## 5 Conclusions  
This paper comprehensively studied a high-speed railway operating problem by using the uncertain mathematics theory. Eight high-speed railway lines in the \"four east-west and four north-south\" railway network were selected for analysis. First, three single-objective models are proposed to optimize the railway network. Through optimization, the total travel time and travel cost for passengers are reduced by 18.04% and 0.063%, respectively. The operation costs and energy consumption costs of the high-speed railway are reduced by 17.08% and 21.99%, respectively. The line efficiency of the high-speed railway is increased by 10.20%. Second, based on the optimization function of the high-speed railway operation, a tri-objective model for line optimization is established. The improved Pareto efficient algorithm was selected to solve the tri-objective model. According to the optimization results, we suggest that the second east-west and the fourth east-west lines remain unchanged, while some stations in the first, second, third, and fourth north-south lines and the first and third east-west lines do not stop. The multi-objective optimization also provides the average speed for each line. Of course, the speed of some lines is quite different from their original speed, such as the first, second, third, and fourth north-south lines. The relevant results obtained by using the improved PE algorithm still need further verification, or even practical investigation, to prove their feasibility. These results match the related national planning policies of the Beijing, Shanghai, Wuhan, Guangzhou, and Zhengzhou main hubs. The new PE railway line-optimizing method proposed in this work is especially suitable for optimization problems, although it can also be used as a general-purpose optimization method. This approach provides decision-makers with a Pareto set of choices. The solutions obtained through this method can reflect the entire Pareto optimal frontier. This approach also automatically captures the Pareto optimal frontier without any formal optimization process. It uses an approximation to guide the sampling process and does not require an accurate approximation model. Through testing and application, the PE method was found to be robust and efficient. This optimization model can be used in future research on high-speed railway network planning problems. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{2}{c}{**Highspeed Railway Lines**} & \multicolumn{2}{c}{**Optimized Speed (km/h)**} & \multicolumn{2}{c}{**Optimized Railway Station**} \\ \hline \multirow{4}{*}{North–South} & 1 st North–South & 263.039 & Beijing–Tianjin–Jinan–Xuzhou–Nanjing–Shanghai \\ \cline{2-4}  & 2 nd North–South & 250 & Beijing–Zhengzhou–Wuhan–Changsha–Guangzhou–Shenzhen \\ \cline{2-4}  & 3 rd North–South & 250.468 & Beijing–Chengde-Shenyang–Changchun–Harbin \\ \cline{2-4}  & 4 th North–South & 274.439 & Hangzhou–Ningbo–Wenzhou–Fuzhou–Xiamen-Shenzhen \\ \hline \multirow{4}{*}{East–West} & 1 st East–West & 251.371 & Xuzhou–Zhengzhou–Xi’an–Lanzhou \\ \cline{2-4}  & 2 nd East–West & 314.710 & Shanghai–Hangzhou–Nanchang–Changsha–Guiyang–Kunming \\ \cline{2-4}  & 3 rd East–West & 250 & Qingdao–Jinan–Shijiazhuang–Taiyuan \\ \cline{2-4}  & 4 th East–West & 250 & Shanghai–Nanjing–Hefei–Wuhan–Chongqing \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Optimized “four east–west and four north–south” railway network. **Author Contributions:** Conceptualization, Q.H. and L.B. ; methodology, J.L. ; resources, M.T. ; writing--original draft preparation, Q.H. ; writing--review and editing, L.B. and J.L. ; supervision, M.T. ; funding acquisition, Q.H. and L.B. All authors have read and agreed to the published version of the manuscript. **Funding:** This research was funded by the National Natural Science Foundation of China, grant number 51178157, High-level Project of the Top Six Talents in Jiangsu Province, grant number JXQC021, Key Science and Technology Program in Henan Province, grant number 182102310004.