α-Pareto Optimal Research Articles

An interactive algorithm for large scale multiple objective programming problems with fuzzy parameters through TOPSIS approach

In this paper, we extend technique for order preference by similarity ideal solution (TOPSIS) for solving large scale multiple objective programming problems involving fuzzy parameters. These fuzzy parameters are characterized as fuzzy numbers. For such problems, the α-Pareto optimality is introduced by extending the ordinary Pareto optimality on the basis of the α-level sets of fuzzy numbers. An interactive fuzzy decision making algorithm for generating α-Pareto optimal solution through TOPSIS approach is provided where the decision maker (DM) is asked to specify the degree α and the relative importance of objectives. Finally, a numerical example is given to clarify the main results developed in the paper.

An interactive algorithm for decomposing the parametric space in fuzzy multiobjective dynamic programming problem

The aim of this paper is to study the stability of multiobjective dynamic programming (MODP) problems with fuzzy parameters in the objective functions and in the constraints. These fuzzy parameters are characterized by fuzzy numbers. For such problems, concept and notion of the stability set of the first kind in parametric nonlinear programming problems are redefined and analyzed qualitatively under the concept of α-Pareto optimality. An interactive fuzzy decision making algorithm for the determination of any subset of the parametric space which has the same corresponding α-Pareto optimal solution is proposed. A numerical example is given to illustrate the method developed in the paper.

Stability on multiobjective dynamic programming problems with fuzzy parameters in the objective functions and in the constraints

The purpose of the paper is to investigate the stability on multiobjective dynamic programming problems with fuzzy parameters in the objective functions and in the constraints. These fuzzy parameters are characterized by fuzzy numbers. For such problems, concept and notion of the stability set of the first kind in parametric nonlinear programming problems are redefined and analyzed qualitatively under the concept of α-Pareto optimality. An algorithm for the determination of any subset of the parametric space which has the same corresponding α-Pareto optimal solution is proposed. A numerical example is given to illustrate the method developed in the paper.

Interactive stability of multiobjective NLP problems with fuzzy parameters in the objective functions and constraints

This paper deals with an interactive stability of multiobjective nonlinear programming problems with fuzzy parameters in the objective functions and constraints. These fuzzy parameters are characterized by fuzzy numbers. The existing results concerning the qualitative analysis set of the first kind are reformulated to study this set under the concept of α-pareto optimality. An interactive algorithm for obtaining the α-best-compromise and obtaining any subset of the parametric space which has the same corresponding α-pareto optimal solution is also presented. An illustrative example is given.

Interactive decision-making for multiobjective linear fractional programming problems with block angular structure involving fuzzy numbers

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. Through the use of the α-level sets of fuzzy numbers, an extended Pareto optimality concept called the α-Pareto optimality is introduced. To generate a candidate for the satisficing solution which is also α-Pareto optimal, the decision maker is asked to specify the degree α and the reference objective values. It is shown that the corresponding α-Pareto optimal solution can be easily 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 decision-making method with decomposition procedures for deriving a satisficing solution for the decision maker efficiently from an α-Pareto optimal solution set is presented. An illustrative numerical example is provided to demonstrate the feasibility of the proposed method.

Stability of multiobjective NLP problems with fuzzy parameters in the objectives and constraints functions

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.

Interactive decision making for large-scale multiobjective linear programs with fuzzy numbers

In this paper, by considering the experts' imprecise or fuzzy understanding of the nature of the parameters in the problem-formulation process, large-scale multiobjective block-angular linear programming problems involving fuzzy numbers are formulated. Through the use of the α-level sets of fuzzy numbers, an extended Pareto optimality concept, called the α-Pareto optimality is introduced. To generate a candidate for the satisficing solution which is also a-Pareto optimal, decision maker is asked to specify the degree α and the reference objective values. It is shown that the corresponding α-Pareto optimal solution can be easily obtained by solving the minimax problems for which the Dantzig-Wolfe decomposition method is applicable. Then a linear programming-based interactive decision-making method for deriving a satisficing solution for the decision maker efficiently from an α-Pareto optimal solution set is presented.

A parametric study of multiobjective NLP problems with fuzzy parameters in the objective functions

This paper deals with the stability of multiobjective nonlinear programming problems with ordinary and fuzzy parameters in the objective 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.

Interactive stability of multiobjective nonlinear programming problems with fuzzy parameters in the constraints

This paper deals with an interactive stability of multiobjective nonlinear programming problems with fuzzy parameters in the constraints. These fuzzy parameters are characterized by fuzzy numbers. The existing results concerning the qualitative analysis of stability set of the first kind are reformulated to study this stability set of first kind under the concept of α-pareto optimality. An interactive algorithm for obtaining the α-best compromise solution and 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.

Stability of multiobjective nonlinear programming problems with fuzzy parameters

This paper deals with multiobjective nonlinear programming problems with fuzzy parameters in the objective functions. These fuzzy parameters are characterized by fuzzy numbers. The existing results concerning the qualitative analysis of some basic notions 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.

Interactive decision making for multiobjective nonlinear programming problems with fuzzy parameters

This paper presents an interactive decision making method for multiobjective nonlinear programming problems with fuzzy parameters. The fuzzy parameters in the objective functions and the constraints are characterized by fuzzy numbers. The concept of α-Pareto optimality is introduced in which the ordinary Pareto optimality is extended based on the α-level sets of fuzzy numbers. In our interactive method, if the decision maker (DM) specifies the degree α of the α-level sets and the reference objective values, a minimax problem is solved and the DM is supplied with the corresponding α-Pareto optimal solution together with the trade-off rates among the values of the objective functions and the degree α. Then by considering the current values of the objective functions and α as well as the trade-off rates, the DM responds by updating his reference objective values and/or the degree α. In this way the satisfying solution for the DM can be derived efficiently from among an α-Pareto optimal solution set. A numerical example illustrates various aspects of the results developed in this paper.

An Interactive Fuzzy Satisficing Method for Multiobjective Linear Programming Problems with Fuzzy Parameters

In this paper, we focus on multiobjective linear programming problems with fuzzy parameters and present a new interactive fuzzy satisficing method for obtaining the satisficing solution of the decision maker (DM) on the basis of the linear programming method. The fuzzy parameters in the description of the objective functions and the constraints, which reflect the experts’ ambiguous understanding of the nature of the parameters in the problem-formulation process, are characterized by fuzzy numbers. The concept of α-multiobjective linear programming and α-Pareto optimality is introduced based on the α-level sets of the fuzzy numbers. Through the interaction with the DM, the fuzzy goals of the DM for each of the objective functions in α-multiobjective linear programming are quantified by eliciting the corresponding membership functions. After determining the membership functions, in order to generate a candidate for the satisficing solution which is also α-Pareto optimal, if the DM specifies the degree a of the a-level sets and the reference membership values, the minimax problem is solved by combined use of the bisection method and the linear programming method, and the DM is supplied with the corresponding α-Pareto optimal solution together with the tradeoff rates among the values of the membership functions and the degree α. Then by considering the current values of the membership functions and α as well as the tradeoff rates, the DM acts on this solution by updating his/her reference membership values and/or the degree α. In this way, the satisficing solution for the DM can be derived efficiently from among an α-Pareto optimal solution set. On the basis of the proposed method, a time-sharing computer program is written and an illustrative numerical example is demonstrated along with the corresponding computer outputs.

Interactive Fuzzy Decisionmaking for Multiobjective Nonlinear Programming Problems with Fuzzy Parameters

Abstract In this paper, in order to deal with the multiobjective nonlinear programming problems with fuzzy parameters characterized by fuzzy numbers, the concept of α-multiobjective nonlinear programming and α-Pareto optimality is introduced on the basis of the a-level sets of the fuzzy numbers. Then by assuming that the fuzzy goals of the decision maker (DM) for each of the objective functions in α-multiobjective nonlinear programming can be quantified by eliciting the corresponding membership functions, a new interactive fuzzy decisionmaking method to derive the satisficing solution of the DM efficiently from among an α-Pareto optimal solution set is presented.

ファジィパラメータを含む多目的非線形計画問題に対する対話型意思決定

In this paper, we focus on multiobjective linear programming problems with fuzzy parameters and present a new interactive decision making method for obtaining the satisficing solution of the decision maker (DM) on the basis of the linear programming method. The fuzzy parameters in the objective functions and the constraints are characterized by fuzzy numbers. The concept of a-Pareto optimality is introduced in which the ordinary Pareto optimality is extended based on the α-level sets of the fuzzy numbers. In our interactive decision making method, in order to generate a candidate for the satisficing solution which is also α-Pareto optimal, if the DM specifies the degree a of the α-level sets and the reference objective values, the minimax problem is solved by making use of the linear programming method, and the DM is supplied with the corresponding α-Pareto optimal solution together with the trade-off rates among the values of the objective functions and the degree α. Then by considering the current values of the objective functions and a as well as the trade-off rates, the DM acts on this solution by updating his/her reference objective values and/or degree α. In this way the satisficing solution for the DM can be derived efficiently from among an α-Pareto optimal solution set. A numerical example illustrates various aspects of the results developed in this paper.

INTERACTIVE DECISION MAKING FOR MULTI-OBJECTIVE LINEAR FRACTIONAL PROGRAMMINGPROBLEMS WITH FUZZY PARAMETERS

Abstract In this paper, we focus on multiobjective linear fractional programming problems with fuzzy parameters and present a new interactive decision making method for obtaining the satisficing solution of the decision maker (DM) on the basis of the linear programming method. The fuzzy parameters in the objective functions and the constraints are characterized by fuzzy numbers. The concept of a-Pareto optimality is introduced in which the ordinary Pareto optimality is extended based on the α-level sets of the fuzzy numbers. In our interactive decision making method, in order to generate a candidate for the satisficing solution which is also a-Pareto optimal, if the DM specifies the degree α of the a-level sets and the reference objective values, the minimax problem is solved by combined use of the bisection method and the linear programming method and the DM is supplied with the corresponding α-Pareto optimal solution together with the trade-off rates among the values of the objective functions and the d...