برنامه ریزی بدون انتظار تولید کارگاهی با استفاده از برنامه ریزی چند هدف خطی فازی

This study develops a fuzzy multi-objective linear programming (FMOLP) model for solving the multi-objective no-wait flow shop scheduling problem in a fuzzy environment. The proposed model attempts to simultaneously minimize the weighted mean completion time and the weighted mean earliness. A numerical example demonstrates the feasibility of applying the proposed model to no-wait flow shop scheduling problem. The proposed model yields a compromised solution and the decision maker's overall levels of satisfaction.

Scheduling is the assigning of a finite number of resources to a number of jobs over time, often with a decision that optimizes one or more objective. In most manufacturing systems it is required that for completion of a job, a set of processes needs to be performed serially. We refer to this as flow shop environment. Emergence of advanced manufacturing systems such as computer-aided design/computer-aided manufacturing (CAD/CAM), flexible manufacturing system (FMS), and computer-integrated manufacturing (CIM) have increased the importance of flow shop scheduling [1]. A flow shop scheduling problem addresses determination of sequencing N jobs needed to be processed on M machines so as to optimize the performance measures such as makespan, tardiness, work in process, number of tardy jobs, idle time, etc. In flow shop scheduling, the processing routes are the same for all the jobs [1]. In the permutation flow shop, passing is not allowed. Thus, the sequencing of different jobs that visit a set of machines is in the same order. In the general flow shop, passing is permitted. Therefore, the job sequence on each machine may be different [2]. Flow shop scheduling problems are popular in the area of scheduling and there are numerous papers that have investigated these problems [3]. Most research is dedicated to single-criterion problems. For example, Pan et al. [4] consider the two-machine flow shop scheduling problem minimizing total tardiness as the objective. Bulfin and M’Hallah [5] propose an exact algorithm to solve the two-machine flow shop scheduling problem with the objective of the weighted number of tardy jobs. Blazewicz et al. [6] analyze different solution procedures for the two-machine flow shop scheduling problem with a common due date and weighted late work criterion. Choi et al. [7] investigate a proportionate flow shop scheduling problem where only one machine is different and job processing times are inversely proportional to machine speeds. The objective is to minimize maximum completion time. Grabowski and Pempera [8] address the no-wait flow shop problem with a makespan criterion and develop and compare various local search algorithms for solving this problem. Wang et al. [9] deal with a two-machine flow shop scheduling problem with deteriorating jobs , minimizing the total completion time. It is well known that the optimal solution of single-objective models can be quite different from the models consisting of multiple objectives. In fact, the decision maker (DM) often wants to minimize the weighted mean completion time or weighted mean earliness. Each of these objectives is valid from a general point of view. Since these objectives conflict with one another, a solution may perform well for one objective but may give inferior results for others. For this reason, scheduling problems have a multi-objective structure. While the above studies treated a single objective, consideration of multiple criteria is practically more realistic [3] and [10]. The multi-objective flow shop scheduling problem has been addressed by some researches. Murata et al. [3] propose a multi-objective genetic algorithm and then apply it to the flow shop scheduling problem with the objective of minimizing makespan and total tardiness. Sayin and Karabati [11] deal with the scheduling problem in a two-machine flow shop environment with the objective of minimizing makespan and sum of completion times simultaneously. To solve this problem, they developed a branch-and-bound procedure that iteratively solves restricted single-objective scheduling problems until the set of efficient solutions is completely enumerated. Danneberg et al. [12] address the permutation flow shop scheduling problem with setup times where the jobs are partitioned into groups or families. Jobs of the same group can be processed together in a batch but the maximum number of jobs in a batch is limited. The setup time depends on the group of jobs. They propose the makespan as well as the weighted sum of completion times of the jobs as the objectives. To solve the problem, they propose and compare various constructive and iterative algorithms. Toktas et al. [10] consider the two-machine flow shop scheduling, minimizing makespan and maximum earliness simultaneously. They develop a branch-and-bound procedure that generates all efficient solutions with respect to the two criteria and also propose a heuristic procedure to generate approximate efficient solutions. Ponnambalam et al. [13] propose a TSP-GA multi-objective algorithm for flow shop scheduling where they use a weighted sum of multiple objectives (i.e., minimizing makespan, mean flow time and machine idle time). The weights are randomly generated for each generation to enable a multi-directional search. Ravindran et al. [14] propose three heuristic algorithms to solve the flow shop scheduling problem using makespan and total flow time criteria. Ishibuchi and Murata [15] present a flow shop scheduling problem with fuzzy parameters such as fuzzy due dates and fuzzy processing times, for which the objectives are to minimize the total flow time, makespan, and the maximum earliness and tardiness of all jobs. A multi-objective genetic algorithm is developed to handle these fuzzy scheduling objectives. In 1978, Zimmermann [16] first extended his fuzzy linear programming (FLP) approach to a conventional multi-objective linear programming (MOLP) problem [17]. For each of the objective functions of this problem, assume that the decision maker (DM) has a fuzzy goal such as ‘the objective functions should be essentially less than or equal to some value’. Then, the corresponding linear membership function is defined and the minimum operator proposed by Bellman and Zadeh [18] is applied to combine all the objective functions. By introducing auxiliary variables, this problem can be transformed into an equivalent conventional LP problem and can easily be solved by the simplex method. Subsequent works on fuzzy goal programming (FGP) include Hannan [19], Leberling [20], Luhandjula [21], and Sakawa [22]. Due to the inherent conflict of the two objectives consisting of the weighted mean completion time and weighted mean earliness, we propose a fuzzy goal programming approach to solve an extended mathematical model of a flow shop scheduling problem. Therefore, the aim of this study is to develop a fuzzy multi-objective linear programming (FMOLP) model for solving the multi-objective no-wait flow shop scheduling problem in a fuzzy environment. First, is the construction of an MOLP model of a multi-objective no-wait flow shop scheduling problem. The model minimizes the weighted mean completion time and weighted mean earliness. Then the model is converted into an FMOLP model by an integration of fuzzy sets and multiple objective programming approaches.

In most real-world flow shop scheduling problems, the DM must simultaneously handle conflicting objectives that usually govern the use of the constrained resources within organizations. We developed an FMOLP method for solving no-wait flow shop scheduling problems involving multiple fuzzy objectives with piecewise linear membership functions. The proposed method aims to simultaneously minimize the weighted mean completion time and the weighted mean earliness. Moreover, the proposed FMOLP method provides a systematic framework that facilitates the fuzzy decision-making process until a satisfactory solution is obtained. A numerical example was worked out to demonstrate the feasibility in applying the proposed FMOLP method to no-wait flow shop scheduling problems. Consequently, the proposed method yielded an effective solution and an overall degree of DM satisfaction with the determined objective values. Accordingly, the proposed method is practically applicable for solving real-world multi-objective no-wait flow shop scheduling problems in a fuzzy environment. The proposed FMOLP method is based on Hannan's method, which implicitly assumes that the minimum operator is an appropriate representation of DM's judgment to combine fuzzy sets by logical ‘and’ operations.