Slot Optimization Based on Coupled Airspace Capacity of Multi-Airport System

Abstract

An airport slot is the core resource in the air transportation system. In most busy airports in China, airline demand significantly exceeds the available slot capacity. Scientific and reasonable slot allocation techniques and methods can improve the operational efficiency and benefits of multi-airport systems. Existing research has predominantly addressed slot allocation optimization for individual airports; however, there are differences in the functional positioning and resource allocation during multi-airport slot optimization, which makes cooperative optimization in the context of multi-airport slot allocation difficult. The dynamic sharing of airspace capacity in multi-airport systems is crucial for optimizing airport slot allocation and improving resource utilization efficiency. This study develops a multi-objective optimization model incorporating coupled airspace capacity relationships within multi-airport systems and the fairness of airlines and airports in order to realize the optimal utilization of multi-airport system resources, considering specialized 24 h airport slot coordination parameter patterns and slot firebreaks in China. Finally, the validity and scalability of the model are verified using real flight data from three airports in the Beijing airport terminal area, and simulations are conducted to verify the model. The findings provide a solid reference for the optimization of airport slot timetables in multi-airport systems, having both important theoretical value and practical significance.

1. Introduction

With the rapid development of the aviation industry and the development strategy of regional economic integration, heightened requirements have emerged regarding the operational efficiency and optimization of regional aviation networks. Single-airport operation resources are limited and often face the phenomenon of limited airport slot resources, leading to supply and demand imbalances. A multi-airport system consists of geographically close airports. Optimizing airport slots from this system perspective allows for the sharing of terminal airspace resources. These resources include airspace capacity assets such as departure/arrival fixtures. This approach maximizes airspace capacity use and reduces underutilization. It also diverts airport slots between system airports to improve operational efficiency, consequently enhancing the quality of air transportation services.

This study focuses on the concept of multi-airport systems. Currently, interactions among airports in these systems increase congestion and delays. Such a system can be configured through airport slot optimization. This approach enhances the interactions between airports and optimizes shared resources. It also decouples resources to reduce flight delays and prevent potential air traffic congestion.

A significant discrepancy persists between theoretical research on flight slot allocation and operational practices in the international aviation domain. China, the United States, and Europe predominantly adopt the airport slot guidelines published by the International Air Transport Association as their operational standard [1,2]. Their methodologies largely focus on individual airport capacity parameters (i.e., slot coordination parameters), neglecting to address the issue from a multi-airport systems perspective that incorporates crucial factors such as interconnected capacity coupling and shared airspace constraints [3]. While pioneering initiatives exist in theoretical frameworks, demonstrating sound conceptual approaches, they have yet to yield substantial contributions concerning the optimization of coupled capacity within multi-airport systems or the enhancement of broader air route network efficiency. The integrated economic situation has evolved rapidly in the last decade. For this reason, researchers have argued that airport slot deviation at a single airport is affected by the regional economy and neighboring airports. Additionally, more studies have focused on analysis and optimization from a multi-airport system perspective. Researchers believe that capacity is an important basis for the optimization and deviation of multi-airport system airport slots—which is the core focus of this study—and most studies consider the demand and restrictions based on the announced capacity as a definite value [4]. On this basis, centering on the capacity research dimension, some studies consider the demand and constraints from the flight transit time [5], the time interval before and after flights [6], the arrival/departure capacity constraints, and the traffic flow constraints [7]. Airport slot optimization involves a large number of research objectives. It also involves a large number of influencing factors. Due to this complexity, model development for multi-airport systems presents heightened complexity. Therefore, some studies have explored the establishment of algorithmic solutions for such models. Multi-airport system airport slot optimization models generally rely on integer linear programming [8,9] while for the more complex model structure of multi-airport systems, most researchers have relied on heuristic algorithms [10], nondominated sequential genetic algorithms [11], asymptotic binary heuristic methods [12], simulated annealing algorithm [13], k-means clustering algorithm [12], and multi-objective particle swarm algorithm [14]. We summarize the relevant literature in

Table 1. A summary of the literature on synergistic optimization of airport slot allocation in multi-airport systems.

Reference	Model Objectives	Constraints
		24 h Coordination Parameters	Airport Capacity	Sector Capacity	Ground Resource	Multi-Airport Coupling Capacity	Corridor Port Capacity	Uncertain Ratio of the Capacity-Flow	Flight Transit Time	Acceptable Slot Deviation
[15]	Minimize total deviation		√	√
[16]	Minimize the weighted sum of total slot deviation and operation delay		√						√
[5]	Minimize total cost and number of slot deviations		√		√				√	√
[7]	Minimize total delay, maximize the sum of average flight satisfaction across all airports, and minimize the deviation of average flight satisfaction across airports		√		√		√		√
[11]	Minimize the weighted sum of the requested slots and slot deviations		√		√	√			√
[17]	Minimize the total delay cost of flights in a multi-airport system				√	√
[12]	Airport slot deviation, total slot deviation, average delay, weighted sum of delay			√			√		√
[13]	Minimize total delays			√					√
[14]	Minimize slot deviation, minimize the total number of flights to be scheduled, and scheduling fairness metrics			√			√			√
[9]	Minimize total delay		√				√	√	√
[18]	Minimize total delay time, minimize total delay cost, minimize total slot deviation, minimize total delay		√	√					√
This Study	Minimize slot deviation and minimize both airport fairness and airline congestion contribution fairness	√	√	√	√	√	√	√	√	√

.

Based on the above research, one of the core problems of airport slot optimization is capacity utilization; however, most of the existing research is based on the announced capacity for analysis, combined with the actual situation of the civil aviation transportation industry at this stage. The interactions between multi-airport systems need to be expanded from the independent capacity of a single airport to the coupled capacity between multiple airports. Therefore, the spatial characteristics of airspace capacity and the temporal characteristics of flights should be fully considered during the research process, and the airspace unit’s capacity should be restrained and refined to propose realistic and practical solutions for airport slot optimization.

Research on the collaborative optimization of airport slot allocation in multi-airport systems has indicated that the objective of optimization generally focuses on a single objective, such as delay or slot deviation minimization, while some studies have explored the fairness of airports and airlines, integrating efficiency and fairness into the objective function. The focus is mainly on nine constraints. The all-day time coordination parameter defines hourly or 15 min airport capacity limits. Single-airport capacity constraints set maximum flight handling limits per airport. Sector capacity controls airspace zone volumes, while corridor opening capacity restricts aircraft flow through designated airways. Ground resource constraints cap apron usage capacity at airports. Additionally, sector and corridor port capacity determine maximum throughput at critical airspace nodes. Crucially, multi-airport coupling constraints govern the shared capacity between system airports. Finally, capacity–flow uncertainty ratios address planning-stage estimation errors in airport slots and taxiing times. These errors induce spatio-temporal resource mismatches, potentially causing flight delays or underutilized airspace capacity.

Although most of the existing models include some of the above constraints, there is room for improvement in terms of their completeness and practical compliance. Firstly, the assumption of singularity of capacity is inconsistent with the characteristics of multi-airport system airspace resources that can be shared, and many studies have failed to consider the variability of airport capacity under the airspace–capacity coupling relationship. Secondly, most of the current fairness considerations stem from the perspective of the interests of the air carrier, and there is a lack of operational-level considerations. The fairness measure based on operational indicators has not yet been integrated. Considering only the number of slots of the carrier at the coordinated airport as a benchmark for measuring fairness is not comprehensive enough and may lead to the unnecessary deviation of flights during off-peak hours. Doing so not only increases the difficulty of slot coordination but is also unreasonable. Thirdly, the uncertainty of capacity–flow matching in the process of flight operation at a multi-airport system may lead to mismatches of other spatial–temporal resources, which can cause, for example, delays, insufficient airspace capacity, and a lack of airspace capacity. This study focuses on nine types of constraints, including coupling capacity, congestion fairness of airports, and uncertainty of capacity–flow matching, to comprehensively analyze and optimize the allocation of airport slots in a multi-airport system.

This study investigates the airport slot co-optimization problem for multi-airport systems, incorporating China’s civil aviation slot management rules. Based on the coupled impact of airspace capacity, a capacity model is developed while interpreting the characteristics of China’s 24 h slot coordination. Focusing on slot deviations, a multi-objective slot optimization model is established, incorporating factors such as system-wide fairness and the levels of congestion contributed by airlines. Finally, using actual flight data, the model is solved using the NSGA-II algorithm. Its efficiency is validated through comparison with a single-airport optimization model, and the superiority of its results is verified using the AirTOp (V2.3.29B54) simulation software. This provides a theoretical and technological support foundation for promoting the construction and coordinated operation of multi-airport systems. This study makes three major contributions to the research field:

(1) By introducing the probabilistic competitiveness of shared resources, it embodies capacity operation coupling among multiple airports within the multi-airport system and realizes the decoupling of the multi-airport system’s spatial–temporal resources. In this study, we optimize the capacity use of the airport itself by fully considering airport slot uncertainty and imposing corridor mouth constraints. The airport coupling relationship is mainly embodied in the multi-airport capacity constraint relationship, describing the constraints and interrelationships relating to each airport’s capacity, airport group capacity, corridor port capacity, and airport parking position group capacity, etc., in order to study the synergistic optimization of airport slots.

(2) It considers the fairness of airport slot optimization of the multi-airport system. This study stands at the level of the airport and airline operation and integrates the fairness of operation indices, allowing us to measure not only the number of slots of airline companies in the coordinated airports as a basis but also the fairness of airlines based on the contribution of flight congestion, combining the amount of slot deviation for each airport within the multi-airport system to assess the fairness of airports.

(3) It includes flexible compatibility with China’s 24 h slot coordination parameters and its slot firebreak. In China’s airport slot management, 24 h coordination parameters are set for each airport, with slot firebreaks introduced at specific times to mitigate potential scheduled delays. In this study, based on the fixed slot firebreak, we further introduce a constraint on the capacity–flow ratio during a peak–valley cycle to accommodate the demand uncertainty.

The subsequent sections of this paper are organized as follows. Section 2 delineates the characteristic parameters of the multi-airport system capacity, encompassing system-wide capacity metrics and 24 h slot coordination parameters. Section 3 presents the proposed slot co-optimization model for multi-airport systems, founded on airspace capacity coupling. Section 4 details the solution algorithm developed for this model. Section 5 details the model evaluation, first analyzing the results and subsequently performing empirical validation using the AirTOp simulation software. Finally, Section 6 concludes the key innovations of this research and discusses prospective directions for future applications.

2. Parameters Characterizing the Capacity of the Multi-Airport System

2.1. 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 $C_u$, the arrival capacity is defined as $C_{u,a}$, and the departure capacity is defined as $C_{u,d}$. 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 Φ:

$C_u=\Phi \left(C_{u,a},C_{u,d}\right)$ Included among these $\Phi :{ℝ}^{+}\times {ℝ}^{+}\to {ℝ}^{+}$ fulfillment:

Convexity: $\lambda \in \left[0,1\right],\exists \hspace{0.33em}\Phi (\lambda x_1 +(1-\lambda )x_2 ,\hspace{0.33em}\lambda y_{1 }+(1-\lambda )y_2 )\le \lambda \Phi (x_1 ,y_1 )+(1-\lambda )\Phi (x_{2 },y_2 )$.

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 λ is a mixing weight, with λ $\in$ [0,1].

Resource competitiveness: ${\displaystyle \frac{\partial \Phi }{\partial C_{u,a}}}>0\hspace{0.33em},\hspace{0.33em}{\displaystyle \frac{\partial \Phi }{\partial C_{u,d}}}>0\hspace{0.33em},\hspace{0.33em}{\displaystyle \frac{{\partial }^2\Phi }{\partial C_{u,a}\hspace{0.33em}\partial C_{u,d}}}<0$.

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 ${ℂ}_U$ is performed as a function of the collective capacities of the airports $\left\{C_1,C_2,\dots C_u\right\}$. The consideration of multi-airport system coupling effects is defined as a normed linear space $\varsigma =\left\{\mathrm{C}=\left(C_1,C_2,\dots C_u\right)\left|C_u=\Phi \left(C_{u,a},C_{u,d}\right),\hspace{0.33em}\forall u\in U\right.\right\}$, and the number of paradigms ${‖\mathrm{C}‖}_p={\left({\displaystyle \sum _{u=1}{C_u}^p}\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$p$}\right.}},\hspace{0.33em}\hspace{0.33em}p\ge 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 $\varsigma$.

$\Psi :\hspace{0.33em}\varsigma \to {ℝ}^{+}$ satisfies subadditivity and homogeneity, where subadditivity characterizes the nonlinear superposition of capacity due to airspace resource sharing: $\Psi \left({\mathrm{C}}_1+{\mathrm{C}}_2\right)\le \Psi \left({\mathrm{C}}_1\right)+\Psi \left({\mathrm{C}}_2\right)$.

Homogeneity reflects a decreasing or constant effect of scale; that is, the increase in capacity of a single airport is greater than the increase in capacity of the entire airport complex: $\Psi \left(\lambda \mathrm{C}\right)={\lambda }^{\alpha }\Psi \left(\mathrm{C}\right),\hspace{0.33em}\hspace{0.33em}\alpha \le 1$.

Thus, as shown in Figure 1, single-airport capacity can be represented by a curve defined by Φ, while multi-airport capacity can be represented by a frontier hypersurface defined by Ψ.

Figure 1 shows a schematic of the capacity curve. The horizontal coordinates in the left figure indicate the number of aircraft arrivals, and the vertical axis indicates the number of aircraft departures. The curve in the figure indicates the capacity frontier. Each coordinate in the right diagram indicates the capacity of each airport in the multi-airport system, the pink surface in the diagram indicates the capacity frontier hypersurface, and the green point is the available capacity on the frontier.

2.2. Twenty-Four-Hourly Slot Coordination Parameters

A firebreak is an innovative mechanism based on dynamic capacity management, the core objective of which is to block the cumulative effect of delays across time by structurally configuring peak and valley periods in airport operations, thus realizing a dynamic balance between supply and demand and system stability, as shown in Figure 2. From the theoretical point of view, the essence of the slot firebreak is to construct a peak and valley structure with time-varying capacity parameters by inserting a capacity buffer “middle period” between neighboring peak periods (e.g., arrival and departure peaks).

This design draws on the idea of “partitioning” in system control theory: by setting up a dynamic buffer with low-capacity valleys, random delays in the preceding peak hours are absorbed, thus preventing delays from spreading through the queuing effect to the following hours (i.e., the chain reaction of congestion evolution). By quantifying the capacity gradient relationship during “peak–valley–peak” hours, a cyclical deviation framework for capacity–demand balance is established such that the airport capacity allocation not only maintains the flexibility to cope with demand fluctuations but also has the systematic resilience to prevent the spread of congestion.

In 24 h time coordination parameterization, the value of the slot firebreak stems from its precise intervention in time-dimensioned congestion dynamics. Traditional static capacity parameters are difficult to adapt to cyclical fluctuations in flight demand, while purely dynamic deviation increases the complexity of time management. Through the static peak and valley structure design, the slot firebreak achieves the purpose of providing “temporary storage space” for the preceding peak delays with the capacity redundancy in the valley time and quantifies the minimum duration of the buffer time through the delay absorption rate model, thus forming the capacity–demand elasticity matching [19,20]. Through the periodic configuration of the peak and valley capacity parameters, a dynamic deviation similar to the “respiratory rhythm” can be formed, such that the system can absorb the delays and gradually restore the steady state at the same time.

3. Multi-Airport System for Cooperative Optimization of Flight Schedules

3.1. Methodology Framework

The methodology framework proposed in this study is shown in Figure 3. The main framework for model operation includes model inputs, flight constraints, capacity constraints, airport resource constraints, multi-objective optimization, solving algorithms, and the Pareto front. This section will elaborate on the model’s construction.

3.2. Model Description

The purpose of airport slot optimization in a multi-airport system is to make full use of public airspace resources and reduce the overall delay rate of flights. However, as a scarce resource, airport slots face a series of problems, such as fairness in the process of adjusting airport slots. Therefore, this study focuses on the actual airport capacity and airport slot management of Chinese civil aviation; conducts overall optimization of airport slots in a multi-airport system, taking into account the needs of air traffic control, airports, and airlines in the utilization of airspace capacity, ground support resources, and acceptable slot deviation; and establishes a mixed-integer multi-objective optimization model in order to improve the fairness of the deviation of airport slots for all airports and airlines in the airport group.

3.3. Multi-Airport System for Cooperative Optimization of Flight Schedules

To construct an optimization model that meets the actual operation of the multi-airport system, this study makes the following assumptions:

Assumption 1: 

A slot is a period of time during which an aircraft may use airport facilities off-block or in-block. All time slots within the same airport are of the same length and are usually set at 5 min.

Assumption 2: 

Airports have the same authority to adjust the timings of arrival/departure flights. Typically, an airport has more power to adjust departure times for departing flights. In this chapter, no distinction is made between arrival and departure flights, and the range of slot deviations depends on the constraints of the model.

Assumption 3: 

Airlines have a maximum acceptable deviation for their requested slot; the maximum acceptable slot deviation is considered to be the same for all airlines, and it is assumed that the maximum deviation threshold can be adjusted based on different levels of resource constraints.

Assumption 4: 

Aircraft transit time is the time interval between the start of an aircraft’s parking in a parking position, acceptance of passenger departures, refueling, maintenance, passenger boarding, etc., and the time until takeoff again, which satisfies the minimum transit time.

3.3.1. Model Notation

To facilitate clarity and comprehension, definitions for the sets, parameters, and variables that feature in the mathematical model were provided in

Table 2. Model notation.

Notation	Description
Sets
M	Set of flights in the multi-airport system
U	Set of airports in the multi-airport system
I	Set of airline companies
Mu	Set of flights in airport u
Mu,a	Set of arrival flights in airport u,u $\in$ U
Mu,d	Set of departure flights in airport u,u $\in$ U
$M_u^2$	Set of transit flights in airport u,u $\in$ U
Mt	Set of flights in the time interval [1, t]
$M_t^k$	Set of flights passing fix k in the time interval [1, t]
$M_t^{k\prime }$	Set of flights passing fix k′ in the time interval [1, t]
$M_t^s$	Set of flights of the s aircraft type in the time interval [1, t]
Q	Set of aircraft types
S	Set of parking position types
T	Set of time slices
K	Set of corridors in airport terminal areas
H	Set of slots in the day, $h\in H$
Tm	Set of possible actual flight time slices for flight m
Parameters
$C_u^h$	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, h $\in$ T
$C_{u,d}^h$	Number of departure flights limit for airport u at time interval h,u $\in$ U, h $\in$ T
${ℂ}_u^a,{ℂ}_u^d,{ℂ}_u$	Airport group arrival capacity, departure capacity, and coupled capacity
$U_t^k$	Number of total flights limit passing fix k at time window t,k $\in$ K, t $\in$ T
tm	The requested slot for flight m
$t_{m_0}$	The actual slot for flight m
$t_{max}^d$	Maximum acceptable deviation for departure flights
$t_{max}^a$	Maximum acceptable deviation for arrival flights
$t_{ma,md}^{u,q}$	Minimum turnaround time for aircraft type q
$T_{ma,md}^{u,q}$	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
$c_u^{firebreak}$	Firebreak period capacity of airport u
$t_u^{firebreak}$	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
Decision variables
$X_m^t=\left\{\begin{array}{c}1 \\ 0\end{array}\right.$	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
Indirect decision variables
${\mu }_i$	Equity indicators of congestion
$a_u$	Number of flight deviations for airport u
${\delta }_{u,k}^a$	The time of arrival flights passing fix k in airport u,u $\in$ U, k $\in$ K
${\delta }_{u,k}^d$	The time of departure flights passing fix k in airport u,u $\in$ U, k $\in$ K

.

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).

$$Z_1=min{\sum }_{u\in U}\left\{{\sum }_{m\in M_u}\left[{\sum }_{t\in T}\left({\sum }_{t_m\in T_m}|t_{m_0}-t_m |\right)X_m^t\right]\right\},$$

(1)

(2) Fairness of airports considering slot deviation proportional to total demand at each airport.

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). Each airport’s unit flight deviation, as shown in Equation (3), is determined by dividing each airport’s deviation by the number of flights and determining fairness by the variance of the airports within the multi-airport system, as shown in Equation (4).

$$a_u={\displaystyle \frac{\sum _{m\in M}\sum _{t\in T}(t_{m_0}-t_m)·X_m^t}{C_u}}$$

(2)

$$\mathrm{N}={\displaystyle \frac{\sum {\mathrm{a}}_{\mathrm{u}}}{\left|\mathrm{U}\right|}}$$

(3)

$$Z_2=min{\displaystyle \frac{\sum _{u\in (0,n)}{(N-a_u)}^2}{\left|U\right|}}$$

(4)

(3) Fairness of airlines’ consideration of congestion contributions

It is obviously unfair for airlines with more off-peak time resources to bear a larger slot deviation in multi-airport system airport slot optimization. Therefore, in the multi-airport system optimization process, the fairness of airlines is reflected by the congestion contribution of each airline, i.e., adding the dimension of “congestion” to “contribution” to define the fairness of considering congestion. Under this rule, airlines with more flights during congested hours will generate more time deviations than stipulated by the original fairness indicator. Therefore, the objective function of the fairness of airlines considering congestion contribution is designed to assess the equilibrium of airport slot allocation by calculating the difference between the fairness index of the airlines and the overall fairness index of the industry. The goal is to minimize the difference between the two, in order to achieve the highest standard of fairness. The fairness objective function expression is shown in Equation (5).

$$Z_3=min\left\{max\left|{\mu }_i-{\displaystyle \frac{\sum _{i\in I}{\mu }_i}{\left|I\right|}}\right|\right\}$$

(5)

$${\mu }_i={\displaystyle \frac{\frac{\sum _{m\in M_i}\sum _{t_m\in T}\left|t_{m_0}-t_m\right|X_m^t}{\sum _{m\in M}\sum _{t_m\in T}\left|t_{m_0}-t_m\right|X_m^t}}{\frac{1}{2}\left(\frac{\sum _{m\in M_i}{v}_m^t}{\sum _{m\in M}{v}_m^t}+\frac{\left|M_i\right|-\sum _{m\in M_i}{v}_m^t}{\sum _{i\in I}\left|M_i\right|-\sum _{m\in M}{v}_m^t}\right)}},\hspace{1em}\forall i\in I$$

(6)

where ${\mu }_i$ indicates the fairness of airline i, with the numerator being the proportion of all slot deviations of airline i to the total slot deviations, as shown in Equation (6). ${ v}_m^t$ in the denominator is a known value of 0–1; $v_m^t$ = 1 indicates that time period t during which flight m occurs is congested (demand is greater than capacity); the flight “contributed” to the congestion at time t; otherwise, it is 0. A congested slot is defined as a period when the demand by flights exceeds the handling capacity of the airport. The denominator is the two components, congested and non-congested slots, and the share of airlines in each of these components is calculated and subsequently weighted and summed as a way of adjusting the weights of the slot deviations.

For the value of ${\mu }_i$, the discussion is illustrated in three cases:

${\mu }_i=1$: Treatment received by airline i is considered fair;

${\mu }_i<1$: Airline i enjoys preferential treatment;

${\mu }_i>1$: Airline i suffers from unfair treatment.

3.3.3. Constraints

(1) Slot uniqueness constraints

$$\sum _{t\in T}{X}_m^t=1,\hspace{1em}m\in M$$

(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

$$\left|t_{m_0}-t_m\le t_{max}\right|,m\in M_u,u\in U,$$

(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.

$${\sum }_{m\in M_{u,a}}{\sum }_{t\in s}^{s+55}{X}_m^t\le C_{u,a}^h,s=0,5,10,\dots 1385,u\in U,$$

(9)

$${\sum }_{m\in M_{u,d}}{\sum }_{t\in s}^{s+55}{X}_m^t\le C_{u,d}^h,s=0,5,10,\dots 1385,u\in U,$$

(10)

$${\sum }_{m\in M_u}{\sum }_{t\in s}^{s+55}{X}_m^t\le C_{u,d}^h,s=0,5,10,\dots 1385,u\in U,$$

(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 Ψ and the multi-airport system capacity $ℂ$ are introduced; these correspond to the multi-airport hourly capacity envelope function relations. In Equation (12), 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.

$$C=<C_1{,C}_2\dots ,C_u>, u\in U,$$

(12)

The airport group capacity constraints are shown in Equation (13).

$$<\sum _{t\in s}^{s+55}{X}_1^t, \sum _{t\in s}^{s+55}{X}_2^t,\dots ,\sum _{t\in s}^{s+55}{X}_m^t, >\le ℂ[C], s \mathrm{=} 0,5,10,\dots ,1385, u\in U$$

(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 constraints

Airport 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., ${\displaystyle \sum _{m\in M_u}{\displaystyle \sum _{t=r}^{r+11}{x}_m^t}}\le c_u^{firebreak}$; when $y_u^s=1$, Equation (16) is equivalent to the airport’s hourly flow being less than its capacity $c_u$.

$${\displaystyle \frac{\sum _{m\in M_u}\sum _{t=s}^{s+t_u^{peak}}-1}{c_u^{peak}{t}_u^{peak}}}=z_u^s,s=1,2,\cdots ,T-t_u^{peak},\forall u\in U$$

(14)

$$y_u^s=min\left(\right(1-z_u^s)\eta ,1),s=1,2,\cdots ,T-t_u^{peak},\forall u\in U$$

(15)

$$\begin{array}{l}{\displaystyle \frac{{\displaystyle \sum _{m\in M_u}{\displaystyle \sum _{t=r}^{r+11}{X}_m^t}}}{c_u^{firebreak}}}\le y_u^s({\displaystyle \frac{c_u}{c_u^{firebreak}}}-1)+1, s=1,2,\cdots T-t_u^{peak},\\ r=s+t_u^{peak},s+t_u^{peak}+1,\cdots s+t_u^{peak}+t_u^{firebreak}-12 , \forall u\in U\end{array}$$

(16)

(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. $∆$ 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).

$${\int }_{t-∆}^{t+∆}{f}_a\left(t\right)·dt=1$$

(17)

$${\int }_{t-∆}^{t+∆}{f}_d\left(t\right)·dt=1$$

(18)

Equation (19) proposes an arrival aircraft corridor judgment variable $Y_m^t.$

$$Y_m^t=\left\{\begin{array}{c}1, X_m^t=1\\ 0, X_m^t=0\end{array}\right.,t\in [t-{\delta }_{u,k}^a-∆,t-{\delta }_{u,k}^a+∆), m\in M_{u,a}$$

(19)

Equation (20) proposes a departure aircraft corridor port judgment variable ${\mathrm{Z}}_{\mathrm{m}}^{\mathrm{t}}$.

$$Z_m^t=\left\{\begin{array}{c}1, X_m^t=1\\ 0, X_m^t=0\end{array}\right.,t\in [t+{\delta }_{u,k}^d-∆,t+{\delta }_{u,k}^d+∆), m\in M_{u,d},$$

(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 $s=\left\{0,60,120\dots 1380\right\},k\in \left\{1,2,\dots ,K\right\}$. $U_t^k$ denotes the capacity of the corridor port at time interval t.

$$\begin{array}{c}{\displaystyle \sum _{t_m\in s}^{t_m+55}}{\displaystyle \sum _{u\in U}}{\displaystyle \sum _{m\in M_{u,a}}}{Y}_m^t·{\int }_{max(t-{\delta }_{u,k}^a-∆,s)}^{min(t-{\delta }_{u,k}^a+∆,s+55)}{f}_a\left(t_m-{\delta }_{u,k}^a\right)dt\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} +{\displaystyle \sum _{t\in s}^{t+55}}{\displaystyle \sum _{u\in U}}{\displaystyle \sum _{m\in M_{u,d}}}{Z}_m^t·{\int }_{max(t+{\delta }_{u,k}^a-∆,s)}^{min(t+{\delta }_{u,k}^d+∆,s+55)}{f}_d\left(t_m+{\delta }_{u,k}^d\right)dt\le U_t^k\end{array}$$

(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).

$${\mathbb{N}}_{\psi }^{+}=\left\{ENS{T}_1\left(s_{\psi }\right),\cdots ENS{T}_m\left(s_{\psi }\right)\right\},m\in M,\psi \in SE{C}_m,$$

(22)

where ${\mathrm{ENST}}_m(s_{\psi })$ denotes the time when flight m enters sector $s_{\psi }$, where $SE{C}_m$ 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).

$${\mu }_{m,\psi }^t=\left\{\begin{array}{c}1, t<ENS{T}_m\left(s_{\psi }\right)<t+59\wedge t+59<EXS{T}_m\left(s_{\psi }\right)\\ 0, otherwise\end{array}\right.,\forall t\in \left\{0,1,2\cdots 1440\right\}, m\in M$$

(23)

At this point, there is a sector access constraint per time interval, where $\overline{s_{\psi }}$ is the access capacity of sector $s_{\psi }$, as shown in Equation (24).

$$\sum _m\sum _t^{t+59}{\mu }_{m,\psi }^t\le \overline{s_{\psi }}\hspace{1em}\forall t\in \left\{0,1,2\cdots 1440\right\}, m\in M$$

(24)

(7) Transit time constraints

The minimum transit time and scheduled transit time constraints are shown in Equation (25) and Equation (26), respectively. The parameters of the minimum transit time constraints refer to the minimum transit schedule in the flight regularity statistics of the Civil Aviation Administration, and the scheduled transit time is published by the airlines.

$$\sum _{t_m\in T_{m_d}}{t}_m{X}_{m_d}^t-\sum _{t_m\in T_{m_a}}{t}_m{X}_{m_a}^t\ge t_{m_a,m_d}^q, (m_a,m_d)\in M_u^2, q\in Q$$

(25)

$$\sum _{t_m\in T_{m_d}}{t}_m{X}_{m_d}^t-\sum _{t_m\in T_{m_a}}{t}_m{X}_{m_a}^t\le T_{m_a,m_d}^q, (m_a,m_d)\in M_u^2, q\in Q$$

(26)

(8) Overnight parking position constraints

The number of overnight aircraft is constrained by the difference between the total number of stops at the airport and the number of overnight aircraft at the airport, as shown in Equation (27).

$${\sum }_{\tau =1}^t{\sum }_{m\in M_t}{a}_m{X}_m^t\le V-V\prime , t\in T{\sum }_{\tau =1}^t{\sum }_{m\in M_t^s}{a}_m{X}_m^t\le V_s-V_s^{\prime }, t\in T, s\in S$$

(27)

4. Algorithm

The objective of airport slot optimization for a multi-airport system is to determine the optimal airport slot allocation scheme by building a multiple-objective optimization model to minimize slot deviations with optimal fairness and aeronautical fairness of airports in the multi-airport system under airspace capacity coupling. In this study, the NSGA-II algorithm (Non-dominated Sorting Genetic Algorithm II) is used to solve the model. The algorithm process is shown in Algorithm 1.

Non-dominated sorting is a mechanism that partitions solutions within a population into distinct hierarchical levels (termed fronts) based on their mutual dominance relations. The Pareto front comprises the set of solutions that are not dominated by any other solution, representing the optimal trade-off solutions attainable for a multi-objective optimization problem [22]. The NSGA-II algorithm efficiently approximates the Pareto-optimal solution set for multi-objective problems by employing non-dominated sorting to stratify and select high-quality solutions (progressing towards the Pareto front) and utilizing crowding distance to maintain diversity within the solution set.

Algorithm 1. NSGA-II (Genetic algorithm partial pseudo-code).
Input: The population size, the maximal generation number, the dataset Output: Non-dominated solution set of slot displacement, airport fairness, airline fairness
1:   2:   3:   4:   5:   6:   7:   8:   9: 10: 11:	Pt = 50 ← Initialize population using proposed variable-length encoding strategy; t ← 0  //Initialize generation zero; m = 1500: Maximal generation number; Crossover probability= 0.8, Mutation probability = 0.3; Repeat Evaluate fitness of each individual in Pt; Qt ← Generate offspring using crossover and proposed mutation operators; Rt ← Pt ∪ Qt  //Combine parent and offspring populations; Pt+1 ← Environmental selection from Rt; t ← t+1; Until (t < m) Return individual with best fitness in Pt.

5. Example Validation

5.1. Introduction to the Arithmetic Example

To validate the proposed model, we conducted simulations using real-world data from Beijing’s multi-airport system, as shown in Figure 4. The multi-airport system in North China includes Beijing Capital International Airport (hereafter referred to as ZBAA), Beijing Daxing International Airport (hereafter referred to as ZBAD), and Tianjin Binhai International Airport (hereafter referred to as ZBTJ).

Beijing’s multi-airport system comprises three Level 3 airports (airports where capacity providers have not developed sufficient infrastructure, or where governments have imposed conditions that make it impossible to meet demand [1]), 19 approach control sectors, and 17 corridors. Within this structure, 11 approach control sectors manage traffic for more than one airport. The terminal area handles over 3000 flights daily, making it a congested terminal area featuring highly shared airspace with coupled airspace units. In the southwestern and eastern portions of the terminal area, in particular, air traffic controllers rely heavily on radar vectoring during actual operations to mitigate potential conflicts. Moreover, unscientific historical flight schedule planning has resulted in significant temporal imbalances between capacity and traffic demand within the current operation. These imbalances further propagate flight delays. Therefore, there is a pressing need to optimize the temporal and spatial distributions and to allocate resources effectively for the shared corridors and coupled sectors within this terminal area.

This study utilizes operational data from a representative day in the schedule of each airport in the 2023 summer and autumn seasons. The capacity parameters are the schedule coordination parameters, corridor capacity, and minimum transit time published by the Civil Aviation Administration of China. The maximum acceptable slot deviation for airlines is 30 min.

5.2. Scene Setting

Taking the multi-airport system airport slot as the main data, and based on the existing model, three experimental scenarios were formulated for system analysis. It should be noted that scenario 1 does not involve solving the equation and only serves as a benchmark for simulation verification.

Scenario 1 (Baseline): 

No optimization is applied; the original timetable is used as a reference.

Scenario 2 (Single-airport task): 

The base timetable of each of the three airports is optimized using a single-airport optimization model with the objective of minimizing slot deviation. The airport capacity constraints are used as the coordinating parameter for each airport, and the corridor capacity is allocated to each airport.

Scenario 3 (Multi-airports task): 

The base schedules of the three airports are combined and optimized using the multi-airport system optimization model, with minimum slot deviation, fairness to airports, and fairness to airline operators as the optimization objectives; airport capacity constraints using the multi-airport system envelope; and corridor port capacity shared by each airport.

Existing models and algorithms are applied using data from the relevant airports and corridor gates, and the data are analyzed and compared based on different scenarios.

5.3. Model Optimization Results

A multiple-objective optimization model with slot deviation, airport fairness, and airline congestion contribution fairness is established, and the NSGA-II algorithm is applied to solve the model to obtain the Pareto frontier solution set for airport slot optimization of the three airports in the Beijing multi-airport system. The Pareto frontier is obtained after the multi-airport schedule optimization, and a non-dominated frontier solution in the figure is selected for analysis. Based on the non-dominated sorting principle of the NSGA-II algorithm, any point on the Pareto front can be approximated as an optimal solution. For the execution of subsequent experimental verification, a distinguished point on the Pareto front was selected as a paradigm optimal solution for analysis in this study, as shown in Figure 5. This non-dominated solution has a slot deviation of 35,535, airport fairness of 0.009, and airline congestion contribution fairness of 13.615.

The optimization results are shown in Figure 5. Airport fairness, slot deviation, and airline congestion contribution fairness represent the three objective values of the model. The blue dots in the figure represent the feasible solutions for solving the output, the red dots represent the Pareto frontier solutions, and the circled red dots serve as the solutions used for data analysis below.

Based on the solution of the multi-airport model, the 24 h dynamic capacity values of each airport are obtained as shown in Figure 6. The obtained dynamic capacity values satisfy three constraints: the 24 h capacity constraints of each airport, the peak–valley single-cycle constraints, and the multi-airport system capacity constraints. The capacity trend of each airport conforms to the regularity of the peak period; for ZBAA, the original flight traffic exceeds the capacity of the airport at 07:00, 10:00, 13:00, 14:00, 17:00, 18:00, and 22:00; the optimization meets the capacity constraints in all cases; and the capacity is maximized. For ZBAD, the raw flight traffic exceeds the airport capacity at 0:00, 08:00, 09:00, 12:00, 14:00, 16:00, and 20:00, and the optimization meets the capacity constraints and maximizes the use of capacity. For ZBTJ, the original flight traffic exceeds the airport capacity at 10:00, 13:00, 15:00, 17:00, and 20:00, and all of them meet the capacity constraints after optimization. The three airports’ flight schedules are optimized to achieve full utilization of airport capacity and peak and trough phase capacity, and the optimization effect is significant.

For the analysis of the selected points in Figure 5 (the points labeled in the Pareto frontier), 20 min has the highest number of slot deviations, followed by 5, 10, and 15 min, and the least number of flights without slot deviations (Figure 7). The cumulative probabilities of the three airports’ patterns were similar, but there was a certain gap in fairness. ZBAA, ZBAD, and ZBTJ have 6.48%, 7.67%, and 10.11% non-adjusted deviations, respectively. According to the picture showing the least number of flights without deviation, 50% of the flights can be adjusted in less than 15 min. 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.

5.4. 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. Statistics of time deviation for each airport.

AIRPORT	ZBAA				ZBAD				ZBTJ
Absolute Time Deviation (Minutes)	SD a for SA b	SA b Probability	SD a for MA c	MA c Probability	SD a for SA	SA b Probability	SD a for MA c	MA c Probability	SD a for Single Airport	SA b Probability	SD a for MA c	MA c Probability
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 deviation	13,715		15,615		12,720		12,985		7005		6935

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

); 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.

Comparing the airport fairness and airline congestion contribution fairness based on different optimization methods (

Table 4. Comparison of results.

Optimization Method	Slot Deviation	Airport Fairness	Airline Congestion Contribution Fairness
Multi-airport	35,535	0.009	13.615
Single airport	33,440	0.018	18.044

) shows that the optimization results of the multi-airport system are all better than the single-airport optimization results, which indicates that more constraints and stronger constraints of the multi-airport model make the optimization of the multi-airport system more reasonable.

5.5. Simulation Verification

5.5.1. Simulation Environment Settings

Due to operational security limitations, computer simulations were used for verification. Using the AirTOp software, a computer simulation model containing the capital, Daxing, and Tianjin airports was set up with structural static parameters such as waypoints, routes, flight procedures, sectors, and other parameters of the multi-airport system airspace, as well as control operation parameters and operation rule parameters in accordance with the operation rules. The simulation model passed the calibration of the air traffic management department, and the simulation airspace environment included the capital city, Daxing, and Tianjin airports, which is a northbound operation and does not contain other airspace user activities.

5.5.2. Implementation of the Simulation

Scenario 1: 

Base timetable as input for simulation.

Scenario 2: 

Three timetables optimized for the single-airport system as input for Scenario 2.

Scenario 3: 

Optimized multi-airport timetable as input for Scenario 3 flight planning. A total of 31 computer simulations were performed for each scenario.

5.5.3. Simulation Results

(1) Simulation delay analysis

The simulated delay results for the three scenarios are presented in

Table 5. Simulation delay results.

Algorithm	Delay Value/Second (Mean ± Standard Deviation)
	Average Delay			Average Approach Delays			Average Departure Delays
	ZBAA	ZBAD	ZBTJ	ZBAA	ZBAD	ZBTJ	ZBAA	ZBAD	ZBTJ
Scenario 1	283.1 ± 19.4	216.4 ± 15.4	403.1 ± 29.1	306.2 ± 19.1	172.3 ± 10.8	300.4 ± 18.9	446.2 ± 19.7	286.3 ± 14.8	551.4 ± 24.6
Scenario 2	211.9 ± 21.1	201.4 ± 20.8	331.0 ± 36.2	174.6 ± 17.3	209.2 ± 18.1	379.1 ± 30.2	244.7 ± 22.6	200.8 ± 21.9	300.4 ± 41.4
Scenario 3	191.3 ± 10.2	157.6 ± 11.2	218.4 ± 9.6	174.1 ± 7.9	111.0 ± 7.8	210.4 ± 8.9	210.1 ± 6.3	199.6 ± 11.2	221.8 ± 13.4

. The main outputs of the simulation are the arrival delay, departure delay, and average delay for each airport. The flight arrival delays refer to the delays caused by aircraft in the terminal area due to sequencing, scheduling, etc.; the flight departure delays refer to the delays caused by aircraft due to runway waiting, slot waiting, etc.; and the average delays refer to the average delays incurred by aircraft at the airport. Table 5 shows the average arrival delay, average departure delay, and average delay of the three scenarios. Based on the results of the multi-airport optimization, all three airports have low delays compared with the single-airport optimization; therefore, the multi-airport optimization simulation is the best.

(2) Sector capacity–flow relationship analysis

Sector flows from the 16th simulation are taken for airport flow–capacity ratio analysis, as shown in Figure 9. In Scenario 1 with no optimization, seven sectors are involved, and overcapacity occurs in 19 time slots; in Scenario 2 with single-airport schedule optimization, seven sectors are involved, and overcapacity occurs in 24 time slots; and in Scenario 3 with multi-airport schedule optimization, no overcapacity occurs in the sectors. The overcapacity results in delays due to the deployment that occurs in the simulation.

(3) Analysis of capacity–flow relationships at corridors

Corridors from the 16th simulation are taken for airport flow–capacity ratio analysis, as shown in Figure 10. In Scenario 1, the AVBOX corridor is overcapacity at 1300, 1400, 2000, and 2100 h; the PEGSO corridor is overcapacity at 0800 h; and the BELAX corridor is overcapacity at 2200 h. In Scenario 2, the AVBOX corridor is overcapacity at 1300, 1400, 2000, and 2100 h; the ELKUR corridor is overcapacity at 0800 and 1700 h; and the PEGSO corridor is overcapacity at 0800 h. Both scenarios show multiple corridor overcapacities at peak hour segments, while Scenario 3 does not show corridor overcapacity periods, indicating better corridor constraint effects and optimization for multi-airport flights.

6. Conclusions

This study established a multi-objective decision-making model for optimizing flight schedules in a multi-airport system on the basis of the comprehensive consideration of the fairness of flight schedules in the multi-airport system, which was achieved through the introduction of probabilistic competitiveness of the shared resources and ensuring flexible compatibility with China’s 24 h schedule coordination parameter and its schedule slot firebreak. Combined with historical data for airport slots in the Beijing multi-airport system, the following conclusions were drawn through computation and analysis using the NSGA-II algorithm:

(1) The multi-airport system, based on the fairness of the flight schedules of the multi-airport system and the single-objective optimization of the single-airport system, demonstrates superior capability in balancing flight schedule slot deviations, the fairness of each airport in the multi-airport system, and the airlines concerned, when compared with the single-objective optimization of the single-airport project.

(2) Through the introduction of shared resource probabilistic competitiveness, this study more fully considers the factors influencing multi-airport system airport slot deviations to solve airport slot uncertainty and corridor constraints, thus developing a more realistic and comprehensive model with full consideration of the multi-airport system in terms of corridor capacity constraints, transit time, parking constraints, and so on. The common single-airport optimization model fails to consider the constraints affecting airport slot deviation fully and simply takes the minimum slot deviation as the goal, preventing it from making full use of the capacity of each airport and corridor to achieve a fairer and more reasonable slot deviation optimization scheme.

(3) The model incorporates China’s 24 h coordination parameters and slot firebreaks by dynamically adjusting the capacity–flow ratio during peak–valley cycles, aligning with real-world operational constraints.

This study demonstrated that the slot deviation optimization model based on airspace capacity coupling can better balance fairness in multi-airport systems and optimize the utilization of shared airspace capacity resources of each airport, corridor, etc. This model is not only limited to the multi-airport system in the Beijing terminal area but can also be applied to multi-airport systems in other regions of China and the world. There are also many possible directions for future research. Firstly, the research example in this study was a multi-airport system, which can be further expanded to incorporate more corridor capacity. Secondly, airspace capacity constraints under the influence of special weather conditions and other influences can also be explored further. Finally, while this study considered a slot firebreak of 1–2 h, future research can involve more in-depth analysis and validation of the setup time and duration.

The proposed model demonstrated high generalizability and extensibility. Regarding capacity parameters, it comprehensively addresses demands across multiple tiers. While encompassing fundamental flight-related parameters, its architecture permits dynamic parameter adjustments (e.g., excluding terminal capacity parameters). This design deliberately accounts for regional bottleneck variations—predominantly flight zones in China versus terminal areas in Europe and North America. Crucially, the model replaces traditional sequential flight coordination (which relies on carrier prioritization systems and suffers from inherent inefficiency) with a global optimization algorithm.

This paradigm shift enables the generation of more equitable and efficient bulk slot allocation solutions (providing bundled slot offers), representing a disruptive innovation to existing manual coordination methodologies. This disruptive potential aligns with China’s ongoing aviation slot management reform agenda (e.g., online seasonal coordination in favor of computational systems), indicating the model’s significant operational viability for use in future slot governance. Engineered explicitly for practical implementation, it contributes to daily slot management, seasonal coordination cycles, and long-term strategic planning (e.g., route network optimization), with primary efficacy in seasonal scheduling. Slot authorities could leverage the model to proactively identify pinned constraints, thereby offering airlines clearer strategic guidance during planning. Furthermore, introduced innovations such as capacity coupling mechanisms and peak/off-peak slot configuration frameworks have substantial research potential. These concepts warrant seamless integration into broader airspace management regimes across tactical, pre-tactical, and flow management phases, extending the model’s academic and operational relevance.