# Optimization Model of Express-Local Train Schedules Under Cross-Line Operation of Suburban Railway
## 4 Space-Time Network Construction  
Rail transit train scheduling optimization involves complex space-time constraints, and traditional methods make it difficult to dynamically portray train and passenger behaviors. For this reason, a space-time network is constructed to uniformly model train operation, passenger flow distribution, and system constraints. The method clearly describes the space-time paths of trains and passengers through a multi-commodity flow model, which is suitable for large-scale problems such as operating map optimization. Space-time network is constructed as shown in Figure 3.  
The space-time network comprises four categories of points, the set of train arrival points \(N_{arr}\), the set of train departure points \(N_{dep}\), the set of passenger access points \(N_{start}\), and the set of passenger termination points \(N_{end}\), of which the set of train arrival points \(N_{arr}=N_{arr}^{1}\cup N_{arr}^{2}\), \(N_{arr}^{1}\), and \(N_{arr}^{2}\) are, respectively, the set of suburban railway train arrivals and the set of urban railway train arrivals. The set of train arrivals is the set of suburban railway train arrivals and the set of urban railway train arrivals. Additionally, there is a set of train departures and an urban  
Figure 3: Space–time network. rail train arrival point set; regarding the train departure point set, \(N_{dep}=N_{dep}^{1}\cup N_{dep}^{2}\), \(N_{dep}^{1}\), and \(N_{dep}^{2}\) are the suburban railway train departure point set and urban rail train departure point set, respectively. In space-time, each point \(i\in N\) is represented as a tuple containing four elements \(\left(\eta_{1}^{i},\eta_{2}^{i},\eta_{3}^{i},\eta_{4}^{i}\right)\) that are associated with the type, train service plan or OD pair, station, and time node corresponding to that point, respectively. Specifically, when \(i\) is a train departure point or a train arrival point, the value of \(\eta_{1}^{i}\) is 1 or 2, and the value of \(\eta_{2}^{i}\) is the suburban railway train service plan corresponding to point \(i\). (If point \(i\) is an urban rail train arrival point or an urban rail train departure point, \(\eta_{2}^{i}=0\).) When point \(i\) is a passenger access point or a passenger termination point, the value of \(\eta_{1}^{i}\) is 3 or 4, and the value of \(\eta_{2}^{i}\) is the number of the OD pair corresponding to point \(i\). The set \(\mathbf{A}\) of arcs in the space-time network consists of six parts: the train traveling arc set \(\mathbf{A}_{mn}\), the train stopping arc set \(\mathbf{A}_{dwell}\), the passenger transferring arc set \(\mathbf{A}_{tra}\), the passenger accessing arc set \(\mathbf{A}_{start}\), the passenger end arc set \(\mathbf{A}_{end}\), and the passenger penalizing arc \(\mathbf{A}_{penalty}\). Similarly, the train traveling arc set \(\mathbf{A}_{mn}\) can be divided into the suburban railway traveling arc set \(\mathbf{A}_{mn}^{1}\) and the urban rail train traveling arc set \(\mathbf{A}_{mn}^{2}\). The set of train stopping arcs \(\mathbf{A}_{dwell}^{1}\) can be divided into the set of suburban railway train stopping arcs \(\mathbf{A}_{dwell}^{1}\) and the set of urban rail train stopping arcs \(\mathbf{A}_{dwell}^{2}\). Each arc connects two space-time points in the space-time network, and the specific construction process of the space-time network is as follows. For a suburban railway train with a train service plan of \(p\) and a first station departure time of \(t,N_{arr}^{1},N_{dep}^{1}\) are, respectively, used to denote the train arrival and train departure points involved. The train traveling arc \(A_{run}^{1}\) represents the running process of the train from the train departure point to the train arrival point, and the train stopping arc \(A_{dwell}^{1}\) represents the stopping process of the train at the station, i.e., the running process from the train arrival point at the station to the train departure point at the station. The cost of train traveling arc and stopping arc \(C_{ij}=c_{train}^{1}\left(\eta_{4}^{i}-\eta_{4}^{i}\right)\).