Research on ATFM Delay Optimization Method Based on Dynamic Priority Ranking

Abstract

Air Traffic Flow Management (ATFM) delay refers to the difference between a flight’s Target Take-Off Time (TTOT) and its Calculated Take-Off Time (CTOT), reflecting congestion levels in the air traffic network. ATFM delays are assigned to balance demand and capacity at key points in the network. The traditional First-Come, First-Served (FCFS) approach allocates delays strictly in the order flights are ready to depart, which is simple but inflexible. This study proposes a dynamic priority-based aircraft sequencing method at critical waypoints under multi-resource scenarios, aiming to reduce ATFM delays. An improved Constrained Position Shifting (CPS) constraint is introduced into the optimization model to enhance the influence of flight priority during decision-making. Additionally, three different priority strategies are designed to compare their respective impacts on ATFM delay. Finally, a dynamic priority-based ATFM delay optimization model is developed to address the identified challenges. Experimental results demonstrate that, compared with the FCFS scheme, the three priority strategies achieve maximum ATFM delay reductions of 30.5%, 44.1%, and 19.9%, respectively. The proposed model effectively allocates shorter delays to critical flights, optimizing resource utilization and improving the operational efficiency of the air route network. The research provides a reference framework for air traffic managers in allocating spatiotemporal resources across multiple congestion hotspots. By aligning priorities with network-wide efficiency goals, it overcomes traditional model limitations, avoids local optima, and supports globally optimal ATFM policy and practice.

1. Introduction

Air Traffic Flow Management (ATFM) is a critical component of Air Traffic Management (ATM). It aims to mitigate imbalances between airspace and airport capacity and demand by evaluating system-wide traffic flows and providing guidance on when, where, and how airspace users should operate. According to the latest forecast from EUROCONTROL, flight volumes are expected to return to pre-pandemic levels by 2025 and continue to grow at an average annual rate of 1.5% thereafter [1]. This suggests that mismatches between airspace demand and available capacity will become increasingly prominent within future ATFM systems.

Addressing regional overloading—defined as demand exceeding capacity within specific time windows [2]—has become a prominent research direction in ATFM studies. In such hotspots (airports or waypoints), ATFM measures such as rerouting, ground and airborne holding, and ATFM Delay assignment are used to balance capacity and demand. In the tactical phase, EUROCONTROL primarily relies on Delay assignment mechanisms and has proposed the Computer-Assisted Slot Allocation (CASA) algorithm, which follows a First-Come, First-Served (FCFS) principle. The system first selects all flights scheduled to enter a constrained sector (hotspot) and sequences them by their estimated arrival times. Flights exceeding the sector’s capacity receive delays accordingly [3]. This forms the basis of ATFM delay computation. After delays are assigned based on the FCFS order, the difference between the Target Take-Off Time (TTOT) and Calculated Take-Off Time (CTOT) becomes the ATFM delay [4].

Although FCFS is often considered fair because it preserves the natural arrival order of flights, it lacks control over the distribution of delays and does not account for traffic conditions across other regions of the network. To address this, the concept of flight priority has been introduced to improve Delay assignment. The Single European Sky ATM Research (SESAR) program proposed the User-Driven Prioritisation Process (UDPP) [5], which enables airspace users (AUs) to assign higher priorities to their costlier flights, thereby gaining flexibility to reorder flight sequences. This concept has been embedded in CASA and expanded through Flight Delay Reordering (FDR) and Selective Flight Protection (SFP) mechanisms [6]. UDPP has been tested in single-resource scenarios. Researchers such as Andrea and Nadine Pilon have developed UDPP models and validated them in specific airports or airline use cases [7,8]. Their results confirm the mechanism’s benefits for AUs, especially in reducing delay costs. SESAR also validated UDPP’s feasibility in practice, reporting significant improvements in on-time performance and reduced missed connections for passengers [9].

However, UDPP is fundamentally based on airline-specific decisions and lacks consideration of system-wide effects across the air traffic network. In practice, capacity–demand imbalances are widespread and are not limited to airports—hotspots frequently occur within the route network and may persist over time. Consequently, the multi-resource nature of these hotspots must be acknowledged. Furthermore, when multiple hotspots are active simultaneously, overlapping ATFM measures and shared temporal-spatial resources may lead to conflicts under priority-based allocation, causing suboptimal local sequencing and inefficient global delay distribution. As a result, network capacity utilization can be compromised. Therefore, to effectively apply ATFM Delay assignment strategies in multi-resource environments, it is essential to explore the interaction mechanisms among flight flows across hotspots and evaluate their collective impact on overall network efficiency.

Therefore, the research gap lies in the absence of an ATFM delay allocation method that integrates dynamic, network-wide flight priorities into multi-resource environments while explicitly accounting for resource coupling across hotspots. Without addressing this, sequencing may yield locally optimal but globally inefficient delay distributions, undermining network capacity utilization and overall operational efficiency.

To fill this gap, this study investigates the integration of a dynamic priority mechanism for more efficient and equitable delay management. Unlike traditional FCFS or UDPP methods that focus on local ordering or airline preferences, the proposed method emphasizes the modeling and optimization of network-wide interdependencies.

Specifically, this paper addresses the following research questions:

• How can a flight prioritization mechanism be designed to reflect network-wide effects, moving beyond airline-centric or localized strategies? How do different priority strategies affect global network performance, in terms of total delay and delay distribution equity?
• How can an ATFM delay optimization model be developed to effectively capture resource coupling in a multi-resource environment, thereby avoiding local optima and enhancing overall capacity utilization and network efficiency?

To answer these questions, this study proposes a dynamic priority-based sequencing method for aircraft passing through waypoints, aiming to allocate ATFM delay across multiple resources by incorporating network-wide dynamic priorities. By doing so, the method effectively reduces overall ATFM delay while balancing fairness and efficiency within the route network. The model is designed for exploratory experiments and is applied at the tactical stage. The main contributions of this study are as follows:

• Introduction of an Improved Constrained Position Shifting (CPS) Constraint with Priority Strategies: The study refines the traditional CPS constraint by integrating priority strategies into the model. In alignment with the Target CASA framework, the modified CPS constraint incorporates restricted displacement strategies under different priority levels. Additionally, to investigate the trade-off between efficiency and fairness in the optimization model, three distinct priority strategies are introduced, examining the relationship between priority levels and allowable displacement.
• Development of a Dynamic Priority-Based ATFM Delay Optimization Model for Multi-Resource Scenarios: A Mixed-Integer Linear Programming (MILP) model is formulated to optimize ATFM delay by dynamically adjusting flight priorities across multiple hotspots. By reordering traffic flows passing through congested areas, the model enhances network-wide delay management through an adaptive priority mechanism.
• Enhancement of the Existing FCFS-Based Sequencing Strategy: Building upon the priority-based approach of the UDPP, this study introduces a global priority-setting mechanism that considers network-wide impacts. This mechanism extends beyond single-resource scenarios and is validated in a multi-resource environment to assess its effectiveness.

This study is organized into seven sections. Section 1 outlines the research background and objectives. Section 2 reviews ATFM delay optimization methods and the progress in ATFM delay assignment research. Section 3 presents the methodological framework. Section 4 constructs the dynamic priority-based ATFM delay optimization model. Section 5 explains the algorithm used in this study. Section 6 presents validation results, including an assessment of overall optimization performance, analysis of priority effects, and evaluation of different priority strategies under various scenarios. Finally, Section 7 summarizes the findings and discusses future research directions.

2. Literature Review

2.1. ATFM Delay Optimization

ATFM delay serves as a metric to investigate airspace congestion patterns and severity, providing a scientific basis and decision support for network planning and operational optimization. In its 2016 Key Performance Indicator (KPI) report, the International Civil Aviation Organization (ICAO) recommended ATFM delay as a critical performance indicator to monitor and identify deficiencies in aviation network performance [10]. Similarly, EUROCONTROL considers ATFM delay a key performance indicator for network capacity, setting a target of 0.5 min for average ATFM delay per flight within the air route network [11]. ICAO ATFM Manual establishes a global benchmark of 1 min for average ATFM delay per flight [12]. Reducing ATFM delay has therefore become a vital objective in improving the operational efficiency of air route network.

International authorities have increasingly prioritized reducing ATFM delays and have issued relevant guidelines. EUROCONTROL introduced the concept of ATFM Saving in its annual reports, setting a target of reducing total annual ATFM delay by 10% [13]. ICAO’s Performance Objective focuses on optimizing ATFM delay assignment, proposing the redistribution of delays to either minimize per-flight delay or concentrate delays on fewer flights [10]. EUROCONTROL’s Flow CONOPS document introduces the Target CASA functionality and the Resourceful Overloading of Slots (RCO) mechanism to optimize resource allocation and flight scheduling during flow management. Target CASA addresses the DCB problem by more precisely applying ATFM regulations to specific flights, while the RCO mechanism offers flexible slot management under dynamic network changes [3]. Meanwhile, SESAR’s ISOBAR project applies AI to address METEO-based DCB imbalances, aiming to balance local and network-level measures while considering user priorities and delay propagation [14].

Recent studies have proposed various ATFM delay optimization methods, including strategic scheduling interventions, rerouting, ground and airborne holding, and ATFM delay assignment. For instance, strategic scheduling interventions have been explored by Jacquillat and Odoni [15]. And Bolić et al., who reduced expected ATFM delays in the strategic phase by adjusting timetables and traffic distribution [16]. Additionally, Wang and Zhao developed a robust slot allocation model to eliminate scheduling conflicts effectively [17]. Presto et al. found that adjusting flight frequencies significantly reduces delays [18]. Regarding ground and airborne holding, Delgado et al. proposed a deceleration strategy in 2011, using the FACET tool to simulate cruise speed reductions, substituting airborne delays for ground delays to save fuel [19]. Similarly, Prats et al. recommended minimizing cruise speed to reduce fuel consumption and execute ATFM delays during flight [20]. In terms of rerouting, Dalmau introduced a gradient-boosted decision tree model to predict tactical-stage rerouting possibilities, alleviating ATFM delays [21]. With the rise of AI, traffic flow prediction enables early detection of capacity–demand imbalances, reducing ATFM delays strategically. Chen et al. proposed BLSTF, a bilinear spatio-temporal fusion network for stable, periodic traffic flows, achieving accurate and interpretable forecasts on four real-world datasets [22]. Chen et al. introduced DST2former, which leverages adaptive embedding and multi-view dynamic feature fusion to enhance spatio-temporal prediction accuracy [23]. As a critical method in the tactical phase, priority-based ATFM Delay Assignment has attracted considerable attention in recent research.

2.2. Priority-Based ATFM Delay Assignment

ATFM delay assignment is based on the CASA framework’s FCFS principle, serving as the fundamental mechanism for delay assignment within the system. To enhance the basic FCFS approach, additional considerations have been incorporated to balance efficiency, fairness, and airline operational needs.

In recent years, the most widely discussed mechanism has been the UDPP proposed by SESAR. Given that the decision-making process for ATFM delay assignment is, to some extent, independent from airspace users, SESAR introduced UDPP in 2015 to enhance user participation. This mechanism allows airspace users to play a more active role in prioritizing flights, thereby improving the flexibility and efficiency of delay management. FDR and SFP mechanisms are included, which allow slot swaps between two flights subject to the same regulation to reduce the cost burden caused by hotspots [6]. In 2016, Bertsimas and Gupta proposed a two-stage method for single-airport scenarios, integrating fairness and airline collaboration through bid-based slot exchanges [24]. In 2020, Yan Xu et al. expanded user involvement by developing a collaborative ATFM framework based on airspace user preferences, improving cost-effectiveness [25]. In 2021, Pilon et al. further validated UDPP by enabling airlines to reorder flights flexibly [8], while Zhang et al. introduced enhanced UDPP with selective flight protection strategies, reducing delay costs and total ATFM delays [26]. In 2025, Gasparin et al. [7] developed a mathematical and optimization model for the UDPP mechanism, termed UDPP-OPT. Their results show that its effectiveness depends on both the number of airlines and the volume of flights within congestion hotspots, providing a new approach to addressing ATFM hotspot challenges [7].

Additionally, the trade-off between efficiency and fairness must also be considered. In 2012, Barnhart et al. proposed a discrete optimization model with five fairness metrics to distribute delays equitably and improve management efficiency [27]. Montlaur and Delgado examined efficiency and fairness from flight and passenger perspectives, suggesting prioritizing passenger satisfaction in delay optimization for passengers are more sensitive to delays in 2020 [28].

In recent years, research has increasingly focused on route networks, expanding its scope to address multi-resource issues within the air traffic network. In 2023, Chen et al. proposed a cooperative coefficient and a heuristic delay prioritization strategy to improve FCFS-based Delay assignment and mitigate DCB issues, both using multi-agent reinforcement learning (MARL) [29,30].

ATFM delay optimization remains challenging, with existing studies focusing on airspace structure and flow adjustments while often neglecting underlying Delay assignment rules. Traditional FCFS methods struggle to meet modern ATFM demands. Although UDPP enhances airline flexibility, it prioritizes local decisions over network-wide efficiency, especially under multi-hotspot constraints. Recent initiatives, such as SESAR’s ISOBAR and EUROCONTROL’s Target CASA, provide network-level policy guidance. This study addresses current limitations by introducing a dynamic priority strategy into a network-wide optimization model to balance fairness and efficiency in ATFM Delay assignment.

3. Methodology

3.1. Demand–Capacity Balancing Network

This study aims to conduct an exploratory investigation of the Target CASA functionality as outlined in the EUROCONTROL Flow CONOPS framework. Building on the precise-control principle of the Target CASA framework, the model integrates dynamic priorities to enable differentiated treatment of flights.

To address the DCB problem in air traffic flow management—particularly the situation where specific regions experience demand exceeding capacity during a specific time window—this study proposes constructing a DCB model. This model aims to precisely describe the distribution of flight demand and capacity in the network, providing a foundation for subsequent ATFM delay optimization strategies. It is assumed that the initial traffic demand (flight sequence) is known. The trajectory of flight $f$ is represented as $d_f$, and the flight sequence consists of a set of binary tuples:

$$d_f=\left\{\left(e_{f,n},{\tau }_{f,n}|n=\mathrm{1,2},\dots ,\left|N\right|\right)\right\}$$

(1)

where $e_{f,n}$ denotes the flight $f$ passes through a waypoint $n$, and ${\tau }_{f,n}$ denotes the time of passing through the waypoint. Therefore, the flight demand of the network is:

$$D^{Initial}=\left\{d_f|fϵF\right\}$$

(2)

where $F$ represents the set of flights.

For the network’s capacity, $C$ denotes the set of capacities, where $N$ is the set of waypoints, and $W$ is the set of time windows:

$$C=\left\{c_{n,w}|nϵN,wϵW\right\}$$

(3)

3.2. Delay Conflict Mechanism Under Multi-Resource Constraints

In ATFM delay assignment, flights through hotspots are sequenced by FCFS, with over-capacity flights allocated CTOTs based on available capacity. For multi-hotspot scenarios, the Most Penalizing Regulation (MPR) strategy, based on CASA principles [31], assigns ATFM delay according to the node causing the longest wait, ensuring flights arrive at the most penalizing node on time while minimizing further delays. Under the MPR strategy, ATFM delay is determined exclusively by the most penalizing ATFM measure, while constraints imposed by other measures are simplified. However, simultaneous or sequential implementation of multiple ATFM measures may lead to conflicts due to overlapping measures and shared use of time–space resources. Although the FCFS-based approach mitigates such conflicts by preserving the natural arrival sequence of flights, it does not fully achieve fairness in resource allocation or the goal of minimizing delays. Flights subjected to ATFM delays may either deviate from the fairness of FCFS or incur additional delays.

A specific example is illustrated in Figure 1. Five flights passing through Waypoint1 and Airport1 are shown in subfigure (a). Suppose at 07:00, Waypoint1 implements ATFM measures, limiting arrivals between 08:30 and 09:00 to one aircraft every five minutes. As a result, flights B, C, and D, which are scheduled to pass through Waypoint1 during this interval, are affected. Their overflight times are sequentially delayed, and each is assigned a CTOT, producing the adjusted schedule in subfigure (b). Subsequently, at 07:30, Airport1 imposes a similar constraint, allowing only one arrival every ten minutes between 08:50 and 09:30. Flights A, B, and E are affected, yielding the updated schedule in subfigure (c). Since flight E arrives later and meets the spacing requirement, no delay is assigned. By combining the FCFS and MPR principles, the final schedule under multiple resource constraints is shown in subfigure (d). Specifically, flight B is constrained by both Waypoint1 and Airport1, and according to the MPR principle, it receives the larger delay of 5 min. Meanwhile, due to the FCFS principle, subsequent flights C and D are delayed accordingly at Waypoint1. In this scenario, the total ATFM delay reaches 20 min. However, if the FCFS principle is not strictly enforced and a globally optimal solution is pursued—namely allowing flight C to overtake flight B at Waypoint1—the total ATFM delay is reduced to 8 min, as shown in subfigure (e).

The example illustrates the inherent trade-off between fairness and efficiency in network-level ATFM. It also serves as the scenario basis for the MILP model in Section 4, where the MILP model, combined with the optimization algorithm, calculates the optimal flight sequence, as shown in Figure 1e. To minimize total ATFM delays, this study selectively relaxes the FCFS rule, allowing globally optimized scheduling across multiple constrained resources. Inspired by the Target CASA concept, a dynamic priority strategy is introduced to guide flight sequencing and Delay assignment, enabling differentiated interventions based on each flight’s network impact, especially under multiple ATFM constraints.

Based on this, the ATFM delay optimization problem is defined as follows: Given a sequence of flights and their estimated times over (ETOs), determine a reordered schedule and Delay assignment strategy across multiple constrained resources, with the objective of minimizing total ATFM delay across the route network. The specific principle is illustrated in Figure 2. To address this, an ATFM delay optimization model is constructed, with the following core methodology: First, dynamic priority strategies for delay assignment are designed. Building upon this, the study investigates resolving DCB problem by adjusting the sequencing of over-capacity flights and allocating delays under relevant constraints. This approach aims to enhance the performance of the entire air route network.

Specifically, this study introduces dynamic priority-based ATFM delay optimization into the ATFM process. The primary goal is to allocate ATFM delays to flights exceeding capacity thresholds, adjust traffic flow, and engage in tactical planning to address demand–capacity imbalances at nodes. A schematic representation of the approach is shown in Figure 3. Following hotspot detection, the system generates explicit flight priorities based on real-time operational status and the departure airport’s handling capability. These priorities determine the schedulable sequence of flights within the optimization model. Subsequently, a MILP solver iteratively optimizes the sequence based on priority, ultimately producing a globally optimal ATFM Delay assignment scheme across multiple hotspots.

3.3. Dynamic Priority Strategy

Flight priority reflects the importance of flights to facilitate preferential allocation of time resources. To more effectively manage flight delays, this study proposes a flight priority calculation framework that integrates flight-specific and airport-specific information while considering the network-wide effects.

3.3.1. Priority Definition

Traditional delay assignment strategies based on the FCFS principle often fail to account for the network-wide effects of flight delays. In reality, the network effects of delays have been explored in previous studies from two key perspectives: on the one hand, when delays occur during the execution of preceding flight tasks, these delays propagate across the flight chain due to the interconnected nature of flights, causing disruptions at multiple airports within the network [32,33,34]. On the other hand, at busier hub airports, tightly packed schedules mean that even minor delays can trigger cascading effects, exponentially increasing the number of affected flights [35]. These findings highlight that a delay in one flight can trigger chain reactions, propagating horizontally through ground delays and vertically through the flight chain, disrupting network stability. Thus, ATFM measures must consider their network-wide effects. Based on the above discussion, it is beneficial to prioritize flights departing from busy airports and those experiencing preceding delays in the ATFM Delay assignment process across the entire network. This prioritization significantly contributes to improving network-wide operational efficiency and optimizing resource utilization.

Accordingly, this section proposes a dynamic priority-setting method based on airport congestion level and preceding flight delays. This qualitative approach assigns a dynamic priority level to each flight according to two criteria: Whether the flight has incurred a preceding delay, and the congestion level of its departure airport. These priority levels are not static—they adapt in real time to changes in the flight’s delay status and the operational status of its departure airport, achieving a truly dynamic prioritization mechanism. The specific indicators and corresponding priority levels are defined as follows:

• Congestion level at the departure airport: Represented by the ratio of traffic to capacity at the airport within a given timeframe, this indicator reflects the airport’s ability to handle operations and determines its congestion level. Airport capacity refers to the published departure capacity, while traffic is defined as the unconstrained demand based on flight plans.
• Occurrence of preceding delays: Flights experiencing preceding delays are given higher priority. Preceding delay is determined using the difference between the Scheduled Off-Block Time (SOBT) and the Actual Off-Block Time (AOBT).

Based on these indicators, four flight priority levels are defined based on two key indicators: whether the departure airport is busy (i.e., the ratio of demand to capacity exceeds 1), and whether there is a preceding delay in the flight chain. The classification of these priority levels is shown in

Table 1. Dynamic Priority-setting Method.

Priority	Departure Airport Congestion Level	Preceding Flight Delays
1	departing from busy airport	experiencing preceding delays
2	departing from busy airport	without preceding delays
3	departing from non-busy airport	experiencing preceding delays
4	departing from non-busy airport	without preceding delays

below.

This approach introduces a flight priority attribute, allowing higher time-value flights to save more time while proportionately delaying lower time-value flights. This differentiated reward-and-penalty mechanism meets operational demands and improves the efficient utilization of limited airspace resources.

3.3.2. Dynamic Priority Strategy

To further incorporate priority into the model, this study treats the priority strategy as a constraint by introducing an improved CPS constraint. The core principle of this constraint is to define a critical parameter that limits the maximum positional shifts (both forward and backward) for each flight relative to its initial FCFS sequence. Dear initially proposed the CPS concept and demonstrated its effectiveness in increasing runway throughput while maintaining a certain level of fairness [36]. Subsequent studies have focused on developing rapid scheduling techniques within the CPS framework, aiming to achieve an optimal balance between fairness and efficiency [37,38]. CPS serves as a reasonable constraint by balancing fairness and efficiency. On the one hand, it prevents excessive delays for certain flights, enhancing fairness. On the other hand, it reduces overall delay by fully utilizing runway resources, improving efficiency.

However, traditional CPS applies uniform constraints to all flights, failing to reflect priority differences among them. Bianco et al. introduced Maximum Position Shifting (MPS), which aims to protect critical flights from excessive delays [39]. Malaek and Naderi proposed Dynamic Position Shifting (DPS), combining flight importance and traffic flow to calculate an optimal sequence [40]. Building on these approaches, this study incorporates CPS strategies with differentiated priority settings: by assigning a maximum positional shift $K$ to flights of different priority levels, flights within the same time window can adjust their positions by up to $K$ slots based on the FCFS order according to their priority. This strategy seeks to reduce delays for critical flights while increasing sequencing flexibility.

When integrating priority settings into delay assignment, this study employs the improved CPS constraint. Before flight sequencing by the central flow management unit, each flight is assigned a clear priority $P_f$, where the highest, high, low, and lowest priorities correspond to 1, 2, 3, and 4, respectively. This priority determines whether a flight takes precedence over others demand–capacity imbalance issues arise. Based on this, the flow management unit screens potential sequencing options according to flight priorities and identifies the globally optimal delay assignment plan through the model, as illustrated in Figure 4.

The relationship between different priority levels and their corresponding maximum positional shifts $K$ plays a critical role in balancing efficiency and fairness within the optimization model. A systematic exploration of optimization performance and adaptability under various priority-to-shift mappings is necessary. Three strategies are inspired by classic scheduling theory, including weighted fair allocation [41], exponential priority amplification [42], and grouped scheduling [43]. They offer a progression from conservative to aggressive prioritization, allowing us to evaluate how different tradeoffs between fairness and efficiency perform under varying congestion conditions.”, as shown in

Table 2. Relationships Between Priority Strategies and Maximum Positional Shifts.

Priority Strategy	$\mathbf{∆}{\mathit{K}}_{\mathit{P}}^{\mathbf{+}}$	$\mathbf{∆}{\mathit{K}}_{\mathit{P}}^{\mathbf{-}}$
PCPS	$\alpha \left(5-P\right)$	$\beta P$
ECPS	$\alpha ·2^{(4-P)}$	$\beta ·2^{(P-1)}$
BCPS	$2·\alpha \left(5-P\right)$	$2·\beta P$

.

• Proportional Constrained Position Shift (PCPS)

In resource-constrained scheduling problems, linear weighting is commonly used to allocate resources among tasks with varying priorities. The Proportional Constrained Priority Strategy (PCPS) adopts the concept of Weighted Fair Queuing (WFQ) [41], ensuring that resource allocation is proportionally distributed, allowing higher-priority tasks to be scheduled earlier. Specifically, the maximum positional shift for a flight is linearly proportional to its priority level. Flights with a priority $P$ can move forward by at most $\alpha \left(5-P\right)$ positions and backward by at most $\beta P$ positions. As shown in Table 2: The feasible range of positional shifts for a flight is $\left[-∆K_P^{+},∆K_P^{-}\right]$, where $\alpha$ and $\beta$ are linear scaling factors that regulate the total allowable forward and backward movements, respectively. This method ensures that high-priority flights are protected while achieving balanced position adjustments, strictly adhering to priority levels while maintaining overall scheduling equilibrium.

2.

Exponential Constrained Position Shift (ECPS)

The concept of Exponential Backoff emphasizes that higher-priority entities should be granted greater flexibility in resource allocation [42]. Building on this principle, the Exponential Constrained Priority Strategy (ECPS) highlights the “scheduling privilege” of high-priority flights, allowing them significantly larger adjustment ranges in Delay assignment. In this method, the maximum positional shift increases exponentially with priority levels. Flights with a priority $P$ can move forward by at most $\alpha ·2^{(4-P)}$ positions and backward by at most $\beta ·2^{(P-1)}$ positions. The formulas are shown in Table 2. This approach provides lower-priority flights with greater flexibility in the optimization process, allowing them to yield resources to high-priority flights. As a result, it accelerates the delay mitigation for critical flights at hotspots.

3.

Balanced Constrained Position Shift (BCPS)

Prakash demonstrated that expanding the adjustment range can enhance the quality of solutions [43]. Based on this insight, the BCPS proportionally enlarges the adjustment range for all flights, providing greater scheduling flexibility. This approach aims to improve optimization performance, particularly in complex route networks with multiple critical waypoints, by significantly enhancing delay distribution and resource utilization across the system.

By implementing dynamic priority strategies, flights in hotspots are scheduled based on priority policies, and waypoint passing sequences are rearranged. This protects high-priority flights, reducing their impact on other flights and alleviating congestion at busy airports. As a result, ATFM delays are redistributed more effectively between busy and less busy airports, as well as between flights with and without preceding delays, optimizing the overall network performance.

4. Dynamic Priority-Based ATFM Delay Optimization Model

The primary objective of this section is to develop an integrated delay assignment model that preserves the existing structure of ATFM measures (including constraints and parameters). The model employs a unified approach to explore feasible delay assignment solutions, aiming to achieve higher operational efficiency (lower delays) and improved applicability. A MILP model is developed to solve the ATFM delay optimization problem based on dynamic priority. The entire analysis period is divided into several time windows for analysis. Key assumptions include:

• The model’s solution is strictly implementable, with flights adhering to their planned overflight times.
• Flights passing through multiple waypoints may have different delays per segment. To ensure operational reliability, the final delay assigned to a flight in the current iteration is determined by adopting the maximum across the segments.
• It is assumed that cruising time between consecutive waypoints is constant for each flight.

4.1. Variable Definition

The variables and constants are defined as

Table 3. Variables and Constants.

Constants	Definitions
$F$	$\mathrm{Set}\mathrm{of}\mathrm{flights}F=\{f_1,f_2,f_3,\dots ,f_i\}$
$N_{f_i}$	$\mathrm{Set}\mathrm{of}\mathrm{waypoints}\mathrm{that}\mathrm{flight}{f}_i$ passes
$F_n$	$\mathrm{Set}\mathrm{of}\mathrm{flights}\mathrm{scheduled}\mathrm{to}\mathrm{pass}\mathrm{waypoint}n$
$T_{f_i,n}$	$\mathrm{Scheduled}\mathrm{time}\mathrm{for}\mathrm{flight}{f}_i$ $\mathrm{to}\mathrm{pass}\mathrm{waypoint}n$
$C_n$	$\mathrm{Capacity}\mathrm{of}\mathrm{waypoint}n$
$S_{f_i,f_j}^n$	$\mathrm{Minimum}\mathrm{separation}\mathrm{interval}\mathrm{at}\mathrm{waypoint}n$
${\widehat{T}}_{f_i}^{n,n^{\prime }}$	$\mathrm{Flying}\mathrm{time}\mathrm{for}\mathrm{flight}{f}_i$ $\mathrm{between}\mathrm{waypoint}n$ $\mathrm{and}\mathrm{waypoint}{n}^{\prime }$
${ET}_n$	$\mathrm{Earliest}\mathrm{departure}\mathrm{time}\mathrm{at}\mathrm{waypoint}n$ after optimization
$R_{f_i,n}$	$\mathrm{Sequence}\mathrm{of}\mathrm{flight}{f}_i$ $\mathrm{at}\mathrm{waypoint}n$, determined by the FCFS rule
Variables
$d_{f_i}^m$	$\mathrm{The}\mathrm{delay}\mathrm{for}\mathrm{flight}{f}_i$ $\mathrm{in}\mathrm{the}\mathrm{iteration}m$
$t_{f_i,n}$	$\mathrm{Actual}\mathrm{time}\mathrm{at}\mathrm{which}\mathrm{flight}{f}_i$ $\mathrm{passes}\mathrm{waypoint}n$
$r_{f_i,n}$	$\mathrm{Actual}\mathrm{sequence}\mathrm{of}\mathrm{flight}{f}_i$ $\mathrm{at}\mathrm{waypoint}n$
$x_{f_i,f_j}^n$	$\mathrm{Binary}\mathrm{variable},\mathrm{equals}1\mathrm{if}\mathrm{flight}{f}_i$ $\mathrm{is}\mathrm{scheduled}\mathrm{before}\mathrm{flight}{f}_j$ $\mathrm{in}\mathrm{the}\mathrm{sorting}\mathrm{process}(\mathrm{both}\mathrm{flights}\mathrm{belong}\mathrm{to}\mathrm{the}\mathrm{set}{F}_n$)
$y_t^{f_i,n}$	$\mathrm{Binary}\mathrm{variable},\mathrm{equals}1\mathrm{if}\mathrm{flight}{f}_i$ $\mathrm{in}\mathrm{set}{F}_p$ $\mathrm{is}\mathrm{scheduled}\mathrm{to}\mathrm{pass}\mathrm{waypoint}n$ $\mathrm{at}\mathrm{separation}\mathrm{time}t$
$z_{ij}^{f_i}$	$\mathrm{Binary}\mathrm{variable},\mathrm{equals}1\mathrm{if}\mathrm{flight}{f}_i$ $\mathrm{travels}\mathrm{from}\mathrm{node}i$ $\mathrm{to}\mathrm{node}i$

.

4.2. Objective Function

The objective function evaluates the quality of the generated solutions. This model adopts an efficiency-oriented approach, aiming to minimize total ATFM delays. Since the MPR strategy, the ATFM delay of a flight equals the difference between the actual and scheduled passing times at its final waypoint. This can be expressed as:

$$min\sum _{f_i\in F}max\left\{t_{f_i,n_{last}}-T_{f_i,n_{last}},0\right\}$$

(4)

where $n_{last}$ is the final waypoint that flight $f_i$ passes.

4.3. Constraints

This section contains a large number of equations, which are divided into three parts: the first two parts relate to network optimization and sequencing problems, while the third part imposes constraints on the variables to facilitate the solution process.

• Network optimization constraints

Uniqueness constraint: This constraint ensures that each flight passes a waypoint exactly once, and is assigned a single passing time:

$$\sum _{t\in W}{y}_t^{f_i,n}\le 1,\forall n\in N_{f_i},\forall f_i\in F$$

(5)

Capacity constraint: This constraint ensures the feasibility of flight passage through waypoints. The entire period is divided into multiple time windows $w$, and the number of flights passing through waypoint $n$ in each time window must not exceed the declared capacity of the waypoint during that window.

$$\sum _{f_i\in F_n}{y}_t^{f_i,n}\le C_n,\forall n\in N$$

(6)

$$\sum _{t\in w}{y}_t^{f_i,n}\le C_n,\forall n\in N$$

(7)

State transition constraint:

$$t_{f_i,n^{\prime }}=t_{f_i,n}+{\widehat{T}}_{f_i}^{n,n^{\prime }},\forall f_i\in F,\forall n,n^{\prime }\in \mathrm{N}$$

(8)

Flow balance constraint: The flow balance constraint ensures that the total number of flights entering a node equals the total number of flights exiting the node.

$$\sum _{jϵN}{z}_{ij}^{f_i}-\sum _{jϵN}{z}_{ji}^{f_i}=0,\forall f_i\in F,\forall i,j\in \mathrm{N}$$

(9)

Route Constraint: Flights are assumed to fly strictly along a predefined sequence of waypoints.

$$y_t^{f_i,n_k}\ge y_t^{f_i,n_{k+1}},\forall f_i,n_k\in N_{f_i}$$

(10)

MPR Constraint: Since the MPR strategy, the ATFM delay experienced by a flight at all nodes will be equal to that of the most restrictive node. Therefore, the delay $d_{f_i}^m$ for flight $f_i$ in the iteration $m$ should be:

$$d_{f_i}^m=max\left\{g_{f_i,1}^m,g_{f_i,2}^m,g_{f_i,3}^m,\dots ,g_{f_i,k}^m\right\},\forall k\in N,\forall f_i\in F_n$$

(11)

where $g_{f_i,k}=max\left\{t_{f_i,k}-T_{f_i,\mathrm{k}},0\right\}$. It should be particularly noted that $t_{f_i,n_{last}}-T_{f_i,n_{last}}$ in the objective function also follows this principle, representing the maximum delay assigned to flight $f_i$ in the final iteration.

In the network, each flight is associated with an earliest departure time. The relationship between the optimized earliest actual departure time and the flight’s actual time over at each waypoint after optimization is described as follows:

$${\mathrm{t}}_{{\mathrm{f}}_{\mathrm{i}},\mathrm{p}\mathrm{o}\mathrm{r}\mathrm{t}}\ge {\mathrm{t}}_{{\mathrm{f}}_{\mathrm{i}},\mathrm{k}}·{\mathsf{\alpha }}_{{\mathrm{f}}_{\mathrm{i}},\mathrm{k}}^{\mathrm{m}}-\sum _{\mathrm{L}}{\widehat{\mathrm{T}}}_{{\mathrm{f}}_{\mathrm{i}}},\forall {\mathrm{f}}_{\mathrm{i}}\in \mathrm{F},\forall \mathrm{L}\in \left\{\left(\mathrm{p}\mathrm{o}\mathrm{r}\mathrm{t},{\mathrm{n}}_{{\mathrm{f}}_{\mathrm{i}},1}\right),\left({\mathrm{n}}_{{\mathrm{f}}_{\mathrm{i}},1},{\mathrm{n}}_{{\mathrm{f}}_{\mathrm{i}},2}\right),\left({\mathrm{n}}_{{\mathrm{f}}_{\mathrm{i}},\mathrm{k}-1},{\mathrm{n}}_{{\mathrm{f}}_{\mathrm{i}},\mathrm{k}}\right)\right\}$$

(12)

$$t_{f_i,\mathrm{p}\mathrm{o}\mathrm{r}\mathrm{t}}\ge M·\left(1-\sum _{kϵN_{f_i}}{\alpha }_{f_i,k}^m\right),\forall f_i\in F$$

(13)

where ${\alpha }_{f_i,k}^m$ is an auxiliary variable, and ${\alpha }_{f_i,k}^m$ takes the value of 1 when $g_{f_i,k}^m=d_{f_i}^m$.

2.

Sequencing and Scheduling Constraints

Separation constraint: To achieve air traffic control safety objectives, a minimum horizontal separation $S$ must be maintained between consecutive flights passing through the same waypoint. In the computation, the flight speed is obtained to convert it into a time-based separation interval $S_{f_i,f_{i-1}}^n=max\{{\displaystyle \frac{S}{V_i}},{\displaystyle \frac{S}{V_{i-1}}}\}$.

$$t_{f_i,n}\ge t_{f_{i-1},n}+S_{f_{i-1},f_i}^n,\forall n\in N,\forall f_i\in F_n$$

(14)

Flight sequence uniqueness constraint:

$$x_{f_i,f_j}^n+x_{f_j,f_i}^n=1,\forall n\in N,\forall f_i,f_j\in F_n:i\ne j$$

(15)

Non-Negative Delay Constraint: The model is proposed to address the demand capacity imbalances through ATFM delay. Under this research context, the actual waypoint passing time of scheduled flights cannot be earlier than the expected time.

$$t_{f_i,n}-T_{f_i,n}\ge 0,\forall f_i\in F,\forall n\in N$$

(16)

Constrained Position Shift (CPS): The CPS constraint limits the maximum positional deviation of flights in the optimized sequence relative to the FCFS sequence. Each flight in the sequence can shift by a maximum of $K$ positions. The objective is to identify a flight sequence that minimizes total delays during the planning period. At the same time, different $K$ values are assigned to flights of different priority levels to achieve a balance between fairness and efficiency. Compared to the traditional CPS constraint, which applies a fixed maximum positional shift, this study introduces an improved CPS constraint by allocating different maximum positional shifts for flights based on their priority levels. Here, $R_{f,n}=R$, where $m$ represents the total number of flights passing through a waypoint.

$$R-{∆K}_{P_f}^{+}\le m-\sum _{f_j\in F_n}{x}_{f_i,f_j}^n,\forall n\in N,\forall f_i,f_j\in F_n:i\ne j$$

(17)

$$m-\sum _{f_j\in F_n}{x}_{f_i,f_j}^n\le R+{∆K}_{P_f}^{-},\forall n\in N,\forall f_i,f_j\in F_n:i\ne j$$

(18)

Additionally, due to the propagation characteristics of traffic flow in the route network, adjustments to the current waypoint sequence of flights must account for their impact on subsequent waypoints.

$$t_{f_i,n}-t_{f_j,n}+x_{f_i,f_j}^{n}M\ge 0,$$

(19)

$$t_{f_i,n}-t_{f_j,n}+\left(x_{f_i,f_j}^n-1\right)M\le 0,$$

(20)

$$r_{f_i,n}-r_{f_j,n}\ge 1-x_{f_i,f_j}^{n}M,$$

(21)

$$r_{f_i,n}-r_{f_j,n}\le \left(1-x_{f_i,f_j}^n\right)\left(m-1\right),\forall n\in N,\forall f_i,f_j\in F_n:i\ne j$$

(22)

3.

Variable Definition and Coupling Constraints

$$y_t^{f_i,n}\left\{\begin{array}{c}1,ift-{\displaystyle \frac{w}{2}}\le t_{f_i,n}\le t+{\displaystyle \frac{w}{2}}\\ 0,otherwise\end{array}\right.$$

(23)

$$t_{f_i,n}\ge (\left|F_n\right|-\sum _{f_j\in F_n}{x}_{f_i,f_j}^n)S_{f_i,f_j}^n+{ET}_n,\forall f\in F_n$$

(24)

$$x_{f_i,f_j}^n\in \left\{\mathrm{0,1}\right\},\forall f_i,f_j\in F_n:i\ne j$$

(25)

$$t_{f_i,n}\ge 0,\forall f_i\in F,\forall n\in N$$

(26)

5. Flight Sequence Optimization Algorithm

This chapter presents a solution approach that integrates neighborhood search and branch-and-bound techniques to solve the optimization model introduced in the previous chapter. The objective is to iteratively improve the sequence of flight overflights at waypoints, minimizing global ATFM delays.

5.1. Algorithm Principles

The order in which flights are sequenced is crucial for generating an effective solution. When solving Mixed Integer Linear Programming (MILP) problems in air traffic management, previous studies have adopted mature solution frameworks and strategies. Durand et al. described metaheuristic algorithms for optimizing air traffic, including route networks and airspace structures [44]. Pawełek and Lichota used the branch-and-cut method in the CPLEX solver—this technique builds a tree of subproblems and prunes regions that cannot yield integer solutions, combining heuristic algorithms to reduce problem size and improve efficiency [45]. This study further explores enhancements to solution methods, employing a strategy that combines the branch-and-bound method with a neighborhood search algorithm to address MILP problems in air traffic flow management. The goal is to leverage the synergy between exact algorithms and heuristic search to improve both efficiency and solution quality in complex scenarios. Specifically, In each iteration, the algorithm optimizes the solution by swapping adjacent flights, with branch selection guided by global delays and pruning strategies to accelerate convergence. This approach optimizes delays at individual waypoints and achieves coordinated optimization across all waypoints through an iteratively updated global delay evaluation mechanism.

• Neighborhood search

Neighborhood search generates a localized solution space, forming the core of this algorithm. By swapping adjacent flights, it creates different sequencing combinations that drive optimization. In each iteration, pairwise swaps of adjacent flights generate new neighboring solutions, impacting the global delay. For example, swapping the initial sequence ABCDEF creates candidate solutions like BACDEF, ACBDEF, ABDCEF, ABCEDF, and ABCDFE. This neighborhood generation supports the branch-and-bound framework and provides diverse solution paths, progressively reducing the search space to optimize delays across all waypoints.

2.

Branch-and-bound

The branch-and-bound method filters for the optimal solution by exploring the solution space with a tree structure. Each reordered sequence from swaps is treated as a branch, and its global total delay is calculated. Sub-branches are created only for solutions with delays better than the current upper bound. A breadth-first search strategy is used to expand nodes level by level, with a FIFO queue ensuring horizontal search before deeper layers. After each iteration, solutions with delays exceeding the upper bound are eliminated, retaining those with lower delays, enhancing search diversity and optimality potential.

5.2. Algorithm Steps

The detailed steps of the algorithm based on the above methods are as follows. The flowchart is shown in Figure 5:

1.

Initialization

(i)

Define variables

Read the flight set $f\in F$, waypoint set $n\in N$, and the initial order of flights. Each flight $f_i$ at the corresponding node $n_j$ has a crossing time of $t_{i,j}$. Define the sorting vector as $\sigma =\left\{{\sigma }_1,{\sigma }_2,{\dots ,\sigma }_m\right\}$.

(ii)

Calculate initial global delay

Set the initial sorting vector $\sigma$ based on the FCFS order and compute the total delay at each waypoint $D_h\left(\sigma \right)$.

$$D_h\left(\sigma \right)=\sum _{i=1}^n{t}_{i,h}-T_{i,h}$$

(27)

Calculate the global total delay:

$$D_{global}\left(\sigma \right)=\sum _{h=1}^m{D}_h\left(\sigma \right)$$

(28)

Record $D_{global}\left(\sigma \right)$ as the current optimal solution $D_{best}$, i.e., the upper bound.

2.

Hotspot detection

The algorithm determines whether a hotspot exists within the current network scope. It retrieves the hotspot queue and checks whether any unresolved hotspots remain. If a hotspot is identified, it proceeds to the next step. Otherwise, if the queue is empty, the process terminates, returning the global optimal solution $D_{best}$.

3

Branch-and-Bound

(i)

Neighborhood solution generation with constraint checking

For the selected candidate solutions that meet the conditions, generate all neighborhood solutions ${\sigma }^{\left(1\right)},{\sigma }^{\left(2\right)},\dots ,{\sigma }^{\left(k\right)}$ through pairwise swaps, where each solution corresponds to one swap operation. For example, if $\sigma =\left\{f_1, f_2, f_3, f_4\right\}$, the generated solutions are: ${\sigma }^{\left(1\right)}=\left\{f_2, f_1, f_3, f_4\right\}$, ${\sigma }^{\left(2\right)}=\left\{f_1, f_3, f_2, f_4\right\},\dots$

Since not every generated neighboring solution is feasible, a feasibility check is required to ensure compliance with priority constraints and minimum separation requirements:

• CPS constraint: The reordered flight sequence must remain within the allowable range defined by the Constrained Position Shifting (CPS) constraint. Otherwise, the solution is discarded.
• Minimum separation constraint: After adjusting the flight sequence at the waypoint, the minimum required separation time must be maintained. Otherwise, the solution is discarded.

(ii)

Calculate lower bound

For each generated neighborhood solution ${\sigma }^{\left(k\right)}$, compute the corresponding global delay value $D_{global}\left({\sigma }^{\left(k\right)}\right)$. For all neighborhood solutions ${\sigma }^{\left(k\right)}$, if $D_{global}\left({\sigma }^{\left(k\right)}\right){\ge D}_{best}$, proceed to Step (v); otherwise, continue to the next step.

(iii)

Prune

For neighborhood solutions that satisfy $D_{global}\left({\sigma }^{\left(k\right)}\right){\ge D}_{best}$, prune them. Otherwise, the remaining high-quality solutions are retained and added to the queue for further exploration.

(iv)

Update upper bound

Compare the global delay value of neighborhood solutions with the current global optimal delay value $D_{best}$:

If $D_{best}{>D}_{global}\left({\sigma }^{\left(k\right)}\right)=min\left\{D_{global}\left({\sigma }^{\left(1\right)}\right),D_{global}\left({\sigma }^{\left(2\right)}\right){\dots \dots ,D}_{global}\left({\sigma }^{\left(k\right)}\right)\right\}$, then update $D_{best}=D_{global}\left({\sigma }^{\left(k\right)}\right)$, and set the corresponding neighborhood solution as the current optimal solution ${\sigma }_{best}$; otherwise keep the upper bound unchanged.

(v)

Termination condition and output

If the queue is not empty, the next solution to be explored is retrieved, and the search continues from step (i). If the queue is empty (i.e., all possible solutions have been expanded or pruned), the branch-and-bound algorithm terminates, and the process returns to step 2 to check for any remaining hotspots.

Figure 5. Flowchart of the Flight Sequencing Optimization Algorithm.

Figure 5. Flowchart of the Flight Sequencing Optimization Algorithm.

6. Validation

ATFM delay serves as a key performance indicator of network efficiency, measured by the “waiting time” at nodes when ATM networks handle scenarios where demand exceeds capacity. To verify the feasibility and efficiency of the proposed model, a series of numerical studies were conducted focusing on the solution’s effectiveness, stability, and performance.

6.1. Experimental Setup

This study focuses on validating the effectiveness of the proposed model by designing a simplified experimental air traffic network. The objective is to verify the core mechanisms of the algorithm—namely the delay assignment strategy and priority-based scheduling logic—within a controlled environment. By leveraging a manageable network topology and tunable parameters, the experiment reduces data complexity while maintaining generalizability to broader airspace scenarios, as it does not rely on any specific data distribution.

To generate experimental instances, three main components are required: network configuration—including the airspace model and flight demand (i.e., number of flights and their schedule); priority assignment; hotspot configuration—including the initial capacity, amplitude of capacity decay, onset and end of hotspots.

For the flight network, the model consists of four hub airports, sixteen waypoints, and traffic data covering the five busiest hours. A 30 min time window is used as the observation unit, chosen based on both the nature of capacity data and algorithm operability. A schematic diagram of the network is shown in Figure 6, where the nodes represent waypoints and the colored curves denote individual trajectories. For clarity, only a subset of trajectories is displayed. Specifically, the flight network is designed as a fully connected grid composed of sixteen waypoints. Each grid node represents a waypoint, with randomized distances between connected nodes. Flights are allowed to move between adjacent grid points to simulate en-route flight segments. Flight trajectories are randomly generated for 200 flights departing over a five-hour simulation horizon. Each flight originates from one of the four hub airports at a random time, traverses a set of randomly selected waypoints, and terminates at one of the other three airports. These trajectories, combined with delay-related Equations (1) and (2), yield a comprehensive demand dataset over the network.

In the priority assignment process, the number of departing flights per airport per time window is used to evaluate the airport’s congestion level. Additionally, each flight is assessed for potential upstream delays based on its flight chain. Together, these factors determine the flight’s dynamic priority level $P$. The distribution across the four priority levels is as follows: Level 1 (18%), Level 2 (20%), Level 3 (35%), and Level 4 (27%). For all three priority strategies evaluated in this study, the parameters $\alpha$ and $\beta$ are both set to 1. Based on the determined priority level $K$ and the formulas in Table 2, the maximum permissible position shift within the flight sequence is calculated for each flight.

Regarding hotspot configuration, the capacity data is generated randomly based on real-world references, with values reflecting typical Chinese airspace capacity ranging from 15 to 30 flights per time window. The capacity sets are then derived according to Formula (3). To evaluate the model’s performance under varying overload conditions, three distinct scenarios are designed: mild overload, moderate overload, and severe overload. Following ATFM implementation practices observed in reality, the approach is as follows: a subset of waypoints is randomly selected to receive hotspot declarations, each specifying a start time, duration, and a capacity reduction percentage. Specifically, capacities are reduced by 10%, 30%, and 50%, respectively, to simulate different levels of airspace overload. This allows for assessment of how various priority strategies perform under these differing stress levels. The performance characteristics of the three overload scenarios after configuration are summarized in

Table 4. The Detailed Characteristics of the Three Overload Scenarios.

Overload Scenario	Total Number of Congested Waypoints (with at Least One Instance of Congestion)	Average Congestion Duration per Waypoint (Time Windows)	Average Waypoint Load Level
Mild overload	4	2.5	1.20
Moderate overload	6	3.5	1.28
Severe overload	7	4.1	1.44

below. The total number of congested waypoints refers to the count of waypoints that experienced congestion at least once; the average congestion duration per waypoint is calculated by dividing the total number of congested time windows by the total number of congested waypoints. The average waypoint load level refers to the mean ratio of actual traffic to maximum capacity across all waypoints within a given time window. This metric intuitively reflects the level of congestion in the airspace under different overload conditions. The experimental environment is Windows 11 64-bit, running on an AMD Ryzen 9 7940HX processor with Radeon Graphics.

6.2. Assessment of Model Performance

This section discusses the results of the proposed ATFM delay optimization method and compares three priority strategies (refer to Section 4). Using the training dataset, four models were trained under the three overload scenarios, corresponding to three priority strategies and the FCFS strategy. These models were named PCPS, ECPS, BCPS, and FCFS, respectively.

6.2.1. Model Stability Analysis

To validate the internal robustness of the proposed model, rather than its generalization across diverse operational scenarios, stability analysis is conducted through multiple simulation experiments. The null hypothesis (H₀) assumes that there is no significant difference in delays between the two methods, while the alternative hypothesis (H₁) states that the proposed model leads to a statistically significant reduction in delays. All experiments are performed under identical constraint parameters, with initial flight sets generated using different random seeds. For each scenario, three independent datasets are employed, and the three dynamic priority strategies are treated as distinct delay data values within that scenario. Consequently, nine independent measurements are obtained per scenario. A paired t-test is then applied to assess the statistical significance of delay reductions. The results demonstrate that, in all scenarios, the proposed method significantly outperforms the FCFS strategy, as shown in

Table 5. Statistical Significance of Delay Reduction across Scenarios.

Overload Scenario	Mean ± SD (Before)	Mean ± SD (After)	Mean Reduction	p-Value
Mild overload	2802 ± 609	1631 ± 136	1172	2.04 × 10−4
Moderate overload	4228 ± 575	2167 ± 496	2061	1.09 × 10−9
Severe overload	4813 ± 418	3412 ± 913	1401	4.57 × 10−4

.

The results demonstrate that the proposed method consistently outperforms the FCFS baseline across all scenarios. As shown in Table 5, all p-values are significantly lower than the 0.05 threshold, confirming the high statistical significance of delay reduction. Furthermore, the model achieves its best optimization performance in the moderate overload scenario. Although the improvement magnitude differs slightly under light and heavy overload conditions, the reductions remain statistically significant, thereby validating the stability and robustness of the method.

6.2.2. Priority Analysis

This section presents the analysis based on one representative dataset from each scenario. To evaluate the impact of different priority strategies on total delay, this study constructed a stacked bar chart of flight delay distributions under various priority strategies, as shown in Figure 7. The results demonstrate that, across all three overload scenarios, the priority-based delay optimization models significantly reduced delays compared to the FCFS strategy. Specifically, the delay times under the optimal priority strategy decreased by 30.5%, 44.1%, and 19.9%, respectively, in mild, moderate, and severe overload scenarios. These findings validate the effectiveness of the proposed delay optimization model, with the best optimization performance observed in the moderate overload scenario. The corresponding computation times are 4410, 6488, and 9715 s. Furthermore, the effects of different priority strategies and their computational times varied across scenarios, a phenomenon further analyzed in Section 6.2.3.

To further analyze the effects of priority strategies, this study visualized individual flight delay distributions under various priorities, as shown in Figure 8. Each row represents an overload scenario, while each column corresponds to a priority strategy. In each subplot, the x-axis represents the sets of flights with different priorities, and each box type represents the data distribution for all flights of that priority level. Compared to FCFS, the median delay duration for flights in all priority levels under the priority-based strategies was reduced to varying extents, and the distributions became more concentrated. This supports the effectiveness of the delay optimization model.

For flights with varying priority levels, delay assignment was improved, significantly reducing the burden on high-priority flights. Specifically, for Priority Levels 1 and 2, the median flight delays decreased significantly across all three scenarios compared to FCFS, with a more concentrated distribution. As for Priority Levels 3 and 4, delays also improved due to the global optimization performance. However, the reduction was less pronounced, and the distributions were more scattered, with some extreme values observed. This reflects the effectiveness of the priority-based strategy in redistributing delays among flights with varying priorities.

To further analyze the effects of different strategies on priority levels, and observe the overall distribution of data below 0, the improvement in delays for each priority level compared to the FCFS strategy is plotted on the vertical axis in Figure 9. Negative values represent reductions in delay relative to FCFS. The results indicate that the ECPS strategy, which broadens the adjustment range for high-priority flights, significantly reduces delays for high-priority flights but concurrently increases delays for lower-priority flights. This effect is particularly pronounced in the severe overload scenario, where the delay changes for Priority Levels 1 and 2 flights show a noticeable downward shift. However, ECPS performs poorly in the moderate overload scenario. Although some Priority 1 flights experienced delay improvements in the range of 30 to 50 min, total delays increased by 114 min. The identical performance of BCPS and PCPS in this scenario suggests that scheduling flexibility among flights was not significantly constrained by the range of priority adjustments. Moreover, the limited occurrence of capacity gaps in the moderate overload scenario implies that the ECPS strategy overly focuses on optimizing delays for high-priority flights, leading to insufficient resource allocation for lower-priority flights.

6.2.3. Efficiency and Fairness Trade-Off

As discussed in Section 3, the inherent trade-off between fairness and efficiency exists in air traffic management models. In scenarios with insufficient capacity, achieving a fair and efficient delay assignment is challenging. The proposed model, compared to traditional CASA’s FCFS, improves network efficiency by sacrificing a degree of overall fairness: from a system optimization perspective, flights with certain priority may be particularly penalized in the allocation process. To quantify the balance between efficiency and fairness under different priority strategies, two metrics are introduced: the Price of Fairness (POF) and the Price of Efficiency (POE). At this point, it is necessary to define the optimal performance, which refers to the scenario where no priority strategies are applied. In this case, the flight times are freely adjusted using the optimization model to achieve the minimum total delay. The resulting total delay under these conditions is denoted as $D_{opt}$.

Price of Fairness (POF): This represents the percentage of performance loss compared to the optimal performance due to fairness considerations. $D_{\phi }$ denotes the total delay under the current priority strategy.

$$POF={\displaystyle \frac{D_{opt}-D_{\phi }}{D_{opt}}}$$

(29)

Price of Efficiency (POE): This represents the percentage of fairness loss compared to the FCFS strategy due to efficiency considerations. $D_{FCFS}$ is the total delay under the FCFS strategy.

$$POE={\displaystyle \frac{D_{\phi }-D_{FCFS}}{D_{FCFS}}}$$

(30)

Table 6. POF and POE for Different Priority Strategies.

Priority Strategy	PCPS		ECPS		BCPS
Overload scenario	POF	POE	POF	POE	POF	POE
Mild overload	−6%	−31%	−6%	−31%	−6%	−31%
Moderate overload	−9%	−44%	−13%	−42%	−9%	−44%
Severe overload	−25%	−13%	−14%	−20%	−14%	−20%

summarizes the POF and POE values for the three priority strategies under different overload scenarios. Compared to FCFS, which assumes equal importance for all flights, the proposed model assigns different priority levels to flights, deviating from the original fairness. The POE values range between 13% and 44%, decreasing as total delays increase. In the severe overload scenario, the POE decreases to 13%, indicating that fairness was sacrificed by 13% to prioritize efficiency.

Compared to the unrestricted movement scenario without priority constraints, this study imposes priority levels on different flights, resulting in a deviation from optimal performance. The POF ranges from 6% to 25% and increases with the growth of total delay, reaching its maximum value of 25% under severe overload conditions. This indicates that considering partial fairness leads to a 25% efficiency loss. The opposite trend observed between the POF and the POE as delays increase further demonstrates that efficiency and fairness cannot be simultaneously achieved in practice.

6.2.4. Effectiveness of Priority Strategies in Different Scenarios

Since ATFM delays essentially reflect the imbalance between demand and capacity at critical nodes, it is necessary to examine how ATFM delays are distributed across nodes under different overload scenarios and priority strategies. The heatmap in Figure 10 illustrates these distributions, where each row represents one of the three overload scenarios, and each column corresponds to a priority strategy. By utilizing the delay assignment model based on priorities, delays were reduced by 30.5% in the mild overload scenario. For the most congested node, Node 6, delays decreased by 23.6% after optimization. The geographic distribution of delays became more uniform, alleviating the congestion pressure on critical nodes. These results demonstrate that the proposed optimization model effectively reallocates resources to mitigate node-specific delays.

In the mild overload scenario, the improvement levels among the three priority strategies were identical. This outcome arises because flight adjustment constraints did not act as bottlenecks under mild overload conditions. Due to the relatively low total delay, even under strict positional shift limitations, most flights could still be adjusted to meet position scheduling requirements. As a result, the advantages of broader adjustment capabilities were not fully realized. In the severe overload scenario, this limitation was mitigated. Priority strategies such as ECPS and BCPS, which allow broader flight adjustments, achieved significant delay reductions. However, in the moderate overload scenario, the performance of the ECPS strategy was relatively inferior. Specifically, a slight increase in total delay was observed, as illustrated in Figure 11. The elevated median delay for ECPS indicates that resources were redistributed to reduce delays for high-priority flights, effectively sacrificing resources for lower-priority flights. This reallocation neutralized the optimization benefits for total delay, and in some cases, led to a small increase in overall delays.

The selection of priority strategies requires comprehensive consideration of multiple factors. This study integrates computational time, delay optimization percentage, POE, and POF, as summarized in

Table 7. Comparison of Priority Strategies Based on Multiple Criteria.

Traffic Scenario	Priority Strategy	POE	Delay (min)	POF	Calculation Time (s)
Mild overload	PCPS	−31%	1473	−6%	4410
	ECPS	−31%	1473	−6%	6488
	BCPS	−31%	1473	−6%	7166
Moderate overload	PCPS	−44%	2754	−9%	6531
	ECPS	−42%	2768	−13%	7864
	BCPS	−44%	2754	−9%	9715
Severe overload	PCPS	−13%	4485	−25%	9807
	ECPS	−20%	4111	−14%	10,689
	BCPS	−20%	4111	−14%	12,670

, and compares different priority strategies using a radar chart, as shown in Figure 12. In terms of computational time, increasing the adjustable range of flights expands the solution space exponentially, thereby increasing computational complexity and significantly prolonging search time. Among the strategies, BCPS has the longest computation time, 1.5 times that of ECPS, due to its larger adjustment range. In the mild overload scenario, all three priority strategies achieved the same delay reduction, but BCPS required more computation time. In the moderate overload scenario, BCPS and PCPS demonstrated the same level of optimization, but the computation time for BCPS was 48.7% higher. However, in the severe overload scenario, BCPS and PCPS both showed significant delay optimization, with BCPS’s computation time increasing by only 18%.

Therefore, PCPS is suitable for mild to moderate overload scenarios characterized by evenly distributed traffic and low resource contention, where it achieves substantial delay reductions with shorter computation times. ECPS performs better in moderate to severe overload scenarios involving a higher proportion of critical flights, where significant improvement in high-priority flight performance is required. BCPS, by allowing broader priority adjustment ranges, balances the optimization needs across different priority levels and is more appropriate for scenarios with fewer high-priority flights and less stringent requirements on computation speed.

7. Conclusions

Building upon existing ATFM measures and delay assignment methods, this study proposes the design and implementation of a model aimed at reducing ATFM delays under multi-resource constraints by utilizing a priority-based sequencing method for flights passing through waypoints. The proposed model is applicable to cross-regional traffic management involving multiple air traffic control units and provides a reference framework for time–space resource allocation among multiple congestion hotspots. The main findings demonstrate that the dynamic priority-based ATFM delay optimization model significantly reduces ATFM delays and redistributes them among flights. The key contributions of this research are summarized as follows:

• Defining flight priorities based on network-wide effects of delays. Drawing on the core concepts of the target CASA framework, two factors are used to determine flight priority: the traffic level of the departure airport and the presence of preceding delays. A dynamic priority-setting method is proposed to categorize flights into four priority levels. This method introduces new attributes to flights, enabling those with higher value to save more time.
• Establishing three distinct priority strategies and incorporating them into the model using improved CPS constraints. By refining the traditional CPS constraint, the study introduces a method that limits the maximum positional shift K for flights of various priority levels. This allows for adjustments of up to K positions relative to the FCFS sequence. To examine the trade-offs between efficiency and fairness in the optimization model, three priority strategies were developed: PCPS, ECPS, and BCPS.
• Developing a dynamic priority-based ATFM delay optimization model. A MILP model was developed to optimize ATFM delays through a dynamic reordering of the air route network’s traffic flows through multiple hotspots. The model demonstrates robust performance in reducing total ATFM delays, redistributing delays across flights, and alleviating demand–capacity imbalances at critical nodes. Validation and testing in various overload scenarios show consistent reductions in total delays, with improvements of 30.5%, 44.1%, and 19.9% under mild, moderate, and severe overload conditions, respectively.
• Exploring the effectiveness of priority strategies under different overload scenarios. A comprehensive evaluation of the three priority strategies was conducted from the perspectives of efficiency and fairness, the delay distribution effects, and computational time. Based on this, the suitable scenarios for each priority strategy were proposed: PCPS is most suitable for light-to-moderate overload scenarios with even flight flow and weak resource competition; ECPS is better suited for severe overload scenarios with a high number of critical flights that require significant optimization of high-priority flight performance; and BCPS, with its more flexible priority adjustment range, balances the optimization needs of different priority flights, making it suitable for complex severe overload scenarios and providing better overall performance.

Regarding the solution approach, this study utilized a branch-and-bound-based flight sequencing optimization algorithm. While this method is precise, it is less efficient for handling large-scale problems. Therefore, future work will explore improved algorithms—such as reinforcement learning—to enhance scalability in solving large-scale DCB problems. Moreover, the current definition of flight priority in this study only accounts for flights with a direct network impact (e.g., those with upstream delays). However, ATFM Delay assignment inevitably causes reactive delays in downstream flights. To address this, future research may incorporate a delay propagation model that treats each flight segment in the flight chain as a weighted edge. By training a surrogate model (e.g., graph neural networks or reinforcement learning policies) on historical delay propagation data, the system can dynamically estimate the downstream impact of delays. This would enable proactive, system-oriented adjustment of flight priority scores, supporting more holistic and anticipatory traffic flow management.