## 3 Two-Echelon Two-Indenture System Modelling
### Mathematical Model of Spare Parts Stock  
Spare parts are components of a system unit that can be dismantled and replaced immediately following a failure. According to the basic METRIC model, Base location \begin{table}
\begin{tabular}{c c c} \hline \hline
**Variable Name** & & **Implication** \\ \hline \multirow{8}{*}{**Input Variables**} & \(i\) & Component number, LRU: \(i=0\); SRU: \(i=1\cdots 1\) \\  & \(h\) & LRU family number, \(h=1\cdots H\) \\  & \(j\) & Location number, Depot: \(j=0\); Base: \(j=1\cdots J\) \\  & \(k\) & Subsystem number, \(k=1\cdots N\) \\  & \(\lambda_{i,j}\) & Yearly demand means value of component \(i\) at location \(j\) \\  & \(q_{i}\) & Conditioned failure rate of a SRU on the parent LRU \\  & \(r_{i,j}\) & Repairing probability of component \(i\) at location \(j\) \\  & \(T_{i,j}\) & Repairing time of component \(i\) at location \(j\) \\  & \(O_{i,j}\) & Order and ship time of component \(i\) from the Base to the location \(j\) \\  & \(fc_{i}\) & Price of component \(i\) \\  & \(M_{i,j}\) & The number of components \(i\) in the system \\ \hline
**Intermediate Variables** & \(N_{i,j}\) & Quantity of component \(i\) in repair at location \(j\) \\ \hline \multirow{2}{*}{**Decision Variables**} & \(X_{i,j,d}\) & Maintenance policy d of component \(i\) at location \(j\) \\  & \(Y_{r,j}\) & Allocation of Maintenance resources \(r\) at location \(j\) \\  & \(s_{i,j}\) & Level of inventory of component \(i\) at location \(j\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: NOTATION. should refill supplies to satisfy the demand with new or fixed LRU, when an LRU is replaced on a subsystem. Two levels including Base and Depot are used to satisfy the demand of LRU, based on the inspection capability at Base level [29]. The aim of inspection is to verify the replacement and demand of SRU. Furthermore, the SRU demand at the Depot is always satisfied by repairing the failed SRU, while at the Base it can be satisfied only if the Base is certified to execute the repairing process, otherwise, the SRU demand is redirected to the Depot [30]. The total demand \(\lambda_{i,j}\) of an SRU is the sum of two contributions:  
\[\lambda_{i,0}=\sum_{j=1}^{I}\lambda_{i,j}\big{(}1-r_{i,j}\big{)}+\lambda_{0,0} q_{i},i=1,2,3\ldots \tag{7}\]  
The demand of SRU deriving from a failure of LRU, inspected at the Bases, not certified to execute the maintenance operations:  
\[\lambda_{i,j}=\lambda_{0,j}\cdot r_{0,j}\cdot q_{i},i=1,2,3\ldots,j=1,2,3\ldots \tag{8}\]  
The demand of SRU deriving from a failure of LRU inspected at the Depot:  
\[\lambda_{0,0}=\sum_{j=1}^{I}\lambda_{0,j}\big{(}1-r_{0,j}\big{)} \tag{9}\]  
[PERSON] points out that the mean variance ratio of the number of repairs is greater than 1, so the Poisson distribution assumption is not applicable [26]. We believe that in the discrete distribution function, the negative binomial distribution satisfies the feature that the variance is greater than the mean. It can be used to describe the discrete feature of the arrival of spare parts demand. At the Depot, the pipeline of any SRU is composed by the components in maintenance and the Expected Backorder. The expected value and the variance of the pipeline are related only to the repairing time and to the level of spare parts in stock:  
\[E[N_{i,0}]=\lambda_{i,0}T_{i,0}+EBO(s_{i,0}|\lambda_{i,0}T_{i,0}),i=1,2,\cdots \tag{10}\]  
For each SRU, any Base offers a pipeline consisting of components under maintenance, previously requested and arriving from the Depot, and the EBO:  
\[E\big{[}N_{i,j}\big{]}=\lambda_{i,j}\big{[}\big{(}(1-r_{i,j})O_{i,j}+r_{i,j}T _{i,j}\big{)}\big{]}+f_{i,j}EBO(s_{i,0}|\lambda_{i,0}T_{i,0}) \tag{11}\]  
\[Var\big{[}N_{i,j}\big{]} =\lambda_{i,j}\big{[}\big{(}(1-r_{i,j})O_{i,j}+r_{i,j}T_{i,j}\big{)} \big{]}+f_{i,j}\big{(}1-f_{i,j}\big{)}EBO(s_{i,0}|\lambda_{i,0}T_{i,0}) \tag{12}\] \[+f_{i,j}^{2}VBO(s_{i,0}|\lambda_{i,0}T_{i,0})\]  
where \(f_{i,j}\) is the fraction of \(i\)-th SRU at the Depot to supply the \(j\)-th Base.