## 7 Conclusions
\\ \(E_{i}\), \(W_{i}^{(tra)},W_{i}^{(bra)}\) & Traction electricity, traction mechanical work, and resistance mechanical work of train \(i\in\mathcal{I}\). \\ \(\eta^{(b)},\eta^{(j)},\eta^{(c)}\) & Traction inverter electrical efficiency, traction motor efficiency, and mechanical transmission efficiency from the output shaft of the traction motor to the wheels. \\ \(p(motor),p(trailer),p(p)\) & Average mass of each motor vehicle, trailer vehicle, and passenger. \\ \(\mu_{\max}\) & The maximum loading rate of a train. \\ \(z_{1},z_{2}\) & Vehicle capacity with and without cab. \\ \(\mu^{(c)},\mu^{(s)}\) & Penalty values. Superscripts \((c)\) and \((s)\) indicate the penalty value of crowding and travel fatigue, respectively. \\ \(\theta,t_{0}\) & Penalty threshold of crowding and travel fatigue. \\ \hline \multicolumn{2}{l}{Auxiliary variables} \\ \hline \multicolumn{2}{l}{Auxiliary variables} \\ \hline \(a_{\mathrm{sw}},a_{\mathrm{sw}}^{(0)}\) & A binary variable, \(\alpha_{\mathrm{sw}}=1\) if the station \(s\in\mathcal{S}\) is a turn-back station, 0 otherwise, i.e., if \\ \(\beta_{s},\beta_{s}^{(0)}\) & \(\exists k\in\mathcal{K}:s_{k}^{(o)}(s_{k}^{(d)})=s\), \(\alpha_{s1}(\alpha_{s2})=1\), otherwise \(\alpha_{s1}(\alpha_{s2})=0\). Superscript \((0)\) indicates the real timetable. \\ \(\beta_{s},\beta_{s}^{(0)}\) & A binary variable, \(\beta_{s}=1\), if the station \(s\in\mathcal{S}\) is an overtaking station, 0 otherwise, i.e., if \\ \(N_{k}^{(velt)},N^{(velt)},N^{(velt)},N^{(velt)}\) & Number of vehicles belong to \(k\in\mathcal{K}\) type train, number of total vehicles in our model and real timetable. \\ \(\mathcal{J}_{s}\) & Train order \(\mathcal{J}_{s}=<J_{s1},J_{s2},\ldots,J_{s1}>\in\mathcal{I}^{l}\), which is an ordered \(l\)-tuple, at station \(s\in\mathcal{S}\) in line plan \(\boldsymbol{\Psi}\), defined as a non-repeated arrangement of train number 1, 2, \(\ldots\), \(l\), where \(\mathcal{I}^{l}\) is the \(l\)-dimensional Cartesian product of train set, and the element \(J_{si}=j\in\mathcal{J}\) denotes the \(i\)-the arriving train is the train \(j\in\mathcal{I}\) at station \(s\in\mathcal{S}\). \\ \(\boldsymbol{\Delta}\) & The path-link incidence matrix in the space-time network. \(\boldsymbol{\Delta}=[\delta_{ap}]_{|\mathcal{P}|\times|\mathcal{W}|}\), where \(\delta_{ap}=1\), if link \(a\in\mathcal{E}\) belongs to path \(p\in\mathcal{P}\), 0 otherwise. \\ \(\boldsymbol{\Gamma}\) & The in-vehicle arc and link incidence matrix in the space-time network. \(\boldsymbol{\Gamma}=[\gamma_{\tilde{a}a}]_{|\boldsymbol{B}|\times|\mathcal{E}|}\), where \(\gamma_{\tilde{a}a}=1\), if in-vehicle arc \(\tilde{a}\in\mathcal{B}\) is the link \(a\in\mathcal{E}\), 0 otherwise. \\ \(\boldsymbol{\Phi}\) & The path-OD incidence matrix in the space-time network. \(\boldsymbol{\Phi}=[\varphi_{pw}]_{|\mathcal{P}|\times|\mathcal{W}|}\), where \(\varphi_{pw}=1\), if path \(p\in\mathcal{P}\) belongs to the OD pair \(w\in\mathcal{W}\), 0 otherwise. \\ \(\Pi\) & The train number and train type incidence matrix. \(\Pi=[\pi_{ik}]_{\times K}\), where \(\pi_{ik}=1\), if train \(i\in\mathcal{I}\) belongs to train type \(k\in\mathcal{K}\), 0 otherwise. \\ \(\boldsymbol{\Gamma}\) & Indicator vector of operation zones. \(\boldsymbol{\Upsilon}=[\tau_{sk}]_{S\times K}\), where \(\tau_{sk}=1\), if \(s\in[s_{k}^{o},s_{k}^{d}]\), \(\forall k\in\mathcal{K}\), 0 otherwise. Note that the operation zones of the train \(i\in\mathcal{I}\) can be calculated by \(\tau_{si}=\sum_{k\in\mathcal{K}}\tau_{\tilde{a}}\tau_{sk},\forall s\in\mathcal{S}\). \\ \(a_{si},d_{si}\) & Arrival and departure time of train \(i\in\mathcal{I}\) at station \(s\in\mathcal{S}\). Let \(d_{si}^{(0)}\) with the superscript \((0)\) indicate the planning departure time. \\ \(T_{sij}\) & A binary variable. \(T_{sij}=1\), if train \(j\in\mathcal{I}\) overtakes train \(i\in\mathcal{I}\) at station \(s\in\mathcal{S}\), 0 otherwise. \\ \hline \multicolumn{2}{l}{Decision variable} \\ \hline \multicolumn{2}{l}{The line planning with the predefined set \(\mathcal{K}\) of train types.