Multi-Strategy Enhanced Crested Porcupine Optimizer: CAPCPO

Abstract

Metaheuristic algorithms are widely used in engineering problems due to their high efficiency and simplicity. However, engineering challenges often involve multiple control variables, which present significant obstacles for metaheuristic algorithms. The Crested Porcupine Optimizer (CPO) is a metaheuristic algorithm designed to address engineering problems, but it faces issues such as falling into a local optimum. To address these limitations, this article proposes three new strategies: composite Cauchy mutation strategy, adaptive dynamic adjustment strategy, and population mutation strategy. The three proposed strategies are then introduced into CPO to enhance its optimization capabilities. On three well-known test suites, the improved CPO (CAPCPO) outperforms 11 metaheuristic algorithms. Finally, comparative experiments on seven real-world engineering optimization problems demonstrate the advantages and potential of CAPCPO in solving complex problems. The multifaceted experimental results indicate that CAPCPO consistently achieves superior solutions in most cases.

1. Introduction

Optimization is the process of seeking the best combination of decision variables for a particular problem. Optimization problems are prevalent in many fields, such as medical image segmentation [1], engineering application problems [2,3,4], and parameter estimation [5,6]. Optimization problem solving methods are generally believed to be divided into deterministic methods and stochastic methods. Deterministic methods include conjugate gradient, dynamic programming, and quasi-Newton methods, etc. [7,8,9], which rely on the derivative of the objective function to solve optimization problems.

Deterministic methods are effective in optimizing linear or convex nonlinear problems. However, real-world optimization problems are often very complex, such as non-convex problems and non-differentiable objective functions, and deterministic optimization algorithms face numerous difficulties in optimizing these problems [10,11]. The incompetence of deterministic algorithms has led researchers to focus on the development of stochastic optimization algorithms. For the majority of stochastic optimization algorithms, when solving optimization problems, candidate solutions are first randomly generated in the search space, then enter the iterative process. During each iteration, these candidate solutions are evaluated and updated. As the iteration progresses, the candidate solutions become closer to the optimal solution. The solution found by stochastic optimization algorithms may not be the optimal solution, but a quasi-optimal solution. To find better quasi-optimal solutions, researchers have developed a large number of stochastic optimization algorithms, among which the metaheuristic algorithms are the most popular and competitive ones. Metaheuristic algorithms are nature-inspired, and are usually classified into multiple subcategories: swarm-based, evolution-based, physics-based, human-based, ancient-based, chemistry-based, plant-based, music-based/art-based, sport-based, mathematical-based, single-solution-based, and hybrid algorithm [12].

Swarm-based algorithms are inspired by the social behaviors of various biological groups in nature. For example, a typical swarm-based metaheuristic algorithm is Particle Swarm Optimization (PSO) [13] which is inspired by the foraging behavior of birds and abstracts the process of birds searching for food into finding the optimal solution to an objective function. Other swarm-based metaheuristic algorithms include Grey Wolf Optimizer (GWO) [14], Sparrow Search Algorithm (SSA) [15], etc.

Evolution-based algorithms are inspired by the process of biological evolution. For example, the Genetic Algorithm (GA) [16] simulates phenomena such as reproduction, crossover, and mutation that occur in natural selection and genetics. Other evolution-based algorithms have also been proposed, such as Differential Evolution (DE) [17], Barnacles Mating Optimizer (BMO) [18], and so on.

Physics-based algorithms are usually inspired by physical laws. Among them, Simulated Annealing (SA) [19] originated the earliest, and its mathematical modeling is derived from the annealing process of solid substances in physics. In addition to SA, other physics-based algorithms have been proposed, including Gravitational Search Algorithm(GSA) [20], Artificial Electric Field Algorithm (AEFA) [21], and so on.

Human-based algorithms simulate human’s social and behavioral activities, including Brain Storm Optimization (BSO) [22], Teaching Learning Based Optimization (TLBO) [23], etc.

Ancient-based algorithms refer to being inspired by the principles and behaviors of ancient civilizations or cultures, including Giza Pyramids Construction (GPC) [24] and Dujiangyan Irrigation System Optimization (DISO) [25], among others.

Chemistry-based algorithms are inspired by the properties of chemical reactions, including Artificial Chemical Process (ACP) [26], Artificial Chemical Reaction Optimization (ACRO) [27], etc.

Plant-based algorithms are inspired by the life processes and competitive behaviors of plants. Among them are Flower Pollination Algorithm (FPA) [28], Plant Competition Optimization (PCO) [29], Waterwheel Plant Algorithm (WWPA) [30], and so on.

Music/art-based algorithms typically draw inspiration from artistic creation, including Melody Search Algorithm (MSA) [31], Musical Composition Algorithm (MMC) [32], Harmony Search Algorithm (HSA) [33], etc.

Sport-based algorithms are inspired by athletic activities, including Golden Ball (GB) [34], Golf Sport Inspired Search (GSIS) [35], etc.

Mathematical-based algorithms are inspired by mathematical theories and methods, including Hyper-Spherical Search (HSS) [36], Arithmetic Optimization Algorithm (AOA) [37], etc.

Single-solution-based algorithms maintain only one solution throughout the search process, including Large Neighborhood Search (LNBS) [38], Variable Neighborhood Search (VNS) [39], etc.

Hybrid algorithms integrate the advantages of two distinct algorithms, including Particle Swarm Optimization and Grey Wolf Optimizer (HPSOGWO) [40], Biogeography-Based Optimization and Grey Wolf Optimizer (HBBOG) [41], etc.

Prior to this paper, there have been numerous metaheuristic algorithms, but according to the “No Free Lunch” (NFL) theorem, there is no algorithm that can solve all optimization problems [42]. In other words, a metaheuristic algorithm may perform well on some optimization problems, but may perform poorly on other types of optimization problems.

The CPO algorithm is a recently proposed excellent swarm intelligence optimization algorithm. CPO introduces a novel mechanism known as the cyclic population reduction mechanism, which enhances population diversity [43], but it still faces some challenges, such as slow convergence rate, excessive parameters, and limited local optimum avoidance ability [44]. To address the shortcomings, three novel strategies are proposed, namely composite Cauchy mutation strategy, adaptive dynamic adjustment strategy, and population mutation strategy. Then, these three strategies are introduced into the original CPO and an enhanced CPO algorithm, CAPCPO, is proposed. The research contributions are summarized as follows:

(1) To enhance the exploration and exploitation capabilities, a novel composite Cauchy mutation strategy is proposed and added to the first and third phases of the original CPO;

(2) To help the algorithm jump out of local optima, a novel adaptive dynamic adjustment strategy is developed and used to the fourth phase of the original CPO;

(3) A novel population mutation strategy is proposed and added to original CPO so as to enhance its diversity of solutions.

The remainder of this paper is arranged as follows: Section 2 introduces the original CPO algorithm. Section 3 describes the three novel strategies and the details of the proposed CAPCPO. Section 4 presents numerical experiments and results, including the comparison of CAPCPO with other highly cited or recently proposed metaheuristic algorithms on three challenging CEC benchmarks. Section 5 tests the effectiveness of the enhanced CAPCPO in solving practical problems by optimizing seven real engineering design problems. Section 6 concludes the paper.

2. Overview of the Crested Porcupine Optimizer

The CPO is a new metaheuristic algorithm proposed by Mohamed Abdel-Basset et al. in 2024 [43]. This algorithm simulates four behaviors of the Crested Porcupine (CP): sight, sound, odor, and physical attack. Similar to other metaheuristic algorithms, the CPO algorithm contains two phases, the exploration phase and the exploitation phase. The exploration phase is used for global search while the exploitation phase is for local search. The initialization, exploration phase, and exploitation phase of CPO are described below.

2.1. Initialization

CPO is a population-based algorithm, where Crested Porcupines (CP) are regarded as the search agents. Each CP is a candidate solution, which is updated during optimization process. The position matrix of search agents is modeled as Equation (1).

$$\mathrm{X}=\left[\begin{array}{c}{X}_1\\ \begin{array}{c}⋮\\ X_i\end{array}\\ \begin{array}{c}⋮\\ X_{N_{max}}\end{array}\end{array}\right]=\left[\begin{array}{ccccc}{x}_{\mathrm{1,1}}& \dots & x_{1,j}& \dots & x_{1,d}\\ ⋮& \ddots & ⋮& \ddots & ⋮\\ x_{i,1}& \dots & x_{i,j}& \dots & x_{i,d}\\ ⋮& \ddots & ⋮& \ddots & ⋮\\ x_{N_{max},1}& \dots & x_{N_{max},j}& \dots & x_{N_{max},d}\end{array}\right]$$

(1)

where $\mathrm{X}$ presents the population, $X_i$ denotes the $i$th CP individual in the population, which represents the location of the $i$th candidate solution, $x_{i,j}$ indicates the $j$th dimension of the $i$th solution (CP), $N_{max}$ is the maximum number of individuals in the CPO population, and $d$ denotes the dimension size of the problem being solved. In the initialization phase, $X_i$ is randomly generated in the search space as shown in Equation (2).

$$X_i=L+rand\times \left(U-L\right)\mid i=\mathrm{1,2},\dots ,N_{max}$$

(2)

where $L$ and $U$ represent the lower bound and upper bound of the optimization problem, respectively, and $rand$ is a randomly generated vector between 0 and 1. During the optimization process, the fitness (objective) function is used to calculate the fitness value of each candidate solution, and the fitness values of all individuals in the population are collected and stored in the fitness matrix $F$. The fitness matrix is given by Equation (3).

$$F=\left[\begin{array}{c}{F}_1\\ \begin{array}{c}⋮\\ F_i\end{array}\\ \begin{array}{c}⋮\\ F_{N_{max}}\end{array}\end{array}\right]=\left[\begin{array}{c}f\left(X_1\right)\\ \begin{array}{c}⋮\\ f\left(X_i\right)\end{array}\\ \begin{array}{c}⋮\\ f\left(X_{N_{max}}\right)\end{array}\end{array}\right]$$

(3)

where $F$ is the matrix that stores all the fitness and $f$ is the objective function. The solution with minimum fitness is the optimal solution.

2.2. Cyclic Population Reduction Technique

CPO introduces a technique known as cyclic population reduction (CPR), wherein the population size ($N_t$) is not constant during the optimization process. Instead, it cyclically decreases with each iteration. The iterative formula for the population, denoted as $N_t$, is depicted in Equation (4).

$$N_t=N_{min}+\left(N_{max}-N_{min}\right)\times \left(1-\left({\displaystyle \frac{t\%\frac{T_{max}}{T}}{\frac{T_{max}}{T}}}\right)\right)$$

(4)

where $N_t$ is the population size in the $t$th iteration, $N_{min}$ and $N_{max}$ are the minimum and maximum population size, respectively, $T$ is a variable determining the number of cycles, $T_{max}$ is the maximum number of iterations, and $\%$ denotes the remainder or modulo operator.

The CPR is shown in Figure 1. $T=i$ means that throughout the entire iteration process, the population size changes $i$ times from $N_{max}$ to $N_{min}$.

In each iteration of the original CPO algorithm, whether the algorithm entered the exploration phase or the exploitation phase is controlled by two random numbers ${\varphi }_1$ and ${\varphi }_2$, which are randomly generated between 0 and 1. When ${\varphi }_1<{\varphi }_2$, the exploration phase is entered; otherwise, the exploitation phase is entered.

2.3. Exploration Phase

The exploration phase of the original CPO provides two position update formulas, simulating two defense strategies for the CP when it is far away from predators: the first defense strategy and the second defense strategy. These two strategies respectively simulate the CP’s sight defense behavior and sound defense behavior.

CPO switches between the first defense strategy and the second defense strategy by comparing the values of two random numbers ${\varphi }_3$ and ${\varphi }_4$ between 0 and 1. When ${\varphi }_3<{\varphi }_4$, enter the first defense strategy; Otherwise, enter the second defense strategy.

The position update formula in the first defensive strategy is shown in Equation (5).

$$X_i^{t+1}=X_i^t+{\tau }_1\times \left|2\times {\tau }_2\times X_{CP}^t-y_i^t\right|$$

(5)

where $X_{CP}^t$ is the best solution before the $t$th iteration, $X_i^t$ represents the current solution, ${\tau }_1$ is a random number based on the normal distribution, and τ₂ is a random value between 0 and 1. $y_i^t$ represents the position of the predator at the $t$th iteration, which is calculated by Equation (6).

$$y_i^t={\displaystyle \frac{X_i^t+X_r^t}{2}}$$

(6)

where $r$ is a random integer between [1, N] and $X_r^t$ represents a random position.

The position update formula in the second defensive strategy is shown in Equation (7).

$$X_i^{t+1}=\left(1-U_1\right)\times X_i^t+U_1\times \left(y_i^t+{\tau }_3\times \left(X_{r1}^t-X_{r2}^t\right)\right)$$

(7)

where $r1$ and $r2$ are two random integers between [1, N], ${\tau }_3$ is a random value between 0 and 1, $U_1$ is a randomly generated vector containing only 0 and 1, and $X_{r1}^t$ and $X_{r2}^t$ denote two random solutions within the population.

2.4. Exploitation Phase

Similar to the exploration phase, the exploitation phase of the original CPO algorithm also provides two position update formulas dedicated to local search, namely, the third defense strategy and the fourth defense strategy. In this phase, first generate a random number ${\varphi }_5$ between 0 and 1. When ${\varphi }_5<0.5$, update the position according to the third defense strategy. Otherwise, update the position according to the fourth defense strategy.

The position update formula in the third defensive strategy is shown in Equations (8)–(11).

$$X_i^{t+1}=\left(1-U_1\right)\times X_i^t+U_1\times \left(X_{r1}^t+S_i^t\times \left(X_{r2}^t-X_{r3}^t\right)-{\tau }_4\times \delta \times {\gamma }_t\times S_i^t\right)$$

(8)

$$S_i^t=exp\left({\displaystyle \frac{f\left(X_i^t\right)}{\sum _{k=1}^{N_t}f\left(X_k^t\right)+ϵ}}\right)$$

(9)

$$\delta =\left\{\begin{array}{c}+1,if rand\le 0.5\\ -1,else\end{array}\right.$$

(10)

$${\gamma }_t=2\times rand\times {\left(1-{\displaystyle \frac{t}{T_{max}}}\right)}^{{\displaystyle \frac{t}{T_{max}}}}$$

(11)

where ${\tau }_4$ is a random value between 0 and 1, $f\left(X_i^t\right)$ represents the objective function value of the $i$th CP at $t$th iteration, $ϵ$ is a small value utilized to circumvent the issue of division by zero, and $rand$ is a randomly generated vector that contains values only between 0 and 1.

The position update formula in the fourth defensive strategy is shown in Equations (12)–(14).

$$X_i^{t+1}=X_{CP}^t+\left(\alpha \left(1-{\tau }_5\right)+{\tau }_5\right)\times \left(\delta \times X_{CP}^t-X_i^t\right)-\beta$$

(12)

$$\beta ={\tau }_6\times \delta \times {\gamma }_t\times D_i^t$$

(13)

$$D_i^t={\tau }_7\times S_i^t\times \left(V_i^{t+1}-V_i^t\right)$$

(14)

where $\alpha$ is a convergence rate factor, ${\tau }_5$ and ${\tau }_6$are random values between 0 and 1, and ${\tau }_7$ is a randomly generated vector that contains values only between 0 and 1. $V_i^{t+1}$ is the final speed of the $i$th individual at the $t+1$th iteration, randomly selected from the current population. $V_i^t$ is the same as $X_i^t$.

3. The Proposed CAPCPO

To address the issues of slow convergence and susceptibility to local optima in the original CPO algorithm, this paper proposes three novel strategies: composite Cauchy mutation strategy, adaptive dynamic adjustment strategy, and population mutation strategy. These three strategies are applied to enhance the original CPO algorithm.

This section first provides a detailed explanation of these three strategies, as well as the proposed CAPCPO.

3.1. Composite Cauchy Mutation Strategy

The diversity of the fitness matrix of the original CPO decreases during the iteration process, causing it to easily fall into a local optimum. To address this issue, this paper proposes composite Cauchy mutation strategy, which is activated when the diversity of the fitness matrix decreases to a certain threshold. In this paper, the diversity of the fitness matrix is defined in Equation (15).

$$A={\displaystyle \frac{std\left(F\right)-mean\left(F\right)}{mean{\left(F\right)}^2}}$$

(15)

where $std\left(F\right)$ and $mean\left(F\right)$ denote the standard deviation and the mean of the fitness matrix, respectively.

When A > 0.01, the composite Cauchy mutation strategy is triggered [45], which is divided into two parts:

The first part is used in the first defensive phase to expand the scope of exploration. At this point, the new positional update formula is as follows:

$$X_i^{t+1}=X_i^t+c_1\times X_i^t$$

(16)

$$c_1=tan\left(\left({\gamma }_1-0.5\right)\times \mathsf{\pi }\right)$$

(17)

where ${\gamma }_1$ is a random value between the interval $\left[0, 1\right)$.

The second part is applied to the third defensive phase to rapidly approximate to the optimum and accelerate the convergence, which is calculated as follows:

$$X_i^{t+1}=X_{CP}^t+c_2\times X_i^t$$

(18)

$$c_2=sin\left(\left({\gamma }_2-0.5\right)\times \mathsf{\pi }\right)$$

(19)

where ${\gamma }_2$ is a random value between the interval $\left[0, 1\right)$.

3.2. Adaptive Dynamic Adjustment Strategy

In the fourth defensive strategy of CPO, the position is updated randomly near the previous optimal position, which may result in either excessively large or small updates. However, small updates in early iterations may lead to slow convergence of the algorithm, while large updates in late iterations may cause the search to exceed the boundary of the search space and miss the optimal solution. To address this issue, this paper proposes an adaptive dynamic adjustment strategy for position updates. The key of this strategy is a position update factor ω, which is calculated according to Equation (20).

$$\omega =exp\left[1-{\left({\displaystyle \frac{t^{1.3}}{T_{max}\times {\gamma }_3\times 0.2}}\right)}^{0.4}\right]\times c^k$$

(20)

$$k=\left\{\begin{array}{c}-1 ,if {\lambda }_1>dim\times {\displaystyle \frac{1}{3}}\\ 1 ,if {\lambda }_2>dim\times {\displaystyle \frac{1}{3}}\\ 0 ,else\end{array}\right.$$

(21)

where ${\gamma }_3$ is random in $\left[0, 1\right)$. $c$ represents the scaling factor, and multiple repeated experiments have demonstrated that $c=10$ yields the most favorable results. ${\lambda }_1$ represents the number of dimensions of $X_i^t$ that exceed the upper bound, while ${\lambda }_2$ indicates the number of dimensions of $X_i^t$ that fall below the lower bound. $k$ is −1 when more than 1/3 of the dimensions in the previous iteration are larger than the upper bound U; $k$ is 1 when more than 1/3 of the dimensions in the previous iteration are smaller than the lower bound L, and $k$ is 0 in the remaining cases.

Obviously, the position update factor ω not only adaptively attenuates with the iterations, but also automatically adjusts the update factor based on whether the previous update results exceed the boundary.

After incorporating the adaptive dynamic adjustment strategy, the final position update formulas for the fourth defensive strategy are shown in Equations (22) and (23).

$$X_i^{t+1}=X_{CP}^t+R\times \left(X_{CP}^t-X_i^t\right)-\beta$$

(22)

$$R=\left\{\begin{array}{c}1, if rand<0.5\\ \omega , else\end{array}\right.$$

(23)

3.3. Population Mutation Strategy

In the original CPO, the size of the population is reduced using the CPR technique, in which the population size changes cyclically in a fixed way without considering diversity of the fitness matrix. In this paper, we propose a novel population mutation strategy that tunes the population size based on the diversity of the fitness matrix. The update formula for the population size in our proposed population mutation strategy is shown in Equation (24).

$$N=\left\{\begin{array}{c}{N}_{min} , if A>0.01\\ N_{max} , else\end{array}\right.$$

(24)

where $N_{min}$ denotes the minimum population size and $N_{max}$ is the maximum population size, in this paper $N_{min}={\displaystyle \frac{1}{2}}{N}_{max}$. $A$ represents the diversity of the fitness matrix defined in Equation (15).

When $A>0.01$, it means that the diversity of the fitness matrix is insufficient, indicating the fitness of individuals in the population is highly similar, and in this case, there is no need for a large population size. Therefore, the population size is set to the minimum value, $N_{min}$. When $A\le 0.01$, the diversity of the fitness matrix is sufficient. For better global search, the population size is set to the maximum value, $N_{max}$. At the same time, a solution is randomly selected for mutation within the population in order to prevent a decrease in the diversity of the fitness matrix. The specific formula is shown in Equation (25).

$$X_r^{t+1}={\displaystyle \frac{\sum _{i=0}^N{X}_i^t}{N}}\times \left({\gamma }_4-0.5\right)\times 2,if A\le 0.01$$

(25)

where $r$ is a random integer between [1, N] and ${\gamma }_4$ is a random value between $\left[0, 1\right)$.

Using population mutation strategy, the average population size is reduced during the iteration process and the running speed of the algorithm is accelerated.

3.4. The Detail of Our Proposed CAPCPO

The pseudocode of the CAPCPO algorithm with three strategies is shown in Algorithm 1. The flowchart is shown in Figure 2, and the green parts are our improvements.

Algorithm 1: The pseudo-code of the CAPCPO algorithm.

Input: Parameters of CAPCPO, such as $N_{max},T_{max}$. Output: The optimal solution

1: Set parameters $T_{max},N_{max},c$. 2: Initialize the population by Equation (1) and Equation (2). 3: While $\left(t<T_{max}\right)$ do 4:   Calculate the fitness of each population. 5:   Determine the best ($X_{CP}^t$) solution so far. 6:   ${\gamma }_t$ is updated based on Equation (11). 7:   $\omega$ is updated based on Equation (20). 8:   Generate ${\varphi }_1, {\varphi }_2, {\varphi }_3,{\varphi }_4 and {\varphi }_5$ five random numbers. 9:   For $i<N$ do 10:   Update parameters $S,D,\delta$. 11:   If ${\varphi }_1<{\varphi }_2$ //Exploration 12:    If$\mathit{\hspace{1em}}{\varphi }_3<{\varphi }_4$  //First defense strategy 13:     If $A>0.01$ 14:      The position of CP is updated based on Equation (16). 15:     Else 16:      The position of CP is updated based on Equation (5). 17:    Else  //Second defense strategy 18:     The position of CP is updated based on Equation (7). 19:   Else //Exploitation phase 20:    If ${\varphi }_5<0.5$  //Third defense strategy 21:     If $A>0.01$ 22:      The position of CP is updated based on Equation (18). 23:     Else 24:      The position of CP is updated based on Equation (8). 25:    Else   //Fourth defense strategy 26:     The position of CP is updated based on Equation (22). 27:   End If 28:   If $f\left(X_i^{t+1}\right)>f\left(X_i^t\right)$ 29:    $X_i^{t+1}=X_i^t$ 30:   If $A>0.01$ 31:    $N=N_{min}$ 32:   Else 33:    $N=N_{max};$ The position of a random CP is updated based on Equation (25). 34:   End If 35:   $t=t+1$ 36:   End For 37: End while 38: Return $X_{CP}^t$ 39: Output the optimal solution

3.5. Time Complexity Analysis

The time complexity of CAPCPO primarily encompasses two key components: population initialization and population update. The main parameters that affect time complexity are the maximum number of iterations $T_{max}$, the dimension of the problem $d$, and the population size $N$. The time complexity of population initialization and population update are respectively $\mathrm{O}(N\times d)$ and $\mathrm{O}(T_{max}\times N\times d)$. Since the population size in CAPCPO is dynamically changing, in the best case, $\mathrm{O}\left(\mathrm{C}\mathrm{A}\mathrm{P}\mathrm{C}\mathrm{P}\mathrm{O}\right)=\mathrm{O}(N_{min}\times d)+\mathrm{O}(T_{max}\times N_{min}\times d)$, and in the worst case, $\mathrm{O}\left(\mathrm{C}\mathrm{A}\mathrm{P}\mathrm{C}\mathrm{P}\mathrm{O}\right)=\mathrm{O}(N_{max}\times d)+\mathrm{O}(T_{max}\times N_{max}\times d)$.

4. Experimental Results and Discussion

In this section, the optimization performance of CAPCPO is critically evaluated from several perspectives using three challenging test suites: CEC2017, CEC2019, and CEC2022.

On CEC2017 test suite: (1) permutation experiments are conducted among all different CPO variants to test the effectiveness of the three strategies; (2) the convergence behavior of CAPCPO is analyzed; (3) the exploration ability, exploitation ability, and local optimum avoidance ability of CAPCPO are compared experimentally with the other 11 algorithms; (4) Finally, the scalability analysis is conducted on the 12 algorithms in order to compare their ability to deal with high-dimensional optimization problems.

Furthermore, to bolster the credibility of the experiment, we also carried out comparative tests among 12 algorithms on CEC2019 and CEC2022.

4.1. Benchmark Functions and Experimental Setup

4.1.1. Benchmark Functions

To test the performance of the proposed CAPCPO, the three test suites CEC2017, CEC2019, and CEC2022 are adopted as the benchmark test functions.

CEC2017 contains two unimodal functions (C1–C2) to test exploitation ability, seven multimodal functions (C3–C9) to test exploration ability, ten hybrid functions (C10–C19) and ten composition functions (C20–C29) to test local optimum avoidance ability. The detail of CEC2017 test suite is shown in

Table 1. Benchmark functions in CEC2017 test suite.

Type	ID	Function	Range	${\mathit{f}}_{\mathit{m}\mathit{i}\mathit{n}}$
Unimodal	C1	Shifted and Rotated Bent Cigar Function	[−100, 100]	100
	C2	Shifted and Rotated Zakharov Function	[−100, 100]	200
Multimodal	C3	Shifted and Rotated Rosenbrock’s Function	[−100, 100]	300
	C4	Shifted and Rotated Rastrigin’s Function	[−100, 100]	400
	C5	Shifted and Rotated Schaffer’s F7 Function	[−100, 100]	500
	C6	Shifted and Rotated Lunacek Bi-Rastrigin’s Function	[−100, 100]	600
	C7	Shifted and Rotated Non-Continuous Rastrigin’s Function	[−100, 100]	700
	C8	Shifted and Rotated Levy Function	[−100, 100]	800
	C9	Shifted and Rotated Schwefel’s Function	[−100, 100]	900
Hybrid	C10	Hybrid Function 1 (N = 3)	[−100, 100]	1000
	C11	Hybrid Function 2 (N = 3)	[−100, 100]	1100
	C12	Hybrid Function 3 (N = 3)	[−100, 100]	1200
	C13	Hybrid Function 4 (N = 4)	[−100, 100]	1300
	C14	Hybrid Function 5 (N = 4)	[−100, 100]	1400
	C15	Hybrid Function 6 (N = 4)	[−100, 100]	1500
	C16	Hybrid Function 7 (N = 5)	[−100, 100]	1600
	C17	Hybrid Function 8 (N = 5)	[−100, 100]	1700
	C18	Hybrid Function 9 (N = 5)	[−100, 100]	1800
	C19	Hybrid Function 10 (N = 6)	[−100, 100]	1900
Composition	C20	Composition Function 1 (N = 3)	[−100, 100]	2000
	C21	Composition Function 2 (N = 3)	[−100, 100]	2100
	C22	Composition Function 3 (N = 4)	[−100, 100]	2200
	C23	Composition Function 4 (N = 4)	[−100, 100]	2300
	C24	Composition Function 5 (N = 5)	[−100, 100]	2400
	C25	Composition Function 6 (N = 5)	[−100, 100]	2500
	C26	Composition Function 7 (N = 6)	[−100, 100]	2600
	C27	Composition Function 8 (N = 6)	[−100, 100]	2700
	C28	Composition Function 9 (N = 3)	[−100, 100]	2800
	C29	Composition Function 10 (N = 3)	[−100, 100]	2900

, where $f_{min}$ is the optimal value, Range is the boundary of the design variable.

The details of CEC2019 and CEC2022 test suites are shown in

Table 2. Benchmark functions in CEC2019 test suite.

ID	Function	d	Range	${\mathit{f}}_{\mathit{m}\mathit{i}\mathit{n}}$
C30	Storn’s Chebyshev Polynomial Fitting Problem	9	[−8192, 8192]	1
C31	Inverse Hilbert Matrix Problem	16	[−16,384, 16,384]	1
C32	Lennard-Jones Minimum Energy Cluster	18	[−4, 4]	1
C33	Rastrigin’s Function	10	[−100, 100]	1
C34	Griewangk’s Function	10	[−100, 100]	1
C35	Weierstrass Function	10	[−100, 100]	1
C36	Modified Schwefel’s Function	10	[−100, 100]	1
C37	Expanded Schaffer’s F6 Function	10	[−100, 100]	1
C38	Happy Cat Function	10	[−100, 100]	1
C39	Ackley Function	10	[−100, 100]	1

and

Table 3. Benchmark functions in CEC2022 test suite.

ID	Function	d	Range	${\mathit{f}}_{\mathit{m}\mathit{i}\mathit{n}}$
C40	Shifted and full Rotated Zakharov Function	10/20	[−100, 100]	300
C41	Shifted and full Rotated Rosenbrock’s Function	10/20	[−100, 100]	400
C42	Shifted and full Rotated Rastrigin’s Function	10/20	[−100, 100]	600
C43	Shifted and full Rotated Non-Continuous Rastrigin’s Function	10/20	[−100, 100]	800
C44	Shifted and full Rotated Levy Function	10/20	[−100, 100]	900
C45	Hybrid Function 1 (N = 3)	10/20	[−100, 100]	1800
C46	Hybrid Function 2 (N = 6)	10/20	[−100, 100]	2000
C47	Hybrid Function 3 (N = 5)	10/20	[−100, 100]	2200
C48	Composition Function 1 (N = 5)	10/20	[−100, 100]	2300
C49	Composition Function 2 (N = 4)	10/20	[−100, 100]	2400
C50	Composition Function 3 (N = 5)	10/20	[−100, 100]	2600
C51	Composition Function 4 (N = 6)	10/20	[−100, 100]	2700

, where $f_{min}$ is the optimal value, range is the boundary of the design variable, and d is dimension.

4.1.2. Experimental Setup

A total of 11 algorithms are compared with our proposed CAPCPO, including CPO [43], PSO [13], GWO [14], Whale Optimization Algorithm (WOA) [46], Sine Cosine Algorithm (SCA) [47], Seagull Optimization Algorithm (SOA) [48], SSA [15], Beluga Whale Optimization (BWO) [49], Dung Beetle Optimizer (DBO) [50], Golden Jackal Optimization (GJO) [51], and Pelican Optimization Algorithm (POA) [52]. These compared algorithms are famous or recently proposed metaheuristic algorithms which are widely used for solving optimization problems. The parameter settings of the compared algorithms and our proposed CAPCPO are presented in

Table 4. Parameter settings of compared algorithms and proposed CAPCPO.

Algorithms	Name of the Parameter	Value of the Parameter
	Population size, maximum number of iterations, number of repeated experiments	$30, 500, 30$
CPO	$\alpha$	$0.1$
	$N_{min}$	$10$
	$T$	$2$
PSO	Cognitive constant	$2$
	Social constant	$2$
	Inertia weight linearly decreased at interval	$\left[0.9, 0.2\right]$
	Velocity limit	10% of the dimensions range of the variables.
GWO	$a$	Linear reduction from 2 to 0
WOA	Probability of encircling mechanism	$0.5$
	Spiral factor	$1$
SCA	$a$	2
SOA	Control parameter	$\left[2, 0\right]$
	$f_c$	$2$
SSA	Leader position update probability	$0.5$
BWO	Probability of whale fall decreased at interval $W_f$	$\left[0.1, 0.05\right]$
DBO	$PballRolling$	$0.2$
	$PbroodBall$	$0.4$
	$PSmall$	$0.2$
	$Pthief$	$0.4$
	$b$	$0.3$
	$k$	$0.1$
	$S$	$0.5$
GJO	$\backslash$	$\backslash$
POA	$I$	Random integer with the value 1 or 2
	$R$	$0.2$
CAPCPO	$c$	$10$

. These values are sourced from the respective original literature or are the optimal values.

For fair comparison, all experiments were conducted according to recognized standards in the optimization field [53], ensuring that the algorithm stands out not because of better computing resources, but because of better performance. Among them, the dimension to be optimized (d) on CEC2017 was set to 30 except for scalability analysis, the initial population size ($N$) of all the algorithms was set to 30, and the maximum number of iterations was set to 500. Moreover, to weaken experimental serendipity, each algorithm was run independently 30 times on each benchmark function to take the average of the results. In this paper, the Friedman test [54] was used to sort and evaluate the results of all algorithms on the benchmark functions. The Wilcoxon signed rank test was employed to determine the statistical significance of dissimilarity between two sets of solutions [55].

All algorithms were written in Python 3.9, and all experiments were performed on an Intel Core i9-13900k, 2.5 GHz CPU, and 16 GB RAM laptop.

4.2. Influence of the Three Strategies

To verify the effectiveness of the three strategies, namely the composite Cauchy mutation strategy, the adaptive dynamic adjustment strategy, and the population mutation strategy, on CPO, a permutation test was performed on all CPO variants on CEC2017. Detail of these CPO variants are shown in

Table 5. Various CPOs from three strategies.

	Composite Cauchy Mutation Strategy (C)	Adaptive Dynamic Adjustment Strategy (A)	Population Mutation Strategy (P)
CPO	0	0	0
CCPO	1	0	0
ACPO	0	1	0
PCPO	0	0	1
CACPO	1	1	0
CPCPO	1	0	1
APCPO	0	1	1
CAPCPO	1	1	1

, where ‘1’ indicates that the strategy is added to the CPO while ‘0’ indicates the opposite. For example, CPCPO indicates the incorporation of the composite Cauchy mutation strategy and the population mutation strategy.

Table 6. Results on benchmark functions (C1–C29).

Fun	Method	CPO	CCPO	ACPO	PCPO	CACPO	CPCPO	APCPO	CAPCPO
C1	Avg	3.3679 × 107	2.0581 × 107	1.4535 × 105	1.1872 × 107	9.3311 × 103	9.9320 × 106	6.4913 × 103	7.5047 × 103
	Std	2.6226 × 107	1.6182 × 107	2.6761 × 105	8.7785 × 106	7.4871 × 103	5.9248 × 106	3.9788 × 103	3.9472 × 103
C2	Avg	7.5399 × 103	6.5058 × 103	2.1949 × 102	4.5374 × 103	2.0893 × 102	4.5018 × 103	2.0093 × 102	2.0239 × 102
	Std	2.4600 × 103	2.8709 × 103	2.5903 × 101	1.6496 × 103	2.1116 × 101	1.4693 × 103	1.1178	4.6230
C3	Avg	4.4097 × 102	4.3703 × 102	3.7751 × 102	4.1060 × 102	3.7724 × 102	4.2609 × 102	3.8089 × 102	3.7701 × 102
	Std	2.9401 × 101	3.2329 × 101	4.0751 × 101	3.4786 × 101	3.5585 × 101	3.8157 × 101	3.4829 × 101	4.3096 × 101
C4	Avg	7.9923 × 102	7.4836 × 102	7.4468 × 102	7.0149 × 102	9.3564 × 102	7.0823 × 102	7.1094 × 102	7.0044 × 102
	Std	9.5745 × 101	5.6495 × 101	1.9502 × 102	3.7700 × 101	1.9752 × 102	2.5075 × 101	9.1386 × 101	1.1498 × 102
C5	Avg	5.0001 × 102	5.0001 × 102	5.0000 × 102	5.0001 × 102	5.0000 × 102	5.0000 × 102	5.0000 × 102	5.0000 × 102
	Std	3.7772 × 10−3	2.3979 ×10−3	6.4887 ×10−4	2.3814 × 10−3	5.4175 × 10−4	2.2044 ×10−3	3.7297 × 10−4	6.9273 ×10−4
C6	Avg	1.7879 × 104	1.7396 × 104	1.6526 × 103	1.8762 × 104	1.2941 × 103	1.5934 × 104	1.2950 × 103	1.2752 × 103
	Std	6.1741 × 103	6.7002 × 103	1.3508 × 103	7.7204 × 103	6.6282 × 102	5.2768 × 103	5.8936 × 102	6.1957 × 102
C7	Avg	7.0083 × 102	7.0071 × 102	7.0004 × 102	7.0068 × 102	7.0005 × 102	7.0068 × 102	7.0005 × 102	7.0003 × 102
	Std	3.5697 ×10−1	2.7847 ×10−1	3.6018 ×10−2	2.9402 ×10−1	8.8472 ×10−2	2.3890 × 10−1	5.3145 × 10−2	2.9987 ×10−2
C8	Avg	8.0277 × 102	8.0250 × 102	8.0487 × 102	8.0249 × 102	8.0527 × 102	8.0241 × 102	8.0511 × 102	8.0499 × 102
	Std	1.3856	1.5187	2.6096	1.5990	2.9744	1.8517	2.7299	2.2539
C9	Avg	7.3005 × 103	7.3907 × 103	4.1465 × 103	7.1449 × 103	4.4234 × 103	7.0424 × 103	4.2585 × 103	4.1004 × 103
	Std	4.3602 × 102	3.8286 × 102	5.6254 × 102	4.2455 × 102	5.4676 × 102	3.4822 × 102	6.4869 × 102	5.4201 × 102
C10	Avg	3.9199 × 104	3.6181 × 104	1.5403 × 104	3.4553 × 104	1.4217 × 104	3.6592 × 104	1.2353 × 104	1.1901 × 104
	Std	1.2805 × 104	1.3472 × 104	9.1676 × 103	1.4831 × 104	1.0931 × 104	1.0431 × 104	1.1680 × 104	8.2810 × 103
C11	Avg	1.2252 × 106	9.1551 × 105	1.6417 × 105	1.5265 × 106	6.8589 × 104	1.3514 × 106	6.3596 × 104	3.8149 × 104
	Std	7.6481 × 105	5.8350 × 105	2.1649 × 105	1.1761 × 106	6.4327 × 104	1.0733 × 106	5.6365 × 104	4.6922 × 104
C12	Avg	7.3457 × 104	6.6576 × 104	1.4459 × 104	4.7929 × 104	1.8177 × 104	3.7313 × 104	7.8905 × 103	1.1056 × 104
	Std	7.8754 × 104	1.3851 × 105	1.0431 × 104	4.5719 × 104	1.7478 × 104	1.9321 × 104	4.6206 × 103	7.0545 × 103
C13	Avg	1.7090 × 105	1.8175 × 105	1.0097 × 105	9.2290 × 104	9.0676 × 104	1.4162 × 105	7.4771 × 104	6.9638 × 104
	Std	1.1397 × 105	1.9385 × 105	8.0831 × 104	3.9448 × 104	7.1650 × 104	7.9404 × 104	6.1910 × 104	6.5020 × 104
C14	Avg	5.7612 × 104	4.9170 × 104	2.0255 × 104	4.8448 × 104	2.1059 × 104	6.2560 × 104	1.6190 × 104	2.1737 × 104
	Std	1.4278 × 104	1.3861 × 104	8.8639 × 103	9.5594 × 103	8.5659 × 103	4.2966 × 104	6.7286 × 103	9.7892 × 103
C15	Avg	1.6265 × 104	7.1302 × 103	2.0061 × 103	3.1516 × 103	1.6731 × 103	3.7963 × 103	1.6126 × 103	1.7085 × 103
	Std	3.3044 × 104	1.6317 × 104	1.1779 × 103	3.3980 × 103	1.8548 × 102	3.8027 × 103	1.3024 × 102	2.8131 × 102
C16	Avg	2.2022 × 104	1.7469 × 104	3.2599 × 103	1.5299 × 104	3.1529 × 103	1.5055 × 104	2.9677 × 103	2.9455 × 103
	Std	6.8442 × 103	9.0722 × 103	5.2573 × 102	7.7648 × 103	5.7551 × 102	6.1570 × 103	5.2382 × 102	4.7358 × 102
C17	Avg	5.0255 × 104	5.5203 × 104	4.5333 × 104	4.5838 × 104	4.1659 × 104	4.4074 × 104	3.4293 × 104	3.2910 × 104
	Std	1.4945 × 104	2.6911 × 104	1.6060 × 104	1.3878 × 104	1.7484 × 104	1.3326 × 104	1.4101 × 104	9.3176 × 103
C18	Avg	4.0902 × 104	2.7445 × 104	1.2381 × 104	3.3524 × 104	1.7734 × 104	2.9905 × 104	1.1319 × 104	1.5701 × 104
	Std	1.6376 × 104	1.0639 × 104	1.1387 × 104	1.3826 × 104	1.7282 × 104	1.4633 × 104	9.5168 × 103	1.6555 × 104
C19	Avg	2.5782 × 103	2.5301 × 103	2.3289 × 103	2.5655 × 103	2.3650 × 103	2.4028 × 103	2.3083 × 103	2.3037 × 103
	Std	2.4538 × 102	2.6290 × 102	1.6885 × 102	2.4476 × 102	2.2067 × 102	2.1638 × 102	2.1499 × 102	2.0642 × 102
C20	Avg	2.4578 × 103	2.3281 × 103	2.2396 × 103	2.3116 × 103	2.3930 × 103	2.3818 × 103	2.3627 × 103	2.3487 × 103
	Std	1.3734 × 102	1.3167 × 102	1.9097 × 102	1.3930 × 102	3.1281 × 102	1.3603 × 102	2.3971 × 102	2.1489 × 102
C21	Avg	2.2784 × 103	2.2793 × 103	2.3337 × 103	2.2713 × 103	2.3285 × 103	2.2734 × 103	2.3136 × 103	2.3028 × 103
	Std	9.4751	1.1339 × 101	3.1230 × 101	6.3853	2.9848 × 101	6.2075	2.0347 × 101	1.6939 × 101
C22	Avg	3.1312 × 103	2.9414 × 103	2.4850 × 103	2.7710 × 103	2.5408 × 103	2.8307 × 103	2.4510 × 103	2.4306 × 103
	Std	2.8441 × 102	1.8950 × 102	1.3745 × 102	1.5331 × 102	2.9750 × 102	1.3740 × 102	1.4068 × 102	1.0901 × 102
C23	Avg	2.9433 × 103	2.7844 × 103	2.5174 × 103	2.6920 × 103	2.5519 × 103	2.7255 × 103	2.5466 × 103	2.5284 × 103
	Std	1.8589 × 102	1.0357 × 102	7.6801 × 101	7.9883 × 101	1.4378 × 102	1.0368 × 102	1.3322 × 102	9.3699 × 101
C24	Avg	2.9002 × 103	2.8903 × 103	2.8539 × 103	2.8807 × 103	2.8611 × 103	2.8911 × 103	2.8422 × 103	2.8426 × 103
	Std	2.8783 × 101	2.2431 × 101	2.5660 × 101	2.4677 × 101	3.2787 × 101	2.7917 × 101	1.3900 × 101	1.2849 × 101
C25	Avg	3.3863 × 103	3.3773 × 103	3.3455 × 103	3.3625 × 103	3.3615 × 103	3.3521 × 103	3.3470 × 103	3.3432 × 103
	Std	3.9417 × 101	2.9357 × 101	3.3370 × 101	3.4726 × 101	3.1748 × 101	2.3228 × 101	2.6757 × 101	3.6732 × 101
C26	Avg	3.2004 × 103	3.1946 × 103	3.2132 × 103	3.1967 × 103	3.2235 × 103	3.1936 × 103	3.1844 × 103	3.1944 × 103
	Std	1.6952 × 101	1.8429 × 101	6.4648 × 101	1.6895 × 101	3.6434 × 101	1.8249 × 101	2.5351 × 101	3.3505 × 101
C27	Avg	3.0372 × 103	2.9975 × 103	2.8909 × 103	2.9337 × 103	2.8871 × 103	2.9227 × 103	2.8075 × 103	2.8258 × 103
	Std	1.1031 × 102	1.3155 × 102	1.3857 × 102	1.2429 × 102	1.3604 × 102	1.1592 × 102	1.1710 × 102	1.0628 × 102
C28	Avg	8.7037 × 105	4.9433 × 105	4.4510 × 105	2.9292 × 105	9.6290 × 104	7.0605 × 105	2.1704 × 105	4.7328 × 104
	Std	1.6842 × 106	1.5240 × 106	1.3889 × 106	3.4502 × 105	2.3445 × 105	1.3067 × 106	8.3157 × 105	1.2348 × 105
C29	Avg	7.5306 × 105	4.4785 × 105	1.3950 × 106	6.1289 × 105	2.0616 × 106	2.0096 × 105	7.4609 × 105	3.9230 × 105
	Std	1.7793 × 106	1.0977 × 106	3.3368 × 106	1.3506 × 106	5.9157 × 106	9.1967 × 104	2.5543 × 106	1.0849 × 106

Bold is the best result of all algorithms.

shows the results of these CPO variants, with Avg and Std being the average and standard deviation of 30 independent experiments of each algorithm on each function, respectively. The algorithm with the best performance is highlighted in bold.

Based on the experimental results in Table 6, the nonparametric Friedman test was performed to obtain the ranking of each algorithm. The results of the Friedman test are presented in

Table 7. Friedman test results on benchmark functions (C1–C29).

	Overall Rank	+/=/−	Average Rank
CPO	7	27/0/2	7.2759
CCPO	8	26/0/3	6.1034
ACPO	3	24/0/5	3.7931
PCPO	6	26/0/3	5.1034
CACPO	4	26/0/3	4.1379
CPCPO	5	25/0/4	5.0345
APCPO	2	19/0/10	2.5172
CAPCPO	1	~	2.0345

, where ‘+/=/−’ indicates the number of benchmark problems where CAPCPO’s performance is superior, inferior, or equal to other algorithms.

From Table 7, the performance of the eight CPO variants from best to worst is: CAPCPO > APCPO > ACPO > CACPO > CPCPO > PCPO > CPO > CCPO. The CAPCPO algorithm which contains the three strategies ranks first, indicating that adding these three strategies simultaneously has the greatest improvement in the performance of the original CPO.

4.3. Qualitative Analysis

The qualitative analysis for the performance of CAPCPO is discussed in this section. The benchmark functions in this section contain four unimodal functions (F1, F3, F4, F6) and four multimodal functions (F8, F9, F10, F14). The details and complete information of these functions are stated in research work of Yao et al. [56].

In this qualitative analysis, four well-known metrics are used to visualize the performance of CAPCPO: (1) search history; (2) trajectory of first dimension; (3) average fitness; (4) convergence curve. The benchmark functions and the results on them are shown in Figure 3.

The search history in Figure 3 (second column) shows the position of each CP in the search space during the iterations, where the green dot represents the location of the global optimum and the black dots represent the locations of the candidate best solutions during the iterations. On the four unimodal functions (F1, F3, F4, F6), most black dots are clustered near the global optimum, indicating that CAPCPO has strong exploitation ability and can achieve fast convergence. On the four multimodal functions (F8, F9, F10, F14), where are obvious multiple local optima, black dots are widely distributed in the search space, which means that CAPCPO has strong exploration ability. These black dots eventually converge near the global optimum, which means that CAPCPO can jump out of the local optimum and converge to the global optimum.

The trajectory of first dimension, as seen in the third column in Figure 3, shows the primary exploratory behavior of CAPCPO. From the trajectory changes, it can be seen that the trajectory of first dimension presents a wide range of fluctuations in early iterations. In the later iterations, it tends to stabilize and eventually settles at the global optimum position, ensuring convergence.

The fourth column in Figure 3 shows the changes in average fitness during iterations. Among them, rapid decline occurred in the initial iterations on all functions, indicating CAPCPO converged quickly when dealing with both unimodal and multimodal problems.

The fifth column in Figure 3 shows the convergence curve of CAPCPO during the iteration process. It can be observed that CAPCPO converges rapidly during the initial iterations, consistently exhibiting a downward trend. This indicates that CAPCPO is capable of widely searching for optimal solutions within the search space and effectively balancing exploration and exploitation.

4.4. IEEE CEC2017: Results and Analysis

In this section, we quantitatively compared our proposed CAPCPO with 11 metaheuristic algorithms on CEC2017 test suite. The comparative experiments are divided into the following perspectives: (1) Comparing their exploitation capabilities on unimodal functions (C1–C2); (2) comparing their exploration capabilities on multimodal functions (C3–C9); (3) comparing the local optimum avoidance capabilities on hybrid functions (C10–C19) and composition functions (C20–C29); (4) scalability analysis to compare their ability in handling high-dimensional optimization problems.

Below are detailed discussions of these comparative experiments.

4.4.1. Exploitation Ability Analysis

To assess the exploitation ability of CAPCPO, we compared it with 11 other metaheuristic algorithms on two unimodal functions (C1–C2). The quantitative results on these two unimodal functions are shown in

Table 8. Comparison results on unimodal functions (C1–C2, d = 30).

ID	Method	CPO	PSO	GWO	WOA	SCA	SOA	SSA	BWO	DBO	GJO	POA	CAPCPO
C1	Avg	3.37 × 107	5.01 × 108	2.76 × 109	1.55 × 1010	1.97 × 1010	5.33 × 1010	1.86 × 1010	7.14 × 109	7.96 × 109	2.07 × 1010	6.05 × 1010	7.50 × 103
	Std	2.62 × 107	1.99 × 109	1.66 × 109	4.92 × 109	3.88 × 109	9.44 × 109	4.68 × 109	1.38 × 109	7.43 × 109	3.40 × 109	6.14 × 109	3.95 × 103
C2	Avg	7.54 × 103	1.24 × 104	2.09 × 104	4.89 × 104	4.39 × 104	1.13 × 105	5.12 × 104	5.79 × 104	4.41 × 104	6.35 × 103	1.08 × 105	2.02 × 102
	Std	2.46 × 103	1.22 × 104	6.21 × 103	8.33 × 103	6.88 × 103	2.17 × 104	1.19 × 104	1.05 × 104	2.58 × 104	7.86 × 103	3.22 × 104	4.62

Bold is the best result of all algorithms.

. It can be seen from Table 8 that the Avg and Std of CAPCPO are the smallest, indicating that CAPCPO outperforms competitive algorithms in the exploitation ability.

Based on the quantitative statistical results in Table 8, a Friedman test was conducted and the results are shown in

Table 9. Results of Friedman test on unimodal functions (C1–C2, d = 30).

Algorithms	Overall Rank	Average Rank
CPO	2	2
PSO	3	3
GWO	4	4
WOA	6	7
SCA	6	7
SOA	11	11.5
SSA	9	8
BWO	6	7
DBO	5	6
GJO	10	10
POA	12	11.5
CAPCPO	1	1

, where overall rank represents the final ranking of all algorithms and average rank represents the average ranking of all algorithms. It can be seen from Table 9 that CAPCPO ranks first.

Table 10. p-values on unimodal functions (C1–C2, d = 30).

ID	Ours vs. CPO	Ours vs. PSO	Ours vs. GWO	Ours vs. WOA	Ours vs. SCA	Ours vs. SOA	Ours vs. SSA	Ours vs. BWO	Ours vs. DBO	Ours vs. GJO	Ours vs. POA
C1	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11
C2	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11

presents the results of the Wilcoxon signed rank test between CAPCPO and other algorithms on unimodal functions, where all p-values are less than 0.05, indicating a significant difference between CAPCPO and other algorithms.

The convergence curves of the 12 algorithms on the two unimodal functions (C1, C2) are shown in Figure 4. Evidently, CAPCPO has the fastest convergence rate and the smallest fitness value among all algorithms.

This results in this subsection indicate that CAPCPO stands out with the best exploitation ability among all 12 algorithms.

4.4.2. Exploration Ability Analysis

In this subsection, we tested the exploration ability of CAPCPO on seven multimodal functions (C3–C9). Quantitative results are presented in

Table 11. Comparison results on multimodal functions (C3–C9, d = 30).

ID	Method	CPO	PSO	GWO	WOA	SCA	SOA	SSA	BWO	DBO	GJO	POA	CAPCPO
C3	Avg	4.41 × 102	1.43 × 103	5.55 × 102	1.55 × 103	3.28 × 103	1.50 × 104	2.76 × 103	1.14 × 103	2.10 × 103	2.07 × 103	1.86 × 104	3.77 × 102
	Std	2.94 × 101	1.05 × 103	1.32 × 102	5.35 × 102	6.58 × 102	4.27 × 103	1.43 × 103	2.17 × 102	1.87 × 103	6.59 × 102	4.17 × 103	4.31 × 101
C4	Avg	7.99 × 102	1.30 × 104	3.97 × 103	1.55 × 104	2.63 × 104	7.11 × 104	3.04 × 104	9.23 × 103	2.02 × 104	2.09 × 104	8.87 × 104	7.00 × 102
	Std	9.57 × 101	1.00 × 104	2.36 × 103	4.78 × 103	4.58 × 103	1.20 × 104	6.82 × 103	2.11 × 103	1.05 × 104	6.09 × 103	6.72 × 103	1.15 × 102
C5	Avg	5.00 × 102	5.00 × 102	5.00 × 102	5.00 × 102	5.00 × 102	5.00 × 102	5.00 × 102	5.00 × 102	5.00 × 102	5.00 × 102	5.00 × 102	5.00 × 102
	Std	3.78 × 10-3	2.33 × 10−3	2.88 × 10−3	4.78 × 10−3	1.01 × 10−2	1.02 × 10−2	5.08 × 10−3	5.97 × 10−3	1.24 × 10−2	5.07 × 10−3	1.18 × 10−2	6.93 × 10−4
C6	Avg	1.79 × 104	1.36 × 104	1.53 × 104	3.38 × 104	4.97 × 104	5.38 × 104	1.68 × 104	4.35 × 104	3.19 × 104	3.08 × 104	7.46 × 104	1.28 × 103
	Std	6.17 × 103	1.53 × 104	8.88 × 103	9.83 × 103	1.52 × 104	1.14 × 104	5.93 × 103	1.24 × 104	1.12 × 104	6.97 × 103	1.45 × 104	6.20 × 102
C7	Avg	7.01 × 102	7.00 × 102	7.00 × 102	7.01 × 102	7.03 × 102	7.03 × 102	7.01 × 102	7.02 × 102	7.02 × 102	7.01 × 102	7.04 × 102	7.00 × 102
	Std	3.57 × 10-1	4.32 × 10−1	6.08 × 10−1	3.79 × 10−1	8.01 × 10−1	8.38 × 10−1	4.56 × 10−1	6.60 × 10−1	1.20	3.50 × 10−1	1.09	3.00 × 10−2
C8	Avg	8.03 × 102	8.11 × 102	8.05 × 102	8.14 × 102	8.19 × 102	8.35 × 102	8.18 × 102	8.11 × 102	8.17 × 102	8.12 × 102	8.30 × 102	8.05 × 102
	Std	1.39	5.85	3.09	5.65	6.27	8.21	3.72	3.38	1.12 × 101	2.50	6.64	2.25
C9	Avg	7.30 × 103	5.77 × 103	4.67 × 103	6.95 × 103	8.45 × 103	9.24 × 103	5.92 × 103	7.96 × 103	7.34 × 103	7.35 × 103	8.13 × 103	4.10 × 103
	Std	4.36 × 102	1.05 × 103	1.18 × 104	6.54 × 102	2.87 × 102	5.59 × 102	9.04 × 102	5.02 × 102	5.45 × 102	4.06 × 102	3.34 × 102	5.42 × 102

. Based on these results, a Friedman test was carried out and the results are shown in

Table 12. Results of Friedman test on multimodal functions (C3–C9, d = 30).

Algorithms	Overall Rank	Average Rank
CPO	3	3.71
PSO	3	3.71
GWO	2	2.57
WOA	5	6.29
SCA	10	10.29
SOA	11	11.00
SSA	6	6.29
BWO	7	6.57
DBO	9	7.86
GJO	8	7.00
POA	12	11.57
CAPCPO	1	1.14

.

From Table 11, CAPCPO ranks first on six benchmark functions (C3–C7, C9) and third on one (C8). As a result, CAPCPO ranks first among all 12 algorithms, as can be seen in Table 12.

Table 13. p-values on multimodal functions (C3–C9, d = 30).

ID	Ours vs. CPO	Ours vs. PSO	Ours vs. GWO	Ours vs. WOA	Ours vs. SCA	Ours vs. SOA	Ours vs. SSA	Ours vs. BWO	Ours vs. DBO	Ours vs. GJO	Ours vs. POA
C3	4.99 × 10−7	5.23 × 10−11	1.04 × 10−10	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11
C4	2.32 × 10−3	8.56 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11
C5	6.37 × 10−11	1.63 × 10−4	2.56 × 10−2	3.51 × 10−11	2.87 × 10−11	2.87 × 10−11	2.73 × 10−10	3.18 × 10−11	3.88 × 10−11	3.18 × 10−11	2.87 × 10−11
C6	2.87 × 10−11	5.77 × 10−11	4.73 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11
C7	2.87 × 10−11	6.42 × 10−10	1.15 × 10−6	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	1.69 × 10−10	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11
C8	2.18 × 10−4	1.84 × 10−4	3.57 × 10−1	8.70 × 10−8	3.01 × 10−10	2.87 × 10−11	6.37 × 10−11	4.27 × 10−6	1.11 × 10−7	5.31 × 10−8	2.87 × 10−11
C9	2.87 × 10−11	4.37 × 10−9	2.46 × 10−2	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	1.94 × 10−9	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11

offers the results of the Wilcoxon signed rank test on the multimodal functions. Obviously, the majority of the p-values are below 0.05, indicating a significant difference between CAPCPO and the other algorithms.

Figure 5 gives the convergence curves of all algorithms on multimodal functions. CAPCPO has the best optimization quality among six benchmark functions (C3–C7, C9).

The results in this subsection shows that CAPCPO outperforms the competitive algorithms in exploration ability.

4.4.3. Local Optimum Avoidance Ability Analysis

In this subsection, to evaluate the local optima avoidance ability of our proposed CAPCPO, we conducted experiments on 10 hybrid functions (C10–C19) and 10 composition functions (C20–C29). The quantitative statistical results are presented in

Table 14. Comparison results on hybrid and composition functions (C10–C29, d = 30).

ID	Method	CPO	PSO	GWO	WOA	SCA	SOA	SSA	BWO	DBO	GJO	POA	CAPCPO
C10	Avg	3.92 × 104	1.52 × 106	1.22 × 105	3.21 × 105	2.71 × 106	8.50 × 107	2.07 × 105	5.34 × 105	2.45 × 106	1.30 × 106	9.55 × 106	1.19 × 104
	Std	1.28 × 104	7.21 × 106	3.39 × 104	3.99 × 105	1.75 × 106	1.02 × 108	1.37 × 105	2.81 × 105	2.48 × 106	1.68 × 106	7.59 × 106	8.28 × 103
C11	Avg	1.23 × 106	5.27 × 108	4.18 × 107	1.02 × 109	3.64 × 109	7.97 × 109	5.95 × 108	6.23 × 108	1.15 × 108	2.59 × 109	1.59 × 1010	3.81 × 104
	Std	7.65 × 105	7.28 × 108	2.86 × 107	7.94 × 108	1.11 × 109	3.51 × 109	7.83 × 108	2.03 × 108	1.35 × 108	1.09 × 109	3.93 × 109	4.69 × 104
C12	Avg	7.35 × 104	1.94 × 108	3.32 × 107	8.67 × 108	2.11 × 109	1.23 × 1010	1.01 × 109	2.95 × 108	5.56 × 108	2.03 × 109	1.54× 1010	1.11 × 104
	Std	7.88 × 104	5.65 × 108	5.08 × 107	4.39 × 108	7.05 × 108	4.49 × 109	1.06 × 109	1.01 × 108	8.24 × 108	7.14 × 108	3.61 × 109	7.05 × 103
C13	Avg	1.71 × 105	7.17 × 105	1.34 × 106	1.71 × 106	3.05 × 106	1.93 × 107	4.65 × 106	1.88 × 106	1.62 × 106	4.90 × 106	1.04 × 106	6.96 × 104
	Std	1.14 × 105	5.38 × 105	1.21 × 106	1.29 × 106	2.00 × 106	1.98 × 107	2.87 × 106	1.30 × 106	1.86 × 106	3.67 × 106	7.39 × 105	6.50 × 104
C14	Avg	5.76 × 104	4.50 × 107	8.38 × 105	1.90 × 108	8.08 × 108	6.15 × 109	3.76× 108	6.05 × 107	1.54 × 108	5.62 × 108	7.87 × 109	2.17 × 104
	Std	1.43 × 104	1.03 × 108	8.20 × 105	1.94 × 108	3.29 × 108	3.76 × 109	4.01 × 108	3.80 × 107	3.50 × 108	3.75 × 108	2.80 × 109	9.79 × 103
C15	Avg	1.63 × 104	3.45 × 105	2.42 × 106	1.43 × 107	3.78 × 107	1.34 × 109	7.10 × 106	1.76 × 106	9.39 × 107	5.49 × 107	2.37 × 109	1.71 × 103
	Std	3.30 × 104	7.07 × 105	4.14 × 106	1.14 × 107	6.94 × 107	1.92 × 109	1.31 × 107	1.26 × 106	1.14 × 108	3.60 × 107	2.37 × 109	2.81 × 102
C16	Avg	2.20 × 104	2.05 × 109	6.48 × 104	2.88 × 107	7.37 × 109	1.87 × 1015	7.10 × 109	9.05 × 107	9.09 × 1010	1.05 × 1010	8.25 × 1011	2.95 × 103
	Std	6.84 × 103	4.00 × 109	2.27 × 104	7.78 × 107	1.02× 1010	5.98 × 1015	2.27 × 1010	1.99 × 108	4.69 × 1011	2.88 × 1010	2.95 × 1012	4.74 × 102
C17	Avg	5.03 × 104	9.72 × 105	1.18 × 106	2.36 × 105	1.15 × 106	1.25 × 106	1.37 × 105	1.14 × 106	4.25 × 106	4.89 × 106	3.96 × 105	3.29 × 104
	Std	1.49 × 104	2.00 × 106	1.96 × 106	1.71 × 105	1.41 × 106	8.54 × 105	5.16 × 104	7.86 × 105	6.66 × 106	4.88 × 106	2.34 × 105	9.32 × 103
C18	Avg	4.09 × 104	7.25 × 105	3.54 × 109	1.46× 1010	9.65 × 1010	2.99 × 1014	2.75 × 1010	2.68 × 109	6.76 × 1011	5.49 × 1011	5.53 × 1012	1.57 × 104
	Std	1.64 × 104	8.90 × 105	3.85 × 109	9.69 × 109	1.76 × 1011	5.74 × 1014	2.87 × 1010	1.63 × 109	1.76 × 1012	5.89 × 1011	1.56 × 1013	1.66 × 104
C19	Avg	2.58 × 103	3.92 × 103	2.80 × 103	5.28 × 103	5.52 × 103	1.01 × 104	1.07 × 104	4.52 × 103	4.95 × 103	5.46 × 103	1.17 × 104	2.30 × 103
	Std	2.45 × 102	7.31 × 102	3.40 × 102	1.05 × 103	1.21 × 103	2.56 × 103	4.11 × 103	5.69 × 102	2.20 × 103	9.06 × 102	2.69 × 103	2.06 × 102
C20	Avg	2.46 × 103	6.81 × 103	4.61 × 103	1.16 × 104	1.53 × 104	5.00 × 104	2.22 × 104	5.97 × 103	1.06 × 104	1.36 × 104	2.53 × 104	2.35 × 103
	Std	1.37 × 102	4.18 × 103	1.82 × 103	4.01 × 103	6.39 × 103	8.55 × 103	7.36 × 103	1.93 × 103	6.03 × 103	6.15 × 103	1.08 × 104	2.15 × 102
C21	Avg	2.28 × 103	2.92 × 103	2.31 × 103	2.49 × 103	2.64 × 103	4.92 × 103	4.55 × 103	2.41 × 103	2.76 × 103	2.54 × 103	4.41 × 103	2.30 × 103
	Std	9.48	4.68 × 102	1.96 × 101	5.74 × 101	6.82 × 101	1.01 × 103	7.52 × 102	2.75 × 101	3.00 × 102	6.20 × 101	1.14 × 103	1.69 × 101
C22	Avg	3.13 × 103	1.97 × 104	8.63 × 103	1.90 × 104	2.64 × 104	6.19 × 104	3.28 × 104	1.40 × 104	1.85 × 104	2.45× 104	5.71 × 104	2.43 × 103
	Std	2.84 × 102	8.77 × 103	3.04 × 103	3.87 × 103	5.02 × 103	8.15 × 103	9.82 × 103	1.95 × 103	6.28 × 103	3.60 × 103	9.96 × 103	1.09 × 102
C23	Avg	2.94 × 103	8.98 × 104	6.33 × 104	1.36 × 104	1.70 × 104	3.38 × 104	1.97 × 104	1.03 × 104	1.11 × 104	1.65 × 104	3.34 × 104	2.53 × 103
	Std	1.86 × 102	4.61 × 103	1.96 × 103	2.30 × 103	1.99 × 103	2.74 × 103	5.33 × 103	1.26 × 103	2.06 × 103	2.58 × 103	2.88 × 103	9.37 × 101
C24	Avg	2.90 × 103	3.23 × 103	3.00 × 103	3.47 × 103	3.91 × 103	7.13 × 103	3.92 × 103	3.20 × 103	3.13 × 103	3.70 × 103	7.65 × 103	2.84 × 103
	Std	2.88 × 101	2.70 × 102	8.57 × 101	2.74 × 102	3.88 × 102	9.91 × 102	3.51 × 102	8.92 × 101	2.94 × 102	2.10 × 102	8.47 × 102	1.28 × 101
C25	Avg	3.39 × 103	3.61 × 103	3.46 × 103	3.72 × 103	4.40 × 103	1.00 × 104	6.52 × 103	3.44 × 103	3.74 × 103	4.40 × 103	9.58 × 103	3.34 × 103
	Std	3.94 × 101	2.39 × 102	8.02 × 101	1.33 × 102	3.40 × 102	3.28 × 103	2.10 × 103	4.98 × 101	3.29 × 102	3.45 × 102	2.86 × 103	3.67 × 101
C26	Avg	3.20 × 103	3.17 × 103	3.16 × 103	3.36 × 103	3.47 × 103	4.12 × 103	4.28 × 103	3.19 × 103	3.27 × 103	3.53 × 103	3.53 × 103	3.19 × 103
	Std	1.70 × 101	1.62 × 102	2.72 × 101	6.58 × 101	6.98 × 101	3.07 × 102	4.04 × 102	2.02 × 101	1.14 × 102	9.74 × 101	1.34 × 102	3.35 × 101
C27	Avg	3.04 × 103	4.32 × 103	3.20 × 103	3.49 × 103	3.94 × 103	6.72 × 103	4.19 × 103	3.38 × 103	4.04 × 103	3.90 × 103	8.15 × 103	2.83 × 103
	Std	1.10 × 102	6.72 × 102	5.44E+01	1.47 × 102	2.19 × 102	1.25 × 103	6.18 × 102	6.51 × 101	3.52 × 102	1.92 × 102	1.26 × 103	1.06 × 102
C28	Avg	8.70 × 105	7.28 × 108	2.58 × 108	9.59 × 108	7.60 × 109	1.00 × 1013	2.22 × 109	4.77 × 108	4.85 × 108	3.67 × 109	3.79 × 1011	4.73 × 104
	Std	1.68 × 106	1.44 × 109	2.73 × 108	6.24 × 108	7.18 × 109	1.95 × 1013	2.58 × 109	3.65 × 108	3.64 × 108	2.18 × 109	6.86 × 1011	1.23 × 105
C29	Avg	7.53 × 105	6.74 × 107	4.89 × 108	2.71 × 109	5.67 × 109	1.07 × 1012	4.40 × 109	6.66 × 108	1.06 × 1010	4.70 × 109	8.31 × 1010	3.92 × 105
	Std	1.78 × 106	3.35 × 108	5.38 × 108	1.27 × 109	4.20 × 109	2.41 × 1012	4.93 × 109	5.11 × 108	1.60 × 1010	1.99 × 109	9.69× 1010	1.08 × 106

Bold is the best result of all algorithms.

. Results of Friedman test on hybrid and composition functions are shown in

Table 15. Results of Friedman test on hybrid and composition functions (C10–C29, d = 30).

Algorithms	Overall Rank	Average Rank
CPO	2	2.1
PSO	5	5.15
GWO	3	3.6
WOA	6	6.25
SCA	10	8.8
SOA	12	11.45
SSA	8	8.35
BWO	4	4.85
DBO	7	7.15
GJO	9	8.5
POA	11	10.6
CAPCPO	1	1.2

.

From Table 14, CAPCPO has the lowest fitness values in 18 out of the 20 functions. As a result, CAPCPO ranks first on C10–C29, as shown in Table 15.

The p-values of the Wilcoxon signed rank test on C10–C29 are shown in

Table 16. p-values on hybrid and composition functions (C10–C29, d = 30).

ID	Ours vs. CPO	Ours vs. PSO	Ours vs. GWO	Ours vs. WOA	Ours vs. SCA	Ours vs. SOA	Ours vs. SSA	Ours vs. BWO	Ours vs. DBO	Ours vs. GJO	Ours vs. POA
C10	4.84 × 10−10	5.77 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11
C11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11
C12	1.19 × 10−10	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11
C13	9.18 × 10−7	1.40 × 10−10	1.40 × 10−10	3.51 × 10−11	2.87 × 10−11	2.87 × 10−11	3.51 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11
C14	2.26 × 10−10	4.29 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11
C15	2.79 × 10−9	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11
C16	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11
C17	2.23 × 10−6	5.29 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11
C18	1.67 × 10−6	2.33 × 10−9	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11
C19	1.73 × 10−4	4.73 × 10−11	1.02 × 10−7	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11
C20	4.60 × 10−2	3.88 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11
C21	1.20 × 10−7	2.87 × 10−11	8.17 × 10−2	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11
C22	5.77 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11
C23	1.87 × 10−10	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11
C24	7.03 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11
C25	1.54 × 10−4	8.49 × 10−10	1.94 × 10−9	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	4.29 × 10−11	4.73 × 10−11	2.87 × 10−11	2.87 × 10−11
C26	1.17 × 10−1	3.71 × 10−5	2.76 × 10−4	6.37 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	5.54 × 10−1	8.34 × 10−4	2.87 × 10−11	2.87 × 10−11
C27	2.10 × 10−8	2.87 × 10−11	5.23 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11
C28	1.02 × 10−9	2.49 × 10−8	5.23 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11
C29	2.20 × 10−5	1.84 × 10−4	4.73 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11

, in which a majority of p-values are below 0.05, suggesting a statistically significant disparity between CAPCPO and the competing algorithms.

The convergence curves of the 12 algorithms on the hybrid functions (C10–C19) and composition functions (C20–C29) are shown in Figure 6 and Figure 7, respectively. As evident from Figure 6, CAPCPO ranks first on 10 hybrid functions(C10–C19), with a particularly pronounced advantage on C10–C18. As illustrated in Figure 7, CAPCPO ranks first on eight composition functions (C20, C22–C25 and C27–C29), second on C21, and third on C26. Over all, CAPCPO outperforms the competing algorithms on hybrid and composition functions. This is attributed to the three newly added strategies, which can enhance the algorithm’s ability to avoid local optima.

The results of this subsection show that, among the 12 algorithms, CAPCPO has the best local optimum avoidance ability.

4.4.4. Scalability Analysis

In this subsection, to assess the scalability of CAPCPO, the dimension of the benchmark functions (d) from CEC2017 is set to 100. The quantitative statistical results are shown in

Table 17. Comparison results on benchmark functions (C1–C29, d = 100).

ID	Method	CPO	PSO	GWO	WOA	SCA	SOA	SSA	BWO	DBO	GJO	POA	CAPCPO
C1	Avg	4.18 × 1010	6.75 × 1010	5.45 × 1010	1.60 × 1011	2.10 × 1011	2.70 × 1011	2.25 × 1011	1.45 × 1011	8.83 × 1010	1.73 × 1011	2.72 × 1011	4.22 × 109
	Std	7.34 × 109	3.63 × 1010	1.20 × 1010	1.75 × 1010	1.35 × 1010	1.22 × 1010	1.50 × 1010	1.00 × 1010	1.82 × 1010	9.40 × 109	8.28 × 109	2.52 × 109
C2	Avg	1.54 × 105	1.57 × 105	1.36 × 105	2.40 × 105	2.95 × 105	4.36 × 105	3.34 × 105	3.31 × 105	3.95 × 105	2.71 × 105	4.78 × 105	5.26 × 104
	Std	1.37 × 104	7.20 × 104	1.73 × 104	2.49 × 104	2.65 × 104	9.78 × 104	2.03 × 104	2.53 × 104	1.02 × 105	1.13 × 104	1.16 × 105	1.30 × 104
C3	Avg	5.14 × 103	3.79 × 104	5.55 × 103	2.78 × 104	6.00 × 104	1.30 × 105	8.07 × 104	2.80 × 104	2.14 × 104	3.48 × 104	1.37 × 105	1.68 × 103
	Std	1.19 × 103	1.05 × 104	1.39 × 103	7.46 × 103	7.53 × 103	1.97 × 104	1.45 × 104	4.22 × 103	8.91 × 103	4.42 × 103	1.26 × 104	3.00 × 102
C4	Avg	4.29 × 104	1.68 × 105	5.34 × 104	1.56 × 105	2.32 × 105	3.43 × 105	2.74 × 105	1.47 × 105	9.47 × 104	1.70 × 105	3.49 × 105	9.20 × 103
	Std	8.83 × 103	3.29 × 104	1.16 × 104	1.93 × 104	1.89 × 104	1.86 × 104	2.49 × 104	1.42 × 104	2.74 × 104	8.85 × 103	1.29 × 104	2.48 × 103
C5	Avg	5.00 × 102	5.00 × 102	5.00 × 102	5.00 × 102	5.00 × 102	5.00 × 102	5.00 × 102	5.00 × 102	5.00 × 102	5.00 × 102	5.00 × 102	5.00 × 102
	Std	9.15 × 10−3	1.10 × 10−2	1.45 × 10−3	5.38 × 10−3	1.30 × 10−2	9.24 × 10−3	1.10 × 10−2	7.70 × 10−3	3.04 × 10−2	5.59 × 10−3	1.27 × 10−2	1.03 × 10−3
C6	Avg	7.42 × 104	5.65 × 104	7.32 × 104	7.18 × 104	1.25 × 105	1.02 × 105	5.26 × 104	9.64 × 104	9.52 × 104	7.60 × 104	1.30 × 105	1.19 × 104
	Std	9.19 × 103	2.81 × 104	1.79 × 104	9.78 × 103	2.14 × 104	1.12 × 104	9.79 × 103	8.74 × 103	1.30 × 104	1.03 × 104	1.28 × 104	7.14 × 103
C7	Avg	7.05 × 102	7.02 × 102	7.01 × 102	7.04 × 102	7.09 × 102	7.07 × 102	7.04 × 102	7.06 × 102	7.08 × 102	7.05 × 102	7.09 × 102	7.00 × 102
	Std	7.61 × 10−1	1.76	1.89	9.11 × 10−1	1.40	5.37 × 10−1	1.33	7.12 × 10−1	1.16	6.51 × 10−1	1.36	1.10 × 10−1
C8	Avg	8.81 × 102	9.47 × 102	8.68 × 102	9.79 × 102	1.07 × 103	1.12 × 103	1.04 × 103	9.86 × 102	1.01 × 103	9.80 × 102	1.09 × 103	8.62 × 102
	Std	1.64 × 101	5.88 × 101	1.54 × 101	1.98 × 101	3.27 × 101	3.16 × 101	2.37 × 101	3.16 × 101	3.73 × 101	1.78 × 101	2.44 × 101	9.17
C9	Avg	3.15 × 104	2.92 × 104	2.07 × 104	2.94 × 104	3.35 × 104	3.29 × 104	2.59 × 104	3.31 × 104	3.15 × 104	3.11 × 104	3.27 × 104	1.70 × 104
	Std	6.36 × 102	2.05 × 103	4.94 × 103	1.12 × 103	5.94 × 102	8.62 × 102	1.57 × 103	5.25 × 102	1.45 × 103	8.34 × 102	6.63 × 102	1.72 × 103
C10	Avg	3.75 × 105	1.36 × 107	1.80 × 107	5.07 × 108	1.02 × 109	1.36 × 1010	5.83 × 109	1.15 × 108	1.22 × 109	9.58 × 108	1.38 × 1010	1.56 × 105
	Std	1.28 × 105	2.07 × 107	2.78 × 107	3.27 × 108	4.61 × 108	6.24 × 109	2.66 × 109	6.28 × 107	1.95 × 109	5.51 × 108	4.89 × 109	3.19 × 104
C11	Avg	3.10 × 109	7.29 × 1010	1.30 × 1010	6.11 × 1010	1.02 × 1011	2.05 × 1011	1.52 × 1011	4.63 × 1010	4.29 × 1010	7.71 × 1010	2.26 × 1011	1.81 × 108
	Std	1.05 × 109	1.89 × 1010	6.04 × 109	1.53 × 1010	1.39 × 1010	1.82 × 1010	2.23 × 1010	8.35 × 109	1.74 × 1010	8.96 × 109	8.30 × 109	8.44 × 107
C12	Avg	4.90 × 109	1.15 × 1011	1.88 × 1010	9.20 × 1010	1.65 × 1011	3.62 × 1011	2.51 × 1011	6.92 × 1010	6.09 × 1010	1.17 × 1011	3.87 × 1011	5.19 × 107
	Std	2.76 × 109	3.74 × 1010	1.04 × 1010	2.29 × 1010	2.14 × 1010	4.99 × 1010	4.00 × 1010	1.26 × 1010	4.27 × 1010	1.93 × 1010	3.12 × 1010	3.50 × 107
C13	Avg	7.03 × 106	4.99 × 107	6.47 × 106	2.50 × 107	9.54 × 107	2.99 × 108	1.87 × 108	4.44 × 107	3.37 × 107	4.20 × 107	1.79 × 108	1.03 × 106
	Std	4.38 × 106	3.42 × 107	4.36 × 106	1.21 × 107	2.89 × 107	1.29 × 108	1.12 × 108	1.62 × 107	1.54 × 107	2.13 × 107	8.20 × 107	4.54 × 105
C14	Avg	1.76 × 108	9.29 × 109	1.81 × 109	1.45 × 1010	2.54 × 1010	5.78 × 1010	4.07 × 1010	9.13 × 109	6.94 × 109	1.48 × 1010	6.36 × 1010	1.70 × 106
	Std	9.70 × 107	6.04 × 109	1.80 × 109	5.37 × 109	4.54 × 109	6.99 × 109	7.50 × 109	2.68 × 109	6.55 × 109	3.00 × 109	3.51 × 109	1.41 × 106
C15	Avg	1.33 × 105	3.33 × 108	7.98 × 107	1.85 × 109	3.08 × 109	1.69 × 1010	8.96 × 109	5.73 × 108	3.46 × 108	2.41 × 109	1.90 × 1010	2.33 × 104
	Std	7.03 × 104	1.11 × 109	1.70 × 108	1.07 × 109	1.14 × 109	3.79 × 109	2.26 × 109	3.45 × 108	4.28 × 108	1.18 × 109	3.35 × 109	9.04 × 103
C16	Avg	4.19 × 107	8.64 × 1013	8.64 × 108	1.20 × 1013	1.86 × 1014	8.96 × 1015	2.81 × 1015	4.77 × 1012	9.99 × 1012	2.46 × 1013	1.09 × 1016	4.33 × 104
	Std	1.92 × 108	1.41 × 1014	2.83 × 109	3.23 × 1013	1.61 × 1014	5.68 × 1015	3.77 × 1015	6.36 × 1012	3.44 × 1013	3.67 × 1013	5.14 × 1015	1.34 × 104
C17	Avg	3.81 × 106	4.92 × 107	7.93 × 106	2.52 × 107	8.16 × 107	8.31 × 108	2.53 × 108	5.36 × 107	7.20 × 107	5.44 × 107	9.05 × 107	1.31 × 106
	Std	2.29 × 106	6.37 × 107	7.22 × 106	1.38 × 107	2.96 × 107	7.08 × 108	3.21 × 108	2.34 × 107	9.68 × 107	3.71 × 107	5.59 × 107	6.27 × 105
C18	Avg	2.03 × 109	1.28 × 1013	1.09 × 1011	2.26 × 1013	4.70 × 1013	6.84 × 1015	1.23 × 1015	1.51 × 1012	2.70 × 1013	4.47 × 1013	5.66 × 1015	1.41 × 105
	Std	2.18 × 109	4.23 × 1013	2.01 × 1011	3.59 × 1013	5.00 × 1013	5.07 × 1015	8.89 × 1014	4.89 × 1012	7.93 × 1013	7.64 × 1013	2.58 × 1015	8.14 × 104
C19	Avg	6.56 × 103	1.33 × 104	6.44 × 103	1.78 × 104	2.00 × 104	3.62 × 104	4.14 × 104	1.39 × 104	1.71 × 104	1.56 × 104	4.09 × 104	5.20 × 103
	Std	1.21 × 103	3.34 × 103	1.65 × 103	3.37 × 103	3.45 × 103	4.62 × 103	4.09 × 103	1.87 × 103	3.91 × 103	2.46 × 103	3.41 × 103	6.27 × 102
C20	Avg	4.36 × 104	8.40 × 104	5.61 × 104	1.63 × 105	2.16 × 105	2.58 × 105	2.28 × 105	1.44 × 105	9.19 × 104	1.66 × 105	2.66 × 105	1.08 × 104
	Std	9.10 × 103	3.87 × 104	1.28 × 104	2.26 × 104	1.46 × 104	7.08 × 103	1.38 × 104	1.46 × 104	2.84 × 104	1.55 × 104	7.68 × 103	2.54 × 103
C21	Avg	2.94 × 103	1.70 × 104	3.55 × 103	7.60 × 103	1.38 × 104	2.74 × 104	2.48 × 104	4.94 × 103	1.10 × 104	9.37 × 103	3.08 × 104	3.02 × 103
	Std	1.24 × 102	2.88 × 103	2.77 × 103	2.30 × 103	1.83 × 103	2.93 × 103	2.01 × 103	7.02 × 102	1.02 × 104	1.00 × 103	1.61 × 103	2.90 × 102
C22	Avg	6.54 × 104	5.83 × 104	4.70 × 104	1.13 × 105	1.28 × 105	1.30 × 105	1.23 × 105	1.07 × 105	7.28 × 104	1.09 × 105	1.30 × 105	1.33 × 104
	Std	1.43 × 104	1.85 × 104	9.79 × 103	3.18 × 103	5.87 × 103	1.95 × 103	2.57 × 103	2.31 × 103	1.37 × 104	8.08 × 103	1.42 × 103	4.79 × 103
C23	Avg	7.90 × 104	7.08 × 104	6.47 × 104	1.42 × 105	1.69 × 105	1.84 × 105	1.70 × 105	1.33 × 105	9.07 × 104	1.38 × 105	1.87 × 105	2.24 × 104
	Std	1.47 × 104	2.69 × 104	1.26 × 104	1.12 × 104	9.16 × 103	4.17 × 103	5.62 × 103	3.85 × 103	2.52 × 104	6.71 × 103	2.31 × 103	5.50 × 103
C24	Avg	6.51 × 103	1.45 × 104	6.65 × 103	1.39 × 104	2.55 × 104	4.36 × 104	2.98 × 104	1.45 × 104	9.10 × 103	1.57 × 104	4.89 × 104	4.72 × 103
	Std	5.65 × 102	4.84 × 103	6.75 × 102	2.09 × 103	3.17 × 103	5.69 × 103	3.78 × 103	1.40 × 103	1.22 × 103	1.52 × 103	3.68 × 103	2.91 × 102
C25	Avg	1.37 × 104	6.05 × 103	1.31 × 104	3.20 × 104	8.36 × 104	2.84 × 105	1.57 × 105	1.10 × 104	2.00 × 104	7.94 × 104	1.88 × 105	1.05 × 104
	Std	1.81 × 103	4.16 × 102	2.05 × 103	8.66 × 103	1.17 × 104	9.97 × 104	5.68 × 104	2.43 × 103	8.79 × 103	1.26 × 104	6.52 × 104	1.39 × 103
C26	Avg	4.40 × 103	5.09 × 103	4.05 × 103	5.81 × 103	7.55 × 103	1.13 × 104	1.09 × 104	4.95 × 103	4.72 × 103	7.00 × 103	7.07 × 103	4.37 × 103
	Std	1.47 × 102	2.16 × 103	1.58 × 102	3.47 × 102	5.58 × 102	1.25 × 103	1.32 × 103	2.49 × 102	5.76 × 102	3.63 × 102	7.82 × 102	2.14 × 102
C27	Avg	4.41 × 103	1.32 × 104	4.72 × 103	9.01 × 103	1.60 × 104	2.90 × 104	2.13 × 104	9.23 × 103	7.28 × 103	1.15 × 104	3.14 × 104	3.45 × 103
	Std	3.11 × 102	2.89 × 103	4.20 × 102	1.11 × 103	2.20 × 103	3.29 × 103	2.79 × 103	1.08 × 103	1.59 × 103	8.47 × 102	3.52 × 103	1.14 × 102
C28	Avg	3.80 × 109	7.05 × 1012	3.48 × 1010	7.42 × 1013	1.49 × 1014	1.02 × 1016	2.05 × 1015	4.06 × 1012	3.82 × 1013	1.76 × 1014	1.37 × 1016	1.73 × 108
	Std	3.33 × 109	3.44 × 1013	2.64 × 1010	9.18 × 1013	1.32 × 1014	7.42 × 1015	1.94 × 1015	6.40 × 1012	6.45 × 1013	1.58 × 1014	8.04 × 1015	1.54 × 108
C29	Avg	4.44 × 109	2.09 × 1011	1.78 × 1011	6.13 × 1013	1.36 × 1014	6.11 × 1015	1.13 × 1015	1.97 × 1012	2.41 × 1014	1.17 × 1014	4.43 × 1015	4.58 × 107
	Std	2.88 × 109	3.71 × 1011	5.27 × 1011	7.71 × 1013	1.79 × 1014	4.87 × 1015	7.73 × 1014	2.69 × 1012	5.73 × 1014	8.96 × 1013	3.02 × 1015	8.13 × 107

Bold is the best result of all algorithms.

. The algorithm with the best performance is highlighted in bold.

As shown in Table 17, CAPCPO outperforms the competitive algorithms for 26 out of 29 test functions.

Based on Table 17, a Friedman test was conducted and the average rank of each algorithm is presented in Figure 8. The overall rank is shown in Figure 9. Obviously, CAPCPO ranks first among 12 algorithms.

Table 18. p-values on benchmark functions (C1–C29, d = 100).

ID	Ours vs. CPO	Ours vs. PSO	Ours vs. GWO	Ours vs. WOA	Ours vs. SCA	Ours vs. SOA	Ours vs. SSA	Ours vs. BWO	Ours vs. DBO	Ours vs. GJO	Ours vs. POA
C1	2.87 × 10−11	4.27 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11
C2	2.87 × 10−11	1.02 × 10−9	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11
C3	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11
C4	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11
C5	2.87 × 10−11	2.87 × 10−11	6.80 × 10−8	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11
C6	2.87 × 10−11	8.56 × 10−11	3.51 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11
C7	2.87 × 10−11	2.87 × 10−11	7.44 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11
C8	3.70 × 10−6	3.18 × 10−11	1.24 × 10−1	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11
C9	2.87 × 10−11	2.87 × 10−11	1.38 × 10−5	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11
C10	3.18 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11
C11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11
C12	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11
C13	4.73 × 10−11	4.73 × 10−11	8.56 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11
C14	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11
C15	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11
C16	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11
C17	8.86 × 10−9	1.15 × 10−10	4.78 × 10−9	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.74 × 10−10	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11
C18	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11
C19	3.21 × 10−6	2.87 × 10−11	3.27 × 10−4	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11
C20	2.87 × 10−11	3.18 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11
C21	6.05 × 10−1	2.87 × 10−11	5.95 × 10−1	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	3.88 × 10−11	7.03 × 10−11	2.87 × 10−11	2.87 × 10−11
C22	3.18 × 10−11	4.29 × 10−11	3.88 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11
C23	2.87 × 10−11	6.81 × 10−9	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11
C24	3.18 × 10−11	2.87 × 10−11	3.51 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11
C25	2.79 × 10−9	2.87 × 10−11	7.89 × 10−7	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	6.36 × 10−1	1.93 × 10−8	2.87 × 10−11	2.87 × 10−11
C26	4.33 × 10−1	3.88 × 10−4	7.32 × 10−7	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	6.42 × 10−10	1.27 × 10−3	2.87 × 10−11	2.87 × 10−11
C27	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11
C28	4.29 × 10−11	2.76 × 10−4	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11
C29	2.87 × 10−11	3.18 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11

presents the results of the Wilcoxon signed rank test on CEC2017 (d = 100), in which a majority of p-values are below 0.05, suggesting a statistically significant disparity between CAPCPO and the competing algorithms.

The box plots for the 12 algorithms on unimodal, multimodal, hybrid, and composition functions are displayed in Figure 10, Figure 11, Figure 12 and Figure 13, respectively. CAPCPO ranks first in all cases except for the second place on three composition functions (C21, C25, and C26).

4.5. IEEE CEC2019: Results and Analysis

In this subsection, we further quantitatively compared CAPCPO with 11 metaheuristic algorithms on CEC2019 test suite. The results are shown in

Table 19. Comparison results on CEC2019 (C30–C39).

ID	Method	CPO	PSO	GWO	WOA	SCA	SOA	SSA	BWO	DBO	GJO	POA	CAPCPO
C30	Avg	3.21 × 103	5.82 × 105	9.30 × 104	8.05 × 102	6.27 × 106	6.62 × 106	7.37 × 102	5.41 × 102	3.48 × 106	5.76 × 102	6.16 × 102	2.91 × 102
	Std	1.07 × 104	2.58 × 106	1.59 × 105	9.70 × 102	9.80 × 106	1.02 × 107	1.15 × 102	9.24 × 101	6.59 × 106	4.33 × 102	3.17 × 102	9.46 × 101
C31	Avg	6.92 × 101	9.06 × 102	5.60 × 102	1.38 × 101	4.56 × 103	2.29 × 103	5.00	4.64	2.07 × 103	4.50	4.80	8.00
	Std	8.74 × 101	1.51 × 103	2.61 × 102	1.68 × 101	1.68 × 103	2.67 × 103	1.02× 10−10	3.43 × 10−1	3.30 × 103	3.29 × 10−1	1.07 × 10−1	1.01 × 101
C32	Avg	4.44	3.23	3.26	4.79	9.67	7.78	4.27	4.88	5.63	6.94	9.07	1.75
	Std	8.31 × 10−1	1.66	1.96	1.47	1.37	1.54	2.41	8.87 × 10−1	1.56	9.62 × 10−1	1.080	1.08
C33	Avg	3.12 × 101	4.05 × 101	9.14 × 101	6.40 × 102	1.63 × 103	1.10 × 104	1.36 × 102	1.67 × 102	1.76 × 103	1.36 × 103	1.27 × 104	1.88 × 101
	Std	6.59	2.92 × 101	1.75 × 102	3.98 × 102	9.02 × 102	5.14 × 103	6.35 × 101	3.34 × 101	1.41 × 103	1.10 × 103	3.96 × 103	5.23
C34	Avg	1.27	1.20	1.40	1.57	2.23	3.69	2.96	1.84	1.92	1.69	4.15	1.11
	Std	8.15 × 10−2	1.36 × 10−1	2.48 × 10−1	1.51 × 10−1	1.07 × 10−1	8.84 × 10−1	8.82 × 10−1	9.55 × 10−2	3.87 × 10−1	3.69 × 10−1	6.46 × 10−1	6.00 × 10−2
C35	Avg	9.74	8.09	1.10 × 101	9.64	1.08 × 101	1.16 × 101	9.88	1.08 × 101	1.07 × 101	9.99	1.01 × 101	8.06
	Std	6.88 × 10−1	1.37	4.56 × 10−1	1.14	8.10 × 10−1	7.30 × 10−1	7.80 × 10−1	7.75 × 10−1	9.06 × 10−1	1.67	7.24 × 10−1	1.11
C36	Avg	1.00	2.06 × 102	9.77 × 101	7.77 × 101	1.63 × 102	9.24 × 102	2.13 × 102	4.20 × 101	2.23 × 102	1.29 × 102	2.48 × 102	1.00
	Std	3.36 × 10−5	2.85 × 102	1.44 × 102	7.49 × 101	8.18 × 101	3.23 × 102	2.29 × 102	8.67 × 101	2.14 × 102	1.19 × 102	1.34 × 102	2.79 × 10−7
C37	Avg	1.00	1.00	1.00	1.04	1.06	1.24	1.00	1.00	1.10	1.08	1.49	1.00
	Std	6.73 × 10−9	1.59 × 10−7	7.61 × 10−3	2.18 × 10−2	1.56 × 10−2	1.36 × 10−1	1.07 × 10−4	1.38 × 10−3	1.21 × 10−1	2.92 × 10−2	1.73 × 10−1	0.00
C38	Avg	1.34	1.32	7.44	1.18 × 101	3.20 × 101	2.35 × 102	2.00	5.68	3.00	4.36 × 101	3.53 × 102	1.22
	Std	7.50 × 10−2	1.47 × 10−1	9.17	1.11 × 101	1.84 × 101	1.28 × 102	3.69 × 10−1	1.47	1.23	2.21 × 101	1.33 × 102	7.84 × 10−2
C39	Avg	2.07 × 101	2.11 × 101	2.15 × 101	2.13 × 101	2.15 × 101	2.17 × 101	2.12 × 101	2.07 × 101	2.13 × 101	2.10 × 101	2.14 × 101	1.98 × 101
	Std	3.46	1.32 × 10−1	1.15 × 10−1	1.83 × 10−1	7.04 × 10−2	5.37 × 10−2	1.03 × 10−1	2.46	8.51 × 10−1	8.45 × 10−1	9.10 × 10−2	4.73

Bold is the best result of all algorithms.

. CEC2019 includes 10 test functions. Except for C31, CAPCPO ranks first on the other nine benchmark functions.

The results of a Friedman test on CEC2019 are shown in

Table 20. Results of Friedman test on CEC2019 (C30–C39).

Algorithms	Overall Rank	Average Rank
CPO	2	3.7
PSO	3	4.5
GWO	8	6.7
WOA	6	5.9
SCA	11	9.6
SOA	12	11.3
SSA	5	5.6
BWO	4	5.1
DBO	9	8.7
GJO	7	6.2
POA	10	9.3
CAPCPO	1	1.4

. The average ranking of CAPCPO is 1.4, and the overall ranking is first.

The p-values of the Wilcoxon signed rank test on CEC2019 are shown in

Table 21. p-values on CEC2019 (C30–C39).

ID	Ours vs. CPO	Ours vs. PSO	Ours vs. GWO	Ours vs. WOA	Ours vs. SCA	Ours vs. SOA	Ours vs. SSA	Ours vs. BWO	Ours vs. DBO	Ours vs. GJO	Ours vs. POA
C30	1.09 × 10−3	1.35 × 10−9	1.15 × 10−10	2.23 × 10−6	2.87 × 10−11	2.87 × 10−11	5.77 × 10−11	5.84 × 10−11	3.88 × 10−11	4.27 × 10−6	9.44 × 10−8
C31	2.47 × 10−7	2.87 × 10−11	2.87 × 10−11	1.81 × 10−5	2.87 × 10−11	2.49 × 10−8	6.36 × 10−1	1.53 × 10−2	3.67 × 10−4	9.27 × 10−4	3.44 × 10−1
C32	4.78 × 10−9	7.39 × 10−8	2.78 × 10−6	1.11 × 10−7	2.87 × 10−11	7.03 × 10−11	1.31 × 10−7	2.33 × 10−9	5.84 × 10−10	4.29 × 10−11	3.18 × 10−11
C33	8.86 × 10−9	1.13 × 10−5	4.73 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11
C34	2.10 × 10−8	3.76 × 10−3	9.67 × 10−9	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11
C35	6.26 × 10−8	5.54 × 10−1	3.18 × 10−11	4.92 × 10−6	4.84 × 10−10	2.87 × 10−11	4.13 × 10−8	2.05 × 10−10	4.40 × 10−10	8.01 × 10−6	2.79 × 10−9
C36	3.18 × 10−11	3.51 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11
C37	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11
C38	3.66 × 10−7	1.05 × 10−2	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	3.18 × 10−11	2.87 × 10−11	1.27 × 10−10	2.87 × 10−11	2.87 × 10−11
C39	1.62 × 10−9	8.83 × 10−1	4.73 × 10−11	4.62 × 10−7	2.87 × 10−11	2.87 × 10−11	9.18 × 10−7	6.26 × 10−8	5.32 × 10−10	2.37 × 10−2	4.29 × 10−11

. The vast majority of p-values are less than 0.05, suggesting the CAPCPO has a statistically significant superiority over the competing algorithm.

The results of this subsection indicate that CAPCPO outperforms competitive algorithms on CEC2019.

4.6. IEEE CEC2022: Results and Analysis

To further test the performance of CAPCPO, this subsection presents comparative experiments on CEC2022, which contains 12 test functions, each with two dimensions: d = 10 and d = 20.

The quantitative results in both dimensions are shown in

Table 22. Comparison results on CEC2022 (C40–C51, d = 10).

ID	Method	CPO	PSO	GWO	WOA	SCA	SOA	SSA	BWO	DBO	GJO	POA	CAPCPO
C40	Avg	3.00 × 102	3.00 × 102	6.46 × 102	1.02 × 103	1.07 × 103	1.05 × 104	3.09 × 102	3.73 × 102	2.10 × 103	2.91 × 103	3.69 × 103	3.00 × 102
	Std	1.63 × 10−3	5.33 × 10−3	6.42 × 102	9.34 × 102	4.69 × 102	1.66 × 103	9.06	3.09 × 101	5.98 × 103	1.46 × 103	1.45 × 103	5.08 × 10−14
C41	Avg	4.01 × 102	4.81 × 102	4.34 × 102	4.45 × 102	4.81 × 102	1.01 × 103	4.40 × 102	4.26 × 102	5.42 × 102	4.85 × 102	1.10 × 103	4.03 × 102
	Std	2.31	4.97 × 101	2.86 × 101	2.17 × 101	2.95 × 101	3.98 × 102	3.18 × 101	1.32 × 101	1.30 × 102	1.51 × 101	5.26 × 102	3.32
C42	Avg	6.00 × 102	6.00 × 102	6.00 × 102	6.00 × 102	6.00 × 102	6.00 × 102	6.00 × 102	6.00 × 102	6.00 × 102	6.00 × 102	6.00 × 102	6.00 × 102
	Std	5.05 × 10−9	1.95 × 10−6	6.20 × 10−3	1.19 × 10−2	2.04 × 10−2	1.56 × 10−1	2.42 × 10−5	9.44 × 10−4	3.95 × 10−2	2.19 × 10−2	1.33 × 10−1	0.00
C43	Avg	8.00 × 102	8.00 × 102	8.00 × 102	8.01 × 102	8.01 × 102	8.01 × 102	8.00 × 102	8.01 × 102	8.01 × 102	8.01 × 102	8.01 × 102	8.00 × 102
	Std	1.11 × 10−1	3.28 × 10−1	1.64 × 10−1	2.06 × 10−1	1.94 × 10−1	5.12 × 10−1	1.97 × 10−1	1.98 × 10−1	2.71 × 10−1	2.98 × 10−1	1.85 × 10−1	8.68 × 10−2
C44	Avg	9.00 × 102	9.01 × 102	9.00 × 102	9.00 × 102	9.01 × 102	9.03 × 102	9.03 × 102	9.00 × 102	9.01 × 102	9.01 × 102	9.03 × 102	9.00 × 102
	Std	8.86 × 10−2	9.87 × 10−1	3.37 × 10−1	2.71 × 10−1	3.68 × 10−1	1.38	1.73	1.76 × 10−1	1.13	4.32 × 10−1	1.45	8.80 × 10−2
C45	Avg	6.73 × 103	3.68 × 104	3.82 × 104	2.25 × 104	6.68 × 106	3.65 × 106	4.02 × 104	3.49 × 105	2.12 × 106	1.97 × 104	1.00 × 105	3.23 × 103
	Std	3.77 × 103	4.73 × 104	1.83 × 104	1.28 × 104	3.86 × 106	3.51 × 106	1.68 × 104	2.23 × 105	1.07 × 107	3.84 × 103	4.56 × 104	2.03 × 103
C46	Avg	2.03 × 103	2.03 × 103	2.04 × 103	2.12 × 103	2.13 × 103	2.25 × 103	2.15 × 103	2.07 × 103	2.08 × 103	2.24 × 103	2.12 × 103	2.02 × 103
	Std	2.52	1.13 × 101	2.03 × 101	4.31 × 101	6.66 × 101	1.00 × 102	8.59 × 101	1.26 × 101	4.72 × 101	6.07 × 101	4.72 × 101	6.63
C47	Avg	2.22 × 103	2.66 × 103	2.84 × 103	2.84 × 103	3.43 × 103	3.31E+06	2.36 × 103	2.69 × 103	3.24 × 103	3.94 × 103	2.46 × 103	2.22 × 103
	Std	4.23	1.08 × 103	7.88 × 102	5.61 × 102	8.88 × 102	1.78E+07	1.56 × 102	3.28 × 102	6.74 × 102	3.47 × 102	2.57 × 102	9.14
C48	Avg	2.32 × 103	2.59 × 103	2.58 × 103	2.60 × 103	2.60 × 103	3.21 × 103	2.71 × 103	2.60 × 103	2.73 × 103	2.74 × 103	2.73 × 103	2.30 × 103
	Std	9.21 × 101	2.06 × 102	1.84 × 102	1.48 × 102	1.44 × 102	4.27 × 102	1.56 × 102	1.48 × 102	1.10 × 102	8.55 × 101	7.38 × 101	3.35 × 10−2
C49	Avg	2.63 × 103	2.70 × 103	2.67 × 103	2.63 × 103	2.64 × 103	3.00 × 103	2.90 × 103	2.66 × 103	2.71 × 103	2.70 × 103	2.76 × 103	2.62 × 103
	Std	5.54 × 101	2.29 × 102	6.78 × 101	4.72 × 101	5.47 × 101	5.05 × 102	4.34 × 102	7.66 × 101	7.86 × 101	7.85 × 101	2.94 × 102	4.96 × 101
C50	Avg	2.60 × 103	2.61 × 103	2.75 × 103	2.77 × 103	2.63 × 103	3.27 × 103	2.72 × 103	2.63 × 103	2.72 × 103	2.69 × 103	2.73 × 103	2.60 × 103
	Std	1.83 × 10−3	5.39 × 101	3.36 × 102	3.07 × 102	6.33	6.33 × 102	2.95 × 102	1.56 × 102	2.14 × 102	2.36 × 102	7.24 × 101	7.83 × 10−9
C51	Avg	2.87 × 103	2.89 × 103	2.87 × 103	2.87 × 103	2.88 × 103	3.03 × 103	3.01 × 103	2.87 × 103	2.92 × 103	2.88 × 103	2.90 × 103	2.87 × 103
	Std	1.85	1.24 × 101	2.05 × 101	2.93	3.18	1.33 × 102	9.34 × 101	4.58 × 10−1	5.74 × 101	1.27 × 101	1.83 × 101	2.13

Bold is the best result of all algorithms.

and

Table 23. Comparison results on CEC2022 (C40–C51, d = 20).

ID	Method	CPO	PSO	GWO	WOA	SCA	SOA	SSA	BWO	DBO	GJO	POA	CAPCPO
C40	Avg	4.13 × 102	6.54 × 102	3.53 × 103	1.18 × 104	9.74 × 103	4.04 × 104	7.93 × 103	2.31 × 104	9.08 × 103	1.51 × 104	4.62 × 104	3.00 × 102
	Std	1.75 × 102	1.25 × 103	2.15 × 103	4.21 × 103	2.40 × 103	1.46 × 104	3.14 × 103	6.97 × 103	8.91 × 103	2.83 × 103	1.53 × 104	1.09 × 10−4
C41	Avg	4.74 × 102	5.92 × 102	5.20 × 102	6.69 × 102	8.10 × 102	2.56 × 103	6.74 × 102	5.80 × 102	6.01 × 102	7.81 × 102	2.97 × 103	4.54 × 102
	Std	1.97 × 101	1.16 × 102	4.47 × 101	8.18 × 101	1.08 × 102	6.61 × 102	1.01 × 102	3.30 × 101	1.50 × 102	9.02 × 101	6.92 × 102	1.25 × 101
C42	Avg	6.00 × 102	6.00 × 102	6.00 × 102	6.00 × 102	6.00 × 102	6.01 × 102	6.00 × 102	6.00 × 102	6.00 × 102	6.00 × 102	6.02 × 102	6.00 × 102
	Std	7.88 × 10−6	2.21 × 10−1	5.63 × 10−2	7.85 × 10−2	1.10 × 10−1	3.32 × 10−1	1.13 × 10−1	2.44 × 10−2	1.75 × 10−1	1.27 × 10−1	1.95 × 10−1	6.38 × 10−11
C43	Avg	8.01 × 102	8.01 × 102	8.01 × 102	8.02 × 102	8.04 × 102	8.04 × 102	8.01 × 102	8.03 × 102	8.02 × 102	8.03 × 102	8.03 × 102	8.01 × 102
	Std	6.72 × 10−1	1.08	3.78 × 10−1	5.98 × 10−1	3.48 × 10−1	7.54 × 10−1	6.63 × 10−1	7.20 × 10−1	6.97 × 10−1	6.09 × 10−1	3.76 × 10−1	2.07 × 10−1
C44	Avg	9.01 × 102	9.03 × 102	9.01 × 102	9.03 × 102	9.04 × 102	9.10 × 102	9.03 × 102	9.02 × 102	9.04 × 102	9.03 × 102	9.06 × 102	9.00 × 102
	Std	5.56 × 10−1	2.39	8.67 × 10−1	1.33	9.55 × 10−1	4.28	1.48	9.89 × 10−1	2.13	6.70 × 10−1	1.71	4.44 × 10−1
C45	Avg	1.85 × 105	5.93 × 107	7.69 × 105	4.11 × 107	3.86 × 108	3.88 × 109	1.36 × 107	6.43 × 107	7.11 × 107	1.28 × 108	2.84 × 109	3.91 × 104
	Std	1.17 × 105	6.62 × 107	3.09 × 106	5.42 × 107	1.71 × 108	2.46 × 109	4.35 × 107	3.09 × 107	2.51 × 108	1.38 × 108	1.64 × 109	1.27 × 104
C46	Avg	2.07 × 103	2.21 × 103	2.08 × 103	2.54 × 103	2.95 × 103	4.29 × 103	3.53 × 103	2.49 × 103	2.51 × 103	2.66 × 103	4.95 × 103	2.04 × 103
	Std	2.24 × 101	1.70 × 102	4.05 × 101	2.88 × 102	4.06 × 102	1.34 × 103	8.45 × 102	2.23 × 102	4.40 × 102	2.96 × 102	1.12 × 103	1.39 × 101
C47	Avg	2.85 × 103	6.34 × 106	4.94 × 103	5.36 × 103	3.03 × 107	8.61× 1010	4.12 × 103	9.36 × 106	6.38 × 103	7.56 × 104	7.76 × 104	2.29 × 103
	Std	3.59 × 102	3.25 × 107	9.72 × 102	1.61 × 103	8.69 × 107	2.71× 1011	1.21 × 103	3.78 × 107	1.96 × 103	2.57 × 105	2.92 × 105	5.83 × 101
C48	Avg	2.67 × 103	3.30 × 103	2.70 × 103	2.92 × 103	3.15 × 103	5.66 × 103	3.29 × 103	2.76 × 103	3.10 × 103	3.08 × 103	3.53 × 103	2.65 × 103
	Std	6.86 × 101	3.35 × 102	4.58 × 101	8.75 × 101	1.26 × 102	1.29 × 103	3.39 × 102	4.42 × 101	3.26 × 102	1.26 × 102	3.29 × 102	4.57
C49	Avg	2.81 × 103	4.31 × 103	4.01 × 103	4.05 × 103	3.84 × 103	7.44 × 103	5.11 × 103	3.22 × 103	5.11 × 103	3.83 × 103	5.97 × 103	2.80 × 103
	Std	7.98 × 101	1.28 × 103	9.34 × 102	1.51 × 103	1.68 × 103	3.92 × 102	1.13 × 103	1.14 × 103	1.68 × 103	1.53 × 103	1.38 × 103	7.68 × 101
C50	Avg	2.60 × 103	3.02 × 103	2.69 × 103	2.70 × 103	3.23 × 103	8.53 × 103	3.56 × 103	2.64 × 103	3.39 × 103	2.82 × 103	8.89 × 103	2.60 × 103
	Std	1.74	6.14 × 102	2.79 × 102	7.33 × 101	7.39 × 102	3.32 × 103	8.99 × 102	2.47 × 101	8.26 × 102	1.58 × 102	4.04 × 103	2.50
C51	Avg	3.00 × 103	2.92 × 103	2.99 × 103	3.05 × 103	3.10 × 103	3.66 × 103	3.69 × 103	2.96 × 103	3.02 × 103	3.14 × 103	3.11 × 103	2.99 × 103
	Std	1.81 × 101	8.45 × 101	2.85 × 101	3.14 × 101	5.23 × 101	3.13 × 102	3.96 × 102	6.92	4.77 × 101	5.78 × 101	7.28 × 101	2.84 × 101

Bold is the best result of all algorithms.

. In the case of d = 10, CAPCPO ranks first on 10 benchmark functions (C40 and C42–C50), and in the case of d = 20, CAPCPO ranks first on 11 benchmark functions (C40–C50).

The results of the Friedman test on CEC2022 are shown in

Table 24. Results of Friedman test on CEC2022 (C40–C51, d = 10/20).

Algorithms	d = 10		d = 20
	Overall Rank	Average Rank	Overall Rank	Average Rank
CPO	2	2	2	2.42
PSO	4	5.17	5	5.92
GWO	5	5.33	3	3.42
WOA	6	6.17	6	6.33
SCA	8	7.67	10	8.83
SOA	12	11.75	12	11.5
SSA	7	7.17	7	7
BWO	3	5	4	5.5
DBO	10	8.67	8	7.5
GJO	9	8.5	8	7.5
POA	11	9.33	11	10.83
CAPCPO	1	1.25	1	1.25

. It is evident that CAPCPO ranks first.

The p-values of the Wilcoxon signed rank test in both dimensions are shown in

Table 25. p-values on CEC2022 (C40–C51, d = 10).

ID	Ours vs. CPO	Ours vs. PSO	Ours vs. GWO	Ours vs. WOA	Ours vs. SCA	Ours vs. SOA	Ours vs. SSA	Ours vs. BWO	Ours vs. DBO	Ours vs. GJO	Ours vs. POA
C40	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11
C41	4.51 × 10−1	2.79 × 10−9	1.12 × 10−9	4.40 × 10−10	2.87 × 10−11	2.87 × 10−11	8.70 × 10−8	4.73 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11
C42	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11
C43	1.04 × 10−1	2.87 × 10−2	2.69 × 10−3	2.79 × 10−9	2.87 × 10−11	2.87 × 10−11	3.96 × 10−5	6.24 × 10−9	1.20 × 10−7	7.03 × 10−11	1.04 × 10−10
C44	3.96 × 10−7	3.64 × 10−10	4.13 × 10−8	3.01 × 10−10	1.04 × 10−10	2.87 × 10−11	4.29 × 10−11	1.93 × 10−8	1.15 × 10−10	3.88 × 10−11	3.18 × 10−11
C45	1.55 × 10−6	1.23 × 10−9	3.51 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11
C46	5.77 × 10−11	7.44 × 10−9	2.33 × 10−9	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11
C47	8.01 × 10−6	1.15 × 10−10	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11
C48	4.40 × 10−10	7.03 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11
C49	6.68 × 10−1	2.46 × 10−4	9.78 × 10−4	3.21 × 10−6	2.23 × 10−6	2.13 × 10−9	4.89 × 10−8	2.47 × 10−7	9.44 × 10−8	1.93 × 10−8	6.28 × 10−7
C50	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11
C51	5.15 × 10−1	1.06 × 10−8	4.53 × 10−3	9.85 × 10−6	3.01 × 10−10	2.87 × 10−11	2.87 × 10−11	7.13 × 10−2	1.38 × 10−5	6.42 × 10−10	3.18 × 10−11

and

Table 26. p-values on benchmark functions (C40–C51, d = 20).

ID	Ours vs. CPO	Ours vs. PSO	Ours vs. GWO	Ours vs. WOA	Ours vs. SCA	Ours vs. SOA	Ours vs. SSA	Ours vs. BWO	Ours vs. DBO	Ours vs. GJO	Ours vs. POA
C40	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11
C41	2.35 × 10−5	1.04 × 10−10	4.78 × 10−9	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	6.37 × 10−11	2.87 × 10−11	2.87 × 10−11
C42	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11
C43	2.29 × 10−7	1.09 × 10−3	3.96 × 10−7	3.18 × 10−11	2.87 × 10−11	2.87 × 10−11	6.80 × 10−8	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11
C44	1.87 × 10−2	4.37 × 10−9	1.48 × 10−5	3.88 × 10−11	2.87 × 10−11	2.87 × 10−11	4.29 × 10−11	1.15 × 10−10	1.69 × 10−10	2.87 × 10−11	2.87 × 10−11
C45	4.37 × 10−9	7.44 × 10−9	1.11 × 10−7	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11
C46	4.37 × 10−9	1.54 × 10−10	5.82 × 10−7	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11
C47	1.69 × 10−10	1.69 × 10−10	2.26 × 10−10	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11
C48	5.84 × 10−10	2.87 × 10−11	4.01 × 10−10	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11
C49	9.53 × 10−1	1.80 × 10−7	3.80 × 10−8	6.24 × 10−9	6.26 × 10−8	2.87 × 10−11	1.27 × 10−10	1.66 × 10−7	7.04 × 10−10	1.77 × 10−9	1.54 × 10−10
C50	5.32 × 10−10	3.88 × 10−11	5.23 × 10−11	3.18 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11	2.87 × 10−11
C51	1.65 × 10−1	5.32 × 10−10	2.31 × 10−1	1.93 × 10−8	4.40 × 10−10	3.18 × 10−11	2.87 × 10−11	6.28 × 10−7	1.10 × 10−2	4.29 × 10−11	6.42 × 10−10

, respectively. The vast majority of p-values are less than 0.05, indicating a significant difference between CAPCPO and other algorithms.

4.7. Computational Effort Analysis

Thirteen test functions were selected from three test suites for memory usage analysis.

Table 27. Results of computational effort analysis.

ID	CPO	PSO	GWO	WOA	SCA	SOA	SSA	BWO	DBO	GJO	POA	CAPCPO
C1	2.965 MB	2.893 MB	2.875 MB	2.880 MB	2.870 MB	2.902 MB	2.882 MB	2.879 MB	2.889 MB	2.889 MB	2.883 MB	2.885 MB
C2	2.962 MB	2.891 MB	2.875 MB	2.879 MB	2.870 MB	2.901 MB	2.881 MB	2.878 MB	2.888 MB	2.888 MB	2.882 MB	2.884 MB
C3	2.886 MB	2.884 MB	2.867 MB	2.876 MB	2.868 MB	2.898 MB	2.870 MB	2.876 MB	2.886 MB	2.885 MB	2.881 MB	2.877 MB
C4	2.884 MB	2.884 MB	2.868 MB	2.877 MB	2.869 MB	2.899 MB	2.871 MB	2.876 MB	2.886 MB	2.886 MB	2.882 MB	2.877 MB
C10	3.474 MB	3.397 MB	3.379 MB	3.384 MB	3.375 MB	3.407 MB	3.387 MB	3.384 MB	3.394 MB	3.394 MB	3.387 MB	3.390 MB
C11	3.452 MB	3.437 MB	3.418 MB	3.429 MB	3.421 MB	3.450 MB	3.422 MB	3.428 MB	3.438 MB	3.437 MB	3.432 MB	3.429 MB
C20	3.577 MB	3.574 MB	3.554 MB	3.567 MB	3.559 MB	3.589 MB	3.559 MB	3.565 MB	3.575 MB	3.576 MB	3.567 MB	3.566 MB
C21	3.623 MB	3.540 MB	3.519 MB	3.528 MB	3.519 MB	3.550 MB	3.530 MB	3.526 MB	3.537 MB	3.537 MB	3.527 MB	3.533 MB
C33	2.507 MB	2.648 MB	2.640 MB	2.638 MB	2.636 MB	2.648 MB	2.431 MB	2.421 MB	2.428 MB	2.645 MB	2.639 MB	2.646 MB
C38	2.445 MB	2.441 MB	2.680 MB	2.439 MB	2.435 MB	2.446 MB	2.436 MB	2.437 MB	2.443 MB	2.442 MB	2.434 MB	2.439 MB
C40	2.506 MB	2.647 MB	2.640 MB	2.638 MB	2.636 MB	2.648 MB	2.431 MB	2.422 MB	2.428 MB	2.645 MB	2.417 MB	2.646 MB
C45	3.071 MB	3.056 MB	3.048 MB	3.054 MB	3.051 MB	3.062 MB	3.051 MB	3.053 MB	3.058 MB	3.057 MB	3.053 MB	3.055 MB
C48	2.791 MB	2.780 MB	2.772 MB	2.778 MB	2.774 MB	2.785 MB	2.775 MB	2.777 MB	2.781 MB	2.781 MB	2.777 MB	2.778 MB
Avg	3.011 MB	3.005 MB	3.010 MB	2.997 MB	2.991 MB	3.014 MB	2.964 MB	2.963 MB	2.972 MB	3.005 MB	2.982 MB	3.000 MB
Rank	11	9	10	6	5	12	2	1	3	8	4	7

presents the memory usage of each algorithm during the optimization process. CAPCPO ranked seventh among the 12 algorithms, which may be attributed to the additional calculations introduced by CACPO, resulting in higher computational effort.

5. CAPCPO for Solving Engineering Problems

In this section, CAPCPO was tested on seven real-world engineering challenges, including the welded beam design problem (WBD) [57], cantilever beam design problem (CBD) [58], step-cone pulley design problem (S-cPD) [59], pressure vessel design problem (PVD) [60], tension/compression spring design problem (T/CSD) [61], three-bar truss design problem (T-bTD) [62], and speed reducer design problem (SRD) [63]. These engineering optimization problems are different from numerical optimization problems as they require obtaining feasible solutions while satisfying multiple constraint conditions. Furthermore, various constraint handling methods have been developed, including static penalty, dynamic penalty, annealing penalty, adaptive penalty, and co-evolutionary penalty, among others [64]. This article utilizes the static penalty method for constraint handling, and the mathematical formulation for this method is presented in Equation (26).

$$F\left(X\right)=f\left(X\right)+\sum _{i=1}^m(R_{k,i}\times max\left[0,g_i\left(X\right){]}^2\right)$$

(26)

where $R_{k,i}$ are the penalty coefficients, $m$ is the total number of constraint conditions, and $g_i\left(X\right)$ are the constraints.

It should be noted that the algorithms participating in the comparative experiment are CPO, PSO, GWO, WOA, SCA, SOA, SSA, BWO, DBO, GJO, and POA. We set the population size of all algorithms to 30 and the maximum number of iterations to 500. To eliminate the influence of randomness, all algorithms were independently experimented 30 times, and the average (Avg), standard deviation (Std), the best (Best), the worst (Worst) were recorded. Then, the rank (Rank) of each algorithm was calculated.

5.1. The Welded Beam Design Problem

Figure 14 shows the schematic representation of WBD. This engineering optimization problem includes seven constraint conditions. The goal is to find the optimal set of design variables ($h, l, t, b$) that minimizes the weight of the welded beam. The mathematical model of this problem is shown below.

• Welded beam design’s mathematical model:
• Solution representation:
• Consider $X$ = [$x_1 x_2 x_3 x_4$] = [$h l t b$]
• Objective function:
• Minimize $f\left(X\right)=1.10471x_1^2{x}_2+0.04811x_3{x}_4(14.0+x_2)$
• Subject to seven constraints:

$$\begin{gathered} g_1\left(X\right)=\tau \left(\mathrm{X}\right)-{\tau }_{max}\le 0 \\ {g}_2\left(X\right)=\sigma \left(\mathrm{X}\right)-{\sigma }_{max}\le 0 \\ {g}_3\left(X\right)=\delta \left(\mathrm{X}\right)-{\delta }_{max}\le 0 \\ {g}_4\left(X\right)=x_1-x_4\le 0 \\ {g}_5\left(X\right)=P-P_c\left(X\right)\le 0 \\ {g}_6\left(X\right)=0.125-x_1\le 0 \\ {g}_7\left(X\right)=1.10471x_1^2+0.04811x_3{x}_4(14.0+x_2)-5.0\le 0 \end{gathered}$$

where:

$$\begin{gathered} \tau (X)=\sqrt{({\tau }^{\prime }{)}^2+2{\tau }^{\prime }{\tau }^{″}{\displaystyle \frac{x_2}{2R}}+({\tau }^{″}{)}^2} \\ {\tau }^{\prime }={\displaystyle \frac{P}{\sqrt{2}{x}_1{x}_2}},{\tau }^{″}={\displaystyle \frac{MR}{J}} \\ M=P(L+{\displaystyle \frac{x_2}{2}}) \\ R=\sqrt{{\displaystyle \frac{x_2^2}{4}}+({\displaystyle \frac{x_1+x_3}{2}}{)}^2} \\ J=2\left\{\sqrt{2}{x}_1{x}_2[{\displaystyle \frac{x_2^2}{4}}+({\displaystyle \frac{x_1+x_3}{2}}{)}^2]\right\} \\ \sigma (X)={\displaystyle \frac{6PL}{x_4{x}_3^2}},\delta (X)={\displaystyle \frac{6P{L}^3}{E{x}_3^2{x}_4}} \\ {P}_c(X)={\displaystyle \frac{4.0134E\sqrt{\frac{x_3^2{x}_4^6}{36}}}{L^2}}\left(1-{\displaystyle \frac{x_3}{2L}}\sqrt{{\displaystyle \frac{E}{4G}}}\right) \\ P=6000 \mathrm{lb},L=14\mathrm{i}\mathrm{n},E=30\times {10}^6 \mathrm{psi},G=12\times {10}^6 \mathrm{psi} \\ {\tau }_{max}=\mathrm{13,600} \mathrm{psi},{\sigma }_{max}=\mathrm{30,000} \mathrm{psi},{\delta }_{max}=0.25 \mathrm{in} \end{gathered}$$

The comparative experimental results are presented in

Table 28. Comparative results for WBD.

Algorithms	Optimum Variables				Avg	Std	Best	Worst	Rank
	${\mathit{x}}_1$	${\mathit{x}}_2$	${\mathit{x}}_3$	${\mathit{x}}_4$
CPO	0.1959	3.4280	9.1635	0.2008	1.6879	0.0142	1.6729	1.7355	4
PSO	0.5276	3.5517	5.6609	0.9236	4.9980	1.1798	3.1022	7.3282	12
BWO	0.1667	4.9576	9.2224	0.2226	1.9869	0.1521	1.7570	2.4451	9
DBO	0.1500	6.0893	9.7929	0.1964	1.9777	0.2724	1.6843	2.3661	8
SSA	0.2052	3.8108	8.9269	0.2215	1.8589	0.0966	1.7421	2.2221	7
WOA	0.1821	3.7605	9.2129	0.2007	1.7169	0.0332	1.6731	1.8218	5
GJO	0.1911	3.5179	9.1967	0.1993	1.6862	0.0084	1.6732	1.7066	3
GWO	0.1960	3.3971	9.1929	0.1989	1.6748	0.0029	1.6718	1.6853	2
POA	0.1493	6.3896	8.6371	0.2359	2.0825	0.2041	1.7401	2.7854	10
SCA	0.1904	3.7698	9.3089	0.2117	1.8289	0.0313	1.7572	1.8956	6
SOA	0.3120	4.0782	5.0601	0.7776	3.4799	0.9360	2.2115	7.1505	11
CAPCPO	0.1988	3.3377	9.1923	0.1988	1.6704	0.0003	1.6702	1.6717	1

Bold is the best result of all algorithms.

, from where it can be observed that CAPCPO ranks first among the 12 algorithms.

5.2. The Cantilever Beam Design Problem

The cantilever beam design (CBD) is an engineering optimization problem to minimize the beam’s weight while meeting five constraint conditions. The schematic representation of the CBD is shown in Figure 15. The mathematical model of the problem is as follows.

• Cantilever beam design’s mathematical model:
• Solution representation:
• Consider $X$ = [$x_1 x_2 x_3 x_{4 }{x}_{5 }$]
• Objective function:
• Minimize $f\left(X\right)=0.6224(x_1+x_2+x_3+x_4+x_5)$
• Subject to one constraint:

$$g(X)={\displaystyle \frac{61}{x_1^3}}+{\displaystyle \frac{37}{x_2^3}}+{\displaystyle \frac{19}{x_3^3}}+{\displaystyle \frac{7}{x_4^3}}+{\displaystyle \frac{1}{x_5^3}}-1\le 0$$

Variable range:

$$0.01\le x_1,x_2,x_3,x_4,x_5\le 100$$

Table 29. Comparative results for CBD.

Algorithms	Optimum Variables					Avg	Std	Best	Worst	Rank
	${\mathit{x}}_1$	${\mathit{x}}_2$	${\mathit{x}}_3$	${\mathit{x}}_4$	${\mathit{x}}_5$
CPO	6.0095	5.2959	4.5133	3.5030	2.1553	1.3402	0.0001	1.3400	1.3404	3
PSO	23.4032	32.8761	27.2915	24.2275	24.2139	8.2376	1.6362	5.4115	11.5122	12
BWO	6.0289	5.3495	4.4908	3.6945	2.2381	1.3604	0.0098	1.3453	1.3825	7
DBO	9.1370	5.2627	4.4891	3.5109	2.1764	1.5335	1.0346	1.3400	7.1049	9
SSA	6.0214	5.2657	4.5266	3.5120	2.2105	1.3439	0.0016	1.3413	1.3475	6
WOA	6.0690	5.3421	4.4856	3.4629	2.1718	1.3436	0.0021	1.3405	1.3495	5
GJO	6.0217	5.3009	4.5105	3.5062	2.1462	1.3407	0.0005	1.3402	1.3421	4
GWO	6.0207	5.3088	4.4970	3.4968	2.1528	1.3401	0.0001	1.3400	1.3404	2
POA	8.1202	7.5237	6.0123	4.7907	4.6840	1.9426	0.3488	1.4411	2.9371	10
SCA	6.1047	5.6379	4.7617	3.6495	2.4151	1.4083	0.0300	1.3444	1.4697	8
SOA	9.3020	8.0132	7.4332	8.0436	7.6762	2.5252	0.7384	1.3884	3.8386	11
CAPCPO	6.2239	5.5391	4.7528	3.7947	2.4896	1.3400	0.0000	1.3400	1.3401	1

Bold is the best result of all algorithms.

presents the comparative results, which show that CAPCPO can provide competitive results compared to other algorithms.

5.3. The Step-Cone Pulley Design Problem

The step-cone pulley design problem (S-cPD) is to find the optimal combination of five design variables under 11 constraints. Figure 16 shows the schematic representation of S-cPD. The mathematical model of the problem is as follows.

• Step-cone pulley design’s mathematical model:
• Solution representation:
• Consider $X$ = [$d_1 d_2 d_3 d_{4 }w$]
• Objective function:
• Minimize

$$\begin{array}{c}f\left(X\right)=\rho w[d_{\mathrm{i}}^2\left\{1+{\left({\displaystyle \frac{N_1}{N}}\right)}^2\right\}+d_2^2\left\{1+{\left({\displaystyle \frac{N_2}{N}}\right)}^2\right\}\\ +d_3^2\left\{\left.1+{\left({\displaystyle \frac{N_3}{N}}\right)}^2\right\}+d_4^2\left\{\left.1+{\left({\displaystyle \frac{N_4}{N}}\right)}^2\right\}\right]\right.\end{array}$$

Subject to 11 constraints:

$$\begin{array}{l}{h}_1\left(x\right)=C_1-C_2=0,h_2\left(x\right)=C_1-C_3=0,\\ h_3\left(x\right)=C_1-C_4=0,\\ g_{1.2.3.4}\left(x\right)=R_i\ge 2, g_{5.6.7.8}\left(x\right)=P_i\ge (0.75\ast 745.6998),\end{array}$$

where:

$$\begin{gathered} C_i={\displaystyle \frac{\pi d_i}{2}}\left(1+{\displaystyle \frac{N_i}{N}}\right)+{\displaystyle \frac{{\left(\frac{N_i}{N}-1\right)}^2}{4a}}+2a, i=(1,2,3,4) \\ {R}_i=\exp \left[\mu \left\{\pi -2{\sin }^{-1}\left\{\left({\displaystyle \frac{N_i}{N}}-1\right){\displaystyle \frac{d_i}{2a}}\right\}\right\}\right], i=(1,2,3,4) \\ {P}_i=stw\left[1-\mathrm{e}\mathrm{x}\mathrm{p}\left[-\mu \left\{\pi -2{\mathrm{s}\mathrm{i}\mathrm{n}}^{-1}\left\{\left({\displaystyle \frac{N_i}{N}}-1\right){\displaystyle \frac{d_i}{2a}}\right\}\right\}\right]\right]{\displaystyle \frac{\pi d_i{N}_i}{60}}, i=(1,2,3,4) \\ \rho =7200 \mathrm{k}\mathrm{g}/{\mathrm{m}}^3,a=3 \mathrm{m},\mu =0.35,s=1.75 \mathrm{M}\mathrm{P}\mathrm{a},t=8 \mathrm{m}\mathrm{m} \end{gathered}$$

Comparative experimental results are presented in

Table 30. Comparative results for S-cPD.

Algorithms	Optimum Variables					Avg	Std	Best	Worst	Rank
	${\mathit{x}}_1$	${\mathit{x}}_2$	${\mathit{x}}_3$	${\mathit{x}}_4$	${\mathit{x}}_5$
CPO	4.023 × 101	5.537 × 101	7.381 × 101	8.850 × 101	8.742 × 101	1.466 × 109	1.882 × 109	1.553 × 108	9.257 × 109	2
PSO	4.911 × 101	6.640 × 101	8.944 × 101	1.082 × 102	1.243 × 102	3.660 × 1016	3.507 × 1016	6.159 × 1014	1.137 × 1017	10
BWO	4.047 × 101	5.569 × 101	7.424 × 101	8.901 × 101	8.814 × 101	2.610 × 1012	3.505 × 1012	1.381 × 1011	1.593 × 1013	4
DBO	4.310 × 101	5.963 × 101	8.269 × 101	9.000 × 101	8.958 × 101	4.052 × 1016	3.813 × 1016	3.480 × 1011	8.403 × 1016	11
SSA	4.025 × 101	5.538 × 101	7.384 × 101	8.853 × 101	8.887 × 101	3.247 × 1012	2.335 × 1012	4.921 × 1011	9.268 × 1012	5
WOA	4.066 × 101	5.597 × 101	7.449 × 101	8.913 × 101	8.847 × 101	1.037 × 1015	3.404 × 1015	1.669 × 1011	1.677 × 1016	7
GJO	4.041 × 101	5.560 × 101	7.413 × 101	8.890 × 101	8.812 × 101	6.356 × 1012	6.049 × 1012	5.167 × 1011	3.367 × 1013	6
GWO	4.051 × 101	5.575 × 101	7.434 × 101	8.912 × 101	8.844 × 101	9.722 × 1011	7.246 × 1011	1.021 × 1011	2.967 × 1012	3
POA	3.997 × 101	5.526 × 101	7.327 × 101	8.781 × 101	8.675 × 101	1.231 × 1015	2.216 × 1015	6.752 × 1011	8.087 × 1015	8
SCA	4.136 × 101	5.814 × 101	7.500 × 101	8.979 × 101	8.979 × 101	1.973 × 1016	7.492 × 1015	3.895 × 1015	3.408 × 1016	9
SOA	5.073 × 101	5.833 × 101	8.734 × 101	8.878 × 101	8.878 × 101	7.825 × 1017	7.504 × 1017	6.480 × 1015	1.731 × 1018	12
CAPCPO	3.997 × 101	5.500 × 101	7.333 × 101	8.792 × 101	8.692 × 101	9.664 × 101	2.347 × 102	1.612 × 101	1.096 × 103	1

Bold is the best result of all algorithms.

, demonstrating that CAPCPO significantly outperforms other algorithms.

5.4. The Pressure Vessel Design Problem

The pressure vessel design (PVD) problem aims at minimizing total vessel cost while satisfying the constraint conditions, as illustrated in Figure 17. This engineering problem aims to find the optimal values for four design variables: the inner radius ($R$), the thickness ($T_s$) of the shell, the thickness ($T_h$) of the head, and the cylindrical portion length ($L$) under four constraints. The mathematical model of the problem is as follows.

• Pressure vessel design’s mathematical model:
• Solution representation:
• Consider $X$ = [$x_1 x_2 x_3 x_4$] = [$T_s T_h R L$]
• Objective function:
• 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 four constraints:

$$\begin{gathered} g_1\left(X\right)=-x_1+0.0193x_3\le 0 \\ {g}_2\left(X\right)=-x_2+0.00954x_3\le 0 \\ {g}_3\left(X\right)=-\pi x_3^2{x}_4-{\displaystyle \frac{4}{3}}\pi x_3^3+129600\le 0 \\ {g}_4\left(X\right)=x_4-240\le 0 \end{gathered}$$

Variable range:

$$\begin{gathered} 0\le x_1\le 99 \\ 0\le x_2\le 99 \\ 10\le x_3\le 200 \\ 10\le x_4\le 200 \end{gathered}$$

Table 31. Comparative results for PVD.

Algorithms	Optimum Variables				Avg	Std	Best	Worst	Rank
	${\mathit{x}}_1$	${\mathit{x}}_2$	${\mathit{x}}_3$	${\mathit{x}}_4$
CPO	8.498 × 10−1	4.253 × 10−1	4.380 × 101	1.603 × 102	6.084 × 103	1.108 × 102	5.931 × 103	6.396 × 103	3
PSO	1.059 × 101	4.267 × 101	7.209 × 101	1.116 × 102	6.689 × 105	3.549 × 105	2.408 × 105	1.592 × 106	12
BWO	9.098 × 10−1	5.128 × 10−1	4.433 × 101	1.654 × 102	6.893 × 103	3.936 × 102	6.231 × 103	7.804 × 103	5
DBO	9.452 × 10−1	7.047	4.862 × 101	1.673 × 102	3.158 × 104	7.760 × 104	5.887 × 103	2.913 × 105	10
SSA	1.136	5.757 × 10−1	5.763 × 101	5.451 × 101	7.096 × 103	3.665 × 102	6.138 × 103	7.643 × 103	6
WOA	1.251	9.561 × 10−1	5.508 × 101	1.104 × 102	1.647 × 104	3.062 × 104	5.913 × 103	1.677 × 105	9
GJO	1.056	5.362 × 10−1	5.452 × 101	7.952 × 101	6.725 × 103	5.139 × 102	5.972 × 103	7.387 × 103	4
GWO	8.460 × 10−1	4.212 × 10−1	4.381 × 101	1.643 × 102	6.058 × 103	2.921 × 102	5.897 × 103	7.164 × 103	2
POA	1.193	6.046 × 10−1	6.141 × 101	3.134 × 101	7.184 × 103	2.947 × 102	6.302 × 103	7.406 × 103	7
SCA	1.101	5.906 × 10−1	5.398 × 101	9.224 × 101	7.638 × 103	8.047 × 102	6.350 × 103	9.469 × 103	8
SOA	3.336	1.359 × 101	5.587 × 101	6.499 × 101	1.188 × 105	1.567 × 105	9.814 × 103	6.442 × 105	11
CAPCPO	8.300 × 10−1	4.107 × 10−1	4.300 × 101	1.685 × 102	5.989 × 103	1.163 × 102	5.886 × 103	6.325 × 103	1

Bold is the best result of all algorithms.

lists the optimal results of 12 algorithms for PVD, and CAPCPO ranks the first.

5.5. The Tension/Compression Spring Design Problem

The tension/compression spring design (T/CSD) is an optimization problem to minimize the weight of tension/compression spring with constraints. Figure 18 shows the schematic representation of T/CSD. There are three variables that require to be optimized: spring wire diameter ($d$), spring coil diameter ($D$), and the number of active coils ($P$). The mathematical model of the problem is presented below.

• Tension/compression spring design’s mathematical model:
• Solution representation:
• Consider $X$ = [$x_1 x_2 x_3$] = [$d D P$]
• Objective function:
• Minimize $f\left(X\right)=(x_3+2)x_2{x}_1^2$
• Subject to four constraints:

$$\begin{gathered} g_1\left(X\right)=1-{\displaystyle \frac{x_2^3{x}_3}{71785x_1^4}}\le 0 \\ {g}_2\left(X\right)={\displaystyle \frac{4x_2^2-x_1{x}_2}{12566(x_2{x}_1^3-x_1^4)}}+{\displaystyle \frac{1}{5108x_1^2}}-1\le 0 \\ {g}_3\left(X\right)=1-{\displaystyle \frac{140.45x_1}{x_2^2{x}_3}}\le 0 \\ {g}_4\left(X\right)=\begin{array}{c}{\displaystyle \frac{x_1+x_2}{1.5}}-1\le 0\end{array} \end{gathered}$$

Variable range:

$$\begin{gathered} 0.05x_1\le 2 \\ 0.25\le x_2\le 1.3 \\ 2\le x_3\le 15 \end{gathered}$$

Table 32. Comparative results for T/CSD.

Algorithms	Optimum Variables			Avg	Std	Best	Worst	Rank
	${\mathit{x}}_1$	${\mathit{x}}_2$	${\mathit{x}}_3$
CPO	5.255 × 10−2	3.784 × 10−1	1.040 × 101	1.276 × 10−2	7.703 × 10−5	1.268 × 10−2	1.299 × 10−2	3
PSO	1.432 × 10−1	9.151 × 10−1	1.043 × 101	2.842 × 1019	3.781 × 1019	1.939 × 10−2	9.808 × 1019	12
BWO	5.036 × 10−2	3.198 × 10−1	1.443 × 101	1.323 × 10−2	3.668 × 10−4	1.291 × 10−2	1.516 × 10−2	8
DBO	5.872 × 10−2	5.959 × 10−1	8.600	7.604 × 1017	4.095 × 1018	1.276 × 10−2	2.281 × 1019	11
SSA	5.055 × 10−2	3.261 × 10−1	1.387 × 101	1.312 × 10−2	1.324 × 10−4	1.279 × 10−2	1.325 × 10−2	6
WOA	5.197 × 10−2	3.627 × 10−1	1.183 × 101	1.304 × 10−2	2.330 × 10−4	1.271 × 10−2	1.346 × 10−2	5
GJO	5.364 × 10−2	4.058 × 10−1	9.967	1.321 × 10−2	5.536 × 10−4	1.275 × 10−2	1.514 × 10−2	7
GWO	5.204 × 10−2	3.668 × 10−1	1.123 × 101	1.276 × 10−2	7.094 × 10−5	1.268 × 10−2	1.298 × 10−2	4
POA	5.224 × 10−2	3.710 × 10−1	1.080 × 101	1.276 × 10−2	8.668 × 10−5	1.268 × 10−2	1.305 × 10−2	2
SCA	5.127 × 10−2	3.454 × 10−1	1.333 × 101	1.327 × 10−2	4.061 × 10−4	1.279 × 10−2	1.493 × 10−2	9
SOA	5.739 × 10−2	5.070 × 10−1	6.400	1.380 × 10−2	4.113 × 10−4	1.316 × 10−2	1.474 × 10−2	10
CAPCPO	1.762 × 10−1	5.080 × 10−1	9.258	1.273 × 10−2	8.662 × 10−5	1.267 × 10−2	1.296 × 10−2	1

Bold is the best result of all algorithms.

presents the experimental results for T/CSD. Based on these results, CAPCPO ranks first.

5.6. The Three-Bar Truss Design Problem

The objective of the three-bar truss design(T-bTD) problem is to minimize the volume of the three-bar truss by adjusting two design variables ($x_1=A_1=A_3,x_2=A_2$). Schematic representation of the T-bTD is shown in Figure 19. The mathematical model for this problem is presented as follows.

• Three bar truss design’s mathematical model:
• Solution representation:
• Consider $X$ = [$x_1 x_2$]
• Objective function:
• Minimize $f\left(X\right)=\left(2\sqrt{2}{x}_1+x_2\right)l$
• Subject to three constraints:

$$\begin{gathered} f\left(X\right)={\displaystyle \frac{\sqrt{2}{x}_1+x_2}{\sqrt{2}{x}_1^2+2x_1{x}_2}}P-\mathsf{\sigma }\le 0 \\ f\left(X\right)={\displaystyle \frac{x_2}{\sqrt{2}{x}_1^2+2x_1{x}_2}}P-\mathsf{\sigma }\le 0 \\ f\left(X\right)={\displaystyle \frac{1}{\sqrt{2}{x}_2+x_1}}P-\mathsf{\sigma }\le 0 \end{gathered}$$

Variable range:

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

where:

$$l=100 \mathrm{c}\mathrm{m},P=2 \mathrm{k}\mathrm{N}/\mathrm{c}{\mathrm{m}}^2,\mathsf{\sigma }=2 \mathrm{k}\mathrm{N}/\mathrm{c}{\mathrm{m}}^2$$

Table 33. Comparative results for T-bTD.

Algorithms	Optimum Variables		Avg	Std	Best	Worst	Rank
	${\mathit{x}}_1$	${\mathit{x}}_2$
CPO	7.886 × 10−1	4.084 × 10−1	2.639 × 102	4.341 × 10−5	2.639 × 102	2.639 × 102	2
PSO	7.807 × 10−1	4.630 × 10−1	2.671 × 102	2.095	2.644 × 102	2.732 × 102	9
BWO	7.894 × 10−1	4.070 × 10−1	2.640 × 102	8.877 × 10−2	2.639 × 102	2.642 × 102	6
DBO	7.899 × 10−1	4.050 × 10−1	2.639 × 102	6.106 × 10−2	2.639 × 102	2.642 × 102	5
SSA	7.838 × 10−1	4.274 × 10−1	2.644 × 102	4.145 × 10−1	2.639 × 102	2.660 × 102	7
WOA	8.324 × 10−1	3.354 × 10−1	2.690 × 102	5.131	2.640 × 102	2.828 × 102	10
GJO	8.036 × 10−1	4.325 × 10−1	2.705 × 102	4.421	2.645 × 102	2.813 × 102	12
GWO	7.889 × 10−1	4.078 × 10−1	2.639 × 102	5.001 × 10−3	2.639 × 102	2.639 × 102	3
POA	7.876 × 10−1	4.115 × 10−1	2.639 × 102	2.910 × 10−2	2.639 × 102	2.640 × 102	4
SCA	8.552 × 10−1	2.786 × 10−1	2.697 × 102	8.581	2.639 × 102	2.828 × 102	11
SOA	7.727 × 10−1	4.795 × 10−1	2.665 × 102	2.680	2.639 × 102	2.734 × 102	8
CAPCPO	7.592 × 10−1	3.915 × 10−1	2.639 × 102	2.173 × 10−7	2.639 × 102	2.639 × 102	1

Bold is the best result of all algorithms.

presents the experimental results, in which CAPCPO ranks first and exhibits the best performance in the experimental comparison.

5.7. The Speed Reducer Design Problem

The speed reducer design (SRD) problem is an engineering optimization problem to minimize the weight of the reducer with 11 constraints. Schematic representation of the SRD is shown in Figure 20. There are seven design variables to be optimized: face width ($b$), module of teeth ($m$), pinion teeth count ($p$), length of the first shaft between bearings ($l_1$), length of the second shaft between bearings ($l_2$), diameter of the first shaft ($d_1$), and diameter of the second shaft ($d_2$). The mathematical model for SRD is presented below.

• Speed reducer design’s mathematical model:
• Solution representation:
• Consider $X$ = $\left[x_1 x_2 x_3 x_4 x_5 x_6 x_7\right]=\left[b m p l_1 l_2 d_{1 }{d}_2\right]$
• Objective function:
• Minimize

$$f\left(X\right)=0.7854x_1{x}_2^2\left(3.3333x_3^2+14.9334x_3-43.0934\right)-1.508x_1\left(x_6^2+x_7^2\right)+7.4777\left(x_6^3+x_7^3\right)+0.7854\left(x_4{x}_6^2+x_5{x}_7^2\right)$$

Subject to 11 constraints:

$$\begin{gathered} g_1\left(X\right)={\displaystyle \frac{27}{x_1{x}_2^2{x}_3}}-1\le 0 \\ {g}_2\left(X\right)=\begin{array}{c}{\displaystyle \frac{397.5}{x_1{x}_2^2{x}_3^2}}-1\le 0\end{array} \\ {g}_3\left(X\right)={\displaystyle \frac{1.93x_4^3}{x_2{x}_3{x}_6^4}}-1\le 0 \\ {g}_4\left(X\right)={\displaystyle \frac{1.93x_5^3}{x_2{x}_3{x}_7^4}}-1\le 0 \\ {g}_5\left(X\right)=\begin{array}{c}{\displaystyle \frac{\sqrt{(\frac{745x_4}{x_2{x}_3}{)}^2+16.9\times {10}^6}}{110.0x_6^3}}-1\le 0\end{array} \\ {g}_6\left(X\right)={\displaystyle \frac{\sqrt{(\frac{754x_5}{x_2{x}_3}{)}^2+157.5\times {10}^6}}{85.0x_7^3}}-1\le 0 \\ {g}_7\left(X\right)=\begin{array}{c}{\displaystyle \frac{x_2{x}_3}{40}}-1\le 0\end{array} \\ {g}_8\left(X\right)={\displaystyle \frac{5x_2}{x_1}}-1\le 0 \\ {g}_9\left(X\right)={\displaystyle \frac{x_1}{12x_2}}-1\le 0 \\ {g}_{10}\left(X\right)={\displaystyle \frac{1.5x_6+1.9}{x_4}}-1\le 0 \\ {g}_{11}\left(X\right)={\displaystyle \frac{1.1x_7+1.9}{x_5}}-1\le 0 \end{gathered}$$

Variable range:

$$\begin{gathered} 2.6\le x_1\le \mathrm{3.6,0.7}\le x_2\le \mathrm{0.8,17.0}\le x_3\le 28.0 \\ 7.3\le x_4\le \mathrm{8.3,7.3}\le x_5\le \mathrm{8.3,2.9}\le x_6\le 3.9 \\ 5.0\le x_7\le 5.5 \end{gathered}$$

The results are shown in

Table 34. Comparative results for SRD.

Algorithms	CPO	PSO	BWO	DBO	SSA	WOA	GJO	GWO	POA	SCA	SOA	CAPCPO
$x_1$	3.501	4.061	3.500	3.557	3.539	3.503	3.506	3.500	3.520	3.542	3.539	3.500
$x_2$	7.001 × 10−1	8.336 × 10−1	7.000 × 10−1	7.000 × 10−1	7.004 × 10−1	7.000 × 10−1	7.000 × 10−1	7.000 × 10−1	7.074 × 10−1	7.001 × 10−1	7.492 × 10−1	7.000 × 10−1
$x_3$	1.700 × 101	2.687 × 101	1.700 × 101	1.737 × 101	1.955 × 101	1.700 × 101	1.700 × 101	1.700 × 101	1.992 × 101	1.700 × 101	2.452 × 101	1.700 × 101
$x_4$	7.984	9.061	8.230	8.117	7.905	8.133	8.124	8.123	7.980	8.212	7.959	7.941
$x_5$	8.096	9.053	8.270	8.278	8.022	8.214	8.198	8.222	8.057	8.259	8.037	8.095
$x_6$	3.894	3.986	3.986	3.870	3.671	3.880	3.873	3.899	3.730	3.845	3.721	3.900
$x_7$	5.494	6.116	5.500	5.500	5.360	5.495	5.493	5.500	5.387	5.493	5.412	5.500
Avg	1.343 × 103	3.083 × 1018	1.342 × 103	1.448 × 103	1.872 × 103	1.344 × 103	1.345 × 103	1.342 × 103	1.105 × 1018	1.361 × 103	9.176 × 1018	1.342 × 103
Std	7.020 × 10−1	5.056 × 1018	9.554 × 10−9	4.545 × 102	4.162 × 102	1.978	3.254	1.564 × 10−1	4.074 × 1018	1.088 × 101	5.925 × 1018	4.600 × 10−3
Best	1.342 × 103	1.650 × 103	1.342 × 103	1.342 × 103	1.352 × 103	1.342 × 103	1.342 × 103	1.342 × 103	1.342 × 103	1.345 × 103	3.476 × 103	1.342 × 103
Worst	1.346 × 103	2.156 × 1019	1.342 × 103	3.893 × 103	2.609 × 103	1.349 × 103	1.355 × 103	1.342 × 103	1.873 × 1019	1.380 × 103	1.825 × 1019	1.342 × 103
Rank	4	11	1	8	9	5	6	3	10	7	12	2

Bold is the best result of all algorithms.

. It can be seen that CAPCPO ranks second, behind BWO.

6. Conclusions and Future Work

In this paper, three new strategies—composite Cauchy mutation strategy, adaptive dynamic adjustment strategy, and population mutation strategy—are first proposed and then introduced into CPO. The improved CPO is named CAPCPO. The composite Cauchy mutation strategy boosts CPO’s exploration and exploitation ability. The adaptive dynamic adjustment strategy accelerates CPO’s convergence rate and boosts the local optimum avoidance ability. The population mutation strategy accelerates CPO’s running speed, while at the same time improves the diversity of the fitness matrix, helping the algorithm jump out of local optimum. We first validated the effectiveness of our proposed strategies on the CEC2017 test suite. Then, we conducted comparative experiments with 11 algorithms on the CEC2017, CEC2019, and CEC2022. The results indicate that our CAPCPO outperforms competitive algorithms in most cases across these three test suites. In a few instances, its performance may not match that of other algorithms, which could be attributed to an ineffective balance between the exploration and exploitation phases. Additionally, due to the introduction of extra calculations, CAPCPO did not achieve the best ranking in the computational effort analysis. Therefore, in the future, we will focus on improving the transition mechanism between the exploration and exploitation phases, as well as reducing the memory usage of the algorithms. Finally, this paper also tested the algorithm’s ability to solve real-world problems on seven engineering optimization problems. Results show that CAPCPO exhibits excellent performance in seven real-world engineering design optimization problems.

For future work, we intend to apply CAPCPO to other real-world optimization problems, such as (1) image segmentation, (2) feature selection, and (3) parameter optimization.