## 5 Case Study
### Calculation Results and Analysis  
Figures 2 and 3 display the relationship between the departure interval and time period number and the relationship between the objective function values and the number of iterations, respectively. According to Figure 2, the average bus departure interval and the total number of bus departures during peak and nonpeak hours of the study route can be obtained, as shown in Table 8.  
The original departure interval of this route was 15 min per trip. According to the operating time of the route, it can be inferred that the original total number of departures per day was 64. The passenger waiting time cost, the bus company's operating cost, and the bus company's exhaust emission cost can be calculated based on the established model before and after optimization, as shown in Table 9. The passenger waiting time cost was reduced by 8.49%, the bus company's operating cost was reduced by 3.13%, and the bus company's exhaust emission cost was reduced by 3.14%. The optimized departure schedule can improve passenger satisfaction and enhance the operational benefits and environmental protection effects of bus company and is conducive to the sustainable development of public transportation enterprises.  
\begin{table}
\begin{tabular}{c c c} \hline
**Formula** & **Weighting Factor** & **Value** \\ \hline Formula (16) & \(\omega_{1}\) & 0.6 \\  & \(\omega_{2}\) & 0.4 \\  & \(\alpha_{p}\) & 0.6 \\ Formula (14) & \(\beta_{p}\) & 0.3 \\  & \(\gamma_{p}\) & 0.1 \\  & \(\alpha_{np}\) & 0.5 \\ Formula (15) & \(\beta_{np}\) & 0.4 \\  & \(\gamma_{np}\) & 0.1 \\ \hline \end{tabular}
\end{table}
Table 7: Model-related parameters.  
Figure 2: The relationship between the departure interval and time period number.  
Due to the differences in people's lives and work and rest situations, the actual departure intervals are usually multiples of 5 for the sake of practical scheduling, and the schedule of departures is generally constant throughout the day. According to the number of people at each station in each time slot, the corresponding departure interval and number of departures are adjusted to obtain the sub-optimal departure interval and number of departures, as shown in Table 10. The total number of trips is 59.  
The costs of the sub-optimal departure schedule are shown in Table 11. The passenger waiting time cost was reduced by 5.49%, the bus company's operating cost was reduced by 7.81%, and the bus company's exhaust emission cost was reduced by 7.82%. Compared with the optimal bus departure plan, the passenger cost reduction in the sub-optimal bus departure plan has been reduced and the reduction in the company's operating costs and pollutant emission costs has increased. Due to further consideration of the company's operating habits and passenger travel habits, the suboptimal solution is appropriate in actual operation and effectively reduces the overall cost of the public transportation system.  
Figure 3: The relationship between the objective function values and the number of iterations.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Time** & **Average Bus Departure** & **Total Number of Bus** \\  & **Interval (min)** & **Departures** \\ \hline Nonpeak hour & 21 & 42 \\ Peak hour & 12 & 16 \\ \hline \hline \end{tabular}
\end{table}
Table 8: The average bus departure interval and the total departure number.