# An Integrated Optimization Method of High-Speed Railway Rescheduling Problem at the Network Level
## 4 NSGA-III Algorithm
### Initial Populations and Infeasible Solution Adjustment
\begin{table}
\begin{tabular}{c c} \hline
**Depot** & **Service Scope** \\ \hline BJN depot & BS Line, TQ Line \\ CZ depot & BS Line, TQ Line, TB Line \\ JN depot & BS Line, TQ Line \\ XZD depot & BS Line, TQ Line \\ NJ depot & BS Line, TQ Line \\ HQ depot & BS Line, TQ Line \\ \hline \end{tabular}
\end{table} 
Table 6: Service scope of the depots. Figure 11: Comparison before and after iteration. (**a**) The solutions before iteration; (**b**) The solutions after iteration. Then, we run our optimization algorithm 10 times and obtain 10 groups of average objectives after optimization. We calculated the mean value, the standard deviation, and the coefficient of variation, as Tables 7 and 8 shows. Figure 12: Convergence curves. We can find that the 10 optimization results have no significant difference; the standard deviation is 3.86, 1787.01, and 4.76, respectively, and the coefficient of variation is 2.82%, 0.35%, and 0.95%, respectively. The small standard deviation and coefficient of variation indicate that our algorithm has considerable reliability and reproducibility. While, the mean computation time of the method is 2691.20 s, which is a relatively long period of time. We consider that the long period of time is due to the large scale of the rescheduling problem, which is the disadvantage and limitation of the method. However, the coefficient of variation is 3.0%, indicating a certain degree of stability for the method. ### Analyze the Bottleneck Section and Key Time

There are a total of 36 sections in the railway network. Generally speaking, if an emergency occurs in different railway sections, it will have different influences on the railway network. Similarly, the emergency occurring at different times also has different influences on the network. Therefore, we try to test the different occasions in order to analyze the characteristics of the network. We designed a group of experiments as follows. For each section, we set 6 one-direction interruptions starting at 9:00, 11:00, 13:00, 15:00, 17:00, and 19:00, respectively, and the duration is 90 mins. Then, we run our algorithm 216(36 \(\times\) 6) times and obtain a Pareto front at each time. For each of the 216 Pareto fronts, we calculate the average of the total delay, respectively, obtain 216 three-dimensional points (X (railway section), Y (occurrence time), Z (total delay)), and fit a three-dimensional figure with these points as follows. In Figure 13, the left part shows the optimization result of the BS Line, and the right part shows the optimization results of the TQ Line and the TB Line. \begin{table}
\begin{tabular}{c c c c} \hline  & **Average Additional Operation Cost** & **Average Total Delay** & **Average Transfer Passengers** \\ \hline Before iteration & 1553.27 & 5,862,896.47 & 553.50 \\ After iteration & 1364.97 & 5,035,295.4 & 495.00 \\ Optimization rate & 12.12\% & 14.12\% & 10.57\% \\ \hline \end{tabular}
\end{table} 
Table 7: Detail comparison before and after iteration. \begin{table}
\begin{tabular}{c c c c c} \hline  & **Average Additional Operation Cost** & **Average Total Delay** & **Average Transfer Passengers** & **Time** \\ \hline
1 & 1368.56 & 5,036,846.36 & 498.20 & 2568 \\
2 & 1361.25 & 5,042,569.52 & 495.01 & 2645 \\
3 & 1364.22 & 5,041,092.68 & 497.32 & 2641 \\
4 & 1365.39 & 5,036,958.25



#### 5.3.1 Bottleneck Section

In the left part of Figure 13, from the X-axis (railway section) point of view, section 15 has the maximum total delay for any interruption occurrence time, indicated by the red dashed line. Therefore, we can call section 15 the bottleneck section of the whole network. It means that if one interruption occurs in section 15, it will have a more significant influence than any other sections on the total delay of the railway network, which should be paid more attention by the operator. We think the reason is that, from the data of the train timetable, there are most trains operated in section 15 and many of them are cross-line trains.