## 3 Two-Echelon Two-Indenture System Modelling
### Mathematical Model of Joint Optimization  
Based on the above service logistics model, a two-echelon two-indenture nonlinear joint optimization model of fleet system is constructed by introducing multiple failure modes and discard decision.  
\[\begin{array}{ccccc}\min&\sum&f{c_{i}}\cdot{s_{i,j}}+\sum&\sum&\sum&vc_{i,j,d} \cdot{X_{i,j,d}}+\sum&\sum&rc_{r,j}\cdot{Y_{r,j}}\\ &s.t.&A\left({s_{i,j}}\right)\geq A_{min},i{=}&0,1,\cdots,j=0,1,\cdots\\ &X_{0,j,repair}+X_{0,j,move}{=}&\lambda_{0,j},j=1,2,\cdots\\ &X_{0,0,repair}+X_{0,0,discard}{=}&\sum\limits_{j=1}^{I}X_{0,j,move}\\ &X_{i,j,repair}+X_{i,j,move}{=}&\lambda_{i,j}\\ &\sum\limits_{j=1}^{I}X_{i,j,move}{=}&\sum\limits_{d\in D}X_{i,0,d}\\ &X_{0,j,repair}{=}&\sum\limits_{i=1}^{I}X_{i,j,repair},i\in\Gamma_{0}\\ &X_{i,j,repair}{\leq}&Y_{r,j}\cdot\lambda_{i,j},i\subset\Omega_{r}\\ &X_{i,j,d}\in&N,Y_{r,j}\in\{0,1\}\end{array} \tag{25}\]  
The objective function consists of three parts: spare parts cost, maintenance decision cost, and maintenance resource cost. The constraints include two parts: availability constraints related to spare parts and LORA related to failure components.  
If multiple failure modes are included in the LORA model, with the increase of component types, more detailed failure mode information is needed as the input of the model. The scale of model solution would be more and more difficult to the solute.  
In the actual project, the serious failure mode usually requires more maintenance resources. In other words, the move ratio R is: \(R=n/N\). Where \(n\) is the number of moves and \(N\) is the number of failed components.  
The move ratio parameter R reflects the maintenance ability r of the Base locations, and its numerical relationship is:  
\[r=1-R=1-n/N=X_{i,j,move}/\lambda_{i,j},j=1,2,\cdots \tag{26}\]  
Discard decision \(X_{i,0,discard}\) means the system needs to order spare parts. Spare parts order lead time is \(L_{i}\), the probability of components being discarded is \(r_{i,0}\), the number of repairs at Depot is:  
\[E[N_{i,0}]=m_{i,0}[((1-r_{i,0})T_{i,0}+r_{i,0}L_{i})],i=0,1,\cdots I \tag{27}\]  
where \(r_{i,0}=X_{i,0,d}/\lambda_{i,j}\).  
To summarize, there would be \(r\) of failure components repaired at Base location; \(r_{i,0}\) of failed components discarded due to the high maintenance cost; and \(1-r-r_{i,0}\) of failed components repaired at Depot location.  
Some authors focus on how to assess system availability with the constraint of resources, especially spare parts and equipment, by analytical method or numerical approach [31; 32; 33]. The practical motivation of this research is optimizing stock allocation targeting maximum availability with the constraint of preassigned maintenance policy. The essential motivation of this technique is on the hypothesis that availability can be dividing into two independent parts, maintenance availability and supply availability, respectively. So, most of the research express the effect by EBO or MDT (mean delay time). However, different maintenance capability factors indicate that maintenance decision-making changes dynamically. The approach of this aircle tries to define the logistic support to ensure a specified availability. In this case, an availability-cost function is created in order to evaluate service logistics costs associated with a required service level.  
The better the maintenance capability of the Base level, the better the system availability. System availability (A) is proportional to maintenance capability (\(r\)). Due to the maintenance resources allocation, the service logistics cost (C) is negatively correlated with \(r\). In summary, A is negatively correlated with C. As two important indicators of the service logistics system, A and C can be connected with each other through \(r\) of Base location.