# Passenger Flow-Oriented Metro Operation without Timetables
## 3 Mathematical Description
### Objective Function
#### 4.1.3 Passenger-Train Interaction Section
By integrating the macroscopic simulator, whenever the strategy needs to be optimised, the best gene sequence generated in the last detecting period would be recorded and compared with the new detecting time in the metro infrastructure section to modify the gene sequence length in real-time. The process to generate the gene sequence can be shown in Figure 3.

**Evaluation:** This step evaluates all the gene sequences from the population generated in the previous stage. Typically, the evaluation process is based on the objective function. First, the binary arrays of gene sequence would be transformed into the variables in the objective function. Then, the evaluation indicators mark a better result with higher (or lower) scores. In this research, the timetable section of the metro simulator could transform the binary arrays into specific operation strategies and ensure the objective function's constraints. The passenger-train interaction section of the simulator could record the waiting time for all passengers, which could be regarded as the evaluation indicator. **Selection:** The gene sequence with a higher score is adopted from this step. Generally, a roulette algorithm is applied to prove the fairness of selection. Then, selected gene sequences will become parent gene sequences for the next step. **Crossover:** In this step, all the parent gene sequences selected in the last step will mate (cross a part of binary arrays) to produce new baby gene sequences which inherit a part of their features, as shown in Figure 4.

Figure 4: Crossover process. Figure 3: Approach to generate gene sequence for GA_RPOM. **Mutation:** A part of the baby gene sequences generated in the last step is mutated. Based on the mutation rate, a part of the gene sequence will switch to opposite numbers (0 to 1, 1 to 0). The combination of mutating gene sequences constitutes the new population for the next period. The flow chart of GA_RPOM in the optimisation process and the interaction with the simulator is shown in Figure 5.

Figure 5: Flow chart of GA_RPOM in the optimisation process. With massive experiments, GA_RPOM could solve most metro operation problems in 30 s. Based on this algorithm, metro systems' operation strategy could be kept generating and optimising in real-time. ## 5 Case Study

For this research, a case study based on Beijing Metro Line 19 is used to demonstrate the new operation method and evaluate the performance of GA_RPOM. The total length of the Beijing Metro Line 19 is 22.4 km with ten stations. The speed limit for the whole line is 120 km/h, and this line will be allocated with new 5G network communication and face recognition technology from Beijing Infrastructure Investment Co Ltd, Beijing, China. Before passengers enter the platforms, they need to choose their origin and destination from the ticket machines. Then, when they pass the platforms' gates, an Automated Fare Collection (AFC) system can collect and generate real-time OD data. Furthermore, the EUHT communication units will be set up in all stations to transfer the OD data. Generally, this detecting and transferring process will take 15 min. Thus, the 15 min is regarded as the detecting period. The case studies use a single-direction line from Mudanyuan to Xingong, whose infrastructure data is shown in Table 4. In macroscopic terms, the simplified line information and running time between stations are shown in Figure 6.

Because this metro line is still under construction, the passenger data in the case studies is mocked and based on the early-stage desk research of the metro operator and the pre-designed capacity target of this line with the consideration of its connections with other commuting systems. Some extreme cases are also applied for testing. The passenger entering speed OD data is shown in Figure 7. The vertical axis represents different OD pairs, while the horizontal axis represents passenger entering speed; different colours represent the different detecting periods.