# Evaluation of the Robustness for Integrated Production Scheduling and Maintenance Planning Problem
## 4 Robustness Measure for JSP with Preventive Maintenance
### Analytical Approximation Calculation  
Since this paper assumes that the job continues to be processed after the machine is repaired, the machine breakdown essentially increases the residence time of the job on a machine. At the operative level, it can be regarded as an increase in operation processing time. In this paper, machine breakdowns are mapped to the operation level. Then, the breakdown probability of each operation can be computed as  
\[pr_{i,j}=1-e^{-\left[\left(S_{i,j}/\theta_{i}\right)^{\beta_{j}}-\left(C_{(i, j)}/\theta_{i}\right)^{\beta_{j}}\;\right]} \tag{7}\]where \(S_{i,j}\) denotes the age of the machine \(M_{j}\) when operation \(O_{i,j}\) starts processing, and \(C_{i,j}\) denotes the age of the machine \(M_{j}\) at the end of the operation \(O_{i,j}\). The expected corrective maintenance time during operation \(O_{i,j}\) is given by Equation (8). \[\overline{tc}_{i,j}=tc_{j}\bigg{[}e^{-[(S_{i,j}/\theta_{j})^{\beta_{j}}-(C_{(i, j)}/\theta_{j})^{\beta_{j}}}\mid\bigg{]} \tag{8}\]  
\(\overline{tc}_{i,j}\) can be regarded as the expected increase of the processing time of operation \(O_{i,j}\) by CM. Machine breakdowns not only affect interrupted operations, but also propagate along with constraints between operations. The influence of failures may be weakened by the idle time between operations or enhanced by superimposed with other failures. Figure 4 shows the influence of RMBs on the single operation and its subsequent operation time arrangement. The expected time spent for the CM activity on the operation \(O_{i,j}\) is illustrated as a red box. As shown in Figure 4a, the completion time of operation \(O_{i,j}\) will be delayed when its processing time increases \(\overline{tc}_{i,j}\), defined by the delayed completion time \(C^{\prime}_{i,j}=C_{i,j}+\overline{tc}_{i,j}\). Once the completion time of operation \(O_{i,j}\) is delayed, the completion time of subsequent operations may be affected. This depends on the idle time between operations, as shown in Figure 4b.The completion time of operation \(O_{i,j^{\prime}}\) will be delayed \(\max\Big{\{}\overline{tc}_{i,j}-Idle^{i,j^{\prime}}_{i,j},0\Big{\}}\). Due to \(\overline{tc}_{i,j}<Idle^{i,j^{\prime}}_{i,j}\), the completion time of operation \(O_{s,j}\) will not be delayed. However, if \(tc_{j}>Idle^{s,j}_{i,j}\), the completion time of operation \(O_{s,j}\) will be delayed. Due to \(tc_{j}>\overline{tc}_{i,j}\), the completion time of partial operations that should be affected may not be changed, which will lead to inaccurate measurement results. Therefore, the expected completion time of operation \(O_{s,j}\) should be expressed as  
\[C^{\prime}_{s,j}=C_{s,j}+\max\Big{\{}pr_{i,j}\Big{(}tc_{j}-Idle^{s,j}_{i,j} \Big{)},0\Big{\}} \tag{9}\]  
Similarly, the expected completion time of operation \(O_{i,j^{\prime}}\) is \(C^{\prime}_{i,j^{\prime}}=C_{s,j}+\max\Big{(}pr_{i,j}\Big{(}tc_{j}-Idle^{i,j^ {\prime}}_{i,j}\Big{)}\Big{)}\), as shown in Figure 4c. The effort of computing the expected completion time of operation with the approach presented in Section 3 is not suitable for the practical utility. Therefore, we propose an approximate method for the expected completion time of operation based on failure propagation at the operative level. Figure 5 shows an example of three jobs and two machines. From the perspective of operation \(O_{3,2}\), \(\{O_{1,1},O_{3,1},O_{2,1}\}\) is the associated operation set of operation \(O_{3,2}\). For the operations in the associated operation set, if breakdowns occur in multiple operations, which may occur due to idle time between operations, or enhanced by superposition with other breakdowns, it makes the calculation of operation delay time very complicated. As shown in Figure 5, two chains can affect the completion time of operation \(O_{3,2}\). Figure 4: The influence of RMBs on the single operation and its subsequent operation time arrangement. (**a**) Operation \(O_{i,j}\) increases expected repair time. (**b**) The effect of the start time of \(O_{i,j}\).