Enhanced crested porcupine optimizer for numerical optimization and wireless sensor network deployment

Abstract

Metaheuristic algorithms are widely used to address complex real-world optimization problems, but many existing algorithms face challenges such as slow convergence, low accuracy, and susceptibility to local optima. The newly proposed Crested Porcupine Optimizer (CPO), while demonstrating effectiveness across domains, still suffers from these limitations in practical applications, restricting its performance in solving complex optimization tasks like wireless sensor network (WSN) deployment. To address these shortcomings, this study aims to enhance CPO’s overall performance by developing an improved version called the Enhanced Crested Porcupine Optimizer (ECPO). The methodology of ECPO integrates four key enhancement strategies: (1) Sobol sequences for population initialization, ensuring uniform distribution of initial solutions to boost global search capability; (2) a guided search strategy based on the global optimal solution, directing the algorithm toward optimal regions to reduce ineffective exploration; (3) an adaptive Lévy flight search strategy, maintaining population diversity and improving convergence accuracy; and (4) a centroid-based reverse learning strategy for population updates, expanding search space coverage and accelerating convergence. The performance of ECPO was validated on four authoritative benchmark suites (CEC2014, CEC2017, CEC2020, CEC2022) by comparing it with classical algorithms (e.g., PSO, DE), recently proposed algorithms (e.g., BKA, SBOA), and CEC-winning algorithms (e.g., LSHADE, AGSK). Statistical results show ECPO outperformed most comparison algorithms in 93.81%, 93.33%, 75.71%, and 86.90% of tests on the four benchmarks, respectively, exhibiting significant advantages in convergence speed, accuracy, and stability. Additionally, when applied to WSN node deployment optimization, ECPO achieved a higher coverage rate (average 84.95%) and better robustness than competing algorithms, with more rational node distribution and minimal resource waste. These findings confirm that ECPO effectively overcomes the limitations of the original CPO and outperforms many state-of-the-art optimizers. As a high-performance metaheuristic algorithm, ECPO not only excels in numerical optimization but also demonstrates broad applicability in practical engineering problems like WSN deployment, providing a reliable tool for solving complex optimization challenges.

Introduction

Optimization is the process of determining the best solution among many alternatives. When addressing optimization problems, it is typically necessary to consider three main elements: decision variables, objective functions, and constraints. The core goal of optimization is to find the optimal values of a set of decision variables under specific constraints to achieve the best outcome for the objective function. Methods for solving these problems are generally categorized into two types: deterministic algorithms and stochastic algorithms^1–3. Deterministic algorithms excel in dealing with problems that are linear, continuous, differentiable, and convex, whereas stochastic algorithms, particularly metaheuristic algorithms, are better suited for handling complex real-world problems⁴. In contrast, metaheuristic algorithms have become the preferred choice for researchers due to their simplicity and efficiency. Although metaheuristic methods cannot guarantee finding a strictly optimal solution, they are designed to approach the optimal solution and are often more efficient in terms of computational resource usage compared to exhaustive search methods. Metaheuristic algorithms iteratively search for satisfactory solutions by randomly adjusting initial candidate solutions until a satisfactory solution is found or a predefined stopping condition is met^5,6.

With advancements in technology, modern society faces increasingly complex optimization problems across various domains such as engineering, resource allocation, and financial risk management⁷. Traditional optimization methods, such as gradient descent and Newton’s method, struggle to address these challenges due to their reliance on specific problem structures. In contrast, metaheuristic algorithms offer a more flexible and effective solution by employing general search strategies to explore the solution space without depending on specific mathematical models⁸. Metaheuristic algorithms primarily include swarm intelligence, evolutionary algorithms, human behavior-inspired algorithms, and physics-based algorithms⁹. Due to their simplicity of implementation, flexibility, strong robustness, and self-organizing characteristics, metaheuristic algorithms are widely applied in fields such as image processing, pattern recognition, data mining, system control, and engineering design¹⁰. For example, Zhou et al.¹¹ proposed a multi-threshold image segmentation method based on the Moth-Flame Optimization (MFO) algorithm to address the issue of increasing thresholds in the process of determining optimal thresholds. Devarapalli et al.¹² introduced an improved Harris Hawks Optimization (MHHO) algorithm to identify the optimal parameters of isolation damping devices in power systems, thereby enhancing power system stability. Jiang et al.¹³ combined the Relevance Vector Machine (RVM) with the Chaotic Gravitational Search Algorithm (CGSA) for the recognition of shearer cutting patterns, among other applications.

Common evolutionary algorithms include Genetic Algorithm (GA)¹⁴, Differential Evolution (DE), which is based on Darwinian evolutionary theory concepts of natural selection and reproduction¹⁵, Tree Seed Algorithm (TSA)¹⁶, Genetic Programming (GP)¹⁷ and Evolution Strategies (ES)¹⁸. Additionally, algorithms based on physical phenomena and rules include Simulated Annealing (SA)¹⁹, Gravitational Search Algorithm (GSA)²⁰, Multi-Verse Optimizer (MVO)²¹, Rime Optimizer (RIME)²², Artificial Electric Field Algorithm (AEFA)²³, Sine-Cosine Algorithm (SCA)²⁴, Kepler Optimization Algorithm²⁵, Snow Melting Optimizer (SAO)²⁶, General Relativity Search Algorithm(GRSA)²⁷, Lightning search algorithm (LSA)²⁸. Heuristic algorithms inspired by human behavior or social phenomena include Teaching–Learning-Based Optimization Algorithm (TLBO)²⁹, Soccer League Competition Algorithm (SLC)³⁰, Human Behavior-based Optimization Algorithm (HBBO)³¹, Gold Rush Optimizer (GRO)³² and War Strategy Optimization Algorithm (WSOA)³³. This paper will focus on heuristic algorithms based on swarm intelligence behavior; thus, the following sections will only cover relevant details about swarm intelligence algorithms.

Swarm Intelligence (SI) algorithms, a significant branch of metaheuristic algorithms, have rapidly developed in recent years. This field has introduced a series of algorithms inspired by the collective behaviors of various species in nature. For example, the Particle Swarm Optimization (PSO)³⁴, algorithm is inspired by the foraging behaviors of bird flocks and fish schools; the Ant Colony Optimization (ACO)³⁵, algorithm is derived from the foraging behavior of ant colonies; the Grey Wolf Optimizer (GWO)³⁶ is based on the social hierarchy and hunting behaviors of grey wolves; the Sea Urchin Optimization Algorithm (SSA)³⁷ is inspired by the swimming and foraging behaviors of sea urchins; the Slime Mould Algorithm (SMA)³⁸ mimics the foraging and spreading behaviors of slime moulds; and the Bat Algorithm (BA)³⁹ is based on bats’ use of echolocation for hunting. Other animal behavior-based algorithms include the Chimpanzee Optimization Algorithm (ChOA)⁹, inspired by the unique social behaviors of chimpanzees; the Gorilla Troop Optimizer (GTO)⁴⁰, based on the social behaviors of gorilla troops; the Chameleon Swarm Algorithm (CSA)⁴¹. which mimics the dynamic behaviors of chameleons searching for food in various environments; the Grasshopper Optimization Algorithm (GOA)⁴², based on the behavior of grasshoppers, mapping larval small-scale movements to local exploitation and adult large-scale movements to global exploration in a “step-by-step” cooperative manner. The Northern Goshawk Algorithm (NGO)⁴³, inspired by the hunting process of northern goshawks; the Marine Predator Algorithm (MPA)⁴⁴, derived from marine predators’ foraging behavior; the Path Finder Algorithm (PFA)⁴⁵, inspired by animal groups searching for optimal food areas or prey; the Harris Hawk Optimization Algorithm (HHO)⁴⁶, based on the hunting process of Harris’s hawks; the Sparrow Search Algorithm (SSA)⁴⁷, derived from the foraging and anti-predation behaviors of sparrows; the Artificial Rabbit Optimizer (ARO), inspired by the survival strategies of rabbits; the Snake Optimizer (SO)⁴⁸, based on the unique mating behaviors of snakes; the Crawfish Optimization Algorithm (COA)⁴⁹, inspired by the foraging, sheltering, and competitive behaviors of crawfish; the Black-winged Kite Optimization Algorithm (BKA)⁵⁰, derived from the migratory and hunting behaviors of black-winged kites; the Coyote Optimization Algorithm (COA)⁵¹, based on the social organization and living conditions of North American coyotes; the Dung Beetle Optimization Algorithm (DBO)⁵², which mimics the rolling, dancing, foraging, stealing, and breeding behaviors of dung beetles; the Remora Optimization Algorithm (ROA)⁵³, inspired by remoras’ feeding behavior on different-sized hosts; the Black Widow Optimization Algorithm (BWO)⁵⁴, based on the unique reproductive behaviors of black widow spiders; The Cape Lynx Optimizer (CLO) is proposed based on the complex behaviors of the Cape lynx in its natural environment, such as auditory localization, group cooperation, predator avoidance, and prey capture. The Big-horned Sheep Optimization Algorithm (BSOA) is a novel approach inspired by the social behaviors of bighorn sheep⁵⁵. the Golden Eagle Optimizer (GEO)⁵⁶, inspired by the speed adjustment behaviors of golden eagles during hunting; and the Secretary Bird Optimization Algorithm (SBOA)⁵⁷, inspired by the survival behaviors of secretary birds; and the Draco Lizard Optimizer (DLO)⁵⁸, developed based on the unique behaviors of the Draco lizard, particularly its skilled gliding mechanisms and adaptive ecological strategies. The Gyroscopic Fireworks Algorithm (GFA) is a novel method proposed by simulating the behavior of gyroscopic fireworks during their display⁵⁹.These algorithms offer diverse and flexible tools for solving optimization problems by simulating collective behaviors in nature.Table 1 summarizes the advantages and disadvantages of some relevant algorithms.

Table 1.

Advantages and disadvantages of different algorithms.

Categories	Algorithm	Proposal time	Advantages	Disadvantages
Evolutionary algorithm	GA	1992	Strong global search capability, wide applicability, high scalability	Slow convergence, poor performance in high-dimensional problems
	DE	1995	Simple implementation, fewer parameters, strong robustness, strong global search capability	Limited local search capability, poor adaptability to discrete problems
	ES	1964	Strong adaptability, suitable for continuous optimization problems	High computational cost, limited local search capability
	GP	1989	Strong global search capability, high adaptability	High computational cost, limited scalability
Physics-inspired algorithms	SA	1983	Global convergence, suitable for various combinatorial optimization problems	Long computation time, slow convergence
	GSA	2009	Simple concept, easy implementation, good global search capability	Weak local search capability, relatively slow convergence, inefficient for large-scale problems
	MVO	2015	Good global search capability and convergence performance	Relatively complex principles, difficult to understand and implement
	RIME	2023	Fast convergence, simple results	Prone to local optima, poor local exploration capability
	SAO	2023	Fast convergence, strong global exploration capability	Poor performance in high-dimensional problems, high computational cost
	KOA	2024	Simple results, strong global exploration capability, easy to extend	Slow convergence, prone to local optima
Human-behavior-inspired algorithms	TLBO	2011	Simple structure, easy implementation	Limited local search capability, prone to local optima
	SLC	2014	Fast convergence, fewer parameters	High computational cost, limited local search capability, parameter sensitivity
	HBBO	2017	Good adaptability, easy to extend, no need for gradient information	High implementation complexity, limited local search capability
	GRO	2022	Fast convergence, simple structure, easy implementation	Poor performance in high-dimensional problems, prone to local optima
Swarm intelligence	PSO	1995	Simple implementation, good scalability	Slow convergence, sensitive to initial values and parameter settings
	ACO	2006	Strong distributed computing capability, effective for discrete optimization problems	Slow convergence, prone to stagnation
	GWO	2014	Simple structure, fewer parameters, easy implementation, fast convergence	Limited local search capability, poor performance in high-dimensional problems
	HHO	2019	Strong global search capability, few parameters, strong robustness	High computational cost, limited local search capability, poor performance in high-dimensional problems
	SSA	2020	Fast convergence, strong robustness, strong global search capability	Weak local search capability, limited local search capability
	SO	2022	Fast convergence, strong global exploration capability	Many fixed parameters, poor local exploration
	SBOA	2024	Fast convergence, strong local search capability, good performance in high-dimensional problems	High computational cost
	DBO	2023	Strong global exploration capability, fewer parameters	High computational cost, poor performance in high-dimensional problems, parameter sensitivity
	CPO	2024	Simple structure, high solving efficiency, good global exploration capability	Prone to local optima, relatively slow convergence

However, according to the No Free Lunch Theorem (NFL)⁶⁰, there exists no algorithm capable of solving all optimization problems. Each optimization algorithm is effective for some optimization problems while ineffective for others. Therefore, researchers continually propose various novel or improved optimization algorithms to address different optimization challenges. For instance, Adegboye et al. introduced a variant of the Pathfinder Algorithm (PFA) called the Quadratic Interpolation Hybrid Pathfinder Algorithm (QHIPFA), which incorporates enhancement techniques to improve the efficiency of both global and local search processes⁶¹. Yanhua Zhang et al. integrated the Salp Navigation Mechanism (SNM) and a worker adaptation mechanism to propose a dynamic Gold Rush Optimizer, addressing the issues of premature convergence and sensitivity to suboptimal solutions in the original Gold Rush Optimizer⁶². To tackle the premature convergence of the Exponential Distribution Optimizer and its lack of diversity when dealing with challenging optimization problems, Adegboye proposed an improved EDO (MEDO), which combines the Salp Swarm Algorithm (SSA) with Quadratic Interpolation (QI). QI enhances the local search efficiency and solution accuracy of MEDO, while SSA helps prevent entrapment in suboptimal solutions by serving as a global migration mechanism⁶³.

Crested Porcupine Optimizer (CPO)⁶⁴ is a recently proposed metaheuristic algorithm based on swarm intelligence. Compared to traditional optimization algorithms, CPO demonstrates superior optimization performance with a simple structure and high problem-solving efficiency. However, like other metaheuristic algorithms, CPO is prone to falling into local optima and has relatively slow convergence speed. These shortcomings have motivated us to seek improvements in its performance. Therefore, we have proposed four enhancement strategies and applied them to CPO, aiming to improve its global search capability, stability, and applicability, thereby more effectively addressing complex optimization problems and real-world challenges.

In ECPO, Sobol sequences are first used to generate random numbers for initializing the population, ensuring that the initial set of solutions is uniformly distributed across the entire search space. This approach enhances the algorithm’s global search capability and reduces the risk of falling into local optima. Next, a guiding search strategy based on the global optimal solution is introduced, enabling the algorithm to converge more effectively towards the optimal solution and minimizing exploration in ineffective regions of the solution space. Additionally, an adaptive Lévy flight search strategy is employed to maintain population diversity and improve convergence accuracy. Finally, a centroid-based reverse learning strategy for population updates further expands the search space coverage, avoids local optima, and accelerates the adjustment speed of solutions, thereby enhancing overall convergence efficiency.

The main contributions of this study are as follows:

1. Introduction of four improvement strategies: Four enhancement strategies have been introduced to improve the overall performance of CPO. These strategies include: Sobol sequence population initialization strategy, Guided search strategy based on global optimal solution, Adaptive Levy flight search strategy and Population renewal strategy based on centroid opposition-based learning.

2. Validation on benchmark suites: The numerical optimization performance of ECPO has been validated using four authoritative and challenging benchmark suites: CEC2014, CEC2017, CEC2020, and CEC2022.

3. Comparative analysis: The superiority of ECPO has been demonstrated through comparisons with widely cited algorithms, recently proposed algorithms, and CEC-winning algorithms.

4. Application to wireless sensor networks: ECPO has been successfully applied to the optimization problem of node deployment in wireless sensor networks, further validating its applicability.

The remainder of this paper is structured as follows: Sect. 2 introduces the inspiration and mathematical model of CPO. Section 3 elaborates on the four improvement strategies of ECPO. Section 4 presents and discusses the experimental results of ECPO on four challenging CEC benchmark suites and the wireless sensor network node deployment optimization problem. Finally, Sect. 5 summarizes the content of the paper and suggests directions for future research.

Crested porcupine optimizer

Inspiration

Crested Porcupine Optimizer (CPO)⁶⁴ is inspired by the various defensive behaviors of the crested porcupine (CP), ranging from minimal to highly aggressive defense mechanisms. The crested porcupine employs four distinct protective strategies: visual, auditory, olfactory, and physical attacks. The first and second defense mechanisms (visual and auditory) reflect the exploratory behavior of CPO, while the third and fourth defense strategies (olfactory and physical attacks) represent its exploitative behavior.

Mathematical model of crested porcupine optimizer

CPO simulates the various defensive behaviors of the crested porcupine. As described in the previous section, the porcupine’s defensive strategies include visual, auditory, olfactory, and physical attacks, arranged in increasing order of intensity. In CPO, it is assumed that the distribution of the entire solution search space is as shown in Fig. 1, with four different regions corresponding to the defensive areas of the porcupine.

Fig. 1.

Schematic diagram of four defense strategies of crested porcupine.

• Region 1 (A) represents the area where the porcupine retreats from predators, corresponding to the first defense zone, which implements the initial defensive strategy (visual).

• Region 2 (B) signifies the second defense zone, where the second defense strategy (auditory) is activated if the predator ignores the first defense mechanism and continues to approach.

• Region 3 (C) represents the third defense zone, where the third defense strategy (olfactory) is triggered if the predator disregards the previous two strategies and continues to close in on the porcupine.

• Region 4 (D) is the final defense zone, used to deploy the ultimate defensive strategy (physical attacks). If all previous defenses fail, the porcupine actively attacks the predator in this area, aiming to repel or even eliminate it to ensure its safety.

Initialization

Like other metaheuristic algorithms, the CPO algorithm first generates a set of candidate solutions randomly within the search space. These solutions are initialized in the search space using the following formula:

1

In this equation, represents the candidate solution in the search space, and are the upper and lower bounds of the problem, and is a randomly initialized vector between 0 and 1. The initial population can be represented as follows:

2

In this equation, represents the position of the solution, is the number of candidate solutions, and denotes the dimensionality of the given problem.

Cyclic population reduction technique

The cyclic population reduction technique, in addition to accelerating convergence speed, also preserves population diversity. This strategy simulates the concept that not all crested porcupines (CPs) will activate their defense mechanisms; only those that are threatened will do so. In this strategy, an attempt is made to periodically remove some CPs from the population during the optimization process to accelerate convergence and then reintroduce them to enhance diversity and avoid local minima. The cyclic process is based on a cyclic variable , and the mathematical model for periodically reducing the population size during the optimization process is as follows:

3

Here, is the variable that determines the number of cycles, represents the current iteration of the function, is the maximum number of function iterations, denotes the modulus operator, and is the minimum number of individuals in the newly generated population, ensuring that the population size cannot fall below.

Exploration

The exploration phase of CPO primarily mimics the first and second defense strategies of the crested porcupine (CP), namely visual and auditory. The mathematical modeling for each strategy is as follows:

First defense strategy

When a CP detects a predator, it starts to raise and fan its quills, giving the impression of being larger. The predator then has two choices: either move closer to it or move away. If a random value generated from a normal distribution is less than 1 or greater than − 1, the CP will move closer to the predator; otherwise, it will move away. This behavior is mathematically simulated as follows:

4

In this equation, represents the best solution at iteration , is the current crested porcupine, and is a randomly selected crested porcupine from the population, where is a random number between . is a random number from a normal distribution, and is a random value within the interval [0, 1].

Second defense strategy

In this strategy, the CP uses sound to create noise and threaten the predator. As the predator gets closer, the porcupine’s noise becomes louder. This behavior is mathematically simulated as follows:

5

In this equation, and are two random integers, and is a random value generated within the interval [0, 1].

Exploitation

In this phase, the exploitation mechanism of CPO is introduced. According to the CP’s defensive behavior, there are two exploitation strategies when a predator approaches: the odor-based strategy and the physical attack strategy. These strategies focus on developing promising areas, dedicated to local exploitation. The CPO utilizes the search space based on the odor and physical attack strategies to approach the optimal solution. The mathematical formulas for these different strategies are as follows:

Third defense strategy

In this strategy, the CP secretes a foul odor that spreads in the surrounding area to deter the predator from approaching. To mathematically simulate this behavior, the following equation is proposed:

6

Here, is a vector composed of random numbers in the interval [0,1], is a value between , is a parameter used to control the search direction, defined by Eq. (7), is the defense factor, defined by Eq. (8), is a random value within the interval [0,1], and is the odor diffusion factor, defined by Eq. (9).

7

8

9

Here, represents the objective function value of the individual at iteration , is a very small value to avoid division by zero, is a random vector containing values generated between 0 and 1, is a randomly generated number between 0 and 1, is the population size, is the current iteration count, and is the maximum number of iterations.

Fourth defense strategy

This strategy involves physical attack. When the predator is very close and strikes with short, thick quills, the CP resorts to physical attack. During the physical attack, two objects undergo a strong inelastic collision on one dimension. This can be expressed by the following equation:

10

In this equation, is the convergence speed factor set to 0.2, and are random values within the interval [0,1], and is the average force affecting the predator. This force is governed by the law of inelastic collisions and is defined by Eq. (11) as follows:

11

In this equation, is the mass of the predator at iteration , denotes the objective function, is the final velocity of the individual in the next iteration , and it is determined based on a randomly selected solution from the current population. is the initial velocity of the individual at iteration , is the number of iterations, and and are vectors containing random values generated between 0 and 1.

In summary, the mathematical formula used to update the position of the CP is given by Eq. (12).

12

In Eq. (12), and are random values generated between 0 and 1.

Enhanced crested porcupine optimizer

Sobol sequence population initialization strategy

The Sobol sequence is a widely used low-discrepancy sequence for generating uniformly distributed pseudo-random numbers, particularly suited for initializing populations in global optimization algorithms. Compared to traditional random initialization methods, the Sobol sequence offers several advantages. Firstly, it generates points that are uniformly distributed across multi-dimensional spaces, ensuring better coverage of the entire search space and avoiding the clustering effects that can occur with traditional random number generation methods. Secondly, as a low-discrepancy sequence, the Sobol sequence minimizes discrepancies in the distribution of sampling points, providing a more uniform distribution than purely random sampling. This feature makes it more effective in exploring the solution space during global optimization. Additionally, the Sobol sequence is generated using a recursive formula, making the process deterministic. This means that under the same initial conditions, the sequence generated will be consistent each time, which enhances the reproducibility of results. Overall, the Sobol sequence is a robust method for population initialization in global optimization, ensuring uniform coverage and consistent results.

Therefore, this paper utilizes random numbers generated by the Sobol sequence for population initialization, ensuring that the initial solution set is uniformly distributed across the entire search space. This approach enhances the algorithm’s global search capability and reduces the risk of getting trapped in local optima. The initialization can be expressed by the following equation:

13

Here, the random number generated by the Sobol sequence, can be represented by the following recursive formula:

14

Here, represents the point in the sequence, denotes the bitwise XOR operation, is the direction vector for the bit, and is the binary representation of .

As shown in Fig. 2, the distributions of the randomly initialized population and the Sobol sequence initialized population can be observed. The randomly initialized population exhibits an uneven distribution with noticeable clustering, whereas the population initialized using the Sobol sequence is uniformly distributed across the entire search space without any clustering. This uniform distribution enhances the algorithm’s global search capability and reduces the risk of getting trapped in local optima.

Fig. 2.

Comparison of population distribution of random and sobol sequence.

Guided search strategy based on global optimal solution

In the second phase of CPO’s defense strategy, the mean of the candidate solution and a random solution is used as the reference point for updating solutions during the generation process. While this approach allows for some exploration of the solution space, it still has shortcomings in terms of convergence speed and solution accuracy. These issues are particularly pronounced in multi-dimensional and complex problems, where excessive randomness and inefficient searches can occur. To address this, an improved strategy is proposed, replacing this mean with the current known global optimal solution . By directly utilizing the global optimal solution, the algorithm can more effectively guide convergence towards the optimal direction, reducing ineffective exploration of other regions in the solution space. This allows for a more focused local search and fine-tuning near the global optimum, enhancing the accuracy of the final solution. With this improvement, the algorithm not only accelerates convergence but also demonstrates greater stability and solution quality in complex problems. The improved equation is shown below.

15

Adaptive levy flight search strategy

Levy flight is a random search strategy used in optimization algorithms, based on the Levy distribution to generate step sizes. The principle behind Levy flight is inspired by the random movement patterns observed in certain animals, such as insects and marine organisms, whose behavior often includes long-distance sudden movements and short-distance fine-tuned adjustments. As shown in Fig. 3a, the Levy distribution is characterized by a heavy tail, meaning that the generated step sizes can be very large, allowing the search process to cover a broader area. This distribution is used to simulate the long-distance sudden movements found in nature, with the corresponding movement trajectory illustrated in Fig. 3b.

Fig. 3.

Levy flight probability density releases and moves trajectories.

To more effectively meet different search requirements, a variation strategy involving both large and small step sizes is employed. In the early stages of the optimization process, large step sizes accelerate the global search, enabling the algorithm to quickly locate potential optimal solutions. This is particularly effective when the initial solution is far from the optimal solution. However, as the optimization process progresses, large step sizes may cause oscillations or even skip over the optimal solution, especially in regions where the solution space is complex or the gradient changes sharply. To address the varying needs of different stages in the search process, an Adaptive Levy Flight Search Strategy is proposed. This strategy adaptively adjusts the step size to meet different search demands. It has been incorporated into the fourth defense phase of the CPO, helping to maintain population diversity to some extent and utilizing smaller step sizes to enhance the optimization capability, thereby improving the algorithm’s convergence accuracy. Consequently, the fourth defense phase can be described by Eq. (16).

16

In this equation, is a randomly chosen positive integer, either 1 or 2 (), and represents the Adaptive Levy Flight function, which is computed using Eq. (17).

17

Here, is a fixed constant of 0.01, and is a fixed constant of 1.5. and are random numbers within the interval [0,1], and represents the Gamma function.

Population renewal strategy based on centroid opposition-based learning

To enhance the global exploration capability and convergence speed of the CPO, this paper employs a centroid-based reverse learning strategy for updating the CPO population, aiming to improve both global search ability and convergence speed. The core of the centroid-based reverse learning population update strategy lies in using the centroid information of the current population to generate new reverse solutions, thereby increasing population diversity and exploration capability. As illustrated in Fig. 4, the process begins by calculating the centroid of all individuals in the current population, representing the overall distribution of the population. Next, through centroid-based reverse learning, each individual is mapped to a symmetric position relative to the centroid, generating new reverse solutions. Finally, these newly generated reverse solutions are compared with the individuals in the original population, with the better solutions selected to advance to the next generation, thus updating the population. In this process, the centroid and the generated reverse solutions are computed using the following formulas:

18

Fig. 4.

Schematic representation of the population update strategy based on centroid opposition-based learning.

Here, represents the individual in the population, and denotes its reverse solution.

This strategy effectively expands the coverage of the search space, reducing the risk of the algorithm getting trapped in local optima. By accelerating the adjustment speed of solutions, it enhances the overall convergence efficiency. This improvement significantly boosts the algorithm’s performance in complex, multi-dimensional optimization problems, enabling it to approach the global optimum more comprehensively and quickly.

Based on the above Sect. 3.1 to 3.4, the pseudocode for ECPO is as shown in Algorithm 1.

Algorithm 1.

Pseudo code of ECPO.

Experimental results and analysis

Test suite and compare algorithm parameter settings

This section evaluates the performance of the proposed optimizer using four recent and challenging numerical optimization benchmark suites: CEC2014⁶⁵, CEC2017⁶⁶, CEC2020⁶⁷ and CEC2022⁶⁸. These test functions are categorized into four types: unimodal functions, multimodal functions, hybrid functions, and composition functions, which strongly challenge both previously proposed and newly developed metaheuristic algorithms. Additionally, a comprehensive comparison is conducted to assess the effectiveness of ECPO against three categories of state-of-the-art optimizers:

(1) Classic and widely used optimizers: Particle Swarm Optimization (PSO)³⁴, Differential Evolution (DE)¹⁵, Grey Wolf Optimizer (GWO)³⁶, Runge Kutta optimizer (RUN)⁶⁹ and Sparrow Search Algorithm (SSA)⁴⁷.

(2) Recently released optimizers: Black-winged Kite Algorithm (BKA)⁵⁰, Secretary Bird Optimization Algorithm(SBOA)⁵⁷, Arctic Puffin Optimization(APO)⁷⁰ and Crested Porcupine Optimizer(CPO)⁶⁴.

(3) CEC competition winners: AGSK⁶⁷, LSHADE⁷¹, LSHADE-SPACMA⁷², LSHADE-cnEpSin⁶⁶ and MadDE⁷³.

The parameter settings for each algorithm are listed in Table 2. To ensure fairness and stability in the experiments, a constant population size of 30 and a maximum iteration count of 500 are used. Each algorithm is independently run 30 times, with the mean (Ave) and standard deviation (Std) recorded and the best results highlighted in bold. All experiments are conducted in a computing environment with Windows 11, hardware configuration of 13th Intel^® Core™ i5-13400 2.5 GHz, 16GB RAM, and MATLAB 2023a software.

Table 2.

Compared algorithms parameter settings.

Algorithms	Name of the parameter	Value of the parameter
PSO		2, 2
DE	F, CR	0.8, 0.1
GWO
RUN		20, 12
SSA		0.1, 0.8
BKA		0.9, [0,1]
SBOA
APO		0.5, 0.5
AGSK		0.05,0.05
LSHADE		0.11, 2.6, 18, 6
LSHADE-SPACMA		0.11, 1.4, 5, 0.5,0.5
LSHADE-cnEpSin		0.11, 1.4, 5
MadDE
CPO		0.1, 80, 0.5, 2
ICPO		0.1, 80, 0.5, 2, {1,2}

Qualitative assessment of ECPO

Exploring convergence patterns

In this section, we use five metrics to describe the performance of ECPO during the optimization process: search history, average fitness value, search trajectory, and convergence curve. The convergence curve shows the best fitness value achieved at each iteration; the average fitness value displays the mean fitness of all individuals at each iteration; the trajectory curve illustrates the evolution of the first dimension of the solution; and the “search history” shows the positions obtained.

As shown in the second column of Fig. 5, the search history of ECPO reveals that ECPO does not follow the same pattern for each test function. Initially, ECPO broadly explores the entire search space and then focuses on searching a smaller region to find solutions closer to the optimal, as illustrated by function F25. The third column of the figure presents the average fitness history curves for CPO and ECPO, demonstrating the competitive performance of ECPO’s solutions. Over time, ECPO’s solutions achieve minimized test functions that are closer to the optimal compared to those of CPO. The fourth column illustrates the trajectory of ECPO in searching for the ideal position in the first dimension. This trajectory curve indicates that ECPO starts with an exploratory trend to find promising solutions and then shifts to a developmental trend, concentrating on promising areas to refine and find better solutions. The fifth column shows that, compared to the original CPO, ECPO’s convergence curve demonstrates an accelerated reduction pattern across various test functions, indicating that ECPO can find solutions closer to the optimal solution more quickly.

Fig. 5.

Convergence behavior of ECPO and CPO.

Population diversity analysis

A population with high diversity indicates a significant degree of variation among individuals, allowing for broader exploration of the search space and reducing the risk of prematurely converging to local optima. Therefore, maintaining good population diversity is a key objective in metaheuristic algorithms. Typically, the diversity of an algorithm’s population is evaluated using the following equation⁷⁴.

19

Here, represents the dispersion of each individual in the population from the population’s centroid, where the centroid is denoted as . indicates the value of the individual in the dimension at each iteration.

The levels of population diversity for ECPO and CPO are shown in Fig. 6. Over time, both ECPO and CPO exhibit a gradual decrease in population diversity, shifting from exploration to exploitation. This characteristic is crucial for random algorithms because, at the beginning of the optimization process, they do not know the location of near-optimal solutions in the search space and must thoroughly explore the space to identify promising regions that may contain these solutions. To accelerate convergence to near-optimal solutions, random algorithms eventually need to reduce the time spent on exploration operators in favor of exploitation operators, which focus on the areas close to the best solution found so far. However, ECPO reduces the step size to a certain level and demonstrates a higher level of population diversity compared to CPO. This indicates that ECPO has a more robust exploratory capability, allowing it to explore effectively at the beginning of the optimization process to avoid local minima and then exploit more effectively towards the end of the process to reach a solution closer to the optimum.

Fig. 6.

Population diversity of ECPO and CPO.

Exploration and exploitation analysis

Exploration and exploitation are crucial metrics for heuristic algorithms. Exploration aims to discover new solutions and optimize unknown areas of the search space, while exploitation focuses on known areas to find better solutions through local search. A good balance between exploration and exploitation can quickly lead to optimal solutions, enhance search efficiency, and adapt to diverse and complex problems, demonstrating robustness. The ratio of exploration to exploitation can be measured using Eq. (20)^57,75.

20

Here, represents the position of the agent in the dimension, and denotes the maximum diversity observed throughout the entire iteration process.

The exploration and exploitation balance ratios for CPO and ECPO are illustrated in Fig. 7. It is evident that the ECPO algorithm shows an improvement in balancing exploration and exploitation throughout the iterations compared to CPO. CPO predominantly focuses on exploitation, with insufficient exploration in the early stages, which leads to suboptimal solutions. In contrast, ECPO increases the proportion of exploration, performing thorough global exploration initially, and then gradually transitioning to local exploitation while maintaining a high exploitation ratio. This approach helps to enhance the convergence speed and search accuracy. By maintaining a dynamic balance between exploration and exploitation, ECPO demonstrates significant advantages in avoiding local optima and preventing premature convergence.

Fig. 7.

Exploration and exploitation of ECPO and CPO.

Ablation experiment

This section provides a detailed analysis of the impact of the four proposed improvement strategies on CPO. These strategies include the Sobol sequence population initialization strategy, the guided search strategy based on the global optimal solution, the adaptive Levy flight search strategy, and the population renewal strategy based on centroid opposition-based learning. Based on these strategies, four variants of CPO were proposed: SCPO, GCPO, ALFCPO, and COBLCPO. The experimental results, as shown in Fig. 8, indicate that all four strategies enhance the convergence accuracy and speed of GWO, with ECPO, which integrates all four strategies, demonstrating the most significant improvement. Specifically, when dealing with unimodal and multimodal functions, the results of SCPO and ALFCPO are relatively consistent, with GCPO and COBLCPO showing more pronounced improvements compared to SCPO. However, when handling more complex hybrid and multimodal functions, the effectiveness of SCPO and ALFCPO diminishes, while GCPO and COBLCPO’s performance becomes more prominent. The integrated ECPO generally exhibits excellent performance across most functions. Overall, ECPO successfully addresses the challenges of local optima and premature convergence, significantly improving the algorithm’s convergence speed and accuracy.

Fig. 8.

Comparison of different improvement strategies.

Quantitative evaluation

This section compares the performance of ECPO with five widely used classic optimizers and four recently proposed optimizers, including PSO, DE, GWO, RUN, SSA, BKA, SBOA, APO, and CPO.

Assessing performance with CEC 2014 test suite

This section evaluates the performance of ECPO using the CEC2014 test suite with a dimension setting of 30. The experimental results are summarized in Table 3, which reports the mean (Ave) and standard deviation (Std) achieved by each optimizer. The convergence speed curves are shown in Fig. 9, demonstrating that ECPO’s convergence curves exhibit an accelerated reduction pattern across various test functions. Compared to CPO and other competing optimizers, ECPO finds solutions closer to the optimal more quickly. Additionally, to better illustrate the performance of each algorithm, the means reported in Table 3 are ranked, and the standard deviation values are averaged, as presented in Fig. 10a&b. These figures display the overall effectiveness of the algorithms. The data confirm that ECPO is the best performer, ranking first in both average standard deviation (Std) and average ranking. Specifically, Fig. 10a shows that ECPO achieves the highest average ranking of 1.27, outperforming all competing optimizers. In contrast, the original CPO ranks fourth with a value of 4.83, which is lower than SBOA and APO with values of 3.5 and 3.83, respectively. Figure 10b indicates that ECPO has the smallest normalized average Std among all competing algorithms, outperforming CPO and others, with BKA being the least effective. This suggests that the proposed ECPO demonstrates superior stability in performance.

Table 3.

Results of ECPO and other metaheuristics for CEC2014 benchmark.

ID	Metric	PSO	DE	GWO	RUN	SSA	BKA	SBOA	APO	CPO	ECPO
F1	Ave	4.5084E + 07	4.3160E + 07	3.2290E + 07	5.1904E + 06	3.3047E + 06	1.0980E + 08	5.1151E + 06	3.0203E + 06	8.7513E + 06	4.5816E + 03
	Std	3.2198E + 07	7.7871E + 06	1.6286E + 07	1.5256E + 06	2.0708E + 06	2.3941E + 08	2.8860E + 06	1.7174E + 06	5.5302E + 06	8.4691E + 03
F2	Ave	3.7195E + 09	4.4915E + 07	3.1355E + 09	1.1473E + 05	2.0881E + 03	1.5312E + 10	5.4877E + 05	3.4324E + 05	1.6835E + 06	2.0000E + 02
	Std	2.4177E + 09	1.2035E + 07	2.0204E + 09	2.6639E + 05	1.6477E + 03	1.6852E + 10	1.0174E + 06	2.2744E + 05	1.1060E + 06	1.9029E-14
F3	Ave	6.6925E + 04	7.3183E + 03	6.6132E + 04	1.5197E + 04	2.3977E + 04	3.4764E + 04	3.2266E + 04	8.2685E + 03	5.1126E + 03	3.0000E + 02
	Std	2.3880E + 04	4.7700E + 03	1.9795E + 04	4.7542E + 03	1.3113E + 04	1.1965E + 04	9.2923E + 03	3.1303E + 03	2.5664E + 03	5.6843E-14
F4	Ave	6.7408E + 02	6.3494E + 02	6.7087E + 02	5.1329E + 02	4.9452E + 02	2.2320E + 03	5.0692E + 02	5.0563E + 02	5.1792E + 02	4.0975E + 02
	Std	1.9566E + 02	2.6951E + 01	1.0118E + 02	1.2971E + 01	3.5627E + 01	3.0023E + 03	2.8802E + 01	1.8517E + 01	1.6083E + 01	2.0140E + 01
F5	Ave	5.2106E + 02	5.2076E + 02	5.2108E + 02	5.2098E + 02	5.2053E + 02	5.2094E + 02	5.2032E + 02	5.2101E + 02	5.2091E + 02	5.2020E + 02
	Std	5.6394E-02	5.8155E-02	6.5442E-02	1.4720E-01	4.8106E-01	8.2611E-02	1.2084E-01	7.6743E-02	5.8545E-02	2.4457E-01
F6	Ave	6.2273E + 02	6.2801E + 02	6.1678E + 02	6.2876E + 02	6.2763E + 02	6.3310E + 02	6.1394E + 02	6.1403E + 02	6.1916E + 02	6.1178E + 02
	Std	2.5753E + 00	1.0876E + 00	3.4123E + 00	2.8659E + 00	2.8854E + 00	3.5353E + 00	3.0575E + 00	4.0556E + 00	4.2341E + 00	4.6837E + 00
F7	Ave	7.2305E + 02	7.0111E + 02	7.3145E + 02	7.0015E + 02	7.0001E + 02	8.2110E + 02	7.0045E + 02	7.0056E + 02	7.0089E + 02	7.0000E + 02
	Std	3.3968E + 01	2.7324E-02	1.9396E + 01	5.4413E-02	9.8053E-03	1.0311E + 02	1.7540E-01	2.1671E-01	1.4743E-01	8.0663E-03
F8	Ave	9.7106E + 02	8.6463E + 02	8.8842E + 02	9.1436E + 02	9.2838E + 02	9.4681E + 02	8.6174E + 02	9.2305E + 02	8.6747E + 02	8.5111E + 02
	Std	3.0293E + 01	7.1844E + 00	2.2437E + 01	1.0880E + 01	9.2027E + 00	4.1552E + 01	1.7576E + 01	4.6189E + 01	7.3605E + 00	2.2556E + 01
F9	Ave	1.1128E + 03	1.0825E + 03	9.9933E + 02	1.0610E + 03	1.0906E + 03	1.0929E + 03	9.9039E + 02	1.0146E + 03	1.0769E + 03	9.8739E + 02
	Std	2.9252E + 01	1.4730E + 01	2.4072E + 01	2.1076E + 01	1.1450E + 01	5.9729E + 01	2.0775E + 01	4.8617E + 01	1.7900E + 01	4.2602E + 01
F10	Ave	5.7684E + 03	2.8059E + 03	4.3784E + 03	2.9574E + 03	4.0829E + 03	5.5044E + 03	2.6562E + 03	6.0605E + 03	3.5779E + 03	1.9720E + 03
	Std	5.9196E + 02	2.2298E + 02	5.2543E + 02	5.6276E + 02	7.7712E + 02	9.2041E + 02	4.5147E + 02	1.3732E + 03	3.5274E + 02	6.2616E + 02
F11	Ave	7.0227E + 03	7.3509E + 03	5.2867E + 03	4.7771E + 03	5.5975E + 03	5.3439E + 03	4.4566E + 03	6.8461E + 03	7.6819E + 03	4.5988E + 03
	Std	5.6955E + 02	3.2074E + 02	1.6532E + 03	6.5086E + 02	5.4262E + 02	9.9024E + 02	5.1426E + 02	1.4708E + 03	4.3426E + 02	6.3798E + 02
F12	Ave	1.2029E + 03	1.2019E + 03	1.2028E + 03	1.2028E + 03	1.2011E + 03	1.2009E + 03	1.2004E + 03	1.2026E + 03	1.2025E + 03	1.2004E + 03
	Std	7.3043E-01	2.7920E-01	1.1017E + 00	1.0009E + 00	6.7851E-01	3.4597E-01	2.2229E-01	7.3838E-01	3.3533E-01	2.8556E-01
F13	Ave	1.3010E + 03	1.3006E + 03	1.3008E + 03	1.3008E + 03	1.3007E + 03	1.3033E + 03	1.3007E + 03	1.3006E + 03	1.3006E + 03	1.3005E + 03
	Std	4.9326E-01	7.9824E-02	3.5701E-01	1.5240E-01	1.7220E-01	1.7378E + 00	1.6334E-01	7.8836E-02	7.9590E-02	1.6837E-01
F14	Ave	1.4021E + 03	1.4003E + 03	1.4020E + 03	1.4003E + 03	1.4004E + 03	1.4402E + 03	1.4003E + 03	1.4003E + 03	1.4003E + 03	1.4002E + 03
	Std	3.7452E + 00	3.9785E-02	3.9445E + 00	1.2706E-01	1.8345E-01	4.3307E + 01	9.3301E-02	5.5302E-02	3.8088E-02	3.5012E-02
F15	Ave	1.6377E + 03	1.5517E + 03	2.6506E + 03	1.6025E + 03	1.5297E + 03	1.3616E + 05	1.5247E + 03	1.5192E + 03	1.5259E + 03	1.5039E + 03
	Std	2.4587E + 02	1.6723E + 01	2.2995E + 03	3.1418E + 01	9.6701E + 00	4.3295E + 05	8.3594E + 00	2.8290E + 00	3.6060E + 00	1.2980E + 00
F16	Ave	1.6128E + 03	1.6128E + 03	1.6125E + 03	1.6120E + 03	1.6130E + 03	1.6128E + 03	1.6120E + 03	1.6131E + 03	1.6130E + 03	1.6114E + 03
	Std	3.3488E-01	3.0684E-01	5.6610E-01	6.0648E-01	4.3395E-01	3.4320E-01	7.8548E-01	2.5034E-01	2.2490E-01	7.8746E-01
F17	Ave	2.6471E + 06	7.2039E + 06	4.0604E + 06	3.3673E + 05	8.1076E + 05	1.6940E + 06	6.8659E + 05	9.0084E + 04	3.0781E + 05	3.0171E + 03
	Std	2.1794E + 06	3.0649E + 06	2.2141E + 06	1.4831E + 05	4.3858E + 05	3.9496E + 06	4.3684E + 05	1.2156E + 05	2.5758E + 05	4.0646E + 02
F18	Ave	1.0248E + 07	2.1676E + 07	2.9607E + 07	4.7804E + 03	1.7415E + 04	3.8458E + 07	1.8285E + 04	1.1870E + 04	1.1179E + 05	2.3035E + 03
	Std	9.1026E + 06	7.3946E + 06	6.3585E + 07	1.9774E + 03	1.4337E + 04	2.1036E + 08	1.8709E + 04	9.3788E + 03	2.0190E + 05	1.1301E + 03
F19	Ave	1.9382E + 03	1.9246E + 03	1.9469E + 03	1.9298E + 03	1.9163E + 03	1.9956E + 03	1.9152E + 03	1.9112E + 03	1.9132E + 03	1.9076E + 03
	Std	6.6006E + 01	3.1463E + 00	2.2826E + 01	1.2442E + 01	3.1009E + 00	1.7229E + 02	2.8662E + 00	2.2454E + 00	2.2330E + 00	2.5525E + 00
F20	Ave	1.5447E + 04	9.4023E + 03	2.4244E + 04	7.5996E + 03	2.0869E + 04	1.5956E + 04	1.1704E + 04	3.6359E + 03	4.2036E + 03	2.0519E + 03
	Std	5.5606E + 03	3.8736E + 03	1.5799E + 04	2.8658E + 03	1.1438E + 04	3.9463E + 04	3.9053E + 03	1.8575E + 03	2.1549E + 03	3.0329E + 01
F21	Ave	2.6185E + 05	1.7758E + 05	5.5700E + 05	4.3048E + 04	6.2032E + 04	1.6904E + 05	6.0427E + 04	8.1464E + 03	1.9628E + 04	2.6675E + 03
	Std	2.4718E + 05	9.0879E + 04	4.6346E + 05	2.1139E + 04	3.9491E + 04	3.0928E + 05	5.2322E + 04	6.1640E + 03	1.3453E + 04	2.5806E + 02
F22	Ave	2.5010E + 03	2.6442E + 03	2.4928E + 03	2.8456E + 03	2.8176E + 03	2.6967E + 03	2.4318E + 03	2.3456E + 03	2.4709E + 03	2.4477E + 03
	Std	1.3466E + 02	1.2560E + 02	1.5109E + 02	2.6898E + 02	2.7651E + 02	2.8937E + 02	1.5309E + 02	9.4819E + 01	1.0351E + 02	1.8281E + 02
F23	Ave	2.6655E + 03	2.6367E + 03	2.6846E + 03	2.6429E + 03	2.6357E + 03	2.7427E + 03	2.6367E + 03	2.6375E + 03	2.6376E + 03	2.6356E + 03
	Std	2.6049E + 01	3.1296E-01	2.2090E + 01	4.3938E + 00	1.5798E-01	1.7716E + 02	1.4153E + 00	1.4088E + 00	1.1588E + 00	1.4054E-12
F24	Ave	2.6336E + 03	2.6018E + 03	2.6047E + 03	2.6026E + 03	2.6124E + 03	2.6485E + 03	2.6011E + 03	2.6017E + 03	2.6020E + 03	2.5969E + 03
	Std	3.8396E + 01	2.6576E-01	5.7837E + 00	3.5876E-01	2.8971E + 01	6.1397E + 01	2.5462E + 00	2.7444E-01	1.9016E-01	1.3286E + 01
F25	Ave	2.7114E + 03	2.7211E + 03	2.7230E + 03	2.7224E + 03	2.7099E + 03	2.7288E + 03	2.7079E + 03	2.7058E + 03	2.7091E + 03	2.7031E + 03
	Std	3.2399E + 00	4.2202E + 00	8.0390E + 00	1.2251E + 01	6.2093E + 00	3.3019E + 01	2.2240E + 00	1.6183E + 00	1.8580E + 00	3.6016E-01
F26	Ave	2.7094E + 03	2.7006E + 03	2.7006E + 03	2.7004E + 03	2.7005E + 03	2.7013E + 03	2.7004E + 03	2.7004E + 03	2.7004E + 03	2.7003E + 03
	Std	4.5984E + 01	7.3217E-02	4.1803E-01	7.8412E-02	1.1720E-01	1.1737E + 00	1.0374E-01	7.2597E-02	6.2952E-02	8.2255E-02
F27	Ave	3.5678E + 03	3.3978E + 03	3.4216E + 03	3.6773E + 03	3.7162E + 03	3.8339E + 03	3.3818E + 03	3.2273E + 03	3.2910E + 03	3.1967E + 03
	Std	7.7816E + 01	1.5979E + 02	7.0258E + 01	8.5047E + 01	1.3560E + 02	1.5454E + 02	1.1063E + 02	7.4037E + 01	9.0269E + 01	1.1575E + 02
F28	Ave	3.3190E + 03	3.4005E + 03	3.3635E + 03	3.5800E + 03	3.3832E + 03	3.7818E + 03	3.2761E + 03	3.2761E + 03	3.3216E + 03	3.2432E + 03
	Std	5.2410E + 01	1.8957E + 01	5.6676E + 01	9.6617E + 01	1.1801E + 02	2.3648E + 02	2.6919E + 01	1.5743E + 01	1.7856E + 01	2.6152E + 01
F29	Ave	4.2558E + 06	1.2890E + 05	1.6369E + 05	9.7545E + 06	6.6111E + 06	1.1069E + 08	1.8294E + 06	7.7000E + 03	1.6613E + 04	2.3454E + 07
	Std	9.3018E + 06	4.6175E + 04	5.6868E + 05	6.0290E + 06	1.1608E + 07	9.3958E + 07	4.7405E + 06	3.7753E + 03	5.0085E + 03	2.1484E + 07
F30	Ave	8.0582E + 03	6.8726E + 03	1.1285E + 04	7.3982E + 03	5.4217E + 03	4.0207E + 04	4.8975E + 03	4.6049E + 03	6.0049E + 03	3.9424E + 03
	Std	2.7864E + 03	7.6695E + 02	5.2130E + 03	1.9083E + 03	1.2306E + 03	1.2196E + 05	6.6966E + 02	3.9145E + 02	7.2704E + 02	4.6967E + 02

The bold font represents the optimal result.

Fig. 9.

Comparison between ECPO and other metaheuristics convergence speed on CEC2014 test suit.

Fig. 10.

Comparison under averaged rank and Std for CEC2014.

Assessing performance with CEC 2017 test suite

This section evaluates the performance of ECPO using the CEC2017 test suite, with a dimension setting of 30. The experimental results are summarized in Table 4, which reports the mean (Ave) and standard deviation (Std) achieved by each optimizer. The convergence speed curves are shown in Fig. 11 indicating that ECPO’s convergence curves exhibit an accelerated reduction pattern across various test functions. ECPO converges to solutions closer to the optimum faster than CPO and other optimizers. Furthermore, Fig. 12a shows that ECPO achieves the best average ranking in this test suite, with a value of 1.27. In contrast, the original CPO ranks fourth with a value of 5.37, which is lower than SBOA and APO, which have values of 3.6 and 2.73, respectively. Figure 12b reveals that ECPO has the smallest normalized average Std among all competing algorithms, outperforming CPO and others. Notably, CPO’s standard deviation is not smaller than that of SSA, indicating that ECPO demonstrates superior stability compared to CPO on the CEC2017 test functions. These results underscore the effectiveness of the four strategies embedded in CPO, leading to improved overall performance of the algorithm.

Table 4.

Results of ECPO and other metaheuristics for CEC2017 benchmark.

ID	Metric	PSO	DE	GWO	RUN	SSA	BKA	SBOA	APO	CPO	ECPO
F1	Ave	1.5289E + 09	2.7051E + 07	2.3916E + 09	1.5015E + 04	4.6123E + 03	9.7558E + 09	1.7718E + 05	2.6443E + 05	8.5309E + 05	1.0000E + 02
	Std	1.4213E + 09	7.0909E + 06	1.5090E + 09	8.5462E + 03	5.8143E + 03	9.2362E + 09	1.6502E + 05	2.1610E + 05	8.5620E + 05	2.6389E-15
F2	Ave	6.3682E + 35	1.6362E + 28	6.8595E + 34	2.8537E + 25	2.2494E + 16	8.4960E + 41	1.2847E + 18	1.1523E + 18	3.2680E + 19	3.0373E + 06
	Std	3.4880E + 36	4.4209E + 28	3.6491E + 35	1.5617E + 26	6.6249E + 16	4.1753E + 42	2.3878E + 18	2.1108E + 18	8.7204E + 19	9.4764E + 06
F3	Ave	6.5310E + 04	1.3235E + 05	6.0092E + 04	8.9797E + 03	4.6490E + 04	3.6462E + 04	3.7144E + 04	1.2740E + 04	6.0187E + 04	3.0000E + 02
	Std	2.6314E + 04	2.3882E + 04	1.3817E + 04	3.4520E + 03	6.9665E + 03	1.8535E + 04	9.5627E + 03	5.0086E + 03	9.4150E + 03	3.5264E-10
F4	Ave	6.5930E + 02	6.3764E + 02	6.5064E + 02	5.1684E + 02	4.9932E + 02	2.2472E + 03	5.1825E + 02	5.0653E + 02	5.2056E + 02	4.1741E + 02
	Std	2.5221E + 02	1.9810E + 01	9.0352E + 01	1.7463E + 01	4.2281E + 01	3.3782E + 03	2.4410E + 01	2.3899E + 01	1.8615E + 01	2.8421E + 01
F5	Ave	7.1878E + 02	6.8797E + 02	6.4786E + 02	7.0925E + 02	7.5333E + 02	7.4405E + 02	6.0108E + 02	6.1988E + 02	6.8899E + 02	6.1187E + 02
	Std	3.1132E + 01	1.3527E + 01	6.0013E + 01	3.8121E + 01	4.9664E + 01	6.3160E + 01	2.7239E + 01	5.1297E + 01	1.5745E + 01	8.6010E + 01
F6	Ave	6.2208E + 02	6.0369E + 02	6.1207E + 02	6.4562E + 02	6.4747E + 02	6.6355E + 02	6.0680E + 02	6.0284E + 02	6.0184E + 02	6.0040E + 02
	Std	7.9161E + 00	5.7252E-01	5.9468E + 00	7.8210E + 00	1.0550E + 01	9.0747E + 00	5.8944E + 00	7.8843E-01	4.4464E-01	1.0810E + 00
F7	Ave	1.0018E + 03	9.4298E + 02	9.1299E + 02	1.0800E + 03	1.2483E + 03	1.2306E + 03	8.9592E + 02	8.9497E + 02	9.4290E + 02	7.9218E + 02
	Std	2.4148E + 01	1.6137E + 01	4.6914E + 01	6.8008E + 01	8.7432E + 01	6.1038E + 01	4.3567E + 01	5.2298E + 01	1.5603E + 01	1.8797E + 01
F8	Ave	1.0065E + 03	9.8780E + 02	8.9288E + 02	9.5668E + 02	9.8754E + 02	9.9081E + 02	8.8524E + 02	9.0814E + 02	9.8122E + 02	8.7603E + 02
	Std	2.7175E + 01	1.3422E + 01	1.9970E + 01	2.9475E + 01	2.4076E + 01	3.8801E + 01	2.4173E + 01	5.1677E + 01	1.0977E + 01	3.4554E + 01
F9	Ave	1.9912E + 03	3.6802E + 03	2.6391E + 03	4.0638E + 03	5.3972E + 03	5.6117E + 03	1.6991E + 03	1.0214E + 03	1.4011E + 03	9.2456E + 02
	Std	8.5317E + 02	6.8327E + 02	1.0475E + 03	7.2420E + 02	1.4459E + 02	1.0821E + 03	5.7539E + 02	9.7536E + 01	4.3012E + 02	1.1059E + 02
F10	Ave	7.4274E + 03	6.9329E + 03	4.7641E + 03	4.8013E + 03	5.4492E + 03	5.3284E + 03	4.5722E + 03	7.6393E + 03	7.7501E + 03	4.5366E + 03
	Std	5.9963E + 02	3.1648E + 02	1.1813E + 03	8.8375E + 02	7.6946E + 02	8.3611E + 02	5.9780E + 02	1.0983E + 03	2.7167E + 02	6.2986E + 02
F11	Ave	1.4816E + 03	2.4841E + 03	2.3593E + 03	1.2135E + 03	1.3083E + 03	1.4439E + 03	1.2458E + 03	1.2022E + 03	1.2735E + 03	1.1530E + 03
	Std	8.2000E + 01	6.4349E + 02	1.0234E + 03	3.3585E + 01	6.7436E + 01	1.6336E + 02	3.3568E + 01	3.3282E + 01	2.2022E + 01	3.2745E + 01
F12	Ave	8.3234E + 07	3.5384E + 07	1.0353E + 08	6.2719E + 06	1.1148E + 06	4.5626E + 08	2.7089E + 06	4.8694E + 05	1.0552E + 06	1.1748E + 04
	Std	7.5201E + 07	1.0121E + 07	9.3009E + 07	4.1844E + 06	1.1101E + 06	1.5365E + 09	1.3987E + 06	3.9530E + 05	7.2933E + 05	1.0150E + 04
F13	Ave	5.1906E + 06	7.5409E + 06	1.5901E + 07	2.8611E + 04	2.4764E + 04	4.9014E + 08	2.0056E + 04	1.1857E + 04	1.9870E + 04	1.3391E + 03
	Std	4.8231E + 06	4.1357E + 06	4.0127E + 07	1.4236E + 04	2.2566E + 04	1.8799E + 09	1.6078E + 04	9.2618E + 03	1.0641E + 04	2.4037E + 01
F14	Ave	1.1004E + 05	5.7602E + 05	8.4274E + 05	2.0269E + 04	5.4103E + 04	9.1276E + 03	3.6414E + 04	1.5180E + 03	2.3591E + 03	1.4494E + 03
	Std	1.4156E + 05	5.0303E + 05	9.5786E + 05	2.1158E + 04	4.6053E + 04	1.2057E + 04	5.1805E + 04	3.4243E + 01	1.3947E + 03	2.4235E + 01
F15	Ave	1.2957E + 05	1.5786E + 06	2.7247E + 06	1.2940E + 04	1.2572E + 04	2.6106E + 05	1.2132E + 04	3.2439E + 03	3.9583E + 03	1.5396E + 03
	Std	9.9236E + 04	1.0817E + 06	5.9989E + 06	3.0617E + 03	1.2702E + 04	1.2846E + 06	1.2260E + 04	1.3980E + 03	1.6133E + 03	3.1868E + 01
F16	Ave	3.0244E + 03	2.9615E + 03	2.6860E + 03	2.8219E + 03	2.9121E + 03	3.0497E + 03	2.4825E + 03	2.6988E + 03	3.0756E + 03	2.3790E + 03
	Std	2.8403E + 02	1.6221E + 02	4.4563E + 02	2.9694E + 02	3.7616E + 02	3.9897E + 02	2.8718E + 02	4.2169E + 02	2.4287E + 02	2.0664E + 02
F17	Ave	2.1791E + 03	2.1752E + 03	2.0960E + 03	2.3020E + 03	2.5536E + 03	2.3092E + 03	1.9556E + 03	1.8670E + 03	2.0821E + 03	1.9647E + 03
	Std	1.6843E + 02	1.1514E + 02	1.9385E + 02	2.2310E + 02	2.7935E + 02	2.1688E + 02	1.2941E + 02	8.8002E + 01	1.2797E + 02	1.2815E + 02
F18	Ave	1.8777E + 06	2.0769E + 06	1.7128E + 06	1.1548E + 05	3.8091E + 05	2.0929E + 05	5.2454E + 05	4.2988E + 04	1.2425E + 05	2.8937E + 03
	Std	1.7524E + 06	1.4240E + 06	2.0660E + 06	7.0314E + 04	3.0899E + 05	2.6462E + 05	5.4136E + 05	1.9987E + 04	6.1601E + 04	2.2575E + 03
F19	Ave	2.8697E + 06	1.0115E + 06	3.1040E + 06	1.5671E + 04	1.4219E + 04	4.5957E + 06	1.5855E + 04	5.0887E + 03	6.6267E + 03	1.9318E + 03
	Std	1.3447E + 07	5.3967E + 05	7.2307E + 06	3.7323E + 04	1.7293E + 04	1.6866E + 07	2.8624E + 04	4.6493E + 03	4.1679E + 03	2.4636E + 01
F20	Ave	2.5530E + 03	2.4946E + 03	2.4648E + 03	2.4822E + 03	2.8044E + 03	2.5893E + 03	2.3273E + 03	2.2494E + 03	2.4472E + 03	2.3039E + 03
	Std	2.0343E + 02	1.3498E + 02	1.4849E + 02	1.6363E + 02	2.8743E + 02	1.8780E + 02	1.4201E + 02	1.4031E + 02	1.2895E + 02	1.9349E + 02
F21	Ave	2.5036E + 03	2.4836E + 03	2.4103E + 03	2.4482E + 03	2.5111E + 03	2.5479E + 03	2.3809E + 03	2.4028E + 03	2.4826E + 03	2.3594E + 03
	Std	2.4712E + 01	1.4765E + 01	3.0758E + 01	3.7116E + 01	6.3839E + 01	5.2460E + 01	2.3895E + 01	5.2251E + 01	1.8027E + 01	2.4872E + 01
F22	Ave	6.4559E + 03	3.5042E + 03	5.0177E + 03	3.6057E + 03	6.3219E + 03	6.1335E + 03	2.5158E + 03	2.3049E + 03	2.3103E + 03	4.2930E + 03
	Std	2.9976E + 03	2.4313E + 02	2.1366E + 03	2.0180E + 03	1.9370E + 03	1.5708E + 03	8.0172E + 02	2.1911E + 00	3.0612E + 00	1.9508E + 03
F23	Ave	2.9379E + 03	2.8385E + 03	2.8031E + 03	2.7968E + 03	2.9144E + 03	3.1348E + 03	2.7543E + 03	2.7637E + 03	2.8412E + 03	2.7090E + 03
	Std	6.2532E + 01	1.2471E + 01	5.8491E + 01	3.6607E + 01	5.3400E + 01	1.4135E + 02	3.2272E + 01	5.0357E + 01	1.5155E + 01	2.1632E + 01
F24	Ave	3.0951E + 03	3.0510E + 03	2.9607E + 03	2.9473E + 03	3.0850E + 03	3.2797E + 03	2.9119E + 03	2.9053E + 03	3.0163E + 03	2.8703E + 03
	Std	6.2871E + 01	1.5144E + 01	6.5728E + 01	3.3624E + 01	8.1477E + 01	9.6469E + 01	2.7720E + 01	4.7123E + 01	2.1148E + 01	1.7823E + 01
F25	Ave	2.9710E + 03	2.9748E + 03	3.0113E + 03	2.9082E + 03	2.9002E + 03	3.3072E + 03	2.9203E + 03	2.8970E + 03	2.9144E + 03	2.8869E + 03
	Std	3.0954E + 01	1.5627E + 01	3.2124E + 01	2.0443E + 01	1.8693E + 01	6.0827E + 02	2.7876E + 01	1.3810E + 01	2.1871E + 01	3.5258E + 00
F26	Ave	5.5338E + 03	5.3431E + 03	4.9633E + 03	5.7988E + 03	6.2167E + 03	7.8945E + 03	4.2233E + 03	4.6133E + 03	5.0773E + 03	3.9054E + 03
	Std	9.4327E + 02	5.1219E + 02	5.4609E + 02	1.3612E + 03	1.3201E + 03	1.5304E + 03	8.9488E + 02	7.5031E + 02	1.1294E + 03	7.6225E + 02
F27	Ave	3.2685E + 03	3.2695E + 03	3.2640E + 03	3.2950E + 03	3.2765E + 03	3.4559E + 03	3.2232E + 03	3.2507E + 03	3.2738E + 03	3.2079E + 03
	Std	4.2787E + 01	8.4520E + 00	2.9852E + 01	2.3289E + 01	5.3641E + 01	1.6533E + 02	1.5240E + 01	1.8105E + 01	1.5329E + 01	1.4100E + 01
F28	Ave	3.3613E + 03	3.3796E + 03	3.5009E + 03	3.2568E + 03	3.2284E + 03	3.8118E + 03	3.2688E + 03	3.2370E + 03	3.2905E + 03	3.1191E + 03
	Std	6.3056E + 01	1.8571E + 01	1.0095E + 02	2.0696E + 01	2.0730E + 01	7.4919E + 02	4.7293E + 01	2.4491E + 01	1.8711E + 01	4.4471E + 01
F29	Ave	4.1476E + 03	4.0211E + 03	3.9318E + 03	4.4026E + 03	4.1859E + 03	4.7392E + 03	3.7214E + 03	3.7174E + 03	4.0080E + 03	3.5643E + 03
	Std	2.2399E + 02	1.2386E + 02	2.0761E + 02	2.4116E + 02	2.3765E + 02	5.3879E + 02	2.4041E + 02	1.9624E + 02	1.4891E + 02	2.0978E + 02
F30	Ave	2.6789E + 06	1.3729E + 06	1.0608E + 07	6.0912E + 05	2.2802E + 04	5.7545E + 06	4.3086E + 04	2.5489E + 04	1.5915E + 05	5.2628E + 03
	Std	1.5311E + 06	7.6736E + 05	8.5575E + 06	5.3060E + 05	3.0178E + 04	1.1691E + 07	8.2289E + 04	2.2132E + 04	9.1084E + 04	2.9995E + 02

The bold font represents the optimal result.

Fig. 11.

Comparison between ECPO and other metaheuristics convergence speed on CEC2017 test suit.

Fig. 12.

Comparison under averaged rank and Std for CEC2017.

Assessing performance with CEC 2020 test suite

In this section, the proposed algorithm and other competing algorithms were further tested on the CEC2020 benchmark suite. These 10 test functions assess the algorithms’ capabilities in exploration, exploitation, and avoiding local minima. Table 5 provides the mean (Ave) and standard deviation (Std) from 30 independent runs of each algorithm on each test function, showing that ECPO achieved the best performance on 6 out of the 10 test functions in CEC2020. According to the convergence curves shown in Fig. 13, ECPO quickly finds solutions close to the optimal, unlike BKA, SSA, and PSO, which exhibit slower convergence speeds and tend to get trapped in local minima after a certain number of iterations, unable to find better solutions. Additionally, as shown in Fig. 14, ECPO has the best average ranking and ranking standard deviation. The experimental results confirm that ECPO falls into the category of powerful optimizers.

Table 5.

Results of ECPO and other metaheuristics for CEC2020 benchmark.

ID	Metric	PSO	DE	GWO	RUN	SSA	BKA	SBOA	APO	CPO	ECPO
F1	Ave	6.8642E + 08	2.5200E + 05	5.8248E + 08	3.6660E + 03	3.3588E + 03	2.0715E + 09	3.9607E + 03	1.7696E + 03	6.6983E + 03	1.0000E + 02
	Std	8.8450E + 08	1.2130E + 05	6.2686E + 08	4.2429E + 03	3.6509E + 03	4.8247E + 09	3.4294E + 03	1.5220E + 03	8.2047E + 03	0.0000E + 00
F2	Ave	4.0759E + 03	2.5522E + 03	2.8158E + 03	2.6544E + 03	3.3314E + 03	3.7357E + 03	2.4222E + 03	3.8472E + 03	3.5921E + 03	2.4022E + 03
	Std	5.2562E + 02	1.9447E + 02	4.7384E + 02	6.5104E + 02	3.5712E + 02	8.0824E + 02	3.6962E + 02	1.0821E + 03	2.3148E + 02	4.1419E + 02
F3	Ave	8.4144E + 02	7.7304E + 02	7.9918E + 02	8.8571E + 02	8.9500E + 02	9.9177E + 02	7.7173E + 02	7.8834E + 02	7.8611E + 02	7.4509E + 02
	Std	2.1542E + 01	5.3351E + 00	2.8379E + 01	4.3026E + 01	7.7750E + 01	4.8033E + 01	1.7812E + 01	2.9207E + 01	9.1658E + 00	1.1488E + 01
F4	Ave	2.3755E + 03	1.9089E + 03	1.9313E + 03	1.9447E + 03	1.9100E + 03	2.8357E + 03	1.9082E + 03	1.9079E + 03	1.9085E + 03	1.9018E + 03
	Std	2.5291E + 03	1.6467E + 00	3.4061E + 01	1.0787E + 01	4.0866E + 00	3.1459E + 03	6.8447E + 00	1.8393E + 00	1.7056E + 00	5.8457E-01
F5	Ave	7.4967E + 03	2.1452E + 03	3.9005E + 05	3.7529E + 03	2.8462E + 03	1.4429E + 05	2.2003E + 03	2.0624E + 03	2.2465E + 03	2.0768E + 03
	Std	5.4125E + 03	8.8605E + 01	9.2834E + 05	3.3301E + 02	3.2134E + 02	4.5907E + 05	2.6693E + 02	2.4025E + 02	1.2080E + 02	1.7596E + 02
F6	Ave	1.6000E + 03	1.6000E + 03	1.6000E + 03	1.6000E + 03	1.6000E + 03	1.6000E + 03	1.6000E + 03	1.6000E + 03	1.6000E + 03	1.6000E + 03
	Std	0.0000E + 00	0.0000E + 00	0.0000E + 00	0.0000E + 00	0.0000E + 00	0.0000E + 00	0.0000E + 00	0.0000E + 00	0.0000E + 00	0.0000E + 00
F7	Ave	7.2686E + 04	2.1048E + 03	2.0004E + 05	1.1088E + 04	3.3126E + 03	3.3058E + 04	2.2562E + 03	2.1307E + 03	2.1189E + 03	2.2401E + 03
	Std	3.6726E + 05	1.1502E + 00	1.1212E + 05	4.7931E + 03	1.3229E + 03	4.4926E + 04	1.5096E + 02	5.4357E + 01	2.4981E + 01	1.3303E + 02
F8	Ave	3.4066E + 03	2.5600E + 03	3.3525E + 03	2.9447E + 03	4.0795E + 03	3.6214E + 03	2.4160E + 03	2.3003E + 03	2.3005E + 03	2.6098E + 03
	Std	1.6796E + 03	7.1083E + 01	1.4103E + 03	1.2419E + 03	1.4438E + 03	1.4291E + 03	6.3150E + 02	5.0764E-01	9.7086E-01	9.6858E + 02
F9	Ave	2.9126E + 03	2.8991E + 03	2.8939E + 03	2.8543E + 03	2.9235E + 03	3.0316E + 03	2.8391E + 03	2.8240E + 03	2.8905E + 03	2.8224E + 03
	Std	8.5779E + 01	3.0387E + 01	4.7349E + 01	2.2978E + 01	4.0752E + 01	7.1460E + 01	1.9554E + 01	1.1417E + 01	1.1597E + 01	1.3027E + 01
F10	Ave	2.9677E + 03	2.9632E + 03	3.0049E + 03	2.9903E + 03	2.9679E + 03	3.1715E + 03	2.9735E + 03	2.9335E + 03	2.9633E + 03	2.9312E + 03
	Std	4.2420E + 01	1.3091E + 01	4.7696E + 01	3.0070E + 01	3.9539E + 01	4.6975E + 02	3.5304E + 01	2.6752E + 01	2.9373E + 01	2.8175E + 01

The bold font represents the optimal result.

Fig. 13.

Comparison between ECPO and other metaheuristics convergence speed on CEC2020 test suit.

Fig. 14.

Comparison under averaged rank and Std for CEC2020.

Assessing performance with CEC 2022 test suite

This section uses the challenging CEC2022 benchmark suite, which consists of 12 test functions categorized into four types: unimodal, multimodal, composite, and hybrid. Figure 15 illustrates the convergence speed of ECPO on this test suite, showing that ECPO rapidly converges to solutions closer to the optimal for most functions, outperforming CPO in terms of speed and accuracy. Table 6 presents the analysis of fitness values for each algorithm over 30 independent trials when solving the CEC2022 test functions. It is evident from the table that ECPO ranks first on 8 test functions, second on 2, and third on 2, without any worst rankings. This performance categorizes ECPO as a high-performance optimizer, demonstrating excellent rankings across the majority of the CEC2022 test functions. Additionally, Fig. 16 confirms that ECPO also achieves the best average ranking and average standard deviation. This section provides further evidence of ECPO’s outstanding performance.

Fig. 15.

Comparison between ECPO and other metaheuristics convergence speed on CEC2022 test suit.

Table 6.

Results of ECPO and other metaheuristics for CEC2022 benchmark.

ID	Metric	PSO	DE	GWO	RUN	SSA	BKA	SBOA	APO	CPO	ECPO
F1	Ave	6.2499E + 03	3.4145E + 04	1.3618E + 04	3.0690E + 02	1.2224E + 04	5.4066E + 03	4.7248E + 03	5.6871E + 02	1.3236E + 04	3.0000E + 02
	Std	2.2413E + 03	5.9086E + 03	4.2498E + 03	9.2809E + 00	7.3798E + 03	5.1319E + 03	2.6046E + 03	2.3571E + 02	3.6401E + 03	0.0000E + 00
F2	Ave	4.8012E + 02	4.7925E + 02	5.2883E + 02	4.5574E + 02	4.5317E + 02	6.2611E + 02	4.6023E + 02	4.5664E + 02	4.6162E + 02	4.3640E + 02
	Std	3.1646E + 01	1.0650E + 01	5.8684E + 01	1.6217E + 01	2.0727E + 01	3.3633E + 02	1.3600E + 01	1.2595E + 01	9.7647E + 00	2.0468E + 01
F3	Ave	6.1339E + 02	6.0063E + 02	6.0730E + 02	6.3449E + 02	6.3467E + 02	6.5175E + 02	6.0123E + 02	6.0028E + 02	6.0028E + 02	6.0044E + 02
	Std	6.0577E + 00	1.3738E-01	4.3691E + 00	9.7001E + 00	1.2926E + 01	8.6831E + 00	1.2656E + 00	1.9642E-01	1.3029E-01	1.3083E + 00
F4	Ave	9.1065E + 02	9.2464E + 02	8.5935E + 02	8.7316E + 02	8.8694E + 02	8.8522E + 02	8.4397E + 02	8.4995E + 02	9.0268E + 02	8.5283E + 02
	Std	1.7019E + 01	1.5364E + 01	2.6380E + 01	8.9706E + 00	1.7326E + 01	2.6663E + 01	1.2120E + 01	2.8773E + 01	1.3822E + 01	3.3743E + 01
F5	Ave	1.0383E + 03	2.0726E + 03	1.3654E + 03	1.9379E + 03	2.3719E + 03	2.2970E + 03	1.0070E + 03	9.0177E + 02	9.1352E + 02	9.0007E + 02
	Std	1.0354E + 02	3.6688E + 02	3.8817E + 02	2.4022E + 02	1.9599E + 02	3.8950E + 02	1.3365E + 02	3.0711E + 00	1.8635E + 01	1.7047E-01
F6	Ave	2.4566E + 06	3.8046E + 06	3.9871E + 06	3.6323E + 03	8.3119E + 03	2.3532E + 07	6.5140E + 03	3.4596E + 03	2.4422E + 04	2.3425E + 03
	Std	4.0310E + 06	2.3076E + 06	8.6673E + 06	7.1037E + 02	7.4167E + 03	1.2882E + 08	6.1952E + 03	2.2972E + 03	1.3489E + 04	2.2495E + 03
F7	Ave	2.1202E + 03	2.0607E + 03	2.0973E + 03	2.1086E + 03	2.1432E + 03	2.1281E + 03	2.0533E + 03	2.0412E + 03	2.0610E + 03	2.0558E + 03
	Std	6.1647E + 01	1.0945E + 01	4.5051E + 01	2.3940E + 01	8.9102E + 01	4.8008E + 01	1.8688E + 01	1.1010E + 01	8.5703E + 00	5.1924E + 01
F8	Ave	2.3130E + 03	2.2293E + 03	2.2747E + 03	2.2387E + 03	2.2891E + 03	2.2962E + 03	2.2277E + 03	2.2291E + 03	2.2308E + 03	2.2265E + 03
	Std	6.5558E + 01	1.9129E + 00	6.4900E + 01	3.9612E + 01	5.8777E + 01	1.0648E + 02	5.0235E + 00	2.3965E + 00	1.7099E + 00	2.1920E + 01
F9	Ave	2.5056E + 03	2.4821E + 03	2.5356E + 03	2.4814E + 03	2.4808E + 03	2.5358E + 03	2.4808E + 03	2.4813E + 03	2.4818E + 03	2.4808E + 03
	Std	3.4078E + 01	5.6894E-01	3.1568E + 01	2.9300E-01	4.1641E-03	1.1652E + 02	5.1761E-02	4.0274E-01	6.7237E-01	0.0000E + 00
F10	Ave	3.9970E + 03	2.5171E + 03	3.4732E + 03	2.7160E + 03	3.7391E + 03	4.5581E + 03	2.6820E + 03	2.5008E + 03	2.5134E + 03	2.6667E + 03
	Std	9.6800E + 02	3.1527E + 01	8.6601E + 02	3.7817E + 02	6.9940E + 02	1.0771E + 03	3.2759E + 02	1.2449E-01	4.7630E + 01	2.7838E + 02
F11	Ave	3.3971E + 03	3.1425E + 03	3.6040E + 03	2.9174E + 03	2.9300E + 03	3.9523E + 03	2.9068E + 03	2.9169E + 03	2.9204E + 03	2.9233E + 03
	Std	3.1238E + 02	1.7362E + 02	4.1661E + 02	3.7640E + 01	4.6609E + 01	1.1300E + 03	1.1285E + 02	3.7880E + 01	7.7937E + 01	5.0401E + 01
F12	Ave	2.9918E + 03	2.9742E + 03	2.9904E + 03	2.9816E + 03	2.9973E + 03	3.0818E + 03	2.9508E + 03	2.9602E + 03	2.9886E + 03	2.9539E + 03
	Std	5.0641E + 01	6.0412E + 00	3.6137E + 01	2.4370E + 01	5.8817E + 01	1.3465E + 02	1.4253E + 01	1.0183E + 01	1.1428E + 01	1.7849E + 01

The bold font represents the optimal result.

Fig. 16.

Comparison under averaged Rank and Std for CEC2022.

Comparison with CEC winners over four challenging CEC benchmarks

Comparison over CEC2014

In this section, the proposed ECPO and the CEC winners were each run independently 30 times, and the fitness values obtained by each algorithm on each CEC2014 test function were analyzed using Ave and Std, as shown in Table 7; Figs. 17 and 18. Among the 30 test functions, ECPO outperforms all competitors on 24, performs comparably to some on 4, and is less effective on the remaining functions. This demonstrates that ECPO is competitive and advantageous compared to some CEC winners. Figure 18 reports the average rankings. It shows that ECPO ranks first with a value of 1.47, LSHADE ranks second with a value of 2.47, while MadDE, as the worst-performing algorithm, ranks last with a value of 5.17. In summary, compared to the CEC winners, the proposed ECPO achieves outstanding results on most CEC2014 test functions.

Table 7.

Comparison between ECPO and CEC winners for CEC2014.

ID	Metric	AGSK	LSHADE	LSHADE-SPACMA	LSHADE-cnEpSin	MadDE	ECPO
F1	Ave	1.1883E + 06	1.0760E + 06	2.0831E + 06	1.9569E + 06	9.0234E + 06	5.2979E + 03
	Std	8.2415E + 05	6.5111E + 05	2.0823E + 06	1.4387E + 06	2.8831E + 06	1.0620E + 04
F2	Ave	1.1005E + 04	1.0313E + 04	2.7256E + 05	1.8585E + 05	2.0152E + 06	2.0000E + 02
	Std	1.6994E + 04	1.2341E + 04	1.1188E + 06	2.6342E + 05	1.5469E + 06	1.9748E-14
F3	Ave	3.2566E + 03	2.7260E + 03	4.0413E + 03	6.6080E + 03	4.6438E + 03	3.0000E + 02
	Std	2.8444E + 03	3.8221E + 03	2.5897E + 03	4.4742E + 03	1.9898E + 03	4.4783E-14
F4	Ave	4.9625E + 02	5.0145E + 02	5.0646E + 02	4.9300E + 02	5.1363E + 02	4.1035E + 02
	Std	3.1562E + 01	2.4693E + 01	2.1191E + 01	3.4165E + 01	1.4228E + 01	2.1391E + 01
F5	Ave	5.2107E + 02	5.2049E + 02	5.2138E + 02	5.2072E + 02	5.2069E + 02	5.2022E + 02
	Std	4.8568E-02	1.2254E-01	7.7828E-02	2.4131E-01	5.0913E-02	2.7504E-01
F6	Ave	6.1072E + 02	6.1445E + 02	6.4795E + 02	6.1603E + 02	6.2459E + 02	6.1002E + 02
	Std	2.5084E + 00	3.5213E + 00	2.8687E + 00	2.9953E + 00	1.9322E + 00	4.8654E + 00
F7	Ave	7.0005E + 02	7.0005E + 02	7.0024E + 02	7.0020E + 02	7.0085E + 02	7.0000E + 02
	Std	5.6672E-02	4.4982E-02	1.9734E-01	1.5511E-01	1.2233E-01	8.4277E-03
F8	Ave	9.2208E + 02	8.0409E + 02	8.1099E + 02	8.2036E + 02	8.5804E + 02	8.5880E + 02
	Std	6.5147E + 01	1.9660E + 00	3.1032E + 00	4.9614E + 00	8.3493E + 00	2.0042E + 01
F9	Ave	1.0891E + 03	9.6365E + 02	9.7798E + 02	9.7198E + 02	1.0498E + 03	9.8421E + 02
	Std	3.2449E + 01	1.2828E + 01	1.5113E + 01	1.7232E + 01	1.5422E + 01	4.3802E + 01
F10	Ave	8.0339E + 03	1.1473E + 03	1.9419E + 03	1.9441E + 03	2.6819E + 03	1.9271E + 03
	Std	4.0918E + 02	6.5319E + 01	4.2796E + 02	1.8147E + 02	3.2572E + 02	4.2963E + 02
F11	Ave	8.5560E + 03	4.5998E + 03	5.9499E + 03	5.3214E + 03	5.8895E + 03	4.5070E + 03
	Std	3.0311E + 02	3.7244E + 02	9.6014E + 02	8.4499E + 02	3.9240E + 02	7.1362E + 02
F12	Ave	1.2032E + 03	1.2008E + 03	1.2062E + 03	1.2017E + 03	1.2015E + 03	1.2005E + 03
	Std	4.3774E-01	3.0534E-01	9.6557E-01	1.1401E + 00	2.5057E-01	4.2492E-01
F13	Ave	1.3005E + 03	1.3005E + 03	1.3006E + 03	1.3005E + 03	1.3006E + 03	1.3004E + 03
	Std	9.5346E-02	1.0936E-01	1.5069E-01	1.2000E-01	9.0832E-02	1.2763E-01
F14	Ave	1.4003E + 03	1.4003E + 03	1.4004E + 03	1.4003E + 03	1.4003E + 03	1.4002E + 03
	Std	6.1156E-02	5.9900E-02	1.1509E-01	1.2909E-01	3.6942E-02	4.4840E-02
F15	Ave	1.5203E + 03	1.5125E + 03	1.5258E + 03	1.5164E + 03	1.5271E + 03	1.5044E + 03
	Std	4.1886E + 00	4.8095E + 00	2.0161E + 01	6.8895E + 00	3.8607E + 00	3.3940E + 00
F16	Ave	1.6132E + 03	1.6120E + 03	1.6131E + 03	1.6125E + 03	1.6127E + 03	1.6116E + 03
	Std	3.0887E-01	4.4512E-01	4.6303E-01	6.0944E-01	2.8887E-01	6.4348E-01
F17	Ave	3.7528E + 03	4.0987E + 03	1.8657E + 04	1.6587E + 04	1.7595E + 06	2.8415E + 03
	Std	7.7784E + 02	1.0397E + 03	4.0111E + 04	5.1766E + 04	9.3767E + 05	2.7263E + 02
F18	Ave	9.3203E + 03	6.4911E + 03	8.9900E + 03	4.2020E + 03	4.2329E + 05	2.4277E + 03
	Std	8.2773E + 03	6.9081E + 03	1.0272E + 04	3.4898E + 03	4.3192E + 05	1.2911E + 03
F19	Ave	1.9134E + 03	1.9168E + 03	1.9175E + 03	1.9185E + 03	1.9167E + 03	1.9066E + 03
	Std	2.1398E + 00	2.3483E + 00	2.6873E + 00	2.7405E + 00	2.4472E + 00	2.2823E + 00
F20	Ave	2.2234E + 03	2.7492E + 03	4.1231E + 03	2.7265E + 03	8.9790E + 03	2.0478E + 03
	Std	1.4542E + 02	2.6999E + 03	5.6878E + 03	7.7811E + 02	2.9667E + 03	2.7791E + 01
F21	Ave	4.1652E + 03	4.6816E + 03	8.0425E + 03	5.1220E + 03	9.5846E + 04	2.7386E + 03
	Std	1.8889E + 03	2.4348E + 03	5.1462E + 03	1.7883E + 03	5.4238E + 04	2.5943E + 02
F22	Ave	2.5671E + 03	2.3763E + 03	2.4894E + 03	2.3588E + 03	2.3791E + 03	2.3789E + 03
	Std	1.8143E + 02	1.1865E + 02	1.8923E + 02	1.0200E + 02	9.8589E + 01	1.5964E + 02
F23	Ave	2.6356E + 03	2.6357E + 03	2.6364E + 03	2.6362E + 03	2.6382E + 03	2.6356E + 03
	Std	7.6142E-02	1.5785E-01	6.6794E-01	5.1087E-01	1.4923E + 00	1.3876E-12
F24	Ave	2.6015E + 03	2.6018E + 03	2.6017E + 03	2.6017E + 03	2.6018E + 03	2.5981E + 03
	Std	3.1298E-01	4.3832E-01	4.7371E-01	2.9437E-01	2.0383E-01	9.9387E + 00
F25	Ave	2.7049E + 03	2.7056E + 03	2.7077E + 03	2.7067E + 03	2.7143E + 03	2.7034E + 03
	Std	1.1959E + 00	1.9364E + 00	3.5423E + 00	2.4875E + 00	2.6534E + 00	7.3611E-01
F26	Ave	2.7004E + 03	2.7004E + 03	2.7004E + 03	2.7004E + 03	2.7004E + 03	2.7003E + 03
	Std	6.5662E-02	6.2583E-02	9.5330E-02	1.3101E-01	5.2748E-02	9.0684E-02
F27	Ave	3.2358E + 03	3.3196E + 03	3.3899E + 03	3.4062E + 03	3.3704E + 03	3.2149E + 03
	Std	5.4261E + 01	6.6552E + 01	6.2113E + 01	9.5481E + 01	1.8572E + 02	1.0934E + 02
F28	Ave	3.2473E + 03	3.2634E + 03	3.2837E + 03	3.2856E + 03	3.3309E + 03	3.2350E + 03
	Std	2.0689E + 01	2.2959E + 01	3.6833E + 01	3.3417E + 01	3.1696E + 01	1.2268E + 01
F29	Ave	4.1099E + 06	4.8263E + 05	3.4272E + 08	1.6141E + 05	3.1685E + 04	5.3207E + 07
	Std	7.1162E + 06	2.6113E + 06	6.0661E + 08	3.7474E + 05	1.6392E + 04	1.0049E + 08
F30	Ave	4.0436E + 03	4.5147E + 03	5.2820E + 03	5.2640E + 03	5.9167E + 03	3.8547E + 03
	Std	4.2573E + 02	5.7004E + 02	1.2182E + 03	1.0320E + 03	6.2355E + 02	2.6969E + 02

The bold font represents the optimal result.

Fig. 17.

Comparison between ECPO and CEC winners under convergence speed on CEC2014.

Fig. 18.

Comparison between ECPO and CEC winners for CEC2014.

Comparison over CEC2017

In this section, the proposed ECPO was compared with the CEC winners on the CEC2017 benchmark suite. Each algorithm’s performance was analyzed based on average fitness values and standard deviations for each test function, as shown in Table 8. Out of 30 test functions, ECPO outperforms all CEC winners on 24 functions, ranks second on 3, third on 2, and fourth on 1, with no worst rankings. This indicates that ECPO achieves excellent results on most functions of CEC2017 and can be classified as a high-performance optimizer. Figure 19 displays the convergence speed curves for ECPO and the CEC winners, clearly showing that ECPO converges faster. Figure 20 reports the average rankings, demonstrating that ECPO is the top-performing algorithm, with LSHADE as the second-best and MadDE as the worst. The experimental results confirm that, compared to CEC winners, ECPO excels in most CEC2017 test functions.

Table 8.

Comparison between ECPO and CEC winners for CEC2017.

ID	Metric	AGSK	LSHADE	LSHADE-SPACMA	LSHADE-cnEpSin	MadDE	ECPO
F1	Ave	1.7259E + 04	1.1329E + 04	8.5100E + 04	9.4541E + 04	1.0241E + 06	1.0000E + 02
	Std	4.1592E + 04	1.2700E + 04	1.2586E + 05	1.5100E + 05	6.2836E + 05	5.2778E-15
F2	Ave	1.7123E + 20	7.6419E + 20	1.0000E + 30	5.8389E + 19	1.6639E + 20	4.2404E + 06
	Std	9.3784E + 20	4.1814E + 21	1.4314E + 14	2.5155E + 20	5.5449E + 20	2.0702E + 07
F3	Ave	6.7372E + 04	3.2626E + 04	5.2895E + 04	2.7917E + 04	8.0406E + 04	3.0000E + 02
	Std	1.8817E + 04	4.9486E + 04	5.6310E + 04	1.7230E + 04	1.3864E + 04	8.7319E-11
F4	Ave	5.0220E + 02	4.9141E + 02	5.0837E + 02	5.0244E + 02	5.1780E + 02	4.0995E + 02
	Std	2.7407E + 01	2.2388E + 01	2.9390E + 01	2.4106E + 01	1.1801E + 01	2.1548E + 01
F5	Ave	6.7639E + 02	5.6601E + 02	5.9006E + 02	5.7709E + 02	6.7514E + 02	5.8792E + 02
	Std	5.7587E + 01	1.4422E + 01	2.5469E + 01	2.0476E + 01	1.9214E + 01	6.4087E + 01
F6	Ave	6.0088E + 02	6.0130E + 02	6.0298E + 02	6.0597E + 02	6.1172E + 02	6.0226E + 02
	Std	6.7065E-01	1.2450E + 00	1.9688E + 00	3.5859E + 00	4.6242E + 00	8.3312E + 00
F7	Ave	9.3541E + 02	8.1786E + 02	8.3470E + 02	8.2836E + 02	9.5568E + 02	7.8752E + 02
	Std	1.8238E + 01	2.2780E + 01	3.1923E + 01	2.8322E + 01	3.1663E + 01	2.3352E + 01
F8	Ave	9.8576E + 02	8.6588E + 02	8.8288E + 02	8.7867E + 02	9.4707E + 02	8.8318E + 02
	Std	2.8889E + 01	1.6891E + 01	1.6340E + 01	1.9753E + 01	1.3334E + 01	3.3825E + 01
F9	Ave	9.5715E + 02	1.0896E + 03	1.3731E + 03	1.3649E + 03	3.6430E + 03	1.0682E + 03
	Std	5.3816E + 01	1.2639E + 02	3.1245E + 02	4.2510E + 02	6.8009E + 02	7.8305E + 02
F10	Ave	8.7103E + 03	4.6906E + 03	5.8339E + 03	5.0232E + 03	5.7660E + 03	4.5626E + 03
	Std	2.7456E + 02	3.2450E + 02	1.0707E + 03	7.8608E + 02	3.6902E + 02	6.1841E + 02
F11	Ave	1.2074E + 03	1.2858E + 03	1.2830E + 03	1.3026E + 03	1.3375E + 03	1.1681E + 03
	Std	3.8007E + 01	1.5041E + 02	5.7067E + 01	7.7381E + 01	3.6437E + 01	3.6788E + 01
F12	Ave	6.8253E + 04	1.5974E + 05	4.8855E + 05	5.1665E + 05	2.3885E + 06	1.1097E + 04
	Std	7.5777E + 04	1.4505E + 05	4.7657E + 05	6.7259E + 05	1.1506E + 06	8.8032E + 03
F13	Ave	1.2382E + 04	1.2275E + 04	2.3675E + 04	1.8601E + 04	8.2149E + 04	1.3942E + 03
	Std	1.2595E + 04	1.2344E + 04	1.9209E + 04	1.5835E + 04	1.1487E + 05	1.5614E + 02
F14	Ave	1.5411E + 03	1.5799E + 03	1.6335E + 03	1.9117E + 03	1.4306E + 05	1.4487E + 03
	Std	4.4786E + 01	6.9736E + 01	1.1542E + 02	1.2617E + 03	1.6546E + 05	2.5581E + 01
F15	Ave	1.7897E + 03	1.9832E + 03	3.5229E + 03	2.9574E + 03	8.8104E + 03	1.5442E + 03
	Std	1.1586E + 02	1.8291E + 02	3.3913E + 03	1.0597E + 03	5.7509E + 03	4.4867E + 01
F16	Ave	3.1226E + 03	2.6080E + 03	3.0709E + 03	2.4396E + 03	2.7269E + 03	2.2498E + 03
	Std	3.4883E + 02	1.8122E + 02	4.9000E + 02	2.9638E + 02	1.9809E + 02	2.6541E + 02
F17	Ave	2.1195E + 03	1.9797E + 03	2.5066E + 03	1.9845E + 03	2.0122E + 03	1.9480E + 03
	Std	2.3055E + 02	1.2491E + 02	3.4711E + 02	9.2996E + 01	9.7725E + 01	1.4652E + 02
F18	Ave	3.4527E + 04	2.9074E + 04	4.8097E + 04	3.7280E + 04	6.1395E + 05	2.1347E + 03
	Std	1.9714E + 04	2.4540E + 04	3.4937E + 04	2.1252E + 04	4.2672E + 05	6.4118E + 02
F19	Ave	2.0092E + 03	2.0617E + 03	3.2223E + 03	2.3559E + 03	1.0501E + 04	1.9298E + 03
	Std	5.0686E + 01	7.5116E + 01	4.8589E + 03	7.2243E + 02	7.1223E + 03	2.2556E + 01
F20	Ave	2.5512E + 03	2.3637E + 03	2.8651E + 03	2.3049E + 03	2.3985E + 03	2.4302E + 03
	Std	2.3840E + 02	1.2640E + 02	3.2533E + 02	1.2771E + 02	9.9023E + 01	1.9607E + 02
F21	Ave	2.4737E + 03	2.3606E + 03	2.3914E + 03	2.3810E + 03	2.4545E + 03	2.3512E + 03
	Std	3.5362E + 01	1.1650E + 01	2.0740E + 01	1.6092E + 01	2.1491E + 01	5.6460E + 01
F22	Ave	3.3158E + 03	2.7975E + 03	3.0870E + 03	3.0650E + 03	2.3210E + 03	3.9980E + 03
	Std	2.6258E + 03	1.1980E + 03	2.3638E + 03	1.4809E + 03	6.7423E + 00	1.9108E + 03
F23	Ave	2.7320E + 03	2.7191E + 03	2.7810E + 03	2.7452E + 03	2.8280E + 03	2.7009E + 03
	Std	7.2375E + 01	1.3798E + 01	6.2632E + 01	2.7444E + 01	2.9665E + 01	1.8441E + 01
F24	Ave	2.9574E + 03	2.8815E + 03	2.9667E + 03	2.9119E + 03	3.0032E + 03	2.8623E + 03
	Std	7.0176E + 01	1.3849E + 01	7.1060E + 01	2.0246E + 01	2.1774E + 01	1.5332E + 01
F25	Ave	2.8909E + 03	2.8929E + 03	2.8942E + 03	2.9030E + 03	2.9250E + 03	2.8866E + 03
	Std	8.0109E + 00	9.7851E + 00	1.0461E + 01	1.7191E + 01	1.8670E + 01	3.7342E + 00
F26	Ave	4.3382E + 03	4.2842E + 03	4.4795E + 03	4.5266E + 03	3.3607E + 03	3.7713E + 03
	Std	7.2391E + 02	5.5664E + 02	7.2095E + 02	2.3325E + 02	7.2307E + 02	7.5111E + 02
F27	Ave	3.2167E + 03	3.2274E + 03	3.4822E + 03	3.2346E + 03	3.2488E + 03	3.2089E + 03
	Std	9.7619E + 00	1.0831E + 01	5.4814E + 02	2.0341E + 01	1.0802E + 01	1.3097E + 01
F28	Ave	3.2299E + 03	3.2326E + 03	4.9679E + 03	3.2606E + 03	3.2806E + 03	3.1141E + 03
	Std	2.1850E + 01	2.2270E + 01	3.1500E + 03	3.9738E + 01	2.6256E + 01	3.6675E + 01
F29	Ave	3.8394E + 03	3.6453E + 03	4.2388E + 03	3.6815E + 03	3.9016E + 03	3.5182E + 03
	Std	3.1041E + 02	1.2342E + 02	5.7775E + 02	1.7236E + 02	1.2487E + 02	1.3075E + 02
F30	Ave	9.5640E + 03	1.4995E + 04	2.6215E + 04	3.8569E + 04	3.1199E + 05	5.2534E + 03
	Std	3.1086E + 03	5.9182E + 03	1.9048E + 04	5.9410E + 04	1.9944E + 05	2.2431E + 02

The bold font represents the optimal result.

Fig. 19.

Comparison between ECPO and CEC winners under convergence speed on CEC2017.

Fig. 20.

Comparison between ECPO and CEC winners for CEC2017.

Comparison over CEC2020

In this section, ECPO and the CEC winners were applied to solve the CEC2020 test suite. The analysis of 30 independent fitness values is shown in Table 9. The results indicate that ECPO achieved the best performance on 6 out of the 10 test functions. Figure 21 displays the convergence speed curves for ECPO and the CEC winners, clearly illustrating that ECPO converges faster and with better precision. As a result, ECPO can be classified as a high-performance optimizer, achieving outstanding rankings on most CEC2020 test functions. Figure 22 further shows that ECPO’s average ranking matches that of LSHADE with a value of 2, outperforming other CEC winners. The experiments in this section further confirm the high performance of ECPO.

Table 9.

Comparison between ECPO and CEC winners for CEC2020.

ID	Metric	AGSK	LSHADE	LSHADE-SPACMA	LSHADE-cnEpSin	MadDE	ECPO
F1	Ave	1.2881E + 03	1.5150E + 03	4.6560E + 03	1.0144E + 06	5.6793E + 04	1.0000E + 02
	Std	2.4454E + 03	1.9515E + 03	3.6748E + 03	1.2947E + 06	5.7825E + 04	0.0000E + 00
F2	Ave	5.2126E + 03	1.5796E + 03	1.8326E + 03	2.9068E + 03	2.1208E + 03	1.9714E + 03
	Std	2.6072E + 02	1.3292E + 02	1.9783E + 02	3.3167E + 02	1.9706E + 02	4.6336E + 02
F3	Ave	8.1940E + 02	7.3347E + 02	7.3563E + 02	7.8944E + 02	7.7096E + 02	7.4450E + 02
	Std	2.0649E + 01	5.4722E + 00	4.7624E + 00	1.8147E + 01	9.2495E + 00	1.2953E + 01
F4	Ave	1.9091E + 03	1.9025E + 03	1.9036E + 03	1.9134E + 03	1.9069E + 03	1.9022E + 03
	Std	1.4000E + 00	4.5817E-01	1.8732E + 00	5.7121E + 00	1.1273E + 00	6.8770E-01
F5	Ave	5.3302E + 03	5.4268E + 03	7.0705E + 03	8.8230E + 04	2.8209E + 05	2.2952E + 03
	Std	3.6978E + 03	5.8610E + 03	5.8457E + 03	1.5022E + 05	1.3758E + 05	2.1724E + 02
F6	Ave	1.8544E + 03	1.8544E + 03	1.8544E + 03	1.8544E + 03	1.8544E + 03	1.8544E + 03
	Std	2.2286E + 02	2.2286E + 02	2.2286E + 02	2.2286E + 02	2.2286E + 02	2.2286E + 02
F7	Ave	2.7274E + 03	2.4407E + 03	3.0523E + 03	6.1199E + 03	7.9618E + 04	2.3808E + 03
	Std	7.2507E + 02	1.4825E + 02	7.2509E + 02	3.0550E + 03	5.4645E + 04	1.3735E + 02
F8	Ave	2.3007E + 03	2.3130E + 03	2.3012E + 03	2.7114E + 03	2.3016E + 03	2.9101E + 03
	Std	8.7396E-01	4.6135E + 01	1.5594E + 00	1.2123E + 03	1.3507E + 00	1.2080E + 03
F9	Ave	2.8673E + 03	2.8218E + 03	2.8534E + 03	2.8706E + 03	2.8356E + 03	2.8263E + 03
	Std	4.1829E + 01	5.2302E + 00	2.0603E + 01	2.2174E + 01	7.9763E + 01	1.3081E + 01
F10	Ave	2.9335E + 03	2.9257E + 03	2.9506E + 03	2.9591E + 03	2.9685E + 03	2.9286E + 03
	Std	2.8356E + 01	2.6781E + 01	3.2454E + 01	3.1574E + 01	2.6969E + 01	2.5023E + 01

The bold font represents the optimal result.

Fig. 21.

Comparison between ECPO and CEC winners under convergence speed on CEC2020.

Fig. 22.

Comparison between ECPO and CEC winners for CEC2020.

Comparison over CEC2022

In this section, we evaluate the performance of ECPO using the challenging CEC2022 benchmark suite, which consists of 12 test functions. Table 10 presents the analysis of fitness values obtained by each algorithm across the experiments. The table shows that ECPO ranks first on 8 test functions, second on 1, third on 1, and does not have the worst ranking on any function. Figure 23 demonstrates that ECPO exhibits faster convergence speed and better convergence precision. However, it also has some drawbacks, such as in the case of CEC2022-F7, where its performance is inferior to LSHADE and AGSK. In this instance, these two winners were able to maintain a faster convergence speed and find better solutions, while ECPO became trapped in a local optimum. Moreover, Fig. 24 reveals that ECPO’s average ranking is superior to all CEC winners, with a value of 1.83. This is followed by LSHADE and AGSK with values of 2.04 and 2.83, respectively, while MadDE has the worst ranking of 5. Overall, this section further confirms the exceptional performance of ECPO.

Table 10.

Comparison between ECPO and CEC winners for CEC2022.

ID	Metric	AGSK	LSHADE	LSHADE-SPACMA	LSHADE-cnEpSin	MadDE	ECPO
F1	Ave	2.4193E + 03	3.2975E + 03	1.2910E + 04	2.2186E + 04	2.0425E + 04	3.0000E + 02
	Std	1.2082E + 03	9.9514E + 03	1.8814E + 04	1.3790E + 04	4.6239E + 03	0.0000E + 00
F2	Ave	4.5229E + 02	4.5147E + 02	4.5073E + 02	4.6435E + 02	4.5776E + 02	4.2616E + 02
	Std	8.9388E + 00	1.2512E + 01	1.5302E + 01	1.5081E + 01	1.0949E + 01	2.4127E + 01
F3	Ave	6.0007E + 02	6.0021E + 02	6.0051E + 02	6.1044E + 02	6.0172E + 02	6.0082E + 02
	Std	8.6984E-02	3.0861E-01	5.4555E-01	6.5345E + 00	7.0094E-01	4.4060E + 00
F4	Ave	9.1102E + 02	8.2593E + 02	8.4035E + 02	8.6532E + 02	8.6839E + 02	8.4802E + 02
	Std	8.9256E + 00	8.5514E + 00	1.4599E + 01	1.3821E + 01	1.1666E + 01	2.4403E + 01
F5	Ave	9.0323E + 02	9.3033E + 02	9.4244E + 02	1.3159E + 03	1.5906E + 03	9.0004E + 02
	Std	2.6831E + 00	3.4536E + 01	7.1663E + 01	5.7025E + 02	3.1628E + 02	8.8205E-02
F6	Ave	6.7402E + 03	3.3709E + 03	2.9103E + 03	7.5112E + 03	5.8448E + 04	2.4829E + 03
	Std	5.9792E + 03	3.2795E + 03	2.3897E + 03	6.0551E + 03	3.2675E + 04	1.5033E + 03
F7	Ave	2.0338E + 03	2.0415E + 03	2.1634E + 03	2.0847E + 03	2.0598E + 03	2.0444E + 03
	Std	6.7598E + 00	1.0659E + 01	9.2983E + 01	2.5174E + 01	1.4559E + 01	3.4149E + 01
F8	Ave	2.2299E + 03	2.2285E + 03	2.3254E + 03	2.2435E + 03	2.2266E + 03	2.2252E + 03
	Std	3.4965E + 00	2.1790E + 01	2.4830E + 02	3.0790E + 01	1.6252E + 00	2.1529E + 01
F9	Ave	2.4808E + 03	2.4808E + 03	2.4809E + 03	2.4821E + 03	2.4810E + 03	2.4808E + 03
	Std	3.7238E-07	4.0501E-05	1.6977E-01	2.3950E + 00	1.5305E-01	0.0000E + 00
F10	Ave	2.7769E + 03	2.4895E + 03	2.5295E + 03	3.1550E + 03	2.5143E + 03	2.7520E + 03
	Std	8.7539E + 02	7.5248E + 01	6.5223E + 01	6.7992E + 02	5.0050E + 01	3.6776E + 02
F11	Ave	2.9210E + 03	2.9274E + 03	2.9211E + 03	3.0233E + 03	2.9048E + 03	2.8951E + 03
	Std	7.7361E + 01	7.0572E + 01	9.1838E + 01	1.4575E + 02	7.7583E + 01	1.4678E + 02
F12	Ave	2.9439E + 03	2.9454E + 03	3.7007E + 03	2.9206E + 03	2.9608E + 03	2.9536E + 03
	Std	8.7751E + 00	8.0625E + 00	3.6191E + 02	4.0136E + 01	9.6153E + 00	3.9077E + 01

The bold font represents the optimal result.

Fig. 23.

Comparison between CPO and CEC winners under convergence speed on CEC2022.

Fig. 24.

Comparison between ECPO and CEC winners for CEC2022.

Computational cost analysis

In this section, we estimate the average computational cost of each algorithm using CPU time before the completion of the optimization process, as shown in Fig. 25. The results indicate that ECPO is significantly faster compared to most other optimizers, being competitive with certain algorithms and noticeably superior to others. For example, LSHADE has the lowest computational cost at 0.166 s, followed by PSO with a cost of 0.179 s. ECPO, proposed in this study, ranks 5th out of 15 algorithms with a cost of 0.235 s, outperforming the winner algorithms AGSK, LSHADE-SPACMA, and LSHADE-cnEpSin, which have costs of 0.236, 0.353, and 0.49 s, respectively. While the computational cost of ECPO is slightly higher by 0.035 s compared to the original CPO (0.2 s), the experimental results and these figures demonstrate that ECPO provides better final precision on most test functions compared to the original CPO and other comparison algorithms. This slight increase in computational cost is negligible given the improved performance of ECPO.

Fig. 25.

Comparison of the average computing cost of each algorithm.

Effectiveness analysis

In this section, the overall effectiveness of the ECPO algorithm and other competitive algorithms is calculated using Eq. (21) and reported in Table 11. The parameters used are as follows: is the total number of test functions, is the number of test functions where ECPO outperforms the competitors, is the number of test functions where ECPO ties with the competitors, and is the number of test functions where ECPO performs worse than the competitors⁷⁶. From Table 11, it can be observed that ECPO achieves outstanding results on the CEC2014, CEC2017, CEC2020, and CEC2022 benchmark suites with overall effectiveness percentages of 93.81%, 93.33%, 75.71%, and 86.90%, respectively. These results significantly surpass those of the other compared algorithms. Based on these outcomes, ECPO can be classified as a high-performance optimizer.

21

Table 11.

Effectiveness of the ECPO and other competitor algorithms.

Test suite	CEC2014(W/T/L)	CEC2017(W/T/L)	CEC2020(W/T/L)	CEC2022(W/T/L)
ECPO VS. PSO	29/0/1	30/0/0	9/1/0	12/0/0
ECPO VS. DE	29/0/1	29/0/1	7/1/2	10/0/2
ECPO VS. GWO	29/0/1	30/0/0	9/1/0	12/0/0
ECPO VS. RUN	29/0/1	29/0/1	9/1/0	9/0/3
ECPO VS. SSA	29/0/1	30/0/0	9/1/0	12/0/0
ECPO VS. BKA	30/0/0	30/0/0	9/1/0	12/0/0
ECPO VS. SBOA	27/0/3	27/0/3	8/1/0	11/0/1
ECPO VS. APO	28/0/2	27/0/3	6/1/3	10/0/2
ECPO VS. CPO	29/0/1	29/0/1	7/1/2	9/0/3
ECPO VS. AGSK	29/0/1	27/0/0	8/1/1	9/0/3
ECPO VS. LSHADE	25/0/5	25/0/5	4/1/5	7/0/5
ECPO VS. LSHADE-SPACMA	28/0/2	28/0/2	6/1/3	9/0/3
ECPO VS. LSHADE-cnEpSin	26/0/4	26/0/4	8/1/1	11/0/1
ECPO VS. MadDE	28/0/2	27/0/3	8/1/1	11/0/1
Total(W/T/L)	394/0/26	392/0/28	106/14/20	146/0/22
	93.81%	93.33%	75.71%	86.90%

ECPO for wireless sensor network node deployment

WSN mathematical model

We consider the sensor’s perception model, as illustrated in Fig. 26, where two circles are centered around the sensor. The first circle represents the sensing area , with being the radius of the disk. is the radius of the communication range. We assume that . It has been shown that if the communication range is at least twice the sensing range, it is sufficient to ensure full coverage of the convex region, which implies connectivity between active nodes^77,78. However, this assumption of omnidirectional sensing capability does not apply to certain types of sensor nodes, such as cameras with directional sensing areas or ultrasonic sensors. As depicted in Fig. 27, if the sensing area is represented as a circle, the diameter of this circle corresponds to the diagonal of a square with the same area.

Fig. 26.

Perception model of sensor node.

Fig. 27.

The sensing area of sensor in terms of square.

Sensors are powered by batteries, which means they have a limited operational time. To extend battery life, energy-efficient and coverage protocols are essential. The probability of event detection is inversely proportional to the Euclidean distance between the sensor and the event. In wireless sensor networks (WSNs), there are two sensor detection models to determine effective coverage: the binary detection model, which assumes no uncertainty, and the probabilistic sensing model, which incorporates random detection probabilities⁷⁹. The probabilistic coverage model better reflects the performance of real sensors in an environment. The binary sensing model is the simplest and most widely analyzed coverage model provided by sensors. In this model, a sensor detects all events within its sensing range. In this paper, we use the binary detection model as described by Eq. (22)⁸⁰:

22

Here, represents the probability that an event occurring at position is detected by a sensor located at position . denotes the Euclidean distance between the sensor and the event.

The objective function proposed in this study primarily focuses on coverage. We first assume that the sensor nodes are denoted as and the deployment space as . The coverage radius of the sensor is , and the covered points are denoted as . The coverage equation is given by Eq. (23):

23

Here, is the Euclidean distance from sensor to point , calculated using Eq. (24).

24

The coverage probability ratio for the deployment area is as follows:

25

where, and represent the length and width of the deployment space. The goal of this problem is to maximize network coverage. This is a maximization problem, so the fitness function is defined as:

26

Experimental results of WSN

In the experiments, nodes of the wireless sensor network are randomly deployed in an area. Specifically, nodes are deployed in a area, with the sensor communication radius . To compare the performance of the algorithms, all parameters in the fitness function are set identically across all algorithms. To ensure fairness, 30 independent runs were conducted using the same environment and parameter settings to eliminate randomness, and the relevant experimental results were recorded. For each comparative algorithm, the minimum (Min), maximum (Max), median (Median), mean (Mean), and standard deviation (Std) were statistically analyzed, with the optimal values highlighted in bold in Table 12. The convergence curves of different algorithms in the WSN are shown in Fig. 28, and the optimized deployments for various algorithms are illustrated in Figs. 29 and 30.

Table 12.

Statistics of WSN experiment results.

Algorithm	Min	Median	Max	Ave	Std	Rank
PSO	0.72	0.763	0.794	0.7632	0.017743	12
DE	0.7448	0.7608	0.7764	0.760373	0.007905	13
GWO	0.8044	0.8318	0.8532	0.833827	0.012981	5
RUN	0.77	0.8148	0.8404	0.811253	0.01525	6
SSA	0.7476	0.7832	0.8152	0.78048	0.014777	9
BKA	0.6808	0.8004	0.8352	0.782027	0.046273	8
SBOA	0.8172	0.8424	0.8612	0.841507	0.010935	2
APO	0.7128	0.7824	0.844	0.775387	0.036882	10
AGSK	0.6844	0.7058	0.8512	0.72328	0.04167	15
LSHADE	0.824	0.8348	0.8488	0.836093	0.014695	3
LSHADE-SPACMA	0.7168	0.774	0.8232	0.768187	0.026252	11
LSHADE-cnEpSin	0.7616	0.7876	0.8216	0.785653	0.014211	7
MadDE	0.8128	0.8364	0.86	0.835907	0.011983	4
CPO	0.7332	0.7484	0.7692	0.747427	0.00833	14
ECPO	0.8124	0.8504	0.8732	0.849533	0.006385	1

The bold font represents the optimal result.

Fig. 28.

WSN convergence curve.

Fig. 29.

Compared with the results obtained by other algorithms in WSN.

Fig. 30.

Compared with the results obtained by CEC winners in WSN.

As observed from Table 12, the proposed ECPO demonstrates superior performance in WSNs. Except for the minimum value (Min), where ECPO does not outperform LSHADE, it significantly surpasses other comparative algorithms in terms of the maximum value (Max), median (Median), and mean (Mean). This indicates that ECPO achieves better results and higher coverage. Furthermore, the mean value is considerably lower than that of other comparative algorithms, demonstrating that ECPO exhibits excellent robustness for this problem, outperforming other methods. These results underscore the outstanding optimization performance of the proposed ECPO, making it a powerful tool for addressing the node placement problem in wireless sensor networks.

As shown in Fig. 28, ECPO converges rapidly to a solution close to the optimal, with a convergence speed superior to that of the CEC winners. Although LSHADE-SPACMA, BKA, and CPO exhibit faster convergence in the early stages, their convergence speed slows down significantly with increasing iterations. This indicates that the proposed ECPO has excellent potential for solving the node placement optimization problem in wireless sensor networks. Figures 29 and 30 reveal that compared to the deployments optimized by other algorithms, the node distribution optimized by ECPO is more rational, with minimal overlap and very uniform placement of each node. In contrast, algorithms like AGSK, LSHADE-SPACMA, BKA, DE, PSO, and CPO result in many overlapping deployed nodes, which is highly inefficient and leads to greater resource wastage. The well-organized layout produced by ECPO improves energy efficiency, coverage, data accuracy, network reliability, cost-effectiveness, and latency, contributing to a more sustainable, reliable, and economical network overall.

Summary and future directions

This paper presents an improved version of the Crested Porcupine Optimizer (ECPO), which addresses the shortcomings of the original CPO, such as slow convergence speed, local optima entrapment, and low convergence accuracy. ECPO enhances CPO’s overall performance by incorporating four improvement strategies: the Sobol sequence population initialization strategy, the guided search strategy based on the global optimal solution, the adaptive Levy flight search strategy, and the population renewal strategy based on centroid opposition-based learning. The performance of ECPO was validated on four benchmark test suites: CEC2014, CEC2017, CEC2020, and CEC2022. Its results were compared with classic algorithms (PSO, DE, GWO, RUN, and SSA), recently proposed algorithms (BKA, SBOA, APO, and CPO), and CEC-winning algorithms (AGSK, LSHADE, LSHADE-SPACMA, LSHADE-cnEpSin, and MadDE). Statistical analysis indicates that ECPO shows significant advantages on most test functions, demonstrating its potential as a high-performance optimizer. Among these four benchmark suites, ECPO achieved outstanding performance on 93.81%, 93.33%, 75.71%, and 86.90% of the test functions, far surpassing other comparative algorithms. Additionally, compared to the CEC benchmark winners, ECPO also outperformed the winning algorithms on most test functions. Finally, the application of ECPO to wireless sensor network node layout optimization further confirms its exceptional performance and broad applicability.

Despite the positive outcomes, this study has certain limitations: ECPO’s key parameters (e.g., the fixed constants and in adaptive Levy flight, minimum population size were determined through empirical trials rather than a systematic adaptive tuning framework, which may restrict its adaptability to diverse problem characteristics—for instance, in high-dimensional functions like CEC2022-F7, fixed parameters occasionally hinder optimal exploration-exploitation balance, leading to local optima. Its applicability is currently limited to continuous single-objective problems (validated via CEC benchmarks and single-objective WSN coverage optimization) and cannot be directly applied to discrete scenarios (e.g., logistics scheduling) or multi-objective problems (e.g., WSN deployment considering both coverage and energy consumption) due to the continuous-oriented design of its core strategies (e.g., centroid-based reverse learning). Moreover, while ECPO’s computational cost (0.235 s) is competitive overall, it is slightly higher than the original CPO (0.2 s) due to integrated improvement strategies, and in large-scale scenarios (e.g., WSN with over 100 nodes or high-dimensional functions over 100 dimensions), repeated calculations (e.g., Gamma function in adaptive Levy flight, centroid computation) may increase runtime and reduce real-time efficiency. Additionally, all experiments were conducted in static environments, and ECPO has not been tested in dynamic scenarios (e.g., WSN with mobile nodes), where its current strategies may struggle to adapt to sudden search space changes, limiting its practicality for dynamic real-world problems.

Future research should focus on developing binary and multi-objective variants of the ECPO algorithm to comprehensively assess its performance. Additionally, this theoretical framework can be applied to problems such as optimizing renewable energy systems, wind farm layouts, photovoltaic models, and fuel cell parameter estimation. The algorithm can also be utilized to address combinatorial optimization problems, including logistics route optimization, production scheduling, and vehicle routing in intelligent transportation systems, as well as traffic signal control and fleet scheduling issues. In the field of communication networks, the ECPO algorithm can be applied to optimize resource allocation and task scheduling, particularly in fog computing and cloud computing for task distribution and load balancing. Path planning, target searching, task allocation, and coordination optimization in drone systems can also benefit from more effective solutions using this algorithm. Moreover, practical and complex problems in blockchain networks, such as joint resource allocation and mining decision-making, diagnostic decision support in healthcare, personalized treatment plan optimization, production line optimization in smart manufacturing, and land use optimization in urban planning, can be studied using this algorithm. By exploring these applications, the potential of the ECPO algorithm across multiple domains can be further uncovered, potentially enhancing its value in solving complex optimization problems.

Author contributions

Feng Zhao: Conceptualization, Methodology, Writing - original draft, Formal analysis, Data curation, Writing - review & editing, Software, Resources, Supervision, Methodology, Visualization Resources. Zenghe Li: Visualization, Formal analysis, Writing - review & editing, Conceptualization, Resources, Supervision, Formal analysis. Peng Guo: Methodology, Writing - original draft, Formal analysis. Aiping Zhao: Writing - review & editing, Supervision, Methodology, Visualization Resources. Youfa Fu: Formal analysis, Writing - review & editing, Conceptualization.

Funding

This research was funded by the Research Project of the Human Resources and Social Security Department in Hebei Province. (JRSHZ-2025-01126).

Data availability

All data generated or analysed during this study are included in this published article.

Declarations

Competing interests

The authors declare no competing interests.