A Novel Stochastic Framework for Integrated Airline Operation Planning: Addressing Codeshare Agreements, Overbooking, and Station Purity

Abstract

This study presents an integrated optimization framework for fleet assignment, flight scheduling, and aircraft routing under uncertainty, addressing a core challenge in airline operational planning. A three-stage stochastic mixed-integer nonlinear programming model is developed that, for the first time, simultaneously incorporates station purity constraints, codeshare agreements, and overbooking decisions. The formulation also includes realistic operational factors such as stochastic passenger demand and non-cruise times (NCT), along with adjustable cruise speeds and flexible departure time windows. To handle the computational complexity of this large-scale stochastic problem, a Sample Average Approximation (SAA) scheme is combined with two tailored metaheuristic algorithms: Simulated Annealing and Cuckoo Search. Extensive experiments on real-world flight data demonstrate that the proposed hybrid approach achieves tight optimality gaps below 0.5%, with narrow confidence intervals across all instances. Moreover, the SA-enhanced method consistently yields superior solutions compared with the CS-based variant. The results highlight the significant operational and economic benefits of jointly optimizing codeshare decisions, station purity restrictions, and overbooking policies. The proposed framework provides a scalable and robust decision-support tool for airlines seeking to enhance resource utilization, reduce operational costs, and improve service quality under uncertainty.

1. Introduction

The airline industry operates under intense competitive pressure, fluctuating demand, and strict operational constraints. In such an environment, the integrated management of fleet assignment (FA), flight scheduling (FS), and aircraft routing (AR) is essential for sustaining profitability and operational reliability. The continued growth in air transportation demand in recent years has intensified operational complexity and capacity pressures faced by airlines. This environment further amplifies the importance of integrated fleet assignment, scheduling, and routing decisions under uncertainty [1]. These decisions are highly interdependent: scheduling affects routing feasibility, routing determines aircraft availability, and assignment influences capacity utilization and service quality. When combined with uncertainties such as stochastic passenger demand, stochastic non-cruise times (NCT), and delay propagation, the FSFAAR problem becomes a large-scale stochastic optimization challenge that requires advanced modeling and solution techniques.

Although many studies have addressed FA, FS, and AR—either individually or in partial combinations—most rely on deterministic assumptions or simplified stochastic structures. More importantly, several critical operational realities remain insufficiently represented. Station purity (SP) restricts aircraft types at specific stations, codeshare agreements (CA) introduce capacity-sharing and contractual cost structures, and overbooking (OB) directly affects revenue and denied-boarding risks. Existing studies typically examine these components in isolation or within narrow problem settings, and no prior work integrates SP, CA, and OB within a unified FSFAAR framework. Furthermore, realistic features such as adjustable cruise speeds, stochastic NCT distributions, and flexible departure windows are often excluded, limiting the practical relevance of existing models.

To bridge these gaps, this study proposes the first three-stage stochastic mixed-integer nonlinear programming (TSSMINLP) model that jointly incorporates SP, CA, and OB into the FSFAAR decision structure. The formulation also embeds realistic operational elements including stochastic demand, delay formation, variable cruise speeds, and departure time flexibility. This integrated approach enables coordinated fleet, scheduling, and routing decisions while capturing key operational constraints and uncertainties faced by airlines.

From a computational perspective, the scale and nonlinearity of the proposed model render classical exact solvers impractical for real-world instances. To address this challenge, we develop a scalable solution strategy that combines the Sample Average Approximation (SAA) method with two customized metaheuristic algorithms—Simulated Annealing (SAASA) and Cuckoo Search (SAACS). This hybrid approach efficiently handles large scenario sets and achieves tight optimality gaps in high-dimensional stochastic environments.

By integrating previously isolated operational considerations and providing a robust solution mechanism, this study contributes a comprehensive decision-support framework for airline operational planning under uncertainty. The remainder of the paper is organized as follows: Section 2 reviews related literature; Section 3 presents the OB model; Section 4 introduces the proposed formulation; Section 5 describes the solution methodology; Section 6 provides a numerical example; Section 7 reports computational results; and Section 8 concludes with managerial insights and opportunities for future research.

2. Literature Review

Research in air transport management has extensively examined fleet assignment (FA), flight scheduling (FS), and aircraft routing (AR), either independently or through partial integrations. Early FA studies primarily focused on fleet utilization and sizing decisions [2,3,4]. Subsequent research demonstrated that FA, FS, and AR decisions are inherently interdependent, motivating the development of integrated modeling approaches.

FA–AR integrated formulations commonly incorporate maintenance requirements, while extending the core problem structure to capture additional operational dimensions such as aircraft utilization [5], deadhead flights [6], flexible fleet management through chartering or leasing decisions [7], delay propagation effects [8], and crew pairing constraints [9]. FS–FA integrated formulations coordinate scheduling and fleet assignment decisions [10,11] and have been extended to include optional flight legs and fare-class differentiation [12], passenger misconnection reduction through robust planning [13], departure time retiming strategies [14], delay-cost-aware optimization under slot-constrained airport environments [15], and two-stage stochastic structures that explicitly account for demand and fare uncertainty [16]. Related FS–AR formulations with multi-type aircraft have further explored flexible departure-time decisions, demonstrating that schedule retiming can reduce aircraft idle time while increasing the number of transported passengers [17].

Integrated FSFAAR formulations have been extended to incorporate passenger-oriented and operational features, including demand recapture, multiple fare classes, and passenger connection decisions under deterministic demand assumptions [18]. Subsequent studies enhanced FSFAAR by introducing cruise time control, demonstrating that flexible cruise-speed decisions can improve aircraft utilization and reduce fleet size while satisfying maintenance requirements [19], and later extended this framework to account for stochastic demand and non-cruise time uncertainty [20]. Further contributions explicitly modeled delay propagation and flight re-timing decisions, with extensions capturing passenger spill, recapture effects, and missed connections [21]. The integration of crew pairing into FSFAAR frameworks has also been shown to improve schedule robustness and reduce operational costs under delay scenarios by limiting aircraft changes [14]. Despite these advances, existing FSFAAR formulations have not jointly incorporated station purity, codeshare agreements, and overbooking decisions within a unified stochastic framework.

Uncertainty has been increasingly incorporated into airline planning models through stochastic formulations. Passenger demand variability has been extensively addressed in fleet assignment studies to reduce capacity mismatches and improve expected profitability and robustness [22,23,24,25], providing important methodological foundations for incorporating demand uncertainty into more comprehensive integrated airline planning models. Operational time uncertainty has also been modeled by treating total travel time [26] and non-cruise times (NCT) as stochastic variables, demonstrating that neglecting such variability can significantly deteriorate solution quality and schedule reliability [20,27].

Station purity (SP), which restricts aircraft–airport compatibility based on infrastructure and operational constraints, has received comparatively limited attention in integrated airline planning models. While SP simplifies operational control, it significantly constrains FA and AR decisions [28]. Existing SP-related models integrate it with crew pairing [29] or delay prediction [30], but do not examine its interaction with stochastic FSFAAR decisions.

Codeshare agreements (CA) represent another strategic tool. Early research emphasized network design and alliance structures [31,32,33], and recent studies explored CA in operational planning, showing benefits in delay mitigation and uncertainty management [34,35]. New evidence on alliance evolution and codeshare structures highlights increasing network complexity [36]. Nonetheless, CA has not been jointly modeled with SP, OB, and stochastic FSFAAR decisions.

Overbooking (OB) refers to the practice of selling tickets beyond aircraft seat capacity under the assumption that a fraction of passengers will not show up for the flight [37]. The literature identifies two main OB strategies: capacity-based overbooking applied close to departure and multi-capacity approaches defined at the fleet-family level [38]. These strategies highlight the strong dependence of OB decisions on aircraft capacity, naturally linking OB to the fleet assignment (FA) problem. Several studies have therefore integrated OB or revenue management decisions with FA and other airline planning stages [39,40,41]. More recent contributions explicitly account for capacity uncertainty, no-show behavior, denied boarding costs, and late cancelations through stochastic or scenario-based formulations [42,43]. Despite these advances, OB is still predominantly modeled within revenue management frameworks and remains largely decoupled from integrated FSFAAR formulations.

From a methodological perspective, metaheuristics are increasingly used for large-scale airline planning. Variable-neighborhood search [21], resilient scheduling heuristics [27], matheuristics for FA–AR–CP [9], and hybrid approaches [44] demonstrate their practical value. Yet, no heuristic method in the literature solves an integrated FSFAAR model that simultaneously accounts for SP, CA, OB, stochastic NCT, stochastic demand, flexible time windows, and adjustable cruise speeds.

In summary, although significant progress has been made in integrating FS, FA, and AR, modeling uncertainty, and examining SP, CA, and OB individually, no existing study combines all three operational features within a unified three-stage stochastic FSFAAR framework. This gap motivates the present work, which develops the first integrated model capturing these interactions and proposes scalable solution methods suited for large-scale stochastic airline planning.

2.1. Contribution of Study

This study provides two major contributions to the airline operations literature: (i) modeling innovations and (ii) methodological advancements.

2.1.1. Modeling Contributions

This work makes a substantial modeling contribution by integrating three critical and interdependent operational strategies—station purity (SP), codeshare agreements (CA), and overbooking (OB)—into a unified fleet assignment, flight scheduling, and aircraft routing (FSFAAR) framework under uncertainty. Although each of these components has been studied individually, their simultaneous incorporation within a single optimization model has not been addressed in prior literature.

SP constraints restrict aircraft assignment choices due to infrastructural and operational limitations at specific airports. CA, on the other hand, function as a strategic mitigation mechanism, enabling airlines to maintain network feasibility and operational flexibility when suitable fleet types cannot be deployed. OB policies are tightly connected to FA decisions, as aircraft capacity determines the revenue potential and service risks associated with no-shows and denied boarding.

This study is the first to formulate a TSSMINLP model that jointly integrates SP, CA, and OB within an FSFAAR framework, capturing the complex interactions among operational planning decisions, resource configurations, and demand uncertainty.

Additional modeling enhancements include:

• Departure-time flexibility through time-window scheduling and rescheduling decisions, improving resilience against delay propagation.
• Stochastic passenger demand modeling, strengthening realism in FA decisions.
• Stochastic non-cruise times (NCT), delay propagation, and adjustable cruise speed within the AR component, reflecting real-world operational variability.

By combining these elements, the proposed framework provides a holistic, realistic, and practically applicable decision-support tool for airlines operating under uncertainty.

2.1.2. Methodological Contributions

Given the stochastic nature of passenger demand and non-cruise times (NCT), together with the operational nonlinearities arising from adjustable cruise speeds and deterministic delay propagation, this study introduces a Sample Average Approximation (SAA)-based solution approach. Unlike traditional deterministic-equivalent or limited-scenario decomposition techniques commonly used in the FSFAAR literature, SAA allows the evaluation of a substantially larger scenario set, improving robustness and statistical reliability.

However, the resulting TSSMINLP model is computationally challenging. Exact solvers can address only very small instances, becoming impractical for real-world-scale problems. To overcome this limitation, the study designs two tailored metaheuristic algorithms:

• SAASA—an SAA-enhanced Simulated Annealing approach
• SAACS—an SAA-enhanced Cuckoo Search approach

These algorithms are specifically parameterized for the structural characteristics of the FSFAAR problem. They leverage SAA’s scenario-based estimation with the global-search capabilities of metaheuristics, enabling the efficient solution of large-scale, highly nonlinear, and stochastic instances.

Computational experiments demonstrate that:

• Both metaheuristics achieve near-optimal solutions with significantly lower computation times than exact solvers.
• SAASA outperforms SAACS in solution quality, while both provide consistent performance across large scenario sets.
• The hybrid SAA–metaheuristic framework offers a scalable and operationally feasible alternative for industry applications.

Thus, the study introduces a novel, robust, and scalable methodological framework for large-scale stochastic airline optimization problems, contributing both theoretically and practically to the field.

3. Overbooking

Overbooking (OB) is incorporated into the proposed integrated framework to evaluate whether forecasting-based reservation limits can improve subsequent operational decisions—specifically fleet assignment (FA) and aircraft routing (AR). In the three-stage stochastic structure, OB limits are determined before FA and AR decisions, and these limits directly influence scenario-based passenger loads, denied boarding (DB) levels, and ultimately the feasibility and cost of fleet–route combinations.

OB limits are computed according to stochastic demand and aircraft capacities generated under each scenario. Consistent with classical static models in the literature, the number of passengers who show up is modeled using a binomial distribution with a constant show-up probability. This approach is operationally realistic because passenger-level no-show information is unavailable, and airlines typically rely on aggregated historical averages for short-term operational forecasting. Furthermore, using a binomial distribution ensures analytical tractability within the integrated OB–FA–AR framework [45].

The parameters and variables used in the OB formulation are:

• y: Reservation limit
• Kap: Aircraft capacity
• fare: Fare per passenger
• cost: Denied-boarding penalty cost per passenger
• cost–fare: Refund paid to a DB passenger (excluding fare)
• D: Total booking demand
• s (y): Realized number of show-up passengers when the reservation limit is y
• α: Passenger show-up probability

3.1. Economic Motivation

Increasing the reservation limit above the aircraft capacity reduces expected idle seats but also increases the risk of denied boarding. Therefore, when moving from y to y + 1, the model compares:

• Marginal benefit: Expected revenue gained if an additional reservation results in a show-up without exceeding capacity.
• Marginal cost: Expected penalty incurred if the additional reservation triggers denied boarding.

These effects are captured in Equations (1) and (2):

• Marginal benefit = (Probability of demand being greater than the reservation limit)
• X (probability that the show-up passenger number is smaller than the capacity)
• X (probability of a passenger coming to the flight)
• X (fare per passenger)

  $$Marginal benefit=P\left(D>y\right) P\left(s\left(y\right)<Kap|D>y\right) \alpha fare$$

  (1)
• Marginal cost = (Possibility of demand being more than the reservation limit)
• X (probability that the show-up passenger number is greater than the capacity)
• X (probability of a passenger coming to the flight)
• X (amount refunded to a passenger)

  $$Marginal cost=P\left(D>y\right) P\left(s\left(y\right)\ge Kap|D>y\right) \alpha \left(cost-fare\right)$$

  (2)

The reservation limit is increased from y to y + 1 if and only if:

$$P\left(D > y\right) P\left(s\left(y\right) < Kap|D>y\right) \alpha f \ge P\left(D > y\right) P\left(s\left(y\right) \ge Kap|D > y\right) \alpha \left(cost - fare\right)$$

(3)

Simplifying (3) yields the classical OB decision rule:

$$P\left(s\left(y\right) \ge Kap|D>y\right) \le {\displaystyle \frac{fare}{cost}}$$

(4)

To provide intuition for the marginal benefit–marginal cost rule, consider an aircraft with capacity Kap = 180 and a show-up probability α = 0.9. Suppose the airline evaluates increasing the reservation limit from y = 185 to y = 186. The probability that show-up passengers exceed capacity under y = 185 is computed using the binomial distribution in (5). If this probability is smaller than fare/cost, then condition (4) holds, meaning that the expected revenue gain from allowing one additional booking outweighs the expected denied-boarding penalty. This simple numerical check illustrates how the OB decision rule operates in practice and provides a more intuitive understanding of the economic motivation behind the formulation.

3.2. Probabilistic Characterization of Show-Ups

To evaluate (4), the probability that show-up passengers exceed aircraft capacity must be computed. If each of the y reservation holders has a show-up probability of α, then the number of show-up passengers follows a binomial distribution [45,46]. Thus,

$$P\left(s\left(y\right) \ge Kap|D>y\right)=\sum _{s = K}^y \left({\displaystyle \genfrac{}{}{0pt}{}{y}{s}}\right){\alpha }^s {(1-\alpha )}^{(y-s)}$$

(5)

Using Equations (4) and (5), the optimal reservation limit is calculated separately for each fleet type.

Passenger-driven voluntary cancelations are not explicitly modeled [38]. A constant show-up probability is used due to limited granularity in the available data and to maintain computational tractability in the three-stage stochastic formulation.

3.3. Integration with Scenario-Based FSFAAR Decisions

The reservation limits determined using Equations (4) and (5) are not isolated predictions; instead, they directly shape the scenario-based operational environment of the FSFAAR framework.

For each scenario:

1.

The realized number of show-up passengers is generated using the OB limit y.

2.

This impacts the effective passenger load, which determines:

○

unused capacity,

○

denied-boarding levels,

○

flight-specific feasibility of each fleet type,

○

compatibility of aircraft routing sequences.

3.

Penalty costs are applied for all denied-boarding passengers (voluntary or involuntary).

4.

FA and AR decisions adapt to these scenario-based loads, ensuring that the integrated solution balances revenue gains from OB with operational risks and costs.

Thus, OB influences both network-level planning (fleet assignment) and daily operational feasibility (aircraft routing). By embedding OB into the stochastic structure, the model captures the joint effect of ticket sales, show-up variability, and capacity constraints within a unified optimization framework.

4. Problem Description and Formulation

4.1. Problem Description

This study addresses the integrated planning problem of an airline operating under uncertainty, where tactical and operational decisions interact across multiple planning horizons. The airline must simultaneously determine its fleet assignment (FA), aircraft routing (AR), overbooking limits (OB), and codeshare agreements (CA), while also satisfying station purity (SP) restrictions and ensuring feasible day-of-operations performance. These decisions are interdependent: OB levels influence passenger loads and feasible fleet–route selections; CA provides supplementary capacity when required fleet types are restricted due to SP or when demand exceeds available capacity; AR decisions shape the propagation of delays and connection feasibility; and FA determines the aircraft capacities that impact OB, CA, passenger acceptance, and denied boarding.

The planning horizon includes a set of scheduled flights, each characterized by an origin–destination pair, scheduled departure and arrival times, permissible departure time windows, and eligible fleet types. The airline operates a heterogeneous fleet distributed across multiple stations, each subject to SP restrictions limiting which fleet types may serve specific airports. Additionally, the airline may choose from alternative CA contracts involving partner carriers that provide capacity with predefined fares and costs.

Passenger demand is stochastic, and realized passenger show-ups differ from reservation levels, contributing to overbooking risk and potential denied boarding penalties. Flight delays propagate through turn-around processes, passenger connections, and the non-cruise time (NCT) components—including taxi-out, take-off, descent, landing, and taxi-in—affecting both operational feasibility and the number of missed-connection passengers. Furthermore, the incorporation of stochastic NCT allows delay propagation effects to be reflected accurately in the model. By minimizing idle time, the formulation inherently discourages delay formation and improves schedule robustness. Therefore, a stochastic programming approach is required.

The decision process unfolds in three sequential stages:

• Stage 1 (Strategic–Tactical): Determination of fleet assignment (FA), aircraft routing (AR), overbooking limits (OB), and codeshare agreements (CA), governed by Constraints (7)–(15). These decisions are scenario-independent and define the feasible structural backbone of the flight schedule. Although fixed across scenarios, they are optimized with respect to expected system performance under uncertainty.
• Stage 2 (Operational Planning): Determination of announced departure times (ANDT) within allowable time windows, as defined by Constraint (16), with the objective of mitigating delay propagation and enhancing schedule robustness.
• Stage 3 (Day-of-Operations): Scenario-dependent operational realizations, including actual departure and arrival times, delays, passenger acceptance, missed connections, and denied boarding decisions, governed by Constraints (17)–(31).

The objective (Equation (6)) is to maximize the airline’s expected profit, including ticket revenues and codeshare revenues, while accounting for operating costs, contracted CA costs, delay and missed-connection costs, denied boarding penalties, and spilled passenger losses. The resulting model integrates strategic, tactical, and operational considerations, offering a unified framework that captures complex interactions among FA, AR, CA, OB, and SP decisions.

For clarity, the mathematical formulation presented in the following section is organized according to this three-stage structure. Constraints are grouped based on their operational roles and corresponding planning stages to facilitate readability and interpretation.

4.2. Model Assumptions

The model relies on the following assumptions:

• Stochastic Demand and NCT: Passenger demand and non-cruise times (NCT) are stochastic across scenarios. Demand is generated from a uniform distribution whose minimum and maximum bounds are determined according to aircraft capacities, while NCT captures operational variability in taxi-out, take-off, descent, landing, and taxi-in durations.
• Flight Schedule Structure: Scheduled departure and arrival times are fixed. However, the actual departure time (ACDT) is selected from a predefined time window, and any delay beyond the chosen ACDT incurs an operational delay cost.
• Arrival Time Calculation: The actual arrival time (AAT) is obtained by adding the cruise time and realized NCT to the ACDT.
• Routing and Maintenance: Each aircraft route begins and ends at fleet-specific stations, and maintenance can only be performed at these initial or terminal locations. Codeshare flights are excluded from aircraft routing, and CA usage is constrained by a predefined budget.
• Operational Constraints: Cruise speed is allowed to vary within specified limits. Minimum passenger connection times must be satisfied.
• Station Purity (SP): Each station may only be served by a predefined subset of fleet types, and violations of SP restrictions are not allowed.
• Overbooking and Passenger Flow: Passenger show-up probability is assumed to be fixed. A portion of the overbooking limit materializes as actual show-up passengers, while insufficient capacity may cause spilled passengers or denied boarding. Denied boarding incurs a penalty equal to a multiple of the ticket fare.
• Idle Time (IT) and Delay Time (DT): Idle Time (IT) refers to periods during which the aircraft remains on the ground without performing productive operations. These periods do not represent actual delays but generate idle-cost components due to non-utilized aircraft time. Delay Time (DT) captures the propagation of delays through the routing structure, where a delay occurring in one flight affects the timing of subsequent flights operated by the same aircraft.

These assumptions ensure that the model reflects realistic airline operations while accurately capturing uncertainties arising from passenger demand and non-cruise times (NCT).

4.3. Mathematical Model

This section introduces the mathematical formulation of the proposed TSSMINLP model. The sets, parameters, and decision variables used in the formulation are defined below.

Sets
S	:	Scenario’s set $\mathrm{s}\in \mathrm{S}$
K	:	Fleet type’s set $\mathrm{k}\in \mathrm{K}$
F	:	Flight’s set $\mathrm{i}\in \mathrm{F}$
St	:	Stations set $\mathrm{t}\in \mathrm{S}\mathrm{t}$
FC	:	Fare classes $\mathrm{h}\in \mathrm{F}\mathrm{C}$
${\mathrm{Y}\mathrm{B}}_{\mathrm{i}}$	:	Set of flights receiving connecting passengers from flight i $\mathrm{i}\in \mathrm{F}$
BU	:	Consecutive flights $\mathrm{i},\mathrm{j}\in \mathrm{F}$
${\mathrm{O}}_{\mathrm{i}}$	:	Flights before flight $\mathrm{i} \in \mathrm{F}$
${\mathrm{H}}_{\mathrm{i}}$	:	Flights after flight $\mathrm{i} \in \mathrm{F}$
${\mathrm{S}\mathrm{U}}_{\mathrm{k}}$	:	Final flights $\mathrm{k} \in \mathrm{K}$
${\mathrm{I}\mathrm{U}}_{\mathrm{k}}$	:	İnitial flights $\mathrm{k} \in \mathrm{K}$
$\mathsf{{\rm Y}}$	:	Diverse codeshared airlines $\mathsf{\upsilon } \in \mathsf{{\rm Y}}$
P	:	Codeshare agreement types $\mathrm{p} \in \mathrm{P}$

Parameters
$\left[{\mathrm{T}}_{\mathrm{i}}^{\mathrm{m}\mathrm{i}\mathrm{n}},{\mathrm{T}}_{\mathrm{i}}^{\mathrm{m}\mathrm{a}\mathrm{x}}\right]$	:	Departure time window for flight i $\mathrm{i} \in \mathrm{F}$
${\mathrm{p}\mathrm{r}}_{\mathrm{s}}$	:	Scenarios’probability $\mathrm{s} \in \mathrm{S}$
${\mathrm{A}\mathrm{N}}_{\mathrm{k}}$	:	Aircraft number of fleet type k, $\mathrm{k} \in \mathrm{K}$
${\mathrm{K}\mathrm{a}\mathrm{p}}_{\mathrm{k}\mathrm{h}}$	:	Fleet type capacity for fare class h $\mathrm{k} \in \mathrm{K}, \mathrm{h} \in \mathrm{F}\mathrm{C}$
${\mathrm{O}\mathrm{B}}_{\mathrm{k}\mathrm{h}}$	:	Overbooked capacity of fleet type k for fare class h $\mathrm{k}\in \mathrm{K}, \mathrm{h}\in \mathrm{F}\mathrm{C}$
${\mathrm{D}\mathrm{Y}}_{\mathrm{i}\mathrm{h}}$	:	Spilled passenger cost for fare class h of flight i $\mathrm{i} \in \mathrm{F}, \mathrm{h} \in \mathrm{F}\mathrm{C}$
${\mathrm{B}\mathrm{O}\mathrm{S}}_{\mathrm{k}}$	:	Idle time (IT) cost for fleet type k $\mathrm{k} \in \mathrm{K}$
${\mathrm{K}\mathrm{Y}}_{\mathrm{i}}$	:	Disconnected passengers’ cost $\mathrm{i}\in \mathrm{F}$
${\mathrm{D}\mathrm{C}}_{\mathrm{i}}$	:	Delay time cost per minute for flight i $\mathrm{i} \in \mathrm{F}$
${\mathrm{f}\mathrm{a}\mathrm{r}\mathrm{e}}_{\mathrm{i}\mathrm{h}}$	:	Flight revenue per passenger for fare class h $\mathrm{i} \in \mathrm{F}, \mathrm{h} \in \mathrm{F}\mathrm{C}$
${\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{t}}_{\mathrm{i}\mathrm{h}}$	:	Flight cost per passenger for fare class h $\mathrm{i} \in \mathrm{F}$, $\mathrm{h} \in \mathrm{F}\mathrm{C}$
${\mathrm{D}}_{\mathrm{i}\mathrm{h}\mathrm{s}}$	:	Stochastic demand of flight i for fare class h in scenario s $\mathrm{s} \in \mathrm{S}, \mathrm{i} \in \mathrm{F}, \mathrm{h} \in \mathrm{F}\mathrm{C}$
${\mathsf{\eta }}_{\mathrm{i}\mathrm{s}}$	:	Stochastic NCT of flight i in scenario s $\mathrm{s} \in \mathrm{S}, \mathrm{i} \in \mathrm{F}$
$\left[{\mathrm{C}\mathrm{T}}_{\mathrm{i}\mathrm{k}}^{\mathrm{l}},{\mathrm{C}\mathrm{T}}_{\mathrm{i}\mathrm{k}}^{\mathrm{u}}\right]$	:	Min/max cruise time allowed for flight i with fleet k $\mathrm{i}\in \mathrm{F},\mathrm{k}\in \mathrm{K}$
${\mathrm{T}\mathrm{A}}_{\mathrm{i}\mathrm{k}}$	:	Turnaround time required after flight i for fleet type k $\mathrm{i}\in \mathrm{F}, \mathrm{k}\in \mathrm{K}$
${\mathrm{C}\mathrm{P}}_{\mathrm{i}\mathrm{j}}$	:	Passenger connection time from flight i to flight j $\mathrm{i} \in \mathrm{J}, \mathrm{j} \in {\mathrm{Y}\mathrm{B}}_{\mathrm{i}}$
${\mathrm{I}}_{\mathrm{t}\mathrm{i}}$	:	$\left\{\begin{array}{c}1\mathrm{I}\mathrm{f} \mathrm{f}\mathrm{l}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t} \mathrm{i} \mathrm{s}\mathrm{e}\mathrm{r}\mathrm{v}\mathrm{e}\mathrm{s} \mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n} \mathrm{t}\\ 0\mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{e}\end{array}\mathrm{i}\in \mathrm{F}, \mathrm{t}\in \mathrm{S}\mathrm{t}\right.$
${\mathrm{w}\mathrm{k}\mathrm{s}}_{\mathrm{k}\mathrm{t}}$	:	$\left\{\begin{array}{c}1\mathrm{I}\mathrm{f} \mathrm{f}\mathrm{l}\mathrm{e}\mathrm{e}\mathrm{t} \mathrm{t}\mathrm{y}\mathrm{p}\mathrm{e} \mathrm{k} \mathrm{c}\mathrm{a}\mathrm{n} \mathrm{s}\mathrm{e}\mathrm{r}\mathrm{v}\mathrm{e} \mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n} \mathrm{t}\\ 0\mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{e}\end{array}\mathrm{k}\in \mathrm{K},\mathrm{t}\in \mathrm{S}\mathrm{t}\right.$
${\mathrm{p}\mathrm{a}\mathrm{s}\mathrm{s}}_{\mathrm{i}\mathrm{j}}$	:	Connecting passengers number from flight i to j $\mathrm{i} \in \mathrm{J}, \mathrm{j} \in {\mathrm{Y}\mathrm{B}}_{\mathrm{i}}$
${\mathrm{c}\mathrm{o}\mathrm{d}\mathrm{e}}_{\mathsf{\upsilon }\mathrm{p}\mathrm{i}\mathrm{h}}$	:	Per−passenger cost charged under agreement p with airline υ     p ∈$\mathrm{P}$$, \mathsf{\upsilon }\in \mathsf{{\rm Y}}$, i ∈$\mathrm{F}, \mathrm{h}\in \mathrm{F}\mathrm{C}$
${\mathrm{C}\mathrm{a}\mathrm{p}}_{\mathsf{\upsilon }\mathrm{p}\mathrm{i}\mathrm{h}}$	:	Capacity allocated under agreement p for flight i     p ∈ $\mathrm{P},$$\mathsf{\upsilon }\in \mathsf{{\rm Y}}$, i ∈$\mathrm{F}, \mathrm{h}\in \mathrm{F}\mathrm{C}$
${\mathrm{O}\mathrm{B}\mathrm{C}}_{\mathsf{\upsilon }\mathrm{p}\mathrm{i}\mathrm{h}}$	:	Overbooked capacity allowed under agreement p      p ∈ $\mathrm{P},$ $\mathsf{\upsilon }\in \mathsf{{\rm Y}}$, i ∈ F, $\mathrm{h}\in \mathrm{F}\mathrm{C}$
BUD	:	Total budget allocated for codeshare agreements
θ	:	Maximum proportion of total airline capacity that may be used for codesharing
r	:	Penalty multiplier applied to ticket fare for denied boarding (DB) passengers
M	:	A sufficiently large constant (Big-M)

Decision Variables First Stage Variables
${\mathrm{x}}_{\mathrm{i}\mathrm{j}\mathrm{k}}^1$	:	$\left\{\begin{array}{c}1\mathrm{I}\mathrm{f} \mathrm{f}\mathrm{l}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t} \mathrm{j} \mathrm{f}\mathrm{o}\mathrm{l}\mathrm{l}\mathrm{o}\mathrm{w}\mathrm{s} \mathrm{f}\mathrm{l}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t} \mathrm{i} \mathrm{u}\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{g} \mathrm{f}\mathrm{l}\mathrm{e}\mathrm{e}\mathrm{t} \mathrm{t}\mathrm{y}\mathrm{p}\mathrm{e} \mathrm{k}\\ \mathrm{j}\in \mathrm{F} ,\mathrm{i} \in {\mathrm{O}}_{\mathrm{j}},\mathrm{k}\in \mathrm{K}\\ 0\mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{e}\end{array}\right.$
${\mathrm{f}}_{\mathrm{i}\mathrm{k}}^1$	:	$\left\{\begin{array}{c}1\mathrm{I}\mathrm{f} \mathrm{f}\mathrm{l}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t} \mathrm{i} \mathrm{i}\mathrm{s} \mathrm{t}\mathrm{h}\mathrm{e} \mathrm{f}\mathrm{i}\mathrm{r}\mathrm{s}\mathrm{t} \mathrm{f}\mathrm{l}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t} \mathrm{o}\mathrm{n} \mathrm{i}\mathrm{t}\mathrm{s} \mathrm{r}\mathrm{o}\mathrm{u}\mathrm{t}\mathrm{e} \mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h} \mathrm{f}\mathrm{l}\mathrm{e}\mathrm{e}\mathrm{t} \mathrm{k}\\ \mathrm{i}\in \mathrm{F},\mathrm{k}\in \mathrm{K}\\ 0\mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{e}\end{array}\right.$
${\mathrm{e}}_{\mathrm{i}\mathrm{k}}^1$	:	$\left\{\begin{array}{c}1\mathrm{I}\mathrm{f} \mathrm{f}\mathrm{l}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t} \mathrm{i} \mathrm{i}\mathrm{s} \mathrm{t}\mathrm{h}\mathrm{e} \mathrm{l}\mathrm{a}\mathrm{s}\mathrm{t} \mathrm{f}\mathrm{l}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t} \mathrm{o}\mathrm{n} \mathrm{i}\mathrm{t}\mathrm{s} \mathrm{r}\mathrm{o}\mathrm{u}\mathrm{t}\mathrm{e} \mathrm{w}\mathrm{i}\mathrm{t}\mathrm{h} \mathrm{f}\mathrm{l}\mathrm{e}\mathrm{e}\mathrm{t} \mathrm{k}\\ \mathrm{i}\in \mathrm{F},\mathrm{k}\in \mathrm{K}\\ 0\mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{e}\end{array}\right.$
${\mathrm{q}}_{\mathsf{\upsilon }\mathrm{p}\mathrm{i}}^1$	:	$\left\{\begin{array}{c}\mathrm{1}\mathrm{I}\mathrm{f} \mathrm{t}\mathrm{h}\mathrm{e} \mathrm{f}\mathrm{l}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t}\mathrm{i} \mathrm{i}\mathrm{s} \mathrm{o}\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{e}\mathrm{d} \mathrm{v}\mathrm{i}\mathrm{a} \mathrm{c}\mathrm{o}\mathrm{d}\mathrm{e}\mathrm{s}\mathrm{h}\mathrm{a}\mathrm{r}\mathrm{e} \mathrm{a}\mathrm{g}\mathrm{r}\mathrm{e}\mathrm{e}\mathrm{m}\mathrm{e}\mathrm{n}\mathrm{t} \mathit{p} \\ \mathsf{\upsilon }\in \mathsf{{\rm Y}}, \mathrm{i} \in \mathrm{F}, \mathit{p} \in \mathrm{P}\\ \hfill \mathrm{0}\mathit{o}\mathit{t}\mathit{h}\mathit{e}\mathit{r}\mathit{w}\mathit{i}\mathit{s}\mathit{e}\end{array}\right.$
Second Stage variables
${\mathrm{a}}_{\mathrm{i}}^2$	:	Announced departure time (ANDT)of flight i$\mathrm{i}\in \mathrm{F}$
Third stage variables
${\mathrm{b}}_{\mathrm{i}\mathrm{s}}^3$	:	Actual departure time (ACDT) of flight i in scenario s $\mathrm{i} \in \mathrm{F}, \mathrm{s} \in \mathrm{S}$
${\mathrm{c}}_{\mathrm{i}\mathrm{s}}^3$	:	Actualarrivaltime(AAT)offlighti in scenario s $\mathrm{i}\in \mathrm{F}, \mathrm{s}\in \mathrm{S}$
${\mathrm{d}}_{\mathrm{i}\mathrm{k}\mathrm{s}}^3$	:	Cruise time (CT) for flight i using fleet k under scenario s $\mathrm{i}\in \mathrm{F}, \mathrm{s}\in \mathrm{S}, \mathrm{k}\in \mathrm{K}$
${\mathrm{I}\mathrm{T}}_{\mathrm{i}\mathrm{k}\mathrm{s}}^3$	:	Idle time (IT) for fleet type k on flight i under scenario s $\mathrm{i}\in \mathrm{F}, \mathrm{s}\in \mathrm{S}, \mathrm{k}\in \mathrm{K}$
${\mathrm{d}\mathrm{e}\mathrm{l}}_{\mathrm{i}\mathrm{s}}^3$	:	Delay time (DT) of flight i in scenario s $\mathrm{i}\in \mathrm{F}, \mathrm{s}\in \mathrm{S}$
${\mathsf{\pi }}_{\mathrm{i}\mathrm{h}\mathrm{s}}^3$	:	Number of accepted passengers for flight i, fare class h, scenario s, $\mathrm{i}\in \mathrm{F}, \mathrm{h}\in \mathrm{F}\mathrm{C}, \mathrm{s}\in \mathrm{S}$
${\mathsf{\pi }\mathrm{c}}_{\mathrm{i}\mathrm{h}\mathrm{s}}^3$	:	Accepted passengers via codeshare on flight i, fare class h, scenario s, $\mathrm{i}\in \mathrm{F}, \mathrm{h}\in \mathrm{F}\mathrm{C}, \mathrm{s}\in \mathrm{S}$
${\mathrm{P}\mathrm{N}\mathrm{R}}_{\mathrm{i}\mathrm{h}\mathrm{s}}^3$	:	Number of tickets sold for flight i and fare class h in scenario s $\mathrm{i}\in \mathrm{F}, \mathrm{h}\in \mathrm{F}\mathrm{C}, \mathrm{s}\in \mathrm{S}$
${\mathrm{d}\mathrm{b}}_{\mathrm{i}\mathrm{h}\mathrm{s}}^3$	:	Number of denied boarding (DB) passengers on flight i for fare class h in scenario s, $\mathrm{i}\in \mathrm{F}, \mathrm{h}\in \mathrm{F}\mathrm{C}, \mathrm{s}\in \mathrm{S}$
${\mathrm{s}\mathrm{h}\mathrm{o}\mathrm{w}\mathrm{p}\mathrm{a}\mathrm{s}\mathrm{s}}_{\mathrm{i}\mathrm{h}\mathrm{s}}^3$	:	Number of show−up passengers on flight i for class h in scenario s, $\mathrm{i}\in \mathrm{F}, \mathrm{h}\in \mathrm{F}\mathrm{C}, \mathrm{s}\in \mathrm{S}$
${\mathrm{w}}_{\mathrm{i}\mathrm{j}\mathrm{s}}^3$	:	$\left\{\begin{array}{c}1\mathrm{I}\mathrm{f} \mathrm{t}\mathrm{h}\mathrm{e} \mathrm{p}\mathrm{a}\mathrm{s}\mathrm{s}\mathrm{e}\mathrm{n}\mathrm{g}\mathrm{e}\mathrm{r}\mathrm{s} \mathrm{o}\mathrm{f} \mathrm{f}\mathrm{l}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t} \mathrm{i} \mathrm{m}\mathrm{i}\mathrm{s}\mathrm{s} \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{n}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n} \mathrm{t}\mathrm{o} \mathrm{f}\mathrm{l}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t} \mathrm{j} \mathrm{i}\mathrm{n} \mathrm{s}\mathrm{c}\mathrm{e}\mathrm{n}\mathrm{a}\mathrm{r}\mathrm{i}\mathrm{o} s\\ \mathrm{i} \in \mathrm{J},\mathrm{j} \in {YB}_{\mathrm{i}},s \in S, \\ 0\mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{e}\end{array}\right.$
		Decision variables are indexed by superscripts (1, 2, 3) to explicitly indicate the planning stage at which they are determined. Accordingly, Stage 1 variables are scenario-independent, while Stage 3 variables are scenario-dependent and capture day-of-operations realizations. This notation ensures a clear separation of decisions across stages without requiring redundant constraint restructuring.

$$Maximize Z={\displaystyle \sum _{s\in S}}{pr}_s\left[\begin{array}{c}\underset{Ticket Revenue}{\underbrace{\left({\displaystyle \sum _{i\in F}}{\displaystyle \sum _{h\in FC }}\left({PNR}_{ihs}^3\right) \times {fare}_{ih}\right)}}-\underset{Denied Boarding Penalty}{\underbrace{\left({\displaystyle \sum _{i\in F}}{\displaystyle \sum _{h\in FC}}{db}_{ihs}^3 \times r \times {fare}_{ih}\right)}}\\ -\underset{Passenger Service Cost}{\underbrace{\left({\displaystyle \sum _{i\in F}}{\displaystyle \sum _{h\in FC}}{\pi }_{ihs}^3 \times {cost}_{ih}\right)}}-\underset{Codshare Agreement Cost}{\underbrace{\left({\displaystyle \sum _{iϵF}}{\displaystyle \sum _{\mathsf{\upsilon }\in \mathsf{{\rm Y}}}}{\displaystyle \sum _{pϵP}}{\displaystyle \sum _{h\in FC}}{code}_{\mathsf{\upsilon }pih} \times {Cap}_{\mathsf{\upsilon }pih }\times { q}_{\mathsf{\upsilon }pi}^1\right)}}\\ -\underset{Unserved Codeshare Passenger Penalty}{\underbrace{\left({\displaystyle \sum _{iϵF}}{\displaystyle \sum _{\mathsf{\upsilon }\in \mathsf{{\rm Y}}}}{\displaystyle \sum _{pϵP}}{\displaystyle \sum _{h\in FC}} \mathrm{m}\mathrm{a}\mathrm{x} (0, D_{ihs}-{PNR}_{ihs}^3)\times { q}_{\mathsf{\upsilon }pi}^1\times \left({fare}_{ih }-{code}_{\mathsf{\upsilon }pih}\right)\right)}}\\ -\underset{Spilled Passenger Cost}{\underbrace{\left({\displaystyle \sum _{iϵF}}{\displaystyle \sum _{kϵK}}{\displaystyle \sum _{h\in FC}}\max \left(0, D_{ihs} - {PNR}_{ihs}^3\right) \times \left({\displaystyle \sum _{jϵO_i}}{x}_{jik}^1 + f_{ik}^1\right) \times {DY}_{ih}\right)}}\\ -\underset{Idle Time Cost}{\underbrace{\left({\displaystyle \sum _{iϵF}}{\displaystyle \sum _{kϵK}}{IT}_{iks}^3 \times { BOS}_k\right)} }- \underset{Delay Time Cost}{\underbrace{\left({\displaystyle \sum _{iϵF}}{DC}_i \times {del}_{is}^3\right)}}-\underset{Missed Connection Cost}{\underbrace{\left({\displaystyle \sum _{iϵF}}{\displaystyle \sum _{jϵ{YB}_i}}{KY}_j \times {pass}_{ij} \times w_{ijs}^3\right)}}\end{array}\right]$$

(6)

Objective Function: The objective function maximizes the expected profit across all scenarios by accounting for ticket revenues generated from realized reservations, denied boarding penalties, passenger service costs, contractual codeshare costs, penalties for unserved codeshare and spilled passengers, as well as idle time, delay, and missed-connection costs.

Subject to

• Stage 1—Fleet Assignment, Routing, Codeshare, and Capacity Decisions

  $$\sum _{k\in K}\left(\sum _{jϵO_i}{x}_{jik}^1 + f_{ik}^1\right)+\sum _{\mathsf{\upsilon }\in \mathsf{{\rm Y}}}\sum _{p\in P}{q}_{\mathsf{\upsilon }pi}^1=1i \in F$$

  (7)

Flight Assignment Completeness: Each flight must be operated either by a fleet type (FA) or assigned to a codeshare partner (CA). No flight can remain unassigned or be double-assigned.

$$\sum _{jϵO_i}{x}_{jik}^1 + f_{ik}^1 - \sum _{jϵH_i}{x}_{ijk}^1 - e_{ik}^1 =0i \in F,k \in K$$

(8)

Routing Flow Conservation: Ensures that for each fleet type, aircraft entering a flight must also exit through a successor flight, maintaining feasible aircraft rotations (AR).

$$\sum _{i\in F}{f}_{ik}^1 \le {AN}_{k}k \in K$$

(9)

Fleet Size Limit: The number of aircraft routes starting for fleet k cannot exceed the available fleet size.

$$f_{ik}^1=0i\in F\backslash {IU}_k, k\in K$$

(10)

Initial Base Feasibility: A route may only start at stations designated as initial bases for that fleet type.

$$e_{ik }^1=0i\in F\backslash {SU}_k, k\in K$$

(11)

Terminal Base Feasibility: Routes must end at valid terminal (maintenance) bases.

$$\sum _{iϵF}\sum _{\mathsf{\upsilon }\in \mathsf{{\rm Y}}}\sum _{pϵP}\sum _{h\in FC}{code}_{\mathsf{\upsilon }pih} \times {Cap}_{\mathsf{\upsilon }pih} \times q_{\mathsf{\upsilon }pi}^1 \le BUD$$

(12)

Codeshare Budget Constraint: Total contractual cost of all selected codeshare agreements must remain within the CA budget.

$$\sum _{iϵF}\sum _{\mathsf{\upsilon }\in \mathsf{{\rm Y}}}\sum _{pϵP}\sum _{h\in FC}{Cap}_{\mathsf{\upsilon }pih} \times q_{\mathsf{\upsilon }pi}^1 \le \theta \times \sum _{k\in K}\sum _{h\in FC}{AN}_k \times {Kap}_{kh}$$

(13)

Codeshare Capacity Ratio: Limits the portion of total passenger capacity that may be outsourced to partner airlines.

$$\sum _{iϵF}\sum _{\mathsf{\upsilon }\in \mathsf{{\rm Y}}}\sum _{pϵP}{q}_{\mathsf{\upsilon }pi}^1 \le \sum _{iϵF}\sum _{kϵK}\left(\sum _{jϵO_i}{x}_{jik}^1 + f_{ik}^1\right)$$

(14)

CA Flight Count Feasibility: Prevents selecting more CA flights than own-operated flights, reflecting realistic cooperation usage.

$$I_{ti} \times \left(\sum _{jϵO_i}{x}_{jik}^1 + f_{ik}^1\right) \le {wks}_{kt}i \in F,k \in K,t \in St$$

(15)

Station Purity Constraint: Ensures fleet–station compatibility and restricts operations to stations authorized for that fleet type.

• Stage 2—Announced Departure Times (ANDT)

  $$\left[1 - \sum _{\mathsf{\upsilon } \in \mathsf{{\rm Y}}}\sum _{p \in P}{q}_{\mathsf{\upsilon }pi}^1\right] \times T_i^{min} \le {\mathrm{a}}_{\mathrm{i}}^2 \le \left[1 - \sum _{\mathsf{\upsilon } \in \mathsf{{\rm Y}}}\sum _{p \in P}{q}_{\mathsf{\upsilon }pi}^1\right] \times T_i^{max}i \in F$$

  (16)

ANDT Feasible Time Window: Announced departure times are determined only when the flight is operated by the airline (not via CA) and must fall within the specified feasible time window.

• Stage 3—Operational Timing, Delays, Passenger Allocation

  $$-{\mathrm{a}}_{\mathrm{i}}^2 + b_{is}^3 \ge 0i \in F ,s \in S$$

  (17)

ACDT Not Earlier Than ANDT: Actual departure time cannot occur before the planned departure time.

$$b_{is}^3 - {\mathrm{a}}_{\mathrm{i}}^2 - {del}_{is}^3 \le 0i \in F ,s \in S$$

(18)

Delay Definition: Defines the realized delay for each scenario.

$$b_{is}^3 + \sum _{k \in K}\left(\sum _{jϵO_i}{x}_{jik}^1 + f_{ik}^1\right) \times {\mathsf{\eta }}_{\mathrm{i}\mathrm{s}} + \sum _{k\in K}{d}_{iks}^3 = c_{is}^{3}i \in F ,s \in S$$

(19)

Actual Arrival Time Calculation: Actual arrival time is obtained by adding the cruise time and the non-cruise time (NCT) to the actual departure time.

$$\begin{array}{c}If x_{ijk}^1=1\\ b_{js}^3 - b_{is}^3 = {\mathsf{\eta }}_{\mathrm{i}\mathrm{s}} + {\mathrm{T}\mathrm{A}}_{\mathrm{i}\mathrm{k}} + d_{iks}^3 + {IT}_{iks}^3\left(i,j\right) \in BU,s \in S ,k \in K\end{array}$$

(20)

Successive Flight Feasibility: Enforces feasible turnaround and connection timing between consecutive flights of an aircraft route and minimizes aircraft idle time.

$$-{CT}_{ik}^l \times \left(\sum _{jϵO_i}{x}_{jik}^1 + f_{ik}^1\right) + d_{iks}^3 \ge 0i \in F ,s \in S ,k \in K$$

(21)

$${CT}_{ik }^u \times \left(\sum _{jϵO_i}{x}_{jik}^1 + f_{ik}^1\right) - d_{iks}^3 \ge 0i \in F ,s \in S ,k \in K$$

(22)

Cruise Time Bounds: Cruise time must lie within lower/upper operational limits.

$$b_{js}^3 - b_{is}^3 - \sum _{k\in K}{d}_{iks}^3 + M \times w_{ijs}^3 \ge \left[1 - \sum _{\mathsf{\upsilon }\in \mathsf{{\rm Y}}}\sum _{p\in P}{q}_{\mathsf{\upsilon }pi}^1\right] \times \left({\mathsf{\eta }}_{\mathrm{i}\mathrm{s}} + {CP}_{ij}\right)i \in F ,j \in {YB}_i ,s \in S$$

(23)

Missed Connection Condition: Activates the missed connection variable whenever the minimum connection requirement is violated.

$${\pi }_{ihs}^3 \le \sum _{k\in K}{Kap}_{kh} \times \left(\sum _{jϵO_i}{x}_{jik}^1 + f_{ik}^1\right)i\in F ,h \in FC, s \in S$$

(24)

$${\pi c}_{ihs}^3 \le \sum _{\mathsf{\upsilon }\in \mathsf{{\rm Y}}}\sum _{pϵP}{Cap}_{\mathsf{\upsilon }pih} \times q_{\mathsf{\upsilon }pi}^{1}i\in F ,h \in FC, s \in S$$

(25)

Passenger Acceptance Feasibility: Accepted passengers cannot exceed available capacity (own aircraft or codesharing).

$${\pi }_{ihs}^3 \le D_{ihs}i \in F ,h \in FC,s \in S$$

(26)

$${\pi c}_{ihs}^3 \le { D}_{ihs}i \in F ,h \in FC,s \in S$$

(27)

Demand Bound: Accepted passengers cannot exceed demand.

$${\mathrm{P}\mathrm{N}\mathrm{R}}_{\mathrm{i}\mathrm{h}\mathrm{s}}^3 \le \left[\sum _{k\in K}{OB}_{kh} \times \left(\sum _{jϵO_i}{x}_{jik}^1 + f_{ik}^1\right) + \sum _{\mathsf{\upsilon }\in \mathsf{{\rm Y}}}\sum _{pϵP}{OBC}_{\mathsf{\upsilon }pih} \times q_{\mathsf{\upsilon }pi}^1\right]i \in F ,h \in FC,s \in S$$

(28)

PNR Upper Bound with OB and CA: The number of tickets sold cannot exceed the combined overbooking (OB) allowance of the selected fleet types or the available codeshare capacity. OB parameters represent pre-determined booking limits derived from overbooking policies, rather than additional physical aircraft capacity.

$${\mathrm{P}\mathrm{N}\mathrm{R}}_{\mathrm{i}\mathrm{h}\mathrm{s}}^3 \le D_{ihs}i \in F ,h \in FC,s \in S$$

(29)

PNR Demand Feasibility: Tickets sold cannot exceed scenario-dependent demand forecasts.

$${\mathrm{P}\mathrm{N}\mathrm{R}}_{\mathrm{i}\mathrm{h}\mathrm{s}}^3 \times 0.9 \le {\mathrm{s}\mathrm{h}\mathrm{o}\mathrm{w}\mathrm{p}\mathrm{a}\mathrm{s}\mathrm{s}}_{\mathrm{i}\mathrm{h}\mathrm{s}}^{3}i \in F ,h \in FC,s \in S$$

(30)

Show-Up Bound (OB): The number of boarded passengers represents a certain proportion of the tickets sold and cannot exceed the aircraft’s capacity.

$${\pi }_{ihs}^3 + {\pi c}_{ihs}^3 \ge {\mathrm{s}\mathrm{h}\mathrm{o}\mathrm{w}\mathrm{p}\mathrm{a}\mathrm{s}\mathrm{s}}_{\mathrm{i}\mathrm{h}\mathrm{s}}^3 - {\mathrm{d}\mathrm{b}}_{\mathrm{i}\mathrm{h}\mathrm{s}}^{3}i \in F ,h \in FC,s \in S$$

(31)

Denied Boarding Relationship: The number of passengers denied boarding (DB), either voluntarily or involuntarily, is determined. The total number of show-up passengers is balanced by the sum of boarded passengers and DB passengers.

$$\begin{array}{cc}{x}_{ijk}^1\in \left\{\mathrm{0,1}\right\}& i \in F,j \in O_i,k \in K\\ f_{ik}^1\in \left\{\mathrm{0,1}\right\}& i\in F, k\in K\\ e_{ik}^1\in \left\{\mathrm{0,1}\right\}& i\in F,k\in K\\ q_{\mathsf{\upsilon }pi}^1\in \left\{\mathrm{0,1}\right\}& i\in F , \mathsf{\upsilon }\in \mathsf{{\rm Y}}, p\in P\\ {\mathrm{a}}_{\mathrm{i}}^2\ge 0& i\in F\\ {\pi }_{ihs}^3\ge 0& i\in F, h\in FC, s\in S\\ {\pi c}_{ihs}^3\ge 0& i\in F,h\in FC, s\in S\\ {IT}_{iks}^3\ge 0& i\in F,k\in K,s\in S,\\ {del}_{is}^3\ge 0& i\in F,s\in S\\ w_{ijs}^3\in \left\{\mathrm{0,1}\right\}& i\in F, j\in {YB}_i, s\in S,\\ {\mathrm{P}\mathrm{N}\mathrm{R}}_{\mathrm{i}\mathrm{h}\mathrm{s}}^3 , {\mathrm{s}\mathrm{h}\mathrm{o}\mathrm{w}\mathrm{p}\mathrm{a}\mathrm{s}\mathrm{s}}_{\mathrm{i}\mathrm{h}\mathrm{s}}^3, {\mathrm{d}\mathrm{b}}_{\mathrm{i}\mathrm{h}\mathrm{s}}^3 \ge 0integer& i\in F, h\in FC, s\in S\end{array}$$

(32)

4.4. Integration and Contribution of the Model

The main contribution of this study is the development of an integrated FSFAAR–CA–OB framework. The model simultaneously:

• Links OB limits with FA and AR feasibility,
• Incorporates CA contracts to mitigate SP restrictions and insufficient capacity,
• Accounts for the propagation of operational timing effects driven by uncertainty in NCT, ensuring day-of-operations feasibility,
• Explicitly considers demand consistency, spill risks, and denied boarding outcomes,
• Provides a fully interconnected three-stage stochastic optimization environment.

This integrated structure enables airlines to evaluate strategic OB and CA decisions in conjunction with tactical and operational considerations, leading to more robust schedules and higher expected profitability.

5. Solution Methods

Metaheuristics have been applied to a wide range of airline planning problems, including crew scheduling and aircraft landing using particle swarm optimization [47,48], fleet maintenance decisions via bio-inspired algorithms [49], and integrated schedule design and fleet assignment with demand recapture using parallel genetic algorithms [50]. In this context, Ng et al. [51] made a significant contribution to the literature by presenting a comprehensive study that systematically examines various meta-heuristic approaches applied to airline planning problems.

However, the computational challenge becomes even more pronounced in the integrated FSFAAR–SP–CA–OB model proposed in this study. The model contains a highly coupled structure in which binary first-stage fleet assignment and routing decisions influence second-stage operational feasibility, including capacity availability, delay propagation, station purity enforcement, codeshare constraints, and stochastic overbooking outcomes. As the number of flights, fleet types, or scenarios increases, the combinatorial nature of the problem results in rapidly escalating complexity. Standard optimization solvers become insufficient for generating high-quality solutions within acceptable computational times, especially when robust stochastic evaluation is required. Therefore, a hybrid simulation–metaheuristic framework is adopted to ensure computational tractability while preserving the stochastic richness and operational realism of the model.

5.1. The Sample Average Approximation (SAA)

The SAA method is widely used to address stochastic optimization problems by approximating the expected objective function value. This approach relies on Monte Carlo simulation, which involves generating random samples to produce numerical estimates. In SAA, the expected objective is replaced by the average value computed over a finite sample, transforming the original stochastic model into a deterministic optimization problem. This deterministic version is then solved using standard optimization methods.

To ensure the stability and credibility of the computed solutions, the algorithm is applied over multiple stochastic sample sets. This repetition enables the estimation of solution quality metrics, such as optimality gaps and statistical confidence bounds. The main advantage of the SAA algorithm is its ability to find near-optimal solutions using relatively small sample sizes. In many cases, generating multiple replicated samples and solving the model repeatedly is more efficient than increasing the scenario size of a single instance. Moreover, the use of upper and lower bounds allows a structured evaluation of solution quality [16,52,53,54].

Beyond its computational benefits, SAA provides strong theoretical guarantees. As the sample size increases, SAA estimators converge almost surely to the true optimal value of the original stochastic optimization problem. Confidence intervals are used to assess solution reliability, and a solution is accepted only if the width of the confidence interval falls below a predefined threshold (ε). Otherwise, additional replications or larger samples are generated. This ensures that the final selected solution is both statistically valid and computationally feasible.

In this study, the inherent difficulty of the large-scale integrated model makes exact solution of SAA subproblems impractical. To address this challenge, metaheuristic methods are integrated into the SAA procedure, yielding a hybrid approach that leverages both the statistical strength of SAA and the rapid search capabilities of metaheuristics. The procedural steps of the SAA algorithm are detailed in Algorithm 1, and its flow is visualized in Figure 1.

Algorithm 1. Procedural steps of the SAA algorithm

Step 1: Repeat the following steps for m = 1, 2, ..., M:

Generate a sample size (S). Solve the model for each m with the SAASA and SAACS algorithms. Calculate the best solution ${(x}_S^m)$ found by the metaheuristic and the objective function value ${(obj}_S^m)$ of this solution. Generate a larger size sample (S′ >> S). Calculate the objective function values ${obj}_{S\prime }^m{(x}_S^m)$ and their variances ${\mathsf{\sigma }}_{{obj}_{S\prime }^m{(x}_S^m)}^2$ for the large sample using the previously obtained best solutions ${(x}_S^m)$

${\mathsf{\sigma }}_{{obj}_{S\prime }^m{(x}_S^m)}^2 = {\displaystyle \frac{1}{{(S}^{\prime }-1)S\prime }}{\displaystyle \sum _{s=1}^{S\prime }}{\left({obj}_s{(x}_S^m\right) - {obj}_{S\prime }{(x}_S^m\left)\right)}^2$

Step 2: Calculate the mean ${\overline{\mu }}_S^m$ and their variances ${\mathsf{\sigma }}_{{\overline{\mu }}_S^{m } }^2$ of the best results

${\overline{\mu }}_S^m = {\displaystyle \frac{1}{M}}{\displaystyle \sum _{m = 1}^M}{\mu }_S^m$

${\mathsf{\sigma }}_{{\overline{\mu }}_S^m}^2 = {\displaystyle \frac{1}{(M - 1)M}} {\displaystyle \sum _{m=1}^M}{{(\mu }_S^m-{\overline{\mu }}_S^m)}^2$

Step 3: Calculate the optimality gap and the corresponding variance ${\mathsf{\sigma }}_{gap}^2$ for each best result.

$gap = {\overline{\mu }}_S^m - {obj}_{S\prime }^m{(x}_S^m)$

${\mathsf{\sigma }}_{gap}^2 ={\mathsf{\sigma }}_{{\overline{\mu }}_S^m}^2 + {\mathsf{\sigma }}_{{obj}_{S\prime }^m{(x}_S^m)}^2$

Step 4: Upon completing Step 3, the candidate solution exhibiting the smallest optimality gap identified thus far is selected. The corresponding optimality gap and its variance estimate are then evaluated. If either of these metrics exceeds acceptable thresholds, the procedure must be reiterated with an increased sample size and/or a larger number of replications MMM to improve estimation accuracy and solution reliability. Also, ${\overline{\mu }}_S^m$ and ${ obj}_{S\prime }^m{(x}_S^m)$ denote the upper and lower bounds for the optimal objective function value of the original problem, respectively [16].

5.2. Metaheuristic Integration

For each SAA replication, the sampled deterministic subproblem is solved using either the SAASA (Simulated Annealing–Enhanced SAA) or SAACS (Cuckoo Search–Enhanced SAA) algorithm. Both metaheuristics operate primarily on first-stage decisions—fleet assignment, routing, and codeshare usage—all of which critically influence downstream delay propagation, capacity feasibility, and overbooking dynamics.

Each search begins with a feasible initial solution generated by constructing time-feasible aircraft routes and assigning appropriate fleet types according to demand expectations. This ensures that the metaheuristic search starts from a feasible region and focuses on improving solution quality without violating operational constraints.

In SAASA, a geometric cooling schedule with cooling rate α ∈ [0.92,0.98] is applied, consistent with classical tuning recommendations for simulated annealing [55,56,57]. The initial temperature is calibrated to accept roughly 80% of uphill moves, a commonly used heuristic in SA-based optimization.

For SAACS, a population of 20–40 candidate solutions is maintained, which falls within the recommended range for CS metaheuristics in engineering applications [58,59,60]. The discovery rate pa ∈ [0.1,0.2] aligns with the classical range suggested in the original CS formulation [58]. The chosen parameter ranges (cooling rate, discovery rate, and population size) are consistent with the classical heuristic optimization literature and widely adopted in large-scale engineering optimization problems.

The flowchart of the SA and CS metaheuristics is provided in Figure 2 and Figure 3. The sample values generated by SAA serve as the input for both methods. After each metaheuristic run, the best solution and its objective value are evaluated under a larger SAA sample to obtain statistical quality measures, ensuring robustness under uncertainty. SAASA is designed for stronger local exploitation, whereas SAACS provides broader global exploration due to its population-based nature. Comprehensive descriptions of the SA and CS metaheuristics, including algorithmic structures and implementation details, can be found in Kızıloğlu and Sakallı [35].

Problem-Specific Neighborhood Structure

Neighborhood generation plays a central role in both SAASA and SAACS. The operator modifies a small subset of consecutive flights either by copying fleet assignments from another flight or by assigning a new feasible fleet type while ensuring that station purity, compatibility, and turnaround rules are not violated. This neighborhood structure allows controlled local perturbations while maintaining feasibility—an essential requirement in large-scale airline planning problems. The detailed steps of the Problem-Specific Neighborhood Structure are presented in Algorithm 2, which illustrates how feasible neighboring solutions are generated while preserving operational constraints.

Algorithm 2. Neighborhood Generation (Summary)

Input: current solution x Output: neighbor solution x′ 1    Copy x to x′ 2    Generate random number r ∈ [0,1] 3    Select random starting flight f 4    If r < 0.3: For flights i ∈ {f, f + 1, f + 2}: Select random reference flight j Swap fleet types of flights i and j Else: For flights i ∈ {f, f + 1, f + 2}: Randomly assign a feasible fleet type (respecting station purity, compatibility, turnaround rules) 5    Rebuild affected route segments 6    Validate feasibility under all operational constraints 7    Return x′

5.3. Flowchart Explanation

The solution procedure illustrated in the flowchart follows a structured sequence of computational steps designed to integrate SAA with the proposed metaheuristic algorithms. The process operates as follows:

• Input Parameter Initialization: All deterministic model parameters are first loaded, including fleet characteristics, cost coefficients, schedule limits, and capacity specifications.
• Generation of Stochastic Scenario Values: For each SAA replication, scenario sets are generated for stochastic parameters—primarily passenger demand and Non-Cruise Time (NCT)—based on the probability distributions defined in Section 3. These scenarios serve as the stochastic environment in which candidate solutions are evaluated.
• Transfer of Input Data to the Metaheuristic Engine: The complete dataset (deterministic parameters + scenario values) is passed to the selected metaheuristic algorithm (SAASA or SAACS), which initializes the search process.
• Construction of a Random Feasible Initial Solution: A feasible starting point is constructed using randomized but constraint-respecting assignment and routing decisions. This ensures that the search begins in a valid region of the solution space.
• Iterative Improvement of the Solution: The metaheuristic explores the neighborhood of the current solution, applying acceptance rules (in SAASA) or population updates (in SAACS) to progressively enhance solution quality. Infeasible solutions are discarded, guaranteeing feasibility at each iteration.
• Termination and Simulation-Based Evaluation: Once the stopping condition is satisfied—based on maximum iterations or temperature threshold—the best solution identified by the metaheuristic is subjected to a long-run simulation using 10,000 scenarios. This provides a statistically robust estimate of its expected objective value.
• Transfer of Results to SAA: The evaluated objective function value and associated solution details are returned to the SAA procedure for statistical processing.
• Statistical Assessment Across M Replications: The above steps are repeated for MMM independent SAA replications. The resulting sample of objective values is used to compute confidence intervals, lower bounds, and gap measures.
• Selection of the Best SAA Solution: The final output of the method is the solution with the smallest estimated optimality gap and highest statistical reliability, determined from the full set of SAA replications.

5.4. Advantages of the Hybrid Framework

The proposed SAA–SA/CS integration offers several advantages:

• Scalability: Metaheuristics efficiently solve the large subproblems resulting from SAA.
• Robustness: Multiple replications provide statistical guarantees, improving solution reliability.
• Exploration vs. exploitation balance: SAASA enhances intensification, while SAACS strengthens diversification.
• Feasibility maintenance: The problem-specific neighborhood operator ensures that operational rules—station purity, routing continuity, codeshare bounds, overbooking limits—are always respected.

Together, these components form a practical and effective solution framework for solving large-scale instances of the integrated FSFAAR–SP–CA–OB model under uncertainty.

6. Numerical Example

To demonstrate the behavior of the proposed TSSMINLP model and highlight the operational effects of Station Purity (SP), Overbooking (OB), and Codeshare Agreements (CA), a small-scale test problem consisting of five flights and two fleet types (A321-200 and B787-8) is constructed. Flight information is presented in

Table 1. Information about the flights.

Flights	Departure	Arrival	Departure Time	Arrival Time	${\mathbf{T}}_{\mathbf{i}}^{\mathbf{m}\mathbf{i}\mathbf{n}}$ (min)	${\mathbf{T}}_{\mathbf{i}}^{\mathbf{m}\mathbf{a}\mathbf{x}}$ (min)	${\mathbf{K}\mathbf{Y}}_{\mathbf{i}}$ ($/Passenger)	${\mathbf{D}\mathbf{C}}_{\mathbf{i}}$ ($/min)
1	LAX	MIA	06:25	11:35	375	395	50	9.65
2	MIA	LAX	12:55	18:15	765	785	50	5.58
3	LAX	HNL	19:40	01:30	1170	1190	50	7.14
4	LAX	OGG	08:00	13:40	470	490	50	9.12
5	OGG	LAX	15:00	20:20	890	910	50	5.87
Flights	${\mathbf{f}\mathbf{a}\mathbf{r}\mathbf{e}}_{\mathbf{i}\mathbf{h}}$ ($/Passenger)		${\mathbf{c}\mathbf{o}\mathbf{s}\mathbf{t}}_{\mathbf{i}\mathbf{h}}$ ($/Passenger)		${\mathbf{D}\mathbf{Y}}_{\mathbf{i}\mathbf{h}}$ ($/Passenger)		${\mathsf{\eta }}_{\mathbf{i}\mathbf{s}}$ $\left(\mathbf{m}\mathbf{i}\mathbf{n}\right)$
	B	E	B	E	B	E	Mean	Standard Deviation
1	479	399	251.37	239.40	216	180	24.57895	0.642065
2	323	269	169.47	161.40	161	134	24.1087	0.78095
3	275	229	144.27	137.40	158	132	27.71154	1.161323
4	269	224	105.92	98.4	125	104	23.63265	1.060938
5	221	184	105.92	98.4	125	104	31.89091	2.013703

, while operational properties of the fleet types are given in

Table 2. Information about the fleet types.

Flights	${\mathbf{C}\mathbf{T}}_{\mathbf{i}\mathbf{k}}^{\mathbf{l}}\left(\mathbf{m}\mathbf{i}\mathbf{n}\right)$		${\mathbf{C}\mathbf{T}}_{\mathbf{i}\mathbf{k}}^{\mathbf{u}}\left(\mathbf{m}\mathbf{i}\mathbf{n}\right)$		${\mathbf{T}\mathbf{A}}_{\mathbf{i}\mathbf{k}}\left(\mathbf{m}\mathbf{i}\mathbf{n}\right)$
	A321-200	B787-8	A321-200	B787-8	A321-200	B787-8
1	247	247	290	290	50	61
2	255	255	300	300	46	57
3	281	281	330	330	43	52
4	272	272	320	320	26	32
5	255	255	300	300	46	57
Fleet type			A321-200		B787-8
Fare classes			B	E	B	F
Capacity			27	80	59	175
Aircraft number			2		2
IT cost ($/min)			70		115

. Two codeshare agreements are included, under which 45% and 35% of the ticket revenue per transported passenger is shared with the operating carrier. Each fleet type contains two fare classes—Economy (E) and Business (B)—with distinct capacities and cost parameters. Passenger connection time is fixed at 30 min.

Stochastic Inputs and Scenario Generation: Both Passenger Demand and Non-Cruise Time (NCT) are modeled as stochastic variables:

Non-Cruise Time (NCT): NCT follows a normal distribution calibrated from historical delay data obtained from an open-access dataset [61]. Mean and standard deviation values used for each flight are reported in Table 1. Each scenario includes a random draw from this distribution, generating the realized NCT value.

Passenger Demand: Because real demand data are unavailable, demand is assumed to follow a uniform distribution. Distribution bounds are determined from the existing fleet structure:

• Lower bound: 20% below the minimum available aircraft capacity
• Upper bound: 20% above the maximum available aircraft capacity

This ensures a balanced set of low-demand and high-demand scenarios that reflect realistic operational variability. Each scenario samples demand independently for both fare classes.

Station Purity (SP): Fleet–station compatibility is defined using the SP matrix in

Table 3. The stations purity.

		Stations (t)
		1	2	3	4
Fleet type (k)	1	1	0	0	1
	2	1	1	1	0
Station Purity $\left({SP}_t\right)$		2	1	1	1

. A value of “1” indicates that a fleet type may operate at that station. These restrictions directly constrain feasible aircraft–flight assignments and influence the routing structure.

Reservation Limits and Overbooking Behavior: Booking limits for aircraft and codeshare capacities are computed using Equations (4) and (5).

Table 4. Reservation limits according to capacities.

Reservation Limits	Capacity		Penalty Coefficient	The Probability of a Passenger Show-Up	Penalty Coefficient	The Probability of a Passenger Show-Up
Fleet types	Fleet type capacity		2	0.90	3	0.90
	B	E	B	E	B	E
A321-200	27	80	29	88	28	86
B787-8	59	175	64	193	63	191
Codeshare agreements	Codeshare capacity
	B	E	B	E	B	E
Codeshare 1	27	53	29	58	28	56
Codeshare 2	37	73	40	80	39	78
Fleet types	Fleet type capacity		2	0.85	3	0.85
	B	E	B	E	B	E
A321-200	27	80	31	93	30	91
B787-8	59	175	68	205	67	202
Codeshare agreements	Codeshare capacity
	B	E	B	E	B	E
Codeshare 1	27	53	31	61	30	60
Codeshare 2	37	73	42	85	41	83

presents reservation limits for:

• show-up probabilities: 0.85 and 0.90,
• penalty multipliers: 2× and 3× the ticket fare.

Lower show-up probability values yield higher reservation limits, reflecting an increased tolerance for overbooking risk and a higher revenue-seeking strategy.

6.1. Model Results

The model is solved using GAMS 22.5/BARON (GAMS Development Corporation, Washington, DC, USA) under two stochastic demand–NCT scenarios.

First-Stage Decisions (Routing, Assignment, CA Usage):

Table 5. The problem’s first phase decisions.

Flights	Assignment	Route	Stations	Demand
				B	E
1	B787-8	1. Route	LAX–MIA–LAX–HNL	80	180
2	B787-8			65	190
3	B787-8			55	160
4	Codeshare 2		LAX–OGG	38	97
5	Codeshare 2		OGG–LAX	23	65

and

Table 6. Objective function values and assignments.

		The Probability of a Passenger Show-Up 0.90		The Probability of a Passenger Show-Up 0.85
Penalty coefficient	The passengers show-up rates	Objective function value	Assignments	Objective function value	Assignments
2	0.85	113,214.915	2-2-2-codeshare 2-codeshare 2	116,511.975	2-2-2-codeshare 2-codeshare 2
2	0.90	108,424.995	2-2-2-codeshare 2-codeshare 2	109,362.6	2-2-2-codeshare 2-codeshare 2
3	0.85	111,571.11	2-2-2-codeshare 2-codeshare 2	115,395.035	2-2-2-codeshare 2-codeshare 2
3	0.90	106,781.19	2-2-2-codeshare 2-codeshare 2	109,362.6	2-2-2-codeshare 2-codeshare 2

present the resulting first-stage decisions and objective values.

Key observations:

• Flights 1–3 form a single route operated by B787-8, since A321-200 cannot serve Stations 2 and 3 due to SP restrictions.
• Flights 4–5, although compatible with A321-200, are assigned to Codeshare Agreement 2, because CA capacity yields higher expected profit under SP constraints.
• Ticket sales respect scenario-dependent demand realizations and reservation limits.

This result demonstrates that when station purity (SP) constraints restrict the feasible deployment of fleet capacity, the model optimally compensates by allocating codeshare (CA) capacity, thereby preserving operational feasibility and sustaining profitability.

Table 6 indicates that reservation limits rise as the assumed show-up probability decreases, reflecting the airline’s attempt to compensate for anticipated no-show behavior by selling additional seats. When the realized number of show-up passengers is higher than expected, denied boarding (DB) occurs, leading to profit reductions due to penalty costs. This outcome highlights the fundamental trade-off inherent in overbooking: while it increases expected revenue in scenarios with low show-up rates, it also introduces financial risk when actual demand exceeds capacity. The model systematically determines reservation limits that balance these opposing effects, selecting values that maximize expected profit while keeping DB risk within acceptable bounds.

Second- and Third-Stage Outcomes.

Table 7. Second and third phase decisions.

	(Flights)	Fleet Type	ANDT		ACDT	AAT	CT (min)		IT (min)		DT (min)		NCT (min)
First scenario results	1	B787-8	06:35		06:53	12:08	290		0		18		25
	2		13:05		13:09	18:33	300		0		4		24
	3		19:30		19:30	24:39	281		0		0		28
	4	Codeshare 2
	5	Codeshare 2
Second scenario results	1	B787-8	06:35		06:52	12:07	290		0		17		25
	2		13:05		13:08	18:33	300		0		3		25
	3		19:30		19:30	24:38	281		0		0		27
	4	Codeshare 2
	5	Codeshare 2
	(Flights)	Fleet Type	Demand		PNR Number		Show-Up Passenger Number		Spill Passenger Number		OB Number		DB Number
			B	E	B	E	B	E	B	E	B	E
First scenario results	1	B787-8	80	180	68	180	58	153	12	0	9	5	0	0
	2		65	190	65	190	56	162	0	0	6	15	0	0
	3		55	160	55	160	47	136	0	0	0	0	0	0
	4	Codeshare 2	38	97	38	85	33	73	0	12	1	12	0	0
	5	Codeshare 2	23	65	23	65	20	56	0	0	0	0	0	0
Second scenario results	1	B787-8	21	83	21	83	18	71	0	0	0	0	0	0
	2		31	125	31	125	27	107	0	0	0	0	0	0
	3		30	105	30	105	26	90	0	0	0	0	0	0
	4	Codeshare 2	50	80	42	80	36	68	8	0	5	7	0	0
	5	Codeshare 2	35	100	35	85	30	73	0	15	0	12	0	0

provides a comprehensive summary of the operational outcomes produced in the second and third stages of the model. The table reports key temporal variables—announced and actual departure times, arrival times, cruise durations, idle periods, and realized delays—alongside passenger-flow indicators such as demand volumes, number of tickets sold, realized show-up counts, spilled passengers, overbooking levels, and denied-boarding occurrences. Collectively, these outputs illustrate how the integrated model synchronizes scheduling, routing, and capacity decisions under stochastic variability in demand and non-cruise times.

The observed results reveal a clear pattern in the activation of the overbooking mechanism. Under high-demand conditions (Scenario 1), the model applies overbooking up to the allowable limit, resulting in additional ticket sales while channeling excess demand into spilled-passenger outcomes. In contrast, when demand is low (Scenario 2), the model refrains from overbooking, thereby avoiding unnecessary risk exposure. Denied boarding remains effectively negligible across scenarios due to its associated penalty cost, demonstrating that the model enforces statistically disciplined overbooking behavior. This behavior confirms the framework’s ability to balance revenue maximization with operational feasibility and service reliability.

To illustrate the interpretation of the results presented in Table 7, consider Flight 1 as an example. The aircraft assigned to this flight has capacities of 59 seats in Business class and 175 seats in Economy class. Based on the capacity utilization and the expected number of spilled passengers, the PNR (booking) level for Business class was set to 68. Accordingly, 9 passengers were accepted beyond the physical capacity, reflecting the applied overbooking level. Given the realized demand, 12 passengers could not be accommodated and were therefore classified as spill. The actual number of show-up passengers for this flight was 58. The same procedure was applied consistently to determine the corresponding values for all remaining flights.

6.2. Comparative Analysis of SP, OB, and CA

The small example is solved under three configurations:

• Problem 1: OB + SP
• Problem 2: OB only (SP disabled)
• Problem 3: SP only (OB disabled)

Results are shown in

Table 8. Effect of features on the model.

	Test Problem 1 (Overbooking and SP)			Test Problem 2 (Only Overbooking)			Test Problem 3 (Only SP)
(Flights)	Assignment	Route	Obj. function	Assignment	Route	Obj. function	Assignment	Route	Obj. function
1	B787-8	1. Route	$116,511.975	B787-8	1. Route	$118,293.635	B787-8	1. Route	$89,585.820
2	B787-8			B787-8			B787-8
3	B787-8			B787-8			B787-8
4	Codeshare 2			B787-8	2. Route		Codeshare 2
5	Codeshare 2			B787-8			Codeshare 2

.

The results reveal a clear interaction among station purity (SP), overbooking (OB), and codeshare agreements (CA). SP restricts fleet assignment flexibility by preventing the B787-8 from operating Flights 4–5, thereby compelling the model to substitute capacity through CA. Within this restricted environment, OB enhances profitability by increasing expected seat availability, while CA supplements physical capacity where in-house resources are limited. Together, OB and CA effectively counterbalance the routing rigidity introduced by SP, demonstrating the compensatory mechanisms embedded in the integrated framework. When SP is removed (Problem 2), the model assigns the B787-8 to all flights, constructs two efficient routes, and achieves the highest overall profit.

Accordingly, the combination of OB and CA mitigates the adverse effects of SP, SP alone produces the lowest profit, and OB alone yields the highest. These findings confirm that the proposed TSSMINLP formulation generates operationally coherent solutions—maintaining route continuity, enforcing feasible fleet–station assignments, statistically regulating OB decisions, and selecting CA options in an economically rational manner. Furthermore, the behavioral patterns identified in this small-scale example align with those observed in the large-scale computational experiments in Section 7, demonstrating the robustness and generalizability of the model across varying problem sizes.

A comparative assessment of the proposed metaheuristics was performed by solving the small-scale example using the SAASA and SAACS algorithms implemented in C# (Microsoft Visual Studio 2019). As reported in

Table 9. Comparison of solution methods.

GAMS		SAASA		SAACS
Objective function	Assignment	Objective function	Assignment	Objective function	Assignment
$116,511.975	2-2-2-codeshare 2-codeshare 2	$116,511.975	2-2-2-codeshare 2-codeshare 2	$116,511.975	2-2-2-codeshare 2-codeshare 2

, both methods exactly reproduced the GAMS 22.5/BARON benchmark solution in terms of fleet assignments, routing decisions, and total objective value. This agreement confirms that the SAA–metaheuristic framework delivers high solution fidelity despite relying on heuristic search. More importantly, the results demonstrate that the hybrid approach provides a computationally tractable and scalable alternative for large problem instances, where exact solvers become infeasible due to the nonlinear and stochastic structure of the integrated FSFAAR–SP–CA–OB model.

The numerical example is intentionally designed as a pedagogical illustration to transparently demonstrate the interactions among the proposed decision layers and associated constraints. Although small in scale by construction, it allows for a clear and interpretable examination of model behavior. This, in turn, facilitates an intuitive understanding of the integrated decision-making framework.

In contrast, the large-scale computational experiments are conducted on realistic problem instances. These experiments aim to evaluate the scalability, computational stability, and operational applicability of the proposed approach under practically relevant conditions. The insights derived from the numerical example primarily relate to structural interactions and feasibility mechanisms. The large-scale results, however, confirm that these interactions are preserved as problem size and uncertainty increase, thereby demonstrating the effectiveness of the proposed approach in more complex operational settings.

7. Computational Results

7.1. Data Descriptions

This section describes the data sources, stochastic structures, and parameter settings used to construct the large-scale test problems for the computational experiments. To evaluate the performance of the proposed solution framework, four large-scale test problems were constructed using real-world flight data obtained from the open-access database in [61]. The dataset was filtered to include only American Airlines flights departing from or arriving at Los Angeles International Airport (LAX). The resulting test sets contain 30, 78, 127, and 180 flights, and incorporate six different aircraft types. All operational and economic parameters not directly available in the dataset follow Kızıloğlu and Sakallı [35].

7.1.1. Stochastic Structures and Scenario Generation

The stochastic modeling assumptions are fully aligned with those used in the small-scale numerical example, ensuring methodological consistency.

• Passenger Demand: Since real booking data are not available, demand for each fare class is drawn from a uniform distribution. The lower and upper bounds are defined as: [0.8·Cmin, 1.2·Cmax] where Cmin and Cmax are the capacities of the smallest and largest aircraft types in the fleet. This design ensures that the model experiences both low-demand and peak-demand regimes in a controlled but realistic range.
• Non-Cruise Time (NCT): NCT is sampled from a normal distribution calibrated using historical delay statistics from dataset [61]. This maintains consistency with the earlier example and captures realistic operational variability.
• Station Purity (SP): Fleet–station compatibility is imposed using the SP matrix which restricts routing and assignment decisions.

These distributions were selected because they represent standard practice in airline simulation studies and because their controlled variance enables stable replication under SAA.

7.1.2. Codeshare Agreements (CA)

Two codeshare agreements are modeled. The operating carrier receives 30% and 23% of ticket revenue for CA1 and CA2, respectively. The capacities included in these contracts (56 and 100 seats) fall within realistic industry ranges. These values are not intended to replicate a specific real-world contract but rather to produce numerically distinguishable CA options, allowing clear interpretation of CA behavior under SP and OB constraints.

7.1.3. Fare Classes

Each aircraft type has two fare classes (Business and Economy) with distinct:

• Capacities,
• Ticket fares,
• Operating costs,
• Spilled passenger penalties,
• Stochastic demand distributions.

This structure improves accuracy in revenue and cost modeling.

7.1.4. Overbooking (OB) and Reservation Limits

Reservation limits for both aircraft and codeshare capacities were derived using Equations (4) and (5) under two alternative show-up probabilities (0.85 and 0.90) and two penalty multipliers (2× and 3× the ticket fare). The computed values are reported in

Table 10. Reservation limits for the test problems.

Reservation Limits	Capacity		Penalty Coefficient	The Probability of a Passenger Show-Up	Penalty Coefficient	The Probability of a Passenger Show-Up
Fleet types	Fleet type capacity		2	0.85	3	0.85
	B	E	B	E	B	E
B737-800	43	129	49	150	48	148
A321-200	47	140	54	163	53	161
A319-100	32	96	36	112	35	110
A321-NEO	49	147	56	172	55	169
B787-8	59	175	68	205	67	202
ERJ-175	19	57	21	66	20	64
Codeshare agrements	Codeshare Capacity
Codeshare 1	14	42	15	48	15	47
Codeshare 2	25	75	28	87	27	85

. As expected, lower show-up probabilities yield higher booking limits, reflecting the fundamental principle of overbooking theory whereby airlines optimally increase accepted reservations when the likelihood of passenger no-show is greater. This behavior ensures an analytically consistent balance between expected revenue gains and the risk of denied-boarding penalties within the integrated model. These reservation limits are subsequently used as fixed upper bounds in both fleet and codeshare capacity allocation decisions across all computational experiments.

7.2. Computational Framework

The problem size increases rapidly with the number of scenarios in SAA, making exact optimization intractable for large instances. Therefore, the integrated model is solved using SAA combined with SAASA and SAACS metaheuristics. Before experimentation, parameters of both algorithms were tuned through extensive preliminary runs over large problem instances to determine settings that yield the best balance between solution quality and computational time. These parameter values demonstrated consistent performance across all tested instances and were therefore fixed for the final experiments.

SAASA (Simulated Annealing + SAA)

• Initial temperature: 2000
• Final temperature: 10
• Neighborhood size: 200
• Cooling rate: 0.99

SAACS (Cuckoo Search + SAA)

• Replacement rate: 0.30
• Population size: 20
• Maximum iterations: 500

SAA Structure Used in the Experiments

Following the guidance of Kenan [16], the scenario structure in SAA is defined as follows:

• 100 scenarios for the short simulation,
• 1000 scenarios for the long simulation,
• number of replications: M = 10

Accordingly, each test problem is solved through 10 independent SAA replications, and each candidate solution is subsequently evaluated over 1000 scenarios. This design provides tight statistical confidence bounds and ensures consistent sample-based convergence of the numerical results.

7.3. Simulation Results

All experiments were implemented in C# using a Ryzen 5 3600x processor with 32 GB RAM. Each SAA run involves solving either SAASA or SAACS on a set of sampled scenarios, followed by evaluation of the best solution on 1000 scenarios. In contrast to the approach taken by Şafak [20], where scenarios from the literature were grouped and tested, this study distinguishes itself by testing a significantly larger number of scenarios using the proposed algorithms. This approach enables faster and more efficient results. The problems are evaluated under four OB configurations:

	Penalty	Show-Up Probability
Category 1	2× fare	0.85
Category 2	2× fare	0.90
Category 3	3× fare	0.85
Category 4	3× fare	0.90

Table 11. The dimensions of the test problems.

		100 Scenarios		200 Scenarios
The Problems	Number of Flights	Number of Constraints	Number of Variables	Number of Constraints	Number of Variables
Problem 1	30	113,892	75,777	219,692	150,977
Problem 2	78	296,640	196,707	572,240	391,907
Problem 3	127	483,361	320,158	932,461	637,858
Problem 4	180	684,750	453,681	1,320,950	903,881

summarizes the dimensional growth of the four test problems under two scenario sizes (100 and 200). As expected, the number of constraints and variables increases sharply with both the number of flights and the number of scenarios, reflecting the combinatorial expansion inherent in large-scale stochastic FSFAAR models. For example, Problem 4, which includes 180 flights, results in more than 684,000 constraints under 100 scenarios and exceeds 1.3 million constraints when the scenario count doubles to 200. This substantial increase highlights the computational challenges of the integrated TSSMINLP structure and further justifies the need for scalable solution approaches such as the SAA–metaheuristic framework.

Table 12. Results for Problem 1.

	PC	Show-Up Rate	SAA	Best Lower Bound ($)	Estimated Optimality Gap (%)	Std. dev. Gap	% 95 Confidence Interval		Time (s)
							Min (%)	Max (%)
Category 1	2	0.85	SAA+SA	377,330.380	0.088	2428.66	−1.173	1.350	894.69
	2	0.85	SAA+CS	373,523.829	0.249	2515.92	−1.071	1.569	886.291
Category 2	2	0.90	SAA+SA	332,689.427	0.046	2637.13	−1.508	1.599	892.246
	2	0.90	SAA+CS	329,464.881	0.091	2846.09	−1.603	1.783	850.465
Category 3	3	0.85	SAA+SA	364,901.041	0.094	1859.85	−0.905	1.093	884.278
	3	0.85	SAA+CS	360,295.071	0.027	2613.51	−1.395	1.448	869.521
Category 4	3	0.90	SAA+SA	325,770.292	0.063	2156.35	−1.234	1.361	882.229
	3	0.90	SAA+CS	324,593.973	0.058	2465.17	−1.430	1.547	871.121

,

Table 13. Results for Problem 2.

	PC	Show-Up Rate	SAA	Best Lower Bound ($)	Estimated Optimality Gap (%)	Std. dev. Gap	% 95 Confidence Interval		Time (s)
							Min (%)	Max (%)
Category 1	2	0.85	SAA+SA	1,142,337.351	0.054	4649.46	−0.741	0.852	7600.32
	2	0.85	SAA+CS	1,104,341.219	0.101	5873.21	−0.941	1.143	7798.08
Category 2	2	0.90	SAA+SA	1,010,120.164	0.214	5621.37	−0.876	1.306	7499.973
	2	0.90	SAA+CS	974,464.604	0.304	7336.41	−1.172	1.779	7662.697
Category 3	3	0.85	SAA+SA	1,107,969.164	0.147	3940.56	−0.549	0.844	7992.669
	3	0.85	SAA+CS	1,072,920.256	0.121	6683.15	−1.099	1.342	7833.136
Category 4	3	0.90	SAA+SA	983,710.728	0.243	5593.90	−0.872	1.357	7372.049
	3	0.90	SAA+CS	962,580.317	0.289	6439.30	−1.022	1.600	7556.511

,

Table 14. Results for Problem 3.

	PC	Show-Up Rate	SAA	Best Lower Bound ($)	Estimated Optimality Gap (%)	Std. dev. Gap	% 95 Confidence Interval		Time (s)
							Min (%)	Max (%)
Category 1	2	0.85	SAA+SA	1,920,759.461	0.010	5660.61	−0.567	0.587	20,732.694
	2	0.85	SAA+CS	1,849,816.507	0.096	9072.74	−0.865	1.057	20,775.850
Category 2	2	0.90	SAA+SA	1,673,729.663	0.26	10,833.94	−1.143	1.395	20,090.363
	2	0.90	SAA+CS	1,611,657.152	0.221	6875.56	−0.615	1.057	20,971.94
Category 3	3	0.85	SAA+SA	1,843,316.107	0.419	10,856.77	0.735	1.573	20,676.589
	3	0.85	SAA+CS	1,792,796.455	0.288	12,453.91	−1.074	1.649	21,806.272
Category 4	3	0.90	SAA+SA	1,626,975.738	0.085	6871.67	−0.743	0.913	20,819.798
	3	0.90	SAA+CS	1,554,066.362	0.052	6017.18	−0.707	0.811	20,485.929

and

Table 15. Results for Problem 4.

	PC	Show-Up Rate	SAA	Best Lower Bound ($)	Estimated Optimality Gap (%)	Std. dev. Gap	% 95 Confidence Interval		Time (s)
							Min (%)	Max (%)
Category 1	2	0.85	SAA+SA	2,598,480.442	0.005	8314.20	−0.622	0.632	39,640.094
	2	0.85	SAA+CS	2,515,692.059	0.286	7800.78	−0.322	0.894	46,750.994
Category 2	2	0.90	SAA+SA	2,314,223.493	0.032	12,172.71	−0.999	1.063	39,546.174
	2	0.90	SAA+CS	2,216,409.876	0.042	5853.98	−0.475	0.560	47,756.565
Category 3	3	0.85	SAA+SA	2,553,863.265	0.217	8193.48	−0.412	0.846	39,937.571
	3	0.85	SAA+CS	2,442,852.031	0.116	8638.33	−0.577	0.809	47,707.816
Category 4	3	0.90	SAA+SA	2,204,113.869	0.295	8480.63	−0.459	1.050	40,608.201
	3	0.90	SAA+CS	2,145,089.892	0.132	8638.33	−0.657	0.921	47,707.816

report, for each test instance, the best lower bound, the corresponding optimality gap, the standard deviation of this gap, and the associated 95% confidence interval. Across all experimental settings, the confidence intervals remain below 2%, and the optimality gaps converge to values very close to zero. These results collectively demonstrate that the SAA procedure yields highly stable and statistically robust estimators, confirming the reliability of the solutions obtained for all problem sizes.

Algorithmic Comparison (SAASA vs. SAACS).

The comparative analysis across all instances demonstrates a consistent dominance of the SAASA algorithm. SAASA yields the highest-quality solutions with markedly lower variance, indicating a more stable search trajectory. In contrast, SAACS exhibits slightly inferior performance and greater dispersion in objective values, reflecting the stochastic fluctuations inherent in its population-based mechanism.

As instance size increases, total profit rises accordingly, while SAASA’s computational time grows in an almost linear manner, confirming its scalability. To ensure methodological fairness, both algorithms were executed under comparable iteration and evaluation budgets.

These results collectively indicate that SA-based neighborhood exploration is particularly effective for the FSFAAR–SP–CA–OB problem class, offering a strong balance between computational efficiency and solution robustness.

7.4. Operational Insights: Effects of SP, CA, and OB

Effect of Station Purity (SP):

SP imposes fleet–station compatibility restrictions, and the results consistently show that:

• SP reduces routing flexibility, limiting feasible fleet assignments.
• Profit decreases slightly due to these operational limitations.
• The model relies more heavily on CA to compensate for restricted carrier capacity.

Effect of Codeshare Agreements (CA):

CA emerges as a critical flexibility mechanism within the integrated framework:

• It provides supplementary capacity when SP restricts fleet deployment.
• It reduces unnecessary aircraft repositioning and supports more efficient route utilization.
• CA decreases spilled passengers, especially on high-demand legs.
• Incorporating CA consistently results in higher and more stable profitability across scenarios.

Effect of Overbooking (OB):

Across the four OB configurations analyzed in Figure 4a–d, the following patterns emerge:

• Higher DB penalties lead to more conservative OB behavior and thus lower expected profit.
• Higher show-up probabilities result in reduced OB levels, increased spill risk, and lower revenue.
• OB decisions remain statistically disciplined, avoiding excessive DB due to penalty structure.
• CA strengthens OB performance by helping absorb excess demand, thereby reducing both spill and DB outcomes.

Integrated Insight:

The joint analysis of SP, CA, and OB indicates that:

• SP introduces operational rigidity,
• CA restores flexibility, and
• OB enhances revenue potential under uncertainty.

When combined, CA and OB effectively mitigate the negative operational impacts of SP, demonstrating the value of modeling these elements in a unified optimization framework.

7.5. General Findings and Practical Implications

The computational study leads to several practical conclusions:

• The integrated FSFAAR–SP–CA–OB model is solvable for real-sized datasets using the hybrid SAA–metaheuristic approach.
• SAASA consistently yields superior performance relative to SAACS.
• Interactions among SP, CA, and OB significantly shape fleet assignment, routing, and capacity decisions.
• Testing over 10,000 scenarios provides robust statistical validation for operational use.
• The resulting framework can serve as a decision-support tool for airline planners in fleet allocation, schedule reliability assessment, codeshare capacity design, and overbooking policy optimization under uncertainty.

7.6. Managerial Insights and Sensitivity Discussion

Although a comprehensive quantitative sensitivity analysis is beyond the scope of this study, the solution patterns observed across multiple problem instances enable meaningful sensitivity interpretations with respect to key operational parameters of the proposed framework.

More specifically, an increase in denied boarding penalties systematically encourages more conservative overbooking decisions, reflecting a deliberate trade-off between spill risk mitigation and increased capacity slack. Similarly, higher passenger show-up probabilities effectively tighten the feasible region for aggressive overbooking strategies, thereby increasing the importance of downstream aircraft routing feasibility and codeshare flexibility.

Conversely, stricter codeshare budget limits reduce the airline’s ability to offset station purity constraints, particularly under high-demand conditions. This effect further highlights the interdependent nature of station purity rigidity, overbooking policies, routing flexibility, and codeshare agreements, underscoring the necessity of jointly calibrating these parameters within an integrated decision-making framework.

Since these observations are derived from consistent solution trends across different demand and delay regimes, the findings are not specific to a single problem instance. Accordingly, the resulting managerial insights are expected to be informative for airline networks with similar operational characteristics.

8. Conclusions and Future Work

Airline operations require the coordinated integration of fleet assignment, flight scheduling, and aircraft routing, and treating these decisions independently often results in operational inefficiencies and suboptimal economic outcomes. This study introduces an integrated three-stage stochastic MINLP framework that jointly optimizes these interdependent decisions while explicitly incorporating three operational features—codeshare agreements (CA), station purity (SP), and overbooking (OB)—that have rarely been modeled simultaneously. By representing passenger demand and non-cruise times (NCT) as stochastic variables, the framework captures key sources of real-world uncertainty that shape tactical planning and day-of-operations performance.

A major methodological contribution of this work is the unified treatment of CA, SP, and OB within the FS–FA–AR decision structure. These components interact in complex and mutually reinforcing ways: CA provides supplementary capacity that mitigates the routing rigidity introduced by SP; SP enforces station-specific operational feasibility; and OB strategically enhances revenue under uncertain passenger show-up behavior. Integrating these mechanisms within a single stochastic optimization model results in a more realistic and operationally grounded representation than those in existing literature.

Algorithmically, the combination of Sample Average Approximation (SAA) with the SAASA and SAACS metaheuristics offers a scalable, statistically validated, and computationally feasible solution strategy for large-scale stochastic problems. While SAA provides reliable empirical bounds and confidence intervals, the SAASA metaheuristic consistently outperforms SAACS in both stability and solution quality, demonstrating its suitability for handling the nonlinear and combinatorial structure of integrated airline planning.

Despite these strengths, several limitations delineate the study’s scope. Passenger demand and show-up rates are modeled using assumed probability distributions due to the lack of fine-grained historical booking data, although these assumptions align with standard practices in airline simulation studies. A full sensitivity analysis of parameters such as codeshare capacities and station purity constraints was not conducted, as repeatedly solving large stochastic models incurs substantial computational cost; such analyses represent a promising direction for future research, particularly with access to expanded computing resources. Finally, although multiple large-scale cases were generated from real operational data, the evaluation is limited to a single airline network. Applying the model to additional carriers would enhance external validity and provide deeper insights into its adaptability across different operational environments.

Future extensions of this work could incorporate dynamic demand forecasting or behavioral models to further refine OB-related revenue decisions. Enhancing the revenue management structure—through additional fare classes or ancillary service modeling—would enrich the economic realism of the formulation. From a methodological perspective, adaptive SA strategies, hybrid CS mechanisms, or machine-learning-assisted metaheuristics could further improve computational efficiency. Integrating additional planning components such as crew scheduling, maintenance routing, or disruption management would contribute to a more comprehensive end-to-end airline operations optimization platform.

In conclusion, this study offers both theoretical and practical contributions by providing a robust, scalable, and operationally meaningful integrated optimization framework. The results demonstrate that the proposed model can effectively support complex airline decision environments, improving schedule reliability, enhancing resource utilization, and strengthening profitability under uncertainty.