# Evaluation of the Robustness for Integrated Production Scheduling and Maintenance Planning Problem
## 1 Introduction
[PERSON] and [PERSON] [15] take both the available uncertainty information and the location of float times into account, and two effective SMs that consider the schedule structure and uncertainty information for flexible JSP with Stochastic failures are proposed. [PERSON] and [PERSON] [19] build training datasets by extracting floating time, the machine age, and failure data from scheduling, and use CNN to predict scheduling robustness. The third category is to directly obtain the approximate robustness based on the analytical approximation method. To improve the accuracy of robustness evaluation, [PERSON] and [PERSON] [20] present two SMs by analyzing the disturbances of the critical operation set and non-critical operation set; the experiment demonstrates that the developed SMs outperform the other three traditional SMs. [PERSON] and [PERSON] [3] develop a scheduling risk evaluative measure based on the relation among the total slack time, available uncertain information, and makespan delay. An analytical approximation algorithm is presented for practical calculation. Two SMs for a single-machine schedule are proposed by [9], which can simultaneously evaluate the robustness and stability criteria. Preventive maintenance can enhance the stability of the production system by reducing the probability of machine failure. In recent years, a number of articles have addressed the integrated problem of preventive maintenance and scheduling jobs on the machine, when failures can occur stochastically [21; 22; 23; 24; 25; 26; 27]. As an extension of scheduling robustness evaluation, the integrated scheduling evaluation to be addressed here has a comparatively higher complexity status. [PERSON] and [PERSON] [28] consider four surrogate measures to evaluate the solution robustness of integrated production scheduling and PM in a single machine, which are based on buffer times, available failure information, and the sequence of preventive maintenance. [PERSON] and [PERSON] [25] extend the approach to evaluate the robustness of integrated production scheduling and maintenance in flow shops. The authors investigated the effect of the predecessor on its successor operation and proposed a mathematical model to approximate the expected delay of each operation. [PERSON] and [PERSON] [1] further extend this research to the case where the interrupted jobs must be repeated right after corrective maintenance (CM) is finished. They derive an exact formula for the expected makespan and propose two approximations for practical calculation. To the best of our knowledge, little research has been conducted on the robustness evaluation of integrated production scheduling and maintenance in job shops. Due to the widespread applications of preventive maintenance in the job shop manufacturing system, we investigate the SMs to provide predictive schedules based on preventive maintenance with satisfactory robustness and lower computational burden. To improve the accuracy of robustness estimation and reduce the computational burden, we will utilize the available failure information and preventive maintenance sequence, as well as the effects of perturbation propagation amongst jobs, to estimate two types of scheduling robustness, i.e., solution robustness and quality robustness. Our proposal is experimentally validated using twelve fault scenarios of different levels. A comparison with three well-known robustness measures shows the significant improvements yielded by our proposal. We also discussed the benefits of our proposal for integrated preventive maintenance scheduling. The remaining paper is organized as follows. The model for the stochastic fault situation, its parameters, and assumptions are introduced in the next section. In Section 3, we analyze the impact of PM activities and CM activities on the completion time of the operation.