# Research on Optimization of Multi-Objective Regional Public Transportation Scheduling
## 5 Example Analysis  
A case study is carried out to test the dispatch approach and algorithm on a regional public transportation network with four bus lines, as shown in Figure 6. The scheduling simulation time window \(T_{1}\) and \(T_{2}\) were set to be 7:00, 7:30, respectively. The minimum and maximum departure interval, \(I_{\min}\) and \(I_{\max}\), were set to be 3 min, 10 min, and the time window for easy transfer, \(\delta_{1}\) and \(\delta_{2}\), were 1 min and 5 min, respectively. Let the waiting cost coefficient \(c_{1}=1.5\), \(c_{2}=0.18\). The passenger capacity of the bus \(V=100\). Based on the investigation of bus transfer stations in line 7, line 15 and line 131 of Lanzhou City, it is found that the more the number of passengers who failed to get on the bus, the higher the proportion of passengers who choose other modes of transportation. The corresponding proportion of choice other transportation modes for those passengers is as follows:  
\[P=\left\{ \begin{array}{c}0.09,1\leq F_{x_{i}s_{i_{d}}}\leq 5,\\ 0.18,6\leq F_{x_{i}s_{i_{d}}}\leq 10,\\ 0.25,11\leq F_{x_{i}s_{i_{d}}}.\end{array} \right. \tag{21}\]  
The size of population is 50, the maximum number of iterations is 150, the maximum mutation rate \(P_{m1}=0.3\), and \(P_{c1}=0.6\), \(P_{c2}=0.4\). The main transfer stations and transfer  
Figure 6: A regional public transportation network. proportions are shown in Table 1, and the average arrival rate, alighting rate of passengers and travel time between stations are listed in Table 2.  
The Pareto solutions obtained and the weighted objective value of them are shown in Table 3, and the Euclidean distance between solutions and positive and negative ideal points and the closeness of solutions are listed in Table 4 after using the entropy-weight TOPSIS method. It can be seen from Table 4 that solution 7 is the most satisfactory operation schedule, and its operation schedule is given in Table 5.  
By comparing solution 7 with the average value of schedules at a fixed departure interval of 3-10 min (as shown in Table 6), it can be seen that the total cost of loss for passengers is reduced by 31.42%, the comprehensive service ratio of buses is increased by 0.04%, and the number of passengers who not achieved easy transfer decreased by 74.29%. For comparing the performance of NSGA-II with the fixed crossover probability and mutation probability (FNSGA-II) and NSGA-II with adaptive adjusted model for crossover and mutation probability (ANSGA-II), the two algorithms solve the test experiment for 10 times. The Pareto solutions obtained of two algorithms after running 10 times are shown in Figure 7, and the number of nondominated set with different iterations of each algorithm is shown in Figure 8. The results show that the obtained solution set by ANSGA is superior to that obtained by FNSGA-II. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Transfer Station Pair** & **Transfer Proportion** & **Transfer Station** & **Transfer Proportion** \\ \hline \(s_{l_{1},2},s_{l_{3},5}\) & 0.35 & \(s_{l_{1},3},s_{l_{2},4}\) & 0.5 \\ \(s_{l_{1},3},s_{l_{3},4}\) & 0.45 & \(s_{l_{1},4},  
+++ ==WARNING: Truncated because of repetitions==
s_{l_{2},5}\) & 0.25 \\ \(s_{l_{1},4},s_{l_{4}}\) & 0.35 & \(s_{l_{1},6},s_{l_{1},6}\) & 0.15 \\ \(s_{l_{1},8},s_{l_{8}}\) & 0.05 & \(s_{l_{1},2},s_{l_{1},2}\) & 0.2 \\ \(s_{l_{2},4},s_{l_{3},4}\) & 0.4 & \(s_{l_{2},5},s_{l_{4},4}\) & 0.3 \\ \(s_{l_{5},6},s_{l_{3},3}\) & 0.45 & \(s_{l_{3},8},s_{l_{3},8}\) & 0.05 \\ \(s_{l_{5},7},s_{l_{2}}\) & 0.35 & \\ \hline \hline \end{tabular} \end{table} 
Table 1: Transfer station and transfer proportion. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Solution** & \(D_{i}^{\star}\) & \(D_{i}^{-}\) & \(F_{i}\) & **Solution** & \(D_{i}^{\star}\) & \(D_{i}^{-}\) & \(F_{i}\) \\
**No. ** & \(D_{i}^{\star}\) & \(D_{i}^{-}\) & \(F_{i}\) & **No.