr υ t r α ] 2.11 since û j g j x = û j υ j, j = 1,2,...,k,and η r µ υr υ t r = η r α.then from which it follows that τ x,w,υ,u,η, υ t Σ m i=1w i f i x τ x,w,υ,û, η, υ t, 2.12 τ x,w,υ,u,η,υ t τ x,w,υ,û, η,υ t τx,w,υ,û, η,υ t Therefore, from the KTSP sufficient optimality theorem, it follows that gx = υ, µ υ υ t = α, that is, w,υ t Sx,υ. Theorem 2.4. For a certain degree of α,iftheproblem isstableforallw,υ t such that Nυ φ, then the set Sx,υ is star shaped with point of common visibility υ = gx, α = µ υ υ t and closed.

4 1998 Characterization of the stability set Proof. From the previous lemma, it follows that w,υ t Sx,υ. Let other point w, υ t Sx,υ, then by the assumption and from the KTSP necessary optimality theorem, it follows that x,υandsomeũ 0, η 0solveKTSP[6]andũ[gx υ] = 0, η[α µ υ υ t ] = 0, then for all x,υ R n+k, u 0, η 0, we have τ x, w,υ,u,η, υ t τx, w,υ,ũ, η, υ t τx,w,υ,ũ, η, υ t, 2.14 and ũ j [g j x υ j ] = 0, η r [α µ υ υ t ] = 0. Putting u j =γu j, ζ r = γη r, w i = γw i, γ 0, and 1 γũ j [g j x υ j ] = 0, 1 γ η r [α µ υ υ t ] = 0. For a certain degree of α = α,wededucefromtherelation2.14that Σ m i=1 w i f i x+σ k [ ] [ j=1u j gj x υ j Σ k t r=1 η r µ υr υ α ] Σ m i=1 w i f i x+σ k [ ] [ ] j=1ũ j gj x υ j Σ k t r=1 η r µ υr υ r α 2.15 Σ m i=1w i f i x+σ k [ ] [ ] j=1ũ j gj x υ j Σ k t r=1 η r µ υr υ r α. That is, for all U 0, ζ 0, and γ 0, we have Σ m [ i=1 w i fi x 1 γ f i x ] + Σ k [ ] j=1u j gj x 1 γg j x γυ j [ Σ k t r=1ζ r µ υr υ α 1 γ t µ υ υ α ] Σ m i=1 w i f i x+σ k [ ] [ j=1ũ j gj x υ j Σ k t r=1 η r µ υr υ α ] Σ m [ i=1w i fi x 1 γ f i x ] + Σ k [ ] j=1ũj gj x 1 γg j x γυ j Σ k ζ [ t r=1 r µ υr υ r α 1 γ t µ υr υ α ] Therefore, it follows from the KTSP sufficient optimality theorem and from the relation 2.16 that1 γw i f i x, 1 γg j x+γυ, andγα 1 γµ υ υ t belongtosx,υ for all γ 0. Hence the set Sx,υ is star shaped with point of common visibility w,υ t. If the set Sx,υ is a one-point set or the whole space, then it is clearly closed. Choose a sequence of points w n,υ tn Sx,υwhichisconvergenttow,υ t, that is, lim n w n, υ tn = w,υ t. Then Σ m i=1w n i 2,... Taking the limit as n,wegetforallgx υ t, µ υ υ t α that f i x Σ m i=1w n i f i x forallgx υ tn, µ υ υ tn α, n = 1, Σ m i=1w i f i x Σ m i=1w i f i x, 2.17 that is, w,υ t Sx,υ, and therefore the set Sx,υisclosed. Theorem 2.5. For a certain degree of α, iftheproblem isstableforallw,υ t such that Nυ φ, then the set Sx,υ is either one-point set or unbounded.

5 Mohamed Abd El-Hady Kassem 1999 Proof. If the set Sx,υ consists only of one point, then it is clear that this point is υ j = g j x, µ υj υ t j α. IfSx,υ contains another point w,υ t, then it is clear from 2.16 and from the KTSP sufficient optimality theorem that { w,υ t R m+3k :1 γw f x,1 γgx+γυ,1 γµ υ υ t γα } Sx,υ, γ 0, 2.18 that is, implying that the set Sx,υ is unbounded. 3. Fuzzy parameters in the objective functions We consider the following convex MONLP problem with fuzzy parameters in the objective functions: min f 1 x, λ 1,..., fm x, λ m 3.1 x X = { x R n : g j x 0, j = 1,2,...,k }, 3.2 where the functions f i x,λ i, i = 1,2,...,m, andg j x, j = 1,2,...,k, areassumedtobe convex on R n and λ R m is the m-dimensional vector space of fuzzy parameters which are characterized by TFNs. The corresponding scalarization problem with fuzzy parameters is minσ m i=1w i f i x, λ i 3.3 x X, 3.4 where w i 0, i = 1,2,...,m, for at least one i satisfying w i > 0andΣ m i=1w i=1 = 1. For a certain degree of α, the above problem can be written in the following nonfuzzy form [9]: minfx,w,λ = Σ m i=1w i f i x,λi 3.5 Mλ = { x,λ R n+m : g j x 0, j = 1,2,...,k; µ λi λi α, i = 1,2,...,m }, 3.6 where µ λi λ i, i = 1,2,...,m, are defined as the previous form of µ υj υ j, which are nondifferentiable on [λ 1 i,λ 3 i ]. Definition 3.1. For a certain degree of α, suppose that the problem issolvable for w,λ = w,λwithanα-optimal point x,λ, then the α-stability set of the first kind of problem corresponding to x,λ denotedbytx,λisdefinedby { w,λ t Tx,λ = } R 4m : Σ m i=1w i f i x,λi = min x,λ Mλ Σm i=1w i f i x,λi, 3.7

6 2000 Characterization of the stability set where R 4m = R m+3m, R m is the m-dimensional vector space of weights, and R 3m is the 3m-dimensional vector space of fuzzy parameters which are characterized by the TFNs. Theorem 3.2. For a certain degree of α, the set Tx,λ is convex and closed. Proof. For a certain degree of α, if the set Tx,λ is a one-point set or the whole space, it is clearly convex and closed. Suppose that w 1,λ t1 andw 2,λ t2 are any two points in Tx,λ, then for all x,λ Mλ, we have τ x,w 1,λ,u,η,λ t1 τ x,w 1,λ,û, η, λ t1 τ x,w 1,λ,û, η, λ t1, τ x,w 2,λ,u,η,λ t2 τ x,w 2,λ,û, η, λ t2 τ x,w 2,λ,û, η, λ t2, 3.8 where τ x,w,λ,u,η,λ t = Σ m i=1w i f i x,λi + Σ k j=1 u j g j x Σ m r=1η r µ λr λr α. 3.9 Therefore, τ x,ŵ,λ,u,η, λ t τ x,ŵ,λ,û, η, λ t τ x,ŵ,λ,û, η, λ t, 3.10 where ŵ = 1 γw 1 + γw 2, λ t = 1 γ λ t1 + γ λ t2,0 γ 1, that is, ŵ, λ t Tx,λ, hence the set Tx,λisconvex. We choose a sequence of points w n,λ tn Tx,λ which is convergent to w,λ t, then τ x,w n,λ,u,η,λ tn τ x,w n,λ,û, η,λ tn x,λ Mλ, n = 1,2, Taking the limit as n,wehave lim τ x,w n,λ,u,η,λ tn lim τ x,w n,λ,û, η,λ tn x,λ Mλ n n From the finiteness of the sum, we get τ x,limw n,λ,u,η,limλ tn τ x,limw n,λ,û, η,limλ tn, x,λ Mλ, n n n n 3.13 therefore, τ x,w,λ,u,η,λ t τ x,w,λ,û, η,λ t x,λ Mλ, 3.14 that is, w,λ t Tx,λ, and hence the set Tx,λisclosed.

7 Mohamed Abd El-Hady Kassem 2001 Theorem 3.3. For a certain degree of α, the set Tx,λ is a cone with vertex at the origin. Proof. It is clear that w,λ t = 0,0 Tx,λ. Suppose that w, λ t Tx,λ, then therefore, τ x, w,λ,u,η, λ t τ x, w,λ,u,η, λ t x,λ Mλ, 3.15 τ x,γ w,λ,u,η,γ λ t τ x,γ w,λ,u,η,γ λ t x,λ Mλ, 3.16 and γ 0, that is, γ w,γ λ t Tx,λforallγ 0, hence the result follows. Lemma 3.4. For a certain degree of α,ifproblem isstableforallw,λ t such that Mλ φ, then α = µ λ λ i, w,λ t Tx,λ. Proof. If Tx,λ ={w,λ t }, then the result is clear. Assume that ŵ, λ t Tw,λ, ŵ, λ t w,λ t, then by the assumption and from the KTSP necessary optimality theorem [6], it follows that x,λ andsome ξ R m, ξ 0solveKTSPand ξ i [α µ λi λ t ] = 0, that is, for all x,λ Mλ, and for all u 0, ξ 0, we have τ x,w,λ,u,ξ, λ t τ x,w,λ,û, ξ, λ t τ x,w,λ,û, ξ, λ t 3.17 and ûgx = 0, τx,w,λ,u,ξ,υ t = Σ m i=1w i f i x,λ+σ k j=1u j g j x Σ k i=1ξ i α µ λi λ t i since ûgx = 0and ξµ λ λ t = ξα for certain degree of α = α.then from which it follows that τ x,w,λ,u,ξ, λ t Σ m i=1w i f i x,λ τ x,w,λ,û, ξ, λ t, 3.18 τ x, w,λ,u,ξ, λ t τ x,ŵ,λ,û, ξ, λ t τ x,w,λ,û, ξ, λ t, 3.19 therefore from the KTSP sufficient optimality theorem, it follows that µ λ λ t = α,ŵ, λ t Tx,λ. Theorem 3.5. For a certain degree of α,iftheproblem isstableforallw,λ t such that Mλ φ, then the set Tx,λ is star shaped with point of common visibility µ λ λ t = α and closed. Proof. From the previous lemma, it follows that w,λ t Tx,λ. Let another point w, λ t Tx,λ, then by the assumption and from the KTSP necessary optimality theorem, it follows that x,λ andsomeũ 0, ξ 0solveKTSPandũgx = 0, ξ[α µ λ λ t ] = 0 for certain degree of α = α,thenforallx,λ Mλandforallu 0, ξ 0, we have τ x,w,λ,u,ξ,λ t τ x,w,λ,ũ, ξ,λ t τ x,w,λ,ũ, ξ,λ t, 3.20 ũ j g j x = 0, ξ i [α µ λi λ t i] = 0forcertaindegreeofα = α.

8 2002 Characterization of the stability set Putting u j = γu j, ξ r = γζ r, w i = γw i, γ 0, and 1 γũ j g j x = 0, 1 γ ξ r [α µ λ λ t ] = 0, we deduce from the relation 3.20andforacertaindegreeofα = α that Σ m i=1w i f i x,λ+σ k j=1u j g j x Σ m [ t ] i=1ξ i µ λi λ i α Σ m i=1w i f i x,λ+σ k j=1ũ j g j x Σ m ξ [ t ] i=1 i µ λi λ i α 3.21 Σ m i=1w i f i x,λ+σ k j=1ũ j g j x Σ m ξ [ t ] i=1 i µ λi λ i α. Then, Σ m [ i=1w i fi x,λ 1 γ f i x,λ ] + Σ k [ j=1ũj gj x 1 γg j x ] Σ m [ t t ] i=1ζ i µ λi λ i α 1 γ µ λi λ i α Σ m i=1w i f i x,λi + Σ k j=1 ũ j g j x Σ m ξ [ t ] i=1 i µ λi λ α Σ m [ ] i=1w i fi x,λi 1 γ fi x,λi + Σ k [ j=1 Ũ j gj x 1 γg j x ] Σ m ξ [ t t ] i=1 i µ λi λ i α 1 γ µ λi λ i α U 0, ξ 0, γ Therefore, it follows from the KTSP sufficient optimality theorem and from the above relation that 1 γw i f i x,λ, 1 γg j x, and γα 1 γµ λ λbelongtotx,λforall γ 0. Hence the set Tx,λ is star shaped with point of common visibility w,λ t. If the set Tx,λ is a one-point set or the whole space, then it is clearly closed. Choose a sequence of points w n,λ tn Tx,λ whichisconvergenttow,λ t, that is, lim n w n,λ tn = w,λ t. Then Σ m i=1w n i f i x,λi Σ m i=1 w n i f i x,λi gx 0, µ λ λ t n α, n = 1,2,..., 3.23 and as n,weget Σ m i=1w i f i x,λi Σ m i=1 w i f i x,λi gx 0, µ λ λ t α, 3.24 that is, w,λ t Tx,λ, and therefore the set Tx,λisclosed. Theorem 3.6. For a certain degree of α, iftheproblem isstableforallw,λ t such that Mλ φ, then the set Tx,λ is either one point set or unbounded. Proof. For a certain degree of α, if the set Tx,λ consists only of one point, then it is clear that this point is w,λ t. If Tx,λ contains another point w 1,λ t1, then it is clear from relation 3.20 and from the KTSP sufficient optimality theorem that { w,λ t R 4m :1 γw f x,λ,1 γgx,1 γµ λ λ t γα } Tx,λ, 3.25 γ 0, that is, the set Tx,λ is unbounded.

9 4. Conclusion Mohamed Abd El-Hady Kassem 2003 The stability set of the first kind for fuzzy parametric multiobjective nonlinear programming, which represents the set of all fuzzy parameters for which an α-pareto optimal point for one fuzzy parameter rests α-pareto optimal for all fuzzy parameters, has been analyzed qualitatively in the author s notes [2, 3, 4], where all the functions are assumed to posses the first-order partial derivation on R n. In this note, no differentiability assumptions are needed and the KTSP necessary and sufficient optimality theorems are used in the derivation of the proposed results. Future extension to this work is the characterization of the stability set of the first kind for MONLP with fuzzy parameters in both the objective functions and the constraints without any differentiability assumptions. Another field of extension is the field of fuzzy parametric nonconvex MONLP, where more difficulties may be found in the characterization of the stability set of the first kind of such problems. References [1] D. Dubois and H. Prade, Fuzzy Sets and Systems. Theory and Applications, Mathematics in Science and Engineering, vol. 144, Academic Press, New York, [2] M. A. Kassem, Interactive stability of multiobjective nonlinear programming problems with fuzzy parameters in the constraints, Fuzzy Sets and Systems , no. 2, [3], Decomposition of the fuzzy parametric space in multiobjective nonlinear programming problems, European J. Oper. Res , [4] M. A. Kassem and E. I. Ammar, Stability of multiobjective nonlinear programming problems with fuzzy parameters in the constraints, Fuzzy Sets and Systems , no. 3, [5] A.KaufmannandM.M.Gupta,Fuzzy Mathematical Models in Engineering and Management Science, North-Holland, Amsterdam, [6] O. L. Mangasarian, Nonlinear Programming, McGraw-Hill, New York, [7] S. A. Orlovski, Multiobjective programming problems with fuzzy parameters, ControlCybernet , no. 3, [8] M. S. Osman and A.-Z. H. El-Banna, Stability of multiobjective nonlinear programming problems with fuzzy parameters, Math. Comput. Simulation , no. 4, [9] M. Sakawa and H. Yano, Interactive decision making for multiobjective nonlinear programming problems with fuzzy parameters, Fuzzy Sets and Systems , no. 3, [10] H. Tanaka and K. Asai, Fuzzy linear programming problems with fuzzy numbers, Fuzzy Sets and Systems , no. 1, Mohamed Abd El-Hady Kassem: Department of Mathematics, Faculty of Science, Tanta University, Tanta, Egypt address: mohd60 371@hotmail.com