# Coordinated Optimization of Feeder Flex-Route Transit Scheduling for Urban Rail Systems
## 3 Mathematical Model
### Flex-Route Transit Operation Organization Model
#### 3.3.2 Constraint Condition  
The model constraints mainly include vehicle operation constraints, departure interval constraints and passenger transfer constraints. The details are as follows:  
(1) Vehicle operation constraints  
Constraints (8) means that when \(x_{ij,k}=1\), the arrival time of shift \(k\) at point \(j\) will not be earlier than the departure time from point \(i\) plus the travel time between two points. When \(x_{ij,k}=0\), this constraint is invalid, where \(M\) is a sufficiently large number.  
\[t_{a,j,k}\geq t_{b,i,k}+x_{ij,k}d_{ij,k}/\
u-M(1-x_{ij,k}),\forall(i,j)\in S,k\in K \tag{8}\]  
Constraints (9) ensure that the departure time of shift \(k\) at the demand point is equal to the time when the vehicle reaches the demand point plus the parking service time.  
\[t_{b,i,k}=t_{a,i,k}+t_{m},\forall i\in R_{m},\;k\in K \tag{9}\]By eliminating all cycles that make other points disjoint, it is ensured that the optimal solution does not contain meaningless subcycles.  
\[\sum_{i\in\varphi}\sum_{j\in\varphi}x_{ijk}\leq|\psi|-1,\forall\psi\subseteq S, \psi\
eq\varnothing,k\in K \tag{10}\]  
(2) Departure interval constraints  
The flex-route transit departure interval is calculated when the vehicle arrives at the feeder service area along the base line, as shown in Figure 3. During the operation of the flex-route transit, the departure interval is constrained by the time interval [\(H_{\text{min}},H_{\text{max}}\)] (the lower and upper limits of the vehicle departure interval). The determination of the departure interval in different periods can be calculated according to the model.  
\[H_{\text{min}}\leq H\leq H_{\text{max}}\,\forall H\in\mathbb{N}^{+} \tag{11}\]  
The value of the optimal departure interval calculated by the model is not fixed. To facilitate the preparation, the initial departure time interval is processed as follows:  
\[k_{m}=h/H,\forall h\in\mathbb{N}^{+},H|h \tag{12}\]  
(3) Passenger transfer constraints  
For passengers, each passenger can only be served once at most, either served or rejected.  
\[\sum_{k\in K}y_{q,k}\leq 1,\forall q\in Q \tag{13}\]  
The Constraints (14), (15) are used to judge whether the transfer passengers can successfully complete the transfer. If the transfer is successful (using the flex-route transit), \(y_{q,k}\) is 1, otherwise is 0.  
\[M\Big{(}y_{q,k}-1\Big{)}\leq\left|P_{p,\phi,k}-t_{p,q}\right|<My_{q,k},\forall q \in Q_{\text{I}},k\in K \tag{14}\]  
\[M\Big{(}y_{q,k}-1\Big{)}\leq\left|P_{d,\phi,k}-t_{d,q}\right|<My_{q,k},\forall q \in Q_{\text{II}},k\in K \tag{15}\]  
\(z_{q,k}\) is {0, 1}. When the waiting time or late arrival time of transfer passengers on the flex-route transit is greater than the acceptable maximum delay time \(\tau\), it is 1; otherwise, it is 0, as shown in Constraints (16), (17).  
\[M\Big{(}z_{q,k}-1\Big{)}\leq\left|P_{p,\phi,k}-t_{p,q}\right|-\tau<Mz_{q,k}, \forall q\in Q_{\text{I}},k\in K \tag{16}\]  
\[M\Big{(}z_{q,k}-1\Big{)}\leq\left|P_{d,\phi,k}-t_{d,q}\right|-\tau<Mz_{q,k}, \forall q\in Q_{\text{II}},k\in K \tag{17}\]  
Figure 3: Flex-route transit feeder service area.