# A Network-Based Method for the EMU Train High-Level Maintenance Planning Problem
## 3 Methodology
### Mathematical Model  
Based on the time-state network, a mathematical model is proposed that inspired by classic arc-path model. The decision variables are defined as follows:  
\[x_{e}^{r}= \begin{cases}1&\text{EMU }e\text{ chooses the path }r,\;e\in E,r\in R\\ 0&\text{otherwise}\end{cases} \]  
And define the associated parameter is as follows:  
\[\delta_{r}^{a}= \begin{cases}1&\text{Arc }a\text{ in the path }r,\;r\in R,a\in A\\ 0&\text{otherwise}\end{cases} \]  
Associated parameter \(\delta_{r}^{a}\) denoted by a matrix, describes the inclusion relation between arc and path. So, \(w_{ik}\) can be describes as follows:  
\[w_{ik}=\sum_{r\in R}\sum_{e\in E}x_{e}^{r}\delta_{r}^{a_{ik}}\;\;\;\forall a_{ ik}\in A_{i},\;i=1,2,3,4,\;k=1,2,\ldots,T \tag{1}\]  
Another set of parameters in the model are listed in Table 3.  
The objective function of the mathematical model is to minimize the costs of HM. We did not consider the optimization of the HM processes, and regarded it as a fixed value in this paper. So, the objective function could be described by decreasing HM frequency. The practical method of decreasing frequency is to increase the interval of two consecutive HMs, i.e., increasing the actual operating mileage between two consecutive HMs. Moreover, according to the maintenance requirements, the actual operating mileage since the last HM cannot exceed the upper limit of the floating range. Therefore, the objective function could be described by minimizing the D-value between \begin{table}
\begin{tabular}{c p{142.3 pt}} \hline \hline
**Parameters** & **Description** \\ \hline \(L_{e}^{upper}\) & The upper bound of the allowable cumulative mileage before HM according to the maintenance requirement. \\ \hline \(L_{e}^{lower}\) & The lower bound of the allowable cumulative mileage before HM according to the maintenance requirement. \\ \hline \(L_{e}^{min}\) & The assumed minimum value of cumulative mileage before HM for convenient calculation. \\ \hline \(L_{e}^{ideal}\) & The ideal cumulative mileage before HM according to the maintenance requirement. \\ \hline \(l_{e}\) & The state variable that indicates the expected operating mileage since the last HM when \(e\) was sent to workshop. \\ \hline \(\overline{l_{e}}\) & The average daily operating mileage for \(e\) that equals to its types. \\ \hline \(l_{e}^{0}\) & The accumulative operating mileage since last HM for \(e\) at the beginning of planning horizon. \\ \hline \(l_{e}^{0}\) & The accumulative operating days since last HM for \(e\) at the beginning of the planning horizon. \\ \hline \(\alpha\) & The size of train-set, if the train-set includes eight cars, then \(\alpha=1\); otherwise, \(\alpha=2\). \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Another set of parameters in the model. HM: high-level maintenance. the maximum cumulative operating mileage of maintenance requirements and the expected operating mileage for all EMUs. As mentioned above, the objective function can be simply described as follows:  
\[\text{min W}= \sum_{e\in E}L_{e}^{\text{upper}}-l_{e} \tag{2}\]  
If a train-set is undergoing inspection at the beginning of the planning horizon, \(l_{e}^{0}\) and \(t_{e}^{0}\) will take a negative value for simplifying the model in this paper, and \(t_{e}^{0}\) presents that \(\left|t_{e}^{0}\right|\) days are remaining to finish the inspection at this moment. So, we have the following formula:  
\[l_{e}^{0}= \begin{cases}l_{e}^{0}&t_{e}^{0}\geq 0\\ t_{e}^{0}l_{e}^{-}&t_{e}^{0}<0\end{cases}  \tag{3}\]  
Then, \(l_{e}\) can be calculated as the following equation:  
\[l_{e}=l_{e}^{0}+(k-1)\cdot\overline{l_{e}}\sum_{r\in R}x_{e}^{r}\delta_{r}^{a_ {ij}^{k}}\quad\forall e\in E,\ j=2,3,4,\ k=1,2,\ldots,T \tag{4}\]  
Constraints of the model are mainly derived from the following several aspects. 1 For safety reasons, the expected operating mileage since last HM have to be less than or equal to the maximum value of cumulative operating mileage, according to HM requirements.