Genetic Algorithm and Mathematical Modelling for Integrated Schedule Design and Fleet Assignment at a Mega-Hub

Abstract

Airline networks are becoming increasingly complex, particularly at mega-hub airports characterized by high transit volumes. Effective schedule design and fleet assignment are critical for an airline, as they directly influence passenger connectivity and profitability. This study addresses the challenge of introducing a new route from a mega-hub to a new destination, while maintaining the existing flight network and leveraging arrivals from spoke airports to ensure connectivity. First, a mixed-integer nonlinear mathematical model was formulated to produce a global optimal solution at a lower time granularity, but it became computationally intractable at higher granularities due to the exponential growth in constraints and variables. Second, a genetic algorithm (GA) was employed to demonstrate scalability and flexibility, delivering near-optimal, high-granularity schedules with significantly reduced computational time. Empirical validation using real-world data from 37 spoke airports revealed that, while the exact model minimized waiting times and maximized profit at lower granularity, the GA provided nearly comparable profit at higher granularity. These findings guide airline managers seeking to optimize passenger connectivity and cost efficiency in competitive global markets.

1. Introduction

In an increasingly competitive global airline market, efficient schedule design and fleet assignment are vital for profitability and meeting passenger demand. Schedule design encompasses both schedule generation and flight frequency determination and affects subsequent decisions concerning resource allocation, cost control, and overall service quality [1]. Fleet assignment ensures that the selected routes are served by the most suitable aircraft types, balancing supply–demand interactions and considering operating expenses [2]. This task is of paramount importance, as effective aircraft utilization directly influences operational performance and cost structure. Consequently, the strategic decisions made in these phases are interrelated, ultimately determining the airline’s ability to remain competitive in a complex environment.

Traditionally, schedule design and fleet assignment problems are solved sequentially; however, this approach often leads to suboptimal solutions by neglecting interdependencies [3]. To overcome the suboptimality problem, integrated schedule design and fleet assignment (ISDFA) has been widely used in the literature as a comprehensive solution that considers flight frequencies, departure times, and aircraft types in an integrated approach to maximize overall system performance [4]. Additionally, ISDFA is often integrated with network optimization to construct a comprehensive framework that captures spatial, temporal, and operational aspects to enhance connectivity, resilience, and profitability in a hub-and-spoke (HS) network [5,6]. However, this integration increases the complexity of the problem due to the high degree of interdependencies between scheduling and fleet allocation [7].

Mega-hub airports concentrate a large volume of transit passengers, often accounting for more than 90% of total traffic, and rely on wave-based scheduling to minimize transfer times and coordinate resources [8,9]. Although this approach supports efficient transfers and economies of scale, it complicates the balance between arrival–departure waves, aircraft rotation, and gate usage [10]. The existing literature frequently focuses on daily planning horizons or incremental schedule design with fixed flight frequencies [3,11], leaving only a few studies addressing comprehensive timetable generation for new outbound routes under fixed inbound schedules [3,12,13,14]. Because wave-based scheduling typically targets a single optimized time window [12], establishing consistent outbound flights aligned with predetermined arrivals requires more specialized approaches to ensure passenger satisfaction and long-term fleet efficiency.

Conventional methods often assume static passenger flows by fixing spill rates and flight frequencies [15], overlooking travelers’ responsiveness to flight departure times and aircraft capacities. Additionally, most scheduling studies employ daily horizons or fixed schedules, limiting insight into the complexities of multi-day dynamics in mega-hub contexts [16]. Only a few publications consider passenger assignment and flight frequency over multiple days [13,14]; this study aims to fill the gap by (1) addressing the challenge of scheduling a new outbound route from a mega-hub to a new destination, while keeping the existing inbound network and capturing transit passengers over a weekly planning horizon, and (2) focusing on optimizing airline profits and passenger waiting times simultaneously for the new outbound route. Specifically, this study explores how an airline can optimize flight frequencies, departure times, and aircraft assignment for a new route from its mega-hub to capture maximum transit passengers and reduce transfer waiting times.

This study proposes a weekly, integrated optimization framework that simultaneously determines flight frequencies, departure times, and fleet assignment, explicitly incorporating weekly passenger flows, spill and recapture dynamics, and passenger reassignment across multiple days. This research thus extends prior works that rely on predetermined frequencies, daily planning horizons, or simplified assumptions about passenger behavior. Our contributions are (1) a new mixed-integer nonlinear programming (MINLP) model for ISDFA incorporating flight frequency decisions that capture the complexities of weekly scheduling and transfer coordination in a specific mega-hub, (2) a GA to solve larger-scale ISDFA problems, which are harder to solve with MINLP models due to computational complexity, and (3) a real-world application demonstrating how our approach supports network expansion and strengthens an airline’s competitive position in new markets. The employed GA features a dynamic gene structure that captures weekly passenger flow, spill, and carryover logic, offering a scalable solution for practitioners seeking more efficient and passenger-centric airline scheduling.

The remainder of this paper is structured as follows. Section 2 reviews the relevant literature. Section 3 presents the methodology. Section 4 describes the empirical study. Section 5 presents the results. Section 6 discusses the results, and Section 7 concludes the paper.

2. Literature Review

2.1. Airline Schedule Planning Process (ASPP)

The Airline Schedule Planning Process (ASPP) typically encompasses four main stages, as follows: schedule design (flight frequency and timetable generation), fleet assignment (allocating aircraft types to flight legs), and aircraft routing (ensuring routes comply with maintenance requirements) [5]. ASPP is particularly complex due to diverse operational constraints and regulatory considerations at each stage [3,17]. These sub-problems are widely recognized as NP-hard (non-deterministic polynomial-time hard), given their distinct planning horizons and tightly interrelated decisions [18]. This study focuses on the integrated schedule ISDFA problem, incorporating flight frequency, which aims to determine which markets to serve, at what frequencies, how to schedule flights accordingly, and which aircraft types to assign. It is typically modeled within a single-objective framework focused on minimizing costs or maximizing revenue [14,19].

There are two primary approaches to generating a flight timetable in schedule design—incremental and comprehensive. Incremental schedule design involves modifying an existing timetable by adding, removing, or shifting flight legs to improve profitability; whereas, comprehensive schedule design creates a new timetable from scratch [7]. The literature mainly focuses on solution methods for incremental schedule design, with limited attention given to comprehensive planning [11]. Several factors contribute to this emphasis on incremental approaches. Most studies prioritize increasing flight frequency on existing routes rather than introducing new destinations [20]. In practice, major airlines often make only minor adjustments to their schedules due to managerial and operational constraints [11]. Their primary objective is to enhance the efficiency and connectivity of existing routes rather than to expand the network [12]. As a result, the literature is mainly focused on optimizing current operations rather than modeling new markets [3]. Additionally, incremental schedule design is inherently unsuitable for entering new markets, as there are no pre-existing flights or historical schedules to modify.

2.2. Strategic Considerations in Mega-Hub Airport Scheduling

The 1978 U.S. airline deregulation authorized carriers’ autonomy over fares, routes, and market entry [15], prompting a shift from point-to-point to hub-and-spoke networks [21]. HS networks link connecting spokes to a central hub, enhancing economies of scale and flight frequency and reinforcing monopolistic advantages at strategic locations [22,23]. However, by the early 2000s, long cycle times at hubs reduced aircraft utilization and performance [23]. In response, mega-hub airports emerged, prioritizing transit passengers over origin–destination traffic [9] and gaining attention for improving transit efficiency and competitiveness [8]. Research on mega-hub networks can be categorized into spatial factors (geographical and structural layout) and temporal factors (airline–airport interconnectivity) [24]. This study specifically focuses on temporal connectivity, emphasizing the role of schedule coordination in enhancing passenger transfers within a mega-hub structure.

Mega-hub network scheduling requires a distinct approach from traditional methods due to the system’s high complexity and strong interdependencies between flights [25]. Conventional scheduling models, generally designed for smaller or less congested networks, are insufficient to handle the arrival/departure patterns and waveform behaviors characteristic of mega-hubs [26]. Moreover, the large volume of flights in such hubs enhances the impact of even minor misalignments in static schedules, often leading to cascading delays. This dynamic necessitates advanced scheduling strategies that incorporate runway capacity, slot coordination, and peak congestion management [27]. Minimizing connection times inherently generates noteworthy schedule peaks, which impose additional resource demands and operational burdens [28]. Consequently, schedule optimization for mega-hubs must extend beyond profit maximization, emphasizing arrival and departure synchronization and overall transit passenger satisfaction.

2.3. ISDFA Problems

Balancing the interaction between supply and demand is a crucial concern in airline scheduling. Spill and recapture dynamics describe how excess passenger demand, constrained by seat capacity, may be partially recovered by assigning “spilled” passengers to alternative itineraries [29]. In static scheduling environments, this mechanism affects essential planning decisions, such as flight frequency, capacity allocation, and network design [6,30]. A fixed spill rate approach is commonly used in the literature, assuming a predefined demand that spills and is then partially recaptured [15]. Although recent studies have incorporated spill and recapture logic for more realistic demand modeling [15,16], they do not fully integrate these features with passenger flow dynamics or frequency planning. In contrast, this study optimizes spilled and recaptured passengers simultaneously, without fixed parameters, and introduces a dispatching mechanism that allows travelers arriving from the same origin to take different connecting flights, thereby capturing behavioral heterogeneity more accurately. To further capture this behavior, discrete choice models are employed to represent passenger preferences among competing itineraries based on attributes such as fares, schedule convenience, and connection times in the literature [31].

While several studies have integrated schedule and fleet assignment decisions, flight frequency is frequently treated as a fixed input derived from demand forecasts or operational norms [11,32]. Specifically, some models adopt a parametric approach where frequency is predefined to estimate market share [33], while others incorporate fixed competitor frequencies in MILP formulations [34]. Among the surveyed literature, only one research paper considers flight frequency as a decision variable, jointly optimizing it with fleet type over a weekly horizon, yet without specifying departure times or allowing multi-day passenger transfers [14]. This method further presumes complete demand satisfaction within each daily shift, excluding spill and recapture dynamics. In contrast, this study jointly optimizes flight frequency and departure times, enabling passengers to be carried over to subsequent days and thus providing both system flexibility and a more realistic representation of demand.

Connectivity is of paramount importance in mega-hub networks, where transfer rates serve as key performance indicators [8]. Recent research largely focuses on enhancing connectivity at major hubs. One study introduces a binary integer programming model combined with simulated annealing and tabu search to reduce transfer waiting times at Istanbul Airport [35], while another proposes a nonlinear model that maximizes connections and minimizes slot disruptions [8]. MINLP is introduced, incorporating heuristic techniques to handle waiting times, seat capacities, buffer intervals, and ground operation constraints, explicitly modeling passenger assignments [36]. However, these studies generally adopt incremental strategies based on existing flight networks and do not address the integration of new routes. Among them, only one study explicitly models passenger assignments [36]; whereas, the others focus primarily on departure schedules or aircraft utilization [8,35].

A limited number of studies address the integration of new routes into mega-hub structures. A speed-constrained multi-objective particle swarm optimization algorithm was applied to optimize route-level decisions at a mega-hub [13], yet detailed flight timing, transfer feasibility, and frequency optimization were not addressed. Similarly, a weekly MILP was introduced to optimize frequencies and fleet types; although, passenger transfer logic was excluded [14]. A Benders decomposition method was utilized to revise schedules for new market opportunities, with the focus primarily on profitability and existing operations rather than mega-hub connectivity [37]. Consequently, these methodologies do not provide a comprehensive integration of passenger flows and scheduling considerations for new route development, emphasizing the necessity of a more comprehensive framework that incorporates connectivity and demand considerations in mega-hub environments.

A summary of key studies, including demand modeling, scheduling approaches, planning horizons, flight frequency decisions, new route integration, and mega-hub focus, is provided in Table 1.

Table 1. Summary of ISDFA problems.

Article	Demand	Schedule	Planning Horizon	Flight Frequency Decision	New Route	Mega-Hub Focus	Model	Algorithm
[34]	S/R 1	Comp.	Week	✓			MILP	Branch and Bound
[37]	S/R 1	Inc.	Day		✓			Benders decomposition
[38]	DC 2	Inc.	Day				BLMM	GA, Simulation
[39]	DC 2	Inc.	Day					CG
[13]		Inc.	Week		✓	✓		Particle Swarm
[11]	DC 2	Comp.	Day				MILP	Branch and Bound
[16]	S/R 1	Inc.	Day					GA
[35]		Inc.	Day			✓	BIP	SA, Tabu Search
[33]	DC 2	Inc.	Day	✓			MINLP	Branch and Bound
[40]	DC 2	Comp.	Day		✓		MINLP	Tabu Search
[8]		Inc.	Day			✓	MILP
[41]	DC 2	Inc.	Day					DA
[32]	S/R 1	Inc.	Day	✓			MILP	CG, VNS
[14]		Comp.	Week	✓	✓	✓	MILP
[36]	S/R 1	Inc.	Day			✓	MINLP	Rule-Based Heuristic
[15]	S/R 1	Comp.	Day					GA
This study	S/R 1	Comp.	Week	✓	✓	✓	MINLP	GA

¹ S/R = spill and recapture; ² DC = discrete choice; Inc. = incremental; Comp. = comprehensive; MILP = mixed-integer linear programming; MINLP = mixed-integer nonlinear programming; BLMM = bi-level mathematical model, BIP = binary integer programming; GA = genetic algorithm; DA = decomposition approach; SA = simulated annealing; VNS = variable neighborhood search; CG = column generation.

Table 1. Summary of ISDFA problems.

Article	Demand	Schedule	Planning Horizon	Flight Frequency Decision	New Route	Mega-Hub Focus	Model	Algorithm
[34]	S/R 1	Comp.	Week	✓			MILP	Branch and Bound
[37]	S/R 1	Inc.	Day		✓			Benders decomposition
[38]	DC 2	Inc.	Day				BLMM	GA, Simulation
[39]	DC 2	Inc.	Day					CG
[13]		Inc.	Week		✓	✓		Particle Swarm
[11]	DC 2	Comp.	Day				MILP	Branch and Bound
[16]	S/R 1	Inc.	Day					GA
[35]		Inc.	Day			✓	BIP	SA, Tabu Search
[33]	DC 2	Inc.	Day	✓			MINLP	Branch and Bound
[40]	DC 2	Comp.	Day		✓		MINLP	Tabu Search
[8]		Inc.	Day			✓	MILP
[41]	DC 2	Inc.	Day					DA
[32]	S/R 1	Inc.	Day	✓			MILP	CG, VNS
[14]		Comp.	Week	✓	✓	✓	MILP
[36]	S/R 1	Inc.	Day			✓	MINLP	Rule-Based Heuristic
[15]	S/R 1	Comp.	Day					GA
This study	S/R 1	Comp.	Week	✓	✓	✓	MINLP	GA

¹ S/R = spill and recapture; ² DC = discrete choice; Inc. = incremental; Comp. = comprehensive; MILP = mixed-integer linear programming; MINLP = mixed-integer nonlinear programming; BLMM = bi-level mathematical model, BIP = binary integer programming; GA = genetic algorithm; DA = decomposition approach; SA = simulated annealing; VNS = variable neighborhood search; CG = column generation.

2.4. Gap and Contribution

Despite extensive research on ISDFA, most existing studies treat flight frequency and departure time independently rather than as joint decision variables. Flight frequencies are often predetermined or embedded within network planning models, while departure times are typically restricted to discrete choices or incrementally adjusted within fixed windows. Furthermore, many models rely on a daily planning horizon, thereby preventing the reassignment of unserved passengers to subsequent flights and rarely incorporating spill and recapture dynamics. Only a limited number of studies address new route integration within a mega-hub network. To address these gaps, this study proposes solution approaches for a specialized ISDFA problem that enables airlines to determine and evaluate new route decisions from a single mega-hub to a new destination, without changing the existing network. This study makes four distinct contributions to the ISDFA literature. First, the framework simultaneously optimizes flight frequency, passenger assignment, aircraft capacity, and departure times, while leveraging inbound arrivals from established spokes. Second, spill and recapture dynamics are fully embedded, allowing passengers to be reassigned within the next day based on capacity and waiting-time constraints rather than fixed parameters. Third, the GA generates weekly, feasible timetables and aircraft assignments for outbound mega-hub flights, using a chromosome structure that encodes both flight schedules and passenger flows. Fourth, it validates the entire framework on real-world mega-hub data from 37 spoke airports, demonstrating practical applicability in the real world.

3. Methodology

3.1. Problem Description

This study examines an airline’s decision-making process for designing a weekly flight schedule from a single mega-hub to a newly introduced destination. All flights in the model depart from this hub and arrive at that same destination, and all passengers come from multiple spokes to this hub but share the same final destination. Since these spoke hub flows are treated independently, the model only handles an aggregate of passenger arrivals at the hub rather than explicit spoke-to-hub transfers or spoke-to-spoke connections.

Passenger arrivals at the hub are aggregated by their arrival time rather than by each passenger’s specific spoke of origin. The model categorizes passengers as assigned, carryover, or spilled, reflecting whether they are successfully assigned to a flight on the same day, assigned the next day, or not assigned at all. The parameter Q marks the start of the evening phase. Passengers arriving after Q may still be assigned to an evening flight if seats are available, but if they remain unassigned at the end of the day, they roll over to the next day as carryover. Spillage, however, can occur at any point, before or after Q, if there is no suitable flight for a given passenger. If a passenger who carries over to the next day also cannot be assigned, they ultimately spill, generating lost revenue for the airline.

The objective is to maximize total revenue by balancing operating costs, passenger fares, spilled passenger costs, and passenger waiting time costs. Because there are no existing flights on this new route, the model constructs a comprehensive schedule from scratch by determining the number of flights, their departure times, aircraft types, and passenger allocations. The problem has several assumptions, which are given as follows:

• All flights depart from a single mega-hub and arrive at the same newly introduced destination.
• The network is simplified to exclude spoke-to-spoke connections, so only point-to-point service from the hub to the new destination is considered.
• Demand is aggregated at the hub rather than differentiated by origin–destination pairs, because passengers from multiple spokes share the same final destination.
• Passengers are allowed to be carried to the following day if they cannot be assigned to a flight on the day of their arrival at the hub.
• Any passenger remaining unassigned after the next day can be spilled permanently, incurring a spill cost for the airline.
• Each day is divided into three equal phases (morning, noon, evening), each defining discrete time windows for flight departures.

The division of days into phases reduces complexity by allowing the model to choose from a finite set of discretized departure windows rather than any possible time within a continuous range. This approach is based on the shift concept introduced by [14] but extends that concept by accounting for both daily time intervals (periods) and subdivisions within each day (phases). The approach enables tracking flight frequency.

3.2. Mathematical Model

The comprehensive mathematical model (CMM) formulation is given as follows:

Sets and Indices
G:	Set of all days within a week, g ∈ G = {1, 2, …|G|}
T:	Set of all time periods within a day, t, p ∈ T = {1, 2, …|T|}
K:	Set of aircraft types, k ∈ K = {1, 2, …|K|}
H:	Set of phases during the day, h ∈ H = {1, 2, …|H|}
I:	Set of waiting time intervals, i ∈ I = {1, 2, …|I|}
Parameters
$\omega$	$\mathrm{Fare} \mathrm{per} \mathrm{passenger}, \omega \in {\mathrm{Z}}^{+}$
α	$\mathrm{Spilled} \mathrm{cost} \mathrm{per} \mathrm{passenger} \mathsf{\alpha }\in {\mathrm{Z}}^{+}$
$C_k$	$\mathrm{Seating} \mathrm{capacity} \mathrm{of} \mathrm{an} \mathrm{aircraft} \mathrm{k}, \mathrm{k} \in \mathrm{K}, {\mathrm{C}}_{\mathrm{k}}\in {\mathrm{Z}}^{+}$
Ok	$\mathrm{Operational} \mathrm{cost} \mathrm{of} \mathrm{an} \mathrm{aircraft} \mathrm{k}, \mathrm{k} \in \mathrm{K}, {\mathrm{O}}_{\mathrm{k}}\in \mathrm{Z}$
TPt, TPp	$\mathrm{Time} \mathrm{periods} \mathrm{at} \mathrm{time} \mathrm{t}, \mathrm{p} \in \mathrm{T}, {TP}_t,{TP}_p\in {\mathrm{Z}}^{+}$
Dt	$\mathrm{Total} \mathrm{demand} \mathrm{of} \mathrm{passenger}\left(\mathrm{s}\right) \mathrm{at} \mathrm{time} \mathrm{t} \in \mathrm{T}, {\mathrm{D}}_{\mathrm{t}}\in {\mathrm{Z}}^{+}$
STh, Eh	$\mathrm{Start} \mathrm{and} \mathrm{end} \mathrm{time} \mathrm{for} \mathrm{phase} \mathrm{h} \in \mathrm{H}, {\mathrm{S}\mathrm{T}}_{\mathrm{h}},{\mathrm{E}}_{\mathrm{h}}\in {\mathrm{Z}}^{+}$
DSg, DEg	$\mathrm{Start} \mathrm{and} \mathrm{end} \mathrm{time} \mathrm{of} \mathrm{day} \mathrm{g}, \mathrm{g} \in \mathrm{G}, {\mathrm{D}\mathrm{S}}_{\mathrm{g}},{\mathrm{D}\mathrm{E}}_{\mathrm{g}}\in {\mathrm{Z}}^{+}$
$T_{max}$	$\mathrm{Maximum} \mathrm{allowable} \mathrm{waiting} \mathrm{time}, T_{max}\in {\mathrm{Z}}^{+}$
ci	$\mathrm{Cost} \mathrm{parameter} \mathrm{for} \mathrm{waiting} \mathrm{time} \mathrm{interval} \mathrm{i}, \mathrm{i} \in \mathrm{I}, {\mathrm{c}}_{\mathrm{i}}\in {\mathrm{Z}}^{+}$
r	$\mathrm{Ready} \mathrm{time} (\mathrm{threshold} \mathrm{for} \mathrm{assignment}), \mathrm{r} \in Z^{+}$
QDg	$\mathrm{Transfer} \mathrm{threshold} \mathrm{time} \mathrm{at} \mathrm{day} \mathrm{g}, \mathrm{g} \in \mathrm{G}, {\mathrm{Q}\mathrm{D}}_{\mathrm{g}}\in {\mathrm{Z}}^{+}$
At	1; if a passenger(s) arrive at time t, 0; otherwise, t ∈ T
$B_0,B_1,B_2$	$\mathrm{Waiting} \mathrm{time} \mathrm{threshold} \mathrm{values}, {\mathrm{B}}_0,{\mathrm{B}}_1,{\mathrm{B}}_2\in {\mathrm{Z}}^{+}$
M	A big number
Decision Variables
${CO}_{g,g+1}$	Total carryover passengers from day g to day g + 1
${{\mathrm{S}\mathrm{P}}_{\mathrm{g}}^{\mathrm{A}\mathrm{Q}},\mathrm{S}\mathrm{P}}_{\mathrm{g}}^{\mathrm{B}\mathrm{Q}}$	Total spilled passengers before and after threshold Q
${AP}_g$	Total assigned passengers on day g
$s_{t,p}$	Total assigned passengers from time t to the flight at time p
$u_t$	Total spilled passengers at time t
$y_{t,k,h}$	1; if flight scheduled at time t (or departure time p) with aircraft k in phase h, 0; otherwise
Xh	Flight departure time for phase h
$w_{t,p},{wp}_{t,p}$	The waiting time and cost for a passenger from time t to the departure flight at time p
${rrw}_{t,p},{rr}_{t,p}$	Total waiting time and cost for the assigned passengers from time t to the departure flight at time p
TW	The total waiting time for all the assigned passengers
S, U	The total number of assigned and spilled passengers during the week
$m_{t,p}$	$1; \mathrm{if} \mathrm{p}-\mathrm{t} \le T_{max}$, 0; otherwise
$n_{t,p}$	1; if p − t > r, 0; otherwise
$v_{t,p}^0$	$1; \mathrm{if}{w}_{t,p}\le B_0$, 0; otherwise
$v_{t,p}^1$	$1; \mathrm{if}{B_0<w}_{t,p}\le B_1,0$; otherwise
$v_{t,p}^2$	$1; \mathrm{if} {B_1<w}_{t,p}\le B_2$, 0; otherwise
$v_{t,p}^3$	$1; \mathrm{if} {B_2<w}_{t,p}\le T_{max}$, 0; otherwise

$$\mathrm{Maximize} \omega {\sum }_{t\in T}{\sum }_{p\in T}{s}_{t,p}-\alpha {\sum }_{tϵT}{u}_t-{\sum }_{tϵT}{\sum }_{kϵK}{\sum }_{hϵH}{O}_k{y}_{t,k,h}-{\sum }_{t\in T}{\sum }_{p\in T}{rr}_{t,p}$$

(1)

Subject to,

$${\sum }_{p\in T|t<p}{s}_{t,p}+u_t=D_t\forall t\in T$$

(2)

$${\sum }_{kϵK}{\sum }_{t={ST}_h}^{E_h}{y}_{t,k,h}\le 1\forall h\in H$$

(3)

$${\sum }_{kϵK}{\sum }_{t=1}^{{ST}_h-1}{y}_{t,k,h}=0\forall h\in H$$

(4)

$${\sum }_{kϵK}{\sum }_{t=E_h+1}^{\left|T\right|}{y}_{t,k,h}=0\forall h\in H$$

(5)

$${\sum }_{t\in T}{s}_{t,p|t<p}\le {\sum }_{k\in K}{\sum }_{h\in H}{y}_{p,k,h}{C}_k\forall p\in T$$

(6)

$$s_{t,p}\le D_t{\sum }_{k\in K}{\sum }_{h\in H}{y}_{p,k,h}\forall t,p\in T|t<p$$

(7)

$${\sum }_{k\in K}{\sum }_{h\in H}{y}_{p,k,h}\le {\sum }_{t\in T|t<p}{s}_{t,p}\forall p\in T$$

(8)

$${\sum }_{t\in T}{\sum }_{kϵK}{y}_{t,k,h}{\mathrm{T}\mathrm{P}}_{\mathrm{t}}=X_h\forall h\in H$$

(9)

$$s_{t,p}\le D_t{m}_{t,p}\forall t,p\in T|t<p$$

(10)

$${TP}_p-{TP}_t\le T_{max}+M(1-m_{t,p})\forall t,p\in T|t<p$$

(11)

$$s_{t,p}\le D_t{n}_{t,p}\forall t,p\in T|t<p$$

(12)

$${TP}_p-{TP}_t\ge r-M(1-n_{t,p})\forall t,p\in T|t<p$$

(13)

$$w_{t,p}\le T_{max}\forall t,p\in T|t<p$$

(14)

$$w_{t,p}=0\forall t,p\in T|t\ge p$$

(15)

$$w_{t,p}=\mathrm{m}\mathrm{a}\mathrm{x}(0,{TP}_p-A_t{TP}_t)\forall t,p\in T|t<p,\forall h\in H$$

(16)

$${rrw}_{t,p}{=w}_{t,p}{s}_{t,p}\forall t,p\in T|t<p$$

(17)

$${wp}_{t,p}=c_0{v}_{t,p}^0+c_1{v}_{t,p}^1+c_2{v}_{t,p}^2+c_3{v}_{t,p}^3\forall t,p\in T|t<p$$

(18)

$$1=v_{t,p}^0+v_{t,p}^1+v_{t,p}^2+v_{t,p}^3\forall t,p\in T|t<p$$

(19)

$${rr}_{t,p}{=wp}_{t,p}{s}_{t,p}\forall t,p\in T|t<p$$

(20)

$${\sum }_{t\in {ST}_1}^{E_3}{\sum }_{p\in T}{s}_{t,p}+{SP}_1^{BQ}+{SP}_1^{AQ}={\sum }_{\mathrm{t}\in {ST}_1\dots E_3}{D}_t$$

(21)

$${\sum }_{t\in {ST}_3}^{E_6}{\sum }_{p\in T}{s}_{t,p}+{SP}_2^{BQ}+{SP}_2^{AQ}={\sum }_{\mathrm{t}\in {ST}_4\dots E_6}{D}_t$$

(22)

$${\sum }_{t\in {ST}_6}^{E_9}{\sum }_{p\in T}{s}_{t,p}+{SP}_3^{BQ}+{SP}_3^{AQ}={\sum }_{\mathrm{t}\in {ST}_7\dots E_9}{D}_t$$

(23)

$${\sum }_{t\in {ST}_9}^{E_{12}}{\sum }_{p\in T}{s}_{t,p}+{SP}_4^{BQ}+{SP}_4^{AQ}={\sum }_{\mathrm{t}\in {ST}_{10}\dots E_{12}}{D}_t$$

(24)

$${\sum }_{t\in {ST}_{12}}^{E_{15}}{\sum }_{p\in T}{s}_{t,p}+{SP}_5^{BQ}+{SP}_5^{AQ}={\sum }_{\mathrm{t}\in {ST}_{13}\dots E_{15}}{D}_t$$

(25)

$${\sum }_{t\in {ST}_{15}}^{E_{18}}{\sum }_{p\in T}{s}_{t,p}+{SP}_6^{BQ}+{SP}_6^{AQ}={\sum }_{\mathrm{t}\in {ST}_{16}\dots E_{18}}{D}_t$$

(26)

$${\sum }_{t\in {ST}_{18}}^{E_{21}}{\sum }_{p\in T}{s}_{t,p}+{SP}_7^{BQ}+{SP}_7^{AQ}={\sum }_{\mathrm{t}\in {ST}_{19}\dots E_{21}}{D}_t$$

(27)

$${SP}_g^{BQ}={\sum }_{\mathrm{t}\in {[DS}_g\dots {QD}_g)}{u}_t\forall g\in G$$

(28)

$${SP}_g^{AQ}={\sum }_{\mathrm{t}\in {[QD}_g\dots {DE}_g]}{u}_t\forall g\in G$$

(29)

$${CO}_{g,g+1}={\sum }_{t\in {DS}_g}^{{DE}_g}{\sum }_{t\in {DS}_{g+1}}^{{DE}_{g+1}}{s}_{t,p}\forall g\in D|g<\left|G\right|-1$$

(30)

$${AP}_1={\sum }_{t\in {DS}_1}^{{DE}_1}{\sum }_{t\in {DS}_1}^{{DE}_1}{s}_{t,p}$$

(31)

$${AP}_g={\sum }_{t\in {QD}_{g-1}}^{{DE}_g}{\sum }_{t\in {DS}_g}^{{DE}_g}{s}_{t,p}\forall g\in D|g\ge 2$$

(32)

$${\sum }_{t\in T|t<p}{\sum }_{p\in T}{rrw}_{t,p}=TW$$

(33)

$${\sum }_{t\in T|t<p}{\sum }_{p\in T}{s}_{t,p}=S$$

(34)

$${\sum }_{t\in T}{u}_t=U$$

(35)

$$s_{t,p}{,w}_{t,p}{,wp}_{t,p}{rr}_{t,p},{rrw}_{t,p}\ge 0\forall t,p\in T|t<p$$

(36)

$$m_{t,p},n_{t,p},v_{t,p}^0,v_{t,p}^1,v_{t,p}^2,v_{t,p}^3ϵ\left(\mathrm{0,1}\right)\forall t,p\in T|t<p$$

(37)

$${CO}_{g,g+1},{SP}_g^{BQ},{SP}_g^{AQ},{AP}_g\ge 0\forall g\in G$$

(38)

$$TW,S,U\ge 0$$

(39)

$$X_h\ge 0\forall h\in H$$

(40)

$$u_t\ge 0\forall t\in T$$

(41)

$$y_{t,k,h}ϵ\left(\mathrm{0,1}\right)\forall t\in T,\forall k\in K,\forall h\in H$$

(42)

The mathematical model aims to maximize profit by fares from assigned passengers, while deducting spill, operating, and waiting costs, as shown in Equation (1). The passenger waiting cost is calculated by multiplying the number of assigned passengers by their corresponding waiting time at the hub based on piecewise cost function. Equation (2) ensures that the total passenger demand equals the sum of the assigned and spilled passengers. Equations (3)–(5) specify aircraft opening conditions. Equation (3) ensures that at most one flight can be opened during each phase. Equation (4) prohibits any aircraft from being opened before the designated start time for that phase. Equation (5) prohibits any aircraft from being opened after the designated end time for that phase. Equations (6)–(8) address aircraft capacity and demand conditions. Equation (6) ensures that aircraft capacity is not exceeded when assigning passengers in any phase. Since only one aircraft can be opened per phase, as specified by Equations (3)–(5), it is automatically satisfied for whichever aircraft is selected. Equation (7) prevents passenger assignments after a flight’s departure and links passengers only to scheduled flights. Equation (8) connects flight “opening” to passenger assignments by requiring at least one passenger to be assigned if the flight is opened. Equations (9)–(20) calculate the waiting time and associated costs. Equation (9) returns the departure time of an open flight in phase h. Equations (10) and (11) ensure that passengers are not assigned to flights if their waiting time at the hub exceeds the maximum allowable limit. Equations (12) and (13) permit passenger allocation only after the defined ready time has elapsed. Equation (14) enforces that the waiting time does not exceed its maximum allowable time. Equation (15) keeps the waiting time within valid bounds and prevents negative waiting times if a flight departs before the passenger arrives. Equation (16) calculates the waiting time for passengers arriving at time t in phase h and assigned to a departing flight in the same phase; it returns zero if the passenger’s arrival occurs after the flight’s departure. Equation (17) determines the total wait time for those passengers. Equations (18) and (19) use a piecewise linear cost function to calculate the waiting cost, and Equation (20) sums the waiting cost for passengers assigned to flights that depart at time p and arrive at time t. Equations (21)–(27) govern passenger flow from Day 1 to Day 7, ensuring that arriving passengers are either allocated to flights or spilled. These constraints balance passenger flow so that inflows match outflows daily, and they also account for carryover passengers, assigned passengers, and spilled passengers. Equations (28)–(32) are passenger assignment constraints. Equation (28) calculates the number of spilled passengers arriving before the transfer threshold Q. Equation (29) calculates the number of spilled passengers arriving after Q. Equation (30) computes the number of carryover passengers arriving on Day g and assigned to a flight on Day g+1. Equation (31) calculates the total number of assigned passengers on Day 1, while Equation (32) calculates the total number of assigned passengers on subsequent days. The distinction between the first day and the rest of the weekdays is necessary, because no carryover passengers can exist before Day 1. Equations (33)–(35) define timetable metrics to evaluate solution quality. Equation (33) calculates the total waiting time of assigned passengers across the week. Equation (34) counts the total number of assigned passengers during the week, while Equation (35) determines the total number of spilled passengers during the week. Finally, Equations (36)–(42) are sign constraints.

The proposed CMM is nonlinear; hence, it requires linearization to optimize both the decision variables and the objective under the mega-hub framework. To address this, a new set of indices and decision variables is introduced as follows:

• L: set of indices used for linearizing waiting cost calculations, where l∈L, l: 0…|L|, the size of L corresponds to the base-2 logarithm of the upper limit of the waiting cost, enabling a binary expansion.
• $z_{t,p}$: 1; if $s_{t,p}\ge 1,0;\mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{e}$.
• ${\partial }_{t,p,l}$: $1;\mathrm{i}\mathrm{f} 2^{\mathrm{l}} \mathrm{i}\mathrm{s} \mathrm{t}\mathrm{h}\mathrm{e} \mathrm{p}\mathrm{a}\mathrm{r}\mathrm{t} \mathrm{d}\mathrm{e}\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{o}\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n} \mathrm{o}\mathrm{f} {wp}_{t,p} \mathrm{a}\mathrm{t} \mathrm{t}\mathrm{i}\mathrm{m}\mathrm{e} \mathrm{t}, 0; \mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{e}$.
• $r_{t,p,l}$: an integer variable used in the linearization of the ${wp}_{t,p}{s}_{t,p}$.

Equation (16) calculates the time difference between the proposed flight’s departure and the passenger’s arrival. If this difference is nonnegative, it is used as the waiting time; otherwise, it becomes zero. Its linearization appears in Equations (43)–(47). The parameter A_t serves as an activation indicator that enables the calculation of the passenger waiting time only if arrivals occur at time t. Multiplying by TP_t returns the arrival time, thus preventing unnecessary or invalid computations for periods with no arrivals and maintaining the logical consistency of the waiting time constraint. Equation (20) computes the aggregated waiting cost for assigned passengers by multiplying the number of assigned passengers by the corresponding cost values. Since this waiting cost is discretized, the binary expansion method [42] is applied as shown in Equations (48)–(53). The linearization of Equation (17) follows the same approach by introducing a set LL, so it is not explicitly discussed.

$$w_{t,p}\le {TP}_p-A_t{TP}_t+\left|T\right|(1-z_{t,p})\forall t,p\in T|t<p$$

(43)

$$w_{t,p}\ge {TP}_p-A_t{TP}_t-\left|T\right|(1-z_{t,p})\forall t,p\in T|t<p$$

(44)

$$w_{t,p}\le \left|T\right|z_{t,p}\forall t,p\in T|t<p$$

(45)

$$w_{t,p}\ge 0\forall t,p\in T|t<p$$

(46)

$$s_{t,p}\le D_t{z}_{t,p}\forall t,p\in T|t<p$$

(47)

$${wp}_{t,p}={\sum }_{l=0}^{\left|L\right|}{2}^l{\partial }_{t,p,l}\forall t,p\in T|t<p$$

(48)

$$r_{t,p,l}\le D_t{\partial }_{t,p,l}t,p\in T|t<p,\forall l\in L$$

(49)

$$r_{t,p,l}\le s_{t,p}t,p\in T|t<p,\forall l\in L$$

(50)

$$r_{t,p,l}\ge s_{t,p}-D_t(1-{\partial }_{\mathrm{t},\mathrm{p},\mathrm{l}})t,p\in T|t<p,\forall l\in L$$

(51)

$$r_{t,p,l}\ge 0t,p\in T|t<p,\forall l\in L$$

(52)

$${rr}_{t,p}={\sum }_{l=0}^{\left|L\right|}{2}^l{r}_{t,p,l}\forall t,p\in T|t<p$$

(53)

By applying a binary expansion approach and the big M method to linearize the nonlinear constraints, the proposed CMM becomes solvable as a mixed-integer linear programming model. This ensures that capacity constraints, flight scheduling, and passenger waiting dynamics are consistently represented under the mega-hub framework. Consequently, this model integrates flight frequency, fleet assignment, and passenger flow management within an integrated mathematical model.

3.3. Genetic Algorithm

In combinatorial optimization, the decision variables are discrete, and the objective is to select or arrange objects in a way that optimizes a given function [43]. Metaheuristics fall into the following two main categories: single-solution and population-based approaches [44]. Single-solution methods explore one candidate’s neighborhood at a time; however, the vast search space and tight constraints of the integrated scheduling problem make finding a high-quality initial solution impractical in this framework. Population-based approaches, primarily evolutionary algorithms and swarm intelligence, maintain and evolve a set of candidates, offering superior global exploration at the expense of greater computational effort [44]. The most widely applied swarm intelligence heuristic in combinatorial optimization is ant colony optimization [43], but it focuses on permutation problems [45] and thus is not suitable for our non-permutation-based problem. Additionally, a single swarm intelligence optimization algorithm cannot handle a vast set of optimization problems [46]. On the other hand, GA combines exploration and exploitation mechanisms and is widely applied in airline scheduling and other combinatorial problems [15,16,43].

GA, introduced by Mitchell [47], is an evolutionary optimization method used across various fields, including engineering, biology, and scheduling. GA evolves a population of candidate solutions (chromosomes) through generations using selection, crossover, and mutation. Fitness values are evaluated based on the model’s objective, and selection methods such as rank-based or proportional selection are used during reproduction. Elitist replacement ensures that high-performing solutions are retained [48]. GA has been widely used in ASPP [49,50,51]; however, its application to ISDFA is limited. Kablan et al. [15] used GA to generate daily feasible timetables and aircraft assignments at hubs by synchronizing inbound and outbound flights, employing a chromosome structure representing round-trip flights. Similarly, Khanmirza et al. [16] employed GA to network-wide scheduling with hybrid route networks (HS and point-to-point), primarily focusing on timetable generation and balanced aircraft usage across entire networks. Recently, Derviş and Demir [52] utilized GA specifically to optimize passenger flow at Istanbul Airport, a global mega-hub, emphasizing the synchronization of domestic and international flights to minimize transfer waiting times.

In contrast to these studies, this study employs a GA specifically designed for outbound flight scheduling from a single mega-hub to a new long-haul destination, assuming fixed inbound arrivals. Unlike traditional approaches emphasizing network-wide scheduling or inbound–outbound synchronization at the hub, our model explicitly tracks passengers who have already arrived at the hub through the airline’s own inbound flights. It distinguishes between assigned, carryover, and spilled passengers, employing temporal reassignment logic to maximize assignment and minimize passenger loss due to excessive waiting. Passenger flow is aggregated into discrete operational slots, with a cost-driven improvement step removing unnecessary flight openings. These enhancements enable the airline to capture and serve transit passengers at the hub more efficiently, optimize capacity, reduce waiting times, and improve overall revenue.

The employed GA incorporates a slot-based chromosome structure to address the complexity of the underlying CMM model, in which the decision variables $s_{t,p,}$ and $u_t$ track assigned and spilled passengers, respectively. Instead of modeling each discrete time period, the algorithm aggregates the day into three slots (morning, noon, and evening), reducing complexity and runtime. Each chromosome consists of 21 genes (7 days × 3 slots per day), encoding flight departure time, aircraft type, and passenger assignment status. The gene structure includes both dependent and independent variables, as illustrated in Figure 1. Genetic operators are applied only to independent variables, while dependent variables are updated afterward to ensure feasibility. This structure improves algorithmic efficiency by limiting the number of variables in each chromosome. To further reduce complexity, spilled and carryover passengers are aggregated and treated as unassigned, allowing their reassignment in the following day’s schedule. This dynamic flow tracking enables the model to distinguish between assigned, carryover, and spilled passengers across the planning horizon. Additionally, a dedicated improvement step removes flights with zero passenger assignments, enhancing cost-efficiency. These features allow the GA to generate feasible, realistic weekly schedules that reflect the operational characteristics of mega-hub environments, offering scalability, fast runtimes, and near-optimal solutions for new route scheduling.

3.4. Algorithm Framework

The employed genetic algorithm (GA) includes the following procedures: initialization, fitness evaluation, rank-based probability assignment, roulette wheel selection, single-point crossover, mutation, improvement, and elitist replacement. The algorithm is governed by the following key control parameters: num_population (initial population size), number_generation (maximum number of generations), crossover_rate (probability threshold for crossover), and mutation_rate (probability threshold for mutation).

The algorithm begins by generating an initial population using flight arrival data, aircraft data, maximum passenger waiting time, waiting time thresholds, cost values for designated intervals, and passenger ready times. In each generation, fitness values are computed based on revenue maximization. Selection is conducted by first assigning selection probabilities using the rank-based method, which moderates selection pressure. Then, the roulette wheel selection probabilistically chooses parents for crossover. For each parent pair, a random value (crossover_rand) is generated and compared to the crossover_rate. If the value is less than or equal to the threshold, single-point crossover is applied to generate two offspring; otherwise, the parents are passed unchanged. In the mutation phase, each offspring is evaluated using a new random value (mutation_rand). If it is less than or equal to the mutation_rate, the mutation is applied by modifying independent variables—flight opening hours and aircraft type. Otherwise, the parents are passed unchanged. All offspring after crossover and mutation operations are updated using the update-dependent variable (UDV) procedure to ensure solution feasibility. Next, offspring are improved by closing flights with zero assigned passengers to reduce unnecessary operating costs. Finally, elitist replacement is used to select the best solutions, while keeping the population size constant. This process is repeated until the maximum number of generations, number_generation, is reached. The termination criterion is based on a predefined number of generations. The best chromosome is returned at the end of the algorithm. The flowchart of the GA is shown in Figure 2.

3.4.1. Initialization

The Initialization Procedure produces feasible solutions by systematically assigning passengers to flights based on predefined constraints. It begins by initializing an empty list to store solutions. For each candidate solution within the predefined initial population, passengers arriving at the hub on the current day are combined with those passengers carried over unassigned from the previous day. Each day within the weekly planning horizon is divided into operational time slots (morning, noon, and evening). Within each slot, the aircraft departure time and aircraft type are randomly selected. Passengers are eligible for assignments if they arrive before the selected departure time. Also, their waiting time, calculated as the difference between departure and arrival, must fall within the predefined minimum and maximum thresholds. If the total number of eligible passengers does not exceed the seating capacity of the chosen aircraft, all passengers are assigned. Otherwise, eligible passengers are sorted by descending waiting times and assigned sequentially until the aircraft’s seating capacity is fully utilized. After assignments, passengers who remain unassigned at the day’s end are categorized according to their arrival times relative to a predefined transfer threshold time, start time of the evening phase, (Q); those arriving at or after Q become “unassigned”, to be reconsidered in the subsequent day’s schedule, while passengers arriving earlier than Q are marked as “spilled” and removed from further assignment. Completed feasible solutions are collected into the solution list, which is returned for further processing in the genetic algorithm. This structured approach ensures the feasibility of each solution. The pseudo-code of the Initialization Procedure is shown below (Algorithm 1).

Algorithm 1: Initialization Procedure

1.    Initialize solution list ← ∅

2.    For solution = 1 to num_population do:

3.            previous unassigned ← ∅

4.            For each day in weekday, do:

5.                    arrivals ← arrivals ∪ previous unassigned

6.                    For each operational slot (morning, noon, evening):

7.                          Randomly select an aircraft departure time and an aircraft type

8.                          Identify passengers eligible based on arrival and waiting constraints

9.                          If total eligible passengers ≤ selected aircraft’s seating capacity:

10.                                  Assign all eligible passengers to the aircraft

11.                          Else:

12.                                  Sort eligible passengers by descending waiting times

13.                                  Assign passengers sequentially until the aircraft seating capacity = 0

14.                          End If

15.                          Remove assigned passengers from arrivals

16.                  End For (slot loop)

17.                  If unassigned passengers ≠ ∅ then

18.                          passengers with arrival time ≥ Q → unassigned

19.                          passengers with arrival time < Q → spilled

20.                  End If

21.                  previous unassigned ← unassigned

22.          End For (day loop)

23.          Add the completed solution to the solution list

24. End For (solution loop)

25. Return the solution list

3.4.2. Genetic Algorithm Operations and Solution Evolution

The GA aligns with the proposed CMM objective of revenue maximization. Within the GA, assigned, unassigned, and spilled passengers are tracked daily and for each time slot. To ensure an accurate calculation of the total spilled passengers over the weekly period, the unassigned passengers from the final time slot of the last day are included in the summation. This adjustment guarantees a precise match with the fitness function in the CMM model. This alignment is the only distinction between the GA’s fitness function and the CMM, ensuring consistency in the revenue objective across both models.

Next, rank-based selection is applied to rank solutions based on their fitness values in descending order. The primary reason for selecting the rank-based method is its ability to handle negative solutions, which may occur when the cost is significantly high in local solutions. Roulette wheel selection is then used to select parents based on proportional probabilities derived from a uniformly random distribution between 0 and the cumulative probability. In the literature, this hybrid strategy is frequently used, leveraging rank-based probability assignment within a roulette wheel selection framework in GA [53].

Single-point crossover is implemented using two vectors representing the independent variables opening time and aircraft type. Two offspring are generated from the selected parents through a randomly chosen single crossover point. As the fitness function is directly proportional to the number of passengers successfully assigned during the week, while other components are cost-related, the mutation procedure is designed to enhance assignments. If any time slot is found to have zero passenger assignments, a new aircraft type and opening time are introduced for that slot, encouraging increased aircraft utilization. After each genetic operator is applied, the update-dependent variable procedure is executed to ensure solution feasibility. This procedure follows the logic of the Initialization Procedure, with one minor distinction; instead of randomly generating the opening hour and aircraft type, these values are retrieved from the chromosome’s independent variables. Passenger assignment and other decision variables are then recalculated accordingly, maintaining feasibility at every step of the algorithm. This procedure enforces feasibility by (1) assigning passengers up to each aircraft’s capacity and within waiting-time limits; (2) transferring any unassigned passengers to the next available time slot; (3) permitting at most one flight per phase; and (4) ensuring total demand is conserved as the sum of assigned and spilled passengers.

The improvement procedure further enhances the revenue by eliminating unused aircraft and flights with zero passenger assignments from the schedule. The associated operating costs are deducted from the revenue. This procedure is applied uniformly to all offspring, regardless of whether crossover or mutation has occurred. Finally, the offspring generation process is completed. These offspring are merged with the current population, and the elitist replacement method is applied. In elitist replacement, the top 50% of solutions from the combined parent, offspring pool are retained for the next generation, and the remaining population is filled using the Initialization Procedure.

4. Empirical Study

The empirical study was conducted on a computer with the following specifications: Intel Core i5 (1.8 GHz), 12 GB of RAM, and Windows 11 Pro. The CMM experiments were run in CPLEX Studio IDE 22.1.1, while the GA experiments were performed in a Python 3.8.5 environment managed through Anaconda 4.12.0.

4.1. Experimental Design and Operational Assumptions

A passenger assignment is conducted when a passenger’s waiting time exceeds 2 h (the ready time, referred to as r or $B_0$) but is less than 10 h (the maximum allowable waiting time, referred to as T_max). An incremental waiting cost is imposed based on the actual waiting time:

• For waiting times ≤ 1 h ($B_0)$, a zero cost (c₀) is applied, making the assignment infeasible.
• For 1 h ${(B}_0)$, <waiting time ≤ 4 h ($B_1)$, a minimal cost (c₁) is applied.
• For 4 h ${(B}_1)$, <waiting time ≤ 7 h ($B_2)$, a slightly higher cost (c₂) is applied.
• For 7 h ${(B}_1)$, <waiting time ≤ 10 h (T_max), a higher cost (c₃) is applied.

The operational day is divided into three equally timed intervals, morning, noon, and evening, whose starting times are 00:00, 08:00, and 16:00, respectively. Both the CMM and GA aim to determine the aircraft type and departure time for each phase in which an aircraft is opened. Importantly, at most one aircraft is allowed to be opened in any given slot, meaning the option of not opening an aircraft is also evaluated within the model framework.

4.2. Data

A one-week timetable from a major international airline operating through its primary hub was analyzed. The dataset covers August 15 to 21, 2021 and includes 5157 international arrivals and departures at the hub. During this week, 37 distinct airports contributed a total of 381 inbound flights, with each origin providing between 2 and 30 flights. Figure 3 shows the distribution of inbound flight frequencies and arriving passengers across the three daily intervals. Given that passenger volume is highest during the evening phase (16:00–24:00), multi-day planning becomes essential to effectively manage demand and schedule outbound routes. Furthermore, annual passenger data from 2021 were used to estimate demand under the assumption of uniform distribution across the year. Due to confidentiality agreements, the name of the airline is not disclosed.

4.3. Experimental Parameters

Parameters used in both the CMM and GA approaches are grouped into the following three categories: model parameters, CMM parameters, and GA control parameters. This study conducts experiments under two distinct time granularities: a 2 h discretization and a 15 min discretization, with 15 min being one of the most common time granularities in the literature [11,32]. In both experiments, arrivals and flight takeoffs are modeled according to the corresponding time granularity. To compare these results with those obtained at a 15 min granularity, the time-related parameter values for the 15 min experiments were multiplied by 8 (since 2 h = 120 min, which is 8 times 15 min), while cost-related parameters were divided by 8 to adjust for the difference in time scale ($B_0$, $B_1$, $B_2$: 4, 16, 28; T_max: 40; r: 4; c₀, c₁, c₂, c₃: 0, 1/8, 4/8, 20/8). Parameter tuning for GA was performed using the 15 min granularity, and the resulting optimal values were applied to both experimental configurations, explained in Section 4.4.

Table 2. Parameter values.

Class	Parameter	Value	Description
Model	$\mathrm{Waiting} \mathrm{Intervals} (B_0$$,B_1$$,B_2$)	0.5, 2, 3.5	Time thresholds for waiting cost calculation
	Aircraft Types (K)	3	Number of aircraft types available
	Aircraft Capacities (Ck)	200, 500, 850	Seating capacity for each aircraft type
	Aircraft Operating Costs (Ok)	50,000, 80,000, 100,000	Fixed operating cost per flight per aircraft type
	$\mathrm{Fare} (\omega$)	500	Revenue per assigned passenger
	$\mathrm{Spilled} \mathrm{Cost} (\alpha$)	100	Penalty for each spilled passenger
	Waiting Costs (c0, c1, c2, c3)	0, 1, 4, 20	Incremental waiting cost coefficients by interval
	Max Waiting (Tmax)	5	Maximum allowable waiting time (in hours)
	Ready Time (r)	0.5	Minimum waiting time before an assignment is allowed
CMM	Waiting Cost Linearization (L)	5	Indices for binary linearization of waiting cost × assigned
	Waiting Time Linearization	2	Indices for linearizing waiting time × assigned
GA	Population Size	30	Number of chromosomes in each generation
	Generation Number	10	Number of generations the GA runs
	Mutation Rate	0.5	Probability of applying a mutation
	Crossover Rate	0.5	Probability of applying crossover

summarizes the parameter values used for the 2 h experiments.

4.4. Parameter Tuning for Control Parameters in GA

Parameter tuning was conducted using the 15 min time granularity. For each GA control parameter (population size, generation number, mutation rate, and crossover rate), the algorithm was run 20 times at each candidate value, while holding the other parameters constant. This tuning strategy, including the chosen parameter ranges and the 20-iteration experimental design, was adapted from Khanmirza et al. [16]. The tuning range and selected values for each control parameter are shown in

Table 3. Tuning control parameters of GA.

Parameter	Tuning Range	Selected Value
Population Size	6, 12, 18, 24, 30	30
Generation Number	5, 10, 15, 20, 25, 30	10
Mutation Rate	0.1, 0.3, 0.5, 0.7, 0.9	0.5
Crossover Rate	0.1, 0.3, 0.5, 0.7, 0.9	0.5

. The performance indicators used were the percentage profit increase (PI) and the percentage runtime increase (RI) relative to the first value in the tuning range.

Figure 4a shows the profit distribution for each tested population size, presented as a box plot after 20 independent executions. The population size of 30 produced the highest profit and the longest runtime, corresponding to a 516% profit increase relative to a population size of 6, along with a 36% improvement in solution consistency. In contrast, solutions for population sizes of 6 and 12 exhibited high variability; whereas, those for populations of 24 and 30 were more consistent, as evidenced by the proximity between the mean and median values. The results indicate that when the population size exceeds 18, the waiting time and the number of assigned passengers become directly proportional; at lower population sizes, they show an inverse relationship.

Figure 4b illustrates the improvement in profit as the generation number increases. For most population sizes, convergence occurs in the early stages of the GA, and by generation 10, the solutions stabilize, indicating that further iterations yield negligible improvements. The best profit obtained across population sizes from 6 to 30 was 1,094,775. This outcome indicates the significant impact of population size on GA performance.

Figure 4c,d show the profit distributions as functions of the crossover rate and mutation rate, respectively, after 20 independent runs. Figure 4c indicates that all values within the crossover tuning range yield consistent solutions, as seen in the close proximity between mean and median profits in each bar chart. Additionally, the charts show the effective exploration of candidate solutions, with evidence from the length of the bars and the presence of outliers. Notably, the relative runtime compared to a crossover rate of 0.1 increases dramatically beyond a rate of 0.3, from 38.9% to 91.2%, and no significant profit increase is observed beyond a crossover rate of 0.5. Similarly, Figure 4d shows that computational time increases sharply for mutation rates above 0.5, while solution quality improves only marginally. Based on the parameter-tuning analysis, the values shown in Table 3 were selected for the GA as providing the best balance between solution quality and run time.

5. Results

The performance of the two solution approaches, CMM and GA, was evaluated under both 2 h and 15 min time discretization. Detailed flight schedules corresponding to each configuration are provided in Appendix A. The CMM (2 h) was solved to global optimality, achieving a weekly profit of USD 3,438,774 with an assignment rate of 76% and an average waiting time of 3 h. In contrast, GA (2 h) made a profit of USD 2,751,850, with a lower assignment rate (72%) and a longer average waiting time of 6 h. When the GA was run with a 15 min granularity, it improved to assign 81% of passengers and achieved a profit of USD 3,412,340, though with a similar average waiting time of approximately 6 h and 7 min.

While the CMM (2 h) model finds the global optimum, the CMM (15 min) model presented high computational complexity in its full-scale form due to the problem size. As shown in

Table 4. Growth in problem size of the CMM as time granularity increases.

	CMM (2 h)	CMM (15 min)
Constraints	335,251	10,422,137
Binary Variables	152,460	4,177,080
Integer Variables	98,919	3,075,795
Other Variables	4	163,972

, the number of constraints increased from 335,251 to over 10 million, and a dramatic increase in the number of variables was observed. Preliminary attempts to reduce the problem size by fixing schedules for one or two days were insufficient. Consequently, flight schedules for Monday, Wednesday, and Friday were fixed to achieve a feasible solution within the available computational resources. These flights were treated as mandatory, an incremental schedule design approach commonly used in the literature to reduce the problem size and decompose it into sub-problems [5]. As a result, only a feasible but not globally optimal solution could be obtained. The partially fixed CMM (15 min) achieved a weekly profit of USD 3,614,559, with a 30.35% MIP gap and a significantly longer runtime (8 h 15 min). In contrast, GA (15 min) took only 145 s to run. Still, the CMM (15 min) model surpasses GA (15 min) in terms of assignment rate and profit, but this comes at a significantly higher computational cost. The summary of the results is shown in

Table 5. Summary of results.

Metric	CMM (2 h)	GA (2 h)	CMM (15 min)	GA (15 min)
Assigned Passengers	11,321	10,641	12,005	11,991
Assignment Rate	76%	72%	81%	81%
Weekly Profit (USD)	3,438,774	2,751,850	3,614,559	3,412,340
Average Waiting Time	3 h	6 h	3 h 12 min	6 h 7 min
Granularity of Time Slots	2 h	2 h	15 min	15 min
Run Time	118 s	40 s	8 h 15 min	145 s

.

5.1. Passenger Assignment and Profit

Figure 5 presents the weekly profit and average passenger waiting time for each method. Among all experiments, the CMM (15 min) model shows the highest profit (USD 3,614,559), while the shortest average waiting time is achieved by the CMM (2 h) model. In contrast, both GA (2 h) and GA (15 min) yield lower performance in terms of profit and passenger waiting time. However, GA (15 min) closely approximates the profit of CMM (15 min), while achieving a similar assignment rate. The small gap in assignment rates between CMM (15 min) and GA (15 min) highlights the trade-off between maximizing profit and minimizing passenger waiting time under high granularity.

The lower performance of the GA at the 2 h resolution is due to multiple interconnected factors. First, the 2 h resolution narrows the search space, enabling the algorithm to become trapped. Second, the GA parameters detailed in Table 3 were tuned for the 15 min scenario and may not be optimal for the 2 h resolution. Finally, at 2 h granularity, certain capacity and connectivity constraints bind more tightly, forcing the algorithm into less-profitable assignments to satisfy feasibility. These factors explain the performance observed at the 2 h level.

5.2. Daily Passenger Assignment

Figure 6 shows the daily passenger assignments generated by the CMM and GA under 2 h and 15 min time discretizations, along with the total number of arrivals per day. Overall, the CMM (15 min) model performs the best, consistently assigning the highest number of passengers on most days, especially Monday and Friday, and closely following the arrival trend. This suggests that its superior results are from the combined strengths of mathematical optimization and high granularity. The GA (15 min) also performs competitively, particularly on Thursday and Sunday, demonstrating the flexibility of the algorithm, despite being slightly less effective than CMM in maximizing assignments. In contrast, both 2 h models, especially GA (2 h), cannot capture daily arrival patterns. GA (2 h) assigned the fewest passengers on all days. The CMM (2 h) model shows moderate results but remains less effective than its 15 min granularity experiment, especially on high-demand days. These results emphasize the combined importance of time granularity and solution approach, with exact optimization and fine time resolution enabling more accurate alignment with fluctuating daily demand.

5.3. Fleet Composition

Figure 7 shows the comparison of aircraft type utilization. The CMM (2 h) solution predominantly employs larger aircraft (Type 3), which maximizes capacity and minimizes unit costs. In contrast, both the GA (2 h) and GA (15 min) solutions utilize a more diverse fleet that includes a small number of small aircraft, potentially leading to higher per-seat costs and underutilization of capacity. The CMM (15 min) solution maintains a fleet composition similar to that of the 2 h model, primarily favoring high-capacity aircraft. Additionally, it opens three more flights than the CMM (2 h) model, showing the increased scheduling flexibility provided by the high time granularity. As a result, the CMM produces flight schedules with higher overall aircraft utilization compared to GA.

6. Discussion

The CMM (2 h) model achieves global optimality with lower passenger waiting times and higher profit using larger time intervals, yet its scalability is severely limited. In real-world long-term schedule planning, airlines require high-granularity schedules, typically with 15 min intervals, to precisely align flight departures and maximize connectivity. However, increasing granularity makes the CMM model computationally intractable, as the number of constraints and variables increases exponentially (from 251,383 decision variables in the 2 h model to approximately 7 million in the 15 min model). To reduce the problem size, we fixed flight schedules on select days (Monday, Wednesday, and Friday) to obtain a feasible 15 min CMM solution. While this approach allowed the model to generate a feasible solution, it resulted in a significant MIP gap and extremely long runtimes (over 8 h), producing only a locally optimal solution.

The development of customized heuristic algorithms for airline scheduling has enabled the effective management of problems characterized by an extremely large and complex search space, yielding significant reductions in computation time. [54]. For example, Xu et al. [32] demonstrated that column generation with variable-neighborhood search produces near-optimal solutions at 15 min granularity in far less time than the exact mathematical model. Our study similarly integrates flight frequency decisions into the scheduling model but focuses specifically on outbound flight planning from the hub, while explicitly modeling spill and recapture dynamics. Similarly, a parallel master–slave GA delivered near-optimal schedules much faster (1000%) than a MILP incorporating a network [16]. The employed GA likewise runs substantially faster (204 times) than the CMM at fine granularity, while recovering a high proportion of the optimal profit.

These findings align with prior studies emphasizing the NP-hard nature of fine-granularity flight scheduling and the need for heuristic methods to solve in an acceptable time. Specifically, while the CMM provides an upper bound on performance under broader discretization, its scalability is limited. Although slightly less effective in reducing average waiting time, the GA offers a practical trade-off. It achieves competitive profit levels, while providing high flexibility, making it suitable for real-time or large-scale scheduling. These results support the hypothesis that higher granularity scheduling can enhance profitability and passenger assignment, but only if accompanied by scalable solution methods. Also, the GA supports the view that integrated heuristic designs can better match schedules to demand trends by adjusting the daily schedule regarding demand dynamics [55].

Several studies have embedded a connection waiting time in their objectives, yet various notations are used. Çiftçi et al. [35] introduce a connection–value parameter computed as passenger demand times fare, divided by connection time, which is maximized alongside slot assignments in their MILP. Cadarso and Marín [56] take an alternate route by maximizing itinerary revenue, while charging a fixed “dummy” cost for each passenger connection. This study aligns with these approaches by embedding waiting time penalties directly in both the mathematical model’s objective and the GA’s fitness function but extends them by modeling the waiting time as a decision variable rather than a fixed parameter and by incorporating frequency optimization with dynamic spill-and-recapture behavior.

There are several limitations in the proposed framework. First, the mathematical model becomes intractable at 15 min granularity given the computational resources available in this study, often resulting in memory-out errors due to the exponential growth in constraints and decision variables. Second, the study relies on annual passenger aggregates for the entire market rather than airline-specific demand profiles. It does not incorporate aircraft load factors, introducing uncertainty into transit passenger volumes. Third, all inbound flights are assumed to arrive on schedule, omitting the impact of stochastic delays on connectivity. Fourth, the framework assumes unlimited fleet capacity for outbound scheduling. Finally, airport slot availability, runway capacity, and ground-time restrictions are not considered.

7. Conclusions

This study addresses the challenge of optimizing flight scheduling and aircraft assignment for outbound flights from a mega-hub to a new destination, while maintaining the existing flight network. Specifically, this study aims to determine the optimal departure times, flight frequencies, and aircraft types that enable an airline to efficiently utilize inbound passenger flows without altering the current flight network. This problem is critical, as it directly affects passenger connectivity and overall profitability. These factors are crucial when entering a competitive new market. To address this problem, a mathematical model (CMM) is proposed, and a customized GA is employed. Real-world data comprising inbound flights from 37 spoke airports were used to apply and evaluate the proposed methods. GA was employed to address the scalability limitations of the large-scale mathematical model. In contrast to CMM, the GA demonstrates scalability and flexibility, producing high-granularity schedules close to CMM’s but within a reasonable run time. For strategic planning, where achieving precise schedules is crucial, the GA proves to be an effective approach for capturing detailed passenger flow dynamics within higher granularity scheduling frameworks. It enables schedule optimization under realistic time constraints with reasonable computational costs.

By offering insights into potential scheduling strategies, our work aims to support airline managers in making rational route expansion decisions, ultimately minimizing passenger wait times at the hub and maximizing revenue. Providing the shortest and most efficient transit passenger waiting time is a critical strategy when entering a new market, especially when competing with airlines already operating in that market. The proposed models offer valuable guidance for aligning outbound schedules with inbound flows, thereby enhancing connectivity and strengthening the airline’s competitive position.

Future Research

Future research can extend this work in several ways. First, a discrete-choice dynamic can be embedded within the mathematical model instead of the current spill-and-recapture dynamics to more realistically mimic passenger flow. Second, the proposed model can be decomposed into a daily planning sub-model, whose solutions are then aggregated to generate a comprehensive weekly flight schedule. Third, a large-neighborhood search (LNS) can be employed, as follows: in the destroy phase, subsets of flight frequency and assignment decisions are removed; in the repair phase, the comprehensive mathematical model (CMM) is applied. This approach is expected to produce near-optimal solutions with finer-granularity scheduling and shorter run times. Finally, the LNS framework can be benchmarked against both the CMM and genetic algorithm (GA) approaches using real mega-hub data, evaluating performance in terms of waiting time, profit, and computational time.