Integrated airline recovery considering passengers’ decision-making under disruptions

1 Introduction

Air transportation, known for its convenience and speed, has become the preferred mode of travel, particularly for long distances. In 2019, global passenger traffic reached 4.5 billion, reflecting a 4.7% increase from the previous year (IATA, 2020). The Federal Aviation Administration projects a 2.5% annual growth in air passenger traffic over the next 20 years (FAA, 2024). Thus, the civil aviation industry is poised for sustained growth and opportunity (Rogovs et al., 2022).

Aircraft are particularly susceptible to various uncertainty factors, including adverse weather, aircraft maintenance, and crew absence (Eshkevari et al., 2024). These factors can cause significant disruptions, not only affecting individual flights but also triggering ripple effects across the interconnected flight network (Omido et al., 2025). Over the past decade, an average of 20.52% of flights have faced delays of more than 15 min due to various uncertainties, while 2.09% of flights have been canceled (BTS, 2023). Therefore, one of the major challenges faced by airport managers and airlines is how to develop effective strategies to ensure the smooth operation of airports.

Reactive strategies, or recovery strategies, play a vital role in addressing disruptions (Wu et al., 2025). In response to these disruptions, airline operations control centers provide real-time information to air traffic controllers about the airline’s operational network, facilitating decisions on resource reallocation and flight schedule adjustments to restore disrupted schedules and resume normal operations.

Scholars have conducted extensive studies on airline recovery problems since Teodorović and Guberinić (1984) published the first paper. While significant progress has been made in developing integrated recovery systems, existing studies predominantly focus on airline-centric approaches while neglecting passenger-oriented perspectives. This oversight limits the applicability of proposed solutions in diverse real-world scenarios and may compromise recovery effectiveness. This paper builds on previous studies by examining integrated recovery during disruptions, incorporating passengers’ decision-making, to better reflect real-world operational conditions. The main contributions of this paper are outlined as follows. (1) Based on passengers’ decision-making, passengers are classified into three types: those who successfully board and complete their itineraries, those who voluntarily opt for a refund, and those who are forced to wait for the original flights due to unsuccessful rebooking attempts. This classification enables a more comprehensive reflection of passengers’ real choices during disruptions and aims to minimize passenger utility loss. (2) Unlike previous studies, which typically restrict substitutions to aircraft of the same type or only allow larger aircraft to replace smaller ones, this paper also considers the possibility of smaller aircraft replacing larger ones. The goal is to strike a balance between minimizing delay time and the costs incurred from aircraft swap.

The remainder of the paper is organized follows. In Section 2, the studies on the airline recovery problem are summarized. Section 3 presents the problem description and the model formulation. Section 4 designs the algorithm to solve the model. In Section 5, the proposed method is verified according to the actual data. Section 6 gives conclusions of the paper and future work.

2 Literature review

In order to better study the airline recovery problem, the relevant studies are reviewed, which is mainly described in three parts: studies on aircraft recovery, studies on aircraft and passenger recovery, and studies applying the Nondominated Sorting Genetic Algorithm II (NSGA-II) to airline recovery. Readers who are interested in other parts can refer to Clausen et al. (2010), Hassan et al. (2021), Su et al. (2021), and Wu et al. (2025).

2.1 Aircraft recovery

Aircraft are regarded as one of the most essential and high-cost resources at airports, given their critical role in operations. As a result, the Aircraft Recovery Problem (ARP) has attracted significant attention from scholars. Teodorović and Guberinić (1984) were the first to study ARP. They proposed delaying flights and aircraft swap as recovery strategies, establishing a related model with the objective of minimizing passenger delay time, and solved it using branch-and-bound method. Subsequent studies have mainly focused on exploring various disruption scenarios, such as Aircraft-on-Ground (AOG), airport closures, and adverse weather. For these scenarios, different recovery strategies have been employed, including traditional strategies (e.g., delaying flights, canceling flights, and aircraft swaps) and novel strategies (e.g., aircraft cruise speed control and flexible aircraft maintenance strategies). For instance, Aktürk et al. (2014) were the first to propose the aircraft cruise speed control recovery strategy, incorporating cruise speed, environmental constraints, and cost as decision variables in an airline recovery optimization model. They formulated an enhanced quadratic mixed-integer programming model to enable efficient solution derivation. Their findings indicated that the proposed method could optimally manage disruptions in real time. Liang et al. (2018) pioneered a flexible aircraft maintenance recovery strategy for three disruption scenarios. They investigated the ARP under airport capacity constraints, constructed a connection network model, and designed a column generation algorithm for solving. The results showed that their method improved recovery costs accuracy by 37.74%. In terms of modeling, the most common approaches are the network-based method, such as time-space network (Yan and Yang, 1996; Thengvall et al., 2003; Aktürk et al., 2014; Lee et al., 2020; Huang et al., 2022; Lee et al., 2022; Zhao et al., 2023; Zang et al., 2024), time-band network (Bard et al., 2001), and connection network (Andersson and Värbrand, 2004; Liang et al., 2018; Haider et al., 2024; Chen and Chen, 2025). Regarding solution algorithms, the main methods used include commercial solvers (Bard et al., 2001; Aktürk et al., 2014; Chen and Chen, 2025), exact algorithms (Yan and Yang, 1996; Andersson and Värbrand, 2004; Liang et al., 2018; Zang et al., 2024), heuristic algorithms (Argüello et al., 1997; Thengvall et al., 2003; Liu et al., 2010; Lee et al., 2020; Zhao et al., 2023), and machine learning techniques (Lee et al., 2022; Haider et al., 2024; Rashedi et al., 2025).

2.2 Aircraft and passenger recovery

In integrated airline recovery problems, the Aircraft and Passenger Recovery Problem (APRP) is the primary focus. Striking a balance between the recovery costs for the airline and the losses incurred by passengers remains a contentious issue and has been a central theme in APRP-related studies. APRP-related studies are summarized in Table 1. Most scholars have employed mathematical modeling approaches to solve the APRP, with the primary modeling methods including time-space network (Bratu and Barnhart, 2006; Sinclair et al., 2014; Arikan et al., 2016; Zhang et al., 2016; Marla et al., 2017; Vink et al., 2020; Lee et al., 2021; Yan and Chen, 2021), time-band network (Hu et al., 2015; Sun et al., 2022; Sun et al., 2024), and connection network (Yan et al., 2018; Wandelt et al., 2023; Xu et al., 2023). Regarding solution methods, some scholars have applied exact methods for solving the problem (Hu et al., 2015; Arikan et al., 2016; Marla et al., 2017; Yan and Chen, 2021; Sun et al., 2022; Xu et al., 2023; Sun et al., 2024). As more factors are considered and the problem scale increases, scholars tend to design various heuristic algorithms for solving the problem to get high-quality solutions (Sinclair et al., 2014; Zhang et al., 2016; Vink et al., 2020; Wandelt et al., 2023).

TABLE 1. Summary of studies on aircraft and passenger recovery.

TSN: time-space network; CN: connection network; TBN: time-band network; ANSGA2-DS: adaptive non-dominated sorting genetic algorithm-II based on dominant strengths.

In real-world scenarios, passengers often exhibit diverse decision-making behaviors when facing flight disruptions. Currently, only a limited number of studies have incorporated passengers’ decision-making into their analyses. Hu et al. (2021) enhanced passengers’ willingness through prospect theory by quantifying psychological losses and employing a multi-directional variable neighborhood search algorithm. Lu et al. (2025) further incorporated overnight stay options and designed a large neighborhood search algorithm for their model. Results demonstrate that considering passenger preferences significantly reduces recovery costs while maintaining satisfactory service levels. Chen et al. (2023) introduced passenger preferences into a bi-objective optimization framework, using total airline delay costs as an efficiency metric and total passenger delay time as an equity metric. Their proposed improved NSGA-II algorithm effectively generated acceptable solutions. Cadarso and Vaze (2023) addressed “phantom passengers” by establishing delay thresholds through regression modeling, then developed a cost-minimization model that was linearized and solved using CPLEX. Results showed their approach achieves operational recovery with reduced costs.

2.3 NSGA-II in airline recovery

Genetic Algorithm (GA) has been widely adopted in airline recovery due to their ability to solve complex combinatorial problems. Early research primarily focused on single-objective formulations, such as minimizing costs in aircraft routing (Kammoun and Rezg, 2018) or crew rescheduling (Chang, 2012). However, the inherent conflicts between stakeholder interests, particularly between airline costs and passenger service quality, necessitated a shift toward multi-objective optimization.

Within this context, NSGA-II, introduced by Deb et al. (2002), has emerged as one of the most influential methods. Its core mechanisms-fast nondominated sorting, crowding distance for diversity preservation, and explicit elitism-make it exceptionally effective for exploring well-distributed Pareto frontiers in complex operational settings. A growing body of research has applied NSGA-II and its variants to airline recovery problems. Chen and Chou (2017) employed an NSGA-II variant for the crew recovery problem, incorporating multiple objectives such as minimizing crew changes and flight time deviations. For the integrated airline recovery problems, Chen et al. (2023) developed an adaptive NSGA-II to balance airline delay costs against passenger delay time, introducing the price of fairness metric to quantify this trade-off; Chen and Wang (2025) proposed an improved NSGA-II for arrival flight rescheduling under ground delay programs, by incorporating passenger satisfaction, they generated more passenger-centric and operationally effective recovery schedules; Chen et al. (2020) solved the aircraft and crew recovery problem using a two-segmented chromosome structure within an NSGA-II variant, demonstrating superior results over traditional sequential planning.

In summary, NSGA-II has proven to be a robust and adaptable framework for multi-objective airline recovery, continuously evolving to accommodate greater operational complexity and a stronger focus on passenger satisfaction.

2.4 Summary

Studies on the airline recovery problem have spanned several decades, resulting in a relatively mature theoretical framework. However, gaps remain in certain areas. Most studies simplify the aircraft swap strategy, overlooking cases where the number of passengers on a large aircraft is smaller than the seating capacity of a small one, as well as the potential for small aircraft to replace larger ones. Additionally, many studies fail to consider passengers’ decision-making during disruptions, assuming unconditional compliance with the airline arrangements, which may not reflect real-world behavior. While some studies have addressed passenger behavior, their classification of passengers’ decisions remains oversimplified, particularly in neglecting detailed consideration of passengers’ tolerance thresholds for delay durations. This paper addresses these issues by enhancing the aircraft swap strategy in the integrated recovery problem, considering the replacement of small aircraft for large ones and incorporating passengers’ decision-making. Passengers are categorized based on their choices, aiming to minimize recovery costs and maintain passenger satisfaction.

3 Problem description and model formulation

3.1 Problem description

3.1.1 Airline recovery strategies

This section investigates the integrated recovery problem for disrupted flights under medium-to-short duration AOG scenarios. Given the substantial costs associated with flight cancellations, the proposed recovery strategies primarily focus on delaying flights and aircraft swaps. Referring to Vink et al. (2020), this paper sets a maximum delay threshold of 6 h when implementing recovery strategy of delaying flights. This limit is imposed because longer delays would incur costs comparable to cancellations. The aircraft swap strategy in this paper extends prior approaches (Zhang et al., 2016) by introducing capacity-constrained small-for-large substitutions (inspired by Lee et al. (2021)), in addition to conventional same-type exchanges and large-to-small swaps. Implementation requires the capacity of smaller aircraft to accommodate all original passengers. The improved aircraft swap recovery strategy is shown in Figure 1. Initially, a large aircraft, a 2 (capacity 180), was scheduled to operate flight f 3 with 100 passengers. However, a sudden malfunction requires a 2 to undergo maintenance. To minimize delays, the airline can implement an aircraft swap strategy, utilizing a smaller available aircraft, a 1 (capacity 120), to replace a 2 for f 3, This allows passengers to reach their destination with minimal delay. This strategy offers airlines flexibility in optimizing recovery while ensuring passenger satisfaction and protecting the reputation of the airline.

3.1.2 Passengers’ decision-making

When flight disruptions occur, passengers exhibit a certain degree of heterogeneity, leading to varying decision-making behaviors in response to delays. Most previous studies assume that passengers fully comply with the airline’s arrangements, overlooking their autonomy in decision-making. This assumption does not align with the realities of passenger behavior. For the convenience of this paper, we focus solely on the different decisions passengers make in response to disruptions, without considering personal attributes such as age, gender, personality, or occupation. When passengers encounter flight disruptions that prevent them from reaching their destination as scheduled within a short timeframe, disrupted passengers may choose to continue waiting, rebook alternative flights, or request refunds. Some passengers with higher delay tolerance may retain their original itineraries, while those less tolerant of delays will opt for rebooking or refunds. For passengers voluntarily seeking refunds, their requests can always be accommodated; however, for those choosing rebooking, insufficient seat capacity may lead to failed rebooking attempts, resulting in being forced to wait for the original flights. Based on passengers’ decision-making willingness and final results, they can be categorized into three types, as shown in Figure 2.

The first type, termed “delayed passengers who successfully board and complete their itineraries,” encompasses two scenarios: (1) when the delay duration is relatively short and dispatchers do not intervene (without implementing aircraft swap strategy), leaving the original aircraft to operate the flight as planned; and (2) when dispatchers employ recovery strategies involving both delaying flights and aircraft swap, deploying other aircraft to replace the disrupted one, ultimately enabling passengers to board and complete their itineraries. Passengers with successful rebooking are accommodated on subsequent flights matching their original itineraries while still experiencing delays, meaning this passenger group effectively includes both those waiting for the original flights and those successfully rebooked, all subjected to comparable delay durations.

The second type, “refunded passengers who voluntarily refund their tickets due to low tolerance for delays,” refers to passengers, typically leisure travelers, who opt for refunds and reschedule their itineraries when faced with lengthy delays. This category mainly includes passengers with lower delay tolerance.

The third type, “denied passengers,” refers to those who are forced to wait due to failed boarding intentions caused by insufficient aircraft capacity to meet rebooking demand. These passengers represent an alternative classification within the same scenario as the first type, where some passengers successfully rebook and complete their itineraries while others, constrained by limited aircraft seating capacity, experience failed rebooking attempts and are forced to wait for their original flights.

It is important to note that the passenger classification in this paper is based on the outcomes of disruptions (e.g., whether they board, refund, or are denied) rather than a behavioral model that captures passengers’ endogenous decision-making processes. This simplification is adopted to maintain tractability of the integrated optimization model and to provide a first-step analysis of how passenger heterogeneity affects recovery decisions. The parameters are used to approximate average passenger responses to delays and aircraft downgrades, and their impact is examined through sensitivity analysis in Section 5.2.3.

3.2 Model formulation

Before constructing the model, several assumptions are made in this paper to facilitate a more comprehensive analysis of the problem, as outlined below.

• The number of crew members is assumed to be sufficient to meet the scheduling requirements of the airline.
• It is assumed that no flight can depart earlier than scheduled.
• The number of refunded passengers is assumed to be directly proportional to the delay time.

The notations used to describe the problem and construct the model are provided in Table 2.

TABLE 2. Notations.

3.2.1 Objective functions

3.2.1.1 Minimize airline recovery costs

Airline recovery costs refer to the economic losses and additional operational management expenses incurred due to flight disruptions caused by unforeseen factors, representing the direct and short-term financial burdens on airlines. The first objective function in this paper seeks to minimize these recovery costs, focusing on aircraft recovery, which is a critical and expensive resource. The costs are primarily categorized into two types: flight delay costs and aircraft swap costs. The first objective function can be represented by Equation 1.

min C = ∑ a ∈ A ∑ f ∈ F C f , a de x f , a + ∑ a , b ∈ A C a , b sw y a , b (1)

The delay costs C f , a de can be obtained by Equation 2.

C f , a de = c ¯ de Δ t f (2)

3.2.1.2 Minimize passenger utility loss

Passenger utility loss refers to the direct economic losses caused by delays or changes to passengers’ itinerary due to disruptions, as well as the indirect losses affecting the passengers’ travel or business activities. These different decision-making results in three possible outcomes, which are defined as three types of passengers: delayed passengers, refunded passengers, and denied passengers. Therefore, the total number of passengers on a disrupted flight can be divided into three types, as shown in Equation 3.

Q f = q 1 f + q 2 f + q 3 f (3)

Correspondingly, the passenger utility loss can be represented by the sum of the utility losses of the three types of passengers. The second objective function can be represented by Equation 4.

min U = u 1 + u 2 + u 3 (4)

• Utility loss of delayed passengers

When implementing aircraft swap strategy during rescheduling to minimize delays and allow passengers to reach their destinations as quickly as possible, smaller aircraft may sometimes replace larger ones. As a result, this paper classifies passengers who successfully board and complete their itineraries into two types. The first type occurs when the aircraft passengers board is not smaller than the original aircraft, meaning passenger utility loss remains unchanged. The second type arises when the aircraft passengers board is smaller than the original one. To quantify this, a parameter α is introduced to represent the proportion of utility loss increase due to the reduction in aircraft size. The value of α is set to 0.1. In summary, the utility loss of delayed passengers can be represented by Equation 5.

u 1 = ∑ a , b ∈ A ∑ f ∈ F d 1 ω a , b α + 1 q 1 f x f , a (5)

In Equation 5, q 1 f needs to be calculated. First, the original passengers are divided into three types, and the number of refunded passengers can be obtained from Equation 6.

q 2 f = λ Δ t f (6)

This linear relationship is adopted as a first-order approximation of passenger sensitivity to delays, enabling tractable integration of passenger behavior into the recovery optimization model. Empirical study in related transportation domains has also observed strong linear correlations between delay duration and passenger refund behavior (Erblich, 2023). To assess the robustness of the model to this assumption, we conduct a sensitivity analysis on λ in Section 5.2.3.

In this paper, λ is defined such that for every 15 minutes of delay beyond the initial 15-min threshold, one passenger will refund their ticket. q 1 f and q 3 f are determined by the number of passengers waiting in the boarding area (Q f − q 2 f) and the seating capacity of the aircraft actually operating the flight. When the seating capacity is sufficient to accommodate all waiting passengers, all of them can successfully board and complete their itinerary. When the seating capacity is insufficient to accommodate all waiting passengers, only a number of passengers equal to the seating capacity of the aircraft can successfully board. That is, q 1 f is determined by comparing the number of waiting passengers with the seating capacity of the aircraft, which is given by Equation 7.

q 1 f = min Q f − λ Δ t f , n f , a (7)

• b. Utility loss of refunded passengers

According to the assumptions, the number of refunded passengers is positively related to the delay duration. Based on Equation 6, the utility loss of refunded passengers can be represented by Equation 8.

u 2 = ∑ a ∈ A ∑ f ∈ F d 2 q 2 f x f , a (8)

• c. Utility loss of denied passengers

While waiting for the flight, due to the implementation of the aircraft swap strategy to mitigate delays, there may be situations where the seating capacity of the aircraft actually operating the flight cannot meet the demand of rebooking passengers. As a result, some passengers may not be able to rebook successfully and are forced to wait for the original flights, leading to willingness failure. The utility loss of denied passengers, u 3, can be expressed by Equation 9.

u 3 = ∑ a ∈ A ∑ f ∈ F d 3 q 3 f x f , a (9)

As previously explained when solving for q 1 f, the number of waiting passengers and the seating capacity of the aircraft actually operating the flight also determine q 3 f, which represents the difference when the seating capacity of the aircraft cannot meet the demand of waiting passengers. Therefore, the expression for q 3 f is given by Equation 10.

q 3 f = max 0 , Q f − λ Δ t f − n f , a (10)

3.2.2 Constraints

3.2.2.1 Corresponding constraint between aircraft and flights

The uniqueness requirement for the correspondence between aircraft and flights, where each flight can be performed by at most one aircraft, is expressed by Equation 11.

∑ a ∈ A x f , a ≤ 1 , ∀ f ∈ F (11)

3.2.2.2 Flight time constraint

The assumption that the actual departure time cannot be earlier than the planned departure time is expressed by Equation 12.

M 1 1 − x f , a + t f ac − t f sc x f , a ≥ 0 , ∀ f ∈ F , a ∈ A (12)

3.2.2.3 Aircraft swap constraints

First, when implementing the aircraft swap strategy, the two aircraft that can be swapped must be at the same airport. Therefore, Equation 13 is used to ensure this condition.

p a = p b ∀ a , b ∈ A , y a , b = 1 (13)

Secondly, alternative aircraft eligible for the swap strategy must meet time requirements. For aircraft with subsequent flight tasks, the scheduled departure time of the subsequent flight must fall within the time range between the scheduled and actual departure times of the disrupted aircraft, as expressed by Equations 14, 15.

t g , b sc y a , b + M 1 1 − y a , b ≥ t f , a sc , ∀ a , b ∈ A , a ≠ b , f , g ∈ F , f ≠ g (14)

t g , b sc y a , b − M 1 1 − y a , b < t f , a ac , ∀ a , b ∈ A , a ≠ b , f , g ∈ F , f ≠ g (15)

For alternative aircraft without subsequent flight tasks, this paper stipulates that the sum of the actual arrival time of the alternative aircraft at the disrupted aircraft’s airport and its turnaround time must be less than the actual departure time of the disrupted aircraft, ensuring the alternative aircraft can replace the disrupted one. This condition is expressed in Equation 16.

t ′ g , b ac y a , b − M 1 1 − y a , b + T u r a < t f , a ac , ∀ a , b ∈ A , a ≠ b , f , g ∈ F , f ≠ g (16)

The above constraints classify the alternative aircraft based on whether they have subsequent flight tasks. To simplify and optimize the constraints, we merge the two types of alternative aircraft, as shown in Equations 17, 18.

1 − τ g , b y a , b t g , b sc + 1 − y a , b M 1 + τ g , b M 2 ≥ t f , a sc , ∀ a , b ∈ A , a ≠ b , f , g ∈ F , f ≠ g (17)

1 − τ g , b y a , b t g , b sc − 1 − y a , b M 1 + τ g , b y a , b t ′ g , b ac − 1 − y a , b M 2 + T u r a < t f , a ac , ∀ a , b ∈ A , a ≠ b , f , g ∈ F , f ≠ g (18)

3.2.2.4 Passenger constraints

This paper classifies flight passengers based on their decision-making. Overall, passenger balance must be maintained. The sum of passengers in different types should not exceed the seating capacity of the aircraft executing the original flight, as shown in Equation 19.

Q f ≤ n f , a f sc , ∀ f ∈ F , a f sc ∈ A (19)

The number of delayed passengers cannot exceed the total number of passengers, both refunded and denied passengers are fewer than the total number of passengers, and the number of passengers in each type is an integer, the above constraints are represented by Equations 20–23.

q 1 f ≤ Q f , ∀ f ∈ F (20)

q m f < Q f , ∀ f ∈ F , m = 2 , 3 (21)

q m f ∈ N , ∀ f ∈ F , m = 1 , 2 , 3 (22)

Q f ∈ N * , ∀ f ∈ F (23)

3.2.2.5 Variable constraints

The constraints on the values of the variables used in the model are shown in Equations 24–28.

x f , a ∈ 0 , 1 , ∀ f ∈ F , a ∈ A (24)

y a , b ∈ 0 , 1 , ∀ a , b ∈ A (25)

τ g , b ∈ 0 , 1 , ∀ g ∈ F , b ∈ A (26)

ω a , b ∈ 0 , 1 , ∀ a , b ∈ A (27)

y a , b ≥ ω a , b , ∀ a , b ∈ A (28)

4 Solution algorithm

The integrated airline recovery problem is NP-hard due to the combinatorial nature of aircraft-flight assignments and swap decisions. As the number of aircraft and flights increases, exact solution methods become computationally prohibitive, especially under the tight time constraints of real-time disruption management. Metaheuristics offer a practical alternative for obtaining high-quality solutions within acceptable time. Among them, NSGA-II (Deb et al., 2002) is a well-established multi-objective evolutionary algorithm that efficiently explores the Pareto frontier and maintains solution diversity. It addresses conflicts among multiple objectives through nondominated sorting, and preserves elite individuals across generations via a dual-criterion selection mechanism that combines nondominated sorting with crowding distance, as illustrated in Figure 3. This elitist strategy prevents the loss of high-quality solutions and enables the accumulation of historically superior individuals. Through fast nondominated sorting, crowding distance computation, and elitist preservation, NSGA-II significantly enhances solving capability and computational efficiency compared to traditional genetic algorithms. These features make it particularly effective for complex integrated airline recovery problems, allowing it to balance competing objectives such as airline recovery costs and passenger utility loss, thereby systematically generating high-quality recovery plans that achieve optimal trade-offs between operational and passenger-centric metrics. Given its successful applications in airline recovery (Chen et al., 2023; Chen and Wang, 2025), NSGA-II is adopted in this paper.

4.1 Population initialization

Based on the characteristics of the aircraft and passenger recovery problem studied in this paper, it is necessary to identify the disrupted aircraft and their respective airports, and then determine the original flight route of the aircraft, scheduled departure time, actual available time after the disruption, and the set of alternative aircraft that can replace the disrupted aircraft to operate the original flight task. After screening the qualified candidates, assume that W alternative aircraft can be selected to replace the disrupted aircraft in operating the original flight task. The initial population can then be randomly generated by sorting the available times of the alternative aircraft. The population consists of N individuals, each representing a specific aircraft swap recovery strategy, and is encoded using chromosome representation. For example, in Figure 4, each individual represents a specific aircraft swap strategy implementation, consisting of genes arranged in a random sequence of 0 and 1. The position of each gene corresponds to whether a particular alternative aircraft is selected for the swap strategy, and the gene’s value indicates the aircraft’s tail number. [0,1,0,1,0,0,0] represents a set of 7 alternative aircraft, where A2 and A4 are selected for the aircraft swap strategy. The specific rule is as follows: A2 will replace the disrupted aircraft to operate the original flight, and A4 will replace A2 to operate its flight. Ultimately, when the disrupted aircraft becomes available, it will be assigned the task originally assigned to A4.

4.2 Fast nondominated sorting

Fast nondominated sorting is the process of comparing all individuals in a population. Each individual i contains two parameters, U i and R i, where U i represents the number of individuals dominated by individual i, and R i represents the number of individuals dominating individual i. In this paper, the dominance principle is defined as follows: individual i dominates individual j if both airline recovery costs and passenger utility loss of individual i are less than those of individual j. Initially, all individuals with U i = 0 are stored in the nondominated set S 1. Then, the set of dominated individuals R i for each individual i in S 1 is iterated. For each individual j in R i, U j is decreased by 1. If U j − 1 = 0, individual j is then added to the nondominated set S 2. This process is repeated until all individuals are assigned to different layers of the nondominated sets.

4.3 Genetic process and termination

For the genetic process of the algorithm, specifically the crossover and mutation operations, this paper adopts the binary crossover and polynomial mutation methods from the NSGA-II algorithm to generate offspring. Specifically, the binary crossover operation crosses over the gene sequences of parent individuals to produce new individuals, thereby enhancing the diversity of the population. Polynomial mutation, on the other hand, randomly changes an individual’s gene values with a certain probability to increase the adaptability of the population and prevent premature convergence.

Regarding the termination criterion of the algorithm, the stopping condition is mainly determined by setting a maximum number of iterations. When the iteration count reaches the preset maximum value, the algorithm will stop and output the optimal solution from the current population. The flowchart of the NSGA-II algorithm used in this paper is shown in Figure 5.

5 Case study

5.1 Data description

This section validates the proposed method using actual flight schedule data from a Chinese airline, focusing on its operations on a typical weekday in April 2019. The dataset, obtained through a collaborative research agreement, comprises 1048 flights operated by 126 aircraft across 32 airports. It includes detailed information such as aircraft tail numbers, flight numbers, scheduled departure and arrival times, origin and destination airports, seat capacity, and actual passenger counts. To ensure data quality, we removed flights with missing or inconsistent records (less than 2% of the original data). Table 3 presents a sample of the data structure.

TABLE 3. Flight schedule data.

PEK: beijing capital international airport; CTU: chengdu shuangliu international airport; SYX: sanya phoenix international airport; CGO: zhengzhou xinzheng international airport.

To evaluate the proposed model under controlled conditions, we focus on AOG scenarios with varying maintenance durations at a single airport. While this represents a common and fundamental disruption type in airline operations, the modeling framework itself is general and can be adapted to other disruption events-such as airport closures, severe weather, or air traffic control restrictions-by modifying input data and constraints accordingly. Based on actual flight data, an aircraft is grounded at PEK. A mechanical failure causes the aircraft to be grounded for maintenance, preventing it from operating as scheduled. Maintenance times are set to 5–8 h. The developed NSGA-II algorithm in this paper was implemented in Python on the PyCharm platform and executed on a desktop computer equipped with an Intel Xeon Gold 6244 CPU, 128GB RAM, and Windows 10 operation system. Referring to Hu et al. (2015) and Sun et al. (2026), the flight delay costs are set at 13.74 yuan/min, with per delayed passenger utility loss of 1.36 yuan/min, average ticket price for refunded passengers, and 696.16 yuan for per denied passenger. Furthermore, referring to Marla et al. (2017) and Lee et al. (2020), the aircraft swap costs are set at 500 dollars per swap, converted to 3560.85 yuan using the 2024 average RMB-USD exchange rate of 7.1217. The aircraft turnaround time is set at 40 min.

In the NSGA-II algorithm, this paper sets the population size to 100, maximum generations to 200, crossover probability to 0.8, and mutation probability to 0.1. Taking disruption scenario 1 as an example for algorithm validation, Figure 6 demonstrates that the airline recovery costs converge at 140 generations while passenger utility loss converges at 135 generations, confirming the effectiveness of the algorithm; these results can consequently serve as reliable inputs for subsequent analysis and discussion.

5.2 Results and discussion

5.2.1 Solution analysis

Figure 7 presents the Pareto solution sets of the objective functions under four disruption scenarios. The results reveal a clear trade-off relationship between airline recovery costs and passenger utility loss, demonstrating that while implementing recovery strategies with additional resources increases operational costs, it concurrently enhances passenger satisfaction by reducing psychological losses, thereby highlighting the critical importance of incorporating passengers’ decision-making. As disruption scale escalates, airlines progressively increase recovery costs to compensate for delays caused by aircraft malfunctions. Notably, disruption scenario 2 exhibits lower passenger utility loss than disruption scenario 1, attributable to the following factors: By employing delaying flights and aircraft swap strategy, airlines mitigate subsequent delays. Passenger utility loss primarily comprises three components: delayed passengers, refunded passengers, and denied passengers. In disruption scenario 2, the implemented recovery strategies enable more passengers to complete their itineraries. Although delayed and refunded passengers incur greater losses compared to disruption scenario 1, the measures significantly reduce the number of denied passengers, resulting in an overall reduction of passenger utility loss. However, as the maintenance time prolongs (disruption scenario 3, 4), the strategies become increasingly inadequate in meeting passenger demands, leading to more frequent rebooking failures and consequently amplified utility loss. Compounded by growing numbers of delayed and refunded passengers, this creates markedly higher utility losses, ultimately degrading travel experiences.

To assess the practicality of the proposed NSGA-II algorithm for real-time disruption management, we recorded the computation time for each disruption scenario under the same environment. Table 4 reports the average computation time over 10 independent runs for the four disruption scenarios. As shown, the time increases with problem complexity but remains within 200 s even for the largest scenario. Given that airline operations control centers typically require recovery decisions within 5 min, our algorithm is sufficiently efficient for practical use. Moreover, the runtime could be further reduced by parallelizing the population evaluation or adopting a warm-start strategy, making it suitable for integration into real-time decision support systems.

TABLE 4. Computation time for different disruption scenarios.

Although both objective functions are influenced by delay time, they capture distinct dimensions of disruption impacts. Airline recovery costs consist of flight delay costs (directly proportional to delay time) and aircraft swap costs (incurred when implementing aircraft substitutions). Passenger utility loss aggregates three components: delayed passenger loss (also delay-dependent), refunded passenger loss (ticket price), and denied passenger loss (fixed compensation). These components respond differently to recovery strategies. For example, implementing an aircraft swap may reduce delays (lowering delay costs and delayed passenger loss) but incurs swap costs (increasing airline costs). Similarly, swapping to a smaller aircraft may allow more passengers to board (reducing denied passenger loss) but introduces downgrade-related psychological loss (increasing delayed passenger loss through parameter α). Therefore, the two objectives are not perfectly correlated. The Pareto frontiers in Figure 7 clearly illustrate this trade-off: points that minimize airline costs often result in higher passenger utility loss, and vice versa. The spread of solutions along the frontier demonstrates that improving one objective typically comes at the expense of the other, confirming the need for multi-objective optimization.

In conclusion, the aircraft swap strategy coupled with passengers’ decision-making can effectively reduce utility loss while maintaining satisfaction levels. Section 5.2.2 will further validate these advantages through comparative analysis with baseline solutions.

5.2.2 Recovery plan analysis

To demonstrate the significance of considering passengers’ decision-making and the aircraft swap strategy, this section compares the proposed recovery strategies with three benchmark plans: B1 (excluding both passengers’ decision-making and improved aircraft swap strategy), B2 (considering passengers’ decision-making but using the traditional aircraft swap strategy), and B3 (using the improved aircraft swap strategy but ignoring passengers’ decision-making). As shown in Table 5, the proposed method consistently achieves better solutions across all disruption scenarios. In disruption scenario 1, it shows relatively modest improvements over B1 (5.57% and 6.54% optimization for the two objective functions). However, its effectiveness increases substantially with larger disruptions, reaching 30.29% and 26.23% optimization in disruption scenario 4, with average optimization rates of 17.15% and 16.59% respectively. These results confirm that integrating passengers’ decision-making with improved aircraft swap strategy significantly improves recovery efficiency, particularly in large-scale disruption scenarios. The methodology demonstrates robust capability in mitigating delays and accelerating airport recovery operations.

TABLE 5. Comparison of different recovery plans.

ARC: airline recovery costs; PUL: passenger utility loss.

Optimization rate = ∣recovery plan-B1∣/B1*100%.

To further investigate the optimization effects of considering passengers’ decision-making and improved aircraft swap strategy, we conducted a comparative analysis between B2 and B3. The results indicate that in disruption scenario 1, B3 demonstrates higher optimization efficiency than B2, with B2 achieving merely 1.72% and 1.19% optimization rates, suggesting that in small-scale disruption scenarios, the enhanced aircraft swap strategy more effectively alleviates airport congestion and meets passenger demands, with its optimization effect being dominant. However, in disruption scenario 2, optimization rates of B2 slightly surpass those of B3, and as the disruption scale increases, optimization rates of B2 progressively improve while B3 remain largely unchanged, demonstrating that in medium-to-large-scale scenarios, merely improving recovery strategies becomes insufficient to satisfy passenger demands and may instead lead to increased passenger utility loss; only by incorporating passengers’ decision-making and implementing targeted aircraft scheduling can costs be effectively reduced. These findings are fully consistent with the conclusions presented in Section 5.2.1.

In summary, the integrated consideration of both passengers’ decision-making and improved aircraft swap strategy proves most effective for cost reduction; while the improved aircraft swap strategy performs well in small-scale scenarios, the incorporation of passengers’ decision-making becomes increasingly crucial as disruption scales expand, enabling further reduction in recovery costs.

5.2.3 Sensitivity analysis

This paper incorporates parameter α to quantify the passenger utility loss when aircraft are downgraded, and conducts a sensitivity analysis to examine its impact on both objective functions. The analysis tests α values over the interval 0 , 1 with a step size of 0.05, with Figure 8 illustrating how the objective functions vary across different disruption scenarios. For small-scale disruptions, the objective functions show modest increases when α is small, but the growth rate accelerates as α rises. Specifically in disruption scenario 1 (Figure 8A), both objectives demonstrate gradual growth when α ∈ 0 , 0.4, but exhibit significantly steeper growth when α ∈ 0.4 , 1. In large-scale disruptions, the influence of α on airline recovery costs still follows this two-phase pattern, while its effect on passenger utility loss differs substantially - showing strong linear correlation from the outset, as seen in disruption scenario 4 (Figure 8D) where passenger utility loss nearly mirrors the increase of α proportionally.

Moreover, the baseline value of λ is set to 1 passenger per 15 min (i.e., 4 passengers per hour). To examine its impact on the two objectives, a sensitivity analysis is conducted by varying λ from 0 to 9 passengers per hour with a step size of 1. Figure 9 illustrates how the objective functions vary across different disruption scenarios. As shown in Figures 9A,B (small-scale scenarios), the airline recovery cost exhibits a monotonically decreasing trend that gradually flattens as λ increases. This is because a higher refund rate reduces the number of waiting passengers, alleviating capacity pressure and allowing more flexible aircraft assignments, which in turn lowers both flight delay costs and aircraft swap costs. Once the swap strategy is fully optimized, the marginal benefit of further increases in λ diminishes, leading to a stabilization of the airline recovery cost. In contrast, the passenger utility loss displays a U-shaped pattern: it first decreases with λ, reaches a minimum at λ = 4, and then increases. This non-monotonic behavior arises from the trade-off between two opposing effects. When λ is small, few passengers refund, resulting in a large number of waiting passengers and a high risk of denied boarding, which incurs substantial utility loss from denied passengers. As λ increases, more passengers opt for refunds, reducing denied boarding incidents and lowering the total passenger utility loss. However, beyond a certain point, the growing number of refunded passengers adds to the total passenger utility loss through the loss from refunded passengers, outweighing the reduction in denied boarding loss and causing the overall passenger utility loss to rise. In large-scale scenarios (Figures 9C,D), the same qualitative trends are observed, but the slopes are steeper. Due to the larger number of affected flights and passengers, even small changes in λ lead to more pronounced fluctuations in both objectives. This heightened sensitivity underscores the need for careful calibration of λ in large disruptions.

The sensitivity analyses above provide further evidence that the two objective functions are not highly correlated, but rather exhibit inherent trade-offs. If the two objectives were perfectly correlated, they would respond to parameter changes in the same direction and with similar magnitudes. However, the results show distinct patterns: For parameter α, Figure 8 reveals that in large-scale disruptions (scenario 4), airline recovery costs show a two-phase response (gradual then steep increase), while passenger utility loss exhibits an immediate and nearly linear increase. This divergence indicates that the two objectives are sensitive to α in fundamentally different ways. For parameter λ, Figure 9 demonstrates even more striking differences: airline recovery costs follow a monotonically decreasing trend that gradually flattens, whereas passenger utility loss displays a U-shaped pattern-first decreasing, reaching a minimum, then increasing. This opposite behavior around the optimal λ value clearly illustrates the trade-off: a higher refund rate reduces airline costs by alleviating capacity pressure, but beyond a certain point, it increases total passenger utility loss due to growing refunded passenger losses.

These divergent responses to parameter variations confirm that the two objectives capture distinct stakeholder perspectives and cannot be simultaneously optimized without compromise. The trade-offs observed in both the Pareto frontiers (Figure 7) and sensitivity analyses (Figures 8, 9) underscore the importance of the multi-objective approach adopted in this paper.

6 Conclusion and future work

This paper investigates the integrated recovery problem based on passengers’ decision-making. It primarily employs delaying flights and aircraft swap as core recovery strategies under disruption scenarios, where the swap strategy specifically permits small aircraft to substitute large ones, thereby enhancing operational flexibility and response capability. It further incorporates passengers’ decision-making when facing disruptions - including waiting for original flights, rebooking, or requesting refunds - which directly lead to three outcomes: successful boarding, voluntary refunds, or forced to wait due to willingness failure, thus comprehensively capturing real-world passenger behaviors. A bi-objective optimization model is formulated to simultaneously minimize airline recovery costs and passenger utility loss. The NSGA-II algorithm is developed to solve the model. Validation using actual airline operational data demonstrates the superiority of propose method over benchmark plan, achieving average reductions of 17.15% in airline recovery costs and 16.59% in passenger utility loss. The improved aircraft swap strategy proves particularly effective for small-scale disruptions, while the passenger-centric approach excels in large-scale disruptions. Sensitivity analysis confirms that parameter calibration requires scenario-specific trade-offs, with minor parameter variations in major disruptions triggering substantial cost fluctuations.

Future work will explore several directions. First, the crew recovery problem will be incorporated into the integrated recovery study. Second, while the current study classifies passengers based on disruption outcomes, future research will develop behavioral models (e.g., discrete choice models) to endogenously represent passengers’ decision-making processes when facing delays, including their choices among waiting, rebooking, and refunding. This will better capture the heterogeneity and rationality of passenger behavior, leading to more realistic assessments of utility loss and more effective recovery strategies. Furthermore, while a proportional relationship between the number of refunded passengers and delay time is assumed, future studies will involve conducting surveys and field investigations to better understand passengers’ decision-making willingness under different disruptions. This will provide more accurate parameter values, improving the practical applicability of the model. Lastly, integrated recovery is a large-scale optimization problem, and efforts will be made to develop more efficient algorithms to enhance both the solution speed and effectiveness.

Generative AI statement

The author(s) declared that generative AI was not used in the creation of this manuscript.

Any alternative text (alt text) provided alongside figures in this article has been generated by Frontiers with the support of artificial intelligence and reasonable efforts have been made to ensure accuracy, including review by the authors wherever possible. If you identify any issues, please contact us.

Publisher’s note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

Footnotes

Edited by: Miguel Jaller, University of California, Davis, United States

Reviewed by: Jiajin Lin, Taizhou Vocational College of Science and Technology, China

Huifen Zhong, Shenzhen Institute of Information Technology, China