## 3 Integrated Model
### Maintenance Situations  
Depending on whether maintenance actions, either failure-induced or preventive, are conducted at the production set-up points, we conclude the following three situations:  
1. If there is neither CM nor PM at time \(t_{i}\), it means that \(X(t_{i})<L_{p}\), neither PM nor CM is used at this time, so the degradation magnitude does not change either, \(X(t_{mt}^{i})=X(t_{i})\), \(t_{mt}^{i}=t_{i}^{\prime}\). Therefore, the magnitude of degradation at time \(t_{i+1}\) is \(X(t_{i+1})=X(t_{mt}^{i})+\Delta X(\Delta t_{s})\). At time \(t_{i+1}\), there are the following three sub-situations. (1) The system has neither CM nor PM: \(X(t_{i+1})=X(t_{mt}^{i})+\lambda(t_{i+1}^{b}-(t_{mt}^{i})^{b})<L_{p}\). The cost at time \(t_{i+1}\) is \(C_{i+1}=C_{i}+\triangle c_{i}=C_{i}+C_{m}+C_{s}\), \(t_{mt}^{i+1}=t_{i+1}^{\prime}\), no change in the number of PM as \(r_{i+1}=r_{i}\), no change in the number of CM as \(f_{i+1}=f_{i}\). (2) The system performs PM only: \(L_{p}\leq X(t_{i+1})=X(t_{mt}^{i})+\lambda(t_{i+1}^{b}-(t_{mt}^{i})^{b})<L_{f}\). The cost at time \(t_{i+1}\) is \(C_{i+1}=C_{i}+\Delta c_{i}=C_{i}+C_{m}+C_{s}+C_{p}\), \(t_{mt}^{i+1}=t_{i+1}^{\prime}+\triangle t_{pm}\), the number of PM changes to \(r_{i+1}=r_{i}+1\), no change in the number of CM as \(f_{i+1}=f_{i}\). (3) The system performs CM only: \(X(t_{i+1})=X(t_{mt}^{i})+\lambda(t_{i+1}^{b}-(t_{mt}^{i})^{b})\geq L_{f}\). The cost at time \(t_{i+1}\) is \(C_{i+1}=C_{i}+C_{m}+C_{s}+C_{f}+C_{f}+\triangle t_{s}\cdot\mathrm{C}_{\min}\)\(t_{mt}^{i+1}=t_{i+1}^{\prime}+\triangle t_{cm}\), the number of CM changes to \(f_{i+1}=f_{i}+1\), no change in the number of PM as \(r_{i+1}=r_{i}\). The minimal repair cost is \(\triangle t_{s}\cdot\mathrm{C}_{\min}\), which ensures that the system can continue to operate from \(t_{mt}^{i}\) to \(t_{i+1}\) even though \(L_{f}\) is reached in \((t_{mt}^{i},t_{i+1})\). 2. If PM is performed at time \(t_{i}\), it means that \(L_{p}\leq X(t_{i})<L_{f}\), the magnitude of degradation at time \(t_{mt}^{i}\) after maintenance becomes \(X(t_{mt}^{i})\), which \(X(t_{mt}^{i})=\gamma\), \(t_{mt}^{i}=t_{i}^{i}+\triangle t_{pm}\). Here, to ensure the effectiveness of PM, we assume that \(\gamma=\Phi(\Psi)\cdot Lp\), \(\Psi\sim N(0,1)\), \(-1<\psi<1\). The magnitude of degradation at time \(t_{i+1}\) is \(X(t_{i+1})=\gamma+\Delta X(\Delta t_{s})\). At time \(t_{i+1}\), there are the following three sub-situations. 1. The system has neither CM nor PM: \(X^{\prime}(t_{i+1})=\gamma+\lambda({t_{i+1}}^{b}-{(t_{mt}^{i})}^{b})<L_{p}\). The cost at time \(t_{i+1}\) is \(C_{i+1}=C_{i}+C_{m}+C_{s}\), \(t_{mt}^{i+1}=t_{i+1}^{\prime}\), no change in the number of PM as \(r_{i+1}=r_{i}\), no change in the number of CM as \(f_{i+1}=f_{i}\). 2. The system performs PM only: \(L_{p}\leq X^{\prime}(t_{i+1})=\gamma+\lambda({t_{i+1}}^{b}-{(t_{mt}^{i})}^{b})<L _{f}\). The cost at time \(t_{i+1}\) is \(C_{i+1}=C_{i}+C_{m}+C_{s}+C_{p}\), \(t_{mt}^{i+1}=t_{i+1}^{\prime}+\triangle t_{pm}\). The number of PM changes to \(r_{i+1}=r_{i}+1\), no change in the number of CM as \(f_{i+1}=f_{i}\). 3. The system performs CM: \(X^{\prime}(t_{i+1})=\gamma+\lambda({t_{i+1}}^{b}-{(t_{mt}^{i})}^{b})\geq L_{f}\). The cost at time \(t_{i+1}\) is \(C_{i+1}=C_{i}+C_{m}+C_{s}+C_{f}+C_{l}+\triangle t_{s}\cdot C_{\min}\), \(t_{mt}^{i+1}=t_{i+1}^{\prime}+\triangle t_{cm}\), the minimal repair cost is \(\triangle t_{s}\cdot C_{\min}\). The number of CM changes to \(f_{i+1}=f_{i}+1\), no change in the number of PM as \(r_{i+1}=r_{i}\). 3. If CM is performed at time \(t_{i}\), it means that \(X(t_{i})\geq L_{f}\). After maintenance, the magnitude of degradation at time \(t_{mt}^{i}\) becomes 0, which is \(X(t_{mt}^{i})=0\). The magnitude of degradation at time \(t_{i+1}\) is \(X(t_{i+1})=X(t_{mt}^{i})+\Delta X(\Delta t_{s})=\lambda({t_{i+1}}^{b}-{(t_{mt }^{i})}^{b})\). At time \(t_{i+1}\), there are the following three sub-situations. 1. The system has neither CM nor PM: \(X^{\prime}(t_{i+1})=\lambda({t_{i+1}}^{b}-{(t_{mt}^{i})}^{b})<L_{p}\).