Optimizing makespan and stability risks in job shop scheduling

Highlights

• Job shop scheduling under random machine breakdowns.

• Multi-objective optimization of makespan, makespan risk and stability risk.

• Design of two operation block-based buffering strategies.

• Development of a two-stage multi-objective predictive scheduling algorithm.

Abstract

In real-world manufacturing environments, the execution of a schedule often encounters uncertain events, which will bring the risks of performance deterioration and production system instability. This study addresses the optimization of risks both in performance and stability for the job shop scheduling under random machine breakdowns, in which three objectives: makespan, makespan risk and stability risk are considered at the same time. The buffering approach under the limited predictive makespan will be proposed and used to generate predictive schedules, which allows inserting additional idle time to control the risks. By utilizing the available information about the relationship between the risks and the random machine breakdowns, we have developed two kinds of operation-block based buffering strategies. In order to meet the decision makers with different risk preferences, a multi-objective predictive scheduling algorithm with the proposed buffering strategies is developed to generate a Pareto solution set. Extensive experimental results indicate that, compared with the existing methods, the proposed method can provide a better Pareto solution set in terms of both the diversity and the convergence.

Introduction

The classic job shop scheduling problem (JSP) has been well studied during the past few decades (Jain and Meeran, 1999; Potts and Strusevich, 2009; Pinedo, 2016), whose primary task is to generate predictive schedules by optimizing some particular performance measures (e.g. makespan, tardiness, etc.) with the assumption that the manufacturing environment is deterministic. A predictive schedule released to the shop floor holds two important functions (Mehta and Uzsoy, 1998): the first is to serve as a basis for planning internal/external activities such as shop resources allocation and material procurement, and the second is that commitments are made to ship orders to customers. However, the execution of a predictive schedule is frequently confronted with uncertain events (e.g. machine breakdowns, new jobs arrival, etc.) in the real-world manufacturing. If an uncertain event occurs, the predictive schedule may be delayed or even infeasible. As a result, the realized schedule that is actually executed on the shop floor may differ from the predictive one, which brings the risks of production system instability and performance deterioration. Thus, it is of great value to deal with the JSP under uncertainties and develop anti-risk predictive schedules.

Researches on the scheduling under uncertainties have attracted substantial attentions (Leon, et al., 1994; Al-Hinai and ElMekkawy, 2011; Xiong, et al., 2013; Yin, et al., 2016). To control the risks, the methodology of robust scheduling can be applied, which generates predictive schedules that are insensitive to uncertainties by optimizing certain measures of robustness. The robustness measures can be mainly classified into two categories: quality robustness and solution robustness. The former is often used to indicate the insensitivity of the performance measure (e.g. makespan) under uncertainties, while the latter refers to the stability of the schedule itself (e.g. starting/completion time) under uncertainties. Obviously, quality robustness can be used to optimize the risk in the schedule performance, while solution robustness can improve the risk in the stability of a production system. Most of the related robust JSP researches often consider either the quality robustness (Wu, et al., 1999; Xiong, et al., 2013) or the solution robustness (Al-Hinai and ElMekkawy, 2011, Nouiri, et al., 2017) with the performance measure. However, Van de Vonder, et al. (2005) has verified that there is a trade-off between the quality robustness and the solution robustness, and a composite robustness measure has been presented in Demeulemeester and Herroelen (2011) for the robust project scheduling. Since the risks in both the schedule performance and the stability are very crucial to the actual production, it may be better to consider both of them at the same time in the optimization of the JSP under uncertainties.

According to whether or not to insert additional idle time, robust scheduling can be divided into the buffering and non-buffering approaches. In the non-buffering approach, the predictive schedule is either directly generated by the optimization algorithm (Xiong, et al., 2013) or the scheduling strategy (Deblaere, et al., 2011) with the robustness measure as one of the optimization objectives. In this way, the predictive schedule with better robustness may be obtained, which will reduce the risks in the actual production. Since the schedules without additional idle time are generally compact, their robustness may be relatively limited, especially in the face of larger uncertainties. By comparison, the buffering approach can provide additional idle time for predictive schedules by using buffering strategies. However, the buffering strategies which determine the location and number of additional idle time are difficult to be designed. Currently, a commonly used strategy for production scheduling is to insert additional idle time for each operation by I_ij = E(D_ij) (Mehta, et al., 1998; Liu, et al., 2007; Wang, et al., 2015), where E(D_ij) is the expected duration of uncertain events during processing of an operation. This strategy can generate predictive schedules with very low risks, but it will lead to the predictive makespan too large. A predictive schedule with too large predictive makespan will significantly reduce the efficiency of internal and external activities, which will not be preferred by decision makers who generally want to take a balance between the performance and the risks. To satisfy the decision makers with different risk preferences, buffering strategies which can effectively allocate additional idle time under the limited predictive makespan need to be studied.

In this paper, we consider the JSP with a given number of jobs which may be affected by random machine breakdowns (RMBs), in which three objectives: makespan, makespan risk and stability risk are considered simultaneously. To address it, the buffering approach under the limited predictive makespan will be proposed to generate predictive schedules. For this purpose, utilizing the available information about the relationship between the risks and the RMBs, two operation-block based buffering strategies with the predictive makespan constraint will be developed, based on the analysis of the surrogate measures for the makespan risk and the stability risk. With the consideration of the makespan, the makespan risk and the stability risk simultaneously, the JSP under RMBs will hold a multi-objective nature. Generally, similar problems can be transferred to a single-objective problem by combing objectives, which, however, is difficult to meet the decision makers with different risk preferences. To facilitate decision makers to make the best trade-off according to the specific circumstance, a multi-objective predictive scheduling algorithm (MA) will be proposed to generate a Pareto solution set. The proposed method is expected to enjoy the following superiorities:

• Compared with the existing non-buffering approach, it can generate the predictive schedules with smaller risks but the same predictive makespan by allocating additional idle time effectively.

• Compared with the commonly used buffering strategy, it can provide a wider range of predictive schedules to meet decision makers with different risk preferences.

The rest of the paper is organized as follows. The next section is devoted to a brief review of the methods on scheduling under uncertainties. In Section 3, the problem formulation for the job shop scheduling subject to random machine breakdowns is presented. In Section 4, two operation block-based buffering strategies which facilitate to allocate additional idle time effectively under the limited predictive makespan are developed based on the surrogate measures for the makespan and stability risks. The details of the proposed multi-objective predictive scheduling algorithm will be presented in Section 5. The results of comparison experiments will be reported in Section 6, and then we conclude in Section 7.