## 3 Maintenance Strategy
### Subsystem Maintenance Planning  
The maintenance strategy proposed in this paper includes minimal maintenance, preventive maintenance, and preventive replacement. When a subsystem failure occurs, minimal maintenance is carried out, with the maintenance effect being as bad as before, i.e., the same level of reliability as before the maintenance. When the reliability of the subsystem reaches a predetermined limit, preventive maintenance is carried out. This article assumes that the subsystem state after preventive maintenance is between before maintenance and brand new. Due to the deterioration in the health status of the system after multiple maintenances, preventive replacement of the subsystem will be carried out after a certain number of preventive maintenances, with the effect of being repaired as new. This article uses the age regression factor and failure rate increasing factor to describe the change in failure rate after preventive maintenance of the subsystem. The failure rate change after one maintenance is:  
\[\lambda_{i+1}(t)=b_{i}\lambda_{i}(t+a_{i}T_{i}) \tag{15}\]  
where \(\lambda_{i}(t)\) is the failure rate function for the \(i\)th maintenance cycle, and \(a_{i}\) and \(b_{i}\) represent the age regression factor and failure rate increasing factor for the \(i\)th maintenance cycle, respectively. \(T_{i}\) is the length of the \(i\)th maintenance cycle, i.e., the maintenance interval. \(0<a_{i}<1\) influences the age after maintenance, which will return to a certain age before the maintenance, and \(b_{i}>1\) indicates the change in the steepness of the failure rate function after maintenance. This means the failure rate will increase faster and subsystems will fail more quickly. The failure rate evolution model is shown in Figure 1.  
When arranging maintenance plans for subsystems, the reliability threshold \(R_{h}\) and number of maintenance cycles \(N\) are used as decision variables, and the cost rate is minimized as the optimization objective. When the reliability of the subsystem reaches the reliability threshold, it is determined whether to perform preventive maintenance or replacement based on the ordinal number of maintenance cycles in which the subsystem is located. When the subsystem is in the \(N\)th maintenance cycle, preventive replacement will be carried out, and preventive maintenance will be carried out in the previous \(N-1\) cycle. The length of each maintenance cycle can be determined by the reliability threshold. Firstly, the relationship between the reliability function \(R(t)\) and the failure rate function \(\lambda(t)\) is:  
\[R(t)=\exp[-\int_{0}^{t}\lambda(t)dt] \tag{16}\]  
Therefore,  
\[R_{\rm h}=\exp[-\int_{0}^{T_{1}}\lambda_{1}(t)dt]=\cdot\cdot\cdot=\exp[-\int_ {0}^{T_{i}}\lambda_{i}(t)dt] \tag{17}\]  
\(\int_{0}^{T_{i}}\lambda_{i}(t)dt\) is the number of failures within \(T_{i}\) time, and it can also be represented by \(-\ln R_{h}\). Maintenance costs include minimum maintenance costs, preventive maintenance costs, preventive replacement costs, downtime loss costs, and failure risk costs. The minimum maintenance cost \(C_{m}\) is the product of the single minimum maintenance cost \(C_{sm}\) and the number of maintenances performed. \[C_{m}=C_{sm}\sum_{i=1}^{N}\int_{0}^{T_{i}}\lambda_{i}(t)dt=N\cdot C_{sm}\cdot (-\ln R_{h}) \tag{18}\]  
As the number of preventive maintenances increases, the cost of preventive maintenance will increase. This article assumes that there is a positive linear increment with increasing maintenance time. The cost of the \(i\)th preventive maintenance is:  
\[C_{pmi}=C_{f}+iC_{v} \tag{19}\]  
where \(C_{f}\) is the fixed cost for preventive maintenance, and \(C_{v}\) indicates the fluctuating cost. Figure 1: Failure rate evolution model (failure rate change after one maintenance). \(C_{r}\) represents the cost of preventive replacement for subsystems.