## 3 Maintenance Strategy
### Opportunistic Maintenance  
From the perspective of only arranging a maintenance plan for subsystems, this will lead to frequent downtime of the machining center. Considering the maintenance time of the subsystems comprehensively, using the maintenance downtime to carry out maintenance of other subsystems that are about to implement preventive maintenance will reduce the downtime of the machining center, improve availability, and reduce costs. Therefore, based on the contents in Section 3.1, this section considers opportunistic maintenance, introduces the threshold value of opportunistic maintenance \(\Delta R\) to judge whether each subsystem carries out preventive opportunistic maintenance or replacement in the maintenance window, establishes the preventive opportunistic maintenance model of machining centers based on the minimum total cost rate, and develops the overall maintenance strategy of machining centers. The schematic diagram of the opportunistic maintenance strategy is shown in Figure 2.  
According to Section 3.1, when the reliability of the subsystems reaches \(R_{h}\), preventive maintenance or replacement is carried out on the subsystems. At this time, a maintenance window will be generated, and an opportunistic maintenance threshold \(\Delta R\) will be used to determine whether to perform opportunistic maintenance or replacement on the other subsystems. Assuming a subsystem has a running time of \(t\), the corresponding reliability is \(R(t)\), and if \(R(t)>R_{h}+\Delta R\), the system will continue to operate without implementing preventive maintenance or replacement. If \(R_{h}<R(t)<R_{h}+\Delta R\), opportunistic maintenance or replacement is performed based on the number of times the subsystem has been maintained. When the system reliability \(R(t)\) reaches the preventive maintenance threshold \(R_{h}\), as described in the previous Section 3.1, preventive maintenance or replacement of the subsystem is carried out. Through the above content, the maintenance actions for the subsystems can be determined based on the reliability of the subsystem at any time. Next, we need to establish an opportunistic maintenance model to determine the maintenance cost rate of the entire machining center after adopting opportunistic mainte  
Figure 2: Opportunistic maintenance threshold \(\Delta R\). nance. By minimizing the cost rate, we can obtain appropriate opportunistic maintenance thresholds as the basis for maintenance decisions. Due to the different maintenance cycle lengths of subsystems, the maintenance cycle of the machining center can be very long. Therefore, when calculating the cost rate of the machine center, this article calculates the finite time cost rate within a suitable time interval. The total cost generated within the finite time of the machining center can be divided into four categories, mainly including direct maintenance costs, downtime loss costs, failure risk costs, and penalty costs caused by incomplete utilization of reliability. The direct maintenance costs of the machining center include the minimum maintenance costs caused by sudden failures, preventive maintenance, and replacement costs. Assuming the number of shutdowns caused by preventive maintenance and replacement in the finite operation time interval \([0,T]\) of subsystem \(k\) is \(m_{k}=n_{rk}N_{k}+q_{k}\), where \(n_{rk}\) is the number of preventive replacements for the subsystem \(k\), \(N_{k}\) is the number of maintenance cycles of the subsystem \(k\), and \(q_{k}\) is the number of preventive maintenances to subsystem \(k\) after the last preventive replacement. The operating time of subsystem \(k\) in each preventive maintenance cycle is \(T_{jk}(j=1,2,\cdots,m_{k})\).