روش پارتو - بهینگی برای مشکلات برنامه ریزی تولید کارگاهی انعطاف پذیر: ترکیب الگوریتم های تکاملی و منطق فازی با همدیگر

Most scheduling problems are complex combinatorial problems and very difficult to solve [[1] and [2]]. That is why, lots of methods focus on the optimization according to a single criterion (makespan, workloads of machines, waiting times, etc.). The combining of several criteria induces additional complexity and new problems. In this paper, we propose a Pareto approach based on the hybridization of fuzzy logic (FL) and evolutionary algorithms (EAs) to solve the flexible job-shop scheduling problem (FJSP). This hybrid approach exploits the knowledge representation capabilities of FL [Fuzzy Sets Syst. 1 (1989)] and the adaptive capabilities of EAs. The integration of these two methodologies for the multi-objective optimization has become an increasing interest. The objective considered is to minimize the overall completion time (makespan), the total workload of machines and the workload of the most loaded machine. Many examples are presented to illustrate some theoretical considerations and to show the efficiency of the suggested methodology.

In most combinatorial optimization problems, we have to simultaneously optimize a set of conflicting objective functions. The literature presents many possible considerations and techniques that can be useful to evaluate solutions [4] and [5]. Mainly, we can distinguish two classes: the Pareto-optimality approaches and the non-Pareto-optimality approaches. In a previous work, we have proposed an aggregative approach for solving multi-objective optimization problems (MOPs) based on the hybridization of fuzzy logic (FL) and evolutionary algorithms (EAs) [6]. This approach makes it possible to construct a set of satisfactory solutions according to the preferences of the decision-maker. In this work, we aim to complete the suggested approach and we propose an extension of the application of such a hybridization. Thereafter, we show how FL can be useful to make EAs more efficient for finding a set of Pareto-optimality solutions. This paper is organized as follows: In Section 2, we shortly describe the Pareto-optimality concepts used for solving MOPs and those especially applied in EAs. Then, the mathematical formulation of FJSP is presented in Section 3. The proposed fuzzy evolutionary approach will be described in Section 4. Section 5 focuses on the illustration of the suggested approach by applying it to solve FJSP and highlights some practical aspects of the application of such an approach for solving hard combinatorial problems. Finally, the last section deals with concluding remarks and shows the efficiency of the hybridization of FL and EAs through different numerical experiments.

In this paper, we have dealt with one of the hardest combinatorial problems (the FJSP) and we have proposed a new Pareto-optimality approach to solve it. This approach is based on a fuzzy evolutionary optimization in which some advanced practical and theoretical considerations are carefully chosen. The multi-objective evaluation of the solution quality is reduced to a single fitness function that measures this quality according to the lower-bound values of the different objective functions. The theoretical expressions of these lower-bounds are presented too. The originality of this approach consists in the application of the novel biological concept of GMO to enhance the final solution quality and in the use of the strong representation capabilities of FL to control EAs. The obtained results show the efficiency of the proposed approach. Although it does not guarantee the optimality, such an approach provides good quality solutions in a reasonable time limit. Moreover, the general aspect of the considered formulation presents a large methodological advantage that makes it possible to solve other particular problems like PMP. As future research direction, the study of the generalization of such an approach for solving other MOPs (real world problems like transportation systems, traffic regulation, etc.) seems an interesting subject which can enrich the proposed approach and give scientific benefits.