# Coordinated Optimization of Feeder Flex-Route Transit Scheduling for Urban Rail Systems
## 3 Mathematical Model
### Flex-Route Transit Operation Organization Model  
The conventional fixed-route public transport timetable compilation method is not suitable for feeder flex-route transit, because flex-route transit has the characteristics of sharing and the uncertainty of vehicle operation, which makes it difficult to provide accurate arrival time for passengers [22; 23]. The commonly used calculation methods for determining departure frequency [24], including two methods based on maximum passenger flow (station survey) and two methods based on cross-sectional passenger flow (following survey data), are not fully applicable to this study. Therefore, this section proposes a method for calculating the departure time interval of flex-route transit based on the urban rail transit timetable, passenger reservation requirements, and departure time window constraints.  
For the two cases of urban rail transit transfer feeder flex-route transit and feeder flex-route transit transfer urban rail transit, the optimal transfer conditions are as follows: When passengers transferring from urban rail transit to feeder flex-route transit arrive at the bus station, the feeder vehicle is already at the station, allowing passengers to transfer without waiting. Similarly, when passengers arrive at the urban rail platform, the train is already at the station, and they can board without delay. This transfer process can be illustrated in a time-space diagram, as shown in Figure 2.  
When the left passenger transfers from the urban rail transit to the feeder flex-route transit, he needs to get off at the time of \(RT_{a}\), and after the transfer time \(\varepsilon\), he reaches the interchange bus station at the time of \(t_{p}\) (before \(t_{b,k}\)) and successfully gets on the bus. Similarly, the right side of the passenger from the feeder flex-route transit transfer urban rail transit success condition is at or before td time to interchange bus station, through the transfer time \(\varepsilon\) at or before \(RT_{d}\) time to reach the urban rail transit station for transfer. According to the above analysis, the conditions of transfer coordination between the two can be obtained, which can be calculated by the following formula:  
\[t_{p}=RT_{a}+\varepsilon \tag{1}\]  
\[t_{d}=RT_{d}-\varepsilon \tag{2}\]  
If passengers arrive too early at the bus station when transferring from urban rail transit to flex-route transit, they must wait for a certain period before being served; if the passenger arrives late and the previous vehicle is scheduled to serve them, the vehicle must wait for the passenger to arrive at the station before picking them up for the next departure. This delay is considered part of the waiting time cost. When passengers transfer from flex-route transit to urban rail transit, arriving early at either the bus or urban rail station does not result in a delay; if the actual arrival time exceeds the expected arrival  
Figure 2: Transfer time–space diagram of urban rail transit and feeder flex-route transit.  
time, the passenger must wait for the next urban rail transit, and the delay is recorded as a late arrival time cost.  
In compiling the flex-route transit timetable for urban rail transit transfers, the focus shifts from a single-vehicle system to a multi-vehicle system. Therefore, the objective function of the timetable compilation model not only considers the passenger perspective but also incorporates the broader interests of both passengers and enterprises, as well as the public welfare aspect of public transit. The primary constraints of the model include the running time and departure interval constraints.