Artificial Bee Colony Algorithm with Pareto-Based Approach for Multi-Objective Three-Dimensional Single Container Loading Problems

Abstract

The ongoing container shortage crisis has presented significant challenges for the freight forwarding industry, requiring companies to implement adaptive measures in order to maintain peak operational efficiency. This article presents a novel mathematical model and artificial bee colony algorithm (ABC) with a Pareto-based approach to solve single-container-loading problems. The goal is to fit a set of boxes with strongly heterogeneous boxes into a container with a specific dimension to minimize the broken space and maximize profits. Furthermore, the proposed algorithm incorporates the bottom-left fill method, which is a heuristic strategy for packing containers. We conducted numerical testing to identify optimal parameters using the $\tilde{C}$ metric method. Subsequently, we evaluated the performance of our proposed algorithm by comparing it to other heuristics and meta-heuristic approaches using the relative improvement (RI) value. Our analysis showed that our algorithm outperformed the other approaches and achieved the best results. These results demonstrate the effectiveness of the proposed algorithm in solving real-world single-container-loading problems for freight forwarding companies.

1. Introduction

The shipping industry is constantly seeking ways to optimize container space and increase profits, which is especially important in the context of the single-container-loading problem (SCLP). The SCLP involves packing strongly heterogeneous rectangular boxes into a rectangular container in a way that maximizes the occupied volume or total profit while adhering to certain constraints, such as no overlaps and that all boxes placed within the container must not exceed their dimensions [1]. Other constraints can also be considered for the SCLP, as they represent practical (real-world) requirements [2]. Since the SCLP cannot be solved in polynomial time, it is classified as an NP-hard problem that is challenging and interesting to solve [3]. The SCLP is an active research topic and has numerous applications in the real world, particularly in the container transportation and distribution industry [4].

The SCLP problem has been studied extensively by many researchers with different objective functions and constraints. The majority of the studies focus on providing solutions with heuristic and metaheuristic approaches. Methods to obtain a feasible solution include the use of different algorithms, such as genetic algorithms (GAs) [5], differential evolution algorithm (DE) [6], particle swarm optimization (PSO), ant colony optimization algorithm (ACO), simulated annealing (SA), and tabu search (TS) [7].

The goal of the resulting SCLP is to maximize the value of the cargo or minimize the broken space while satisfying a number of practical constraints to ensure safety and positioning; meet non-overlapping constraints; and facilitate cargo handling, including customer priorities, load balancing, cargo stability, and stacking constraints [8]. However, the practical limitations mentioned above show the importance of the heuristic filling procedure and will be explained in the next section.

The basic heuristic filling procedure to generate and complete loading strongly heterogeneous boxes into the fixed container with specific dimensions is a wall-building-based method that decomposes the container structure into layers, which are further split into strips [9]. The packing of a strip is formulated as a container-loading problem [10]. The second approach is the caving degree approach (CDA), which prioritizes packing items into corners to reduce space waste by keeping items close to each other [11]. The heuristic filling procedures, namely sequence triple, have been presented for three-dimensional layouts. In each iteration, the sequence triple is transformed into a packing solution to evaluate its objective value, with the goal of maximizing the utilization of space [12]. Moreover, there is also a diversity-minded box placement method, a tree search algorithm specifically designed for three-dimensional container-loading problems (3DCLP) that uses a special form of tree search to ensure a balance between its search range and diversity [13]. Although some heuristic filling procedures have been considered in the literature, this is the first time a filling method has been proposed for a 3DCLP, and it is a multi-objective problem. The heuristic filling procedure called ‘the bottom-left fill algorithm (BLF)’ [14] has been presented for three-dimensional SCLP and applied multi-objective functions. This BLF consists of sequentially placing the pieces in the position as far as possible to the bottom-left place in the examined strip without overlapping them with those previously positioned. The coordinate of the bottom left-hand side of the boxes and containers is represented by a three-dimensional coordinate. Moreover, the bottom-left fill algorithm was developed in conjunction with a meta-heuristics approach to determine the optimal solution for arranging boxes within containers.

As shown in

Table 1. Literature relevant to the knapsack problem and container-loading problem.

Author and Year	Topic Research	Problem	The Dimension			Objective Function		Rotations
			1D	2D	3D	Single	Multi	Rotate	No
Huang et al., 2016 [15]	An effective placement method for the single-container-loading problem	Single-container-loading problem	-	✓	-	✓	-	-	✓
Araya et al., 2017 [16]	VCS: a new heuristic function for selecting boxes in the single-container-loading problem	Single-container-loading problem	-	✓	-	✓	-	-	✓
Ali et al., 2021 [17]	Novel binary differential evolution algorithm for knapsack problems	Knapsack problem	✓	-	-	✓	-	-	✓
Abdel-Basset et al., 2021 [18]	A binary equilibrium optimization algorithm for the 0–1 knapsack problem	Knapsack problem	✓	-	-	✓	-	-	✓
Patra et al., 2022 [19]	GWO-based simulated annealing approach for load balancing in the cloud for hosting containers as a service	Load balancing in the containerized cloud	-	-	✓	✓	-	-	✓
Junqueira et al., 2022 [20]	Solving the integrated multi-port stowage planning and container relocation problems with a genetic algorithm and simulation	Container relocation problem	-	-	✓	✓	-	-	✓
Yan et al., 2023 [21]	Multi-objective scheduling strategy of mine transportation robot based on three-dimensional loading constraints	Multiple-container-loading problem	-	-	✓	-	✓	-	✓
This paper	Artificial bee colony algorithm with pareto-based approach for multi-objective three-dimensional single-container-loading problems	Single-container-loading problem	-	-	✓	-	✓	✓	-

, this paper presents a review of the knapsack problem and container-loading problem, focusing on the dimensions, objective function, and rotations. The reviewed literature indicates that most of the research has focused on one or two dimensions and single-objective functions and has not considered rotation conditions in their research. Overall, the reviewed literature highlights the lack of research on the use of box rotation techniques to optimize box arrangement within a single container. This provides an opportunity for future research to examine the efficacy of box rotation techniques in optimizing the multi-objective 3D container-loading process.

The literature reviewed in

Table 2. The review of heuristic methods and meta-heuristics for solving container-loading problems.

Author and Year	Heuristic Method	Meta-Heuristic						Objectives Function		Pareto-Based Approach
		PSO	GA	DE	ACO	ABC	Hybrid Method	Single	Multi
Liu et al., 2017 [22]	Heuristic algorithm	-	-	-	-	-	✓	✓	-	-
Ramos et al., 2018 [23]	-	-	✓	-	-	-	-	✓	-	-
Castellucci et al., 2019 [24]	Dynamic programming algorithm	-	-	-	-	-	-	✓	-	-
Ghesmi and Brindley, 2021 [25]	A nonlinear finite element method	-	-	-	-	-	-	✓	-	-
Erbayrak et al., 2021 [26]	Family unity concept	-	-	-	-	-	-	-	✓	✓
Huang et al., 2022 [27]	-	✓	-	-	-	-	-	✓	-	-
Zhang et al., 2022 [28]	-	-	-	-	✓	-	✓	✓	-	-
Erkalkan et al., 2023 [29]	-	-	-	✓	-	-	-	✓	-	-
Krebs et al., 2023 [30]	Adaptive large neighborhood search tackling	-	-	-	-	-	✓	✓	-	-
This paper	ABC with Pareto-based approach	-	-	-	-	✓	✓	-	✓	✓

reveals the use of various heuristic methods and meta-heuristics to address the container-loading problem, including PSO, GA, ABC, ACO, DE, and hybrid methods. Notably, some studies proposed unique approaches, such as a multi-population biased random-key genetic algorithm (BRKGA) or an ant colony stacking method (ASM). However, a gap remains in the literature on combining artificial bee colony algorithms with a Pareto-based approach to tackle this problem. Thus, this research introduces the artificial bee colony algorithm with the Pareto-based approach. To our knowledge, this is the first study to present the aforementioned method to simultaneously maximize profit and space utilization for a single container of fixed size.

In summary, the reviewed literature suggests a lack of research exploring the effectiveness of box rotation techniques in optimizing the multi-objective 3D container-loading problem. There is a need for research to investigate the use of combining artificial bee colony algorithms with a Pareto-based approach to tackle a multi-objective problem. Additionally, the proposed bottom-left fill with rotation heuristic algorithm incorporating a hybrid method is the first method to propose simultaneously maximizing profit and space utilization for a single container with fixed dimensions.

Thus, in this research, we combine the artificial bee colony algorithm with a Pareto-based approach to solve the SCLP for strongly heterogeneous rectangular boxes. The ABC algorithm is commonly used by researchers because of its simplicity, flexibility, robustness, ability to explore local solutions, ability to handle objective costs, and broad applicability with complex functions [31]. However, one of its well-known limitations is its tendency to get trapped in local optima, which is ascribed to its exploration process. The proposed algorithm avoids this issue by utilizing a powerful meta-heuristics approach with a Pareto-based method and a heuristic addition process called the bottom-left filling algorithm. The research contributions of this article are as follows:

• A novel mathematical model for solving the multi-objectives SCLP is developed.
• This article proposes a new hybrid approach that combines the artificial bee colony algorithm with the Pareto-based approach to solve single-container-loading problems.
• The performance of the ABC algorithm with the Pareto-based approach is compared with that of the best-performing meta-heuristic algorithms in terms of solution quality and computation time.

The rest of this article is organized as follows. In Section 2, we present the problem formulation and describe our novel mathematical model. Section 3 provides a brief overview of the ABC algorithm with the Pareto-based approach for solving the SCLP. In Section 4, we present the results of computational experiments, including comparative analyses and illustrative examples. The empirical study and discussion of the results are shown in Section 5, while Section 6 summarizes the work and its future directions.

2. Mathematical Formulation

In this research, a novel mathematical model was created to solve the multi-objective three-dimensional container-loading problem. To solve this problem in the context of the arrangement of boxes of goods in a container, the aim was to arrange the boxes in the container such that the container was at its maximum weight threshold. The box arrangement was required, therefore, to minimize empty space [32]. An additional aim was to arrange the boxes such that profit was maximized. In this research, the global optimal solution for each objective was found using the exact method. The problem formulation is given below, and the parameter and variable definitions are listed in Abbreviations at below.

2.1. Assumptions for Mathematical Formulation

The following assumptions were made to establish the mathematical formulation:

• Each box and the knapsack were required to have features parallel to one another;
• The total weight was set to the container weight limit;
• Each box was required to be perfectly stackable with other boxes;
• Each box could be rotated along two axes only;
• Each box and the knapsack were required to have rectangular parallelepiped features;
• The starting point of the knapsack and each box were required to be at the bottom-left rear corner;
• The knapsack position was mapped using a 3D Cartesian coordinate system with a starting point of $\left(0,0,0\right)$;
• The knapsack and box sizes were not necessarily required to be expressed as negative rational numbers.

2.2. Mathematical Model for Solving Multi-Objective Three-Dimensional Container-Loading Problems

In the proposed mathematical model, the objective functions maximize the profit and minimize the empty space in the knapsack.

$$\mathrm{Max}{\sum }_{i\in J}{p}_i{t}_i$$

(1)

$$\min \left(WDH-{\sum }_{i\in J}{w}_i{d}_i{h}_i{t}_i\right)$$

(2)

In addition, the following constraints are applied. The total number of chosen boxes must not be wider than the knapsack space:

$${\sum }_{i\in J}{w}_i{d}_i{h}_i{t}_i\le WDH$$

(3)

The boxes must not differ:

$$b{w}_{ij}+b{w}_{ji}+b{d}_{ij}+b{d}_{ji}+b{h}_{ij}+b{h}_{ji}\ge t_i+t_j-1;\hspace{1em}i<j,\hspace{1em}i\in J,\hspace{1em}j\in J$$

(4)

The total sum of the coordinate points of the bottom-left corner and box dimensions must not exceed that of the knapsack:

$$x{w}_i+{\sum }_{r\in R}s{w}_{ir}{q}_{ir}\le W;\hspace{1em}\forall i$$

(5)

$$x{d}_i+{\sum }_{r\in R}s{d}_{ir}{q}_{ir}\le D;\hspace{1em}\forall i$$

(6)

$$x{h}_i+{\sum }_{r\in R}s{h}_{ir}{q}_{ir}\le H;\hspace{1em}\forall i$$

(7)

If box $i$ precedes box $j$, the total sum of the coordinates of the bottom-left back of box $i$ with the given dimension of box $i$ must be less than or equal to the position of box $j$:

$$x{w}_i+{\sum }_{r\in R}s{w}_{ir}{q}_{ir}\le x{w}_j+T\left(1-b{w}_{ij}\right);\hspace{1em}i<j,\forall i,\forall j$$

(8)

$$x{d}_i+{\sum }_{r\in R}s{d}_{ir}{q}_{ir}\le x{d}_j+T\left(1-b{d}_{ij}\right);\hspace{1em}i<j,\forall i,\forall j$$

(9)

$$x{h}_i+{\sum }_{r\in R}s{h}_{ir}{q}_{ir}\le x{w}_j+T\left(1-b{h}_{ij}\right);\hspace{1em}i<j,\forall i,\forall j$$

(10)

If box $j$ antecedes box $i$, the total sum of the coordinates of the bottom-left back of box $j$ with the given dimension of box $j$ must be less than or equal to the position of box $i$:

$$x{w}_j+{\sum }_{r\in R}s{w}_{jr}{q}_{jr}\le x{w}_i+T\left(1-b{w}_{ji}\right);\hspace{1em}i<j,\forall i,\forall j$$

(11)

$$x{d}_j+{\sum }_{r\in R}s{d}_{jr}{q}_{jr}\le x{d}_i+T\left(1-b{d}_{ji}\right);\hspace{1em}i<j,\forall i,\forall j$$

(12)

$$x{h}_j+{\sum }_{r\in R}s{h}_{jr}{q}_{jr}\le x{h}_i+T\left(1-b{h}_{ji}\right);\hspace{1em}i<j,\forall i,\forall j$$

(13)

If a coordinate point is not chosen for box $i$, it must be equal to 0:

$$x{w}_i\le T{t}_i;\hspace{1em}\forall i$$

(14)

$$x{d}_i\le T{t}_i;\hspace{1em}\forall i$$

(15)

$$x{h}_i\le T{t}_i;\hspace{1em}\forall i$$

(16)

If box $i$ is chosen to precede box $j$, box $j$ cannot be placed before box $i$:

$$b{w}_{ij}+b{w}_{ji}\le 1;\hspace{1em}\forall i,\forall j$$

(17)

$$b{d}_{ij}+b{d}_{ji}\le 1;\hspace{1em}\forall i,\forall j$$

(18)

$$b{h}_{ij}+b{h}_{ji}\le 1;\hspace{1em}\forall i,\forall j$$

(19)

If a box is not chosen, that box cannot precede any other box:

$$b{w}_{ij}\le t_i;\hspace{1em}\forall i,\forall j$$

(20)

$$b{d}_{ij}\le t_i;\hspace{1em}\forall i,\forall j$$

(21)

$$b{h}_{ij}\le t_i;\hspace{1em}\forall i,\forall j$$

(22)

$$b{w}_{ji}\le t_i;\hspace{1em}\forall i,\forall j$$

(23)

$$b{d}_{ji}\le t_i;\hspace{1em}\forall i,\forall j$$

(24)

$$b{h}_{ji}\le t_i;\hspace{1em}\forall i,\forall j$$

(25)

The total sum of the rotation of any box is less than or equal to $1$:

$${\sum }_{r\in R}{\rho }_{ir}\le 1;\hspace{1em}\forall i$$

(26)

The total sum of the rotation of box $i$ is less than or equal to the binary box selection value:

$${\sum }_{r\in R}{\rho }_{ir}\le t_i;\hspace{1em}\forall i$$

(27)

The box and knapsack starting points are at $\left(\mathrm{0,0},0\right)$:

$$x{w}_i\ge 0;\hspace{1em}\forall i$$

(28)

$$x{d}_i\ge 0;\hspace{1em}\forall i$$

(29)

$$x{h}_i\ge 0;\hspace{1em}\forall i$$

(30)

The total sum of the weights of all chosen boxes is less than or equal to the overall mass:

$${\sum }_{i\in J}{m}_i{t}_i\le M;\hspace{1em}\forall i$$

(31)

The binary variable equations are as follows:

$${\rho }_{ir}=\left\{0,1\right\};\hspace{1em}\forall i,\forall r$$

(32)

$$t_i=\left\{0,1\right\};\hspace{1em}\forall i,\forall r$$

(33)

$$b{w}_{ij}=\left\{0,1\right\};\hspace{1em}\forall i,\forall r$$

(34)

$$b{d}_{ij}=\left\{0,1\right\};\hspace{1em}\forall i,\forall r$$

(35)

$$b{h}_{ij}=\left\{0,1\right\};\hspace{1em}\forall i,\forall r$$

(36)

The mathematical model was developed to ensure that the problem was solved according to the following setup. First, the objective function equations indicate that the profit is to be maximized and the empty space in the knapsack is to be minimized. The container is placed in a coordinate system with its origin at the bottom-left rear corner. The constraints listed above are then applied using the given contrasting function equations. In detail, Constraints (3)–(7) ensure that the total number of chosen boxes does not exceed the knapsack space. Constraints (8)–(13) ensure that no box overlap is permitted. Constraints (14)–(16) force zero values on coordinate points that are not chosen for boxes. Constraints (17)–(19) guarantee that boxes already selected for the knapsack are not re-selected. Constraints (20)–(25) verify that unselected boxes are not loaded in the knapsack. Constraints (26) and (27) ensure that the box’s rotation value is a binary variable, i.e., $\left\{0,1\right\}$. Constraints (28)–(30) state that the box and knapsack start points are at (0,0,0), and Constraint (31) guarantees that the total sum of the weights of all chosen boxes is less than or equal to the knapsack weight. Finally, Constraints (32)–(36) specify the binary variables.

3. Methods

The optimization algorithm proposed in this article is a hybrid of the ABC and Pareto-based approaches. This section presents a brief description of the hybridization process of the ABC algorithm with the Pareto-based approach. The main reason behind the high suitability of the ABC algorithm for this work is its good performance for difficult optimization problems, which is achieved via effective search mechanisms in a single cycle and easily implementable bee phases [33]. As noted above, the problem considered here involves a multi-objective function that requires a collection of a nondominated set of optimal values, and the Pareto approach is the most suitable approach for comparing such optimal values. This method also solves the problems of a lack of use of secondary information, the high number of objective function evaluations, and slowness during sequential processing that beset the classic ABC algorithm. The ABC algorithm modification with the Pareto approach proposed herein is a meta-heuristics application involving temporal memory, where the original answer is memorized, and unique or different optimal solutions are then found. The Pareto optimal solution set is a group of Pareto optimal solutions to the given problems that are nondominated compared with all solutions in the search space, as shown in Figure 1. Here, the optimal values obtained by the algorithm are represented by the solutions in the Pareto-optimal set.

The meta-heuristic approach shown in the flowchart of Figure 2 illustrates the procedure for obtaining accurate results using the proposed hybrid ABC algorithm with the Pareto approach.

The algorithm consists of seven main processes: problem formulation encoding, initialization, box arrangement using the BLF algorithm and the wall-building approach, implementation of the ABC operators, calculation of the nectar amount, roulette wheel selection, and Pareto optimal front checking. The ABC operators consist of swaps, rotations, insertions, crossovers, and mutations, through which the algorithm finds the optimal values in the local search area. These processes are described in

Table 3. Procedure in the ABC algorithm with Pareto-based approach.

Step 1: Initialization step: a set of initial solutions is randomly generated according to Equation (1). Step 2: Nectar amount calculation step: the fitness value for the box arrangement in the container is evaluated according to Equation (2). The outcome values are then compared using the Pareto-based approach, and fitness value collection continues. Step 3: Neighborhood search determination or scout-bee step: the set of initial answers from the initialization step is used to find neighboring answers via the crossover, swap and insertion, mutation, or rotation operators, by adjusting the answer in each of the rounds through the random selection of one procedure. The results are then submitted to the Pareto-based approach for comparison, and fitness value collection continues. Step 4: Roulette wheel selection or onlooker-bee step: An answer is randomly selected from the answer set. This answer is then refined, with the onlooker bees randomly choosing a value between 0 and 1. Subsequently, the answers are compared by the Pareto-based approach, and the fitness values are collected. Step 5: Check-limited or limited-scout step: the repetitions of the answer sets that have not yet improved the fitness value are examined. In Step 3, the value of the limited scout is fixed; if the answer set exceeds this value, it is discarded, and a new answer set is randomly created in Step 1. Step 6: Termination criterion step: the number of cycles required to fix the answer sets is examined. When the fixed cycles are complete, all optimal answer sets are presented, and the process is stopped. Step 7: Pareto optimal front checking step: the fitness answer values obtained from the Step 2 calculations are collected and compared; thus, the nondominated set of fitness values is produced.

.

Note that, for the roulette wheel selection step, an answer is randomly selected from the answer set. The answer is then refined as the onlooker bees randomly choose a value between 0 and 1 and select the answer from Equation (3), as shown in Figure 3. Subsequently, the solutions obtained via the Pareto-based approach are compared, and the fitness values are collected [34].

Most meta-heuristic algorithms were originally proposed for solving continuous optimization problems. To solve combinatorial problems such as the container-loading problem, a solution representation is required to transform the dimension values of each box into a practical utilization solution. In this research, a solution representation with encoding and decoding schemes was developed for loading the boxes into a single container to minimize the empty space and maximize profit. The steps in the proposed heuristic filling procedure are presented in

Table 4. Steps in the proposed modified BLF heuristic filling procedure.

Step 1: Input the problem data, including the coordinate points and dimensions of boxes and containers. Step 2: Calculate the priority of each box according to the determined identification number of each box. The priority is calculated by grouping boxes of similar sizes together. Step 3: Check available space inside a container for a new layer. If there is enough space, then go to the next step; otherwise, go to Step 11. Step 4: Input the width, length, height, and profit data of the boxes with the highest priority among the set of available boxes. Step 5: Select the box with the highest priority in the set of available boxes. Step 6: Calculate the possibility of packing the selected boxes in the current layer. If possible, go to the next step; otherwise, go to Step 4. Step 7: Load the selected boxes into the container and remove the packed boxes from the set of available boxes. Step 8: Update the available space data in the layer of the container. Step 9: Update the available box dataset. Step 10: Check the available space for the new box in the current layer. If there is sufficient space, then go to the next step; otherwise, go to Step 3. Step 11: Check the available box datasets. If there are some boxes, go to Step 3; otherwise, go to the last step. Step 12: Calculate the utilization rate and profit of the container and stop.

.

In the proposed heuristic filling procedure for the solution of the SCLP, the input data include coordinate points and the dimensions of boxes and containers. The calculation step is identified based on the number of boxes, and the priority is calculated by grouping boxes of similar sizes together. The next step is to check the available space inside a container for a new layer. This is the process of checking the possibility of packing the selected boxes into the current layer. The most important step is to update the container layer regarding empty space and update the remaining box dataset to calculate the container utilization and profit margin. In the last step, the nondominated set of Pareto solutions is checked.

Operators for the ABC Algorithm with Pareto-Based Approach

As noted above, the ABC optimization algorithm involves calculating or problem-solving to find values, the most appropriate process from the values found, or all possible ways to find the expected solution. The key parameters of the proposed ABC algorithm with the Pareto-based approach are listed in

Table 5. Key parameters of the ABC algorithm with Pareto-based approach [35].

No.	Key Parameter	Value
1	Colony size $\left(C_s\right)$	1, …, 100
2	Number of employed bees $\left(E_o\right)$	$\left(C_s\right)$/2
3	Number of onlooker bees $\left(O_r\right)$	$\left(C_s\right)$/2
4	Mutation rate $\left(M_r\right)$	0, …, 1
5	Swap rate $\left(S_r\right)$	0, …, 1
6	Crossover rate $\left(C_r\right)$	0, …, 1
7	Number of iterations $\left(I_o\right)$	1, …, 50,000

.

In the first stage, the initial population is randomly generated for the employed bees. The employed bees create a new vector parameter from the old parameter when it is subjected to crossover, swap and insert, or mutation. The results are later submitted to the Pareto-based approach for comparison of the optimal values, and the fitness value collection is continued.

4. Experimental Results and Evaluation

The proposed ABC algorithm with the Pareto-based approach was implemented in MATLAB version R2020. Calculations were performed using an AMD Ryzen 7 3700U processor with Radeon Vega Mobile GFX 2.30 GHz, a 64-bit Windows 10 operating system, and 8 GB of RAM. Three-step sequential experiments were conducted. Experiment A was designed to identify the optimal parameters for the ABC algorithm with the Pareto-based approach, and the $\tilde{C}$ metric method was used to compare its performance. Experiment B aimed to evaluate the performance of the ABC algorithm with the Pareto-based approach by a LINGO optimization solver on a small problem dataset sourced from the OR-Library database [36]. To compare the results with previous studies and provide a clear comparison of solutions, the well-known Loh and Nee (LN) test cases [37] for medium to large problems were employed. We compared the results of the proposed algorithm with heuristics and meta-heuristics. The algorithms compared included spatial representation [37], layering algorithm [38], development heuristics [39], genetic algorithm [40], and tabu search [41]. As a result, the relative improvement (RI) value was utilized to facilitate the comparison process. The RI value was estimated using two equations: the first Equation (37) evaluated the number of boxes left (BL), and the second Equation (38) evaluated the volume utilization (VU). It should be noted that a higher RI value indicates that the ABC algorithm with the Pareto-based approach outperforms the comparative research solution.

RI = ((Sol_orig − Sol_ABC)/Sol_orig) × 100

(37)

RI = −((Sol_orig − Sol_ABC)/Sol_orig) × 100

(38)

where RI represents the relative improvement (%) between Sol_orig and Sol_ABC, and Sol_orig represents the solution obtained from the comparative research. Sol_ABC represents the solution obtained from the ABC algorithm with the Pareto-based approach.

As a final step, Experiment C aims to evaluate the performance of the proposed algorithm on the largest and most complex problems by testing it on a set of generated instances. In addition, the complexity indicator (CI) will be utilized to measure the complexity of the problem. This will facilitate the identification of the proposed algorithm’s capabilities to solve complex and real-world problems efficiently and effectively.

4.1. Experiment A

As noted above, this experiment aimed to investigate and identify the optimal parameters for the ABC algorithm with the Pareto-based approach (including the number of bees or the bee colony size (C_s), the loop quantity (I_o), mutation rate (M_r), and crossover rate (C_r). The $\tilde{C}$ metric was used for efficiency comparison. An experimental design utilizing a three-level factorial design was conducted to determine the optimal setting of the parameter values, as outlined in

Table 6. Key parameters of ABC algorithm with Pareto-based approach.

No.	Levels	Function Evaluation		Crossover Rate $\left({\mathit{C}}_{\mathit{r}}\right)$	Mutation Rate $\left({\mathit{M}}_{\mathit{r}}\right)$
		Bee Colony Size $\left({\mathit{C}}_{\mathit{s}}\right)$	Loop Quantity $\left({\mathit{I}}_{\mathit{o}}\right)$
1	1	10	5000	0.1	0.1
2	2	50	1000	0.5	0.5
3	3	100	500	1.0	1.0

.

The computational runs were replicated 20 times using a different case study. There are two case studies available to assess the performance of the parameters: one with a small knapsack problem in “Problem1” and the other with a large knapsack problem in “Problem2”. In practice, the available computation time is limited. Therefore, the iterations were fixed at 50,000, as test runs indicated that this number was sufficient to achieve convergent results. In this research, the $\tilde{C}$ metric was used to compare the non-dominant answer groups of the methods individually. For example, to measure the effectiveness of levels 1 and 2, the denotation $\tilde{C}$(1,2) measures the number of members of 2 dominated by 1. Thus, |2| is the number of solutions in this method. The smaller the value of $\tilde{C}$(1,2), the more strongly the indication is that level 2 is better than level 1. The $\tilde{C}$ metric results are presented in

Table 7. Function evaluation results given by $\tilde{C}$ metric.

Problem	C(1,2)	C(2,1)	C(1,3)	C(3,1)	C(2,3)	C(3,2)	Best
Problem 1	0.286	1.000	1.000	0.200	1.000	0.286	2 Size × Iter 50 × 1000
Problem 2	0.200	0.750	0.667	0.500	1.000	0.600
Average	0.243	0.875	0.833	0.350	1.000	0.443

,

Table 8. Crossover rate evaluation results given by $\tilde{C}$ metric.

Problem	C(1,2)	C(2,1)	C(1,3)	C(3,1)	C(2,3)	C(3,2)	Best
Problem 1	0.200	1.000	0.667	0.000	1.000	0.200	2 ${\mathbf{C}\mathbf{s}\mathbf{s}\mathbf{s}}_{\mathbf{r}}$ = 0.5
Problem 2	0.333	0.750	0.750	0.250	0.500	0.333
Average	0.267	0.875	0.708	0.125	0.750	0.267

and

Table 9. Mutation rate evaluation results given by $\tilde{C}$ metric.

Problem	C(1,2)	C(2,1)	C(1,3)	C(3,1)	C(2,3)	C(3,2)	Best
Problem 1	0.750	0.200	1.000	0.000	0.250	0.500	1 ${\mathbf{M}}_{\mathbf{r}}$ = 0.1
Problem 2	1.000	0.000	0.600	0.167	0.200	0.500
Average	0.875	0.100	0.800	0.083	0.225	0.500

for the function evaluation, crossover rate, and mutation rate, respectively.

In Table 7, Table 8 and Table 9, the $\tilde{C}$ metric values that are less than or close to 0 are chosen. We evaluated the function evaluation values presented in Table 7 and found the optimal parameter for function evaluation is “levels2” with a bee colony size of 50 and a loop quantity of 1000. As in Table 8, we evaluated the crossover rate values and found that the optimal parameter for crossover rate values is “levels2,” which is equal to 0.5. Finally, we evaluated the mutation rate in Table 9, and it was found that the optimal parameter for the mutation rate was “levels1”, which was 0.1. Therefore, we summarize the results of the function evaluation, crossover rate, and mutation rate using values of 50 × 1000, 0.5, and 0.1, respectively, in

Table 10. Optimal parameters as indicated by $\tilde{C}$ metric.

Function Evaluation (${\mathit{C}}_{\mathit{s}}$ × ${\mathit{I}}_{\mathit{o}}$)	50 × 1000
Crossover rate (${\mathit{C}}_{\mathit{r}})$	0.5
Mutation rate $\left({\mathit{M}}_{\mathit{r}}\right)$	0.1

.

4.2. Experiment B

The aim of this experiment was to benchmark the performance of the proposed method using the optimal parameter settings, which were identified in Experiment A. To test whether the non-dominant set provided by the meta-heuristic method could yield an answer close to that of an exact method, the LINGO optimization solver was implemented, and its optimal solutions were compared with those of the proposed method. That is, the closeness or consistency of the optimal solutions given by the proposed method to the best solutions from the LINGO optimization solver was determined. If closeness or a match was achieved, it was inferred that the non-dominant set obtained from the meta-heuristic method was suitable and of appropriate quality. The comparative results are presented in

Table 11. Comparison of optimal solutions given by the LINGO solver and ABC algorithm with Pareto-based approach.

No.	Case Study	OR-Library Responses		Exact Method (LINGO V.20)		ABC with Pareto
		Total Profit (USD)	Total Empty Space (M2)	Total Profit (USD)	Total Empty Space (M2)	Total Profit (USD)	Total Empty Space (M2)
1	P01/1	309	58%	309	58%	309	58%
2	P01/2	309	85%	309	85%	309	85%
3	P02/1	51	75%	51	75%	51	75%
4	P02/2	51	91%	51	91%	51	91%
5	P07/1	1458	29%	1458	29%	1458	29%
6	P08/1	265	36%	265	36%	265	36%
7	P08/2	265	50%	265	50%	265	50%
8	P10/1	1268	22%	1268	22%	1268	22%
9	P10/2	1268	65%	1268	65%	1268	65%
10	P10/3	1268	42%	1268	42%	1268	42%

. In this work, a dataset from the OR-Library database, which is a collection of test datasets for various Operations Research (OR) problems, was used.

An optimal solution given by the Pareto approach that was equal to the LINGO optimization solver solution was chosen for comparison. Table 11 shows that the proposed method provided equivalent answers to the exact method for data from the OR-Library. Hence, it can be concluded that the proposed ABC algorithm with the Pareto-based approach is effective for solving the problem.

Table 12. Computation runtimes of the LINGO solver and ABC algorithm with Pareto-based approach.

No.	Case Study	Computation Runtimes (Average Time (s))
		LINGO	ABC with Pareto
1	P01/1	0.01	11
2	P01/2	0	12
3	P02/1	0	12
4	P02/2	0	11
5	P07/1	0	98
6	P08/1	97,204	73
7	P08/2	88,126	72
8	P10/1	112,063	106
9	P10/2	98,762	118
10	P10/3	99,211	116

presents the average runtimes of the ABC algorithm using the Pareto-based approach for solving problems from the OR-Library. These values are compared with the resolution times of the exact method using the LINGO optimization solver. While the proposed method is slower than the exact method in small-scale problems, it should be noted that for more complex problems, the exact method is not able to solve the problem within an acceptable computational runtime. In such cases, the ABC algorithm with the Pareto-based approach takes more time to optimize but still delivers a solution within a reasonable computational runtime.

The performance of the proposed algorithm was also evaluated using the well-known Loh and Nee (LN) test cases, and

Table 13. Comparative results of the Loh and Nee (LN) test cases with other approaches.

Volume Utilization (%)
Problem	Ngoi et al. [37]		Bischoff et al. [38]		Bischoff and Ratcliff [39]		Gehring and Bortfeldt [40]		Bortfeldt and Gehring [41]		This Work
	Spatial Representation		Layering Algorithm		Development Heuristics		GA		Tabu Search		ABC with Pareto
	BL	VU	BL	VU	BL	VU	BL	VU	BL	VU	BL	VU
LN01	0	62.5	0	62.5	0	62.5	0	62.5	0	62.5	0	62.5
LN02	54	80.7	23	89.7	35	90	39	89.5	28	96.6	35	90
LN03	0	53.4	0	53.4	0	53.4	0	53.4	0	53.4	0	53.4
LN04	0	55	0	55	0	55	0	55	0	55	0	55
LN05	0	77.2	0	77.2	0	77.2	0	77.2	0	77.2	0	77.2
LN06	48	88.7	24	89.5	77	83.1	32	91.1	49	91.2	20	92.2
LN07	10	81.8	1	83.9	18	78.7	7	83.3	0	84.7	0	84.7
LN08	0	59.4	0	59.4	0	59.4	0	59.4	0	59.4	0	59.4
LN09	0	61.9	0	61.9	0	61.9	0	61.9	0	61.9	0	61.9
LN10	0	67.3	0	67.3	0	67.3	0	67.3	0	67.3	0	67.3
LN11	0	62.2	0	62.2	0	62.2	0	62.2	0	62.2	0	62.2
LN12	0	78.5	3	78.5	0	78.5	0	78.5	0	78.5	0	78.5
LN13	2	84.1	5	82.3	20	78.1	0	85.6	4	84.3	0	85.6
LN14	0	62.8	0	62.8	0	62.8	0	62.8	0	62.8	0	62.8
LN15	0	59.5	0	59.5	0	59.5	0	59.5	0	59.5	0	59.5
Average	7.6	69.0	3.7	69.7	10.0	68.6	5.2	69.9	5.4	70.4	3.7	70.1

BL = no. of boxes left, and VU = volume utilization.

shows that the proposed algorithm was able to solve the SCLP. However, the performance of the proposed algorithm will need to be further measured with RI values to facilitate a comparison of results.

To ensure a comprehensive and unbiased evaluation of our proposed algorithms, we conducted a comparative study that included both heuristics and meta-heuristics from scholarly works. In addition, we integrated widely accepted meta-heuristic methods, such as the genetic algorithm and tabu search, into our evaluation. Furthermore, we included other heuristics that have been developed to address the same problem, such as the spatial representation, layering algorithm, and development heuristics, for comparative analysis.

Table 14. Comparative results of the relative improvement indicator for each approach.

Test Case	Ngoi et al. [37]		Bischoff et al. [38]		Bischoff and Ratcliff [39]		Gehring and Bortfeldt [40]		Bortfeldt and Gehring [41]
	Spatial Representation		Layering Algorithm		Development Heuristics		GA		Tabu Search
	BL	VU	BL	VU	BL	VU	BL	VU	BL	VU
LN01	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0
LN02	35.2	11.5	−52.2	0.3	0.0	0.0	10.3	0.6	−25.0	−6.8
LN03	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0
LN04	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0
LN05	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0
LN06	58.3	3.9	16.7	3.0	74.0	11.0	37.5	1.2	59.2	1.1
LN07	100.0	3.5	100.0	1.0	100.0	7.6	100.0	1.7	0.0	0.0
LN08	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0
LN09	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0
LN10	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0
LN11	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0
LN12	0.0	0.0	100.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0
LN13	100.0	1.8	100.0	4.0	100.0	9.6	0.0	0.0	100.0	1.5
LN14	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0
LN15	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0	0.0

BL = no. of boxes left, and VU = volume utilization.

presents the results of the relative improvement (RI) of different approaches. It has been observed that the proposed method outperforms its competitors only when the RI effect is positive and negative in the worst-case scenario. For the case of zero, it provides the same result. In comparison with the studies by Ngoi et al., Bischoff and Ratcliff, and Gehring and Bortfeldt, our proposed algorithm obtained better results with positive RI results, except in some cases for which our method yields the same results. Regarding Bischoff et al.’s study, our approach yielded superior or comparable outcomes, except for LN02, where our method exhibited a more effective utilization of volume but a lower efficiency in terms of the number of boxes left. Compared with Bortfeldt and Gehring’s research, our method generated better or similar results, except for one case. In the case of LN02, we experienced a loss both in the number of boxes left and volume utilization.

In summary, the results of the reliability index (RI) analysis clearly demonstrate that the proposed method outperformed or matched other methods in 14 out of 15 case studies, with only 1 case performing worse than some studies. Therefore, based on these findings, it can be concluded that the proposed method represents the best-performing approach.

4.3. Experiment C

In large-scale experiments, we demonstrated the effectiveness of the proposed algorithm in terms of computational time and the optimal solution to the most difficult and complex problems. We generated a dataset of container sizes and boxes consisting of 225 cases with variations in dimensions and profit per box, with only one container size being set. To test the performance of the proposed method, we designed datasets with different levels of problem complexity: low, medium, and high. We defined the complexity according to the complexity indicator (CI), which was calculated as follows:

-

CI = 100%, where the size of every box arranged in the container was unique.

-

CI = 50%, where the size of every box arranged in the container was 50% unique.

-

CI = 25%, where the size of every box arranged in the container was 25% unique.

Table 15. The Pareto optimal solutions and computational runtimes of the ABC algorithm with Pareto-based approach.

Method/ Problem	Complexity Indicator (CI)	ABC with Pareto		Computation Runtimes
		Broken Space (M2)	Profit (USD)	Average Runtimes (second)	Standard Deviation (SD)
1	100%	0.550806	1038	113.26	2.21
		0.531041	990
		0.56424	1055
		0.589491	1356
2	100%	0.533096	4809	112.34	2.22
		0.547748	4992
3	100%	0.412006	3310	111.62	4.86
		0.410012	3302
		0.432743	3400
4	100%	0.492349	2193	112.99	2.53
		0.509934	2354
		0.500023	2312
5	100%	0.573422	1203	116.79	2.10
		0.579743	1219
6	50%	0.316389	3201	114.19	2.97
		0.314329	3007
		0.323278	3219
		0.316194	3198
7	50%	0.3104	2845	116.39	2.77
		0.310001	2703
8	50%	0.421	3200	114.59	3.64
		0.38891	2704
		0.42109	3432
		0.39234	3102
9	50%	0.352539	2500	115.59	4.19
		0.412403	2583
		0.321111	2397
10	50%	0.312853	2735	116.59	6.03
		0.32222	3205
		0.31532	3134
11	25%	0.200343	1237	119.19	4.01
		0.200098	1200
12	25%	0.322469	1290	111.39	5.82
		0.212	1228
		0.332034	1434
13	25%	0.270933	1625	119.19	5.41
		0.289043	1627
		0.262393	1555
		0.264829	1587
14	25%	0.312032	890	114.19	7.18
		0.273421	850
15	25%	0.17402	1520	115.79	8.01
			Average	114.94	4.26

presents the Pareto front results of the ABC algorithm with the Pareto-based approach. It is important to note that LINGO cannot find a solution within a reasonable computing time (10 h) for small problem sizes, thereby restricting its ability to handle large and complex problems. In contrast, the ABC algorithm with the Pareto approach has proven to be capable of solving optimization problems of varying complexities and sizes, with an average resolution time of only 114.94 s and a very low average standard deviation of 4.26, which is significantly shorter than the 15 min average required for manual planning. This has resulted in an 87.23% improvement in time efficiency. Based on the results of small problems (instances 11–15), the ABC algorithm with the Pareto-based approach easily finds the optimal solution and gives the best answer regarding the empty space and appropriate profit in every case. As a result of medium problems (instances 6–10), the proposed method shows a significant or approximately double gain of profit, even though the empty space is slightly increased. Finally, in the case of large or complex problems of maximum box size (instances 1–5), the proposed method shows a significant increase in profit while the computational time remains the same, although the remaining space increases slightly. Consequently, it can be concluded that the proposed approach is effective at solving real-world problems based on the solution quality and computation cost.

5. Discussion

This article presents a novel mathematical model and artificial bee colony algorithm (ABC) with a Pareto-based approach to solve multi-objective three-dimensional single-container loading problems. The mathematical model is constructed as a multi-objective three-dimensional problem under complex constraints such as box rotation and non-overlapping boxes. To investigate the optimal solution, the LINGO optimization solver was implemented. To obtain accurate results using the Pareto-based approach with the ABC algorithm, optimal parameters were identified as per Table 6. Based on the findings, it can be concluded that setting the function evaluation, crossover rate, and mutation rate to 50 × 1000, 0.5, and 0.1, respectively (as shown in Table 10), leads to optimal solution outcomes. In Experiment B, the performance of the proposed algorithm was evaluated using the optimal parameter settings identified in Experiment A. The LINGO optimization solver was utilized to compare the optimal solutions provided by the proposed method with the exact solutions. An open dataset from the OR-Library database [36] was employed, and the comparison results from Table 11 indicated that, in all cases, one solution obtained from the Pareto optimal set is identical to that obtained from the LINGO optimizer. These findings suggest that the proposed algorithm can be an effective alternative to the LINGO optimization solver, particularly when the computational time required to solve the problem is significantly lower than that of the exact method. This advantage is evident from the results presented in Table 12. Table 14 presents the results of the relative improvement (RI) of different approaches for medium to large problems of the well-known Loh and Nee (LN) test cases [37]. The comparison results indicate that the proposed method outperformed or was equal to its competitors, with only 3 cases resulting in a worse solution out of 150. These findings suggest that the proposed method is significantly more effective than the other methods. However, it should be noted that the exact method cannot provide a solution for all cases. For the large instances in Experiment C, we generated a dataset of container sizes and boxes consisting of 225 cases with variations in dimensions and profit per box, with only container size being set. It is important to note that LINGO cannot find a solution within a reasonable computing time (10 h) for small problem sizes, thereby restricting its ability to handle large and complex problems. Table 15 presents the Pareto front results of the ABC algorithm with the Pareto-based approach. It has proven capable of solving optimization problems of varying complexities and sizes, with an average resolution time of only 114.94 s and a very low average standard deviation of 4.26, which is significantly shorter than the 15-minute average required for manual planning. This has resulted in an 87.23% improvement in time efficiency. Consequently, it can be concluded that the proposed approach is effective at solving real-world problems based on the solution quality and computation cost.

6. Conclusions and Future Works

This study proposed a hybrid algorithm, the ABC algorithm with the Pareto-based approach, which combines the advantages of both algorithms to optimize the multi-objective three-dimensional single-container-loading problem. A mathematical model was successfully developed using this approach, and the $\tilde{C}$ metric method was used to identify key parameter values to reduce the computational burden. The performance of the proposed algorithm was compared to that of an exact method, and the results indicated that the proposed algorithm provides the same objective values as the exact method for small-scale problems within a satisfactory computer runtime. However, for more complex problems, the exact method may not be able to solve the problem within an acceptable computational runtime, whereas the ABC algorithm with the Pareto-based approach still delivers a solution within a reasonable computational runtime. Moreover, the presented algorithm was evaluated using the Loh and Nee (LN) test cases, demonstrating superior performance compared to other heuristic and meta-heuristic methods. The proposed algorithm was also tested on a challenging case study involving complex real-world problems, and the results showed that the algorithm effectively solved these problems within a reasonable computational runtime. These findings demonstrate that the proposed algorithm is comparable to the best-performing heuristic algorithm and can be applied to real-world problems.

However, in future directions, the practical limitations of the problem will tend to become significantly more complex. Therefore, this meta-heuristic method may need to be redesigned to support, for example, a hybrid of meta-heuristic methods, or a two-stage approach using clustering methods such as k-means can be considered to group box sizes before considering the order of packing for containers while taking into account the center of gravity during the loading process. Additionally, keeping the computation time as low as possible can further improve the efficiency of the proposed algorithm.