An improved meta-heuristic for makespan minimization of a single batch machine with non-identical job sizes

Abstract

We consider the problem of minimizing the makespan on a single batch machine with non-identical job sizes, where several jobs can be simultaneously processed as a batch. We formulate makespan minimization as a problem of minimizing the wasted space. Applying a candidate set strategy to narrow the search space, combined with a wasted-space-based heuristic to update the pheromone information, an improved max–min ant system algorithm is presented. A specific local search method is incorporated to gain better performance. Appropriate parameter settings in the proposed algorithm are determined by extensive experiments. The experimental results show that the proposed algorithm outperforms several previously studied algorithms.

Introduction

A parallel batch machine can process several jobs simultaneously as a batch. Such machines are widely used in many manufacturing industries such as the semiconductor industry, casting industry, metal industry, aeronautical industry, pharmaceutical industry, and logistics freight [1]. In the semiconductor industry, there are four main steps in the processing of very large-scale integrated circuits: wafer fabrication, wafer probe, assembly and final testing. Integrated circuits are subjected to burn-in operations that take place in the final testing stage. The purpose of the burn-in operations is to subject the integrated circuit to thermal stress for a certain period of time in order to bring out the latent defects (if there is any). The burn-in time of an integrated circuit is typically specified by the customers and is quite lengthy. A typical burn-in time is 120 h, as opposed to a few hours in the other operations in the manufacturing process. Since the burn-in process is time-consuming, it emerges as a major bottleneck of the entire manufacturing process. Thus, efficient execution of the burn-in operation is critical for the success of the company.

The burn-in operation can be described as follows. Each integrated circuit constitutes a job. The jobs need to be loaded onto boards which are then put into an oven for an extended period of time. Each job has a prespecified burn-in time and a certain size (which is the number of boards the job occupies). The oven has a finite capacity; i.e., the number of boards that can be put into the oven. Several jobs can be put into the oven at the same time, provided that the capacity of the oven is not exceeded. A job must be put into the oven for its prespecified burn-in time, but it can be longer. Therefore, the processing time of a batch is simply the longest burn-in time of all the jobs in the batch. The problem is to determine how the jobs should be batched together so as to minimize the makespan. We call this problem the Single Batch Machine with Non-identical Job Sizes, in short, the SBMN problem.

The SBMN problem has received tremendous attention in the last two decades. Many research papers have been written on this subject, see the literature review in the next section. Its popularity is due largely to the following facts. First, since the burn-in operation is a lengthy process, a reduction in the makespan represents a significant increase in the throughput. Second, a reduction in the makespan represents a significant saving of energy. The oven typically maintains a temperature of 120 °C, and hence consumes a lot of energy. Third, the SBMN problem is an intriguing problem because it involves two dimensions – time and space.

The SBMN problem is known to be NP-hard. Motivated by the computational complexity of the problem, we propose an improved max–min ant system (MMAS) algorithm to solve it. Utilizing the equivalence between the minimization of the wasted space and the minimization of makespan, we construct a candidate set, comprising the unscheduled jobs that can decrease the wasted space of the current batch, to narrow the search space. We also provide a novel definition of heuristic information to make the ants move faster towards better solutions. In addition, to further improve the solution quality, a local optimization strategy is used in the algorithm. Experimental results show that our algorithm outperforms several previously studied algorithms.

Our approach differs from previous approaches to the SBMN problem in several aspects. First, although some meta-heuristics have been developed for the SBMN problem, little effort has been made to apply the MMAS algorithm to the problem and compare comprehensively its performance with other ant colony optimization (ACO) algorithms. Second, once a solution is obtained, we apply a local optimization algorithm to further enhance the solution quality.

The rest of the paper is organized as follows. In Section 2, we review previous work related to the SBMN problem and MMAS. Section 3 describes the SBMN problem. In Section 4, an improved MMAS algorithm is presented in detail. The experimental framework and the computational results are provided in Section 5. Finally, we draw some concluding remarks and suggest some future lines of research in Section 6.