Fuzzy Satisficing Method Research Articles

An interactive fuzzy satisficing method for structured multiobjective linear fractional programs with 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. 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.

多目的整数計画問題に対する遺伝的アルゴリズムによる対話型ファジィ満足化手法

This paper deals with multiobjective integer programming problems by considering fuzzy goals of the decision maker for objective functions. After determining the fuzzy goals of the decision maker, if the decision maker specifies the reference membership values, the corresponding Pareto optimal solution can be obtained by solving the augmented minimax problem which becomes an integer programming problem. For solving the problem, decoding algorithms for 0-1 programming problems are revised and ringed double strings are also introduced. Then an interactive fuzzy satisficing method is presented together with an illustrative numerical example.

ファジィ数を含む多目的非凸計画問題に対する浮動小数点型遺伝的アルゴリズムによる対話型ファジィ満足化手法

ファジィ数を含む多目的非凸計画問題に対する浮動小数点型遺伝的アルゴリズムによる対話型ファジィ満足化手法

AN INTERACTIVE FUZZY SATISFICING METHOD FOR MULTIOBJECTIVE LINEAR FRACTIONAL PROGRAMS WITH BLOCK ANGULAR STRUCTURE

In this paper, we focus on multiobjective linear fractional programming problems with block angular structure. By considering the vague nature of human judgments, we assume that the decision maker may have a fuzzy goal for each of the objective functions. Having elicited the corresponding membership functions including nonlinear ones, if the decision maker specifies the reference membership values, the corresponding 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 a Pareto optimal solution set is presented. An illustrative numerical example is provided to demonstrate the feasibility of the proposed method.

A three-level fuzzy satisficing method for large-scale multiobjective nonlinear programming problems

A three-level fuzzy satisficing method for large-scale multiobjective nonlinear programming problems

角型構造の大規模多目的ファジィ線形計画問題に対する分解手法の有効性

In this paper, we focus on large-scale multiobjective fuzzy linear programming problems with the block angular structure and examine the efficiency of the Dantzig-Wolfe decomposition method in the interactive fuzzy satisficing method proposed by M. Sakawa et al. After overviewing the Dantzig-Wolfe decomposition method (1961) and the interactive fuzzy satisficing method, three-objective linear programming problems with 15 coupling constraints are considered in order to show the efficiency of the Dantzig-Wolfe decomposition method over the revised simplex method. Through a lot of computational experiments on workstation for numerical examples with both 50 and 200 variables, the advantages of the Dantzig-Wolfe decomposition method are discussed with respect to processing time and required memory storage.

A fuzzy satisficing method for multiobjective linear optimal control problems

In this paper, we propose a fuzzy satisficing method for the solution of multiobjective linear continuous optimal control problems. To solve these multiobjective linear continuous optimal control problems, we first discretize the time and replace the system of differential equations by difference equations. By introducing suitable auxiliary variables, approximate linear multiobjective programming problems are formulated. Then by considering the vague nature of human judgements, we assume that the decision maker may have fuzzy goals for the objective functions. Having elicited the corresponding linear membership functions through the interaction with the decision maker, if we adopt the fuzzy decision for combining them, it is shown that the formulated problem can be reduced to a linear programming problem and the satisficing solution for the decision maker can be obtained through the simplex method of linear programming. An Illustrative numerical example is provided to indicate the efficiency of the proposed method.

A fuzzy satisficing method for large-scale multiobjective linear programming problems with block angular structure

In this paper, we focus on large-scale multiobjective linear programming problems with the block angular structure. By considering the imprecise nature of human judgements, we assume that the decision maker may have a fuzzy goal for each of the objective functions. Having elicited the corresponding linear membership functions through the interaction with the decision maker, we adopt the fuzzy decision for aggregating them. Then it is shown that the formulated problem can be reduced to one master problem and a number of linear subproblems and the satisficing solution for the decision maker can be obtained by directly applying the Dantzig-Wolfe decomposition method.

<特集>線形分布定数系における多目的最適制御問題に対する対話型ファジィ満足化手法(ファジィ意思決定)

Abstract In this paper, we propose an interactive fuzzy satisficing method for the solution of a multiobjective optimal control problem in a linear distributed-parameter system governed by a heat conduction equation. In order to reduce the control problem of this distributed-parameter system to an approximate multiobjective linear programming problem, we use a numerical integration formula and introduced the suitable auxiliary variables. By considering the vague nature of human judgements, we assume that the decision maker may have fuzzy goals for the objective functions. Having elicited the corresponding linear membership functions through the interaction with the decision maker, if the decision maker specifies the reference membership values, the corresponding Pareto optimal solution can be obtained by solving the minimax problems. Then a linear programming-based interactive fuzzy satisficing method for deriving a satisficing solution for the decision maker efficiently from a Pareto optimal solution set is presented. An illustrative numerical example is worked out to indicate the efficiency of the proposed method.

An Interactive Fuzzy Satisficing Method for Multiobjective 0-1 Programming Problems Through Revised Genetic Algorithms

In this paper, an interactive fuzzy satisficing method for a multiobjective 0-1 programming problem is presented by incorporating the desirable features of both the interactive fuzzy programming methods and genetic algorithms. By considering the imprecise nature of human judgements, the fuzzy goals of the decision maker (DM) for objective functions are quantified by eliciting linear membership functions. If the DM specifies the reference membership levels for all of the membership functions, the corresponding Pareto optimal solution which is, in the minimax sense, nearest to the requirement can be obtained by solving the minimax problem. To generate Pareto optimal solutions by applying the proposed genetic algorithm which is modified to generate only feasible solutions, the algorithm is further revised by providing some new mechanism for forming an initial population after the first interaction with the DM. illustrative numerical examples demonstrate the both feasibility and efficiency of the proposed methods.

Fuzzy Programming for Large-Scale Multiobjective Linear Programming Problems with Fuzzy Parameters

In this paper, a large-scale multiobjective block-angular linear programming problem involving fuzzy parameters is formulated by considering the experts' vague understanding of the nature of the parameters in the problem-formulation process. Using the α-level sets of fuzzy numbers, the corresponding nonfuzzy α-programming problem is introduced and the fuzzy goals of the decision maker are quantified by eliciting the membership functions. Through the introduction of an extended Pareto optimality concept, 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 is applicable. Then a linear programming-based interactive fuzzy satisficing method for deriving a satisficing solution for the decision maker from an extended Pareto optimal solution set is presented.

A fuzzy satisficing method for large-scale linear programming problems with block angular structure

In this paper, we focus on large-scale linear programming problems with block angular structure for which the Dantzig-Wolfe decomposition method has been successfully applied. By considering the vague nature of human judgements, we assume that the decision maker may have a fuzzy goal for the objective function and fuzzy constraints for the coupling constraints. Having elicited the corresponding linear membership functions through the interaction with the decision maker, if we adopt the convex fuzzy decision for combining them, it is shown that, under some appropriate conditions, the formulated problem can be reduced to a number of independent linear subproblems and the overall satisficing solution for the decision maker is directly obtained just only solving the subproblems.

An interactive fuzzy satisficing method for large-scale multiobjective linear programming problems with block angular structure

In this paper, we present an interactive fuzzy satisficing method for large-scale multiobjective linear programming problems with the block angular structure. By considering the vague nature of human judgements, we assume that the decision maker (DM) may have a fuzzy goal for each of the objective functions. Having elicited the corresponding linear membership functions, if the DM specifies the reference membership levels for all the membership functions, the corresponding Pareto optimal solution which is, in the minimax sense, nearest to the requirement or better than that if the reference membership levels are attainable can be obtained by solving the minimax problem. Here it is shown that the formulated minimax problem can be reduced to one master problem and a number of linear subproblems and the Pareto optimal solution together with the trade-off rate information between the membership functions can be obtained by applying the Dantzig-Wolfe decomposition method. In this way, the satisficing solution for the DM can be derived from Pareto optimal solutions by updating the current reference membership levels on the basis of the current levels of the membership functions together with the trade-off rates between the membership functions.

角型構造の大規模線形計画問題に対する凸ファジィ決定に基づくファジィ満足化手法

角型構造の大規模線形計画問題に対する凸ファジィ決定に基づくファジィ満足化手法

ファジィ多目的計画法による熱延仕上げミルのパススケジュール設計

ファジィ多目的計画法による熱延仕上げミルのパススケジュール設計

An interactive fuzzy satisficing method for generalized multiobjective linear programming problems with fuzzy parameters

A new interactive fuzzy satisficing method for multiobjective linear programming problems with fuzzy parameters is proposed. In general, two types of fuzziness of human judgements should be incorporated in multiobjective programming problems. One is the experts' ambiguous understanding of the nature of the parameters in the problem-formulation process, and the other is the fuzzy goals of the decision maker for each of the objective functions. In order to cope with both types of fuzziness, multiobjective linear programming problems with fuzzy parameters which reflect the experts' ambiguous understanding in the problem-formulation are formulated and the concept of generalized α-multiobjective linear programming and M-α-Pareto optimality is introduced. In our interactive fuzzy satisficing method, the satisficing solution of the decision maker is derived efficiently from among M-α-Pareto optimal solutions. On the basis of the proposed method, an interactive computer program is written to implement man-machine interactive procedures. Finally, an illustrative numerical example for multiobjective linear programming problems with fuzzy parameters is demonstrated along with the corresponding computer outputs.

An interactive fuzzy satisficing method for multiobjective nonlinear programming problems with fuzzy parameters

This paper presents an interactive fuzzy satisficing method for multiobjective nonlinear programming problems with fuzzy numbers. On the basis of the α-level sets of the fuzzy numbers, the concept of α-multiobjective nonlinear programming and (local) M-α-Pareto optimality is introduced. Through interaction with the decision maker (DM), the fuzzy goals of the DM for each of the objective functions in α-multiobjective nonlinear 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 (locally) M-α-Pareto optimal, if the DM specified the degree α of the α-level sets and the reference membership values, the augmented minimax problem is solved and the DM is supplied with the corresponding (local) M-α-Pareto optimal solution together with the trade-off 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 trade-off rates, the DM responds by updating his reference membership values and/or the degree α. In this way the (local) satisfying solution for the DM can be derived efficiently from among an M-α-Pareto optimal solution set. Based on the proposed method, an interactive computer program has been written and an illustrative numerical example is given along with the corresponding computer outputs.

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.

An Interactive Fuzzy Satisficing Method for Multiobjective Linear-Programming Problems and Its Application

A new interactive fuzzy satisficing method is presented for solving multiobjective linear-programming problems by assuming that the decisionmaker (DM) has fuzzy goals for each of the objective functions. Through the interaction with the DM the fuzzy goals of the DM are quantified by eliciting the corresponding membership functions, including nonlinear functions. After determining the membership functions to generate a candidate for the satisficing solution which is also Pareto optimal, if the DM specifies reference membership values, the minimax problem is solved by combined use of the bisection and linear-programming methods, and the DM is supplied with the corresponding Pareto-optimal solution together with the trade-off rates between the membership functions. Then by considering the current values of the membership functions as well as the trade-off rates, the DM responds by updating his/her reference membership values. In this way the satisficing solution for the DM can be derived efficiently from among a Pareto-optimal solution set by updating his/her reference membership values. On the basis of the proposed method, a time-sharing computer program is written and an application to an optimal operation problem in a package system in automated warehouses is demonstrated along with the computer outputs.

An interactive fuzzy satisficing method for multiobjective nonlinear programming problems with fuzzy parameters

AbstractThe fuzziness induced in the decision making in the actual system in general, comprises fuzziness in the formulation of the problem and that in the judgments of the decision maker (DM) as a human being. This paper proposes an interactive fuzzy satisficing method, which takes into consideration the fuzziness in problem formulation as well as that in the judgments of DM. The multiobjective nonlinear programming problem containing fuzzy parameters is transformed into the α ‐multiobjective nonlinear programming problem. After determining the fuzzy goal of DM for each objective function by eliciting the membership function, the following reference values are set: the fuzziness index α of DM for the fuzzy parameter; and the reference membership values, which are reference for the membership functions. For the specified α and the reference membership values, the corresponding augmented minimax problem is solved, and DM is supplied with the α‐Pareto optimal, together with the trade‐off ratios among membership functions and the trade‐off ratios between α and the membership functions. Obtaining the result, the DM takes the information concerning current α‐Pareto optimal solution and the trade‐off ratios into consideration, and respond by updating the reference membership values and the value of α if necessary. In this way the satisficing solution finally is derived for DM, thereby ensuring the α‐Pareto optimality. This is the proposed interactive fuzzy satisficing method. The interactive process is demonstrated for a numerical example by the interactive computer program.