## 5 Numerical Example
### Sensitivity Analysis  
In this section, sensitivity analyses are performed to investigate the impact of the number of designed vehicle routes on the model performance. Table 5 shows a detailed comparison of model performance among three scenarios. We found that both total satisfaction and mileages slightly increased, as the number of vehicles increases. As the former increases more than the latter, the objective  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Vehicle**} & \multicolumn{2}{c}{**The Sequence of Demand Points**} & \multirow{2}{*}{**Travel Distance (km)**} & \multirow{2}{*}{**Travel Time (min)**} & **Number of** \\  & & **Visited by Vehicle** & & **Passengers** \\ \hline V1 & D1(8:15)-C5(8:17)-C6(8:18)-C7(820) & & & \\  & -C8(8:21)-C5(8:25)-M (8:27) & & & \\ V2 & D4(8:17)-C14(8:20)-C11(8:22) & & & \\  & -C1(2:82)-C10(8:27)-M(6:29) & & & \\ V3 & D4(8:03)-C13(8:05)-C2(8:10)-C4(8:11) & & & \\  & -C3(8:12)-C18(8:13)-C9(8:15)-M(8:16) & & & \\ \hline \hline \end{tabular}
\end{table}
Table 4: Routing and scheduling plan of each vehicle.  
Figure 4: Result comparison of proposed model and traditional DRTs.  
value is slightly reduced. This raise can be attributed to the increase in mileage from vehicles dispatched from the depot, which results in a reduced number of demand points visited by each truck and an increase in the invalid mileage. Similarly, it causes the vehicle to reach the rail station faster, as the number of the customer points visited by each vehicle decreases. In this case, a shorter time spent riding in the vehicle caused an increase in total passenger satisfaction. Therefore, there is the contradiction between total passenger satisfaction and total mileage, leading to the appropriate number of vehicles in the actual operation being decided by both operation cost and service level.  
Furthermore, Figure 5 reveals how the changes of weight coefficient \(c_{1}/c_{2}\) affect a trade-off between total mileage and the total average satisfied demand. As the target weight coefficient \(c_{1}/c_{2}\) gradually increases, the change of the weight coefficient does not reach the critical value and the total mileage and the passenger satisfaction remain the same, except for \(c_{1}/c_{2}=3.5\) changing the objective value at 0.1 km to be 0.3. When these changes happen, the scheduling objective focuses on sacrificing passenger satisfaction with a reduction in mileage.  
Furthermore, the solutions of the improved BA are compared with the standard BA, standard GSO, and Cplex in order to verify the effectiveness. The results are shown in Table 6, from which we can see the following:  
1. When the scale of the problem is small, all three heuristic algorithms can find the optimal solution. As the scale of the problem becomes larger, the quality of the solutions worsens.
2. The quality and the robustness of the improved BA are better than those of the standard BA and GSO. This shows that there is effective improvement of the algorithm when introducing the idea that GSO's \"small part-segmented rogues walk at random\" into standard BA to improve the speed and position updates in the formulas of bats, which can maintain the diversity of groups.  
\begin{table}
\begin{tabular}{c} **Probability** \\ **(yuan)** \\ \end{tabular} \\ \hline
15 & \(-\)36.4 & 100\% & \(-\)36.4 & 86.7\% & \(-\)36.4 & 74.7\% & \(-\)36.4 & 73.4\% \\
30 & \(-\)41.8 & 100\% & \(-\)40.6 & 77.1\% & \(-\)38.8 & 66.2\% & \(-\)37.7 & 62.8\% \\
60 & \(-\)76.5 & 100\% & \(-\)69.9 & 62.3\% & \(-\)63.3 & 51.8\% & \(-\)61.5 & 47.6\% \\ \hline \hline \end{tabular}
\end{table}
Table 6: Comparison of different algorithms.  
Figure 5: Influence of weight coefficient \(c_{1}/c_{2}\) on the scheduling result.