## 4 Numerical Example
\(D_{n2}<D_{n1}<D_{n,\alpha}\); therefore, it is believed that the distribution conforms to the hypothesis and the improved average rank method is more effective in fitting. There is only one fault in the automatic tool changer and servo subsystem, so it is assumed that they obey negative exponential distribution. The final results of parameter estimation for the other machining center subsystems are shown in Table 5. The reliability functions of these subsystems are shown in Figure 6.  
The shape parameters of the electrical subsystem and cooling subsystem are less than 1, so only the minimum maintenance is carried out in the event of a failure. This strategy is also applied to the automatic tool changer and servo subsystem. These four subsystems are not included in the following text when calculating costs. A total of six subsystems will participate in the development of the maintenance strategy mentioned above, and their cost parameters are shown in Table 6. Based on experience [30; 31], the age regression factor and the failure rate growth factor are \(a_{i}=i/(5i+7)\), \(b_{i}=(12i+1)/(11i+1)\). \begin{table}
\begin{tabular}{c c c} \hline \hline
**Subsystem** & \(\hat{\eta}\) & \(\hat{\beta}\) \\
**Symbol** & \(\hat{\eta}\) & \(\hat{\beta}\) \\ \hline T & 3826.095 & 1.875 \\ F & 2168.142 & 2.6717 \\ H & 2161.748 & 1.7714 \\ L & 2816.645 & 2.6985 \\ NC & 2672.429 & 2.9124 \\ E & 271,521.5 & 0.3857 \\ C & 6254.379 & 0.9874 \\ A & 5647.764 & 1.6973 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Parameter estimation of the reliability model for the machining center subsystems. According to Section 3.1, the results of the single subsystem maintenance strategy arrangement are shown in Table 7. The optimal maintenance cycle for the hydraulic subsystem, auxiliary subsystem, feed subsystem, and lubrication subsystem is 4, and the optimal maintenance cycle for the numerical control subsystem and main transmission subsystem is 5. Next, the opportunistic maintenance of the entire machining center is considered. The finite operating time is selected as 6000 h. Using the particleswarm function in MATLAB \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Cost** & **H** & **A** & **F** & **NC** & **T** & **L** \\
**Parameters** & & & & & & \\ \hline \(C_{sm}\) & 550 & 850 & 405 & 450 & 1425 & 180 \\ \(C_{f}\) & 380 & 380 & 380 & 380 & 380 & 380 \\ \(C_{v}\) & 30 & 25 & 55 & 65 & 35 & 30 \\ \(C_{r}\) & 10,500 & 6375 & 3050 & 6800 & 15,500 & 3500 \\ \(t_{cm}\) & 3.12 & 3.85 & 6.94 & 2.79 & 2.34 & 3 \\ \(t_{r}\) & 6 & 8 & 7 & 4 & 7 & 5 \\ \(t_{f}\) & 1.267 & 2.533 & 1.52 & 0.95 & 1.9 & 1.9 \\ \(t_{v}\) & 0.1 & 0.167 & 0.22 & 0.1625 & 0.175 & 0.15 \\ \(c_{d}\) & 200 & 200 & 200 & 200 & 200 & 200 \\ \(c_{risk}\) & 8170 & 4010 & 11,044 & 7126 & 12,404 & 5926 \\ \(V_{k}\) & 15,000 & 13,500 & 16,000 & 15,500 & 15,600 & 14,000 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Cost parameters of machining center subsystems. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Subsystem** & \(N\) & \(R_{h}\) & \(T_{1}\) & \(T_{2}\) & \(T_{3}\) & \(T_{4}\) & \(T_{5}\) \\ \hline H & 4 & 0.609 & 1455.009 & 1236.057 & 1049.594 & 896.7964 & \\ A & 4 & 0.497 & 4574.119 & 3890.057 & 3314.518 & 2843.579 & \\ F & 4 & 0.921 & 851.3723 & 726.153 & 607.5688 & 505.635 & \\ NC & 5 & 0.854 & 1428.052 & 1218.195 & 1019.239 & 848.0675 & 707.4946 \\ T & 5 & 0.72 & 2113.055 & 1793.646 & 1517.095 & 1289.705 & 1105.025 \\ L & 4 & 0.855 & 1417.086 & 1208.969 & 1011.509 & 841.5174 & \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Optimal maintenance arrangement for subsystems. Figure 6: Reliability functions of machining center subsystems. R2016a to solve the model, the minimum cost is obtained as \(C=100899\), which corresponds to the opportunistic maintenance threshold \(\Delta R=(0.03,0.037,0.038,0.032,0.033,0.057)\).