Large-Scale Competitive Learning-Based Salp Swarm for Global Optimization and Solving Constrained Mechanical and Engineering Design Problems

Abstract

The Competitive Swarm Optimizer (CSO) has emerged as a prominent technique for solving intricate optimization problems by updating only half of the population in each iteration. Despite its effectiveness, the CSO algorithm often exhibits a slow convergence rate and a tendency to become trapped in local optimal solutions, as is common among metaheuristic algorithms. To address these challenges, this paper proposes a hybrid approach combining the CSO with the Salp Swarm algorithm (SSA), CL-SSA, to increase the convergence rate and enhance search space exploration. The proposed approach involves a two-step process. In the first step, a pairwise competition mechanism is introduced to segregate the solutions into winners and losers. The winning population is updated through strong exploitation using the SSA algorithm. In the second step, non-winning solutions learn from the winners, achieving a balance between exploration and exploitation. The performance of the CL-SSA is evaluated on various benchmark functions, including the CEC2017 benchmark with dimensions 50 and 100, the CEC2008lsgo benchmark with dimensions 200, 500 and 1000, as well as a set of seven well-known constrained design challenges in various engineering domains defined in the CEC2020 conference. The CL-SSA is compared to other metaheuristics and advanced algorithms, and its results are analyzed through statistical tests such as the Friedman and Wilcoxon rank-sum tests. The statistical analysis demonstrates that the CL-SSA algorithm exhibits improved exploitation, exploration, and convergence patterns compared to other algorithms, including SSA and CSO, as well as popular algorithms. Furthermore, the proposed hybrid approach performs better in solving most test functions.

1. Introduction

Metaheuristic search techniques are frequently employed to address various difficult or nondifferentiable optimization issues. It has demonstrated the ability to solve many complex optimization problems. Furthermore, the metaheuristic search algorithm can readily reach an optimal solution in a reasonable amount of time [1]. However, optimization problems have grown in number and complexity in recent years. As a result, better optimization methods are needed to solve these problems. Several powerful algorithms can be used to solve a given problem. However, we cannot name one of them as the best without comparing it to the others on the problem. Consequently, optimization algorithms can be used to solve various problems efficiently.

In the literature, metaheuristics can be classified into two groups according to their structures: evolutionary-based, trajectory-based, and nature-inspired algorithms [2]. The evolutionary model is a theoretical framework used to comprehend biological evolution. In evolutionary models, candidate populations go through natural selection and genetic variety to discover solutions to problems. These models, which reconstruct the process of species evolution by retracing the course of populations’ history, are founded on the principle of competition. In these models, populations are produced randomized, with each individual standing in for a different potential solution. Those with a higher fitness level are chosen for reproduction based on a pattern of choice after an objective function has been applied to evaluate which solution is the most appropriate. After this, a new generation of offspring is produced using genetic operators such as crossover and mutation to reproduce the previously chosen individuals. Finally, a different strategy is utilized to select the surviving progeny and parents from the original group. Genetic Algorithm (GA) [3], Memetic Algorithm (MA), Differential Evolution (DE) [4], Harmony Search (HS) algorithm [5], and Clonal Selection Algorithm (CSA) [6] are examples of popular evolutionary-based algorithms and methodologies.

The search space is traversed by a single solution in the trajectory-based approach. Utilizing randomization and exact guidance, such as the greedy criterion in a limited number of repetitions, is typically required while transitioning to and improving the solution. Simulated Annealing (SA) [7] Tabu Search (TS) [8], and Iterated Local Search (ILS) [9] are the most common algorithms in this category. This group has the benefit of requiring few function evaluations and few calculations.

Nature-inspired algorithms produce and develop many alternative solutions within the limits of the issue iteratively. Knowledge transmission (information sharing), collaboration, and interaction among candidate solutions are all features of population-based algorithms for identifying the optimal solution in a search space.

The most well-known algorithms in this category are Particle Swarm Optimization (PSO) [10], Ant Colony Optimization [11], Grey Wolf Optimizer (GWO) [12], Bat Algorithm (BA) [13], Artificial Bee Colony (ABC) [14], Cuckoo Search (CS) algorithm [15], and Competitive Swarm Optimizer (CSO) [16]. Nature-inspired algorithms have the advantage of performing a global search because they can produce several solutions [17] in each iteration, reducing the risk of local optima stagnation. In addition, running multiple search agents in parallel to find the best solution is normally beneficial for enhancing overall performance. Furthermsore, population-based algorithms can make use of a variety of random operators, such as crossover [18], mutation [19], and selection [20], to achieve superior exploration and exploitation capabilities. Because of the benefits mentioned above, population-based metaheuristics are very common and commonly used today.

Cheng and Jin [16] introduced a PSO variation known as a Competitive Swarm Optimizer (CSO) for large-scale optimization. CSO performs admirably on various continuous test tasks with sizes up to 5000. The pair-wise competitive scenario CSO typically employs divides the solutions into winners and losers, with the winners going directly to the next iteration. The CSO approach has several advantages in addition to simplicity and quick convergence. However, the CSO algorithm still has two obvious defects, which may lead to low search efficiency. First, the particles are randomly paired in the competition, and the fitness differences between the winners and losers are often too large or too small, affecting global exploration [21]. The CSO does not develop a solution to the premature problem. It is challenging for the algorithm to depart from the local optimal answer if the population experiences a premature problem. Second, only half of the particles (i.e., losers) are updated in each generation, while the other half (i.e., winners) are not, which is incompatible with local exploitation [22].

Mirjalili et al. [23] suggested the modern population-based SSA algorithm. This algorithm mimics salps’ swarming and foraging activity in the deep sea. Despite the simplicity of the mathematical model of SSA, its optimization capabilities in complex engineering optimization problems are superior to other recent algorithms (GWO, ABC, CS, CSA, etc.). Furthermore, it has been shown that SSA shows less probability of falling into local optimal solutions, particularly for complex landscapes and modes, by integrating several random operators [24]. However, SSA has two potential disadvantages. First, the convergence rate of SSA is insufficient for generating solutions with great precision. Second, it lacks the exploration ability of crossover operators in the evolutionary algorithms (EAs). As a result, several researchers have attempted to suggest some SSA variants that would improve the ability of SSA.

The authors of [25] improved the SSA using chaos theory, known as the chaotic Salp Swarm algorithm (CSSA), to enhance the convergence rate and resulting precision of feature selection. While ref. [26] improved the SSA by combining it with PSO (SSAPSO) to enhance the efficacy of the exploration and exploitation and apply it to feature selection also. The authors of [27] proposed a novel SSA variation termed harmonized salp chain-built optimization. In this variation, levy-flight search and opposition-based learning accelerate convergence and prevent salps from settling for suboptimal solutions. In addition, the authors of [28] hybridized the Sine Cosine (SCA) search strategy into the SSA to improve exploration capabilities and the convergence rate. Moreover, the authors of [29] proposed a hybridization between SSA as a population-based algorithm and ββ-hill climbing optimizer balance the diversification and intensification of the economic load dispatch problem search space. The authors of [30] integrate ensemble mutation strategy and new restart mechanisms into SSA, demonstrating robust efficacy in enhancing SSA’s exploration and exploitation capacity. These numerous applications show how broadly applicable the SSA is. The literature review at [31] gives readers a thorough overview of SSA.

This paper presents a hybridized SSA and CSO called CL-SSA to resolve the inherent limitations of SSA and CSO. The hybridization of two basic algorithms aims to boost the novel optimizer’s exploitation and exploration abilities. Using a set of test functions, we first demonstrate the effects of the proposed modifications on CSO and SSA, respectively. Then, the scalability test of the CL-SSA is confirmed using 50 and 100-dimensional CEC2017 benchmarks and 200, 500 and 1000-dimensional CEC2008lsgo benchmark test functions. Moreover, solving seven engineering design problems (EDPs) examined the CL-SSA’s applicability. The tests’ results demonstrated that CL-SSA is capable of successful optimization.

The main contributions of this paper are:

• Hybridizing a Competitive Swarm Optimization with a Salp Swarm algorithm allows exploration and exploitation to be efficiently balanced.
• Various use cases, such as the CEC 2017 benchmark functions with dimensions 50 and 100, the CEC 2008LSGO benchmarks with dimensions 200, 500 and 1000, and the solution of seven common EDPs were used to validate the proposed CL-SSA.
• Regarding scalability, the proposed CL-SSA beats its competitors in large-scale global optimization problems.
• Wilcoxon rank-sum and Friedman determine whether the performance differences between algorithms found are statistically significant.
• According to two statistical tests, Friedman and Wilcoxon rank-sum, CL-SSA outperforms numerous PSO and SSA versions and other advanced algorithms.

The remaining sections of this paper can be categorized as follows: Section 2 introduces the large-scale global optimization problem. Section 3 contains a full summary of CL-related SSA’s works. We cover original SSA and CSO algorithms here. A detailed description of the CL-SSA is given in Section 4. The results of the algorithm across CEC2017 benchmark functions and CEC2008LSGO benchmark functions are presented in Section 5, and the results are discussed. Section 6 includes experimental findings related to the use of CL-SSA to solve EDPs. The conclusion of this paper and the possible course of work are presented in Section 7.

2. Large-Scale Global Optimization

Large-Scale Global Optimization (LSGO) is a term used to describe how most real-world optimization issues deal with many decision variables. Due to increasing multimodality and many decision variables, these problems become difficult to address. The local search space often becomes so constrained that finding the global best solution becomes quite challenging. Many population-based algorithms have been presented to deal with these throughout the previous few decades. In addition to the performance degradation, such techniques greatly increase computational complexity during simulation.

Consequently, solving large-scale global optimization problems in various engineering and scientific domains becomes challenging. In the literature, there has been a great variety of algorithms proposed to handle these problems. Based on the decomposition of the problem dimensions, such algorithms can be categorized into two types—decomposition-based and non-decomposition-based. In decomposition-based algorithms, called Cooperative Coevolution (CC), high-dimensional problems are decomposed into low-dimensional subproblems. Potter and De Jong of [30] were the first to develop this concept in 1994. A round-robin strategy’s standard optimization process for a predetermined number of generations handles each subproblem. The final n-dimensional solution is then constructed by merging solutions from all subproblems. Yang et al. [32,33,34] used a DE-based cooperative coevolution (CC) approach called DECC-G to solve LSGO problems with 500 and 1000 dimensions, employing the random grouping of decision variables. The multilevel CC method (MLCC) refinement uses a decomposer pool that utilizes the dynamic group size based on the decomposer’s past performance. Similar algorithms, such as CCPSO [35] and CC-CMA-ES [36], are being presented as a gradual improvement. CC has been demonstrated to enhance the efficiency of large-scale optimizations, but it does have some shortcomings [30]:

• The decomposition approach utilized has an impact on its performance
• As the number of interacting factors increases, its performance will deteriorate.
• When an optimization problem is completely non-separable, the performance is solely determined by the EA used.
• A high degree of computational complexity exists.

On the other hand, non-decomposition-based algorithms ignore the divide-and-conquer technique and instead employ a variety of successful ways to improve the algorithms’ performance. Swarm intelligence-based algorithms [37], evolutionary computation-based algorithms [38,39], and local search-based algorithms [40,41] are the most common categories. For example, large-scale problems can be handled with a modified CSO (MCSO) [42] and quadratic interpolation by the Whale Optimization Algorithm (WOA) [43]. To solve high-dimensional problems with 200 to 5000 decision variables, Li Y et al. [44] proposed a modified Sine Cosine algorithm (SCA) known as DSCA. Using the Grey Wolf method (GWO), Qaraad et al. [45] developed an improved Salp algorithm for dealing with high-dimensional functions.

In light of the limits of CCEAs, researchers attempt to design novel search algorithms for standard evolutionary algorithms to increase their diversity, allowing them to solve a wider variety of large-scale optimization issues while utilizing limited function evaluations more effectively. Therefore, this work applies the non-decomposition algorithm to handle large-scale problems (swarm intelligence).

3. Related Work

Nature-inspired algorithms offer distinct properties that have attracted researchers from other domains to solve various challenges. However, the No Free Lunch theorem states that no optimum optimization algorithm exists to solve all optimization problems. Consequently, developing novel algorithms to handle issues in real-world applications remains difficult.

Combining basic metaheuristic algorithms to create a new optimization method is becoming increasingly frequent. Countless modifications to the original CSO algorithm have been made since its inception.

CSO improvement and hybridizations with other techniques are listed chronologically, with the oldest listed first (see

Table 1. Review of earlier modification of basic CSO.

Approaches	Enhancement Type, and Year	Problems	Methodology
An Entropy-based Swarm Optimizer for large-scale problem. DCSO [46].	Modifications, 2016	Solve large-scale problems CEC2008lsgo	The proposed approach utilizes population entropy to provide a quantitative description of the population’s diversity and to partition the population into two subgroups dynamically. To accelerate convergence during the early stages of the execution process, sub-groups with higher fitness will have a smaller size, and sub-groups with lower fitness will learn from each other to speed up the algorithm’s convergence.
Enhanced CSO for large-scale problems. MCSO [42].	Modifications, 2017	Solve large-scale problems CEC2008lsgo	Unlike CSO, this work proposes a modified CSO (MCSO) in which a tri-competitive criterion updates two-thirds of population swarms. A tiny modification in CSO significantly impacts the quality of the solution.
A Competitive Swarm Optimizer for Large-Scale Multi-Objective Optimization. LMOCSO [47].	Modifications, 2019	Solve large-scale multi-objectives problems	It is suggested that the position be updated in two stages. First, before learning from the leader, the leader pre-updates each individual’s position based on its initial velocity. Then the leader updates each pre-updated individual’s position. All modified individuals are mixed with the original particles; only half of the combined population lives to the next generation.
Competitive Swarm algorithm with orthogonal learning. OLCSO [48].	Modifications, 2019.	Solve 24 test functions and power system economic dispatch (ED).	As a way to quickly uncover more meaningful information in the winning and loser individuals, the orthogonal learning (OL) technique was created for CSO.
Competitor-leading Swarm Optimizer using dynamic Gaussian mutation. WLCSODGM [49].	Modifications, 2020.	Extracting parameters from solar photovoltaic models	To address the shortcomings of CSO, two new features are included in WLCSODGM. Winner-leading search strategies are first presented to encourage exploration and aid the losers in identifying more potential areas. Secondly, a dynamic Gaussian mutation operator with adaptive mutation probability and adjustable mutation amplitude is integrated to boost exploration further and enable individuals to jump out of bad local optima.
A simplified CSO optimizer SCSO [50].	Modifications, 2020.	Solid oxide fuel cells parameter identification.	CSO uses two simplified components: a simplified learning equation, in which losers only learn from winners, and a revised random number generation method, in which new random numbers are generated for each loser individually rather than on a per-dimension basis.
Enhanced Competitive Swarm optimization, ImCSO [51].	Hybridizations, 2022	Solve Multi-area economic dispatch (MAED) issue.	The ImCSO method adds two performance-enhancing enhancements to competitive swarm optimization. First, a ranking-paired learning technique was implemented to improve the learning effectiveness of the loser individuals; then, a differential evolution (DE) method was employed to update and enhance the winning individuals.
Improved CSO For MOPs [52].	Modifications, 2022	Solve multi-objective optimization problems (MOPs)	To broaden the applicability of CSO, a competitive mechanism comprised of Pareto dominance and numerical comparison is provided, and the relevant components are adjusted to make the proposed algorithm acceptable for MOPs. In addition, a revised learning scheme is proposed to increase the optimization’s effectiveness and speed up the optimization process.
A Hybrid Genetic Algorithm/Competitive Swarm Optimizer [22].	Hybridizations, 2022	Feature selection problem	To speed up the production of new particles in the algorithm and minimize premature population growth, this study proposes to include the genetic algorithm’s crossover and mutation operators in the competitive swarm optimization.

).

Following the discussion thus far, it can be concluded that:

• CSO has been the topic of a vast amount of research.
• The efficiency of the CSO optimization algorithm has been demonstrated in various contexts. However, despite its superiority to other well-established optimizers, local solutions will probably impede CSO’s performance.
• Several alternative CSO variations have been created to avoid local optima and maintain a balance between exploratory and exploitative operations by altering the CSO mechanism to increase convergence speed.

Table 1 highlights various discoveries and advancements, indicating that the algorithm should initiate its quest for optimal solutions as soon as possible. However, CSO has some significant drawbacks, such as early convergence, incapability to escape local optima, lack of population diversity, and an inadequate balance between exploitation and exploration. To overcome these limitations, this article presents an improved version of CSO, named CL-SSA. The distinguishing features of CL-SSA from the previously cited works are:

• CL-SSA incorporates competition-based methods to classify the possible solutions into winners and losers. Then, learning procedures update the loser particles, enabling them to explore their search space. The Salp Swarm technique is utilized to enhance and modernize the winning particles of each generation.
• The competitive learning process enhances the exploitation phase, leading to more precise solutions and assisting the algorithm in rapidly and effectively discovering the global optimum.
• The proposed CL-SSA technique was tested on various dimensions and limited engineering design test functions. The results indicate that the suggested approach is a competent method for addressing the challenges of CSO. Furthermore, the researchers investigated several SSA variants and optimization approaches, and the outcomes indicate that the proposed technique performs significantly better in most cases.

3.1. Testbench

In optimization, the objective function is a mathematical function that needs to be optimized or maximized, subject to a set of constraints. The objective function represents the quantity that needs to be optimized or minimized, and it is typically denoted as a function of one or more decision variables.

The objective function can take many forms, depending on the specific problem being solved. For example, it could be a linear or nonlinear function of the decision variables, or it could be a more complex function that involves integrals, derivatives, or other mathematical operations. In all cases, the objective function represents the quantity that is being optimized, and the optimization algorithm seeks to find the set of decision variables that maximize or minimize this quantity.

The objective function is a crucial component of any optimization problem because it defines the goal of the optimization process. Without an objective function, there would be no way to determine the optimal values of the decision variables that satisfy the constraints of the problem. The objective function is typically defined by the problem owner or domain expert, who has a good understanding of the problem and the goals of the optimization process.

In this research, the objective functions are expressed as minimization problems, all CEC 2017 testbench functions are expressed as minimization problems, the objective is to minimize the value of $Minf\left(\mathit{x}\right)$ while $\mathit{x}$ is a vector with D dimensions, denoted by

$$\mathit{x}={\left[x_1,x_2,\dots ,x_D\right]}^T$$

(1)

The search range is defined as ${\left[-100,100\right]}^D$.

For the purposes of illustration, let us consider F₁ as a representative example:

The F₁ function is a Shifted and Rotated Bent Cigar Function that is unimodal in nature. The optimal value of F₁ is denoted by $F_i^{\ast }=F_i\left(x^{\ast }\right)=100$, where $x^{\ast }$ represents the global minimum.

The function $F_1\left(x\right)$ is obtained by applying a transformation function $f_1$ to $M\left(x-o_1\right)$, which is then added to $F_1^{\ast }$.

$$F_1\left(x\right)=f_1\left(M\left(x-o_1\right)\right)+F_1^{\ast }$$

(2)

Here, $o_1$ represents the randomly distributed shifted global optimum within ${\left[-80,80\right]}^D$.

The $F_1$ function is characterized by being unimodal, non-separable, and exhibiting a smooth but narrow ridge. Additionally, each function and basic function is assigned a different rotation matrix, denoted by $M_i$.

For comprehensive information on the CEC2017 benchmark functions, including their ranges, global optima, and other pertinent parameters, please refer to Section 5 and the CEC2017 benchmark function definitions document [53].

3.2. Salp Swarm Algorithm

The Salp Swarm algorithm, also known as SSA, is a recently published swarm optimization technique that imitates the swarming behavior of salps in water. The SSA method utilizes the concept of a salp chain, which is the swarming behavior of the barrel-shaped salps. Salps move and have tissues that are quite similar to jellyfish. In the ocean, salps engage in a particular form of swarming activity known as “salp chain” activity, which helps them locate food.

In SSA, individuals are designated as either leaders or followers. The leader of the salp chain is responsible for defining movement directions, finding food sources, leading the SSA chain to the food, and regularly updating the sites. The remaining population members are called “followers”, They take turns obeying the leader to construct the chainlike structure.

To provide compelling proof of the effectiveness of SSA, it can be simulated using a salp chain in search of optimal food sources, denoted as F within the search space. In this swarm, salps are classified as leaders or followers based on their position in the food chain. The chain is established with a leader initiating it, and all followers subsequently adopt the leader’s actions to gain direction for their behavior. It should be noted that SSA is user-friendly and shares similarities with other swarm intelligence algorithms. The method begins with establishing a salp population as the starting point. In Equation (1), the swarm X, consisting of n salps, is represented as a two-dimensional matrix.

$$X_i=\left[\begin{array}{cccc}{x}_1^1& x_2^1& \cdots & x_d^1\\ x_1^2& \ddots & \ddots & x_d^2\\ ⋮& \ddots & \ddots & ⋮\\ x_1^n& x_2^1& \cdots & x_d^n\end{array}\right]$$

(3)

$$x_i^1=\left\{\begin{array}{ll}{F}_i+c_1\left(\left(u{b}_i-l{b}_i\right)c_2+l{b}_i\right)& c_3\ge 0.5\\ F_i-c_1\left(\left(u{b}_i-l{b}_i\right)c_2+l{b}_i\right)& c_3<0.5\end{array}\right.$$

(4)

$x_i^1$ denotes the position of the chain salps leader with $j$th dimension; $F_i$ is the food position with $j$th dimension, $l{b}_j$ and $u{b}_j$ are the lower and upper bound of salps positions components, respectively, $c_2$ and $c_3$ are two random scalars from the range $\left[0,1\right]$.

During the iteration process, the essential control parameter is $c_1$, which stabilizes the exploitation and exploration phases. The following is the expression of $c_1$:

$$c_1=2e^{-{\left(\frac{4t}{Tmax}\right)}^2}$$

(5)

where t is the current iteration and Tmax is the maximum number of iterations. The salps chain of followers’ positions is updated by the following equation, such that:

$$x_i^j=\frac{1}{2} k\ast tim{e}^2+s_0 time$$

(6)

where $i\ge 2$, $x_i^j$ refers to the position of the jth salp in the ith dimension. $time$ represents the time, $s_0$ is an initial speed, having $k=\frac{s_{final }}{s_0}$ where $s_{final}=\frac{x-x_0}{time}$. In optimization, the number of iterations represents the amount of time taken. Thus, a difference in outcomes between iterations is equivalent to one unit of time. Assuming that, $s_0$ is equal to zero, the solution to this problem can be expressed as the following equation.

$$x_i^j=\frac{1}{2}(x_i^j+x_i^{j-1})$$

(7)

where $i\ge 2$.

3.3. Competitive Swarm Optimization Algorithm

The particles in the CSO learn from competitors chosen randomly rather than from the global or personal best position. First, in each cycle, the population of particles is randomly split into two groups of equal size. Then competition is established between the particles in each group. The particle with the highest fitness value at the end of the competition is declared the winner, and it is then instantly advanced to the following iteration. Conversely, after learning from the winner, the loser adjusts its position and velocity. The location and velocity of the loser are updated mathematically as follows:

$$V_l^{t+1}=R_1^t{V}_l^t+R_2^t\left(X_w^t-X_l^t\right)+\varphi R_3^t\left(X^{-t}-X_l^t\right)$$

(8)

$$X_l^{t+1}=X_l^t+V_l^{t+1}$$

(9)

where $t$ represent the current iteration, $R_1^t, R_2^t, R_3^t$ are randomly generated vectors within ${\left[0 , 1\right]}^n$, $X_w^t, X_l^t$ represent the winner and loser particle, respectively, $X^{-t}$ indicate the mean position of the current swarm in iteration $t$ and $\varphi$ is the control parameter of $X^{-t}$ influence. The pseudo-code for CSO is represented in algorithm (1).

The pseudo-code shows that CSO is easy to implement; however, two issues may contribute to low search efficiency. First, the particles are randomly matched in the competition, which may affect global exploration. Second, CSO only updates the worst particles while leaving the winners alone, which is not conducive to local exploitation. On the other hand, SSA suffers from low convergence and stagnation in local optima and rates. Therefore, this work combines SSA and CSO into a single algorithm to increase the overall performance of the heuristic by combining both models. The idea is maintaining a higher convergence rate while expanding the search space’s exploration rate.

4. The Proposed Algorithm

When solving and coping with optimization problems, an algorithm may get stuck in the local optimal value. In this case, the algorithm must search the solution space extensively to prevent local optimization. The innovative learning methodologies give the particles varied learning orientations by learning from possible particles rather than from the globally best particle and historically best particle as in the classic PSO. The particle that loses the competition in the CSO algorithm will modify its position by learning from its more active particle (winner). As was previously stated in this article, there is no evasive mechanism in the competitive swarm optimizer to deal with the problem of too few new particle generations in a single iteration. In addition, slow convergence and stagnation during continuous iteration problems.

Updating algorithms that are simple to calculate based on the original algorithm are being considered by the authors as possible solutions to these problems. Considering that the winning individual is not updated in the original method, it is logical to wonder if a simple update for the winning individual may be proposed. As a result, the authors suggest that the winners may be updated by employing the Salp Swarm approach in the SSA algorithm, guaranteeing a simple update of winners and a high-quality new search agent.

So, competition mechanisms are included in CL-SSA that separate solutions into winners and losers. Competition-based learning strategies are then used to update the loser particles, allowing the losers to explore their search space. Finally, a Salp Swarm (SA) approach is utilized to update the winning particles to enhance and update each generation’s winning particles. The enhanced CSO (CL-SSA) algorithm presented in this research is described in detail in this sub-section.

4.1. A Primitive Stage in CL-SSA Methodology

CL-SSA begins with an initial population of (N) individuals, as do most optimization methods. An individual’s search space is limited by the lower and upper bounds, as depicted in Equation (10).

$$X_i^j=l{b}_i+rand\times \left(u{b}_i-l{b}_i\right)$$

(10)

$X_i^j$ are the initial search agent, $i$ represents a number of random candidate solutions $i\in \left\{1,2,\dots , n\right\}$, in the search space. $j$ denotes the problem dimension $j\in \left\{1,2,\dots , d\right\}$. The lower and upper bounds are represented by $u{b}_i,l{b}_i,$ respectively.

4.2. Updating Solution Scenarios in CL-SSA

Within each iteration of the CL-SSA algorithm, a pairwise competition mechanism is utilized to segregate the solutions into distinct categories of winners and losers. The victor is distinguished by a superior fitness level, while the vanquished party exhibits inferior fitness levels. The losers with weak fitness are assigned to learn from the winners with good fitness. Figure 1 depicts the unique learning process. A pairwise competition mechanism is implemented as follows:

$$V_l^{t+1}=R_1^t{V}_l^t+R_2^t\left(X_w^t-X_l^t\right)+\varphi R_3^t\left(X^{-t}-X_l^t\right)$$

(11)

$$X_l^{t+1}=X_l^t+V_l^{t+1}$$

(12)

where $t$ represent the current iteration, $R_1^t,R_2^t,R_3^t$ are randomly generated vectors within ${\left[0 , 1\right]}^n$, $X_w^t, X_l^t$ represent the winner and loser particle, respectively, $X^{-t}$ indicate the mean position of the current swarm in iteration $t$ and $\varphi$ is the control parameter of $X^{-t}$ influence. Moreover, implicit multiplication refers to Hadamard’s element-wise vector multiplication operation.

4.3. Salp Swarm Mechanism to Update the Winners’ Position

After updating the loser individuals, the Salp Swarm (SA) strategy updates the winner individuals.

$$x_w^{t+1}=\left\{\begin{array}{ll}Xbest+c_1\left(\left(u{b}_j-l{b}_j\right)c_2+l{b}_j\right)& c_3\ge 0.5\\ Xbest-c_1\left(\left(u{b}_j-l{b}_j\right)c_2+l{b}_j\right)& c_3<0.5\end{array}\right.$$

(13)

$x_w^{t+1}$ represents the winner position with $\mathrm{j}$th dimension; $l{b}_j$ and $u{b}_j$ are the lower and upper bound of the position’s components, respectively. $Xbest$ is the best position, $c_2$ and $c_3$ are two random scalars from the range $\left[0,1\right]$. The essential control parameter is $c_1$, which stabilizes the exploitation and exploration phases. The pseudo-code of the CL-SSA based on the previous explanations is presented in the algorithm.

4.4. CL-SSA Stopping Condition

After completing the optimization scenarios and iterating until the stopping conditions are satisfied, the proposed CL-SSA obtains the best possible solution. A flowchart of the CL-SSA algorithm is depicted in Figure 2, which includes the pseudocode for the algorithm (Algorithms 1 and 2).

4.5. Computational Complexity

In practice, assessing the computing complexity of metaheuristic approaches is critical. The proposed CL-SSA and other algorithms (e.g., SSA, CSO, PSO, GWO, WOA and SCA) take O ($n_{o }\times n_{p }$) time to initialize the population, where $n_{p }$ denotes the size of the population, and the symbol $n_{o }$ is used to represent the dimension of the problem being solved, corresponding to the number of variables involved. The formula $O\left(Maximu{m}_{iterations }\times O_{f }\right)$ is used to calculate the fitness of search agents for all algorithms, where $O_{f }$ is the objective function for a given issue. The complete procedure takes $O\left(N\right)$ time to reproduce. The computational complexity of CL-SSA is $O\left(s\right)+O\left(Maximu{m}_{iterations }\times \left(S\times dim+S\right)\right)$ where dim is the problem dimension and $O\left(S\right)$ is the number of Search agents. Depending on this, the overall complexity $O\left(Maximu{m}_{iterations }\times S\times Dim\right)$. SSA, CSO, PSO, GWO, and SCA algorithms have a computational complexity of $O\left(N\times Maximu{m}_{iterations }\times n_{o }\times n_{p }\times O_{f }\right)$. As can be seen in

Table 2. Comparison between the execution time (in seconds) of CL-SSA and other algorithms in solving CEC2017 benchmarks functions with 100 dimensions, 30 times individually run.

F	CL-SSA	SSA	CSO	PSO	HHO	BAT	WOA	MFO	EO	SCA
F1	287.21	195.09	220.68	255.41	22.54	192.03	232.36	286.04	8.37	159.79
F2	201.89	203.66	231.04	264.92	40.11	203.31	241.71	265.97	21.96	164.05
F3	211.27	216.4	243.13	275.4	63.43	213.22	253.71	248.28	34.11	171.37
F4	297.32	205.38	231.76	264.15	39.5	202.74	240.56	235.56	21.93	164.13
F5	238.38	238.28	265.65	296.77	111.37	236.26	274.57	269.59	56.02	186.98
F6	224.25	224.59	250.52	282.18	77.43	223.46	258.58	255.09	41.45	174.89
F7	287.21	268.15	295.71	324.94	171.25	265.44	304.51	298.01	86.05	203.9
F8	274.86	242.64	267.79	297.46	115.24	237.1	274.74	271.08	57.75	190.03
F9	246.62	251.64	279.1	307.93	138.14	253.58	288.41	355.74	62.98	197.32
F10	256.08	216.27	241.59	273.2	65.15	214.17	251.11	261.55	29.77	202.89
F11	213.79	220.52	248.09	279.76	73.52	221.6	256.38	253.95	35.65	175.08
F12	221.36	218.85	246.55	276.12	69.79	221.48	255.18	249.81	32.84	171.78
F13	285.77	218.76	222.32	269.44	67.82	219.23	252.52	248.48	31.57	174.08
F14	213.46	213.82	195.34	224.99	57.49	203.31	248.61	245.2	27.91	174.03
F15	213.05	235.14	261.91	298.65	108.65	191.19	271.9	268.26	8.37	187.23
F16	238.39	392.74	420.72	459.88	430.48	392.07	433.06	389.03	54.26	285.85
F17	433.06	210.42	237.22	270.22	50.53	208.26	243.08	213.17	215	167.63
F18	208	262.79	290.94	324.83	158.65	260.59	298.34	370.8	27.36	202.48
F19	287.28	385.95	414.25	448.76	410.16	384.61	421.5	426.97	76.67	279.44
F20	422.17	274.48	301.33	334.03	145.53	271.83	307.45	302.59	177.31	207.33
F21	281.26	308.37	336.94	369.18	205.29	251.05	344.85	339.88	61.02	233.22
F22	328.08	343.02	367.96	408.49	267.22	230.64	382.52	375.93	84.34	286.25
F23	373.47	278.03	307.84	343.85	159.44	277.39	318.53	310.41	109.24	209.76
F24	298.62	308.73	309.59	373.95	202.27	305.01	346.63	275.55	65.77	226.62
F25	328.73	360.93	323.83	390.93	383.51	415.91	423.34	450.93	86.21	291.03
F26	462.95	283.29	286.84	333.96	384.99	421.18	346.27	448.95	144.74	245.84
F27	461.16	245.29	265.08	310.95	319.64	322.74	292.5	411.95	131.07	224.61
F28	370.26	325.03	340.37	387.9	511.23	319.61	369.95	499.72	107.2	290.05
F29	332.38	220.81	201.27	229.03	247.36	304.86	261.24	313.21	178.57	204.14

, the proposed CL-SSA and other algorithms have a relatively short operating time.

Algorithm 1: Pseudo-Code of CSO

Initialize position$X_i^t, V_{i }^t$according to upper and lower bounds. $t \leftarrow 0$ while (termination condition is not met) for (each particle $P_i^t$) Evaluate the New $\left(X_i^t\right)$ and recorded them as fitness if ($Fitness< Best\_Fitness$) then update the Best_Position and Best_Fitness end-if end for $P_{}^{t+1} \leftarrow \varnothing$ for (each Swarm in (Search Agents/2) Randomly select particles $P_{r1}^t and P_{r2}^t$ from $P_{}^t$ If $f\left(X_{r1}^t\right)$ is better than $f\left(X_{r2}^t\right)$ then $P_w^t \leftarrow P_{r1}^t , P_l^t \leftarrow P_{r2}^t$ else $P_w^t \leftarrow P_{r2}^t , P_l^t \leftarrow P_{r1}^t$ end if for (each dimension) $V_l^{t+1} = R_1^t{V}_l^t+R_2^t\left(X_w^t- X_l^t\right)+ \varphi R_3^t\left(X^{-t}- X_l^t\right)$ $X_l^{t+1}= X_l^t+ V_l^{t+1}$ $P_{}^{t+1} \leftarrow P_{}^{t+1} \cup \left\{P_w^t, P_l^{t+1}\right\}$  $P_{}^t \leftarrow P_{}^t \backslash \left\{P_{r1}^t, P_{r2}^t\right\}$ end for end for $t \leftarrow t+1$ end while

Algorithm 2: Pseudo-Code of CL-SSA

Initialize position$X_i^t, V_{i }^t$according to lower and upper bounds. $t \leftarrow 0$ while (termination condition is not met) for (each particle $P_i^t$) Evaluate the New $\left(X_i^t\right)$ and recorded them as fitness if ($Fitness< Best\_Fitness$) then update the Best_Position and Best_Fitness end-if end $P_{}^{t+1} \leftarrow \varnothing$ for (each Swarm in (Search Agents/2))   Randomly select particles $P_{r1}^t and P_{r2}^t$ from $P_{}^t$ If $f\left(X_{r1}^t\right)$ is better than $f\left(X_{r2}^t\right)$ then $P_w^t \leftarrow P_{r1}^t , P_l^t \leftarrow P_{r2}^t$ else $P_w^t \leftarrow P_{r2}^t , P_l^t \leftarrow P_{r1}^t$ end if for (each dimension) Update Loser Position $X_l^{t+1}$ By Equations (4) and (5) Update Winner Position $X_w^{t+1}$ By Equation (1) $P_{}^{t+1} \leftarrow P_{}^{t+1} \cup \left\{P_w^{t+1}, P_l^{t+1}\right\}$  $P_{}^t \leftarrow P_{}^t \backslash \left\{P_{r1}^t, P_{r2}^t\right\}$ end for end for $t \leftarrow t+1$ end while

In summary, the idea of the proposed algorithm is to maintain a higher convergence rate while increasing the search space’s exploration rate by utilizing the benefits of the two algorithms. There are two steps in the modified model. First, a pairwise competition mechanism is used, which divides the solutions into winners and losers. A chain population’s winners’ position is updated through the strong exploitation of SSA. Second, those particles which do not win the competition learn from the winners, leading to a good balance between exploration and exploitation. Two sets of large-scale benchmark functions were used to test the proposed algorithm’s general performance and compare it to existing metaheuristic algorithms.

5. Experimental Results and Analysis

This section tests CL-SSA’s performance on 50- and 100-dimensional CEC2017 [53] and LSGO problems from CEC2008 [54] with 200, 500, and 1000 dimensions.

Table 3. CEC 2017 benchmark function.

Type	Fun	Function Name	Fmin
U	F1	Shifted and Rotated Bent Cigar Function	100
U	F2	Shifted and Rotated Zakharov	300
M	F3	Shiftedv and Rotated Rosenbrock’s	400
M	F4	Shifted and Rotated Rastrigin’s	500
M	F5	Shifted and Rotated Expanded Scaffer’s F6	600
M	F6	Shifted and Rotated Lunacek Bi_Rastrigin	700
M	F7	Shifted and Rotated Non-Continuous Rastrigin’s	800
M	F8	Shifted and Rotated Levy	900
M	F9	Shifted and Rotated Schwefel’s	1000
H	F10	Hybrid Function 1 (N = 3)	1100
H	F11	Hybrid Function 2 (N = 3)	1200
H	F12	Hybrid Function 3 (N = 3)	1300
H	F13	Hybrid Function 4 (N = 4)	1400
H	F14	Hybrid Function 5 (N = 4)	1500
H	F15	Hybrid Function 6 (N = 4)	1600
H	F16	Hybrid Function 6 (N = 5)	1700
H	F17	Hybrid Function 6 (N = 5)	1800
H	F18	Hybrid Function 6 (N = 5)	1900
H	F19	Hybrid Function 6 (N = 6)	2000
C	F20	Composition Function 1 (N = 3)	2100
C	F21	Composition Function 2 (N = 3)	2200
C	F22	Composition Function 3 (N = 4)	2300
C	F23	Composition Function 4 (N = 4)	2400
C	F24	Composition Function 5 (N = 5)	2500
C	F25	Composition Function 6 (N = 5)	2600
C	F26	Composition Function 7 (N = 6)	2700
C	F27	Composition Function 8 (N = 6)	2800
C	F28	Composition Function 9 (N = 3)	2900
C	F29	Composition Function 10 (N = 3)	3000
		Range [−100, 100] D

and

Table 4. CEC 2008lsgo benchmark function.

Fun	Function Name	Characteristics
F1	Shifted Sphere	Separable
F2	Schwefel Problem	Non-Separable
F3	Shifted Rosenbrock	Non-Separable
F4	Shifted Rastrigin	Separable
F5	Shifted Griewank	Non-Separable/Separable
F6	Shifted Ackley	Separable
F7	Fast Fractal	Non-Separable

provide more information, respectively. The purpose of using such sets is to test the robustness of CL-SSA’s in solving a variety of benchmark functions. CL-SSA’s performance is compared to that of numerous state-of-the-art swarm intelligence algorithms, including SSA, CSO, PSO, HHO [55], BAT, WOA [56], MFO [57], EO [58], SMA [59], and SCA [60]. The performance and convergence behavior of CL-SSA was then compared to SSA variants, CPSO [61], PPSO_W [62], HIWOA [63], RW- GWO [64], CLPSO [65], WFOA [66], LJA [67], LNMRA [68], PPSO [62], and HPSO_TVAC [69]. In addition, the Friedman and the Wilcoxon rank-sum tests are used to evaluate the overall effectiveness of the algorithms using the performance data.

5.1. Experimental Setup

The characteristics of swarm intelligence algorithms make them stochastic, so they should be compared in a neutral environment, and all tests should be conducted fairly [69]. As a result, all algorithms were implemented on Python 3, and experiments were run on an Intel Core i5-7300U CPU (2.50 GHz) with 8 GB memory. The trials were carried out using CEC2017 benchmark functions with dimensions 50 and 100, including unimodal, multimodal, hybrid, and composite problems and 200, 500 and 1000 dimensional CEC2008lsgo benchmark functions. A 30-times independent experiment is conducted on each function. Setting the population size (Npop = 30) and the maximum number of iterations (Tmax = 2500) ensures consistency in programming across all experiments and fair comparisons.

5.2. Performance Metric and Parameter Setting

Three metrics are used in this study to evaluate the performance of the CL-SSA: the average of the optimization results, which can be determined as follows:

$$Mean=\frac{1}{n} {\displaystyle \sum }_{i=1}^n{S}_i$$

(14)

where $S_i$ is the result of each optimization. The standard deviation (std) of the optimization results can be determined by:

$$std=\sqrt{\frac{1}{n-1} {\displaystyle \sum }_{i=1}^n{\left(S_i-Mean \right)}^2}$$

(15)

and median (MED). Intuitively, the average value can illustrate how well algorithms perform and their ability to avoid errors generated in calculations. On the other hand, the std shows how dispersed the data is. The lower the std, the more robust and reliable the method is in practice.

Table 5. Settings of the parameters.

Algorithm	Parameter	Range/Value
CL-SSA	Coefficient ($\varphi$)	0.3
SSA	Coefficient (c1)	[2/e, 2]
GWO	Convergence constant (a)	[0, 2]
PSO	Inertia weight (wmin, wmax) Cognitive coefficient (c1, c2)	0.04, 0.092 2
HHO	beta	1.5
SCA	Convergence constant(r1)	[0, 2]
WOA	Convergence constant (a) Coefficient (b)	[0, 2] 1
LNMRA	Step Beta Sigma_v	0.001 1 1
SHADE	Miu_f Miu_cr	0.5 0.5
RW_GWO	Convergence constant (a)	[0, 2]
HIWOA	feedback_max Convergence constant (a) Coefficient (b) Coefficient (p)	10 [0, 2] 1 0.5
LJA	Sigma_v Multiplier beta	1 0.001 1
PPSO	v_max	0.5
PSO_W	v_max	0.5
DESAP_abs	Miu_f Miu_cr	0.5 0.5
HPSO_TVAC	Coefficient (ci) Coefficient (cf)	0.5 0.0
ESSA	Coefficient (r1) Coefficient (c1)	50 × random [2/e, 2]
IWSSA	Cmax, Cmin Coefficient (c1)	1, 0.00003 [2/e, 2]
ISSA	Cmax, Cmin Coefficient (c1)	1, 0.00003 [2/e, 2]
IWSSA	Cmax, Cmin Coefficient (c1)	1, 0.00003 [2/e, 2]
STS-SSA	Coefficient (c1) Coefficient (r)	[2/e, 2] Random
HSSASCA	Coefficient (c1) Coefficient (r)	[2/e, 2] 2 × π × Random
ISSA_OBL	Coefficient (c1) Max_local_iteration	[2/e, 2] 10
TVSSA	Coefficient (c1)	[2/e, 2]

gives the values of CL-SSA’s parameter and the other methods, derived from the original articles. By selecting parameters in this manner, it becomes possible for each algorithm to operate at an optimal level of efficacy, thus guaranteeing fair comparisons. The best trials will be highlighted in bold.

5.3. Numerical Performance Evaluation on CEC2017

Table 6. Comparison of CL-SSA results on CEC 2017 unimodal functions with traditional algorithms during 2500 iterations.

Fun	Dim	Criteria	CL-SSA	SSA	CSO	PSO	HHO	BAT	WOA	MFO	EO	SSA
F1	50	Mean	1.1998E+04	3.9718E+09	1.8024E+12	1.5226E+11	7.0994E+11	1.3576E+12	1.1616E+10	4.8803E+11	8.2585E+10	3.9718E+09
		Std	1.0175E+04	2.5374E+09	2.1917E+11	4.9522E+10	7.9216E+10	3.2370E+10	4.4716E+09	1.9694E+11	4.5875E+10	2.5374E+09
		Median	5.1813E+11	3.4296E+09	1.7786E+12	1.5494E+11	7.3182E+11	7.7654E+10	1.0101E+10	5.1813E+11	7.3088E+10	3.4296E+09
	100	Mean	1.2695E+04	2.3573E+11	4.6180E+12	1.1253E+12	2.0656E+12	3.4885E+12	2.2612E+11	1.3768E+12	6.1263E+11	2.3573E+11
		Std	1.4874E+04	4.5722E+10	2.6313E+11	1.8567E+11	1.2105E+11	4.6928E+11	4.4300E+10	5.6156E+11	1.0773E+11	4.5722E+10
		Median	1.3444E+12	2.3452E+11	4.5837E+12	1.1540E+12	2.0805E+12	3.4536E+12	2.3080E+11	1.3444E+12	5.8255E+11	2.3452E+11
F2	50	Mean	5.2200E+04	1.1225E+05	7.2610E+05	1.6300E+05	2.0354E+05	4.1816E+06	1.9314E+05	3.0085E+05	1.0838E+05	1.1225E+05
		Std	1.7077E+04	2.5059E+04	1.5853E+06	2.9673E+04	2.7572E+04	1.5051E+04	6.0011E+04	1.0468E+05	1.7539E+04	2.5059E+04
		Median	2.7934E+05	1.0950E+05	3.8059E+05	1.6160E+05	2.0033E+05	8.6020E+04	1.8246E+05	2.7934E+05	1.1072E+05	1.0950E+05
	100	Mean	4.5659E+05	3.6710E+05	9.5294E+05	4.6890E+05	3.5304E+05	5.7655E+06	8.4087E+05	9.2292E+05	4.1302E+05	3.6710E+05
		Std	6.1241E+04	5.8214E+04	3.6471E+05	5.1678E+04	9.9450E+03	1.4749E+07	1.5496E+05	1.6262E+05	3.1601E+05	5.8214E+04
		Median	9.3209E+05	3.6332E+05	8.5414E+05	4.5531E+05	3.5674E+05	1.1743E+06	8.4470E+05	9.3209E+05	3.2705E+05	3.6332E+05
Rank	50	W-T-L	02/00/00	00/00/02	00/00/02	00/00/02	00/00/02	00/00/02	00/00/02	00/00/02	00/00/02	00/00/02
	100	W-T-L	01/00/01	00/00/02	00/00/02	00/00/02	01/00/01	00/00/02	00/00/02	00/00/02	00/00/02	00/00/02

Bold indicates Best values.

,

Table 7. Comparison of CL-SSA results on CEC 2017 multimodal functions with traditional algorithms during 2500 iterations.

Fun	Dim	Criteria	CL-SSA	SSA	CSO	PSO	HHO	BAT	WOA	MFO	EO	SCA
F3	50	Mean	5.6439E+02	8.0769E+02	6.4537E+04	4.7532E+03	2.1139E+04	5.2325E+04	1.1536E+03	5.3589E+03	1.3770E+03	8.6304E+03
		Std	4.5872E+01	9.7707E+01	1.4021E+04	1.2961E+03	4.2301E+03	4.6405E+02	1.4601E+02	3.4754E+03	5.4146E+02	1.7564E+03
		Median	4.3417E+03	7.9184E+02	6.6024E+04	4.9295E+03	2.0775E+04	1.1903E+03	1.1226E+03	4.3417E+03	1.1984E+03	8.6066E+03
	100	Mean	6.9476E+02	3.5834E+03	1.9418E+05	2.5358E+04	6.5536E+04	1.3810E+05	4.4686E+03	3.5353E+04	7.1286E+03	3.8574E+04
		Std	3.6103E+01	1.0589E+03	3.1389E+04	3.9858E+03	9.5881E+03	3.2215E+04	9.5637E+02	1.4718E+04	2.7111E+03	6.7270E+03
		Median	3.4185E+04	3.4503E+03	1.9109E+05	2.5663E+04	6.7425E+04	1.3143E+05	4.3586E+03	3.4185E+04	6.8168E+03	3.8365E+04
F4	50	Mean	7.3945E+02	8.5057E+02	1.4813E+03	9.7399E+02	9.5115E+02	1.1761E+03	9.7640E+02	9.9080E+02	7.7653E+02	1.0918E+03
		Std	5.4397E+01	7.1369E+01	6.6096E+01	6.1558E+01	3.6029E+01	4.0567E+01	7.4720E+01	9.1509E+01	3.1363E+01	3.3506E+01
		Median	9.7027E+02	8.4573E+02	1.4863E+03	9.7285E+02	9.4841E+02	7.3393E+02	9.5886E+02	9.7027E+02	7.8363E+02	1.0953E+03
	100	Mean	9.4326E+02	1.5435E+03	2.6773E+03	1.7927E+03	1.6936E+03	2.1506E+03	1.7317E+03	1.8728E+03	1.3261E+03	1.9699E+03
		Std	6.5659E+01	1.0227E+02	1.1495E+02	7.5827E+01	5.9818E+01	1.6425E+02	1.4248E+02	1.6220E+02	6.4067E+01	6.4611E+01
		Median	1.8701E+03	1.5232E+03	2.6829E+03	1.8010E+03	1.6866E+03	2.1578E+03	1.7322E+03	1.8701E+03	1.3211E+03	1.9848E+03
F5	50	Mean	6.4519E+02	6.8390E+02	7.7410E+02	6.8480E+02	7.0210E+02	7.1176E+02	7.1845E+02	6.8939E+02	6.4733E+02	6.9978E+02
		Std	1.4066E+01	1.1133E+01	1.1936E+01	8.1966E+00	6.5216E+00	1.0412E+01	1.5680E+01	1.2821E+01	1.0023E+01	5.9072E+00
		Median	6.8712E+02	6.8335E+02	7.7338E+02	6.8565E+02	7.0246E+02	6.3397E+02	7.1904E+02	6.8712E+02	6.4863E+02	6.9806E+02
	100	Mean	6.3656E+02	6.9087E+02	7.6776E+02	7.0414E+02	6.9365E+02	7.1141E+02	7.1100E+02	7.0228E+02	6.6936E+02	7.1383E+02
		Std	1.0267E+01	6.8905E+00	8.6535E+00	6.7014E+00	4.3198E+00	1.2306E+01	1.1620E+01	1.0443E+01	7.0990E+00	6.9836E+00
		Median	7.0111E+02	6.9128E+02	7.6862E+02	7.0363E+02	6.9392E+02	7.0921E+02	7.0734E+02	7.0111E+02	6.6915E+02	7.1414E+02
F6	50	Mean	1.0744E+03	1.4831E+03	4.0768E+03	1.5149E+03	1.7497E+03	3.0027E+03	1.7913E+03	2.1734E+03	1.1440E+03	1.7071E+03
		Std	8.6499E+01	1.6259E+02	2.8262E+02	7.0157E+01	5.7989E+01	1.0037E+02	9.9501E+01	5.1998E+02	8.4218E+01	7.8656E+01
		Median	2.1207E+03	1.4348E+03	4.1110E+03	1.4962E+03	1.7734E+03	1.0981E+03	1.7802E+03	2.1207E+03	1.1441E+03	1.7140E+03
	100	Mean	1.3542E+03	3.3039E+03	9.0612E+03	3.1808E+03	3.3356E+03	6.1288E+03	3.5676E+03	5.4031E+03	2.3115E+03	3.7154E+03
		Std	8.6771E+01	1.7992E+02	5.7278E+02	1.6058E+02	9.9968E+01	1.1528E+03	1.8337E+02	9.7577E+02	1.9286E+02	1.5484E+02
		Median	5.6745E+03	3.3170E+03	9.0759E+03	3.2097E+03	3.3565E+03	5.8336E+03	3.5718E+03	5.6745E+03	2.2700E+03	3.6979E+03
F7	50	Mean	1.0465E+03	1.1845E+03	1.7719E+03	1.2504E+03	1.2060E+03	1.7460E+03	1.2533E+03	1.4034E+03	1.0700E+03	1.4059E+03
		Std	4.9501E+01	7.8451E+01	7.0720E+01	5.2763E+01	3.7096E+01	8.7391E+01	5.4483E+01	7.9131E+01	3.3495E+01	3.0096E+01
		Median	1.3732E+03	1.1791E+03	1.7741E+03	1.2460E+03	1.2005E+03	1.0367E+03	1.2551E+03	1.3732E+03	1.0742E+03	1.4110E+03
	100	Mean	1.2313E+03	1.9669E+03	3.1342E+03	2.1341E+03	2.0172E+03	3.0101E+03	2.0936E+03	2.5702E+03	1.5877E+03	2.3392E+03
		Std	6.8454E+01	1.1068E+02	1.1098E+02	9.2826E+01	6.1783E+01	1.7068E+02	1.2279E+02	1.9038E+02	5.9831E+01	7.5155E+01
		Median	2.5610E+03	1.9887E+03	3.1105E+03	2.0954E+03	2.0153E+03	2.9646E+03	2.0981E+03	2.5610E+03	1.5779E+03	2.3421E+03
F8	50	Mean	1.0629E+04	1.4864E+04	7.2084E+04	1.9896E+04	1.4882E+04	1.6487E+04	2.8484E+04	1.8674E+04	7.9795E+03	2.5729E+04
		Std	5.5501E+03	3.0426E+03	1.0019E+04	3.9645E+03	1.1714E+03	4.1021E+03	8.0053E+03	5.4503E+03	1.9059E+03	4.0333E+03
		Median	1.7816E+04	1.4591E+04	7.2679E+04	2.0257E+04	1.4634E+04	1.1895E+04	2.6412E+04	1.7816E+04	7.9439E+03	2.5422E+04
	100	Mean	1.0659E+04	3.9503E+04	1.6315E+05	6.7153E+04	3.2090E+04	3.3981E+04	5.8392E+04	5.0896E+04	3.1658E+04	8.2355E+04
		Std	4.1685E+03	5.1322E+03	1.5433E+04	9.6433E+03	2.8940E+03	6.2995E+03	1.1831E+04	7.8591E+03	4.1729E+03	8.0208E+03
		Median	5.0552E+04	3.9880E+04	1.6455E+05	6.6113E+04	3.1426E+04	3.3349E+04	5.7792E+04	5.0552E+04	3.1664E+04	8.1893E+04
F9	50	Mean	7.6133E+03	8.1752E+03	1.5695E+04	1.4572E+04	1.1396E+04	1.1470E+04	1.1941E+04	8.9003E+03	8.5931E+03	1.5019E+04
		Std	1.0814E+03	8.4831E+02	5.9362E+02	7.0817E+02	1.3035E+03	2.5879E+03	1.4001E+03	1.2283E+03	1.0440E+03	3.6635E+02
		Median	8.8946E+03	8.0082E+03	1.5781E+04	1.4688E+04	1.1109E+04	7.0758E+03	1.1961E+04	8.8946E+03	8.6631E+03	1.5061E+04
	100	Mean	1.4408E+04	2.0005E+04	3.3422E+04	3.2142E+04	2.5823E+04	2.6739E+04	2.5949E+04	1.8797E+04	2.0995E+04	3.2214E+04
		Std	1.4399E+03	1.6465E+03	6.9795E+02	5.8905E+02	1.7891E+03	2.0880E+03	2.2590E+03	2.3036E+03	1.6122E+03	5.9194E+02
		Median	1.9340E+04	2.0084E+04	3.3462E+04	3.2126E+04	2.5795E+04	2.7179E+04	2.5891E+04	1.9340E+04	2.1033E+04	3.2377E+04
Rank	50	W-T-L	06/00/01	00/00/07	00/00/07	00/00/07	00/01/06	00/00/07	00/00/07	00/00/07	01/00/05	00/00/07
	100	W-T-L	07/00/00	00/00/07	00/00/07	00/00/07	00/00/07	00/00/07	00/00/07	00/00/07	01/00/06	00/00/07

,

Table 8. Comparison of CL-SSA results on CEC 2017 hybrid functions with traditional algorithms during 2500 iterations.

Fun	Dim	Criteria	CL-SSA	SSA	CSO	PSO	HHO	BAT	WOA	MFO	EO	SCA
F10	50	Mean	1.4681E+03	2.5243E+03	4.9765E+04	4.5045E+03	1.6366E+04	9.2219E+04	2.7826E+03	2.2277E+04	3.7610E+03	8.8287E+03
		Std	8.1378E+01	5.2605E+02	1.4261E+04	1.1911E+03	2.6789E+03	1.9235E+03	6.2324E+02	1.6461E+04	1.4568E+03	1.9562E+03
		Median	1.7249E+04	2.4019E+03	4.8166E+04	4.1196E+03	1.7185E+04	4.6620E+03	2.6769E+03	1.7249E+04	4.0441E+03	8.5790E+03
	100	Mean	4.1092E+03	6.7060E+04	7.9919E+05	1.1680E+05	2.6486E+05	2.2139E+06	1.5208E+05	1.7267E+05	5.9696E+04	1.2000E+05
		Std	5.2293E+02	1.4812E+04	1.8637E+06	1.7481E+04	8.5476E+04	6.5401E+06	7.0243E+04	9.7593E+04	1.6276E+04	2.0472E+04
		Median	1.7809E+05	6.8557E+04	4.3373E+05	1.1959E+05	2.3515E+05	5.9234E+05	1.3497E+05	1.7809E+05	6.1605E+04	1.2443E+05
F11	50	Mean	4.3991E+08	2.8427E+09	8.7154E+11	3.7629E+10	3.8376E+11	7.0702E+11	6.6106E+09	6.2771E+10	6.5190E+09	1.2244E+11
		Std	2.8169E+08	2.6407E+09	2.1318E+11	1.6754E+10	1.1125E+11	1.3012E+10	3.6491E+09	4.4554E+10	1.3511E+10	2.9309E+10
		Median	4.8341E+10	1.8342E+09	8.3733E+11	3.4483E+10	3.9675E+11	5.6990E+09	5.7552E+09	4.8341E+10	8.3411E+08	1.2433E+11
	100	Mean	1.6010E+09	2.0037E+10	2.3313E+12	2.4344E+11	1.1610E+12	2.0457E+12	3.2977E+10	4.2754E+11	7.5556E+10	6.4331E+11
		Std	8.0595E+08	1.0242E+10	3.8233E+11	6.7627E+10	1.9599E+11	3.4340E+11	1.1660E+10	2.5471E+11	5.3742E+10	1.0672E+11
		Median	4.3611E+11	1.7447E+10	2.3884E+12	2.2957E+11	1.1553E+12	2.0770E+12	3.0185E+10	4.3611E+11	6.3779E+10	6.3042E+11
F12	50	Mean	1.7051E+05	1.1362E+05	5.7140E+11	5.6417E+09	1.7003E+11	4.9892E+11	1.7950E+08	2.3321E+10	2.1894E+08	4.2119E+10
		Std	1.4031E+05	6.6878E+04	2.0472E+11	3.4595E+09	1.1734E+11	1.1307E+09	2.4990E+08	2.8751E+10	4.1634E+08	1.5629E+10
		Median	7.4904E+09	9.4230E+04	5.8955E+11	4.5182E+09	1.4086E+11	1.1385E+09	1.1254E+08	7.4904E+09	7.4045E+07	3.7503E+10
	100	Mean	9.0617E+04	1.5242E+05	7.0462E+11	4.4845E+10	2.9775E+11	5.8092E+11	5.5065E+08	9.2866E+10	5.0401E+09	1.2810E+11
		Std	3.7540E+04	1.4096E+05	1.2916E+11	1.5357E+10	4.9243E+10	1.3604E+11	2.9915E+08	6.0448E+10	6.0453E+09	2.3883E+10
		Median	8.1160E+10	9.1384E+04	7.3085E+11	4.1476E+10	2.8204E+11	5.9749E+11	4.5559E+08	8.1160E+10	2.9355E+09	1.3108E+11
F13	50	Mean	1.2008E+05	7.7740E+05	1.3423E+08	1.1785E+06	3.0409E+07	1.3900E+08	2.2916E+06	2.3597E+06	6.0765E+05	4.8600E+06
		Std	7.1381E+04	7.6717E+05	9.5201E+07	1.2007E+06	3.1658E+07	8.4840E+05	1.7075E+06	4.0136E+06	4.1725E+05	3.3729E+06
		Median	1.0330E+06	5.7270E+05	1.0402E+08	6.7662E+05	1.9990E+07	6.4380E+05	2.1273E+06	1.0330E+06	4.3257E+05	3.7367E+06
	100	Mean	1.1320E+06	1.0147E+07	2.6889E+08	1.2765E+07	3.2072E+07	2.4361E+08	1.0287E+07	2.7774E+07	3.4174E+06	3.7005E+07
		Std	6.8693E+05	7.2673E+06	1.2851E+08	8.6537E+06	1.7790E+07	1.6703E+08	4.3293E+06	3.4254E+07	2.0685E+06	1.5343E+07
		Median	1.4734E+07	9.2296E+06	2.4549E+08	9.5284E+06	2.9672E+07	2.1031E+08	1.0289E+07	1.4734E+07	2.9409E+06	3.3851E+07
F14	50	Mean	1.1646E+05	6.1851E+04	1.5473E+11	1.0299E+08	1.9376E+10	1.0830E+11	2.6692E+07	1.5315E+09	1.0017E+07	6.0028E+09
		Std	7.3090E+04	3.1378E+04	6.3059E+10	9.2472E+07	1.4683E+10	1.3302E+09	4.6381E+07	2.9099E+09	1.8452E+07	2.9843E+09
		Median	3.6516E+05	5.3678E+04	1.4639E+11	6.1667E+07	1.7076E+10	5.6777E+06	6.9735E+06	3.6516E+05	6.3119E+06	5.5749E+09
	100	Mean	8.0538E+04	8.7903E+04	3.1412E+11	2.0396E+09	1.2474E+11	2.9806E+11	9.4876E+07	2.9843E+10	1.5343E+08	4.0447E+10
		Std	2.8115E+04	5.0676E+04	7.4723E+10	1.1441E+09	3.2832E+10	7.6292E+10	1.1027E+08	2.8454E+10	3.4253E+08	1.0127E+10
		Median	2.1269E+10	7.3956E+04	3.1500E+11	1.7431E+09	1.2729E+11	2.9760E+11	5.9699E+07	2.1269E+10	9.6757E+07	4.0855E+10
F15	50	Mean	3.2502E+03	4.0044E+03	1.1277E+04	4.5017E+03	7.4021E+03	1.0050E+04	5.5156E+03	4.4713E+03	4.3088E+03	5.8587E+03
		Std	4.5413E+02	5.6240E+02	1.7481E+03	6.5274E+02	1.7532E+03	5.1707E+02	8.2213E+02	5.1265E+02	5.0526E+02	4.3894E+02
		Median	4.4675E+03	3.8421E+03	1.0868E+04	4.5738E+03	7.0329E+03	3.1450E+03	5.4215E+03	4.4675E+03	3.2075E+03	5.9157E+03
	100	Mean	5.7302E+03	8.3449E+03	3.1151E+04	1.1853E+04	1.8556E+04	2.5482E+04	1.3820E+04	8.6144E+03	7.0167E+03	1.3767E+04
		Std	7.4299E+02	9.7690E+02	5.0418E+03	1.0479E+03	3.3873E+03	4.6551E+03	1.8683E+03	9.4573E+02	7.3247E+02	8.2636E+02
		Median	8.6344E+03	8.3358E+03	2.9385E+04	1.1640E+04	1.7780E+04	2.5823E+04	1.3382E+04	8.6344E+03	6.9794E+03	1.3646E+04
F16	50	Mean	3.2088E+03	3.6578E+03	1.1841E+05	3.4267E+03	5.2162E+03	1.4824E+05	4.1929E+03	4.5350E+03	2.8172E+03	4.7099E+03
		Std	4.8759E+02	3.7987E+02	1.6342E+05	3.7043E+02	1.0849E+03	2.4953E+02	4.7261E+02	1.5261E+03	3.4580E+02	2.9107E+02
		Median	4.1822E+03	3.5684E+03	7.3252E+04	3.3747E+03	4.9515E+03	2.8549E+03	4.1133E+03	4.1822E+03	2.7939E+03	4.7159E+03
	100	Mean	4.9906E+03	6.3522E+03	3.5467E+07	7.6711E+03	1.2458E+06	2.9129E+07	9.0697E+03	1.3964E+04	6.0923E+03	3.1784E+04
		Std	5.5442E+02	7.0946E+02	3.8771E+07	7.8138E+02	1.7074E+06	2.9679E+07	1.4551E+03	8.8489E+03	9.0458E+02	3.4417E+04
		Median	1.0276E+04	6.4461E+03	2.1059E+07	7.6258E+03	7.7396E+05	1.5652E+07	8.7400E+03	1.0276E+04	5.9459E+03	1.7575E+04
F17	50	Mean	1.0508E+06	5.8773E+06	3.5688E+08	7.8895E+06	7.1213E+07	4.5271E+08	1.6329E+07	1.2940E+07	3.3766E+06	2.7543E+07
		Std	1.1657E+06	4.1666E+06	2.3422E+08	5.3734E+06	4.1172E+07	1.1304E+07	1.2742E+07	1.5007E+07	3.0159E+06	1.3971E+07
		Median	9.1585E+06	4.5470E+06	3.4075E+08	6.2030E+06	5.3916E+07	3.4996E+06	1.1674E+07	9.1585E+06	2.2264E+06	2.3506E+07
	100	Mean	2.2456E+06	9.4022E+06	6.3412E+08	1.2765E+07	4.4600E+07	5.7997E+08	6.7821E+06	1.5582E+07	4.9296E+06	6.8871E+07
		Std	1.0847E+06	6.5439E+06	2.6073E+08	5.6760E+06	2.9050E+07	3.7541E+08	3.3439E+06	2.2018E+07	2.3412E+06	2.7359E+07
		Median	8.6561E+06	6.9434E+06	5.9068E+08	1.1905E+07	3.9400E+07	4.7834E+08	5.7711E+06	8.6561E+06	4.5522E+06	6.3998E+07
F18	50	Mean	1.1831E+07	1.8537E+07	7.4824E+10	1.5669E+08	7.6594E+09	1.7953E+04	2.9073E+07	8.1900E+08	1.0586E+07	3.7441E+09
		Std	1.0640E+07	1.9845E+07	2.6950E+10	2.1936E+08	8.2028E+09	8.1862E+07	6.0493E+07	2.3042E+09	4.6176E+07	1.6665E+09
		Median	3.9884E+07	8.9217E+06	7.2336E+10	6.8679E+07	6.0827E+09	5.0090E+06	1.0580E+07	3.9884E+07	1.6186E+06	3.4072E+09
	100	Mean	3.8830E+07	9.4704E+07	3.4352E+11	6.0974E+09	1.2598E+11	2.9056E+11	1.3396E+08	2.3635E+10	2.5645E+08	3.7390E+10
		Std	2.6479E+07	9.2518E+07	9.1658E+10	2.1388E+09	3.8086E+10	8.8460E+10	9.1640E+07	2.5503E+10	2.4401E+08	1.2168E+10
		Median	1.7162E+10	5.8316E+07	3.4896E+11	6.0070E+09	1.2332E+11	3.0659E+11	1.1066E+08	1.7162E+10	1.6820E+08	3.6621E+10
F19	50	Mean	3.0415E+03	3.2123E+03	4.4799E+03	3.8124E+03	3.5217E+03	4.2371E+03	3.7654E+03	3.8433E+03	2.7373E+03	4.0073E+03
		Std	3.0214E+02	2.8691E+02	2.3923E+02	3.2409E+02	2.8057E+02	4.8895E+02	3.5123E+02	2.9218E+02	1.8773E+02	1.6109E+02
		Median	3.7971E+03	3.2352E+03	4.4991E+03	3.8840E+03	3.5962E+03	3.1441E+03	3.8130E+03	3.7971E+03	2.7515E+03	4.0069E+03
	100	Mean	4.9088E+03	5.3169E+03	8.1496E+03	7.3618E+03	6.1294E+03	6.5275E+03	6.4749E+03	5.8325E+03	4.9299E+03	7.4805E+03
		Std	6.2352E+02	4.9608E+02	3.3625E+02	3.1514E+02	4.5139E+02	6.2945E+02	6.2752E+02	4.9779E+02	5.8303E+02	2.9552E+02
		Median	5.9831E+03	5.3501E+03	8.1779E+03	7.3775E+03	6.0798E+03	6.3653E+03	6.4573E+03	5.9831E+03	4.7970E+03	7.5185E+03
Rank	50	W-T-L	05/00/05	02/00/08	00/00/10	00/00/10	00/00/10	01/00/09	00/00/10	00/00/10	02/00/08	00/00/10
	100	W-T-L	10/00/00	00/00/10	00/00/10	00/00/10	00/00/10	00/00/10	00/00/10	00/00/10	00/00/10	00/00/10

and

Table 9. Comparison results of the CL-SSA on CEC 2017 composite functions with traditional algorithms during 2500 iterations.

Fun	Dim	Criteria	CL-SSA	SSA	CSO	PSO	HHO	BAT	WOA	MFO	EO	SCA
F20	50	Mean	2.5119E+03	2.6337E+03	3.4272E+03	2.8132E+03	3.0354E+03	3.1505E+03	2.9591E+03	2.7880E+03	2.5549E+03	2.9080E+03
		Std	6.9664E+01	6.5141E+01	1.3392E+02	4.5956E+01	8.5156E+01	5.1253E+01	1.0125E+02	7.4204E+01	4.0495E+01	4.4603E+01
		Median	2.7805E+03	2.6461E+03	3.4129E+03	2.8191E+03	3.0136E+03	2.5296E+03	2.9529E+03	2.7805E+03	2.5501E+03	2.9030E+03
	100	Mean	2.7576E+03	3.4894E+03	5.2211E+03	3.8893E+03	4.5762E+03	4.9836E+03	4.2348E+03	3.7608E+03	3.1810E+03	4.0482E+03
		Std	5.3408E+01	1.5515E+02	2.3706E+02	1.0796E+02	2.3923E+02	2.3724E+02	1.5350E+02	1.5180E+02	1.0243E+02	9.8909E+01
		Median	3.7430E+03	3.4769E+03	5.2287E+03	3.8850E+03	4.5362E+03	5.0220E+03	4.2096E+03	3.7430E+03	3.1723E+03	4.0420E+03
F21	50	Mean	9.0056E+03	1.0456E+04	1.7535E+04	1.5929E+04	1.3399E+04	1.3899E+04	1.3274E+04	1.0489E+04	1.0430E+04	1.6654E+04
		Std	9.8797E+02	1.8497E+03	6.7861E+02	1.7448E+03	1.2077E+03	2.5284E+03	1.3236E+03	1.0101E+03	1.1499E+03	4.3188E+02
		Median	1.0681E+04	1.0309E+04	1.7459E+04	1.6385E+04	1.3331E+04	8.7901E+03	1.3483E+04	1.0681E+04	1.0284E+04	1.6743E+04
	100	Mean	1.6362E+04	2.2767E+04	3.5695E+04	3.4442E+04	2.8812E+04	2.9013E+04	2.9336E+04	2.0740E+04	2.3329E+04	3.4620E+04
		Std	1.5260E+03	3.9441E+03	5.9943E+02	9.0043E+02	1.8223E+03	2.1403E+03	1.4424E+03	1.8117E+03	2.0662E+03	4.8249E+02
		Median	2.0743E+04	2.3048E+04	3.5696E+04	3.4536E+04	2.8888E+04	2.9524E+04	2.9309E+04	2.0743E+04	2.3727E+04	3.4696E+04
F22	50	Mean	2.9966E+03	3.1652E+03	4.9413E+03	3.4429E+03	4.2338E+03	4.6178E+03	3.7212E+03	3.2306E+03	3.0158E+03	3.5900E+03
		Std	6.2754E+01	9.9767E+01	3.5815E+02	8.0966E+01	2.1316E+02	9.5482E+01	1.8827E+02	7.5640E+01	7.6110E+01	7.4917E+01
		Median	3.2190E+03	3.1466E+03	4.9722E+03	3.4505E+03	4.2068E+03	2.9887E+03	3.7509E+03	3.2190E+03	2.9988E+03	3.5781E+03
	100	Mean	3.2692E+03	4.0517E+03	7.8410E+03	4.8826E+03	6.1546E+03	6.5157E+03	5.0337E+03	3.9574E+03	3.8096E+03	5.0469E+03
		Std	6.2223E+01	1.9241E+02	7.6971E+02	1.6030E+02	4.1342E+02	3.2219E+02	2.2157E+02	1.5333E+02	1.1032E+02	1.1976E+02
		Median	3.9539E+03	4.0224E+03	7.8317E+03	4.8976E+03	6.0059E+03	6.4910E+03	5.0372E+03	3.9539E+03	3.7903E+03	5.0380E+03
F23	50	Mean	3.1114E+03	3.2941E+03	5.4531E+03	3.6635E+03	4.4963E+03	4.8639E+03	3.7762E+03	3.2456E+03	3.2548E+03	3.7748E+03
		Std	5.2938E+01	9.1873E+01	5.3837E+02	7.9836E+01	2.4054E+02	1.3758E+02	1.7234E+02	5.0906E+01	5.3650E+01	6.0641E+01
		Median	3.2428E+03	3.2697E+03	5.4339E+03	3.6529E+03	4.4862E+03	3.1655E+03	3.7733E+03	3.2428E+03	3.2531E+03	3.7681E+03
	100	Mean	3.7076E+03	4.7885E+03	1.3579E+04	6.5373E+03	9.4448E+03	1.0324E+04	6.2601E+03	4.5921E+03	4.6739E+03	6.8837E+03
		Std	8.7531E+01	2.2635E+02	1.1507E+03	4.2588E+02	8.3798E+02	9.4264E+02	4.9438E+02	2.1200E+02	1.6028E+02	2.3409E+02
		Median	4.5432E+03	4.8186E+03	1.3644E+04	6.4859E+03	9.2670E+03	1.0264E+04	6.2316E+03	4.5432E+03	4.7030E+03	6.9293E+03
F24	50	Mean	3.0398E+03	3.3119E+03	3.1935E+04	5.7477E+03	1.0165E+04	2.5086E+04	3.4868E+03	6.3543E+03	3.4880E+03	7.3824E+03
		Std	1.9911E+01	8.8150E+01	6.1081E+03	6.9769E+02	9.7346E+02	4.2855E+02	1.3352E+02	3.8485E+03	3.0011E+02	7.7618E+02
		Median	4.6978E+03	3.2901E+03	3.2585E+04	5.6523E+03	1.0114E+04	3.6081E+03	3.4616E+03	4.6978E+03	3.4318E+03	7.1342E+03
	100	Mean	3.3596E+03	6.0008E+03	7.2258E+04	1.1799E+04	1.9723E+04	4.7479E+04	5.5848E+03	1.2524E+04	6.7010E+03	1.7955E+04
		Std	6.1848E+01	6.1195E+02	1.1334E+04	1.3336E+03	1.5921E+03	9.3177E+03	4.3271E+02	4.5619E+03	1.3861E+03	2.0896E+03
		Median	1.1606E+04	6.0001E+03	7.1537E+04	1.1688E+04	1.9814E+04	4.6036E+04	5.5343E+03	1.1606E+04	6.3627E+03	1.7465E+04
F25	50	Mean	6.2166E+03	8.1864E+03	2.5739E+04	1.0853E+04	1.4923E+04	2.1654E+04	1.3985E+04	9.0482E+03	7.1443E+03	1.2815E+04
		Std	7.0977E+02	2.6747E+03	2.8204E+03	7.2209E+02	7.0111E+02	7.7652E+02	1.2147E+03	8.4580E+02	8.8017E+02	4.8472E+02
		Median	8.9413E+03	8.4747E+03	2.5622E+04	1.0808E+04	1.4790E+04	6.6293E+03	1.4167E+04	8.9413E+03	7.0630E+03	1.2770E+04
	100	Mean	1.0423E+04	2.5297E+04	7.5461E+04	2.9229E+04	4.4998E+04	6.9185E+04	3.4082E+04	2.0671E+04	2.0258E+04	3.7638E+04
		Std	9.5108E+02	4.3512E+03	5.3480E+03	2.0919E+03	2.5397E+03	1.0123E+04	3.8023E+03	1.9358E+03	2.2299E+03	2.2810E+03
		Median	2.0980E+04	2.5646E+04	7.5748E+04	2.8806E+04	4.5047E+04	6.8035E+04	3.3852E+04	2.0980E+04	1.9809E+04	3.7229E+04
F26	50	Mean	3.4094E+03	3.8778E+03	8.0075E+03	4.6061E+03	6.3506E+03	3.2000E+03	4.2860E+03	3.6462E+03	3.2000E+03	4.5789E+03
		Std	8.0394E+01	1.6882E+02	1.1129E+03	1.8671E+02	9.0943E+02	1.0179E+02	4.6912E+02	1.2301E+02	4.5234E−04	1.7429E+02
		Median	3.6575E+03	3.8635E+03	8.1302E+03	4.6837E+03	6.3388E+03	3.6400E+03	4.1515E+03	3.6575E+03	3.2000E+03	4.6083E+03
	100	Mean	3.5006E+03	4.5605E+03	1.5352E+04	6.5106E+03	1.2079E+04	3.2000E+03	5.2365E+03	4.1596E+03	3.2000E+03	7.7914E+03
		Std	6.5125E+01	3.0469E+02	1.4324E+03	5.6751E+02	1.7550E+03	8.4589E−05	7.1517E+02	2.1109E+02	6.7268E−04	4.4099E+02
		Median	4.1055E+03	4.4969E+03	1.5578E+04	6.4281E+03	1.2251E+04	3.2000E+03	5.0389E+03	4.1055E+03	3.2000E+03	7.7985E+03
F27	50	Mean	3.3032E+03	3.8314E+03	1.8491E+04	5.6108E+03	9.8906E+03	3.3000E+03	4.2640E+03	8.5074E+03	3.3972E+03	7.3097E+03
		Std	2.6293E+01	2.5852E+02	2.1954E+03	5.3861E+02	8.8391E+02	5.2113E+02	2.7414E+02	1.1168E+03	3.0357E+02	6.6932E+02
		Median	8.8647E+03	3.7505E+03	1.8472E+04	5.7168E+03	9.8491E+03	4.4045E+03	4.2451E+03	8.8647E+03	3.3000E+03	7.2954E+03
	100	Mean	3.4456E+03	7.7512E+03	5.5906E+04	1.3335E+04	2.3305E+04	3.3000E+03	7.1873E+03	2.0466E+04	7.5013E+03	2.2540E+04
		Std	4.4425E+01	1.4427E+03	5.3016E+03	1.8049E+03	1.5677E+03	7.4312E−05	6.7414E+02	3.7780E+03	2.5215E+03	1.7854E+03
		Median	2.0413E+04	7.9106E+03	5.6877E+04	1.3312E+04	2.3460E+04	3.3000E+03	7.0248E+03	2.0413E+04	7.6957E+03	2.2195E+04
F28	50	Mean	4.7548E+03	6.2617E+03	7.2083E+05	7.0347E+03	2.8188E+04	4.1215E+05	8.3133E+03	5.7523E+03	6.1007E+03	8.0068E+03
		Std	5.3035E+02	7.3720E+02	1.0621E+06	8.1032E+02	2.4483E+04	3.5471E+02	9.8718E+02	6.1505E+02	2.6972E+02	8.4545E+02
		Median	5.6224E+03	6.2149E+03	2.6408E+05	7.1574E+03	1.9691E+04	4.8124E+03	8.2636E+03	5.6224E+03	4.0220E+03	7.9432E+03
	100	Mean	7.9012E+03	1.2600E+04	3.1638E+06	1.5366E+04	1.6406E+05	3.3225E+06	1.6504E+04	4.6390E+04	7.0358E+03	2.3573E+04
		Std	5.1372E+02	1.6671E+03	2.3415E+06	2.0827E+03	1.0297E+05	4.1866E+06	2.6686E+03	1.0414E+05	6.9421E+02	6.1400E+03
		Median	1.2531E+04	1.2690E+04	2.6916E+06	1.5229E+04	1.3196E+05	1.5849E+06	1.5969E+04	1.2531E+04	6.9530E+03	2.1914E+04
F29	50	Mean	1.9487E+08	5.3017E+08	1.0441E+11	8.9709E+08	1.2465E+10	8.8602E+10	5.8251E+08	1.8660E+09	9.3325E+06	6.1046E+09
		Std	8.7167E+07	2.3030E+08	4.3974E+10	5.7533E+08	7.9202E+09	2.3990E+08	3.0970E+08	3.8939E+09	2.3090E+07	1.9829E+09
		Median	1.1812E+08	5.1165E+08	9.8974E+10	7.1469E+08	9.5543E+09	3.3991E+08	5.2992E+08	1.1812E+08	3.2918E+06	6.2343E+09
	100	Mean	4.5824E+08	1.7985E+09	5.2550E+11	2.0740E+10	2.1234E+11	4.4762E+11	2.8979E+09	3.7843E+10	3.1983E+09	7.8699E+10
		Std	2.0376E+08	1.0242E+09	9.5427E+10	7.1268E+09	6.4807E+10	1.0144E+11	1.2956E+09	2.5353E+10	7.4087E+09	1.6810E+10
		Median	3.3752E+10	1.5934E+09	5.2241E+11	1.9132E+10	2.0877E+11	4.3014E+11	2.5355E+09	3.3752E+10	1.7117E+08	7.5722E+10
Rank	50	W-T-L	07/00/03	02/00/08	00/00/10	00/00/10	00/00/10	02/00/08	00/00/10	00/00/10	01/00/09	00/00/10
	100	W-T-L	08/00/02	00/00/10	00/00/10	00/00/10	00/00/10	02/00/08	00/00/10	00/00/10	00/00/10	00/00/10

provide the numerical results, with bold letters denoting the best results. Furthermore, the tables’ last rows show the number of wins (W), ties (T), and losses (L) for each method. CL-SSA’s performance was evaluated using several functions from the CEC2017 benchmark with dimensions 50 and 100 and 200 and 500, and 1000 dimensional CEC2008lsgo benchmark functions. Moreover, it was compared to other algorithms in terms of exploitation and exploration and capacity to escape from local optima. In addition, the overall efficacy of CL-SSA’s performance is compared to that of other competing techniques.

5.3.1. Exploitation and Exploration Competence Analysis

Due to their unimodal nature, F1 and F2 can be used to measure exploitation capabilities. Based on these benchmark functions, the CL-SSA method is empirically tested on 50 and 100 dimensions. Table 6 presents the results obtained by the CL-SSA and the other competitor algorithms to demonstrate the suggested CL-SSA’s exploitation capabilities. It is important to test the optimization algorithms’ exploration ability with F3–F9. Because the local optima increase exponentially with dimension, these are excellent for testing exploration algorithms. Table 7 demonstrates the proposed CL-SSA algorithm’s capacity to solve these benchmark functions on dimensions 50 and 100. The results in Table 6 and Table 7 clearly illustrate that implementing a competitive learning technique based on the Salp Swarm strategy enhances the ability of exploitation and exploration.

5.3.2. Competence Analysis of Escape Ability from Local Optima

The Composite and Hybrid functions are essential for evaluating the optimal balance of exploitation and exploration for the algorithm to avoid local optima trapping. According to Table 8, the CL-SSA algorithm outperforms competing algorithms in the hybrid functions F10–F19. The benchmark functions F20–F29 are composite problems, and Table 9 shows that the CL-SSA algorithm outperforms the comparison algorithms in solving these issues.

5.3.3. CL-SSA Overall Effectiveness (OE)

This part compares the CL-SSA’s overall effectiveness (OE) [70] to other competitors, based on their performance in Table 6, Table 7, Table 8 and Table 9. Equation (16) shows the OE of the comparison algorithms, where N and L are the total numbers of test functions and losses for each algorithm. On all test functions with dimensions 50 and 100, the CL-SSA with OE = 96.55 percent shows that it is the most effective algorithm, according to

Table 10. Overall effectiveness OE of the CL-SSA with traditional algorithms.

Dimensions	Criteria	CL-SSA	SSA	CSO	PSO	HHO	BAT	WOA	MFO	EO	SCA
50	W-T-L	20/00/09	02/00/27	00/00/29	00/00/29	00/00/29	03/00/26	00/00/29	00/00/29	04/00/25	00/00/29
	OE	68.96%	6.89%	0.0%	0.0%	0.0%	10.34%	0.0%	0.0%	13.79%	0.0%
100	W-T-L	26/0/3	00/00/29	00/00/29	00/00/29	00/00/29	02/00/27	00/00/29	00/00/29	01/00/28	00/00/29
	OE	89.65%	0.0%	0.0%	0.0%	0.0%	6.89%	0.0%	0.0%	3.44%	0.0%

.

$$\mathrm{OE}=\left(\frac{\mathrm{N}-\mathrm{L}}{\mathrm{N}}\right)\ast 100$$

(16)

5.4. Convergence Analysis

Figure 3 depicts the convergence curves of the CL-SSA and conventional algorithms on CEC2017 with dimensions 100 over 2500 iterations. The results demonstrate that the CL-SSA outperforms traditional algorithms because it vigorously explores the search space in the early iterations and converges slowly to the global optimum in the latter rounds. Moreover, Figure 3 indicates that the CL-SSA converges almost as quickly as the other algorithms. The convergence curves in hybrid and composite functions similarly demonstrate the ability of CL-SSA to balance exploitation and exploration while avoiding being stuck in local optima.

5.5. Statistical Results Test Analysis

The nonparametric statistical analysis is used in this section to compare the proposed CL-SSA and the other traditional algorithms. The results are scientifically valid when statistical tests are used [71]. CL-SSA performance is statistically compared to other techniques using the Wilcoxon rank-sum and Friedman tests.

5.5.1. Wilcoxon Rank-Sum Test

The Wilcoxon rank-sum [72] is a nonparametric statistical test used to examine the CL-SSA’s results against competitors in terms of significance. The samples (results of each algorithm) in the rank-sum experiment are assigned ranks, and the total of the ranks is then calculated. The null hypothesis (H0) indicates no significant difference between the compared algorithms employing the benchmark functions at a 0.05 level of significance.

Table 11. Wilcoxon rank-sum of the CL-SSA vs. other traditional algorithms on CEC2017.

Fun	Dim	SSA	CSO	PSO	HHO	BAT	WOA	MFO	EO	SCA
F1	50	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05
	100	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05
F2	50	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05
	100	<0.05	<0.05	0.2938503	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05
F3	50	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05
	100	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05
F4	50	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05
	100	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05
F5	50	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	0.3966910	<0.05
	100	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05
F6	50	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05
	100	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05
F7	50	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05
	100	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05
F8	50	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	0.0945708	<0.05
	100	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05
F9	50	0.0609399	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05
	100	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05
F10	50	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05
	100	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05
F11	50	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05
	100	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05
F12	50	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05
	100	0.1639705	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05
F13	50	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05
	100	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05
F14	50	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05
	100	0.9318383	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05
F15	50	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	0.7735772	<0.05
	100	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05
F16	50	<0.05	<0.05	0.0700285	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05
	100	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05
F17	50	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05
	100	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05
F18	50	0.5086571	<0.05	<0.05	<0.05	<0.05	0.7147714	0.0775516	<0.05	<0.05
	100	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05
F19	50	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05
	100	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	0.5808997	<0.05
F20	50	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05
	100	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05
F21	50	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05
	100	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05
F22	50	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	0.4987350	<0.05
	100	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05
F23	50	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05
	100	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05
F24	50	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05
	100	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05
F25	50	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05
	100	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05
F26	50	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05
	100	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05
F27	50	<0.05	<0.05	<0.05	<0.05	0.1144607	<0.05	<0.05	0.4142468	<0.05
	100	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05
F28	50	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	0.3083881	<0.05	<0.05
	100	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05
F29	50	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	0.3083882	<0.05	<0.05
	100	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	0.4231943	<0.05

show the p-values on dimensions 50 and 100, with p-values greater than 0.05 denoted by underline style. According to the results, the null hypothesis is rejected for most functions, and CL-SSA delivers statistically significant results compared to other methods.

5.5.2. Non-Parametric Friedman Test

The non-parametric Friedman test [73] establishes whether significant differences exist between the proposed CL-SSA’s results and competitors’. First, each method is ranked independently, with the best first and second results assigned numbers 1 and 2, and the poorest results are given number k. The average rank is then determined to obtain each algorithm’s final rank. Next, Equation (17) calculates the Friedman test, where k signifies the number of swarm intelligence algorithms; Rj and n denote the average rank of algorithm j and the number of swarm intelligence algorithms, respectively. Finally,

Table 12. The results of the Freidman test on CEC2017’s test functions with dimensions 50 and 100.

Algorithm	Dim	Average Rank	Overall Rank
CL-SSA	50	1.7099	1
	100	1.4007‘	1
SSA	50	3.1986	3
	100	3.2699	3
CSO	50	9.6932	10
	100	9.7396	10
PSO	50	5.3508	5
	100	5.6861	6
HHO	50	7.2176	8
	100	6.8228	7
BAT	50	8.0000	9
	100	8.0476	9
WOA	50	5.4114	6
	100	4.9631	4
MFO	50	51237	4
	100	5.2675	5
EO	50	2.4007	2
	100	2.7420	2
SCA	50	6.8942	7
	100	7.0606	8

presents the Friedman test results on dimensions 50 and 100, showing that the CL-SSA ranks first among other algorithms and outperforms the competition.

$$F_f= \frac{12n}{k\left(k+1\right)} \left[{{\displaystyle \sum }}_j{R}_j^2-\frac{k{\left(k+1\right)}^2}{4}\right]$$

(17)

5.6. Run Time Analysis

This section outlines the time required to solve the benchmark functions for CEC2017 and CEC2008lsgo. A comparison was made between CL-SSA results and those of the other participants. Table 2 records the results of each benchmark run by each participant 30 times. Considering the losers and winners updating strategies, CL-SSA computation takes longer, as the table demonstrates. On the other hand, CL-SSA can outperform several algorithms, such as PSO, CSO, GWO, and WOA, while taking less time. In general, even though it is time-consuming, CL-SSA has significant efficiency advantages over other algorithms.

5.7. An Analysis of CL-SSA’s Effectiveness

This paper outlines a competition technique that divides solutions into winners and losers. Losers are then updated using a competition-based learning strategy, allowing them to explore their search area more extensively. The winning particles are updated using the Salp Swarm (SA) method. As a result, this section will focus on how the two mechanisms interact with the CSO. We selected five functions from CEC 2017 (F3, F5, F9, F15 and F22) as examples because they are representative and have larger effects. These functions are unimodal as well as multimodal.

Figure 4 shows the qualitative findings of CL-SSA in handling unimodal and multimodal functions to analyze the location and fitness changes during the optimization process intuitively. CL-SSA’s trajectory in the first dimension, CL-SSA’s anticipated exploitation and exploration stages and CL-SSA’s average global best fitness xis all shown in the chart. The trajectory shows how CL-SSA’s position shifts in the first-dimensional space. The average global best fitness shows how the CL-SSA’s moderate fitness changes as the iteration progress. Graphics illustrating the planned CL-SSA’s exploitation and exploration phases are demonstrated during the iteration process.

Core exploratory behavior is demonstrated by using the initial trajectory of the CL-SSA in the first dimension to represent additional components of this system. To ensure fast convergence and an accurate search for optimal solution, CL-SSA particle oscillations in both prophase and anaphase must be fast [74]. Early on, CL-SSA’s location curve is extremely high, with an amplitude of up to 50% of the exploration space, as shown in the figure. Particle position amplitude decreases as iteration time goes on if the function is smooth, but it increases considerably as iteration time goes on when the function’s amplitude swings dramatically. When it comes to diverse roles, CL-SSA is a fantastic choice. Changes can be large or small, In general. The CL-SSA has good search capabilities based on the high early fluctuations. In contrast, later changes are more gradual but still noticeable. Regarding finding the best possible solution, CL-SSA is the best in this field.

To gain a deeper grasp of the current trends in exploration and exploitation while also looking for the best possible solution. An illustration of the proposed CL-SSA’s exploration and exploitation stages can be seen in Figure 4c. There are two curves in each figure. The blue curve shows the algorithm’s exploration, while the orange curve represents its exploitation. At the outset, the proposed CL-SSA method exhibits a high ratio of exploration coupled with a low proportion of exploitation. However, as the algorithm proceeds through the majority of the selected functions, it promptly transitions to an exploitation strategy for most iterations. Consequently, the proposed CL-SSA strikes an appropriate balance between the phases of exploitation and exploration.

The average global fitness curves are shown in Figure 4, illustrating the iterative approach’s tendency to change CL-SSA’s fitness. In order to secure rapid convergence during the prophase and precise search outcomes in the anaphase of CL-SSA, the average fitness value commences a decremental trend while the oscillation frequency exhibits an inverse correlation with the number of iterations.

5.8. Compare CL-SSA to the Most Modern Optimization Techniques

The proposed CL-SSA method has a greater search efficiency than common algorithms such as SSA, PSO, CSO, BAT, WOA, MFO, HHO, EO, and SCA. The following parts will compare various contemporary and cutting-edge algorithms with the proposed methodology.

5.8.1. Evaluation of CL-SSA in Comparison to a Few SSA Variations

Using the 29 CEC 2017 benchmark functions, we will evaluate the proposed CL-SSA approach against eight different improved SSA variations, including ISSA [75], ESSA [76], HSSASCA [28], TVSSA [77], STS-SSA [78], IWSSA [76], SSA-FGWO, and ISSA_OBL. Improved versions of SSAs offer unique ways to improve the original SSA, which shows their major advantages. All algorithms have a maximum population size of 30 and Tmax = 2500 to ensure fairness. The mean, std, and median of all methods are compared after 30 runs to assess their performance. There is a summary statistics table in Appendix A. Table 5 lists the algorithm’s most important parameters. Results of the Wilcoxon signed-rank and Freidman tests are shown in

Table 13. Results of Wilcoxon rank-sum and Friedman tests on CEC2017 test functions for CL-SSA against other SSA methodologies.

Fun	ESSA		HSSASCA		ISSA		ISSA_OBL		IWSSA		STS-SSA		TVSSA		SSA-FGWO		CL-SSA
	We.	Fr.	We.	Fr.	We.	Fr.	We.	Fr.	We.	Fr.	We.	Fr.	We.	Fr.	We.	Fr.	We.	Fr.
F1	<0.05	5.034	<0.05	7.759	0.9195	2.379	<0.05	5.966	<0.05	7.241	<0.05	9.000	0.7736	2.414	0.3158	2.897	----	2.310
F2	<0.05	4.000	<0.05	9.000	<0.05	2.793	<0.05	1.517	<0.05	6.034	<0.05	6.448	<0.05	4.517	<0.05	2.793	----	7.897
F3	<0.05	5.138	<0.05	7.138	<0.05	2.931	<0.05	5.862	<0.05	7.862	<0.05	9.000	<0.05	3.276	<0.05	2.172	----	1.621
F4	<0.05	5.483	<0.05	7.034	<0.05	2.931	<0.05	4.862	<0.05	8.207	<0.05	8.759	<0.05	3.828	<0.05	2.897	----	1.000
F5	<0.05	5.345	<0.05	7.345	<0.05	2.724	<0.05	5.172	<0.05	8.103	<0.05	8.552	<0.05	3.276	<0.05	3.448	----	1.034
F6	<0.05	4.414	<0.05	7.621	<0.05	4.431	<0.05	5.862	<0.05	6.966	<0.05	8.741	<0.05	3.345	<0.05	2.621	----	1.000
F7	<0.05	3.345	<0.05	6.724	<0.05	3.966	<0.05	4.966	<0.05	7.586	<0.05	8.862	<0.05	4.103	<0.05	4.448	----	1.000
F8	<0.05	6.000	<0.05	8.207	<0.05	2.552	<0.05	4.724	<0.05	7.828	<0.05	7.966	<0.05	3.552	<0.05	3.172	----	1.000
F9	<0.05	5.517	<0.05	8.069	<0.05	2.621	<0.05	5.000	<0.05	7.862	<0.05	8.069	<0.05	2.655	<0.05	3.345	----	1.862
F10	<0.05	5.931	<0.05	7.724	<0.05	3.483	<0.05	5.069	<0.05	7.655	<0.05	8.621	<0.05	2.034	<0.05	3.310	----	1.172
F11	<0.05	5.345	<0.05	7.793	<0.05	2.931	<0.05	5.621	<0.05	7.207	<0.05	9.000	<0.05	2.379	<0.05	2.966	----	1.759
F12	<0.05	5.897	<0.05	7.862	0.4054	3.448	<0.05	2.207	<0.05	7.000	<0.05	8.966	0.2662	2.690	0.0886	3.828	----	3.103
F13	<0.05	5.966	<0.05	7.552	<0.05	3.034	<0.05	5.034	<0.05	7.724	<0.05	8.690	<0.05	2.586	<0.05	2.517	----	1.897
F14	<0.05	6.000	<0.05	7.897	0.2163	3.724	<0.05	1.034	<0.05	7.241	<0.05	8.862	0.0609	3.000	<0.05	3.966	----	3.276
F15	<0.05	4.000	<0.05	7.172	0.0588	2.828	<0.05	6.000	<0.05	7.828	<0.05	9.000	<0.05	3.310	<0.05	2.862	----	2.000
F16	<0.05	4.690	<0.05	7.828	<0.05	2.931	<0.05	4.759	<0.05	7.241	<0.05	8.931	<0.05	3.828	<0.05	2.897	----	1.897
F17	<0.05	5.897	<0.05	7.138	0.3158	3.034	<0.05	3.483	<0.05	7.966	<0.05	8.897	0.2050	2.759	<0.05	3.690	----	2.138
F18	<0.05	4.552	<0.05	7.931	0.0588	4.000	<0.05	1.103	<0.05	7.138	<0.05	8.931	0.8948	3.345	<0.05	4.483	----	3.517
F19	<0.05	4.276	<0.05	8.276	0.1548	3.138	<0.05	3.483	<0.05	7.931	<0.05	7.793	0.1548	3.345	<0.05	4.241	----	2.517
F20	<0.05	4.517	<0.05	7.586	<0.05	2.931	<0.05	5.828	<0.05	7.414	<0.05	9.000	<0.05	4.172	<0.05	2.552	----	1.000
F21	<0.05	4.379	<0.05	7.276	<0.05	4.517	<0.05	4.621	<0.05	7.379	<0.05	7.966	<0.05	4.793	<0.05	3.000	----	1.069
F22	<0.05	3.379	<0.05	7.759	<0.05	3.000	<0.05	6.034	<0.05	7.207	<0.05	9.000	<0.05	4.517	<0.05	3.103	----	1.000
F23	<0.05	4.655	<0.05	7.586	<0.05	3.034	<0.05	6.000	<0.05	7.414	<0.05	9.000	<0.05	3.586	<0.05	2.724	----	1.000
F24	<0.05	5.448	<0.05	7.690	<0.05	2.276	<0.05	5.552	<0.05	7.414	<0.05	8.897	<0.05	2.793	<0.05	3.517	----	1.414
F25	0.4142	3.000	<0.05	7.103	<0.05	3.517	<0.05	5.759	<0.05	7.897	<0.05	9.000	<0.05	3.828	<0.05	3.310	----	1.586
F26	<0.05	3.345	<0.05	7.414	<0.05	2.724	<0.05	6.000	<0.05	7.586	<0.05	9.000	<0.05	4.448	<0.05	3.276	----	1.207
F27	<0.05	5.276	<0.05	7.552	<0.05	2.414	<0.05	5.724	<0.05	7.448	<0.05	9.000	<0.05	3.310	<0.05	2.690	----	1.586
F28	<0.05	1.621	<0.05	7.690	<0.05	3.345	<0.05	5.897	<0.05	7.379	<0.05	8.931	<0.05	4.586	<0.05	3.448	----	2.103
F29	<0.05	3.466	<0.05	7.966	<0.05	3.500	<0.05	5.138	<0.05	7.103	<0.05	8.931	<0.05	3.483	<0.05	3.414	----	2.000
Fr. Average		4.687	----	7.644	----	3.143	----	4.768	----	7.478	----	8.683	----	3.440	----	3.227	----	1.930
Fr. Rank		5.000	----	8.000	----	2.000	----	6.000	----	7.000	----	9.000	----	4.000	----	3.000	----	1.000
We.: Wilcoxon rank-sum p-value					Fr.: Friedman test result

. Figure 5 depicts the algorithmic convergence graphs. For unimodal functions F1 and F2, CL-SSA is superior to other variants, as shown in Appendix A. The Salp Swarm movement strategy has been shown to significantly enhance the original CSO’s exploitation potential compared to other SSAs. Among all methods, CL-SSA provides the smallest solutions for multimodal functions such as F3 through F9. It is also demonstrated in Appendix A that the suggested CL-SSA algorithm can solve these benchmark problems. Salp’s strategy-based competitive learning approach helps improve exploration and exploitation capabilities, according to the data in Appendix A. Hybrid functions F10–F19 and composite functions F20–F29 are competitive areas for CL-SSA’s research. The Salp movement strategy has also increased the original CSO’s ability to escape local optima trapping, according to data from Appendix A. Results in Appendix A reveal that the overall CL-SSA’s effectiveness (OE) is superior to that of other improved SSAs by more than half of the functions. With an OE of 82.75%, the CL-SSA algorithm is the most efficient for all test functions tested. Figure 5 shows how the suggested CL-SSA and other approaches perform over 2500 iterations. The results show that the CL-SSA outperforms the other algorithms because it first explores the search space before settling gradually to the global optimum in the later iterations.

Furthermore, as seen in Figure 5, the CL-SSA converges almost as quickly as the other approaches. The convergence curves for composite and hybrid functions reveal that CL-SSA can maintain an even balance between exploitation and exploration while avoiding trapping local optima. p-values larger than 0.05 are denoted by underlining in Table 13. As a result, the H0 is rejected for the great majority of functions, and CL-SSA yields results that are significant statistically when compared to other approaches. Using the Friedman test, Table 13 demonstrates that CL-SSA outperforms its competitors and ranks first among the others.

5.8.2. Comparison of CL-SSA with Other Algorithms

More advanced algorithms are compared to CL-SSA’s performance, including HIWOA, LJA, WFOA, RW-GWO, and LNMRA. The efficiency of CL-SSA is also compared to different PSO algorithm variants, including CPSO, CLPSO, PPSO, PPSO_W, and HPSO_TVAC. These experiments employ the CEC2017 benchmark functions, which may be found in this sub-section. The population size (N) is 30, and the Tmax = 2500. The mean, std, and median are compared after 30 runs to determine the overall performance of all techniques. Table 5 lists the algorithm’s most important parameters.

Table 14. CL-SSA’s performance in comparison to other efficient algorithms and PSO algorithm variants.

Fun	Critria	CL-SSA	LJA	PPSO	RW-GWO	WFOA	PPSO-W	NIMRA	HI-WOA	HPSO-TVAC	CLPSO	CPSO	SMA
F1	Mean	1.2695E+04	3.3161E+12	2.5678E+09	1.2826E+10	2.4077E+12	1.2364E+09	2.3821E+12	1.6391E+12	1.5490E+09	1.8605E+09	3.9739E+12	5.5385E+05
	Std	1.4874E+04	6.4745E+11	4.3714E+09	2.5666E+09	1.3965E+11	3.2377E+09	1.4659E+11	1.2174E+11	4.0015E+08	3.4082E+08	6.8253E+11	5.5695E+05
	Median	4.1577E+12	3.3186E+12	9.5675E+08	1.2966E+10	2.3875E+12	2.3634E+08	2.4127E+12	1.6468E+12	1.5946E+09	1.7904E+09	1.7904E+09	4.1577E+12
F2	Mean	4.5659E+05	1.3872E+06	1.6378E+05	8.5598E+05	2.6618E+12	1.8443E+05	2.9018E+05	3.5714E+05	4.1920E+05	5.1265E+05	7.7746E+05	2.9213E+05
	Std	6.1241E+04	4.8879E+05	2.7573E+04	3.0649E+05	9.5760E+12	9.2338E+04	2.2192E+04	5.0429E+03	5.8206E+04	4.4093E+04	1.0566E+05	4.2281E+04
	Median	7.5826E+05	1.2829E+06	1.5882E+05	8.0316E+05	1.0197E+11	1.5447E+05	2.9088E+05	3.5866E+05	4.2257E+05	5.1959E+05	5.1959E+05	7.5826E+05
F3	Mean	6.9476E+02	1.2025E+05	9.8675E+02	1.3544E+03	9.2216E+04	9.5582E+02	6.8519E+04	4.4207E+04	1.0066E+03	9.5009E+02	1.3225E+05	7.3570E+02
	Std	3.6103E+01	3.8870E+04	8.8774E+01	1.4357E+02	1.3375E+04	9.0829E+01	1.2322E+04	7.3501E+03	9.4630E+01	5.2737E+01	3.5729E+04	4.8774E+01
	Median	1.3868E+05	1.0967E+05	9.7118E+02	1.3207E+03	8.9029E+04	9.3974E+02	7.1146E+04	4.4123E+04	9.8511E+02	9.4485E+02	9.4485E+02	1.3868E+05
F4	Mean	9.4326E+02	2.2911E+03	1.3929E+03	1.5225E+03	2.1420E+03	1.4106E+03	1.9024E+03	1.8898E+03	1.1953E+03	1.5072E+03	2.3897E+03	1.2354E+03
	Std	6.5659E+01	1.3541E+02	6.8244E+01	8.8565E+01	5.4324E+01	7.8142E+01	4.7337E+01	9.1984E+01	1.8384E+02	5.1861E+01	1.3403E+02	8.7990E+01
	Median	2.3930E+03	2.2641E+03	1.3931E+03	1.5227E+03	2.1456E+03	1.3979E+03	1.9174E+03	1.8585E+03	1.1521E+03	1.5149E+03	1.5149E+03	2.3930E+03
F5	Mean	6.3656E+02	7.3670E+02	6.8396E+02	6.9407E+02	7.2645E+02	6.8254E+02	7.1032E+02	7.1189E+02	6.6672E+02	6.7411E+02	7.4808E+02	6.7774E+02
	Std	1.0267E+01	7.2137E+00	5.3424E+00	5.6574E+00	4.6667E+00	4.9362E+00	3.6556E+00	6.2374E+00	9.5454E+00	5.8344E+00	1.7824E+01	7.2528E+00
	Median	7.5036E+02	7.3693E+02	6.8582E+02	6.9501E+02	7.2664E+02	6.8180E+02	7.1058E+02	7.1212E+02	6.6579E+02	6.7340E+02	6.7340E+02	7.5036E+02
F6	Mean	1.3542E+03	5.3997E+03	3.2418E+03	3.0381E+03	4.1138E+03	3.2191E+03	3.7211E+03	3.8364E+03	2.0360E+03	2.0921E+03	8.6242E+03	2.1264E+03
	Std	8.6771E+01	1.1921E+03	1.4694E+02	2.0286E+02	7.4425E+01	1.6872E+02	1.0219E+02	7.4437E+01	7.7863E+01	6.6241E+01	9.1548E+02	2.0826E+02
	Median	8.6953E+03	5.1908E+03	3.2559E+03	3.0075E+03	4.1001E+03	3.2477E+03	3.7371E+03	3.8323E+03	2.0398E+03	2.0861E+03	2.0861E+03	8.6953E+03
F7	Mean	1.2313E+03	2.6432E+03	2.4357E+03	1.9029E+03	2.6946E+03	2.1037E+03	1.9930E+03	2.2224E+03	1.5123E+03	1.8161E+03	2.8217E+03	1.5439E+03
	Std	6.8454E+01	1.6530E+02	1.1724E+02	9.0182E+01	6.4812E+01	1.0482E+02	8.9275E+01	1.2155E+02	2.2888E+02	4.7588E+01	1.7689E+02	1.1790E+02
	Median	2.7956E+03	2.6452E+03	2.4222E+03	1.9011E+03	2.6842E+03	2.0927E+03	1.9671E+03	2.1773E+03	1.4287E+03	1.8183E+03	1.8183E+03	2.7956E+03
F8	Mean	1.0659E+04	1.0593E+05	2.8699E+04	5.0438E+04	7.6710E+04	2.7683E+04	6.1396E+04	6.2216E+04	2.9299E+04	4.3004E+04	1.2422E+05	3.1471E+04
	Std	4.1685E+03	1.8627E+04	3.3634E+03	5.1198E+03	3.5889E+03	3.2075E+03	2.4540E+03	5.6377E+03	1.0337E+04	5.7758E+03	2.1420E+04	4.6320E+03
	Median	1.2584E+05	1.0117E+05	2.8123E+04	4.9407E+04	7.6655E+04	2.7166E+04	6.1222E+04	6.2104E+04	2.6996E+04	4.3745E+04	4.3745E+04	1.2584E+05
F9	Mean	1.4408E+04	3.2342E+04	1.7797E+04	2.0045E+04	3.2200E+04	1.7785E+04	2.8554E+04	2.9687E+04	3.1391E+04	2.8894E+04	3.3735E+04	1.6639E+04
	Std	1.4399E+03	1.1262E+03	1.5941E+03	1.5046E+03	1.1264E+03	1.9474E+03	1.0252E+03	2.1381E+03	5.4552E+02	7.1170E+02	1.3839E+03	1.6400E+03
	Median	3.3738E+04	3.2236E+04	1.7818E+04	2.0288E+04	3.2297E+04	1.7982E+04	2.8353E+04	3.0077E+04	3.1537E+04	2.8883E+04	2.8883E+04	3.3738E+04
F10	Mean	4.1092E+03	5.2568E+05	5.6727E+03	4.8198E+04	2.8200E+05	9.6015E+03	1.1360E+05	1.8643E+05	3.4775E+04	5.8145E+04	4.5652E+05	3.2163E+03
	Std	5.2293E+02	2.5139E+05	4.4777E+03	1.8519E+04	9.1700E+04	1.2701E+04	2.0037E+04	2.7638E+04	7.3469E+03	7.7484E+03	1.5329E+05	3.4316E+02
	Median	4.4818E+05	5.0172E+05	4.5379E+03	4.3298E+04	2.6801E+05	3.9713E+03	1.1554E+05	1.8751E+05	3.3599E+04	5.8132E+04	5.8132E+04	4.4818E+05
F11	Mean	1.6010E+09	1.5910E+12	2.6390E+09	3.3429E+09	1.5435E+12	3.7617E+09	1.1400E+12	5.6840E+11	3.7227E+08	1.0597E+09	1.8683E+12	7.6459E+08
	Std	8.0595E+08	3.1554E+11	8.3166E+09	1.7755E+09	1.2718E+11	5.9728E+09	2.2303E+11	1.0856E+11	1.8273E+08	4.2736E+08	4.1655E+11	3.4794E+08
	Median	1.7599E+12	1.5962E+12	1.0344E+09	2.9391E+09	1.5446E+12	1.4228E+09	1.1992E+12	5.5767E+11	3.3002E+08	9.1495E+08	9.1495E+08	1.7599E+12
F12	Mean	9.0617E+04	4.4490E+11	2.9440E+07	8.3809E+07	3.7552E+11	5.0170E+08	2.9025E+11	1.2718E+11	7.3204E+04	2.8697E+06	4.5758E+11	1.1241E+05
	Std	3.7540E+04	1.1855E+11	1.6096E+08	1.3570E+08	5.3500E+10	1.4970E+09	7.7652E+10	2.7067E+10	2.1612E+04	1.2244E+06	1.5120E+11	1.0696E+05
	Median	4.6102E+11	4.5023E+11	5.0663E+04	2.1247E+07	3.7902E+11	6.6103E+04	2.8910E+11	1.2780E+11	7.1737E+04	2.5469E+06	2.5469E+06	4.6102E+11
F13	Mean	1.1320E+06	2.0604E+08	5.4733E+05	6.0618E+06	6.3340E+08	4.4013E+05	3.2071E+06	8.7333E+06	2.6484E+06	7.2262E+06	2.3637E+08	1.7300E+06
	Std	6.8693E+05	1.6724E+08	2.8898E+05	2.5074E+06	4.7810E+08	3.2313E+05	2.4666E+06	2.9851E+06	1.0088E+06	2.3886E+06	2.0024E+08	8.8188E+05
	Median	1.6989E+08	1.6668E+08	5.0482E+05	5.1722E+06	5.7065E+08	3.4512E+05	2.1926E+06	8.1864E+06	2.3744E+06	7.1102E+06	7.1102E+06	1.6989E+08
F14	Mean	8.0538E+04	2.0545E+11	2.9343E+04	8.5542E+07	3.0108E+11	3.9061E+04	9.8076E+10	5.0478E+10	1.8820E+04	6.5290E+05	2.3724E+11	2.7634E+04
	Std	2.8115E+04	7.4086E+10	1.3233E+04	2.9751E+08	3.8751E+10	1.5676E+04	4.0580E+10	1.2943E+10	8.1053E+03	4.5268E+05	1.0949E+11	1.7932E+04
	Median	2.1485E+11	1.9534E+11	2.8594E+04	6.0041E+06	2.9161E+11	3.6095E+04	9.0077E+10	5.0809E+10	1.5255E+04	4.8606E+05	4.8606E+05	2.1485E+11
F15	Mean	5.7302E+03	2.0659E+04	6.9244E+03	7.2032E+03	2.8509E+04	6.8893E+03	1.4590E+04	1.7109E+04	5.7580E+03	8.2867E+03	5.5980E+03	6.2474E+03
	Std	7.4299E+02	3.4590E+03	8.7708E+02	7.3170E+02	2.8217E+03	8.0828E+02	2.4214E+03	1.5789E+03	7.7167E+02	3.9257E+02	7.7167E+02	6.7674E+02
	Median	5.6380E+03	2.0139E+04	7.0973E+03	7.1983E+03	2.8580E+04	6.8286E+03	1.3927E+04	1.6775E+04	5.6380E+03	8.3776E+03	8.3776E+03	5.6380E+03
F16	Mean	4.9906E+03	9.3921E+06	6.3158E+03	7.8518E+03	4.5673E+07	6.3905E+03	2.6510E+05	3.4327E+04	5.0204E+03	5.5037E+03	5.0204E+03	5.7975E+03
	Std	5.5442E+02	2.1651E+07	7.6765E+02	1.8691E+03	4.0600E+07	7.1055E+02	4.9346E+05	3.7633E+04	6.0053E+02	3.7417E+02	6.0053E+02	6.6691E+02
	Median	4.9109E+03	2.2140E+06	6.3699E+03	7.1101E+03	2.4410E+07	6.2890E+03	6.2230E+04	2.0932E+04	4.9109E+03	5.5162E+03	5.5162E+03	4.9109E+03
F17	Mean	2.2456E+06	3.6618E+08	8.0066E+05	7.6161E+06	2.8295E+08	7.7582E+05	4.0715E+06	1.3649E+07	4.6950E+06	8.2002E+06	3.9148E+08	2.8611E+06
	Std	1.0847E+06	2.2811E+08	3.5574E+05	3.2855E+06	2.4888E+08	1.1239E+06	4.8480E+06	4.2834E+06	2.1381E+06	2.2623E+06	2.4555E+08	1.2705E+06
	Median	2.8720E+08	3.1671E+08	7.9885E+05	6.9028E+06	2.0392E+08	4.3015E+05	2.3515E+06	1.3124E+07	4.4693E+06	8.3666E+06	8.3666E+06	2.8720E+08
F18	Mean	3.8830E+07	1.8192E+11	8.8499E+05	4.0206E+07	2.9095E+11	1.3151E+06	8.9733E+10	4.8374E+10	1.7937E+04	1.1665E+06	2.3775E+11	7.9805E+06
	Std	2.6479E+07	8.4576E+10	1.3659E+06	2.9360E+07	5.3348E+10	1.7856E+06	4.1759E+10	1.4272E+10	1.0487E+04	5.8364E+05	7.4554E+10	5.7612E+06
	Median	2.2708E+11	1.7161E+11	3.5192E+05	3.4004E+07	2.7654E+11	5.5973E+05	9.8805E+10	4.5478E+10	1.5712E+04	1.0379E+06	1.0379E+06	2.2708E+11
F19	Mean	4.9088E+03	8.0982E+03	5.5662E+03	5.4831E+03	7.5695E+03	5.7024E+03	6.1897E+03	6.4290E+03	6.3286E+03	5.9330E+03	6.3286E+03	5.0795E+03
	Std	6.2352E+02	7.0482E+02	6.4658E+02	4.7856E+02	5.5112E+02	6.0038E+02	3.4409E+02	6.0036E+02	1.0973E+03	3.6082E+02	1.0973E+03	5.6951E+02
	Median	6.8623E+03	7.9573E+03	5.5009E+03	5.3538E+03	7.5359E+03	5.6488E+03	6.2708E+03	6.3082E+03	6.8623E+03	5.9871E+03	5.9871E+03	6.8623E+03
F20	Mean	2.7576E+03	4.6020E+03	3.8117E+03	3.5538E+03	5.0403E+03	3.8469E+03	4.2847E+03	4.2363E+03	2.7873E+03	3.3145E+03	4.5660E+03	3.0861E+03
	Std	5.3408E+01	2.1551E+02	2.2213E+02	1.6141E+02	1.8223E+02	1.6627E+02	1.9725E+02	1.3853E+02	7.0029E+01	4.0909E+01	2.0775E+02	1.0595E+02
	Median	4.5257E+03	4.6159E+03	3.7987E+03	3.5026E+03	5.0154E+03	3.7802E+03	4.2906E+03	4.2544E+03	2.7890E+03	3.3101E+03	3.3101E+03	4.5257E+03
F21	Mean	1.6362E+04	3.4261E+04	2.2094E+04	2.2548E+04	3.4334E+04	2.0633E+04	3.0977E+04	3.1416E+04	3.1792E+04	3.1488E+04	3.5623E+04	1.8481E+04
	Std	1.5260E+03	1.3473E+03	2.2735E+03	1.3579E+03	1.1052E+03	1.7641E+03	7.1274E+02	2.3722E+03	5.3215E+03	1.8833E+03	1.3925E+03	1.4672E+03
	Median	3.5828E+04	3.4079E+04	2.1823E+04	2.2539E+04	3.4620E+04	2.0211E+04	3.0895E+04	3.0850E+04	3.3656E+04	3.1871E+04	3.1871E+04	3.5828E+04
F22	Mean	3.2692E+03	5.4558E+03	5.1487E+03	4.2098E+03	7.4475E+03	5.2134E+03	5.4337E+03	5.0818E+03	3.4499E+03	3.7196E+03	7.2673E+03	3.3815E+03
	Std	6.2223E+01	2.9135E+02	5.1305E+02	1.5787E+02	2.7257E+02	4.4713E+02	2.8316E+02	3.1350E+02	9.2967E+01	5.3846E+01	6.6305E+02	8.4208E+01
	Median	7.3306E+03	5.4046E+03	5.0648E+03	4.2322E+03	7.4298E+03	5.2037E+03	5.4305E+03	5.0175E+03	3.4516E+03	3.7279E+03	3.7279E+03	7.3306E+03
F23	Mean	3.7076E+03	7.8571E+03	8.2110E+03	5.0791E+03	1.4218E+04	8.4074E+03	7.2367E+03	6.6390E+03	4.0313E+03	4.3904E+03	1.2647E+04	3.9756E+03
	Std	8.7531E+01	7.8506E+02	1.4229E+03	2.7372E+02	1.2733E+03	1.4618E+03	4.7922E+02	5.5394E+02	1.8426E+02	7.0554E+01	1.4041E+03	1.2879E+02
	Median	1.2586E+04	7.8607E+03	8.2001E+03	5.0622E+03	1.3850E+04	8.6242E+03	7.0590E+03	6.6090E+03	3.9918E+03	4.3890E+03	4.3890E+03	1.2586E+04
F24	Mean	3.3596E+03	4.3871E+04	3.6411E+03	3.9272E+03	2.7522E+04	3.5438E+03	2.4630E+04	1.5497E+04	3.6671E+03	3.6655E+03	6.3107E+04	3.4583E+03
	Std	6.1848E+01	1.4191E+04	7.4168E+01	1.3074E+02	3.0290E+03	9.0738E+01	2.3463E+03	1.3541E+03	9.1891E+01	5.3373E+01	1.6089E+04	5.8326E+01
	Median	6.0039E+04	3.7089E+04	3.6508E+03	3.9197E+03	2.7012E+04	3.5358E+03	2.4728E+04	1.5483E+04	3.6673E+03	3.6615E+03	3.6615E+03	6.0039E+04
F25	Mean	1.0423E+04	5.0103E+04	2.8645E+04	2.1940E+04	6.3601E+04	2.6744E+04	4.7986E+04	3.9307E+04	1.6421E+04	1.9156E+04	7.2470E+04	1.3162E+04
	Std	9.5108E+02	6.7687E+03	5.0767E+03	2.9514E+03	2.9866E+03	7.1788E+03	3.4971E+03	2.1362E+03	4.8204E+03	6.6996E+02	1.1321E+04	1.1469E+03
	Median	7.2188E+04	5.0761E+04	2.8881E+04	2.2418E+04	6.4249E+04	2.7506E+04	4.8768E+04	3.8918E+04	1.6428E+04	1.9158E+04	1.9158E+04	7.2188E+04
F26	Mean	3.5006E+03	1.0041E+04	4.4989E+03	3.2000E+03	1.8237E+04	4.6832E+03	6.0862E+03	7.6436E+03	3.7728E+03	4.0337E+03	1.3378E+04	3.6120E+03
	Std	6.5125E+01	1.7820E+03	5.6375E+02	4.3558E−04	1.8847E+03	6.6048E+02	7.1880E+02	7.7483E+02	1.2166E+02	1.0704E+02	1.7483E+03	7.4657E+01
	Median	1.3107E+04	9.7827E+03	4.3724E+03	3.2000E+03	1.8194E+04	4.5442E+03	6.0404E+03	7.8043E+03	3.7492E+03	4.0269E+03	4.0269E+03	1.3107E+04
F27	Mean	3.4456E+03	4.0964E+04	3.7102E+03	3.3000E+03	1.6101E+04	3.6791E+03	3.0763E+04	1.6101E+04	3.7867E+03	3.9006E+03	5.2879E+04	4.7640E+03
	Std	4.4425E+01	9.1270E+03	7.1728E+01	4.8835E−04	1.1571E+03	2.5100E+02	2.3650E+03	1.1571E+03	8.2888E+01	5.9388E+01	8.7330E+03	3.8195E+03
	Median	5.0614E+04	3.9560E+04	3.6990E+03	3.3000E+03	1.6147E+04	3.6120E+03	3.0692E+04	1.6147E+04	3.7809E+03	3.8869E+03	3.8869E+03	5.0614E+04
F28	Mean	7.9012E+03	9.2768E+05	1.0102E+04	8.0853E+03	3.3799E+04	1.0650E+04	3.9732E+04	3.3799E+04	8.1899E+03	9.6972E+03	1.0650E+04	8.0056E+03
	Std	5.1372E+02	1.5456E+06	9.3869E+02	1.1433E+03	1.6389E+04	1.3430E+03	2.7419E+04	1.6389E+04	6.3525E+02	4.5190E+02	1.3430E+03	8.3991E+02
	Median	1.0551E+04	3.7866E+05	1.0040E+04	7.8472E+03	2.9998E+04	1.0551E+04	2.8429E+04	2.9998E+04	7.3009E+03	9.7152E+03	9.7152E+03	1.0551E+04
F29	Mean	4.5824E+08	2.8058E+11	2.4063E+07	8.8386E+07	1.7515E+11	7.5903E+07	1.7515E+11	9.9196E+10	2.2825E+06	9.0387E+07	3.1133E+11	1.6783E+08
	Std	2.0376E+08	1.0408E+11	1.7551E+07	3.4667E+07	7.8347E+10	2.7578E+08	7.8347E+10	2.4649E+10	1.3117E+06	3.9002E+07	1.1842E+11	9.0220E+07
	Median	2.8544E+11	2.5120E+11	1.7967E+07	8.2467E+07	1.6746E+11	1.7482E+07	1.6746E+11	1.0185E+11	1.8942E+06	8.7516E+07	8.7516E+07	2.8544E+11
Rank W-T-L		17/0/12	00/00/29	01/00/28	02/00/27	00/00/29	00/00/29	00/00/29	00/00/29	05/00/25	00/00/29	01/00/28	01/00/28
OE		72.41%	0.00%	3.44%	6.89%	0.00%	6.89%	0.00%	0.00%	17.24%	0.00%	3.44%	3.44%

demonstrates that the CL-SSA algorithm compares favorably to and outperforms various advanced methods. The effectiveness of the proposed algorithm is compared to that of more advanced algorithms using the p-value of the Wilcoxon rank-sum test. In

Table 15. The Wilcoxon rank-sum comparison of the CL-SSA to the other advanced methods used in CEC2017.

Fun	LJA	PPSO	RW-GWO	WFOA	PPSO-W	NIMRA	HI-WOA	HPSO-TVAC	CLPSO	CPSO	SMA
1	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05
2	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05
3	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05
4	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05
5	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05
6	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05
7	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05
8	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05
9	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05
10	<0.05	<0.05	<0.05	<0.05	0.870291194	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05
11	<0.05	<0.05	<0.05	<0.05	0.785508454	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05
12	<0.05	<0.05	<0.05	<0.05	0.498735035	<0.05	<0.05	0.114460738	<0.05	<0.05	0.773577158
13	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05
14	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05
15	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	0.657612423	<0.05	0.657612423	<0.05
16	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	0.845869569	<0.05	0.845869569	<0.05
17	<0.05	<0.05	<0.05	<0.05	<0.05	0.253030147	<0.05	<0.05	<0.05	<0.05	<0.05
18	<0.05	<0.05	0.907149516	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05
19	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	0.183638262
20	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	0.085720339	<0.05	<0.05	<0.05
21	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05
22	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05
23	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05
24	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05
25	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05
26	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05
27	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05
28	<0.05	<0.05	0.657612423	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	0.907149516
29	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05

, underlining indicates p-values more than 0.05. Since CL-SSA yields statistically significant findings compared to other approaches, we can conclude that the null hypothesis does not hold for most functions. According to Friedman test findings in

Table 16. The Friedman test comparison of the CL-SSA to the other advanced methods used in CEC2017.

Fun	CL-SSA	LJA	PPSO	RW-GWO	WFOA	PPSO-W	NIMRA	HI-WOA	HPSO-TVAC	CLPSO	CPSO	SMA
1	1.0000	11.2069	4.3448	6.9310	9.5517	3.3448	9.5862	8.0000	4.9310	5.4483	11.6552	2.0000
2	6.9310	10.7241	1.5862	9.5517	11.8276	1.8621	3.4483	5.1724	6.0690	7.7931	9.6207	3.4138
3	1.2414	11.1724	4.7586	6.9655	10.2414	4.2759	9.0690	8.0690	5.0345	3.9655	11.4483	1.7586
4	1.0690	11.1379	4.6207	6.2414	10.1034	4.5862	8.6552	8.3448	2.7931	6.0000	11.7241	2.7241
5	1.0690	11.1724	5.1379	6.7931	10.1379	4.9655	8.3448	8.7241	2.4828	3.3448	11.6207	4.2069
6	1.0000	10.8966	6.3448	5.4828	10.1379	6.1724	8.1724	8.8621	2.7241	3.2069	11.9310	3.0690
7	1.0000	10.4483	9.0345	4.9310	10.8621	7.0000	6.0000	7.8276	2.7586	4.0000	11.5172	2.6207
8	1.0345	11.2069	3.4138	6.8621	10.0000	3.1724	8.4828	8.4138	3.5517	5.8621	11.7931	4.2069
9	1.3103	10.4138	3.3793	4.4828	10.2759	3.2414	6.6552	7.8966	9.2414	7.0000	11.5172	2.5862
10	2.7241	11.3448	3.2414	5.9655	10.1724	2.9310	7.9655	9.0690	5.3103	6.6552	11.4138	1.2069
11	5.1034	10.8966	4.0345	6.4138	10.6552	4.8966	9.0690	8.0345	1.2759	3.5862	11.3448	2.6897
12	3.7241	11.0690	2.0690	6.8276	10.5862	3.5517	9.3103	8.0345	2.9655	5.7931	11.0000	3.0690
13	3.3103	10.7586	1.9655	7.2414	11.3793	1.6207	5.3448	8.3448	5.1724	7.8966	10.8621	4.1034
14	4.8966	10.4483	2.5517	7.0000	11.6897	3.2759	8.9655	8.1379	1.7931	6.0000	10.7586	2.4828
15	2.7586	10.7931	5.4138	5.8966	11.9655	5.2759	9.1724	10.0690	2.4138	7.7586	2.5862	3.8966
16	2.4138	11.0690	6.0000	7.2414	11.8966	6.1724	9.7586	9.2069	2.6552	3.9310	2.6207	5.0345
17	3.8966	11.1724	1.9310	6.8621	10.6207	1.5172	4.4828	8.6552	5.6207	7.4828	11.2069	4.5517
18	6.3103	10.2069	2.7586	6.3793	11.5517	2.8966	9.0000	8.2069	1.0345	3.5862	11.0345	5.0345
19	2.4828	11.6552	4.3448	4.3793	10.9655	5.1724	6.8621	7.8276	7.6379	6.0690	7.6379	2.9655
20	1.3793	10.2414	6.3793	5.3103	11.9310	6.5517	8.7931	8.3103	1.6207	4.0345	10.4138	3.0345
21	1.2414	10.0345	4.1724	4.5517	10.4483	3.3793	6.9310	7.7241	8.3103	7.5862	11.4828	2.1379
22	1.1379	8.7931	7.4138	5.0345	11.5862	7.7241	8.8276	7.2069	2.5517	4.0000	11.4138	2.3103
23	1.0690	8.3793	8.7241	5.0000	11.7241	8.8966	7.4483	6.5862	2.5862	3.9310	11.2414	2.4138
24	1.1379	11.2069	4.4828	6.8966	9.7931	3.3793	9.2414	8.0000	5.0000	5.0000	11.7586	2.1034
25	1.2069	9.7241	6.3448	5.0345	11.2414	6.0000	9.2759	7.9655	3.2759	3.8966	11.6897	2.3448
26	2.1724	9.9655	6.3103	1.0000	11.8966	6.5517	7.9655	9.0690	4.0000	5.2069	11.0000	2.8621
27	2.1379	10.9655	4.8276	1.0000	8.4655	4.2759	10.1724	8.4655	5.6897	6.7931	11.8621	3.3448
28	2.9310	12.0000	6.3793	2.7931	9.9483	6.8103	10.1034	9.9483	1.8966	5.5172	6.8103	2.8621
29	6.8621	11.1034	2.5172	4.3793	9.6724	2.8276	9.6724	8.3103	1.0345	4.6207	11.2414	5.7586
Mean.	2.5707	10.6968	4.6373	5.6361	10.7354	4.5630	8.1647	8.2235	3.8424	5.3781	10.4209	3.1308
Rank	1.0000	11.0000	5.0000	7.0000	12.0000	4.0000	8.0000	9.0000	3.0000	6.0000	10.0000	2.0000

, most functions are performed by the CL-SSA, demonstrating that the CL-SSA ranks first for most functions. Figure 6 shows that CLSSA outperforms its competitors in terms of solution quality. In this way, exploration and exploitation can work together more effectively in the long run. As a result, CL-SSA’s dominance in composition functions can be seen in Figure 6.

5.9. An Investigation into the Scalability of the Solution to the High-Dimensional Function Optimization Problem

It is common practice to have high-dimensional and large-scale functions whenever a problem requires optimization. Because of this, the search space becomes more complicated, making it more difficult to optimize. Consequently, problems of varying dimensions may be used to evaluate the impact of scalability on the efficiency of the proposed approach to optimization. This study uses CL-SSA to find solutions to CEC2008 LSGO problems with dimensions of 200, 500 and 1000, respectively. Similarly, CSO, SSA, and various other algorithms are utilized for comparative experimentation. The parameter settings for those algorithms are fully compatible with Section 5.2, and

Table 17. CL-SSA’s performance in comparison to other efficient algorithms on CEC2008lsgo.

Fun	Dim	Criteria	CL-SSA	SSA	CSO	PSO	WOA	GWO
F1	200	Mean	−4.50E+02	1.86E+05	9.41E+05	4.08E+04	1.21E+05	1.88E+05
		Std	1.22E−02	2.47E+04	4.48E+04	1.65E+04	1.54E+04	2.11E+04
		Median	−4.50E+02	1.84E+05	9.44E+05	3.73E+04	1.22E+05	1.84E+05
	500	Mean	2.52E+04	1.20E+06	6.63E+05	1.50E+06	8.07E+05	8.26E+05
		Std	4.39E+03	5.38E+04	4.87E+04	5.44E+04	3.41E+04	4.87E+04
		Median	2.49E+04	1.20E+06	6.59E+05	1.49E+06	8.08E+05	8.24E+05
	1000	Mean	7.60E+05	2.93E+06	2.41E+06	3.15E+06	2.25E+06	2.18E+06
		Std	5.24E+04	6.83E+04	1.11E+05	4.44E+04	4.83E+04	5.55E+04
		Median	7.56E+05	2.93E+06	2.40E+06	3.16E+06	2.26E+06	2.18E+06
F2	200	Mean	−3.56E+02	−3.49E+02	−2.95E+02	−3.96E+02	−3.76E+02	−3.68E+02
		Std	8.03E+00	3.60E+00	4.68E+00	1.95E+00	8.57E+00	1.51E+00
		Median	−3.54E+02	−3.49E+02	−2.95E+02	−3.96E+02	−3.74E+02	−3.68E+02
	500	Mean	−3.74E+02	−3.38E+02	−3.15E+02	−3.90E+02	−3.20E+02	−3.56E+02
		Std	1.06E+01	2.71E+00	4.75E+00	2.13E+00	5.83E+00	5.71E−01
		Median	−3.74E+02	−3.39E+02	−3.15E+02	−3.90E+02	−3.20E+02	−3.56E+02
	1000	Mean	−3.04E+02	−3.34E+02	−3.06E+02	−3.87E+02	−3.70E+02	−3.52E+02
		Std	2.57E+00	2.62E+00	3.23E+00	1.28E+00	9.16E+00	8.67E−01
		Median	−3.04E+02	−3.34E+02	−3.06E+02	−3.87E+02	−3.68E+02	−3.53E+02
F3	200	Mean	1.30E+04	4.58E+10	7.16E+11	1.18E+11	1.60E+10	3.90E+10
		Std	2.60E+04	8.97E+09	6.83E+10	1.84E+10	3.02E+09	8.72E+09
		Median	4.49E+03	4.72E+10	7.26E+11	1.21E+11	1.61E+10	3.80E+10
	500	Mean	3.28E+09	3.88E+11	2.75E+11	5.04E+11	1.86E+11	2.68E+11
		Std	9.36E+08	2.92E+10	4.38E+10	2.56E+10	1.47E+10	1.54E+10
		Median	3.22E+09	3.89E+11	2.75E+11	5.04E+11	1.86E+11	2.65E+11
	1000	Mean	2.35E+11	1.15E+12	1.24E+12	1.19E+12	6.80E+11	7.72E+11
		Std	3.18E+10	3.96E+10	9.17E+10	3.06E+10	2.16E+10	2.66E+10
		Median	2.30E+11	1.15E+12	1.24E+12	1.18E+12	6.76E+11	7.71E+11
F4	200	Mean	1.47E+03	2.19E+03	3.92E+03	2.20E+03	2.20E+03	1.53E+03
		Std	2.64E+02	1.04E+02	1.39E+02	2.37E+02	1.48E+02	1.25E+02
		Median	1.38E+03	2.18E+03	3.93E+03	2.14E+03	2.20E+03	1.54E+03
	500	Mean	6.08E+03	7.03E+03	4.89E+03	7.94E+03	6.87E+03	5.71E+03
		Std	3.86E+02	1.74E+02	2.47E+02	1.72E+02	3.17E+02	1.51E+02
		Median	6.05E+03	7.02E+03	4.86E+03	7.94E+03	6.88E+03	5.71E+03
	1000	Mean	1.49E+04	1.55E+04	1.35E+04	1.69E+04	1.48E+04	1.34E+04
		Std	3.11E+02	2.57E+02	3.17E+02	2.10E+02	3.31E+02	2.30E+02
		Median	1.48E+04	1.54E+04	1.35E+04	1.69E+04	1.48E+04	1.34E+04
F5	200	Mean	−1.80E+02	1.39E+03	7.58E+03	−1.77E+02	8.05E+02	1.24E+03
		Std	1.94E−02	2.21E+02	4.06E+02	9.48E−01	1.03E+02	1.67E+02
		Median	−1.80E+02	1.40E+03	7.54E+03	−1.78E+02	7.96E+02	1.23E+03
	500	Mean	5.79E+01	9.37E+03	5.31E+03	1.17E+04	6.30E+03	6.46E+03
		Std	4.38E+01	4.88E+02	4.38E+02	3.20E+02	2.71E+02	3.70E+02
		Median	4.81E+01	9.39E+03	5.31E+03	1.17E+04	6.27E+03	6.39E+03
	1000	Mean	6.75E+03	2.55E+04	2.13E+04	2.76E+04	1.98E+04	1.91E+04
		Std	5.51E+02	8.31E+02	7.81E+02	4.44E+02	5.08E+02	4.41E+02
		Median	6.57E+03	2.55E+04	2.13E+04	2.76E+04	1.99E+04	1.91E+04
F6	200	Mean	−1.26E+02	−1.19E+02	−1.19E+02	−1.19E+02	−1.21E+02	−1.21E+02
		Std	3.88E+00	4.27E−02	2.70E−02	9.04E−02	1.49E−02	4.00E−01
		Median	−1.27E+02	−1.19E+02	−1.19E+02	−1.19E+02	−1.21E+02	−1.21E+02
	500	Mean	−1.21E+02	−1.19E+02	−1.20E+02	−1.19E+02	−1.20E+02	−1.20E+02
		Std	3.34E−03	2.77E−02	1.43E−01	5.02E−02	2.31E−02	1.75E−01
		Median	−1.21E+02	−1.19E+02	−1.20E+02	−1.19E+02	−1.20E+02	−1.20E+02
	1000	Mean	−1.21E+02	−1.19E+02	−1.19E+02	−1.19E+02	−1.19E+02	−1.20E+02
		Std	1.12E−03	1.41E−02	4.12E−02	2.36E−02	1.54E−02	3.33E−01
		Median	−1.21E+02	−1.19E+02	−1.19E+02	−1.19E+02	−1.19E+02	−1.20E+02
F7	200	Mean	−4.62E+05	−3.63E+05	−2.31E+05	−1.64E+05	−4.17E+05	−5.39E+05
		Std	4.58E+04	3.97E+04	1.33E+04	2.09E+02	5.31E+04	4.47E+04
		Median	−4.64E+05	−3.63E+05	−2.28E+05	−1.64E+05	−4.13E+05	−5.42E+05
	500	Mean	−1.44E+06	−6.43E+05	−7.84E+05	−4.31E+05	−9.15E+05	−9.26E+05
		Std	7.73E+04	4.89E+04	7.94E+04	1.68E+04	9.65E+04	5.44E+04
		Median	−1.43E+06	−6.42E+05	−7.90E+05	−4.34E+05	−9.21E+05	−9.19E+05
	1000	Mean	−2.07E+06	−1.07E+06	−1.25E+06	−7.26E+05	−1.96E+06	−1.45E+06
		Std	8.00E+04	5.82E+04	8.32E+04	6.95E+03	2.47E+05	7.89E+04
		Median	−2.08E+06	−1.06E+06	−1.25E+06	−7.27E+05	−1.97E+06	−1.45E+06
Rank	200	W/T/L	5/0/2	0/0/7	0/0/7	1/0/6	0/0/7	5/0/2
	500	W/T/L	5/0/2	0/0/7	1/0/6	0/0/7	0/0/7	5/0/2
	1000	W/T/L	5/0/2	0/0/7	0/0/7	1/0/6	0/0/7	5/0/2
Overall OE			71.42%	0%	4.76	9.52%	0%	9.52%

displays the results of the calculations performed. According to Table 16, CL-SSA is the most efficient algorithm compared to other state-of-the-art algorithms for the majority of functions. According to Table 17, the competitive learning technique improved the efficacy of the initial algorithm by making its exploitation more effective in determining the appropriate combination of exploration to avoid becoming stuck in a local optimal solution. The results reported in Table 17 show that CL-SSA outperforms other algorithms in more than half of the functions when evaluating the overall efficacy (OE) of various methods. With an overall efficiency (OE) of 71.42 percent, the CL-SSA is the most efficient method for all test functions with dimensions of 200, 500 and 1000. The results of the Friedman test are shown in

Table 18. The Friedman test comparison of the CL-SSA to the other methods used in CEC2008lsgo.

Algorithm	Dim	Mean Rank	Overall Rank
CL-SSA	200	1.76	1
	500	1.39	1
	1000	2.08	1
SSA	200	3.87	4
	500	4.78	5
	1000	4.77	5
CSO	200	5.86	6
	500	3.02	3
	1000	4.02	4
PSO	200	4.56	5
	500	5.30	6
	1000	5.15	6
WOA	200	2.44	2
	500	3.59	4
	1000	2.56	3
GWO	200	2.52	3
	500	2.92	2
	1000	2.41	2

, demonstrating that the CL-SSA ranks top for most functions. The p-values of the Wilcoxon rank-sum test are displayed in

Table 19. The Wilcoxon rank-sum comparison of the CL-SSA to the other methods used in CEC2008lsgo.

Fun	Dim	SSA	CSO	PSO	WOA	GWO
F1	200	<0.05	<0.05	<0.05	<0.05	<0.05
	500	<0.05	<0.05	<0.05	<0.05	<0.05
	100	<0.05	<0.05	<0.05	<0.05	<0.05
F2	200	<0.05	<0.05	<0.05	<0.05	<0.05
	500	<0.05	<0.05	<0.05	<0.05	<0.05
	100	<0.05	<0.05	<0.05	<0.05	<0.05
F3	200	<0.05	<0.05	<0.05	0.29385	<0.05
	500	<0.05	<0.05	<0.05	<0.05	<0.05
	100	<0.05	<0.05	<0.05	<0.05	<0.05
F4	200	<0.05	<0.05	<0.05	<0.05	<0.05
	500	<0.05	<0.05	<0.05	<0.05	<0.05
	100	<0.05	<0.05	<0.05	<0.05	<0.05
F5	200	<0.05	<0.05	<0.05	<0.05	<0.05
	500	<0.05	<0.05	<0.05	<0.05	<0.05
	100	<0.05	<0.05	<0.05	<0.05	<0.05
F6	200	<0.05	<0.05	<0.05	<0.05	<0.05
	500	<0.05	<0.05	<0.05	<0.05	<0.05
	100	<0.05	<0.05	<0.05	<0.05	<0.05
F7	200	<0.05	<0.05	<0.05	<0.05	<0.05
	500	<0.05	<0.05	<0.05	<0.05	<0.05
	100	<0.05	<0.05	<0.05	<0.05	<0.05

, and p-values greater than 0.05 are indicated by underlining in the table.

Consequently, the findings indicate that the null hypothesis cannot be accepted for any of the functions. In comparison to other algorithms, the CL-SSA’s results are significant statistically. Furthermore, Figure 7 shows that the CL-SSA converges quite close to the speed of the other methods. As a result, one may conclude that the enhanced method can still maintain a high level of robustness and optimization accuracy even when used to solve problems on a massive scale. The experiments’ findings demonstrate that CL-SSA can avoid dimensional catastrophe and possesses high optimization efficiency when handling problems involving high-dimensional functions.

Comparison of CL-SSA with Some Advanced Algorithms on CEC2008lsgo

This section uses CMA-ES [79], LM-CMA [80], SHADE [81], DESAP_ABS [82], large-scale QIWOA, large-scale SSA-FGWO, large-scale DSCA, and other advanced algorithms for comparative experiments. Section 5.2 and Section 5.7 are fully consistent with the parameter selections of these methods. Table 19 displays the results of the computations. CL-SSA outperforms other advanced algorithms for most functions, as shown in

Table 20. CL-SSA’s performance in comparison to other efficient algorithms on CEC2008lsgo.

Fun.	Dim	Criteria.	CL-SSA	PPSO	PPSO_W	DESAP-abs	SHADE	CMA-ES	Large-Scale LM-CMA	Large-Scale DSCA	Large-Scale QIWOA	Large-Scale SSA-FGWO
F1	200	Mean	−4.4998E+02	2.6772E+03	5.4006E+03	−2.7512E+02	2.2708E+02	−4.3967E+02	7.0964E+05	6.5031E+05	1.1638E+06	−4.4993E+02
		Std	1.2219E−02	8.5605E+02	3.4502E+03	2.6943E+02	1.1299E+03	1.2216E+01	3.3163E+03	1.1424E+04	5.8877E+04	4.2897E−02
		Median	−4.4999E+02	2.4525E+03	4.7828E+03	−3.6937E+02	−3.2287E+02	−4.4454E+02	7.0976E+05	6.5013E+05	1.1749E+06	−4.4994E+02
	500	Mean	2.5190E+04	1.3313E+05	7.5489E+04	1.2108E+05	1.3365E+05	3.9292E+06	1.7502E+06	1.6861E+06	3.0315E+06	4.6255E+04
		Std	4.3899E+03	1.2619E+04	7.5530E+03	2.3364E+04	2.0715E+04	1.5634E+06	1.1832E+03	1.1061E+04	9.5840E+04	7.7301E+03
		Median	2.4897E+04	1.3097E+05	7.3522E+04	1.1849E+05	1.3324E+05	3.7134E+06	1.7501E+06	1.6886E+06	3.0411E+06	4.4608E+04
	1000	Mean	6.8014E+05	1.0015E+06	7.6006E+05	9.2483E+05	9.0875E+05	4.2327E+07	3.3889E+06	3.3340E+06	6.1819E+06	9.3810E+05
		Std	3.1489E+04	3.6436E+04	5.2365E+04	6.4763E+04	6.2037E+04	8.4355E+05	1.3849E+03	1.2298E+04	1.7538E+05	5.6351E+04
		Median	6.7782E+05	9.9824E+05	7.5591E+05	9.2253E+05	9.0474E+05	4.2267E+07	3.3887E+06	3.3354E+06	6.2061E+06	9.3789E+05
F2	200	Mean	−3.5596E+02	−3.6047E+02	−3.6246E+02	−3.5706E+02	−3.5777E+02	−3.3232E+02	−3.5266E+02	−3.5369E+02	−3.1996E+02	−3.8271E+02
		Std	8.0307E+00	2.1336E+00	2.3069E+00	3.8655E+00	2.9115E+00	4.9002E+01	3.1757E−01	2.3425E−01	2.6497E+01	3.3156E+01
		Median	−3.5446E+02	−3.6040E+02	−3.6224E+02	−3.5708E+02	−3.5812E+02	−3.3832E+02	−3.5263E+02	−3.5362E+02	−3.1171E+02	−4.0575E+02
	500	Mean	−3.2036E+02	−3.5376E+02	−3.5426E+02	−3.4315E+02	−3.4279E+02	1.2047E+02	−3.5076E+02	−3.5136E+02	−3.0426E+02	−4.0315E+02
		Std	5.8323E+00	8.0592E−01	6.3533E−01	2.5032E+00	2.6893E+00	2.0736E+01	1.2333E−01	2.8219E−01	2.4546E+01	3.8410E−01
		Median	−3.2011E+02	−3.5353E+02	−3.5421E+02	−3.4315E+02	−3.4340E+02	1.2060E+02	−3.5077E+02	−3.5124E+02	−3.0266E+02	−4.0312E+02
	1000	Mean	−3.0408E+02	−3.5180E+02	−3.5205E+02	−3.3592E+02	−3.3460E+02	1.6946E+02	−3.5020E+02	−3.5045E+02	−2.8790E+02	−4.0160E+02
		Std	2.5745E+00	3.8920E−01	4.1104E−01	2.2646E+00	2.4557E+00	1.8947E+01	1.0407E−01	5.4511E−02	2.6450E+01	2.4682E−01
		Median	−3.0435E+02	−3.5181E+02	−3.5206E+02	−3.3589E+02	−3.3431E+02	1.6919E+02	−3.5019E+02	−3.5046E+02	−2.7888E+02	−4.0170E+02
F3	200	Mean	1.3032E+04	9.3831E+07	1.7735E+08	4.1573E+06	5.3790E+06	1.2703E+06	2.2873E+11	2.0390E+11	1.1503E+12	1.6072E+04
		Std	2.6001E+04	1.4134E+08	3.3095E+08	5.7731E+06	1.0039E+07	2.8249E+06	2.7017E+09	7.8054E+09	1.0305E+11	2.1138E+04
		Median	4.4884E+03	5.9751E+07	5.3990E+07	2.3844E+06	1.8954E+06	2.8398E+05	2.2899E+11	2.0324E+11	1.1625E+12	6.4294E+03
	500	Mean	3.2778E+09	1.8499E+10	7.0075E+09	2.2040E+10	2.3608E+10	7.8263E+13	6.2994E+11	5.9848E+11	3.4239E+12	7.5370E+09
		Std	9.3632E+08	2.9282E+09	1.2707E+09	5.9385E+09	9.8310E+09	6.8808E+13	9.2016E+08	5.8070E+09	2.5231E+11	1.1762E+09
		Median	3.2180E+09	1.7845E+10	6.7025E+09	2.2496E+10	2.1734E+10	4.6474E+13	6.2997E+11	5.9954E+11	3.4407E+12	7.5690E+09
	1000	Mean	2.3484E+11	2.3570E+11	1.4491E+11	2.6598E+11	2.7297E+11	5.0307E+14	1.2800E+12	1.2498E+12	7.3545E+12	2.1300E+11
		Std	3.1829E+10	2.1792E+10	1.3619E+10	2.8582E+10	2.3764E+10	2.7203E+13	1.0472E+09	6.9337E+09	4.7647E+11	1.5409E+10
		Median	2.3026E+11	2.3700E+11	1.4636E+11	2.6148E+11	2.7135E+11	5.0759E+14	1.2799E+12	1.2507E+12	7.5223E+12	2.1131E+11
F4	200	Mean	1.4656E+03	1.4943E+03	1.4537E+03	9.4315E+01	9.1132E+01	1.7684E+03	2.8460E+03	3.2038E+03	4.4070E+03	1.4387E+03
		Std	2.6406E+02	6.8208E+01	6.1215E+01	2.9267E+01	3.4613E+01	4.7737E+01	6.9609E+01	3.9045E+01	1.3897E+02	6.5995E+01
		Median	1.3792E+03	1.4918E+03	1.4449E+03	9.2123E+01	8.9181E+01	1.7830E+03	2.8473E+03	3.1983E+03	4.4002E+03	1.4262E+03
	500	Mean	6.0838E+03	5.1695E+03	4.8718E+03	3.3826E+03	3.4492E+03	1.4643E+04	7.9342E+03	8.5034E+03	1.2030E+04	4.7854E+03
		Std	3.8583E+02	1.1067E+02	1.1263E+02	1.3219E+02	1.2362E+02	3.6013E+03	9.0520E+01	6.7729E+01	3.6042E+02	1.8835E+02
		Median	6.0464E+03	5.1678E+03	4.8472E+03	3.3906E+03	3.4816E+03	1.5264E+04	7.9427E+03	8.5198E+03	1.2052E+04	4.8332E+03
	1000	Mean	1.4855E+04	1.2436E+04	1.1864E+04	9.6151E+03	9.6849E+03	1.1570E+05	1.6776E+04	1.7509E+04	2.4672E+04	1.1479E+04
		Std	3.1104E+02	2.0438E+02	1.9346E+02	8.5960E+02	8.8780E+02	2.2747E+03	1.1014E+02	7.3639E+01	5.7674E+02	1.5736E+02
		Median	1.4835E+04	1.2466E+04	1.1865E+04	9.4773E+03	9.6464E+03	1.1613E+05	1.6776E+04	1.7508E+04	2.4747E+04	1.1470E+04
F5	200	Mean	−1.7989E+02	−1.5175E+02	−1.4168E+02	−1.7760E+02	−1.7685E+02	−1.7894E+02	5.4735E+03	4.9057E+03	9.4118E+03	−1.7986E+02
		Std	1.9410E−02	7.7510E+00	2.4995E+01	2.1159E+00	4.1867E+00	3.2656E−01	4.8962E+00	9.1108E+01	6.2377E+02	2.6025E−02
		Median	−1.7990E+02	−1.5262E+02	−1.5142E+02	−1.7839E+02	−1.7806E+02	−1.7903E+02	5.4749E+03	4.9243E+03	9.4015E+03	−1.7986E+02
	500	Mean	5.7945E+01	1.0218E+03	4.8369E+02	8.5119E+02	8.4989E+02	3.2068E+04	1.3805E+04	1.3236E+04	2.5389E+04	2.3384E+02
		Std	4.3771E+01	1.2334E+02	8.0174E+01	2.0654E+02	2.0018E+02	1.4824E+04	1.6961E+00	1.1390E+02	8.6929E+02	5.1319E+01
		Median	4.8053E+01	1.0256E+03	4.4503E+02	8.3238E+02	8.0955E+02	3.0153E+04	1.3805E+04	1.3251E+04	2.5659E+04	2.2156E+02
	1000	Mean	6.7488E+03	8.6033E+03	7.7488E+03	7.9741E+03	7.7369E+03	3.8061E+05	2.9913E+04	2.9339E+04	5.4695E+04	8.0817E+03
		Std	5.5120E+02	4.9872E+02	3.0262E+02	5.8388E+02	4.8391E+02	9.4707E+03	1.7698E+00	9.7727E+01	1.5176E+03	3.4661E+02
		Median	6.5735E+03	8.5387E+03	5.8777E+03	7.9070E+03	7.9142E+03	3.8025E+05	2.9913E+04	2.9337E+04	5.4944E+04	8.1490E+03
F6	200	Mean	−1.2603E+02	−1.2020E+02	−1.2017E+02	−1.2947E+02	−1.2912E+02	−1.1852E+02	−1.1930E+02	−1.2071E+02	−1.1968E+02	−1.2069E+02
		Std	3.8841E+00	3.0494E−01	2.9172E−01	1.5076E+00	1.6057E+00	1.8932E−02	4.3721E−02	1.5212E−02	4.8867E−01	2.6335E−07
		Median	−1.2714E+02	−1.2002E+02	−1.2002E+02	−1.2927E+02	−1.2922E+02	−1.1851E+02	−1.1930E+02	−1.2070E+02	−1.1998E+02	−1.2069E+02
	500	Mean	−1.1981E+02	−1.2014E+02	−1.2014E+02	−1.2580E+02	−1.2568E+02	−1.1843E+02	−1.1923E+02	−1.2060E+02	−1.1974E+02	−1.2070E+02
		Std	2.3079E−02	2.5109E−01	2.5021E−01	5.5969E−01	7.0056E−01	8.2699E−03	1.8152E−02	4.2895E−01	4.7190E−01	1.0274E−04
		Median	−1.1981E+02	−1.2002E+02	−1.2002E+02	−1.2586E+02	−1.2571E+02	−1.1843E+02	−1.1923E+02	−1.2071E+02	−1.2001E+02	−1.2070E+02
	1000	Mean	−1.2062E+02	−1.2012E+02	−1.2020E+02	−1.1983E+02	−1.1982E+02	−1.1838E+02	−1.1906E+02	−1.2051E+02	−1.1963E+02	−1.2062E+02
		Std	1.1205E−03	2.1407E−01	2.6054E−01	7.6812E−02	7.3583E−02	6.0834E−03	1.9392E−02	4.2733E−01	5.5379E−01	1.4068E−06
		Median	−1.2062E+02	−1.2002E+02	−1.2002E+02	−1.1983E+02	−1.1982E+02	−1.1838E+02	−1.1906E+02	−1.2063E+02	−1.1996E+02	−1.2062E+02
F7	200	Mean	−4.6169E+05	−1.6203E+05	−1.6396E+05	−1.5003E+05	−1.5078E+05	−7.3856E+04	−1.1327E+05	−2.1300E+05	−1.9382E+05	−3.7819E+05
		Std	4.5769E+04	8.3409E+04	9.3961E+04	1.7668E+04	1.1841E−10	3.8741E+03	2.7790E+02	2.0781E+04	1.7263E+04	7.4017E+04
		Median	−4.6435E+05	−1.4681E+05	−1.4681E+05	−1.4681E+05	−1.5078E+05	−7.3149E+04	−1.1332E+05	−2.0734E+05	−1.9066E+05	−3.9328E+05
	500	Mean	−7.8410E+05	−2.4764E+05	−2.7922E+05	−2.5692E+05	−4.4433E+05	−1.2414E+05	−2.2791E+05	−4.1784E+05	−4.1758E+05	−6.9082E+05
		Std	7.9399E+04	8.8804E−11	1.7299E+05	6.3422E+03	1.5607E+03	1.2644E+03	3.0043E+01	1.7342E+04	5.8683E+04	9.9641E+04
		Median	−7.9021E+05	−2.4764E+05	−2.4764E+05	−2.5577E+05	−4.4404E+05	−1.2437E+05	−2.2791E+05	−4.1903E+05	−4.1207E+05	−7.2319E+05
	1000	Mean	−2.0737E+06	−4.5666E+05	−4.3494E+05	−9.3312E+05	−3.8930E+05	−3.8930E+05	−3.7510E+05	−6.9783E+05	−7.3218E+05	−1.0994E+06
		Std	8.0016E+04	3.9182E+05	2.7285E+05	7.5706E+04	2.2836E+04	2.2836E+04	3.6810E+03	3.8523E+04	8.4488E+04	7.9636E+04
		Median	−2.0791E+06	−3.8513E+05	−3.8513E+05	−9.5793E+05	−3.8513E+05	−3.8513E+05	−3.7443E+05	−6.8944E+05	−7.1352E+05	−1.1124E+06
Rank	200	W-T-L	04/00/03	00/00/07	00/00/07	01/00/06	01/00/06	00/00/07	00/00/07	00/00/07	00/00/07	01/00/06
	500	W-T-L	04/00/03	00/00/07	00/00/07	02/00/05	00/00/07	00/00/07	00/00/07	00/00/07	00/00/07	01/00/06
	1000	W-T-L	04/00/03	00/00/07	01/00/06	01/00/06	00/00/07	00/00/07	00/00/07	00/00/07	00/00/07	01/00/06
Overall OE			57.14%	00.00%	04.76%	19.04%	04.76%	00.00%	00.00%		00.00%	14.28%

. This test results in the CL-SSA coming out on top for various tasks, as shown in

Table 21. The Friedman test comparison of the CL-SSA to the other advanced methods used in CEC2008lsgo.

Algorithm	Dim	Mean Rank	Overall Rank
CL-SSA	200	2.5123	1
	500	3.7537	2
	1000	3.0591	1
PPSO	200	5.7882	6
	500	5.1749	6
	1000	4.9828	5
PPSO_W	200	5.7438	5
	500	3.9236	3
	1000	3.1847	2
DESAP-abs	200	3.9704	4
	500	4.2709	5
	1000	4.4335	4
SHADE	200	3.8177	3
	500	3.9754	4
	1000	5.1010	6
CMA-ES	200	6.1724	7
	500	9.8424	10
	1000	9.6429	10
Large-scale LM-CMA	200	8.6700	9
	500	7.6798	8
	1000	7.8177	8
Large-scale DSCA	200	6.6700	8
	500	5.8374	7
	1000	5.5813	7
Large-scale QIWOA	200	8.7635	10
	500	8.1527	9
	1000	7.9261	9
Large-scale SSA-FGWO	200	3.000	2
	500	2.3892	1
	1000	3.2709	3

. The p-values of the Wilcoxon rank-sum test are also included in

Table 22. The Wilcoxon rank-sum comparison of the CL-SSA to the other advanced methods used in CEC2008lsgo.

Fun	Dim	PPSO	PPSO_W	DESAP-abs	SHADE	CMA-ES	Large-Scale LM-CMA	Large-Scale DSCA	Large-Scale QIWOA	Large-Scale SSA-FGWO
>F1	200	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05
	500	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05
	1000	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05
F2	200	<0.05	<0.05	0.272918441	0.088592954	0.060939894	<0.05	0.301060782	<0.05	<0.05
	500	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05
	1000	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05
F3	200	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	0.097678662
	500	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05
	1000	0.469596754	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05
F4	200	0.077551628	0.199498378	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	0.388084142
	500	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05
	1000	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05
F5	200	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05
	500	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05
	1000	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05
F6	200	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05
	500	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05
	1000	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	0.498735035
F7	200	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05
	500	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05
	1000	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05	<0.05

, with p-values more than 0.05 being underlined. CL-SSA yields significant statistical results compared to other approaches because the null hypothesis is rejected for all functions. Consequently, the improved method may maintain outstanding optimization precision and resilience when addressing large-scale problems. The experimental results showed that CL-SSA could solve high dimensional functions without dimensional catastrophe and has good optimization efficiency.

5.10. Parameter Sensitivity Analysis

In swarm intelligence algorithms, the number of control parameters impacts their performance. The impact of the control parameter for social component $\varphi$ on the CL-SSA’s performance is evaluated in this section. By analyzing the proposed CL-SSA method, we aim to identify the best setting for the social component parameter $\varphi$. The control parameter for the social component $\varphi$ ranges from 0 to 0.3, with a 0.1 increment. The social component parameter is evaluated utilizing the CEC 2017 benchmark functions, each of dimension 50.

Table 23. CL-SSA $\varphi$ parameter sensitivity on CEC2017 with 50 dimensions during 2500 iterations.

Fun	$\mathbf{CL}-\mathbf{SSA}\mathit{\varphi }$ = 0.0	Freidman Test Rank	$\mathbf{CL}-\mathbf{SSA}\mathit{\varphi }$ = 0.1	Freidman Test Rank	$\mathbf{CL}-\mathbf{SSA}\mathit{\varphi }$ = 0.2	Freidman Test Rank	$\mathbf{CL}-\mathbf{SSA}\mathit{\varphi }$ = 0.3	Freidman Test Rank
F1	1.16E+08	3.38	8.06E+07	2.83	8.34E+07	2.79	1.20E+04	1.00
F2	2.59E+05	2.79	2.78E+05	3.17	2.71E+05	3.03	5.22E+04	1.00
F3	6.80E+02	3.03	6.64E+02	2.86	6.74E+02	2.93	5.64E+02	1.17
F4	7.94E+02	2.48	8.12E+02	2.90	8.03E+02	2.83	7.39E+02	1.79
F5	6.70E+02	2.97	6.65E+02	2.76	6.70E+02	2.93	6.45E+02	1.34
F6	1.21E+03	2.72	1.22E+03	2.86	1.21E+03	2.79	1.07E+03	1.62
F7	1.11E+03	2.66	1.09E+03	2.55	1.13E+03	3.14	1.05E+03	1.66
F8	1.40E+04	2.34	1.77E+04	3.17	1.68E+04	2.97	1.06E+04	1.52
F9	8.30E+03	2.62	8.27E+03	2.59	8.34E+03	2.79	7.61E+03	2.00
F10	2.52E+03	3.24	2.45E+03	2.83	2.52E+03	2.93	1.47E+03	1.00
F11	1.93E+09	2.93	1.95E+09	3.28	1.23E+09	2.48	4.40E+08	1.31
F12	2.41E+05	2.97	1.74E+05	2.55	1.92E+05	2.41	1.71E+05	2.07
F13	4.68E+05	2.69	5.91E+05	3.00	4.96E+05	2.83	1.20E+05	1.48
F14	8.80E+04	2.38	9.40E+04	2.52	8.68E+04	2.41	1.16E+05	2.69
F15	3.71E+03	2.38	3.72E+03	2.76	4.00E+03	3.28	3.25E+03	1.59
F16	3.69E+03	3.14	3.52E+03	2.45	3.62E+03	2.72	3.21E+03	1.69
F17	4.32E+06	2.55	4.58E+06	2.93	5.87E+06	3.10	1.05E+06	1.41
F18	3.69E+07	2.72	2.15E+07	2.31	3.02E+07	2.83	1.18E+07	2.14
F19	3.48E+03	2.72	3.52E+03	2.90	3.53E+03	2.93	3.04E+03	1.45
F20	2.57E+03	2.90	2.56E+03	2.62	2.60E+03	3.03	2.51E+03	1.45
F21	9.80E+03	2.62	9.99E+03	2.76	9.92E+03	2.93	9.01E+03	1.69
F22	3.02E+03	2.52	3.03E+03	2.62	3.04E+03	2.86	3.00E+03	2.00
F23	3.15E+03	2.72	3.17E+03	2.62	3.15E+03	2.66	3.11E+03	2.00
F24	3.14E+03	2.97	3.15E+03	2.93	3.14E+03	3.07	3.04E+03	1.03
F25	6.86E+03	2.86	6.84E+03	2.45	6.99E+03	3.00	6.22E+03	1.69
F26	3.62E+03	2.69	3.65E+03	3.17	3.63E+03	2.97	3.41E+03	1.17
F27	3.57E+03	3.14	3.50E+03	2.93	3.54E+03	2.90	3.30E+03	1.03
F28	5.52E+03	3.03	5.41E+03	2.79	5.43E+03	2.76	4.75E+03	1.41
F29	3.75E+08	2.72	4.19E+08	3.03	3.63E+08	2.72	1.95E+08	1.52
Overall Freidman test rank		2.79	----	2.80	----	2.86	----	1.55

depicts the mean error values corresponding to the functions with different parameters. In addition, The Friedman test is used to assess the results statistically. The best CL-SSA performance is attained by value 0.3, as shown in bold letters in Table 23.

6. Engineering Design Experiments

It is common practice to employ optimization strategies to find solutions to engineering design challenges, including [83,84,85,86]. The benchmark set, consisting of seven well-known constrained design problems in diverse engineering fields, is described in this section. These challenges were listed in the CEC2020 conference benchmark set of real-world issues (CEC2020) [87] and were chosen as the starting point for this discussion. The next part of this section analyzes whether or not the CL-SSA is suitable for overcoming technical hurdles and difficulties.

6.1. Engineering Design Challenges Benchmark

Optimal engineering design naturally gives rise to problems of constraint optimization since specific design restrictions have to be considered whenever attempting to minimize or maximize the cost function. The CL-SSA model is applied to seven well-known constrained design challenges in several engineering fields in this context. These problems serve as benchmarks for real-world optimization objectives as part of the 2020 Competitions on Evolutionary Computation, more commonly referred to as CEC 2020 [87].

Table 24. The particulars about the seven real-world engineering design problems. The total number of decision variables in the problem is donated by D, the number of equality constraints and inequality constraints is denoted by h and g, respectively, and the value of the best-known feasible objective function is $f\left(x^{\ast }\right)$.

Fun	Name	$\mathit{D}$	$\mathit{h}$	$\mathit{g}$	$\mathit{f}\left({\mathit{x}}^{\ast }\right)$
EDP 1	Process design problem	5	0	3	2.6887000000E+04
EDP 2	Weight Minimization of a Speed Reducer	7	0	11	2.9944244658E+03
EDP 3	Multiple disk clutch brake design problem	5	0	7	2.3524245790E−01
EDP 4	Welded beam design	4	0	5	1.6702177263E+00
EDP 5	Three-bar truss design problem	2	0	3	2.6389584338E+02
EDP 6	Pressure vessel design	4	0	4	5.8853327736E+03
EDP 7	Tension/compression spring design	3	0	3	1.2665232788E−02

includes a brief explanation of several engineering issues.

6.2. Numerical Evaluation of Performance

The constrained violation addresses these constrained engineering design difficulties since each must fight with various natural constraints. The outcomes of different metaheuristic algorithms in dealing with specific design problems were obtained from the literature to make fair evaluations. As many times as possible, the recommended process was carried out. With a population of 20, the maximum number of iterations is 2500. Numerical findings from CL-SSA are presented here for the engineering design issues described thus far.

6.2.1. Process Design Problem

It is a matter of minimizing, which can be explained in the following manner.

$Minimizef\left(x\right)=5.357854x_1^2+40792.141-37.29329x_4+0.835689x_4{x}_3$

Subject to:

$$\begin{gathered} g_1\left(x\right)=-92+a_3{x}_4{x}_2+a_1+a_2{x}_4{x}_3-a_4{x}_4{x}_3 \le 0 \\ {g}_1\left(x\right)=-110+a_7{x}_4{x}_2+a_5+a_6{x}_5{x}_3-a_8{x}_1^2 \le 0 \\ {g}_1\left(x\right)=a_9 +a_{11}{x}_4{x}_1+a_{10}{x}_4{x}_3-25+a_{12}{x}_1{x}_2\le 0 \end{gathered}$$

With bounds:

$$\begin{gathered} 27\le x_1,x_2,x_3\le 45 \\ {x}_4\in \left\{78,79,\dots .,102\right\} \\ {x}_5\in \left\{78,79,\dots .,102\right\} \end{gathered}$$

where the values of $a_1 to a_{12}$ are given in

Table 25. The constants of the process design problem.

$a_1=85.334407$ $a_2=0.00556858$ $a_3=0.0006262$ $a_4=0.00222053$

$a_5=80.51249$ $a_6=0.0071317$ $a_7=0.0029955$ $a_8=0.0021813$

$a_9=9.300961$ $a_{10}=0.0047026$ $a_{11}=$0.0012547 $a_{12}=$0.0019085

.

From

Table 26. An analysis of various previously published research compared to the CL-SSA technique for solving engineering design challenges.

Problem	Algorithm	Mean	Best	Worst	Std.
Process design problem	CL-SSA	2.67E+04	2.67E+04	2.67E+04	7.93E−10
	SASS [88]	2.69E+04	2.69E+04	2.69E+04	1.11E−11
	BiPopEpsMAgES [89]	2.69E+04	2.69E+04	2.69E+04	1.11E−11
	COLSHADE [90]	2.69E+04	2.69E+04	2.69E+04	1.11E−11
	$\mathrm{iLSHADE}ϵ$ [91]	2.69E+04	2.69E+04	2.69E+04	1.11E−11
	$ϵ$MAgES [90]	2.69E+04	2.69E+04	2.69E+04	1.11E−11
	IUDE [92]	2.69E+04	2.69E+04	2.69E+04	1.11E−11
Speed reducer design problem	CL-SSA	3.00E+03	2.99E+03	3.01E+03	5.20E+00
	SASS	3.00E+03	2.99E+03	3.05E+03	6.29E+00
	BiPopEpsMAgES	2.99E+03	2.99E+03	2.99E+03	4.64E+13
	COLSHADE	2.99E+03	2.99E+03	2.99E+03	4.64E+13
	$\mathrm{iLSHADE}ϵ$	2.99E+03	2.99E+03	2.99E+03	4.64E+13
	$ϵ$MAgES	2.99E+03	2.99E+03	2.99E+03	4.64E+13
	IUDE	2.99E+03	2.99E+03	2.99E+03	4.64E+13
Multiple disk clutch brake design problem	IUDE	0.235242	0.235242	0.235242	1.69E−16
	$ϵ$MAgES	0.235242	0.235242	0.235242	1.69E−16
	$\mathrm{iLSHADE}ϵ$	0.235242	0.235242	0.235242	1.13E−16
	COLSHADE	0.235242	0.235242	0.235242	2.83E−17
	BiPopEpsMAgES	0.235242	0.235242	0.235242	5.84E−16
	SASS	0.235242	0.235242	0.235242	8.55E−07
	CL-SSA	0.235242	0.235242	0.235242	4.27E−08
Welded beam design	CL-SSA	1.69E+00	1.67E+00	1.74E+00	1.71E−02
	SASS	1.68E+00	1.67E+00	1.79E+00	2.01E−02
	BiPopEpsMAgES	1.67E+00	1.67E+00	1.67E+00	2.30E−03
	COLSHADE	1.67E+00	1.67E+00	1.67E+00	2.27E−16
	$\mathrm{iLSHADE}ϵ$	1.67E+00	1.67E+00	1.67E+00	7.59E−07
	$ϵ$MAgES	1.67E+00	1.67E+00	1.85E+00	3.95E−02
	IUDE	1.67E+00	1.67E+00	1.67E+00	1.20E−16
Three-bar truss design problem	CL-SSA	2.63E+02	2.63E+02	2.63E+02	1.16E−13
	SASS	2.64E+02	2.64E+02	2.64E+02	5.69E−14
	BiPopEpsMAgES	2.64E+02	2.64E+02	2.64E+02	8.71E− 05
	COLSHADE	2.64E+02	2.64E+02	2.64E+02	5.8E− 14
	$\mathrm{iLSHADE}ϵ$	2.64E+02	2.64E+02	2.64E+02	1.99E−02
	$ϵ$MAgES	2.65E+02	2.64E+02	2.64E+02	2.88E+00
	IUDE	2.64E+02	2.64E+02	2.64E+02	0.00E+00
Pressure vessel design	CL-SSA	4.93E+03	4.66E+03	5.32E+03	1.46E+02
	SASS	6.41E+03	6.06E+03	8.96E+03	6.28E+02
	BiPopEpsMAgES	6.17E+03	6.06E+03	7.46E+03	2.10E+02
	COLSHADE	6.06E+03	6.06E+03	6.09E+03	8.53E+00
	$\mathrm{iLSHADE}ϵ$	8.48E+03	6.06E+03	1.49E+04	3.14E+03
	$ϵ$MAgES	7.38E+03	6.06E+03	1.19E+04	1.93E+03
	IUDE	6.06E+03	6.06E+03	6.06E+03	6.16E+00
Tension/compression spring design	CL-SSA	1.27E−02	1.26E−02	1.42E−02	2.73E−04
	SASS	1.27E−02	1.27E−02	1.27E−02	2.62E−06
	BiPopEpsMAgES	1.27E−02	1.27E−02	1.37E−02	1.09E−04
	COLSHADE	1.27E−02	1.27E−02	1.27E−02	1.08E−07
	$\mathrm{iLSHADE}ϵ$	1.30E−02	1.27E−02	1.78E−02	1.06E−03
	$ϵ$MAgES	1.27E−02	1.27E−02	1.37E−02	2.16E−04
	IUDE	1.27E−02	1.27E−02	1.27E−02	1.08E−05

, we can conclude that the CL-SSA method outperforms the vast majority of other cutting-edge approaches to the problem of process design by using optimal decision variables to achieve optimal minimization of the weights associated with process design.

6.2.2. Weight Minimization of a Speed Reducer

The weight of the speed reducer must be kept to a minimum while also considering the bending stress of the gear teeth, surface stress, shaft transverse deflections, and shaft stresses. There are seven parameters in total: x₁, x₂, x₃, x₄, x₅, x₆, x₇, which represent the face width (b), the tooth module (m), the number of pinion teeth (z), l1, l2, d1, d2, and d2 which represent the length of the first shaft between bearings.

$$\begin{array}{c}Minimization f\left(x\right)\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}=0.7854x_1{x}_2^2\left(3.3333x_3^2+14.9334x_3-43.0934\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}-1.508x_1\left(x_6^2+x_7^2\right)+7.4777\left(x_6^3+x_7^3\right)\end{array}$$

Subject to:

$$\begin{array}{l}{g}_1\left(x\right)=\frac{27}{x_1{x}_2^2{x}_3}-1\le 0\\ g_2\left(x\right)=\frac{397.5}{x_1{x}_2^2{x}_3^2}-1\le 0\\ g_3\left(x\right)=\frac{1.93x_4^3}{x_2{x}_3{x}_6^4}-1\le 0\\ g_4\left(x\right)=\frac{1.93x_5^3}{x_2{x}_3{x}_7^4}-1\le 0\\ g_5\left(x\right)=\frac{\sqrt{\left(\frac{{745}_{x_4}}{x_2{x}_3}{)}^2+16.9X {10}^6\right.}}{110.0x_6^3}-1\le 0\\ g_6\left(x\right)=\frac{\sqrt{\left(\frac{{745}_{x_4}}{x_2{x}_3}{)}^2+157.5 X {10}^6\right.}}{85.0 x_6^3}-1\le 0\\ g_7\left(x\right)=\frac{x_2{x}_3}{40}-1\le 0\\ g_8\left(x\right)=\frac{5x_2}{x_1}-1\le 0\\ g_9\left(x\right)=\frac{x_1}{12x_2}-1\le 0\\ g_{10}\left(x\right)=\frac{1.5x_6+1.9}{x_4}-1\le 0\\ g_{11}\left(x\right)=\frac{1.1x_7+1.9}{x_5}-1\le 0\end{array}$$

where 2.6 ≤ $x_1$ ≤3.6, 0.7 ≤ $x_2$ ≤ 0.8, 17 ≤ $x_3$ ≤ 28, 7.3 ≤ $x_4$ ≤ 8.3, 7.8 ≤ $x_5$ ≤ 8.3, 2.9 ≤ $x_6$≤ 3.9, 5.0 ≤ $x_7$≤ 5.5.

According to the results in Table 26. The most cutting-edge approaches were shown to be less effective than the CL-SSA method in tackling the speed reducer design challenge.

6.2.3. Multiple Disk Clutch Brake Design Problem

A constrained mechanical design problem, the many disk clutch brake (see Figure 8e). The following is an example of how to formulate a mathematical problem:

$$Minimize f\left(x\right)=\pi \left(r_o^2-r_i^2\right)t\left(Z+1\right)\rho$$

Subject to:

$$\begin{array}{l}{g}_1\left(x\right)=r_o-r_i-\Delta r\ge 0,\\ g_2\left(x\right)=l_{max}-\left(Z+1\right)\left(t+\delta \right)\ge 0,\\ g_3\left(x\right)=p_{max}-p_{rz}\ge 0,\\ g_4\left(x\right)=p_{max}{v}_{srmax}-p_{rz}{v}_{sr}\ge 0,\\ g_5\left(x\right)=v_{srmax}-v_{sr}\ge 0,\\ g_6\left(x\right)=T_{max}-T\ge 0,\\ g_7\left(x\right)=M_h-s{M}_s\ge 0,\\ g_8\left(x\right)=T\ge 0,\end{array}$$

where:

$$\begin{array}{l}{M}_h=\frac{2}{3}\mu FZ\frac{r_o^3-r_i^3}{r_o^2-r_i^2},\\ p_{rz}=\frac{F}{\pi \left(r_o^2-r_i^2\right)},\\ v_{sr}=\frac{2\pi n\left(r_o^3-r_i^3\right)}{90\left(r_o^2-r_i^2\right)},\\ T=\frac{I_z\pi n}{30\left(M_h+M_f\right)}.\end{array}$$

And $\Delta r=20 \mathrm{mm}, t_{max}=3 \mathrm{mm},$ $t_{max}=1.5 \mathrm{mm},$ $l_{max}=30 \mathrm{mm}, Z_{max}=10, v_{srmax}=10 \mathrm{m}/\mathrm{s},$ $\mu =0.5, s=1.5, M_s=40 \mathrm{N} \mathrm{m}, M_f=3 \mathrm{N} \mathrm{m}, n=250 \mathrm{rpm}, p_{max}=1 \mathrm{MPa}, I_z=55{ \mathrm{Kg} \mathrm{mm}}^2, T_{max}=15 s, F_{max}=1000 \mathrm{N}, r_{imin}=55 \mathrm{mm}, r_{omax}=110 \mathrm{mm}.$

Table 26 shows that the CL-SSA method outperformed most current metaheuristics in handling this engineering problem by using optimal decision factors to reduce the truss weight.

6.2.4. Welded Beam Design Problem

Welded beam design must take into account the shear stress (τ), bending stress in the beam ($\mathsf{\sigma }$), buckling load on the bar (Pc), end deflection of the beam (δ), and side limitations. There are four design variables in this equation h ($x_1$), l ($x_2$), t ($x_3$) and b ($x_4$) (see Figure 8c). It is possible to state this issue quantitatively as follows:

$$Minimize f\left(x\right)=01.10471x_1{}^2{x}_2+0.04811x_3{x}_4\left(14.0+x_2\right)$$

$Subject to:$

$$\begin{array}{l}{g}_1\left(x\right)=\tau \left(x\right)-{\tau }_{max}\le 0\\ g_2\left(x\right)=\sigma \left(x\right)-{\sigma }_{max}\le 0\\ g_3\left(x\right)=x_1-x_4\le 0\\ g_4\left(x\right)=0.10471x_1{}^2+0.04811x_3{x}_4\left(14.0+x_2\right)-5.0\le 0\\ g_5\left(x\right)=0.125-x_1\le 0 \\ g_6\left(x\right)=\delta \left(x\right)-{\delta }_{max}\le 0\\ g_6\left(x\right)=P-P_c\left(x\right)\le 0\end{array}$$

where $\tau \left(x\right)=\sqrt{\left({\tau }^{’}){}^2+2{\tau }^{’}{\tau }^{″}\frac{x_2}{2R}\right.+\left({\tau }^{″}{)}^2\right.}$

• ${\tau }^{’}=\frac{P}{2^{0.5 }{x}_1{x}_2}$
• ${\tau }^{″}=\frac{MR}{J}$
• $M=P\left(L+\frac{x_2}{2}\right)$
• $R=\sqrt{\frac{x_2^2}{4}+\left(\frac{x_1+x_3}{2}\right.{)}^2}$
• $J=2\left\{2^{0.5}{x}_1{x}_2\left[\frac{x_2^2}{12}+\left(\frac{x_1+x_3}{2}\right)\left(\frac{x_1+x_3}{2}\right)\right]\right\}$
• $\sigma \left(x\right)=\frac{6PL}{x_4{x}_3^2}$
• $\delta \left(x\right)=\frac{4P{L}^3}{E{x}_3^3{x}_4}$
• $P_c\left(x\right)=\frac{4.013E\sqrt{\frac{x_3^2{x}_4^6}{36}}}{L^2}\left(1-\frac{x_3}{2L}\sqrt{\frac{E}{4G}}\right)$

where δmax = 0.25 in, L = 14 in, G = 12 × 106 psi E = 30 × 106 psi, σmax = 30,000 psi, τmax = 13,600 psi, P = 6000 lb, 0.1 ≤ x₁ ≤ 2, 0.1 ≤ x₂ ≤ 10, 0.1 ≤ x₃ ≤ 10, 0.1 ≤ x₄ ≤ 2.

Table 26 shows the results of a cost-optimal welded beam structure design problem solved with the help of CL-SSA and other cutting-edge algorithms. As shown in Table 26, the CL-SSA is more efficient than the comparing algorithms and can handle hybrid decision variables faster.

6.2.5. Pressure Vessel Problem

It is necessary to minimize the pressure vessel’s cost f(x) while also considering the costs of materials, production, and welding. A vessel’s internal radius ($x_1$), shell thickness ($x_2)$, head thickness ($x_3)$, and length ($x_4$) are all factors to consider throughout the design process (see Figure 8b). Continuous variables such as Th, Ts, L, and R, for example, reflect the likely thickness of rolled steel plates. The following is a numerical representation of the pressure vessel design challenge:

$$Minimize f\left(x\right)=0.6224x_1{x}_3{x}_4+1.7781x_2{x}_3^2+3.1661x_1^2{x}_4+19.84x_1^2{x}_{3 }$$

$Subject to:$

$$g_1\left(x\right)=-x_1+0.0193x_3\le 0$$

$$g_2\left(x\right)=-x_2+0.00954x_3\le 0$$

$$g_3\left(x\right)=-Пx_3^2{x}_4-\frac{4}{3}Пx_3^2+1296000\le 0$$

$$g_4\left(x\right)=x_4-240\le 0$$

where 1 ≤ x₁ ≤ 99, 1 ≤ x₂ ≤ 99, 10 ≤ x₃ ≤ 200, 10 ≤ x₄ ≤ 200.

Table 26 shows the results of a cost-optimal pressure vessel design problem solved with the help of CL-SSA and other cutting-edge algorithms. Unlike previous algorithms, the CL-SSA can solve hybrid decision variables more quickly.

6.2.6. Three-Bar Truss Design Problem

It is a well-known problem with constrained engineering problems for testing novel optimization approaches, such as constructing a three-bar truss. For the most part, its primary purpose is to reduce weight by employing three stress levels. (Take a look at Figure 8a). For this optimization technique, the following formulas are needed:

$$Minimize f\left(x\right)=\left(2\sqrt{2x_1}+x_2 \right)l$$

$$Subject to:{ \mathrm{g}}_1\left(x\right)=\frac{\left(\sqrt{2x_1}+x_2 \right)}{\left(\sqrt{2x_1^2}+2x_1{x}_2 \right) }P\le \sigma$$

$${\mathrm{g}}_2\left(x\right)=\frac{x_2}{\left(\sqrt{2x_1^2}+2x_1{x}_2 \right) }P\le \sigma$$

$${\mathrm{g}}_3\left(x\right)=\frac{1}{\left(x_1+\sqrt{2x_2^{}} \right) }P\le \sigma$$

$$0\le x_1,x_2\le 1,$$

where $P=2 \mathrm{kN}/{\mathrm{cm}}^2$, $l=100 \mathrm{cm}$, $\sigma =2 \mathrm{kN}/{\mathrm{cm}}^2$.

The CL-SSA algorithm outperformed many cutting-edge approaches when solving a three-bar truss problem, as shown in Table 26.

6.2.7. Tension/Compression Spring Design Problem

This tension/compression spring design problem aims to reduce the spring’s weight f(x) while maintaining constraints like minimum deflection, shear stress, surge frequency, outer diameter limits, and design factors in mind. The design factors (x₃) are the mean coil diameter D (x₂), the wire diameter d (x₁), and the number of active coils P. (see Figure 8d). The mathematical version of this problem is as follows:

$$\min \mathrm{f}\left(\mathrm{x}\right)=\left({\mathrm{x}}_3+2\right){\mathrm{x}}_2{\mathrm{x}}_1^2$$

$$\mathrm{s}.\mathrm{t}. {\mathrm{g}}_1\left(\mathrm{x}\right)=1-\frac{{\mathrm{x}}_2^3{\mathrm{x}}_3}{71785{\mathrm{x}}_1^4}\le 0$$

$${\mathrm{g}}_2\left(\mathrm{x}\right)=\frac{4{\mathrm{x}}_2^2-{\mathrm{x}}_1{\mathrm{x}}_2}{12,566\left({\mathrm{x}}_2{\mathrm{x}}_1^3-{\mathrm{x}}_1^4\right)}+\frac{1}{5108{\mathrm{x}}_1^2}-1\le 0$$

$${\mathrm{g}}_3\left(\mathrm{x}\right)=1-\frac{140.45{\mathrm{x}}_1}{{\mathrm{x}}_2^2{\mathrm{x}}_3}\le 0$$

$${\mathrm{g}}_4\left(\mathrm{x}\right)=\frac{{\mathrm{x}}_1+{\mathrm{x}}_2}{1.5}-1\le 0$$

where 0.05 ≤ x₁ ≤ 2, 0.25 ≤ x₂ ≤ 1.3, 2 ≤ x₃ ≤ 15.

In Table 26, CL-SSA and other cutting-edge algorithms were used to solve a cost-optimal tension/compression spring design problem. According to the data, The suggested algorithm outperforms the alternatives and can quickly handle hybrid decision variables.

7. Conclusions and Future Work

This paper offers a large-scale Salp Swarm Optimizer based on competitive learning (CL-SSA). The suggested algorithm’s major goal is to maintain a higher search space exploration rate while maintaining a faster convergence rate. There are two steps in the proposed updated algorithm. A pairwise competition mechanism is first implemented, which sorts the solutions into winners and losers. The winners’ places in the chain population are then updated using the powerful SSA exploitation. Second, the particle that loses the competition will adjust its position based on the one that won. The proposed approach has been created specifically for large-scale global optimization issues. CL-SSA was first tested by solving typical benchmark problems (CEC2017 benchmark functions with 50 and 100 decision variables and seven CEC2008lsgo benchmarks with 200, 500 and 1000 decision variables). Later, the CL-SSA’s applicability was tested by solving seven common EDPs listed in the “CEC2020 conference benchmark set of real-world issues (CEC2020)”. Finally, the proposed CL-SSA was evaluated against the original SSA, CSO, and other state-of-the-art algorithms. The proposed CL-SSA performs better in terms of convergence time and ability to avoid local optima than the original SSA and CSO algorithms. Compared to previous metaheuristic algorithms, the suggested approach produces very competitive results in the efficiency of solutions. The results indicate that the CL-SSA is superior, and binary versions of the algorithm can be created in the future as future works.