# Slot Optimization Based on Coupled Airspace Capacity of Multi-Airport System
## 4 Algorithm
The model solution is set to 30 min, but all the slot deviations can be solved within 20 min, showing that the model can optimize the flight layout in the shortest possible time. Figure 6: (a) Schematic diagram of supply and demand relationship before and after ZBAA optimization; (b) schematic diagram of supply and demand relationship before and after ZBAD optimization; (c) schematic diagram of supply and demand before and after ZBTJ optimization. (The vertical axis represents the planned number of slots. The horizontal axis represents the time period of the day, including 0 to 23, with odd-numbered periods not displayed. The amount of time in the column corresponds to the time period. The gray line represents the capacity of the corresponding time period). ### Comparison of the Optimization Effect of the Multi-Airport Model and the Single-Airport Model

The results of the single-airport and multi-airport optimizations were compared, with the single-airport constraints being more homogeneous (Table 3); multi-airport optimization increased airport and airline fairness, incorporating constraints related to airspace coupling. To further compare the optimization results, Scenario 2 and Scenario 3 for each airport were compared and analyzed, as shown in Table 3 and Figure 8. It can be seen that there are slot deviations in units of 0, 5, and 10 min, with the multi-airport slot deviations being less than the single-airport slot deviations, showing that the multi-airport system can optimize slot deviations and fairness with the least number of adjusted flights under the condition of satisfying the minimum slot deviations. \begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline
**AIRPORT** & \multicolumn{4}{c}{**ZBAA**} & \multicolumn{4}{c}{**ZBAD**} & \multicolumn{4}{c}{**ZBTJ**} \\ \hline
**Absolute Time** & **SD a** & **SA b** & **SD a** & **MA c** & **SD a** & **SA b** & **SD a** & **MA c** & **SD a** & **SD a** & **MA c** & **SD a** & **MA c** \\
**Deviation** & **for** & **Probab-** & **for** & **Probab-** & **for** & **SA b** & **Probab-** & **for** & **Probab-** & **for** & **Probab-** & **for** & **Probab-** \\
**(Minutes)** & **SA b** & **SD a** & **SD a** & **SD a** & **SD a** & **SD a** & **SD a** & **SD a** & **SD a** & **SD a** & **SD a** & **MA c** & **Probab-** \\ \hline
0 & 112 & 10.08\% & 72 & 6.48\% & 83 & 8.60\% & 74 & 7.67\% & 47 & 8.80\% & 54 & 10.11\% \\
5 & 257 & 23.13\% & 185 & 16.65\% & 197 & 20.41\% & 171 & 17.72\% & 104 & 19.48\% & 100 & 18.73\% \\
10 & 176 & 15.84\% & 169 & 15.21\% & 185 & 19.17\% & 162 & 16.79\% & 85 & 15.92\% & 79 & 14.79\% \\
15 & 130 & 11.70\% & 140 & 12.60\% & 131 & 13.58\% & 130 & 13.47\% & 65 & 12.17\% & 75 & 14.04\% \\
20 & 436 & 39.240\% & 545 & 49.05\% & 369 & 41.04\% & 428 & 44.35\% & 233 & 43.63\% & 226 & 42.32\% \\ Total & 1111 & & 1111 & & 965 & & 965 & & 534 & & 534 & \\ Total & 13,715 & & 15,615 & & 12,720 & & 12,985 & & 7005 & & 6935 & \\ \hline \hline \end{tabular}

* SD stands for slot deviation. b SA stands for single airport. c MA stands for multi-airport.

\end{table} 
Table 3: Statistics of time deviation for each airport. Figure 7: Frequency and cumulative probability plot for slot deviation times. (The horizontal axis represents the absolute slot deviation. The left vertical axis represents the frequency of deviation, measured in number of times. The right vertical axis represents cumulative probability, measured in percentages. The columns in the figure refer to the left vertical axis, representing the current absolute slot deviation quantity. The dotted lines in the graph refer to the right vertical axis and represent the cumulative probability of the current absolute slot deviation).