Freight aircraft boxing and stowage method based on grey wolf genetic hybrid optimization

Abstract

The air freight loading efficiency and the flight safety are crucial to the efficient operation of a transportation system. Therefore, aiming at the problem of multi-cargo loading optimization under the complicated constraint condition of the cargo hold, a multi-objective optimization model containing business load limitation, gravity center shift, volume utilization rate and rotational inertia is constructed, and a solution method based on a grey wolf genetic hybrid algorithm is provided. A high-quality initial population is generated by adopting a two-stage initialization strategy, and evaluation is performed through a fitness function. And dynamically adjusting a step length coefficient and a mutation probability to realize self-adaptive adjustment of the search capability. In the selection stage, excellent individuals are reserved by using a tournament strategy, and meanwhile, the population quality is improved through an alpha wolf direct inheritance mechanism. In the breeding stage, the feasibility and diversity of solutions are enhanced through single-point crossing and mutation operation based on the gravity center distance. And after iterative updating, outputting the scheme with the highest evaluation value as the optimal loading scheme. According to the algorithm, the loading balance and the calculation efficiency are improved, and the high loading rate of the space in the cabin is ensured.

Description

Technical Field

[0001] The present invention belongs to the technical field of aircraft loading planning and intelligent algorithms, and in particular relates to a cargo aircraft packing and loading method based on grey wolf genetic hybrid optimization. Background Art

[0002] In recent years, with the rapid development of global air cargo operations and the growing demand for efficient, safe, and low-carbon transportation, aircraft cargo hold loading optimization has become a hot topic in aviation logistics research. A reasonable loading plan is not only directly related to the airline's transportation efficiency and economic benefits, but also to the operational stability and flight safety of the aircraft. With the widespread deployment of large, long-range cargo aircraft such as the Boeing 777F, scientific and systematic loading decisions are particularly critical. Existing air cargo loading decisions are typically divided into two relatively independent stages: the first is the packing stage, which involves the rational allocation of bulk cargo into unit load devices (ULDs); the second is the stowage stage, which involves the optimal positioning of the ULDs in the aircraft's cargo hold. While this "segmented" loading decision-making process simplifies the operational process, in practice it often leads to problems such as uneven distribution of ULD mass and contour, excessive aircraft center of gravity offset, operational imbalance, and increased fuel consumption. This makes it difficult to meet the comprehensive requirements of modern air cargo operations for high loading rates, high balance, high safety, and low carbon emissions. Especially in the operational practice of air cargo, the loading plan needs to be completed quickly within two hours before takeoff, and must simultaneously meet multiple constraints such as weight, center of gravity, volume, and loading and unloading efficiency. In this process, a large amount of reliance on manual experience for packing results in significant quality differences between containers; and in the loading stage, due to the uncertainty of prior information on packing, it is difficult to achieve optimal loading. More importantly, the vast majority of current optimization models only focus on maximizing load or minimizing center of gravity balance, while ignoring the moment of inertia indicator that significantly affects aircraft controllability, thereby limiting the overall loading solution's ability to further improve flight safety and fuel economy.

[0003] In existing research, extensive progress has been made in the exploration of three-dimensional packing algorithms and loading balance strategies. Some studies have adopted heuristic or hybrid heuristic algorithms, such as simulated annealing, genetic algorithms, and local search algorithms, and have achieved good space utilization and computational performance; some studies have also attempted to transform the loading process into a multi-objective optimization problem, comprehensively considering factors such as loading efficiency, center of gravity position, and loading and unloading sequence. It is worth noting that existing studies have introduced moment of inertia as a key indicator to measure flight controllability in cargo aircraft loading models, and verified its optimization value through modeling and experiments. However, overall, research on incorporating moment of inertia into air cargo loading optimization models is still relatively scarce. Most existing methods are theoretical model constructions and lack efficient solution strategies combined with engineering applications. On the other hand, when facing the high-dimensional, strongly constrained, and nonlinear combinatorial optimization problem of air cargo loading, existing optimization algorithms generally have the following problems: (1) the algorithm converges slowly and is difficult to meet the needs of fast loading; (2) the solution space is huge and is prone to falling into local optimality; (3) the algorithm model has weak generalization ability and is difficult to adapt to changes in different routes, aircraft models, and cargo characteristics. Although the Genetic Algorithm (GA) has good local search capabilities and adaptability, it still has limitations in handling complex constraints; and the Grey Wolf Algorithm (GWO) has strong global development capabilities, but it is prone to convergence stagnation in the later stages. Therefore, how to integrate the advantages of different algorithms and construct a hybrid optimization algorithm that combines convergence speed, global search capabilities and constraint solution adaptability has become an important direction for improving the level of intelligent loading in air cargo. In response to the above technical difficulties and industry needs, some studies in recent years have attempted to open up a collaborative optimization mechanism for the "packing-stowage" link. Existing research has also proposed step-by-step optimization, combination optimization and improved combination optimization models, which provide an effective framework for comprehensively considering payload and center of gravity offset. However, there are still significant gaps in the introduction of moment of inertia, joint modeling of complex constraints, and efficient and intelligent solutions.

[0004] In summary, the current air cargo loading technology still has the following deficiencies: (1) There is a lack of a systematic optimization model that simultaneously considers load, center of gravity, volume, and moment of inertia; (2) There is no efficient solution strategy that combines the advantages of the gray wolf optimization algorithm and the genetic algorithm, taking into account both convergence speed and global optimality; (3) The traditional "packing-stowage" separation model fails to achieve overall coordinated optimization of the loading plan, affecting flight safety and economy. Therefore, it is urgent to propose a more scientific, comprehensive, and engineering-practical three-dimensional air cargo loading optimization method and its hybrid intelligent solution strategy to effectively improve loading efficiency, balance performance, and flight controllability, and help build a new generation of efficient, green, and safe air logistics system. Summary of the Invention

[0005] In light of this, the present invention aims to overcome the shortcomings of the prior art by constructing a four-element composite objective function encompassing payload capacity, center of gravity offset, volume constraint, and moment of inertia, incorporating moment of inertia as an optimization target into the aircraft loading optimization problem. Furthermore, a hybrid optimization algorithm is designed that combines the global search capabilities of the Grey Wolf Algorithm with the adaptive mechanism of the Genetic Algorithm, effectively addressing the vulnerability of traditional algorithms to local optimality in loading optimization. This approach achieves comprehensive improvements in loading efficiency, balancing performance, and computational efficiency, providing new theoretical methods and practical guidance for air cargo loading optimization.

[0006] To achieve the above object, the technical solution of the present invention is achieved as follows:

[0007] A cargo aircraft packing and stowing method based on gray wolf genetic hybrid optimization includes building a multi-objective optimization model, and the optimization objective function is

[0008]

[0009] Among them, f load is the load objective function, f cg is the center of gravity shift objective function, f inertia is the moment of inertia objective function, P penalty is the constraint penalty objective function, α, β, γ, and δ are adjustable objective weight coefficients;

[0010] A hybrid optimization algorithm that integrates the global search capability of the grey wolf algorithm and the adaptive mechanism of the genetic algorithm is designed to solve the multi-objective optimization model.

[0011] Furthermore, the load objective function normalizes the ratio between the total cargo weight of the current solution and the maximum allowable payload of the aircraft so that the target value is always between [0, 1]. The specific form is as follows:

[0012]

[0013] Among them, w i is the weight of cargo i, W max is the maximum allowable payload of the aircraft, x ij is the decision variable, defined as

[0014] Furthermore, the center of gravity offset objective function sets the target center of gravity position and minimizes the deviation between the center of gravity of the current loading solution and the target center of gravity. The center of gravity offset is expressed as a percentage of the mean aerodynamic chord and is converted into an actual deviation distance using the following formula:

[0015]

[0016] in is the center of gravity of the current loading solution, a t is the target center of gravity position, Δa max For the maximum permissible deviation, the center of gravity position is converted to the actual distance using the following formula:

[0017]

[0018] where a m is the actual center of gravity position, a ％MAC is the center of gravity position expressed as a percentage of the mean aerodynamic chord, L MAC is the length of the mean aerodynamic chord, L LE is the distance of the leading edge of the mean aerodynamic chord relative to the aircraft reference zero point.

[0019] Furthermore, the moment of inertia objective function is in the form of:

[0020]

[0021] Where I is the moment of inertia of the current loading scheme, expressed as

[0022]

[0023] I max It is the preset theoretical maximum moment of inertia.

[0024] Furthermore, the constraint penalty objective function is in the form of

[0025] P penalty =λ w P w +λ v P v +λ l P l

[0026] Among them, P w P is the penalty value caused by exceeding the allowable load of the cargo hold. v is the penalty value for exceeding the container volume limit, P l is the penalty value caused by the unbalanced loading on the left and right sides. The calculation method of each penalty item is as follows

[0027]

[0028]

[0029] Among them, W total is the total weight of the current load, W max is the maximum payload of the aircraft, V total is the total volume currently loaded, V max is the maximum volume limit of the aircraft, W Land W R are the total weight of the left and right cabins, ΔW threshold is the weight threshold.

[0030] Furthermore, the following constraint system is also set up:

[0031] Cargo unique distribution constraint, cargo hold capacity constraint, left and right hold balance constraint, main cargo hold symmetry constraint, maximum payload limit, and center of gravity moment limit.

[0032] Furthermore, the design of a hybrid optimization algorithm that integrates the global search capability of the gray wolf algorithm and the adaptive mechanism of the genetic algorithm to solve the multi-objective optimization model specifically includes:

[0033] The social hierarchy structure in the gray wolf optimization algorithm is adopted to divide the population individuals into α wolves, β wolves, δ wolves and ω wolves. In the initialization phase, a two-stage initialization strategy is adopted to generate a high-quality initial population, which is evaluated by the fitness function.

[0034] Dynamically adjust the step size coefficient and mutation probability to achieve adaptive adjustment of search capability;

[0035] In the selection phase, the tournament strategy is used to retain excellent individuals, while the population quality is improved through the α-wolf direct inheritance mechanism;

[0036] In the reproduction stage, the feasibility and diversity of solutions are enhanced through single-point crossover and mutation operations based on centroid distance;

[0037] After iterative updating, the solution with the highest evaluation value is output as the optimal loading solution.

[0038] Furthermore, the two-stage initialization strategy includes

[0039] In the first stage, cargo is allocated to spaces close to the target center of gravity in priority according to cargo density;

[0040] The second stage dynamically adjusts the distribution plan of the remaining cargo based on the current center of gravity position to achieve preliminary center of gravity balance control.

[0041] Furthermore, the fitness function takes maximizing the loading rate as the main goal, while penalizing the center of gravity deviation, excessive moment of inertia and constraint violation. The fitness function is as follows: F(X) = α·f load (X)+β·f cg (X)+γ·f inertia (X)-P penalty (X)+loading_bonus(X)-P inertia (X)

[0042] Among them, f load (X), f cg(X) and f inertia (X) are the normalized loading rate, center of gravity offset and moment of inertia objective functions; P penalty (X) is a penalty for violation of the constraint;

[0043] loading_bonus(X) is the loading bonus, which encourages the algorithm to load as much cargo as possible;

[0044] P inertia (X) is the moment of inertia penalty term, which is imposed when the moment of inertia exceeds the threshold.

[0045] Furthermore, it also includes the continuous spatial position update of the gray wolf algorithm:

[0046] Check whether α wolf, β wolf, and δ wolf have been defined, create an empty dictionary as the updated solution, obtain the union of all cargo indices contained in the three dominant wolf solutions, calculate the center of gravity position of the current solution as the reference point for position update, and for each cargo index in the union, the algorithm calculates random coefficients A and C to control the step size and direction of the position update:

[0047] A1, A2, A3=random(-a,a),random(-a,a),random(-a,a)

[0048] C1, C2, C3=random(0,2),random(0,2),random(0,2)

[0049] Among them, a is a parameter that decreases linearly with the number of iterations, from 2 to 0, controlling the balance between exploration and exploitation. The algorithm obtains the allocation position of the goods by the three dominant wolves. If a dominant wolf does not contain the goods, it is skipped. For valid positions, the algorithm assigns different weights: α wolf has the highest weight (1.0), β wolf has the second highest weight (0.7), and δ wolf has the lowest weight (0.5). The algorithm combines the weight of the distance to the center of gravity to calculate the comprehensive weight:

[0050]

[0051] combined_weights[i]=valid_weights[i]×distance_weights[i]

[0052] A new location is randomly selected based on the combined weights and the capacity constraints are checked, and the goods are allocated to that location if satisfied.

[0053] Compared with the existing technology, the cargo aircraft packing and loading method based on gray wolf genetic hybrid optimization described in the present invention has the following advantages:

[0054] (1) This paper integrates the global search capability of the gray wolf optimization algorithm with the local evolution mechanism of the genetic algorithm to construct a dual evolution process. Combining the hierarchical guidance of the gray wolf and the diversity variation of the genetic operator, it effectively improves the efficiency of solution space exploration and utilization, avoids falling into the local optimum, and enhances the quality, stability and robustness of the solution. At the same time, by introducing the α wolf retention and the elite strategy in the genetic algorithm, it ensures that the current optimal solution is retained and inherited during the iteration process, improving the consistency and repeatability of the results of multiple runs.

[0055] (2) The present invention is oriented towards the multi-objective optimization requirements of aircraft loading tasks, and constructs a composite fitness function that comprehensively considers the maximization of loading rate, minimization of center of gravity offset and moment of inertia control, and is supplemented by the design of constraint penalty items and reward items to coordinate the conflict relationship among multiple objectives, thereby improving the feasibility and engineering practicality of the loading scheme and meeting the actual loading constraints and safety requirements in air transportation.

[0056] (3) To improve the quality of the optimization starting point, the present invention combines cargo density and center of gravity position characteristics to design a structured initial population generation strategy, effectively guiding the initial search direction and reducing the center of gravity deviation of the initial solution. At the same time, to meet the center of gravity balance requirements, a mutation strategy based on the cargo center of gravity contribution is introduced, focusing on adjusting the distribution of cargo that has a significant impact on the center of gravity, significantly improving the stability of the loading plan and the center of gravity control accuracy.

[0057] (4) In view of the discrete nature of the loading problem, the present invention discretizes the continuous update mechanism of the gray wolf algorithm and proposes a joint weighted sampling strategy based on the center of gravity distance and the wolf level to achieve efficient adaptation to the discrete cabin space. At the same time, a dynamic adjustment mechanism for key parameters is designed to timely adjust the mutation rate and convergence factor according to the evolutionary stage, expand the search range in the early stage, and strengthen local convergence in the later stage, thereby comprehensively improving the efficiency and accuracy of the algorithm at different stages. BRIEF DESCRIPTION OF THE DRAWINGS

[0058] The accompanying drawings, which constitute part of the present invention, are provided to provide a further understanding of the present invention. The exemplary embodiments of the present invention and their descriptions are provided to explain the present invention and do not constitute an undue limitation of the present invention. In the accompanying drawings:

[0059] Figure 1 This is a flow chart of a method for packing and stowing air cargo provided by an embodiment of the present invention;

[0060] Figure 2 Schematic diagram of the main cargo hold and lower cargo hold layout and lever arms of a B777F aircraft provided by an embodiment of the present invention;

[0061] Figure 3 This is a bar chart comparing the performance of the two algorithms under different types of test examples provided by the embodiment of the present invention;

[0062] Figure 4 This is a schematic diagram of the loading distribution of the main cargo hold and the lower cargo hold of a B777F aircraft provided by an embodiment of the present invention;

[0063] Figure 5 This is a schematic diagram of the loading distribution of the main cargo hold and lower cargo hold of the B777F aircraft provided by the improved combined optimization method. DETAILED DESCRIPTION

[0064] It should be noted that, in the absence of conflict, the embodiments of the present invention and the features in the embodiments may be combined with each other.

[0065] In the description of the present invention, it should be understood that the terms "center", "longitudinal", "lateral", "up", "down", "front", "back", "left", "right", "vertical", "horizontal", "top", "bottom", "inside", "outside" and the like indicate orientations or positional relationships based on the orientations or positional relationships shown in the accompanying drawings, and are only for the convenience of describing the present invention and simplifying the description, rather than indicating or implying that the device or element referred to must have a specific orientation, be constructed and operated in a specific orientation, and therefore cannot be understood as limiting the present invention. In addition, the terms "first", "second", etc. are only used for descriptive purposes and cannot be understood as indicating or implying relative importance or implicitly indicating the number of the indicated technical features. Therefore, features defined as "first", "second", etc. may explicitly or implicitly include one or more of the features. In the description of the present invention, unless otherwise specified, "multiple" means two or more.

[0066] In the description of the present invention, it should be noted that, unless otherwise expressly specified or limited, the terms "mounted," "connected," and "connected" should be understood in a broad sense. For example, they may refer to fixed connections, detachable connections, or integral connections; mechanical connections or electrical connections; direct connections or indirect connections through an intermediate medium; and internal communication between two components. Those skilled in the art will understand the specific meanings of the above terms in the present invention based on specific circumstances.

[0067] The present invention will be described in detail below with reference to the accompanying drawings and in conjunction with embodiments.

[0068] In one embodiment of the present invention, a three-dimensional aircraft cargo loading method based on multi-objective optimization is proposed. This method constructs a four-element composite objective function that comprehensively considers the aircraft's payload, center of gravity offset, moment of inertia, and penalty values ​​for multiple constraints to improve cargo hold space utilization, loading balance, and flight safety.

[0069] First, we model the tasks that the present invention intends to achieve. Given an aircraft, its cargo hold contains m position sets P = {p1, ..., pm}, each position p j With maximum load capacity Maximum volume and the arm length a relative to the aircraft reference point j ; At the same time, the set of goods to be loaded C={c1,...,c n}, each item c i With weight w i and volume v i In one possible implementation, the decision variable is defined as

[0070]

[0071] The composite objective function includes the following four sub-objectives:

[0072] (1) Loading capacity objective function: This function is used to maximize the total cargo weight. This objective is achieved by normalizing the ratio between the total cargo weight of the current solution and the aircraft's maximum payload capacity, ensuring that the target value is always between [0, 1], making it easier to balance with other objectives. The specific form is as follows:

[0073]

[0074] Among them, w i is the weight of cargo i, W max The maximum allowable payload of the aircraft.

[0075] (2) Center of gravity offset objective function: To ensure that the loaded aircraft is in a safe flight state, a target center of gravity position is set and the deviation between the center of gravity of the current loading solution and the target center of gravity is minimized. The center of gravity offset is usually expressed as a percentage of the mean aerodynamic chord (MAC) and is converted to the actual deviation distance using the following formula:

[0076]

[0077] in is the center of gravity of the current loading solution, a t is the target center of gravity position, Δa max The center of gravity position is usually expressed as a percentage of the mean aerodynamic chord (MAC) and can be converted to actual distance using the following formula:

[0078]

[0079] where a m is the actual center of gravity position, that is, the actual physical distance of the aircraft center of gravity relative to the aircraft reference zero point, a ％MAC is the center of gravity position expressed as a percentage of the mean aerodynamic chord (MAC), LMAC is the length of the mean aerodynamic chord (MAC), L LE is the distance of the leading edge of mean aerodynamic chord (LEMAC) relative to the aircraft reference zero point.

[0080] (3) Moment of inertia objective function: For the first time, the distribution of cargo relative to the center of mass under the loading scheme is included in the optimization objective, and the objective function is defined to minimize the moment of inertia. A smaller moment of inertia helps improve aircraft maneuverability and fuel economy. The objective function form is:

[0081]

[0082] Where I is the moment of inertia of the current loading scheme, which can be obtained by the formula

[0083]

[0084] I max is the preset theoretical maximum moment of inertia. In addition, to prevent extreme solutions from causing control risks, a penalty term is further introduced for moments of inertia exceeding the threshold:

[0085]

[0086] Where λ is the penalty weight coefficient, I threshold is the moment of inertia threshold.

[0087] (IV) Constraint penalty objective function: For various hard constraints in the model, such as weight, volume, balance and other constraints, a unified penalty function form is introduced to avoid infeasible solutions from interfering with the overall goal. The constraint penalty function form is as follows:

[0088] P penalty =λ w P w +λ v P v +λ l P l

[0089] Among them, P w P is the penalty value caused by exceeding the allowable load of the cargo hold. v is the penalty value for exceeding the container volume limit, P l is the penalty value caused by unbalanced loading on the left and right sides. The calculation method of each penalty item is as follows:

[0090]

[0091] Among them, W total is the total weight of the current load, W max is the maximum payload of the aircraft, V total is the total volume currently loaded, Vmax is the maximum volume limit of the aircraft, W L and W R are the total weight of the left and right cabins, ΔW threshold is the weight threshold. In summary, the optimization objective function proposed in the present invention is as follows:

[0092]

[0093] Among them, α, β, γ, and δ are adjustable target weight coefficients, which are used to make trade-offs according to the actual loading scenario requirements. In order to ensure the solution effect of the above optimization model and the feasibility of the loading scheme, the following constraint system is further designed: (1) Cargo unique allocation constraint, that is, each cargo can only be loaded into one cargo hold position; (2) Cargo hold capacity constraint, that is, the loading weight and volume of each cargo hold position must not exceed its capacity limit; (3) Left and right hold balance constraint, that is, the difference in cargo weight between the left and right sides of the main cargo hold must not exceed the preset allowable difference; (4) Main cargo hold symmetry constraint, that is, the cargo on the left and right sides of the cargo holds arranged side by side must meet the asymmetric linear load limit; (5) Maximum payload limit, that is, the total loaded weight must not exceed the maximum payload value allowed by the aircraft during takeoff, landing, and no fuel state; (6) Center of gravity moment limit, that is, the overall center of gravity position obtained after loading must be within the allowable range and meet the moment boundary determined by multiple factors such as the minimum / maximum center of gravity position and the fuel weight center of gravity position.

[0094] To solve the above complex multi-objective optimization problem, this paper proposes a hybrid Grey Wolf Genetic Algorithm (GWGA) that integrates the Grey Wolf Optimization Algorithm (GWO) and the Genetic Algorithm (GA). It fully utilizes the global search capability of GWO and the adaptive evolution mechanism of GA to improve the search efficiency of the algorithm while ensuring the convergence of the algorithm and the quality of the solution. Its overall framework is as follows: Figure 1 shown.

[0095] Population initialization

[0096] Population initialization is a fundamental step in the algorithm. This method creates a population of random solutions, each representing a feasible loading plan. Unlike previous algorithms, we do not use completely random solutions to join the population. Instead, we employ a heuristic strategy to generate random loading plans with a certain quality. In implementation, the algorithm first creates an empty dictionary to represent the solutions, where the keys are cargo indices and the values ​​are slot indices. The cargoes are then sorted from highest to lowest density (weight / volume), a common packing heuristic that helps improve loading efficiency. In the first phase, the algorithm prioritizes cargoes with high density and prioritizes slots close to the target center of gravity. Specifically, for each cargo, the algorithm first checks whether the total weight and volume constraints are exceeded. Then, it identifies all feasible slots that meet the weight and volume constraints, sorts them by distance to the target center of gravity, and selects the slot closest to the center of gravity for allocation. In the second phase, the algorithm calculates the center of gravity of the current solution and then allocates the remaining cargoes in a targeted manner based on the direction of center of gravity deviation to improve center of gravity balance. If the current center of gravity is very close to the target center of gravity (with a deviation of less than 0.1 meter), a random cabin is selected. Otherwise, the cabin that can balance the center of gravity is prioritized based on the direction of the center of gravity deviation. After generating the initial population, the algorithm evaluates the fitness of each solution and determines the initial alpha, beta, and delta values. This process lays the foundation for subsequent optimization iterations.

[0097] Tournament Selection

[0098] The selection operation is a key step in the genetic algorithm, used to select high-quality individuals from the current population as the fathers of the next generation. The specific steps of tournament selection are as follows:

[0099] 1. Initialize an empty list to store the selected individuals.

[0100] 2. Loop population size times, each time performing the following operations: 1) Randomly select two individuals with indices idx1 and idx2 from the current population. 2) Compare the fitness values ​​of these two individuals. 3) Add a deep copy of the individual with higher fitness to the selection list.

[0101] 3. Return the selection list as the parent population.

[0102] Through tournament selection, the algorithm can gradually improve the average quality of the population while maintaining population diversity, providing good parent individuals for subsequent crossover and mutation operations.

[0103] Crossover Operation

[0104] The crossover operation combines the characteristics of two parent individuals to generate a child individual with new characteristics. The detailed steps of the crossover operation are as follows:

[0105] 1. First, the crossover probability is judged. If the randomly generated probability value is greater than the preset crossover rate, the deep copy of the parent generation 1 is directly returned without performing crossover.

[0106] 2. Otherwise, create an empty dictionary as the child solution.

[0107] 3. Get the union of all cargo indices contained in the two parent solutions:

[0108] cargo_indices1=set(parent1.keys())

[0109] cargo_indices2=set(parent2.keys())

[0110] all_indices=cargo_indices1U cargo_indices2

[0111] 4. Randomly select an intersection point in the range 0 to the length of all_indices.

[0112] 5. Convert all indices into lists and sort them to ensure determinism of the intersection operation.

[0113] 6. Initialize the slot capacity tracking dictionary poSition_weights and

[0114] position_volumes is used to ensure that the generated child solutions satisfy the capacity constraints.

[0115] 7. For the index before the intersection, the allocation scheme of parent 1 is preferred:

[0116] 1) Traverse the first crossover_point indexes.

[0117] 2) If the current index cargo_idx exists in parent 1, get its assigned cabin position_idx.

[0118] 3) Check whether the capacity constraint of the space is met:

[0119] position_weights[position_idx]+cargo.weight≤position.max_weight

[0120] position_volumes[position_idx]+cargo.volume≤18.9

[0121] If the constraints are satisfied, the assignment is added to the child solution and the slot capacity tracking is updated.

[0122] 8. For the indexes after the intersection, the allocation scheme of parent 2 is preferred, and the process is similar.

[0123] 9. Return the generated sub-generation solution.

[0124] This crossover strategy effectively combines the strengths of both parent generations while ensuring the feasibility of the solution through constraint checking and avoiding invalid solutions. Through single-point crossover, the algorithm can retain some of the structure of the parent solutions while introducing new combinations, increasing population diversity and boosting the algorithm's exploration capabilities.

[0125] mutation operation

[0126] The mutation operation increases population diversity by randomly changing certain characteristics of individuals, helping the algorithm escape from local optimality. The specific steps of the mutation operation are as follows:

[0127] 1. First, the mutation probability is judged. If the randomly generated probability value is greater than the preset mutation rate mutation_rate, the original solution is returned directly without performing mutation.

[0128] 2. Otherwise, create a deep copy of the original solution as the basis for mutation.

[0129] 3. If the solution is not empty, calculate the center of gravity of the current solution:

[0130] total_weight=∑ i∈mutated cargos[i].weight

[0131] total_moment=∑ i∈mutated cargos[i].weight×cargo_positions[mutated[i]].arm

[0132] current_cg=total_moment / total_weight

[0133] 4. Find the items farthest from the center of gravity, calculate the distance from each item to the center of gravity, and multiply it by its weight to use as the impact indicator:

[0134] cargo_distances

[0135] =[(i,|cargo_positions[mutated[i]].arm-current_cg|)for i inmutated.keys()]

[0136] Sort in descending order of distance multiplied by weight to find the cargo that has the greatest impact on the center of gravity.

[0137] 5. Randomly select one of the top 20% of the most influential goods to mutate.

[0138] 6. Calculate the current usage of each cabin and find all feasible alternative cabins:

[0139] Initialize the position_weights and position_volumes dictionaries to record the space occupied by other cargoes except the cargo to be mutated.

[0140] Iterate over all slots, check capacity constraints, and find all feasible slots.

[0141] 7. If there are feasible spaces, calculate the probability weight of each feasible space based on the distance to the center of gravity:

[0142]

[0143] The closer the cabin is to the center of gravity, the higher the probability of being selected.

[0144] 8. Select a new slot based on the calculated probability and reallocate the selected cargo to that slot.

[0145] 9. Return the mutated solution.

[0146] Through this mutation strategy, the algorithm can improve the center of gravity balance of the solution in a targeted manner while maintaining the basic structure of the solution, thereby improving the quality of the solution.

[0147] Gray Wolf Location Update

[0148] Gray wolf position update is the core innovation of this hybrid algorithm. This mechanism simulates the social hierarchy and collaborative hunting behavior in the gray wolf group, guiding the search direction of the algorithm. The algorithm sorts all individuals in the current population in descending order according to the fitness value, and determines the three dominant wolves in the group: alpha wolf (optimal solution), beta wolf (second best solution) and delta wolf (third best solution). The algorithm applies the gray wolf optimized position update mechanism to the discrete solution space of the loading optimization problem. The method first checks whether the three dominant wolves are defined, then creates an empty dictionary as the updated solution, obtains the union of all cargo indexes contained in the three dominant wolf solutions, and calculates the center of gravity position of the current solution as the reference point for position update. For each cargo index in the union, the algorithm calculates random coefficients A and C to control the step size and direction of the position update:

[0149] A1, A2, A3=random(-a,a),random(-a,a),random(-a,a)

[0150] C1, C2, C3=random(0,2),random(0,2),random(0,2)

[0151] Here, a is a parameter that decreases linearly with the number of iterations, from 2 to 0, controlling the balance between exploration and exploitation. The algorithm then obtains the locations assigned to the item by the three dominant wolves. If a dominant wolf does not have the item, it is skipped. The algorithm assigns different weights to valid locations: the alpha wolf has the highest weight (1.0), the beta wolf has the second highest weight (0.7), and the delta wolf has the lowest weight (0.5). Next, the algorithm combines this weight with the distance from the center of gravity to calculate the overall weight:

[0152]

[0153] combined_weights[i]=valid_weights[i]×distance_weights[i]

[0154] A new location is randomly selected based on the combined weights, and the capacity constraints are checked. If satisfied, the cargo is assigned to that location. Through this location update mechanism, the algorithm is able to leverage information from the three dominant wolves to guide its search direction, while also considering center-of-gravity balance to optimize load balance, maintaining randomness to enhance the algorithm's exploration capabilities, and strictly adhering to constraints to ensure the feasibility of the solution.

[0155] Optimization process

[0156] The complete optimization process coordinates the operation of various components to achieve the iterative optimization process of the Grey Wolf Genetic Hybrid Optimization Algorithm. The main steps of the optimization process are as follows:

[0157] 1. Record the start time for subsequent calculation of the running time.

[0158] 2. For each iteration (from 0 to max_generations-1):

[0159] 1) Linearly decrease from 2 to 0:

[0160] a=2-generation×(2 / max_generations)

[0161] This parameter controls the exploration and exploitation balance of the Grey Wolf Algorithm. Larger values ​​in the early stages promote global exploration, while smaller values ​​in the later stages promote local exploitation.

[0162] 2) Perform the tournament selection operation to generate the parent population selected.

[0163] 3) Initialize the new population new_population and first add a deep copy of the alpha wolf (elite retention strategy) to ensure that the optimal solution will not be lost during the iteration process.

[0164] 4) Generate new individuals in a loop until the population size reaches the required size:

[0165] 4.1) Randomly select two parent individuals parent1 and parent2.

[0166] 4.2) Perform crossover operation to generate offspring child.

[0167] 4.3) Perform mutation operations on the offspring.

[0168] 4.4) Apply the gray wolf position update mechanism to generate updated offspring.

[0169] 4.5) Add the updated offspring to the new population.

[0170] 5) Replace the old population with the new population:

[0171] population=new_population[:population_size]

[0172] 6) Evaluate the fitness of each individual in the new population.

[0173] 7) Update alpha, beta and delta wolves.

[0174] 8) Regularly output optimization progress information, including the current number of iterations, optimal fitness, number of loaded goods, and elapsed time.

[0175] 3. After the optimization is completed, the best solution (alpha wolf) is returned.

[0176] Through this iterative optimization process, the algorithm is able to gradually improve the quality of the solution, ultimately finding a high-quality loading solution that meets multiple objectives. The elite retention strategy ensures that the optimal solution is not lost, while the gray wolf position update mechanism provides additional search directions, enhancing the algorithm's exploration capabilities.

[0177] In the Gray Wolf Genetic Hybrid Optimization Algorithm, fitness calculation is a core step in evaluating solution quality, directly influencing the algorithm's search direction and convergence. The algorithm incorporates a multi-objective fitness evaluation system that comprehensively considers three key metrics: maximum load capacity, center of gravity control accuracy, and load balance. Specifically, the algorithm first calculates the total load weight and volume and compares them against maximum limits. If the limits are exceeded, a penalty coefficient is applied proportionally. The actual center of gravity position is then calculated by summing the weight moments of the cargo and dividing them by the total weight. The deviation from the target center of gravity is then measured, with smaller deviations indicating more accurate center of gravity control. Furthermore, the moment of inertia is calculated by summing the square of each cargo's distance to the actual center of gravity and multiplying its weight to assess load balance. The algorithm also prioritizes lateral balance by calculating the weight difference between the left and right cargo sides and applying a penalty when the difference exceeds a set threshold. Furthermore, the algorithm incorporates a penalty and reward mechanism, imposing a nonlinear penalty for excessive moment of inertia and providing additional rewards for high loads (within the limits). Finally, after normalizing all indicators, the comprehensive fitness value is calculated according to the preset weights to achieve a reasonable trade-off between multiple objectives and guide the algorithm to search for high-quality solutions.

[0178] The present invention also provides a multi-objective composite fitness function design method, wherein the fitness function takes maximizing the loading rate as the main objective, while penalizing the center of gravity offset, excessive moment of inertia, and constraint violation. The fitness function is in the following form:

[0179] F(X)=α·f load (X)+β·f cg (X)+γ·f inertia (X)-P penalty (X)+loading_bonus(X)-P inertia (X)

[0180] Among them, f load (X), f cg (X) and f inertia (X) are the normalized loading rate, center of gravity offset and moment of inertia objective functions; P penalty (X) is a penalty for violation of the constraint;

[0181] loading_bonus(X) is the loading bonus item, which encourages the algorithm to load as much cargo as possible; P inertia (X) is the moment of inertia penalty term, which is imposed when the moment of inertia exceeds the threshold.

[0182] Experimental results

[0183] In order to comprehensively evaluate the performance of the proposed algorithm, this paper constructed 20 test cases, covering loading scenarios of different scales and constraints. The test data includes various parameter configurations: aircraft maximum payload Maximum volume load V max =651.2m 3 , the maximum number of containers m = 37; the maximum container load Volume V j =18.9m 3 ; Cargo quantity range n∈[340,1100], cargo weight range w i ∈[50,600]kg, cargo volume range v i ∈[0.02,2.71]m 3 These parameter settings are based on actual air cargo scenarios to ensure the practicality and representativeness of the test data. Test cases can be divided into four categories according to the constraints: Type I means that both weight and volume are within the limits, i.e. And V total ≤V max ; Type II means the weight exceeds the limit but the volume does not exceed the limit, that is And V total ≤V max ; Type III means that the weight is within the limit but the volume exceeds the limit, i.e. And V total >V max ; Type IV means that both weight and volume exceed the limit, i.e. And V total >V max Among them, W total and V total Represent the total weight and total volume of all cargo respectively. The detailed parameters of some typical test cases are shown in Table 1. These cases cover loading scenarios under different constraints and can fully verify the adaptability and robustness of the algorithm. Figure 2 The mean aerodynamic chord is 7.08 m and the center of gravity of the aircraft is at 28% MAC (mean aeronautical chord).

[0184] Table 1 Typical test case data

[0185]

[0186]

[0187] The comprehensive performance comparison of the GWGA algorithm and the improved combination optimization model on four types of test cases, such as Figure 3As shown in the figure, it can be clearly seen that the GWGA algorithm shows significant advantages in the two key indicators of moment of inertia and calculation time, which fully verifies the effectiveness of incorporating moment of inertia into the optimization target and the superiority of integrating the gray wolf algorithm and the genetic algorithm. Table 2 further provides detailed data support. While maintaining a high loading rate and volume utilization rate, the GWGA algorithm significantly reduces the moment of inertia (an average reduction of about 28% compared to the improved combination optimization model) and greatly improves the calculation efficiency (an average improvement of about 91% compared to the improved combination optimization model). In particular, in the Type III (volume-constrained) test case, the GWGA algorithm also shows obvious advantages in the control of the center of gravity offset. This shows that the comprehensive optimization model proposed in the present invention can achieve a good balance between multiple objectives, and the hybrid optimization algorithm can effectively avoid falling into local optimality, providing a more efficient and balanced solution for aircraft three-dimensional loading optimization.

[0188] In order to more intuitively demonstrate the performance differences of the algorithms, a typical test case (case 14) in type III was selected for visualization analysis. The GWGA algorithm and the improved combination optimization model were respectively displayed. From the visualization results, we can clearly observe the essential differences between the two algorithms in loading strategies. Figure 4 and Figure 5 As shown in the figure, the GWGA algorithm concentrates heavier cargo near the aircraft's center of gravity, forming a "core-shell" structure centered on the center of gravity, with heavy cargo concentrated in the core area and light cargo distributed on the periphery. The improved combined optimization model, on the other hand, tends to evenly distribute the weight of each cargo hold, without considering the relative position of the cargo to the center of gravity. This difference in loading strategy directly leads to a significant difference in moment of inertia performance. The GWGA algorithm has a moment of inertia of 10,582,935 kg·m 2 , which is higher than 16132813kg·m of the improved combined optimization model 2 This is approximately 34.4% lower, due to the GWGA algorithm's strategy of concentrating the weight near the center of gravity. This "core-shell" loading strategy not only improves the aircraft's balance, but also enhances maneuverability and fuel efficiency, making it particularly suitable for practical operational scenarios requiring high flight safety and economy.

[0189] Table 2 Performance comparison of GWGA algorithm and improved combination optimization model on different types of test cases

[0190]

[0191]

[0192] The above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, improvements, etc. made within the spirit and principles of the present invention should be included in the scope of protection of the present invention.

Claims

1. A cargo aircraft packing and stowing method based on gray wolf genetic hybrid optimization, characterized by: include Construct a multi-objective optimization model, and the optimization objective function is Among them, f load is the load objective function, f cg is the center of gravity shift objective function, f inertia is the moment of inertia objective function, P penalty is the constraint penalty objective function, α, β, γ, and δ are adjustable objective weight coefficients; A hybrid optimization algorithm that integrates the global search capability of the grey wolf algorithm and the adaptive mechanism of the genetic algorithm is designed to solve the multi-objective optimization model.

2. The cargo aircraft packing and stowing method based on gray wolf genetic hybrid optimization according to claim 1 is characterized by: The load objective function normalizes the ratio between the total cargo weight of the current solution and the maximum allowable payload of the aircraft so that the target value is always between [0, 1]. The specific form is as follows: Among them, w i is the weight of cargo i, W max is the maximum allowable payload of the aircraft, x ij is the decision variable, defined as 3. The cargo aircraft packing and stowing method based on gray wolf genetic hybrid optimization according to claim 1 is characterized by: The center of gravity offset objective function sets the target center of gravity position and minimizes the deviation between the center of gravity of the current loading solution and the target center of gravity. The center of gravity offset is expressed as a percentage of the mean aerodynamic chord and is converted to the actual deviation distance using the following formula: in is the center of gravity of the current loading solution, a t is the target center of gravity position, Δa max For the maximum permissible deviation, the center of gravity position is converted to the actual distance using the following formula: where a m is the actual center of gravity position, a ％MAC is the center of gravity position expressed as a percentage of the mean aerodynamic chord, L MAC is the length of the mean aerodynamic chord, L LE is the distance of the leading edge of the mean aerodynamic chord relative to the aircraft reference zero point.

4. The method for packing and stowing cargo aircraft based on gray wolf genetic hybrid optimization according to claim 2, characterized in that: The moment of inertia objective function is in the form of: Where I is the moment of inertia of the current loading scheme, expressed as I max It is the preset theoretical maximum moment of inertia.

5. The cargo aircraft packing and stowing method based on gray wolf genetic hybrid optimization according to claim 1 is characterized by: The constraint penalty objective function is in the form of P penalty =λ w P w +λ v P v +λ l P l Among them, P w P is the penalty value caused by exceeding the allowable load of the cargo hold. v is the penalty value for exceeding the container volume limit, P l is the penalty value caused by the unbalanced loading on the left and right sides. The calculation method of each penalty item is as follows Among them, W total is the total weight of the current load, W max is the maximum payload of the aircraft, V total is the total volume currently loaded, V max is the maximum volume limit of the aircraft, W L and W R are the total weight of the left and right cabins, ΔW threshold is the weight threshold.

6. The cargo aircraft packing and stowing method based on gray wolf genetic hybrid optimization according to claim 1 is characterized by: It also includes setting up the following constraint system: Cargo unique distribution constraint, cargo hold capacity constraint, left and right hold balance constraint, main cargo hold symmetry constraint, maximum payload limit, and center of gravity moment limit.

7. The cargo aircraft packing and stowing method based on gray wolf genetic hybrid optimization according to claim 1 is characterized by: The hybrid optimization algorithm designed to integrate the global search capability of the gray wolf algorithm and the adaptive mechanism of the genetic algorithm to solve the multi-objective optimization model specifically includes: The social hierarchy structure in the gray wolf optimization algorithm is adopted to divide the population individuals into α wolves, β wolves, δ wolves and ω wolves. In the initialization phase, a two-stage initialization strategy is adopted to generate a high-quality initial population, which is evaluated by the fitness function. Dynamically adjust the step size coefficient and mutation probability to achieve adaptive adjustment of search capability; In the selection phase, the tournament strategy is used to retain excellent individuals, while the population quality is improved through the α-wolf direct inheritance mechanism; In the reproduction stage, the feasibility and diversity of solutions are enhanced through single-point crossover and mutation operations based on centroid distance; After iterative updating, the solution with the highest evaluation value is output as the optimal loading solution.

8. The method for packing and stowing cargo aircraft based on gray wolf genetic hybrid optimization according to claim 7, characterized in that: The two-phase initialization strategy includes In the first stage, cargo is allocated to spaces close to the target center of gravity in priority according to cargo density; The second stage dynamically adjusts the distribution plan of the remaining cargo based on the current center of gravity position to achieve preliminary center of gravity balance control.

9. The method for packing and stowing cargo aircraft based on gray wolf genetic hybrid optimization according to claim 7, characterized in that: The fitness function aims to maximize the loading rate while penalizing the center of gravity offset, excessive moment of inertia, and constraint violation. The fitness function is as follows: F(X)=α·f load (X)+β·f cg (X)+γ·f inertia (X)-P penalty (X)+loading_bonus(X)-P inertia (X) Among them, f load (X), f cg (X) and f inertia (X) are the normalized loading rate, center of gravity offset and moment of inertia objective functions; P penalty (X) is a penalty for violation of the constraint; loading_bonus(X) is the loading bonus, which encourages the algorithm to load as much cargo as possible; P inertia (X) is the moment of inertia penalty term, which is imposed when the moment of inertia exceeds the threshold.

10. The cargo aircraft packing and stowing method based on gray wolf genetic hybrid optimization according to claim 1, characterized in that: It also includes the continuous spatial position update of the gray wolf algorithm: Check whether α wolf, β wolf, and δ wolf have been defined, create an empty dictionary as the updated solution, obtain the union of all cargo indices contained in the three dominant wolf solutions, calculate the center of gravity position of the current solution as the reference point for position update, and for each cargo index in the union, the algorithm calculates random coefficients A and C to control the step size and direction of the position update: A1,A2,A3=random(-a,a),random(-a,a),random(-a,a) C1,C2,C3=random(0,2),random(0,2),random(0,2) Among them, a is a parameter that decreases linearly with the number of iterations, from 2 to 0, controlling the balance between exploration and exploitation. The algorithm obtains the allocation position of the goods by the three dominant wolves. If a dominant wolf does not contain the goods, it is skipped. For valid positions, the algorithm assigns different weights: α wolf has the highest weight (1.0), β wolf has the second highest weight (0.7), and δ wolf has the lowest weight (0.5). The algorithm combines the weight of the distance to the center of gravity to calculate the comprehensive weight: combined_weights[i]=valid_weights[i]×distance_weights[i] A new location is randomly selected based on the combined weights and the capacity constraints are checked, and the goods are allocated to that location if satisfied.

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