Total tardiness minimization on unrelated parallel machine scheduling with auxiliary equipment constraints
Introduction
This paper deals with the problem of scheduling jobs on unrelated parallel machines. Each job has a due date and requires a single operation. A setup that includes detaching one die and attaching another from the appropriate die type is incurred if there is a switch from processing one type of job to another type. Due to the attributes of the machines and the fitness of dies to each, the processing time for a job depends on both the job and the machine, each job being restricted to processing on certain machines. Moreover, the detaching (attaching) time may have different values depending on both the die type and the machine on which the die is detached (attached). Such a production environment can be found in injection-molding departments where parallel machines are used to process different components and for which setups are required for auxiliary equipment (e.g., molds). This scheduling problem is also frequently encountered in the die-casting industry, since many different machines are used to produce castings. The dies (or molds) are quite expensive; thus, the number of dies of each type available is limited. Therefore, dies must be treated as restricted secondary resources, the fact of which distinguishes this study from many past studies in unrelated parallel-machine scheduling in which secondary resources are unrestricted.
It is known that this class of scheduling problems is NP-hard [1]. In this paper, an effective heuristic combining the threshold-accepting (TA) method, tabu list techniques from the tabu search method, and designed improvement procedures is proposed to minimize total tardiness. Computational characteristics of the presented heuristic are evaluated in empirical comparisons with an apparent-tardiness-cost-with-setup (ATCS) procedure [2], a basic simulated annealing (SA) method from the literature [3], and optimal solutions. Computational experiences demonstrate that the proposed heuristic significantly outperforms the ATCS procedure and the SA method, and is capable of obtaining optimal solutions for small-sized problems.
The rest of this paper is organized in five sections. The related research is reviewed in Section 2; the TA method and tabu lists are briefly discussed in Section 3; the proposed heuristic is described in Section 4; computational experiments and results are reported in Section 5; conclusions and suggestions for future research are discussed in Section 6.
Section snippets
Related research
Parallel-machine scheduling with tardiness-oriented performance measures has attracted a lot of research attention in recent years. For identical parallel-machine scheduling, Root [4] considered that all jobs have the same due dates and presented a constructive algorithm to minimize the total tardiness. Lawler [5] assumed that all jobs have the same processing times and formulated the total tardiness problem as a transportation problem. Elmaghraby and Park [6] developed a branch-and-bound
TA method and tabu lists
The TA method and tabu lists are briefly described in this section.
Heuristic H1
The proposed heuristic, H1, is described in this section. First, the following notations are defined: group a set of jobs that are allocated to the same machine and require the same type of die index for jobs index for machines index for die types Inner_max maximum number of consecutive iterations without improvement to the best known solution by a specific procedure Outer_max maximum number of consecutive iterations without improvement to the best known solution
Computational experiments
In this section, the computational characteristics and effectiveness of H1 are evaluated by using two sets of newly generated test problems, since there are no “standard” or benchmarking test data in the open literature. The first set of test problems, consisting of problems in smaller sizes, is designed mainly for demonstrating that H1 is capable of obtaining optimal solutions. As for the second set of test problems comprising problems in larger sizes, H1 is compared with an ATCS procedure, a
Conclusions and suggestions for future research
This research has dealt with scheduling jobs on unrelated parallel machines with auxiliary equipment constraints. An effective heuristic based on threshold-accepting method, tabu lists, and improvement procedures has been proposed to minimize total tardiness. Two sets of test problems have been generated to evaluate the computational characteristics of the proposed heuristic. Computational results have indicated that the proposed heuristic is capable of obtaining optimal solutions for problems
Acknowledgements
We wish to thank the anonymous referees for improving this paper through their useful comments.
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