## 3 Mathematical Model
### Maintenance Cost  
The concerned urban distribution center system (UDC) is subject to random failures, repair actions and maintenance activities. The strategy adopted for maintenance here is the preventive maintenance policy with minimal repair at failure. Therefore, a perfect preventive maintenance action (overhaul) is performed \(P\) times periodically at times \(\Delta t\times T\), restores the degradation impact and makes the system as good as new (AGAN) at each \(j\). \(T\) (\(j=1,2,\ldots,P\)). When the site fails between successive perfect maintenance, a minimal repair is performed. It restores the system to the same state before failure (As Bad As Old). We assume that preventive maintenance and minimal repair durations are negligible. The correlation of the distribution plan and the system degradation is presented with an increase of the failure rate as a function of time, transportation and deliveries [15; 16; 17]. Each maintenance action (preventive and/or corrective actions) consumes a part of the capacity for the delivery rate of the concerned UDC. Therefore, the preventive maintenance and corrective maintenance consumes respectively \(L_{p}=\text{a}\ \times\ C_{\text{max}}\) and \(L_{r}=\text{b}\ \times\ C_{\text{max}}\)[18]. The idea is to develop an optimal maintenance strategy that considers the UDC's degradation according to the distribution plan. It consists of determining the optimal number \(P^{\star}\) of preventive maintenance actions during the finite distribution horizon. In order to know the best time intervals when a preventive maintenance action must be carried out, it is important to minimize the total cost of maintenance. The analytical expression of the total maintenance cost for the concerned UDC is expressed as follows:  
\[C_{M}(x,P)=M_{p}\times(P-1)+M_{c}\times\zeta_{M}(x,P) \tag{9}\]  
where \(\zeta_{M}(x,P)\) corresponds to the average number of failures that occur during the horizon \(H\), considering the distribution rate in each period. In this part, we are inspired by the work of [PERSON] et al. [12], where the failure rate in each interval \(k\) is formulated as follows:  
\[\lambda_{t}(k)=\lambda_{t-1}(\Delta t)+\frac{x_{t}}{C_{\text{max}}}\lambda_{n }(k)\ \ \forall k\in[0,\Delta t] \tag{10}\]  
with \(\lambda_{0}(k=0)=\lambda_{0}\), and \(\Delta\lambda_{t}(k)=\frac{x_{t}}{C_{\text{max}}}\lambda_{n}(k)\) is the nominal failure rate corresponding to the failure of the concerned UDC with its maximal delivery rate during the distribution horizon. The average number of failures over the horizon \(H\times\Delta t\) using the failure rate can be presented as follows:  
\[ \begin{split}\resizebox{100.0 pt}{}{\includegraphics[scale=100.0 pt]{fig/L100.pdf}}\end{split}  \tag{11}\]  
Therefore, the total cost of maintenance strategy is expressed as follows:  
\[ \begin{split} C_{M}(x,P)=M_{p}\times(P-1)+M_{c}\times\sum\limits_{ j=0}^{P-1}\int\limits_{\begin{subarray}{c}m((j+1)\times\frac{T}{2})+1\end{subarray}}^{ \begin{subarray}{c}m((j+1)\times\frac{T}{2})+1\end{subarray}}\int\limits_{ \begin{subarray}{c}\Delta n(k)dk+\frac{(j+1)\times L-In((j+1)\times\frac{T}{2}) \times\Delta t}{0}\end{subarray}}^{\begin{subarray}{c}(j+1)\times\frac{T}{2} )+1\end{subarray}}\lambda_{m}(k)dk+\int\limits_{0}^{(j+1)\times\frac{T}{2})+1 }\lambda_{m((j+1)\times\frac{T}{2})+1}(k)dk\\ +\int\limits_{(j+1)\times T}^{(ln((j+1)\times\frac{T}{2})+1)\times \Delta t}\left(\frac{\left(ln\left((j+1)\times\frac{T}{2})+1\right)\right)}{C_{ \max}}\times\lambda_{n}(k)dk\right]\end{split}  \tag{12}\]  
As discussed previously, the maintenance actions (corrective and/or preventive maintenance) will affect the maximum available capacity for the delivery rate of the UDC. Let \(P\) the number of preventive maintenance actions where \(P\) is equal to \(H/T^{\star}\) with \(T^{\star}=k\times\Delta t\).