Masatoshi Sakawa

Fuzzy programming for multiobjective job shop scheduling with fuzzy processing time and fuzzy duedate through genetic algorithms

Abstract In this paper, by considering the imprecise or fuzzy nature of the data in real-world problems, job shop scheduling with fuzzy processing time and fuzzy duedate is introduced. On the basis of the agreement index of fuzzy duedate and fuzzy completion time, multiobjective fuzzy job shop scheduling problems are formulated as three-objective ones which not only maximize the minimum agreement index but also maximize the average agreement index and minimize the maximum fuzzy completion time. Having elicited the linear membership functions for the fuzzy goals of the decision maker, we adopt the fuzzy decision of Bellman and Zadeh. By incorporating the concept of similarity among individuals into the genetic algorithms using the Gannt chart, a genetic algorithm which is suitable for solving the formulated problems are proposed. As illustrative numerical examples, both 6×6 and 10×10 three-objective job shop scheduling problems with fuzzy duedate and fuzzy processing time are considered, and the feasibility and effectiveness of the proposed method are demonstrated by comparing with the simulated annealing method.

Interactive decision making for multiobjective nonlinear programming problems with fuzzy parameters

This paper presents interactive decision making methods for multiobjective linear, linear fractional and nonlinear programming problems with fuzzy parameters. On the basis of the α-level sets of the fuzzy numbers, the concept of α-multiobjective programming and (local) M-α-Pareto optimality is introduced. Through the interaction with the decision maker (DM), the fuzzy goals of the DM for each of the objective functions in α-multiobjective programming are quantified by eliciting the corresponding membership functions. After determining the membership functions, in order to generate a candidate for the (local) satisficing solution which is also (local) M-α-Pareto optimal, if the DM specifies 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/her reference membership values and/or the degree α. In this way the (local) satisficing solution for the DM can be derived efficiently from among an M-α-Pareto optimal solution set. Based on the proposed methods for multiobjective linear, linear fractional and nonlinear programming problems with fuzzy parameters, interactive computer programs are developed and an illustrative numerical example for nonlinear case is demonstrated.

Theory and methodologyMinimax regret solution to linear programming problems with an interval objective function

In this paper, a linear programming problem with an interval objective function is treated. First, the previous approaches to this problem are reviewed and the drawbacks are pointed out. To improve the drawbacks, a new approach to this problem is proposed by introducing the minimax regret criterion as used in decision theory. The properties of minimax regret solution are investigated. In order to obtain the minimax regret solution, a method of solution by a relaxation procedure is proposed. It is shown that the solution is obtained by repetitional use of the simplex method. A numerical example is given to illustrate the proposed solution method.

An efficient genetic algorithm for job-shop scheduling problems with fuzzy processing time and fuzzy duedate

Abstract In this paper, by considering the imprecise or fuzzy nature of the data in real-world problems, job-shop scheduling problems with fuzzy processing time and fuzzy duedate are formulated and a genetic algorithm which is suitable for solving the formulated problems is proposed. On the basis of the agreement index of fuzzy duedate and fuzzy completion time, the formulated fuzzy job-shop scheduling problems are interpreted so as to maximize the minimum agreement index. For solving the formulated fuzzy job-shop scheduling problems, an efficient genetic algorithm is proposed by incorporating the concept of similarity among individuals into the genetic algorithms using the Gannt chart. As illustrative numerical examples, both 6×6 and 10×10 job-shop scheduling problems with fuzzy duedate and fuzzy processing time are considered. Through the comparative simulations with simulated annealing, the feasibility and effectiveness of the proposed method are demonstrated.

Interactive computer programs for fuzzy linear programming with multiple objectives

Abstract In this article, we present interactive computer programs fof solving fuzzy linear programming problems with multiple objectives. Through the use of five types of membership functions including non-linear functions, the fuzzy of imprecise goals of the decision maker are quantified. Although the formulated problem becomes a nonlinear programming problem, it can be reduced to a set of linear inequalities if some variable is fixed. Based on this idea, we propose a new method by combined use of bisection method and linear programming method. On the basis of the proposed method, FORTRAN programs that run in conversational mode are developed to implement man-machine interactive procedures. The commands in our programs and major prompt messages are also explained. An illustrative numerical example for the interactive processes is demonstrated together with the computer outputs

Interactive fuzzy programming for multi-level linear programming problems with fuzzy parameters

This paper presents interactive fuzzy programming for multi-level linear programming problems with fuzzy parameters. In fuzzy programming for multi-level linear programming problems, recently developed by Lai et al., since the fuzzy goals are determined for both an objective function and decision variables at the upper level, undesirable solutions are produced when these fuzzy goals are inconsistent. In order to overcome such problems, after eliminating the fuzzy goals for decision variables, interactive fuzzy programming for multi-level linear programming problems with fuzzy parameters is presented. In our interactive method, after determining the fuzzy goals of the decision makers at all levels, a satisfactory solution is derived efficiently by updating the satisfactory degrees of decision makers with considerations of overall satisfactory balance among all levels. Illustrative numerical examples for two-level and three-level linear programming problems are provided to demonstrate the feasibility of the proposed method.

Interactive multiobjective fuzzy random linear programming : Maximization of possibility and probability

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.

Fuzzy programming and profit and cost allocation for a production and transportation problem

Abstract In this paper, we deal with a real problem on production and transportation in a housing material manufacturer, and consider a production and transportation planning under the assumption that the manufacturer makes multiple products at factories in multiple regions and the products are in demand in each of the regions. First, we formulate mixed zero–one programming problems such that the cost of production and transportation is minimized subject to capacities of factories and demands of regions. Second, to realize stable production and satisfactory supply of the products in fuzzy environments, fuzzy programming for the production and transportation problem is incorporated. Finally, under the optimal planning of production and transportation, we show a profit and cost allocation by applying a solution concept from game theory. Using actual data, we show usefulness of the fuzzy programming and a rational allocation scheme of the profit and cost.

Fuzzy programming for multiobjective 0–1 programming problems through revised genetic algorithms

Abstract Recently, genetic algorithms (GAs), a new learning paradigm that models a natural evolution mechanism, have received a great deal of attention regarding their potential as optimization techniques for solving combinatorial optimization problems. In this paper, we focus on multiobjective 0–1 programming problems as a generalization of the traditional single objective ones. By considering the imprecise nature of human judgements, we assume that the decision maker may have fuzzy goal for each of the objective functions. After eliciting the linear membership functions through the interaction with the decision maker, we adopt the fuzzy decision of Bellman and Zadeh or minimum-operator for combining them. In order to investigate the applicability of the conventional GAs for the solution of the formulated problems, a lot of numerical simulations are performed by assuming several genetic operators. Then, instead of using the penalty function for treating the constraints, we propose three types of revised GAs which generate only feasible solutions. Illustrative numerical examples demonstrate both feasibility and efficiency of the proposed methods.

Robust optimization under softness in a fuzzy linear programming problem

Abstract In this paper, we discuss the softness and the robustness of the optimality in the setting of linear programming problems with a fuzzy objective function. A fuzzy goal defined on the deviation from the optimal value is introduced in order to define the soft-optimal solution. Fuzzy coefficients are regarded as possibility distributions. A necessity measure based on the possibility distribution is used for defining a necessarily optimal solution, i.e., a robust-optimal solution. Since a necessarily optimal solution does not exist in many cases, a necessarily soft-optimal solution is defined. A solution algorithm for the best necessarily soft-optimal solution is proposed.