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

This note presents the characterization of the stability set of the ﬁrst kind for multi-objective nonlinear programming (MONLP) problems with fuzzy parameters either in the constraints or in the objective functions without any di ﬀ erentiability 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.


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.
Proof.From the previous lemma, it follows that (w,υ t ) ∈ S(x,υ).Let other point ( w, υ t ) ∈ S(x,υ), then by the assumption and from the KTSP necessary optimality theorem, it follows that (x,υ) and some u ≥ 0, η ≥ 0 solve KTSP [6] and u[g(x For a certain degree of α = α, we deduce from the relation (2.14) that (2.15) That is, for all U ≥ 0, ζ ≥ 0, and γ ≥ 0, we have (2.16) Therefore, it follows from the KTSP sufficient optimality theorem and from the relation x,υ) for all γ ≥ 0. Hence the set S(x,υ) is star shaped with point of common visibility (w,υ t ).
If the set S(x,υ) is a one-point set or the whole space, then it is clearly closed.Choose a sequence of points (w (n) ,.... Taking the limit as n → ∞, we get for all g(x) that is, (w,υ t ) ∈ S(x,υ), and therefore the set S(x,υ) is closed.
Theorem 2.5.For a certain degree of α, if the problem (2.7)- (2.8) is stable for all (w,υ t ) such that N(υ) = φ, then the set S(x,υ) is either one-point set or unbounded.

Mohamed Abd El-Hady Kassem 1999
Proof.If the set S(x,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.If S(x,υ) contains another point (w ,υ t ), then it is clear from (2.16) and from the KTSP sufficient optimality theorem that that is, implying that the set S(x,υ) is unbounded.

Fuzzy parameters in the objective functions
We consider the following convex MONLP problem with fuzzy parameters in the objective functions: where the functions f i (x,λ i ), i = 1,2,...,m, and g j (x), j = 1,2,...,k, are assumed to be 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 where w i ≥ 0, i = 1,2,...,m, for at least one i satisfying w i > 0 and Σ m i=1 w i=1 = 1.For a certain degree of α, the above problem can be written in the following nonfuzzy form [9]: 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 (3.5)-(3.6) is solvable for (w,λ) = (w,λ) with an α-optimal point (x,λ), then the α-stability set of the first kind of problem (3.5)-(3.6)corresponding to (x,λ) denoted by T(x,λ) is defined by 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 T(x,λ) is convex and closed.

Mohamed Abd El-Hady Kassem 2001
Theorem 3.3.For a certain degree of α, the set T(x,λ) is a cone with vertex at the origin.
Theorem 3.6.For a certain degree of α, if the problem (3.5)- (3.6) is stable for all (w,λ t ) such that M(λ) = φ, then the set T(x,λ) is either one point set or unbounded.

Conclusion
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.