# Optimization Model of Express-Local Train Schedules Under Cross-Line Operation of Suburban Railway
## 4 Space-Time Network Construction
Similarly, a collection of urban rail train arrival points, a collection of urban rail train departure points, a collection of urban rail train traveling arcs, and a collection of urban rail train stopping arcs are constructed based on known urban rail train timetables, with the cost of the arcs \(C_{ij}=c_{train}^{2}\left(\eta_{4}^{i}-\eta_{4}^{i}\right)\). The construction of points and arcs is shown in Equations (1)-(4):  
\[N_{pt}^{1,arr}=\Big{\{}i=\left(1,p,s,t+\tau_{ps}^{arr}\right),\forall t\in T,p \in P,\left(s,s^{\prime}\right)\in L_{p}\Big{\}}. \tag{1}\]  
\[N_{pt}^{1,dep}=\Big{\{}i=\left(2,p,s^{\prime},t+\tau_{ps}^{dep}\right),\forall t \in T,p\in P,\left(s,s^{\prime}\right)\in L_{p}\Big{\}}. \tag{2}\]  
\[A_{pt}^{1,run}=\Bigg{\{}(i,j),\forall t\in T,p\in P,\left(s,s^{\prime}\right) \in L_{p}\Bigg{|} \begin{matrix}i=\left(2,p,s,t+\tau_{ps}^{dep}\right),\\ j=\left(1,p,s^{\prime},t+\tau_{ps}^{arr}\right)\end{matrix} \Bigg{\}}. \tag{3}\]  
\[A_{pt}^{1,dwell}=\Bigg{\{}(i,j),\forall t\in T,p\in P,\left(s,s^{\prime}\right) \in L_{p}\Bigg{|} \begin{matrix}i=\left(1,p,s,t+\tau_{ps}^{arr}\right),\\ j=\left(2,p,s,t+\tau_{ps}^{dep}\right)\end{matrix} \Bigg{\}}. \tag{4}\]  
Each OD pair and departure time combination corresponds to a passenger access point \(N_{start}\) and a passenger termination point \(N_{end}\), which, respectively, denote the passenger's start and end points in the space-time network under the combination. Each passenger access arc\((i,j)\in A_{start}\) denotes a passenger waiting for a train at the starting point, and the cost of the arc \(C_{ij}=c_{wait}\left(\eta_{4}^{j}-\eta_{4}^{i}\right)\). Each termination arc\((i,j)\in A_{end}\) denotes the process of passengers getting off the train after reaching the endpoint, and the cost of the arc \(C_{ij}=0\). Each penalty arc\((i,j)\in A_{penalty}\) denotes the scenario in which a passenger is not served,and the cost of the arc \(C_{ij}=c_{penalty}\). The specific point and arc construction process is described in Equations (5)-(9):  
\[N_{wt}^{start}=\{i=(3,w,o_{w},t),\forall t\in T,w\in W\}. \tag{5}\]  
\[N_{wt}^{end}=\{i=(4,w,d_{w},t),\forall t\in T,w\in W\}. \tag{6}\]  
\[A_{wt}^{start}=\Bigg{\{}(i,j),\forall j\in N_{pt}^{dep},w\in W,t\in T\Bigg{|} \begin{matrix}i=(3,w,o_{w},t),\\ \eta_{3}^{j}=o_{w},0\leq\eta_{4}^{j}-t\leq\Delta t\Bigg{\}}. \tag{7}\]  
\[A_{wt}^{end}=\Big{\{}(i,j),\forall i\in N_{pt}^{arr},w\in W,t\in T\Big{|}j=(4, w,d_{w},t),\eta_{3}^{j}=d_{w},t\leq\eta_{4}^{j}\Big{\}}. \tag{8}\]  
\[A_{wt}^{penalty}=\{(i,j),\forall w\in W,t\in T|i=(3,w,o_{w},t),j=(4,w,d_{w},t)\}. \tag{9}\]  
The passenger transfer arc \(A_{tra}\) considers both passenger transfers from suburban railway trains to urban rail trains and transfers between suburban railway trains and does not consider passenger transfers from urban rail trains to suburban railway trains. The total transfer time is the sum of the transferring walk time and the waiting time for a passenger to transfer from the transfer station from the current train to a train on another line, and the cost of the arc \(C_{ij}=c_{walk}\tau_{\eta_{3}^{tra}}^{tra}+c_{wait}\bigg{(}\eta_{4}^{j}-\eta_{4}^ {i}-\tau_{\eta_{3}^{i}}^{tra}\bigg{)}\). The transfer arc construction is shown in Equation (10):  
\[A_{ptptt}^{tra} \tag{10}\] \[=\Bigg{\{}(i,j),\forall i\in N_{pt}^{1,arr},j\in N_{ptt}^{1,dep} \cup N_{ptt}^{2 dep}\Bigg{|} \begin{matrix}\eta_{2}^{i}\in\{p_{1},p_{3}\},\eta_{ 2}^{j}\in\{p_{0},p_{2},p_{4}\},\\ \eta_{3}^{i}=\eta_{3}^{j}=s_{r+g},0\leq\eta_{4}^{j}-\eta_{4}^{i}-\tau_{\eta_{3 }^{i}}^{tra}\leq\Delta t\Bigg{\}}.\end{matrix} \]  
In addition to the above arcs used by trains and passengers, this paper also proposes conflict arcs that consider the train spacing case.