# Slot Optimization Based on Coupled Airspace Capacity of Multi-Airport System
## 4 Algorithm
The airport capacity constraints are used as the coordinating parameter for each airport, and the corridor capacity is allocated to each airport._
**Scenario 3 (Multi-airports task):**: _The base schedules of the three airports are combined and optimized using the multi-airport system optimization model, with minimum slot deviation, fairness to airports, and fairness to airline operators as the optimization objectives; airport capacity constraints using the multi-airport system envelope; and corridor port capacity shared by each airport._

Existing models and algorithms are applied using data from the relevant airports and corridor gates, and the data are analyzed and compared based on different scenarios. ### Model Optimization Results

A multiple-objective optimization model with slot deviation, airport fairness, and airline congestion contribution fairness is established, and the NSGA-II algorithm is applied to solve the model to obtain the Pareto frontier solution set for airport slot optimization of the three airports in the Beijing multi-airport system. The Pareto frontier is obtained after the multi-airport schedule optimization, and a non-dominated frontier solution in the figure is selected for analysis. Based on the non-dominated sorting principle of the NSGA-II algorithm, any point on the Pareto front can be approximated as an optimal solution. For the execution of subsequent experimental verification, a distinguished point on the Pareto front was selected as a paradigm optimal solution for analysis in this study, as shown in Figure 5. This non-dominated solution has a slot deviation of 35,535, airport fairness of 0.009, and airline congestion contribution fairness of 13.615. The optimization results are shown in Figure 5. Airport fairness, slot deviation, and airline congestion contribution fairness represent the three objective values of the model. The blue dots in the figure represent the feasible solutions for solving the output, the red dots represent the Pareto frontier solutions, and the circled red dots serve as the solutions used for data analysis below. Based on the solution of the multi-airport model, the 24 h dynamic capacity values of each airport are obtained as shown in Figure 6. The obtained dynamic capacity values

Figure 5: Pareto frontier convergence graph. (The color bar on the right represents the generation. The red dot represents the Pareto frontier point). satisfy three constraints: the 24 h capacity constraints of each airport, the peak-valley single-cycle constraints, and the multi-airport system capacity constraints. The capacity trend of each airport conforms to the regularity of the peak period; for ZBAA, the original flight traffic exceeds the capacity of the airport at 07:00, 10:00, 13:00, 14:00, 17:00, 18:00, and 22:00; the optimization meets the capacity constraints in all cases; and the capacity is maximized. For ZBAD, the raw flight traffic exceeds the airport capacity at 0:00, 08:00, 09:00, 12:00, 14:00, 16:00, and 20:00, and the optimization meets the capacity constraints and maximizes the use of capacity. For ZBTJ, the original flight traffic exceeds the airport capacity at 10:00, 13:00, 15:00, 17:00, and 20:00, and all of them meet the capacity constraints after optimization. The three airports' flight schedules are optimized to achieve full utilization of airport capacity and peak and trough phase capacity, and the optimization effect is significant. For the analysis of the selected points in Figure 5 (the points labeled in the Pareto frontier), 20 min has the highest number of slot deviations, followed by 5, 10, and 15 min, and the least number of flights without slot deviations (Figure 7). The cumulative probabilities of the three airports' patterns were similar, but there was a certain gap in fairness. ZBAA, ZBAD, and ZBTJ have 6.48%, 7.67%, and 10.11% non-adjusted deviations, respectively. According to the picture showing the least number of flights without deviation, 50% of the flights can be adjusted in less than 15 min.