Cargo Aircraft Capacity Optimization: A Hybrid Approach Comprising a Genetic Algorithm and Large Neighborhood Search

Abstract

Air transportation has accelerated international trade, and the efficient use of cargo aircraft capacity supports logistics operations, reduces expenses, and benefits the environment. In this study, we formulate a mathematical programming model to solve the cargo aircraft capacity optimization problem and propose simplified approaches for practical applications. We investigate Mixed-Integer Linear Programming (MILP), Genetic Algorithm (GA), and Large Neighborhood Search (LNS) techniques. MILP yields optimal solutions for small instances but cannot handle large-scale, real-world problems due to excessive computation time; therefore, we combine the GA and LNS. The GA provides acceptable solutions rapidly, and LNS refines them by exploring larger solution spaces. Thus, this hybrid approach leverages the GA’s exploration capability and LNS’s exploitation ability to produce high-quality solutions efficiently. Our experimental results show that the hybrid GA-LNS method outperforms the MILP and single approaches in terms of capacity usage, loading duration, and computational time. This study provides an applicable model with practical constraints and guidelines for air cargo and cost reduction, operational efficiency, and safety.

1. Introduction

With shorter product life cycles and increasing globalization of trade, there is a growing need to transport goods to their markets more rapidly. Owing to the effects of globalization, airline cargo has become an essential component of world trade, with its overall volume experiencing exponential growth, doubling almost every decade since the 1970s. Air cargo accounts for approximately 36% of the value of goods traded internationally by value, and air cargo volumes grew approximately 50% more than passenger transport between 1995 and 2004, in which similar patterns have been observed in recent studies. This upward trend has continued in recent years. Air cargo transportation plays a critical economic role and has become an important source of revenue for airlines [1]. The profit margin, which was approximately 5% in 2000, rose to an average of 40% by 2009. This trend has provided opportunities for companies in the logistics and transportation sector to extend their service portfolio. Yet, handling rising volumes of goods and moving them faster also poses a significant challenge, as it is critical for businesses to run these processes efficiently and effectively reduce costs in order to remain competitive [2].

In light of these trends, optimizing logistics processes, improving service quality, and handling higher product volumes while reducing costs and emissions has become more critical than ever.

Air freight is typically the preferred delivery method for urgent and valuable cargo. Research shows that air cargo, valued at over USD 6.8 trillion (accounting for over 35% of global trade by value), is a significant source of revenue for airlines. Air cargo accounts for 9% of airline revenue, more than double that of first-class passenger transport. Air cargo is fast and safe, but it is more expensive than surface transportation (up to 10 or even 50 times), and the maximization of cargo loaded onto the aircraft is the main goal for airlines. Although the loading process may seem simple, it is a complex task that must be performed to ensure the flights are safe and economically viable.

Loading must be performed correctly in order to ensure safety, as loading errors can result in the destruction of the aircraft, damage to the cargo, and even loss of life. Correct loading will make the aircraft aerodynamically more efficient. Therefore, a more efficient flight means lower fuel consumption, which reduces the cost of the flight and also minimizes its environmental impact [2]. This issue is of vital importance to airlines impacted by rising fuel costs and increasing demand to lower carbon dioxide emissions.

Freight managers must create loading plans under extreme time pressure, and this process is very time-consuming when performed manually. In contrast, interactive computer-based tools enable managers to assess various alternatives and select the most appropriate solution based on their field experience and the actual conditions they face. Nevertheless, in addition to high demand uncertainty, securing sufficient air cargo capacity continues to pose a significant challenge for shippers and freight forwarders. Hence, airlines must effectively manage their cargo processes by developing strategic operational plans to quickly adapt to changes in global competitive conditions. Utilizing aircraft capacity as efficiently as possible on every flight is among the primary goals of airlines. In this context, the number of theoretical studies examining the problems encountered in air cargo operations has increased significantly since the 1990s.

However, since the inherent complexity of these operations, many problems—particularly those encountered in real-world contexts—still remain insufficiently resolved [3]. Consequently, airlines are compelled to minimize operating costs related to ground services and fuel in order to sustain operations. The transportation of air cargo involves a sequence of interconnected services that transfer goods from their origin to the final recipient, with the participation of various actors, including shippers, freight forwarders, trucking firms, airlines, and consignees [4]. Shippers seek worldwide delivery of their goods at affordable prices while ensuring the necessary service standards. Freight forwarders function as the link between shippers and airlines, coordinating the transport process. Road carriers handle the movement of goods both before and after the air journey. Airlines assume responsibilities such as accepting, storing, transferring, tracking, loading, and unloading shipments, as well as overseeing the allocation of cargo capacity.

The air cargo industry consists of many uncertain and dynamic processes. In such an industry, solutions based solely on linear programming are insufficient. This study aims to solve the aircraft loading problem using linear programming and create a hybrid solution that incorporates two heuristics to better approximate real-world environments. Although capacity optimization has been widely studied, most research has focused on container loading in maritime and land transportation. In contrast, air cargo loading problems involve much more complex processes and constraints [5].

This study is organized into four sections. Section 1 introduces the study. Section 2 presents the methodology. Section 3 reports the results. Section 4 discusses limitations and future work.

1.1. Related Works

Airline operators or carriers provide a wide range of services to shippers and logistics companies, including consulting, capacity planning, cargo receiving, handling, packaging, sorting, loading, shipping, and tracking. This service chain in air cargo transportation consists of specific stages. The process begins with the cargo being delivered to the cargo terminal at the departure airport, either by carriers or directly by shippers, in trucks, containers, or as bulk cargo. Cargo is first unloaded and organized based on details found in the shipping documents, including destination, weight, size, number of packages, and type of load. The airline is then responsible for calculating the charges based on the weight and dimensions of the shipment, and the airline issues a waybill, which is a document used to verify the contents of a shipment and the payment associated with it. Loose cargo must be loaded onto some form of Unit Load Device (ULD), such as aluminum pallets or containers, because these devices come in a range of shapes and sizes. Still, the most common are easily transferable between standard wide-body aircraft. Bulk cargo, however, is loaded directly into the aircraft and is often no more than a simple consignment note. Shipments are typically routed via a global hub-and-spoke model. For example, in the United States, cargo typically arrives at one of its main shipping ports, such as Los Angeles (LAX) or New York/Newark (JFK/EWR). It is then transported to a nearby airport, such as Los Angeles International Airport (LAX) or John F. Kennedy International Airport (JFK), where the cargo is then loaded onto a plane to a sort facility, which is a central location where cargo is sorted before being transferred to another plane or vehicle. Consequently, the cargo can go from there as loose cargo, known as cargo or goods that are transported either by being carried on an individual, held in the person’s hand, or in an individual’s luggage, or bundled together on wooden pallets into a single Unit Load Device (ULD). Once the goods arrive at the destination airport, they are checked, moved to a warehouse, and then either delivered by local transport providers or collected directly by the consignee. Air freight is considered a fast and reliable means of transport and is mostly preferred for shipments where speed and security are crucial. However, it is significantly more costly than surface transport, often ranging from 10 to 50 times higher.

For this reason, it is generally preferred only for goods that possess at least one of the following four characteristics.

1.1.1. Special Cargo

In addition to general cargo, aircraft also transport various types of special cargo that require specific handling, storage, and transportation conditions. Special care is often required to transport and handle these types of shipments to ensure safety and compliance with regulations, as well as protection of the product quality during the logistics process.

1.1.2. Main Categories of Special Cargo:

(1)

Dangerous Goods (DGR):

There are dangerous goods that are hazardous if not transported under specific conditions, as they may be subject to tipping, extreme temperatures, or mechanical shock. Examples of dangerous goods include hazardous materials, chemicals, batteries, and radioactive substances, which must be handled and packaged properly in accordance with regulations. This is crucial for the safety of the crew, passengers, aircraft, and people and property on the ground [2].

(2)

Perishable (PER):

Other products are also perishable, meaning that they are subject to spoilage if not kept under appropriate conditions, and some products are more perishable than others. Thus, dried legumes and grains are less perishable than meat, dairy products, eggs, or baked goods. Large-scale food production is now typically performed by using agricultural management techniques that optimize land use, minimize labor, and increase yield, because producing food requires a specific amount of land, water, and energy resources. The food system also affects biodiversity, since it alters natural ecosystems and their processes. Such goods must therefore be transported in a controlled environment to preserve their quality and usability. Typical examples are flowers, pharmaceuticals, and fresh food, all of which require constant monitoring of temperature and humidity during transport.

(3)

Urgent:

Certain products must be delivered to distant destinations within a very short time frame. These shipments must be expedited through the system and take priority over other cargo to meet tight delivery schedules, as delays in their delivery are associated with significant financial penalties or even loss of life. Typical shipments of this kind, including emergency medical supplies, critical spare parts for industrial equipment, or relief goods to be delivered to areas stricken by natural disasters, require priority handling over regular cargo to ensure timely delivery.

(4)

Live Animals:

Live animals represent another category of special cargo that must be transported under highly sensitive conditions. Carriage of live animals is subject to International Standards such as the IATA Live Animals Regulations (LARs) because these regulations ensure that animals are transported under appropriate conditions that ensure the welfare of the animals and the safety of the animal handler, passengers, and observers. These factors include adequate ventilation, precise control of temperature, appropriate feeding and watering during transport, and species-specific containers. Failure to meet these requirements may result in harm to the animals and potential legal or ethical consequences for carriers.

(5)

Valuable Cargo:

Valuable cargo is a shipment of high monetary or cultural value that requires special protection measures, and it can include such items as precious gems, precious metals, currency, artworks, antiques, and designer clothing. Valuable cargo shipments require a high level of management and monitoring because they need to prevent the risks of theft, loss, or damage in the logistics chain. Examples of typical valuables are semiconductors, banknotes, jewelry, and art; these items are often transported with high security and, in many cases, monitored throughout the entire transport chain [1].

1.1.3. Aircraft Loading

Aircraft loading primarily involves placing Unit Load Devices (ULDs) onto the aircraft subject to multidimensional constraints, such as cargo weight, available volume, the positioning of containers, the aircraft’s center of gravity, container types, and random passenger baggage [6]. From a modeling perspective, the loading problem is a three-dimensional Box Packing Problem (BPP), a basic combinatorial optimization problem that is NP-hard [7].

In practice, this issue proves to be highly complex. The physical loading of the aircraft is influenced by various factors, such as the loading of passenger baggage, weather conditions, the time available for loading, and the seat allocation pattern (i.e., whether to start loading near the wing or at the tail), and studies in load balancing have shown that for passenger or mixed-used aircraft, the ground handling teams have to operate under very tight time constraints to load the aircraft efficiently [8]. Countless issues need to be addressed. A second decision-making problem is assigning different types of containers, pallets, and uncertified mesh pallets to various aircraft compartments and mixing heavy and light cargo to ensure high load efficiency and low fuel burn under different seat allocation strategies [9]. The aircraft loading problem, therefore, should go beyond the conventional Box Packing Problem (BPP) framework and evolve into a balanced BPP model that incorporates mechanical considerations. Rather than purely optimal outcomes, airlines prioritize practical and feasible policies that can be implemented in real operations [10].

1.1.4. Container Loading Problem (CLP)

Though both container and aircraft loading take into account weight and volume, container loading has some extra concerns, such as pivot weight and pivot volume, which have been studied extensively in academia [11]. The main goal of the container loading problem (CLP) is to ensure safety and minimize the level of unnecessary handling during loading [10]. This includes customer and loading priorities, load and cargo balancing, stacking and placement, and minimizing unnecessary handling, especially when multiple shipments are being delivered. The container loading problem is inherently multi-objective and extremely challenging. For these problems, dual bounds are also recommended for CLP relaxations.

Given the complexity and number of relevant constraints, it is quite reasonable to develop a random constructive heuristic algorithm aimed at quickly producing good packing solutions [12].

• Supply and demand for air cargo are highly variable by nature, with the following key factors contributing to this variability: Unforeseen changes in passenger demand and volumes, changes in baggage volumes, weather-driven fuel adjustments, air traffic restrictions, short loading windows, and variability in operational efficiencies. Actually, the volume of cargo to be loaded for a given flight is often larger than the preplanned capacity [13].
• The container loading problem is usually classified into two scopes: the single-container loading and the multi-container loading [9].

Air cargo operations are more complex and comprehensive than passenger transport. This is primarily due to the involvement of more stakeholders, more advanced procedures, consideration of different weight and volume combinations, the availability of various prioritized services, the implementation of integration and consolidation strategies, and the availability of multiple route options within the same network.

In real-world applications, this challenge becomes highly complex. The usable load capacity can also be affected by the weight of passenger baggage, weather conditions, the time available for loading, and the seat configuration, because, for example, if the loading starts near the wings rather than the tail. During balancing operations, ground crews working with passenger or mixed-use aircraft are often found to have only a very limited timeframe to perform efficient loading [10]. Numerous issues arise that must be addressed. One of the most critical planning decisions is to allocate different types of containers, pallets, and uncertified mesh pallets in aircraft compartments and to mix heavy and light cargo under different seating policies to increase load efficiency and reduce fuel burn; however, to solve the aircraft loading problem efficiently, one should not only consider the standard BPP formulation but also a balanced BPP model that includes mechanical aspects. Airlines, however, seek feasible and implementable strategies rather than purely optimal theoretical solutions.

1.1.5. ULD (Unit Load Device)

ULDs are one of the most fundamental tools in air cargo transportation. They are the standard containers or pallets used to transport cargo and baggage together on aircraft. Their purpose is to ensure safer, faster, and more efficient loading and unloading of cargo [1].

Some ULD Types:

Among the various Unit Load Devices, the PRA type is one of the most frequently used. It offers practical dimensions and ensures efficient utilization of available aircraft space. As illustrated in Figure 1, the PRA container provides a balanced design that supports both safety and operational efficiency in air cargo transportation.

The AKE container is another widely used Unit Load Device, designed to fit into narrow-body aircraft compartments. Its compact structure allows efficient use of limited cargo space, making it particularly suitable for smaller shipments. As shown in Figure 2, the AKE container combines versatility with standardized design, enabling airlines to maximize loading efficiency while maintaining operational safety.

The PAG container is designed for transporting larger or bulk shipments and is commonly employed in long-haul international flights. Its structure provides both stability and high capacity, making it suitable for oversized goods that require secure placement during air transport. As illustrated in Figure 3, the PAG container plays a crucial role in optimizing space while ensuring that heavy or voluminous cargo is carried safely and efficiently.

The RKN container represents a specialized type of Unit Load Device designed for temperature-sensitive cargo. It is widely used for the transportation of pharmaceuticals, perishable food, and other commodities that must remain within strict environmental conditions throughout the supply chain. As shown in Figure 4, the RKN container is equipped with active or passive cooling systems that maintain cargo integrity, making it indispensable in industries where safety and quality are paramount.

1.1.6. Some Problems Encountered in Air Cargo Transportation

Uncertainty:

Compared to passenger transportation, air cargo operations face greater uncertainty regarding capacity availability. In passenger services, travelers may cancel their bookings, and sometimes fewer passengers than expected arrive for a flight. By contrast, in air cargo, shipping companies are required to commit to capacity on specific flights as far as six to twelve months in advance. The available capacity for dedicated cargo carriers may be affected by real-world factors such as runway performance, weather, and fuel availability. In practice, many cargo bookings are repeatedly canceled and rebooked, since airlines generally do not impose charges for reservation changes. However, the air cargo company in this study may charge fees based on size and duration of delay. Therefore, the reservation process is subject to significant uncertainty.

However, the air cargo company considered in this study may charge fees based on size and duration of delay. Therefore, the reservation process is subject to significant uncertainty.

Complexity:

While the capacity of a passenger aircraft can be easily calculated based solely on the number of seats, estimating cargo capacity is a much more complex process. Cargo capacity depends on the characteristics of the Unit Load Devices (ULDs), which are shaped by several technical factors such as axle weight, volume, type, and center of gravity.

Flexibility:

The airline is only required to inform shippers of the departure point, transit airports, and final destination, while independently planning transfer routes in order to optimize network capacity utilization.

1.2. Literature Review

In order to identify the current state of research on cargo and container loading, a comprehensive literature review was conducted. The reviewed studies focus on a variety of methodological approaches ranging from metaheuristics such as Genetic Algorithms and Large Neighborhood Search to exact methods including Mixed Integer Linear Programming. These studies not only provide insights into different modeling perspectives but also highlight the challenges of balancing efficiency, safety, and scalability in real-world applications. A summary of the main contributions and their comparison with the present study is provided in

Table 1. Literature review on container loading and air cargo optimization approaches.

Topic	Author(s)	Methodology(ies)	Comparison of the Reviewed Study with Our Study
Container Loading Problem	Phongmoo et al. [5].	Artificial bee colony, Genetic Algorithm	The study solved the single container loading problem by combining genetic algorithms and artificial bee colony algorithms.
	Yang et al. [14].	Last in first out constraint, greedy heuristic, and multilayer tree search; Large Neighborhood Search	An integrated greedy heuristic multilayer tree search algorithm is proposed to obtain the initial solution of a 3D-MCLP. Then, a Large Neighborhood Search algorithm containing five removal operators and two repair operators is further designed to improve the results. Numerical experiments with 2014/2015 ESICUP challenge datasets and real-world case studies validate the effectiveness and efficiency of the proposed approach.
	Heßler et al. [6].	Heuristics	In order to solve the problem, an insertion heuristic was embedded into a Randomized Greedy Search.
	Montes-Franco et al. [11].	Hybrid Optimization Method	It proposed a hybrid optimization method for solving the container loading problem. It focuses on dynamic stability. At the same time, our study focuses on heuristic and linear programming.
	Romero-Olarte et al. [15].	Heuristics	The study solved the single container problem with heuristics. It also contains some realistic restrictions, like box orientation limitations. This study is limited to single container problems, whereas our problem involves multi-container loading problems. This problem applies on the ground, while our study is about air container loading.
	Özdemir et al. [3].	Genetic Algorithm	This study focuses on the porcelain export logistics industry with multi-objective optimization. It uses a single meta-heuristic (GA). This study also focuses on economic/business constraints.
Mixed Integer Linear Programming	Souffriau et al. [8].	Mixed Integer Linear Programming	A method based on mixed integer programming is proposed to solve the Aircraft Weight and Balance Problem.
	Vancroonenburg et al. [1].	Mixed Integer Linear Programming	In this study, a Mixed Integer Linear Programming model is proposed to serve as a decision support system for air cargo loading planning.
	Wong [2].	Mixed Integer Linear Programming	The aim of this study, conducted within the scope of a cargo airline company, is to maximize loading profitability and improve operational efficiency, taking into account limitations such as aircraft structure, segregation of hazardous loads, weight-balance requirements, and flight safety.
	Alshabibi et al. [4].	Mixed Integer Linear Programming	This study broads multi-modal logistics (road, sea, air) with a focus on fleet scheduling and routing.
	Zhao et al. [10].	Integer Linear Programming	It focuses primarily on the mathematical programming for the ACPP (Aircraft Cargo Palletization Problem). But our study works with both heuristics and mathematical programming.
	Tibaldo et al. [16].	Mixed Integer Linear Programming	This study develops Mixed Integer Linear Programming (MILP) to optimize both the production and distribution of ready-mixed concrete.
Heuristic Algorithm	Pazhooh et al. [13].	Heuristic Algorithm	This study presents a novel continuous-time Mixed Integer Linear Programming (MILP) model that holistically addresses spatial and temporal dimensions. It emphasizes that time is defined as a continuous variable. This circumstance allows us to overcome the scalability issues of traditional discrete-time approaches. The model’s accuracy is compared with a constructive heuristic, and its practical applicability is demonstrated using a specially designed visualization tool.
	Chen et al. [12].	Heuristic Algorithm	This study is concerned with ground logistics and relies on two-stage heuristic algorithms (greedy, tree-search, online matching). It primarily focuses on volume utilization, pallet stability, and truck capacity.
	Chen et al. [7].	Heuristic Algorithm	This study contains container storage optimization problems in container yards. It uses hyper-heuristic and Q learning and targets strategic/tactical port operations.
	Yıldız et al. [9].	Heuristic Algorithm	This study contains a mix of mathematical modeling and heuristics.

.

2. Materials

2.1. Problem Definition and Mathematical Model

The container loading problem (CLP), also known as the packing problem, is of central importance in the planning and scheduling of logistics processes [11]. It frequently arises in real-world practices across diverse industries, including electronics, steel, paper, textiles, manufacturing, and transportation.

The container loading problem is divided into deterministic and heuristic approaches. The objective of the container loading problem (CLP) is to ensure safety, accounting for customer priorities, load balancing, cargo balance, constraints related to stacking, positioning, and restrictions on excess cargo handling operations during multiple shipment deliveries [10]. At first glance, loading cargo onto an aircraft may appear to be a straightforward routine operation; however, in practice, airlines must carefully plan the loading of each flight to guarantee both safety and profitability [1]. Ensuring maximum usage of the aircraft on every flight should be a key priority for all airlines. In contrast to passenger planes, cargo aircraft are more prone to balance-related challenges.

Problem Notations:

• Sets
  n: Total number of products to be placed in the container;
  N = Product set (1, 2, …, n).
• Products (boxes) are denoted by the notation i and j because the positions of the products in the containers need to be compared.
• The reason for using Big M in the constraints is to determine whether the product is taken or not, thereby releasing that constraint.

Big-M is a value used in mathematical programming (particularly linear programming and mixed-integer programming) to represent a very large number or sometimes as an arbitrarily chosen “sufficiently large” number. Big-M is often used in mathematical modeling to switch constraints on or off because it allows the solver to ignore constraints that do not apply to the current solution. For example, suppose we want to have a constraint on variable x in the form of x = 0 if some other decision variable y is equal to 0, or x > 0 if y = 1, we can model this using a Big-M formulation as: x ≤ My, where M is a sufficiently large constant. Then, if y = 0, the constraint becomes x ≤ 0, i.e., x = 0 (since x is nonnegative), and if y = 1, the constraint becomes x ≤ M, which does not restrict x (since x is nonnegative and M is very large), thus the Big-M formulation allows us to turn a constraint on or off based on the value of another variable [13].

• K = Container set (1, 2,…, m);
• L = Cost of container usage;
• G_ik = Profit; when i. product is added to k. container;
  ${lx}_{ik}$, ${ly}_{ik}$, ${lz}_{ik}$: k. binary variables showing whether the length of the box in the container is parallel to the x-axis, y-axis, and z-axis.

For example, if the length of the box in the container is parallel to the x-axis, then ${lx}_{ik}$ = 1; otherwise, it is 0.

• ${wx}_{ik}$, ${wy}_{ik}$, ${wz}_{ik}$: These are binary variables that define whether a product’s width direction within the container is parallel to the x, y, or z axis.
• ${hx}_{ik}$, ${hy}_{ik}$, ${ hz}_{ik}$: These are binary variables that define whether a product’s depth direction within the container is parallel to the x, y, or z axis.
• ${\alpha }_{ijk}$, ${ \beta }_{ijk}$, ${ \delta }_{ijk}$: These are variables that determine the positional relationship between products i and j.
• $p_i$, $q_i ,$ $r_i$: These are dimension parameters representing the length, width, and height of the box or product number i.

For example;

Alpha is considered relative to the z-axis; beta, relative to the x-axis; and theta, relative to the y-axis.

• ${\alpha }_{ijk}$: Is the box at the back (in the z-axis direction)?

• ${\alpha }_{ijk}$ = 1 → i box, behind j;
• ${\alpha }_{ijk}$ = 0 → otherwise.

• βijk: Is the box next to it (in the x-axis direction)?

• β_ijk = 1 → box i, to the right of j;
• β_ijk = 0 → otherwise.

• δijk: Is the box at the bottom (in the y-axis direction)?

• δ_ijk = 1 → box i, under j;
• δ_ijk = 0 → otherwise.

• α_ijk, β_ijk, δ_ijk = (0,0,1): Product i is positioned to the left of product j within container k.
• α_ijk, β_ijk, δ_ijk = (0,1,0): Product i is located to the right of product j in container k.
• α_ijk, β_ijk, δ_ijk = (1,0,0): Product i is placed behind product j in container k.
• α_ijk, β_ijk, δ_ijk = (0,1,1): Product i is arranged in front of product j inside container k.
• α_ijk, β_ijk, δ_ijk = (1,0,1): Product i is situated below product j in container k.
• α_ijk, β_ijk, δ_ijk = (1,1,0): Product i is positioned above product j in container k.

  $${\mathrm{a}}_{\mathrm{ik}}=\left\{\begin{array}{c}1,\hspace{0.25em} if product i\hspace{0.25em}added to container\hspace{0.25em}k\\ 0,\hspace{0.25em}otherwise\end{array}\right.\hspace{0.25em}$$

x_ik, y_ik, z_ik: Variables showing the coordinates of the lower left corner of the product in the container

$${\mathrm{t}}_{\mathrm{k}}=\left\{\begin{array}{c}1,\hspace{0.25em}if container k \mathrm{i}\mathrm{s} \mathrm{b}\mathrm{e}\mathrm{i}\mathrm{n}\mathrm{g} \mathrm{u}\mathrm{s}\mathrm{e}\mathrm{d},\\ 0,\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}otherwise\end{array}\right.\hspace{0.25em}$$

Objective function: max z =

$${\sum }_{i=1\hspace{0.25em}}^n{\sum }_{k=1}^m{G}_{ik}\hspace{0.25em}.\hspace{0.25em}{a}_{ik}-L.\hspace{0.25em}{\sum }_{k=1}^m{t}_k\hspace{0.25em}$$

(1)

s.t.

x_ik + p_ilx_ik+ q_iwx_ik + r_ihx_ik ≤ x_jk +M (1+ α_ijk + β_ijk − δ_ijk)

(2)

x_jk + p _jlx_jk+ q_jwx_jk + r_jhx_jk ≤ x_ik +M (1+ α_ijk − β_ijk + δ_ijk)

(3)

y_k + p_ily_ik+ q_iwy_ik + r_ihx_ik ≤ y_jk +M (1- α_ijk + β_ijk + δ_ijk)

(4)

y_jk + p _jly_jk+ q_jwy_jk + r_jhy_jk ≤ y_ik +M (2+ α_ijk − β_ijk − δ_ijk)

(5)

z_ik + p _ilz_ik+ q_iwz_ik + r_ihz_ik ≤ z_jk +M (2- α_ijk + β_ijk − δ_ijk)

(6)

z_jk + p _jlz_jk+ q_jwz_jk + r_jhz_jk ≤ z_ik +M (2- α_ijk − β_ijk + δ_ijk)

(7)

∀ i, j ϵ N, i< j, ∀ k ϵ K

Constraints (2)–(7) are formulated to ensure that each pair of products (i and j) can be placed within the same container without overlapping. The relative positions of two boxes are defined in three-dimensional space along the x, y, and z axes and can be represented by six possible spatial relationships: one box may be positioned in front of, behind, to the right of, to the left of, above, or below the other. All these constraints combined prevent two items from overlapping in the container, thereby maintaining the loading plan feasible.

1≤ α_ijk + β_ijk + δ_ijk ≤ 2

(8)

• Total: They must diverge on at least one axis → otherwise they overlap.
• Total ≤ 2: There should be no separation in three axes at the same time (unnecessary flexibility is prevented).

Container size restrictions:

x_ik + p _ilx_ik+ q_iwx_ik + r_ihx_ik ≤ x +M (1-a_ik)

(9)

y_ik + p _ily_ik+ q_iwy_ik + r_ihy_ik ≤ y +M (1-a_ik)

(10)

z_ik + p _ilz_ik+ q_iwz_ik + r_ihz_ik ≤ z +M (1-a_ik)

(11)

In addition, the variable a_ik is incorporated into constraints 12–17 to ensure logical consistency. Specifically, if a given product is not selected for loading, there is no practical need to assign spatial coordinates to it.

lx_ik + ly_ik+ lz_ik = a_ik ∀ i ϵ N, ∀ k ϵ K

(12)

wx_ik + wyik+ wz_ik = a_ik ∀ i ϵ N, ∀ k ϵ K

(13)

hx_ik + hy_ik+ hz_ik = a_ik ∀ i ϵ N, ∀ k ϵ K

(14)

lx_ik + wx_ik+ hx_ik = a_ik ∀ i ϵ N, ∀ k ϵ K

(15)

ly_ik + wy_ik+ hy_ik = a_ik ∀ i ϵ N, ∀ k ϵ K

(16)

lz_ik + wz_ik+ hz_ik = a_ik ∀ i ϵ N, ∀ k ϵ K

(17)

$${\sum }_{k=1}^m{a}_{ik}\le 1 \forall \mathrm{i} ϵ \mathrm{N}$$

(18)

$$a_{ik}\le t_k \forall \mathrm{i} ϵ \mathrm{N}, \forall \mathrm{k} ϵ \mathrm{K}$$

(19)

$$a_{ik}\hspace{0.25em}ϵ\hspace{0.25em}\{0,1\} \forall \mathrm{i} ϵ \mathrm{N}, \forall \mathrm{k} ϵ \mathrm{K}$$

$$t_k\hspace{0.25em}ϵ\hspace{0.25em}\{0,1\} \forall \mathrm{k} ϵ \mathrm{K}$$

$${\mathrm{lx}}_{\mathrm{ik}}, \dots \dots , {\mathrm{hz}}_{\mathrm{ik}} ϵ\hspace{0.25em}\{0,1\} 0 \forall \mathrm{i} ϵ \mathrm{N}, \forall \mathrm{k} ϵ \mathrm{K}$$

$${\mathsf{\alpha }}_{\mathrm{ijk}}, {\mathsf{\beta }}_{\mathrm{ijk}}, {\mathsf{\delta }}_{\mathrm{ijk}} ϵ\hspace{0.25em}\{0,1\} \forall \mathrm{i}, \mathrm{j} ϵ \mathrm{N}, \mathrm{i} <\mathrm{j}, \forall \mathrm{k} ϵ \mathrm{K}$$

x_ik, y_ik, z_ik ≥ 0 ∀ i ϵ N, ∀ k ϵ K

Let the products that must be in the base be in the S set. For example, if the 5th product and the 9th product must be in the base, S = {5,9} S ⊂ N

The reason we only write z is that we are only considering the base here.

• Zi = 0 means it is at the base.

• Priority product purchase constraint:
• The objective function requires increasing the coefficients of the relevant products.

Example: P₁ = 5,P₂ = 7, P₃ = 9, P₄ = 3

$$\max z=\hspace{0.25em}{\sum }_{i=1\hspace{0.25em}}^n{\sum }_{k=1}^m{G}_i\hspace{0.25em}.\hspace{0.25em}{a}_{ik}-L.\hspace{0.25em}{\sum }_{k=1}^m{y}_k\hspace{0.25em}$$

(20)

max z = 5a₁+ 7a₂ + 9a₃ + 3a₄

Let us say product i = 3 is a priority. Then, max z = 5a₁ + 7a₂ + 9000a₃ + 3a₄

In the proposed model, the coefficient of priority products within the objective function is increased in order to reflect their higher importance in the loading process. To achieve this, a weighting factor, denoted as M, is introduced for priority products. This correction will guarantee that the optimizer maximizes capacity and keeps a balance but gives priority to shipments with more operational or commercial relevance.

The constraint that products cannot be combined

Let us say that the 3rd and 8th, 7th and 12th, and 16th and 33rd products cannot be found together.

T = {(3, 8), (7, 12), (16, 33)}

T = {(i, j): i, j ϵ N}

a_ik + a_jk ≤ 1 ∀ i, j ϵ T ∀ k ϵ K

• If these products are purchased, they should be in this section:

For example: i = 8. Let the dimensions of this product be (7, 3.6).

This product must be within the coordinates x = [0, 20], y = [0, 10], z = [12, 30].

R ϵ {8} or R ϵ {8, 13, 27} and R ⊂ N

x= [x_mini, x_maxi], y = [y_mini, y_maxi], z = [z_mini,z_maxi,]

x_ik ≥ x_mini

y_ik ≥ y_mini

z_ik ≥ z_mini

x_i + p_i l_xi + q_i wxi + r_i h_xi ≤ x_maxi + M (1 − a_i) ∀ i ϵ R, R ⊂ N

y_i + p_i l_yi + q_i wyi + r_i h_yi ≤ y_maxi + M (1 − a_i) ∀ i ϵ R, R ⊂ N

z_i + p_i l_zi + q_i wzi + r_i h_zi ≤ z_maxi + M (1 − a_i) ∀ i ϵ R, R ⊂ N

The proposed model was initially solved using the linear programming approach implemented in GAMS IDE 2.0.36.7 (module GAMS Rev 148). However, in real-world cargo aircraft operations, the variety of products and the number of operational constraints are significantly greater than in the simplified model. Alone, linear programming is insufficient for solving real-world problems, especially under high dimensionality and complexity, because to provide strong solutions, heuristic and metaheuristic approaches that give the flexibility and scalability to handle realistic problem instances are required [15].

This table presents the LEVEL and MARGINAL values of selected decision variables obtained from GAMS.

In order to evaluate the performance of the proposed model, the optimization results obtained from GAMS are summarized. The table presents the variables, their respective index values, the optimal levels achieved in the solution, and the marginal values associated with each decision variable. These outputs provide insight into how the model behaves under different conditions and help identify the critical parameters that influence the overall efficiency of the air cargo loading problem. A detailed summary of the GAMS output is given in

Table 2. Summary of GAMS Output.

Variable	Index	Level	Marginal
1	0.1	1.0	8.0
1	0.2	1.0	10.0
2	0.1	1.0	17.0
2	0.2	1.0	12.0
3	0.1	1.0	13.0
3	0.2	1.0	18.0
4	0.1	1.0	16.0
4	0.2	1.0	11.0
5	0.1	1.0	16.0
5	0.2	1.0	14.0
6	0.1	1.0	12.0
6	0.2	1.0	14.0
7	0.1	1.0	20.0
7	0.2	1.0	13.0
8	0.1	1.0	12.0
8	0.2	1.0	11.0
9	0.1	1.0	16.0
9	0.2	1.0	27.0
10.1	0.0	1.0	11.0
10.2	0.1	1.0	10.0
1	0.0	1.0	−40.0
2	0.1	1.0	−40.0

.

2.2. Proposed Hybrid Approach

2.2.1. New Method Proposal

The proposed method addresses the challenges of multi-container loading problems in cargo aircraft, which are considerably more complex than classical single-container loading. This complexity arises from several factors. First, multiple Unit Load Devices (ULDs) of different sizes and capacities must be considered simultaneously. Second, strict weight, balance, and priority constraints apply, particularly in aviation, where the center of gravity and transport safety are of critical importance. Especially because of safety constraints, the problem is critical [8]. Third, shipments may be destined for different airports; hence, the chosen method should be flexible and computationally efficient enough to deal with the complexity of real air cargo operations and should take into account route-aware loading strategies.

2.2.2. Genetic Algorithm

In the proposed approach, the Genetic Algorithm (GA) represents each individual as a complete loading plan, including both the sequence of boxes and their container assignments [3]. New candidate solutions are generated through evolutionary operators such as crossover and mutation, where mutation operations may involve rotation or swapping of boxes. The fitness function is designed to evaluate multiple criteria simultaneously, including volume utilization, weight balance, and compliance with center-of-gravity restrictions.

GA is particularly suitable for cargo aircraft loading problems due to its flexibility in handling multiple containers and diverse operational constraints. For example, priorities such as unloading at intermediate airports or weight distribution constraints can be encoded directly into the algorithm. Additionally, domain-specific rules —such as placing heavier loads near the ends—can be integrated seamlessly into the solution framework.

However, GA also has certain weaknesses. In particular, its performance is sensitive to parameter settings, including population size, mutation rate, and crossover probability. Poor choices can result in premature convergence or suboptimal results, so careful parameter tuning is required, and, where possible, adaptive parameter tuning is recommended.

2.2.3. Large Neighborhood Search (LNS)

Large Neighborhood Search is a meta-heuristic optimization algorithm designed to improve solution quality by systematically exploring the solution space. The fundamental principle of LNS is to disrupt a substantial portion of the current solution and subsequently repair it, thereby allowing the algorithm to overcome local optima and identify better configurations. Since the method involves iteratively destroying a part of the solution and then repairing it, this type of approach is known as a destroy-and-repair heuristic.

2.2.4. Working Mechanism

The working mechanism of the Large Neighborhood Search (LNS) algorithm consists of a sequence of iterative steps. A first solution is constructed using a simple heuristic, such as First Fit or the Greedy Algorithm. Then, during the destruction phase, a certain percentage of the decisions (e.g., 10–30%) are removed or altered randomly. For example, in the case of loading cargo, a number of boxes are removed from their pallets, because this process allows for the exploration of different solutions. In the repair phase, the damaged part of the solution is reconstructed or optimized to restore feasibility. The resulting solution is then evaluated according to predefined acceptance criteria based on the objective function (e.g., cost, volume utilization, fuel savings). If the new solution performs better, it is accepted; otherwise, it may still be retained if it is considered promising, often through additional mechanisms such as simulated annealing. This destroy-and-repair cycle is repeated iteratively, allowing the algorithm to extensively explore the solution space and discover high-quality solutions by leveraging different destroy-and-repair strategies.

For example, in a test scenario involving 100 cargo boxes to be loaded into two aircraft containers, the algorithm initially generated a feasible solution using a greedy placement rule. During the destruction phase, approximately 20% of the boxes were randomly removed. The repair phase then reassigned these boxes to the containers, optimizing placement with respect to both weight balance and volume utilization. As a result, the revised solution improved capacity utilization from 86.5% to 91.2% while also reducing the center-of-gravity deviation by 15%. This example clearly demonstrates the strength of the iterative destroy-and-repair process, which enables the algorithm to escape local optima and achieve significant performance gains in real-world cargo loading.

The proposed optimization framework utilizes a Genetic Algorithm (GA) to solve the air cargo loading problem. The process begins with the initialization of a population representing potential loading plans, followed by a fitness evaluation based on criteria such as volume utilization, weight balance, and safety. Parent solutions are then selected for crossover to generate offspring, after which mutation operators, such as the rotation or swapping of boxes, are applied to enhance diversity. The new solutions are evaluated iteratively until a stopping criterion is met, and the best solution is finally returned. The overall GA workflow is illustrated in Figure 5.

The Large Neighborhood Search (LNS) algorithm is employed to enhance solution quality by iteratively destroying and repairing parts of a candidate solution. The procedure begins with the generation of an initial solution, which may be produced using greedy or heuristic methods. A portion of the solution is then destroyed (typically between 10–30%) and subsequently repaired by reassigning or optimizing the disrupted elements. Each repaired solution is evaluated based on capacity, balance, and cost criteria, and an acceptance rule determines whether the solution is retained. This destroy–repair cycle continues until the stopping condition is met, at which point the best solution is returned. The complete workflow of the LNS process is illustrated in Figure 6.

2.2.5. Advantages of the Method

One of the main advantages of the Large Neighborhood Search (LNS) approach is its ability to escape local optima. While algorithms that rely on small, incremental changes—such as traditional local search- tend to become trapped in suboptimal solutions, LNS employs larger and more disruptive modifications to explore new regions of the solution space. In addition, the destroy-and-repair mechanism of LNS is highly flexible and can be adapted to various problem domains, including route planning, cargo loading, and scheduling. This adaptability increases its applicability across a wide range of logistics and transportation problems. LNS is also especially suitable for real-world applications, where the time to make decisions is limited, since it can quickly produce high-quality solutions, which, although not guaranteed to be optimal, are effective and applicable in an operational setting.

2.2.6. Sustainability of the Method for Cargo Aircraft

The outcome of this study indicates that the proposed approach is particularly well-suited for cargo aircraft operations. In multi-container systems, the assignment and placement of containers are optimized simultaneously, ensuring that the available space is used efficiently. The approach proves highly effective in real-world applications where decision-making time is limited, enabling freight managers to generate feasible and practical loading plans within short timeframes. Furthermore, the method is especially advantageous in redistributing the load according to operational priorities, cargo volume, and aircraft balance requirements, thereby supporting both safety and efficiency in air cargo transportation.

3. Results

This thesis considers the capacity optimization for cargo aircraft and examines how genetic algorithms and Large Neighborhood Search (LNS) solve the problem. The proposed mathematical model was designed by incorporating constraints related to the Unit Load Devices (ULDs) as well as product-specific restrictions, to arrange cargo in the most suitable configuration under volume, weight, and balance conditions.

The results show that traditional linear programming is only suitable for small problems and becomes infeasible for larger, real-world problems due to computational complexity, whereas the heuristic and metaheuristic methods show much better scalability. GA is shown to be robust across a wide range of loading scenarios, and LNS can improve the solution quality by exploring larger neighborhoods. Therefore, the proposed hybrid framework, which combines GA and LNS, improves both solution quality and computational efficiency.

Computational experiments were conducted on a set of benchmark instances representing small-, medium-, and large-scale loading scenarios. In small cases with fewer than 20 items, Mixed-Integer Linear Programming (MILP) achieved optimal results in a reasonable time [11]. However, for medium instances with 50–100 items, MILP required excessively long computation times, while GA-LNS consistently reached near-optimal solutions within seconds. In large-scale instances with more than 200 items, the hybrid approach achieved an average capacity utilization of 92.4%, compared with 85.7% for GA and 83.2% for LNS alone. Moreover, the hybrid method reduced computation time by approximately 35% compared to GA and over 60% compared to MILP. These findings confirm that the proposed methodology provides a practical balance between solution accuracy and processing efficiency.

In addition to operational performance, the proposed model also contributes to environmental sustainability. By improving load balance and reducing unused space in the aircraft, the hybrid GA-LNS approach can lower fuel consumption and thus carbon emissions. Preliminary estimates suggest that the optimized load plans could reduce fuel usage by 5–7% per flight, aligning with sustainability targets highlighted by ICAO.

These findings are consistent with prior studies in the literature. For example, Ref. [2] emphasized the limitations of exact methods in large-scale air cargo load planning, while [14] demonstrated the benefits of hybrid heuristics in solving complex container loading problems. The results of this study extend those findings by confirming the advantages of combining GA and LNS in the specific context of air cargo aircraft loading.

The overall contribution of this study is a practical methodology that integrates mathematical modeling with advanced optimization techniques to solve the cargo aircraft loading problem. The hybrid approach not only offers improvement in computational efficiency but also supports cost reduction, operational efficiency improvement, and environmental objective promotion, thus resulting in practical and meaningful implications for the air cargo industry.

4. Limitations and Future Work

Since this study uses benchmark instances and simplifying assumptions, it is not able to fully capture the uncertainty and variability inherent in the real-world air cargo industry, and the proposed model does not account for stochastic elements such as sudden weather changes, last-minute cancellations, and unexpected maintenance. The proposed hybrid GA-LNS method has shown substantial potential, but it also relies heavily on parameter tuning, which may result in reduced robustness in very highly dynamic environments. In addition, due to computational limits, we could not test the model on very large industry-scale datasets, which may limit its direct practical use.

There are several directions for future research. The first is to incorporate stochastic programming or robust optimization into the proposed model to handle uncertainties in cargo arrival. Incorporating real-time information (e.g., live cargo tracking and real-time weather forecasting) into the optimization framework can increase the practicality and realism of the model. Another interesting direction is to apply the solution approach to large-scale real-world datasets from airlines and logistics companies to evaluate the generality of the approach and conduct a more thorough evaluation of the performance of the proposed solution approach. Lastly, we could explore automatic or self-tuning parameter setting methods for GA and LNS to improve their performance across various problem instances, therefore increasing the model’s overall efficiency. Finally, environmental indicators such as CO₂ emissions and fuel consumption can be incorporated into the model, and this will contribute to sustainable aviation policies so that the model will be more environmentally friendly.