An ACO algorithm for makespan minimization in parallel batch machines with non-identical job sizes and incompatible job families
Graphical abstract
Introduction
Batch scheduling problems occur in many industries, such as semiconductor manufacturing, cargo handling in port, transportation, storage systems, and so on. The problem is motivated by the diffusion operation in the wafer fabrication of semiconductor manufacturing. Due to the chemical nature of the process, only jobs with the same recipe can be processed together [1]. All jobs requiring the same recipe can be viewed as a job family, and jobs in the same family have the same processing times. Effective scheduling of these operations is particularly important because of their long processing time requirements, generally 10 h as opposed to 1 or 2 h for most other processes.
In contrast to the classical machine scheduling, a batch processing machine (BPM) can process several jobs in a batch simultaneously. There are two types of batch scheduling problems: s-batch and p-batch. In s-batch, the jobs in a batch are processed in serial and the processing time of a batch is the sum of the processing times of all the jobs in the batch, while in p-batch, the jobs in a batch are processed in parallel and the processing time of a batch is the longest processing time of the jobs in the batch. P-batch scheduling is more important than s-batch scheduling in semiconductor manufacturing [2]. Additionally, p-batch scheduling is often encountered in many modern manufacturing industries such as food, chemical and mineral processing, pharmaceutical and metalworking industries as well as environmental stress screening chamber fabrication [3].
In this paper, we consider p-batch scheduling on parallel BPMs with identical machine capacity, non-identical job sizes and incompatible job families. Jobs with non-identical sizes come from different job families. All jobs in the same family have the same processing times, while jobs from different families have different processing times. The jobs have to be grouped into batches such that the total size of the jobs in the batch does not exceed the machine capacity. Moreover, jobs from different families cannot be grouped together in the same batch. The jobs are assumed to be ready at time zero. The processing time of a batch is equal to the longest processing time of all the jobs in the batch [4]; in our case the processing times of all the jobs in a batch are the same. The batches are then scheduled on the machines to minimize the makespan. The problem of minimizing the makespan on a single BPM with non-identical job sizes has been shown to be NP-hard [5]. Thus, our problem is also NP-hard.
Like other batch scheduling problems, the problem in this paper can be addressed by solving two independent subproblems: group the jobs into batches and schedule the batches on the parallel BPMs. We propose a Max–Min Ant System (MMAS) algorithm to group the jobs into batches, and then apply the Multi-Fit (MF) algorithm [6] to schedule the batches on the machines.
In the literature, there are several articles that deal with incompatible job families [1], [30], [31], [35], [36]. The article in [1] assumes that the job sizes are identical, while we deal with arbitrary job sizes. The articles in [30], [31] deal with the total weighted tardiness objective, while we study the makespan objective. The articles [35], [36] study exactly the same problem as ours. Article [35] studies parallel BPMs while article [36] studies a single BPM. We will be comparing the performance of our heuristic against the heuristics of [35]. As we shall see later, our heuristic outperforms all of the heuristics in [35].
The paper is organized as follows. In the following section we review previous related work on BPM scheduling problems as well as the MMAS algorithm. Section 3 formally defines the problem, and a lower bound is provided. The proposed algorithm and its implementation are described in Section 4, and the results of the computational experiment are reported in Section 5. Finally, in Section 6, we conclude the paper with a summary and some directions for future research.
Section snippets
Previous related work
Recently, a lot of research has been done on scheduling on BPMs as well as the MMAS algorithm. Previous work related to these two areas will be presented in the next two subsections.
Problem statement and notation
Using the three-field notation in [55], our problem can be denoted by P ∣ p-batch, B, F ∣ Cmax. A set of N jobs, denoted by , coming from F different families, denoted by , is to be grouped into batches. Thus, . All jobs in family have the same processing times, denoted by pj. Let ZN×F record the relationship between the jobs and the families, where zij = 1 if job belongs to family ; otherwise, zij = 0. The batch set, denoted by , is to be
Description of the algorithm
Our problem can be solved in two steps. First, group the job set into batches. Second, assign the batches to the parallel BPMs. For the first subproblem, we propose a MMAS algorithm to batch the jobs. For the second subproblem, we use the MF algorithm to schedule the batches on the machines. In the classical scheduling model, Leung [11] has given an O(log M * N2(F−1))-time algorithm to find a minimum length schedule for a set of N jobs with F different processing times on M identical and parallel
Computational experiments
To evaluate the performance of the proposed algorithm, a series of computational experiments are performed, where our algorithm is compared with some previously studied algorithms.
Conclusions
In this paper, we study batch scheduling on parallel BPMs with non-identical job sizes and incompatible job families to minimize the makespan. We present a novel MMAS-based meta-heuristics coupled with the MF algorithm to solve the problem. The jobs are grouped into batches by the MMAS heuristic. The batches are then scheduled on the BPMs by the MF algorithm. The correlation between minimizing the unused space in a batch and minimizing the makespan enables us to concentrate on reducing the
Acknowledgements
The work of the first author is supported by grants from the National Natural Science Foundation of China (71171184), the Science Foundation of Anhui University (33050044) and the Foundation of China Scholarship Council (201206505002).
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