# Slot Optimization Based on Coupled Airspace Capacity of Multi-Airport System
## 3 Multi-Airport System for Cooperative Optimization of Flight Schedules
### Multi-Airport System for Cooperative Optimization of Flight Schedules
#### 3.3.2 Objective  
Based on the characteristics of airspace capacity coupling between multiple airports in the multi-airport system, the objective function is derived from three aspects: (1) minimizing flight diversions, (2) minimizing the ratio of flight diversions to total airport demand to reflect fairness, and (3) evaluating the contribution of airline congestion to slot allocation fairness. (1) Minimum slot deviation  
The objective is to minimize the total slot deviation in the multi-airport system, as shown in Equation (1). \[Z_{1} =\textit{min}\sum_{u\in U}\bigg{\{}\sum_{m\in M_{u}}\bigg{[}\sum_{l \in T}\bigg{(}\sum_{u\in T_{m}}|t_{m_{0}}-t_{m}\mid\bigg{)}X_{m}^{t}\bigg{]} \bigg{\}}, \tag{1}\]  
(2) Fairness of airports considering slot deviation proportional to total demand at each airport. \begin{table}
\begin{tabular}{l l} \hline \hline
**Notation** & **Description** \\ \hline
**Parameters** & \\ \hline \(C_{u}^{h}\) & Number of total flights limit for airport u at time interval \(h\),\(u\in U\), \(h\in T\) \\ \(C_{u,a}^{h}\) & Number of arrival flights limit for airport u at time interval \(h\),\(u\in U\), \\ \(C_{u,d}^{h}\) & Number of departure flights limit for airport u at time interval \(h\),\(u\in U\), \\ \(C_{u^{\prime}}^{a},C_{u^{\prime}}^{d},C_{u}\) & Airport group arrival capacity, departure capacity, and coupled \\ \(U_{t}^{k}\) & Number of total flights limit passing fix \(k\) at time window \(t\),\(k\in K\), \(t\in T\) \\ \(t_{m}\) & The requested slot for flight \(m\) \\ \(t_{m_{0}}\) & The actual slot for flight \(m\) \\ \(t_{max}^{t}\) & Maximum acceptable deviation for departure flights \\ \(t_{max}^{t_{max}}\) & Maximum acceptable deviation for arrival flights \\ \(t_{max}^{t_{max}^{t_{max}^{t_{max}^{t_{max}^{t_{max}^{t_{max}^{t_{max}^{t_{max} ^{t_{max}}}}}}}}}}\) & Minimum turnaround time for aircraft type \(q\) \\ \(T_{m,nd}^{t_{max}^{t_{max}^{t}}}\) & Scheduled turnaround time for aircraft type \(q\) \\ \(c_{u}^{peak}\) & Peak period capacity of airport \(u\) \\ \(t_{u}^{peak}\) & Peak period duration at airport \(u\), in units of 5 min \\ \(c_{u}^{check}\) & Firebreak period capacity of airport \(u\) \\ \(t_{u}^{check}\) & Firebreak period duration at airport \(u\), in units of 5 min \\ \(z_{u}^{s}\) & Auxiliary variable to determine if a wave period occurs. If a wave occurs, \(z_{u}^{s}\) is 1; otherwise, it is 0 \\ \(y_{u}^{s}\) & Auxiliary variable to determine if a trough constraint needs to be imposed. When \(y_{u}^{s}=0\), the trough constraint is added; otherwise, 1 \\ \hline \hline \multicolumn{2}{l}{**Decision variables**} \\ \hline \(X_{m}^{t}=\bigg{\{}\begin{matrix}1\\ 0\end{matrix}\bigg{.}\) &
\begin{tabular}{l} Binary variables, where \(X_{m}^{t}=1\) indicates flight m is assigned to \\ arrive/depart no earlier than slot \(t\) and \(X_{m}^{t}=0\); otherwise, \(t\in T\), \(m\in M\) \\ \end{tabular} \\ \hline \multicolumn{2}{l}{**Indirect decision variables**} \\ \hline \(\mu_{i}\) & Equity indicators of congestion \\ \(a_{u}\) & Number of flight deviations for airport \(u\) \\ \(\vartheta_{u,k}^{a}\) & The time of arrival flights passing fix \(k\) in airport \(u\),\(u\in U\), \(k\in K\) \\ \(\vartheta_{u,k}^{a}\) & The time of departure flights passing fix \(k\) in airport \(u\),\(u\in U\), \(k\in K\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: _Cont._During the actual operation of the multi-airport system, it is unrealistic to guarantee absolute fairness of the airports, and the degree of fairness is usually measured by comparing the deviation of the fairness indicator of any airport from the mean value. Therefore, the objective function that minimizes the maximum value of the deviation of the fairness indicator of one airport from the mean of the fairness indicators of all airports aims to reduce the least fair situation to a minimum, as shown in Equation (2).