HAGWO: A Hierarchical Adversarial Grey Wolf Optimizer and Its Application in the 3D Bin Packing Problem

Abstract

The Grey Wolf Optimizer (GWO) is a popular metaheuristic, yet it often suffers from premature convergence and rapid diversity loss in complex, high-dimensional, or highly constrained optimization problems. This paper introduces HAGWO, a novel Hierarchical Adversarial Grey Wolf Optimizer that addresses these limitations through three synergistic enhancements: dynamic hierarchical population stratification, adaptive Levy flight perturbation, and hierarchical adversarial-like position updating. These mechanisms enable adaptive balancing of global exploration and local exploitation while preserving population diversity throughout the search process. Extensive experiments on the CEC 2017 bound-constrained benchmark suite across 30D, 50D, and 100D dimensions demonstrate that HAGWO achieves superior overall performance among eight state-of-the-art algorithms, with statistically significant advantages confirmed by Friedman mean ranks and Wilcoxon signed-rank tests. When adapted to the strongly NP-hard three-dimensional bin packing problem with identical bins (3D-SBSBPP), HAGWO delivers highly competitive results, outperforming the well-established BRKGA and most other metaheuristics while closely approaching the original GWO in solution quality and exhibiting exceptional run-to-run stability. By rigorously evaluating HAGWO across both high-dimensional continuous benchmarks and a practical constrained combinatorial application, this study validates the effectiveness of its hierarchical adversarial-like framework and provides valuable insights into algorithm design and transferability across different problem domains.

1. Introduction

Optimization problems permeate science, engineering, and industry, ranging from continuous parameter tuning to discrete combinatorial decision-making. Among these, high-dimensional continuous optimization [1] and strongly constrained combinatorial tasks—such as the three-dimensional bin packing problem (3D-BPP)—remain particularly challenging due to multimodal landscapes, the curse of dimensionality, and strict feasibility constraints [2]. While exact methods are often intractable for large-scale instances, metaheuristic algorithms, especially swarm intelligence (SI) techniques inspired by collective natural behaviors, have emerged as powerful tools for efficiently navigating complex search spaces.

Over recent decades, numerous SI algorithms have been developed, including Particle Swarm Optimization (PSO) [3], Differential Evolution (DE) [4], Artificial Bee Colony (ABC) [5], and Whale Optimization Algorithm (WOA) [6]. These methods have achieved considerable success across diverse applications. Nevertheless, many still suffer from premature convergence, rapid loss of population diversity, and inadequate exploration–exploitation balance, particularly in high-dimensional or highly constrained scenarios [7,8,9].

The Grey Wolf Optimizer (GWO), introduced by Mirjalili et al. [10], is among the most widely adopted SI algorithms. Drawing inspiration from the social hunting hierarchy of grey wolves, GWO assigns leadership roles to alpha, beta, and delta wolves, with omega wolves updating positions based on these leaders. Its simplicity, minimal parameter tuning, and strong global search capability have led to successful applications in both continuous and discrete problems [11]. However, the original GWO exhibits notable limitations: (1) premature convergence and diversity loss in later stages, especially on multimodal or high-dimensional landscapes; (2) a rigid, uniform update strategy guided solely by the top three leaders, limiting phase-specific adaptability; and (3) reduced flexibility in highly constrained combinatorial problems such as 3D-BPP, where geometric and capacity constraints demand more nuanced local adjustments.

To address these shortcomings, this paper proposes HAGWO (Hierarchical Adversarial Grey Wolf Optimizer), a novel enhancement of GWO incorporating three synergistic innovations: (1) dynamic hierarchical population stratification, inspired by concepts such as Tianji’s horse racing optimization [12], which adaptively partitions the population into superior, medium, and inferior groups with evolving proportions across search phases; (2) hierarchical adversarial-like position updating, assigning distinct update rules to each group to foster opposition-based learning and directed information flow; and (3) an adaptive Levy flight perturbation mechanism that provides controlled long-range exploration with intensity decreasing over iterations [13]. The term “adversarial” in HAGWO refers to an opposition-based perturbation strategy inspired by adversarial learning principles, rather than a game-theoretic minimax formulation.

The effectiveness of HAGWO is rigorously validated through two complementary experimental tracks. On the CEC 2017 benchmark suite [1] in 30D, 50D, and 100D dimensions, HAGWO achieves superior overall performance among eight state-of-the-art algorithms, with statistical significance confirmed by Friedman and Wilcoxon rank-sum tests. When adapted to the strongly NP-hard three-dimensional single-bin-size bin packing problem (3D-SBSBPP), HAGWO delivers highly competitive results, outperforming the established BRKGA [14] and several other metaheuristics while achieving the second-best overall performance in terms of total bin count and exhibiting exceptional run-to-run stability.

The main contributions of this study are as follows:

• We propose HAGWO, an effective GWO variant that integrates dynamic hierarchical stratification, adversarial-like position updating, and adaptive Levy flight perturbation to enhance diversity, adaptability, and search efficiency.
• We conduct comprehensive high-dimensional benchmarking on CEC 2017, demonstrating statistically significant superiority in continuous optimization.
• We adapt HAGWO to 3D-SBSBPP and provide a detailed comparative analysis, offering insights into algorithm transferability across continuous benchmarks and constrained combinatorial applications.

The remainder of the paper is organized as follows. Section 2 reviews related work. Section 3 details the proposed HAGWO algorithm. Section 4 describes its adaptation to 3D-BPP. Section 5 presents the experimental setup and results. Section 6 discusses key findings, implications, and limitations. Finally, Section 7 concludes the paper.

2. Related Work

SI algorithms have emerged as powerful tools for solving complex optimization problems, drawing inspiration from natural collective behaviors. Among them, GWO [10] has gained significant attention due to its intuitive social hierarchy and strong global search capabilities. The algorithm’s explicit division into alpha, beta, delta, and omega wolves provides a natural structural substrate for tiered extensions, making it particularly amenable to enhancements that require role-differentiated update strategies. However, the original GWO often suffers from premature convergence in high-dimensional or multimodal landscapes [11].

To enhance GWO’s performance, numerous variants have been proposed. Nadimi-Shahraki et al. [11] introduced the Improved Grey Wolf Optimizer (I-GWO), which incorporates a dimension learning-based hunting (DLH) search strategy to enhance both global and local search capabilities; it was evaluated on the CEC 2018 benchmark suite and demonstrated strong competitiveness. Jiang et al. [15] proposed the Enhanced GWO with Elite Inheritance and Balance Search Mechanisms (EBGWO), which inherits elite solutions across generations and stochastically balances exploration and exploitation. More recent advancements include the Improved Adaptive GWO (IAGWO) that employs multi-strategy ensemble and adaptive mechanisms to significantly enhance exploration [9], and the Scale-Free Network-based GWO (SFGWO), which divides the population into network-structured layers [16]. Hybrid strategies have also been introduced, such as the Genetic Learning-based Grey Wolf Optimizer (GL-GWO) [17]. Despite these advances, few studies have focused on hierarchical adversarial-like enhancements of GWO on the CEC 2017 suite, and their efficacy in constrained combinatorial problems remains underexplored.

The CEC 2017 benchmark suite [1] is a standard testbed for evaluating metaheuristics on single-objective real-parameter optimization. It includes 30 functions categorized into unimodal, multimodal, hybrid, and composition types. Several SI algorithms, including CCO [18], WOA [6], and EAO [19], have been benchmarked on CEC 2017, with statistical tests like Friedman and Wilcoxon being used to validate significance [20].

In the domain of combinatorial optimization, the three-dimensional bin packing problem (3D-BPP) [2] is a classic NP-hard problem with significant applications in logistics, manufacturing, and supply chain management. The objective is to pack a set of rectangular items into a minimum number of identical rectangular bins while respecting constraints such as item orientations, non-overlapping placements, and stability requirements. Unlike its two-dimensional counterpart, 3D BPP introduces additional complexity due to the need for three-dimensional spatial reasoning and rotation possibilities (up to six orientations per item), making exact methods computationally infeasible for large instances [21].

Heuristic and metaheuristic approaches have been extensively explored to tackle 3D BPP. Traditional heuristics include the First-Fit Decreasing (FFD) and Best-Fit Decreasing (BFD) strategies, often combined with space-filling rules like the Lowest Bottom-Left (LBL) or Extreme Point-Based Space (EMS) [22]. Metaheuristics enhance these heuristics by providing global search capabilities. For example, Genetic Algorithms (GAs) have been adapted with biased random-key encodings in BRKGA [14], which decodes chromosomes into item sequences and orientations for effective packing. The Grey Wolf Optimizer (GWO) [10] has been applied to packing problems with notable success [11]. However, advanced modifications incorporating hierarchical or adversarial-like mechanisms in GWO remain scarce for 3D BPP. This study bridges these gaps by proposing HAGWO and evaluating it on both CEC 2017 and 3D BPP.

3. Proposed HAGWO

The proposed Hierarchical Adversarial Grey Wolf Optimizer (HAGWO) builds upon the foundational principles of the original GWO while addressing its key limitations through novel enhancements. In this section, we first provide an overview of the original GWO algorithm and analyze its shortcomings, followed by a detailed description of HAGWO’s three synergistic innovations: dynamic hierarchical population stratification, adaptive Levy flight perturbation mechanism, and hierarchical adversarial-like position updating.

3.1. Original Grey Wolf Optimizer (GWO)

The Grey Wolf Optimizer (GWO), introduced by Mirjalili et al. [10], is a metaheuristic inspired by the social hunting hierarchy and behavior of grey wolves. The algorithm divides the wolf pack into four levels: alpha ($\alpha$) as the leader, beta ($\beta$) as the second-in-command, delta ($\delta$) as the third, and omega ($\omega$) as the followers. The search process mimics encircling, hunting, and attacking prey.

The encircling behavior is modeled as

$$\overrightarrow{D}=|\overrightarrow{C}·{\overrightarrow{X}}_p\left(t\right)-\overrightarrow{X}\left(t\right)|,$$

(1)

where ${\overrightarrow{X}}_p\left(t\right)$ represents the prey position (approximated by $\alpha$, $\beta$, or $\delta$), $\overrightarrow{X}\left(t\right)$ is the current wolf position, and $\overrightarrow{C}$ is a coefficient vector that adds randomness to the encircling.

The position update is given by

$$\overrightarrow{X}(t+1)={\overrightarrow{X}}_p\left(t\right)-\overrightarrow{A}·\overrightarrow{D},$$

(2)

where $\overrightarrow{A}$ controls the step size and direction, allowing for both exploration (when $|\overrightarrow{A}|>1$) and exploitation (when $|\overrightarrow{A}|<1$).

The coefficients are defined as

$$\overrightarrow{A}=2\overrightarrow{a}·{\overrightarrow{r}}_1-\overrightarrow{a},$$

(3)

$$\overrightarrow{C}=2·{\overrightarrow{r}}_2,$$

(4)

with $\overrightarrow{a}$ linearly decreasing from 2 to 0 over iterations to transition from exploration to exploitation, and ${\overrightarrow{r}}_1$, ${\overrightarrow{r}}_2$ being random vectors in [0, 1] for stochasticity.

The hunting phase averages the positions guided by the top three wolves:

$$\overrightarrow{X}(t+1)={\displaystyle \frac{{\overrightarrow{X}}_1+{\overrightarrow{X}}_2+{\overrightarrow{X}}_3}{3}},$$

(5)

where ${\overrightarrow{X}}_1$, ${\overrightarrow{X}}_2$, and ${\overrightarrow{X}}_3$ are positions updated toward $\alpha$, $\beta$, and $\delta$, respectively. The pseudocode of GWO is presented in Algorithm 1.

Algorithm 1 Pseudocode of the Original GWO.

Require: Population size N, maximum iterations $T_{\max }$, dimension $dim$, bounds $lb$, $ub$

Ensure: Optimal solution $X_{\mathrm{best}}$, fitness $f\left(X_{\mathrm{best}}\right)$

1: Initialize population X and evaluate fitness  2: Identify $\alpha$, $\beta$, $\delta$  3: for $t=1$ to $T_{\max }$ do  4:     Update $a=2-2\times (t/T_{\max })$  5:     for each wolf i do  6:         Compute $X_1$, $X_2$, $X_3$ with Equations (1) and (2) toward $\alpha$, $\beta$, $\delta$  7:         Update $X(i,:)$ using Equation (5)  8:         Apply boundary handling  9:     end for 10:     Reevaluate fitness and update $\alpha$, $\beta$, $\delta$ 11: end for 12: Return $X_{\mathrm{best}}$, $f\left(X_{\mathrm{best}}\right)$

3.2. Limitations of Original GWO

Despite its effectiveness, the original GWO exhibits several limitations, particularly in complex optimization landscapes. First, the uniform update strategy for all wolves leads to rapid loss of population diversity, causing premature convergence in multimodal or high-dimensional problems [11]. Second, the fixed leadership hierarchy (relying solely on the top three wolves) lacks adaptability across different search phases, often resulting in insufficient exploration in the early stages and poor exploitation later on [23]. These issues motivate the development of enhanced variants that incorporate hierarchical structures, adversarial-like mechanisms, and advanced disturbance strategies.

3.3. HAGWO: Hierarchical Adversarial Grey Wolf Optimizer

To overcome the above limitations while leveraging the structural plasticity of GWO’s hierarchy, HAGWO introduces three synergistic enhancements: (1) dynamic hierarchical population stratification, (2) adaptive Levy flight perturbation mechanism, and (3) hierarchical adversarial-like position updating. These mechanisms work in concert to achieve adaptive balance between global exploration and local exploitation while maintaining population diversity throughout the search process. The population is dynamically stratified into superior ($\varsigma$), medium ($\zeta$), and inferior ($\xi$) groups based on fitness rankings.

3.3.1. Dynamic Hierarchical Population Stratification

HAGWO employs a dynamic hierarchical population stratification mechanism based on phase-aware search progression. The population is partitioned into superior ($\varsigma$), medium ($\zeta$), and inferior ($\xi$) groups according to fitness rankings. The group proportions adaptively evolve based on the search phase:

In the early phase ($t\le 0.5\times T_{\max }$), a larger proportion of inferior wolves promotes broad global exploration:

$$\left\{\begin{array}{c}{n}_{\varsigma }=0.1\times N\hfill \\ n_{\zeta }=0.3\times N\hfill \\ n_{\xi }=N-n_{\varsigma }-n_{\zeta }\hfill \end{array}\right.$$

(6)

In the late phase ($t>0.5\times T_{\max }$), an increased share of superior wolves enhances focused local exploitation:

$$\left\{\begin{array}{c}{n}_{\varsigma }=0.3\times N\hfill \\ n_{\zeta }=0.3\times N\hfill \\ n_{\xi }=N-n_{\varsigma }-n_{\zeta }\hfill \end{array}\right.$$

(7)

Design rationale and parameter justification. The stratification ratios $(0.1, 0.3, 0.6)$ in the early phase and $(0.3, 0.3, 0.4)$ in the late phase are determined by the following design principles. First, the medium group ratio is fixed at $0.3$ to ensure a stable transitional layer between superior and inferior wolves throughout the optimization process. Second, in the early phase, the superior ratio of $0.1$ ensures that at least one elite wolf exists for any population size $N\ge 10$, while the remaining $0.6$ inferior wolves maximize exploration coverage. In the late phase, the superior ratio increases to $0.3$ to intensify exploitation around promising regions, while the inferior ratio decreases to $0.4$ to maintain residual diversity. The threshold at $0.5\times T_{\max }$ represents a standard phase transition point that balances early exploration and late exploitation, consistent with established practices in phase-aware metaheuristic design [9,16]. We acknowledge that these ratios constitute a manually designed schedule rather than a fully adaptive fuzzy mechanism; the term “phase-aware” is used to describe this deliberate phase-dependent structural control.

3.3.2. Adaptive Levy Flight Perturbation Mechanism

To augment the algorithm’s capability to escape local optima and enhance long-range exploration, HAGWO integrates an adaptive Levy flight perturbation mechanism. The disturbance factor is formulated as

$$df=0.1\times p\times \mathrm{Levy}\left(\lambda \right),\lambda =1.5,$$

(8)

where $p=1-t/T_{\max }$ serves as a linearly decreasing scaling coefficient, ensuring stronger perturbation in early-to-mid stages (promoting diversity via long jumps) and gradual attenuation in later stages (supporting precise local refinement).

The Levy flight step $\mathrm{Levy}\left(\lambda \right)$ is generated using the Mantegna algorithm [24]:

$$L={\displaystyle \frac{u}{{\left|v\right|}^{1/\lambda }}},u\sim \mathcal{N}(0,{\sigma }_u^2),v\sim \mathcal{N}(0,1),$$

(9)

where

$${\sigma }_u={\left({\displaystyle \frac{\Gamma (1+\lambda )\sin (\pi \lambda /2)}{\Gamma ((1+\lambda )/2)\lambda 2^{(\lambda -1)/2}}}\right)}^{1/\lambda },$$

(10)

and $\Gamma (·)$ is the gamma function.

Parameter justification. The Levy flight exponent $\lambda =1.5$ is adopted following established practice in Levy-enhanced swarm intelligence methods [25,26]. This value lies in the theoretically recommended range $1<\lambda \le 3$ and has been empirically validated across numerous metaheuristic algorithms including cuckoo search [26] and various GWO variants [27]. The scaling coefficient $0.1$ in Equation (8) ensures that the disturbance magnitude remains a small fraction of the search space, preventing excessive disruption of promising solutions while providing sufficient perturbation to escape local optima. The linearly decreasing term $p=1-t/T_{\max }$ naturally attenuates the disturbance intensity as the search progresses, aligning with the widely accepted principle that exploration should dominate early and exploitation should dominate late [10]. These parameter choices require no additional problem-specific tuning beyond the standard GWO parameters.

3.3.3. Hierarchical Adversarial-like Position Updating

HAGWO implements hierarchical adversarial-like position updating, where each stratified group employs a distinct opposition-based strategy to drive competition and learning against higher-tier guidance. We clarify that the term “adversarial” herein refers to an opposition-based perturbation mechanism inspired by the concept of adversarial learning, rather than a game-theoretic minimax formulation or an explicit multi-objective adversarial game. Specifically, lower-tier wolves generate candidate positions by actively opposing the direction suggested by higher-tier guidance, creating reverse-search trajectories that enhance exploration. This mechanism ensures directed information flow from superior to inferior groups while actively opposing suboptimal directions through perturbation terms.

Let $\overline{X}=(X_{\alpha }+X_{\beta }+X_{\delta })/3$ denote the mean position of the current top three leaders. The group-specific candidate position update rules are as follows:

- Superior wolves ($\varsigma$): focus on elite refinement and internal information exchange:

$$X_{\varsigma ,i}^{\prime }=p·\overline{X}+(1-p)·X_{\varsigma \left(\mathrm{rand}\right)},$$

(11)

where $X_{\varsigma \left(\mathrm{rand}\right)}$ is the position of a randomly selected superior wolf (different from the current individual).

- Medium wolves ($\zeta$): integrate superior guidance with an adversarial-like perturbation scaled by the disturbance factor:

$$\begin{array}{cc}\hfill X_{\zeta ,i}^{\prime }& =p·\overline{X}+(1-p)·X_{\varsigma \left(\mathrm{rand}1\right)}\hfill \\ \hfill & +df·\left(X_{\zeta ,i}-\left[p·\overline{X}+(1-p)·X_{\varsigma \left(\mathrm{rand}2\right)}\right]\right),\hfill \end{array}$$

(12)

where $X_{\varsigma \left(\mathrm{rand}1\right)}$ and $X_{\varsigma \left(\mathrm{rand}2\right)}$ are independently sampled superior wolves. The perturbation term generates an opposition direction relative to a superior-guided position.

- Inferior wolves ($\xi$): perform aggressive opposition learning against medium wolves to drive global exploration:

$$\begin{array}{cc}\hfill X_{\xi ,i}^{\prime }& =p·\overline{X}+(1-p)·{\displaystyle \frac{X_{\zeta \left(\mathrm{rand}1\right)}+X_{\zeta \left(\mathrm{rand}2\right)}}{2}}\hfill \\ \hfill & +df·\left(p·\overline{X}+(1-p)·{\displaystyle \frac{X_{\zeta \left(\mathrm{rand}1\right)}+X_{\zeta \left(\mathrm{rand}2\right)}}{2}}-X_{\xi ,i}\right).\hfill \end{array}$$

(13)

Once the candidate positions $X_{·,i}^{\prime }$ have been computed for all wolves, a boundary correction mechanism is executed to maintain feasibility. Subsequently, an elitist selection criterion is invoked: a newly generated position is adopted solely when it demonstrates superior or equivalent fitness compared with the incumbent; otherwise, the current position is preserved.

Component interaction and synergistic design. The three mechanisms in HAGWO are designed to function as an integrated system rather than independent modules. The dynamic stratification determines which wolves receive which update rules; the adversarial-like updating provides the directional structure for each group’s search behavior; and the Levy flight perturbation supplies the stochastic magnitude for exploration. Removing any single mechanism would break this structural coherence: without stratification, all wolves would use identical update rules, reverting to the original GWO’s uniform strategy; without the adversarial-like component, the stratified groups would lack directional opposition; and without Levy perturbation, the escape capability from local optima would be substantially weakened. Consequently, the synergistic interaction among the three components constitutes the core contribution of HAGWO. The individual contribution of each mechanism is empirically validated through the ablation study presented in Section 5.5.

3.3.4. Pseudocode

The complete pseudocode and flowchart of HAGWO are detailed in Algorithm 2 and Figure 1, respectively.

Algorithm 2 Pseudocode of the proposed HAGWO algorithm.

Require: Population size N, maximum iterations $T_{\max }$, dimension $dim$, bounds $lb$, $ub$

Ensure: Optimal solution $X_{\mathrm{best}}$, fitness $f\left(X_{\mathrm{best}}\right)$

1: Initialize population X and evaluate fitness  2: for $t=1$ to $T_{\max }$ do  3:     Sort population by fitness  4:     Determine group sizes based on Equations (6) or (7)  5:     Update leaders $\alpha$, $\beta$, $\delta$  6:     Compute $a=2-2\times (t/T_{\max })$  7:     Compute $p=1-t/T_{\max }$  8:     Compute $df=0.1\times p\times \mathrm{Levy}\left(\lambda \right)$  9:     for each superior wolf do 10:         Update using Equation (11) 11:     end for 12:     for each medium wolf do 13:         Update using Equation (12) 14:     end for 15:     for each inferior wolf do 16:         Update using Equation (13) 17:     end for 18:     Apply boundary handling and greedy selection 19: end for 20: Return $X_{\mathrm{best}}$, $f\left(X_{\mathrm{best}}\right)$

3.4. Computational Complexity Analysis

We conduct a rigorous complexity analysis to investigate the computational implications of the proposed enhancements, directly comparing HAGWO against the original GWO. In what follows, N signifies the population size, D denotes the problem dimension, $T_{\max }$ indicates the maximum iteration count, and $f_{\mathrm{eval}}$ stands for the cost incurred by a single fitness evaluation.

Original GWO. In each iteration, GWO performs (i) fitness evaluation for all N wolves, costing $O(N·f_{\mathrm{eval}})$; (ii) sorting to identify $\alpha$, $\beta$, $\delta$, costing $O\left(N\log N\right)$; and (iii) position updates for all N wolves across D dimensions, costing $O(N·D)$. The total time complexity is therefore

$$O\left(T_{\max }·(N\log N+N·D+N·f_{\mathrm{eval}})\right).$$

(14)

The space complexity is $O(N·D)$ for storing the population matrix.

Proposed HAGWO. In each iteration, HAGWO additionally performs (i) dynamic stratification, which requires only index-based partitioning after sorting and thus adds $O\left(N\right)$—negligible compared to the $O\left(N\log N\right)$ sorting already required for leader identification; (ii) Levy flight generation via the Mantegna algorithm, which produces D random variates per wolf at $O\left(1\right)$ per dimension, yielding $O(N·D)$; and (iii) hierarchical adversarial-like updates, which compute candidate positions for all N wolves across D dimensions, also yielding $O(N·D)$. The greedy selection step evaluates one candidate position per wolf, adding $O(N·f_{\mathrm{eval}})$, which is identical to the fitness re-evaluation cost already present in GWO. Consequently, the asymptotic time complexity of HAGWO remains

$$O\left(T_{\max }·(N\log N+N·D+N·f_{\mathrm{eval}})\right),$$

(15)

strictly of the same order as GWO. The space complexity is likewise unchanged at $O(N·D)$.

Practical runtime overhead. While HAGWO preserves the same asymptotic complexity as GWO, the practical per-iteration overhead includes Levy random number generation (Mantegna algorithm), subgroup assignment, and the additional arithmetic operations in the adversarial-like update equations. To quantify this overhead, we measured the average runtime per iteration for both GWO and HAGWO on the CEC 2017 F1 function at $D=30$ dimensions with $N=30$ wolves. The results show that HAGWO incurs approximately 8–12% additional runtime per iteration compared to GWO, primarily attributable to the Levy flight generation and the more complex position update calculations. For the 3D-BPP experiments, where fitness evaluation dominates the computational cost (due to the expensive DFTRC-2 placement heuristic), the relative overhead of HAGWO’s additional operations is less than 3% of the total execution time. These observations confirm that HAGWO’s enhanced search capabilities are achieved with only modest practical computational cost.

4. Adaptation of HAGWO to the 3D Bin Packing Problem

To investigate the generalization capability of HAGWO to discrete combinatorial optimization problems, we adapt it to the three-dimensional single-bin-size bin packing problem (3D-SBSBPP). The adaptation retains the core HAGWO evolutionary framework while integrating problem-specific encoding, decoding, and placement components to ensure packing feasibility and solution quality.

4.1. Problem Formulation: Three-Dimensional Single-Bin-Size Bin Packing

The three-dimensional single-bin-size bin packing problem (3D-SBSBPP) requires packing a set of n rectangular boxes, each with dimensions $(l_i,w_i,h_i)$ for $i=1,\dots ,n$, into the minimum number of identical rectangular bins of fixed dimensions $(L,W,H)$. All dimensions are positive integers, and without loss of generality, each box satisfies $l_i\le L$, $w_i\le W$, $h_i\le H$ after possible rotations. Boxes may be rotated in any of the six orthogonal orientations but must remain axis-aligned with the bin edges. No overlapping is allowed, and each box must lie completely within a bin.

The problem is strongly NP-hard [2] and has wide industrial applications, including container loading, palletizing, cargo arrangement, warehouse optimization, and arises as a subproblem in cutting, scheduling, and logistics planning [28,29].

Due to the combinatorial complexity and strong constraints, exact mixed-integer programming approaches become intractable for large instances. Instead, we adopt a metaheuristic approach based on biased random-key encoding and deterministic decoding, which guarantees feasible packings through a greedy placement heuristic while allowing efficient search in a continuous space.

4.2. Encoding and Decoding Framework

Each candidate solution in HAGWO is represented as a real-valued vector of length $2n$ with components uniformly distributed in $[0, 1]$. The vector is semantically divided into two parts:

• The first n keys encode the box packing sequence (BPS), i.e., the priority order in which boxes are considered for placement.
• The second n keys encode the vector of box orientation biases (VBO), which guides biased selection of orientations during placement.

The decoding procedure transforms an individual into a feasible packing configuration in a deterministic manner:

• Decoding of BPS: Sort the first n keys in ascending order to obtain permutation $\pi$. Then, BPS = $\left[\pi \right(1),\pi (2),\dots ,\pi (n\left)\right]$, where BPS_i is the box index placed at stage i. Lower key values indicate higher placement priority.
• Decoding of VBO: The second n keys directly form VBO = $[v_1,\dots ,v_n]\in {[0,1]}^n$. For each box i in the BPS order, $v_i$ is used to select a biased orientation from the feasible set in the chosen maximal space.
• Placement and feasibility: Boxes are placed sequentially using Algorithm 4. For each box, (1) attempt placement in existing bins using the DFTRC-2 heuristic (Algorithm 3); (2) if no feasible maximal space is found, open a new bin and place the box at the origin $(0,0,0)$; and (3) update maximal spaces using the difference process (DP) [30] with added pruning rules for efficiency. The placement process guarantees no overlap, axis-alignment, and boundary compliance.
• Fitness evaluation: After all boxes are placed, compute the number of used bins NB and the adjusted fitness value (detailed in Section 4.3.5).

This encoding–decoding framework ensures every individual produces a feasible solution, while HAGWO evolves high-quality sequences and orientation biases in the continuous space ${[0, 1]}^{2n}$. Figure 2 illustrates the overall architecture of the proposed approach.

4.3. Common Components

The following components support the decoding and placement process, ensuring efficient and feasible packings.

4.3.1. Maximal-Space Representation

We adopt the maximal-space representation to track available empty spaces in bins, using a list S of empty maximal spaces (EMSs). Each EMS is defined by its minimum and maximum coordinates ($x_i, y_i, z_i$ and $X_i, Y_i, Z_i$). To manage EMSs, we employ the difference process (DP) from Lai and Chan [30], which involves (1) placing a box in an EMS; (2) generating new EMSs from intersections with the placed box and removing intersected EMSs; and (3) eliminating EMSs with infinite thinness or those inscribed by others.

Figure 3 illustrates the DP process.

To reduce the computational overhead of inscribed EMS elimination, we incorporate two pruning rules:

• Discard new EMSs with volume smaller than any remaining box (automatically removes thin EMSs).
• Discard new EMSs with smallest dimension less than any remaining box’s smallest dimension (targets large-volume unfit EMSs).

4.3.2. Placement Heuristics

When determining where to place each box, we exclusively evaluate candidate positions located at the origin vertices $(x_i, y_i, z_i)$ of every feasible empty maximal space. We adopt the DFTRC-2 heuristic from [14], which measures compactness by the squared Distance to the Front-Top-Right Corner (DFTRC) of the bin after placement. Higher DFTRC indicates greater compactness. DFTRC-2 evaluates all feasible EMSs and orientations, selecting the one maximizing DFTRC.

An illustrative example is provided in Figure 4. The figure depicts a partial packing configuration containing three already-placed boxes: box 1 (beige), box 3 (green), and box 4 (red). Box 2 (cyan, enlarged in the upper-left corner) is the next item to be packed. Hatched rectangles labeled “2” indicate candidate placements within different EMSs, whose minimum-coordinate corners are marked by black dots A, B, C, and D. The solid arrow demonstrates that DFTRC-2 ultimately selects position C (from EMS corner D) for placing box 2.

The pseudocode of the DFTRC-2 placement heuristic is detailed in Algorithm 3.

Algorithm 3 DFTRC-2 Heuristic Procedure for Box Placement.

Require: Box to pack b, bin $bin$

Ensure: Winning EMS $EM{S}^{\ast }$ and orientation $B{O}^{\ast }$, or 0 if not possible

1: Let $N_{ems}$ be the number of available EMSs in bin $bin$  2: Initialize $maxDis{t}^2\leftarrow -1$, $B{O}^{\ast }\leftarrow -1$, $EM{S}^{\ast }\leftarrow 0$  3: for $i=1$ to $N_{ems}$ do  4:     Let $x,y,z$ be the min coordinates of $EM{S}_i$  5:     for all possible orientations $BO$ of b do  6:         Let $d^{BO},w^{BO},h^{BO}$ be the dimensions of b in orientation $BO$  7:         if b fits in $EM{S}_i$ of bin $bin$ using $BO$ then  8:            $Dis{t}^2\leftarrow {(D-x-d^{BO})}^2+{(W-y-w^{BO})}^2+{(H-z-h^{BO})}^2$  9:            if $Dis{t}^2>maxDis{t}^2$ then 10:                $maxDis{t}^2\leftarrow Dis{t}^2$ 11:                $B{O}^{\ast }\leftarrow BO$ 12:                $EM{S}^{\ast }\leftarrow EM{S}_i$ 13:            end if 14:         end if 15:     end for 16: end for 17: return $EM{S}^{\ast }$            ▹ returns 0 if b cannot be packed in $bin$

4.3.3. Box Orientation

The orientation of each box is determined by the VBO vector generated by HAGWO. Let EMS denote the empty maximal space chosen by DFTRC-2 for accommodating box i, and let BOs represent the vector containing all admissible orientations of box i within EMS. The selected orientation, $B{O}^{\ast }$, is then computed as

$$B{O}^{\ast }=BOs\left(\lceil VB{O}_i\times nBOs\rceil \right)$$

(16)

where $nBOs$ indicates the total number of orientations stored in BOs, and $\lceil x\rceil$ denotes the ceiling of x (i.e., the least integer greater than x).

4.3.4. Placement Procedure

The packing process operates sequentially, with exactly one box being assigned to a bin at each step. The packing sequence of boxes is determined by the BPS generated through HAGWO evolution. This procedure integrates four components: the BPS and VBO vectors produced by HAGWO, the list $S_b$ recording all empty maximal spaces for each active bin b, and the DFTRC-2 placement heuristic. Every step of the process consists of five principal substeps:

(1)

Box selection;

(2)

Bin and empty maximal space selection;

(3)

Box orientation selection;

(4)

Box packing;

(5)

State information update.

The pseudocode of the placement procedure is presented in Algorithm 4.

Algorithm 4 Placement Procedure for 3D Bin Packing.

Require: Box packing sequence $BPS=[b_1,\dots ,b_n]$, orientation bias vector $VBO\in {[0,1]}^n$, initial empty bin template

Ensure: Final set of packed bins B with updated EMS lists

1: Initialize $B\leftarrow \left\{1\right\}$, $NB\leftarrow 1$  2: for $i=1$ to n do  3:     $b\leftarrow BP{S}_i$  4:     $ems\leftarrow 0$, $bin\leftarrow 0$  5:     for $k=1$ to $NB$ do  6:         $ems\leftarrow \mathrm{DFTRC}\text{-}2(b,k,B{O}^{\ast })$  7:         if $ems>0$ then  8:            $bin\leftarrow k$  9:            break 10:         end if 11:     end for 12:     if $bin=0$ then 13:         $bin\leftarrow NB+1$ 14:         $B\leftarrow B\cup \left\{bin\right\}$ 15:         $NB\leftarrow NB+1$ 16:         $ems\leftarrow 1$ 17:     end if 18:     Let $BOs$ be the feasible orientations of b in $ems$ of bin $bin$ 19:     Let $n_{BO}\leftarrow \left|BOs\right|$ 20:     $B{O}^{\ast }\leftarrow BOs\left(\lceil VB{O}_b\times n_{BO}\rceil \right)$ 21:     Pack b at origin of $ems$ in bin $bin$ using $B{O}^{\ast }$ 22:     Update EMS list of bin $bin$ via DP procedure [30] 23: end for

4.3.5. Fitness Function

An evolutionary algorithm necessitates a criterion for evaluating solution quality. For bin packing problems, the total number of bins utilized, denoted as NB, serves as an intuitive quality indicator. Nevertheless, because multiple solutions may yield identical NB values, this metric lacks sufficient discriminatory power when comparing solutions with equivalent bin counts.

To address this limitation, we introduce an enhanced quality measure termed the adjusted number of bins (aNB). This metric integrates NB with an auxiliary indicator—ranging from 0 to 1—that quantifies the improvement potential inherent in a given packing configuration. The underlying principle is as follows: given two solutions with equal NB, the one containing the most underutilized bin offers greater scope for optimization, since the residual capacity in that bin could potentially absorb items currently assigned to other bins, ultimately facilitating a decrease in the overall bin consumption. While a least-loaded bin may sometimes reflect poor utilization, in the context of the aNB fitness function it serves as an indicator of improvement potential rather than a direct measure of packing quality. By combining NB with the normalized least-load ratio, aNB provides a finer gradient for the optimizer to discriminate among solutions with identical bin counts.

We define LeastLoad as the total weight (or volume) contained in the least-loaded bin of a packing configuration. For the 3D variant, the capacity of every bin is given by BinCap = $W\times H\times D$. The adjusted number of bins is subsequently obtained by augmenting NB with the ratio of LeastLoad to BinCap:

$$aNB=NB+{\displaystyle \frac{LeastLoad}{BinCap}}.$$

(17)

This fitness encourages both minimizing NB and improving space utilization within bins. The fractional term LeastLoad/BinCap provides differentiation granularity below the integer bin count, guiding the search toward configurations where spare capacity is concentrated in the least-loaded bin, thereby increasing the likelihood of future bin reduction through box rearrangement.

4.4. HAGWO for 3D-BPP

We adapt the HAGWO framework to 3D-BPP, using the same individual representation and decoding procedure described above. Each individual is a real-valued vector of length $2n\in [0,1]$, divided into two semantic groups of size n (packing sequence and orientation bias). The complete pseudocode is given in Algorithm 5.

The hierarchical adversarial-like mechanism and Levy flight disturbance give HAGWO a greater ability to escape local optima than BRKGA. All other components (maximal spaces, DFTRC-2 placement, aNB fitness) are identical to the BRKGA implementation for fair comparison.

Algorithm 5 HAGWO for 3D-BPP.

Require: Number of boxes n, bin dimensions, box data, population size N, max iterations $T_{\max }$

Ensure: Best fitness, best packing

1: Generate initial population X ($N\times 2n$) uniformly in $[0,1]$  2: Evaluate each individual using evaluate (BPS + VBO → placement → aNB)  3: Set $BestP\leftarrow$ best individual, $BestValue\leftarrow$ its aNB  4: for $t=1$ to $T_{\max }$ do  5:     Sort population by fitness to identify $\alpha$, $\beta$, $\delta$  6:     Compute $a=2-2\times (t/T_{\max })$  7:     Compute $p=1-t/T_{\max }$  8:     Compute $df=0.1\times p\times \mathrm{Levy}\left(\lambda \right)$  9:     Determine dynamic group sizes: superior ($n_{\varsigma }$), medium ($n_{\zeta }$), inferior ($n_{\xi }$) 10:     for each wolf in superior group do 11:         Update position using elite averaging and internal exchange (Equation (11)) 12:     end for 13:     for each wolf in medium group do 14:         Update position using superior guidance with adversarial-like Levy disturbance (Equation (12)) 15:     end for 16:     for each wolf in inferior group do 17:         Update position using medium adversarial-like learning with Levy flight (Equation (13)) 18:     end for 19:     Apply boundary handling 20:     Evaluate new positions; apply greedy selection 21:     Update $\alpha$, $\beta$, $\delta$ and $BestP$ if improved 22:     Record best fitness of generation 23: end for 24: Return $BestValue$, packing of $BestP$

4.4.1. Distinguishing Optimizer from Decoder Contributions

We acknowledge that all algorithms compared on 3D-BPP—including HAGWO, GWO, PSO, WOA, BSA, CCO, EAO, and BRKGA—share the same encoding–decoding framework (biased random-key encoding, DFTRC-2 placement heuristic, maximal-space management, and aNB fitness function). Therefore, performance differences among these algorithms must be attributed to the optimizer’s ability to navigate the continuous search space ${[0,1]}^{2n}$, rather than to the decoding or placement components. The encoding–decoding framework serves as a common “sandbox” in which the metaheuristics compete to discover high-quality packing sequences and orientation biases. HAGWO’s competitive performance in this shared framework demonstrates that its hierarchical adversarial-like search mechanism is effective at finding better solutions within the same representational space, not that the framework itself provides an unfair advantage.

4.4.2. Parallel Implementation

The fitness evaluation phase, which includes random-key decoding, sequential DFTRC-2 placement, and aNB computation, accounts for the majority of the computational cost in the proposed HAGWO algorithm. To accelerate this bottleneck, we employ MATLAB R2020a’s parfor construct for shared-memory parallelization on multi-core CPUs. In each generation, the evaluation of the entire population is distributed across available cores using three separate parfor loops—one for each hierarchical subgroup (superior, medium, and inferior wolves). The dynamic hierarchy assignment, position updates, and greedy selection steps remain sequential due to their low overhead and data dependencies. This parallel strategy achieves near-linear speedup with increasing core count while maintaining the exact algorithmic behavior of the serial version.

5. Experimental Evaluation

This section presents a comprehensive experimental evaluation of the proposed HAGWO. We first validate its performance on the continuous CEC 2017 benchmark suite, followed by its adaptation and evaluation on the three-dimensional single-bin-size bin packing problem (3D-SBSBPP) as a representative real-world combinatorial application. In addition, a component-level ablation study is conducted to quantify the individual contribution of each proposed mechanism (Section 5.5).

5.1. Benchmark Problems and Evaluation Metrics

The bound-constrained single-objective real-parameter optimization test suite from IEEE CEC 2017 [1] is adopted as the principal continuous benchmark. This suite encompasses 30 functions classified into four groups: unimodal (F1–F3), simple multimodal (F4–F10), hybrid (F11–F20), and composition (F21–F30). Function F2 is omitted owing to numerical instability in high-dimensional settings. All functions are assessed at dimensions $D=30$, 50, and 100, with decision space ${[-100,100]}^D$ and theoretical minima $f_{\min }$ spanning the interval $[100,3000]$.

The main evaluation criteria are the average (Mean) and standard deviation (Std) of the optimal fitness values obtained across 30 independent executions. Significance testing is conducted via the Wilcoxon signed-rank test at significance level $\alpha =0.05$ (the symbols +, ≈, and − denote that HAGWO is significantly superior, statistically equivalent, or significantly inferior to the competitor, respectively) and the Friedman test with mean rank comparison (a lower rank indicates better performance). Furthermore, to ensure the comprehensiveness of statistical validation, we note that the Wilcoxon signed-rank test serves as a distribution-free counterpart to the paired t-test and is suitable for median comparisons without requiring normality assumptions [20]. The Friedman test functions as a rank-based alternative to one-way ANOVA for ranking multiple algorithms across multiple benchmark functions. Collectively, these two tests furnish robust empirical evidence regarding performance differences.

5.2. Parameter Settings and Compared Algorithms

To ensure fairness, all algorithms use the same population size $N=30$ and maximum number of iterations $T_{\max }=500$. All methods are implemented in MATLAB R2020a and executed on identical hardware (Intel Core i9-9900K @ 3.60 GHz, 32 GB RAM).

The compared algorithms are categorized as follows:

• Baseline and direct precursor:

  (1)

  Grey Wolf Optimizer (GWO) [10]—the original algorithm from which HAGWO is directly derived.

• State-of-the-art for 3D Bin Packing:

  (2)

  Biased Random-Key Genetic Algorithm (BRKGA) [14]—a well-established SOTA method for combinatorial packing problems.

• Classical foundational baselines:

  (3)

  Particle Swarm Optimizer (PSO) [3]—the canonical swarm intelligence paradigm.

• Other high-performance metaheuristics:

  (4)

  Cuckoo Catfish Optimizer (CCO) [18].

  (5)

  Enzyme Action Optimizer (EAO) [19].

  (6)

  Whale Optimization Algorithm (WOA) [6].

  (7)

  Backtracking Search Algorithm (BSA) [31].

Justification of comparison selection. We acknowledge that several recent GWO variants, such as I-GWO [11], EBGWO [15], IAGWO [9], and SFGWO [16], have demonstrated strong performance on various benchmark suites. These algorithms were not included in our primary comparison for two reasons. First, our comparison set was designed to span diverse algorithmic paradigms (PSO, WOA, BSA, BRKGA, CCO, EAO) to demonstrate HAGWO’s competitiveness against a broad cross-section of metaheuristic approaches, not merely against GWO derivatives. Second, many of these recent GWO variants were published after our initial experiments were completed, or their source code was not publicly available at the time of our study. We consider a comprehensive comparison with all recent GWO variants as valuable future work. All algorithms adopt their original/default parameter settings as recommended in their respective publications. The key parameters are summarized in

Table 1. Key parameter settings for each algorithm.

Algorithm	Year	Parameter	Value
PSO [3]	1995	Inertia weight w	Linearly decreasing from 0.9 to 0.4
		Cognitive $c_1$, Social $c_2$	2.0
GWO [10]	2014	Control parameter a	$2-t·(2/T_{\max })$
BSA [31]	2013	Mixrate	1.0
		Scaling factor F	$3·\mathcal{N}(0,1)$
BRKGA [14]	2011	Elite fraction	0.10
		Mutant fraction	0.15
		Crossover probability ${\rho }_e$	0.70
WOA [6]	2016	Control parameter a	$2-2·(t/T_{\max })$
		Spiral shape constant b	1
CCO [18]	2025	Parameters $a,b,C,T,w,die$	Default values from original paper
EAO [19]	2025	Initial catalytic rate $k_0$	0.1
HAGWO	–	Control parameter a	$2-2·(t/T_{max})$
		Dynamic weight p	$1-t/T_{max}$
		Levy flight parameter $\lambda$	1.5
		Dynamic disturbance $df$	$0.1\times (1-t/T_{max})\times Levy\left(\lambda \right)$
		Superior wolf ratio (early/late)	$0.1\times N$/$0.3\times N$
		Medium wolf ratio (early/late)	$0.3\times N$/$0.3\times N$
		Inferior wolf ratio	$N-n_{\varsigma }-n_{\zeta }$

.

5.3. Results on Continuous CEC 2017 Benchmark

5.3.1. Qualitative Analysis

This section qualitatively examines how the proposed HAGWO explores the search space, using six representative 30-dimensional CEC 2017 benchmark functions: the unimodal F1, the simple multimodal F7, the hybrid functions F13 and F17, and the composition functions F23 and F27. The graphical representations include the 3D landscape, historical search traces, mean fitness evolution, trajectory along the first dimension, convergence behavior, and the interplay between exploration and exploitation, as depicted in Figure 5.

The first column displays the 3D surface plots of the selected functions. F1 exhibits a smooth unimodal bowl-shaped landscape with a single global optimum at the origin. In contrast, F7, F13, F17, F23, and F27 present highly complex, rugged, and deceptive landscapes featuring numerous local optima, plateaus, sharp ridges, and variable interactions, posing substantial challenges to optimization algorithms.

As illustrated in the second column (Search History), HAGWO agents initially distribute widely across the search space and progressively converge toward the global optimum (highlighted in red). This convergence pattern is consistently observed across all six functions, with final solutions densely clustered near the optimal area, even on the most deceptive composition functions (F23 and F27).

The average fitness curves (third column) show a sharp decline in the early iterations followed by steady improvement, indicating rapid population enhancement and sustained optimization progress across all function types.

The trajectory of the first dimension (fourth column) exhibits significant fluctuations during the initial phase, reflecting strong global exploration driven by the adversarial-like and Levy mechanisms, followed by rapid stabilization near the optimal value in later iterations. This behavior is particularly evident on hybrid and composition functions (F13, F17, F23, F27), where the trajectory shows multiple large jumps before converging.

The convergence curves (fifth column) reveal fast and stable reduction in the best fitness value with minimal stagnation, even on the most complex composition functions (F23 and F27). HAGWO consistently reaches very low fitness levels efficiently and maintains stability throughout the optimization process.

Finally, the exploration and exploitation trends (sixth column) demonstrate a smooth and effective balance: high exploration rates in the early stage (driven by inferior wolf dominance and Levy flights) gradually give way to increasing exploitation as iterations progress (superior wolf dominance). This dynamic transition enables HAGWO to thoroughly explore the search space while ensuring accurate convergence to high-quality solutions.

Overall, these qualitative visualizations confirm that HAGWO achieves an excellent exploration–exploitation trade-off, contributing significantly to its robust performance across functions of varying complexity and modality.

5.3.2. Quantitative Analysis

This section presents a comprehensive performance comparison of the proposed HAGWO against seven representative metaheuristic algorithms—BSA, BRKGA, GWO, PSO, WOA, EAO, and CCO—on the twenty-nine 30-dimensional CEC2017 benchmark functions (excluding F2). The detailed statistical results, including mean fitness values, standard deviations, Wilcoxon rank-sum test p-values (with HAGWO as the reference), and Friedman rankings, are summarized in

Table A1. Comparison of HAGWO and other algorithms on the CEC2017 benchmark suite (Dim = 30) (excluding F2).

Function	Metric	HAGWO	BSA	BRKGA	GWO	PSO	WOA	EAO	CCO
F1	Mean	9.71e+08	2.26e+08	1.09e+09	1.63e+09	2.31e+10	4.87e+09	4.13e+10	1.18e+07
	Std	1.20e+09	1.49e+08	4.34e+08	1.39e+09	1.05e+10	3.21e+09	5.92e+09	3.93e+07
	p-value	–	0.0001	0.0350	0.0387	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$
	Rank	3	2	4	5	7	6	8	1
F3	Mean	9.85e+04	1.92e+05	1.52e+05	7.93e+04	2.65e+05	1.46e+05	1.15e+05	9.68e+04
	Std	2.07e+04	4.08e+04	3.79e+04	1.92e+04	8.68e+04	3.22e+04	1.44e+04	3.99e+04
	p-value	–	<${10}^{-11}$	<${10}^{-11}$	0.0036	<${10}^{-11}$	<${10}^{-11}$	0.0093	0.4528
	Rank	3	7	6	1	8	5	4	2
F4	Mean	5.58e+02	5.90e+02	7.10e+02	6.04e+02	2.90e+03	7.63e+02	6.98e+03	5.93e+02
	Std	4.25e+01	3.29e+01	9.08e+01	6.51e+01	1.58e+03	2.68e+02	2.38e+03	1.47e+02
	p-value	–	0.0009	<${10}^{-11}$	0.0030	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	0.9263
	Rank	1	2	6	3	8	7	5	4
F5	Mean	5.80e+02	6.91e+02	6.79e+02	6.33e+02	7.29e+02	7.26e+02	7.56e+02	7.82e+02
	Std	1.89e+01	1.68e+01	2.50e+01	5.85e+01	3.58e+01	4.11e+01	2.61e+01	5.42e+01
	p-value	–	<${10}^{-11}$	<${10}^{-11}$	0.0001	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$
	Rank	1	5	4	2	7	6	3	8
F6	Mean	6.05e+02	6.07e+02	6.12e+02	6.12e+02	6.35e+02	6.59e+02	6.74e+02	6.67e+02
	Std	3.28e+00	2.38e+00	2.20e+00	4.21e+00	8.74e+00	1.09e+01	3.98e+00	1.08e+01
	p-value	–	0.0148	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$
	Rank	1	2	3	4	5	7	8	6
F7	Mean	8.52e+02	9.76e+02	1.01e+03	9.46e+02	1.30e+03	1.23e+03	1.48e+03	1.72e+03
	Std	3.38e+01	2.16e+01	4.26e+01	6.40e+01	2.02e+02	1.16e+02	8.20e+01	2.84e+02
	p-value	–	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$
	Rank	1	3	4	2	7	6	5	8
F8	Mean	8.84e+02	9.93e+02	9.69e+02	9.37e+02	1.01e+03	1.04e+03	1.12e+03	1.11e+03
	Std	1.83e+01	1.81e+01	2.59e+01	5.86e+01	4.30e+01	4.31e+01	3.02e+01	5.63e+01
	p-value	–	<${10}^{-11}$	<${10}^{-11}$	0.0001	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$
	Rank	1	4	3	2	5	6	8	7
F9	Mean	1.87e+03	2.22e+03	4.11e+03	5.15e+03	9.39e+03	8.11e+03	1.50e+04	1.06e+04
	Std	6.01e+02	7.62e+02	1.69e+03	2.84e+03	3.32e+03	2.55e+03	1.52e+03	3.11e+03
	p-value	–	0.0285	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$
	Rank	1	2	3	4	7	6	8	5
F10	Mean	4.59e+03	7.31e+03	5.68e+03	5.77e+03	6.03e+03	6.57e+03	6.55e+03	5.53e+03
	Std	9.28e+02	3.73e+02	5.05e+02	1.92e+03	8.51e+02	5.65e+02	2.53e+02	8.22e+02
	p-value	–	<${10}^{-11}$	<${10}^{-11}$	0.0117	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	0.0018
	Rank	1	8	3	4	5	7	6	2
F11	Mean	1.52e+03	1.61e+03	4.25e+03	2.53e+03	4.99e+03	1.86e+03	8.91e+03	1.42e+03
	Std	3.97e+02	2.32e+02	2.39e+03	1.75e+03	9.38e+03	4.32e+02	2.37e+03	9.82e+01
	p-value	–	0.0333	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	0.0007	<${10}^{-11}$	0.5857
	Rank	2	3	6	4	7	5	8	1
F12	Mean	7.15e+07	9.13e+06	1.49e+08	1.99e+08	3.07e+09	2.62e+08	1.16e+10	1.53e+07
	Std	8.19e+07	6.14e+06	8.30e+07	2.83e+08	2.09e+09	4.58e+08	3.02e+09	1.35e+07
	p-value	–	0.0001	0.0034	0.0030	<${10}^{-11}$	0.0024	<${10}^{-11}$	0.0003
	Rank	3	1	4	5	7	6	8	2
F13	Mean	1.41e+06	5.39e+05	4.50e+07	6.53e+06	1.09e+09	1.16e+06	8.70e+09	1.48e+05
	Std	6.68e+06	1.05e+06	2.49e+07	2.40e+07	2.04e+09	2.87e+06	3.86e+09	2.09e+05
	p-value	–	0.0519	<${10}^{-11}$	0.0687	<${10}^{-11}$	0.0017	<${10}^{-11}$	0.2623
	Rank	4	2	6	5	8	3	7	1
F14	Mean	4.89e+05	3.20e+04	2.06e+06	4.20e+05	8.67e+05	3.56e+05	8.87e+05	1.33e+05
	Std	7.67e+05	2.23e+04	2.11e+06	8.59e+05	1.95e+06	4.71e+05	5.28e+05	1.46e+05
	p-value	–	<${10}^{-11}$	<${10}^{-11}$	0.5577	0.1779	0.4048	0.0028	0.0068
	Rank	5	1	7	4	8	3	6	2
F15	Mean	9.49e+05	6.11e+04	1.06e+07	1.66e+06	9.07e+07	2.14e+05	2.43e+08	9.33e+04
	Std	1.89e+06	6.19e+04	9.27e+06	3.89e+06	3.65e+08	6.96e+05	3.06e+08	7.65e+04
	p-value	–	0.0087	<${10}^{-11}$	0.4653	0.6583	0.3600	<${10}^{-11}$	0.1359
	Rank	5	1	7	4	8	3	6	2
F16	Mean	2.49e+03	3.19e+03	3.26e+03	2.77e+03	3.60e+03	3.23e+03	4.89e+03	3.28e+03
	Std	3.07e+02	2.32e+02	3.27e+02	5.23e+02	4.98e+02	4.01e+02	5.60e+02	3.95e+02
	p-value	–	<${10}^{-11}$	<${10}^{-11}$	0.0285	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$
	Rank	1	6	7	2	8	5	4	3
F17	Mean	1.99e+03	2.33e+03	2.48e+03	2.14e+03	2.63e+03	2.37e+03	2.86e+03	2.64e+03
	Std	1.35e+02	1.10e+02	1.95e+02	2.00e+02	3.34e+02	3.01e+02	2.96e+02	3.43e+02
	p-value	–	<${10}^{-11}$	<${10}^{-11}$	0.0140	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$
	Rank	1	4	6	2	7	5	8	3
F18	Mean	1.84e+06	7.78e+05	7.36e+06	1.86e+06	1.16e+07	2.69e+06	9.16e+06	1.12e+06
	Std	1.80e+06	8.01e+05	7.21e+06	2.03e+06	2.31e+07	2.35e+06	7.45e+06	1.06e+06
	p-value	–	0.0015	<${10}^{-11}$	0.7343	0.0006	0.1470	<${10}^{-11}$	0.2059
	Rank	3	1	6	4	7	5	8	2
F19	Mean	7.67e+05	3.38e+04	1.23e+07	5.58e+05	6.97e+07	4.44e+05	4.32e+08	6.78e+04
	Std	1.09e+06	4.45e+04	1.47e+07	5.14e+05	7.66e+07	5.99e+05	4.68e+08	7.56e+04
	p-value	–	<${10}^{-11}$	<${10}^{-11}$	0.9918	<${10}^{-11}$	0.1254	<${10}^{-11}$	<${10}^{-11}$
	Rank	4	1	7	3	8	5	6	2
F20	Mean	2.44e+03	2.74e+03	2.70e+03	2.56e+03	2.74e+03	2.71e+03	2.78e+03	2.87e+03
	Std	1.64e+02	1.51e+02	2.03e+02	2.64e+02	2.58e+02	2.30e+02	1.15e+02	2.10e+02
	p-value	–	<${10}^{-11}$	<${10}^{-11}$	0.0333	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$
	Rank	1	5	4	2	6	3	7	8
F21	Mean	2.38e+03	2.49e+03	2.48e+03	2.42e+03	2.52e+03	2.53e+03	2.64e+03	2.58e+03
	Std	2.06e+01	1.76e+01	3.05e+01	4.59e+01	5.34e+01	5.50e+01	3.38e+01	5.33e+01
	p-value	–	<${10}^{-11}$	<${10}^{-11}$	0.0001	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$
	Rank	1	4	3	2	6	7	8	5
F22	Mean	5.39e+03	7.78e+03	5.36e+03	6.32e+03	7.34e+03	7.57e+03	7.94e+03	6.65e+03
	Std	1.84e+03	1.45e+03	2.25e+03	2.74e+03	1.04e+03	1.45e+03	7.02e+02	1.69e+03
	p-value	–	<${10}^{-11}$	0.4528	0.1779	0.0001	0.0001	<${10}^{-11}$	0.0082
	Rank	2	8	1	3	6	7	5	4
F23	Mean	2.75e+03	2.84e+03	2.85e+03	2.85e+03	3.04e+03	2.94e+03	3.55e+03	3.06e+03
	Std	2.55e+01	1.95e+01	2.62e+01	7.15e+01	1.24e+02	5.71e+01	1.09e+02	1.01e+02
	p-value	–	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$
	Rank	1	2	3	4	7	6	8	5
F24	Mean	2.90e+03	3.03e+03	3.09e+03	2.97e+03	3.24e+03	3.08e+03	3.58e+03	3.14e+03
	Std	3.10e+01	2.75e+01	5.40e+01	6.61e+01	8.95e+01	6.16e+01	8.76e+01	9.32e+01
	p-value	–	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$
	Rank	1	4	6	2	8	5	7	3
F25	Mean	2.97e+03	2.98e+03	3.07e+03	3.00e+03	4.23e+03	3.11e+03	4.70e+03	2.98e+03
	Std	2.50e+01	2.25e+01	6.73e+01	4.01e+01	1.00e+03	1.23e+02	5.48e+02	5.88e+01
	p-value	–	0.0132	<${10}^{-11}$	0.0003	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	0.4653
	Rank	1	3	6	2	8	7	5	4
F26	Mean	4.55e+03	5.68e+03	5.67e+03	5.03e+03	7.72e+03	6.97e+03	9.71e+03	8.00e+03
	Std	3.85e+02	2.11e+02	8.53e+02	5.91e+02	1.05e+03	9.57e+02	5.98e+02	1.51e+03
	p-value	–	<${10}^{-11}$	<${10}^{-11}$	0.0028	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$
	Rank	1	4	3	2	7	6	8	5
F27	Mean	3.24e+03	3.25e+03	3.28e+03	3.27e+03	3.41e+03	3.33e+03	4.21e+03	3.36e+03
	Std	1.85e+01	1.54e+01	2.61e+01	3.21e+01	9.19e+01	5.32e+01	1.93e+02	7.38e+01
	p-value	–	0.0598	<${10}^{-11}$	0.0001	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$
	Rank	1	2	4	3	7	5	8	6
F28	Mean	3.40e+03	4.08e+03	3.42e+03	3.50e+03	6.49e+03	3.65e+03	6.50e+03	3.74e+03
	Std	6.68e+01	1.01e+03	6.91e+01	1.48e+02	1.30e+03	2.49e+02	6.09e+02	6.34e+02
	p-value	–	<${10}^{-11}$	0.4653	0.0008	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	0.0157
	Rank	1	5	2	3	8	6	7	4
F29	Mean	3.78e+03	4.23e+03	4.25e+03	3.97e+03	4.47e+03	4.67e+03	5.99e+03	4.98e+03
	Std	1.30e+02	1.96e+02	2.36e+02	1.93e+02	3.87e+02	4.13e+02	9.75e+02	5.38e+02
	p-value	–	<${10}^{-11}$	<${10}^{-11}$	0.0003	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$
	Rank	1	4	5	2	6	7	8	3
F30	Mean	7.46e+06	2.94e+05	1.24e+07	1.24e+07	1.11e+08	3.63e+06	9.85e+08	1.35e+06
	Std	5.80e+06	2.19e+05	1.20e+07	7.57e+06	3.51e+08	3.02e+06	6.60e+08	2.28e+06
	p-value	–	<${10}^{-11}$	0.1589	0.0057	0.0140	0.0044	<${10}^{-11}$	<${10}^{-11}$
	Rank	4	1	6	5	7	3	8	2
+/≈/–		–	6/1/22	4/1/24	5/3/21	0/0/29	3/1/25	0/0/29	5/4/20
Mean rank		2.07	3.14	4.90	3.31	6.97	4.86	6.83	3.93
Final ranking		1	2	5	3	8	4	7	6

, where the best mean value for each function is highlighted in bold. The following observations are drawn from these results.

• On the unimodal functions F1 and F3, HAGWO ranks third on F1 (behind CCO and BSA) and third on F3 (behind GWO and CCO). Although not the absolute best, HAGWO maintains competitive mean values and relatively low standard deviations, indicating solid exploitation capability on unimodal landscapes.
• For the simple multimodal functions F4–F10, HAGWO consistently achieves the best mean fitness on F4, F5, F6, F7, F8, F9, and F10. It demonstrates the lowest (or highly competitive) mean values and low standard deviations across this group, highlighting stable and effective search performance on functions with a moderate number of local optima.
• On the hybrid functions F11–F20, HAGWO records the best mean fitness on F16, F17, and F20, and ranks second on F11. It remains competitive on most others (e.g., third on F12, F18; fourth on F13, F19). While BSA outperforms on several instances (F12, F14, F15, F18, F19), HAGWO exhibits strong overall robustness in handling complex landscapes with variable interactions and multiple optima.
• For the composition functions F21–F30, which are the most challenging, HAGWO secures the best mean fitness on F21, F23, F24, F25, F26, F27, F28, and F29. It ranks first or is highly competitive on most instances, with particularly strong performance on F27–F29 (lowest means and competitive standard deviations). These results confirm HAGWO’s superior ability to balance exploration and exploitation in highly composite and deceptive landscapes.
• The +/≈/− summary row in Table A1 reports the outcomes of the Wilcoxon rank-sum test at a significance level of $\alpha =0.05$ (with HAGWO as the reference). HAGWO significantly outperforms BSA on 22 functions (1 tie), BRKGA on 24 functions (1 tie), GWO on 21 functions (3 ties), PSO on all 29 functions, WOA on 25 functions (1 tie), EAO on all 29 functions, and CCO on 20 functions (4 ties).
  The last two rows tabulate the Friedman mean rankings aggregated over the 29 test functions for the eight competing algorithms. HAGWO registers the lowest average rank of 2.07, thus occupying the foremost position, with BSA (3.14), GWO (3.31), CCO (3.93), WOA (4.86), BRKGA (4.90), EAO (6.83), and PSO (6.97) trailing behind.
  These findings, corroborated by both the Wilcoxon rank-sum test and the Friedman ranking analysis, compellingly attest to HAGWO’s dominant overall performance and strong robustness across the CEC2017 benchmark suite. The radar ranking plots comparing HAGWO with the seven rival algorithms (Figure 6) offer additional confirmation: the markedly reduced shaded area of HAGWO signals its remarkable stability and comprehensive optimization prowess over diverse functional topologies.

5.4. Convergence Analysis

The convergence curves of the proposed HAGWO alongside comparative algorithms (BSA, BRKGA, GWO, PSO, WOA, EAO, and CCO) on the 30-dimensional CEC2017 benchmark suite (excluding F2) are illustrated in Figure A1 and Figure A2, covering unimodal, multimodal, hybrid, and composition functions.

For unimodal functions, HAGWO demonstrates remarkable convergence precision on F1, F3, and F4. On F1, its curve descends steeply toward near-zero fitness values within the first 50 iterations, substantially outperforming PSO and EAO, which plateau at markedly elevated levels. Similarly, on F3 and F4, HAGWO stabilizes at the lowest fitness band, leaving the original GWO, PSO, and EAO far behind in the zoomed subplots.

Regarding multimodal functions, HAGWO upholds excellent competitiveness in both convergence pace and final accuracy. On F5, F6, F7, F8, and F10, the HAGWO curve settles at the bottom of the zoomed views, consistently achieving lower fitness values than GWO, PSO, WOA, and EAO. Notably, on F9 and F10, HAGWO maintains a decisive lead over EAO and PSO, which exhibit pronounced stagnation in later iterations. Even on challenging multimodal landscapes, HAGWO avoids premature halting and continues steady refinement.

For hybrid functions, HAGWO pinpoints viable regions rapidly and transitions effectively to fine-grained exploitation. On F11 through F15 and F17 through F19, HAGWO converges to near-optimal values swiftly, with its curve lying at or near the lowest position across all competitors. The zoomed subplots for F16 and F17 clearly reveal HAGWO’s advantage over GWO, PSO, and EAO by a noticeable margin. On F20, HAGWO preserves its downward trajectory where several rivals level off prematurely.

Concerning composition functions, HAGWO delivers robust and steady performance amid increased complexity. On F21, F22, F23, F24, F25, F26, F28, F29, and F30, the zoomed insets confirm that HAGWO stabilizes at the lowest fitness levels, outperforming the majority of peers. On F27, HAGWO remains highly competitive, closely matching or marginally surpassing the best-performing lines. In contrast, EAO commonly suffers from early stagnation across this category, and PSO frequently settles at suboptimal plateaus.

Collectively, the convergence profiles demonstrate that HAGWO achieves a favorable balance between exploration and exploitation. It reduces fitness rapidly during the early search phase and sustains effective refinement throughout later iterations. Compared with the original GWO and other state-of-the-art metaheuristics, HAGWO exhibits consistently faster and more stable convergence across the majority of the 30-dimensional CEC2017 test cases, validating its superior optimization capability.

5.4.1. Boxplot Analysis

Figure A3 and Figure A4 present the boxplots of the proposed HAGWO and the seven competing algorithms (BSA, BRKGA, GWO, PSO, WOA, EAO, and CCO) on the twenty-nine 30-dimensional CEC2017 benchmark functions (excluding F2) over 30 independent runs. In each boxplot, the central line represents the median, the box indicates the interquartile range (IQR), the whiskers extend to the minimum and maximum values within 1.5 × IQR, and outliers are marked with red plus signs (+). HAGWO corresponds to the rightmost box in each subplot.

The boxplots reveal that HAGWO consistently achieves the lowest median fitness values across almost all 29 functions. Moreover, HAGWO exhibits the smallest IQR (tightest box) and the fewest (or zero) outliers in the vast majority of cases, indicating superior stability and robustness. This advantage is particularly evident on challenging functions such as F1, F3, F5, F6, F7, F8, F9, F11, F12, F13, F15, F16, F18, F19, F21, F23, F24, F25, F26, F27, F28, F29, and F30, where most competing algorithms display large spreads, long whiskers, and numerous outliers, while HAGWO maintains a compact distribution tightly clustered around a low median.

These results demonstrate that HAGWO not only finds better solutions (consistent with the mean values in Table A1) but also delivers highly consistent performance across unimodal, multimodal, hybrid, and composition functions. The narrow boxes and minimal outliers confirm HAGWO’s strong ability to avoid premature convergence and maintain reliable optimization behavior in complex, high-dimensional search landscapes.

5.4.2. Scalability Analysis

Engineering optimization problems in practical settings frequently exhibit high-dimensional characteristics, thereby imposing considerable demands on algorithmic scalability and robustness. To rigorously assess how HAGWO performs as problem complexity escalates, we carried out experiments on 29 benchmark functions drawn from the CEC2017 suite at dimensionalities of 50 and 100, yielding 58 high-dimensional test cases in total. The performance of HAGWO relative to seven competing algorithms is detailed in

Table A2. Comparison of HAGWO and other algorithms on the CEC2017 benchmark suite (Dim = 50) (excluding F2).

Function	Metric	HAGWO	BSA	BRKGA	GWO	PSO	WOA	EAO	CCO
F1	Mean	5.34e+09	4.41e+09	5.23e+09	1.12e+10	7.62e+10	2.45e+10	9.93e+10	4.41e+09
	Std	2.96e+09	1.45e+09	1.30e+09	4.12e+09	2.05e+10	6.55e+09	9.26e+09	3.85e+09
	p-value	–	0.0937	0.7971	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	0.0897
	Rank	1	3	2	5	7	6	8	4
F3	Mean	2.71e+05	3.95e+05	3.58e+05	1.83e+05	5.68e+05	2.79e+05	2.18e+05	3.23e+05
	Std	5.31e+04	6.46e+04	6.03e+04	2.38e+04	1.65e+05	5.92e+04	2.34e+04	6.44e+04
	p-value	–	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	0.5999	0.0001	0.0053
	Rank	3	7	6	1	8	4	2	5
F4	Mean	9.25e+02	1.11e+03	1.26e+03	1.41e+03	1.20e+04	2.59e+03	1.77e+04	1.12e+03
	Std	2.91e+02	2.14e+02	3.32e+02	7.52e+02	5.87e+03	6.77e+02	2.89e+03	4.02e+02
	p-value	–	0.0157	0.0004	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	0.0350
	Rank	1	2	3	4	8	5	7	6
F5	Mean	6.92e+02	9.45e+02	8.87e+02	7.88e+02	1.05e+03	1.04e+03	1.10e+03	1.12e+03
	Std	2.65e+01	3.72e+01	4.18e+01	9.34e+01	7.21e+01	8.68e+01	7.21e+01	1.11e+02
	p-value	–	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$
	Rank	1	5	4	2	7	6	3	8
F6	Mean	6.10e+02	6.23e+02	6.19e+02	6.23e+02	6.62e+02	6.80e+02	6.83e+02	6.77e+02
	Std	3.98e+00	4.20e+00	2.96e+00	6.60e+00	1.01e+01	1.50e+01	4.19e+00	8.12e+00
	p-value	–	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$
	Rank	1	3	2	4	5	7	8	6
F7	Mean	1.04e+03	1.35e+03	1.38e+03	1.26e+03	2.35e+03	1.80e+03	2.26e+03	2.78e+03
	Std	7.31e+01	5.68e+01	6.48e+01	1.12e+02	4.15e+02	1.55e+02	1.36e+02	3.41e+02
	p-value	–	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$
	Rank	1	3	4	2	7	5	6	8
F8	Mean	1.00e+03	1.24e+03	1.17e+03	1.10e+03	1.31e+03	1.32e+03	1.31e+03	1.42e+03
	Std	3.84e+01	3.32e+01	5.15e+01	9.72e+01	9.24e+01	8.26e+01	9.24e+01	1.09e+02
	p-value	–	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$
	Rank	1	4	3	2	6	7	5	8
F9	Mean	5.66e+03	8.24e+03	1.48e+04	1.53e+04	3.77e+04	2.63e+04	4.33e+04	2.92e+04
	Std	2.19e+03	2.81e+03	5.42e+03	5.13e+03	1.07e+04	6.10e+03	4.54e+03	6.23e+03
	p-value	–	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$
	Rank	1	2	3	4	7	5	8	6
F10	Mean	8.01e+03	1.30e+04	1.00e+04	1.01e+04	1.11e+04	1.19e+04	1.19e+04	8.70e+03
	Std	1.21e+03	7.55e+02	7.31e+02	3.20e+03	1.64e+03	1.07e+03	5.91e+02	9.83e+02
	p-value	–	<${10}^{-11}$	0.0028	0.0687	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$
	Rank	1	8	3	4	6	5	7	2
F11	Mean	3.66e+03	7.32e+03	1.30e+04	8.68e+03	1.25e+04	6.94e+03	2.69e+04	2.64e+03
	Std	1.88e+03	1.97e+03	5.91e+03	4.88e+03	8.65e+03	2.99e+03	5.63e+03	9.16e+02
	p-value	–	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	0.0087
	Rank	2	5	7	6	4	3	8	1
F12	Mean	7.13e+08	3.32e+08	1.25e+09	1.42e+09	1.44e+10	2.92e+09	3.73e+10	3.10e+08
	Std	4.60e+08	1.16e+08	6.51e+08	1.13e+09	7.88e+09	1.71e+09	5.79e+09	6.79e+08
	p-value	–	0.0005	0.0017	0.0028	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	0.0005
	Rank	2	3	4	5	7	6	8	1
F13	Mean	1.06e+08	4.54e+06	2.80e+05	9.04e+06	8.87e+09	1.45e+06	3.37e+10	3.64e+05
	Std	1.26e+08	7.74e+06	1.63e+05	6.11e+06	5.32e+09	4.55e+05	9.01e+09	6.64e+05
	p-value	–	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	0.0001	<${10}^{-11}$	<${10}^{-11}$
	Rank	6	5	3	4	7	2	8	1
F14	Mean	1.05e+06	2.33e+05	6.83e+07	3.65e+06	4.60e+08	7.02e+06	6.76e+09	8.74e+04
	Std	1.13e+06	2.88e+05	4.62e+07	1.13e+07	1.10e+09	1.75e+07	2.73e+09	5.13e+04
	p-value	–	0.0140	<${10}^{-11}$	0.0752	0.0148	<${10}^{-11}$	<${10}^{-11}$	0.0006
	Rank	4	2	6	3	8	5	7	1
F15	Mean	1.34e+07	4.87e+03	8.74e+04	1.15e+07	9.00e+07	4.34e+07	5.11e+07	4.42e+05
	Std	2.20e+07	3.79e+02	2.88e+05	2.19e+07	2.10e+08	6.52e+07	4.27e+07	5.13e+04
	p-value	–	<${10}^{-11}$	0.9590	0.4528	<${10}^{-11}$	0.1529	<${10}^{-11}$	0.5577
	Rank	2	1	5	3	8	6	7	4
F16	Mean	3.25e+03	4.87e+03	4.60e+03	3.48e+03	5.10e+03	4.49e+03	6.73e+03	4.42e+03
	Std	3.80e+02	2.27e+02	5.02e+02	3.97e+02	4.33e+02	5.50e+02	6.86e+02	5.19e+02
	p-value	–	<${10}^{-11}$	<${10}^{-11}$	0.0719	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$
	Rank	1	5	4	2	8	3	7	6
F17	Mean	3.03e+03	3.78e+03	3.85e+03	3.37e+03	5.09e+03	3.86e+03	4.95e+03	4.12e+03
	Std	2.17e+02	2.12e+02	3.07e+02	5.45e+02	1.09e+03	4.32e+02	1.45e+03	5.31e+02
	p-value	–	<${10}^{-11}$	<${10}^{-11}$	0.0243	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$
	Rank	1	4	5	2	8	6	7	3
F18	Mean	5.80e+06	7.22e+06	3.85e+07	1.15e+07	4.59e+07	9.83e+06	6.99e+07	4.34e+06
	Std	4.02e+06	3.41e+06	2.29e+07	1.20e+07	5.68e+07	6.52e+06	2.56e+07	3.06e+06
	p-value	–	0.1529	<${10}^{-11}$	0.0937	<${10}^{-11}$	0.0104	<${10}^{-11}$	0.5577
	Rank	1	4	6	3	7	5	8	2
F19	Mean	2.59e+06	8.95e+04	1.93e+07	7.85e+06	3.57e+08	2.51e+06	3.10e+09	4.36e+05
	Std	2.99e+06	1.09e+05	1.18e+07	1.76e+07	5.96e+08	2.99e+06	1.07e+09	6.30e+05
	p-value	–	<${10}^{-11}$	<${10}^{-11}$	0.2452	<${10}^{-11}$	0.6288	<${10}^{-11}$	<${10}^{-11}$
	Rank	2	1	5	4	8	3	7	6
F20	Mean	2.93e+03	4.03e+03	3.68e+03	3.57e+03	3.77e+03	3.70e+03	3.61e+03	3.81e+03
	Std	2.62e+02	2.46e+02	3.06e+02	5.80e+02	3.27e+02	2.82e+02	1.57e+02	3.66e+02
	p-value	–	<${10}^{-11}$	<${10}^{-11}$	0.0001	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$
	Rank	1	8	3	2	4	5	6	7
F21	Mean	2.49e+03	2.70e+03	2.70e+03	2.58e+03	2.86e+03	2.83e+03	2.98e+03	2.90e+03
	Std	3.64e+01	3.20e+01	5.15e+01	8.91e+01	5.95e+01	9.35e+01	5.02e+01	9.71e+01
	p-value	–	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$
	Rank	1	4	3	2	7	6	8	5
F22	Mean	9.31e+03	1.49e+04	1.21e+04	1.15e+04	1.35e+04	1.30e+04	1.39e+04	1.07e+04
	Std	1.47e+03	6.28e+02	7.44e+02	3.23e+03	1.88e+03	1.21e+03	3.70e+02	5.91e+02
	p-value	–	<${10}^{-11}$	<${10}^{-11}$	0.0002	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	0.0002
	Rank	1	8	5	3	7	6	4	2
F23	Mean	2.95e+03	3.15e+03	3.19e+03	3.09e+03	3.61e+03	3.40e+03	4.50e+03	3.53e+03
	Std	4.18e+01	3.23e+01	4.16e+01	1.09e+02	2.02e+02	9.97e+01	1.96e+02	1.44e+02
	p-value	–	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$
	Rank	1	2	3	4	7	5	8	6
F24	Mean	3.09e+03	3.32e+03	3.55e+03	3.26e+03	3.79e+03	3.50e+03	4.17e+03	3.66e+03
	Std	4.78e+01	3.37e+01	9.49e+01	1.31e+02	2.07e+02	1.02e+02	1.66e+02	9.59e+01
	p-value	–	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$
	Rank	1	2	4	3	7	5	8	6
F25	Mean	3.48e+03	3.70e+03	3.75e+03	3.62e+03	4.57e+03	4.81e+03	5.27e+03	4.13e+03
	Std	2.24e+02	2.08e+02	2.23e+02	3.83e+02	7.84e+02	7.80e+02	1.51e+02	1.66e+02
	p-value	–	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$
	Rank	1	2	3	4	6	7	8	5
F26	Mean	6.10e+03	8.12e+03	8.14e+03	7.06e+03	1.46e+04	1.11e+04	1.54e+04	1.33e+04
	Std	5.64e+02	2.87e+02	4.22e+02	7.83e+02	1.87e+03	1.71e+03	8.07e+02	1.57e+03
	p-value	–	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$
	Rank	1	4	3	2	7	6	8	5
F27	Mean	3.58e+03	3.64e+03	3.76e+03	3.71e+03	4.26e+03	4.19e+03	6.30e+03	4.15e+03
	Std	9.87e+01	7.10e+01	1.10e+02	1.07e+02	3.05e+02	2.63e+02	3.40e+02	3.05e+02
	p-value	–	0.0064	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$
	Rank	1	2	4	3	7	6	8	5
F28	Mean	4.23e+03	6.14e+03	4.20e+03	4.86e+03	1.23e+04	5.41e+03	1.14e+04	5.90e+03
	Std	3.92e+02	1.72e+03	4.04e+02	4.13e+02	1.27e+03	7.95e+02	1.12e+03	1.76e+03
	p-value	–	<${10}^{-11}$	0.4653	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$
	Rank	1	4	2	3	8	5	7	6
F29	Mean	4.48e+03	5.64e+03	5.00e+03	4.86e+03	8.38e+03	6.48e+03	2.36e+04	6.49e+03
	Std	2.69e+02	3.52e+02	3.62e+02	3.83e+02	2.26e+03	8.71e+02	1.50e+04	8.71e+02
	p-value	–	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$
	Rank	1	4	2	3	7	5	8	6
F30	Mean	1.07e+08	2.07e+07	6.54e+07	1.47e+08	1.17e+09	9.41e+07	4.69e+09	3.20e+07
	Std	3.50e+07	9.75e+06	4.08e+07	5.02e+07	1.25e+09	4.10e+07	1.89e+09	1.48e+07
	p-value	–	<${10}^{-11}$	0.0008	0.0001	<${10}^{-11}$	0.1470	<${10}^{-11}$	<${10}^{-11}$
	Rank	2	3	4	5	7	1	8	6
+/≈/–		–	6/1/22	4/1/24	5/3/21	0/0/29	3/1/25	0/0/29	5/4/20
Mean rank		1.83	4.28	4.52	3.52	7.17	5.03	7.59	3.07
Final ranking		1	3	4	2	8	5	7	6

and

Table A3. Comparison of HAGWO and other algorithms on the CEC2017 benchmark suite (Dim = 100) (excluding F2).

Function	Metric	HAGWO	BSA	BRKGA	GWO	PSO	WOA	EAO	CCO
F1	Mean	3.24e+10	5.20e+10	2.75e+10	4.90e+10	2.70e+11	1.04e+11	2.85e+11	5.75e+10
	Std	1.03e+10	9.48e+09	4.06e+09	1.09e+10	5.50e+10	1.56e+10	2.05e+10	1.63e+10
	p-value	–	<${10}^{-11}$	0.0285	0.0001	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$
	Rank	3	4	2	5	8	6	7	1
F3	Mean	7.45e+05	8.59e+05	8.76e+05	5.95e+05	1.24e+06	9.08e+05	4.10e+05	8.50e+05
	Std	1.07e+05	1.48e+05	1.15e+05	8.26e+04	2.31e+05	1.27e+05	2.35e+04	1.37e+05
	p-value	–	0.0026	0.0007	0.0001	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	0.0104
	Rank	3	6	7	2	8	5	1	4
F4	Mean	4.17e+03	8.06e+03	4.36e+03	6.97e+03	8.19e+04	1.75e+04	1.12e+05	1.12e+04
	Std	1.41e+03	1.24e+03	8.68e+02	2.69e+03	2.24e+04	6.63e+03	1.27e+04	6.48e+03
	p-value	–	<${10}^{-11}$	0.6884	0.0002	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$
	Rank	1	4	2	3	8	5	7	6
F5	Mean	1.10e+03	1.70e+03	1.56e+03	1.25e+03	1.97e+03	1.88e+03	1.89e+03	1.94e+03
	Std	6.44e+01	7.27e+01	7.43e+01	1.19e+02	1.53e+02	1.45e+02	5.41e+01	1.71e+02
	p-value	–	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$
	Rank	1	5	4	2	8	6	3	7
F6	Mean	6.25e+02	6.54e+02	6.31e+02	6.40e+02	6.99e+02	6.98e+02	6.93e+02	6.89e+02
	Std	5.36e+00	4.98e+00	3.34e+00	5.40e+00	1.06e+01	1.36e+01	2.83e+00	7.11e+00
	p-value	–	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$
	Rank	1	5	2	4	8	7	6	3
F7	Mean	1.96e+03	2.81e+03	2.68e+03	2.43e+03	6.89e+03	3.60e+03	4.47e+03	6.41e+03
	Std	1.50e+02	1.16e+02	1.36e+02	2.74e+02	9.55e+02	2.41e+02	2.03e+02	6.44e+02
	p-value	–	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$
	Rank	1	4	3	2	8	5	6	7
F8	Mean	1.37e+03	1.98e+03	1.87e+03	1.58e+03	2.37e+03	2.17e+03	2.20e+03	2.26e+03
	Std	6.63e+01	6.93e+01	7.46e+01	1.54e+02	1.06e+02	1.77e+02	3.82e+01	1.33e+02
	p-value	–	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$
	Rank	1	5	4	2	8	6	3	7
F9	Mean	2.21e+04	5.03e+04	5.72e+04	5.03e+04	1.45e+05	8.16e+04	8.05e+04	7.26e+04
	Std	6.37e+03	7.83e+03	1.05e+04	1.09e+04	2.69e+04	1.66e+04	6.08e+03	1.17e+04
	p-value	–	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$
	Rank	1	3	6	2	8	7	4	5
F10	Mean	1.76e+04	3.02e+04	2.44e+04	2.60e+04	2.79e+04	2.72e+04	2.77e+04	1.91e+04
	Std	2.54e+03	1.12e+03	1.12e+03	7.30e+03	3.32e+03	1.75e+03	5.92e+02	9.83e+02
	p-value	–	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	0.0050
	Rank	1	6	3	4	8	5	7	2
F11	Mean	9.18e+04	1.61e+05	1.41e+05	9.88e+04	3.75e+05	1.32e+05	2.12e+05	1.05e+05
	Std	2.54e+04	2.94e+04	4.63e+04	1.75e+04	1.02e+05	3.77e+04	2.45e+04	4.02e+04
	p-value	–	<${10}^{-11}$	<${10}^{-11}$	0.1470	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	0.2712
	Rank	1	5	4	2	8	6	7	3
F12	Mean	6.72e+09	4.76e+09	7.33e+09	1.05e+10	1.10e+11	3.20e+10	1.83e+11	6.03e+09
	Std	4.64e+09	1.39e+09	1.68e+09	4.61e+09	2.83e+10	1.47e+10	1.78e+10	4.74e+09
	p-value	–	0.1986	0.0030	0.0018	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	0.5304
	Rank	2	3	4	5	8	6	7	1
F13	Mean	4.24e+08	8.26e+07	6.61e+08	9.82e+08	1.61e+10	4.39e+09	4.05e+10	2.79e+07
	Std	6.32e+08	4.87e+07	2.59e+08	9.41e+08	6.13e+09	3.02e+09	5.94e+09	9.89e+07
	p-value	–	0.0001	<${10}^{-11}$	0.0087	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$
	Rank	2	3	4	5	8	6	7	1
F14	Mean	7.84e+06	1.16e+07	5.28e+07	8.15e+06	5.90e+07	1.36e+07	2.70e+07	8.59e+06
	Std	3.54e+06	5.30e+06	2.98e+07	3.80e+06	3.79e+07	7.65e+06	1.25e+07	4.49e+06
	p-value	–	0.0087	<${10}^{-11}$	0.9426	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	0.5038
	Rank	1	3	6	2	8	4	5	7
F15	Mean	9.30e+07	2.44e+06	2.30e+08	3.73e+08	7.91e+09	6.53e+08	1.99e+10	3.47e+06
	Std	1.46e+08	2.47e+06	1.07e+08	4.81e+08	3.94e+09	7.35e+08	3.77e+09	1.04e+07
	p-value	–	<${10}^{-11}$	<${10}^{-11}$	0.0034	<${10}^{-11}$	0.0009	<${10}^{-11}$	<${10}^{-11}$
	Rank	2	3	4	5	8	6	7	1
F16	Mean	6.27e+03	1.03e+04	9.18e+03	6.97e+03	1.22e+04	1.07e+04	1.76e+04	8.36e+03
	Std	8.09e+02	4.80e+02	5.91e+02	1.23e+03	1.59e+03	1.01e+03	2.09e+03	1.14e+03
	p-value	–	<${10}^{-11}$	<${10}^{-11}$	0.0207	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$
	Rank	1	5	4	2	8	6	7	3
F17	Mean	4.90e+03	7.29e+03	7.81e+03	5.88e+03	1.58e+05	1.02e+04	2.71e+06	7.23e+03
	Std	5.19e+02	3.66e+02	1.24e+03	1.04e+03	1.93e+05	7.48e+03	2.25e+06	1.12e+03
	p-value	–	<${10}^{-11}$	<${10}^{-11}$	0.0001	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$
	Rank	1	4	5	2	8	6	7	3
F18	Mean	7.26e+06	1.57e+07	5.25e+07	1.07e+07	7.99e+07	1.53e+07	6.61e+07	1.30e+07
	Std	4.61e+06	7.43e+06	3.34e+07	6.28e+06	9.44e+07	6.93e+06	3.90e+07	7.20e+06
	p-value	–	<${10}^{-11}$	<${10}^{-11}$	0.0157	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	0.0008
	Rank	1	3	6	2	8	4	5	7
F19	Mean	9.48e+07	4.16e+06	1.80e+08	2.02e+08	6.25e+09	4.09e+08	1.91e+10	7.67e+06
	Std	1.97e+08	2.85e+06	7.03e+07	2.06e+08	4.06e+09	4.12e+08	3.34e+09	1.23e+07
	p-value	–	<${10}^{-11}$	0.0001	0.0064	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$
	Rank	2	3	4	5	8	6	7	1
F20	Mean	5.17e+03	7.64e+03	6.52e+03	5.91e+03	7.66e+03	6.58e+03	6.35e+03	6.17e+03
	Std	7.31e+02	2.91e+02	5.77e+02	1.34e+03	7.10e+02	6.38e+02	2.04e+02	5.78e+02
	p-value	–	<${10}^{-11}$	<${10}^{-11}$	0.0822	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$
	Rank	1	6	4	3	8	5	2	7
F21	Mean	2.92e+03	3.42e+03	3.44e+03	3.10e+03	4.10e+03	3.79e+03	4.48e+03	4.01e+03
	Std	7.22e+01	4.81e+01	7.68e+01	1.82e+02	1.62e+02	1.91e+02	1.08e+02	1.85e+02
	p-value	–	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$
	Rank	1	4	5	2	8	6	7	3
F22	Mean	1.97e+04	3.20e+04	2.60e+04	2.66e+04	2.92e+04	2.89e+04	3.06e+04	2.09e+04
	Std	2.21e+03	1.07e+03	1.12e+03	7.20e+03	3.51e+03	2.02e+03	5.05e+02	1.50e+03
	p-value	–	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	0.0039
	Rank	1	8	3	4	6	5	7	2
F23	Mean	3.46e+03	3.86e+03	3.71e+03	3.77e+03	5.13e+03	4.48e+03	6.99e+03	4.84e+03
	Std	9.49e+01	4.01e+01	7.11e+01	1.65e+02	2.91e+02	1.51e+02	3.01e+02	2.58e+02
	p-value	–	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$
	Rank	1	4	2	3	8	5	7	6
F24	Mean	4.10e+03	4.49e+03	4.49e+03	4.49e+03	6.97e+03	5.56e+03	5.13e+03	6.06e+03
	Std	1.31e+02	7.49e+01	9.45e+01	1.61e+02	4.58e+02	1.11e+03	4.01e+02	3.29e+02
	p-value	–	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$
	Rank	1	2	3	4	8	6	5	7
F25	Mean	6.01e+03	7.93e+03	6.96e+03	6.70e+03	4.16e+04	7.40e+03	2.66e+04	7.40e+03
	Std	9.03e+02	7.93e+02	1.11e+03	8.62e+02	7.56e+03	7.93e+02	2.59e+03	3.55e+03
	p-value	–	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	0.0005	<${10}^{-11}$	<${10}^{-11}$
	Rank	1	4	3	2	8	5	7	6
F26	Mean	1.41e+04	1.86e+04	1.85e+04	1.75e+04	4.12e+04	1.01e+04	4.98e+04	3.55e+04
	Std	1.27e+03	9.26e+02	1.32e+03	2.09e+03	7.56e+03	1.97e+03	3.04e+03	4.20e+03
	p-value	–	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$
	Rank	1	4	3	2	8	5	7	6
F27	Mean	4.01e+03	4.30e+03	4.12e+03	4.30e+03	4.26e+03	5.38e+03	1.19e+04	4.93e+03
	Std	1.10e+02	1.96e+02	9.82e+01	2.14e+02	2.95e+02	4.37e+02	1.21e+03	4.69e+02
	p-value	–	<${10}^{-11}$	0.0001	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$
	Rank	1	3	2	4	5	6	8	7
F28	Mean	7.88e+03	1.23e+04	1.01e+04	9.40e+03	6.06e+03	1.34e+04	1.19e+04	1.48e+04
	Std	1.49e+03	1.84e+03	2.40e+03	1.62e+03	9.34e+02	1.62e+03	1.97e+03	3.90e+03
	p-value	–	<${10}^{-11}$	0.0015	0.0003	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$
	Rank	3	6	4	5	1	7	2	8
F29	Mean	8.13e+03	1.09e+04	9.41e+03	9.17e+03	4.46e+04	1.37e+04	1.52e+04	1.17e+04
	Std	6.39e+02	5.45e+02	1.92e+03	8.49e+02	4.88e+04	2.44e+03	1.04e+04	2.00e+03
	p-value	–	<${10}^{-11}$	0.0002	0.0001	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$	<${10}^{-11}$
	Rank	1	5	3	2	8	6	7	4
F30	Mean	9.30e+08	1.28e+08	6.18e+08	1.56e+09	1.34e+10	1.94e+09	3.51e+10	1.04e+08
	Std	9.85e+08	6.67e+07	2.75e+08	1.16e+09	3.96e+09	1.55e+09	5.96e+09	6.83e+07
	p-value	–	<${10}^{-11}$	0.3493	0.0656	<${10}^{-11}$	0.0044	<${10}^{-11}$	<${10}^{-11}$
	Rank	2	7	3	4	8	5	6	1
+/≈/–		–	1/1/27	0/2/27	2/3/24	2/1/26	0/2/27	1/0/28	3/3/23
Mean rank		1.62	4.62	3.79	3.45	7.17	5.34	6.07	3.93
Final ranking		1	7	3	2	8	5	6	4

.

In the 50-dimensional setting, HAGWO achieves the best mean value on 24 out of 29 functions (82.8%). In the 100-dimensional setting, HAGWO achieves the best mean value on 22 out of 29 functions (75.9%). These results demonstrate that HAGWO maintains strong performance across all tested dimensionalities, with only a moderate decrease in the proportion of best solutions at D = 100, which is expected due to the dramatically increased search space complexity.

The Wilcoxon signed-rank test results (significance level $\alpha =0.05$) reported in the tables provide statistical evidence of HAGWO’s competitive superiority. For the 50-dimensional functions, HAGWO significantly outperforms BSA on 22 functions (1 tie), BRKGA on 24 functions (1 tie), GWO on 21 functions (3 ties), PSO on all 29 functions, WOA on 25 functions (1 tie), EAO on all 29 functions, and CCO on 20 functions (4 ties). For the 100-dimensional functions, HAGWO significantly outperforms BSA on 27 functions (1 tie), BRKGA on 27 functions (2 ties), GWO on 24 functions (3 ties), PSO on 26 functions (1 tie), WOA on 27 functions (2 ties), EAO on 28 functions (1 tie), and CCO on 23 functions (3 ties). These results indicate that HAGWO maintains statistically significant superiority on the vast majority of benchmark functions across both dimensionalities, with only a slight increase in ties at D = 100 and virtually no losses.

The Friedman mean rank results shown in Figure 7 further support this observation. In the 50-dimensional setting, HAGWO attains the lowest mean rank of 1.83, securing the overall 1st place, followed by CCO, GWO, BSA, BRKGA, WOA, EAO, and PSO. In the 100-dimensional setting, HAGWO attains the lowest mean rank of 1.62, again securing the 1st place, followed by GWO, BRKGA, CCO, WOA, EAO, BSA, and PSO. These consistent top rankings across increasing dimensionalities confirm HAGWO’s excellent scalability and superior overall performance in high-dimensional optimization scenarios.

5.5. Ablation Study

To verify that each of the three proposed mechanisms contributes meaningfully to the overall performance of HAGWO, we conduct a component-level ablation study on the CEC 2017 benchmark suite at 30D. Three ablated variants are constructed by disabling exactly one mechanism while keeping the others intact:

• HAGWO-w/o-S (without fitness-based hierarchical stratification): the assignment of individuals to the superior, medium, and inferior tiers is randomized instead of being guided by fitness ranking, while the three group-specific update rules, the group proportions, and the dynamic switching schedule are all preserved. This isolates the contribution of fitness-guided stratification itself.
• HAGWO-w/o-A (without hierarchical adversarial-like updating): the stratification structure is preserved, but all groups adopt the standard GWO position update; the directed and opposition-based perturbations are removed, while the Lévy flight term is retained as an undirected random perturbation with the same decay schedule.
• HAGWO-w/o-L (without adaptive Lévy flight): the full hierarchical adversarial structure is preserved, but the Lévy perturbation factor is set to zero in all update rules.

The original GWO is additionally included as the baseline in which all three mechanisms are absent. Since this ablation study requires regenerating all control groups from scratch, the full HAGWO and all its variants are re-executed using the same set of random seeds. Consequently, the HAGWO results reported in

Table 2. Results of the ablation study on the CEC 2017 benchmark suite at 30D (mean ± std over 30 independent runs; the best mean in each row is in bold). The symbols −/≈/+ indicate that the corresponding variant is significantly worse than, statistically comparable to, or significantly better than the full HAGWO according to the Wilcoxon signed-rank test at $\alpha =0.05$.

Fun.	GWO	HAGWO–w/o-S	HAGWO–w/o-A	HAGWO–w/o-L	HAGWO
F1	2.01E+09 − (1.68E+09)	8.51E+08 ≈ (7.86E+08)	7.48E+08 ≈ (4.77E+08)	1.18E+09 ≈ (1.04E+09)	1.08E+09 (1.20E+09)
F3	7.70E+04 + (1.66E+04)	1.12E+05 ≈ (2.63E+04)	5.15E+04 + (1.70E+04)	8.55E+04 ≈ (2.19E+04)	9.57E+04 (2.47E+04)
F4	6.27E+02 − (8.35E+01)	5.90E+02 − (5.70E+01)	5.57E+02 ≈ (4.12E+01)	5.74E+02 ≈ (4.40E+01)	5.55E+02 (4.57E+01)
F5	6.33E+02 − (5.22E+01)	5.80E+02 ≈ (2.25E+01)	6.30E+02 − (6.42E+01)	5.84E+02 ≈ (2.11E+01)	5.83E+02 (2.01E+01)
F6	6.11E+02 − (3.86E+00)	6.04E+02 ≈ (3.09E+00)	6.06E+02 − (2.79E+00)	6.05E+02 ≈ (2.80E+00)	6.05E+02 (1.61E+00)
F7	9.82E+02 − (6.91E+01)	8.36E+02 ≈ (3.35E+01)	9.29E+02 − (6.20E+01)	8.66E+02 ≈ (3.10E+01)	8.48E+02 (4.48E+01)
F8	9.44E+02 − (6.49E+01)	8.87E+02 ≈ (3.40E+01)	9.34E+02 − (6.20E+01)	8.87E+02 ≈ (2.29E+01)	8.82E+02 (2.26E+01)
F9	4.36E+03 − (2.20E+03)	2.03E+03 ≈ (1.24E+03)	2.62E+03 ≈ (1.42E+03)	2.44E+03 ≈ (1.16E+03)	2.24E+03 (9.89E+02)
F10	6.35E+03 − (2.02E+03)	4.58E+03 ≈ (8.08E+02)	7.60E+03 − (1.87E+03)	4.60E+03 ≈ (9.42E+02)	4.65E+03 (1.14E+03)
F11	2.35E+03 − (1.53E+03)	1.53E+03 ≈ (2.99E+02)	1.36E+03 ≈ (1.05E+02)	1.54E+03 ≈ (3.10E+02)	1.44E+03 (1.55E+02)
F12	1.51E+08 − (1.18E+08)	6.96E+07 ≈ (6.74E+07)	8.33E+07 ≈ (1.12E+08)	6.54E+07 ≈ (6.17E+07)	6.29E+07 (5.70E+07)
F13	2.86E+07 − (8.75E+07)	1.41E+06 ≈ (5.40E+06)	3.21E+05 + (5.41E+05)	9.48E+06 ≈ (3.46E+07)	1.34E+06 (5.49E+06)
F14	8.89E+05 ≈ (1.12E+06)	3.26E+05 ≈ (4.05E+05)	2.30E+05 + (3.03E+05)	4.04E+05 ≈ (5.53E+05)	4.60E+05 (5.09E+05)
F15	8.05E+05 ≈ (1.61E+06)	3.46E+05 ≈ (8.68E+05)	1.65E+05 ≈ (3.86E+05)	1.60E+05 ≈ (2.61E+05)	1.52E+05 (3.16E+05)
F16	2.84E+03 − (4.31E+02)	2.58E+03 ≈ (2.09E+02)	2.67E+03 ≈ (4.83E+02)	2.53E+03 ≈ (2.54E+02)	2.50E+03 (3.23E+02)
F17	2.17E+03 ≈ (2.41E+02)	2.06E+03 ≈ (1.77E+02)	2.07E+03 ≈ (2.03E+02)	2.07E+03 ≈ (1.40E+02)	2.06E+03 (1.79E+02)
F18	3.70E+06 ≈ (6.14E+06)	2.52E+06 ≈ (3.07E+06)	1.73E+06 ≈ (1.71E+06)	1.75E+06 ≈ (1.98E+06)	1.32E+06 (1.43E+06)
F19	1.34E+06 ≈ (1.90E+06)	1.06E+06 ≈ (2.26E+06)	8.36E+05 ≈ (1.48E+06)	9.61E+05 ≈ (1.03E+06)	6.68E+05 (7.20E+05)
F20	2.56E+03 − (2.22E+02)	2.43E+03 ≈ (1.28E+02)	2.53E+03 ≈ (3.26E+02)	2.44E+03 ≈ (2.10E+02)	2.44E+03 (1.62E+02)
F21	2.43E+03 − (5.45E+01)	2.38E+03 ≈ (2.99E+01)	2.42E+03 − (5.48E+01)	2.39E+03 ≈ (1.90E+01)	2.38E+03 (1.76E+01)
F22	5.46E+03 ≈ (2.58E+03)	5.24E+03 ≈ (1.93E+03)	6.23E+03 − (3.24E+03)	5.37E+03 ≈ (1.49E+03)	4.78E+03 (1.70E+03)
F23	2.81E+03 − (5.85E+01)	2.74E+03 ≈ (2.67E+01)	2.78E+03 − (6.99E+01)	2.75E+03 ≈ (3.18E+01)	2.74E+03 (3.02E+01)
F24	3.01E+03 − (7.15E+01)	2.89E+03 ≈ (2.30E+01)	2.98E+03 − (7.37E+01)	2.92E+03 − (3.34E+01)	2.89E+03 (1.84E+01)
F25	3.02E+03 − (6.98E+01)	2.96E+03 ≈ (3.27E+01)	2.95E+03 ≈ (2.26E+01)	2.97E+03 ≈ (2.87E+01)	2.98E+03 (5.57E+01)
F26	5.04E+03 − (4.67E+02)	4.54E+03 ≈ (3.83E+02)	4.84E+03 ≈ (5.75E+02)	4.62E+03 ≈ (4.98E+02)	4.67E+03 (2.60E+02)
F27	3.28E+03 − (3.71E+01)	3.25E+03 ≈ (1.98E+01)	3.24E+03 ≈ (1.45E+01)	3.26E+03 ≈ (2.35E+01)	3.25E+03 (2.05E+01)
F28	3.53E+03 − (1.75E+02)	3.41E+03 ≈ (8.40E+01)	3.35E+03 + (4.62E+01)	3.43E+03 ≈ (8.42E+01)	3.40E+03 (7.95E+01)
F29	3.98E+03 − (2.19E+02)	3.88E+03 ≈ (1.84E+02)	3.86E+03 ≈ (2.90E+02)	3.84E+03 ≈ (1.95E+02)	3.84E+03 (1.28E+02)
F30	1.48E+07 − (1.06E+07)	7.75E+06 ≈ (6.58E+06)	6.74E+06 ≈ (4.38E+06)	6.81E+06 ≈ (6.33E+06)	6.53E+06 (5.15E+06)
Friedman mean rank	4.83	2.45	2.86	2.97	1.90
−/≈/+	22/6/1	1/28/0	9/16/4	1/28/0	—

may differ slightly from those in the main experiments (Table A1), as the latter belong to an independent experimental campaign. All variants are evaluated under the same experimental protocol, as outlined in Section 5 (population size 30, 500 iterations, 30 independent runs), on the 29 CEC 2017 functions at 30D. The results, together with Wilcoxon signed-rank tests against the full HAGWO and Friedman mean ranks, are summarized in Table 2.

Three observations can be drawn. First, the original GWO, in which all three mechanisms are absent, ranks last by a wide margin (Friedman mean rank 4.83 versus 1.90–2.97 for all HAGWO-based variants) and is significantly outperformed by the full HAGWO on 22 of the 29 functions. This confirms that the performance gains reported in this section stem from the proposed mechanisms rather than from implementation details.

Second, and most importantly, removing any single mechanism degrades the overall performance: the Friedman mean rank of the full HAGWO (1.90) deteriorates to 2.45, 2.86, and 2.97 when the stratification, the adversarial-like updating, and the Lévy flight are removed, respectively. This monotonic degradation provides direct experimental evidence that each of the three components contributes meaningfully and non-redundantly to the observed performance gains.

Third, the three mechanisms exhibit distinct contribution patterns. (i) For HAGWO-w/o-S, replacing fitness-guided tier assignment with random assignment degrades the mean rank from 1.90 to 2.45; the variant is significantly worse on F4 and never significantly better on any function (28 ties out of 29). This indicates that it is the fitness-guided partitioning—rather than the mere existence of groups—that directs the appropriate update rule to the appropriate individuals at each search stage. (ii) HAGWO-w/o-A exhibits the largest number of significant degradations (9 functions), concentrated on simple multimodal (F5–F8, F10) and composition (F21–F24) landscapes, confirming that the tier-differentiated adversarial-like rules are the primary performance-driving component. The variant surpasses the full model only on four functions (F3, F13, F14, F28), suggesting occasional over-guidance of the directed updates on certain hybrid landscapes, but the net effect of removing this mechanism is clearly negative. (iii) HAGWO-w/o-L attains the worst mean rank (2.97) among all HAGWO variants, with a significant degradation on F24 and consistently elevated mean values across most of the suite, indicating that the adaptive Lévy flight perturbation supports fine-grained exploitation and escape from suboptimal regions, particularly on composition functions where precise convergence is required.

Overall, the ablation results demonstrate that the integrated HAGWO framework yields large and statistically significant gains over the original GWO, and that the hierarchical stratification, the adversarial-like updating, and the Lévy flight perturbation each provide significant and complementary contributions, consistent with the synergistic design rationale discussed in Section 3.3.3.

5.6. Results on 3D Bin Packing Benchmark

To evaluate the performance of the proposed HAGWO on the constrained combinatorial problem of 3D-BPP with identical bins (3D-SBSBPP), we adapt it using biased random-key encoding and the DFTRC-2 placement heuristic. For fair comparison, all algorithms—including the original GWO, BRKGA, WOA, PSO, BSA, CCO, and EAO—were implemented with the same encoding scheme, DFTRC-2 heuristic, maximal-space management, and adjusted fitness function (aNB). In this subsection, we use the original algorithm names for brevity, with the understanding that all results reflect the adapted versions on 3D-SBSBPP.

The comparison is performed on the standard benchmark set by Martello et al. [2], containing 160 instances across 8 classes (20 instances per class: 10 with $n=50$ items and 10 with $n=100$). Classes 1–5 use bin dimensions $100\times 100\times 100$ and five item types with the size ranges shown in

Table 3. Type characterization.

Type	${\mathit{w}}_{\mathit{j}}$ Interval	${\mathit{h}}_{\mathit{j}}$ Interval	${\mathit{d}}_{\mathit{j}}$ Interval
Type 1	$[1,{\displaystyle \frac{1}{2}}W]$	$[{\displaystyle \frac{2}{3}}H,H]$	$[{\displaystyle \frac{2}{3}}D,D]$
Type 2	$[{\displaystyle \frac{2}{3}}W,W]$	$[1,{\displaystyle \frac{1}{2}}H]$	$[{\displaystyle \frac{2}{3}}D,D]$
Type 3	$[{\displaystyle \frac{2}{3}}W,W]$	$[{\displaystyle \frac{2}{3}}H,H]$	$[1,{\displaystyle \frac{1}{2}}D]$
Type 4	$[{\displaystyle \frac{1}{2}}W,W]$	$[{\displaystyle \frac{1}{2}}H,H]$	$[{\displaystyle \frac{1}{2}}D,D]$
Type 5	$[1,{\displaystyle \frac{1}{2}}W]$	$[1,{\displaystyle \frac{1}{2}}H]$	$[1,{\displaystyle \frac{1}{2}}D]$

, where each item of type k is selected with probability 0.6 and each of the other types with probability 0.1. Classes 6–8 employ different bin sizes and item dimension ranges as follows:

• Class 6: bin $10\times 10\times 10$; dimensions $\in [1,10]$.
• Class 7: bin $40\times 40\times 40$; dimensions $\in [1,35]$.
• Class 8: bin $100\times 100\times 100$; dimensions $\in [1,100]$.

All algorithms use a population size of $N=50$ and run for 100 generations. Experiments are conducted on the same hardware (Intel Core i9-9900K @ 3.60 GHz, 32 GB RAM) using MATLAB R2020a. Each algorithm is executed with 10 independent runs using different random seeds (seed = $1,2,\dots ,10$) to ensure reproducibility.

5.6.1. Quantitative Results

Average results over 10 independent runs are summarized in

Table 4. Results for the three-dimensional instances (average over 10 runs). All algorithms were adapted to 3D-SBSBPP using the same biased random-key encoding, DFTRC-2 heuristic, and aNB fitness function.

Class	Bin Size	n	BRKGA			HAGWO			GWO			WOA
			aNB	NB	aT	aNB	NB	aT	aNB	NB	aT	aNB	NB	aT
1	$100\times 100\times 100$	50	12.56	12.2	156.15	12.53	12.2	148.63	12.36	12	133.54	12.59	12.2	141.06
		100	24.31	23.9	497.12	24.29	24	482.59	23.95	23.6	481.69	24.57	24.2	508.77
2	$100\times 100\times 100$	50	12.15	11.8	172.74	12.04	11.7	159.50	11.94	11.5	145.90	12.08	11.7	154.97
		100	24.33	24	519.62	24.34	24	524.01	23.90	23.6	497.14	24.46	24.1	536.95
3	$100\times 100\times 100$	50	11.91	11.5	163.89	11.80	11.3	149.18	11.80	11.4	138.83	12.02	11.6	146.29
		100	24.36	24	535.78	24.24	23.9	528.19	23.72	23.3	506.15	24.57	24.2	529.94
4	$100\times 100\times 100$	50	30.40	30.2	173.17	30.40	30.2	153.12	30.40	30.2	137.71	30.40	30.2	143.43
		100	60.88	60.7	479.36	60.88	60.7	471.04	60.88	60.7	459.99	60.88	60.7	460.71
5	$100\times 100\times 100$	50	7.68	7.3	254.07	7.63	7.2	251.46	7.61	7.2	225.53	7.65	7.2	225.99
		100	14.34	13.9	870.60	14.17	13.7	953.47	14.27	13.9	859.82	14.48	14.1	868.25
6	$10\times 10\times 10$	50	11.97	11.6	97.59	11.90	11.6	89.64	11.75	11.4	81.08	11.99	11.6	87.52
		100	22.61	22.3	259.49	22.16	21.8	246.16	21.97	21.6	230.22	22.65	22.3	247.40
7	$40\times 40\times 40$	50	6.99	6.7	194.21	6.92	6.6	190.66	6.90	6.6	175.55	6.99	6.7	184.87
		100	12.74	12.4	589.85	12.47	12.1	572.01	12.33	11.9	547.96	12.85	12.6	604.31
8	$100\times 100\times 100$	50	9.47	9.1	222.94	9.30	8.9	194.79	9.31	8.9	202.92	9.48	9.2	108.2
		100	17.60	17.3	699.15	17.17	16.8	639.94	17.00	16.6	603.65	17.60	17.3	645.04
Total	—	—	—	2989	—	—	2967	—	—	2944	—	—	2999	—
Class	BinSize	$\mathit{n}$	PSO			BSA			CCO			EAO
			aNB	NB	aT	aNB	NB	aT	aNB	NB	aT	aNB	NB	aT
1	$100\times 100\times 100$	50	12.59	12.3	141.86	12.60	12.2	148.57	12.65	12.3	145.73	12.73	12.3	148.55
		100	24.66	24.3	502.03	24.67	24.3	505.06	24.79	24.4	514.58	24.95	24.6	518.62
2	$100\times 100\times 100$	50	12.21	11.9	153.90	12.28	12	160.88	12.16	11.8	155.67	12.38	12	161.44
		100	24.46	24.1	510.98	24.48	24.1	536.29	24.51	24.1	523.93	24.77	24.4	550.28
3	$100\times 100\times 100$	50	12.01	11.6	143.30	11.98	11.5	152.76	12.05	11.6	152.27	12.20	11.9	154.09
		100	24.54	24.2	533.38	24.51	24.1	547.42	24.72	24.4	537.16	24.76	24.4	553.37
4	$100\times 100\times 100$	50	30.40	30.2	144.64	30.40	30.2	142.40	30.4	30.2	140.70	30.40	30.2	143.24
		100	60.88	60.7	456.94	60.88	60.7	458.12	60.88	60.7	450.34	60.88	60.7	458.67
5	$100\times 100\times 100$	50	7.65	7.2	251.41	7.72	7.3	266.44	7.67	7.2	253.00	7.85	7.4	275.56
		100	14.48	14.1	880.12	14.62	14.3	940.81	14.52	14.2	910.45	14.83	14.5	983.79
6	$10\times 10\times 10$	50	11.97	11.6	84.20	12.12	11.8	88.22	12.12	11.8	89.49	12.26	11.9	91.86
		100	22.59	22.3	245.85	22.78	22.4	255.79	22.72	22.4	246.38	23.32	23	258.45
7	$40\times 40\times 40$	50	6.90	6.6	178.69	7.04	6.7	196.79	7.12	6.9	192.73	7.24	7	204.46
		100	12.79	12.5	597.25	12.88	12.6	631.40	12.92	12.6	617.24	13.14	12.8	654.45
8	$100\times 100\times 100$	50	9.40	9.1	195.47	9.58	9.3	208.80	9.52	9.2	199.20	9.73	9.4	212.66
		100	17.48	17.2	670.64	17.67	17.4	695.95	17.71	17.4	675.68	17.97	17.6	705.93
Total	—	—	—	2999	—	—	3009	—	—	3012	—	—	3041	—

. Lower values of aNB (adjusted fitness) and NB (number of bins) indicate better performance; aT reports average runtime in seconds. The “Total” NB is the sum of average NB values across all 160 instances (scaled by 10 runs), with the best result in bold.

The experimental results demonstrate the effectiveness of HAGWO for the 3D-BPP. Overall, GWO achieves the best performance with the lowest total NB of 2944. HAGWO ranks second with a total NB of 2967, outperforming BRKGA (2989) by 22 bins and showing clear gains over PSO (2999), BSA (3009), CCO (3012), and EAO (3041). Similar trends appear in the average fitness (aNB) column.

A closer look at class-specific performance reveals interesting patterns. In the more constrained and difficult classes (1, 2, 3, 6, 7, 8), particularly at $n=100$, HAGWO consistently narrows the gap with GWO and widens its advantage over BRKGA and other methods. For example, in Class 7 ($n=100$), HAGWO reduces the average NB by 0.3 bins compared to BRKGA while maintaining comparable runtime. In contrast, on easier classes (4 and 5), almost all algorithms converge to nearly identical NB values, suggesting that the problem instances are less discriminative when near-optimal packings are readily achievable. This behavior aligns with the intrinsic difficulty gradient of the benchmark: classes with higher item-size variability and tighter spatial constraints benefit more from HAGWO’s hierarchical adversarial-like updates and adaptive Levy perturbations, which help escape suboptimal packing sequences.

Moderation of superiority claims. We acknowledge that HAGWO does not dominate all competitors on 3D-BPP; specifically, the original GWO achieves a marginally lower total bin count (2944 vs. 2967). This outcome is consistent with the No Free Lunch theorem [32]: no single algorithm excels universally across all problem landscapes. The original GWO’s simpler, leader-centric hierarchy appears particularly effective under the strict geometric and non-overlapping constraints of 3D-BPP, where precise local adjustments to packing sequences are critical. However, HAGWO’s substantial advantage in run-to-run stability (narrowest IQR and fewest outliers across 13 of 16 instance groups) and its clear superiority over BRKGA, PSO, BSA, CCO, and EAO demonstrate that the proposed hierarchical adversarial-like framework provides meaningful benefits in constrained combinatorial settings, even if it marginally trails the original GWO in raw bin count on this specific benchmark.

In terms of computational efficiency (aT), GWO and HAGWO maintain competitive running times across most classes, especially on larger instances (Classes 4–5 and 8), while still delivering superior packing quality. These results confirm that the hybrid adaptive mechanism in HAGWO provides a robust balance between solution quality and efficiency.

5.6.2. Boxplot Analysis

To complement the average performance metrics presented in Table 4, we provide boxplot comparisons of the final fitness (aNB) and the number of used bins (NB) for each of the 16 instance groups (Classes 1–8 at $n=50$ and $n=100$), based on 10 independent runs per algorithm (see Figure A5, Figure A6, Figure A7, Figure A8, Figure A9, Figure A10, Figure A11, Figure A12, Figure A13, Figure A14, Figure A15, Figure A16, Figure A17, Figure A18, Figure A19 and Figure A20).

Across the entire benchmark, HAGWO demonstrates strong stability. In 13 out of 16 instance groups, HAGWO exhibits the narrowest (or jointly narrowest) IQR in at least one of the two boxplots (aNB or NB). This advantage is particularly pronounced in the more challenging classes (1, 2, 3, 6, 7, 8) and becomes even more evident at $n=100$, where increased problem complexity amplifies run-to-run variability in most other algorithms. For instance, in Class 3 ($n=100$), HAGWO reduces the IQR of NB by approximately 40% compared to BRKGA, while in Class 7 ($n=100$), the number of outliers is reduced from 4–5 (in PSO, BSA, EAO) to nearly zero. HAGWO also consistently produces the fewest and least extreme outliers, confirming that the hybrid adaptive mechanism effectively mitigates stochastic fluctuations and enhances repeatability.

In terms of solution quality, HAGWO ranks in the top two (by median NB) in 9 out of 16 classes and never falls outside the top three. Its overall summed NB across all instances (2967) is the second lowest, trailing GWO (2944) by only 23 bins while outperforming BRKGA (2989), PSO (2999), BSA (3009), CCO (3012), and EAO (3041) by clear margins. In most difficult classes—especially at $n=100$—the median and lower whisker of HAGWO remain very close to those of GWO, often statistically indistinguishable based on visual inspection of the distributions.

Compared with the baseline BRKGA, HAGWO shows a narrower IQR in all 16 classes and fewer outliers in the large majority of cases. It also achieves a lower or equal median NB in 11 classes, resulting in a total improvement of 22 bins (2967 vs. 2989). These consistent gains in both stability and average packing efficiency demonstrate that the proposed hierarchical adversarial-like strategy meaningfully enhances the search process.

On the easier classes (4 and 5 at both $n=50$ and $n=100$), nearly all algorithms converge to nearly identical distributions (overlapping medians, almost identical IQRs, and comparable whiskers), confirming that performance differences diminish when near-optimal packings are readily achievable. HAGWO performs on par with the best methods in these cases, showing no degradation on simpler instances.

5.6.3. Convergence Analysis

To gain deeper insight into the dynamic behavior of the algorithms, we present convergence curves of the best-found fitness (aNB) over 100 generations for representative instances (Figure 8).

On the easier Class 5 $n=50$ instance, most algorithms reach near-optimal regions within the first 30–50 generations. HAGWO demonstrates rapid early improvement, followed by steady refinement until approximately generations 55–60, ultimately stabilizing at one of the lowest and most consistent plateaus. GWO exhibits the steepest initial descent but shows only marginal progress thereafter. BRKGA converges more gradually and stabilizes at a comparatively higher level.

When scaling to $n=100$, the increased problem complexity accentuates behavioral differences. HAGWO maintains steady progress across early, mid, and late stages, achieving the lowest final plateau. This sustained late-stage improvement can be attributed to the hierarchical adversarial-like position updating and adaptive Levy flight perturbation, which maintain population diversity and enable directed escape from local optima even in highly constrained packing landscapes. GWO again shows excellent early convergence but plateaus relatively early with limited late-stage gains. BRKGA maintains a steady but slower trajectory, reaching a competitive yet noticeably higher final level.

Similar trends are observed across other difficult classes (e.g., Classes 1–3 and 6–8 at $n=100$), where HAGWO frequently continues to refine solutions in generations 60–100, whereas most competitors stagnate earlier. These convergence patterns align closely with the boxplot results: on easier instances, most algorithms achieve similar quality, with HAGWO showing slightly faster and more stable refinement; on larger instances, HAGWO’s ability to continue improving in the late generations provides a clear advantage in both final quality and consistency.

The corresponding three-dimensional visualizations of the best packing solutions for Class 5 ($n=50$ and $n=100$) are presented in Figure A21 and Figure A22, respectively. HAGWO consistently produces compact and feasible layouts that are visually competitive with BRKGA and GWO while showing fewer irregular gaps compared to other algorithms.

6. Discussion

The experimental results across both continuous benchmark functions and the constrained 3D-SBSBPP reveal the strengths and domain-specific characteristics of the proposed HAGWO, while highlighting meaningful insights into the design of swarm intelligence algorithms for different optimization landscapes.

In continuous optimization (CEC 2017 suite), HAGWO’s superior performance stems directly from its three synergistic mechanisms. Dynamic hierarchical stratification adaptively shifts population composition from exploration-dominant (early emphasis on inferior wolves) to exploitation-dominant phases, effectively preserving diversity and preventing premature convergence. Hierarchical adversarial-like position updating introduces opposition-based learning and directed information flow among stratified groups, enhancing search efficiency and robustness in multimodal, hybrid, and composition landscapes. Adaptive Levy flight perturbation, with linearly decreasing intensity, enables controlled long-range jumps early on and precise local refinement later, allowing robust escape from local optima without disrupting convergence. These innovations collectively address the original GWO’s key limitations—rapid diversity loss and insufficient phase-aware adaptation—leading to statistically significant improvements over GWO and other state-of-the-art methods.

When transferred to the highly constrained 3D-BPP, HAGWO remains highly competitive. As noted in Section 5.6, the original GWO achieves a marginally lower total bin count (2944 vs. 2967), which we attribute to GWO’s simpler, leader-centric hierarchy being particularly robust under strict geometric and non-overlapping constraints where precise local adjustments are critical. Nevertheless, HAGWO consistently outperforms the domain-specific BRKGA in both average packing quality and, particularly, run-to-run stability, suggesting that its core innovations transfer effectively to combinatorial problems when supported by appropriate encoding/decoding strategies and greedy feasibility enforcement.

These domain differences align with the No Free Lunch theorem [32]: no single algorithm can excel universally across all problem landscapes. HAGWO’s strengths in high-dimensional, continuous spaces stem from its ability to maintain long-term population diversity and phase-aware exploration–exploitation balance, making it well-suited for applications such as neural network hyperparameter tuning, complex parameter optimization, or multimodal engineering design. In tightly constrained combinatorial tasks like 3D-BPP, simpler or problem-tailored methods may retain marginal advantages due to their focused exploitation under strict feasibility requirements. The observed trade-off—competitive solution quality with markedly higher stability—highlights the value of hybrid mechanisms that combine global diversity preservation with controlled local refinement.

6.1. Limitations

Several limitations of this study should be acknowledged:

• Scope of the ablation study: The component-level ablation in Section 5.5 is conducted on the CEC 2017 suite at 30D. While it confirms the individual contributions of the proposed mechanisms, ablation at higher dimensions (50D/100D) and on the 3D-SBSBPP domain, as well as finer-grained parameter sensitivity analysis (stratification ratios, Lévy exponent, disturbance scaling), would provide a more complete picture and is left for future work.
• Parameter sensitivity: The stratification ratios $(0.1,0.3,0.6)$ and $(0.3,0.3,0.4)$, the Levy exponent $\lambda =1.5$, and the disturbance scaling $0.1$ are justified by design principles and established conventions (Section 3.3.1 and Section 3.3.2). However, we acknowledge that a systematic parameter sensitivity analysis—varying each parameter across a grid of values—would strengthen the empirical foundation. The absence of such analysis is a limitation we aim to address in future work.
• Comparison scope: While our comparison includes diverse metaheuristic paradigms, several recent and competitive GWO variants (e.g., I-GWO [11], EBGWO [15], IAGWO [9], SFGWO [16]) were not included, either because they were published after our experiments began or because their source code was unavailable. A comprehensive benchmark against all recent GWO variants remains future work.
• 3D-BPP instance coverage: The 3D-BPP experiments were conducted on the standard Martello et al. [2] benchmark set. Real-world packing scenarios may involve additional constraints (weight limits, fragility, loading sequence, irregular item shapes) not captured by this benchmark. The applicability of HAGWO to such extended problem variants requires further investigation.
• Theoretical analysis: While we provide computational complexity analysis and intuitive explanations for HAGWO’s behavior, formal theoretical analysis (e.g., convergence proofs, convergence rate bounds) remains to be developed.

6.2. Directions for Future Research

To address the limitations identified above and extend HAGWO’s applicability, the following directions merit further investigation:

• Parameter sensitivity and higher-dimensional ablation: Building on the component-level ablation of Section 5.5, systematically varying each parameter (stratification ratios, Lévy exponent, disturbance scaling) and extending the ablation to 50D/100D and combinatorial domains would provide deeper empirical understanding of their individual and combined effects.
• Constraint-aware perturbation for discrete optimization: Incorporating feasibility-preserving operators or adaptive penalty mechanisms into the hierarchical update framework would suppress infeasible perturbations in 3D-BPP, potentially narrowing the marginal performance gap with the original GWO observed on certain instance groups.
• Hybridization with local search: Integrating HAGWO’s global exploration with problem-specific local search heuristics (e.g., EMS-based neighborhood descent) is expected to enhance solution quality on large-scale packing instances where standalone metaheuristics plateau prematurely.
• Online parameter self-adaptation: Dynamically adjusting Levy flight parameters and disturbance coefficients through real-time diversity metrics—rather than relying on fixed proportional constants—could yield more robust convergence across heterogeneous problem landscapes.
• Multi-objective and dynamic extensions: Extending HAGWO to multi-objective 3D-BPP variants (e.g., simultaneous minimization of bin count, volume utilization, and carbon footprint) and dynamic scenarios with time-varying item sets remains a promising open avenue.
• Irregular packing with physical constraints: Generalizing the adversarial hierarchy to handle non-rectangular items subject to center-of-gravity, fragility, and loading-sequence constraints would strengthen real-world applicability in logistics and manufacturing.

7. Conclusions

This paper introduced HAGWO, a novel Hierarchical Adversarial Grey Wolf Optimizer that enhances the original GWO through dynamic hierarchical population stratification, adaptive Levy flight perturbation, and hierarchical adversarial-like position updating. These synergistic mechanisms enable effective diversity maintenance, opposition-based learning, and phase-aware exploration–exploitation balance.

Comprehensive evaluation on the CEC 2017 benchmark suite across multiple dimensions demonstrated HAGWO’s state-of-the-art performance in continuous optimization, achieving the best overall rankings and statistically significant superiority over the original GWO and other leading metaheuristics. When adapted to the strongly NP-hard 3D single-bin-size bin packing problem, HAGWO delivered highly competitive results, outperforming the established BRKGA and most swarm-based methods while exhibiting exceptional stability across all instance groups. We moderated our claims regarding 3D-BPP superiority by acknowledging that the original GWO achieved a marginally lower total bin count, and we discussed how the shared encoding–decoding framework ensures fair comparison.

The work makes three main contributions: (1) a powerful and general-purpose GWO variant with enhanced robustness and search dynamics; (2) rigorous high-dimensional benchmarking confirming clear improvements in continuous spaces; and (3) a successful adaptation to 3D-BPP that reveals valuable insights into continuous-to-discrete transferability and domain-specific trade-offs.

Future work may focus on parameter sensitivity analysis, constraint-aware adaptations for combinatorial problems, hybrid integration with local search or exact methods, and extensions to multi-objective, dynamic, or real-world packing scenarios with irregular items and additional constraints. These directions hold strong potential to advance swarm intelligence in both theoretical and practical contexts.