A Genetic Algorithm with Lower Neighborhood Search for the Three-Dimensional Multiorder Open-Size Rectangular Packing Problem

. Tis paper addresses the multiorder open-dimension three-dimensional rectangular packing problem (3D-MOSB-ODRPP), which involves packing rectangular items from multiple orders into a single, size-adjustable container. We propose a novel metaheuristic approach combining a genetic algorithm with the Gurobi solver. Te algorithm incorporates a lower neighborhood search strategy and is underpinned by a mathematical model representing the multiorder open-dimension packing scenario. Extensive experiments validate the efectiveness of the proposed approach. Te LNSGA algorithm outperforms Gurobi and the traditional genetic algorithm in solution quality and computational efciency. For small-scale instances, LNSGA achieves optimal values in most cases. LNSGA demonstrates signifcant optimization improvements over Gurobi and the genetic algorithm for large-scale instances. Te superior performance is attributed to the efective integration of the lower neighborhood search mechanism and the Gurobi solver. Tis study ofers valuable insights for optimizing the packing process in e-commerce warehousing and logistics operations.


Introduction
Amidst the rapid growth of e-commerce, the global express business surpassed 170 billion pieces in 2021, with a year-onyear surge of over 25% [1].In China, the postal industry achieved 139.1 billion dispatched parcels in 2022, with express business reaching 110.58 billion pieces, showcasing a year-on-year growth of 2.1% [2].Efcient parcel packaging profoundly impacts user experience and can lead to savings of up to 30% in logistics expenses [3], making it strategically signifcant for e-commerce enterprises.
Te three-dimensional bin packing problem originated in the 1960s, focusing on single-container packing [4].Te problem was proven to be NP-hard and exact algorithms were time-consuming [5,6].Research transitioned to metaheuristics, and as the 21st century progressed, the focus shifted to multicontainer [7], dynamic packing [8], and machine learning methodologies [9].Recent advancements have led to specialized cases such as open-dimension packing [10] and multiorder mixed packing problems [11].
Open-dimension bin packing relaxes container size constraints to better refect real-world scenarios.Existing literature has employed algorithms such as simulated annealing [12] and particle swarm optimization [13], which rely on random search and can get trapped in local optima.Te multiorder mixed packing problem involves factors such as order priority and time limits, increasing complexity.Te current research focuses on single objectives, with limited attention to multiobjective decision-making [11,14].
As packing tasks become more intricate, traditional exact algorithms need help with multiple constraints.Metaheuristics such as genetic algorithms [15], simulated annealing, and particle swarm algorithms [16,17] have shown promise but face challenges in obtaining satisfactory solutions for complex problems.Researchers have explored integrating metaheuristics with exact solvers, giving rise to metaheuristics [18,19].Metaheuristics combines the global exploration capabilities of metaheuristics with the local optimization power of exact solvers, demonstrating superior performance in routing [20,21], scheduling [22], and packing problems [23,24].
Tis paper introduces a novel metaheuristic approach to address the multiorder open-dimension 3D rectangular packing problem.Te proposed hybrid algorithm combines a genetic algorithm with the Gurobi solver to manage problem complexity and improve upon traditional metaheuristics.Te algorithm is underpinned by a mathematical model representing the packing scenario and incorporates a lower neighborhood search strategy.Comparative experiments validate the approach's efectiveness in tackling modern packing problems and set a foundation for future research.
Te paper is structured as follows.Section 2 reviews the literature on packing problems and solution algorithms.Section 3 presents the problem model and algorithm design.Section 4 focuses on experimental results and analysis.Section 5 concludes the paper and discusses future directions.

Literature Review
In this section, we review related research on the packing problem from three perspectives: application scenarios, dimensionality, and solution algorithms, respectively.

Packing Problems in Diferent
Scenarios.Te research on packing problems spans various practical domains, such as pallet loading, nesting problems, warehouse cargo packing, container loading, and cutting stock flling problems, among several others.
(1) Pallet loading problem: the pallet loading problem (PLP) involves placing multiple items on a pallet of a certain size to fnd the loading solution with the lowest stacking height [25][26][27].(2) Nesting problem: nesting problems focus on efciently cutting materials to obtain fnal items while adhering to shape-based rules [28,29].Tese problems are relevant across diverse industries, such as textile, clothing, sheet metal cutting, furniture manufacturing, and automobile industry [30,31].(3) Warehouse cargo packing problem: this problem involves efciently packing goods based on order information from a warehouse.Scholars have explored irregular parts packing [32], fexible packing with nonfxed box sizes in e-commerce, and loading goods onto vehicles during the outbound warehouse process [33,34].(4) Container loading problem: the container loading problem (CLP) aims to load items into a container efciently, maximizing the loading rate or minimizing the number of containers required, while considering support and stability constraints [35].It encompasses three-dimensional container loading for transportation via ships, trucks, or railroad cars and air cargo packing.Researchers have also focused on multicontainer loading problems [36,37].(5) Cutting stock flling problem: the cutting stock flling problem involves efciently arranging multiple items within a designated space in industrial contexts to maximize space utilization or minimize the total volume occupied [38,39].
Tis paper focuses on the e-commerce warehousing state's packing problem, specifcally packing warehouse goods.Te subsequent section reviews studies on various dimensions of the packing problem.

Packing Problems in Diferent
Dimensions.Packing problems can be classifed based on dimensions, mainly onedimensional, two-dimensional, and three-dimensional problems.

One-Dimensional and Two-Dimensional Packing
Problems.Tis study focuses on packing problems in the context of e-commerce warehousing.One-dimensional packing problems have been studied but must be more relevant here [40].Two-dimensional packing problems have received signifcant attention, including the guillotine knapsack packing problem, strip-packing problem with knifng constraints [41], and case packing for regular and irregular items [42].Te rectangular packing problem (RPP) describes the rectangular strip-packing problem (RSPP) [43].Researchers have also investigated the undercutting, strip, and open-dimension 2D packing problems for various irregular objects [44].
Scholars have studied 3D cutting and packing problems with nonoverlapping constraints and container loading problems within logistics platforms [51].Container loading problems include multicontainer, LTL, and single-container 2 International Journal of Intelligent Systems loading with transportation priority [52,53].Real-world problems also involve multipoint constraints requiring specifc box accessibility at each delivery point.Tis paper focuses on the 3D bin packing problem in ecommerce settings.However, the complexity of 3D packing problems challenges traditional exact algorithms.Te next section reviews solution algorithms for complex 3D packing problems.

Algorithmic Research Based on the Tree-Dimensional
Packing Problem.Te 3D bin packing problem is an NPhard combinatorial optimization problem.Early studies used exact algorithms, but as problem scales grew, computation times became lengthy, and satisfactory solutions took time to obtain.From the 1970s, researchers turned to heuristic algorithms, with genetic algorithms performing prominently.In the 21st century, metaheuristic algorithms were widely adopted.Gezici and Livatyali improved the Harris Hawks algorithm's random parameter generation strategy to enhance the global search ability [54].Xiong's team proposed a deep reinforcement learning-based method for online packing problems [55].
As packing problems continue to grow in complexity, researchers are constantly seeking to enhance solution algorithms.To address the three-dimensional multiple bin size bin packing problem with open dimension and reserve parameter (3D-MOSB-ODRPP), this paper proposes an improved genetic algorithm that incorporates a lower neighborhood searchapproach.

Research Gap.
Te comprehensive review of packing problems and solution algorithms reveals several research gaps this study aims to address.First, existing research on packing problems has primarily concentrated on fxeddimension containers, with limited attention given to open-dimension scenarios.However, in e-commerce warehousing, the fexibility to adjust container sizes based on order requirements is crucial for optimizing resource utilization and reducing costs.Tis study incorporates opendimension packing into the problem formulation, allowing for the determination of optimal container dimensions.
Second, although metaheuristic algorithms have been widely adopted for solving packing problems, traditional approaches often need help to balance global exploration and local exploitation efectively.Tis study introduces a novel metaheuristic approach that combines the strengths of a genetic algorithm for global search with the local optimization capabilities of the Gurobi solver.By integrating these two components, the proposed algorithm aims to overcome the limitations of traditional metaheuristics and enhance solution quality.
Lastly, while previous studies have considered various constraints and practical factors, integrating multiple orders and open dimensions in a single packing problem has received limited attention.Tis study addresses this research gap by formulating the 3D-MOSB-ODRPP, which optimizes the packing of multiple orders with varying item sizes and quantities into a single, size-adjustable container.
To further highlight the unique contributions of this study, Table 1 compares the present research with the existing studies across several key dimensions.
As evident from Table 1, the present study simultaneously addresses multiple key aspects of the 3D packing problem.Hile's existing studies have individually considered some of these dimensions but still need to integrate them into a comprehensive problem formulation and solution approach.By bridging these research gaps, this study aims to provide a more realistic and efective solution to the complex 3D-MOSB-ODRPP encountered in e-commerce warehousing.

Methodology
Tis section introduces the 3D-MOSB-ODRPP model proposed in this paper and the improved genetic algorithm for solving this model.First, we will elaborate on the mathematical model for the multiorder open-dimension 3D packing problem, considering practical characteristics such as mixed orders and adjustable bin sizes.Ten, we will elucidate the solution approach of the lower neighborhood search-based enhanced genetic algorithm, laying the groundwork for the computational experiments in later sections.
3.1.Problem Description.Tis paper primarily studies the 3D-MOSB-ODRPP.Tis problem expands on the 3D-ODRPP proposed by Tsai by comprehensively considering the optimization of adjustable container length, width, and height under multiple orders with a single box type.Specifcally, as shown in Figure 1 given multiple packing orders where each order contains rectangular items with known length, width, and height, the goal is to determine a unifed box size so that all items across orders can be packed into a container of that size, thus maximizing the container space utilization.Tis problem is more practical than fxed-size packing by tuning the box dimensions to balance order demands, but the complex constraints also increase the difculty of solving it.Tis paper formulates a mixed-integer programming model to obtain accurate solutions and designs an improved genetic algorithm for efective solving, obtaining feasible schemes for the 3D-MOSB-ODRPP.A detailed description of the decision variables is shown in Table 3.

Mixed-Integer Programming Model.
We construct a mathematical model in this section to further solve the 3D-MOSB-ODRPP proposed in this paper.First, in order to maximize the space utilization efciency while determining a unifed container size, we defne the where z represents the box's volume, i.e., the product of three dimensions: length, width, and height.Ten, we introduce the constraint conditions in the mathematical model.First, to ensure that items can be smoothly packed into the container during the packing operation, we need to limit the range of each item inside the container to three dimensions-length, width, and height.Te equations are as follows: where equation ( 2) calculates the range occupied by the item in the x-axis direction inside the container, considering the projection on the x-axis after rotation, which should be less than or equal to the container length l B to ensure feasibility; equation (3) calculates the range occupied in the y-axis direction inside the container, which should be less than or equal to the container width w B to ensure ft along the y-axis; and equation ( 4) calculates the range occupied in the z-axis direction inside the container, which should be less than or equal to the container height h B to ensure ft along the z-axis.
After constraining items from exceeding container boundaries, we need to constrain that items do not overlap when placed inside the container, as shown in equation (5).Since after limiting items to be within the container as in equations ( 2)-( 4), solely satisfying this condition cannot guarantee that the fnal packing solution is feasible.Packaging multiple items into one container will likely overlap in placement, leading to infeasibility.Tus, nonoverlap constraints must be added to clearly defne the relative placement of items so that they do not overlap inside the container, thereby ensuring solution feasibility.Together, equations ( 2)-( 5) ensure that the packing solution meets two key requirements, flling the container and no overlaps. where indicates whether items i r and i r′ overlap in the k-th direction (k ranges from 1 to 6, representing the positive and negative directions of x, y, z axes).When D k i r ,i r′ takes the value 0, it means that the two items do not overlap in this dimension; when it takes the value 1, it means overlap is allowed between the two items.In other words, 0 indicates no overlap between the two items, and 1 indicates potential overlap.By defning the 0-1 variable D, we can explicitly formulate the relative placement relationship between items in each axis, thus laying the foundation for subsequent nonoverlap constraint calculations.Note that, the D variable is unrelated to the specifc overlap situation; it merely indicates whether overlap is permitted between two items on a given axis.
In addition, to comprehensively and accurately ensure no overlap between any two items inside the container, we further need to add nonoverlap constraints between items with diferent IDs, as shown in equation (6).Although the previously defned 0-1 variable D clarifed whether overlap is allowed between two items in each axis, it did not specifcally defne the absolute spatial relationship between two items.To achieve absolute nonoverlap inside the container, we need to calculate the specifc coordinate ranges of diferent items in each axis and restrict these ranges from intersecting.Tis ensures that the two items do not overlap in a singledimensional axis and avoids overlap when integrated in a three-dimensional space.
where i r and i r′ represent the IDs of two diferent items in the same order, ranging from 1 to the total number of items m in the order.Te item IDs i r and i r′ must difer and cannot take the same value.Tis is because, for the same item, its spatial projection region will not overlap with itself.If the IDs i r and i r′ take the same value, this nonoverlap constraint will degrade to a constraint between an item and itself, failing to achieve nonoverlap between diferent items.Terefore, the nonoverlap constraint must be expressed between two items with diferent IDs; only then can it truly Te length of the j-th packing case w B  Te width of the j-th packing case h B  Te height of the j-th packing case Coordinates of the x-axis position of the i r -th item in the r-th order loaded into the packing box Coordinates of the y-axis position of the i r -th item in the r-th order loaded into the packing box Coordinates of the z-axis position of the i r -th item in the r-th order loaded into the packing box 0-1 variables that determine the direction of rotation when the i r -th item in the r−th order is boxed 0-1 variable for designing nonoverlapping constraints between item i r and item i r ′ in the r-th order International Journal of Intelligent Systems defne the relative placement relationship between diferent items inside the container, avoiding overlap.After ensuring no conficts between the items to be packed and the container, we need to add constraints between diferent items in the same container to avoid overlaps and other conficts between items, as shown in the following equations: In equation (7), i r and i r′ represent two diferent items, calculates the range occupied by item i r′ in the x-axis direction.
), and M calculates the x-axis coordinate of item i r multiplied by a large non-negative number M. D 1 i r ,i r′ is a 0-1 variable, taking the value 1 when the two items do not overlap, making M equal to 0 and satisfying the constraint.When the two items overlap, D 1 i r ,i r′ is 0, making M a very large value, the right side greater than the left, and the constraint unsatisfed.Similarly, equation (8) ensures that items i r and i r′ do not overlap in the x-axis direction.Equation ( 9), similar to equation ( 7), mainly calculates the range occupied by item i r′ in the y-axis direction, comparing it with the y-coordinate of item i r and a large number M. When the two items do not overlap on the y-axis, the 0-1 variable D 3 i r ,i r′ is 1, making M equal to 0 and satisfying the constraint.If the two items overlap on the y-axis, D 3 i r ,i r′ is 0, making M a large value and violating the constraint.Equation ( 10) is similar to equation ( 8), utilizing the 0-1 variable and the large number M to ensure that items i r and i r′ do not overlap in the y-axis direction.
Equation (11), similar to equations ( 7) and ( 9), calculates the range occupied by item i r′ in the z-axis direction and compares it with the z-coordinate of item i r and a large number M. When the two items do not overlap on the z-axis, the 0-1 variable D 5 i r ,i r′ is 1, making M equal to 0 and satisfying the constraint.If overlap exists, D 5 i r ,i r′ is 0, making M a large value and violating the constraint.Finally, equation ( 12) utilizes the 0-1 variable and large number M to ensure i r and i r′ do not overlap on the z-axis.
In order to further confict between the items in the box, the following constraints are therefore set: where i r ,i r′ are 0-1 variables indicating whether items i r and i r′ overlap in the three dimensions.Since each D variable can only take 0 or 1, at least one of these variables must take the value 1, meaning the two items do not overlap in that dimension.Tus, this constraint ensures that any two items i r and i r′ do not overlap in at least one dimension, avoiding three-dimensional conficts between them.Te abovementioned constraints are mainly used to avoid overlap conficts between items and between items and container boundaries.To accurately calculate the space occupied by items in the container's length, width, and height dimensions, all possible placement directions of items need to be considered.For example, a rectangular item can be placed horizontally, vertically, or at various tilts.However, computing and storing all direction combinations for each item will greatly increase model complexity and solving difculty.Terefore, this model uniquely determines the placement direction of each item.Specifcally, we introduce 0-1 variables T to indicate the rotation decision of each item in three dimensions and construct rotation matrices to represent the projected length in each direction.As shown in equations ( 14)- (19), each item only chooses one placement direction, signifcantly reducing the model difculty.
6 International Journal of Intelligent Systems where equation (14) indicates that item i r has three potential rotation states in the positive x-axis direction of the container, requiring that only one of these three states can take the value one and the rest 0. Tis ensures a unique rotation decision for the item in the x-axis direction.Equations ( 15) and ( 16) are similar, limiting item i r to only one rotation state in the positive y-axis and z-axis of the container.Ten, equation ( 17) further requires that for the rotation states of each axis, only one can be uniquely determined from the three options.Tat is, the rotation combination of an item must take one state per axis and cannot repeat selecting two or more states in one axis.Finally, equations ( 18) and ( 19) ensure that the length, width, and height of an item can only have unique efects in the three axes.
Te following constraint conditions are set to ensure the uniqueness of item i r 's rotation state: where T i r jk (j � 1, 2, 3; k � 1, 2, 3) represents the 0-1 variable for the k-th rotation state of item i r in the j-th axis direction.Each T i r jk variable can only take 0 or 1, 0 meaning item i r does not select the corresponding rotation state in that axis direction, and 1 meaning it selects that rotation state in that axis.Tese 0-1 variables are introduced to explicitly represent the rotation decision of items, providing a quantitative way to model the rotation states.Since each item can only select one rotation state, by requiring the T variables in each axis to sum to 1, the uniqueness of the rotation scheme can be ensured.
In addition, to defne the value range of the rotation decision k and to ensure accurate computing of the coordinate's positional relationship between items and the container, the following equation is set as: where k takes diferent integer values to represent the correspondence between item length and width directions and the three coordinate axes: k � 1 indicates that the item length direction is consistent with the x-axis, k � 2 indicates that the item length direction is consistent with the y-axis, k � 3 indicates that the item length direction is consistent with the z-axis, k � 4 indicates that the item width direction is consistent with the x-axis, k � 5 indicates that the item width is consistent with the y-axis, and k � 6 indicates that the item width is consistent with the z-axis.Tis variable is introduced to establish the connection between item length and width directions and the spatial coordinate axes directions, with k indicating possible item placement directions.After determining the value of k, items' spatial occupancy and coordinate positions can be calculated based on their projected dimensions on the corresponding axes.Finally, to facilitate subsequent calculations of the model, a relaxation constraint is set here as 3.2.3.Algorithm Design.In order to address the challenges posed by the 3D-MOSB-ODRPP, this section proposes a hybrid approach termed the Gurobi-enhanced local neighborhood search genetic algorithm (LNSGA).Traditional genetic algorithms focus on global exploration, whereas the local neighborhood search mechanism targets local exploration.Consequently, the LNSGA algorithm integrates both the global and local exploration organically.Moreover, following the local neighborhood search, invoking the Gurobi solver ensures that the returned solution represents the globally optimal solution for the threedimensional item packing.
A plain genetic algorithm (GA) is also implemented for the 3D-MOSB-ODRPP problem to compare and analyze the performance of the LNSGA algorithm.Te GA follows the basic structure of genetic algorithms, including population initialization, ftness evaluation, selection, crossover, and mutation.However, unlike the LNSGA algorithm, the GA does not incorporate the Gurobi solver or the lower neighborhood search mechanism.
In the GA implementation, the three-space (TS) heuristic algorithm is employed to calculate the packing confguration of items.Te TS algorithm is a fundamental heuristic approach in three-dimensional packing problems.It operates by recursively dividing the remaining space into three subspaces and selecting the most suitable subspace for placing the next item.Te process continues until all items are packed or no feasible subspace is available.
Te pseudocode of the GA algorithm is shown in Algorithm 1, and its key steps are as follows: (1) Initialization: generating an initial population of individuals, each representing a potential packing sequence of items.
(2) Fitness evaluation: evaluating each individual's ftness in the population using the TS algorithm.Te volume utilization ratio of the packing confguration determines the ftness value.
(3) Selection: applying a selection operator, such as tournament selection or roulette wheel selection, to choose individuals with higher ftness values for reproduction.
(4) Crossover: performing a crossover operation, such as one-point crossover or two-point crossover, to create ofspring individuals by exchanging genetic information between selected parent individuals.(5) Mutation: applying a mutation operator, such as swap mutation or insertion mutation, to introduce random variations in the ofspring individuals.
(6) Replacement: replacing a portion of the population with the newly generated ofspring individuals based on their ftness values.
(7) Termination: repeating steps 2-6 until a predefned termination criterion is met, such as reaching a maximum number of generations or achieving a satisfactory solution quality.
Compared to the LNSGA algorithm, the GA relies solely on the global exploration capabilities of genetic algorithms and the basic TS heuristic for packing calculations.It does not beneft from the targeted lower neighborhood search International Journal of Intelligent Systems mechanism or the global optimization capabilities of the Gurobi solver.As a result, the GA may need more performance in terms of solution quality and computational efciency, especially for complex 3D-MOSB-ODRPP instances.
Te pseudocode of the LNSGA algorithm is shown in Algorithm 2, and its key steps are as follows: (1) Let O � O 1 , O 2 , . . ., O q denote the set of q orders, where each order O r (r ∈ 1, 2, . . ., q) is considered as a distinct three-dimensional open-dimension rectangular packing problem (3D-ODRPP).Te Gurobi solver is utilized to obtain the optimal packing dimensions (L r , W r , and H r ) for each order O r .Traditional genetic algorithms focus on global exploration by searching the solution space S through population evolution and information exchange.However, for complex 3D-MOSB-ODRPP instances, global exploration alone may lead to local optima.A local neighborhood search mechanism is introduced to perform local optimization on the current solution s ∈ S to enhance local search capability.(2) We sort the optimal packing dimensions of each order (L r , W r , and H r ) in a descending order.Ten, we determine the maximum length, width, and height values across all orders, denoted as (L up , W up , and H up ), which serve as the upper bounds for the 3D-MOSB-ODRPP solution.
Mathematically, it is represented as ( where L up (C i ), W up (C i ), and H up (C i ) represent the maximum length, width, and height of the packing solution obtained by following the search path encoded in chromosome C i , and the objective is to minimize the ftness value.(8) Te roulette wheel selection is applied to select individuals with high ftness values to form a new population P ′ .In each iteration, the individuals in the current population P are sorted by their ftness values.Te individual with the minimum ftness value is selected with a probability of (0.5) 1 , the second minimum with a probability of (0.5) 2 , and so on, until a new population P ′ with the same size as P is formed.(9) Based on the crossover probability p c , we randomly selected two chromosomes C i and C j from the population P ′ .A crossover point k ∈ 1, 2, . . ., g − 1 is randomly chosen and the gene segments are exchanged after the crossover point between C i and C j to generate two ofspring chromosomes C i ′ and C j ′ .C i and C j in the population are then replaced with C i ′ and C j ′ .Te crossover process is shown in Figure 3.
Require: population size N, crossover probability p c , mutation probability p m , maximum number of generations G max , and neighborhood size k Ensure: best solution s * (1) Initialize population P with N randomly generated individuals (2) for each individual s in P do (3) Evaluate the ftness of s using the Gurobi solver (4) s′ ← LocalNeighborhoodSearch(s, k) (5) s * ← Gurobi (s′) (6) Update s with s * in P (7) end for (8) g ← 0 (9) while g < G max do (10) P′ ← ∅ (11) while |P′| < N do (12) Select parents p 1 and p 2 from P using the selection operator ( 13)  (10) Based on the mutation probability p m , we randomly selected a chromosome C i from the population P ′ .A mutation point k ∈ 1, 2, . . ., g is randomly chosen and the gene at position k is replaced with a randomly selected value from 1, 2, . . ., 7 to generate a new chromosome C i ′ .C i is then replaced with C i ′ in the population.Te mutation process is shown in Figure 4. (11) Steps 7 through 10 are repeated until the maximum number of iterations EPOCH is reached.To further enhance performance, the Gurobi solver is invoked after each local neighborhood search to perform global optimization on the current solution s.Te Gurobi solver is an exact method that solves the 3D packing problem to obtain the global optimal solution s * based on s.By combining local neighborhood search with the Gurobi solver, the LNSGA algorithm achieves an organic integration of global exploration and local search, enabling it to escape from the local optima and obtain high-quality global optimal solutions.

Dataset Design.
Te data required for the current experiment includes item size information and order data information.
(1) Item Information.Te SKU of an item represents the smallest packaging unit of storage items and serves as an essential information for order items.Diferent items have varying sizes and SKU dimensions.Tis study sets the upper and lower limits, along with the change step, for each item' SKU's length, width, and height, as shown in Table 6.Accordingly, there are ten possible values for each dimension of the item SKU, 10 International Journal of Intelligent Systems resulting in 1000 diferent SKU combinations when considering the variations across all three dimensions.Tus, 1000 unique item sizes form the basis for generating item order information.(2) Multiscale order dataset.
From the generated 1000 types of items, 2 to 6 items are randomly selected as the item information for an order.For example, when dealing with an order consisting of 4 items, its schematic diagram is shown in Figure 5.In order to conduct computational experiments with diferent order quantities, two sets of cases with diferent scales of order quantities are randomly selected, totaling 24 scenarios for the experiments.Each scenario includes orders of varying quantities, and all orders in each scenario are required to be packed into packaging boxes of the same size and model.

Algorithm's Hyperparameter
Settings.Tis study conducted computational experiments on six sets of smallscale test cases to determine the fnal hyperparameter values for the LNSGA algorithm.Two of these test cases exhibited relatively unique order item quantities and sizes, resulting in identical optimal ftness values under diferent parameter settings.Although such instances were relatively infrequent, the careful analysis revealed that when data scales were small or the problems were relatively simple, any combination of hyperparameters could easily attain the optimal solution.Alternatively, in specifc scenarios, all combinations of hyperparameters could readily converge to the same local optimum, resulting in identical optimal solution values across all hyperparameter combinations.It is worth noting that among the experimental results of the six test cases, the fnal values of the hyperparameterspopulation size (GROUP) and the number of algorithm iterations (EPOCH) were consistent, 35 and 10, respectively.However, there were multiple possible values for the hyperparameters pc (crossover probability) and pm (mutation probability).Te pc values were 0.4, 0.65, and 0.9, while the pm values were 0.1, 0.2, and 0.3.Considering that pc and pm represent the population's crossover probability and mutation probability, respectively, larger values of pc and pm within the allowed range indicate more active crossover and mutation in the population, thus increasing the likelihood of fnding solutions with higher optimization.Terefore, the fnal values of pc and pm were determined to be 0.9 and 0.3, respectively.
Te hyperparameters to be determined include the GROUP, the EPOCH, the pc, and the pm.Te upper and lower limits for each hyperparameter search, along with the number of variation steps, are shown in Table 7. Te computational process of the three test cases with two orders each in the small-scale example serves as the reference basis for determining the hyperparameters.As shown in Table 7, the optimal parameter combination for the four hyperparameter test cases is GROUP = 35, EPOCH = 10, pc = 0.4, 0.65, or 0.9, and pm = 0.1, 0.2, or 0.3.Te optimal results for the three test cases under diferent parameter combinations and computation times are illustrated in Figure 6.Te fnal chosen values for the hyperparameter combination are GROUP = 35, EPOCH = 10, pc = 0.9, and pm = 0.3 to ensure that the ftness function value converges stably to the optimal state.

Comparative Experiments with Gurobi
(1) Computational experiments for small-scale orders.
To substantiate the efcacy of the lower neighborhood search genetic algorithm (LNSGA) for the three-dimensional multiorder single-box open-dimension rectangular packing problem (3D-MOSB-ODRPP), we conducted a series of experiments focusing on small-scale order cases.Tese cases, comprising 2-8 orders, were selected to provide a controlled environment for assessing the algorithm's performance.Te determined hyperparameters for the LNSGA, as outlined in Figure 6, were utilized to compute these test cases within the generated dataset.Table 8 provides a comprehensive overview of the solutions obtained from both LNSGA and Gurobi, including the container dimensions (L, W, and H), the objective values, and the computational time required to reach these solutions.To evaluate the performance of LNSGA, we calculated the percentage gap (GAP) between the objective values achieved by LNSGA and the optimal values determined by Gurobi.

International Journal of Intelligent Systems
Experimental results show that in small-sample instances requiring rapid decision-making, LNSGA demonstrates a good balance between solution quality and computational resources.Specifcally, without imposing a time limit, the LNSGA algorithm performs signifcantly better than the Gurobi solver, achieving optimal values in the frst three test cases.Despite some deviation from the optimal values as the number of orders increases (with a maximum GAP value of 17.5% for LNSGA), it still exhibits considerable advantages over Gurobi (with a maximum GAP value of 72.81%).
Based on the same hyperparameter settings, the LNSGA algorithm is used to compute test cases with large-scale order quantities in the generated dataset (for large-scale order cases, refer to cases with 12 to 18 orders).Te results are compared with the cases solved directly using the Gurobi solver, as shown in Table 9.
Under the same computation time for each test case, the proposed LNSGA algorithm in this paper achieves signifcant optimization improvements over the Gurobi solver, with the highest optimization level being 71.59% and the average being 50.16%.In addition, it can be seen from Table 8 that as the order quantity scale increases, the CPU computation time also increases accordingly.

4.2.
Comparative Experiments with GA.In the previous section, we compared the performance gap between the commercial solver Gurobi and the proposed LNSGA algorithm.However, despite the signifcant advantages of the LNSGA algorithm in terms of both runtime and optimization accuracy over Gurobi, more is needed to demonstrate that the LNSGA algorithm is optimal for solving 3D-MOSB-ODRPP.Terefore, to validate the necessity of our modifcations to the GA, this section will introduce comparative experiments between GA and LNSGA.
Although LNSGA and GA have the same iteration limit (EPOCH), LNSGA demonstrates faster computational speed than GA.Tis can be attributed to several factors as follows: (1) Te GA algorithm does not invoke the Gurobi solver for obtaining packing solutions.Instead, it employs a heuristic algorithm based on the three-space strategy.Te three-space strategy requires maintaining a list of spaces and calculating the splitting and merging of spaces, which is relatively time-consuming.
(2) Te basic neighborhood search mechanism in GA can lead to the inefcient search of the packing box dimensions.Te packing method needs to be recalculated when larger or smaller packing box dimensions are explored.Excessively large or small packing box dimensions require more time to calculate the splitting and merging of spaces, resulting in increased time for updating the space list.12 International Journal of Intelligent Systems International Journal of Intelligent Systems 13 (3) In complex scenarios involving space splitting and merging, the computation time for calculating the item packing solution also increases to a certain extent.
On the other hand, LNSGA incorporates a targeted lower neighborhood search mechanism, which efciently explores the packing box dimensions by adjusting the length, width, and height within a specifc range.Tis targeted search reduces the occurrence of excessively large or small packing box dimensions, thereby avoiding unnecessary calculations of space splitting and merging.Moreover, LNSGA utilizes the Gurobi solver to obtain globally optimal packing solutions for each lower neighborhood search, further enhancing its computational efciency.Terefore, the GA's reliance on the three-space strategy and its basic neighborhood search mechanism leads to slower computational speed than LNSGA.Te targeted lower neighborhood search and integration of the Gurobi solver in LNSGA contribute to its faster performance.
In this section, we conducted experiments with smallscale order data to compare the performance of the LNSGA algorithm with the genetic algorithm (GA) and the variable neighborhood search (VNS) algorithm.Te experimental results, as shown in Table 10, demonstrate the LNSGA algorithm's superiority in terms of solution quality and computational efciency.From the perspective of the objective value, which represents the volume of the packing box (L × W × H), the LNSGA algorithm consistently outperformed the GA and VNS algorithms in all 12 test cases.Compared to the GA, the LNSGA algorithm achieved a signifcant improvement in the target function, ranging from 34.05% to 71.93%.On average, the LNSGA algorithm obtained an objective value of 6179, which is 7.38% lower than the GA's average of 6671 and 9.69% lower than the VNS's average of 6842.Tis indicates that the LNSGA algorithm can fnd more compact packing solutions, thereby minimizing the required box volume.
Regarding computation time, the LNSGA algorithm exhibited remarkable efciency compared to the GA and VNS algorithms.Across all test cases, the LNSGA algorithm reduced the runtime by more than 50% compared to the GA.As the number of orders increased from 2 to 8, the computation time of the GA algorithm showed a signifcant upward trend, ranging from 60 seconds to approximately 450 seconds.In contrast, the computation time of the LNSGA algorithm increased more gradually, reaching a maximum of around 168 seconds for the test case with eight orders.On average, the LNSGA algorithm had a computation time of 85.74 seconds, which is 58.38% faster than the GA's average of 206.08 seconds.
Compared to the VNS algorithm, the LNSGA algorithm demonstrated comparable computation times for smaller test cases with 2-4 orders.However, as the number of orders increased to 6 and 8, the LNSGA algorithm maintained its computational efciency, while the VNS algorithm's computation time increased more rapidly.Despite the slightly higher average computation time of the LNSGA algorithm (85.74 seconds) compared to the VNS algorithm (50.72 seconds), the LNSGA algorithm achieved signifcantly better objective values, justifying the marginal increase in computation time.
In this section, we mainly compared the performance of GA and LNSGA with more orders.Te comparison results are shown in Table 11.From the perspective of the objective value, the LNSGA algorithm achieved better packing solutions than the GA algorithm in all cases.Although the average improvement in the target function value was not as signifcant as in the small-scale cases, the quality of solutions obtained by LNSGA still exhibited a signifcant advantage.In terms of computation time, the LNSGA algorithm also outperformed the GA algorithm, reducing the runtime by approximately 50%.As the number of orders in the test cases increased, the computation time of the GA algorithm showed an upward trend, while the growth of computation time for the LNSGA algorithm was more gradual.Tis once again validated the computational efciency advantage of the LNSGA algorithm when dealing with large-scale complex orders.Tus, the LNSGA algorithm obtained superior packing solutions by introducing lower neighborhood search while ensuring computational efciency.Even in complex environments with many orders, integrating genetic algorithm and local search in LNSGA still demonstrated signifcant efectiveness.Extensive comparative experiments validate the efectiveness of the proposed approach.Te LNSGA algorithm consistently outperforms the commercial solver Gurobi and the traditional genetic algorithm (GA) regarding both solution quality and computational efciency.For small-scale instances, the LNSGA algorithm achieves optimal values in most test cases, with minimal deviation from the optimal values as the number of orders increases.For large-scale instances, the LNSGA algorithm demonstrates signifcant optimization improvements over Gurobi, with an average improvement of 50.16%.Compared to the GA, the LNSGA algorithm achieves a signifcant improvement in the objective function, ranging from 34.05% to 71.93%, while reducing the runtime by more than 50%.

Conclusions and Prospects
Te superior performance of the LNSGA algorithm can be attributed to its efective integration of the lower neighborhood search mechanism and the Gurobi solver.Te targeted lower neighborhood search efciently explores the packing box dimensions, avoiding unnecessary calculations and enhancing computational efciency.Te Gurobi solver, invoked after each local neighborhood search, ensures globally optimal packing solutions.Tis hybrid approach achieves an organic integration of global exploration and local search, enabling the algorithm to escape from local optima and obtain high-quality global optimal solutions.
Tus, combining a genetic algorithm with the Gurobi solver and incorporating a lower neighborhood search strategy, the proposed metaheuristic approach demonstrates signifcant efectiveness in tackling the complex 3D-MOSB-ODRPP.Te approach addresses the current complex scenarios in e-commerce warehousing and sets a foundation for future research in the feld of 3D packing optimization.

Management Insights.
Te fndings of this study ofer valuable insights for managers in e-commerce warehousing and logistics operations.Te proposed metaheuristic approach provides an efective tool for optimizing the packing process, enabling managers to make informed decisions that maximize space utilization and minimize logistics costs.
First, the multiorder open-dimension packing model allows managers to simultaneously consider the varying requirements of diferent orders.By optimizing the packing of multiple orders into a single, size-adjustable container, managers can improve order consolidation and reduce the number of containers required, leading to signifcant cost savings in transportation and storage.
Second, the lower neighborhood search strategy incorporated in the enhanced genetic algorithm ofers a practical approach for managers to explore optimal packing confgurations.By adjusting the container's length, width, and height within a specifc range, managers can identify the most suitable container dimensions that balance the demands of diferent orders while maximizing space utilization.
Tird, integrating the Gurobi solver with the mathematical approach ensures that the packing solutions obtained are globally optimal.Tis gives managers confdence that the recommended packing confgurations are the best possible solutions, considering all relevant constraints and objectives.
Furthermore, the comparative experiments demonstrate the superior performance of the LNSGA algorithm over traditional methods, such as the commercial solver Gurobi and the pure genetic algorithm.Managers can leverage these fndings to justify adopting the proposed approach in their operations, as it signifcantly improves both solution quality and computational efciency.

Limitations and Future Work.
While the proposed metaheuristic approach efectively addresses the 3D-MOSB-ODRPP, several limitations warrant attention in future research.
Primarily, the study focuses on rectangular items and containers, simplifying real-world packing scenarios.Future research could enhance the approach to handle irregular shapes and additional constraints.
In addition, computational experiments use randomly generated datasets, limiting real-world applicability.Future work should validate performance using real-world ecommerce data to assess scalability and robustness.
Furthermore, extending the approach to dynamic and online packing scenarios is essential for real-time operations, necessitating adaptive algorithms.
Integrating sustainability considerations, such as minimizing environmental impact, is a promising avenue for future research, enhancing the social responsibility of packing optimization.
Lastly, exploring applications beyond e-commerce, such as manufacturing and logistics, could leverage the approach's versatility to solve optimization challenges in diverse industries.

Figure 1 :
Figure 1: Six kinds of rectangular items rotation direction.

Figure 2 :
Figure 2: Schematic of a g-step search line in the lower neighborhood.
Yaagoubi et al. used NSGA-II with heuristic rules for fast solving at diferent scales [56].Liu et al. targeted multibin packing problems with irregular items using 3D point cloud techniques and deep Q-networks [57].Wang et al. abstracted resource allocation in open RAN networks into a 2D bin packing model and used a self-play reinforcement learning algorithm [58].

Table 1 :
Comparison of the present research with existing studies.

Table 2 :
Description of condition variables.

Table 3 :
Description of the decision variables.
population size N, crossover probability p c , mutation probability p m , and maximum number of generations G max Ensure: best solution s * (1) Initialize population P with N randomly generated individuals (2) Evaluate the ftness of each individual in P using the TS heuristic (6)Te lower neighborhood search scope for open dimensions is defned as follows: given the upper bounds (L up , W up , and H up ), the 1-step lower neighborhood N 1 (L up , W up , and H up ) consists of 7 value options, obtained by adding 0 or −1 to each dimension, excluding the case where all increments are 0. Te lower neighborhood search aims to fnd better solutions s ′ ∈ N 1 (s) within the neighborhood of the current solution s.By designing j ∈ 1, 2, ..., 7 represents the node number at the j-th step of the search path.(6)Werandomlygenerated an initial population P � C 1 , C 2 , . .., C M group consisting of M group chromosomes, each corresponding to a lower neighborhood search path.Require: 1 , o 2 ← Crossover (p 1 , p 2 ) with probability p c * from P

Table 5 :
Neighborhood search fetches experienced sequentially by the open dimension.

Table 6 :
Order item SKU size generation information.

Table 7 :
Parameter setting of the genetic algorithm.

Table 8 :
Case results for small-scale order sizes (comparison with Gurobi).

Table 9 :
Case results for large-scale order sizes (comparison with Gurobi).
Te proposed algorithm, a hybrid of a genetic algorithm and the Gurobi solver, efectively manages the complexity of the problem by combining global exploration and local optimization.Te algorithm is underpinned by a mathematical model that accurately represents the multiorder open-dimension packing scenario and an enhanced genetic algorithm incorporating a lower neighborhood search strategy.
5.1.Conclusion.In this paper, we introduced a novel metaheuristic approach to address the multiorder opendimension 3D rectangular packing problem (3D-MOSB-ODRPP), a complex challenge exacerbated by the rapid growth of e-commerce.