Double-Swarm Grey Wolf Optimizer with Covariance and Dimension Learning for Engineering Optimization Problems

Abstract

Grey Wolf Optimizer (GWO) has been widely applied in many fields due to its advantages of fast convergence speed, simple parameter settings, and easy implementation. However, with the deepening of algorithm research, GWO has also exposed some problems, such as being prone to becoming stuck in local optima and insufficient convergence accuracy. To address the above issues, a double-swarm Grey Wolf Optimizer with covariance and dimension learning (CDL-DGWO) is proposed. In the CDL-DGWO, firstly, chaotic grouping is used to divide the grey wolf population into two sub-swarms, forming a symmetric cooperative search framework, thereby improving the diversity of the population. Meanwhile, covariance and dimension learning strategies are utilized to improve the hunting behavior of grey wolves; the global search capability and the stability of the algorithm are thereby enhanced. Moreover, CDL-DGWO is validated on 23 benchmark problems and the CEC2017 test set. The results indicate that the CDL-DGWO algorithm outperforms swarm intelligence optimization algorithms such as Particle Swarm Optimization (PSO), Moth Flame Optimization (MFO), and other variants of GWO. Finally, the CDL-DGWO algorithm addresses three engineering design problems that are representative of real-world scenarios. The statistical analysis of the experimental outcomes demonstrates the feasibility and practicality of the proposed methodology.

1. Introduction

Optimization problems are prevalent across diverse real-world applications, such as image segmentation [1], economic load dispatch [2], and feature selection [3]. Traditional optimization methods often fail to solve complex, nonlinear, non-convex, discontinuous, or discrete problems effectively. In contrast, metaheuristic algorithms have gained attention due to their avoidance of unnecessary assumptions and their powerful global search capabilities. Over the past few decades, numerous metaheuristic algorithms have been developed, including Particle Swarm Optimization (PSO) [4], Grey Wolf Optimizer [5], Moth Flame Optimization (MFO) [6], Slime Mold Algorithm (SMA) [7], Whale Optimization Algorithm (WOA) [8], Mayfly Algorithm (MA) [9], and Harris Hawks Optimization (HHO) [10].

The Grey Wolf Optimizer (GWO) is a population-based algorithm that emulates the leadership hierarchy and hunting behavior of grey wolves [5]. In GWO, the population is categorized into alpha, beta, delta, and omega wolves, with the first three leading the optimization process by steering the search direction. GWO is recognized as an effective metaheuristic, with applications across various domains such as fuel cell technology [11], localization technology [12], and robotics [13].

Despite its successful applications across various fields, GWO encounters challenges including reduced population diversity, imbalanced exploration and exploitation, and premature convergence. These limitations are attributed to its dependence on $\alpha$, $\beta$ and $\delta$ wolves for location updates. To overcome these limitations, researchers have conducted extensive studies. In [14,15], GWO is combined with SCA and GSA to create two hybrid algorithms, integrating the strengths of both to enhance search performance. In [16], the GWO is hybridized with the Jaya algorithm to enhance task scheduling in fog computing, improving load balancing, response time, and resource efficiency. In [17], an enhanced GWO incorporating Lévy mutation (LGWO) was introduced, utilizing Lévy mutation and greedy selection strategies to refine the hunting phases. In [18], a modified version of GWO, termed RWGWO, was proposed to enhance its optimization capabilities. In [19], the “survival of the fittest” (SOF) principle and differential evolution (DE) were employed to improve GWO. This approach increases the likelihood of GWO escaping local optima. Ref. [20] introduced an improved GWO variant leveraging information entropy for dynamic position updates and a nonlinear convergence strategy, aiming to balance exploration and exploitation. Ref. [21] introduced a fuzzy strategy Grey Wolf Optimization algorithm, utilizing fuzzy mutation, crossover operators, and a non-inferior selection strategy to refine wolf positions and enhance search accuracy. Ref. [22] introduced DI-GWOCD, a discrete version of the Improved Grey Wolf Optimizer, for effectively detecting communities in complex networks. It enhances community detection through local search strategies and binary distance calculations, demonstrating superior performance over existing algorithms in community quality assessment. However, increasing population diversity in solving practical problems requires further attention. In [23], an Improved Adaptive Grey Wolf Optimization (IAGWO) algorithm was introduced by integrating concepts from Particle Swarm Optimization (PSO), using an Inverse Multiquadratic Function (IMF) for inertia weight adjustment, and employing a Sigmoid-based adaptive updating mechanism. Extensive experiments show that IAGWO outperforms the compared algorithms on benchmark test sets and effectively addresses 19 real-world engineering challenges, demonstrating its robustness and broad application potential in optimization problems.

Ref. [24] introduced an Artificial Bee Colony Algorithm with Adaptive Covariance Matrix(ACoM-ABC), which incorporates an adaptive covariance matrix to enhance the performance of the original Artificial Bee Colony (ABC) in addressing problems with high variable correlations. The effectiveness of ACoM-ABC was validated through extensive testing on multiple benchmark platforms. Ref. [25] proposed Cumulative Covariance Matrix Artificial Bee Colony (CCoM-ABC) algorithm accumulates covariance matrix information from each generation to construct an eigen-coordinate system, thereby guiding the search direction. Additionally, it dynamically selects between the eigen-coordinate and natural coordinate systems during the search process to balance exploration and exploitation capabilities, thereby improving performance on non-separable problems. The Covariance Matrix Adapted Grey Wolf Optimizer (CMA-GWO) is an enhanced version of the Grey Wolf Optimizer (GWO) [26]. It employs a Covariance Matrix Adaptation (CMA) strategy in the initial phase to optimize the initial positions of the prey, thereby augmenting the algorithm’s exploration capability and convergence rate. This research not only provides a novel approach for modeling and optimizing the direct metal deposition additive manufacturing process but also enhances the predictive accuracy and generalization ability of the model through the improved GWO. The I-GWO algorithm effectively balances global and local search capabilities by integrating the Grey Wolf Optimizer (GWO) with a Dimension Learning-based Hunting (DLH) search strategy [27]. This integration prevents premature convergence to local optima and maintains population diversity. I-GWO demonstrates its potential in solving real-world engineering optimization problems by effectively handling constraints and identifying optimal solutions.

In light of the “No Free Lunch” theorem [28], which posits that no single optimization algorithm can be universally optimal for all problems, this study addresses several issues of the GWO, including insufficient population diversity, premature convergence, and an imbalance between exploitation and exploration. To overcome these challenges, a double-swarm Grey Wolf Optimizer with covariance and dimension learning (CDL-DGWO) is introduced. In CDL-DGWO, chaotic grouping is used to divide the grey wolf population into two sub-swarms, thereby improving the population diversity. Meanwhile, covariance and dimension learning are utilized to improve the hunting behavior of grey wolves, thereby enhancing both global search capability and algorithm stability. In summary, the main contributions of this paper are as follows:

(1)

Chaotic grouping is utilized to generate two sub-swarms of grey wolves. This strategy can improve population diversity of CDL-DGWO algorithm.

(2)

Covariance and dimension learning strategies are utilized to improve the hunting behavior of grey wolves, which can enhance the global search capability and algorithm stability.

(3)

The performance of the CDL-DGWO algorithm is validated on 23 benchmark problems and the CEC2017 test suite. The results indicate that the CDL-DGWO outperforms the compared swarm intelligence algorithms such as PSO, MFO, and GWO variants in terms of solving optimal solutions and convergence performance. Additionally, the CDL-DGWO is applied to three engineering design problems, which fully demonstrate the practicality of the proposed methodology.

The structure of this article is as follows: Section 2 introduces the GWO; Section 3 details the CDL-DGWO; Section 4 presents simulation verification and comparative results; Section 5 applies CDL-DGWO to engineering problems; the article concludes in the final section.

2. Grey Wolf Optimizer (GWO)

The Grey Wolf Optimizer (GWO) is a metaheuristic algorithm inspired by the leadership and social behavior of grey wolves, designed to simulate predation process [5,29]. This article provides a brief introduction to GWO.

In GWO, the population of wolves is divided into four categories based on their hunting capabilities: $\alpha$, $\beta$ and $\delta$, and $\omega$. The $\alpha$, $\beta$ and $\delta$ wolves are considered the leaders and have the best fitness, guiding the rest of the pack in searching for prey. In the context of optimization, the prey represents the optimal solution.

The hunting process of grey wolves is simulated through three main steps: encircling, hunting, and attacking the prey. The mathematical model for encircling prey is given by the following formulas:

$$\mathit{D}\left(t\right)=\mathrm{abs}\left({\mathit{X}}_{\mathrm{p}}\left(t\right)\mathit{C}-\mathit{X}\left(t\right)\right)$$

(1)

$$\mathit{X}(t+1)={\mathit{X}}_{\mathrm{p}}\left(t\right)-\mathit{D}\left(t\right)\mathit{A}$$

(2)

Here, t denotes the current iteration count. $\mathit{X}$ signifies any grey wolf within the population, while ${\mathit{X}}_{\mathrm{p}}$ represents the position vector of the prey. $\mathit{A}$ and $\mathit{C}$ are diagonal coefficient matrices that dictate the search dynamics, and their calculation methods are described in [30]. $\mathit{D}$ denotes the distance between the grey wolf and its prey. The notation “$\mathrm{abs}(\ast )$” signifies the transformation of each vector element into its absolute value.

The hunting process is led by the $\alpha$, $\beta$ and $\delta$ wolves, with the position update formulas for the other wolves being:

$${\mathit{D}}_{\alpha }\left(t\right)=\mathrm{abs}\left({\mathit{X}}_{\alpha }\left(t\right){\mathit{C}}_1-{\mathit{X}}^v\left(t\right)\right)$$

(3)

$${\mathit{D}}_{\beta }\left(t\right)=\mathrm{abs}\left({\mathit{X}}_{\beta }\left(t\right){\mathit{C}}_2-{\mathit{X}}^v\left(t\right)\right)$$

(4)

$${\mathit{D}}_{\delta }\left(t\right)=\mathrm{abs}\left({\mathit{X}}_{\delta }\left(t\right){\mathit{C}}_3-{\mathit{X}}^v\left(t\right)\right)$$

(5)

$${\mathit{X}}_1(t+1)={\mathit{X}}_{\alpha }\left(t\right)-{\mathit{D}}_{\alpha }\left(t\right){\mathit{A}}_1$$

(6)

$${\mathit{X}}_2(t+1)={\mathit{X}}_{\beta }\left(t\right)-{\mathit{D}}_{\beta }\left(t\right){\mathit{A}}_2$$

(7)

$${\mathit{X}}_3(t+1)={\mathit{X}}_{\delta }\left(t\right)-{\mathit{D}}_{\delta }\left(t\right){\mathit{A}}_3$$

(8)

$${\mathit{X}}^v(t+1)={\displaystyle \frac{{\mathit{X}}_1(t+1)+{\mathit{X}}_2(t+1)+{\mathit{X}}_3(t+1)}{3}}$$

(9)

Among them, ${\mathit{X}}^v\left(t\right)$ represents the position of the v-th wolf in the previous iteration. ${\mathit{X}}^v(t+1)$ signifies the position of the current solution. Subsequently, other candidate grey wolves randomly adjust their positions near the prey, guided by the optimal information from $\alpha$, $\beta$ and $\delta$.

Finally, when the prey is cornered, the wolves attack and capture it. This phase in GWO is simulated by updating the position of the wolves when the coefficients $\left|\mathit{A}\right|$ and $\left|\mathit{C}\right|$ are less than 1, forcing the wolves to converge towards the prey.

The main steps and pseudo-code of GWO are presented as Algorithm 1:

Algorithm 1: Grey Wolf Optimizer (GWO).

1: Initialize $t=1$, dimension D, population size N, $max\_FEs$, $FEs=0$; %   $max\_FEs$ and $FEs$ represent the maximum number of function evaluations and   the current number of function evaluations respectively. 2: Initialize population $\mathit{X}$ of wolves; 3: while$FEs\le max\_FEs$ 4:        Calculate the fitness value of wolves and update $FEs=FEs+N$; 5:        ${\mathit{X}}_{\alpha }$, ${\mathit{X}}_{\beta }$ and ${\mathit{X}}_{\delta }$ are the first three wolves with the best fitness; 6:        for $i=1:N$ 7:                for $j=1:D$ 8:                        Updata parameters a, $\mathit{A}$ and $\mathit{C}$; 9:                        Update the variables in ${\mathit{X}}_i$ using Equations (3)–(9); 10:               end 11:       end 12: end while

3. Double-Swarm Grey Wolf Optimizer with Covariance and Dimension Learning (CDL-DGWO)

Inspired by previous research on GWO, a double-swarm Grey Wolf Optimizer with covariance and dimension learning (CDL-DGWO) is proposed. In CDL-DGWO, chaotic grouping is used to divide the grey wolf population into two sub-swarms, thereby the diversity of the population is improved. Meanwhile, covariance and dimension learning strategies are utilized to improve the hunting behavior of grey wolves, thereby enhancing the global search capability and algorithm stability. The CDL-DGWO and its pseudocode are described in detail as follows.

3.1. Chaotic Grouping and Dynamic Regrouping

In this section, the entire population is grouped using a chaos grouping mechanism [31]. The mechanism divides the population by generating a set of highly correlated sequences through a chaos function, thereby improving the grouping quality and search performance. The iterative chaos map is used as the chaos function, with the mapping formula as follows:

$$x_i=sin\left({\displaystyle \frac{b\pi }{x_{i-1}}}\right)$$

(10)

where b is a constant with range $(0,1)$.

The process of chaotic grouping is given in Algorithm 2, where $N_d$ denotes a chaotic sequence.

Algorithm 2: Chaotic grouping mechanism.

1: The first chaotic value ($N_d\left(1\right)$) is initialized randomly in (0,1); 2: for  $i=2:N$ 3:          $b=rand$; 4:          $N_d\left(i\right)=sin\left({\displaystyle \frac{b\pi }{N_d(i-1)}}\right)$; 5: end for 6:$[A,I]=sort\left(N_d\right);$ 7: The grey wolf population is divided into two sub-swarms according to I;

To strengthen the information exchange between the two sub-swarms in the search process, the two sub-swarms are reorganized every $R_d$ generations. If the value of $R_d$ is too large, timely information exchange between sub-swarms is hindered. Conversely, if the value of $R_d$ is too small, the sub-swarms cannot conduct an adequate search. In the early stage of the search, the sub-swarms should be given sufficient time to search independently. In the later stage of the search, maximizing population diversity enhances the ability of the algorithm to locate the global optimum. This paper employs the following dynamic regrouping mechanism:

$$\begin{array}{c}{R}_d=ceil\left[n_s-\left(n_s-n_d\right)\times {\displaystyle \frac{FEs}{max\_FEs}}\right]\hfill \end{array}$$

(11)

where $n_s=40$ and $n_d=10$ represent the maximum and minimum interval iterations respectively, and $ceil$ indicates rounding up to the nearest integer.

The choice of dividing the population into two sub-swarms establishes a symmetrical cooperative search architecture, which is based on a balance between algorithmic complexity and performance enhancement. A dual-swarm structure provides a clear and computationally efficient framework for implementing complementary search strategies: one swarm focused on exploration (via covariance learning) and the other on exploitation (via dimension learning). This dichotomy has been effectively demonstrated in other multi-swarm optimizers [32], proving sufficient to introduce beneficial population diversity and mitigate premature convergence.

3.2. Learning Strategies

In GWO, the positions of individual grey wolves are updated through hunting activities. However, the inter-variable interactions are typically not considered, which makes GWO perform poorly in solving complex optimization problems. The update of individual grey wolf positions using a covariance matrix aims to transform the original space into a feature space, thereby enhancing information sharing among variables and improving the algorithm’s overall performance. The detailed explanation is as follows:

The covariance matrices (${\Sigma }_{{\mathit{X}}^{\ast }}$) for the populations of wolves and ${\mathit{X}}_{\alpha }$, ${\mathit{X}}_{\beta }$ and ${\mathit{X}}_{\delta }$ are computed, with the calculation process for ${\Sigma }_{{\mathit{X}}^{\ast }}$ detailed below:

$$\begin{array}{c}{\Sigma }_{{\mathit{X}}^{\ast }}=\left[cov({\mathit{X}}_j^{\ast },{\mathit{X}}_k^{\ast })\right]\hfill \end{array}$$

(12)

where ${\mathit{X}}^{\ast }=[\mathit{X},{\mathit{X}}_{\alpha },{\mathit{X}}_{\beta },{\mathit{X}}_{\delta }]$, $j=1,2,\dots ,d$, $k=1,2,\dots ,d$, ${\mathit{X}}_j^{\ast }={\left[x_{1j}^{\ast },x_{2j}^{\ast },\dots ,x_{nj}^{\ast }\right]}^T$, ${\mathit{X}}_k^{\ast }={\left[x_{1k}^{\ast },x_{2k}^{\ast },\dots ,x_{nk}^{\ast }\right]}^T$.

$$\begin{array}{c}cov({\mathit{X}}_j^{\ast },{\mathit{X}}_k^{\ast })={\displaystyle \frac{{\sum }_{i=1}^n({x^{\ast }}_{ij}\left(t\right)-{\overline{X^{\ast }}}_j\left(t\right))({x^{\ast }}_{ik}\left(t\right)-{\overline{X^{\ast }}}_k\left(t\right))}{n-1}}\hfill \end{array}$$

(13)

where $i=1,2,\dots ,n$. The formula for ${\overline{X^{\ast }}}_j\left(t\right)$, ${\overline{X^{\ast }}}_k\left(t\right)$ are as follows:

$$\begin{array}{c}{\overline{X^{\ast }}}_j\left(t\right)=\begin{array}{c}\sum _{i=1}^n{x^{\ast }}_{ij}\left(t\right)/n\end{array}\hfill \end{array}$$

(14)

$$\begin{array}{c}{\overline{X^{\ast }}}_k\left(t\right)=\begin{array}{c}\sum _{i=1}^n{x^{\ast }}_{ik}\left(t\right)/n\end{array}\hfill \end{array}$$

(15)

To rotate the original space into the eigenspace, eigen decomposition is applied to ${\Sigma }_{{\mathit{X}}^{\ast }}$ as follows:

$$\begin{array}{c}{\Sigma }_{{\mathit{X}}^{\ast }}={\mathit{Q}}_{{\mathit{X}}^{\ast }}{\Lambda }_{{\mathit{X}}^{\ast }}{\mathit{Q}}_{{\mathit{X}}^{\ast }}^T\hfill \end{array}$$

(16)

where ${\mathit{Q}}_{{\mathit{X}}^{\ast }}$ is orthogonal matrices, which is composed of eigenvectors of ${\Sigma }_{{\mathit{X}}^{\ast }}$. ${\Lambda }_{{\mathit{X}}^{\ast }}$ is diagonal matrices that composed of eigenvalues of ${\Sigma }_{{\mathit{X}}^{\ast }}$.

Then, the wolf individuals are transformed from the original space to the eigenspace. The conversion formula is as follows:

$$\begin{array}{c}{\mathit{X}}_i^{\ast eig}\left(t\right)={\mathit{X}}_i^{\ast }\left(t\right)\times {\mathit{Q}}_{{\mathit{X}}^{\ast }}\hfill \end{array}$$

(17)

Since ${\mathit{X}}^{\ast eig}\left(t\right)=[{\mathit{X}}^{eig}\left(t\right),{\mathit{X}}_{\alpha }^{eig}\left(t\right),{\mathit{X}}_{\beta }^{eig}\left(t\right),{\mathit{X}}_{\delta }^{eig}\left(t\right)]$, ${\mathit{X}}^{eig}\left(t\right)$, ${\mathit{X}}_{\alpha }^{eig}\left(t\right)$, ${\mathit{X}}_{\beta }^{eig}\left(t\right)$ and ${\mathit{X}}_{\delta }^{eig}\left(t\right)$ can be obtained.

In the eigenspace, wolves update their positions with the following form:

$${\mathit{D}}^{eig}\left(t\right)=\mathrm{abs}\left({\mathit{X}}_{\mathrm{p}}^{eig}\left(t\right)\mathbf{C}-{\mathit{X}}^{eig}\left(t\right)\right)$$

(18)

$${\mathit{X}}^{eig}(t+1)={\mathit{X}}_{\mathrm{p}}^{eig}\left(t\right)-{\mathit{D}}^{eig}\left(t\right)\mathit{A}$$

(19)

$${\mathit{X}}^{eig}(t+1)={\displaystyle \frac{{\mathit{X}}_1^{eig}(t+1)+{\mathit{X}}_2^{eig}(t+1)+{\mathit{X}}_3^{eig}(t+1)}{3}}$$

(20)

Among them, $\mathrm{p}\in [\alpha ,\beta ,\delta ]$. ${\mathit{X}}^{eig}\left(t\right)$ represents the position of the wolf in the previous iteration in eigenspace. ${\mathit{X}}^{eig}(t+1)$ is the position of the current solution, and ${\mathit{X}}_1^{eig}$, ${\mathit{X}}_2^{eig}$ and ${\mathit{X}}_3^{eig}$ denote the position of the leaders $\alpha$, $\beta$ and $\delta$.

After the wolves’ positions have been updated, the wolves are transformed from the eigenspace to the original space by the following formula:

$${\mathit{X}}_i^{\ast }(t+1)={\mathit{X}}_i^{\ast eig}(t+1)\times {\mathit{Q}}_{{\mathit{X}}^{\ast }}^T$$

(21)

While covariance learning enhances global search capability, dimension learning provides local refinement. The combination ensures a balanced approach to optimization.

In GWO, $\alpha$, $\beta$ and $\delta$ guide the other wolves to update their position in each iteration, resulting in strong convergence. Dimension learning involves updating positions based on interactions with different neighbors and randomly selected wolves. The detailed process is as follows [27]:

First, the radius $R_i$ is obtained by calculating the Euclidean distance between the current position $\mathit{X}\left(t\right)$ and the candidate position ${\mathit{X}}^v(t+1)$:

$$\begin{array}{c}{R}_i\left(t\right)=∥X_i\left(t\right)-X_i^v(t+1)∥\hfill \end{array}$$

(22)

Then, the neighborhood of $\mathit{X}\left(t\right)$ is defined using the radius $R_i$. $D_i$ is the Euclidean distance between ${\mathit{X}}_i$ and ${\mathit{X}}_j$.

$$\begin{array}{c}{N}_i\left(t\right)=\left\{{\mathit{X}}_{\mathrm{j}}\left(t\right)\left|D_i\left({\mathit{X}}_i\left(t\right),{\mathit{X}}_j\left(t\right)\right)\right.\le R_i\left(t\right)\right\}\hfill \end{array}$$

(23)

Finally, calculate the candidate solution $X_i^{DL}(t+1)$ obtained through dimension learning:

$$\begin{array}{c}{X}_{i\eta }^{DL}(t+1)=X_{i\eta }(t+1)+rand\times \left(X_{n\eta }\left(t\right)-X_{r\eta }\left(t\right)\right)\hfill \end{array}$$

(24)

where ${\mathit{X}}_{r\eta }\left(t\right)$ is a random wolf, and ${\mathit{X}}_{n\eta }\left(t\right)$ is d-th dimension of a random neighbor. The better solution between ${\mathit{X}}_i^{DL}(t+1)$ and ${\mathit{X}}_i^v(t+1)$ is selected as the final candidate ${\mathit{X}}_i(t+1)$ for the next iteration.

3.3. Framework of the CDL-DGWO

This subsection presents the detailed pseudo-code and a comprehensive flowchart that elucidate the workings of the CDL-DGWO. These are delineated in Algorithm 3 and Figure 1, respectively.

To ensure clarity and facilitate understanding, the steps of the CDL-DGWO are outlined as follows:

Step 1:

Initialize the system parameters, which include population size N, individual dimension d, random parameters a, $\mathit{A}$ and $\mathbf{C}$, as well as the maximum number of function evaluations ($max\_FEs$) and the current number of function evaluations ($FEs$).

Step 2:

Generate the initial population $\mathit{X}$ of wolves randomly within the defined upper and lower bounds of $\mathit{X}$.

Step 3:

Divide the population into two sub-swarms using Algorithm 2.

Step 4:

Check the stopping criteria: Determine whether $FEs\le max\_FEs$ or the best fitness value meets the accuracy requirements. If conditions are met, output the position of ${\mathit{X}}_{\alpha }$ as the best approximated optimum, otherwise, proceed to step 5.

Step 5:

Calculate and sort the fitness of each individual, updating $FEs=FEs+n$. Select the top three individuals as ${\mathit{X}}_{\alpha }$, ${\mathit{X}}_{\beta }$ and ${\mathit{X}}_{\delta }$.

Step 6:

Sub-swarm 1 enhances search efficiency by integrating a dimension learning mechanism with GWO’s original position update mechanism. Updating the position of every grey wolf individual using Equations (3)–(9). Conduct dimension learning operation using Equations (22)–(24).

Step 7:

For the Sub-swarm 2, introduce a covariance matrix to enhance information sharing among individual variables, thereby improving the overall performance of the algorithm.

Step 8:

Perform dynamic regrouping using Equation (11) and Algorithm 2, then return to Step 4.

Step 9:

Return the best solution.

Algorithm 3: Pseudocode of CDL-DGWO.

1: Initialize $t=1$, d, N, $max\_FEs$, $FEs=0$; 2: Initialize population $\mathit{X}$ of wolves; 3: The population is divided into 2 sub-swarms by Algorithm 2; 4: while$FEs\le max\_FEs$ 5:        Calculate the fitness value of wolves, $FEs=FEs+N$; 6:        ${\mathit{X}}_{\alpha }$, ${\mathit{X}}_{\beta }$ and ${\mathit{X}}_{\delta }$ are the first three wolves with the best fitness;        %% The Sub-swarm 1 7:        for $i=1:N/2$ 8:                for $j=1:d$ 9:                        Updata parameters a, $\mathit{A}$ and $\mathbf{C}$; 10:                        Update the variables in ${\mathit{X}}_i$ using Equations (3)–(9); 11:               end 12:       end 13:       Conduct dimension learning operation using Equations (22)–(24);        %% The Sub-swarm 2 14:       Calculate the covariance matrices ${\Sigma }_{{\mathit{X}}^{\ast }}$ using Equations (12)–(15); 15:       ${\mathit{Q}}_{{\mathit{X}}^{\ast }}$ is obtained, which is composed of eigenvectors of ${\Sigma }_{{\mathit{X}}^{\ast }}$,           based on eigen decomposition relation Equation (16); 16:        The individuals of wolves are transformed into eigenspace based on           eigenvector using Equation (17); 17:        for $i=1:N/2$ 18:                for $j=1:d$ 19:                       Updata parameters a, $\mathit{A}$ and $\mathbf{C}$; 20:                       Update the variables in ${\mathit{X}}_i^{eig}$ using Equations (18)–(20); 21:                end 22:        end 23:        Convert positions of wolves to original space using Equation (21); 24:        If mod(t,$R_d$)==0 25:            Regrouping using Algorithm 2; 26:        end 27: end while

3.4. Computational Complexity of the CDL-DGWO

The computational complexity of CDL-DGWO is determined by the original GWO and the hierarchical guidance strategy. The original GWO primarily consists of the following stages: initialization with a complexity of $O(N·D)$, fitness evaluation with a complexity of $O\left(1\right)$, the first three wolfs update with a complexity of $O\left(N\right)$, and position update with a complexity of $T·O(N·D)$. Based on these stages, the overall complexity of GWO can be expressed as $O(N·D)+T·O(N·D)=O(T·N·D)$, where N represents the population size, D denotes the dimensionality of the function, and T is the maximum number of iterations.

The computational complexity of CDL-DGWO is primarily determined by the following components per iteration:

(1)

Fitness Evaluation: $O\left(N\right)$.

(2)

Standard GWO Operations (population update, leader selection): $O(N·D)$.

(3)

Covariance Matrix Calculation (for Sub-swarm 1) The covariance matrix computation using Equations (12)–(15) requires $O({\displaystyle \frac{1}{2}}N·D^2)$ operations. The subsequent eigen decomposition in Equation (16) has a complexity of $O\left(D^3\right)$.

(4)

Dimension Learning (for Sub-swarm 2): The neighborhood search and candidate solution update in Equations (22)–(24) contribute $O({\displaystyle \frac{1}{2}}N·D)$.

(5)

Chaotic Grouping: The generation and sorting of chaotic sequences incur $O(NlogN)$, but when amortized over all iterations, this becomes negligible.

Therefore, the complexity per iteration is: $O(N·D)$ + $O({\displaystyle \frac{1}{2}}N·D^2+D^3$ + $O({\displaystyle \frac{1}{2}}N·D)$. Over T iterations, Big O notation ignores constant coefficients and lower-order terms. Therefore, the computational complexity of the CDL-DGWO can be simplified to: $O\left(T·(N·D^2+D^3)\right)$.

4. Test Results and Analysis

4.1. Test Suites, Test Methods and Performance Index

To assess the performance of the CDL-DGWO presented in this article, 23 basic functions [33] and the CEC2017 test set [34] are utilized for evaluation. The former includes unimodal, multimodal, and fixed-dimensional multimodal problems. The CEC2017 test suite is characterized by high nonlinearity, multimodality, high dimensionality, and non-convexity. These characteristics pose greater challenges for algorithms in solving problems, as they need to overcome issues such as local optima and curse of dimensionality in the solution space.

Based on the above two test sets, a comprehensive performance evaluation of CDL-DGWO is conducted. Additionally, CDL-DGWO is compared with and validated against other improved GWO and heuristic algorithms. The comparison algorithms include: GWO [5], MIGWO [30], IGWO [19], LGWO [17], HGWOSCA [14], RWGWO [18], EGWO [35], PSOGWO [36], PSO [37], MFO [6], SSA [38], WOA [8], HSCA [39], HCLPSO [40], PSOGSA [41], mSCA [42]. The parameters of comparative algorithms in the experiment are given in

Table 1. Parameter settings.

Algorithm	Population Size (n)	Parameters	Reference
CDL-DGWO	30	$r_1=rand,r_2=rand,n_s=40,n_d=10$	~
GWO	30	$r_1=rand,r_2=rand$	[5]
MIGWO	30	$r_1=rand,r_2=rand,n_s=40,n_d=10$	[30]
IGWO	30	$r_1=rand,r_2=rand$	[27]
LGWO	30	$r_1=rand,r_2=rand,\beta =3/2$	[17]
HGWOSCA	30	$r_1=rand,r_2=rand,r=rand$	[14]
RWGWO	30	$r_1=rand,r_2=rand$	[18]
EGWO	30	$r_1=rand,r_2=rand$	[35]
PSOGWO	30	$r_1=rand,r_2=rand,c_1=c_2=2$	[36]
PSO	30	$c_1,c_2=2,\omega =0.9$	[37]
MFO	30	$t\in [-1,1],b=1$	[6]
SSA	30	$c_2=rand,c_3=rand$	[38]
WOA	30	$r_1=rand,r_2=rand,p=rand$	[8]
HSCA	30	a = 2, $r_2=2·\pi ·rand$, $r_3=2·rand$, $r_4=rand$	[39]
HCLPSO	30	$w=0.99-0.2,c_1=2.5-0.5,c_2=0.5-2.5$	[40]
PSOGSA	30	$elitistcheck=1,rpower=1,w=0.3,c_1=0.5,c_2=1.5$	[41]
mSCA	30	$J_R=0.1,r=rand$	[42]

. The parameters listed in Table 1 are defined as follows, adhering to their original sources: $r_1$, $r_2$, r, p are uniformly distributed random numbers in [0,1]; a is a control parameter that decreases linearly from 2 to 0; $c_1$, $c_2$ are acceleration coefficients; w is the inertia weight; b, t are constants specific to MFO; $eltistcheck$, $rpower$ are control parameters as defined in [41]. All other unspecified parameters follow the standard definitions from the corresponding references.

During the experiment, the maximum number of function evaluations ($max\_FEs$) for the two test suites are set at 30,000 and 300,000, respectively. The dimension D of both test suites is set to 30. To ensure a fair comparison of the algorithms’ core search capabilities while maintaining statistical reliability, all algorithms were evaluated using the following procedure: For each test function, an identical set of 30 initial populations was generated. Each algorithm was then independently run 30 times, with each run starting from a corresponding population in this shared set. Different random seeds were employed across these runs to ensure the robustness of results, while the consistent initialization eliminates any performance bias that could arise from specific starting positions. The maximum number of function evaluations ($max\_FEs$) for the 23 benchmark functions is set to 30,000. This value is chosen in accordance with common practices in the metaheuristic optimization literature [33], providing a sufficient budget for algorithms to demonstrate convergence behaviors while maintaining computational tractability for extensive comparative studies involving multiple algorithms and independent runs.

In this paper, Mean, Std, the multiple-problem Wilcoxon test and Friedman test are used to evaluate the performance of CDL-DGWO algorithm. Mean and Std denote the average and standard deviation of 30 independent runs, respectively. the multiple-problem Wilcoxon test is applied to each test set.

4.2. Effects of Proposed Strategies

The proposed CDL-DGWO algorithm builds upon the original GWO and enhances its performance through the introduction of hybrid chaotic grouping and dynamic regrouping. These mechanisms effectively divide the population into two distinct sub-swarms. The first sub-swarm employs a covariance strategy, while the second sub-swarm utilizes a dimension learning strategy. Both strategies are designed to improve the overall performance of the algorithm.

To evaluate the effectiveness of the proposed strategies, a comparative experiment is designed in this section. Specifically, the C-GWO algorithm represents an enhanced variant of the original GWO. In this variant, one sub-swarm incorporates the covariance learning mechanism, whereas the other sub-swarm retains the original GWO update mechanism. Similarly, the D-GWO algorithm is another variant of GWO. Here, one sub-swarm employs the dimension learning strategy, while the other sub-swarm maintains the original GWO update mechanism.

Table 2. Various GWOs with three strategies.

Algorithm	Chaotic Grouping and Dynamic Regrouping	Covariance	Dimension Learning
GWO	0	0	0
C-GWO	1	1	0
D-GWO	1	0	1
CDL-DGWO	1	1	1

illustrates the configuration of these strategies. Within this matrix, the binary indicator “1” signifies that a particular strategy is applied within the algorithm, while “0” indicates that the strategy is not utilized.

To evaluate the effectiveness of the introduced policies, the performance of four algorithms (GWO, C-GWO, D-GWO, and CDL-DGW) was assessed across 23 benchmark functions. In the simulation tests, the population size was set to 30 for all algorithms, and the stopping criterion was defined as a maximum of 30,000 function evaluations. Each algorithm was independently executed 30 times on each function. The results, including the average values (Mean) and standard deviations (Std) for each algorithm, are presented in

Table 3. The test results of 4 algorithms.

	GWO		C-GWO		D-GWO		CDL-DGWO
	Mean	Std	Mean	Std	Mean	Std	Mean	St
F1	$2.516\times {10}^{-58}$	$8.746\times {10}^{-58}$	$1.130\times {10}^{-59}$	$2.335\times {10}^{-59}$	$9.641\times {10}^{-30}$	$9.372\times {10}^{-30}$	$\mathbf{2}.\mathbf{351}\times {\mathbf{10}}^{-\mathbf{218}}$	$\mathbf{0}.\mathbf{000}\times {\mathbf{10}}^{\mathbf{0}}$
F2	$9.422\times {10}^{-35}$	$6.918\times {10}^{-35}$	$2.913\times {10}^{-32}$	$2.318\times {10}^{-32}$	$5.036\times {10}^{-19}$	$3.248\times {10}^{-19}$	$\mathbf{2}.\mathbf{307}\times {\mathbf{10}}^{-\mathbf{111}}$	$\mathbf{2}.\mathbf{865}\times {\mathbf{10}}^{-\mathbf{111}}$
F3	$9.987\times {10}^{-14}$	$5.107\times {10}^{-13}$	$1.057\times {10}^{-22}$	$2.884\times {10}^{-22}$	$1.174\times {10}^{-4}$	$3.338\times {10}^{-4}$	$\mathbf{9}.\mathbf{554}\times {\mathbf{10}}^{-\mathbf{191}}$	$\mathbf{0}.\mathbf{000}\times {\mathbf{10}}^{\mathbf{0}}$
F4	$1.887\times {10}^{-14}$	$4.129\times {10}^{-14}$	$4.749\times {10}^{-22}$	$1.054\times {10}^{-21}$	$1.303\times {10}^{-6}$	$1.043\times {10}^{-6}$	$\mathbf{2}.\mathbf{285}\times {\mathbf{10}}^{-\mathbf{103}}$	$\mathbf{3}.\mathbf{825}\times {\mathbf{10}}^{-\mathbf{103}}$
F5	$2.713\times {10}^{+1}$	$9.697\times {10}^{-1}$	$2.573\times {10}^{+1}$	$2.279\times {10}^{-1}$	$2.540\times {10}^{+1}$	$5.165\times {10}^{-1}$	$\mathbf{2}.\mathbf{507}\times {\mathbf{10}}^{+\mathbf{1}}$	$\mathbf{1}.\mathbf{789}\times {\mathbf{10}}^{-\mathbf{1}}$
F6	$6.378\times {10}^{-1}$	$3.760\times {10}^{-1}$	$\mathbf{6}.\mathbf{129}\times {\mathbf{10}}^{-\mathbf{5}}$	$\mathbf{1}.\mathbf{657}\times {\mathbf{10}}^{-\mathbf{5}}$	$1.822\times {10}^{-1}$	$2.162\times {10}^{-1}$	$1.415\times {10}^{-3}$	$3.174\times {10}^{-4}$
F7	$9.346\times {10}^{-4}$	$4.688\times {10}^{-4}$	$6.456\times {10}^{-4}$	$3.687\times {10}^{-4}$	$2.159\times {10}^{-3}$	$1.093\times {10}^{-3}$	$\mathbf{1}.\mathbf{278}\times {\mathbf{10}}^{-\mathbf{4}}$	$\mathbf{7}.\mathbf{789}\times {\mathbf{10}}^{-\mathbf{5}}$
F8	$-5.899\times {10}^{+3}$	$8.466\times {10}^{+2}$	$-7.327\times {10}^{+3}$	$9.019\times {10}^{+2}$	$-8.240\times {10}^{+3}$	$8.214\times {10}^{+2}$	$-\mathbf{9}.\mathbf{084}\times {\mathbf{10}}^{+\mathbf{3}}$	$\mathbf{4}.\mathbf{477}\times {\mathbf{10}}^{+\mathbf{2}}$
F9	$1.956\times {10}^{-1}$	$\mathbf{1}.\mathbf{071}\times {\mathbf{10}}^{\mathbf{0}}$	$3.553\times {10}^{-1}$	$1.462\times {10}^0$	$1.370\times {10}^0$	$2.146\times {10}^0$	$\mathbf{0}.\mathbf{000}\times {\mathbf{10}}^{\mathbf{0}}$	$\mathbf{6}.\mathbf{245}\times {\mathbf{10}}^{\mathbf{0}}$
F10	$1.735\times {10}^{-14}$	$4.710\times {10}^{-15}$	$4.678\times {10}^{-15}$	$9.014\times {10}^{-16}$	$3.784\times {10}^{-14}$	$6.018\times {10}^{-15}$	$\mathbf{8}.\mathbf{882}\times {\mathbf{10}}^{-\mathbf{16}}$	$\mathbf{0}.\mathbf{000}\times {\mathbf{10}}^{\mathbf{0}}$
F11	$1.511\times {10}^{-3}$	$3.457\times {10}^{-3}$	$\mathbf{0}.\mathbf{000}\times {\mathbf{10}}^{\mathbf{0}}$	$\mathbf{0}.\mathbf{000}\times {\mathbf{10}}^{\mathbf{0}}$	$1.505\times {10}^{-3}$	$3.984\times {10}^{-3}$	$\mathbf{0}.\mathbf{000}\times {\mathbf{10}}^{\mathbf{0}}$	$\mathbf{0}.\mathbf{000}\times {\mathbf{10}}^{\mathbf{0}}$
F12	$3.828\times {10}^{-2}$	$2.342\times {10}^{-2}$	$\mathbf{8}.\mathbf{960}\times {\mathbf{10}}^{-\mathbf{6}}$	$\mathbf{2}.\mathbf{886}\times {\mathbf{10}}^{-\mathbf{6}}$	$8.510\times {10}^{-3}$	$6.385\times {10}^{-3}$	$1.060\times {10}^{-4}$	$1.963\times {10}^{-5}$
F13	$5.752\times {10}^{-1}$	$1.917\times {10}^{-1}$	$\mathbf{8}.\mathbf{718}\times {\mathbf{10}}^{-\mathbf{4}}$	$2.772\times {10}^{-3}$	$1.423\times {10}^{-1}$	$1.072\times {10}^{-1}$	$1.827\times {10}^{-3}$	$\mathbf{4}.\mathbf{214}\times {\mathbf{10}}^{-\mathbf{4}}$
F14	$3.681\times {10}^0$	$3.527\times {10}^0$	$2.528\times {10}^0$	$2.604\times {10}^0$	$9.980\times {10}^1$	$2.545\times {10}^{-11}$	$\mathbf{9}.\mathbf{980}\times {\mathbf{10}}^{\mathbf{1}}$	$\mathbf{7}.\mathbf{377}\times {\mathbf{10}}^{-\mathbf{12}}$
F15	$2.313\times {10}^3$	$6.120\times {10}^3$	$6.712\times {10}^4$	$1.472\times {10}^3$	$\mathbf{3}.\mathbf{075}\times {\mathbf{10}}^{\mathbf{4}}$	$3.944\times {10}^8$	$\mathbf{3}.\mathbf{075}\times {\mathbf{10}}^{\mathbf{4}}$	$\mathbf{3}.\mathbf{863}\times {\mathbf{10}}^{\mathbf{8}}$
F16	$-1.032\times {10}^0$	$5.130\times {10}^9$	$-1.032\times {10}^0$	$1.682\times {10}^8$	$-1.032\times {10}^0$	$2.646\times {10}^9$	$-\mathbf{1}.\mathbf{032}\times {\mathbf{10}}^{\mathbf{0}}$	$\mathbf{3}.\mathbf{824}\times {\mathbf{10}}^{-\mathbf{10}}$
F17	$3.979\times {10}^1$	$4.034\times {10}^7$	$3.979\times {10}^1$	$4.539\times {10}^7$	$3.979\times {10}^1$	$3.274\times {10}^7$	$\mathbf{3}.\mathbf{979}\times {\mathbf{10}}^{\mathbf{1}}$	$\mathbf{1}.\mathbf{914}\times {\mathbf{10}}^{\mathbf{8}}$
F18	$3.000\times {10}^0$	$1.042\times {10}^5$	$3.000\times {10}^0$	$2.561\times {10}^5$	$3.000\times {10}^0$	$8.063\times {10}^9$	$\mathbf{3}.\mathbf{000}\times {\mathbf{10}}^{\mathbf{0}}$	$\mathbf{2}.\mathbf{785}\times {\mathbf{10}}^{\mathbf{9}}$
F19	$-3.862\times {10}^0$	$2.669\times {10}^3$	$-3.863\times {10}^0$	$5.728\times {10}^4$	$-3.863\times {10}^0$	$1.479\times {10}^6$	$-\mathbf{3}.\mathbf{863}\times {\mathbf{10}}^{\mathbf{0}}$	$\mathbf{4}.\mathbf{499}\times {\mathbf{10}}^{\mathbf{7}}$
F20	$-3.305\times {10}^0$	$5.126\times {10}^2$	$-3.291\times {10}^0$	$5.303\times {10}^2$	$-3.307\times {10}^0$	$3.831\times {10}^2$	$-\mathbf{3}.\mathbf{322}\times {\mathbf{10}}^{\mathbf{0}}$	$\mathbf{1}.\mathbf{335}\times {\mathbf{10}}^{\mathbf{5}}$
F21	$-9.984\times {10}^0$	$9.225\times {10}^1$	$-7.590\times {10}^0$	$2.037\times {10}^0$	$-9.491\times {10}^0$	$1.715\times {10}^0$	$-\mathbf{1}.\mathbf{015}\times {\mathbf{10}}^{\mathbf{1}}$	$\mathbf{6}.\mathbf{410}\times {\mathbf{10}}^{\mathbf{4}}$
F22	$-\mathbf{1}.\mathbf{040}\times {\mathbf{10}}^{\mathbf{1}}$	$3.609\times {10}^4$	$-6.997\times {10}^0$	$1.887\times {10}^0$	$-1.040\times {10}^1$	$8.830\times {10}^4$	$-\mathbf{1}.\mathbf{040}\times {\mathbf{10}}^{\mathbf{1}}$	$\mathbf{7}.\mathbf{553}\times {\mathbf{10}}^{\mathbf{4}}$
F23	$-1.027\times {10}^1$	$1.481\times {10}^0$	$-7.457\times {10}^0$	$2.201\times {10}^0$	$-1.053\times {10}^1$	$1.030\times {10}^3$	$-\mathbf{1}.\mathbf{054}\times {\mathbf{10}}^{\mathbf{1}}$	$\mathbf{9}.\mathbf{043}\times {\mathbf{10}}^{\mathbf{4}}$

.

The results of the multiple comparisons using the Wilcoxon signed-rank test are presented in

Table 4. Results of multiple-problem Wilcoxon test.

CDL-DGWO VS.	+	−	≈	R+	R−	p-Value	a = 0.05
GWO	22	1	0	267	9	0.000087	+
C-GWO	19	3	1	240	36	0.003302	+
D-GWO	22	1	0	269	7	0.000068	+

. Additionally,

Table 5. Friedman test results of the 4 algorithms.

	GWO	C-GWO	D-GWO	CDL-DGWO
Ave	3.1739	2.7609	2.8261	1.2391
Rank	4	2	3	1

displays the Friedman test results for each algorithm across 23 benchmark functions.

Compared to the original GWO, the C-GWO algorithm achieved the smallest values on 14 out of the 23 benchmark functions, including F1 and F3. This demonstrates the effectiveness of the covariance strategy employed by C-GWO. Similarly, D-GWO outperformed GWO on 14 functions, such as F5 and F6, by achieving the smallest values. This highlights the efficacy of the dimension learning strategy utilized by D-GWO. According to the results of the Friedman test, CDL-DGWO, C-GWO, and D-GWO all ranked higher than the original GWO, with CDL-DGWO achieving the highest overall ranking. This indicates that the integration of both covariance and dimension learning strategies in CDL-DGWO yields the best performance.

4.3. Comparisons on Classical Benchmark Problems

This section presents the comparison results between CDL-DGWO and 15 other algorithms. Specifically,

Table 6. The test results of 16 algorithms.

	CDL-DGWO		GWO		IGWO		LGWO
	Mean	Std	Mean	Std	Mean	Std	Mean	Std
F1	$\mathbf{2}.\mathbf{351}\times {\mathbf{10}}^{-\mathbf{218}}$	$\mathbf{0}.\mathbf{000}\times {\mathbf{10}}^{\mathbf{0}}$	$2.516\times {10}^{-58}$	$8.746\times {10}^{-58}$	$1.590\times {10}^{-28}$	$2.301\times {10}^{-28}$	$9.719\times {10}^{-37}$	$2.487\times {10}^{-36}$
F2	$\mathbf{2}.\mathbf{307}\times {\mathbf{10}}^{-\mathbf{111}}$	$\mathbf{2}.\mathbf{865}\times {\mathbf{10}}^{-\mathbf{111}}$	$9.422\times {10}^{-35}$	$6.918\times {10}^{-35}$	$8.090\times {10}^{-18}$	$7.145\times {10}^{-18}$	$1.191\times {10}^{-22}$	$2.191\times {10}^{-22}$
F3	$\mathbf{9}.\mathbf{554}\times {\mathbf{10}}^{-\mathbf{191}}$	$\mathbf{0}.\mathbf{000}\times {\mathbf{10}}^{\mathbf{0}}$	$9.987\times {10}^{-14}$	$5.107\times {10}^{-13}$	$9.682\times {10}^4$	$2.175\times {10}^3$	$1.720\times {10}^6$	$5.233\times {10}^6$
F4	$\mathbf{2}.\mathbf{285}\times {\mathbf{10}}^{-\mathbf{103}}$	$\mathbf{3}.\mathbf{825}\times {\mathbf{10}}^{-\mathbf{103}}$	$1.887\times {10}^{-14}$	$4.129\times {10}^{-14}$	$1.825\times {10}^5$	$1.473\times {10}^5$	$3.475\times {10}^9$	$4.666\times {10}^9$
F5	$2.507\times {10}^1$	$1.789\times {10}^1$	$2.713\times {10}^1$	$9.697\times {10}^1$	$\mathbf{2}.\mathbf{437}\times {\mathbf{10}}^{\mathbf{1}}$	$\mathbf{8}.\mathbf{492}\times {\mathbf{10}}^{\mathbf{1}}$	$2.669\times {10}^1$	$2.875\times {10}^1$
F6	$1.415\times {10}^3$	$3.174\times {10}^4$	$6.378\times {10}^1$	$3.760\times {10}^1$	$2.225\times {10}^2$	$6.807\times {10}^2$	$1.252\times {10}^0$	$3.846\times {10}^1$
F7	$\mathbf{1}.\mathbf{278}\times {\mathbf{10}}^{\mathbf{4}}$	$\mathbf{7}.\mathbf{789}\times {\mathbf{10}}^{\mathbf{5}}$	$9.346\times {10}^4$	$4.688\times {10}^4$	$2.904\times {10}^3$	$9.196\times {10}^4$	$1.748\times {10}^3$	$1.266\times {10}^3$
F8	$-9.084\times {10}^3$	$4.477\times {10}^2$	$-5.899\times {10}^3$	$8.466\times {10}^2$	$-8.626\times {10}^3$	$1.475\times {10}^3$	$-3.666\times {10}^3$	$3.742\times {10}^2$
F9	$\mathbf{0}.\mathbf{000}\times {\mathbf{10}}^{\mathbf{0}}$	$\mathbf{6}.\mathbf{245}\times {\mathbf{10}}^{\mathbf{0}}$	$1.956\times {10}^1$	$1.071\times {10}^0$	$2.407\times {10}^1$	$1.285\times {10}^1$	$1.895\times {10}^{-15}$	$1.038\times {10}^{-14}$
F10	$\mathbf{8}.\mathbf{882}\times {\mathbf{10}}^{-\mathbf{16}}$	$\mathbf{0}.\mathbf{000}\times {\mathbf{10}}^{\mathbf{0}}$	$1.735\times {10}^{-14}$	$4.710\times {10}^{-15}$	$6.093\times {10}^{-14}$	$9.014\times {10}^{-15}$	$1.024\times {10}^{-14}$	$3.021\times {10}^{-15}$
F11	$\mathbf{0}.\mathbf{000}\times {\mathbf{10}}^{\mathbf{0}}$	$\mathbf{0}.\mathbf{000}\times {\mathbf{10}}^{\mathbf{0}}$	$1.511\times {10}^3$	$3.457\times {10}^3$	$7.142\times {10}^3$	$1.088\times {10}^2$	$4.529\times {10}^4$	$2.481\times {10}^3$
F12	$\mathbf{1}.\mathbf{060}\times {\mathbf{10}}^{\mathbf{4}}$	$\mathbf{1}.\mathbf{963}\times {\mathbf{10}}^{\mathbf{5}}$	$3.828\times {10}^2$	$2.342\times {10}^2$	$3.472\times {10}^3$	$1.894\times {10}^2$	$1.090\times {10}^1$	$9.437\times {10}^2$
F13	$1.827\times {10}^3$	$4.214\times {10}^4$	$5.752\times {10}^1$	$1.917\times {10}^1$	$1.238\times {10}^1$	$1.057\times {10}^1$	$1.167\times {10}^0$	$1.611\times {10}^1$
F14	$9.980\times {10}^1$	$7.377\times {10}^{-12}$	$3.681\times {10}^0$	$3.527\times {10}^0$	$\mathbf{9}.\mathbf{980}\times {\mathbf{10}}^{\mathbf{1}}$	$\mathbf{1}.\mathbf{271}\times {\mathbf{10}}^{-\mathbf{16}}$	$3.356\times {10}^0$	$3.294\times {10}^0$
F15	$3.075\times {10}^4$	$3.863\times {10}^8$	$2.313\times {10}^3$	$6.120\times {10}^3$	$\mathbf{3}.\mathbf{075}\times {\mathbf{10}}^{\mathbf{4}}$	$\mathbf{3}.\mathbf{332}\times {\mathbf{10}}^{\mathbf{9}}$	$5.226\times {10}^3$	$8.495\times {10}^3$
F16	$-1.032\times {10}^0$	$3.824\times {10}^{-10}$	$-1.032\times {10}^0$	$5.130\times {10}^9$	$-\mathbf{1}.\mathbf{032}\times {\mathbf{10}}^{\mathbf{0}}$	$\mathbf{6}.\mathbf{046}\times {\mathbf{10}}^{-\mathbf{16}}$	$-1.032\times {10}^0$	$6.109\times {10}^6$
F17	$3.979\times {10}^1$	$1.914\times {10}^8$	$3.979\times {10}^1$	$4.034\times {10}^7$	$\mathbf{3}.\mathbf{979}\times {\mathbf{10}}^{\mathbf{1}}$	$\mathbf{0}.\mathbf{000}\times {\mathbf{10}}^{\mathbf{0}}$	$3.982\times {10}^1$	$3.379\times {10}^4$
F18	$3.000\times {10}^0$	$2.785\times {10}^9$	$3.000\times {10}^0$	$1.042\times {10}^5$	$\mathbf{3}.\mathbf{000}\times {\mathbf{10}}^{\mathbf{0}}$	$\mathbf{1}.\mathbf{385}\times {\mathbf{10}}^{-\mathbf{15}}$	$3.000\times {10}^0$	$1.649\times {10}^5$
F19	$-3.863\times {10}^0$	$4.499\times {10}^7$	$-3.862\times {10}^0$	$2.669\times {10}^3$	$-\mathbf{3}.\mathbf{863}\times {\mathbf{10}}^{\mathbf{0}}$	$\mathbf{2}.\mathbf{494}\times {\mathbf{10}}^{-\mathbf{15}}$	$-3.858\times {10}^0$	$3.204\times {10}^3$
F20	$-3.322\times {10}^0$	$1.335\times {10}^5$	$-3.305\times {10}^0$	$5.126\times {10}^2$	$-3.322\times {10}^0$	$2.500\times {10}^7$	$-3.247\times {10}^0$	$7.994\times {10}^2$
F21	$-1.015\times {10}^1$	$6.410\times {10}^4$	$-9.984\times {10}^0$	$9.225\times {10}^1$	$-9.788\times {10}^0$	$1.196\times {10}^0$	$-6.674\times {10}^0$	$2.127\times {10}^0$
F22	$-1.040\times {10}^1$	$7.553\times {10}^4$	$-1.040\times {10}^1$	$3.609\times {10}^4$	$-1.040\times {10}^1$	$1.396\times {10}^8$	$-7.482\times {10}^0$	$1.355\times {10}^0$
F23	$-1.054\times {10}^1$	$9.043\times {10}^4$	$-1.027\times {10}^1$	$1.481\times {10}^0$	$-1.054\times {10}^1$	$1.717\times {10}^9$	$-7.474\times {10}^0$	$1.178\times {10}^0$
	HGWOSCA		RWGWO		EGWO		PSOGWO
	Mean	Std	Mean	Std	Mean	Std	Mean	Std
F1	$2.110\times {10}^{-60}$	$4.918\times {10}^{-60}$	$3.610\times {10}^{-59}$	$6.021\times {10}^{-59}$	$1.148\times {10}^{-66}$	$5.831\times {10}^{-66}$	$2.294\times {10}^3$	$8.690\times {10}^3$
F2	$2.210\times {10}^{-35}$	$2.325\times {10}^{-35}$	$1.042\times {10}^{-34}$	$1.029\times {10}^{-34}$	$7.641\times {10}^{-40}$	$2.453\times {10}^{-39}$	$2.611\times {10}^4$	$1.429\times {10}^5$
F3	$3.240\times {10}^{-15}$	$8.878\times {10}^{-15}$	$1.299\times {10}^{-14}$	$3.241\times {10}^{-14}$	$5.041\times {10}^{-11}$	$2.690\times {10}^{-10}$	$7.001\times {10}^3$	$1.655\times {10}^4$
F4	$7.559\times {10}^{-15}$	$9.375\times {10}^{-15}$	$5.423\times {10}^{-14}$	$1.210\times {10}^{-13}$	$2.669\times {10}^{-4}$	$9.189\times {10}^{-4}$	$8.557\times {10}^0$	$1.774\times {10}^1$
F5	$2.702\times {10}^1$	$7.320\times {10}^{-1}$	$2.677\times {10}^1$	$8.616\times {10}^{-1}$	$2.781\times {10}^1$	$8.242\times {10}^{-1}$	$3.980\times {10}^6$	$1.045\times {10}^7$
F6	$6.026\times {10}^{-1}$	$3.574\times {10}^{-1}$	$4.173\times {10}^{-1}$	$2.537\times {10}^{-1}$	$3.419\times {10}^0$	$5.527\times {10}^{-1}$	$1.841\times {10}^3$	$5.270\times {10}^3$
F7	$7.093\times {10}^{-4}$	$3.457\times {10}^{-4}$	$8.766\times {10}^{-4}$	$4.871\times {10}^{-4}$	$3.693\times {10}^{-3}$	$2.237\times {10}^{-3}$	$1.075\times {10}^{-1}$	$4.728\times {10}^{-1}$
F8	$-5.981\times {10}^3$	$9.382\times {10}^2$	$-8.065\times {10}^3$	$6.192\times {10}^2$	$-7.408\times {10}^3$	$6.512\times {10}^2$	$-6.790\times {10}^3$	$1.839\times {10}^3$
F9	$1.556\times {10}^{-1}$	$8.523\times {10}^{-1}$	$4.169\times {10}^{-14}$	$1.558\times {10}^{-13}$	$1.666\times {10}^2$	$4.093\times {10}^1$	$3.975\times {10}^1$	$2.266\times {10}^1$
F10	$1.451\times {10}^{-14}$	$1.638\times {10}^{-15}$	$1.652\times {10}^{-14}$	$3.178\times {10}^{-15}$	$2.065\times {10}^{-1}$	$7.863\times {10}^{-1}$	$4.575\times {10}^0$	$6.102\times {10}^0$
F11	$5.283\times {10}^{-4}$	$2.894\times {10}^{-3}$	$2.089\times {10}^{-3}$	$9.132\times {10}^{-3}$	$8.740\times {10}^{-3}$	$1.466\times {10}^{-2}$	$5.300\times {10}^1$	$1.058\times {10}^2$
F12	$3.033\times {10}^{-2}$	$1.568\times {10}^{-2}$	$2.456\times {10}^{-2}$	$9.966\times {10}^{-3}$	$2.328\times {10}^0$	$3.159\times {10}^0$	$2.177\times {10}^6$	$6.982\times {10}^6$
F13	$4.724\times {10}^{-1}$	$2.215\times {10}^{-1}$	$4.490\times {10}^{-1}$	$2.012\times {10}^{-1}$	$2.464\times {10}^0$	$3.606\times {10}^{-1}$	$2.975\times {10}^6$	$1.629\times {10}^7$
F14	$3.124\times {10}^0$	$3.398\times {10}^0$	$1.097\times {10}^0$	$3.033\times {10}^{-1}$	$9.538\times {10}^0$	$4.624\times {10}^0$	$2.578\times {10}^0$	$2.078\times {10}^0$
F15	$2.985\times {10}^{-3}$	$6.933\times {10}^{-3}$	$6.018\times {10}^{-4}$	$1.607\times {10}^{-3}$	$1.763\times {10}^{-3}$	$5.070\times {10}^{-3}$	$4.553\times {10}^{-3}$	$8.066\times {10}^{-3}$
F16	$-1.032\times {10}^0$	$8.592\times {10}^{-9}$	$-1.032\times {10}^0$	$5.860\times {10}^{-9}$	$-1.032\times {10}^0$	$1.593\times {10}^{-9}$	$-1.032\times {10}^0$	$3.153\times {10}^{-6}$
F17	$3.979\times {10}^{-1}$	$4.925\times {10}^{-7}$	$3.979\times {10}^{-1}$	$2.171\times {10}^{-5}$	$3.979\times {10}^{-1}$	$5.481\times {10}^{-8}$	$3.980\times {10}^{-1}$	$5.329\times {10}^{-4}$
F18	$3.000\times {10}^0$	$1.419\times {10}^{-5}$	$3.000\times {10}^0$	$1.353\times {10}^{-5}$	$3.000\times {10}^0$	$5.462\times {10}^{-7}$	$3.001\times {10}^0$	$5.172\times {10}^{-3}$
F19	$-3.861\times {10}^0$	$3.020\times {10}^{-3}$	$-3.862\times {10}^0$	$1.336\times {10}^{-3}$	$-3.861\times {10}^0$	$3.446\times {10}^{-3}$	$-3.860\times {10}^0$	$3.767\times {10}^{-3}$
F20	$-3.285\times {10}^0$	$6.915\times {10}^{-2}$	$-3.293\times {10}^0$	$6.277\times {10}^{-2}$	$-3.306\times {10}^0$	$8.792\times {10}^{-2}$	$-3.173\times {10}^0$	$2.244\times {10}^{-1}$
F21	$-9.744\times {10}^0$	$1.580\times {10}^0$	$-9.648\times {10}^0$	$1.542\times {10}^0$	$-5.706\times {10}^0$	$2.432\times {10}^0$	$-8.159\times {10}^0$	$2.934\times {10}^0$
F22	$-1.040\times {10}^1$	$3.213\times {10}^{-4}$	$-1.040\times {10}^1$	$2.113\times {10}^{-4}$	$-1.040\times {10}^1$	$6.554\times {10}^{-4}$	$-8.947\times {10}^0$	$2.706\times {10}^0$
F23	$-1.036\times {10}^1$	$9.872\times {10}^{-1}$	$-1.027\times {10}^1$	$1.481\times {10}^0$	$-9.454\times {10}^0$	$2.805\times {10}^0$	$-1.002\times {10}^1$	$1.309\times {10}^0$
	PSO		MFO		SSA		WOA
	Mean	Std	Mean	Std	Mean	Std	Mean	Std
F1	$6.502\times {10}^{-8}$	$3.198\times {10}^{-7}$	$6.667\times {10}^2$	$2.537\times {10}^3$	$1.248\times {10}^{-8}$	$3.180\times {10}^{-9}$	$1.561\times {10}^{-150}$	$8.464\times {10}^{-150}$
F2	$3.143\times {10}^{-4}$	$4.980\times {10}^{-4}$	$4.333\times {10}^1$	$1.953\times {10}^1$	$7.704\times {10}^{-1}$	$9.766\times {10}^{-1}$	$4.748\times {10}^{-101}$	$2.535\times {10}^{-100}$
F3	$1.457\times {10}^1$	$6.441\times {10}^0$	$1.792\times {10}^4$	$1.309\times {10}^4$	$2.644\times {10}^2$	$1.757\times {10}^2$	$2.948\times {10}^3$	$1.683\times {10}^3$
F4	$6.037\times {10}^{-1}$	$1.161\times {10}^{-1}$	$6.781\times {10}^1$	$8.798\times {10}^0$	$6.767\times {10}^0$	$2.862\times {10}^0$	$4.604\times {10}^1$	$3.510\times {10}^1$
F5	$5.793\times {10}^1$	$4.544\times {10}^1$	$2.559\times {10}^6$	$1.383\times {10}^7$	$9.254\times {10}^1$	$1.034\times {10}^2$	$2.715\times {10}^1$	$5.562\times {10}^{-1}$
F6	$3.142\times {10}^{-8}$	$6.443\times {10}^{-8}$	$2.693\times {10}^3$	$5.261\times {10}^3$	$\mathbf{1}.\mathbf{330}\times {\mathbf{10}}^{-\mathbf{8}}$	$\mathbf{2}.\mathbf{226}\times {\mathbf{10}}^{-\mathbf{9}}$	$8.895\times {10}^{-2}$	$9.105\times {10}^{-2}$
F7	$6.551\times {10}^{-2}$	$1.837\times {10}^{-2}$	$8.570\times {10}^{-1}$	$2.102\times {10}^0$	$1.108\times {10}^{-1}$	$4.311\times {10}^{-2}$	$1.488\times {10}^{-3}$	$1.826\times {10}^{-3}$
F8	$-5.933\times {10}^3$	$7.651\times {10}^2$	$-8.022\times {10}^3$	$7.484\times {10}^2$	$-7.696\times {10}^3$	$8.065\times {10}^2$	$-\mathbf{1}.\mathbf{139}\times {\mathbf{10}}^{\mathbf{4}}$	$\mathbf{1}.\mathbf{664}\times {\mathbf{10}}^{\mathbf{3}}$
F9	$4.461\times {10}^1$	$1.290\times {10}^1$	$2.029\times {10}^2$	$3.750\times {10}^1$	$5.519\times {10}^1$	$1.460\times {10}^1$	$3.790\times {10}^{-15}$	$1.442\times {10}^{-14}$
F10	$5.493\times {10}^{-2}$	$3.005\times {10}^{-1}$	$1.986\times {10}^1$	$2.037\times {10}^{-1}$	$2.161\times {10}^0$	$8.076\times {10}^{-1}$	$4.086\times {10}^{-15}$	$2.351\times {10}^{-15}$
F11	$9.273\times {10}^{-3}$	$9.641\times {10}^{-3}$	$6.057\times {10}^0$	$2.295\times {10}^1$	$9.597\times {10}^{-3}$	$1.184\times {10}^{-2}$	$2.400\times {10}^{-3}$	$1.315\times {10}^{-2}$
F12	$6.911\times {10}^{-3}$	$2.630\times {10}^{-2}$	$8.095\times {10}^{-1}$	$1.280\times {10}^0$	$6.713\times {10}^0$	$2.662\times {10}^0$	$9.515\times {10}^{-3}$	$7.152\times {10}^{-3}$
F13	$5.494\times {10}^{-3}$	$5.588\times {10}^{-3}$	$4.101\times {10}^7$	$1.251\times {10}^8$	$3.199\times {10}^0$	$9.046\times {10}^0$	$2.356\times {10}^{-1}$	$2.071\times {10}^{-1}$
F14	$3.891\times {10}^0$	$2.418\times {10}^0$	$1.655\times {10}^0$	$1.719\times {10}^0$	$9.980\times {10}^{-1}$	$2.387\times {10}^{-16}$	$1.847\times {10}^0$	$2.497\times {10}^0$
F15	$8.333\times {10}^{-4}$	$1.316\times {10}^{-4}$	$1.723\times {10}^{-3}$	$3.547\times {10}^{-3}$	$1.569\times {10}^{-3}$	$3.559\times {10}^{-3}$	$6.127\times {10}^{-4}$	$2.876\times {10}^{-4}$
F16	$-1.032\times {10}^0$	$6.712\times {10}^{-16}$	$-1.032\times {10}^0$	$6.775\times {10}^{-16}$	$-1.032\times {10}^0$	$7.214\times {10}^{-15}$	$-1.032\times {10}^0$	$2.096\times {10}^{-11}$
F17	$3.979\times {10}^{-1}$	$0.000\times {10}^0$	$3.979\times {10}^{-1}$	$0.000\times {10}^0$	$3.979\times {10}^{-1}$	$2.024\times {10}^{-14}$	$3.979\times {10}^{-1}$	$6.162\times {10}^{-6}$
F18	$3.000\times {10}^0$	$7.142\times {10}^{-16}$	$3.000\times {10}^0$	$1.662\times {10}^{-15}$	$3.000\times {10}^0$	$6.351\times {10}^{-14}$	$3.000\times {10}^0$	$2.203\times {10}^{-5}$
F19	$-3.863\times {10}^0$	$2.710\times {10}^{-15}$	$-3.863\times {10}^0$	$2.710\times {10}^{-15}$	$-3.863\times {10}^0$	$2.058\times {10}^{-14}$	$-3.861\times {10}^0$	$2.064\times {10}^{-3}$
F20	$-\mathbf{3}.\mathbf{322}\times {\mathbf{10}}^{\mathbf{0}}$	$\mathbf{1}.\mathbf{355}\times {\mathbf{10}}^{-\mathbf{15}}$	$-3.310\times {10}^0$	$4.740\times {10}^{-2}$	$-3.274\times {10}^0$	$6.015\times {10}^{-2}$	$-3.321\times {10}^0$	$9.200\times {10}^{-4}$
F21	$-5.869\times {10}^0$	$2.297\times {10}^0$	$-6.576\times {10}^0$	$3.687\times {10}^0$	$-7.052\times {10}^0$	$3.066\times {10}^0$	$-9.982\times {10}^0$	$9.305\times {10}^{-1}$
F22	$-\mathbf{1}.\mathbf{040}\times {\mathbf{10}}^{\mathbf{1}}$	$\mathbf{8}.\mathbf{727}\times {\mathbf{10}}^{-\mathbf{16}}$	$-9.512\times {10}^0$	$2.309\times {10}^0$	$-9.461\times {10}^0$	$2.472\times {10}^0$	$-7.908\times {10}^0$	$2.913\times {10}^0$
F23	$-1.036\times {10}^1$	$9.787\times {10}^{-1}$	$-4.470\times {10}^0$	$3.448\times {10}^0$	$-9.020\times {10}^0$	$2.867\times {10}^0$	$-7.142\times {10}^0$	$3.358\times {10}^0$
	HSCA		HCLPSO		PSOGSA		mSCA
	Mean	Std	Mean	Std	Mean	Std	Mean	Std
F1	$3.397\times {10}^{-12}$	$1.276\times {10}^{-11}$	$1.443\times {10}^{-7}$	$1.468\times {10}^{-7}$	$1.000\times {10}^3$	$3.051\times {10}^3$	$3.046\times {10}^{-43}$	$1.078\times {10}^{-42}$
F2	$2.560\times {10}^{-9}$	$9.389\times {10}^{-9}$	$7.538\times {10}^{-5}$	$6.356\times {10}^{-5}$	$1.235\times {10}^1$	$3.115\times {10}^1$	$1.776\times {10}^{-26}$	$9.294\times {10}^{-26}$
F3	$2.468\times {10}^4$	$2.294\times {10}^4$	$2.144\times {10}^2$	$1.209\times {10}^2$	$6.785\times {10}^3$	$6.192\times {10}^3$	$8.700\times {10}^{-8}$	$4.606\times {10}^{-7}$
F4	$6.826\times {10}^1$	$3.238\times {10}^1$	$2.609\times {10}^0$	$7.573\times {10}^{-1}$	$6.713\times {10}^1$	$2.102\times {10}^1$	$1.282\times {10}^{-12}$	$6.887\times {10}^{-12}$
F5	$2.779\times {10}^1$	$6.040\times {10}^{-1}$	$5.561\times {10}^1$	$3.740\times {10}^1$	$6.034\times {10}^3$	$2.283\times {10}^4$	$2.699\times {10}^1$	$6.103\times {10}^{-1}$
F6	$2.206\times {10}^0$	$2.999\times {10}^{-1}$	$5.383\times {10}^{-7}$	$1.465\times {10}^{-6}$	$1.677\times {10}^3$	$3.813\times {10}^3$	$8.297\times {10}^{-1}$	$4.832\times {10}^{-1}$
F7	$9.681\times {10}^{-3}$	$1.017\times {10}^{-2}$	$1.960\times {10}^{-2}$	$9.357\times {10}^{-3}$	$7.324\times {10}^{-2}$	$3.546\times {10}^{-2}$	$1.982\times {10}^{-4}$	$1.572\times {10}^{-4}$
F8	$-4.159\times {10}^3$	$3.152\times {10}^2$	$-1.096\times {10}^4$	$3.394\times {10}^2$	$-7.161\times {10}^3$	$8.738\times {10}^2$	$-6.056\times {10}^3$	$8.557\times {10}^2$
F9	$8.641\times {10}^0$	$2.245\times {10}^1$	$1.657\times {10}^1$	$5.686\times {10}^0$	$1.609\times {10}^2$	$3.848\times {10}^1$	$0.000\times {10}^0$	$0.000\times {10}^0$
F10	$1.755\times {10}^1$	$7.000\times {10}^0$	$1.927\times {10}^{-4}$	$4.538\times {10}^{-4}$	$1.745\times {10}^1$	$2.875\times {10}^0$	$6.112\times {10}^{-12}$	$2.815\times {10}^{-11}$
F11	$1.150\times {10}^{-2}$	$2.371\times {10}^{-2}$	$7.714\times {10}^{-3}$	$9.874\times {10}^{-3}$	$6.094\times {10}^0$	$2.282\times {10}^1$	$0.000\times {10}^0$	$0.000\times {10}^0$
F12	$2.377\times {10}^{-1}$	$4.844\times {10}^{-2}$	$6.911\times {10}^{-3}$	$2.630\times {10}^{-2}$	$7.466\times {10}^0$	$4.759\times {10}^0$	$3.678\times {10}^{-2}$	$2.148\times {10}^{-2}$
F13	$1.801\times {10}^0$	$3.950\times {10}^{-1}$	$\mathbf{1}.\mathbf{100}\times {\mathbf{10}}^{-\mathbf{3}}$	$\mathbf{3}.\mathbf{352}\times {\mathbf{10}}^{-\mathbf{3}}$	$1.367\times {10}^7$	$7.487\times {10}^7$	$6.555\times {10}^{-1}$	$2.280\times {10}^{-1}$
F14	$9.981\times {10}^{-1}$	$2.886\times {10}^{-4}$	$9.980\times {10}^{-1}$	$0.000\times {10}^0$	$4.042\times {10}^0$	$3.535\times {10}^0$	$1.197\times {10}^0$	$6.054\times {10}^{-1}$
F15	$7.572\times {10}^{-4}$	$1.921\times {10}^{-4}$	$4.352\times {10}^{-4}$	$9.822\times {10}^{-5}$	$4.955\times {10}^{-3}$	$7.725\times {10}^{-3}$	$7.723\times {10}^{-4}$	$3.876\times {10}^{-4}$
F16	$-1.032\times {10}^0$	$7.172\times {10}^{-6}$	$-1.032\times {10}^0$	$6.454\times {10}^{-16}$	$-1.032\times {10}^0$	$5.608\times {10}^{-16}$	$-1.032\times {10}^0$	$3.856\times {10}^{-8}$
F17	$3.981\times {10}^{-1}$	$2.335\times {10}^{-4}$	$3.979\times {10}^{-1}$	$0.000\times {10}^0$	$3.979\times {10}^{-1}$	$0.000\times {10}^0$	$3.979\times {10}^{-1}$	$3.559\times {10}^{-6}$
F18	$3.000\times {10}^0$	$7.177\times {10}^{-6}$	$3.000\times {10}^0$	$7.283\times {10}^{-16}$	$3.000\times {10}^0$	$2.438\times {10}^{-15}$	$3.000\times {10}^0$	$2.343\times {10}^{-5}$
F19	$-3.856\times {10}^0$	$2.984\times {10}^{-3}$	$-3.863\times {10}^0$	$2.696\times {10}^{-15}$	$-3.863\times {10}^0$	$2.470\times {10}^{-15}$	$-3.862\times {10}^0$	$1.838\times {10}^{-3}$
F20	$-3.029\times {10}^0$	$2.301\times {10}^{-1}$	$-3.322\times {10}^0$	$1.372\times {10}^{-15}$	$-3.322\times {10}^0$	$1.412\times {10}^{-15}$	$-3.193\times {10}^0$	$5.201\times {10}^{-2}$
F21	$-3.390\times {10}^0$	$2.672\times {10}^0$	$-\mathbf{1}.\mathbf{015}\times {\mathbf{10}}^{\mathbf{1}}$	$\mathbf{6}.\mathbf{019}\times {\mathbf{10}}^{-\mathbf{15}}$	$-3.487\times {10}^0$	$1.161\times {10}^0$	$-4.978\times {10}^0$	$1.894\times {10}^0$
F22	$-6.127\times {10}^0$	$1.968\times {10}^0$	$-1.040\times {10}^1$	$7.376\times {10}^{-16}$	$-9.067\times {10}^0$	$2.717\times {10}^0$	$-8.098\times {10}^0$	$2.678\times {10}^0$
F23	$-5.985\times {10}^0$	$2.076\times {10}^0$	$-\mathbf{1}.\mathbf{054}\times {\mathbf{10}}^{\mathbf{1}}$	$\mathbf{1}.\mathbf{234}\times {\mathbf{10}}^{-\mathbf{15}}$	$-2.441\times {10}^0$	$8.119\times {10}^{-2}$	$-8.732\times {10}^0$	$2.592\times {10}^0$

shows the average and standard deviation of the 16 algorithms after 30 independent runs.

Table 7. Results of multiple-problem Wilcoxon test.

CDL-DGWO VS.	+	−	≈	R+	R−	p-Value	a = 0.05
GWO	22	1	0	267	9	0.00009	+
IGWO	13	10	0	178	98	0.22376	≈
LGWO	23	0	0	276	0	0.00003	+
HGWOSCA	22	1	0	267	9	0.00009	+
RWGWO	22	1	0	265	11	0.00011	+
EGWO	22	1	0	268	8	0.00008	+
PSOGWO	23	0	0	276	0	0.00003	+
PSO	16	7	0	240	36	0.00192	+
MFO	19	4	0	266	10	0.00010	+
SSA	17	6	0	251	25	0.00059	+
WOA	21	2	0	249	27	0.00074	+
HSCA	23	0	0	276	0	0.00003	+
HCLPSO	11	12	0	165	111	0.41153	≈
PSOGSA	18	5	0	261	15	0.00018	+
mSCA	21	0	2	276	0	0.00006	+

presents the results of the multiple-problem Wilcoxon test.

Table 8. Friedman test results of the 16 algorithms.

	Rank	Ave		Rank	Ave
CDL-DGWO	1	3.2	PSO	6	7.3
GWO	8	7.7	MFO	13	11.2
IGWO	2	4.4	SSA	11	9.7
LGWO	12	10.1	WOA	7	7.3
HGWOSCA	5	7.3	HSCA	15	12.4
RWGWO	4	6.5	HCLPSO	3	5.7
EGWO	10	9.6	PSOGSA	14	11.9
PSOGWO	16	13.4	mSCA	9	8.3

shows the Friedman test results for each algorithm across the benchmark problems.

The statistical outcomes for “R+” represent a positive rank, indicating the superiority of CDL-DGWO compared to its counterparts. Conversely, “R−” denotes a negative rank, suggesting that CDL-DGWO is worse in performance relative to other algorithms. The “+/−/≈” represent that the CDL-DGWO is significantly better than, worse than, and similar to its compared algorithm on the associated function, respectively. Friedman test is used to provide a ranking of all algorithms on a group of test problems, thereby providing a more transparent assessment of the algorithm’s efficacy. “Ave” represents the average ranking, and “Rank” denotes the final ranking.

At the 0.05 level, when p < 0.05, it indicates that CDL-DGWO is significantly better than the compared algorithm, while p > 0.05 indicates that there is no significant difference in performance between the two compared algorithms.

Regarding the 23 benchmark problems, Table 6 indicates that CDL-DGWO outperforms other algorithms on F1-F4, F7, F9, F10, F11, and F12. CDL-DGWO also exhibits strong competitiveness in the remaining test problems. IGWO demonstrates superior results to other algorithms on problems F5 and F14–F19. SSA outperforms other algorithms on problem F6; WOA does so on F8; HCLPSO does so on F13, F21, and F23. Table 7 reveals no significant difference between CDL-DGWO, IGWO, and HCLPSO for the 23 test problems, but a significant difference when compared to the other 13 algorithms. In addition, Table 8 presents the ranking of 16 algorithms, with an average ranking of 3.2 and a final ranking of 1 for CDL-DGWO.

For a more intuitive demonstration of the CDL-DGWO’s performance, convergence curves have been plotted, comparing it with other swarm intelligence algorithms and GWO variants on selected functions. Figure 2 shows the convergence curves of CDL-DGWO for problems F1, F4, F10, F12, F21, and F23. These include problems with single peaks, multiple peaks, and fixed-dimensional multimodal peaks.

On unimodal functions (F1, F4), CDL-DGWO does not exhibit a superior convergence rate initially but outperforms other algorithms in the later stages, achieving higher precision. For multimodal functions (F10, F12), it demonstrates rapid early convergence, though the pace of improvement slows considerably thereafter. A similar two-phase convergence pattern—characterized by a fast start followed by gradual refinement—is also observed on functions F21 and F23.

To sum up, CDL-DGWO can find competitive solutions to benchmark testing problems. Double swarm technology and dimensional learning can maintain population diversity and enhance the search space exploration efficiency. Covariance strategies can improve information exchange between variables, thereby enhancing the performance of GWO.

4.4. Comparisons on CEC 2017

To further assess the performance of CDL-DGWO, this study analyzes its performance based on the CEC 2017 benchmark and compares it with various algorithms, including GWO, IGWO, LGWO, HGWOSCA, RWGWO, EGWO, PSOGWO, MIGWO, PSO, MFO, SSA, WOA, HSCA, PSOGSA, and mSCA.

Table 9. Comparisons of CDL-DGWO and other algorithms on CEC 2017.

	CDL-DGWO		GWO		IGWO		LGWO
	Mean	Std	Mean	Std	Mean	Std	Mean	Std
f1	$7.183\times {10}^4$	$2.616\times {10}^4$	$2.508\times {10}^9$	$2.050\times {10}^9$	$2.958\times {10}^3$	$3.985\times {10}^3$	$4.265\times {10}^9$	$1.211\times {10}^9$
f3	$4.513\times {10}^2$	$9.020\times {10}^1$	$4.053\times {10}^4$	$1.031\times {10}^4$	$2.794\times {10}^3$	$2.100\times {10}^3$	$2.628\times {10}^4$	$7.301\times {10}^3$
f4	$4.895\times {10}^2$	$1.962\times {10}^1$	$5.800\times {10}^2$	$6.143\times {10}^1$	$4.808\times {10}^2$	$2.247\times {10}^1$	$7.119\times {10}^2$	$1.234\times {10}^2$
f5	$5.489\times {10}^2$	$1.563\times {10}^1$	$5.981\times {10}^2$	$2.627\times {10}^1$	$\mathbf{5}.\mathbf{415}\times {\mathbf{10}}^{\mathbf{2}}$	$\mathbf{3}.\mathbf{279}\times {\mathbf{10}}^{\mathbf{1}}$	$7.197\times {10}^2$	$2.275\times {10}^1$
f6	$6.007\times {10}^2$	$4.127\times {10}^{-1}$	$6.069\times {10}^2$	$3.709\times {10}^0$	$6.000\times {10}^2$	$7.742\times {10}^{-2}$	$6.288\times {10}^2$	$4.654\times {10}^0$
f7	$\mathbf{7}.\mathbf{722}\times {\mathbf{10}}^{\mathbf{2}}$	$\mathbf{9}.\mathbf{598}\times {\mathbf{10}}^{\mathbf{0}}$	$8.602\times {10}^2$	$4.584\times {10}^1$	$7.830\times {10}^2$	$4.624\times {10}^1$	$1.017\times {10}^3$	$3.702\times {10}^1$
f8	$8.441\times {10}^2$	$9.207\times {10}^0$	$8.914\times {10}^2$	$2.529\times {10}^1$	$\mathbf{8}.\mathbf{361}\times {\mathbf{10}}^{\mathbf{2}}$	$\mathbf{9}.\mathbf{587}\times {\mathbf{10}}^{\mathbf{0}}$	$9.893\times {10}^2$	$2.155\times {10}^1$
f9	$9.033\times {10}^2$	$2.461\times {10}^0$	$1.859\times {10}^3$	$7.011\times {10}^2$	$\mathbf{9}.\mathbf{000}\times {\mathbf{10}}^{\mathbf{2}}$	$\mathbf{1}.\mathbf{415}\times {\mathbf{10}}^{-\mathbf{2}}$	$3.101\times {10}^3$	$6.791\times {10}^2$
f10	$\mathbf{3}.\mathbf{178}\times {\mathbf{10}}^{\mathbf{3}}$	$\mathbf{8}.\mathbf{812}\times {\mathbf{10}}^{\mathbf{2}}$	$3.819\times {10}^3$	$5.983\times {10}^2$	$4.781\times {10}^3$	$2.322\times {10}^3$	$7.377\times {10}^3$	$6.386\times {10}^2$
f11	$1.215\times {10}^3$	$3.289\times {10}^1$	$1.752\times {10}^3$	$6.526\times {10}^2$	$\mathbf{1}.\mathbf{150}\times {\mathbf{10}}^{\mathbf{3}}$	$\mathbf{3}.\mathbf{044}\times {\mathbf{10}}^{\mathbf{1}}$	$1.681\times {10}^3$	$3.622\times {10}^2$
f12	$2.901\times {10}^6$	$1.654\times {10}^6$	$6.447\times {10}^7$	$7.641\times {10}^7$	$6.921\times {10}^5$	$5.769\times {10}^5$	$3.042\times {10}^8$	$1.507\times {10}^8$
f13	$1.052\times {10}^5$	$4.797\times {10}^4$	$1.891\times {10}^7$	$7.078\times {10}^7$	$5.122\times {10}^4$	$2.883\times {10}^4$	$1.069\times {10}^8$	$8.052\times {10}^7$
f14	$1.067\times {10}^4$	$7.038\times {10}^3$	$1.954\times {10}^5$	$3.830\times {10}^5$	$3.309\times {10}^3$	$1.880\times {10}^3$	$2.006\times {10}^5$	$2.569\times {10}^5$
f15	$3.917\times {10}^4$	$2.745\times {10}^4$	$3.476\times {10}^5$	$7.603\times {10}^5$	$1.135\times {10}^4$	$1.270\times {10}^4$	$1.647\times {10}^6$	$1.848\times {10}^6$
f16	$1.981\times {10}^3$	$1.949\times {10}^2$	$2.355\times {10}^3$	$2.193\times {10}^2$	$\mathbf{1}.\mathbf{855}\times {\mathbf{10}}^{\mathbf{3}}$	$2.410\times {10}^2$	$2.952\times {10}^3$	$3.035\times {10}^2$
f17	$1.813\times {10}^3$	$7.537\times {10}^1$	$1.962\times {10}^3$	$1.326\times {10}^2$	$\mathbf{1}.\mathbf{805}\times {\mathbf{10}}^{\mathbf{3}}$	$6.059\times {10}^1$	$2.109\times {10}^3$	$1.296\times {10}^2$
f18	$1.910\times {10}^5$	$1.739\times {10}^5$	$7.865\times {10}^5$	$9.127\times {10}^5$	$9.726\times {10}^4$	$5.664\times {10}^4$	$1.033\times {10}^6$	$6.360\times {10}^5$
f19	$1.604\times {10}^4$	$1.626\times {10}^4$	$2.001\times {10}^6$	$5.366\times {10}^6$	$6.462\times {10}^3$	$5.908\times {10}^3$	$6.249\times {10}^6$	$3.526\times {10}^6$
f20	$2.171\times {10}^3$	$7.226\times {10}^1$	$2.391\times {10}^3$	$1.260\times {10}^2$	$\mathbf{2}.\mathbf{160}\times {\mathbf{10}}^{\mathbf{3}}$	$\mathbf{6}.\mathbf{996}\times {\mathbf{10}}^{\mathbf{1}}$	$2.496\times {10}^3$	$1.274\times {10}^2$
f21	$2.342\times {10}^3$	$1.074\times {10}^1$	$2.387\times {10}^3$	$2.914\times {10}^1$	$\mathbf{2}.\mathbf{337}\times {\mathbf{10}}^{\mathbf{3}}$	$\mathbf{3}.\mathbf{047}\times {\mathbf{10}}^{\mathbf{1}}$	$2.500\times {10}^3$	$2.145\times {10}^1$
f22	$2.685\times {10}^3$	$8.483\times {10}^2$	$4.696\times {10}^3$	$1.603\times {10}^3$	$2.843\times {10}^3$	$1.484\times {10}^3$	$7.773\times {10}^3$	$2.549\times {10}^3$
f23	$2.693\times {10}^3$	$2.557\times {10}^1$	$2.765\times {10}^3$	$4.404\times {10}^1$	$\mathbf{2}.\mathbf{680}\times {\mathbf{10}}^{\mathbf{3}}$	$\mathbf{2}.\mathbf{167}\times {\mathbf{10}}^{\mathbf{1}}$	$2.886\times {10}^3$	$2.032\times {10}^1$
f24	$\mathbf{2}.\mathbf{855}\times {\mathbf{10}}^{\mathbf{3}}$	$\mathbf{1}.\mathbf{071}\times {\mathbf{10}}^{\mathbf{1}}$	$2.927\times {10}^3$	$5.373\times {10}^1$	$2.858\times {10}^3$	$3.882\times {10}^1$	$3.060\times {10}^3$	$2.195\times {10}^1$
f25	$2.887\times {10}^3$	$1.075\times {10}^0$	$3.005\times {10}^3$	$7.306\times {10}^1$	$2.887\times {10}^3$	$2.033\times {10}^0$	$3.013\times {10}^3$	$3.559\times {10}^1$
f26	$\mathbf{3}.\mathbf{462}\times {\mathbf{10}}^{\mathbf{3}}$	$\mathbf{5}.\mathbf{632}\times {\mathbf{10}}^{\mathbf{2}}$	$4.696\times {10}^3$	$4.182\times {10}^2$	$3.802\times {10}^3$	$1.139\times {10}^2$	$5.797\times {10}^3$	$1.934\times {10}^2$
f27	$3.207\times {10}^3$	$8.679\times {10}^0$	$3.251\times {10}^3$	$2.794\times {10}^1$	$3.198\times {10}^3$	$1.052\times {10}^1$	$3.262\times {10}^3$	$1.598\times {10}^1$
f28	$3.213\times {10}^3$	$1.505\times {10}^1$	$3.409\times {10}^3$	$8.442\times {10}^1$	$3.187\times {10}^3$	$3.844\times {10}^1$	$3.436\times {10}^3$	$6.555\times {10}^1$
f29	$3.466\times {10}^3$	$8.141\times {10}^1$	$3.727\times {10}^3$	$1.568\times {10}^2$	$\mathbf{3}.\mathbf{400}\times {\mathbf{10}}^{\mathbf{3}}$	$\mathbf{6}.\mathbf{621}\times {\mathbf{10}}^{\mathbf{1}}$	$4.046\times {10}^3$	$2.016\times {10}^2$
f30	$1.901\times {10}^5$	$1.010\times {10}^5$	$7.589\times {10}^6$	$5.473\times {10}^6$	$5.246\times {10}^4$	$3.458\times {10}^4$	$2.331\times {10}^7$	$8.817\times {10}^6$
	HGWOSCA		RWGWO		EGWO		PSOGWO
	Mean	Std	Mean	Std	Mean	Std	Mean	Std
f1	$1.892\times {10}^9$	$1.124\times {10}^9$	$8.026\times {10}^4$	$3.752\times {10}^4$	$1.739\times {10}^{10}$	$7.129\times {10}^9$	$2.801\times {10}^9$	$7.624\times {10}^9$
f3	$3.063\times {10}^4$	$9.958\times {10}^3$	$3.348\times {10}^2$	$2.855\times {10}^1$	$5.511\times {10}^4$	$7.082\times {10}^3$	$3.447\times {10}^4$	$2.447\times {10}^4$
f4	$5.927\times {10}^2$	$8.847\times {10}^1$	$4.871\times {10}^2$	$1.515\times {10}^1$	$3.540\times {10}^3$	$2.152\times {10}^3$	$5.957\times {10}^2$	$1.945\times {10}^2$
f5	$5.924\times {10}^2$	$2.079\times {10}^1$	$5.693\times {10}^2$	$2.064\times {10}^1$	$8.281\times {10}^2$	$6.851\times {10}^1$	$6.037\times {10}^2$	$6.556\times {10}^1$
f6	$6.065\times {10}^2$	$3.887\times {10}^0$	$6.005\times {10}^2$	$5.531\times {10}^{-1}$	$6.733\times {10}^2$	$1.018\times {10}^1$	$6.152\times {10}^2$	$1.771\times {10}^1$
f7	$8.498\times {10}^2$	$3.531\times {10}^1$	$8.043\times {10}^2$	$1.804\times {10}^1$	$1.155\times {10}^3$	$7.159\times {10}^1$	$9.662\times {10}^2$	$2.634\times {10}^2$
f8	$8.848\times {10}^2$	$2.094\times {10}^1$	$8.620\times {10}^2$	$1.592\times {10}^1$	$1.046\times {10}^3$	$4.306\times {10}^1$	$9.244\times {10}^2$	$8.061\times {10}^1$
f9	$1.649\times {10}^3$	$5.717\times {10}^2$	$9.425\times {10}^2$	$1.698\times {10}^2$	$1.192\times {10}^4$	$2.390\times {10}^3$	$2.528\times {10}^3$	$1.897\times {10}^3$
f10	$3.991\times {10}^3$	$5.381\times {10}^2$	$4.014\times {10}^3$	$4.445\times {10}^2$	$5.432\times {10}^3$	$7.264\times {10}^2$	$4.242\times {10}^3$	$9.858\times {10}^2$
f11	$1.673\times {10}^3$	$6.090\times {10}^2$	$1.204\times {10}^3$	$3.410\times {10}^1$	$4.927\times {10}^3$	$1.481\times {10}^3$	$1.712\times {10}^3$	$8.868\times {10}^2$
f12	$6.010\times {10}^7$	$6.450\times {10}^7$	$1.722\times {10}^6$	$1.463\times {10}^6$	$1.845\times {10}^9$	$1.918\times {10}^9$	$7.160\times {10}^7$	$3.061\times {10}^8$
f13	$1.768\times {10}^7$	$5.356\times {10}^7$	$7.785\times {10}^4$	$4.263\times {10}^4$	$2.908\times {10}^9$	$3.760\times {10}^9$	$1.327\times {10}^7$	$5.070\times {10}^7$
f14	$1.744\times {10}^5$	$3.292\times {10}^5$	$2.243\times {10}^4$	$1.653\times {10}^4$	$2.430\times {10}^5$	$2.373\times {10}^5$	$2.531\times {10}^5$	$4.549\times {10}^5$
f15	$2.311\times {10}^5$	$5.384\times {10}^5$	$3.955\times {10}^4$	$2.229\times {10}^4$	$2.543\times {10}^7$	$6.438\times {10}^7$	$5.483\times {10}^6$	$2.109\times {10}^7$
f16	$2.368\times {10}^3$	$2.703\times {10}^2$	$2.268\times {10}^3$	$3.120\times {10}^2$	$4.010\times {10}^3$	$7.070\times {10}^2$	$2.689\times {10}^3$	$6.333\times {10}^2$
f17	$1.885\times {10}^3$	$9.808\times {10}^1$	$1.919\times {10}^3$	$1.124\times {10}^2$	$2.287\times {10}^3$	$1.912\times {10}^2$	$2.132\times {10}^3$	$2.230\times {10}^2$
f18	$5.731\times {10}^5$	$1.235\times {10}^6$	$2.136\times {10}^5$	$2.494\times {10}^5$	$9.151\times {10}^5$	$2.240\times {10}^6$	$2.272\times {10}^6$	$6.050\times {10}^6$
f19	$1.591\times {10}^6$	$3.131\times {10}^6$	$2.508\times {10}^4$	$1.981\times {10}^4$	$1.023\times {10}^7$	$9.289\times {10}^6$	$8.880\times {10}^6$	$4.481\times {10}^7$
f20	$2.365\times {10}^3$	$1.200\times {10}^2$	$2.311\times {10}^3$	$1.057\times {10}^2$	$2.795\times {10}^3$	$1.916\times {10}^2$	$2.390\times {10}^3$	$1.907\times {10}^2$
f21	$2.387\times {10}^3$	$2.226\times {10}^1$	$2.361\times {10}^3$	$1.730\times {10}^1$	$2.553\times {10}^3$	$4.808\times {10}^1$	$2.417\times {10}^3$	$6.833\times {10}^1$
f22	$4.902\times {10}^3$	$1.335\times {10}^3$	$4.818\times {10}^3$	$1.346\times {10}^3$	$7.612\times {10}^3$	$7.756\times {10}^2$	$3.875\times {10}^3$	$1.808\times {10}^3$
f23	$2.742\times {10}^3$	$2.798\times {10}^1$	$2.731\times {10}^3$	$2.302\times {10}^1$	$3.099\times {10}^3$	$6.047\times {10}^1$	$2.801\times {10}^3$	$1.500\times {10}^2$
f24	$2.915\times {10}^3$	$4.728\times {10}^1$	$2.911\times {10}^3$	$1.978\times {10}^1$	$3.278\times {10}^3$	$8.485\times {10}^1$	$2.985\times {10}^3$	$1.059\times {10}^2$
f25	$2.962\times {10}^3$	$3.752\times {10}^1$	$\mathbf{2}.\mathbf{886}\times {\mathbf{10}}^{\mathbf{3}}$	$\mathbf{1}.\mathbf{430}\times {\mathbf{10}}^{\mathbf{0}}$	$3.464\times {10}^3$	$3.304\times {10}^2$	$3.018\times {10}^3$	$2.433\times {10}^2$
f26	$4.472\times {10}^3$	$3.028\times {10}^2$	$4.282\times {10}^3$	$2.011\times {10}^2$	$9.010\times {10}^3$	$1.005\times {10}^3$	$5.100\times {10}^3$	$1.090\times {10}^3$
f27	$3.234\times {10}^3$	$1.203\times {10}^1$	$3.217\times {10}^3$	$8.677\times {10}^0$	$3.593\times {10}^3$	$1.141\times {10}^2$	$3.268\times {10}^3$	$1.573\times {10}^2$
f28	$3.340\times {10}^3$	$5.101\times {10}^1$	$3.212\times {10}^3$	$1.852\times {10}^1$	$3.586\times {10}^3$	$2.501\times {10}^2$	$3.318\times {10}^3$	$6.933\times {10}^1$
f29	$3.680\times {10}^3$	$1.222\times {10}^2$	$3.510\times {10}^3$	$1.191\times {10}^2$	$4.872\times {10}^3$	$3.584\times {10}^2$	$3.908\times {10}^3$	$3.214\times {10}^2$
f30	$4.908\times {10}^6$	$3.804\times {10}^6$	$2.699\times {10}^5$	$1.556\times {10}^5$	$1.117\times {10}^7$	$3.277\times {10}^6$	$4.281\times {10}^6$	$9.186\times {10}^6$
	MIGWO		PSO		MFO		SSA
	Mean	Std	Mean	Std	Mean	Std	Mean	Std
f1	$2.958\times {10}^3$	$2.963\times {10}^3$	$\mathbf{2}.\mathbf{750}\times {\mathbf{10}}^{\mathbf{3}}$	$\mathbf{3}.\mathbf{950}\times {\mathbf{10}}^{\mathbf{3}}$	$1.730\times {10}^{10}$	$1.017\times {10}^{10}$	$5.821\times {10}^3$	$6.783\times {10}^3$
f3	$1.019\times {10}^4$	$2.951\times {10}^3$	$\mathbf{3}.\mathbf{000}\times {\mathbf{10}}^{\mathbf{2}}$	$\mathbf{6}.\mathbf{908}\times {\mathbf{10}}^{-\mathbf{8}}$	$9.905\times {10}^4$	$6.987\times {10}^4$	$3.000\times {10}^2$	$1.080\times {10}^{-8}$
f4	$4.913\times {10}^2$	$1.788\times {10}^1$	$\mathbf{4}.\mathbf{619}\times {\mathbf{10}}^{\mathbf{2}}$	$\mathbf{2}.\mathbf{545}\times {\mathbf{10}}^{\mathbf{1}}$	$1.437\times {10}^3$	$8.620\times {10}^2$	$4.879\times {10}^2$	$1.858\times {10}^1$
f5	$5.860\times {10}^2$	$2.386\times {10}^1$	$7.112\times {10}^2$	$2.259\times {10}^1$	$7.114\times {10}^2$	$5.572\times {10}^1$	$6.218\times {10}^2$	$3.161\times {10}^1$
f6	$\mathbf{6}.\mathbf{000}\times {\mathbf{10}}^{\mathbf{2}}$	$\mathbf{8}.\mathbf{228}\times {\mathbf{10}}^{-\mathbf{7}}$	$6.321\times {10}^2$	$8.016\times {10}^0$	$6.388\times {10}^2$	$9.977\times {10}^0$	$6.267\times {10}^2$	$9.452\times {10}^0$
f7	$8.921\times {10}^2$	$1.109\times {10}^1$	$8.297\times {10}^2$	$2.944\times {10}^1$	$1.181\times {10}^3$	$2.654\times {10}^2$	$8.709\times {10}^2$	$4.392\times {10}^1$
f8	$8.566\times {10}^2$	$1.908\times {10}^1$	$9.437\times {10}^2$	$1.461\times {10}^1$	$1.006\times {10}^3$	$4.301\times {10}^1$	$9.111\times {10}^2$	$3.050\times {10}^1$
f9	$9.001\times {10}^2$	$3.152\times {10}^{-1}$	$3.710\times {10}^3$	$6.253\times {10}^2$	$6.880\times {10}^3$	$1.894\times {10}^3$	$2.482\times {10}^3$	$7.975\times {10}^2$
f10	$4.327\times {10}^3$	$1.448\times {10}^3$	$4.424\times {10}^3$	$6.930\times {10}^2$	$5.329\times {10}^3$	$6.262\times {10}^2$	$4.712\times {10}^3$	$8.184\times {10}^2$
f11	$1.246\times {10}^3$	$2.813\times {10}^1$	$1.189\times {10}^3$	$1.962\times {10}^1$	$3.839\times {10}^3$	$3.764\times {10}^3$	$1.287\times {10}^3$	$5.469\times {10}^1$
f12	$5.974\times {10}^6$	$1.627\times {10}^6$	$\mathbf{3}.\mathbf{525}\times {\mathbf{10}}^{\mathbf{4}}$	$\mathbf{1}.\mathbf{418}\times {\mathbf{10}}^{\mathbf{4}}$	$5.578\times {10}^8$	$9.341\times {10}^8$	$1.163\times {10}^6$	$1.036\times {10}^6$
f13	$1.543\times {10}^6$	$9.175\times {10}^5$	$\mathbf{5}.\mathbf{506}\times {\mathbf{10}}^{\mathbf{3}}$	$\mathbf{4}.\mathbf{310}\times {\mathbf{10}}^{\mathbf{3}}$	$2.877\times {10}^7$	$1.675\times {10}^8$	$9.051\times {10}^4$	$6.699\times {10}^4$
f14	$1.562\times {10}^4$	$1.018\times {10}^4$	$\mathbf{3}.\mathbf{063}\times {\mathbf{10}}^{\mathbf{3}}$	$\mathbf{1}.\mathbf{242}\times {\mathbf{10}}^{\mathbf{3}}$	$1.158\times {10}^5$	$2.291\times {10}^5$	$4.578\times {10}^3$	$2.643\times {10}^3$
f15	$7.833\times {10}^4$	$5.184\times {10}^4$	$\mathbf{5}.\mathbf{369}\times {\mathbf{10}}^{\mathbf{3}}$	$\mathbf{2}.\mathbf{742}\times {\mathbf{10}}^{\mathbf{3}}$	$6.489\times {10}^4$	$5.183\times {10}^4$	$6.134\times {10}^4$	$4.507\times {10}^4$
f16	$2.047\times {10}^3$	$1.981\times {10}^2$	$2.637\times {10}^3$	$2.209\times {10}^2$	$3.105\times {10}^3$	$4.018\times {10}^2$	$2.424\times {10}^3$	$3.006\times {10}^2$
f17	$1.830\times {10}^3$	$4.314\times {10}^1$	$2.190\times {10}^3$	$2.112\times {10}^2$	$2.606\times {10}^3$	$3.060\times {10}^2$	$2.055\times {10}^3$	$1.558\times {10}^2$
f18	$3.322\times {10}^5$	$2.010\times {10}^5$	$\mathbf{9}.\mathbf{476}\times {\mathbf{10}}^{\mathbf{4}}$	$\mathbf{4}.\mathbf{135}\times {\mathbf{10}}^{\mathbf{4}}$	$2.585\times {10}^6$	$4.165\times {10}^6$	$1.281\times {10}^5$	$9.244\times {10}^4$
f19	$8.312\times {10}^4$	$5.671\times {10}^4$	$\mathbf{2}.\mathbf{481}\times {\mathbf{10}}^{\mathbf{3}}$	$\mathbf{5}.\mathbf{620}\times {\mathbf{10}}^{\mathbf{2}}$	$2.183\times {10}^7$	$5.219\times {10}^7$	$2.747\times {10}^5$	$1.671\times {10}^5$
f20	$2.184\times {10}^3$	$7.148\times {10}^1$	$2.615\times {10}^3$	$1.537\times {10}^2$	$2.757\times {10}^3$	$1.819\times {10}^2$	$2.379\times {10}^3$	$1.203\times {10}^2$
f21	$2.372\times {10}^3$	$2.163\times {10}^1$	$2.449\times {10}^3$	$2.126\times {10}^1$	$2.483\times {10}^3$	$4.310\times {10}^1$	$2.400\times {10}^3$	$2.430\times {10}^1$
f22	$\mathbf{2}.\mathbf{300}\times {\mathbf{10}}^{\mathbf{3}}$	$\mathbf{2}.\mathbf{650}\times {\mathbf{10}}^{-\mathbf{7}}$	$4.150\times {10}^3$	$2.313\times {10}^3$	$7.021\times {10}^3$	$1.126\times {10}^3$	$5.703\times {10}^3$	$1.314\times {10}^3$
f23	$2.771\times {10}^3$	$6.283\times {10}^1$	$3.195\times {10}^3$	$1.096\times {10}^2$	$2.842\times {10}^3$	$3.893\times {10}^1$	$2.747\times {10}^3$	$2.801\times {10}^1$
f24	$2.952\times {10}^3$	$2.206\times {10}^1$	$3.251\times {10}^3$	$7.596\times {10}^1$	$2.981\times {10}^3$	$3.764\times {10}^1$	$2.915\times {10}^3$	$2.325\times {10}^1$
f25	$2.889\times {10}^3$	$8.318\times {10}^0$	$2.887\times {10}^3$	$1.496\times {10}^1$	$3.052\times {10}^3$	$2.541\times {10}^2$	$2.902\times {10}^3$	$2.257\times {10}^1$
f26	$3.509\times {10}^3$	$7.992\times {10}^2$	$5.495\times {10}^3$	$1.931\times {10}^3$	$6.007\times {10}^3$	$4.835\times {10}^2$	$4.487\times {10}^3$	$8.828\times {10}^2$
f27	$3.204\times {10}^3$	$1.078\times {10}^1$	$\mathbf{3}.\mathbf{173}\times {\mathbf{10}}^{\mathbf{3}}$	$\mathbf{8}.\mathbf{508}\times {\mathbf{10}}^{\mathbf{0}}$	$3.252\times {10}^3$	$2.324\times {10}^1$	$3.235\times {10}^3$	$1.768\times {10}^1$
f28	$\mathbf{3}.\mathbf{173}\times {\mathbf{10}}^{\mathbf{3}}$	$\mathbf{4}.\mathbf{698}\times {\mathbf{10}}^{\mathbf{1}}$	$3.184\times {10}^3$	$5.199\times {10}^1$	$3.503\times {10}^3$	$1.564\times {10}^2$	$3.189\times {10}^3$	$4.191\times {10}^1$
f29	$3.562\times {10}^3$	$6.350\times {10}^1$	$3.570\times {10}^3$	$1.953\times {10}^2$	$4.163\times {10}^3$	$3.262\times {10}^2$	$3.883\times {10}^3$	$1.947\times {10}^2$
f30	$7.462\times {10}^5$	$3.776\times {10}^5$	$\mathbf{7}.\mathbf{687}\times {\mathbf{10}}^{\mathbf{3}}$	$\mathbf{2}.\mathbf{490}\times {\mathbf{10}}^{\mathbf{3}}$	$1.016\times {10}^6$	$1.519\times {10}^6$	$1.082\times {10}^6$	$8.138\times {10}^5$
	WOA		HSCA		PSOGSA		mSCA
	Mean	Std	Mean	Std	Mean	Std	Mean	Std
f1	$2.902\times {10}^6$	$1.834\times {10}^6$	$6.485\times {10}^9$	$1.327\times {10}^9$	$6.453\times {10}^9$	$7.820\times {10}^9$	$4.234\times {10}^8$	$4.429\times {10}^8$
f3	$1.677\times {10}^5$	$6.343\times {10}^4$	$6.422\times {10}^4$	$1.927\times {10}^4$	$5.010\times {10}^4$	$5.324\times {10}^4$	$4.436\times {10}^4$	$6.226\times {10}^3$
f4	$5.307\times {10}^2$	$3.171\times {10}^1$	$1.036\times {10}^3$	$1.542\times {10}^2$	$1.304\times {10}^3$	$1.120\times {10}^3$	$5.319\times {10}^2$	$2.516\times {10}^1$
f5	$8.284\times {10}^2$	$5.966\times {10}^1$	$7.783\times {10}^2$	$2.375\times {10}^1$	$8.090\times {10}^2$	$6.117\times {10}^1$	$5.822\times {10}^2$	$1.582\times {10}^1$
f6	$6.741\times {10}^2$	$9.136\times {10}^0$	$6.359\times {10}^2$	$4.100\times {10}^0$	$6.611\times {10}^2$	$1.006\times {10}^1$	$6.027\times {10}^2$	$1.392\times {10}^0$
f7	$1.218\times {10}^3$	$7.427\times {10}^1$	$1.061\times {10}^3$	$2.761\times {10}^1$	$1.408\times {10}^3$	$2.361\times {10}^2$	$8.237\times {10}^2$	$2.059\times {10}^1$
f8	$1.011\times {10}^3$	$3.564\times {10}^1$	$1.049\times {10}^3$	$2.060\times {10}^1$	$1.055\times {10}^3$	$5.163\times {10}^1$	$8.713\times {10}^2$	$1.558\times {10}^1$
f9	$8.313\times {10}^3$	$2.662\times {10}^3$	$4.892\times {10}^3$	$1.339\times {10}^3$	$1.057\times {10}^4$	$2.606\times {10}^3$	$1.028\times {10}^3$	$1.221\times {10}^2$
f10	$5.967\times {10}^3$	$7.782\times {10}^2$	$8.582\times {10}^3$	$3.595\times {10}^2$	$5.216\times {10}^3$	$7.810\times {10}^2$	$3.976\times {10}^3$	$5.503\times {10}^2$
f11	$1.419\times {10}^3$	$6.189\times {10}^1$	$1.813\times {10}^3$	$1.887\times {10}^2$	$3.020\times {10}^3$	$3.723\times {10}^3$	$1.313\times {10}^3$	$4.849\times {10}^1$
f12	$2.933\times {10}^7$	$2.175\times {10}^7$	$6.253\times {10}^8$	$1.488\times {10}^8$	$9.344\times {10}^8$	$1.460\times {10}^9$	$2.654\times {10}^7$	$4.546\times {10}^7$
f13	$1.473\times {10}^5$	$7.488\times {10}^4$	$1.816\times {10}^8$	$6.742\times {10}^7$	$4.712\times {10}^8$	$8.209\times {10}^8$	$4.060\times {10}^6$	$2.109\times {10}^7$
f14	$3.196\times {10}^5$	$2.016\times {10}^5$	$1.612\times {10}^5$	$1.042\times {10}^5$	$5.418\times {10}^4$	$1.028\times {10}^5$	$3.805\times {10}^4$	$2.521\times {10}^4$
f15	$3.609\times {10}^4$	$2.452\times {10}^4$	$3.762\times {10}^6$	$4.250\times {10}^6$	$2.260\times {10}^7$	$1.428\times {10}^8$	$6.007\times {10}^4$	$2.998\times {10}^4$
f16	$3.621\times {10}^3$	$4.487\times {10}^2$	$3.702\times {10}^3$	$3.142\times {10}^2$	$3.241\times {10}^3$	$3.878\times {10}^2$	$2.176\times {10}^3$	$2.395\times {10}^2$
f17	$2.620\times {10}^3$	$2.216\times {10}^2$	$2.474\times {10}^3$	$2.065\times {10}^2$	$2.811\times {10}^3$	$2.727\times {10}^2$	$1.862\times {10}^3$	$9.638\times {10}^1$
f18	$1.973\times {10}^6$	$1.616\times {10}^6$	$5.933\times {10}^6$	$5.500\times {10}^6$	$5.804\times {10}^6$	$1.424\times {10}^7$	$2.806\times {10}^5$	$2.313\times {10}^5$
f19	$4.442\times {10}^6$	$2.701\times {10}^6$	$1.984\times {10}^7$	$1.306\times {10}^7$	$9.239\times {10}^7$	$3.052\times {10}^8$	$9.513\times {10}^5$	$1.878\times {10}^6$
f20	$2.705\times {10}^3$	$1.913\times {10}^2$	$2.671\times {10}^3$	$1.742\times {10}^2$	$2.817\times {10}^3$	$2.240\times {10}^2$	$2.225\times {10}^3$	$7.728\times {10}^1$
f21	$2.558\times {10}^3$	$3.666\times {10}^1$	$2.562\times {10}^3$	$2.436\times {10}^1$	$2.541\times {10}^3$	$3.982\times {10}^1$	$2.362\times {10}^3$	$2.690\times {10}^1$
f22	$7.391\times {10}^3$	$1.375\times {10}^3$	$9.922\times {10}^3$	$3.785\times {10}^2$	$7.329\times {10}^3$	$8.299\times {10}^2$	$2.476\times {10}^3$	$2.541\times {10}^2$
f23	$3.070\times {10}^3$	$1.055\times {10}^2$	$2.943\times {10}^3$	$2.290\times {10}^1$	$3.048\times {10}^3$	$1.014\times {10}^2$	$2.718\times {10}^3$	$1.887\times {10}^1$
f24	$3.172\times {10}^3$	$8.501\times {10}^1$	$3.118\times {10}^3$	$2.220\times {10}^1$	$3.143\times {10}^3$	$7.963\times {10}^1$	$2.883\times {10}^3$	$1.533\times {10}^1$
f25	$2.934\times {10}^3$	$2.213\times {10}^1$	$3.083\times {10}^3$	$4.636\times {10}^1$	$3.179\times {10}^3$	$4.275\times {10}^2$	$2.927\times {10}^3$	$1.478\times {10}^1$
f26	$7.456\times {10}^3$	$1.271\times {10}^3$	$6.522\times {10}^3$	$2.646\times {10}^2$	$7.821\times {10}^3$	$1.310\times {10}^3$	$4.243\times {10}^3$	$1.634\times {10}^2$
f27	$3.349\times {10}^3$	$5.435\times {10}^1$	$3.278\times {10}^3$	$1.870\times {10}^1$	$3.305\times {10}^3$	$5.616\times {10}^1$	$3.219\times {10}^3$	$1.055\times {10}^1$
f28	$3.341\times {10}^3$	$5.283\times {10}^1$	$3.630\times {10}^3$	$1.076\times {10}^2$	$3.531\times {10}^3$	$2.805\times {10}^2$	$3.282\times {10}^3$	$3.395\times {10}^1$
f29	$4.878\times {10}^3$	$3.360\times {10}^2$	$4.645\times {10}^3$	$2.693\times {10}^2$	$4.331\times {10}^3$	$2.916\times {10}^2$	$3.549\times {10}^3$	$1.044\times {10}^2$
f30	$1.941\times {10}^7$	$9.071\times {10}^6$	$3.772\times {10}^7$	$1.686\times {10}^7$	$9.832\times {10}^5$	$1.629\times {10}^6$	$3.246\times {10}^6$	$2.402\times {10}^6$

provides the average and standard deviation of 16 algorithms running independently for 30 times.

Table 10. Results of multiple-problem Wilcoxon test.

CDL-DGWO VS.	+	−	≈	R+	R−	p-Value	a = 0.05
GWO	29	0	0	435	0	0.00000	+
IGWO	7	22	0	94	341	0.00757	−
LGWO	29	0	0	435	0	0.00000	+
HGWOSCA	29	0	0	435	0	0.00000	+
RWGWO	21	8	0	347	88	0.00511	≈
EGWO	29	0	0	435	0	0.00000	+
PSOGWO	29	0	0	435	0	0.00000	+
MIGWO	23	6	0	369	66	0.00105	+
PSO	16	13	0	205	230	0.78694	≈
MFO	29	0	0	435	0	0.00000	+
SSA	21	8	0	293	142	0.10256	≈
WOA	28	1	0	417	18	0.00002	+
HSCA	29	0	0	435	0	0.00000	+
PSOGSA	29	0	0	435	0	0.00000	+
mSCA	28	1	0	417	18	0.00002	+

presents the results of the multiple-problem Wilcoxon test.

Table 11. Test results of the 16 algorithms.

	Rank	Ave		Rank	Ave
CDL-DGWO	2	3.03	MIGWO	4	4.76
GWO	9	8.55	PSO	6	5.79
IGWO	1	2.21	MFO	12	12.21
LGWO	11	11.45	SSA	7	6.45
HGWOSCA	8	7.55	WOA	13	12.28
RWGWO	3	4.21	HSCA	15	13.69
EGWO	16	14.66	PSOGSA	14	13.66
PSOGWO	10	9.93	mSCA	5	5.59

shows the Friedman test results for each algorithm on CEC 2017. This test assesses the performance of each algorithm across multiple problems, providing a ranking that highlights the overall effectiveness of the CDL-DGWO relative to the other contenders.

For CEC2017, it can be seen from Table 9 that CDL-DGWO performs better than other compared algorithms on problems f7, f10, f24, and f26. CDL-DGWO also has strong competitiveness in other test cases. According to Table 10, there is no significant difference between CDL-DGWO, IGWO, SSA, and PSO on the 23 test instances, but a notable difference is observed when compared to the other algorithms. “+” indicates the superiority of CDL-DGWO over the 11 comparative algorithms. In addition, Table 11 presents the rankings for the 16 algorithms, with CDL-DGWO achieving an average ranking of 3.03 and a final ranking of 2. In summary, CDL-DGWO is effective in solving complex optimization problems.

The achievement of near-optimal solutions on several fixed-dimension multimodal functions (e.g., F14–F16) primarily demonstrates CDL-DGWO’s efficacy in locating the global optimum on these specific static benchmarks. To assess the risk of overfitting and generalization capability, the algorithm was further tested on the more complex and modern CEC2017 test suite, which comprises functions characterized by multimodality, asymmetry, noise, and rotation properties, along with intricate variable interactions and noisy fitness landscapes. The competitive performance of CDL-DGWO on CEC2017 (as shown in Table 9, Table 10 and Table 11) suggests that its strengths—maintained population diversity and adaptive search strategies—generalize effectively beyond the simpler classical benchmarks. Furthermore, its success on the diverse set of engineering design problems in Section 5, which involve non-linear constraints and real-world variable dependencies, provides additional evidence of its robustness and general applicability.

4.5. Analysis of Population Diversity in CDL-DGWO

Population diversity reflects the distribution of solutions within a population and can be used to assess the exploration and exploitation capabilities of an algorithm during the iterative process. In this subsection, based on the method described in [43], the population diversity index of CDL-DGWO is calculated and compared with that of the GWO. The calculation method is as follows:

$${\overline{X}}_d={\displaystyle \frac{1}{N}}\sum _{s=1}^N{X}_{s,d}$$

(25)

$$Di{v}_d={\displaystyle \frac{1}{N}}\sum _{s=1}^N\left|X_{s,d}-{\overline{X}}_d\right|$$

(26)

$$Div={\displaystyle \frac{1}{D}}\sum _{d=1}^{D}Di{v}_d$$

(27)

where ${\overline{X}}_d$ represents the average value of solutions in the d-th dimension.

Figure 3 illustrates the dynamic changes in population diversity with respect to the number of function evaluations for the CDL-DGWO and GWO on four typical functions (unimodal function f3, simple multimodal function f7, hybrid function f14, and composite function f26) from the 30-dimensional CEC2017 benchmark test suite. These selected functions represent different types of optimization problems, including unimodal, multimodal, hybrid, and composite functions, and thus can reflect the performance of the algorithms in various scenarios. As observed from the figure, the population diversity of the GWO algorithm remains relatively stable and consistent throughout the search process for all four test functions. This indicates that GWO maintains a relatively stable level of population diversity during the search process. However, this relatively uniform diversity pattern limits the algorithm’s ability to escape local optima in complex optimization problems, revealing its limitations in exploring new solution spaces. Consequently, the global search capability of GWO is somewhat compromised. In contrast, the population diversity of the proposed CDL-DGWO algorithm fluctuates significantly throughout the search process and consistently maintains a higher level of diversity. This dynamic diversity pattern confers several notable advantages. This ability to balance exploration and exploitation significantly enhances the algorithm’s capacity to escape local optima, resulting in stronger adaptability and robustness when tackling complex optimization problems.

5. Testing of Engineering Design Problems

This section employs three practical engineering optimization problems to evaluate CDL-DGWO. In the experiment, the maximum number of function evaluations ($max\_FEs$) is set to 15,000, and the population size is set to 30. Each experiment is conducted independently 30 times.

5.1. Constrained Problem

For the constrained problem, the objective function and constraint of this problem are as follows [44]:

$$\mathbf{Min}{F}_1\left(x\right)=5.3578547x_3^3+0.8356891x_1{x}_5+37.293239x_1+40729.141$$

(28)

$$\begin{array}{cc}\mathbf{s}.\mathbf{t}.\hfill & g_1\left(x\right)=85.334407+0.0056858x_2{x}_5+0.0006262x_1{x}_4-0.0022053x_3{x}_5-92⩽0\hfill \\ \hfill & g_2\left(x\right)=-85.334407-0.0056858x_2{x}_5-0.0006262x_1{x}_4-0.0022053x_3{x}_5⩽0\hfill \\ \hfill & g_3\left(x\right)=80.51249+0.0071317x_2{x}_5+0.0029955x_1{x}_2+0.0021813x_1^2-110⩽0\hfill \\ \hfill & g_4\left(x\right)=-80.51249-0.0071317x_2{x}_5-0.0029955x_1{x}_2-0.0021813x_1^2+90⩽0\hfill \\ \hfill & g_5\left(x\right)=9.300961+0.0047026x_3{x}_5+0.0012547x_1{x}_3+0.0019085x_3{x}_4-25⩽0\hfill \\ \hfill & g_6\left(x\right)=-9.300961-0.0047026x_3{x}_5-0.0012547x_1{x}_3-0.0019085x_3{x}_4+20⩽0\hfill \end{array}$$

(29)

$$\begin{array}{cc}& 78⩽x_1⩽102\hfill \\ & 33⩽x_2⩽45\hfill \\ & 27⩽x_i⩽45i=3,4,5\hfill \end{array}$$

(30)

For this minimization problem,

Table 12. Comparison of results on welded beam design problem.

	Mean	Std	Best	Best Values for Variables
				x1	x2	x3	x4	x5
CDL-DGWO	−30,939.86	240.84	−31,453.63	58.1663	30.2820	30.9084	46.29	42.19
GWO	−30,657.62	3.37	−30,663.58	78	33	30.01993	45	36.72181
MIGWO	−30,621.15	19.16	−30,652.95	78.1843	36.3522	32.8490	35.8049	34.0235
IGWO	−30,653.64	5.15	−30,660.87	78.0330	33.0005	30.0320	44.9935	36.6943
LGWO	−30,514.90	84.48	−30,631.71	78	33	30.5260	45	35.8899
HGWOSCA	−30,655.91	3.64	−30,662.48	78	33	30.0244	45	36.7473
RWGWO	−30,659.20	3.56	−30,664.49	78	33	30.0067	45	36.7748
EGWO	−30,658.42	5.88	−30,664.79	78	33.0182	30.0358	45	36.6768
PSOGWO	−30,650.96	27.82	−30,665.28	78	33	30.0208	44.9763	36.7329
MFO	−30,642.89	43.32	−30,665.54	78	33	29.9953	45	36.7758
SSA	−29,739.53	$1.1053\times {10}^{-11}$	−29,739.53	78.0427	35.6896	33.9524	41.2558	30.1411
WOA	−29,913.76	142.60	−30,323.91	78	35.5690	33.7554	40.9303	29.9783
HSCA	−30,280.57	188.77	−30,566.36	78	33	30.4282	45	36.1814
PSOGSA	−30,660.77	19.52	−30,665.54	78	33	29.9953	45	36.7758
mSCA	−30,607.12	34.17	−30,647.25	78	33.1531	30.1047	44.9989	36.5289

provides comparative results for the constrained problem. It is evident that CDL-DGWO yields the optimal solution. The results in Table 12 demonstrate that CDL-DGWO is competitive for the constrained problem.

5.2. Tension/Compression Spring Design Problem

For the tension/compression spring design problem, the goal is to achieve the lowest-cost spring by selecting a set of values, as illustrated in Figure 4. This optimization must consider constraints such as shear stress, resonant frequency, and minimum deflection requirements.

The objective function and constraint for this problem are as follows [5,45]:

$$\mathbf{Min}{F}_2\left(x\right)=\left(x_3+2\right)x_2{x}_1^2$$

(31)

$$x=\left(x_1,x_2,x_3\right)=(d,D,N)$$

(32)

$$\begin{array}{cc}\mathbf{s}.\mathbf{t}.\hfill & g_1\left(x\right)=1-{\displaystyle \frac{x_2^3{x}_3}{71785x_1^4}}⩽0\hfill \\ & g_2\left(x\right)={\displaystyle \frac{4x_2^2-x_1{x}_2}{12566\left(x_2{x}_1^3-x_1^4\right)}}+{\displaystyle \frac{1}{5108x_1^2}}⩽0\hfill \\ & g_3\left(x\right)=1-{\displaystyle \frac{140.45x_1}{x_2^2{x}_3}}⩽0\hfill \\ & g_4\left(x\right)={\displaystyle \frac{x_1+x_2}{1.5}}-1⩽0\hfill \end{array}$$

(33)

where N represents the number of active coils, D is mean coil diameter, and d is the wire diameter. $0.05⩽x_1⩽2.00$, $0.25⩽x_2⩽1.30$, $2.00⩽x_3⩽15.0$.

Table 13. Comparison of results on tension/compression spring design.

	Mean	Std	Best	Best Values for Variables
				x1	x2	x3
CDL-DGWO	0.0127	0.0000	0.0127	0.0519	0.3599	11.1646
GWO	0.0128	0.0000	0.0127	0.0546	0.4303	8.0324
MIGWO	0.0128	0.0000	0.0127	0.0535	0.3814	11.7483
IGWO	0.0127	0.0000	0.0127	0.0517	0.3562	11.3484
LGWO	0.0132	0.0005	0.0128	0.0539	0.4088	9.1575
HGWOSCA	0.0128	0.0002	0.0127	0.0500	0.3172	14.0638
RWGWO	0.0128	0.0002	0.0127	0.0500	0.3172	14.0646
EGWO	0.0132	0.0005	0.0127	0.0585	0.5413	5.2880
PSOGWO	0.0128	0.0001	0.0127	0.0501	0.3187	13.9319
MFO	0.0131	0.0008	0.0127	0.0500	0.3174	14.0278
SSA	$7.39\times {10}^{10}$	$3.35\times {10}^{11}$	0.0185	0.0777	1.3657	1.4074
WOA	0.0142	0.0011	0.0127	0.0657	0.7960	2.6487
HSCA	0.0139	0.0009	0.0130	0.0500	0.3115	15.0000
PSOGSA	0.0135	0.0009	0.0127	0.0528	0.3828	9.9089
mSCA	0.0128	0.0002	0.0127	0.0507	0.3318	12.9811

demonstrates the comparison results of comparison among CDL-DGWO and competitors, CDL-DGWO achieves the best performance. Table 13 provides the statistical results for multiple algorithms in terms of average and standard deviation. It is evident that CDL-DGWO achieves the minimum weight design compared to other optimization algorithms.

5.3. Welded Beam Design Problem

Welded beam design is a standard engineering benchmark problem, as introduced in the literature [5,45]. As shown in Figure 5, the objective is to design a welded beam at the lowest possible manufacturing cost by finding the global optimal solution.

The design involves four optimization variables: the weld thickness (h), length of the clamped bar (l), the bar height (t), and thickness of the bar (b). The constraints for the welded beam design include the shear stress ($\tau$) and bending stress ($\sigma$) in the beam, buckling load ($P_b$) on the bar, and end deflection ($\delta$) of the beam. Let $\mathit{X}=\left\{x_1,x_2,x_3,x_4\right\}=\left\{h,l,t,b\right\}$, the specific mathematical model is described as follows:

$$\mathbf{Min}{F}_3\left(x\right)=1.10471x_1^2{x}_2+0.04811x_3{x}_4(14+x_2)$$

(34)

$$\begin{array}{cc}\mathbf{s}.\mathbf{t}.\hfill & g_1\left(x\right)=\tau \left(x\right)-{\tau }_{max}⩽0\hfill \\ \hfill & g_2\left(x\right)=\sigma \left(x\right)-{\sigma }_{max}⩽0\hfill \\ \hfill & g_3\left(x\right)=x_1-x_4⩽0\hfill \\ \hfill & g_4\left(x\right)=0.10471x_1^2+0.04811x_3{x}_4(14+x_2)⩽0\hfill \\ \hfill & g_5\left(x\right)=0.125-x_1⩽0\hfill \\ \hfill & g_6\left(x\right)=\delta \left(x\right)-{\delta }_{max}⩽0\hfill \\ \hfill & g_7\left(x\right)=P-P_c\left(x\right)⩽0\hfill \end{array}$$

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$$\begin{array}{c}0.1⩽x_m⩽2,m=1,4,0.1⩽x_m⩽10,m=2,3\\ \tau \left(\mathit{x}\right)=\sqrt{{\left({\tau }^{\prime }\right)}^2+{\left({\tau }^{\prime \prime }\right)}^2+\left(l{\tau }^{\prime }{\tau }^{\prime \prime }\right)/\sqrt{0.25(l^2+{(h+t)}^2)}}\\ {\tau }^{\prime }={\displaystyle \frac{P}{\sqrt{2}{x}_1{x}_2}},{\tau }^{\prime \prime }={\displaystyle \frac{MR}{J}},M=P(L+{\displaystyle \frac{x_2}{2}}),R=\sqrt{{\displaystyle \frac{x_2^2}{4}}+{\left({\displaystyle \frac{x_1+x_3}{2}}\right)}^2}\\ J=2\left\{\sqrt{2}{x}_1{x}_2\left[{\displaystyle \frac{x_2^2}{12}}+{\left({\displaystyle \frac{x_1+x_3}{2}}\right)}^2\right]\right\},\sigma \left(\mathit{x}\right)={\displaystyle \frac{6PL}{x_4{x}_3^2}},\delta \left(x\right)={\displaystyle \frac{4P{L}^3}{E{x}_3^3{x}_4}},\\ P_c\left(\mathit{x}\right)={\displaystyle \frac{4.013E\sqrt{\frac{x_3^2{x}_4^6}{36}}}{L^2}}(1-{\displaystyle \frac{x_3}{2L}}\sqrt{{\displaystyle \frac{E}{4G}}}),P=6000lb,L=14in,E=30·{10}^6\mathrm{psi},\\ G=12·{10}^6\mathrm{psi},{\tau }_{max}=13,600\mathrm{psi},{\sigma }_{max}=30000\mathrm{psi},{\delta }_{max}=0.25in\end{array}$$

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Table 14. Comparison of results on welded beam design problem.

	Mean	Std	Best	Best Values for Variables
				x1	x2	x3	x4
CDL-DGWO	1.7249	$\mathbf{1}.\mathbf{12}\times {\mathbf{10}}^{\mathbf{15}}$	1.7249	0.2057	3.4705	9.0366	0.2057
GWO	1.7295	0.0032	1.7262	0.2046	3.4990	9.0382	0.2058
MIGWO	1.7410	0.0081	1.7314	0.2351	3.2464	8.4059	0.2436
IGWO	1.7295	0.0015	1.7267	0.2053	3.4809	9.0614	0.2058
LGWO	1.8406	0.0297	1.7622	0.2064	3.5079	9.0255	0.2170
HGWOSCA	1.7302	0.0030	1.7263	0.2056	3.4777	9.0453	0.2060
RWGWO	1.7290	0.0017	1.7266	0.2039	3.5142	9.0429	0.2057
EGWO	1.7472	0.0404	1.7264	0.2046	3.4962	9.0349	0.2059
PSOGWO	1.7444	0.0352	1.7251	0.2054	3.6264	9.0816	0.2088
MFO	1.8213	0.1587	1.7249	0.3087	2.5470	7.3754	0.3088
SSA	9.5473	$3.60\times {10}^{15}$	9.5473	1.1900	3.1444	3.4581	1.6228
WOA	4.2099	0.9787	1.9154	0.1654	4.1512	10.0000	0.2050
HSCA	1.8732	0.0489	1.7835	0.2130	4.0698	8.7929	0.2195
PSOGSA	2.2723	0.2732	1.7472	0.2990	2.6560	7.3623	0.3099
mSCA	1.7415	0.0111	1.7277	0.2026	3.5556	9.0790	0.2081

provides comparative results for the welded beam design problem. It is evident that CDL-DGWO achieved smaller design solutions in terms of both the average and optimal values compared to the other 14 algorithms, and also obtained the smallest standard deviation. The results in Table 14 demonstrate that CDL-DGWO is competitive in solving the welded beam design problem.

Based on the strong performance of CDL-DGWO demonstrated across 23 benchmark functions, the CEC2017 test suite, and three engineering design problems, the algorithm shows particular promise in complex, real-world optimization scenarios characterized by high dimensionality, non-linearity, and multi-modality. Its enhanced population diversity, achieved through chaotic double-swarm grouping and dynamic regrouping, makes it exceptionally suitable for problems where traditional GWOs are prone to premature convergence. The integration of covariance matrix learning allows CDL-DGWO to effectively handle problems with correlated variables by transforming the search space into an eigenspace, thereby improving information sharing among dimensions. This is especially beneficial in fields such as mechanical design, structural optimization, and energy systems—where design variables often exhibit strong interactions. Furthermore, the dimension learning strategy enhances local exploration, making CDL-DGWO a robust choice for fine-tuning solutions in precision-critical applications like aeronautical engineering, robotics trajectory planning, and embedded system design.

6. Conclusions

In this study, a novel Double-swarm Grey Wolf Optimizer with Covariance and Dimension Learning (CDL-DGWO) has been proposed. This enhancement of the traditional Grey Wolf Optimizer (GWO) utilizes chaotic grouping to partition the population into two sub-swarms, which has been found to improve population diversity. Furthermore, covariance and dimension learning strategies have been incorporated to refine the hunting behavior of grey wolves, thereby significantly enhancing the global search capability and stability of the algorithm. The performance of CDL-DGWO was evaluated using 23 benchmark problems and the CEC2017 test suite. It was demonstrated that CDL-DGWO surpasses the compared swarm intelligence algorithms such as PSO, MFO, and GWO variants in terms of optimal solution identification and convergence performance. Moreover, according to the results of the Friedman test, CDL-DGWO ranks first compared to the comparison algorithms. The efficacy of CDL-DGWO in addressing real-world challenges was further confirmed through its successful application to three practical engineering design problems. The results indicate that the CDL-DGWO algorithm exhibits significant performance advantages in handling complex engineering design optimization problems, particularly in the search for global optima and the maintenance of algorithmic stability. Future research may focus on further refinements in parameter tuning, multi-group dynamic grouping strategy, integration with other optimization techniques, and testing across a broader range of practical applications.