# Slot Optimization Based on Coupled Airspace Capacity of Multi-Airport System
## 2 Parameters Characterizing the Capacity of the Multi-Airport System  
### Multi-Airport System Capacity  
Multi-airport system capacity refers to the maximum number of aircraft landings and takeoffs that can be handled by a multi-airport system consisting of several geographically close and functionally complementary airports within a certain period of time, under the premise of ensuring safety and efficiency. Multi-airport coupling capacity is an important parameter in the capacity evaluation of the multi-airport system in China and has a certain constraining influence on airport slot optimization, slot deviation, and route optimization. The capacity value of each airport is obtained under the condition of maximum airspace capacity, and the 24 h slot coordination parameter is compiled by the slot management department using special methods, such as the maximum value of utilization efficiency. In the following section, arrival and departure competition and inter-airport capacity coupling within a single airport will be detailed by constructing a generalized analytical framework that has no explicit constraints but naturally contains resource competition relationships. Firstly, the capacity of a single airport's airfield is defined as \(C_{u,t}\), the arrival capacity is defined as \(C_{u,t}\), and the departure capacity is defined as \(C_{u,t}\). The airport capacity cannot simply be obtained by adding the arrival and departure capacities; it is necessary to define the total airport capacity as a convex nonlinear combination of arrival and departure capacities, that is, a convex function \(\Phi\):  
\(C_{u}=\Phi(C_{u,t},C_{u,t})\) Included among these \(\Phi:\mathbb{R}^{+}\times\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}\) fulfillment:  
Convexity: \(\lambda\in[0,1],\exists\)\(\Phi(\lambda x_{1}\ +(1-\lambda)x_{2}\,\ \lambda y_{1}\ +(1-\lambda)y_{2}\ )\leq \lambda\Phi(x_{1}\,y_{1}\ )+(1-\lambda)\Phi(x_{2}\,y_{2}\ )\). The principle of convexity is that runways cannot simultaneously maximize arrival and departure capacities. This reflects diminishing returns when reallocating resources. In the convexity formula, \(x\) and \(y\) are capacity vectors (or represent specific capacity combinations). The parameter \(\lambda\) is a mixing weight, with \(\lambda\in[0,1]\). Resource competitiveness: \(\frac{\partial\Phi}{\partial C_{u,t}}>0\,\ \frac{\partial\Phi}{ \partial C_{u,t}}>0\,\ \frac{\partial^{2}\Phi}{\partial C_{u,t}}<0\). The principle of resource competitiveness is that arrivals and departures compete for shared infrastructure capacity. The first partial derivative represents the marginal gain in airport capacity per additional unit of arrival capacity. The second partial derivative represents the marginal gain in airport capacity per additional unit of departure capacity. The cross-partial derivative represents the interaction effect, indicating how the marginal gain from one type of capacity changes when the other type changes. For a multi-airport system, due to resource allocation constraints among airports and airspace resource coupling, the capacity relationship among airports can be expressed as a generalized function. Constructing the overall capacity of the multi-airport system \(\mathbb{C}_{U}\) is performed as a function of the collective capacities of the airports \(\{C_{1},C_{2},\ldots C_{u}\}\). The consideration of multi-airport system coupling effects is defined as a normed linear space \(\varsigma\ =\ \{\mathbb{C}=(C_{1},C_{2},\ldots C_{u})|C_{u}=\Phi(C_{u,t},C_{u,t})\), \(\forall u\in U\}\), and the number of paradigms \(\left\|C\right\|_{p}\ =\ \left(\sum\limits_{u=1}C_{u}r^{p}\right)^{\frac{1}{p}}\), \(p\ \geq 1\) is assigned. The system capacity scale is quantified by the \(p\)-norm. The role of \(p\) is to control sensitivity to imbalances, with the purpose of measuring the \"effective size.\"The multi-airport system capacity is a nonlinear generalized function defined on \(\varsigma\).