Aviation container stowage optimization model and optimization method considering actual operation control and operation targets of airline

Abstract

According to the aviation container stowage optimization model and optimization method considering the actual operation and control target of the airline, the rising range of model complexity caused by the increase of constraint conditions is reduced, and the solving efficiency and quality of an optimal solution are improved. The model comprises a decision variable, a target function, a constraint condition, a fitness function and a termination condition; optimization is carried out in three stages, wherein optimization is carried out with the maximum load capacity as an objective function, and a to-be-optimized individual with the maximum load capacity and a first-stage optimal solution are obtained; carrying out second-stage optimization by taking the maximum pitching rotational inertia as an objective function and taking the to-be-optimized individual with the maximum loading capacity and the optimal solution as initial solutions to obtain the maximum reachable pitching rotational inertia under the condition of the maximum loading capacity; and carrying out third-stage optimization by taking the normalized minimum pitching rotational inertia and minimum gravity center deviation as objective functions and taking the to-be-optimized individual with the maximum loading capacity and the optimal solution as initial solutions to obtain an optimal loading plan with the minimum pitching rotational inertia and gravity center deviation under the condition of the maximum loading capacity.

Description

Technical Field

[0001] This invention relates to an aircraft container loading optimization model and method, specifically, an aircraft container loading optimization model and method that takes into account the actual operation and control objectives of airlines. Background Technology

[0002] Currently, the aviation industry has gradually entered the Industry 4.0 era, prioritizing improved flight operation efficiency, procedure optimization, and enhanced safety. However, improper unit loading devices (ULDs) can negatively impact aircraft safety, maneuverability, and economy, leading to a persistently high number of ground and air accidents or incident symptoms. Therefore, a practical and in-depth study of the weight and balance problem (WBP) of aircraft units is both urgent and necessary.

[0003] Current research on WBP mainly focuses on two aspects: mathematical models and algorithms. In terms of mathematical models, researchers mostly use MILP (Mixed-Integer Linear Planning) and ILP (Integer Linear Planning) models, with 0-1 MILP being the dominant approach. In terms of algorithmic research, researchers primarily use heuristic algorithms and artificial intelligence algorithms.

[0004] Mathematical models for aircraft containers are discussed. Paquay et al. treat air cargo loading as a three-dimensional container loading problem and propose a 0-1 MILP model to solve it. Limbourg et al. developed an MILP model to reduce container stacking at the target center of gravity of the aircraft. They set a zero-fuel center of gravity deviation of 0.01 as a constraint, aiming to minimize rotational inertia, thus reducing container stacking in the center of gravity direction, improving aircraft maneuverability, reducing fuel consumption, and lowering costs. Kaluzny et al. discussed the loading of non-standard air cargo in a militarized context, proposing an MILP for locating non-standard cargo. They considered container spacing and center of gravity envelope constraints as optimization objectives, but lacked consideration for other structural and safety constraints. Desai et al. studied the loading of aircraft containers on passenger-to-freighter conversions, considering four types of constraints: allocation constraints, maximum position weight limits, zero-fuel weight limits, and traditional constraints related to the aircraft structure, to reduce deviations from the optimal center of gravity (CG), and established a 0-1 MILP model.

[0005] This shows that most researchers use MILP and ILP models, with 0-1 MILP being the dominant one.

[0006] Algorithm research on the optimization problem of aircraft containers. Larsen et al. were the first to propose a computerized program to solve flight loading plans using heuristic algorithms. This heuristic method can establish a preliminary feasible plan within a ten-minute time limit. Feng et al. proposed a rule-based optimization method to improve operational efficiency in the context of multi-segment flights of passenger aircraft. This method is a heuristic method constructed from high-level business logic and optimization algorithms. Li proposed a large-scale neighborhood search heuristic method to optimize the total cost of cargo loading.

[0007] Regarding artificial intelligence algorithms: Wodziak et al. used traditional genetic algorithms to solve cargo loading problems to find the optimal loading position; Ma designed a hybrid genetic algorithm to optimize the cargo space in the belly of passenger aircraft; Zhao Hongfei et al. proposed a tabu neighborhood search optimization algorithm based on weight difference priority to solve the multi-objective multi-vehicle loading optimization problem; Dahmani et al. designed a multi-objective particle swarm optimization algorithm suitable for real-world case studies based on the optimization objectives of maximizing the total amount of cargo loaded and the overall priority. Experiments showed that the particle swarm algorithm is an effective method for solving the dual-objective aircraft cargo loading problem; Zhang Hong et al. used an improved ant colony algorithm to solve the container loading problem of the C919.

[0008] Limbourg's paper on container loading from the University of Liège in Belgium is one of the classic documents in the study of container loading problems in aviation, and it has been practically applied in airlines.

[0009] Limbourg developed a novel mixed-integer linear procedure for optimally loading a set of containers and pallets into a compartmented freighter. It is based on a real-world problem provided by an aviation partner, taking into account stringent technical and safety constraints. With the objective of minimizing rotational inertia, a zero-fuel center-of-gravity deviation of 0.01 is set as a constraint, reducing container stacking in the center-of-gravity direction, improving aircraft maneuverability, reducing fuel consumption, and lowering costs. Furthermore, cargo loading is a time-consuming manual task, or at best computer-aided. This paper also develops fully automated software to rapidly calculate the optimal solution.

[0010] The results show that Limbourg's solution can achieve better results in a few seconds to hundreds of seconds compared to manual planning.

[0011] The objective function, constraints, and solution methods in this paper are summarized in Table 1 below.

[0012]

[0013] Table 1. Summary of objective function, constraints, and solution methods in Limbourg's paper.

[0014] The following are the optimal load plans for the Boeing 747 aircraft under different real-world conditions.

[0015]

[0016] These existing technologies can be found in the following reference: Limbourg S, Schyns M, Laporte G. Automatic aircraft cargo load planning[J]. Journal of the Operational Research Society, 2012, 63(9): 1271-1283.

[0017] Based on the above analysis, the existing technology has the following shortcomings:

[0018] (1) The airline's operational control objective is to optimize and improve the aircraft's maneuverability and operational economy and profitability, with flight safety as the premise. The existing WBP solution does not define the corresponding optimization model objective function and optimization strategy according to the airline's operational control and operation objectives and their priorities, which leads to a decrease in the practicality of the optimal solution and makes it difficult to meet the needs of actual problems;

[0019] (2) The constraints considered in the publicly available WBP solutions are mostly focused on the ULD position, aircraft center of gravity and weight, while few solutions consider the matching relationship between the ULD and the aircraft cabin. This leads to the optimization model not matching the actual situation, thus reducing the practicality of the optimal solution.

[0020] (3) The WBP problem is quite complex due to the large number of ULDs and cabins, and its heterogeneous nature, generally belonging to the medium-to-large-scale NP problem. Existing WBP problem solving algorithms with good optimization performance are mostly heuristic methods and intelligent methods. The former includes branch and bound, greedy algorithms, etc., while the latter includes genetic algorithms and their various extensions. Heuristic methods are based on finite rules and mainly rely on human experience, and the optimization results are basically approximate optimal solutions; intelligent methods are easily affected by the initial solution and exhibit defects such as premature convergence and unstable convergence. Summary of the Invention

[0021] The purpose of this invention is to provide an aircraft container loading optimization model and method that considers the actual operational control objectives of airlines, so that the objective function and priority of the optimization model can better reflect the real situation of WBP (Warranty Business Plan). An adaptation matrix with binary elements is used to define the adaptation relationship between all types of ULDs and cabin classes. On the one hand, the type and number of 0-1 decision variables remain unchanged, allowing the new optimization model to reuse the objective function and constraint expressions from the original MILP model, thereby mitigating the increase in model complexity caused by the increase in constraints and reducing the negative impact on the efficiency of finding the optimal solution. On the other hand, using the ULD-cabin class adaptation relationship as a constraint, an initial solution generator based on random sampling is designed to improve the quality of the initial solution, thereby improving the efficiency and quality of finding the optimal solution.

[0022] The present invention adopts the following technical solution:

[0023] An optimization model for aircraft container loading that takes into account the actual operational control and operational objectives of airlines.

[0024] The optimization model includes decision variables, objective function, constraints, fitness function, and termination condition;

[0025] 1) Decision variables

[0026] A 0-1 matrix is ​​used to define the loading relationship between the container set and the bay set; the element values ​​of the 0-1 matrix are shown in the following formula:

[0027]

[0028] In the formula, It is the ULD number. It is the cabin number; Indicates the number of ULDs to be loaded; Indicates the number of locations that need to be loaded; It is a decision variable, representing the first... The first container and the first The loading relationship between each cabin; if the first... The container was loaded into the first In each cabin, The value is 1 if the value is 1 otherwise; the decision variable matrix is ​​defined as follows:

[0029]

[0030] 2) Objective function

[0031] The objective function and optimization are defined and carried out in three stages according to priority.

[0032] A. Objective function of the first stage

[0033] The objective function for maximizing load capacity is shown in the following formula:

[0034]

[0035] For the first The weight of each container;

[0036] With the maximum load capacity as the objective function, optimization is carried out under the condition that the constraints remain unchanged, to obtain the individual to be optimized with the maximum load capacity and the optimal solution in the first stage;

[0037] B. Objective function of the second stage

[0038] The objective function for maximizing the pitch rotation moment of inertia is shown in the following equation:

[0039]

[0040] In the formula It is the column vector of the balancing lever arm of the cabin. It is the first The balancing arm of each cabin; The core of the goal The lever arm value;

[0041] With the objective function being the maximum pitch moment of inertia, and under unchanged constraints, the second stage of optimization is carried out using the individual to be optimized with the largest load and the optimal solution obtained in the first stage as the initial solution, to obtain the maximum achievable pitch moment of inertia under the condition of the maximum load.

[0042] C. Objective function of the third stage

[0043] The objective functions for minimizing the normalized pitch inertia and the takeoff center of gravity deviation are shown in the following equation:

[0044] ,

[0045] in, These are weight values;

[0046] ,

[0047] This represents the maximum pitch inertia obtained in the second stage.

[0048] ,

[0049]

[0050] and These are the actual takeoff center of gravity and the target center of gravity, respectively, expressed as a percentage of mean aerodynamic chord length.

[0051] The maximum difference between each boundary of the takeoff center of gravity envelope and the target center of gravity; with the minimum normalized pitch inertia and the minimum center of gravity deviation as the objective functions, and with the individual to be optimized and the optimal solution with the maximum payload obtained in the first stage as the initial solution, the third stage of optimization is carried out under the condition that the constraints remain unchanged, to obtain the optimal loading plan with the minimum pitch inertia and center of gravity deviation under the condition of maximum payload.

[0052] 3) Constraints

[0053] The constraints include: position constraints, compartment and container compatibility constraints, maximum load limits, structural constraints, center of gravity envelope constraints, and lateral imbalance constraints.

[0054] As a further improvement, the aforementioned aircraft container loading optimization model,

[0055] When using a genetic algorithm to solve the optimization problem, the objective function in the first stage becomes:

[0056]

[0057] In the formula This refers to the maximum payload capacity of the aircraft under the given flight plan conditions.

[0058] The objective function for the second stage is transformed into:

[0059]

[0060] In the formula The pitch inertia of the entire aircraft when loading containers of maximum weight that meet adaptation constraints in each compartment.

[0061] As a further improvement, the aforementioned aircraft container loading optimization model,

[0062] Regarding the adaptation constraints between cargo slots and containers, the adaptation constraints between the container set and the cargo slot set are defined as follows:

[0063]

[0064] In the formula Indicates the first The first container and the first The compatibility relationship between cabin classes; if the two are compatible, then The value is 1 if it is not 1, and 0 otherwise; and if =0, It must be 0; if If it is 1, then It can be 0 or 1, and the constraint relationship between them can be expressed as:

[0065]

[0066] The above formula can be improved using relatively low-cost matrix operations, array element picking, and addition operations as follows:

[0067]

[0068] In the formula, SUM() and DIAG() are the array summation function and the diagonal element picking function of the two-dimensional array, respectively.

[0069] As a further improvement, the aforementioned aircraft container loading optimization model,

[0070] The fitness functions for the first, second, and third stages are as follows:

[0071] (45)

[0072] in,

[0073] (32)

[0074] (33)

[0075] (34)

[0076] (35)

[0077] (36)

[0078] (37)

[0079] (38)

[0080] (39)

[0081] (40)

[0082] (41)

[0083] (42)

[0084] (43)

[0085] (44)

[0086]

[0087] .

[0088] As a further improvement, the aforementioned aircraft container loading optimization model,

[0089] The optimization is terminated using no significant change in fitness and the maximum number of iterations, as described mathematically below:

[0090]

[0091] In the formula Let the threshold be the average change in the optimal fitness value over consecutive generations. The characteristic parameter for fitness change is continuous. The average change in the optimal fitness value across generations The calculation method is as follows:

[0092]

[0093] In the formula , , These are the current iteration number, the optimization stage number, and the maximum number of iterations, respectively. It is the first The first optimization phase The optimal fitness value of the generation. Set as 10%-20%.

[0094] As a further improvement, the aforementioned aircraft container loading optimization model,

[0095] For the center of gravity envelope constraint, the center of gravity envelope with the vertical axis representing gross weight and the horizontal axis representing %MAC is surrounded by four envelopes based on the center of gravity boundaries under four different weights: taxiing weight (TW), takeoff weight (TOW), landing weight (LW), and zero fuel weight (ZFW).

[0096] This invention also provides an aircraft container loading optimization method that considers the actual operational control and operational objectives of airlines. Using the aforementioned aircraft container loading optimization model, an improved genetic algorithm is employed to solve the model according to the following steps:

[0097] Step 1: Establish the initial population using an initial value generation method based on adaptation constraints;

[0098] Step 2: Calculate the fitness function and arrange the calculated fitness function values ​​in descending order of fitness.

[0099] Step 3: Adopt the optimal preservation strategy to retain the best individuals;

[0100] Step 4: When the random floating-point number is greater than the crossover probability, perform PBX crossover; when the random floating-point number is less than the mutation probability, perform uniform crossover and use a position-constrained mutation method to carry out individual gene mutation.

[0101] Step 5: Determine if the optimization termination condition is met. If it is met, output the optimal solution; otherwise, return to step 2.

[0102] As a further improvement, the initial population generation method for the aforementioned aircraft container loading optimization method is as follows:

[0103] A. when hour

[0104] ① Define the initial population as ,in The number of individuals in each generation of the population. The first individual in the initial population, with dimension 1. Assign the adaptation matrix to ,Right now ; ;

[0105] ②From The Extract the elements with a value of 1 from the column and construct a two-dimensional array; the first column of the array is the row number of the extracted element in the adaptation matrix, and the elements in the second column are all 1;

[0106] ③ Randomly select a row from the two-dimensional array in step ②, and set it as follows: ,in To randomly select elements in No. Row number in the column; The In the column, the first All other elements are set to zero;

[0107] ④ In The In the process, remove All other elements are set to zero; at this time, if The to If all elements in any column are zero, then remove steps ③ and ④. No. Column and number If the operation fails, return to step ③ and perform random sampling again; otherwise, proceed to step ⑤.

[0108] ⑤If If the condition is met, the process terminates, and the first individual of the initial population is generated; otherwise, proceed. Return to step ②;

[0109] The other individuals in the initial population are generated sequentially by referring to steps ①-⑤ above;

[0110] B. When hour

[0111] ① Define the initial population as ,in The number of individuals in each generation of the population. The first individual in the initial population, with dimension 1. Assign the adaptation matrix to ,Right now ; ;

[0112] ②From The Extract the elements with a value of 1 from the rows and construct a two-dimensional array as shown in the following formula; the first column of the array is the column number of the extracted element in the adaptation matrix, and all elements in the second column are 1;

[0113]

[0114] ③ Randomly select a row from the two-dimensional array in step ②, and set it as follows: ,in To randomly select elements in No. Column numbering in a row; The In the row, except for the first All other elements are set to zero;

[0115] ④ In The In the list, except All other elements are set to zero; at this time, if The to If all elements in any row are zero, then remove the steps in steps ③ and ④. No. row and number If the column operation fails, return to step ③ and perform random sampling again; otherwise, proceed to step ⑤.

[0116] ⑤If If the condition is met, the process terminates, and the first individual of the initial population is generated; otherwise, proceed. Return to step ②;

[0117] Following steps ①-⑤ above, generate the other individuals in the initial population in sequence.

[0118] As a further improvement, the initial solution generation methods for the second and third stages of the above-mentioned aircraft container loading optimization method are as follows:

[0119] The successful operation of the first phase yielded a solution that satisfies all constraints and achieves optimal load utilization, with the following dimension:

[0120] 1× ( (Based on the number of cabins), construct the initial population for the second phase. And after the second phase is completed, construct the initial solution for the third phase. Initial solutions for the second and third phases and An elite individual retention strategy is adopted, ensuring that the first individual retains the optimal solution from the first stage to guarantee at least one high-quality solution in the population. Let the individual that obtains the optimal solution from the first stage be denoted as . ,

[0121] The generation of the initial solution in the second and third stages can be expressed as follows: , ,in, This represents the first individual in the initial population of the second and third stages, which is also the first individual in the optimal solution of the first stage. Indicates the number of individuals in each generation of the population;

[0122] for and remove individual The solution is obtained by using a random perturbation generation strategy on the optimal solution of the first stage.

[0123] As a further improvement, the aforementioned aircraft container loading optimization method employs a position-constrained mutation method to carry out the specific process of individual gene mutation:

[0124] A.When hour

[0125] ① Convert the 0-1 decision variable matrix into a one-dimensional array The elements of this vector are ULD numbers, and the order of the elements is the cabin number;

[0126] ② Gene mutation, that is, when the ULD number in the one-dimensional array mutates, it needs to follow positional constraints. When the mutated gene is in the order of the one-dimensional array... The element is changing the cabin position. When placing a ULD, the potential value of that element should be a ULD number that does not appear in the one-dimensional array, i.e., satisfying the condition. serial number;

[0127] ③ Finally, in A value is randomly selected from the data as the value of the mutated gene.

[0128] B. When hour

[0129] ① Convert the 0-1 decision variable matrix into a one-dimensional array The elements of this vector are cabin numbered, and the order of the elements is ULD number;

[0130] ② Gene mutation, that is, when the cabin number in the one-dimensional array mutates, it needs to follow positional constraints. When the mutated gene is in the order of the one-dimensional array... The element that changes the first When placing a ULD in a compartment, the potential value of this element should be a compartment number that does not appear in the one-dimensional array, i.e., satisfying the condition... serial number;

[0131] ③ Finally, in A value is randomly selected as the value of the mutated gene.

[0132] In summary, the aircraft container loading optimization model and optimization method of the present invention, which considers the actual operational control objectives of airlines, includes:

[0133] (1) Quantification of airlines’ actual operational control objectives and their priorities

[0134] 1) First-stage objective function: Maximum load capacity

[0135]

[0136] 2) Second-stage objective function: Maximize pitch rotation inertia

[0137]

[0138] 3) Objective function for the third stage:

[0139] The objective functions for minimizing the normalized pitch inertia and the takeoff center of gravity deviation are shown in the following equation:

[0140] ,

[0141] in, These are weight values;

[0142] ,

[0143] This represents the maximum pitch inertia obtained in the second stage.

[0144] ,

[0145]

[0146] and These are the actual takeoff center of gravity and the target center of gravity, respectively, expressed as a percentage of mean aerodynamic chord length.

[0147] It represents the maximum difference between each boundary of the takeoff center of gravity envelope and the target center of gravity.

[0148] (2) Calculation of binary matrix reflecting the fit between ULD and cabin class and constraint compliance.

[0149] A ULD (Unified Loading Disposal Container) is a container designed by aircraft manufacturers to improve the efficiency and safety of air cargo loading and unloading, ensure flight safety, and prevent cargo loss, taking into account constraints such as aircraft load-bearing capacity, cargo hold internal space, and center of gravity offset limitations. The ULD-cargo space fit relationship is provided by the aircraft manufacturer and is a result of comprehensively considering constraints such as aircraft load-bearing capacity and cargo hold internal space; that is, the ULD-cargo space fit relationship is a comprehensive reflection of other common constraints. Therefore, a WBP solution that satisfies the ULD-cargo space fit relationship belongs to the feasible region.

[0150] 1) First, set the decision variables.

[0151]

[0152] If the The container was loaded into the first If the cabin class is a specific cabin, the decision variable is 1; otherwise, it is 0.

[0153] 2) Compatibility constraints between cargo hold and container

[0154] If container cabin The types that can be loaded, then =1, otherwise, Then it is 0.

[0155] The adaptation matrix Y is represented as:

[0156]

[0157] The matching relationship between aircraft cargo hold spaces and containers to be loaded, if =0, It must be 0, if If it is 1, then It can be 0 or 1, and can be expressed as:

[0158]

[0159] (3) Initial solution generator based on random sampling with ULD and cabin adaptation relationship as constraints.

[0160] The optimization model in this method is a 0-1 integer programming model, where the decision variables can only take the values ​​0 or 1. Therefore, all individuals in the population are numerically encoded for ease of solution. An initial value generator is established based on the fitness matrix, and the generated values ​​serve as the initial population, ensuring the fit between the container and the cabin. This improves the efficiency of the genetic algorithm and avoids generating many invalid solutions. Error genes may be generated during crossover and mutation; these errors can be penalized using a penalty function to improve the overall quality of the population.

[0161] (4) The genetic algorithm was improved in terms of initial value generation and convergence calculation. The overall process and flowchart of the improved genetic algorithm are as follows.

[0162] Step 1: Establish the initial population using an initial value generation method based on adaptation constraints;

[0163] Step 2: Calculate the fitness function and arrange the calculated fitness function values ​​in descending order of fitness.

[0164] Step 3: Adopt the optimal preservation strategy to retain the best individuals;

[0165] Step 4: When the random floating-point number is greater than the crossover probability, perform PBX crossover; when the random floating-point number is less than the mutation probability, perform uniform crossover and use a position-constrained mutation method to carry out individual gene mutation.

[0166] Step 5: Determine if the optimization termination condition is met. If it is met, output the optimal solution; otherwise, return to step 2.

[0167] The beneficial effects of this invention are as follows: By quantifying the flight safety, maneuverability, and economic objectives of airlines, this invention designs optimization strategies for multiple operational control objectives, enabling the objective function and its priority in the optimization model to better reflect the true situation of WBP (Warfare Bypass). Without increasing the types of decision variables or model complexity, a binary adaptation matrix is ​​used to define the adaptation relationship between all types of ULDs and specific aircraft cabins. Aircraft manufacturers have already comprehensively considered physical and geometric constraints such as aircraft load capacity and cargo hold internal space when defining the ULD-cabin adaptation relationship. Utilizing this prior knowledge, on the one hand, the binary ULD-cabin adaptation relationship matrix replaces spatial constraints such as aircraft load line density, cumulative load constraints, non-overlapping, and non-boundary constraints, reducing the complexity and difficulty of solving the optimization model; on the other hand, using the ULD-cabin adaptation relationship as a constraint, an initial solution generator based on random sampling is designed to improve the quality of the initial solution. Attached Figure Description

[0168] Figure 1This is a three-stage optimization strategy diagram;

[0169] Figure 2 It is the feasibility envelope of the aircraft's center of gravity;

[0170] Figure 3 This is a schematic diagram of the lateral unbalance constraint and the center of gravity of the ULD;

[0171] Figure 4 This is a flowchart of the population initial value generation process;

[0172] Figure 5 This is a diagram illustrating the cross operation;

[0173] Figure 6 It is a cross-flowchart;

[0174] Figure 7 This is a diagram illustrating the mutation operation;

[0175] Figure 8 This is a flowchart of the improved genetic algorithm;

[0176] Figure 9 , Figure 10 , Figure 11 These are the iterative curves of the three-stage improved genetic algorithm in cases 1, 2, and 3, respectively.

[0177] Figure 12 , Figure 13 , Figure 14 These are the cargo hold loading diagrams for Case 1, 2, and 3, respectively.

[0178] Figure 15 This is a load limit diagram for the main cargo hold area;

[0179] Figure 16 This is a load limit diagram for the lower cargo hold area;

[0180] Figure 17 , 18 Both are load limit diagrams for the upper and lower compartment joint line;

[0181] Figure 19 , Figure 20 These are the cumulative load limit diagrams for the front and rear cargo holds, respectively.

[0182] Figure 21 This is a centroid envelope verification diagram.

[0183] Figure 22 , Figure 23 , Figure 24 These are comparison charts showing the initial and optimized fitness values ​​for each run in the three stages of Case 1, 2, and 3.

[0184] Figure 25 These are the first-stage load capacity optimization results for Cases 1, 2, and 3;

[0185] Figure 26 , Figure 27 , Figure 28 These are the three-stage iteration diagrams for Case 1, 2, and 3, respectively. Detailed Implementation

[0186] This invention provides an optimized method for loading aircraft containers that takes into account the actual operational control and operational objectives of airlines. The invention is described below.

[0187] (1) The transformation between the actual operation control and operational objectives of airlines

[0188] In general, airlines' cargo transportation objectives are primarily to maximize revenue by improving transportation economics while ensuring flight safety, through optimizing aircraft maneuverability, deviation from the target center of gravity, and maximum load utilization. To ensure that the aircraft container loading optimization model and its results objectively and scientifically reflect the actual operational control and management objectives of airlines, this invention defines and quantifies flight safety, aircraft maneuverability and economics, and transportation revenue through the objective function and constraints of the optimization model, as follows:

[0189] 1) Flight safety

[0190] Flight safety is the primary objective of airline operations control and the foundation for ensuring normal operations. This invention uses a series of strong constraints to keep the aircraft within its normal performance range at all times, ensuring safety throughout the entire process from pushback, taxiing to landing. These strong constraints include, but are not limited to, aircraft payload limitations, volume constraints, cabin and container fit constraints, center of gravity envelope constraints (covering takeoff, zero-fuel, and landing center of gravity envelopes), structural constraints, and lateral imbalance constraints. These strong constraints are applied throughout all stages of the optimization strategy, ensuring that both the stage-optimal solution and the global optimal solution satisfy the aforementioned strong constraint conditions.

[0191] 2) Aircraft Directional Maneuverability: Aircraft maneuverability refers to an aircraft's ability to perform flight maneuvers to quickly change its state of motion within a short period of time. It is divided into speed maneuverability, altitude maneuverability, and directional maneuverability. The maneuverability described in this invention mainly relates to the pitch directional maneuverability of the aircraft. For transport aircraft, under the condition of constant load and engine performance, the better its pitch directional maneuverability, the more effective angle of attack it can obtain by changing the pitch angle in a short time, thereby obtaining greater lift to achieve obstacle clearance, which is crucial for the go-around of transport aircraft. This invention optimizes the pitch directional maneuverability of transport aircraft by defining and quantifying the objective function of minimizing pitch (yaw) rotational inertia. In addition, by using lateral imbalance constraints to limit the left-right load difference of the aircraft within a normal range, the aircraft can be guaranteed to have normal roll directional maneuverability.

[0192] 3) Transportation Economy and Revenue: Optimal transportation revenue is achieved by defining an objective function that maximizes aircraft payload. Based on this, transportation economy is optimized by setting a target takeoff center of gravity that aligns with the airline's economic interests and minimizing the deviation between the actual and target center of gravity. Simultaneously, based on the principles of energy conversion and work, minimizing the pitch inertia, while maintaining the same pitch maneuverability, further reduces the energy consumed by changes in aircraft pitch attitude. In other words, minimizing pitch inertia further reduces fuel consumption, improving transportation economy and ultimately reducing transportation costs and increasing airline revenue.

[0193] (2) Optimization strategies for airline operation control and operational objectives

[0194] This addresses the problem that existing methods transform multiple objective functions into a single objective through weighted summation. However, due to inconsistencies in the dimensions of the objectives and difficulties in determining the weights, the optimization process becomes insensitive to the objective function, and the optimization results fail to objectively reflect the true optimization intent. For example... Figure 1 As shown, this invention designs a three-stage optimization strategy based on the priorities of flight safety, profitability, maneuverability, and economy in the actual operation and control of airlines, as well as their relationship with aircraft load attribute parameters.

[0195] This strategy first aims to maximize the aircraft's load capacity, and uses safety-related constraints such as aircraft load capacity and center of gravity offset as strong constraints to obtain a ULD loading scheme that satisfies all strong constraints and maximizes load utilization.

[0196] Secondly, taking the ULD in the optimal loading scheme of the previous step as the loading object, and aiming at maximizing the pitch rotation inertia, the maximum pitch rotation inertia under the condition of maximizing load utilization was optimized while satisfying all safety constraints. Based on this maximum value, the pitch rotation inertia objective function was normalized. In this way, it can be ensured that the value of the normalized pitch rotation inertia objective function is accurately located in the [0,1] interval and has the same order of magnitude as other normalized objective functions;

[0197] Finally, with the goal of minimizing the normalized pitch inertia and the average aerodynamic chord length percentage of the center of gravity deviation from the target position, and using the ULD set in the maximum load utilization loading scheme as the loading object, under the condition of satisfying all safety constraints, the ULD loading scheme with the minimum pitch inertia and center of gravity deviation under the condition of maximum load utilization was optimized, thus realizing the actual operation control and operational goals of the airline to maximize maneuverability, economy and profitability with flight safety as the cornerstone.

[0198] (3) Optimize model construction

[0199] An optimization model is a mathematical description of the air cargo container loading problem. Formally, it includes functional relationships, equations, inequalities, vectors, matrices, and mathematical operations between these elements. From an operations research perspective, an optimization model consists of three parts: decision variables, objective function, and constraints. The correctness, completeness, and accuracy of the optimization model directly affect whether the details of the air cargo container loading problem are objectively, comprehensively, and accurately expressed, and also influence the accuracy and practicality of the optimal solution. For the convenience of subsequent descriptions of the solutions, the variables and symbols involved in this invention are defined as follows.

[0200] Number of ULDs to be loaded

[0201] Number of locations to be loaded

[0202] ULD Number

[0203] Cabin number

[0204] Adaptation

[0205] Adaptation Matrix

[0206] The decision variable represents the relationship between the i-th container and the j-th compartment, and takes the value 0 or 1.

[0207] No. The weight of each container

[0208] No. The balancing arm of each cabin

[0209] Target focus lever arm value

[0210] weight value

[0211] Target focus

[0212] Aircraft takeoff weight

[0213] No. Weight limit per cabin

[0214] Number of load-limiting zones

[0215] The first load-limiting area Weight limits in each area

[0216] No. The ULD falls in the first Line load in the line load limiting area

[0217] No. Limits for individual line load regions

[0218] No. The ULD fell in the front cargo hold. Regional load

[0219] Forward cargo hold Load limits for each region

[0220] No. The ULD fell into the rear cargo hold. Regional load

[0221] Rear cargo hold Load limits for each region

[0222] Aircraft empty weight lever arm

[0223] aircraft fuel tank lever arm

[0224] aircraft spare fuel tank lever arm

[0225] 1) Decision variables

[0226] A 0-1 matrix is ​​used to define the loading relationship between the container set and the bay set. The element values ​​of the 0-1 matrix are shown in the following formula:

[0227] (1)

[0228] In the formula, Indicates the first The first container and the first The loading relationship between the compartments. If the first... The container was loaded into the first In each cabin, The value is 1 if the value is true, and 0 otherwise. Therefore, the decision variable matrix can be defined as follows:

[0229]

[0230] 2) Objective function

[0231] The objective functions are defined in the three stages of the optimization strategy below.

[0232] A. Objective function of the first stage

[0233] The objective function for maximizing load utilization is shown in the following equation.

[0234]

[0235] Since an improved genetic algorithm is used to solve the optimization problem, the objective function needs to be transformed into minimizing the objective function to facilitate the subsequent definition of the fitness function. The objective function is transformed into:

[0236]

[0237] In the formula This is the maximum payload capacity of the aircraft under the given flight plan conditions. The calculation method can be found in the "Maximum Payload Limitation" section of "Constraints".

[0238] B. Objective function of the second stage

[0239] The objective function for maximizing pitch rotation inertia is shown in the following equation.

[0240]

[0241] In the formula This is the column vector of the balancing force arms of the cabin. For the same purpose, referring to the transformation of the objective function in the first stage, the objective function in the second stage is transformed as follows:

[0242]

[0243] In the formula The pitch inertia of the entire aircraft when loading containers of maximum weight that meet adaptation constraints in each compartment.

[0244] C. Objective function of the third stage

[0245] The objective function for minimizing the takeoff center of gravity deviation is shown in the following equation:

[0246]

[0247] In the formula and These are the actual takeoff center of gravity and the target center of gravity, respectively, expressed as a percentage of the mean aerodynamic chord length.

[0248] The objective functions of pitch rotation inertia and center of gravity deviation are normalized as follows:

[0249]

[0250] In the formula This represents the maximum pitch inertia obtained in the second stage.

[0251]

[0252] In the formula This is the maximum difference between each boundary of the takeoff center of gravity envelope and the target center of gravity, calculated based on the piecewise function of the center of gravity envelope of the taxiing weight in the "Center of Gravity Envelope Constraint".

[0253] In summary, the objective function for the third stage is defined as follows:

[0254]

[0255] 3) Constraints

[0256] A. Position constraints

[0257] Each compartment can only carry a maximum of one container, and the constraints are defined as follows:

[0258]

[0259] In the formula, SUM() is the array summation function. for The Column vector.

[0260] Each container can be placed in at most one compartment, and the constraints are defined as follows:

[0261]

[0262] In the formula for The Column vector.

[0263] B. Compatibility constraints between cargo space and container

[0264] The adaptation constraints between the container set and the bay set are defined as follows:

[0265]

[0266] In the formula Indicates the first The first container and the first The compatibility between cabin classes. If they are compatible, then... The value is 1 if it is not 1, and 0 otherwise. Furthermore, if... =0, It must be 0; if If it is 1, then It can be 0 or 1, and the constraint relationship between them can be expressed as:

[0267]

[0268] The fit constraint inequality expressed in the above formula has a time-consuming drawback, mainly in two aspects:

[0269] ① The constraint formula involves pure algebraic operations, requiring a complete iteration through all constraints to verify compliance. That is, the constraints expressed in the above formula are: The number of constraints increases with the number of containers and bays. When the problem reaches a large scale, this constraint expression will reduce optimization efficiency.

[0270] ② From a computer programming perspective, loop statements and comparison operations are needed to traverse and determine the compliance of adaptation constraints, with loop statements requiring double nesting. In terms of computational time, addition, subtraction, multiplication, and comparison operations are of the same order of magnitude and are fast; loop statements are relatively time-consuming, typically taking four to five times longer per loop than addition, subtraction, multiplication, and comparison operations, and this time increases dramatically with the number of containers and bays (the computational increase is equal to the increase in bays for each additional container).

[0271] Therefore, in order to reduce the time consumption of optimization calculations, this invention combines the advantages of built-in operation functions in computing program development tools such as Python, Visual Studio, and Matlab, which are more efficient than loop statements such as for and while. It uses relatively low-time matrix operations, array element picking, and addition operations to improve the above formula as follows:

[0272]

[0273] In the formula, SUM() and DIAG() are the array summation function and the diagonal element picking function of a two-dimensional array, respectively. To verify the rationality of the above formula, the proof is as follows:

[0274] The diagonal elements can be expressed by the following algebraic expression:

[0275]

[0276] because for A 3D matrix, the above expression has a total of elements. The elements participating in the above addition operation are... The possible values ​​can be summarized into three cases:

[0277]

[0278] The second condition in the above formula is not satisfied. Therefore, the compatibility constraint between the container and the compartment is not satisfied. For a pair of container and compartment combinations, the compatibility constraint can be further defined as follows:

[0279]

[0280] The above expression is an algebraic expression involving subtraction and multiplication operations. Since it considers the container and cabin combinations established using permutations and combinations as the object for constraint satisfaction judgment, it also involves traversal issues. Therefore, to simplify the number of operations, we define the following adaptation constraints for a single cabin:

[0281]

[0282] In the above formula, for The diagonal elements can be obtained using the function DIAG( To pick it up. The above formula can be further rewritten as:

[0283]

[0284] The number of diagonal elements in the above formula is Compared to previous adaptation constraint expression methods, it reduces The adaptation constraints can be uniformly defined using the following formula when all cabin class adaptation constraints are satisfied:

[0285]

[0286] When the fit constraint is not satisfied, the above equation does not hold, and The value of is negative. Therefore, in addition to serving as a constraint expression for the fit between the container and the cabin, the above formula can also be directly used to define the corresponding penalty function. See the "Fitness Function and Termination Condition" section of "Optimization Algorithms" for details.

[0287] C. Maximum load limit

[0288] Maximum payload limits are typically determined by the maximum zero-fuel weight (MZFW), maximum takeoff weight (MTOW), and maximum landing weight (MLW), and these three weights satisfy the following constraints:

[0289]

[0290] Where OEW is the aircraft's operating empty weight, FW represents the aircraft's takeoff fuel capacity, and RFW represents the aircraft's reserve fuel capacity.

[0291] The actual maximum load limit is;

[0292]

[0293]

[0294] in, It is a system set by airlines based on their specific circumstances and in conjunction with the aircraft's load and balance manual, which differs from [previous system]. The maximum workload.

[0295] Similarly, to improve computational efficiency, the above algebraic expression is rewritten as a matrix operation formula as follows:

[0296]

[0297] D. Structural constraints

[0298] Aircraft structural constraints can be categorized into cabin load constraints, area load constraints, upper and lower cabin joint line load constraints, forward cargo hold cumulative load constraints, and aft cargo hold cumulative load constraints.

[0299] ① Compartment load constraints: The weight of the container placed in each compartment must not exceed the maximum load of that compartment;

[0300]

[0301] In the formula for The Column vector.

[0302] ②Regional load constraints: The total weight of containers loaded on a compartment in a certain region shall not exceed the maximum load of that region;

[0303]

[0304] In the formula, It is the first Submatrices of decision variables corresponding to all cabin classes in a region. Definition It is the first The cabin number vector in each region can be determined according to... The cabin class numbers are sequentially from Extract the corresponding column vectors to construct .

[0305] ③ Upper and lower cabin joint line load constraints: Based on the joint line load area division provided in the aircraft manual, establish a cabin number vector for each line load area, using... Indicates. The subscript is used to indicate. This refers to the numbering of the line load constraint area. Based on the correspondence between the actual loaded compartments of each container and the line load constraint area, these are sequentially numbered from... Extract the corresponding column vectors to construct The load constraints for the upper and lower compartment joint line are defined as follows:

[0306]

[0307] In the formula All containers fall into the first Line load vector of each line load region It refers to the number of load constraint areas of the upper and lower compartments combined.

[0308] ④ Cumulative load constraint in the forward cargo hold (from nose to rear): The cumulative load limit along the longitudinal axis of the fuselage is considered for both the upper and lower cargo holds, and is specifically defined as follows:

[0309]

[0310] In the formula All containers landed in the forward cargo hold. The load vector of the region. Definition It is the first The cargo hold number vector within the cumulative load constraint area of ​​each forward cargo hold (from nose to rear) is used sequentially, based on the correspondence between the actual loaded cargo hold of each container and the cumulative load constraint area of ​​the forward cargo hold. Extract the corresponding column vectors to construct . It is the number of cumulative load constraint areas in the forward cargo hold (from the nose to the rear).

[0311] ⑤ Cumulative load constraint in the aft cargo hold (from tail to front): The cumulative load limit along the longitudinal axis of the fuselage is considered for both the upper and lower cargo holds, and is specifically defined as follows:

[0312]

[0313] In the formula All containers were placed in the rear cargo hold. The load vector of the region. Definition It is the first The cargo hold number vectors within the cumulative load constraint area of ​​each aft cargo hold (from tail to front) are sequentially determined based on the correspondence between the actual loaded cargo hold of each container and the cumulative load constraint area of ​​the aft cargo hold. Extract the corresponding column vectors to construct . It is the number of cumulative load constraint areas in the rear cargo hold (from tail to front).

[0314] E. Centroid envelope constraint (centroid deviation constraint)

[0315] ① Calculation of MAC centroid

[0316] The takeoff center of gravity balance arm, landing balance arm, and zero-fuel balance arm of the aircraft are calculated as follows:

[0317]

[0318]

[0319]

[0320]

[0321]

[0322] In the formula The weight of fuel consumed by the aircraft during taxiing from pushback to takeoff; , , and These are the aircraft's taxiing balance arm, takeoff balance arm, landing balance arm, and zero-fuel balance arm. It is the balance arm corresponding to the operating empty machine weight. It is the overall balancing lever arm of the fuel. This refers to the total weight of the fuel. and The center of gravity changes throughout the entire process from pushback and taxiing to safe landing, and this change has a significant and non-negligible impact on the aircraft's center of gravity, especially during flight and landing. However, the above formula does not define the center of gravity during flight. Therefore, to simulate the impact of this change on the overall aircraft center of gravity, this invention extracts several key points between takeoff fuel weight and landing fuel weight, defining a fuel mass vector. (Fuel mass), calculated based on the initial fuel distribution. , in the assumption Assuming the above takeoff center of gravity expression remains unchanged, the overall aircraft center of gravity balance lever vector at key fuel weight points during flight can be calculated using the above takeoff center of gravity expression. , where s is the number of key points, and is substituted into the subsequently defined center of gravity offset constraint inequality to limit the aircraft's center of gravity offset during flight and ensure flight safety.

[0323] To ensure aircraft balance and safety, the aircraft's center of gravity must be kept within a safe range of its center of gravity envelope. In practical use, the horizontal axis of the aircraft's center of gravity envelope is usually represented as %MAC:

[0324]

[0325] In the formula, MAC represents the mean aerodynamic chord length, %MAC represents the aircraft's center of gravity as a percentage of the mean aerodynamic chord length; BA represents the balance arm length of the aircraft's center of gravity after container loading, namely the balance arm length for taxiing, takeoff, landing, zero-fuel, and flight; LEMAC is the balance arm length of the leading edge of the mean aerodynamic chord. Substituting the previously calculated takeoff, landing, zero-fuel, and flight balance arm lengths into the above formula yields the corresponding values. , , , and ,in .

[0326] ② Definition of centroid envelope constraint

[0327] Figure 2 This is the center of gravity envelope of a certain type of aircraft. The vertical axis represents gross weight, and the horizontal axis represents %MAC. The center of gravity boundaries under four different weights—taxi weight (TW), takeoff weight (TOW), landing weight (LW), and zero fuel weight (ZFW)—form four envelopes.

[0328] In order to Figure 2 The center of gravity envelope is converted into a center of gravity offset constraint. The piecewise function of the center of gravity (%MAC) with respect to the total aircraft weight is calculated for the four weight cases as follows:

[0329]

[0330]

[0331]

[0332]

[0333] In the formula, the weight intervals in the four function expressions are determined by the inflection points on the corresponding envelope lines. For each weight interval, the function expression for the center of gravity is divided into an upper limit function and a lower limit function. By substituting the known actual TW, TOW, LW, and ZFW values ​​into the piecewise function expressions for the center of gravity of the three different weight aircraft, the corresponding upper and lower limit %MAC of the center of gravity can be calculated. , , , and , , , .

[0334] Based on this, the envelope constraints for the takeoff center of gravity, landing center of gravity, and zero-fuel center of gravity are defined as follows:

[0335]

[0336]

[0337]

[0338]

[0339]

[0340] F. Lateral Imbalance Constraints

[0341] This invention assumes that all loaded containers are standard containers, and that the center of gravity is the same as the centroid. Therefore, in the horizontal plane, the geometric center of gravity of the compartment coincides with the center of gravity of the container. Figure 3 As shown. Therefore, the lateral balance of an aircraft can be controlled by limiting the weight difference between the left and right sides.

[0342]

[0343] and These are the sets of numbers for the right-hand cargo hold and the set of numbers for the left-hand cargo hold, respectively. This is a loading relationship matrix between the right-side compartments and the container, which can be viewed... Based on The result is obtained by setting all column elements to zero. This is a matrix showing the loading relationship between the left-side compartments and the container, which can be viewed... Based on The result is obtained by setting all column elements to zero. For lateral unbalanced weight limitations.

[0344] (4) Optimization algorithm improvement

[0345] This invention improves the initial population generation method, fitness function definition method, and mutation operation method of the genetic algorithm to prevent premature convergence and enhance the convergence speed and quality of the optimal solution. The specific solution is as follows:

[0346] 1) Initial population generation method

[0347] See Figure 4 The flowchart shown illustrates the process of generating initial values ​​for the human population.

[0348] To improve the speed of generating high-quality initial solutions, this invention generates individuals in the initial population based on the fitness matrix in two different cases, with the specific steps as follows:

[0349] A. when hour

[0350] ① Define the initial population as ,in The number of individuals in each generation of the population. The first individual in the initial population, with dimension 1. Assign the adaptation matrix to ,Right now ;

[0351] ②From The (Initially, Extract the elements with a value of 1 from the column, such as Figure 4 As shown, a two-dimensional array is constructed. The first column of this array is the row number of the extracted element in the adaptation matrix, and the elements in the second column are all 1s.

[0352] ③ Randomly select a row from the two-dimensional array in the previous step, such as... ,in To randomly select elements in No. The row number in the column. For example... Figure 4 As shown, in order to ensure that individuals satisfy constraint 1 and constraint 2, in The In the column, the first All other elements are set to zero;

[0353] ④ Similarly, in order to satisfy the first two constraints, in The In the process, remove All other elements are set to zero. At this point, if... The to If all elements in any column are zero, then remove the steps in steps three and four. No. Column and number If the operation fails, return to step 3 and perform random sampling again; otherwise, proceed to the next step.

[0354] ⑤If If the process succeeds, the process terminates, and the first individual of the initial population is generated. The other individuals in the initial population are then generated sequentially using the same method; otherwise, the process continues. Now, let's go back to step two.

[0355] B. When hour

[0356] ① Define the initial population as ,in The number of individuals in each generation of the population. The first individual in the initial population, with dimension 1. Assign the adaptation matrix to ,Right now ;

[0357] ②From The (Initially, Extract the elements with a value of 1 from the rows, and construct a two-dimensional array as shown in the first case. The first column of this array is the column number of the extracted element in the adaptation matrix, and all elements in the second column are 1.

[0358]

[0359] ③ Randomly select a row from the two-dimensional array in the previous step, such as... ,in To randomly select elements in No. Column number within a row. For example... Figure 4 As shown, in order to ensure that individuals satisfy constraint 1 and constraint 2, in The In the row, except for the first All other elements are set to zero;

[0360] ④ Similarly, in order to satisfy the first two constraints, in The In the list, except All other elements are set to zero. At this point, if... The to If all elements in any row are zero, then remove the steps in steps three and four. No. row and number If the column operation fails, return to step 3 and perform random sampling again; otherwise, proceed to the next step.

[0361] ⑤If If the process succeeds, the process terminates, and the first individual of the initial population is generated. The other individuals in the initial population are then generated sequentially using the same method; otherwise, the process continues. Now, let's go back to step two.

[0362] 2) Fitness function and termination condition

[0363] As can be seen from the previous definition of constraints, different types of constraints involve different numbers of restrictions. For example, the number of constraints in an adaptation constraint is... The maximum payload limit has only one condition. Since all the above constraints are strong constraints, they must be satisfied simultaneously to ensure container loading and flight safety; therefore, the importance of each type of constraint tends to be equal. In this case, if the inconsistent number of constraints is ignored and the penalty functions for each type of constraint are defined independently, it will inevitably lead to significant differences in the value ranges of each penalty function, and consequently, differences in the weights of each type of constraint. Therefore, based on a comprehensive consideration of the objective function's value range and the number of constraints, this invention normalizes the definition of the penalty functions for each type of constraint as follows:

[0364] (32)

[0365] (33)

[0366] (34)

[0367] (35)

[0368] (36)

[0369] (37)

[0370] (38)

[0371] (39)

[0372] (40)

[0373] (41)

[0374] (42)

[0375] (43)

[0376] (44)

[0377]

[0378]

[0379] In summary, based on the definitions of the objective functions for each stage outlined above, the fitness functions for the three stages are defined as follows:

[0380] (45)

[0381] To prevent getting trapped in local optima and to improve the stability and quality of the optimal solution, this invention uses both no significant change in fitness and the maximum number of iterations as termination conditions for optimization. The parameter representing the change in fitness is continuous. The average change in the optimal fitness value across generations The calculation method for this parameter is as follows:

[0382]

[0383] In the formula , , These are the current iteration number, the optimization stage number, and the maximum number of iterations, respectively. It is the first The first optimization phase The optimal fitness value of the generation. Therefore, the mathematical description of the optimization termination condition is defined as follows:

[0384]

[0385] In the formula This parameter is set as the threshold for the average change in the optimal fitness value over consecutive generations. Considering the range of values ​​for the objective function and the penalty function, this parameter is generally set to... And will Set as 10%-20%.

[0386] 3) Selecting Operators

[0387] Selection operations are based on the evaluation of individual fitness, and their purpose is to improve the computational efficiency and global convergence of the algorithm. Commonly used selection operators include proportional selection, optimal preservation strategy, and sorting selection strategy.

[0388] To improve the global convergence and computational efficiency of the algorithm and avoid destroying the chromosome with the highest fitness during evolution due to genetic operations, the improved genetic algorithm of this invention adopts an optimal preservation strategy to perform a survival-of-the-fittest operation on the model. The specific operation process is as follows:

[0389] A. Sort the current group according to fitness, and the top 10% with the best fitness directly enter the next generation;

[0390] B. Randomly generate 10% of the individuals and combine them with the remaining 90% of the individuals to form a new population. Sort the individuals in the population from best to worst fitness value, and select the top 90% of the individuals by fitness value for crossover and mutation operations.

[0391] C. Offspring individuals consist of the top 10% of the fittest individuals from A and individuals resulting from crossover mutations in B.

[0392] 4) Crossover and position-constraint-based variation

[0393] To avoid duplicate genes during the crossover process, position-based crossover (PBX) is chosen. This method results in higher fitness of each new chromosome and avoids duplicate genes in the chromosomes after crossover. See also Figure 5 The specific operation process is as follows:

[0394] Step 1: Randomly generate a floating-point number between 0 and 1. If the number is greater than the given crossover probability, then the parent chromosome and the paired chromosome will undergo position-based crossover.

[0395] Step 2: As Figure 5 As shown, several genes from the parent generation are randomly selected as non-crossover genes, and their positions do not have to be consecutive;

[0396] Step 3: Generate a child generation. In the child generation, genes at non-crossover positions are the same as those in the parent generation.

[0397] Step 4: Retain the elements in the paired parent generation that are the same as the non-crossing genes, and move the other elements to the crossing gene positions in the offspring in their original order.

[0398] The program's execution logic is as follows: Figure 6 As shown.

[0399] Positional constraints exist between row vector elements and column vector elements of decision variables. Therefore, directly altering individual genes (elements of the decision variable matrix) through random mutation in existing genetic algorithms may lead to a decrease in the quality of mutated individuals, resulting in fewer high-quality individuals in the population, causing slow convergence and reduced optimal solution quality. To address this issue, this invention proposes a mutation method based on positional constraints. This method determines the value of mutated genes according to positional constraints, effectively preventing situations where mutated genes do not meet positional constraints. This method has a positive effect on improving the quality of individuals in the population. See [link to related documentation]. Figure 7 The specific plan is as follows:

[0400] A.When hour

[0401] ①For example Figure 7 As shown, the 0-1 decision variable matrix is ​​transformed into a one-dimensional array. The elements of this vector are ULD numbers, and the order of the elements is the cabin number;

[0402] ② Gene mutation, that is, when the ULD number in a one-dimensional array mutates, it needs to follow positional constraints, such as Figure 7 As shown, when the mutated genes are in the order of a one-dimensional array... Elements (i.e., changing cabin class) When placing a ULD on top, the potential value of the element should be the ULD number that does not appear in the one-dimensional array, i.e., satisfying the condition. Numbering ensures that mutation operations simultaneously satisfy the constraints. and ;

[0403] ③ Finally, in A value is randomly selected as the value of the mutated gene.

[0404] B. When hour

[0405] ①For example Figure 7 As shown, the 0-1 decision variable matrix is ​​transformed into a one-dimensional array. The elements of this vector are cabin numbered, and the order of the elements is ULD number;

[0406] ② Genetic mutation, that is, when the cabin number in a one-dimensional array mutates, it needs to follow positional constraints, such as Figure 7 As shown, when the mutated genes are in the order of a one-dimensional array... The element (i.e., changing the first) When placing a ULD in a specific compartment, the potential value of this element should be a compartment number that does not appear in the one-dimensional array, i.e., satisfying the condition... Numbering ensures that mutation operations simultaneously satisfy the constraints. and ;

[0407] ③ Finally, in A value is randomly selected as the value of the mutated gene.

[0408] 5) Initial solution generation methods for the second and third stages

[0409] The successful operation of the first phase yielded a solution that satisfies all constraints and achieves optimal load utilization, with the following dimension:

[0410] 1× ( (Based on the number of cabins), the initial population for the second phase begins to be constructed. And after the second phase is completed, construct the initial solution for the third phase. To prevent the degradation of the initial solution quality in the second and third stages, the initial solutions in the second and third stages... and An elite individual retention strategy is adopted to ensure that the first individual retains the optimal solution from the first stage, thus guaranteeing at least one high-quality solution in the population. The individual that yielded the optimal solution in the first stage of this paper is... .

[0411] The generation of the initial solution in the second and third stages can be expressed as follows: , ,in, This represents the first individual in the initial population of the second and third stages, which is the first individual in the optimal solution of the first stage. This indicates the number of individuals in each generation of the population. In this paper, the number of individuals in the three stages is 100.

[0412] for and remove individual The solution is obtained by using a random perturbation generation strategy on the optimal solution of the first stage.

[0413]

[0414] In the formula, For the initial population of the second and third stages Individual, For the perturbation operator, For the first Individual perturbation parameters.

[0415] Perturbation operator Under the premise of ensuring the feasibility and quality of the solution, differentiated individuals are generated through local exchanges with limited settings. In the context of container loading, this means that, under the constraint of the adaptation matrix between the compartment and the container, the positions of any two containers are randomly swapped multiple times, and the perturbation operator is used. Defined as:

[0416]

[0417] In the formula, For swap operation, For parameter-based The generated swap set.

[0418] Disturbance parameters Including disturbance intensity and disturbance range The relationship is as follows:

[0419]

[0420] In the formula, In the second phase In the third stage . , Indices representing two cabin classes. This means that both indexes are between 1 and 1. Within the range, This indicates that the two cabin classes are different and an exchange must be made between the two different cabin classes.

[0421] 6) Improve the overall process of genetic algorithms

[0422] Step 1: Establish the initial population using an initial value generation method based on adaptation constraints;

[0423] Step 2: Calculate the fitness function and arrange the calculated fitness function values ​​in descending order of fitness.

[0424] Step 3: Adopt the optimal preservation strategy to retain the best individuals;

[0425] Step 4: When random floating-point numbers Greater than the crossover probability When the random floating-point number is less than the mutation probability, perform PBX crossover; At that time, uniform crossover was performed, and a position-constrained mutation method was used to carry out individual gene mutations;

[0426] Step 5: Determine whether the optimization termination condition is met. If it is met, output the optimal solution; otherwise, return to step 2.

[0427] Draw the overall flowchart of the improved genetic algorithm based on each step of the above design, as follows: Figure 8 As shown.

[0428] The following examples further illustrate the present invention's method for optimizing aircraft container loading that takes into account the actual operational control objectives of airlines.

[0429] I. Instance-related data

[0430] The cargo aircraft used for loading the container in this invention is a B747-400F, and the data used are all derived from the B747-400F weight and balance control and loading manual. In the B747-400F, , The weight limits for the B747-400F are shown in Table 2.

[0431]

[0432] Table 2 Weight Limitation Data for B747-400F

[0433] The load limits for the upper and lower compartment joint line are shown in Table 3.

[0434]

[0435] Table 3 Load Limits for Upper and Lower Cabin Joint Line

[0436] The cumulative load limits for the front and rear cargo holds are shown in Table 4.

[0437]

[0438] Table 4 Cumulative Load Limits for Fore and Aft Cargo Compartments

[0439] A specific example of this invention is as follows: A courier company books a cargo plane with a cooperating airline to load goods. The plane is a one-way transport from city A to city B, and the aircraft model is B747-400F. Taking a container from a certain cargo manifest as an example, we randomly select the transport data of the airline's B747-400F on a certain day. The takeoff fuel volume is 101,500 kg, and the range fuel volume is 82,300 kg. We take three containers of different sizes (large, medium, and small) as examples to verify the ability of the model and algorithm of this invention to handle the container loading problem. Each container type is applicable to loading on a B747-400F.

[0440] This example requires loading three batches of containers of different sizes into the cargo hold and performing three optimizations: maximum load capacity experiment, maximum pitch inertia experiment, minimum pitch inertia experiment, and minimum center of gravity offset experiment. Then, the impact on aircraft safety, maneuverability, and economy is analyzed.

[0441] The optimization process is as follows:

[0442] (1) First stage: Taking the maximum load capacity as the objective function, optimization is carried out under the condition that the constraints remain unchanged, and the individual to be optimized with the maximum load capacity and the optimal solution of the first stage are obtained to enter the next stage of optimization.

[0443] (2) Second stage: Taking the maximum pitch moment of inertia as the objective function, under the condition that the constraints remain unchanged, the second stage of optimization is carried out with the individual to be optimized and the optimal solution with the maximum load obtained in the first stage as the initial solution, so as to obtain the maximum achievable pitch moment of inertia under the maximum load.

[0444] (3) Third stage: The maximum achievable pitch moment of inertia under the maximum load obtained in the second stage can be used for objective function normalization in the third stage. The third stage takes the minimum pitch moment of inertia and the minimum center of gravity deviation as objective functions, and takes the individual to be optimized and the optimal solution with the maximum load obtained in the first stage as the initial solution, and carries out the optimization in the third stage under the unchanged constraints. Finally, the optimal loading plan with the minimum pitch moment of inertia and center of gravity deviation under the maximum load is obtained.

[0445] In the calculations of this invention, the target centroid %MAC was set to 26. Key program variables were set as follows: population size was 100. To improve the accuracy of the results, the maximum number of iterations was increased to 3000, the crossover probability was 0.9, the mutation probability was 0.1, and each case was run independently 100 times, with the average time taken. Optimization was performed alternately with the objectives of maximizing load capacity, maximizing pitch inertia, minimizing pitch inertia, and minimizing centroid shift.

[0446] II. Results and Analysis

[0447] To further illustrate the effectiveness of the model and algorithm proposed in this invention, loading optimization was performed on standard containers of different sizes. The optimization results for three objectives (Case 1, 2, and 3) are shown in Tables 5-7. The iterative curves of the improved genetic algorithm are shown in the figure. Figure 9-11 As shown.

[0448]

[0449] To verify the correctness of the optimized loading scheme, we need to verify whether the results meet the constraints.

[0450] (1) Position constraints

[0451] according to Figure 12-14 As shown in the loading scheme, each compartment loads one container, and each container is loaded onto one compartment. The loading scheme is verified to meet the adaptation constraints using the adaptation matrix Y generated during loading. According to the adaptation relationship comparison, when Y is 0, the result is 0; when Y is 1, the result is either 0 or 1, indicating no conflict and thus meeting the adaptation constraints.

[0452] (2) Regional load constraints, combined upper and lower compartment load constraints, cumulative load constraints of front and rear cargo compartments, and center of gravity envelope constraints.

[0453] like Figure 15 , 16 As shown, the regional load limits are 11450kg, 36624kg, 63140kg, 56907kg, 2041kg, 27669kg and 26081kg. The actual regional loads loaded in the three sets of experiments are all below the maximum regional load limit, thus satisfying the constraints.

[0454] like Figure 17 , 18 As shown, the combined line load limits for the upper and lower compartments are 38.5 kg / in, 90.7 kg / in, 131.5 kg / in, 90.7 kg / in, 77.1 kg / in and 16.3 kg / in. The actual line loads loaded in the three sets of experiments did not exceed the combined line load limits for the upper and lower compartments, thus satisfying the constraints.

[0455] like Figure 19 ,20 As shown, the limits for cumulative load in the forward cargo hold are 9176 kg, 13675 kg, 49916 kg, 76044 kg and 83057 kg, and the limits for cumulative load in the aft cargo hold are 83057 kg, 56930 kg, 13825 kg and 2041 kg. The actual loading calculations for the three sets of experiments show that the cumulative loads in both the forward and aft cargo holds do not exceed the limits, thus satisfying the constraints.

[0456] like Figure 21 As shown, the three sets of experimental results and the three centroids are all within the envelope formed by MTOW, MLW and MZFW and the %MAC limit, satisfying the constraints.

[0457] (3) Lateral imbalance constraint

[0458] according to Figure 12-14 As shown in the loading diagram, the weight differences of the three loading experiments are 6335kg, 242kg, and 3574kg, respectively, which do not exceed 12648kg, thus satisfying the constraints.

[0459] 1. Initial solution quality

[0460] To demonstrate the effectiveness of improving the quality of the initial solution in enhancing the stability and efficiency of the genetic algorithm optimization process, this invention designs three case sizes: large, medium, and small. For each case, optimization is performed 100 times in three optimization stages. The initial and optimized fitness function values ​​corresponding to the optimal solution in each stage of each case are compared as follows: Figure 22-24 As shown:

[0461] To evaluate the improvement of the fitness function value of the optimal solution relative to the fitness function value of the initial solution, the improvement can be calculated separately for each run of each stage of each case. The maximum percentage improvement of the fitness function value and the average percentage improvement of each optimization stage of each case are statistically analyzed, as shown in the table below. The table shows the maximum percentage improvement of the fitness function value of cases 1, 2, and 3 and the average percentage improvement of each optimization stage of cases 1, 2, and 3.

[0462]

[0463]

[0464] Table 10. Coefficients of variation of fitness function values ​​for Cases 1, 2, and 3

[0465] *Note: Since the coefficient of variation is calculated as standard deviation / mean, the average value of the optimization result of the final fitness function value in the first stage of Cases 2 and 3 tends to 0. This will cause the denominator in the coefficient of variation calculation to approach zero, making the coefficient of variation value tend to infinity, thus losing its statistical significance and comparability. Therefore, in evaluating the stability of the first stage of Cases 2 and 3, this invention uses the load capacity as the evaluation index for calculating the coefficient of variation. By calculating the coefficient of variation of the load capacity results obtained from 100 independent runs, as shown by the *marked area and the figure below, the coefficient of variation calculated based on the load capacity shows a small degree of dispersion, which can effectively reflect the stability performance of the algorithm.

[0466] comprehensive Figure 22-24 As can be seen from Tables 8, 9, and 10:

[0467] ①Case 1

[0468] In the first stage, optimizing the objective function for maximizing payload, since the number of containers is less than the number of compartments, neither the maximum nor the average percentage increase in the fitness function value increased. In the second stage, optimizing the objective function for maximizing pitch inertia, the maximum percentage increase in the fitness function value was 46.97%, and the average percentage increase was 41.32%. The difference between the highest and average percentage increases was only 5.65%, indicating that the quality of the initial solution resulted in a small difference between the final optimized fitness function values, reflecting the good quality of the initial solution in this invention. In the third stage, optimizing the objective functions for minimizing pitch inertia and center of gravity deviation, the maximum percentage increase in the fitness function value was 37.91%, and the average percentage increase was 23.31%. The difference between the highest and average percentage increases was only 14.6%. High optimization performance was maintained in all three stages, and the small difference between the highest and average percentage increases in the fitness function value indicates that a good initial solution improved the optimization results in subsequent stages, demonstrating the good quality of the initial solution in this invention.

[0469] ②Case 2

[0470] In the first stage, optimizing the objective function for maximizing payload, the fitness function achieved a maximum improvement percentage of 75.20% and an average improvement percentage of 63.47% due to the slightly larger number of containers (60) compared to the number of compartments (40). As more excellent initial solutions were selected, the difference between the highest and average improvement percentages of the fitness function increased by 11.73%. In the second stage, optimizing the objective function for maximizing pitch inertia, the fitness function achieved a maximum improvement percentage of 50.92% and an average improvement percentage of 45.69%, with a difference of 5.23% between the highest and average improvement percentages. In the third stage, optimizing the objective function for minimizing pitch inertia and center of gravity deviation, the fitness function achieved a maximum improvement percentage of 26.12% and an average improvement percentage of 21.20%, with a difference of 4.92% between the highest and average improvement percentages.

[0471] High optimization performance was maintained in all three stages, and the difference between the highest and average percentage improvement in the fitness function value was small. This indicates that a good initial solution led to improved optimization results in subsequent stages, reflecting the good quality of the initial solution in this invention.

[0472] ③Case 3

[0473] In the first stage, the objective function for maximizing payload capacity is optimized. Since the number of containers (100) is much greater than the number of compartments (40), the maximum percentage increase in fitness function value is 100.00%, and the average percentage increase is 99.96%. The difference between the highest and average percentage increase in fitness function value is 0.04%, and the rich selection space produces high-quality and diverse initial solutions.

[0474] In the second stage, optimizing the objective function for the maximum pitch inertia, the fitness function value achieved a maximum improvement percentage of 66.34% and an average improvement percentage of 57.15%. The difference between the highest and average improvement percentages was only 9.19%, indicating a significant optimization effect. In the third stage, optimizing the objective function for the constraints of minimizing pitch inertia and center of gravity deviation, the fitness function value achieved a maximum improvement percentage of 22.80% and an average improvement percentage of 19.26%. The difference between the highest and average improvement percentages was only 3.54%, demonstrating that the excellent initial solution quality enabled the algorithm to reach the global optimum. Furthermore, the small difference between the highest and average improvement percentages indicates that a good initial solution leads to improved optimization results in subsequent stages, reflecting the high quality of the initial solution in this invention.

[0475] ④ Coefficient of variation of fitness function values ​​in cases 1, 2, and 3

[0476] The coefficient of variation, also known as the coefficient of dispersion, is a relative statistic that measures the degree of variation in data. It is mainly used to compare the dispersion of data. Generally, the smaller the coefficient of variation, the smaller the degree of variation in the data; conversely, the larger the coefficient of variation, the greater the degree of variation in the data. As shown in Table 10, the final coefficients of variation are all below 10%, indicating that the dispersion of the optimized result is small. This shows that the optimized fitness function value of this invention tends to be stable, which reflects the good quality of the initial solution of this invention.

[0477] In summary, the results show that a high-quality initial solution not only performs well in the current stage, but also provides a better starting point in subsequent stages, forming a positive cycle of "high-quality initial solution → significant optimization effect → better intermediate solution → continuous optimization and improvement".

[0478] 2. Algorithm effectiveness

[0479] Depend on Figure 26-28 As can be seen, Cases 1, 2, and 3, which use the improved genetic algorithm, have better convergence results and can converge quickly.

[0480] 3. Robustness:

[0481] The three groups of experiments all had different container types and weights, exhibiting strong heterogeneity. However, the experimental results show that regardless of the number of containers, the established mathematical model can quickly yield experimental results, demonstrating strong robustness.

[0482] 4. Optimization effect

[0483] This invention is optimized in three progressive stages.

[0484] (1) Case 1: The number of containers is 30, which is less than the number of aircraft compartments (40).

[0485] Phase 1: Maximum Load Capacity Test

[0486] Since the number of containers is less than the number of aircraft cabins, there will be 10 empty cabins in the end. At this time, the aircraft's payload is 69,040 kg and the load factor is 58.75%.

[0487] Phase 2: Experiment on the Maximum Pitch Moment of Inertia

[0488] The maximum pitch moment of inertia was calculated by normalizing the pitch moment of inertia calculated in the three stages. Compared to stages one and three, the pitch moment of inertia in stage two is 2.43E10, which is 33.74% and 43.21% higher than stages one and three, respectively.

[0489] ③ Phase Three: Experiments to minimize pitch rotational inertia and center of gravity shift

[0490] Payload: Since the number of containers is less than the number of aircraft cabins, the aircraft's payload remains unchanged. At this point, the aircraft's payload is 69,040, and the payload ratio is 58.75%.

[0491] Center of gravity deviation: The target value of the center of gravity in this invention is set at 26% MAC. The deviation values ​​in the three experiments were 3.06, 0.54, and 0, respectively, with deviation rates of 11.77%, 2.08%, and 0%. It can be seen that the optimization effect of minimizing the center of gravity deviation is significant.

[0492] Pitch moment of inertia: Under the maximum load, the pitch moment of inertia in stage three was reduced by 43.21% compared to stage two, resulting in the minimum pitch moment of inertia and the minimum center of gravity shift.

[0493] (2) Case 2: The number of containers is 60, which is slightly greater than the number of aircraft cabins (40).

[0494] Phase 1: Maximum Load Capacity Test

[0495] In this case, there are 60 containers and 40 aircraft bays. The number of containers is slightly greater than the number of aircraft bays, increasing the selection of containers for each bay. The optimized maximum payload of the aircraft is 110,508 kg, with a payload ratio of 94.04%.

[0496] Phase 2: Experiment on the Maximum Pitch Moment of Inertia

[0497] The maximum pitch moment of inertia was calculated by normalizing the pitch moment of inertia calculated in the three stages. Compared to stages one and three, the pitch moment of inertia in stage two is 3.10E10, which is 8.71% and 9.03% higher than stages one and three, respectively.

[0498] ③ Phase Three: Experiments to minimize pitch rotational inertia and center of gravity shift

[0499] Payload: Since the number of containers is slightly greater than the number of aircraft cabins, after optimization by the genetic algorithm, the maximum payload of the aircraft is 110,508 kg, and the aircraft load factor is 94.04%. The maximum payload remains unchanged in subsequent stages.

[0500] Center of gravity deviation: The target value of the center of gravity in this invention is set at 26% MAC. The deviation values ​​in the three experiments were 3.45, 2.82, and 0, respectively, with deviation rates of 13.27%, 10.85%, and 0.00%. It can be seen that the optimization effect of minimizing the center of gravity deviation is significant.

[0501] Pitch moment of inertia: Under the maximum load, the pitch moment of inertia in stage three was reduced by 9.03% compared to stage two, resulting in the minimum pitch moment of inertia and the minimum center of gravity shift.

[0502] (3) Case 3: The number of containers is 100, which is much greater than the number of aircraft cabins (40).

[0503] Phase 1: Maximum Load Capacity Test

[0504] In this case, there are 100 containers and 40 aircraft bays, meaning the number of containers far exceeds the number of bays, offering more options for container selection. The optimized maximum payload of the aircraft is 117,510 kg, with a load factor of 100%.

[0505] Phase 2: Experiment on the Maximum Pitch Moment of Inertia

[0506] The maximum pitch moment of inertia was calculated by normalizing the value when the minimum pitch moment of inertia was calculated among the three stages. Compared to stages one and three, the pitch moment of inertia in stage two is 3.22E10, which is 6.83% and 8.70% higher than that in stages one and three, respectively.

[0507] ③ Phase Three: Experiments to minimize pitch rotational inertia and center of gravity shift

[0508] Payload: Since the number of containers is much greater than the number of aircraft cabins, after optimization by the genetic algorithm, the maximum payload of the aircraft is 117,510 kg, and the aircraft load factor is 100%. The maximum payload remains unchanged in subsequent stages.

[0509] Center of gravity deviation: The target value of the center of gravity in this invention is set at 26% MAC. The deviation values ​​in the three experiments were 2.82, 4.81, and 0, respectively, with deviation rates of 10.85%, 18.50%, and 0.00%. It can be seen that the optimization effect of minimizing the center of gravity deviation is significant.

[0510] Pitch moment of inertia: Under the maximum load, the pitch moment of inertia in stage three was reduced by 8.70% compared to stage two, resulting in the minimum pitch moment of inertia and the minimum center of gravity shift.

[0511] IV. Technical Features and Positive Effects of the Invention

[0512] 1. Technical features of the present invention

[0513] (1) Unlike previous studies that focused on a single objective function, this invention quantifies the airline's flight safety, maneuverability, and economic objectives, and designs optimization strategies for multiple operational control objectives. Tests are conducted to maximize payload, maximize pitch inertia, minimize pitch inertia, and minimize center of gravity shift, optimizing each objective function sequentially. Airlines can choose the objective function based on different cargo loads to adapt to actual conditions.

[0514] (2) Unlike previous studies that had fewer constraints on position, center of gravity and weight, the loading model of this invention includes multiple constraints, taking into account all aspects of position, center of gravity and weight. Specific constraints include: position constraints, compartment and container adaptation constraints, maximum load limit, structural constraints, center of gravity envelope constraints, and lateral imbalance constraints.

[0515] (3) Without increasing the types of decision variables and model complexity, a binary adaptation matrix is ​​used to define the adaptation relationship between all types of ULDs and specific aircraft cabins. Based on comprehensive consideration of physical and geometric constraints such as aircraft load capacity and cargo hold internal space, a binary ULD-cabinet adaptation relationship matrix is ​​proposed to replace traditional constraints such as aircraft load line density, cumulative load constraints, and non-overlapping and non-boundary spatial constraints. At the same time, based on the ULD-cabinet adaptation relationship, an initial solution generator based on random sampling is designed to improve the quality of the initial solution.

[0516] 2. Positive effects of the invention

[0517] (1) This invention focuses on the optimization of loading problems of standard containers and proposes an optimization method for aircraft containers that comprehensively considers the safety, maneuverability and economy of aircraft flight.

[0518] (2) The results of this invention show that, while ensuring the safety of the center of gravity, the payload of the aircraft is reasonably planned and increased, the accumulation of cargo within the center of gravity range is reduced, and the maneuverability and economy of the aircraft are improved.

Claims

1. An aircraft container loading optimization model that considers the actual operational control and operational objectives of airlines, characterized by: The optimization model includes decision variables, objective function, constraints, fitness function, and termination condition; 1) Decision variables A 0-1 matrix is ​​used to define the loading relationship between the container set and the bay set; the element values ​​of the 0-1 matrix are shown in the following formula: In the formula, It is the ULD number. It is the cabin number; Indicates the number of ULDs to be loaded; Indicates the number of locations that need to be loaded; It is a decision variable, representing the first... The first container and the first The loading relationship between each cabin; if the first... The container was loaded into the first In each cabin, The value is 1 if the value is true and 0 otherwise; the decision variable matrix is ​​defined as follows: ； 2) Objective function The objective function and optimization are defined and carried out in three stages according to priority. A. Objective function of the first stage The objective function for maximizing load capacity is shown in the following formula: ； For the first The weight of each container; With the maximum load capacity as the objective function, optimization is carried out under the condition that the constraints remain unchanged, to obtain the individual to be optimized with the maximum load capacity and the optimal solution in the first stage; B. Objective function of the second stage The objective function for maximizing the pitch rotation moment of inertia is shown in the following equation: ； In the formula It is the column vector of the balancing lever arm of the cabin. It is the first The balancing arm of each cabin; The core of the goal The lever arm value; With the objective function being the maximum pitch moment of inertia, and under unchanged constraints, the second stage of optimization is carried out using the individual to be optimized with the largest load and the optimal solution obtained in the first stage as the initial solution, to obtain the maximum achievable pitch moment of inertia under the condition of the maximum load. C. Objective function of the third stage The objective functions for minimizing the normalized pitch inertia and the takeoff center of gravity deviation are shown in the following equation: ， in, These are weight values; ， This represents the maximum pitch inertia obtained in the second stage. ， ； and These are the actual takeoff center of gravity and the target center of gravity, respectively, expressed as a percentage of mean aerodynamic chord length. The maximum difference between each boundary of the takeoff center of gravity envelope and the target center of gravity; with the minimum normalized pitch inertia and the minimum center of gravity deviation as the objective functions, and with the individual to be optimized and the optimal solution with the maximum payload obtained in the first stage as the initial solution, the third stage of optimization is carried out under the condition that the constraints remain unchanged, to obtain the optimal loading plan with the minimum pitch inertia and center of gravity deviation under the condition of maximum payload. 3) Constraints The constraints include: position constraints, compartment and container compatibility constraints, maximum load limits, structural constraints, center of gravity envelope constraints, and lateral imbalance constraints.

2. The aircraft container loading optimization model as described in claim 1, characterized in that: When using a genetic algorithm to solve the optimization problem, the objective function in the first stage becomes: ； In the formula This refers to the maximum payload capacity of the aircraft under the given flight plan conditions. The objective function for the second stage is transformed into: ； In the formula The pitch inertia of the entire aircraft when loading containers of maximum weight that meet adaptation constraints in each compartment.

3. The aircraft container loading optimization model as described in claim 1, characterized in that: Regarding the adaptation constraints between cargo slots and containers, the adaptation constraints between the container set and the cargo slot set are defined as follows: ； In the formula Indicates the first The first container and the first The compatibility relationship between cabin classes; if the two are compatible, then The value is 1 if it is not 1, and 0 otherwise; and if =0, It must be 0; if If it is 1, then It can be 0 or 1, and the constraint relationship between them can be expressed as: ； The above formula can be improved using relatively low-cost matrix operations, array element picking, and addition operations as follows: ； In the formula, SUM() and DIAG() are the array summation function and the diagonal element picking function of the two-dimensional array, respectively.

4. The aircraft container loading optimization model as described in claim 1, characterized in that: The fitness functions for the first, second, and third stages are as follows: (45) in, (32) (33) (34) (35) (36) (37) (38) (39) (40) (41) (42) (43) (44) ； 。 5. The aircraft container loading optimization model as described in claim 4, characterized in that: The optimization is terminated using no significant change in fitness and the maximum number of iterations, as described mathematically below: ； In the formula Let the threshold be the average change in the optimal fitness value over consecutive generations. The characteristic parameter for fitness change is continuous. The average change in the optimal fitness value across generations The calculation method is as follows: ； In the formula , , These are the current iteration number, the optimization stage number, and the maximum number of iterations, respectively. It is the first The first optimization phase The optimal fitness value of the generation. Set as 10%-20%.

6. The aircraft container loading optimization model as described in claim 1, characterized in that: For the center of gravity envelope constraint, the center of gravity envelope with the vertical axis representing gross weight and the horizontal axis representing %MAC is surrounded by four envelopes based on the center of gravity boundaries under four different weights: taxiing weight (TW), takeoff weight (TOW), landing weight (LW), and zero fuel weight (ZFW).

7. An optimization method for aircraft container loading that considers the actual operational control and operational objectives of airlines, characterized by: Using the aircraft container loading optimization model as described in claim 2, the model is solved using an improved genetic algorithm according to the following steps: Step 1: Establish the initial population using an initial value generation method based on adaptation constraints; Step 2: Calculate the fitness function and arrange the calculated fitness function values ​​in descending order of fitness. Step 3: Adopt the optimal preservation strategy to retain the best individuals; Step 4: When the random floating-point number is greater than the crossover probability, perform PBX crossover; when the random floating-point number is less than the mutation probability, perform uniform crossover and use a position-constrained mutation method to carry out individual gene mutation. Step 5: Determine if the optimization termination condition is met. If it is met, output the optimal solution; otherwise, return to step 2.

8. The aircraft container loading optimization method as described in claim 7, characterized in that: The initial population generation method is as follows: A. when hour ① Define the initial population as ,in The number of individuals in each generation of the population. The first individual in the initial population, with dimension 1. Assign the adaptation matrix to ,Right now ; ; ②From The Extract the elements with a value of 1 from the column and construct a two-dimensional array; the first column of the array is the row number of the extracted element in the adaptation matrix, and the elements in the second column are all 1; ③ Randomly select a row from the two-dimensional array in step ②, and set it as follows: ,in To randomly select elements in No. Row number in the column; The In the column, the first All other elements are set to zero; ④ In The During the process, the exception will be removed. All other elements are set to zero; at this time, if The to If all elements in any column are zero, then remove steps ③ and ④. No. Column and number If the operation fails, return to step ③ and perform random sampling again; otherwise, proceed to step ⑤. ⑤If If the condition is met, the process terminates, and the first individual of the initial population is generated; otherwise, proceed. Return to step ②; The other individuals in the initial population are generated sequentially by referring to steps ①-⑤ above; B. When hour ① Define the initial population as ,in The number of individuals in each generation of the population. The first individual in the initial population, with dimension 1. ; Assign the adaptation matrix to ,Right now ; ; ②From The Extract the elements with a value of 1 from the rows and construct a two-dimensional array as shown in the following formula; the first column of the array is the column number of the extracted element in the adaptation matrix, and all elements in the second column are 1; ； ③ Randomly select a row from the two-dimensional array in step ②, and set it as follows: ,in To randomly select elements in No. Column numbering in a row; The In the row, except for the first All other elements are set to zero; ④ In The In the list, except All other elements are set to zero; at this time, if The to If all elements in any row are zero, then remove the steps in steps ③ and ④. No. row and number If the column operation fails, return to step ③ and perform random sampling again; otherwise, proceed to step ⑤. ⑤If If the condition is met, the process terminates, and the first individual of the initial population is generated; otherwise, proceed. Return to step ②; Following steps ①-⑤ above, generate the other individuals in the initial population in sequence.

9. The aircraft container loading optimization method as described in claim 7, characterized in that: The methods for generating the initial solutions in the second and third stages are as follows: The successful operation of the first phase yielded a solution that satisfies all constraints and achieves optimal load utilization, with the following dimension: 1× , To determine the number of cabins, construct the initial population for the second phase. And after the second phase is completed, construct the initial solution for the third phase. ; Initial solutions for the second and third phases and An elite individual retention strategy is adopted, ensuring that the first individual retains the optimal solution from the first stage to guarantee at least one high-quality solution in the population. Let the individual that obtains the optimal solution from the first stage be denoted as . , The generation of the initial solution in the second and third stages can be expressed as follows: , ,in, This represents the first individual in the initial population of the second and third stages, which is also the first individual in the optimal solution of the first stage. Indicates the number of individuals in each generation of the population; for and remove individual The solution is obtained by using a random perturbation generation strategy on the optimal solution of the first stage.

10. The aircraft container loading optimization method as described in claim 7, characterized in that: The specific process of conducting individual gene variation using a location-constrained variation method: A.When hour ① Convert the 0-1 decision variable matrix into a one-dimensional array The elements of this vector are ULD numbers, and the order of the elements is the cabin number; ② Gene mutation, that is, when the ULD number in the one-dimensional array mutates, it needs to follow positional constraints. When the mutated gene is in the order of the one-dimensional array... The element is changing the cabin position. When placing a ULD, the potential value of that element should be a ULD number that does not appear in the one-dimensional array, i.e., satisfying the condition. serial number; ③ Finally, in A value is randomly selected from the data as the value of the mutated gene. B. When hour ① Convert the 0-1 decision variable matrix into a one-dimensional array The elements of this vector are cabin numbered, and the order of the elements is ULD number; ② Gene mutation, that is, when the cabin number in the one-dimensional array mutates, it needs to follow positional constraints. When the mutated gene is in the order of the one-dimensional array... The element that changes the first When placing a ULD in a compartment, the potential value of this element should be a compartment number that does not appear in the one-dimensional array, i.e., satisfying the condition... serial number; ③ Finally, in A value is randomly selected as the value of the mutated gene.

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