# An Integrated Optimization Method of High-Speed Railway Rescheduling Problem at the Network Level
## 4 NSGA-III Algorithm
### Initial Populations and Infeasible Solution Adjustment
As for gene fragment IV, we randomly select two coupled trains and decouple them. The corresponding values in the matrix of the two trains should be changed from 1 to 0. Then, we randomly select two short trains and couple them. The corresponding values in the matrix of the two trains should be changed from 0 to 1. ### Termination Conditions

The algorithm ends and outputs the result when there are no new individuals being added to the Pareto front for continuous 30 iterations or when the number of iterations reaches the predetermined value. ### Algorithm

The framework of the algorithmic procedure is summarized as follows in Figure 9.

## 5 Case Study

In this section, we set up a series of experiments to test our model and algorithm. All the computational tests are performed by an Intel (R) Xeon (R) processor with 32 cores and 64 GB RAM. ### Basic Data

We use a whole day's train timetable, ticket selling data, and train set circulation plan of a Chinese railway network as the basic data. All the input data above is from our team's engineering project. The railway network structure is shown in Figure 10 as follows. It is composed of Beijing-Shanghai HSR (BS Line, the red line), Tianjin-Qinhuangdao HSR (TQ Line, the yellow line), and Tianjin-Baoding HSR (TB Line, the blue line). BS Line starts from BJS station and ends at SHHQ station, with a total of 23 stations; IQ Line starts from TJW station and ends at QHD station, totaling 9 stations; TB Line starts from TJW station and ends at BD station, totaling 7 stations. Figure 9: Flow chart of the solution approach. The basic data for the three lines are shown in Table 4:

There are 6 train set depots in the network, and Table 5 shows the service scope of the depots. The CZ depot serves all three lines, and the operator can dispatch standby train sets from the depot to the TB Line and TQ Line by TJX railway station or to the BS Line by TJN railway station. The other 5 depots serve the BS Line and TQ Line. ### Adaptability Analysis of NSGA-III Algorithm

We set a one-direction interruption from station 19 to station 18, which starts at 9:00 am. and lasts for 90 min. We use the algorithm in Section 4 with 30 population sizes and 1000 iterations, and the solving time is 2673 s. The comparison before and after iteration is plotted in Figure 11. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Line**} & \multirow{2}{*}{**Length**} & \multicolumn{2}{c}{**All Trains**} \\ \cline{3-4}  & & **Mainline Trains** & **Cross-Line Trains** \\  & & **(Short Trains/Normal Trains)** & **(Short Trains/Normal Trains)** \\ \hline BS Line & 1318 km & 558 (156/402) & 67 (8/59) \\ TQ Line & 287 km & 25 (14/11) & 104 (59/45) \\ TB Line & 157 km & 28 (16/12) & 37 (11/26) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Basic data of the three lines. Figure 10: Railway network structure. As Figure 11 shows, the left part is the 30 solutions before iteration, and the right part is the result of the 30 solutions after iteration. The detailed comparison of them is shown in Table 6.

We can see that the average value of all three objectives declines to different degrees, which are 14.12%, 12.12%, and 10.57%, respectively, indicating that our algorithm can improve the quality of the individuals effectively. In order to test and verify the effectiveness of our model and algorithm, we collected the values of three objectives (additional costs, total delay, and transfer number of affected passengers) for 1000 generations and plotted three convergence curves based on these values. The convergence curves are as follows:

In Figure 12, we can see that all three objectives reach a relatively stable value during the iteration, which indicates the effectiveness of our model and algorithm.