# A Network-Based Method for the EMU Train High-Level Maintenance Planning Problem
## 3 Methodology
### Mathematical Model
On the other hand, the expected operating mileage should be greater than or equal to the minimum value. To search the feasible solution effectively and to control the solution space, we have the assumed minimum value instead of the minimum value from HM requirements, i.e.,  
\[L_{e}^{min}\leq l_{e}\leq L_{e}^{upper}\quad\forall e\in E \tag{5}\]  
2 According to the proposed approach, any \(e\) can choose one and only one path on the time-state network, which is the uniqueness constraint of the path. Its formulation is listed as follows:  
\[\sum_{r\in R}x_{e}^{r}=1\ \ \forall e\in E \tag{6}\]  
3 The minimum number of train-sets in available state is variable according to travel demand, and it should be guaranteed. To describe this requirement, a set of constraints established as follows:  
\[\sum_{r\in R}\sum_{e\in E}x_{e}^{r}\delta_{r}^{a_{1}k}\alpha_{e}\geq b_{1}(a_{ 1,k})\ \ \forall a_{1,k}\in A_{1},\ k=1,2,\ldots,T \tag{7}\]  
4 The number of train-sets in any level of maintenance state should not exceed the capacity of simultaneously maintenance service. Similarly, a set of constraints established as follows:  
\[\sum_{r\in R}\sum_{e\in E}x_{e}^{r}\delta_{r}^{a_{ik}}\alpha_{e}\leq b_{i}\ \ \forall a_{ik}\in A_{i},\ i=2,3,4,\ k=1,2,\ldots,T \tag{8}\]  
5 Only a certain number of train-sets can be accepted in the workshop over several days, which is due to the limited maintenance resources in some workstations. An associated parameter \(\theta_{r}^{a}\) is introduced for conveniently describing this relationship. The definition of \(\theta_{r}^{a}\) is as follows:  
\[\theta_{r}^{a}= \begin{cases}1&\text{Arc $a$ in the path $r$, $r\in R$, $a\in A$}\\ 0&\text{otherwise}\end{cases} \]The arc \(a\) in \(\theta_{r}^{a}\) can be regarded as the first few days of the maintenance process, and the duration is equal to the interval for the receiving train-set. The associated parameter \(\theta_{r}^{a}\) can be calculated beforehand, and its meaning can be visually represented, as shown in Figure 5.  
\(\mathrm{U}_{e}^{i}\) (\(i=2\), \(3\), \(4\)) is the duration of the minimum interval for receiving the another train-set after train-set \(e\) be received in the \(i\)-th state, e.g., \(\mathrm{U}_{15}^{2}=2\) describes a train-set (No. 15) entering into the workshop for undergoing third-level maintenance, and the next train-set could be sent to the workshop at two or more days later. Therefore, the constraints of receiving train-sets in the workshop can be presented as follows:  
\[\sum_{r\in R}\sum_{e\in E}x_{e}^{r}\theta_{r}^{a_{k}}\leq N_{i}\ \ \ \forall a_{ik}\in A_{i},\ i=2,3,4,\ k=1,2,\ldots,T \tag{9}\]  
where \(N_{i}\) is the number of acceptable train-sets at the same time in \(i\)-th state. Although each train-set can choose one path in the path set \(R\), the choice is not discretionary. The train-set can choose the path that involves corresponding information of maintenance level and maintenance service time. To this end, we introduced a parameter \(\lambda_{e}^{r}\), which describes a consistent one-to-one match between each train-set and each path, as follows:  
\[\lambda_{e}^{r}= \begin{cases}1&\text{path $r$ could be chosen by $e$, $e$ $\in E,r\in R$}\\ 0&\text{otherwise}\end{cases} \]  
The value of \(\lambda_{e}^{r}\) should be calculated beforehand. Then, the valid inequality constraints are established:  
\[x_{e}^{r}\leq\lambda_{e}^{r}\ \ \ \forall e\in E,\ r\in R \tag{10}\]  
From the above analysis, the model is summarized as follows. Model I:  
\[\text{min }Z= \sum_{e\in E}L_{e}^{upper}-l_{e}\]  
\(s.t.\) constraints (4)-constraints (10) and:  
\[x_{e}^{r}\ \in\ \{0,\ 1\}\ \ \ \forall e\in E,\ r\in R \tag{11}\]