## 3 Mathematical Model
### Mathematical Model  
To maximize the revenue of train operation, on the one hand, we should optimize the structure of train operation and reduce the cost of train operation; on the other hand, we need to improve the level of train service and serve more passengers. Therefore, maximizing the revenue of train operation will help to improve the social and economic benefits of railway transportation simultaneously. This paper aims to maximize the revenue of train operations. Suppose the revenue of train operation is obtained by calculating the difference between the train ticket price revenue and train operation cost. In that case, the cost of \begin{table}
\begin{tabular}{c c} \hline \hline
**Notations** & **Description** \\ \hline \multicolumn{2}{c}{**Sets**} \\ \hline \(S\) & Set of stations, \(i,j\in S\) \\ \(A\) & Set of sections, \(a\in A\) \\ \(T\) & Set of the time periods, \(t\in T\) \\ \(P_{ij}\) & Set of possible passenger paths from station \(I\) to station \(j\), \(p\in P_{i,j}\) \\ \(P\) & Set of all passenger paths, \(p\in P\) \\ \(\mathcal{L}^{up}\) & Set of possible \(lines\) with train OD, route, and stops in the upstream direction, \(l\in\mathcal{L}^{up}\) \\ \(\mathcal{L}^{down}\) & Set of possible \(lines\)with train OD, route, and stops in the downstream direction, \(l\in\mathcal{L}^{up}\) \\ \(K\) & Set of the possible number of carriages of a \(line,\ k\in K\) \\ \(F\) & Set of operation frequency of a \(line,\ f\in F\) \\ \(D\) & Set of OD-pairs, \((i,j)\in D\) \\ \(\mathcal{A}^{up}_{L}\) & Set of riding arcs in the upstream direction, \(\hat{a}\in\mathcal{A}^{up}_{L}\) \\ \(\mathcal{A}^{down}_{L}\) & Set of riding arcs in the downstream direction, \(\hat{a}\in\mathcal{A}^{up}_{L}\) \\ \(\mathcal{A}^{l}_{L}(p)\) & Set of riding arcs passed by path \(p,\ \hat{a}\in\mathcal{A}_{L}(p)\) \\ \hline \multicolumn{2}{c}{**Parameters**} \\ \hline \(\eta_{l}\) & 0–1 parameter, if \(line\) 1 stops at all stations, \(\eta_{l}=1\); otherwise, \(\eta_{l}=0\). \\ \(\rho\) & The passenger-kilometer fare rate of the train \\ \(\tau_{p}\) & Travel distance of path \(p\) \\ \(C_{g}\) & Fixed organization cost of one _carriage_ \\ \(C_{r}\) & Operation cost per unit mileage of one _carriage_ \\ \(L\) & The length of the rail line \\ \(\varphi\) & Passenger time value \\ \(q_{ij}(t)\) & Number of passengers from station \(i\) to station \(j\) at time period \(t\) \\ \(\varepsilon\) & The probability that unserved passengers choose to travel in the next time period \\ \(\delta\) & The capacity of one carriage \\ \(\mathcal{U}^{up}\) & Maximum number of lines that can be operated in the upstream direction of each period \\ \(\mathcal{U}^{down}\) & Maximum number of lines that can be operated in the downstream direction of each period \\ \(\frac{f}{\mu}\) & Minimum number of operate lines of each period \\ \(N\) & Number of reserved carriages \\ \(l(\hat{a})\) & Mapping between riding arc \(\hat{a}\) and _line_\(l\) \\ \hline \multicolumn{2}{c}{**Variables**} \\ \hline \(x^{up}_{l,f,k}(t)\) &
\begin{tabular}{c} Binary variable, if \(line\) \(l\) with \(k\) \\ \(\text{carriages are operated}\) with frequency \(f\) in the upstream direction at \(t\) \\ time period, \(x^{up}_{l,f,k}(t)=1\) ; otherwise \(x^{up}_{l,f,k}(t)=0\) \\ Binary variable, if \(line\) \(l\) with \(k\) \\ \(x^{down}_{l,f,k}(t)\) & Binary variable, if \(line\) \(l\) with \(k\) \\ \(x^{up}_{l,f,k}(t)\) & Energy \(x^{up}_{l,f,k}(t)=0\) \\ \(y_{p}(t)\) & Integer variable, the number of passengers traveling on path \(p\) at time period \(t\) \\ \(z_{i,j}(t)\) & Integer variable, number of unserved passengers from station \(i\) to station \(j\) at time period \(t\) \\ \(n^{down}(t)\) & Integer variable, number of carriages remaining in the downstream direction at time period \(t\) \\ \(n^{up}(t)\) & Integer variable, number of carriages remaining in the upstream direction at time period \(t\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters, decision variables, and notations.