A Stochastic Optimization Model for Multi-Airport Flight Cooperative Scheduling Considering CvaR of Both Travel and Departure Time

Abstract

By assuming that both travel and departure time are normally distributed variables, a multi-objective stochastic optimization model for the multi-airport flight cooperative scheduling problem (MAFCSP) with CvaR of travel and departure time is firstly proposed. Herein, conflicts of flights from different airports at the same waypoint can be avoided by simultaneously assigning an optimal route to each flight between the airport and waypoint and determining its practical departure time. Furthermore, several real-world constraints, including the safe interval between any two aircraft at the same waypoint and the maximum allowable delay for each flight, have been incorporated into the proposed model. The primary objective is minimization of both total carbon emissions and delay times for all flights across all airports. A feasible set of non-dominated solutions were obtained using a two-stage heuristic approach-based NSGA-II. Finally, we present a case study of four airports and three waypoints in the Beijing–Tianjin–Hebei region of China to test our study.

1. Introduction

At present, China has formed world-class airport clusters such as the Beijing–Tianjin–Hebei region, the Guangdong–Hong Kong–Macao Greater Bay Area, the Yangtze River Delta and the Chengdu–Chongqing region. These are MAFCSP systems with dense aviation networks, well-developed ground transportation systems, complete comprehensive functions and coordinated development. They directly or indirectly serve world-class urban agglomerations and promote regional economic development and international exchanges. In the MAFCSP network, there are some nodes (i.e., airports and waypoints) and edges linking them. Each flight at an airport with its alternative paths between the airport and waypoint at a planned departure time at the planning level needs a specific scheduling scheme designed according to the actual situation when it is actually executed. Obviously, if any two flights at different airports pass through the same waypoint, there are generally two strategies in order to avoid possible collision between them. On the one hand, if their departure times are unchanged, they can adjust their routes. On the other hand, if their routes do not change, their departure times can be adjusted. Compared with the single airport flight scheduling problem (SAFSP), MAFCSP can reduce the total travel time they hover in the air to avoid conflicts between them by coordinating the routes and times of related flights in each airport [1,2,3,4,5]. The solution scale of MAFCSP is larger than that of SAFSP, which has been proved to be an NP-hard problem [6]. Therefore, MAFCSP has attracted the attention of many scholars and engineers.

The main motivation of this paper is to integrate the route design and departure time setting process of each flight in MAFCSP to find the relationship between safety, efficiency, and carbon emissions. At present, existing research studies them separately, resulting in more travel mileages and times for all flights across all airports. Further, carbon emissions are a concern in civil aviation operations. The carbon emissions of different flights on different routes vary depending on their aircraft performance. Hence, it is necessary to study green MAFCSP with a combination of assigning an optimal route of each flight in candidates and determining its actual departure time to reduce delays and carbon emissions.

Another motivation of this paper is to study stochastic MAFCSP with CVaR of travel and departure times. Most studies have focused on MAFCSP with certain travel and departure times [1,2,3,4] and neglected their uncertainty in the real world [5,6,7]. Although some studies have studied MAFCSP with uncertain departure times or travel times separately, no studies have considered MAFCSP with both of these variables. In general, stochastic MAFCSP such as expected value model [8] and chance constraints based on their probability distributions (i.e., mean and variance) [4] have been used to deal with such uncertainty. However, drawbacks of these stochastic models lie in (1) decision-makers’ aversion to tail risk is neglected; and (2) extreme event optimization at specific confidence levels being ignored, while the robustness of the results to tail risk cannot be analyzed. CVaR is a risk measurement tool based on random distribution, which focuses on the average of tail losses and is used to quantify extreme losses under specific risk levels. Hence, it is necessary to study stochastic MAFCSP with CVaR of travel and departure times.

The key contribution of this paper lies in putting forward a multi-objective stochastic optimization model for MAFCSP with uncertain travel and departure times. The two tasks of this study are as follows: (1) To determine the optimal schedules for MAFCSP by concurrently assigning an optimal route of each flight in candidates between the airport and waypoint and determining its actual departure time to prevent potential conflicts between all flights from different airports at all shared waypoints, by taking into account some real-world constraints such as the safe interval between any two aircrafts at the same waypoint and the maximum allowable delay for each flight. (2) To establish a stochastic optimization model for MAFCSP by presuming CVaR of travel and departure time to examine the impact of specific confidence level on the schedules. (3) To design a two-stage heuristic approach-based NSGA-II to find a feasible set of non-dominated solutions. Finally, a case study of four airports and three waypoints in the Beijing–Tianjin–Hebei region of China is employed to generate optimal schedules, thereby validating the feasibility of this research. This study can be used as a diagnostic tool to shift decision-making from “avoiding unlikely losses” to “preparing for their average severity.” in designing the MAFCSP scheme, ensuring decisions account for both the probability and impact of extreme outcome in complex, high-stakes environments.

The remainder of this paper is organized as follows: Section 2 provides a concise overview of the relevant literature pertaining to MAFCSP. Section 3 introduces the concepts of MAFCSP with uncertain travel speed and elaborates on the formulation of our robust model. Section 4 applies a two-stage heuristic approach-based NSGA-II to obtain a feasible set of non-dominated solutions. Section 5 presents a case study of four airports and three waypoints in the Beijing–Tianjin–Hebei region of China, aimed at validating the feasibility of this research. Finally, concluding remarks and directions for future work are discussed in Section 6.

2. Related Work

The rapid growth of air traffic, especially in metropolitan regions with multiple airports, has introduced significant operational challenges in terms of flight scheduling, slot allocation, and resource management. A multi-airport system (MAS) consists of two or more airports within a shared airspace or terminal area, and coordinating operations across these airports requires overcoming various challenges such as competing for airspace, runway capacities, and passenger transfers [1,2,3,4,5]. Traditional scheduling models have primarily focused on addressing single-airport operational challenges. Such models typically concentrate on resource allocation within individual airports, with objectives like minimizing schedule deviations or maximizing airline revenue. However, they overlook the spatio-temporal interdependencies between airports. Particularly in congested airspace, flights may suffer from cascading delays due to shared route points or airspace capacity constraints. Furthermore, uncertainties in flight times and dynamic capacity limitations further complicate scheduling decisions. In multi-airport system (MAS) operations, the interdependencies between airports are critical, often resulting in inefficiencies and delays [6,7,8,9].

Despite significant research on single-airport scheduling, studies addressing multi-airport coordination remain limited. Existing work mainly focuses on specific operational aspects such as slot allocation [10,11,12], airspace structure optimization [13,14,15], and robust scheduling mechanisms under uncertainty [16,17]. However, a comprehensive scheduling model that integrates airspace constraints, dynamic capacity allocation, and inter-airport cooperation is still lacking. The Civil Aviation Administration of China’s 2025 strategic plan for MAS aims to address these challenges by improving capacity management and transfer connectivity [18]. Nonetheless, gaps persist in developing unified MAS scheduling models that can handle uncertain conditions such as weather disruptions, fluctuating runway capacity, and passenger flow dynamics. This study proposes a robust scheduling framework integrating genetic algorithms with predictive airspace capacity forecasting, aiming to reduce delays while optimizing slot allocation and passenger transfers in MAS.

In recent years, airport slot scheduling has transitioned from traditional single-airport models to more integrated frameworks that account for the dynamics of multi-airport systems (MASs). These systems present unique challenges due to shared airspace, interconnected airport operations, and increased uncertainty in both demand and capacity.

Early studies predominantly focused on optimizing slot allocation from a centralized, single-airport perspective. However, with the growing complexity of air transportation networks, researchers began to explore multi-objective scheduling frameworks that can balance efficiency, fairness, and operational constraints. For example, Feng et al. [19] introduced a bi-objective airport slot scheduling model that explicitly considers both scheduling efficiency and environmental noise abatement, marking a shift toward incorporating sustainability in scheduling decisions. Similarly, Katsigiannis et al. [20] and Katsigiannis and Zografos [21] developed multi-level optimization models that account for flight flexibility and total capacity constraints, reflecting a deeper understanding of the trade-offs in real-world slot coordination.

As attention shifted toward MAS, researchers recognized the need to incorporate inter-airport dependencies and the uncertainties in airspace capacity and travel time. Wang et al. [22] proposed a novel slot allocation model that considers both airspace capacity and flying time uncertainty, providing a realistic framework for MAS coordination. Corolli et al. [23] and Delahaye and Wang [24] further enhanced this line of work by employing stochastic models to evaluate how capacity uncertainty affects slot feasibility and system reliability. Liu et al. [25] extended this approach to MAS slot allocation under uncertainty, emphasizing the importance of integrated models for network-wide performance.

Uncertainty modeling has become a cornerstone in modern slot allocation research. Techniques such as chance-constrained programming and robust optimization have been widely adopted. Clare and Richards [26] applied chance constraints to air traffic flow management, allowing for probabilistic capacity violations, while Wang and Zhao [27] developed a simultaneous optimization framework that jointly considers airport network slot assignment and uncertain capacity constraints. Agogino and Rios [28] evaluated the robustness of air traffic scheduling mechanisms under departure uncertainty, suggesting that resilient models are essential to mitigate cascading delays across interconnected airports.

To support these modeling approaches, foundational scheduling and timetabling methodologies have also been adapted. For instance, Brucker et al. [29] presented a comprehensive classification of resource-constrained project scheduling models, many of which underpin slot allocation algorithms. Burke et al. [30,31] further contributed heuristic methods for timetable generation, which have been translated into slot assignment contexts for large-scale MAS networks.

As the field evolves, there is increasing emphasis on aligning operational objectives with principles of fairness and accessibility. Zografos et al. [32] and Androutsopoulos and Madas [33] explored fairness-driven extensions to strategic scheduling, offering mechanisms to balance stakeholder priorities in constrained environments. Jiang and Zografos [34] developed decision-making frameworks to integrate fairness directly into MAS slot allocation policies. These approaches are critical to ensure the equitable distribution of limited capacity, particularly in congested and competitive airspace systems.

Finally, to enhance practical applicability, several studies have proposed IATA-compliant mechanisms and decision-support tools. Ribeiro et al. [35] introduced an optimization approach that respects existing guidelines while improving allocation outcomes. Fairbrother et al. [36] and Castelli et al. [37] evaluated the performance of scheduling systems that simultaneously incorporate efficiency, fairness, and airline preferences, providing a bridge between theoretical development and implementation.

Together, these studies have laid a solid foundation for the cooperative scheduling of MAS operations. Yet, most approaches still treat the assignment of all flights to their optimal route and determination of departure times for these flights in isolation in uncertain environments (i.e., uncertain travel and departure time). The integration of these elements into a unified, stochastic scheduling framework based on CvaR of travel and departure times remains an open research challenge that this study aims to address.

3. Methodology

3.1. Research Framework

An MAFCSP network contains many nodes such as some airports and waypoints as well as edges linking them. There are a set of flights between their airports and waypoints with their corresponding planned departure times. At the planning level, the routes and schedules of all flights should be optimized, i.e., each flight leaves its airport at the actual departure time and sequentially selects a route to reach its related waypoint at the actual arrival time. If two adjacent flights are routed through the same airport or waypoint, they must maintain a safe interval time. Obviously, no coordination of all routes and schedules for these flights may cause them to pass though such the same node within a safe interval, resulting in some flying flights at different airports hovering for a while to pass through the same waypoint, or some flights at the same airport waiting at its gate must run for a period of time to drive off the runway. Furthermore, the no-flight time window for each waypoint, related to control and weather, is also considered in this model. In addition, CvaR is used to depict uncertain travel times between two adjacent nodes, caused by bad weather or the driving behavior of a pilot, etc., and the departure time of each flight, caused by hesitancy of an airline to choose when a flight should take off. The main aim of this paper is to find the optimal relationship between the network layout of MAFCSP, the space–time distribution of flights, their routes and schedules, the uncertainty of departure and arrival times, total carbon emissions, and delay times for all flights across all airports.

Figure 1 depicts the principles and scope of a small MAFCSP instance, including three airports (A1–A3), two waypoints (P1–P2), six routes between them, and seven flights. In this example, the routes and schedules of these flights (F1–F7) are yielded using our model. Take F5 and F6 as an example: if both of them select the shortest routes of 5 min, they will pass through P2 at 6:05, resulting in the two flights will conflicting.

The purpose of this study is to present a multi-objective stochastic mixed integer programming model for MAFCSP to simultaneously design the routes of these flights at each airport and determine their actual departure times to avoid conflicts between them to minimize both total carbon emissions and delay times for all flights across all airports [38,39]. To ensure that our model could be in line with real-world situations, the main assumptions are given as follows:

(1)

The no-fly status of the air traffic network can be obtained in advance.

(2)

The safe interval time of two adjacent flights passing through the same waypoint is assumed to be a fixed interval independent of their altitude level.

(3)

The safe interval time of any two adjacent flights leaving the same airport, with no relation to the number of runways, is a fixed interval.

(4)

The uncertain statistical characteristics for travel times between nodes and departure times of all flights can also be estimated through big data analysis of flight operations.

(5)

The carbon emission per unit time/mileage for each flight can be obtained in advance.

3.2. Notations

A few preliminary definitions and notations are given in

Table 1. Definitions and notations of our model.

Set:
$A$	Set of airports
$W$	Set of waypoints
$F_a$	Set of flights leaving from airport $a$
$F_w$	Set of flights passing though waypoint $w$
$R_{aw}$	Set of routes between airport $a$ and waypoint $w$
Index:
$a$	Airport
$k,v$	Flight
$w$	Waypoint
$r$	Route
Index:
$d_r$	Length of route $r$ between airport $a$ and waypoint $w$
$c_k$	The carbon emission per hour of flight $k$
$v_k$	The speed of flight $k$
$T_s^k$	The planned departure time of flight $k$
$T_o$	The maximum delay time for each flight
$T_p$	The safe departure intervals for two aircrafts at any airport
$T_s$	The safety flighting intervals of two aircrafts at any waypoint
$[T_a^L,T_a^U]$	The control time window of any airport $a$
$[T_w^L,T_w^U]$	The control time window of any waypoint $w$
$H$	A relatively large fixed value
Parameter:
$x_r^k$	Whether flight $k$ is assigned route $r$
$t_w^k$	The actual arrival time of flight k passing through the waypoint $w$
$t_s^k$	The actual departure time of flight $k$

.

3.3. Formulation

In this section, we present a novel 0–1 mixed linear integer programming model for MAFCSP that incorporates uncertain travel and departure times. The model is outlined as follows:

$$\min {\mathrm{f}}_1={\sum }_{\mathrm{a}\in \mathrm{A}}{\sum }_{\mathrm{w}\in \mathrm{W}}{\sum }_{\mathrm{k}\in {\mathrm{F}}_{\mathrm{a}}}{\sum }_{\mathrm{r}\in {\mathrm{R}}_{\mathrm{a}\mathrm{w}}}{\mathrm{x}}_{\mathrm{r}}^{\mathrm{k}}.{\displaystyle \frac{{\mathrm{d}}_{\mathrm{r}}}{{\mathrm{v}}_{\mathrm{k}}}}{.\mathrm{c}}_{\mathrm{k}}$$

(1)

$$\min {\mathrm{f}}_2={\sum }_{\mathrm{a}\in \mathrm{N}}{\sum }_{\mathrm{k}\in {\mathrm{F}}_{\mathrm{a}}}{[\mathrm{t}}_{\mathrm{s}}^{\mathrm{k}}-{\mathrm{T}}_{\mathrm{s}}^{\mathrm{k}}]$$

(2)

$${\sum }_{\mathrm{r}\in {\mathrm{R}}_{\mathrm{a}\mathrm{w}}}{\mathrm{x}}_{\mathrm{r}}^{\mathrm{k}}=1,\mathrm{w}\in \mathrm{W},\mathrm{a}\in \mathrm{A},\mathrm{k}\in {\mathrm{F}}_{\mathrm{a}}$$

(3)

$${\mathrm{t}}_{\mathrm{s}}^{\mathrm{k}}+{\displaystyle \frac{{\mathrm{d}}_{\mathrm{r}}}{{\mathrm{v}}_{\mathrm{k}}}}-\left(1-{\mathrm{x}}_{\mathrm{r}}^{\mathrm{k}}\right).\mathrm{H}\le {\mathrm{t}}_{\mathrm{w}}^{\mathrm{k}},\mathrm{w}\in \mathrm{W},\mathrm{a}\in \mathrm{A},\mathrm{k}\in {\mathrm{F}}_{\mathrm{a}}$$

(4)

$${\mathrm{t}}_{\mathrm{s}}^{\mathrm{k}}+{\displaystyle \frac{{\mathrm{d}}_{\mathrm{r}}}{{\mathrm{v}}_{\mathrm{k}}}}+\left(1-{\mathrm{x}}_{\mathrm{r}}^{\mathrm{k}}\right).\mathrm{H}\ge {\mathrm{t}}_{\mathrm{w}}^{\mathrm{k}},\mathrm{w}\in \mathrm{W},\mathrm{a}\in \mathrm{A},\mathrm{k}\in {\mathrm{F}}_{\mathrm{a}}$$

(5)

$${0\le \mathrm{t}}_{\mathrm{s}}^{\mathrm{k}}-{\mathrm{T}}_{\mathrm{s}}^{\mathrm{k}}\le {\mathrm{T}}_{\mathrm{o}},\mathrm{k}\in {\mathrm{F}}_{\mathrm{a}},\mathrm{a}\in \mathrm{A}$$

(6)

$${|\mathrm{t}}_{\mathrm{s}}^{\mathrm{k}}-{\mathrm{t}}_{\mathrm{s}}^{\mathrm{v}}|\ge {\mathrm{T}}_{\mathrm{p}},\mathrm{k},\mathrm{v}\in {\mathrm{F}}_{\mathrm{a}},\mathrm{a}\in \mathrm{A}$$

(7)

$$\left|{\mathrm{t}}_{\mathrm{w}}^{\mathrm{k}}-{\mathrm{t}}_{\mathrm{w}}^{\mathrm{v}}\right|\ge {\mathrm{T}}_{\mathrm{s}},\mathrm{k},\mathrm{v}\in {\mathrm{F}}_{\mathrm{w}},\mathrm{w}\in \mathrm{W}$$

(8)

$${{\mathrm{T}}_{\mathrm{a}}^{\mathrm{L}}\le \mathrm{t}}_{\mathrm{s}}^{\mathrm{k}}\le {\mathrm{T}}_{\mathrm{a}}^{\mathrm{U}},\mathrm{k}\in {\mathrm{F}}_{\mathrm{a}},\mathrm{a}\in \mathrm{A}$$

(9)

$${{\mathrm{T}}_{\mathrm{w}}^{\mathrm{L}}\le \mathrm{t}}_{\mathrm{w}}^{\mathrm{k}}\le {\mathrm{T}}_{\mathrm{w}}^{\mathrm{L}},\mathrm{k}\in {\mathrm{F}}_{\mathrm{w}},\mathrm{w}\in \mathrm{W}$$

(10)

The objective function (1) is to minimize total carbon emissions related to fuel consumption for the mileage of all flights assigned routes. The objective function (2) is to minimize the total deviation between the planned and actual departure times for all flights at different airports. Constraint (3) ensures that each flight must be assigned to one of the routes between its airport and waypoint. Constraints (4) and (5) are used to calculate the relationship between the departure time of each flight leaving the airport and the arrival time of each flight arriving at the waypoint. Constraint (6) guarantees that the deviation between the planned and actual departure times of each flight is within a certain range. Constraint (7) guarantees that the difference in departure times of any two flights at each airport is greater than the safety interval. Constraint (8) guarantees that the difference in the arrival times of any two flights at each waypoint is greater than the safe interval. Constraint (9) ensures each flight at each airport can only take off within the feasible time window affected by military control or weather activities, etc. Constraint (10) ensures some flights arriving at each waypoint from different airports must pass through this airspace within the feasible time window affected by military control or weather activities, etc.

As above mentioned, objective function (2) and constraint (5) involve the uncertainty in the planned departure time ${\mathrm{T}}_{\mathrm{s}}^{\mathrm{k}}$ of each flight. To describe such uncertainty of ${\mathrm{T}}_{\mathrm{s}}^{\mathrm{k}}$ using CVaR, assuming the random distribution of uncertain ${\mathrm{T}}_{\mathrm{s}}^{\mathrm{k}}(\mathrm{y})$ is $\mathrm{G}({\mathrm{x}}_{\mathrm{r}}^{\mathrm{k}},{\mathrm{t}}_{\mathrm{s}}^{\mathrm{k}},{\mathrm{t}}_{\mathrm{w}}^{\mathrm{k}},\mathrm{y})$, the possibility within a certain threshold $\propto$ could be calculated as follows:

$$\mathsf{\Psi }\left({\mathrm{x}}_{\mathrm{r}}^{\mathrm{k}},{\mathrm{t}}_{\mathrm{s}}^{\mathrm{k}},{\mathrm{t}}_{\mathrm{w}}^{\mathrm{k}},\mathsf{\alpha }\right)\triangleq {\int }_{\mathrm{G}({\mathrm{x}}_{\mathrm{r}}^{\mathrm{k}},{\mathrm{t}}_{\mathrm{s}}^{\mathrm{k}},{\mathrm{t}}_{\mathrm{w}}^{\mathrm{k}},\mathrm{y})\le \propto }{\mathrm{T}}_{\mathrm{s}}^{\mathrm{k}}(\mathrm{y})\mathrm{d}\mathrm{y}$$

(11)

Based on Equation (11), the VaR of our scheme with consideration of the confidence level $\mathsf{\beta }$ can be defined as follows:

$${\mathrm{V}\mathrm{a}\mathrm{R}}_{\mathsf{\beta }}\left({\mathrm{x}}_{\mathrm{r}}^{\mathrm{k}},{\mathrm{t}}_{\mathrm{s}}^{\mathrm{k}},{\mathrm{t}}_{\mathrm{w}}^{\mathrm{k}}\right)\triangleq \mathrm{m}\mathrm{i}\mathrm{n}\{\mathsf{\alpha }\in \mathfrak{R}:\mathsf{\Psi }\left(({\mathrm{x}}_{\mathrm{r}}^{\mathrm{k}},{\mathrm{t}}_{\mathrm{s}}^{\mathrm{k}},{\mathrm{t}}_{\mathrm{w}}^{\mathrm{k}},\mathsf{\alpha }\right)\ge \mathsf{\beta }\}$$

(12)

By considering that the expected loss value exceeds ${\mathrm{V}\mathrm{a}\mathrm{R}}_{\mathsf{\beta }}\left({\mathrm{x}}_{\mathrm{r}}^{\mathrm{k}},{\mathrm{t}}_{\mathrm{s}}^{\mathrm{k}},{\mathrm{t}}_{\mathrm{w}}^{\mathrm{k}}\right)$, the conditional risk level CVaR of our scheme can be obtained accordingly:

$$\mathrm{C}{\mathrm{V}\mathrm{a}\mathrm{R}}_{\mathsf{\beta }}\left({\mathrm{x}}_{\mathrm{r}}^{\mathrm{k}},{\mathrm{t}}_{\mathrm{s}}^{\mathrm{k}},{\mathrm{t}}_{\mathrm{w}}^{\mathrm{k}}\right)\triangleq {\displaystyle \frac{1}{1-\mathsf{\beta }}}{\int }_{\mathrm{G}({\mathrm{x}}_{\mathrm{r}}^{\mathrm{k}},{\mathrm{t}}_{\mathrm{s}}^{\mathrm{k}},{\mathrm{t}}_{\mathrm{w}}^{\mathrm{k}},\mathrm{y})\ge {\mathrm{V}\mathrm{a}\mathrm{R}}_{\mathsf{\beta }}\left({\mathrm{x}}_{\mathrm{r}}^{\mathrm{k}},{\mathrm{t}}_{\mathrm{s}}^{\mathrm{k}},{\mathrm{t}}_{\mathrm{w}}^{\mathrm{k}}\right)}{\mathrm{G}({\mathrm{x}}_{\mathrm{r}}^{\mathrm{k}},{\mathrm{t}}_{\mathrm{s}}^{\mathrm{k}},{\mathrm{t}}_{\mathrm{w}}^{\mathrm{k}},\mathrm{y})\mathrm{T}}_{\mathrm{s}}^{\mathrm{k}}(\mathrm{y})\mathrm{d}\mathrm{y}$$

(13)

In order to approximate the calculate ${\mathrm{V}\mathrm{a}\mathrm{R}}_{\mathsf{\beta }}\left({\mathrm{x}}_{\mathrm{r}}^{\mathrm{k}},{\mathrm{t}}_{\mathrm{s}}^{\mathrm{k}},{\mathrm{t}}_{\mathrm{w}}^{\mathrm{k}}\right)$, ${\mathrm{T}}_{\mathrm{s}}^{\mathrm{k}}(\mathrm{y})$ will be divided into a set L of possibilities, i.e., ${\mathrm{T}}_{\mathrm{s}\left[\mathrm{l}\right]}^{\mathrm{k}}$. Its linearization can be approximated, such as how Equation (14) is solved by a solver.

$$\underset{{\mathrm{x}}_{\mathrm{r}}^{\mathrm{k}},{\mathrm{t}}_{\mathrm{s}}^{\mathrm{k}},{\mathrm{t}}_{\mathrm{w}}^{\mathrm{k}},\mathsf{\theta },\mathsf{\alpha }}{\min }\{\mathsf{\theta }:\mathsf{\alpha }+{\displaystyle \frac{1}{\mathrm{L}(1-\mathsf{\beta })}}{\sum }_{\mathrm{l}=1}^{\mathrm{L}}{[\mathrm{G}\left({\mathrm{x}}_{\mathrm{r}}^{\mathrm{k}},{\mathrm{t}}_{\mathrm{s}}^{\mathrm{k}},{\mathrm{t}}_{\mathrm{w}}^{\mathrm{k}},{\mathrm{T}}_{\mathrm{s}\left[\mathrm{l}\right]}^{\mathrm{k}}\right)-\mathrm{a}]}^{+}\le \mathsf{\theta }\}$$

(14)

Similarly to CVaR of ${\mathrm{T}}_{\mathrm{s}}^{\mathrm{k}}$, the CVaR of uncertain travel time ${\displaystyle \frac{{\mathrm{d}}_{\mathrm{r}}}{{\mathrm{v}}_{\mathrm{k}}}}$ can be approximately calculated like Equations (11)–(14).

4. Two-Stage Heuristic Approach-Based NSGA-II for Solving MAFCSP

In this study, we aim to optimize two potentially conflicting objectives between total carbon emissions and delay times for all flights across all airports. To address the limitations of the weighted-sum approach, which requires assigning weights to reflect decision-makers’ preferences, we have chosen to utilize NSGA-II to identify a set of Pareto optimal solutions for the proposed model [40].

As previously mentioned, our model encompasses three core variables, including ${\mathrm{x}}_{\mathrm{r}}^{\mathrm{k}}$, ${\mathrm{t}}_{\mathrm{w}}^{\mathrm{k}}$, and ${\mathrm{t}}_{\mathrm{s}}^{\mathrm{k}}$, where ${\mathrm{x}}_{\mathrm{r}}^{\mathrm{k}}$ determines ${\mathrm{t}}_{\mathrm{w}}^{\mathrm{k}}$ and ${\mathrm{t}}_{\mathrm{s}}^{\mathrm{k}}$. Obviously, once the routes of these flights can be selected in candidates (i.e., ${\mathrm{y}}_{\mathrm{i}}^{\mathrm{k}}$), their actual departure times (i.e., ${\mathrm{x}}_{\mathrm{i}\mathrm{j}}^{\mathrm{k}}$) are also easily obtained according to the principle of “first-come-first-served”. Hence, a two-stage heuristic approach-based NSGA-II, shown in Figure 2, is developed to solve MAFCSP.

In the first stage, NSGA-II is employed to assign a route in candidates for each flight. In the second stage, a polynomial algorithm is integrated into NSGA-II to find the actual departure time for each flight. Through these two operations, for the solution of MAFCSP with some constraints, the safe interval between any two aircrafts at the same waypoint and the maximum allowable delay for each flight are also established within the objective function.

4.1. NSGA-II in the First Stage

A one-dimensional vector $\mathrm{U}=({\mathrm{u}}_1,{\mathrm{u}}_2,\dots ,{\mathrm{u}}_{\mathrm{I}})$ represents the chromosome of a feasible solution in MAFCSP, where each element ${\mathrm{u}}_{\mathrm{i}}$ in $\mathrm{U}$, being a natural number in 1, 2, …, $\mathrm{K}$, denotes route ${\mathrm{u}}_{\mathrm{i}}$ of flight $\mathrm{i}$ ($\mathrm{i}$ = 1, 2, …, $\mathrm{I}$). For example, a chromosome vector $\mathrm{U}$ = {1 1 2 2 1 2} of six flights could be coded as follows: flights 1, 2, and 5 are assigned to route 1; flights 3, 4, and 6 are assigned to route 2.

Figure 3 illustrates the optimization procedure of the NSGA-II. An initial population, comprising a set of individuals, is generated randomly. At each iteration, each individual is first decoded to assign routes in candidate for all flights, and a polynomial algorithm is employed to determine the actual departure times for these flights. Subsequently, all objective functions are computed for fitness evaluation. Selected individuals from the parent population undergo gene exchange through crossover and mutation operators to produce new individuals. The current population, which includes both older and newly generated individuals, is then sorted and selected again based on non-domination criteria to obtain offspring. This selection process relies on both rank and crowding distance associated with each individual. The crowding distance—calculated by averaging the distances between individuals within a front—is utilized to ensure a consistent distribution of solutions along the Pareto front. When solutions reside in the same non-dominated front, those with higher crowding distances are prioritized; conversely, if they differ in rank, selections favor those with lower ranks first. The algorithm continues until it reaches the maximum number of iterations specified.

4.2. Polynomial Algorithm in Second Stage

In the first stage, routes in candidate are assigned to all flights. Based on the output of the first stage algorithm, the actual departure times for these flights would be found based on the principle of “first-come-first-served”. The detail process for a polynomial algorithm to calculate the actual departure times for these flights is described as follows.

Step 1: Sort all departing flights at each airport by their departure times.

Step 2: Based on the safe departure intervals of flights at each airport, calculate the departure times of all departing flights at each airport.

Step 3: Based on the assigned routes and departure times of each flight, calculate the time it takes for all flights to reach the route points.

Step 4: For all departing flights at each route point, sort them by their arrival time.

Step 5: Check whether any adjacent flights meet the safety interval. If the safety interval is not met, adjust the departure time of the corresponding flight and return to step 3.

Step 6: When adjacent flights passing through any route point meet the safety interval, output the result.

5. Case Study

5.1. Example Description

A realistic airport group network comprising four airports and three waypoints in the Beijing–Tianjin–Hebei region of China, as illustrated in Figure 3, is employed to validate the feasibility and accuracy of the proposed model. As shown in

Table 2. Flight information.

Airport	Waypoint	Number of Flights
ZBAA	P52	36
	VAGBI	21
	EKEAT	3
ZBAD	P52	29
	VAGBI	20
	EKEAT	1
ZBTJ	P52	6
	VAGBI	11
	EKEAT	8
ZBSJ	P52	3
	VAGBI	11
	EKEAT	12

, there are a total of 160 departure flights at all airports during 7:00–9:00 on 1 May 2022, where 36.8%, 31.3%, 15.6%, and 16.3% of these flights leaving from these airports and 46.3%, 39.4%, and 14.3% of these flights pass through these busy waypoints. The other parameters are set as follows: $T_o$ = 180 min,$T_p$ = 3 min,$T_s$ = 2 min, $\beta$ = 0.85.

5.2. Results

The computational time required to identify a set of Pareto optimal solutions typically does not exceed two minutes. As previously discussed, the proposed model is capable of generating nine feasible meta-optimal solutions across three dimensions, involving its actual departure time, assigned route, and fuel consumption. The upper and lower bounds for objective function 1 (i.e., total carbon emissions) are 87,648.75 kg and 84,294 kg, respectively, while those for objective function 2 (i.e., total delay times) are 11,726 min and 10,745 min. Figure 4 reveals the changing relationship between these two goals, i.e., total fuel consumption for the mileage of all flights assigned routes and the total deviation between the planned and actual departure times for all flights at different airports. As the value of total carbon emissions related to mileage becomes larger, that of total flight delays becomes smaller. This is because if two flights with their shortest routes conflict at the same waypoint, one flight can avoid passing through this waypoint simultaneously with the other flight by taking a detour.

Table 3. Optimal schedule.

Flight	Airport	Planned Departure Time and Delays	Waypoint	Assigned Route	Actual Arrive Time	Total Carbon Emissions (kg)
1	ZBAA	7:00 (0 min)	P52	2	7:17	354.4 (0)
2	ZBAD	7:00 (2 min)	P52	2	7:22	472.5 (−118.1)
3	ZBAA	7:00 (4 min)	P52	1	7:19	354.3 (0)
4	ZBAD	7:00 (10 min)	VAGBI	2	7:35	590.6 (−118.1)
5	ZBAA	7:00 (10 min)	VAGBI	2	7:39	472.5 (−118.1)
6	ZBAD	7:00 (14 min)	VAGBI	2	7:39	590.6 (−118.1)
7	ZBAA	7:00 (17 min)	VAGBI	1	7:42	590.6 (0)
8	ZBAD	7:00 (20 min)	VAGBI	2	7:45	590.6 (−114.1)
9	ZBAA	7:00 (23 min)	VAGBI	1	7:48	590.6 (0)
10	ZBAA	7:00 (31 min)	VAGBI	2	7:51	472.5 (−115.1)
…	…	…	…	…	…
160	ZBSJ	8:50 (172 min)	EKEAT	2	12:10	472.5 (0)

is used to provide an optimal schedule related to the meta-optimal solution (84,813.7, 11,397), from which each flight is assigned a route between the airport and the waypoint to calculate the carbon emissions; the actual departure time of each flight from the airport and the arrival time at the waypoint are also obtained. Take flight 1 as an example: it left ZBAA at the actual departure time of 7:00 with no delays, selected route 2 between ZBAA and P52 to emit carbon emissions amounting to 354.4 kg (no alternative to the shortest path that results in any additional carbon emissions), and arrived at P52 at an arrival time of 7:17. Compared with the path before optimization, the carbon emissions of the optimized path have been reduced by 5%. Figure 5 and Figure 6 describe the time-varying flight flow of each waypoint or airport before and after optimization. Based on these findings, we can draw the following conclusions: (1) Before optimization, the flight flow of each airport or waypoint based on the planned departure time of flights in some periods is greater than its capacity, which makes the scheduling scheme infeasible. (2) After optimization, the flight flow of each airport or waypoint is smaller than its capacity at all times; however, more delay times are required for all flights to complete the departure process when the number of flights exceeds the capacity of each airport or waypoint.

5.3. Comparative Analysis and Parameter Sensitivity

Table 4. Comparison of optimal solutions based on specific confidence level.

Scenarios	CVaR	${\mathit{f}}_1$	${\mathit{f}}_2$
$\beta =0.75$	1.1	83,512.3 kg	11,432 min
$\beta =0.85$	1.2	84,813.7 kg	11,397 min
$\beta =0.95$	1.3	85,314.4 kg	11,421 min

shows how specific confidence levels influence model performance. The increase in the specific confidence level results in a bigger value of CvaR. It may cause an increase in flight time between airports and waypoints, as well as more delays in the departure time of each flight. In this case, the increase in the specific confidence level leads to higher total carbon emissions, but no obvious pattern for the variation in flight delay time. However, the overall departure completion time of the entire flight has also been postponed due to the delay in the departure time of the flight.

Table 5. Comparison of optimal solutions according to different safe departure intervals for two aircrafts at any airport.

Scenarios	${\mathit{f}}_1$	${\mathit{f}}_2$
$T_p$ = 1 min	86,420.2 kg	8486 min
$T_p$ = 2 min	86,538.3 kg	8747 min
$T_p$ = 3 min	84,813.7 kg	11,397 min

illustrates the impact of varying safe departure intervals for two aircrafts at any airport on model performance. The capacity for departures at an airport is primarily influenced by the safety interval established between departing aircrafts. As this interval is gradually reduced, the total flight delay times correspondingly decrease. This phenomenon can be attributed to the fact that increasing airport capacity alleviates the imbalance between runway supply and demand, given that the number of flights remains constant. Once a critical threshold is reached, total flight delay times will diminish; otherwise, they will remain unchanged. However, the process of flight path selection is also constrained by the capacity of waypoints. Therefore, the increase in airport capacity has not led to significant changes in flight path patterns.

Table 6. Comparison of optimal solutions according to different safety flight intervals of two aircrafts at any waypoint.

Scenarios	${\mathit{f}}_1$	${\mathit{f}}_2$
$T_s$ = 1 min	84,530.2 kg	8893 min
$T_s$ = 2 min	84,813.7 kg	11,397 min
$T_s$ = 3 min	85,522.57 kg	15,046 min

indicates how different safety flight intervals of two aircrafts at any waypoint influence model performance. The increase in waypoint capacity allows more flights to pass through the location at the same time, resulting in a reduction in the total flight delay time. Furthermore, the increased capacity of the waypoint may lead to more flexible route selection for flights, making it easier to choose shorter paths to reduce carbon emissions.

6. Conclusions

This study introduced a multi-objective stochastic optimization model for MAFCSP with CvaR of travel and departure times to coordinate the routes and their departure times of flights at each airport to avoid conflicts between them in the same shared waypoint. This was carried out in order to find the optimal relationship between the layout and spatio-temporal distribution of the assigned flight traffic of the air route network, specific confidence level, total carbon emissions, and delay times for all flights across all airports. Practical constraints such as the safe interval time between adjacent flights at the same airport or waypoint and maximum deviation between planned and actual flight mileage and time for each flight were also considered in the proposed model. A two-stage heuristic approach-based NSGA-II was used to solve our model. Finally, a practical case of four airports and three waypoints in the Beijing–Tianjin–Hebei region of China was conducted to prove the correctness and effectiveness of this study.

The main imaging findings included the following points:

(1)

Before optimization, due to the unreasonable planned departure time, the capacity of airports and route points in some periods was less than the number of flights. After optimization, the capacity of airports and route points at all times is greater than the number of flights, but more queuing dissipation time is required for all flights to complete the departure process.

(2)

An increase in total carbon emissions related to mileages results in a reduction in total flight delays. This phenomenon occurs because when two flights on their shortest routes may conflict at the same waypoint, one flight can circumvent simultaneous passage through that waypoint by opting for a detour.

(3)

The increase in the specific confidence level results in a bigger value of CvaR, causing an increase in flight time between airports and waypoints, as well as more delays in the departure time of each flight. Although a schedule with a larger CvaR leads to more total carbon emissions, it is robust and easily applied in practice.

However, the limitations of our model lie in three aspects: (1) it relies on precise probability distribution assumptions for travel and departure time, which makes the distributed robust model better able to cope when not enough data can be used to accurately estimate the probability distribution; (2) some realistic constraints, such as the priority and fairness of flights at each airport or airline and the workload of the air controller, are neglected; (3) disturbances in the individual phase of the take-off procedure caused by irregular situations such as weather, etc., are not considered in our model. By extending this study, our future research aims to develop a robust model for MAFCSP with more realistic constraints.