An ant colony optimization for single-machine tardiness scheduling with sequence-dependent setups

Abstract

In many real-world production systems, it requires an explicit consideration of sequence-dependent setup times when scheduling jobs. As for the scheduling criterion, the weighted tardiness is always regarded as one of the most important criteria in practical systems. While the importance of the weighted tardiness problem with sequence-dependent setup times has been recognized, the problem has received little attention in the scheduling literature. In this paper, we present an ant colony optimization (ACO) algorithm for such a problem in a single-machine environment. The proposed ACO algorithm has several features, including introducing a new parameter for the initial pheromone trail and adjusting the timing of applying local search, among others. The proposed algorithm is experimented on the benchmark problem instances and shows its advantage over existing algorithms. As a further investigation, the algorithm is applied to the unweighted version of the problem. Experimental results show that it is very competitive with the existing best-performing algorithms.

Introduction

The operations scheduling problems have been studied for over five decades. Most of these studies either ignored setup times or assumed them to be independent of job sequence [1]. However, an explicit consideration of sequence-dependent setup times (SDST) is usually required in many practical industrial situations, such as in the printing, plastics, aluminum, textile, and chemical industries [2], [3]. As Wortman [4] indicates, the inadequate treatment on SDST will hinder the competitive advantage.

On the other hand, a survey of US manufacturing practices indicates that meeting due dates is the single most important scheduling criterion [5]. Among the due-date criteria, the weighted tardiness is the most flexible one as it can be used to differentiate between customers.

While the importance of the weighted tardiness problem with SDST has been recognized, the problem has received little attention in the scheduling literature, mainly because of its complexity. This inspires us to develop a heuristic to obtain a near-optimal solution for this practical problem in the single-machine environment. It is noted that the single-machine problem does not necessarily involve only one machine; a complicated machine environment with a single bottleneck may be treated as a single machine problem.

We now give a formal description of the problem. We have n jobs which are all available for processing at time zero on a continuously available single machine. The machine can process only one job at a time. Associated with each job j is the required processing time $(p_j)$, due date $(d_j)$, and weight $(w_j)$. In addition, there is a setup time $(s_{\mathit{ij}})$ incurred when job j follows job i immediately in the processing sequence. Let Q be a sequence of the jobs, $Q=[Q(0),Q(1),\dots ,Q(n)]$, where $Q(k)$ is the index of the kth job in the sequence and $Q(0)$ is a dummy job representing the starting setup of the machine. The completion time of $Q(k)$ is $C_{Q(k)}={\sum }_{l=1}^k\{s_{Q(l-1)Q(l)}+p_{Q(l)}\}$, the tardiness of $Q(k)$ is $T_{Q(k)}=\max \{C_{Q(k)}-d_{Q(k)},0\}$, and the (total) weighted tardiness for sequence Q is ${\mathit{WT}}_Q={\sum }_{k=1}^n{w}_{Q(k)}{T}_{Q(k)}$. The objective of the problem is to find a sequence with minimum weighted tardiness of jobs. Using the three-field notation, this problem can be denoted by $1/s_{\mathit{ij}}/\sum w_j{T}_j$ and its unweighted version by $1/s_{\mathit{ij}}/\sum T_j$.

Lawler et al. [6] show that the $1//\sum w_j{T}_j$ problem is strongly NP-hard. Since the incorporation of setup times complicates the problem, the $1/s_{\mathit{ij}}/\sum w_j{T}_j$ problem is also strongly NP-hard. The unweighted version $1/s_{\mathit{ij}}/\sum T_j$ is strongly NP-hard because $1/s_{\mathit{ij}}/C_{\max }$ is strongly NP-hard [7, p. 79] and $C_{\max }$ reduces to $\sum T_j$ in the complexity hierarchy of objective functions [7, p. 27]. For such problems, there is a need to develop heuristics for obtaining a near-optimal solution within a reasonable computation time.

Scheduling heuristics can be broadly classified into two categories: the constructive type and the improvement type. In the literature, the best constructive-type heuristic for the $1/s_{\mathit{ij}}/\sum w_j{T}_j$ problem is apparent tardiness cost with setups (ATCS), proposed by Lee et al. [8]. Like other constructive-type heuristics, ATCS can derive a feasible solution quickly but the solution quality is usually unsatisfactory, especially for large-sized problems. On the other hand, the improvement-type heuristic can produce better solutions but with much more computational efforts. For the $1/s_{\mathit{ij}}/\sum w_j{T}_j$ problem, Cicirello [9] develops four different improvement-type heuristics, including limited discrepancy search (LDS), heuristic-biased stochastic sampling (HBSS), value-biased stochastic sampling (VBSS), and hill-climbing using VBSS (VBSS-HC), to obtain solutions for a set of 120 benchmark problem instances each with 60 jobs. Recently, an simulated annealing (SA) algorithm has been used to update 27 such instances in the benchmark library. To the best of our knowledge, the work of Cicirello [9] is the only research that develops improvement-type heuristics for the $1/s_{\mathit{ij}}/\sum w_j{T}_j$ problem. The importance of the problem in real-world production systems and its computational complexity justify us to challenge the problem using a recent metaheuristic, the ant colony optimization (ACO). On the other hand, there exist several heuristics of the improvement type for the unweighted problem $1/s_{\mathit{ij}}/\sum T_j$ [10], [11], [12].

The remainder of this paper is organized as follows. In Section 2, we briefly describe the background of ACO heuristic and then give the detailed steps of the proposed ACO algorithm. Computational experiments are conducted and reported in Section 3. Finally, conclusions are given in Section 4.