# Evaluation of the Robustness for Integrated Production Scheduling and Maintenance Planning Problem
## 1 Introduction  
In literature, job shop scheduling problems are usually investigated under the assumption that machines are always available. However, machine failures could occur at any time in the actual production process and are difficult to prevent. Although preventive maintenance (PM) is an efficient approach to lower the risk of machine failure, it also takes up the production time of the machine. Lack of, or excessive maintenance, will reduce the performance of the production. Even if PM activities are scheduled, machine failures are also difficult to avoid altogether. Conversely, if no PM activities are planned, once unexpected machine failure occurs, any production process on the machine will be rejected during its downtime. To solve this problem, an unplanned maintenance activity is required to restore the machine to its original state, which always causes longer downtime of the machine, resulting in degraded production system stability and performance. Therefore, the impact of stochastic failures and PM activities should be considered when the job shop schedule is formulated. However, it remains an open challenge. To keep up with the better scheduling performance of the job shop with stochastic failures, some research uses the expected schedule performance as an optimization criterion to select a schedule [1; 2]. This method can guarantee satisfactory practical performance on average, but the deviation is neglected [3]. Robust scheduling is always used to deal with uncertainties in the field of job shop scheduling. A representative survey by [PERSON] and [PERSON] [4] discusses robust scheduling with uncertainties. The prerequisite of robust scheduling is to define the robustness of the schedule, specify the robustness criterion, and define its measurement method. In general, scheduling robustness is the ability of a schedule to maintain state and/or performance in an uncertain environment [5; 6; 7; 8; 9], but they have different emphases and even differ significantly. [PERSON] and [PERSON] [6] define a robustness schedule as a schedule that is insensitive to stochastic disturbances givenan assumed control policy. They proposed to measure robustness by the expectation of deviation between the actual scheduling makespan and the initial scheduling makespan and introduced a measurement algorithm. [PERSON] and [PERSON] [5] classify robustness measures into two categories: quality robustness and solution robustness. Quality robustness is used to indicate the deterioration of the scheduling performance under uncertainty, whereas solution robustness, also known as stability, refers to the insensitivity of the starting times of operations to uncertainty. Robustness evaluation is another critical characteristic of robust scheduling. However, it is difficult to directly assess the impact of a random failure on the baseline schedule, due to the fact that the occurrence time of failure is unknown. One approach for evaluating the effects of uncertainties is to simulate a large number of uncertainty scenarios [10; 11; 12; 13] at the cost of intensive computation [14]. To tackle this problem, slack-time-based surrogate measures (SMs) have been employed to approximate the robustness of a schedule [15]. Two typical slacks, total slack and free slack, have been widely investigated for robust scheduling [15; 16]. A surrogate measure base on the mean of total slack was originally stated by [6], but ignored the available uncertainty information. [PERSON] and [PERSON] [17] present a surrogate robustness measure based on the total amount of free slack for all activities. [PERSON] and [PERSON] [18] extend the robustness measure in [17] and also discuss the deficiencies of those measures. However, slack-based measures for the robustness of a schedule only focus on the structure of the schedule and ignore the available uncertain information, which makes it insensitive to the settings changes of uncertainty. To tackle this problem.