# Evaluation of the Robustness for Integrated Production Scheduling and Maintenance Planning Problem
## 5 Computational Results
### Comparison between the Integrated PM Schedule and Non-PM Schedule  
The performance difference between the integrated PM schedule and the non-PM schedule is demonstrated. Three evaluation criteria are defined to investigate the difference between the two schedules \(\delta_{p}\) and \(\delta_{pm}\) in schedule performance. \(SRIR\big{(}\delta_{p},\delta_{pm}\big{)}\) indicates the solution robustness improvement ratio of \(\delta_{pm}\) to \(\delta_{p}\), as shown in Equation (18). If \(SRIR\big{(}\delta_{p},\delta_{pm}\big{)}\) is greater than zero, it indicates that the solution robustness of \(\delta_{pm}\) is better than \(\delta_{p}\), and vice versa. \(PRIR(\delta_{p},\delta_{pm})\) is defined to investigate the performance improvement ratio of \(\delta_{pm}\) to \(\delta_{ma}\), as shown in Equation (19). If \(PIR\big{(}\delta_{p},\delta_{pm}\big{)}\) is greater than zero, it indicates that \(\delta_{pm}\) has advantages to \(\delta_{p}\) in scheduling performance, and vice versa. \(PRIR\big{(}\delta_{p},\delta_{pm}\big{)}\) indicates the quality robustness improvement ratio of \(\delta_{pm}\) to \(\delta_{p}\). If \(PIR\big{(}\delta_{p},\delta_{pm}\big{)}\) is greater than zero, it indicates that the \(\delta_{pm}\) has advantages to \(\delta_{p}\) in quality robustness. \[SRIR\big{(}\delta_{p},\delta_{pm}\big{)}=\frac{SR\ \big{(}\delta_{p}\big{)}-SR\ \big{(}\delta_{pm}\big{)}}{SR\ \big{(}\delta_{p}\big{)}}\times 100\% \tag{18}\]  
\[PIR\big{(}\delta_{p},\delta_{pm}\big{)}=\frac{E\big{[}C_{max}(\delta_{p}) \big{]}-E\big{[}C_{max}(\delta_{pm})\big{]}}{E\big{[}C_{max}(\delta_{p}) \big{]}}\times 100\% \tag{19}\]  
\[QRIR\big{(}\delta_{p},\delta_{pm}\big{)}=\frac{QR\ \big{(}\delta_{p}\big{)}-QR \ \big{(}\delta_{pm}\big{)}}{QR\ \big{(}\delta_{p}\big{)}}\times 100\% \tag{20}\]  
In Section 3.1, we pointed out that PM activities can increase scheduling stability and may also reduce scheduling performance. Excellent PM insertion strategies can enhance scheduling stability and performance at the same time. As shown in Table 5, most \(SRIR\) values are greater than zero, which indicates that the scheduling stability improved after PM activities. However, \(SRIR\) values varied greatly under different fault settings. In the  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \multirow{2}{*}{_tc,\(\theta\)_} & \multicolumn{2}{c}{\(RM1\)} & \multicolumn{2}{c}{\(RM2\)} & \multicolumn{2}{c}{\(RM3\)} & \multicolumn{2}{c}{\(QR_{AC}\)} \\ \cline{2-9}  & \(R^{2}\) & \(Sig.\) & \(R^{2}\) & \(Sig.\) & \(R^{2}\) & \(Sig.\) & \(R^{2}\) & \(Sig.\) \\ \hline
20,0.5 & 0.747 & \(<\)0.001 & 0.655 & \(<\)0.001 & 0.719 & \(<\  
case of (20,1.0), (20,1.5), (40,1.0), (40,1.5), and (60,1.5), the values of \(PIR\) are less than 0, which indicates that scheduling performance after inserting PM activities is worse than that of initial schedule under the condition of low failure probability or little CM time. For \(QRIR\), the experimental data have the same characteristics as \(QRIR\) values. One possible reason for this case is that the QR value is part of SR and is more sensitive to changes in fault parameters. To show the above analysis more clearly, the experimental data are also shown in Figure 9. We have a few key findings. Figure 9a shows the quartile graph of SRIR. When \(\theta\) is the same, the SRIR values increase with the increase of CM time. This indicates that, when the machine failure probability is constant, SRIR values are positively correlated with CM time. It also shows that it is more effective to insert PM activities when the CM time is too long. At the same time, when the CM time is fixed, the SRIR values decrease gradually with the reduction of fault probability, indicating that the improvement effect on the stability of the scheduling scheme is limited when the fault probability is small. Figure 9b shows the quartile graph of PIR. Figure 9b shows the same trend as Figure 9a, but some PIR values are less than 0 when the fault probability is low, and the CM time is short.