# Coordinated Optimization of Feeder Flex-Route Transit Scheduling for Urban Rail Systems
## 3 Mathematical Model
### Notation  
A directed graph \(G=(S,\,A,\,T)\) was defined to represent the route planning network of vehicles, where \(S\) represents the set of all optional stops (including interchange station and flexible demand points), \(A=S^{2}\) represents the edge set formed by all optional stops, and \(T=(t_{ij})\) is the set of vehicles driving time corresponding to each edge. The following symbols are used in this paper:  
\(Q_{1}\)--set of type I passengers  
\(Q_{1}\)--set of type II passengers  
Q--\(Q_{1}\cup Q_{1}\), set of all passengers (demand), quantity \(q_{m}\), \(q\in Q\)  
\(K\)--set of feeder flex-route transit vehicle departure shift, \(k\in K\)  
\(R_{n}\)--set of non-interchange stations (demand points outside the base line) that need to be passed by feeder flex-route transit  
\(\psi\)--proper subset of \(S\) other than empty set  
\(RT_{a}\)--urban rail transit arrival time  
\(RT_{d}\)--urban rail transit departure time  
\(\varepsilon\)--time required for passengers to get in and out of the urban rail transit station for transfer  
\(t_{p,q}\)--time when passenger \(q\) transfers from urban rail transit to flex-route transit to reach the bus station  
\(t_{d,q}\)--scheduled bus station arrival time of passenger \(q\) who transfer from flex-route transit to urban rail transit  
\(\theta_{W}\)--unit value of waiting time cost of transfer passengers/flex-route transit waiting at interchange station  
\(\theta_{L}\)--unit value of time cost for passengers transferring from flex-route transit to urban rail transit to arrive late at interchange station  
\(\theta_{F}\)--unit conversion value (number of people converted to cost) of the cost of transfer failure (passengers are not served by flex-route transit service)  
\(\theta_{O}\)--unit value of vehicle/enterprise operation costs  
\(q_{K}\)--actual number of passengers taking feeder flex-route transit shift \(k\)  
\(d_{ij,k}\)--Manhattan distance from stop \(i\) to stop \(j\) for feeder flex-route transit shift \(k\)  
\(p\)--vehicle running speed of feeder flex-route transit  
\(H\)--departure interval during operation cycle of feeder flex-route transit  
\(h\)--operation cycle of feeder flex-route transit  
\(k_{m}\)--number of departures within the operating period  
\(\tau\)--maximum tolerable waiting or late arrival time for passengers  
\(P_{p,q,k}\)--time when feeder flex-route transit shift \(k\) picks up passenger \(q\)  
\(P_{d,q,k}\)--time when feeder flex-route transit shift \(k\) drops off passenger \(q\)  
\(t_{a,j,k}\)--arrival time of feeder flex-route transit shift \(k\) at station \(i\)  
\(t_{b,j,k}\)--departure time of feeder flex-route transit shift \(k\) at station \(i\)  
\(t_{m}\)--service time of feeder flex-route transit at flexible demand points  
\(x_{ij,k}=\{0,1\}\)--indicates if an arc \((i,j)\) is chosen by feeder flex-route transit shift \(k\) (\(x_{ij,k}=1\)) or not (\(x_{ij,k}=0\))  
\(y_{q,k}=\{0,1\}\)--indicates if passenger \(q\) taking feeder flex-route transit shift \(k\) (\(y_{q,k}=1\)) or not (\(y_{q,k}=0\))  
\(z_{q,k}=\{0,1\}\)--indicates if waiting time or late arrival time of transfer passenger \(q\) taking feeder flex-route transit shift \(k\) is greater than the acceptable maximum waiting time \(\tau\) (\(z_{q,k}=1\)) or not (\(z_{q,k}=0\))