# Passenger Flow-Oriented Metro Operation without Timetables
## 3 Mathematical Description
### Objective Function
#### 4.1.3 Passenger-Train Interaction Section
The case study compares the quality of service based on all passengers' waiting time in the research horizon between two traditional periodic timetables and the real-time passenger-oriented operation method presented in this paper. From Table 3, there are two optional dwelling times and two optional departure intervals; one periodic timetable is applied with the short dwelling time and short departure interval, named'short periodic timetable', and the other one uses the long dwelling time and long departure interval, named 'long periodic timetable'. Contrarily, the RPOM method will keep generating operation strategies and optimising based on detecting passenger flow data in real-time. \begin{table}
\begin{tabular}{l l} \hline \hline
**Parameter** & **Data** \\ \hline Number of trains & 9 \\ Number of stations & 10 stations \\ Minimum interval & 2 min \\ Research time horizon & 90 min \\ Capacity of each train & 2000 \\ Optional dwell time & Short dwelling: 0.5 min, \\  & Long dwelling: 1.5 min \\ Optional departure interval & Short interval: 4 min, \\  & Long interval: 5 min \\ The number of solutions & 289 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Infrastructure parameters of Beijing Metro Line 19. The case study was carried out on a Windows 10 laptop with a Core i5 8 th generation CPU and 8 GB of RAM. The GA_RPOM program is coded in the Java language based on JDK 1.8. The population size is 200, the mutation rate is 0.2 with a maximum mutation number of 5, and the maximum iteration number is 600. The parameters of the GA_RPOM for this case study are shown in Table 5. We can obtain the operation strategies in 30 s based on the algorithm program. The convergence graph of the GA_RPOM based on the dynamic passenger flow data in the research time horizon is shown in Figure 8. The operation strategies based on RPOM and the comparison of all passenger waiting times between RPOM and periodic timetables are shown in Table 6.

Figure 6: Simplified line information and the running time of Beijing Metro Line 19. Figure 7: Passenger entering speed OD data. \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline \multicolumn{2}{c}{**Detecting Time**} & \multicolumn{8}{c}{**0 to 15 min**} \\ \hline \multicolumn{2}{c}{**Strategy**} & \multicolumn{2}{c}{**RPOM**} & \multicolumn{2}{c}{**ST**} & \multicolumn{2}{c}{**LT**} \\ \hline \multicolumn{2}{c}{**All passengers waiting time**} & \multicolumn{2}{c}{293,716} & \multicolumn{2}{c}{572,956} & \multicolumn{2}{c}{303,212} \\ \hline  & **DI** & **DW1** & **DW2** & **DW3** & **DW4** & **DW5** & **DW6** & **DW7** & **DW8** & **DW9** \\ \hline
**T1** & **X** & L & S & S & S & S & S & S & S \\ \hline
**T2** & L & L & S & L & S & S & S & L & S \\ \hline
**T3** & L & L & S & L & S & S & L & L & S \\ \hline
**T4** & L & L & S & S & L & S & L & L & S & L \\ \hline
**T5** & L & L & L & S & L & S & S & S & L \\ \hline
**T6** & L & L & L & L & S & S & L & S & L & S \\ \hline
**T7** & L & L & L & L & L & L & L & S & L & L \\ \hline
**T8** & L & L & L & L & L & L & S & S & L & L \\ \hline
**T9** & L & L & L & L & L & L & L & L & S & S \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Operation strategies generated by RPOM and comparison of all passenger waiting times. Figure 8: Convergence graph of GA_RPOM for the case study. \begin{table}
\begin{tabular}{l l} \hline Length of the gene sequence & 89 \\ \hline Population size & 200 \\ \hline Generations & 600 \\ \hline Mutation probability & 0.2 \\ \hline Computing time & 30 s \\ \hline Number of solutions & 2\({}^{89}\) \\ \hline \end{tabular}
\end{table} 
Table 5: The parameters of the GA_RPOM for this case study.