## 5 Case Study and Discussion
### Analysis of Results
From the perspective of timetables, departure times decide when passengers can use the transit service, whereas departure intervals denote the maximum waiting time for passengers. Therefore, to further analyze the performance of different algorithms in solving WT models, we calculate the average departure time bias (DB) and the average interval bias (IB) between the original and the optimized schedules. The computations are described by Formulas (28) and (29), where \(T_{i}^{op}\) and \(T_{i}^{or}\) are the departure times of the \(i\)-th trip in the optimized and original schedules, respectively. \[DB=\frac{1}{\|\mathbf{V}_{self}^{l}\|}\sum_{i\in\mathbf{V}_{self}^{l}}\mid T_{ i}^{op}-T_{i}^{or}\mid\qquad\forall l \tag{28}\]  
\[IB=\frac{1}{\|\mathbf{E}_{self}^{l}\|}\sum_{(i,j)\in\mathbf{E}_{self}^{l}}\mid T _{j}-T_{i}-\bar{l}^{l}\mid\qquad\forall l \tag{29}\]  
Table 4 shows the computational results of different algorithms, where a larger value means a greater difference between the optimized and original schedules. From the results, we can conclude that the DP-LNS algorithm not only effectively ensures overall service regularity but also controls the range of the variation, with about a 23.27% bias reduction. Moreover, the service frequency provided by the DP-LNS algorithm is more similar to the original schedule compared to the schedules generated by the other algorithms. This saves passengers from disruptions caused by excessive waiting times while also guaranteeing the reliability of public transit services. Figure 5 illustrates the advantages of the DP-LNS algorithm, where the x-axis indicates the departure time and the y-axis indicates the cumulative number of departed trips. The curves in the figure represent the implementation of different departure schedules. Each point (x, y) signifies the departure time of the y-th trip at time x under the respective scheme. Therefore, when two curves have points closer together at the same y-coordinate, it signifies a closer alignment of the departure times for the two schemes. As observed in Figure 5, the red curve aligns more closely with the blue curve, indicating that the rescheduling scheme derived from the DP-LNS algorithm exhibits higher consistency with the original schedule. This significantly contributes to the maintenance of bus service stability. In addition to the solution, the optimization process can provide guidance for the actual operation of a public transportation system. Figure 6 shows the variation curve of the departure intervals, where the x-axis is the sequential number of intervals and the y-axis is the value of the corresponding interval. The greedy algorithm solves problem (23) without considering service frequency. Thus, it may select some trips within the time-delay windows to guarantee sufficient trips, generating disproportionate intervals. For instance, as highlighted by the arrow in Figure 6, an interval of over 30 min occurs for downward trips, significantly reducing the reliability of the public transit service and passenger satisfaction. However, this also means that there is a shortage of vehicle resources during the corresponding period, suggesting that the use of additional vehicles could improve the quality of the transit service. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & \multicolumn{3}{c}{**Upward**} & \multicolumn{3}{c}{**Downward**} \\ \cline{2-7}  & \multicolumn{2}{c}{**DB (min)**} & \multicolumn{2}{c}{**IB (min)**} & \multicolumn{2}{c}{**DB (min)**} & \multicolumn{2}{c}{**IB (min)**} \\ \cline{2-7}  & **Average** & **Total** & **Average** & **Total** & **Average** & **Total** & **Average** & **Total** \\ \hline Greedy & 6.73 & 188.33 & 3.04 & 173.08 & 7.49 & 200.52 & 2.73 & 155.52 \\ Greedy-LNS & 5.79 & 127.43 & 2.19 & 124.78 & 6.31 & 179.0 & 2.22 & 126.50 \\ DP-LNS & 5.25 & 110.23 & 2.10 & 119.64 & 6.32 & 185.52 & 2.06 & 117.20 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Analysis of the regularity of transit services. Figure 5: Temporal variations in trip departures.