# 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  
(1) Slot uniqueness constraints  
\[\sum_{t\in T}X_{m}^{t}=1,\ m\in M \tag{7}\]  
Limit each flight to one time slot at the same airport on the same day, as shown in Equation (7). (2) Flight continuity constraints  
\[|t_{m_{0}}-t_{m}\leq t_{max}|,m\in M_{u},u\in U, \tag{8}\]  
Limit the maximum acceptable slot deviation per airline in each airport, as shown in Equation (8). (3) Airport group capacity constraints  
The traditional single-airport arrival, departure, and total capacity constraints are shown in Equation (9), Equation (10), and Equation (11), respectively. \[\sum_{m\in M_{u,d}}\sum_{l\in s}^{s+55}X_{m}^{t}\leq C_{u,d}^{h},s=0,5,10,\dots 1 385,u\in U, \tag{9}\]  
\[\sum_{m\in M_{u,d}}\sum_{l\in s}^{s+55}X_{m}^{t}\leq C_{u,d}^{h},s=0,5,10,\dots 1 385,u\in U, \tag{10}\]  
\[\sum_{m\in M_{u}}\sum_{l\in s}^{s+55}X_{m}^{t}\leq C_{u,d}^{h},s=0,5,10,\dots 1 385,u\in U, \tag{11}\]  
On the basis of the single-airport capacity constraint, the multi-airport system capacity constraint is proposed based on the multi-airport system capacity (airport group capacity). As in Section 2.1, first, the mapping of the multi-airport system capacity in the fugitive linear space is defined, and the multi-airport system capacity envelope generalization \(\mathbb{V}\) and the multi-airport system capacity \(\mathbb{C}\) are introduced; these correspond to the multi-airport hourly capacity envelope function relations. In Equation (12), \(\mathbf{C}\) denotes a tuple of multi-airport system capacities combined together. Among them, \(<C_{1},C_{2}\dots,C_{u}>\) represents the combination of the functional relationship between the arrival capacity and departure capacity of each airport to form \(C\). \[C=<C_{1},C_{2}\dots,C_{u}>,\ u\in U, \tag{12}\]  
The airport group capacity constraints are shown in Equation (13). \[<\sum_{l\in s}^{s+55}X_{1}^{t},\ \sum_{l\in s}^{s+55}X_{2}^{t},\dots,\sum_{l \in s}^{s+55}X_{m}^{t},>\leq\ \mathbb{C}[C],\ s=0,5,10,\dots,1385,\ u\in U \tag{13}\]  
Constraints are imposed on the arrival, departure, and total capacities of individual airports within the multi-airport system, as well as the system-level capacity. (4) Twenty-four-hour coordinate parameter constraints and peak-valley single-cycle constraintsAirport operational delays are more significant with the platform structure approach to airport slot allocation than with the peak-and-valley structure. The platform structure approach may lead to a gradual increase in operational delays, whereas the peak-and-valley structure approach is more effective in strategically reducing the inherent delays associated with the scheduling of flights. The basic principle is as follows. Wave identification is performed in a fixed step, rolling in the flight schedule, and when the airport throughput reaches the specified peak flow in the consecutive time period \([s,s+t_{u}^{peak}]\), according to Equation (14), then \(z_{u}^{s}\) is 1; otherwise, it is 0. Equation (15) constructs an auxiliary variable \(y_{u}^{s}\) such that when \(z_{u}^{s}=1\), then we have \(y_{u}^{s}=0\); when \(z_{u}^{s}=0\), then \(y_{u}^{s}=1\). Equation (16) is the trough capacity constraint; when \(y_{u}^{s}=0\), which implies that the peak hour starts at s, the trough hourly capacity constraint is imposed by rolling over the \([s+t_{u}^{peak},s+t_{u}^{peak}+t_{u}^{firebreak}]\) hour, i.e., \(\sum\limits_{m\in M_{u}}\sum\limits_{t=r}^{r+11}x_{m}^{t}\leq c_{u}^{firebreak}\); when \(y_{u}^{s}=1\), Equation (16) is equivalent to the airport's hourly flow being less than its capacity \(c_{u}\).