Scheduling of nonresumable jobs and flexible maintenance activities on a single machine to minimize makespan

Abstract

This study addresses a single machine scheduling problem with periodic maintenance, where the machine is assumed to be stopped periodically for maintenance for a constant time w during the scheduling period. Meanwhile, the maintenance period [u, v] is assumed to have been previously arranged and the time w is assumed not to exceed the available maintenance period [u, v] (i.e. w ⩽ v − u). The time u(v) is the earliest (latest) time at which the machine starts (stops) its maintenance. The objective is to minimize the makespan. Two mixed binary integer programming (BIP) models are provided for deriving the optimal solution. Additionally, an efficient heuristic is proposed for finding the near-optimal solution for large-sized problems. Finally, computational results are provided to demonstrate the efficiency of the models and the effectiveness of the heuristics. The mixed BIP model can optimally solve up to 100-job instances, while the average percentage error of the heuristic is below 1%.

Introduction

Pinedo (2002, p. 393) emphasized the need for research on real-world scheduling problems: “Most theoretical models do not take machine availability constraints into account; usually it is assumed that machines are available at all times. In practice, machines are usually not continuously available.” Research on scheduling, with machine availability constraints, focuses mainly on machine breakdowns, maintenance activities, and tool changes. Two types of processing cases exist for machine availability constraint problems, namely: nonresumable and resumable. If a job cannot be completed before the unavailable period, then it must be restarted rather than continued, a case called nonresumable. The resumable case is that where a job can continue after the machine becomes available again. This study deals with a single machine nonresumable jobs scheduling problem with multiple periodic maintenance activities.

The motivation for studying the problem with maintenance and scheduling comes from production practice. For example, a machine may be designed to be examined, refueled or maintained after working for a certain period of time. The problem encountered here belongs to such a case. Although the problem with maintenance and scheduling is important as it happens frequently in industry, it has been relatively unstudied in previous literature. This problem has been classified as machine scheduling with availability constraints. Schmidt, 2000, Sanlaville and Schmidt, 1998 provided surveys of this topic. Lee (1996) presented analytical results for the resumable and nonresumable cases considering a fixed unavailability period in the planning horizon. Notably, all of these analytical results assumed that the machine unavailability was known and fixed in advance. Lee and Liman (1992) established a simpler proof of NP-hardness and demonstrated a tight error bound for the shortest processing time (SPT) heuristic for the single machine deterministic maintenance problem. Moreover, Yang et al. (2002) were the first to study problems of single machine scheduling with a flexible maintenance activity, and considered that machines should be stopped for constant maintenance or resetting during the scheduling period. They termed flexible maintenance since the actual maintenance time is less than or equal to the available maintenance time. The objective was to minimize the makespan. They demonstrated that the proposed problem is NP-hard and provided a heuristic algorithm with complexity O(n log n).

The above studies assume that the machine has a single maintenance activity. Liao and Chen (2003) investigated a single machine scheduling problem in which multiple and periodic maintenance activities are required in a complete schedule. However, Liao and Chen (2003) assumed that each maintenance activity was fixed and known in advance. Qi et al. (1999) considered a problem in which multiple maintenance activities must be scheduled together with jobs on a single machine. They decided the starting time of each maintenance activity and the job sequence. Graves and Lee (1999) clarified the complexity of various single machine problems using this model. Finally, Lee and Leon (2001) investigated a single machine problem using this model in which job processing time may change following a maintenance activity.

Lee (1996) considered resumable and nonresumable cases without flexible maintenance. Meanwhile, Yang et al. (2002) only considered just one flexible maintenance activity and nonresumable case. In practice multiple maintenance activities are required for the machine. This study deals with the problem of scheduling nonresumable jobs on a single machine with flexible and multiple maintenance activities. The mathematical programming formulation is a natural method of solving machine scheduling problems (Rinnooy Kan, 1976). Two mixed binary integer programming (BIP) models are provided to derive the optimal solution. Additionally, an efficient heuristic is proposed for finding the near-optimal solution for large-sized problems. Computational results are provided to demonstrate the efficiency of the models and the effectiveness of the heuristic. The criterion considered is to minimize the makespan.

The remainder of this paper is organized as follows. Section 2 describes the problem. Section 3 then presents the two integer programming models. Furthermore, Section 4 presents an efficient heuristic scheduling algorithm. Computational experiments and results then are given in Section 5, followed by a summary in Section 6.