# Passenger Flow-Oriented Metro Operation without Timetables
## 3 Mathematical Description
### Objective Function
#### 4.1.3 Passenger-Train Interaction Section
In the 90 min research time horizon, the operation strategies' optimisation process has been implemented six times, or 53 dwelling times, and the departure intervals have been rescheduled. Because the waiting time of passengers who have been left is also counted, from time 0 to 45 min, RPOM set each train's departure interval to 'long' to take more passengers, and it kept adjusting the strategies to lead trains to stop longer in the station with high passenger flow entering speed. Between 45 and 60 min, an extreme situation happened: only two passengers entered every minute in the last station. Thus, RPOM changes the strategy of Train 8 rapidly to a short departure interval. After 60 min, it kept the long dwelling strategies to take more passengers to their destination. Typically, these modifications were implemented after noticeable passenger flow variations occurred, especially when a sudden, massive passenger flow occurred. Unlike the fixed dwelling times strategies, with the GA_RPOM method, trains aim to dwell shorter in the stations with lower passenger entering speeds and stop longer in the stations with high passenger entering speeds. Furthermore, GA_RPOM keeps modifying its operation strategies, making the massive passenger flows in the busy stations embark earlier and preventing these passengers from being stranded in these stations in real-time. With high flexibility, in most situations, the performance of RPOM was better than the periodic timetables, which is reflected by all of the passenger waiting time data. Though in some extreme situations, with the number of modifiable strategies decreased, the performance of RPOM was slightly worse than the short periodic timetable, which occurred once from 45 to 60 min, the \begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \multicolumn{2}{c}{**Detecting time**} & \multicolumn{8}{c}{**60 to 75 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}{107,819} & \multicolumn{2}{c}{120,759} & \multicolumn{2}{c}{117,163} \\ \multicolumn{2}{c}{**(minutes)**} & & & & & & & & & & \\ \hline  & **DI** & **DW1** & **DW2** & **DW3** & **DW4** & **DW5** & **DW6** & **DW7** & **DW8** & **DW9** \\ \hline



& L & L & L & L & L & L & L & L & L \\ \hline \end{tabular}

* Abbreviation: DW + number: dwelling time + station number; DI: departure interval; T + number: train’s number; S: short time strategy; L: long time strategy; ST: short periodic timetable; LT: long periodic timetable. Annotation: The strategies in the White box are modifiable; The strategies in the Grey box are fixed.

\end{table} 
Table 6: _Cont._overall performance of RPOM is much better than two periodic timetables. Moreover, as passenger flow variations are always in real-life operation, a flexible operation method is more valuable than a perfectly prepared fixed timetable. With a fixed number of trains, compared with the periodic strategies such as the ST and LT, there is no significant increase in trains' running costs of GA_RPOM, and it is easily applied in practice. Of course, extra operational training for the metro operators and drivers is necessary. However, compared with the experience-based operation method, more accurate indicators could be provided to the operators and drivers by GA_RPOM. ## 6 Conclusions

In this paper, an innovative passenger flow-oriented metro operation method without timetables has been presented. A mathematical model is proposed to minimise all passenger waiting times with a limited rolling stock number in the research horizon. A new algorithm program based on a GA, GA_RPOM, integrated with a simulator, has been developed to solve the model and optimise the metro operation plan in an acceptable computational time. Different from the traditional GA for other dynamic scheduling problems, GA_RPOM could modify the gene sequence length to optimise the operation strategies for the metro system in real-time.