# Coordinated Optimization of Feeder Flex-Route Transit Scheduling for Urban Rail Systems
## 5 Result Analysis
### Case Analysis  
Based on the system's relevant parameter values and the spatio-temporal characteristics of passenger demand, the optimal departure interval for different demand levels can be  
Figure 4: Case study of bus and subway station. determined. The departure interval discussed here specifically applies to feeder flex-route transit within the feeder service area and does not imply that all M3 bus services operate at this frequency. In system cost calculations, the default weight coefficient assigned to each time-related cost is set to 1. If greater emphasis is placed on passenger waiting time and late arrival costs, their corresponding weight coefficients are increased while maintaining all other conditions constant. For instance, when the weight coefficient ratio for waiting time cost, late arrival time cost, transfer failure cost, and enterprise operation cost is set to 2:2:1:1, the optimal departure interval is adjusted accordingly, as shown in Table 1.  
As shown in Table 1, when greater emphasis is placed on passenger waiting costs and late arrival time, the optimal departure interval adjusts accordingly as the corresponding weight coefficient increases. This requires reducing the departure interval, increasing the departure frequency, minimizing passenger-related travel time costs, and optimizing overall system operation costs. Using the departure interval under the same weight as an example, the system operation cost \(U\), average vehicle running time, number of departures, and rejection rate (percentage of passengers not served due to constraints) at each demand level are calculated, as shown in Table 2.  
Table 2 shows that when passenger demand is high and service frequency is relatively low, the average vehicle running time increases. However, since increasing service frequency raises operating costs, this scheme represents the optimal solution under these conditions. If greater emphasis is placed on passenger waiting costs and late arrival time, the average vehicle running time and system running cost under different demand levels will also change, as shown in Figures 5 and 6. As shown in Figure 5, as the departure frequency increases, the number of vehicles operating per cycle rises, the average number of passengers per shift declines, and the corresponding average running distance shortens, leading to a reduction in the average vehicle running time per cycle. \begin{table}
\begin{tabular}{l c c c c c c c c c c} \hline \hline \multirow{2}{*}{**H (min)**} & \multicolumn{8}{c}{**Demand Levels (Passengers/h)**} \\ \cline{2-10}  & **10** & **20** & **30** & **40** & **50** & **60** & **70** & **80** & **90** & **100** \\ \hline Equal weights & 30 & 30 & 20 & 20 & 15 & 15 & 12 & 10 & 6 & 6 \\ Attach importance to passengers & 30 & 20 & 20 & 15 & 12 & 10 & 10 & 6 & 5 & 5 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Optimal departure intervals corresponding to different demand levels. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multicolumn{1}{c}{**Demand Levels**} & \multirow{2}{*}{\(\mathbf{H\) (min)} & \(\mathbf{U\) (USD)} & \begin{tabular}{c} **Average Vehicle** \\ **Running Time (min)** \\ \end{tabular} & \
\begin{tabular}{c} **Number of** \\ **Departure Shifts** \\ \end{tabular} &
\begin{tabular}{c} **Rejection Rate** \\ **(\%)** \\ \end{tabular} \\ \hline
10 & 30 & 31.46 & 11.86 & 2 & 0 \\
20 & 30 & 82.71 & 19.17 & 2 & 5 \\
30 & 20 & 103.16 & 20.27 & 3 & 0 \\
40 & 20 & 160.18 & 28.73 & 3 & 7.5 \\
50 & 15 & 231.41 & 25.21 & 4 & 0 \\
60 & 15 & 308.76 & 32.55 & 4 & 6.67 \\
70 & 12 & 362.57 & 32.30 & 5 & 5.71 \\
80 & 10 & 421.44 & 29.19 & 6 & 2.5 \\
90 & 6 & 447.48 & 18.35 & 10 & 0 \\
100 & 6 & 525.12 & 20.92 & 10 & 3 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: System simulation results corresponding to different demand levels.