## 3 Description of the MRP Mathematical Model with Remanufacturing Parts
### Maintenance Plan  
According to optimal dates founded as \(T_{PR}^{*}\) and \(X^{*}\), we determine the optimal disassembly-assembly plan. This plan allows determination of the system's solicitation rate, which has an influence on the effectiveness of the system; likewise, it leads to failures. The objective is to find the optimal number of preventive maintenance \(N^{*}\). We assume that, after each corrective maintenance, the system is in the state just before breakdown. In addition, preventive and corrective actions have negligible times. Between two preventive maintenance actions, the failure rate depends on the disassembly or assembly processes. Each disassembly or assembly process solicits the machine at a solicitation rate, which depends on the process type and relative to the highest level of system solicitation. We propose in this paper an example of solicitation levels \(u\) for disassembly types in Table 1 and assembly types in Table 2. We take the example of five types of components which can be assembled together or disassembled from each other; each disassembly or assembly operation of a specific type gives a machine solicitation rate. We assume that the maximum level of machine solicitation is \(U=2\). The following analytical expression of the total maintenance cost includes preventive and corrective maintenance actions:  
\[C_{M}=C_{P}\times(N-1)+C_{c}\times\varphi(A,N) \tag{13}\] \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Disassembled at** & **Type 1** & **Type 2** & **Type 3** & **Type 4** & **Type 5** \\ \hline Type 1 & - & 0.5 & 1 & 0.5 & 1 \\ Type 2 & - & - & 0.5 & 0.5 & 0.5 \\ Type 3 & - & - & - & 0.5 & 0.5 \\ Type 4 & - & - & - & - & 1 \\ Type 5 & - & - & - & - & - \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Example of the different levels of machine solicitation related to the type of disassembly for the proposed system. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Assembled at** & **Type 1** & **Type 2** & **Type 3** & **Type 4** & **Type 5** \\ \hline Type 1 & - & 0.5 & 0.5 & 1 & 0.5 \\ Type 2 & - & - & 0.5 & 1 & 1 \\ Type 3 & - & - & - & 0.5 & 1.5 \\ Type 4 & - & - & - & - & 2 \\ Type 5 & - & - & - & - & - \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Example of the different levels of machine solicitation related to the type of assembly for the proposed system. where \(N\in\{1,\,2,\,3,\,\ldots\}\) and \(\varphi(A,N)\) which represents the expected function of failures number during the horizon \(H\cdot\Delta t\) depending on the disassembly-assembly plan. The failure rate \(\lambda_{k}(t)\) is cumulative at the beginning of each interval \(k\) and is relative to the level of machine solicitation \(u(k)\). \(u(k)\) is determined from the optimal disassembly-assembly plan. We give the failure rate in the interval \(k\) as follows:  
\[\lambda_{k}(t)=\lambda_{k-1}(\Delta t)+\Delta\lambda_{k}(t)\qquad\forall t\in [0,\Delta t] \tag{14}\]  
with:  
\[\Delta\lambda_{k}(t)=\frac{u(k)}{U}\lambda_{n}(t)\quad\text{with}\quad k=1\, \ldots\,\,H \tag{15}\]  
\[A=\{u(1),\,\ldots,u(k),\,\ldots,\,u(H)\},\,k\in\{1\,\ldots\,\,H\}\]  
where \(\lambda_{n}(t)\) is the failure rate for nominal conditions is equivalent to the failure rate of the maximum level of machine solicitation during all horizon \(H\cdot\Delta t\). Our work is based on [PERSON] et al. [33] where they established an optimal integrated maintenance-production policy under subcontractor constraint, considering the influence of the production rate variation on the manufacturing system degradation and consequently on the maintenance strategy adopted. Based on the work of [PERSON] et al. [33], the average failure number expressed as follows:  
\[