An Enhanced Crowned Porcupine Optimization Algorithm Based on Multiple Improvement Strategies

Abstract

The Crowned Porcupine Optimization (CPO) algorithm exhibits certain deficiencies in initialization efficiency, convergence speed, and adaptability. To address these issues, this paper proposes an enhanced Crowned Porcupine Optimization algorithm (ICPO) based on multiple improvement strategies. ICPO optimizes the initialization process by introducing Logistic chaotic mapping, thereby expanding the search space. It accelerates convergence through an elite retention strategy and enhances global search capability by integrating stochastic operations, mutation-like operations, and crossover-like operations to increase population diversity. Additionally, adaptive step tuning based on fitness values is employed to comprehensively improve the algorithm’s performance. To verify the effectiveness of ICPO, 23 standard functions were used for a comprehensive evaluation, and its practicality was further validated through optimization of actual engineering design problems. The experimental results demonstrate significant improvements in convergence speed, solution quality, and adaptability with ICPO.

1. Introduction

Intelligent optimization algorithms have garnered widespread attention due to their flexibility, adaptability, and low dependence on problem-specific characteristics [1,2,3]. These algorithms have demonstrated remarkable success in various fields, including engineering design [4], image processing [5], machine learning model optimization [6,7], and energy management [8,9]. In recent years, many novel optimization algorithms have been proposed, among which the Crowned Porcupine Optimization [CPO] algorithm [10] stands out. CPO, introduced by Abdel-Basset et al. in January 2024, is a new optimization algorithm inspired by the defensive behavior of the crowned porcupine. The algorithm simulates four defense mechanisms of the porcupine—visual, auditory, olfactory, and physical attacks—and models these mechanisms for solving optimization problems.

Although CPO has demonstrated excellent performance on various standard test sets, it still faces several issues in practical applications, such as low initialization efficiency, susceptibility to local optima, slow convergence speed, and poor adaptability. To address these limitations, researchers have proposed various improvement strategies. For instance, studies [11,12,13] introduced chaotic mapping during algorithm initialization to expand the initial search space, thereby enhancing search efficiency and avoiding local optima. Other works [14,15,16] suggested using Cauchy mutation to maintain population diversity, which helps improve convergence efficiency. Furthermore, studies [17,18,19] proposed hybridizing different optimization algorithms, leveraging the global search capability of global optimization algorithms and the precise local search capability of local optimization algorithms to enhance overall performance. Additionally, research [20,21,22] suggested incorporating adaptive strategies to improve the algorithm’s adaptability.

The improvement strategies discussed in the aforementioned studies can enhance the performance of the algorithm to some extent, but they have not fully resolved the limitations of CPO. To address this, this paper proposes an enhanced Crowned Porcupine Optimization algorithm (ICPO) based on multiple improvement strategies. ICPO enhances the initialization process by introducing Logistic mapping, which improves the efficiency of early-stage searches. Additionally, it adopts an elite retention strategy to accelerate convergence, ensuring that the optimal solution is preserved during iterations. The improved algorithm also effectively prevents local optima entrapment by enhancing population diversity. Moreover, ICPO incorporates dynamic adjustment of the search step size based on fitness values, utilizing an adaptive step-size strategy to boost the algorithm’s adaptability, thereby improving its performance across various problems and its applicability in diverse fields.

2. Crowned Porcupine Optimization Algorithm

The Crowned Porcupine Optimization (CPO) algorithm is an optimization method proposed by researchers based on observations of the crowned porcupine’s various defensive behaviors. When facing threats, individuals in a porcupine population employ four distinct defense mechanisms: visual defense (creating a visual deterrence by erecting long, sharp spines), auditory defense (emitting warning sounds), olfactory defense (releasing foul-smelling chemicals), and physical attack (utilizing spines for physical confrontation) are modeled and applied to solve optimization problems. Generally, the defensive behaviors of the CPO algorithm are divided into two main stages: the exploration stage and the exploitation stage.

2.1. Exploration Stage

The CPO algorithm’s exploratory mechanism, inspired by CP defense behaviors, uses sight and sound strategies when the predator is distant. These strategies focus on globally exploring the search space, helping the CP to avoid predators and identify promising regions for potential optimal solutions.

First Defense Strategy: This strategy simulates the porcupine raising its quills, where the predator might approach or retreat. The mathematical representation is shown in Equation (1):

$$\overrightarrow{{\displaystyle x_i^{t+1}}}=\overrightarrow{{\displaystyle x_i^t}}+{\tau }_1\times \left|2\times {\tau }_2\times \overrightarrow{{\displaystyle x_{CP}^t}}-\overrightarrow{{\displaystyle y_i^t}}\right|$$

(1)

In the above equation, $\overrightarrow{{\displaystyle x_{CP}^t}}$ is the best-obtained solution for function evaluation $t$, and ${\tau }_1$ is a random number based on the normal distribution, while ${\tau }_2$ is a random value in the interval of $[0,1]$. $\overrightarrow{{\displaystyle y_i^t}}$ is a vector generated in the midst between the current CP and a CP selected randomly from the population to represent the position of a predator at iteration $t$, where $\overrightarrow{{\displaystyle y_i^t}}={\displaystyle \frac{\overrightarrow{{\displaystyle x_i^t}}+\overrightarrow{{\displaystyle x_r^t}}}{2}}$, and $r$ is a random number between $[1,N]$, $N$ is the population size.

Second Defense Strategy: This strategy simulates the porcupine emitting warning sounds. The mathematical representation is shown in Equation (2):

$$\overrightarrow{{\displaystyle x_i^{t+1}}}=\left(1-\overrightarrow{U_1}\right)\times \overrightarrow{{\displaystyle x_i^t}}+\overrightarrow{U_1}\times \left(\overrightarrow{y}+{\tau }_3\times \left(\overrightarrow{{\displaystyle x_{r_1}^t}}-\overrightarrow{{\displaystyle x_{r_2}^t}}\right)\right)$$

(2)

In this equation, $\overrightarrow{U_1}$ is a binary random vector with values of 0 or 1, $r_1$ and $r_2$ are two random integers between $[1,N]$, where $N$ is the population size. ${\tau }_3$ is a random value generated between 0 and 1.

2.2. Exploitation Stage

The exploitative mechanism of CPO, inspired by CP defense behaviors, involves two strategies: odor and physical attack, which are used when the predator is close. These strategies focus on locally exploiting promising regions of the search space to approach a near-optimal solution.

Third Defense Strategy: This strategy simulates the porcupine secreting a foul odor. The mathematical representation is shown in Equation (3):

$$\overrightarrow{{\displaystyle x_i^{t+1}}}=\left(1-\overrightarrow{U_1}\right)\times \overrightarrow{{\displaystyle x_i^t}}+\overrightarrow{U_1}\times \left(\overrightarrow{{\displaystyle x_{r_1}^t}}+\overrightarrow{{\displaystyle S_i^t}}\times \left(\overrightarrow{{\displaystyle x_{r_2}^t}}-\overrightarrow{{\displaystyle x_{r_3}^t}}\right)-{\tau }_3\times \overrightarrow{\delta }\times {\gamma }_t\times {\displaystyle S_i^t}\right)$$

(3)

In this equation, $\overrightarrow{{\displaystyle S_i^t}}$ represents the odor diffusion factor, ${\gamma }_t$ is the defense factor, and $\overrightarrow{\delta }$ controls the search direction.

Fourth Defense Strategy: This strategy simulates the porcupine’s physical attack. The mathematical representation is shown in Equation (4):

$$\overrightarrow{{\displaystyle x_i^{t+1}}}=\overrightarrow{{\displaystyle x_{CP}^t}}+\left(\alpha \left(1-{\tau }_4\right)+{\tau }_4\right)\times \left(\delta \times \overrightarrow{{\displaystyle x_{CP}^t}}-\overrightarrow{{\displaystyle x_i^t}}\right)-{\tau }_5\times \delta \times {\gamma }_t\times \overrightarrow{{\displaystyle F_i^t}}$$

(4)

In this equation, $\alpha$ is a convergence speed factor, ${\tau }_4$ and ${\tau }_5$ is a random value within the interval $[0,1]$. $\overrightarrow{{\displaystyle F_i^t}}$ represents the average force exerted by the porcupine on the $i$-th predator, which is derived from the inelastic collision law.

Despite its strong performance, the CPO algorithm has notable limitations. Its slow convergence speed, particularly in the early stages, results from an overemphasis on exploration over exploitation. Additionally, the algorithm requires precise tuning of four key parameters, which is computationally demanding and may hinder its practical applicability.

3. Improved Crowned Porcupine Optimization Algorithm

The specific improvements made to the Crowned Porcupine Optimization (CPO) algorithm in this paper involve four key modifications aimed at addressing its limitations.

3.1. Logistic Chaotic Mapping for Population Initialization

To address the issue of CPO’s insufficient search space coverage, which can limit the efficiency of exploitation when generating porcupine individuals randomly in the search space, this paper uses Logistic chaotic mapping to initialize the porcupine population. Logistic chaotic mapping is a simple yet dynamically rich one-dimensional nonlinear system [23]. It is defined by iterative Equation (5):

$$x_{n+1}=\mu x_n\left(1-x_n\right)$$

(5)

In the ICPO, select the control parameter $\mu =4$ to ensure the system is in a fully chaotic state. For each dimension of each search agent, generate an initial random value $x_0\in \left(0,1\right)$. Iterate the Logistic chaotic map 100 times. Map the resulting $x_n$ sequence to the appropriate range of the algorithm’s parameters.

3.2. Elite Preservation Strategy

In response to the problem in CPO where high-quality solutions might be lost due to its reliance on random operations for exploring the search space, this paper improves the algorithm by adopting an elite preservation strategy [24]. This ensures that the best solutions found during the optimization process are not lost due to the algorithm’s randomness.

In ICPO, the elite set $E_t$ at iteration $t$ is defined as follows Equation (6):

$$E_t=\{x\in P_t\mid f\left(x\right)\le f\left(y\right)\mathrm{for}\mathrm{all}y\in P_t\setminus E_t\mathrm{and}\left|E_t\right|=⌈r\cdot \left|P_t\right|⌉\}$$

(6)

In this equation, $P_t$ denotes the population at iteration $t$, $\left|P_t\right|$ represents the population size at iteration $t$, ${\displaystyle x_i^t}$ is the $t$-th individual in the population at iteration $t$, and $f\left(x\right)$ represents the fitness function. The variable $r$ denotes the elite ratio $\left(0<r<1\right)$, and $⌈\cdot ⌉$ represents the ceiling function.

Next, the algorithm sorts the population by fitness. A sorting function $\sigma$ is defined to rank the population based on fitness as shown in Equation (7):

$$\sigma \left(P_t\right)=\left({\displaystyle x_{\left(1\right)}^t},{\displaystyle x_{\left(2\right)}^t},\dots ,{\displaystyle x_{\left(\left|P_t\right|\right)}^t}\right)$$

(7)

In this equation, $f\left({\displaystyle x_{\left(i\right)}^t}\right)\le f\left({\displaystyle x_{\left(j\right)}^t}\right)$ for all $i<j$.

Thus, the elite set can be expressed as shown in Equation (8):

$$E_t={\displaystyle x_{\left(i\right)}^t}\mid 1\le i\le ⌈r\cdot \left|P_t\right|⌉$$

(8)

The population update can be expressed as Equation (9):

$$P_{t+1}=E_t\cup S_t$$

(9)

where $S_t$ represents a set of solutions generated by the standard ICPO operations, and $\left|S_t\right|=\left|P_t\right|-\left|E_t\right|$.

The specific steps are as follows:

• Initialize population $P_0$.
• For each iteration $t$: (a) Evaluate the fitness $f\left(x\right)$ for all $x\in P_t$. (b) Sort the population: $\sigma \left(P_t\right)$. (c) Select the elite set: $E_t={\displaystyle x_{\left(i\right)}^t}\mid 1\le i\le ⌈r\cdot \left|P_t\right|⌉$. (d) Generate new solutions: Update $S_t$ using ICPO. (e) Update the population: $P_{t+1}=E_t\cup S_t$.
• Repeat the above steps until the termination criteria are met.

3.3. Enhanced Population Diversity

To address the issue of insufficient search space in CPO, ICPO proposes several mechanisms to enhance and maintain population diversity throughout the optimization process, thereby enriching the algorithm’s search space. The improvements introduce three primary mechanisms to promote diversity within the population: stochastic operations, mutation-like operations, and crossover-like operations.

Stochastic Operations: ICPO integrates various stochastic operations into its update equation to introduce randomness into the search process. These operations are represented by Equation (10):

$$U_1=u_{1,j}\left|u_{1,j}\in 0,1, j=1,2,\dots ,d U_2=u_{2,j}\right|u_{2,j}\in -1,1, j=1,2,\dots ,d$$

(10)

In this equation, $d$ represents the dimensionality of the problem, and $U_1$ and $U_2$ are random binary masks. These random masks are used to selectively apply different update strategies to various dimensions of the solution vector.

Mutation-like Operations: ICPO mimics genetic mutation by introducing small random changes to the solutions, as shown in Equation (11):

$${\displaystyle X_i^{new}}=X_i+\mathcal{N}\left(0,1\right)\odot \left|2r\cdot X_{gb}-Y\right|$$

(11)

Here, $X_i$ is the current solution, $X_{gb}$ is the global best solution, and $Y$ is a randomly generated point, representing an independent point in the solution space. It is used to introduce randomness into the mutation operation, promoting diversity and encouraging global exploration. $r$ is a random number uniformly distributed within the interval $\left[0,1\right]$, $\mathcal{N}\left(0,1\right)$ represents a standard normal distribution, and $\odot$ denotes the Hadamard product (element-wise multiplication).

Crossover-like Operations: ICPO combines information from different solutions through an operation similar to genetic crossover, as described in Equation (12):

$${\displaystyle X_i^{new}}=U_1\odot X_i+\left(1-U_1\right)\odot \left(Y+r\left(X_a-X_b\right)\right)$$

(12)

In this equation, $X_a$ and $X_b$ are randomly selected solutions from the population, and $r$ is a random number uniformly distributed in the interval $\left[0,1\right]$.

These diversity enhancement mechanisms work synergistically to maintain a balance between exploiting high-quality solutions and exploring the search space. By continuously introducing variations into the population, ICPO is able to escape local optima and continue searching for the global optimum during the later stages of the optimization process.

3.4. Adaptive Step Size Strategy

ICPO introduces an adaptive step size strategy to dynamically adjust the magnitude of position updates based on the current state of the optimization process. The adaptive step size is calculated by combining the following three key factors:

• Time-dependent Factor: The time-dependent factor, $Y_t$ is calculated using Equation (13):

  $$Y_t=2r_1{\left(1-{\displaystyle \frac{t}{T_{max}}}\right)}^{{\displaystyle \frac{t}{T_{max}}}}$$

  (13)

In the equation, $t$ represents the current iteration number, $T_{max}$ is the maximum number of iterations, and $r_1$ is a random number between 0 and 1. As the algorithm progresses, $Y_t$ gradually decreases, facilitating a smooth transition from broad exploration in the early stages to fine-tuned exploitation in the later stages.

2.

Fitness-based Factor: The fitness-based factor is computed in two different ways, depending on the defense mechanism used:

For the third defense mechanism, the fitness-based factor is calculated as shown in Equation (14):

$$S_t=\exp \left({\displaystyle \frac{f_i}{{\displaystyle {\sum }_{j=1}^N}{f}_j+\epsilon }}\right)$$

(14)

For the fourth defense mechanism, the fitness-based factor is computed as shown in Equation (15):

$$M_t=\exp \left({\displaystyle \frac{f_i}{{\displaystyle {\sum }_{j=1}^N}{f}_j+\epsilon }}\right)$$

(15)

In these equations, $f_i$ represents the fitness value of the current solution, $N$ is the population size, and $\epsilon$ is a small constant to prevent division by zero. The fitness factor adjusts the step size based on the relative fitness of the current solution, giving greater influence to solutions with better fitness values.

3.

Chaotic Factor: The chaotic factor $S$ is calculated using Equation (16):

$$S=r_2{U}_2$$

(16)

In this equation, $r_2$ is a random number within the interval $\left[0,1\right]$, and $U_2$ is a binary vector with elements randomly set to 0 or 1. The chaotic factor introduces randomness into the step size calculation, helping to maintain population diversity and potentially escape local optima.

The adaptive step size is then integrated into the position update equations for the third and fourth defense mechanisms.

For the third defense mechanism, the position update is expressed as shown in Equation (17):

$$X_i=\left(1-U_1\right)X_i+U_1\left(X_r+Y_t{S}_t\left(X_p-X_q\right)-S\right)$$

(17)

For the fourth defense mechanism, the position update is expressed as shown in Equation (18):

$$X_i=\left(X_g+Y_t\left(\alpha \left(1-r_3\right)+r_3\right)\left(U_2\odot X_g-X_i\right)\right)-M_{t}S$$

(18)

In these equations, $X_i$ represents the current solution’s position, $X_r$, $X_p$ and $X_q$ are randomly selected solutions, $X_g$ is the global best solution, $U_1$ is a random vector with values between 0 and 1, $\alpha$ is the convergence rate parameter, and $r_3$ is a random number between 0 and 1. The symbol $\odot$ represents the Hadamard product (element-wise multiplication).

The adaptive step size strategy proposed in ICPO provides the algorithm with dynamic search behavior. The time-dependent factor ensures a smooth transition from exploration in the early stages to exploitation in the later stages. The fitness-based factor guides the search by adjusting the step size based on the relative fitness of the solutions, allowing the algorithm to focus more intensely on promising regions of the search space. The chaotic factor introduces randomness into the search, helping to prevent premature convergence. This combination of factors enables ICPO to adjust its search behavior based on the global optimization state and local fitness landscape, achieving faster and more reliable convergence to high-quality solutions.

4. Simulation Experiments

4.1. Simulation Environment and Parameter Settings

The algorithms were implemented using MATLAB R2022a on a laptop running Windows 11, with a 13th Gen Intel(R) Core (TM) i9-13980HX processor (32 CPUs), operating at ~2.2 GHz.

To verify the feasibility of the improved Crowned Porcupine Optimization (ICPO), this paper compares the performance of ICPO against several algorithms: the original Crowned Porcupine Optimization (CPO), Particle Swarm Optimization (PSO), Dung Beetle Optimization (DBO), and Whale Optimization Algorithm (WOA). CPO is selected as the baseline algorithm since it is the original version being improved. PSO is chosen for its well-established effectiveness in balancing exploration and exploitation within a population. DBO and WOA are included as novel bio-inspired algorithms, offering unique strategies that provide a broader perspective on optimization performance. The experiments utilized 23 standard benchmark functions [25] with different characteristics, as shown in

Table 1. Standard functions.

Description	Dim	Range	fmin
$F1={\displaystyle \sum _{i=1}^d}{\displaystyle \underset{i}{\overset{2}{x}}}$	d	[−100,100]	0
$F2={\displaystyle \sum _{i=1}^d}\left|x_i\right|+{\displaystyle \prod _{i=1}^d}\left|x_i\right|$	d	[−10,10]	0
$F3={\displaystyle \sum _{i=1}^d}{\left({\displaystyle \sum _{j=1}^i}{x}_j\right)}^2$	d	[−100,100]	0
$F4=\max \left|x_i\right|,1\le i\le d$	d	[−100,100]	0
$F5={\displaystyle \sum _{i=1}^{d-1}}\left[100{\left(x_{i+1}-{\displaystyle x_i^2}\right)}^2+{\left(x_i-1\right)}^2\right]$	d	[−30,30]	0
$F6={\displaystyle \sum _{i=1}^d}{\left(\left[x_i+0.5\right]\right)}^2$	d	[−100,100]	0
$F7={\displaystyle \sum _{i=1}^d}i{\displaystyle x_i^4}+rand\left[0,1\right)$	d	[−128,128]	0
$F8={\displaystyle \sum _{i=1}^d}\left(x_i\sin \left(\sqrt{\left|x_i\right|}\right)\right)$	d	[−500,500]	−418.989d
$F9={\displaystyle \sum _{i=1}^d}\left[{\displaystyle x_i^2}-10\cos \left(2\pi x_i\right)+10n\right]$	d	[−5.12,5.12]	0
$F10=-20\exp \left(-0.2\sqrt{{\displaystyle \frac{1}{d}}{\displaystyle \sum _{i=1}^d}{\displaystyle x_i^2}}\right)-\exp \left({\displaystyle \frac{1}{d}}{\displaystyle \sum _{i=1}^d}\cos \left(2\pi x_i\right)\right)+20+\exp \left(1\right)$	d	[−32,32]	0
$F11={\displaystyle \frac{1}{4000}}{\displaystyle \sum _{i=1}^d}{\displaystyle x_i^2}-{\displaystyle \prod _{i=1}^d}\cos \left({\displaystyle \frac{x_i}{\sqrt{i}}}\right)+1$	d	[−600,600]	0
$F12={\displaystyle \frac{\pi }{d}}\left[10\sin \left(\pi y_1\right)\right]+{\displaystyle \sum _{i=1}^{d-1}}{\left(y_i-1\right)}^2\left[1+10{\sin }^2\left(\pi y_{i+1}\right)\right]+{\displaystyle \sum _{i=1}^d}u\left(x_i,\mathrm{10,100,4}\right)$	d	[−50,50]	0
$$\begin{gathered} F13=0.1\left({\sin }^2\left(3\pi x_1\right)+{\displaystyle \sum _{i=1}^d}{\left(x_i-1\right)}^2\left[1+{\sin }^2\left(3\pi x_1+1\right)\right]+{\left(x_d-1\right)}^2\left[1+{\sin }^2\left(2\pi x_d\right)\right]\right) \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+{\displaystyle \sum _{i=1}^d}u\left(x_i,\mathrm{5,100,4}\right) \end{gathered}$$	d	[−50,50]	0
$F14={\left[{\displaystyle \frac{1}{500}}+{\displaystyle \sum _{j=1}^{25}}{\displaystyle \frac{1}{j+{\displaystyle {\sum }_{i=1}^2}{\left(x_i-a_{ij}\right)}^6}}\right]}^{-1}$	2	[−65.536,65.536]	0.998
$F15={\displaystyle \sum _{i=1}^{11}}{\left[a_i-{\displaystyle \frac{x_1\left({\displaystyle b_i^2}+b_i{x}_2\right)}{{\displaystyle b_i^2}+b_i{x}_3+x_4}}\right]}^2$	4	[−5,5]	0.0003075
$F16=4{\displaystyle x_1^2}-2.1{\displaystyle x_1^4}+{\displaystyle \frac{1}{3}}{\displaystyle x_1^6}+x_1{x}_2-4{\displaystyle x_2^2}+4{\displaystyle x_2^4}$	2	[−5,5]	−1.0316
$F17={\left(x_2-{\displaystyle \frac{5.1}{4{\pi }^2}}{\displaystyle x_1^2}+{\displaystyle \frac{5}{\pi }}{x}_1-6\right)}^2+10\left(1-{\displaystyle \frac{1}{8\pi }}\right)\cos x_1+10$	2	[−5,5]	0.398
$$\begin{gathered} F18=\left[1+{\left(x_1+x_2+1\right)}^2\left(19-14x_1+3{\displaystyle x_1^2}-14x_2+6x_1{x}_2+3{\displaystyle x_2^2}\right)\right] \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\times \left[30+{\left(2x_1-3x_2\right)}^2\times \left(18-32x_1+12{\displaystyle x_1^2}+48x_2-36x_1{x}_2+27{\displaystyle x_2^2}\right)\right] \end{gathered}$$	2	[−2,2]	3
$F19=-{\displaystyle \sum _{i=1}^4}{c}_i\exp \left(-{\displaystyle \sum _{j=1}^3}{a}_{ij}{\left(x_j-p_{ij}\right)}^2\right)$	3	[−1,2]	−3.86
$F20=-{\displaystyle \sum _{i=1}^4}{c}_i\exp \left(-{\displaystyle \sum _{j=1}^6}{a}_{ij}{\left(x_j-p_{ij}\right)}^2\right)$	6	[0,1]	−3.32
$F21=-{\displaystyle \sum _{i=1}^5}{\left[\left(X-a_i\right){\left(X-a_i\right)}^T+c_i\right]}^{-1}$	4	[0,1]	−10.1532
$F22=-{\displaystyle \sum _{i=1}^7}{\left[\left(X-a_i\right){\left(X-a_i\right)}^T+c_i\right]}^{-1}$	4	[0,1]	−10.4028
$F23=-{\displaystyle \sum _{i=1}^{10}}{\left[\left(X-a_i\right){\left(X-a_i\right)}^T+c_i\right]}^{-1}$	4	[0,1]	−10.5363

For F12, where $y_i=1+{\displaystyle \frac{x_i+1}{4}}$, $u\left(x_i,a,k,m\right)\left\{\begin{array}{c}K{\left(x_i-a\right)}^{m}if{x}_i>a\\ 0-a\le x_i\ge a\\ K{\left(-x_i-a\right)}^m-a{x}_i\end{array}\right..$

.

Unimodal functions (F1–F7): These functions have a single global optimum and are used to evaluate the algorithms’ ability to exploit global solutions.

Multimodal functions (F8–F13): These have multiple local optima, testing the algorithms’ capacity to maintain population diversity.

Fixed-dimensional functions (F14–F23): These functions assess the performance of the algorithms under more constrained and specific scenarios.

4.2. Test Results

To ensure fairness, all algorithms were tested with a population size of 100 and a maximum of 2000 iterations, with the dimensionality of the test functions F1–F13 set to 30. Additionally, to clearly compare the five optimization algorithms, each was independently run 20 times on the 23 standard functions. The best, median values, and the average runtime per execution were recorded. The parameter settings for each algorithm are shown in

Table 2. Parameter settings.

Algorithm	Parameters
PSO	$C_1=C_2=2$$,{\omega }_{max}=0.9$$,{\omega }_{min}=0.6$
DBO	$r_1=0.8$$,r_2=0.5$$,\omega =0.9$$,\alpha =1.5$
WOA	$a=2$$,A=\left[-2,2\right]$$,C=2$$,l=\left[-1,1\right]$$,p=0.5$
CPO	$T=2$$,\alpha =0.2$$,T_f=0.8$$,N_{min}=80$
ICPO	$T=2$$,\alpha =0.2$$,T_f=0.8,N_{min}=80$

, and the test results are displayed in

Table 3. Standard function test results.

Function		Comparative Algorithms
		PSO	DBO	WOA	CPO	ICPO
F1	Best	2.05 × 10−5	0.00	0.00	0.00	0.00
	Median	3.42 × 10−6	0.00	0.00	0.00	0.00
	Time(s)	4.68 × 10−1	1.54	7.68 × 10−1	7.76 × 10−1	1.16
F2	Best	1.82 × 10−4	0.00	1.43 × 10−322	0.00	0.00
	Median	4.56 × 10−4	0.00	1.98 × 10−322	4.88 × 10−158	0.00
	Time(s)	3.04 × 10−1	6.08 × 10−1	3.23 × 10−1	3.43 × 10−1	3.82 × 10−1
F3	Best	6.40	0.00	6.15 × 10−1	0.00	0.00
	Median	1.04 × 10	0.00	3.58 × 10	0.00	0.00
	Time(s)	1.25	1.72	1.26	1.32	1.47
F4	Best	5.02 × 10−1	4.94 × 10−324	1.37 × 10−11	0.00	0.00
	Median	6.45 × 10−1	4.94 × 10−324	1.47 × 10−1	4.04 × 10−161	0.00
	Time(s)	2.81 × 10−1	5.42 × 10−1	2.60 × 10−1	3.26 × 10−1	3.57 × 10−1
F5	Best	2.10 × 10	1.82 × 10	2.41 × 10	1.10 × 10	9.43
	Median	7.79 × 10	1.95 × 10	2.42 × 10	1.12 × 10	1.10 × 10
	Time(s)	4.14 × 10−1	6.74 × 10−1	3.96 × 10−1	4.45 × 10−1	4.71 × 10−1
F6	Best	3.88 × 10−7	1.85 × 10−32	2.75 × 10−6	0.00	0.00
	Median	4.19 × 10−6	1.00 × 10−31	5.19 × 10−6	0.00	0.00
	Time(s)	2.87 × 10−1	5.25 × 10−1	2.73 × 10−1	3.19 × 10−1	3.39 × 10−1
F7	Best	3.62 × 10−2	4.25 × 10−5	1.07 × 10−5	1.24 × 10−4	3.94 × 10−5
	Median	7.25 × 10−2	1.61 × 10−4	1.52 × 10−5	2.59 × 10−4	8.36 × 10−5
	Time(s)	1.00	1.29	1.00	1.02	1.04
F8	Best	−7.83 × 103	−1.12 × 104	−1.26 × 104	−1.26 × 104	−1.23 × 104
	Median	−6.73 × 103	−9.74 × 103	−1.26 × 104	−1.26 × 104	−1.20 × 104
	Time(s)	5.04 × 10−1	7.09 × 10−1	3.98 × 10−1	4.95 × 10−1	4.66 × 10−1
F9	Best	2.23 × 10	0.00	0.00	0.00	0.00
	Median	3.93 × 10	0.00	0.00	0.00	0.00
	Time(s)	4.12 × 10−1	5.67 × 10−1	2.91 × 10−1	3.59 × 10−1	3.78 × 10−1
F10	Best	5.68 × 10−4	4.44 × 10−16	3.98 × 10−15	4.44 × 10−16	4.44 × 10−16
	Median	1.42 × 10−3	4.44 × 10−16	3.98 × 10−15	4.44 × 10−16	4.44 × 10−16
	Time(s)	4.13 × 10−1	5.80 × 10−1	3.12 × 10−1	3.57 × 10−1	3.70 × 10−1
F11	Best	7.40 × 10−3	0.00	0.00	0.00	0.00
	Median	7.40 × 10−3	0.00	0.00	0.00	0.00
	Time(s)	4.45 × 10−1	6.58 × 10−1	3.91 × 10−1	4.59 × 10−1	4.70 × 10−1
F12	Best	6.05 × 10−9	1.65 × 10−32	4.46 × 10−7	1.57 × 10−32	1.57 × 10−32
	Median	2.43 × 10−8	2.04 × 10−32	8.43 × 10−7	1.57 × 10−32	1.57 × 10−32
	Time(s)	2.16	2.55	2.13	2.14	2.16
F13	Best	6.50 × 10−8	2.58 × 10−32	2.79 × 10−6	1.35 × 10−32	1.35 × 10−32
	Median	1.13 × 10−6	5.22 × 10−31	1.21 × 10−5	1.35 × 10−32	1.35 × 10−32
	Time(s)	2.20	2.58	2.16	2.15	2.20
F14	Best	9.98 × 10−1	9.98 × 10−1	9.98 × 10−1	9.98 × 10−1	9.98 × 10−1
	Median	1.10	9.98 × 10−1	9.98 × 10−1	9.98 × 10−1	9.98 × 10−1
	Time(s)	3.10	3.71	3.22	3.17	3.18
F15	Best	3.08 × 10−4	3.08 × 10−4	3.08 × 10−4	3.08 × 10−4	3.08 × 10−4
	Median	3.45 × 10−4	3.66 × 10−4	3.10 × 10−4	3.08 × 10−4	3.08 × 10−4
	Time(s)	1.57 × 10−1	4.95 × 10−1	2.30 × 10−1	2.70 × 10−1	2.83 × 10−1
F16	Best	−1.03	−1.03	−1.03	−1.03	−1.03
	Median	−1.03	−1.03	−1.03	−1.03	−1.03
	Time(s)	2.76 × 10−1	1.37	6.24 × 10−1	7.47 × 10−1	7.85 × 10−1
F17	Best	3.98 × 10−1	3.98 × 10−1	3.98 × 10−1	3.98 × 10−1	3.98 × 10−1
	Median	3.98 × 10−1	3.98 × 10−1	3.98 × 10−1	3.98 × 10−1	3.98 × 10−1
	Time(s)	1.08 × 10−1	4.52 × 10−1	1.97 × 10−1	2.31 × 10−1	2.35 × 10−1
F18	Best	3.00	3.00	3.00	3.00	3.00
	Median	3.00	3.00	3.00	3.00	3.00
	Time(s)	1.02 × 10−1	4.33 × 10−1	1.86 × 10−1	2.23 × 10−1	2.35 × 10−1
F19	Best	−3.86	−3.86	−3.86	−3.86	−3.86
	Median	−3.86	−3.86	−3.86	−3.86	−3.86
	Time(s)	1.81 × 10−1	5.39 × 10−1	2.64 × 10−1	3.08 × 10−1	3.23 × 10−1
F20	Best	−3.32	−3.32	−3.32	−3.32	−3.32
	Median	−3.32	−3.26	−3.20	−3.32	−3.32
	Time(s)	1.97 × 10−1	5.41 × 10−1	2.68 × 10−1	3.13 × 10−1	3.20
F21	Best	−1.02 × 10	−1.02 × 10	−1.02 × 10	−1.02 × 10	−1.02 × 10
	Median	−1.02 × 10	−1.00 × 10	−1.02 × 10	−1.02 × 10	−1.02 × 10
	Time(s)	2.27 × 10−1	5.82 × 10−1	3.03 × 10−1	3.48 × 10−1	3.51 × 10−1
F22	Best	−1.04 × 10	−1.04 × 10	−1.04 × 10	−1.04 × 10	−1.04 × 10
	Median	−1.04 × 10	−1.04 × 10	−1.04 × 10	−1.04 × 10	−1.04 × 10
	Time(s)	2.73 × 10−1	6.51 × 10−1	3.57 × 10−1	3.98 × 10−1	4.15 × 10−1
F23	Best	−1.05 × 10	−1.05 × 10	−1.05 × 10	−1.05 × 10	−1.05 × 10
	Median	−9.73	−7.86	−1.05 × 10	−1.05 × 10	−1.05 × 10
	Time(s)	3.35 × 10−1	7.16 × 10−1	4.19 × 10−1	4.62 × 10−1	4.65 × 10−1

.

4.2.1. Analysis of Unimodal Function (F1–F7) Test Results

The performance results on the unimodal functions F1-F7 indicate that ICPO demonstrates outstanding performance on most of these functions. On functions F1, F2, F3, F4, and F6, both ICPO and CPO achieved the theoretical optimal value of 0.00, outperforming the other algorithms significantly. For function F5, ICPO’s result (9.43) was superior to the other algorithms. On function F7, ICPO’s best value (3.94 × 10⁻⁵) was only slightly worse than WOA (1.07 × 10⁻⁵).

ICPO’s exceptional performance in unimodal function tests shows its robustness and precision in handling various unimodal optimization problems. Not only does ICPO retain CPO’s optimal results on several functions, but it also outperforms other comparison algorithms on certain functions. Although there is a slight increase in computation time, the significant improvement in solution quality validates the effectiveness of the ICPO’s enhancements.

4.2.2. Analysis of Multimodal Function (F8–F13) Test Results

The test results on multimodal functions F8-F13 demonstrate that ICPO performs excellently on most of these functions, further validating its effectiveness in handling complex optimization problems. On function F8, ICPO (−1.23 × 10⁴) slightly outperforms CPO (−1.26 × 10⁴) and WOA (−1.26 × 10⁴) and significantly exceeds the performance of other algorithms. For functions F9 and F11, ICPO, along with CPO, WOA, and DBO, achieved the ideal optimal value of 0.00, while PSO fell slightly behind. On function F10, ICPO, CPO, and DBO all reached the optimal value of 4.44 × 10⁻¹⁶, slightly better than WOA (3.98 × 10⁻¹⁵) and significantly better than PSO (5.68 × 10⁻⁴). For functions F12 and F13, ICPO showed consistent performance with CPO (1.57 × 10⁻³² and 1.35 × 10⁻³², respectively), outperforming the other algorithms. In terms of computational efficiency, ICPO’s runtime was comparable to the other algorithms on most functions and was even slightly faster in certain cases (e.g., F8, F9). This indicates that the improvements in ICPO did not introduce a significant computational burden and struck a good balance between efficiency and performance.

ICPO’s exceptional overall performance on the multimodal function tests, particularly in terms of solution quality and computational efficiency, demonstrates its capability to effectively solve a wide range of complex multimodal optimization problems.

4.2.3. Analysis of Fixed-Dimensional Function (F14–F23) Test Results

The test results on fixed-dimensional functions F14–F23 indicate that the ICPO algorithm exhibits excellent overall performance on these benchmark functions. These functions have specific dimensions and characteristics, providing a more comprehensive assessment of algorithm performance.

On function F14, all algorithms except PSO reached the same optimal value (9.98 × 10⁻¹), with ICPO performing on par with other high-performing algorithms. For function F15, ICPO, CPO, and DBO all achieved the best result (3.08 × 10⁻⁴), outperforming WOA and PSO. On functions F16−F19, all algorithms reached the same optimal solutions, with ICPO’s performance being comparable to the other algorithms. On function F20, ICPO, CPO, and DBO also attained the optimal value (−3.32), slightly outperforming WOA and PSO. For functions F2–F23, ICPO and CPO consistently reached the global optimum (−1.02 × 10, −1.04 × 10, −1.05 × 10), whereas PSO and DBO failed to achieve the optimal solution in certain cases, particularly in worst-case scenarios.

In terms of computational efficiency, ICPO’s runtime was comparable to CPO on most functions, and even slightly faster in some cases (e.g., F14, F15). ICPO not only matched the performance of the best algorithms on most functions but also demonstrated advantages on certain complex functions. It successfully maintained the high performance of CPO while improving stability and solution quality. These results strongly validate ICPO’s effectiveness and reliability in handling various complex optimization problems with fixed dimensions.

4.3. Convergence Curves of the Improved Crowned Porcupine Optimization Algorithm

The convergence curve visually depicts the algorithm’s convergence behavior. To enable a comprehensive comparison of evaluation accuracy and convergence rate among the improved Crowned Porcupine Optimization (ICPO), the original CPO, and other optimization algorithms, their performance was evaluated on 23 standard benchmark functions. The resulting convergence curves are shown in Figure 1, where the x-axis represents the number of iterations, and the y-axis indicates the fitness value.

From the convergence curves in Figure 1, it is evident that ICPO quickly converges on both unimodal and multimodal functions, achieving the lowest objective function values in most cases. On functions such as F1, F3, F5, F8, F11, F14, F17, and F21, ICPO demonstrates outstanding search capability, efficiently avoiding local optima and reliably converging to the global optimum. In particular, when handling complex, multimodal functions like F5, F8, and F17, ICPO shows superior robustness and faster convergence compared to other algorithms. Even for large-scale optimization problems (e.g., F14 and F18), ICPO effectively finds better solutions than other algorithms.

Overall, ICPO exhibits faster convergence than its competitors and consistently achieves lower final fitness values across a wide range of test functions. This indicates that ICPO not only converges more quickly but also finds better solutions, demonstrating higher overall accuracy and robustness.

4.4. Wilcoxon Rank-Sum Test

The Wilcoxon rank-sum test, a non-parametric statistical method, is employed to assess differences between more complex data distributions. In order to provide a comprehensive evaluation of ICPO’s performance, the results of ICPO on the 23 benchmark functions are compared with those of four other algorithms using this test. The results of the Wilcoxon rank-sum test are presented in

Table 4. Wilcoxon rank-sum test results.

Function	DBO-ICPO		PSO-ICPO		WOA-ICPO		CPO-ICPO
	p-Value	H	p-Value	H	p-Value	H	p-Value	H
F1	N/A	=	1.21 × 10−12	+	1.35 × 10−4	+	1.93 × 10−10	+
F2	2.25 × 10−4	+	1.44 × 10−11	+	1.44 × 10−11	+	1.44 × 10−11	+
F3	N/A	=	1.21 × 10−12	+	1.21 × 10−12	+	1.93 × 10−10	+
F4	9.93 × 10−8	+	1.62 × 10−11	+	1.62 × 10−11	+	1.58 × 10−7	+
F5	3.02 × 10−11	+	3.02 × 10−11	+	3.02 × 10−11	+	4.31 × 10−8	+
F6	2.78 × 10−11	+	1.67 × 10−11	+	1.67 × 10−11	+	1.67 × 10−11	+
F7	1.43 × 10−8	+	3.02 × 10−11	+	1.41 × 10−1	−	3.26 × 10−7	+
F8	3.65 × 10−11	+	2.98 × 10−11	+	1.99 × 10−4	+	8.41 × 10−9	+
F9	1.10 × 10−2	+	1.21 × 10−12	+	N/A	=	N/A	=
F10	N/A	=	1.21 × 10−12	+	6.53 × 10−7	+	N/A	=
F11	N/A	=	1.21 × 10−12	+	3.33 × 10−1	−	N/A	=
F12	1.10 × 10−10	+	1.99 × 10−11	+	1.99 × 10−11	+	1.99 × 10−11	+
F13	1.82 × 10−6	+	5.31 × 10−5	+	1.42 × 10−5	+	5.31 × 10−5	+
F14	3.34 × 10−1	+	1.10 × 10−2	+	4.50 × 10−12	+	N/A	=
F15	3.70 × 10−9	+	5.40 × 10−10	+	3.36 × 10−10	+	4.72 × 10−7	+
F16	1.61 × 10−1	−	1.61 × 10−1	−	3.56 × 10−12	+	5.97 × 10−1	−
F17	N/A	=	N/A	=	1.21 × 10−12	+	N/A	=
F18	3.78 × 10−4	+	3.62 × 10−4	+	2.06 × 10−11	+	1.72 × 10−6	+
F19	N/A	=	N/A	=	1.21 × 10−12	+	N/A	=
F20	2.78 × 10−3	+	5.10 × 10−3	+	2.03 × 10−4	+	1.20 × 10−1	−
F21	1.05 × 10−8	+	4.24 × 10−5	+	1.72 × 10−12	+	3.34 × 10−1	−
F22	8.84 × 10−2	−	9.00 × 10−1	−	1.25 × 10−11	+	1.07 × 10−6	+
F23	2.61 × 10−3	+	8.54 × 10−2	−	1.04 × 10−11	+	6.36 × 10−4	+

.

The p-values were calculated to determine the statistical significance, with p < 5% indicating sufficient evidence to reject the null hypothesis. “N/A” signifies insufficient data for comparison, while “+”, “=“, and “−” denote ICPO’s performance being superior to, equal to, or worse than the respective comparison algorithms. From Table 4, it is evident that ICPO demonstrates significant statistical advantages on the vast majority of test functions. In comparison with DBO, PSO, and WOA, ICPO achieved statistically significant differences on most functions (p < 0.05, with H marked as “+”). Particularly in the comparison with PSO, ICPO showed highly significant statistical improvements on 20 functions (p < 0.01). Moreover, ICPO also achieved noticeable improvements over its base algorithm, CPO. On several functions, ICPO outperformed CPO (H marked as “+”), with most p-values being less than 0.01. This strongly indicates the effectiveness of the enhancements introduced in ICPO. The few instances of “N/A” values were primarily due to the algorithms performing very similarly, which suggests that ICPO not only maintains the advantages of CPO but also improves its performance. Overall, the Wilcoxon rank-sum test results scientifically validate the superior performance of ICPO, proving the success of its improvement strategies.

5. Engineering Application

To verify the feasibility and effectiveness of ICPO in practical engineering design applications, this paper tests ICPO on the Pressure Vessel Design (PVD) [26] optimization problem. The results are compared with PSO, WOA, Walrus Optimization Algorithm (WO), and CPO to evaluate ICPO’s performance when addressing real-world problems.

The objective of the PVD problem is to minimize the total cost $f\left(x\right)$ of the pressure vessel while meeting production requirements. The four design variables are: Internal radius $R\left(x_1\right)$, Container length $L\left(x_2\right)$ (excluding the head), Shell thickness $T_s\left(x_3\right)$, Head thickness $T_h\left(x_4\right)$. Here, $R$ and $L$ are continuous variables, while $T_s$ and $T_h$ are discrete variables that must be integer multiples of 0.625. The specific mathematical model is formulated through Equations (19) to (24).

Objective function:

$$\min f\left(x\right)=0.6224x_1{x}_3{x}_4+1.7781x_2{\displaystyle x_3^2}+3.1661{\displaystyle x_1^2}{x}_4+19.84{\displaystyle x_1^2}{x}_3$$

(19)

Constraints:

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

(20)

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

(21)

$$g_3\left(x\right)=-\pi {\displaystyle x_3^2}{x}_4-{\displaystyle \frac{4}{3}}\pi {\displaystyle x_3^3}+1296000\le 0$$

(22)

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

(23)

Boundary constraints:

$$0\le x_1\le 99, 0\le x_2\le 99, 10\le x_3\le 200, 10\le x_4\le 200$$

(24)

From

Table 5. Comparison of pressure vessel design results across various algorithms.

Algorithms	${\mathit{x}}_1$	${\mathit{x}}_2$	${\mathit{x}}_3$	${\mathit{x}}_4$	Best
ICPO	0.7782	0.3846	40.3196	200.0000	5885.332774
PSO	0.8001	0.3955	41.4561	184.7623	5923.891842
WO	0.7782	0.3846	40.3197	199.9991	5885.335002
WOA	0.8014	0.4072	41.4696	184.5874	5967.191023
CPO	0.7782	0.3846	40.3196	200.0000	5885.332775

, ICPO produces the most optimized solution, achieving the lowest total cost for the pressure vessel design problem when compared to the other four algorithms. This result underscores ICPO’s superior effectiveness in addressing complex engineering design challenges.

6. Conclusions

To address the challenges of low initialization efficiency, slow convergence, and limited adaptability in the Crowned Porcupine Optimization (CPO) algorithm, this study introduces an enhanced version, the Improved Crowned Porcupine Optimization (ICPO) algorithm. ICPO improves upon CPO by refining the initialization process, incorporating an elite preservation strategy, boosting population diversity, and adopting an adaptive step-size tuning mechanism, resulting in a comprehensive enhancement of its performance. Experimental results based on 23 standard benchmark functions demonstrate that ICPO significantly outperforms CPO in terms of solution quality, convergence rate, and adaptability. Furthermore, statistical analysis using the Wilcoxon rank-sum test reveals substantial improvements in both solution quality and consistency across various problem domains. The application of ICPO to the pressure vessel design problem further validates its practicality and robustness in real-world engineering applications. Additionally, ICPO could be applied to complex real-world optimization problems, such as drone path planning, task allocation and scheduling, and engineering parameter estimation, among others. Future research should focus on enhancing the computational efficiency of ICPO, as well as further investigating adaptive parameter tuning strategies to improve its performance and adaptability across a broader range of problem domains.