# Optimization Model of Express-Local Train Schedules Under Cross-Line Operation of Suburban Railway
## 3 Problem Statement
### Symbol Definition  
We define the following sets, parameters, and variables:  
**Parameters and sets:**  
\(T\): Set of time nodes where \(T=\{t|t=1,\ldots,|T|\}\);  
\(P\): Set of train service plans for suburban railway trains, \(P=\{p|p=p_{1},p_{2},p_{3},p_{4}\}\);  
\(P^{*}\): Set of cross-line train service plans for suburban railway trains, \(P^{*}=\{p|p=p_{2},p_{4}\}\),  
\(P^{*}\subseteq P\);  
S: Set of all stations on suburban railway and urban rail transit lines, \(\forall s\in S\), \(S=\{s_{1},s_{2},\ldots,s_{n}\}\), where \(n\) is the total number of all stations on suburban railway and urban transit lines;  
\(S_{urb}\): Set of all stations on the suburban railway line, \(S_{urb}=\{s|s=s_{1},s_{2},\ldots,s_{r}\}\), where \(r\) is the total number of all stations on the suburban railway line, \(S_{urb}\subseteq S\);  
\(S_{sub}\): Set of all stations on the urban rail transit line, \(S_{sub}=\{s|s=s_{r+1},\ldots,s_{r+g},\ldots,s_{n}\}\),  
\(S_{sub}\subseteq S\);  
\(S*:\) Set of urban rail stations to be covered by suburban railway trains after cross-line operation, i.e., the set of stations in the co-located section, \(S*\subseteq S,s_{r+g}\) is the set of urban rail transit stations where the suburban railway line meets the urban rail transit line;  
\(L_{p}:\) Set of station intervals covered by the suburban railway train service program \(p\), \((s,s^{\prime})\in L_{p}\), \(\forall p\in P\);  
\(S_{p}:\) Set of station covered by the suburban railway train service program \(p\), \(\forall s\in S_{p}\), \(\forall p\in P\);  
\(W:\) OD pairs set, \(\forall w\in W\);  
\(U_{wt}\): Number of passengers departing at time \(t\) for OD pair \(w\), \(\forall w\in W,t\in T\);  
\(N:\) Set of points in space-time network, \(\forall i\in N\);  
\(A:\) Set of arcs in space-time networks, \(\forall(i,j)\in A\);  
\(o_{w}/d_{w}\): Origin and destination of OD pair \(w\), \(\forall w\in W\);  
\(\tau_{p}^{app}\): Operating hours for suburban railway train service plan \(p\), \(\forall p\in P\);  
\(\tau_{p}^{arp}/\tau_{p}^{dep}\): Suburban railway train service plan \(p\) arrival time from first station to station \(s\), departure time, \(\forall p\in P,s\in S\);  
\(\tau_{s}^{tra}\): Transfer walk time for passengers transferring at station \(s\), \(\forall s\in S\);  
\(\Delta t\): Maximum single waiting time tolerated by passengers;  
\(\delta_{s,1}^{arp}/\delta_{s,1}^{dep}/\delta_{s,1}^{da}\): Minimum arrival headway, minimum departure headway, and minimum arrival-departure headway at station \(s\) after both suburban railway trains operate in cross-line mode, \(\forall s\in S\);  
\(\delta_{s,2}^{arr}/\delta_{s,2}^{dep}/\delta_{s,2}^{da}\): Minimum arrival headway, minimum departure headway, and minimum arrival-departure headway at station \(s\) between suburban railway trains after cross-line operation and urban rail transit trains, \(\forall s\in S\);  
\(C_{ij}\): Cost of arc\((i,j)\), \(\forall(i,j)\in A\);  
\(c_{train}^{1}/c_{train}^{2}\): Passenger cost per unit travel time on suburban railway trains, passenger cost per unit travel time on urban rail transit trains;  
\(c_{wait}/c_{walk}/c_{penalty}/c_{op}\): Passenger cost per unit waiting time, passenger cost per unit transfer walking time, unit penalty cost for unserved passengers, and cost per unit operating time of suburban railway trains;  
\(Q\): Capacity of suburban railway trains;  
\(K\): Passenger flow thresholds for triggering cross-line trains.