# 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
#### 3.3.3 Constraints
\[\frac{\sum\limits_{m\in M_{u}}\sum\limits_{t=s}^{s+t_{u}^{peak}}-1}{c_{u}^{peak }t_{u}^{peak}}=z_{u}^{s},s=1,2,\cdots,T-t_{u}^{peak},\forall u\in U \tag{14}\]  
\[y_{u}^{s}=min((1-z_{u}^{s})\eta,1),s=1,2,\cdots,T-t_{u}^{peak},\forall u\in U \tag{15}\]  
\[\frac{\sum\limits_{m\in M_{u}}\sum\limits_{t=r}^{r+11}X_{m}^{t}}{c_{u}^{firebreak }}\leq y_{u}^{s}(\frac{c_{u}}{c_{u}^{firebreak}}-1)+1,\;s=1,2,\cdots T-t_{u}^ {peak}, \tag{16}\]  
\[|t_{m_{0}}-t_{m}\leq t_{max}|,m\in M_{u},u\in U, \tag{8}\]0  
(5) Corridor capacity of airport terminal area constraints  
There is an aircraft flight time difference between the off-block time and the corridor times. The corridor capacity constraint requires the introduction of aircraft flight time parameters. \(\delta_{u,k}^{a}\) and \(\delta_{u,k}^{d}\) denote the arrival and departure flight times from airport \(u\) to corridor \(k\), respectively. \(\Delta\) denotes the upper and lower bounds of the probability. The flight time parameters can be obtained from the analysis of historical data, and the runtime parameters of the aircraft within the terminal area obey the probability distribution function, as shown in Equation (17) and Equation (18). \[|t_{m_{0}}-t_{m}\leq t_{max}|,m\in M_{u},u\in U, \tag{8}\]1  
\[|t_{m_{0}}-t_{m}\leq t_{max}|,m\in M_{u},u\in U, \tag{8}\]2  
Equation (19) proposes an arrival aircraft corridor judgment variable \(Y_{m}^{t}\). \[Y_{m}^{t}= \begin{cases}1,\;X_{m}^{t}=1\\ 0,\;X_{m}^{t}=0\end{cases} ,t\in\left[t-\delta_{u,k}^{a}-\Delta,t-\delta_{u,k} ^{a}+\Delta\right),\;m\in M_{u,a} \tag{19}\]  
Equation (20) proposes a departure aircraft corridor port judgment variable \(Z_{m}^{t}\). \[Z_{m}^{t}= \begin{cases}1,\;X_{m}^{t}=1\\ 0,\;X_{m}^{t}=0\end{cases} ,t\in\left[t+\delta_{u,k}^{d}-\Delta,t+\delta_{u,k} ^{d}+\Delta\right),\;m\in M_{u,d}, \tag{20}\]At this point, the arrival and departure flow rates of the corresponding corridor can be obtained by accumulating \(Y_{m}^{t}\) and \(Z_{m}^{t}\) over the time interval; thus, the corridor capacity constraint can be written as shown in Equation (21), with \(s=\{0,60,120\ldots 1380\}\), \(k\in\{1,2,\ldots,K\}\). \(U_{t}^{k}\) denotes the capacity of the corridor port at time interval \(t\). \[ \begin{array}{l}\sum\limits_{t_{m}\in\overline{s}}^{t_{m}+55}\sum\limits_{u \in U}\sum\limits_{m\in M_{u,d}}Y_{m}^{t}.\int_{max(t-\delta_{u,u}^{u}-\lambda,s)}^{min(t-\delta_{u,u}^{u}+\lambda,s+55)}f_{a}\Big{(}t_{m}-\delta_{u,k}^{a} \Big{)}dt\\ +\sum\limits_{t\in s}^{t+55}\sum\limits_{u\in U}\sum\limits_{m\in M_{u,d}}Z_{m }^{t}.\int_{max(t+\delta_{u,k}^{u}-\lambda,s)}^{min(t+\delta_{u,k}^{u}+\lambda, s+55)}f_{d}\Big{(}t_{m}+\delta_{u,k}^{d}\Big{)}dt\leq U_{t}^{k}\end{array}  \tag{21}\]  
(6) Sector capacity constraints  
Sector capacity refers to the ability of an airspace sector to serve, accommodate, and handle aircraft traffic per time interval, using the number of flights as the unit (usually counted in unit-hours). According to the ICAO's statistical requirements for sector capacity [21], this study uses the number of aircraft entering the sector per unit of time as the statistical sector capacity. Similar to the corridor constraints, the sector flight time counting method is therefore omitted here. For an aircraft between sectors, establish the set of sector entry times \(\mathbb{N}_{\psi}^{+}\), defined in Equation (22). \[\mathbb{N}_{\psi}^{+}=\big{\{}ENST_{m}\big{(}s_{\psi}\big{)},\cdots ENST_{m} \big{(}s_{\psi}\big{)}\big{\}},m\in M,\psi\in SEC_{m}, \tag{22}\]  
where \(\text{ENST}_{m}\big{(}s_{\psi}\big{)}\) denotes the time when flight m enters sector \(s_{\psi}\), where \(SEC_{m}\) is the set of all sectors that flight m has to pass through. The passage of aircraft \(m\) in sector \(s_{\psi}\) at time t is calculated as shown in Equation (23). \[\mu_{m,\psi}^{t}=\left\{ \begin{array}{l}1,\;t<ENST_{m}\big{(}s_{\psi}\big{)} <t+59\wedge t+59<EXST_{m}\big{(}s_{\psi}\big{)},\forall t\in\{0,1,2\cdots 1440\},\;m\in M\\ 0,\;otherwise\end{array} \right.