Optimizing e-commerce warehousing through open dimension management in a three-dimensional bin packing system

Single-bin 3D packing problem with fixed dimensions

For the single-bin 3D packing problem with fixed dimensions, researchers have primarily focused on studying it through heuristic algorithms to address the challenge of solving large-scale 3D packing problems within a reasonable timeframe. Mahvash, Awasthi & Chauhan (2018) employed a heuristic algorithm based on the column generation (CG) technique to tackle the single-bin 3D packing problem. Di Puglia Pugliese, Guerriero & Calbi (2019) developed a simple and effective heuristic approach to address a 3D bin packing problem that considers various operational constraints with the goal of inserting objects with the same destination into a new container. Recently, Yang et al. (2023) investigated the applicability of heuristics for online 3D binning problems and explored how to effectively integrate heuristics with deep reinforcement learning techniques. It should be noted that, given the nature of 3D packing studies, heuristic algorithms do not guarantee to find the globally optimal solution. Therefore, researchers often resort to small-scale exact algorithms such as column generation algorithms and branch-and-bound algorithms. Elhedhli, Gzara & Yildiz (2019) proposed a novel formulation and a column generation-based solution for solving the 3D binning problem. Similarly, Silva & Wauters (2019) focused on exact methods for comparing 3D cutting and packing problems. They adapted the chosen method to the specific problem analyzed and conducted experiments using classical benchmark datasets as well as newly generated examples from cutting and packing generators in the literature, enabling a comprehensive comparison.

Furthermore, Borges et al. (2020) proposed two dynamic programming algorithms and a branch-and-bound algorithm to address the 3D binning problem. These algorithms aim to find optimal solutions by considering different constraints and variations within the problem. Gzara, Elhedhli & Yildiz (2020) employed a layer-based column generation method to enhance the loading rate of a single-bin 3-D packing problem. Erbayrak, Özkır & Yıldırım (2021) employed a weighted objective function to determine the optimal packing solution that minimizes the number of bins used while also balancing the deviation from the ideal center of gravity and maximizing the family uniform ratio. Küçük & Yildiz (2022) proposed a constraint planning-based solution to the three dimensional loading capacity vehicle path problem (3l-CVRP) in distribution logistics, which involves solving a combination of a vehicle route and a three-dimensional loading problem. Tresca et al. (2022) combined a mixed integer linear programming (MILP) formulation and layer construction heuristics to solve the single bin fixed-size 3D packing problem. Tole et al. (2023) proposed an algorithm consisting of two phases: an initialization phase and a refinement phase for the fixed-size single-bin 3D packing problem. A first-time fit grid search (FGS) algorithm to generate the initial solution, and then a simulated annealing (SA) algorithm was designed to explore the solution space of the initial solution by the proposed novel local neighborhood search strategy known as circular quadrant perturbation.

Several researchers have also combined algorithms to address the 3D container loading problem. Huang et al. (2022) introduced a novel technique called Ternary Search Tree Differential Evolutionary algorithm (TSTDE) to solve the 3D container loading problem (3D-CLP). Zhao et al. (2022a) proposed a novel stacking tree approach to optimize the packing strategy by analyzing the stability of the packing online. Su et al. (2021) combined a chemical reaction optimization algorithm with a greedy algorithm to solve the 3D bin packing problem, demonstrating superior performance compared to existing algorithms. Shuai et al. (2023) investigated the robotic three-dimensional bin packing problem (R-3DBP) and presented a new bin packing algorithm that considers sensor and planner machine errors, allowing self-correction of the position of boxed items. Moura et al. (2023) proposed a deep reinforcement learning (DRL) model utilizing the Transformer architecture as a policy network and approximate policy optimization (PPO) to train the network for solving the 3D packing problem. Finally, Romero et al. (2023) introduced a hybrid quantum–classical framework to tackle real-world 3D modeling problems.

Open-size single-bin 3D packing issues

For the open-size single-bin 3D bin packing problem, several studies have been conducted to address different aspects of the problem. Litvinchev, Pankratov & Romanova (2019) investigated the challenge of packing a set of irregular objects into a rectangular container with variable height. They devised a solution algorithm that combines a fast starting point algorithm with an efficient local optimization process. Araújo et al. (2019) focused on additive manufacturing in the context of the irregular 3D bin packing problem. They introduced a new taxonomy and dataset using the 3D Irregular Items Problem (3DIP) model algorithm. Wang (2022) undertook research on the online constrained variable size solving 3D bin packing problem, employing computer simulation techniques. Harrath (2022) explored the three-dimensional single bin size bin packing problem (3D-SBSPP) and proposed a three-stage heuristic algorithm. Their approach involved creating a minimum number of bins, allowing for six possible item rotation directions while ensuring that the crated items satisfy equilibrium constraints. The algorithm demonstrated superiority over other heuristics in various problem examples. Mungwattana, Piyachayawat & Janssens (2022) proposed a two-stage optimization method for solving the Dealer Pallet Loading Problem (DPLP) with multiple sizes. Their approach utilized an evolutionary algorithm and a differential evolutionary algorithm. The results indicated a 0.42% higher total number of pallets compared to the exact algorithm for the benchmark problem set, along with a significant 78.54% reduction in average computation time. In a recent study, Que, Yang & Zhang (2023) proposed a deep reinforcement learning model to tackle the 3D packing problem. They leveraged deep learning techniques to address the challenge of variable container height, enabling them to obtain superior solutions in shorter timeframes.

Fixed size multiple-bin type three-dimensional bin packing problems

For the problem of packing multiple bins of different sizes into fixed-size 3D containers, researchers have predominantly relied on heuristic algorithms for their investigations. Paquay, Limbourg & Schyns (2018) introduced a fast constructive heuristic to address the 3D multiple different size bin packing problem. Sangchooli & Sajadifar (2021) developed a mathematical model for the 3D multi-bin packing problem and employed a novel heuristic algorithm along with the GRASP algorithm to solve it. Li, Chen & Huo (2022) proposed a hybrid adaptive large neighborhood search (HALNS) algorithm that encompasses a collection of primitive damage-repair operators and incorporates a heuristic packing algorithm specifically designed to tackle large-scale heterogeneous container loading problems within a limited time frame. Zhao et al. (2022b) conducted a study on the multi-carton 3D packing problem involving heterogeneous squeezable items in an e-commerce retail department store. They designed a heuristic algorithm aimed at selecting the most suitable packing materials to pack all items in an order, using the minimum number of packing materials while maximizing space utilization. Trivella & Pisinger (2016) used a multistage local search heuristic algorithm with the objective of packing objects of different sizes into a finite number of similar bins/containers to minimize the number of bins used. Vieira et al. (2021) investigated the problem of packing a wide range of bin sizes, and proposed a two-stage crating algorithm, which firstly uses a 0–1 plan to assign to the least number of bines, and then loaded into three-dimensional containers by mixed integer nonlinear planning to minimize the total container volume, and validated the effectiveness and efficiency of the method on data. Alonso et al. (2019) loaded onto trucks to solve the multiple container loading problem for companies providing services for customer orders and solved it by an Integer Linear Programming (ILP) model.

All of the aforementioned articles suggest the utilization of heuristic algorithms to address the 3D bin packing problem. While heuristics are relatively straightforward to implement, none of these approaches can guarantee global optimality for 3D bin packing. As a result, Li et al. (2022) proposed a recurrent conditional query learning (RCLL) method to solve the 3D bin packing problem. Hernández-Hernández & Castillo-García (2021) proposed a fuzzy logic classification (FLC) approach to successfully assign bines to destinations that are as close as possible, addressing the 3D packing problem. Dell’Amico, Furini & Iori (2020) introduced a holistic algorithm that utilizes a hybrid approach to minimize the amount of packing required. Agarwal et al. (2020) presented a reliable and efficient bin packing system called Jampacker, along with an offline 3D bin packing algorithm (Jampack) that achieves higher packing efficiency.

Multiple-bin-type 3D bin packing problems in open sizes

For the complex challenge of open-dimensional 3D bin packing involving multiple-bin types, Tsai & Li (2006) proposed a highly effective global optimization method. Their approach converts the 3D-ODRPP (3D open-dimensional rectangular items problem) into a hybrid integer linear programming problem, enabling efficient solutions to the problem. Expanding on this research, Tsai, Wang & Lin (2015) focused on the rectangular items problem, which entails placing a given number and size of rectangular items inside a large rectangle with the minimum possible area. Another notable contribution in this field was made by Junqueira & Morabito (2017), who introduced the 3D open-dimensional rectangular packing problem. The objective of this problem is to determine the minimum volume of a rectangular container needed to pack a set of rectangular items. To solve this problem, an optimization solver, location-free hybrid-integer programming (MIP) formulation, and a segmented linearization method were employed. Furthermore, Lin, Tsai & Chang (2017) investigated the 3D-ODRPP. Their innovative approach involves transforming the original problem into a hybrid-integer linear programming problem. This reformulation reduces the number of constraints by utilizing discrete variables to linearize symbolic terms.

In summary, the current research primarily focuses on single-bin size variable and non-variable problems, as well as multi-bin size non-variable problems. For the fixed-size multi-bin 3D bin packing problem, researchers have proposed various heuristic algorithms and mathematical models to find optimal bin packing solutions. Similarly, for the open-size single-bin 3D bin packing problem, different algorithms and models have been employed to address different variations and achieve goals such as reducing packing volume and creating economic and ecological benefits. Additionally, for the problem of multiple-bin types with fixed dimensions, researchers have proposed diverse heuristic algorithms and mathematical models to find optimal bin-packing solutions.

However, the research on 3D packing problems with multiple bin types in the field of e-commerce warehousing has been more limited so far, and the existing research suffers from the problems of high model complexity and long computation time, although some models and methods have been proposed to solve the 3D packing problem with a single bin type in the previous researches. However, facing the open-size 3D packing problem (3D-MODRPP) with many different bin types, the existing methods have high complexity and long computation time, which makes it difficult to be effectively solved in practical applications. This leads to the problem of inefficiency and excessive computational cost in dealing with multiple bin types and open-size scenarios in logistics enterprises such as the e-commerce warehousing field, despite the existence of some mature solutions.

Therefore, in this article, by constructing a mixed integer planning model and adopting a new 0–1 integer variable assumption method, we conduct an in-depth study on the open-size 3D packing problem for multiple bin types in the e-commerce context on the basis of existing research. This study successfully reduces the number of constraints, which effectively reduces the complexity of the model and accelerates the computational process. It also achieves more excellent results based on a dataset that is closer to the 3D-MODRPP problem. Table 1 illustrates a comparison between the existing research on the open 3D bin packing problem and the proposed study on the open-size multiple-bin type bin packing problem.

Problem description

In this study, we investigate the 3D-MODRPP, which involves arranging a batch of rectangular items into various packing bins categorized by their dimensions. The 3D-MODRPP specifically focuses on the organization of rectangular items within a designated bin, considering different bin types based on their dimensions. The application scenario for this problem lies within the realm of e-commerce warehouse packing, where multiple-bin sizes are typically available. When processing different sets of orders, the packing operator must select the appropriate bin to accommodate all the items in the order, aiming to maximize space utilization. Our optimization objective in this study is to achieve the highest loading rate for the packing bin. We allow the rotation of items in six different directions, as illustrated in Fig. 1. Where, l^B, w^B, h^B denote the length, width, and height of the packing bin respectively, and l^I, w^I, h^I denote the length, width, and height of the item respectively. It is worth noting that in the e-commerce storage bin packing process, fillers are commonly utilized to secure or support the items. However, this research does not focus on the investigation of items’ support conditions. The constraints considered in this article’s research problem are as follows:

1. Each shipment requires considering six different rotation directions for the bin.

2. It is essential to ensure that the bin remains within the boundaries of the packing bin and does not exceed them.

3. To maintain proper packing, it is necessary to prevent any overlap between the items inside the packing bin.

Variable assumptions

Variables can be categorized into two types: conditional variables and decision variables. Conditional variables are those that are known or determined in advance, while decision variables are those that require experimental calculations or decision-making processes.

1. Conditional variables

   n: Indicates the number of bin type categories;

   q: Indicates the number of orders;

   m_r: Indicates the number of items in the first order, where r = 1, 2, q;

   $l_{i_r}^I,w_{i_r}^I,h_{i_r}^I$: Denote the length with, and height of the first item in the first order, respectively, where i_r =1,2,…, m_r;

   $l_{i_r}^T,w_{i_r}^T,h_{i_r}^T$: Denotes the dimensions values in the axes, axes, and axes dimensions after rotation of the first item in the first order into the bin, i.e., the length, width, and height of them after rotation, respectively, where i_r =1,2,…, m_r;

   M: Denotes a sufficiently large positive number.

1. Decision variables

   x_ir, y_ir, z_ir: The coordinates of the packing position of the first item in the axes, axes, and axes dimensions, respectively, using the lower left and right corners of the bin into which the first order is loaded as the origin and establishing a three-dimensional right-angle coordinate system;

   $l_j^B,w_j^B,h_j^B$: Denote the length, width, and height of the first type of bin, respectively, j = 1, 2,…,n; R_r,j: Indicates whether all items in the first order are loaded into the first bin, and is a 0–1 variable, where 1 means loaded and 0 means not loaded, where r = 1, 2,…, q, j = 1, 2,…,n;

   $T_{11}^{i_r},T_{12}^{i_r},T_{13}^{i_r},T_{21}^{i_r},T_{22}^{i_r},T_{23}^{i_r},T_{31}^{i_r},T_{32}^{i_r},T_{33}^{i_r}$: Indicates the direction of rotation of the item, as a 0–1 variable;

   $D_{i_r,i_r^{\prime }}^k$: A 0–1 variable indicating whether two items overlap with each other, $,I_r\ne i_r^{{}^{\prime }},I_r,I_r^{{}^{\prime }}=1,2,\dots ,m,$ k = 1, 2, 3, 4, 5, 6,  The equation determines whether every two items overlap by using six 0–1 variables;

   $O_{i_r,i_r^{\prime }}^1,O_{i_r,i_r^{\prime }}^2,O_{i_r,i_r^{\prime }}^3$: A 0–1 variable is used to indicate whether there is an overlap between two items. To determine whether each pair of items overlaps, only three 0–1 variables are needed; $U_1^{i_r},U_2^{i_r},U_3^{i_r},U_4^{i_r},U_3^{i_r},U_6^{i_r}$: A 0–1 variable representing the six directions of item rotation.

Model construction

1. The objective function of the model is to maximize the volume of all items within various types of bines, aiming to achieve the highest loading rate. (1)${\displaystyle maxz=\frac{\sum _{r=1}^q\sum _{i_r=1}^{m_r}{l}_{i_r}^I\cdot w_{i_r}^I\cdot h_{i_r}^I}{\sum _{r=1}^q\sum _{j=1}^n{R}_{r,j}\cdot l_j^B\cdot w_j^B\cdot h_j^B}}$

1. Constraints

   (a) Rotation constraints

   Assume that the item can be rotated in 6 directions. (2)${\displaystyle l_{i_r}^T=T_{11}^{i_r}\cdot l_{i_r}^I+T_{12}^{i_r}\cdot w_{i_r}^I+T_{13}^{i_r}\cdot h_{i_r}^I}$ (3)${\displaystyle w_{i_r}^T=T_{21}^{i_r}\cdot l_{i_r}^I+T_{22}^{i_r}\cdot w_{i_r}^I+T_{23}^{i_r}\cdot h_{i_r}^I}$ (4)${\displaystyle h_{i_r}^T=T_{31}^{i_r}\cdot l_{i_r}^I+T_{32}^{i_r}\cdot w_{i_r}^I+T_{33}^{i_r}\cdot h_{i_r}^I}$ (5)${\displaystyle T_{11}^{i_r}+T_{12}^{i_r}+T_{13}^{i_r}=1}$ (6)${\displaystyle T_{21}^{i_r}+T_{22}^{i_r}+T_{23}^{i_r}=1}$ (7)${\displaystyle T_{31}^{i_r}+T_{32}^{i_r}+T_{33}^{i_r}=1}$ (8)${\displaystyle T_{11}^{i_r}+T_{21}^{i_r}+T_{31}^{i_r}=1}$ (9)${\displaystyle T_{12}^{i_r}+T_{22}^{i_r}+T_{32}^{i_r}=1}$ (10)${\displaystyle T_{13}^{i_r}+T_{23}^{i_r}+T_{33}^{i_r}=1}$ The rotation constraints for items described in Eqs. (2) to (10) require representing the rotation direction of each item using nine 0–1 variables and nine constraints. It is possible to simplify the model by reducing the number of 0–1 variables and constraints, as outlined below. (2' )${\displaystyle l_{i_r}^T=U_1^{i_r}\cdot l_{i_r}^I+U_2^{i_r}\cdot l_{i_r}^I+U_3^{i_r}\cdot w_{i_r}^I+U_4^{i_r}\cdot w_{i_r}^I+U_5^{i_r}\cdot h_{i_r}^I+U_6^{i_r}\cdot h_{i_r}^I}$ (3' )${\displaystyle w_{i_r}^T=U_1^{i_r}\cdot w_{i_r}^I+U_2^{i_r}\cdot h_{i_r}^I+U_3^{i_r}\cdot l_{i_r}^I+U_4^{i_r}\cdot h_{i_r}^I+U_5^{i_r}\cdot l_{i_r}^I+U_6^{i_r}\cdot w_{i_r}^I}$ (4' )${\displaystyle h_{i_r}^T=U_1^{i_r}\cdot h_{i_r}^I+U_2^{i_r}\cdot w_{i_r}^I+U_3^{i_r}\cdot h_{i_r}^I+U_4^{i_r}\cdot l_{i_r}^I+U_{i_r}^{\dot{r}}\cdot W_{i_r}^{i_r}\cdot w_{i_r}^I\cdot w_{i_r}^I+U_{\dot{6}}^{i_r}\cdot l_{\dot{r}}^I}$ (5' )${\displaystyle U_1^{i_r}+U_2^{i_r}+U_3^{i_r}+U_4^{i_r}+U_5^{i_r}+U_6^{i_r}=1}$

   (b) Transboundary constraints

   The cross-boundary constraint ensures that the boxed items cannot extend beyond the boundaries of the packing bin, and all items must remain inside the packing bin. (11)${\displaystyle x_{i_r}+l_{i_r}^T\le \sum _{j=1}^n{R}_{r,j}\cdot l_j^B}$ (12)${\displaystyle y_{i_r}+w_{i_r}^T\le \sum _{j=1}^n{R}_{r_{,j}}\cdot w_j^B}$ (13)${\displaystyle z_{i_r}+h_{i_r}^T\le \sum _{j=1}^n{R}_{r,j}\cdot h_j^B}$ After substituting Eqs. (2) into (11) and (3) into (12) and (4) into (13), respectively, the following equation can be obtained: (11' )${\displaystyle x_{i_r}+U_1^{i_r}\cdot l_{i_r}^I+U_2^{i_r}\cdot l_{i_r}^I+U_3^{i_r}\cdot w_{i_r}^I+U_4^{i_r}\cdot w_{i_r}^I+U_5^{i_r}\cdot h_{i_r}^I+U_6^{i_r}\cdot h_{i_r}^I\le \sum _{j=1}^n{R}_{r,j}\cdot l_j^B}$ (12' )${\displaystyle y_{i_r}+U_1^{i_r}\cdot w_{i_r}^I+U_2^{i_r}\cdot h_{i_r}^I+U_3^{i_r}\cdot l_{i_r}^I+U_4^{i_r}\cdot h_{i_r}^I+U_5^{i_r}\cdot l_{i_r}^I+U_6^{i_r}\cdot w_{i_r}^I\le \sum _{j=1}^n{R}_{r,j}\cdot w_j^B}$ (13' )${\displaystyle z_{i_r}+U_1^{{\dot{ı}}_r}\cdot h_{i_r}^I+U_2^{{\dot{ı}}_r}\cdot w_{i_r}^I+U_3^{{\dot{ı}}_r}\cdot h_{i_r}^I+U_4^{i_r}\cdot l_{i_r}^I+U_5^{{\dot{ı}}_r}\cdot w_{i_r}^I+U_6^{{\dot{ı}}_r}\cdot l_{i_r}^I\le \sum _{j=1}^n{R}_{r,j}\cdot h_j^B}$ In conjunction with Eqs. (5) to (10), these equations represent the non-transgression constraint following item rotation.

   (c) Overlapping constraints

   Overlap constraint means that two items within the same order cannot overlap in spatial extent. (14)${\displaystyle x_{i_r^{\prime }}+l_{i_r^{\prime }}^T\le x_{i_r}+\left(1-D_{i_r,i_r^{\prime }}^1\right)\cdot M}$ (15)${\displaystyle x_{i_r}+l_{i_r}^T\le x_{i_r^{\prime }}+\left(1-D_{i_r,i_r^{\prime }}^2\right)\cdot M}$ (16)${\displaystyle y_{i_r^{\prime }}+w_{i_r^{\prime }}^T\le y_{i_r}+\left(1-D_{i_r,i_r^{\prime }}^3\right)\cdot M}$ (17)${\displaystyle y_{i_r}+w_{i_r}^T\le y_{i_r^{\prime }}+\left(1-D_{i_r,i_r^{\prime }}^4\right)\cdot M}$ (18)${\displaystyle z_{i_r^{\prime }}+h_{i_r^{\prime }}^T\le z_{i_r}+\left(1-D_{i_r,i_r^{\prime }}^5\right)\cdot M}$ (19)${\displaystyle z_{i_r}+h_{i_r}^T\le z_{i_r^{\prime }}+\left(1-D_{i_r,i_r^{\prime }}^6\right)\cdot M}$ (20)${\displaystyle D_{i_r,i_r^{\prime }}^1+D_{i_r,i_r^{\prime }}^2+D_{i_r,i_r^{\prime }}^3+D_{i_r,i_r^{\prime }}^4+D_{i_r,i_r^{\prime }}^5+D_{i_r,i_r^{\prime }}^6\ge 1}$ According to Eqs. (2) to (4), the length, width, and height dimensions of the rotated items in Eqs. (14) to (20) are obtained by substituting. (14' )${\displaystyle x_{i_r^{\prime }}+U_1^{i_r^{\prime }}\cdot l_{i_r^{\prime }}^I+U_2^{i_r^{\prime }}\cdot l_{i_r^{\prime }}^I+U_3^{i_r^{\prime }}\cdot w_{i_r^{\prime }}^I+U_4^{i_r^{\prime }}\cdot w_{i_r^{\prime }}^I+U_5^{i_r^{\prime }}\cdot h_{i_r^{\prime }}^I+U_6^{i_r^{\prime }}\cdot h_{i_r^{\prime }}^I\le x_{i_r}+\left(1-D_{i_r,i_r^{\prime }}^1\right)\cdot M}$ (15' )${\displaystyle x_{i_r}+U_1^{i_r}\cdot l_{i_r}^I+U_2^{i_r}\cdot l_{i_r}^I+U_3^{i_r}\cdot w_{i_r}^I+U_4^{i_r}\cdot w_{i_r}^I+U_5^{i_r}\cdot h_{i_r}^I+U_6^{i_r}\cdot h_{i_r}^I\le x_{i_r^{\prime }}+\left(1-D_{i_r,i_r^{\prime }}^2\right)\cdot M}$ (16' )${\displaystyle y_{i_r^{\prime }}+U_1^{i_r^{\prime }}\cdot w_{i_r^{\prime }}^I+U_2^{i_r^{\prime }}\cdot h_{i_r^{\prime }}^I+U_3^{i_r^{\prime }}\cdot l_{i_r^{\prime }}^I+U_4^{i_r^{\prime }}\cdot h_{i_r^{\prime }}^I+U_5^{i_r^{\prime }}\cdot l_{i_r^{\prime }}^I+U_6^{i_r^{\prime }}\cdot w_{i_r^{\prime }}^I\le y_{i_r}+\left(1-D_{i_r,i_r^{\prime }}^3\right)\cdot M}$ (17' )${\displaystyle y_{i_r}+U_1^{i_r}\cdot w_{i_r}^I+U_2^{i_r}\cdot h_{i_r}^I+U_3^{i_r}\cdot l_{i_r}^I+U_4^{i_r}\cdot h_{i_r}^I+U_5^{i_r}\cdot l_{i_r}^I+U_6^{i_r}\cdot w_{i_r}^I\le y_{i_r^{\prime }}+\left(1-D_{i_r,i_r^{\prime }}^4\right)\cdot M}$ (18' )${\displaystyle z_{i_r^{\prime }}+U_1^{i_r^{\prime }}\cdot h_{i_r^{\prime }}^I+U_2^{i_r^{\prime }}\cdot w_{i_r^{\prime }}^I+U_3^{i_r^{\prime }}\cdot h_{i_r^{\prime }}^I+U_4^{i_r^{\prime }}\cdot l_{i_r^{\prime }}^I+U_5^{i_r^{\prime }}\cdot w_{i_r^{\prime }}^I+U_6^{i_r^{\prime }}\cdot l_{i_r^{\prime }}^I\le z_{i_r}+\left(1-D_{i_r,i_r^{\prime }}^5\right)\cdot M}$ (19' )${\displaystyle z_{i_r}+U_1^{i_r}\cdot h_{i_r}^I+U_2^{i_r}\cdot w_{i_r}^I+U_3^{i_r}\cdot h_{i_r}^I+U_4^{i_r}\cdot l_{i_r}^I+U_5^{i_r}\cdot w_{i_r}^I+U_6^{i_r}\cdot l_{i_r}^I\le z_{i_r^{\prime }}+\left(1-D_{i_r,i_r^{\prime }}^6\right)\cdot M}$ Equations (14’) to (19’) and (20) collectively establish a non-overlapping constraint between items when packing the same order of items into a single bin. Tsai, Wang & Lin (2015) proposed a method to reduce the number of 0–1 decision variables when representing the non-overlapping constraints between two items. Instead of using six 0–1 variables, Tsai’s approach utilizes only three 0–1 variables. Although Tsai’s study focuses on the 3D-ODRPP problem, which differs slightly from the problem addressed in this article, we can adapt (Tsai, Wang & Lin, 2015)’s research ideas to reduce the non-overlapping constraint in this article. Consequently, Eqs. (14’) to (19’) and (20’) are modified as follows: (14'' )${\displaystyle x_{i_r^{\prime }}+U_1^{i_r^{\prime }}\cdot l_{i_r^{\prime }}^I+U_2^{i_r^{\prime }}\cdot l_{i_r^{\prime }}^I+U_3^{i_r^{\prime }}\cdot w_{i_r^{\prime }}^I+U_4^{i_r^{\prime }}\cdot w_{i_r^{\prime }}^I+U_5^{i_r^{\prime }}\cdot h_{i_r^{\prime }}^I+U_6^{i_r^{\prime }}\cdot h_{i_r^{\prime }}^I\le x_{i_r}+O_{i_r,i_r^{\prime }}^1+O_{i_r,i_r^{\prime }}^2+\left(1-O_{i_r,i_r^{\prime }}^3\right)\cdot M}$ (15'' )${\displaystyle x_{i_r}+U_1^{i_r}\cdot l_{i_r}^I+U_2^{i_r}\cdot l_{i_r}^I+U_3^{i_r}\cdot w_{i_r}^I+U_4^{i_r}\cdot w_{i_r}^I+U_5^{i_r}\cdot h_{i_r}^I+U_6^{i_r}\cdot h_{i_r}^I\le x_{i_r^{\prime }}+O_{i_r,i_r^{\prime }}^1+\left(1-O_{i_r,i_r^{\prime }}^2\right)\cdot M+O_{i_r,i_r^{\prime }}^3}$ (16'' )${\displaystyle y_{i_r^{\prime }}+U_1^{i_r^{\prime }}\cdot w_{i_r^{\prime }}^I+U_2^{i_r^{\prime }}\cdot h_{i_r^{\prime }}^I+U_3^{i_r^{\prime }}\cdot l_{i_r^{\prime }}^I+U_4^{i_r^{\prime }}\cdot h_{i_r^{\prime }}^I+U_5^{i_r^{\prime }}\cdot l_{i_r^{\prime }}^I+U_6^{i_r^{\prime }}\cdot w_{i_r^{\prime }}^I\le y_{i_r}+\left(1-O_{i_r,i_r^{\prime }}^1\right)\cdot M+O_{i_r,i_r^{\prime }}^2+O_{i_r,i_r^{\prime }}^3}$ (17'' )${\displaystyle y_{i_r}+U_1^{i_r}\cdot w_{i_r}^I+U_2^{i_r}\cdot h_{i_r}^I+U_3^{i_r}\cdot l_{i_r}^I+U_4^{i_r}\cdot h_{i_r}^I+U_5^{i_r}\cdot l_{i_r}^I+U_6^{i_r}\cdot w_{i_r}^I\le y_{i_r^{\prime }}+O_{i_r,i_r^{\prime }}^1+\left(1-O_{i_r,i_r^{\prime }}^2\right)\cdot M+\left(1-O_{i_r,i_r^{\prime }}^3\right)\cdot M}$ (18'' )${\displaystyle z_{i_r^{\prime }}+U_1^{i_r^{\prime }}\cdot h_{i_r^{\prime }}^I+U_2^{i_r^{\prime }}\cdot w_{i_r^{\prime }}^I+U_3^{i_r^{\prime }}\cdot h_{i_r^{\prime }}^I+U_4^{i_r^{\prime }}\cdot l_{i_r^{\prime }}^I+U_5^{i_r^{\prime }}\cdot w_{i_r^{\prime }}^I+U_6^{i_r^{\prime }}\cdot l_{i_r^{\prime }}^I\le z_{i_r}+\left(1-O_{i_r,i_r^{\prime }}^1\right)\cdot M+O_{i_r,i_r^{\prime }}^2+\left(1-O_{i_r,i_r^{\prime }}^3\right)\cdot M}$ (19'' )${\displaystyle z_{i_r}+U_1^{i_r}\cdot h_{i_r}^I+U_2^{i_r}\cdot w_{i_r}^I+U_3^{i_r}\cdot h_{i_r}^I+U_4^{i_r}\cdot l_{i_r}^I+U_5^{i_r}\cdot w_{i_r}^I+U_6^{i_r}\cdot l_{i_r}^I\le z_{i_r^{\prime }}+\left(1-O_{i_r,i_r^{\prime }}^1\right)\cdot M+\left(1-O_{i_r,i_r^{\prime }}^2\right)\cdot M+O_{i_r,i_r^{\prime }}^3}$ (20' )${\displaystyle O_{i_r,i_r^{\prime }}^1+O_{i_r,i_r^{\prime }}^2+O_{i_r,i_r^{\prime }}^3\ge 1}$ (20'' )${\displaystyle O_{i_r,i_r^{\prime }}^1+O_{i_r,i_r^{\prime }}^2+O_{i_r,i_r^{\prime }}^3\le 2}$ As depicted in Eqs. (21) to (24), certain variables possess specific values, while others are subject to constraints: (21)${\displaystyle x_{i_r},y_{i_r},z_{i_r}{l}_j^B,w_j^B,h_j^B⩾0}$ (22)${\displaystyle R_{r,j},D_{i_r,i_r^{\prime }}^k,O_{i_r,i_r^{\prime }}^1,O_{i_r,i_r^{\prime }}^2,O_{i_r,i_r^{\prime }}^3,U_1^{i_r},U_2^{i_r},U_3^{i_r},U_4^{i_r},U_5^{i_r},U_6^{i_r}=0\vee 1}$ (23)${\displaystyle j=1,2,\dots ,n;r=1,2,\dots ,q}$ (24)${\displaystyle i_r\ne i_r^{{}^{\prime }},i_r,i_r^{\prime }=1,2,\dots ,m;k=1,2,3,4,5,6}$

Example data

The information regarding all item SKU sizes is presented in Table 2, encompassing a total of 16 distinct sizes. These sizes have been derived using the arithmetic method of Tsai, Wang & Lin (2015).

Ten different problem arithmetic bins were developed based on the types of items, and each problem arithmetic bin was loaded into a packing bin model. Table 3 provides detailed information on the type and number of items present in each arithmetic bin. Questions 1 through 4 represent scenarios involving strongly heterogeneous items, while Questions 5 and 6 are for weakly heterogeneous items. Questions 7 through 10 are derived from actual production data obtained from different firms. The arithmetic examples are from Tsai, Wang & Lin (2015).

Algorithm solving

1. Strong heterogeneous packing problem

   Problems 1 to 4 fall under the category of strongly heterogeneous item packing problems, where “strongly heterogeneous” implies that there are very few items with identical shapes and sizes. In this study, we employ a hybrid integer programming model and utilize the GUROBI solver to compute and compare the results with the approach presented by Tsai and Li, Chen & Huo (2022), as well as Tsai, Wang & Lin (2015). The obtained results are summarized in Table 4.

   As shown in Table 4, the computational time of the proposed model in this article is significantly reduced compared to the CPU computation time while achieving the same experimental results. Additionally, the present model reduces the number of 0–1 variables in both overlap and rotation constraints. In the original Tsai, Wang & Lin (2015) method, problem 1 was reduced from 75 to 43 variables, problem 2 from 96 to 60 variables, problem 3 from 120 to 81 variables, and problem 4 from 147 to 105 variables. Consequently, the number of constraints is also reduced. For instance, problem 3, which consists of six items with different SKU types, saw a decrease in CPU computation time from 20,641 s in Tsai, Wang & Lin (2015) method to 20.69 s in the proposed method. This reduction in variables simplifies the model’s complexity and enhances its ability to solve the highly heterogeneous bin packing problem.

2. Weak heterogeneous packing problem

   Problems 5 and 6 are categorized as weakly heterogeneous item bin packing problems, where “weakly heterogeneous item bin packing problem” refers to the classification of items into a relatively small number of categories, with each category consisting of identical items in terms of shape and size. Table 5 presents the computational results for these two weakly heterogeneous bin packing problems and compares them with the method proposed by Tsai and Li, Chen & Huo (2022), as well as Tsai, Wang & Lin (2015). As shown in Table 5, our model significantly reduces the utilization of 0–1 variables. Furthermore, in terms of computational time, the proposed method in this article demonstrates the shortest CPU time for solving the model. The complexity of solving these problems is also lower compared to solving the strongly heterogeneous classification packing problem, mainly due to the reduced need for combinatorial strategies when rotating the bins. The computational results indicate that the proposed method effectively addresses the challenges posed by the weakly heterogeneous packing problem.

1. Actual bin packing problem

   Problems 7 to 10 in this article are based on the packing example of the enterprise and were solved using the GUROBI solver. GUROBI is a highly efficient mathematical optimization solver used for solving various optimization problems, including linear programming, integer programming, and quadratic programming. It is widely employed in operations research, industrial optimization, logistics planning, and similar fields. In this study, computational results were obtained using GUROBI and compared with those obtained from LINGO and CPLEX. Table 6 presents the comparative findings. GUROBI demonstrates exceptional computational performance on the packing instance data, requiring less CPU time than LINGO and CPLEX to solve the reconfigured hybrid integer planning model. For problem 9, involving five items, GUROBI produces a solution within 5 s, while both LINGO and CPLEX require CPU times exceeding 60 min. As the proposed model in this article is linear and of low complexity, the computation time is significantly lower compared to the nonlinear model employed by LINGO.

1. Bin packing problems for multiple-bin open sizes

   The 3D bin packing problem with multiple bin sizes focuses on the application of strong heterogeneous bin packing. Table 7 displays eight randomly generated orders, each comprising two to four items.

   The calculation results for the corresponding orders were obtained under various designs of open sizes for different bin types, as presented in Table 8. Orders 01–04 were tested with 2 and 3 bin models, respectively, achieving an average space utilization of 80.16% and 91.92%, respectively, the CPU calculation time remained within 20 s for both orders 01–04. Visual representations of the results for orders 01–04 are displayed, with Fig. 2 showcasing the results for the 2-bin type and Fig. 3 illustrating the results for the 3-bin type.

Orders 01–06 underwent experimentation with packing bin types 2, 3, and 4 individually. Notably, the average space utilization achieved impressive rates of 62.72%, 81.59%, and 90.93% for each bin type, respectively. Moreover, the CPU computation time remained well within the 15-minute limit. Regarding the visualization of the results for orders 01–06, Fig. 4 displays the outcomes obtained using bin type 2, Fig. 5 exhibits the results with bin type 3, and Fig. 6 illustrates the results with bin type 4.

The utilization and performance of orders 01–08 were evaluated using different types of bins: two, three, and four. The average space utilization achieved was 62.24%, 80.43%, and 89.35% respectively. As expected, the loading efficiency improved with an increasing number of bin types. However, it should be noted that the CPU computation time also increased accordingly, staying within 7 min. To visually represent these results, Fig. 7 illustrates the visualization for bin type 2, Fig. 8 for bin type 3, and Fig. 9 for bin type 4. These figures provide a clear depiction of the optimization achieved in terms of loading rate and space utilization as the number of bin types increased, while also indicating the corresponding increase in CPU computation time.

System design

To effectively integrate the e-commerce warehouse bin packing problem-solving model with the current industry trends of “unmanned” and “intelligent,” this study presents a refined design for a fully automatic digital intelligent bin packing system. Leveraging mature hardware and software, the proposed system aims to enhance operational efficiency and cater to the evolving needs of the industry. Figure 10 illustrates the layout of the fully automatic digital intelligent bin packing system. Comprising three key modules, the digital automatic packing system initiates its operation sequence by inputting order items in module 3. Subsequently, module 2, utilizing the model developed in ‘Variable assumptions and model construction’ and the algorithm outlined in ‘Example data’–‘Algorithm solving’, adjusts the order and orientation of the packing items. Finally, module 1 completes the packing operation for the order items, concluding the process.

In Fig. 10, it is illustrated that once the picking process is completed, the order items proceed to module 3. Utilizing a chassis equipped with a gravity-sensing device, the system verifies the alignment between the items positioned on each chassis and the corresponding order information. This step serves as the initial review of the items picked for the order. By combining the efforts of robot arm No. 1 and camera No. 1, the items belonging to the current order are removed from the chassis and sequentially placed onto the conveyor belt in module 2. Consequently, the order items are then directed toward the order and direction adjustment link. The infrared scanning device is positioned next to the conveyor belt in module 2. Its purpose is to scan the items as they pass through in a sequential manner, thereby obtaining the SKU information associated with each item.

Simultaneously, it captures the current sequence of items on the conveyor belt. By comparing this information with the original order details, the device completes the review of the second-order information. In module 2, the intelligent packing strategy calculates a packing scheme. Based on this scheme, the first mechanical arm adjusts the order of items and their placement direction to align with the requirements of the packing scheme. This ensures that the items are properly arranged and oriented before they proceed to module 1 for the packing operation. In module 1, the order items are carefully boxed, and the appropriate type of bin must be selected based on the predetermined packing scheme. The types of bins have been predetermined, and there is an unlimited supply of each type. Spare bines are neatly folded and stored until they are needed, at which point mechanical devices unfold them for use. Within module 1, the second mechanical arm collaborates with camera No. 2 to smoothly and accurately grab the order items from the conveyor belt and place them inside the designated packing bins. Furthermore, the packing bin employed in module 1 is positioned on a chassis equipped with a weight sensor. Once all the items for an order have completed the packing process, the weight-sensing chassis within module 1 can verify the correctness of the packing operation, thereby fulfilling the role of a third review. This comprehensive approach ensures that the entire order-packing operation is completed with utmost precision. Throughout the process, the order information undergoes three rounds of review, guaranteeing a high level of accuracy. Moreover, the packing scheme strictly adheres to the intelligent packing strategy’s calculation results, enabling the implementation of a digitized packing scheme.

Implications for practical

A mixed integer planning model is constructed to solve the 3D-MODRPP and a 3D packing system with multiple bin sizes of open dimensions is proposed. In the model, a new method of assuming 0–1 integer variables is used to reduce the number of constraints, thus reducing the computational complexity of the model. A dataset for the 3D-MODRPP problem has also been generated based on the classical dataset for the single bin 3D packing problem, and superior results have been achieved. The practical impacts of this study are as follows:

1. Reduced complexity and computational time: Compared with the existing methods, the models and methods studied in this article can reduce the model complexity and computational time when solving 3D packing problems with multiple bin types. This will improve the efficiency of logistics companies in practice and reduce operational costs.

2. Improve resource utilization: The research presented in this article significantly contributes to optimizing bin dimensions, thereby maximizing bin space utilization while minimizing wastage. The findings of this study have valuable implications for reducing warehouse space requirements and enhancing overall resource utilization efficiency.

3. Drive transport efficiency: Optimized 3D packing solutions will help to reduce the volume of goods, thereby reducing wasted space in transport, increasing packing efficiency, and further improving transport efficiency.

4. Expand the scope of research: The research presented in this article offers a novel solution to the complex 3D packing problem involving multiple bin types within the context of e-commerce warehousing. By addressing this intricate challenge, the article significantly expands the boundaries of existing packing problem research. This expansion not only broadens the horizons of academic exploration but also holds the potential to revolutionize practical logistics optimization.

Implications for theory

This study explores the solution of multiple bin 3D packing problems by applying it to the field of e-commerce warehousing. This provides a typical bin for the application of combinatorial optimization problems in different application areas, which is also of guiding significance for the application of problems in other areas. It is mainly reflected in the following aspects:

1. Algorithmic and optimization method innovation: In this article, we present a novel approach aimed at minimizing the count of 0–1 variables associated with overlapping constraints and rotational constraints. Our method achieves this by considering 0–1 integer variables, leading to a reduction in both constraint quantity and computational complexity within the model. This algorithmic advancement not only yields tangible outcomes in effectively addressing diverse challenges within 3D packing problems but also introduces fresh insights and methodologies applicable to a broader spectrum of combinatorial optimization issues.

2. Extension of application area: The research presented in this article opens up broader avenues and directions in the field of 3D packing. We aspire for these findings to serve as a catalyst, inspiring researchers to delve deeper into intricate packing challenges and thereby fostering the advancement of the field.

3. Dataset generation and verification of authenticity: The dataset produced within this article closely aligns with the challenges posed by the 3D-MODRPP problem, thereby furnishing a notably dependable resource for tackling real-world predicaments. Furthermore, the devised methodology for dataset generation and the meticulous process employed for validating outcomes hold considerable significance in the realms of dataset construction and ensuring credibility across diverse domains.

4. Combination of theoretical models and practical problems: This research synergistically integrates a mixed integer programming model with established hardware and software technologies, effectively addressing real-world challenges within the realm of e-commerce warehousing. This harmonious fusion of theoretical constructs and pragmatic issues serves to enhance the symbiotic relationship between academia and industry, fostering collaborative efforts and driving the resolution of tangible problems in a concerted manner.