## 5 Case Studies  
### Basic Data  
The Beijing-Shanghai HSR is the main corridor of the \"eight vertical and eight horizontal\" high-speed railway network in China. The Beijing-Shanghai high-speed railway company has been optimally adjusting ticket prices since the end of 2020. Among the 27 trains on the Beijing-Shanghai HSR, some of them only stop at a few large stations, with a shorter running time and higher ticket price, which are more suitable for dynamic pricing. Therefore, one of these trains (G19) was chosen as the case study. The G19 train stops at four stations, namely, the Beijing South Railway Station, Jinan West Railway Station, Nanjing South Railway Station, and Shanghai Hongqiao Railway Station, thus forming three segments and six ticket purchasing OD sections--(1,2), (1,3), (1,4), (2,3), (2,4), and (3,4)--for the corresponding passengers to choose. The seats occupied according to the number of tickets is shown in Figure 3. The G19 train uses CR400 BF EMU, and the rated passenger capacity of the second-class seats is 1,113 passengers. According to the survey, the known ticket pre-allocation scheme of each OD section is shown in Table 2.  
Since the actual passenger ticket-purchasing demand of the Beijing-Shanghai HSR cannot be accurately obtained, we use the actual passenger ticket-purchasing data as an approximate substitute. Moreover, due to the large differences in the number of tickets allocated for each OD section, the direct statistics on the number of daily tickets sold are ambiguous; therefore, we introduce the daily ticket-purchasing rate \(\mathrm{G}_{\mathrm{ij}}^{\mathrm{t}}\) for analysis, as shown in Equation (20). Currently, the pre-sale period for China Railway passenger tickets is 15 days. We collected passengers' ticket-purchasing data of partial OD sections during the pre-sale period on a certain date in January 2022. According to the method of testing random numbers based on an exponential distribution [40], we verified that the statistics satisfied the randomness requirement. \[\mathrm{G}_{\mathrm{ij}}^{\mathrm{t}}=\frac{\mathrm{Tickets\ left\ on\ the\ (t-1)th\ day\ -\ Tickets\ left\ on\ the\ tth\ day\ -\ Tickets\ left\ on\ the\ tth\ day\ }}{\mathrm{Total\ tickets\ sold\ during\ the\ pre\ -\ sale\ period}}\times 100\% \tag{20}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline
**OD Section** & **Origin Station** & **Destination Station** &
\begin{tabular}{c} **Ticket** \\ **Pre-Allocation** \\ \end{tabular} \\ \hline (1,2) & Beijing South & Jinan West & 131 \\ (1,3) & Beijing South & Nanjing South & 234 \\ (1,4) & Beijing South & Shanghai Hongqiao & 548 \\ (2,3) & Jinan West & Nanjing South & 136 \\ (2,4) & Jinan West & Shanghai Hongqiao & 173 \\ (3,4) & Nanjing South & Shanghai Hongqiao & 134 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Ticket pre-allocation scheme. Figure 3: Diagram of the seats occupied by number of tickets. The daily ticket-purchasing rates of Beijing South Station-Shanghai Hongqiao Station, Jinan West Station-Nanjing South Station, and Nanjing South Station-Shanghai Hongqiao Station are shown in Figure 4.  
Figure 4 indicates that the daily ticket-purchasing rates are different in different OD sections. In addition, for each OD section, the daily ticket-purchasing rates for the former 14 days of the pre-sale period increased over time. While for the long-distance OD section (Beijing South-Shanghai Hongqiao and Jinan West-Nanjing South), the daily ticket-purchasing rate decreased on the 15 th day, i.e., the day of departure. The daily ticket-purchasing rate of the short-distance OD section (Nanjing South-Shanghai Hongqiao) continued to increase. Therefore, we selected the data of the daily tickets sold of the former 14 days of the pre-sale period of this train on a certain date in January 2022 for statistics. The parameter values of each OD section are shown in Table 3.  
In this paper, we only consider the second-class seat ticket price, and the actual price of a G19 train ticket is CNY 626.