# Optimization of Multi-Day Flexible EMU Routing Plan for High-Speed Rail Networks
## 2 Model Description
### Model Construction
#### 2.2.2 Symbol Definition  
The sets, indexes, parameters, intermediate variables, and decision variables defined in the model are shown in Tables 2-4.  
Figure 4: EMU connection network.  
\begin{table}
\begin{tabular}{c l} \hline \hline
**Variable** & **Definition** \\ \hline \(L^{v,k}_{ij}\) & Continuous variable; cumulative operating distance of EMU \(v\) on arc \((i,j)\) on day \(k\) \\ \(T^{v,k}_{ij}\) & Continuous variable; cumulative operating time of EMU \(v\) on arc \((i,j)\) on day \(k\) \\ \(x^{v,k}_{ij,j}\) & Binary variable; equals 1 if \(v\) passes arc \((i,j)\) on day \(k\) and is 0 otherwise \\ \(y^{p,k}_{i,j}\) & Binary variable; equals 1 if \(v\) completes first-level maintenance while traversing arc \((i,j)\) on day \(k\) and is 0 otherwise \\ \(SD_{v,k}\) & Binary variable; equals 1 if \(v\) selects conventional fast train service on day \(k\) and is 0 otherwise \\ \(SG_{v,k}\) & Binary variable; equals 1 if \(v\) selects high-speed train service on day \(k\) and is 0 otherwise \\ \hline \hline \end{tabular}
\end{table}
Table 4: Decision variable definitions.  
\begin{table}
\begin{tabular}{c l} \hline \hline
**Symbol** & **Definition** \\ \hline \(i,j\) & Vertex index \\ \((i,j)\) & Arc index \\ \(s\) & Station index \\ \(k\) & Days in the planning horizon index \\ \(v\) & EMU index \\ \(V\) & Set of EMUs; \(v\in V\) \\ \(K\)/day & Set of days in the planning horizon; \(K=\{1,2,\ldots,|K|\}\) \\ \(O\) & Set of virtual source and sink vertices \\ \(D\) & Set of EMU depot vertices \\ \(S\) & Set of station stabling yard vertices \\ \(\mathcal{C}\) & Set of all train service vertices in the timetable; \(\mathcal{C}=\mathcal{C}^{1}_{1}\cup\mathcal{C}^{1}_{2}\cup\cdots\cup\mathcal{ C}^{k}\) \\ \(\mathcal{C}^{k}\) & Set of all train service vertices on day \(k\), including both high-speed and conventional fast trains; \\ \(\mathcal{C}^{k}=\mathcal{C}^{k}_{1}\cup\mathcal{C}^{k}_{2}\) & \(\mathcal{C}_{1}\) is the set of high-speed train service vertices; \(\mathcal{C}_{2}\) is the set of conventional fast train service vertices \\ \(\mathcal{C}^{k}_{1,\text{long}},\mathcal{C}^{k}_{1,\text{short}}\) & Long- and short-formation high-speed train service vertices on day \(k\) \\ \(\mathcal{C}^{k}_{2,\text{long}},\mathcal{C}^{k}_{2,\text{short}}\) & Long- and short-formation conventional fast train service vertices on day \(k\) \\ \(\mathcal{N}\) & Set of all vertices in the connection network; \(N=O\cup D\cup S\cup\mathcal{C},i,j\in\mathcal{N}\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Index and set descriptions.  
\begin{table}
\begin{tabular}{c l} \hline \hline
**Symbol** & **Definition** \\ \hline \(P^{\text{store}}_{i}\) & Nighttime stabling capacity at vertex \(i\) (station or depot); \(i\in\mathcal{S}\cup\mathcal{D}\) \\ \(P^{\text{main}}_{i}\) & Maintenance capacity at depot \(i\); \(i\in\mathcal{D}\) \\ \(L_{\text{cycle}}\)/km & Maximum cumulative mileage for EMU first-level maintenance cycle \\ \(T_{\text{cycle}}\)/km & Maximum cumulative time for EMU first-level maintenance cycle \\ \(T_{\text{con}}\) & Minimum train connection time \\ \(T_{\text{cou}}\), \(T_{\text{decou}}\) & Minimum technical operation time required for coupling and decoupling \\ \(l_{i_{i}}l_{j}\) & Distance between train service vertices \(i\) and \(j\); \(i,j\in\mathcal{C}\) \\ \(t_{i_{i}}t_{j}\) & Travel duration between train service vertices \(i\) and \(j\); \(i,j\in\mathcal{C}\) \\ \(t^{a}_{i_{j}}\), \(t^{a}_{i_{j}}\) & Arrival time at train service vertices \(i\) and \(j\) \\ \(t^{d}_{i_{j}}\), \(t^{d}_{j}\) & Departure time at train service vertices \(i\) and \(j\) \\ \(t_{ij}\) & Connection time between vertices \(i\) and \(j\) \\ \(s^{a}_{i_{j}}s^{d}_{j}\) & Arrival station of train service vertices \(i\) and \(j\) \\ \(s^{a}_{i_{j}}s^{d}_{j}\) & Departure station of train service vertices \(i\) and \(j\) \\ \(\zeta\) & EMU idle cost coefficient \\ \(\tilde{F}_{ij}\) & Empty run cost from \(i\) to \(j\) \\ \hline \hline \end{tabular}
\end{table}
Table 3: Parameter descriptions.