A branch-and-price algorithm for scheduling of deteriorating jobs and flexible periodic maintenance on a single machine

Highlights

• Study a scheduling problem with deteriorating jobs and flexible maintenance.

• Propose two integer programming formulations and a branch-and-price algorithm.

• Develop an efficient label-setting algorithm to solve the pricing problem.

• Evaluate the algorithm through extensive computational results.

• Introduce a myopic heuristic to practitioners for the online case.

Abstract

In many production systems, maintenance activities including preventive maintenance, repairs and tool changes are periodically scheduled. The activities can revert the machine from a sub-normal processing rate to a normal one.

In this paper, we study a single machine scheduling problem where deteriorating jobs and flexible periodic maintenance are considered. The single machine is operated to process a set of jobs with alternating processing periods and maintenance periods. In a processing period, a subset of jobs is sequentially processed and the completion time of the last job cannot exceed the allowed maximum duration. The actual processing time of each job grows at a linear job-specific deterioration rate and depends on its starting time within the period. Between two processing periods, a maintenance period with a fixed duration exists and the maintenance activities are carried out so that the processing rate of the machine is reverted to the normal rate. The objective is to schedule all the jobs to a set of processing periods and to minimize the makespan of the schedule.

We formulate the problem using a set-partitioning model and, for a solution method, we make use of a branch-and-price algorithm. A label-setting algorithm with a dominance rule is designed to solve the pricing problem in column generation. Computational experiments are conducted on a set of randomly generated test instances to evaluate the performance of the proposed method.

Introduction

Machine scheduling has a very wide range of applications from various industries including manufacturing and production, fire fighting, pharmaceutical industry, chemical industry, queueing and communication systems. In many real-world applications, the processing times of jobs are not deterministic but grow because of deterioration effects. Such deterioration effects are brought by the jobs that await service or the operating machine whose condition becomes worse. Delays of service would increase efforts. As pointed out by Kunnathur and Gupta (1990), the time to control problem situations such as fire fighting and hospital emergency ward scheduling increases if efforts are not started at the proper time. In many production systems, the physical conditions of the machine change after running for some time. Because of the degradations due to the changes such as overheat, the machine is losing efficiency and increases processing times (Mosheiov, 1991).

The actual processing times of the jobs depend on the deterioration rates and the starting times. Compared to the normal processing times, which are measured on the machine at its best condition, the actual processing times are larger. The deteriorating jobs thus occupy the machine much longer and cause more delays. To revert the machine to its best condition, maintenance activities (Lee & Leon, 2001) such as preventive maintenance, repairs and tool changes are operated. In some real-world manufacturing and production systems, such maintenance activities are periodically operated, for instance, periodic repairs (Liao & Chen, 2003).

We study the problem of scheduling a set of deteriorating jobs on a single machine by considering flexible periodic maintenance. In the problem, the machine is run to process jobs in the processing periods. Between two processing periods, the machine is stopped and maintenance activities take place to revert the machine in a maintenance period. Whenever the processing of a job completes, it is flexible to terminate the processing period and start a maintenance period. For each processing period, the machine is allowed to run for a maximum duration. That is, the completion time of the last job processed in the period cannot exceed the maximum duration. Meanwhile, the maintenance periods have the same fixed duration. The problem is to schedule all the jobs to some processing periods and minimize the makespan of the schedule. The jobs assigned to the same processing period form a batch and are sequentially processed by the machine. In addition, the jobs are considered to be non-resumable. That is, once a non-resumable job is started, the processing of the job must be finished without any interruption.

Liao and Chen (2003) and Ji, He, and Cheng (2007) were the first to investigate periodic maintenance in a single-machine scheduling problem with the objectives to minimize the maximum tardiness and the makespan, respectively. In both papers, the maintenance activities are known in advance because the processing periods and the maintenance periods are fixed. Chen (2008) pointed out that a maintenance activity can be flexibly scheduled within a time window and studied the single-machine scheduling problem with flexible periodic maintenance that minimizes the makespan. However, to the best of our knowledge, none of them and the other related work have considered both flexible periodic maintenance and deteriorating jobs as our problem, where the decision of when to start a maintenance period is a critical problem. Chen (2008) showed that the single-machine problem with flexible periodic maintenance and makespan minimization is strongly NP-complete. The problem considered in this paper reduces to their problem if the deterioration rates are zeros, therefore it is also NP-complete.

Because of the deterioration effect, many problems with deterioration effect in the literature are generally formulated using mixed nonlinear integer programming models. In this paper, we introduce a set-partitioning model for the problem. Given the set of jobs to form a batch, the optimal index policy of Kunnathur and Gupta (1990) is used to sort the jobs and identify the optimal sequence of the jobs, which obtains the minimum completion time of the last job. With the optimal sequence, the processing time of the batch of jobs is conveniently calculated. The makespan of a schedule is computed by just summing up the processing times of the batches and the maintenance times in the schedule.

Each feasible batch corresponds to a variable or column in the set-partitioning model. The resulting integer linear programming formulation has a large number of binary variables for the large number of possible batches. We thus propose a branch-and-price (BP) algorithm for the problem (Barnhart, Johnson, Nemhauser, Savelsbergh, Vance, 1998, Desrosiers, Lübbecke, 2005). Because the deterioration effect increases the actual processing times, if the accumulated increment of some of the jobs is larger than the maintenance time, two new batches can be created by splitting the current batch, thus reducing the makespan. Based on this observation, we propose a dominance rule to detect whether a batch is dominated by two or more smaller batches; a dominated batch can then be decomposed into multiple batches together with the necessary maintenance periods. Therefore, we strengthen the set-partitioning model by eliminating all the dominated batches and forbidding to generate such batches in the pricing problem.

In the BP algorithm, to generate columns with negative reduced costs, we reduce the pricing problem to a variant of the shortest path problem with resource constraints - a dominance rule is also used to speed up the pricing procedure. The branching scheme is a two-level scheme where we first branch on the number of batches and then on the sequences of the jobs. The branching rule is chosen in order to preserve the structure of the pricing problem.

To the best of our knowledge, our algorithm is the first column generation-based exact method for scheduling problems with deterioration effect. We also consider the extension of our problem to the online case, where jobs arrive in real time, and we describe a maintenance policy to improve practical decisions.

The paper is organized as follows. The next section reviews the related literature. The descriptions of the problem and two mathematical formulations, including a set-partitioning model, are given in Section 3; the section also describes the dominance rule used to reduce the number of variables of the set-partitioning model. Section 4 describes the BP algorithm. Section 5 reports the computational results. Section 6 investigates the extension to the online case. Concluding remarks are given in Section 7.