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1.
The Karush-Kuhn-Tucker optimality conditions for the optimization problem with fuzzy-valued objective function 总被引:1,自引:0,他引:1
Hsien-Chung Wu 《Mathematical Methods of Operations Research》2007,66(2):203-224
The Karush-Kuhn-Tucker (KKT) conditions for an optimization problem with fuzzy-valued objective function are derived in this
paper. A solution concept of this optimization problem is proposed by considering an ordering relation on the class of all
fuzzy numbers. The solution concept proposed in this paper will follow from the similar solution concept, called non-dominated
solution, in the multiobjective programming problem. In order to consider the differentiation of a fuzzy-valued function,
we use the Hausdorff metric to define the distance between two fuzzy numbers and the Hukuhara difference to define the difference
of two fuzzy numbers. Under these settings, the KKT optimality conditions are elicited naturally by introducing the Lagrange
function multipliers. 相似文献
2.
Hsien-Chung Wu 《Fuzzy Optimization and Decision Making》2009,8(3):295-321
The optimality conditions for multiobjective programming problems with fuzzy-valued objective functions are derived in this paper. The solution concepts for these kinds of problems will follow the concept of nondominated solution adopted in the multiobjective programming problems. In order to consider the differentiation of fuzzy-valued functions, we invoke the Hausdorff metric to define the distance between two fuzzy numbers and the Hukuhara difference to define the difference of two fuzzy numbers. Under these settings, the optimality conditions for obtaining the (strongly, weakly) Pareto optimal solutions are elicited naturally by introducing the Lagrange multipliers. 相似文献
3.
《Optimization》2012,61(3):473-489
The optimality conditions for an optimization problem with fuzzy-valued objective function are derived in this article. The solution concept of this optimization problem will follow the similar solution concept, called nondominated solution, in multiobjective programming problem. In order to consider the differentiation of fuzzy-valued function, we invoke the Hausdorff metric to define the distance between two fuzzy numbers and the Hukuhara difference to define the difference of two fuzzy numbers. Under these settings, the optimality conditions for obtaining the nondominated solutions are elicited naturally by introducing the Lagrange multipliers. 相似文献
4.
Hsien-Chung Wu 《Fuzzy Optimization and Decision Making》2007,6(3):179-198
The weak and strong duality theorems in fuzzy optimization problem based on the formulation of Wolfe’s primal and dual pair
problems are derived in this paper. The solution concepts of primal and dual problems are inspired by the nondominated solution
concept employed in multiobjective programming problems, since the ordering among the fuzzy numbers introduced in this paper
is a partial ordering. In order to consider the differentiation of a fuzzy-valued function, we invoke the Hausdorff metric
to define the distance between two fuzzy numbers and the Hukuhara difference to define the difference of two fuzzy numbers.
Under these settings, the Wolfe’s dual problem can be formulated by considering the gradients of differentiable fuzzy- valued
functions. The concept of having no duality gap in weak and strong sense are also introduced, and the strong duality theorems
in weak and strong sense are then derived naturally. 相似文献
5.
The KKT conditions in multiobjective programming problems with interval-valued objective functions are derived in this paper. Many concepts of Pareto optimal solutions are proposed by considering two orderings on the class of all closed intervals. In order to consider the differentiation of an interval-valued function, we invoke the Hausdorff metric to define the distance between two closed intervals and the Hukuhara difference to define the difference of two closed intervals. Under these settings, we are able to consider the continuity and differentiability of an interval-valued function. The KKT optimality conditions can then be naturally elicited. 相似文献
6.
Hsien-Chung Wu 《Fuzzy Optimization and Decision Making》2003,2(3):261-273
The fuzzy-valued Lagrangian function of fuzzy optimization problem via the concept of fuzzy scalar (inner) product is proposed. A solution concept of fuzzy optimization problem, which is essentially similar to the notion of Pareto solution in multiobjective optimization problems, is introduced by imposing a partial ordering on the set of all fuzzy numbers. Under these settings, the saddle point optimality conditions along with necessary and sufficient conditions for the absence of a duality gap are elicited. 相似文献
7.
Hsien-Chung Wu 《Fuzzy Optimization and Decision Making》2004,3(4):345-365
A solution concept of fuzzy optimization problems, which is essentially similar to the notion of Pareto optimal solution (nondominated solution) in multiobjective programming problems, is introduced by imposing a partial ordering on the set of all fuzzy numbers. We also introduce a concept of fuzzy scalar (inner) product based on the positive and negative parts of fuzzy numbers. Then the fuzzy-valued Lagrangian function and the fuzzy-valued Lagrangian dual function for the fuzzy optimization problem are proposed via the concept of fuzzy scalar product. Under these settings, the weak and strong duality theorems for fuzzy optimization problems can be elicited. We show that there is no duality gap between the primal and dual fuzzy optimization problems under suitable assumptions for fuzzy-valued functions. 相似文献
8.
H. C. Wu 《Journal of Optimization Theory and Applications》2008,139(2):361-378
Scalarization of fuzzy multiobjective programming problems using the embedding theorem and the concept of convex cone (ordering
cone) is proposed in this paper. Since the set of all fuzzy numbers can be embedded into a normed space, this motivation naturally
inspires us to invoke the scalarization techniques in vector optimization problems to evaluate the a multiobjective programming
problem. Two solution concepts are proposed in this paper by considering different convex cones. 相似文献
9.
H. C. Wu 《Journal of Optimization Theory and Applications》2004,121(2):397-417
A solution concept for fuzzy multiobjective programming problems based on ordering cones (convex cones) is proposed in this paper. The notions of ordering cones and partial orderings on a vector space are essentially equivalent. Therefore, the optimality notions in a real vector space can be elicited naturally by invoking a concept similar to that of the Pareto-optimal solution in vector optimization problems. We introduce a corresponding multiobjective programming problem and a weighting problem of the original fuzzy multiobjective programming problem using linear functionals so that the optimal solution of its corresponding weighting problem is also the Pareto-optimal solution of the original fuzzy multiobjective programming problem. 相似文献
10.
The KKT conditions in an optimization problem with interval-valued objective function are derived in this paper. Two solution concepts of this optimization problem are proposed by considering two partial orderings on the set of all closed intervals. In order to consider the differentiation of an interval-valued function, we invoke the Hausdorff metric to define the distance between two closed intervals and the Hukuhara difference to define the difference of two closed intervals. Under these settings, we derive the KKT optimality conditions. 相似文献
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12.
《European Journal of Operational Research》1998,107(3):575-589
In this paper, by considering the experts' vague or fuzzy understanding of the nature of the parameters in the problem formulation process, multiobjective linear fractional programming problems with block angular structure involving fuzzy numbers are formulated. Using the a-level sets of fuzzy numbers, the corresponding nonfuzzy a-multiobjective linear fractional programming problem is introduced. The fuzzy goals of the decision maker for the objective functions are quantified by eliciting the corresponding membership functions including nonlinear ones. Through the introduction of extended Pareto optimality concepts, if the decision maker specifies the degree a and the reference membership values, the corresponding extended Pareto optimal solution can be obtained by solving the minimax problems for which the Dantzig-Wolfe decomposition method and Ritter's partitioning procedure are applicable. Then a linear programming-based interactive fuzzy satisficing method with decomposition procedures for deriving a satisficing solution for the decision maker efficiently from an extended Pareto optimal solution set is presented. An illustrative numerical example is provided to demonstrate the feasibility of the proposed method. 相似文献
13.
Hsien-Chung Wu 《Fuzzy Optimization and Decision Making》2003,2(1):61-73
The concept of fuzzy scalar (inner) product that will be used in the fuzzy objective and inequality constraints of the fuzzy primal and dual linear programming problems with fuzzy coefficients is proposed in this paper. We also introduce a solution concept that is essentially similar to the notion of Pareto optimal solution in the multiobjective programming problems by imposing a partial ordering on the set of all fuzzy numbers. We then prove the weak and strong duality theorems for fuzzy linear programming problems with fuzzy coefficients. 相似文献
14.
Hsien-Chung Wu 《Fuzzy Optimization and Decision Making》2003,2(1):13-29
The solution concepts of the fuzzy optimization problems using ordering cone (convex cone) are proposed in this paper. We introduce an equivalence relation to partition the set of all fuzzy numbers into the equivalence classes. We then prove that this set of equivalence classes turns into a real vector space under the settings of vector addition and scalar multiplication. The notions of ordering cone and partial ordering on a vector space are essentially equivalent. Therefore, the optimality notions in the set of equivalence classes (in fact, a real vector space) can be naturally elicited by using the similar concept of Pareto optimal solution in vector optimization problems. Given an optimization problem with fuzzy coefficients, we introduce its corresponding (usual) optimization problem. Finally, we prove that the optimal solutions of its corresponding optimization problem are the Pareto optimal solutions of the original optimization problem with fuzzy coefficients. 相似文献
15.
Elsaid Ebrahim Ammar 《Fuzzy Sets and Systems》1997,90(3):5479
This paper deals with the stability of multiobjective nonlinear programming problems with fuzzy parameters in the objectives and constraints functions. These fuzzy parameters are characterized by fuzzy numbers. The existing results concerning the qualitative analysis of the notions (solvability set, stability sets of the first kind and of the second kind) in parametric nonlinear programming problems are reformulated to study the stability of multiobjective nonlinear programming problems under the concept of α-pareto optimality. An algorithm for obtaining any subset of the parametric space which has the same corresponding α-pareto optimal solution is also presented. An illustrative example is given to clarify the obtained results. 相似文献
16.
《European Journal of Operational Research》1998,107(3):564-574
In this paper, by considering the experts' vague or fuzzy understanding of the nature of the parameters in the problem-formulation process, multiobjective 0–1 programming problems involving fuzzy numbers are formulated. Using the a-level sets of fuzzy numbers, the corresponding nonfuzzy α-programming problem is introduced. The fuzzy goals of the decision maker (DM) for the objective functions are quantified by eliciting the corresponding linear membership functions. Through the introduction of an extended Pareto optimality concept, if the DM specifies the degree α and the reference membership values, the corresponding extended Pareto optimal solution can be obtained by solving the augmented minimax problems through genetic algorithms with double strings. Then an interactive fuzzy satisficing method for deriving a satisficing solution for the DM efficiently from an extended Pareto optimal solution set is presented. An illustrative numerical example is provided to demonstrate the feasibility and efficiency of the proposed method. 相似文献
17.
Hideki Katagiri Masatoshi SakawaKosuke Kato Ichiro Nishizaki 《European Journal of Operational Research》2008
This paper considers multiobjective linear programming problems with fuzzy random variables coefficients. A new decision making model is proposed to maximize both possibility and probability, which is based on possibilistic programming and stochastic programming. An interactive algorithm is constructed to obtain a satisficing solution satisfying at least weak Pareto optimality. 相似文献
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Several fuzzy approaches can be considered for solving multiobjective transportation problem. This paper presents a fuzzy goal programming approach to determine an optimal compromise solution for the multiobjective transportation problem. We assume that each objective function has a fuzzy goal. Also we assign a special type of nonlinear (hyperbolic) membership function to each objective function to describe each fuzzy goal. The approach focuses on minimizing the negative deviation variables from 1 to obtain a compromise solution of the multiobjective transportation problem. We show that the proposed method and the fuzzy programming method are equivalent. In addition, the proposed approach can be applied to solve other multiobjective mathematical programming problems. A numerical example is given to illustrate the efficiency of the proposed approach. 相似文献