Time Using Ant Colony Optimization

We describe an Ant Colony Optimization (ACO) algorithm for solving a single machine scheduling problem. In the operating situation modeled, setup times are sequence dependent and the objective is to minimize total tardiness. This problem has previously been treated by Rubin & Ragatz [1995] and by Tan et al. [2000] among others. A new feature using look-ahead information in the transition rule of the ACO algorithm shows an improvement in performance. A comparison with other solution approaches indicates that the ACO that we describe is competitive and has a certain advantage for larger problems. Keywords : scheduling, metaheuristic, ant colony optimization, single machine, total tardiness, sequence dependent setups. 1. Introduction Industrial production scheduling constitutes a fertile field for both researchers and practitioners of operational research. The interest in this field is generated not only by the problem-solving challenge that it offers but also by the practical results that can be achieved. However, researchers such as Maccarthy & Liu [1993] and McKay & Wiers [1999] have remarked on the sometimes wide gap between the theoretical problems treated and those met in practice. The development of efficient solution procedures for the scheduling of orders in a casting center belonging to a Canadian multinational firm is the principal aim of the research reported in this paper. The authors have previously reported [Gravel et al., 2000] [Gravel et al., 2001] details of successful work in this metaheuristics area. The aim of this paper is to report on the performance of some extensions to the "ant colony optimization" (ACO) algorithm [Dorigo, 1992] which has already demonstrated its usefulness in this industrial situation and is presently incorporated in software used by the firm. In the industrial application, the holding furnaces may require certain draining and cleaning operations of varying durations between the casting of two successive orders for different metal alloys. These operations may be seen as the setup operations dealt with in the literature. We seek a schedule for current released orders that takes into account these sequence dependent setup times as well as multiple objectives. We validate the performance of the new elements that we have introduced by solving a known problem from the literature, the single machine problem with sequence dependent setups. This allows us to compare our results with those previously published [Tan et al., 2000] for various metaheuristics. Single machine scheduling is a classic problem that has been well covered in the literature [Koulamas, 1997]. This problem offers a lower level of complexity than that of other configurations often treated in scheduling publications, such as parallel and serial machines, or cellular shops. It is, however, possible to achieve interesting practical results through the study of single machine shops. For example, some shops may have a bottleneck machine that strongly influences performance and which therefore allows the shop to be studied as a single machine Scheduling a single machine with sequence-dependent ...; C. Gagné, W.L. Price, M. Gravel 2 [Graves, 1981] [Hax & Candea, 1984]. We have already referred to our own workon an application treating complex operations having sequence dependent setup times as a single machine shop. Various objective functions may be useful in the scheduling of a single machine shop. Among these, we find the minimization of total tardiness, an objective that seeks to improve customer service. Meeting target delivery dates has been declared as the most important scheduling objective by Wisner & Siferd [1995] who also found that 58% of production planners actively seek to meet delivery dates. Even in the case of a single machine, minimizing tardiness is a difficult objective to attain since there are no simple sequencing rules that apply, save in two cases described by Emmons [1969], and in these two cases, the setups are sequence independent. In general, the problem of scheduling a single machine with sequence dependent setup times has generally been presented with the objective of minimizing the total production time (makespan) for the set of released orders [Baker, 1992] [Morton & Pentico, 1993]. In this case, the problem may be represented as a traveling salesman problem. Where the objective is to meet delivery dates where setup times are sequence dependent, the literature is not extensive. One of the conclusions of Allahverdi et al. [1999] is that there is a need for further research in this area, and in scheduling in general. Formally, the problem of scheduling a single machine having sequence dependent setup times where the objective is the minimization of total tardiness can be defined as follows [Rubin & Ragatz, 1995]: let there be n jobs to produce, all released at time zero, and which must be completed without interruption on a single machine. Each job j has as attributes its production duration pj, its delivery date dj, and its setup time sij, which is incurred when job j is undertaken following job i in an job sequence Q. We define Q = {Q(0), Q(1), …, Q(n)} as the job sequence where Q(j) is the subscript of the j job in the sequence and where Q(0) = 0. The machine is continuously available through the planning period and can process only one job at a time. Once a job is started it must be completed without interruption. The end time of job j is expressed as: