EAPO: A Multi-Strategy-Enhanced Artificial Protozoa Optimizer and Its Application to 3D UAV Path Planning

Abstract

Three-dimensional unmanned aerial vehicle (UAV) path planning presents a challenging optimization problem characterized by high dimensionality, strong nonlinearity, and multiple constraints. To address these complexities, this study proposes an Enhanced Protozoan Optimizer (EAPO) by refining the initialization, behavioral decision-making, environmental perception, and population diversity preservation mechanisms of the original Protozoan Optimizer. Specifically: Latin hypercube sampling enriches initial population diversity; a behavior adaptation mechanism based on historical success dynamically adjusts the exploration-exploitation balance; environmental structure modeling using perception fields enhances local exploitation capabilities; an adaptive hibernation-reconstruction strategy boosts global escape ability. Ablation experiment validates the effectiveness of each enhancement module, while exploration-exploitation analysis demonstrates EAPO maintains an optimal balance throughout the optimization process. Comprehensive evaluations using CEC2022 and CEC2020 benchmark datasets, ten real-world engineering design problems, and four drone path planning scenarios of varying scales and complexities further validate its excellent performance. Experimental results demonstrate that EAPO outperforms the baseline APO and twelve advanced optimizers in convergence accuracy, stability, and robustness. In UAV path planning applications, paths generated by EAPO satisfy all constraints and outperform APO-generated paths across multiple path quality evaluation metrics concerning safety, smoothness, and energy consumption. Compared to APO, EAPO achieved average fitness improvements of 14.0%, 4.5%, 8.7%, and 31.42% across the four maps, respectively, fully demonstrating its practical value and formidable capability in tackling complex engineering optimization problems.

1. Introduction

In recent years, unmanned aerial vehicles (UAVs) have emerged as a new type of aircraft, finding extensive applications in numerous fields such as power line inspections, complex terrain exploration, disaster relief, and military reconnaissance. In these diverse and complex mission scenarios, UAV path planning is a core component for ensuring autonomous navigation. Its objective is to devise the safest and most efficient optimal flight trajectory at the lowest possible cost, while satisfying flight constraints and avoiding obstacles and threat zones [1]. However, path planning faces multiple challenges due to terrain variations in three-dimensional space, the distribution of threat sources, and the UAV’s own physical characteristics. The objective is to plan a safe, efficient, and optimal flight trajectory at the lowest possible cost while satisfying flight constraints and avoiding obstacles and threat zones [2]. However, due to terrain variations in three-dimensional space, the distribution of threat sources, and the UAV’s own physical constraints, path planning is inherently a complex optimization problem characterized by high dimensionality, nonlinearity, and multiple constraints. This places extremely high demands on the global search capability and convergence accuracy of algorithms [3].

In path planning research, traditional algorithms such as Dijkstra’s algorithm (1959) [4], A-star (1968) [5], Artificial Potential Field (APF, 1985) [6], and Rapidly Exploring Random Tree (RRT, 1996) [7] possess theoretical foundations. However, they often face issues such as low computational efficiency and susceptibility to local optima in high-dimensional complex environments. In recent years, meta-heuristic algorithms such as Particle Swarm Optimization (PSO, 1995) [8], Differential Evolution Algorithm (DE, 1995) [9], The Whale Optimization Algorithm (WOA, 2016) [10], Harris Hawks Optimization (HHO, 2019) [11], Arithmetic Optimization Algorithm (AOA, 2021) [12], Dung Beetle Optimizer (DBO, 2023) [13], and their variants have become effective tools for solving such NP-hard problems due to their robust global optimization capabilities and adaptability. Zhao et al. [14] proposed an unmanned vehicle path planning method based on the Adaptive Particle Swarm Optimization (APSO) algorithm. Through simulation experiments conducted on grid maps preprocessed by a map simplification strategy, they demonstrated that this method outperforms the original algorithm and comparison algorithms. Freitas et al. [15] introduced a novel path planner, DE3D-NURBS, based on the differential evolution (DE) algorithm for three-dimensional path planning with maximum and minimum ascent/descent angles and maximum curvature for robots. Simulation results demonstrated the proposed planner’s capability to generate feasible paths. Li et al. [16] introduced a learning-based approach through constructing a reverse population, incorporating a random convergence factor and endpoint random neighborhood perturbation. This yielded an improved whale optimization algorithm (IWOA) considering a global perspective, applied to trajectory planning for small unmanned aerial vehicle (UAV) swarms. You et al. [17] proposed a hybrid strategy Harris-Hawthorne optimization algorithm (MSHHO) by integrating four innovative strategies to enhance HHO’s effectiveness in local development and global exploration. Experiments demonstrated MSHHO’s efficacy in overcoming complex optimization challenges. To conserve UAV resources, reduce computational time, and enhance solution accuracy, Wang et al. [18] introduced compact and parallel techniques into the AOA algorithm, resulting in a parallel and compact AOA algorithm (PCAOA). Chen et al. [19] applied multiple strategies, including an adaptive Cauchy mutation policy, to the DBO algorithm, proposing an improved dung beetle optimizer (IDBO) for UAV path planning. Simulations validated that the proposed improved algorithm effectively solves UAV path planning problems.

The Artificial Protozoa Optimizer (APO) is a novel meta-heuristic algorithm proposed by Wang et al. [20] in 2024. It guides the search process by simulating the foraging, hibernation, and reproduction behaviors of protozoa, demonstrating outstanding performance in solving diverse and complex optimization problems. Nevertheless, the APO algorithm still exhibits deficiencies in convergence accuracy and stability when handling complex problems, necessitating further refinement and enhancement. In this regard, numerous valuable explorations have been conducted by scholars worldwide. Elamy et al. [21] proposed a machine learning approach utilizing the Artificial Protozoa Optimizer to predict the tribological properties of Cu-ZrO_2 nanocomposites. Liao et al. [22] integrated the Sine-Cosine Algorithm (SCA) and Leader Selection Strategy within the APO framework, thereby proposing the Sine-Cosine Multi-Objective APO (LSCMOAPO) algorithm. Its performance was validated on benchmark functions and five practical engineering problems. Wang et al. [23] introduced a novel membrane algorithm based on the Artificial Protozoa Optimizer (MAAPO), incorporating a parallel distributed paradigm within the Membrane Computation (MC) framework and an autotrophic model enhanced within APO. They applied MAAPO to solve multi-level threshold image segmentation problems using Otsu and Kapur entropy as objective functions. Bo et al. [24] enhanced APO’s efficiency and solution accuracy by implementing cyclone foraging strategies during the foraging phase and hybrid mutation strategies during the reproduction phase. These works lay a crucial foundation for APO algorithm improvements and applications, while also indicating that introducing more advanced search strategies and adaptive mechanisms holds promise for further performance enhancement.

This paper aims to systematically enhance the APO algorithm by proposing an Enhanced Artificial Protozoa Optimizer (EAPO). The core improvements include integrating a Latin hypercube sampling initialization strategy to ensure population diversity at the outset, introducing a behaviorally adaptive selection mechanism based on past success experience to dynamically balance exploration and exploitation, and combining a perceptual field-based environmental modeling strategy with an adaptive hibernation strategy to enhance the algorithm’s optimization efficiency and robustness in complex environments. To comprehensively validate EAPO’s effectiveness, this study conducts systematic testing across three benchmarks: First, evaluating its numerical optimization performance on the CEC2022 and CEC2020 benchmark suite; second, applying it to ten real-world engineering design problems; and finally, performing simulation validation in the more challenging scenario of UAV 3D path planning.

The primary contribution of this paper lies in proposing a systematic set of APO enhancement strategies and constructing the EAPO algorithm. Through comprehensive multi-dimensional and multi-scenario experimental validation, the superiority and practicality of EAPO in handling complex optimization problems have been fully demonstrated. The structure of this paper is as follows: Section 1 provides an introduction; Section 2 details the problem modeling for UAV path planning; Section 3 explains the fundamental principles of the original APO algorithm; Section 4 focuses on the various improvement strategies of the EAPO algorithm; Section 5 and Section 6 present experimental results and analyses of the algorithm on the CEC2022 benchmark suite, CEC2020 benchmark suite, engineering optimization problems, and UAV path planning, respectively; the Section 8 summarizes the research work and offers future directions.

2. Modeling of UAV Path Planning

This section addresses the modeling issues in UAV path planning from the perspectives of decision variables, objective functions, and cost function components [25,26,27,28].

2.1. Decision Variables

The UAV path is represented by $n$ waypoints. For planning convenience, each waypoint is described using spherical coordinates, with its decision variable vector $x$ defined as follows:

$$\begin{array}{c} x=\left[r_1,r_2,\cdots ,r_n,{\theta }_1,{\theta }_2,\cdots ,{\theta }_n,{\phi }_1,{\phi }_2,\cdots ,{\phi }_n,v\right]\end{array}$$

(1)

where $r_i$ represents the radial distance of the $i$th waypoint; ${\theta }_i$ represents the elevation angle of the $i$th waypoint, primarily used for vertical obstacle avoidance, ${\theta }_i\in \left[-{\displaystyle \frac{\pi }{4}},{\displaystyle \frac{\pi }{4}}\right]$; ${\phi }_i$ represents the azimuth angle of the $i$th waypoint, primarily used for horizontal detouring; $v$ represents the flight speed of the UAV.

The schematic diagram of the decision variable space is shown in Figure 1. Through the conversion function from spherical coordinates to Cartesian coordinates, the actual position sequence of the drone in three-dimensional space can be obtained as follows:

$$\begin{array}{c} \left\{\begin{array}{c}{x}_i=x_0+{\displaystyle \sum _{j=1}^i}{r}_j·cos{\theta }_j·cos{\phi }_i\\ y_i=y_0+{\displaystyle \sum _{j=1}^i}{r}_j·cos{\theta }_j·cos{\phi }_i\\ z_i=z_0+{\displaystyle \sum _{j=1}^i}{r}_j·cos{\theta }_j·cos{\phi }_i\end{array}\right.\end{array}$$

(2)

where $x_0$, $y_0$, and $z_0$ represent the initial positions in the $x$, $y$, and $z$ directions, respectively.

2.2. Objective Function

The goal of path planning is to find an optimal solution that minimizes the total objective function value. The total objective function $F_T$ consists of a weighted cost function $F_W$ and an additional penalty term $F_P$, as shown in Equation (3).

$$\begin{array}{c} F_T=F_W+F_P\end{array}$$

(3)

2.2.1. Weighted Cost Function

The weighted cost function is a linearly weighted sum of the sub-cost functions, expressed as:

$$\begin{array}{c} F_W={\omega }_1·f_1+{\omega }_2·f_2+{\omega }_3·f_3+{\omega }_4·f_4+{\omega }_5·f_5+{\omega }_6·f_6\end{array}$$

(4)

where $f_1$ to $f_6$ and ${\omega }_1$ to ${\omega }_6$ respectively represent flight cost, altitude cost, threat cost, angle constraint, trajectory segment constraint, and no-fly zone constraint along with their respective weights, with the sum of all weights equaling 1. Threat cost and altitude cost are assigned higher weights to emphasize obstacle avoidance and safe flight.

2.2.2. Penalty Term

To strictly enforce avoidance of threat zones and no-fly zones, an additional penalty term is imposed beyond the weighted cost, as shown in Equation (5):

$$\begin{array}{c} F_P=100·\left(N_{rc}+N_{ac}\right)+10·f_6\end{array}$$

(5)

where $N_{rc}$ and $N_{ac}$ represent the number of times the path traverses the radar and artillery threat zones, respectively. The penalty coefficients (100 and 10) are fixed values determined through empirical analysis. The coefficient 100 for threat violations ensures that any infeasible solution (crossing threat zones) has a fitness value significantly worse than any feasible solution, effectively creating a barrier in the search space. The coefficient 10 for no-fly zones reflects their lower criticality compared to threat zones. These values remain constant throughout the optimization process to maintain consistent constraint prioritization.

2.3. Cost Function Components

2.3.1. Flight Path Cost

The flight path cost function $f_1$ aims to generate the shortest or near-shortest flight path to minimize mission duration and energy consumption. $f_1$ is defined as follows:

$$\begin{array}{c} f_1={\displaystyle \frac{L}{L_{max}}}\end{array}$$

(6)

$$\begin{array}{c} L={\displaystyle \sum _{i=1}^{n-1}}‖P_{i+1}-P_i‖\end{array}$$

(7)

$$\begin{array}{c} L_{max}=5·‖P_n-P_1‖\end{array}$$

(8)

where $P_i$ is the Cartesian coordinate of the $i$th waypoint; $L$ is the total actual path length; $L_{max}$ is the maximum allowable path length.

2.3.2. Altitude Cost

Altitude Cost $f_2$ ensures the UAV operates within safe altitude limits, avoiding collisions with terrain and complying with airspace management regulations. $f_2$ is defined as follows:

$$\begin{array}{c} f_2={\displaystyle \sum _{k=1}^n}{(f}_{co}\left(k\right)+f_{bo}\left(k\right))\end{array}$$

(9)

$$f_{co\left(k\right)}=\left\{\begin{array}{ll}100,& if z_k\le H_{terrian}\left(x_k,y_k\right)+10\\ 0,& otherwise\end{array}\right.$$

(10)

$$f_{bo\left(k\right)} = \left\{\begin{array}{ll}10,& if z_k<H_{min} or z_k>H_{max}\\ 0,& otherwise\end{array}\right.$$

(11)

where $f_{co\left(k\right)}$ represents the terrain collision penalty cost; $f_{bo}\left(k\right)$ denotes the altitude over-limit penalty cost; $z_k$ signifies the absolute elevation of the $k$th waypoint; $H_{terrian}\left(x_k,y_k\right)$ indicates the ground elevation at the horizontal coordinate; $H_{max}$ and $H_{min}$ respectively denote the upper and lower limits of absolute flight altitude.

2.3.3. Threat Cost

Threat cost $f_3$ compels the UAV to avoid radar and artillery threat zones, modeled as spheres in three-dimensional space. The cost comprises a fixed penalty and a distance-based inverse penalty. $f_3$ is defined as follows:

$$\begin{array}{c} f_3={\displaystyle \sum _{j=1}^{N_r}}\left({\displaystyle \frac{1}{d_{ij}^2}}+50\right)+{\displaystyle \sum _{j=1}^{N_a}}\left({\displaystyle \frac{R_j^2}{d_{ij}^2+1}}+50\right)\end{array}$$

(12)

where $d_{ij}$ denotes the shortest three-dimensional Euclidean distance from the $i$th track segment to the $j$th threat center; ${\displaystyle \frac{1}{d_{ij}^2}}$ represents the radar range penalty term, where the penalty value increases sharply with decreasing distance; $R_j$ signifies the effective radius of the $j$th artillery threat zone.

2.3.4. Angular Constraint Cost

The angular constraint cost $f_4$ limits the turning angle of the UAV between consecutive waypoints, ensuring the planned path aligns with the UAV’s physical maneuverability. $f_4$ is defined as follows:

$$\begin{array}{c} f_4=\left\{\begin{array}{c}\begin{array}{cc}10,& if {\theta }_k>{\theta }_{max}\end{array}\\ \begin{array}{cc}0,& otherwise\end{array} \end{array}\right.\end{array}$$

(13)

where ${\theta }_k$ is the turning angle at the kth path point; ${\theta }_{max}$ is the maximum allowable turning angle.

2.3.5. Trajectory Segment Constraint Cost

To prevent planning overly dense and impractical paths, each trajectory segment must have sufficient length to meet the minimum step size requirement for flight control. trajectory segment constraint cost $f_5$ is defined as follows:

$$\begin{array}{c}{f}_5=\left\{\begin{array}{c}\begin{array}{cc}5,& if ‖P_{i+1}-P_i‖<L_{min}\end{array}\\ \begin{array}{cc}0,& otherwise\end{array} \end{array}\right.\end{array}$$

(14)

where $L_{min}$ is the minimum allowable track segment length.

2.3.6. No-Fly Zone Constraint Cost

The no-fly zone constraint cost $f_6$ compels the UAV to avoid specific prohibited areas, modeled as vertical cylinders extending from the ground to a specified height. $f_6$ is defined as follows:

$$\begin{array}{c} f_6={\displaystyle \sum _{i=1}^{n-1}}{\displaystyle \sum _{j=1}^{N_{nf}}}{f}_{nf}\left(i,j\right)\end{array}$$

(15)

where $N_{nf}$ is the number of no-fly zones; $f_{nf}(i,j)$ is the penalty cost resulting from the interaction between the $i$th flight segment and the $j$th no-fly zone. When the 2D projection of the $i$th flight segment intersects the horizontal projection of the $j$th no-fly zone and the altitude range of the $i$th flight segment overlaps with the vertical range of the $j$th no-fly zone, $f_{nf}(i,j)$ is set to 50; otherwise, it is 0.

3. Artificial Protozoa Optimizer (APO)

The Artificial Protozoa Optimizer is a swarm intelligence optimization algorithm inspired by the survival behaviors of natural protozoa, such as Euglena. It simulates three fundamental behaviors—foraging, dormancy, and reproduction—executed by protozoa in their environment to seek optimal solutions to problems.

3.1. Foraging

Foraging behavior is categorized into autotrophic and heterotrophic modes, governed by the probability parameter $p_{ah}$, defined as:

$$\begin{array}{c} p_{ah}={\displaystyle \frac{1}{2}}·\left(1+\cos \left({\displaystyle \frac{FES}{{MAX}_{FES}}}·\pi \right)\right)\end{array}$$

(16)

where $FES$ and ${MAX}_{FES}$ represent the current function evaluation count and the maximum function evaluation count, respectively.

If $p_{ah}$ is greater than the random number, execute the autotrophic mode; otherwise, execute the heterotrophic mode. The value of $p_{ah}$ changes from 1 to 0 during iteration, so the algorithm favors autotrophy (exploration) in the early stages and heterotrophy (eploitation) in the later stages.

3.1.1. Autotrophic Mode

The autotrophic mode simulates euglena performing photosynthesis under light. If light is too intense, it moves away; if light is moderate, it moves toward it. In the algorithm, this corresponds to exploration, helping escape local optima. The position update formula in autotrophic mode is as follows:

$$\begin{array}{c} X_i^{new}=X_i+f·\left(X_j-X_i+{\displaystyle \frac{1}{np}}{\displaystyle \sum _{k=1}^{np}}{w}_a·\left(X_{k-}-X_{k+}\right)\right)\odot M_f\end{array}$$

(17)

$$\begin{array}{c} X_i=\left[x_i^1,x_i^2,\cdots ,x_i^{dim}\right], X_i=sort\left(X_i\right)\end{array}$$

(18)

$$\begin{array}{c} f=rand·\left(1+cos\left({\displaystyle \frac{FES}{{MAX}_{FES}}}·\pi \right)\right)\end{array}$$

(19)

$$\begin{array}{c} {np}_{max}=⌊{\displaystyle \frac{ps-1}{2}}⌋\end{array}$$

(20)

$$\begin{array}{c} w_a=e^{-\left|{\displaystyle \frac{f\left(x_{k-}\right)}{f\left(x_{k+}\right)+eps}}\right|}\end{array}$$

(21)

$$M_f\left[d_i\right] = \left\{\begin{array}{ll}1,& if d_i is in randperm\left(dim,⌈dim·{\displaystyle \frac{i}{ps}}⌉ \right)\\ 0,& otherwise\end{array}\right.$$

(22)

where $X_i$ represents the current individual’s position; $X_j$ denotes another randomly selected individual providing a random direction; $f$ is the foraging factor controlling the movement step size; $np$ is the logarithm of neighbors in external factors, influencing the number of individuals referenced during search; ${np}_{max}$ denotes the maximum value of $np$; $X_{k-}$ and $X_{k+}$ represent neighbors ranked below and above the current individual, respectively. Their difference introduces population diversity to enhance exploration; $rand$ indicates a random number selected from the interval [0, 1] uniformly distributed; $ps$ denotes the population size; $w_a$ is the weight factor, whose value correlates with the fitness of neighbors, biasing the search toward more favorable regions; $M_f$ is the foraging mapping vector. For the individual ranked $i$, the number of dimensions updated is $⌈dim·{\displaystyle \frac{i}{ps}}⌉$, meaning poorer individuals update more dimensions (exploration) while better individuals update fewer dimensions (exploitation); $d_i$ denotes the dimension index, and $d_i\in \{\mathrm{1,2},\cdots ,dim\}$; $\odot$ represents the Hadamard product.

3.1.2. Heterotrophic Mode

The heterotrophic mode simulates paramecia absorbing surrounding organic matter in darkness, moving toward areas rich in food. In the algorithm, this corresponds to exploitation, aiding in fine-grained search near good solutions. The position update formula in heterotrophic mode is as follows:

$$\begin{array}{c}{X}_i^{new}=X_i+f·\left(X_{near}-X_i+{\displaystyle \frac{1}{np}}{\displaystyle \sum _{k=1}^{np}}{w}_h·\left(X_{i-k}-X_{i+k}\right)\right)\odot M_f\end{array}$$

(23)

$$\begin{array}{c}{X}_{near}=\left(1\pm Rand·\left(1-{\displaystyle \frac{FES}{{MAX}_{FES}}}\right)\right)\odot X_i\end{array}$$

(24)

$$\begin{array}{c}{w}_h=e^{{\displaystyle \frac{f\left(X_{i-k}\right)}{f\left(X_{i+k}\right)}}}\end{array}$$

(25)

$$\begin{array}{c}Rand=\left[{rand}_1,{rand}_2,\cdots ,{rand}_{dim}\right]\end{array}$$

(26)

where  $X_{near}$ represents a position generated near the current individual, whose value gradually converges toward $X_i$ with iterations, facilitating the transition from exploration to exploitation. $X_{i-k}$ and $X_{i+k}$ denote fixed neighbors based on ranking, specifically the $(i-k)$th and $(i+k)$th protozoa selected from $k$ paired neighbors. Notably, if $X_i$ equals $X_{ps}$, $X_{i\pm k}$ is also set to $X_{ps}$. $w_h$ denotes the weight factor.

3.2. Dormancy

When environmental conditions deteriorate, paramecia enter dormancy by forming protective cysts. In the algorithm, if a solution shows no improvement for an extended period, it is reinitialized. This increases population diversity and prevents getting stuck in local optima. The position update formula during dormancy is as follows:

$$\begin{array}{c} X_i^{new}=X_{min}+Rand\odot \left(X_{max}-X_{min}\right)\end{array}$$

(27)

$$\begin{array}{c} X_{min}=\left[{lb}_1,{lb}_2\cdots ,{lb}_{dim}\right],X_{min}=\left[{lb}_1,{lb}_2\cdots ,{lb}_{dim}\right]\end{array}$$

(28)

where $X_{max}$ and $X_{min}$ denote the upper bound vector and lower bound vector, respectively, and ${lb}_{di}$ and ${ub}_{di}$ represent the lower bound and upper bound of the $d_i$th variable, respectively.

3.3. Reproduction

Euglena reproduce through binary fission, where one organism divides into two similar individuals. In the algorithm, this corresponds to copying the current solution and introducing a small perturbation to generate a new solution, aiding further exploration near good solutions. The position update formula during reproduction is as follows:

$$\begin{array}{c}{X}_i^{new}=X_i\pm rand\cdot \left(X_{min}+Rand\odot \left(X_{max}-X_{min}\right)\right)\odot M_r\end{array}$$

(29)

$$M_r\left[d_i\right] = \left\{\begin{array}{ll}1,& if d_i is in randperm\left(dim,⌈dim·{\displaystyle \mathit{rand}}⌉ \right)\\ 0,& otherwise\end{array}\right.$$

(30)

where $M_r$ denotes the mapping vector during reproduction, which randomly determines which dimensions undergo variation.

The proportion fraction $pf$ determines what percentage of individuals in the current population will enter hibernation or reproduce, while the remaining individuals will forage. Its value is defined as:

$$\begin{array}{c}pf={pf}_{max}·rand\end{array}$$

(31)

where ${pf}_{max}$ represents the maximum value of $pf$.

Probability parameter $p_{dr}$ controls the probability of executing hibernation and reproduction, defined as:

$$\begin{array}{c}{p}_{dr}={\displaystyle \frac{1}{2}}·\left(1+cos\left(\left(1-{\displaystyle \frac{i}{ps}}\right)·\pi \right)\right)\end{array}$$

(32)

If $p_{dr}$ is greater than the random number, hibernate; otherwise, perform reproduction. Lower-ranked individuals are more inclined to hibernate (explore) to escape local optima; higher-ranked individuals are more inclined to reproduce (exploit) to conduct detailed searches in high-quality regions.

4. Enhanced Artificial Protozoan Optimization Algorithm (EAPO)

Although APO demonstrates competitive performance, our preliminary analysis reveals four inherent limitations when addressing complex problems: (1) random initialization leads to uneven population distribution; (2) fixed behavior-selection probabilities fail to adapt to different optimization stages; (3) search directions rely heavily on stochastic perturbations without exploiting local landscape information; and (4) population diversity rapidly degrades in later iterations, increasing the risk of premature convergence.

To address these issues, this study proposes an Enhanced Artificial Protozoa Optimizer (EAPO) by introducing four complementary strategies, each explicitly designed to overcome a specific limitation of the original APO. Specifically, Latin hypercube sampling is used to improve the uniformity of the initial population. This strengthens global exploration in the early stage: a behaviorally adaptive selection strategy based on historical success is designed. It adjusts the selection probabilities of different behaviors according to their past performance. This helps balance exploration and exploitation during the search; A sensory field–based environmental modeling strategy is introduced. It provides local landscape information for guiding the search. The search direction becomes more informative instead of purely random; an adaptive hibernation–reconstruction mechanism is applied. It restores population diversity when stagnation occurs. This helps the algorithm escape local optima. These four strategies are mutually supportive and collectively form a coherent enhancement framework.

4.1. Latin Hypercube Sampling

In meta-heuristic algorithms, the quality of the initial population critically impacts both the convergence speed and the quality of the final solution. Traditional random initialization methods uniformly and independently sample within the solution space. In high-dimensional spaces, this approach is prone to issues such as uneven population distribution, incomplete coverage, and exploration blind spots. Consequently, the algorithm may converge slowly in the early stages or prematurely settle into local optima.

To address this issue, Latin hypercube sampling (LHS) [29,30] is adopted for population initialization. LHS is a stratified sampling method that promotes uniform coverage in each dimension. For a D-dimensional optimization problem, the search range of each dimension is first divided into $ps$ equal intervals, where $ps$ denotes the population size. A $ps\times D$ matrix is then constructed, in which each column is a random permutation of integers from 1 to $ps$. This matrix ensures that each interval index appears exactly once in every column.

Next, one sample point is randomly generated within each interval of every dimension. According to the matrix arrangement, these sample points are combined across dimensions to generate $ps$ initial D-dimensional individuals.

Through this approach, the initial population created by Latin hypercube sampling uniformly covers the entire search space across every dimension. This strategy directly addresses the uneven spatial coverage caused by random initialization in APO, thereby improving early-stage exploration efficiency.

4.2. Behavioral Adaptive Selection Strategy Based on Historical Success

This strategy differs from APO’s reliance on fixed formulas by dynamically adjusting the selection probability of each behavior (autotrophy, heterotrophy, dormancy, reproduction) in subsequent iterations based on its historical success rate during algorithm execution. Behaviors with higher success rates receive greater selection probabilities, thereby achieving adaptive exploration-exploitation adjustments within the algorithm. A schematic illustration of its specific mechanism is shown in Figure 2. In this figure, the success rate $SR\left(a\right)$ of each behavior is first mapped to a confidence value using the confidence-weighted mechanism defined in Equations (35) and (36), which prevents frequently selected behaviors from dominating the selection process. The exploration–exploitation balance mechanism further adjusts these confidence values according to the current iteration stage, as described in Equations (37)–(44). Finally, the normalized probabilities $\widehat{p}\left(a\right)$ are obtained and used to stochastically select the corresponding behavior for each individual.

4.2.1. Historical Feedback Mechanism

First, define four basic behaviors:

$$\begin{array}{c} A=\left\{\begin{array}{ccc}{a}_1& a_2& \begin{array}{cc}{a}_3& a_4\end{array}\end{array}\right\}\end{array}$$

(33)

where $a_1$, $a_2$, $a_3$, and $a_4$ represent autotrophy, heterotrophy, dormancy, and reproduction, respectively.

Second, for each behavior $a$, set up two counters $S\left(a\right)$ and $C(a$), representing the total number of successful attempts for action a and the total number of times action a was attempted, respectively.

Finally, define the base success rate $SR\left(a\right)$ for action a as:

$$\begin{array}{c}SR\left(a\right)={\displaystyle \frac{S\left(a\right)+e}{C\left(a\right)+m·e}}\end{array}$$

(34)

where $m=4$ represents the total number of behaviors; $e={10}^{-6}$ serves to prevent the denominator from becoming zero.

4.2.2. Sample Confidence Mechanism

To prevent over-reliance on behaviors with few attempts and better balance historical experience with exploring new behaviors, a sample confidence mechanism [31] is implemented.

First, the confidence of each behavior $a$ is calculated using a logarithmic function. This design prevents behaviors with many attempts from gaining excessive advantage and preserves opportunities for behaviors with fewer attempts. The confidence calculation formula for each behavior is given as follows:

$$\begin{array}{c} conf\left(a\right)={\displaystyle \frac{\ln \left(C\left(a\right)+1\right)}{\ln \left(C_{max}+1\right)}}\end{array}$$

(35)

where $C_{max}$ is the value attempted most frequently across all behaviors.

For new behaviors with few attempts, the $conf\left(a\right)$ value is low. This means that even with limited historical data, the behavior still receives a baseline $p\left(a\right)$ value, encouraging the algorithm to explore. For mature behaviors with many attempts, the $conf\left(a\right)$ value is high. This indicates the algorithm fully trusts the historical performance of this behavior, encouraging the use of known effective actions.

Next, the confidence is converted into a final confidence factor that influences decision-making, calculated using the formula:

$$\begin{array}{c} CF\left(a\right)=\left(1-\omega \right)+\omega ·conf\left(a\right)\end{array}$$

(36)

where $CF\left(a\right)$ is the confidence factor, used to account for the reliability of the sample size; $\omega$ is the confidence weight parameter, used to control the influence of the confidence level, $\omega \in \left[0,1\right]$.

4.2.3. Adaptive Mechanism for Exploration/Exploitation Phases

Since algorithms exhibit varying demands for exploration/exploitation during iteration, different weights are assigned to the four behaviors of APO across different iteration phases. The specific definitions are as follows:

$$\begin{array}{c} PF\left(a\right)={\omega }_r·{\beta }_r\left(a\right)+{\omega }_{oi}·{\beta }_{oi}\left(a\right)\end{array}$$

(37)

$$\begin{array}{c} {\omega }_r=0.2+0.8·\mathit{cos}\left({\displaystyle \frac{FES}{{MAX}_{FES}}}·{\displaystyle \frac{\pi }{2}}\right)\end{array}$$

(38)

$$\begin{array}{c} {\omega }_{oi}=0.2+0.8·\mathit{sin}\left({\displaystyle \frac{FES}{{MAX}_{FES}}}·{\displaystyle \frac{\pi }{2}}\right)\end{array}$$

(39)

$$\begin{array}{c} {\beta }_r=\left\{\begin{array}{ccc}1.0& 0.2& \begin{array}{cc}1.0& 0.4\end{array}\end{array}\right\}\end{array}$$

(40)

$$\begin{array}{c} {\beta }_{oi}=\left\{\begin{array}{ccc}0.2& 1.0& \begin{array}{cc}0.3& 0.8\end{array}\end{array}\right\}\end{array}$$

(41)

where $PF\left(a\right)$ represents the stage factor; ${\omega }_r$ and ${\omega }_{oi}$ denote the weights for exploration and exploitation, respectively; ${\beta }_r\left(a\right)$ and ${\beta }_{oi}\left(a\right)$ denote the weights for the exploratory and exploitative aspects of the four behaviors. In the early phase, exploratory behaviors $(a_1, a_3)$ receive higher weights, while in the later phase, exploitative behaviors ${(a}_2, a_4)$ receive higher weights.

4.2.4. Calculation of Combined Action Probability

For behaviors when $C\left(a\right)>0$, the combined probability of each behavior being selected in the next iteration is:

$$\begin{array}{c} p\left(a\right)=SR\left(a\right)·CF\left(a\right)·PF\left(a\right)\end{array}$$

(42)

For behaviors when $C\left(a\right)=0$, $p\left(a\right)$ is defined by Formula (43) to assign different initial probabilities to untried behaviors based on the current stage, thereby encouraging attempts at corresponding behavior types at appropriate stages.

$$\begin{array}{c} p\left(a\right)=\left\{\begin{array}{c}\begin{array}{cc}0.3+0.4·{\omega }_r,& a\in \left\{a_1,a_3\right\}\end{array}\\ \begin{array}{cc}0.3+0.4·{\omega }_{oi},& a\in \left\{a_2,a_4\right\}\end{array}\end{array}\right.\end{array}$$

(43)

Ultimately, the probability $\widehat{p}\left(a\right)$ that behavior $a$ is selected in the next iteration is normalized with respect to $p\left(a\right)$:

$$\begin{array}{c} \widehat{p}\left(a\right)={\displaystyle \frac{p\left(a\right)}{\sum _{i=1}^{m}p\left(a_i\right)}}\end{array}$$

(44)

4.2.5. Behavior Selection Mechanism

A roulette wheel is used to select four behaviors:

$$\begin{array}{c} {Action}_1=\left\{\begin{array}{c}\begin{array}{cc}{a}_1,& if\end{array} rand<{\displaystyle \frac{\widehat{p}\left(a_1\right)}{\widehat{p}\left(a_1+a_2\right)}}\\ \begin{array}{cc}{a}_2,& otherwise\end{array} \end{array}\right.\end{array}$$

(45)

$$\begin{array}{c} {Action}_2=\left\{\begin{array}{c}\begin{array}{cc}{a}_3,& if\end{array} rand<{\displaystyle \frac{\widehat{p}\left(a_3\right)}{\widehat{p}\left(a_3+a_4\right)}}\\ \begin{array}{cc}{a}_4,& otherwise \end{array}\end{array}\right.\end{array}$$

(46)

To avoid over-reliance on early history, apply decay at the end of each iteration, that is:

$$\begin{array}{c} S_{t+1}\left(a\right)=\lambda ·S_t\left(a\right)\end{array}$$

(47)

$$\begin{array}{c} C_{t+1}\left(a\right)=\lambda ·C_t\left(a\right)\end{array}$$

(48)

where $S_{t+1}\left(a\right)$ and $C_{t+1}\left(a\right)$ represent the total number of successful occurrences of behavior $a$ and the total number of attempts at behavior $a$ during the $t+1$ iteration, respectively; $\lambda \in (0, 1)$ is the decay factor.

4.3. Environmental Structure Modeling Strategy Based on Sensory Field

The original APO algorithm relies on random mapping for its search direction, lacking utilization of local terrain structures. This results in significant jumps and insufficient local exploration capabilities. To address this, this paper introduces a local sensory field. By constructing an approximate gradient direction based on the quality differences in neighboring individuals, the search is guided toward potential optimal regions. This mechanism equips APO with basic environmental perception and shifts the search direction from blind perturbation to information-guided movement. Consequently, search stability and convergence accuracy are improved. The schematic diagram of sensory field modeling is shown in Figure 3. In this figure, the current solution is denoted by a star, with neighboring individuals distributed throughout the local search space and represented by dots with green outlines. The horizontal and vertical axes represent two dimensions of the decision space. The perceptual direction indicated by the arrow corresponds to the aggregated guidance vector defined in Equation (49), which is computed based on the fitness differences between the current individual and its neighbors.

We define the environmental perception model as:

$$\begin{array}{c} \nabla f\left(X_i\right)={\displaystyle \frac{1}{\left|N_i\right|}}{\displaystyle \sum _{X_j\in N_i}}\left(f_j-f_i\right)\left(X_j-X_i\right)\end{array}$$

(49)

where $N_i$ denotes the neighborhood set of the $i$th individual (which may consist of the $k$ best individuals or the $k$ nearest neighbors); $f_j$ represents the fitness of neighborhood individual $X_j$; $X_j-X_i$ is the relative position vector, indicating the directional trend of the local landscape.

Autotrophic and heterotrophic behaviors are now updated via guided search, so Equations (17) and (23) become as follows:

$$\begin{array}{c} X_i^{new}=X_i+f·\left(X_j-X_i+{\displaystyle \frac{1}{np}}{\displaystyle \sum _{k=1}^{np}}{w}_a·\left(X_{k-}-X_{k+}\right)\right)\odot M_f-\alpha \left(t\right)·Normalize\left(\nabla f\left(X_i\right)\right)\end{array}$$

(50)

$$\begin{array}{c} X_i^{new}=X_i+f·\left(X_{near}-X_i+{\displaystyle \frac{1}{np}}{\displaystyle \sum _{k=1}^{np}}{w}_h·\left(X_{i-k}-X_{i+k}\right)\right)\odot M_f-\alpha \left(t\right)·Normalize\left(\nabla f\left(X_i\right)\right)\end{array}$$

(51)

$$\begin{array}{c}\alpha \left(t\right)={\alpha }_{max}-{\displaystyle \frac{FES}{{MAX}_{FES}}}\left({\alpha }_{max}-{\alpha }_{min}\right)\end{array}$$

(52)

where $\alpha \left(t\right)$ is the stride factor, ${\alpha }_{max}$ and ${\alpha }_{min}$ are the initial stride and final stride, respectively; $Normalize(·)$ is the vector normalization function, which prevents the stride from becoming excessively large.

The neighborhood-based gradient estimation relies on a fixed-size local neighborhood. Therefore, the computational complexity scales linearly with the neighborhood size and does not increase exponentially with the problem dimension. Consequently, the sensory field strategy does not significantly affect the overall runtime of the algorithm.

4.4. Adaptive Hibernation Reconstruction Strategy

To prevent premature convergence caused by population diversity degradation and enhance the ability to escape local optima, individuals with low energy or stuck in stagnation will transition from direct random reset to guided reconstruction. This introduces directionality, reduces blind jumping, and improves both population diversity and search efficiency. Consequently, Equation (27) is modified as follows:

$$\begin{array}{c} X_i^{new}=X_{worst}+Rand\odot \left(X_{best}-X_{worst}\right)\end{array}$$

(53)

4.5. EAPO Algorithm Pseudocode and Flowchart

The flowchart of the EAPO algorithm (Algorithm 1), which integrates Latin hypercube sampling initialization, behavior adaptation selection strategy based on historical success, environmental structure modeling strategy based on perceptual fields, and adaptive hibernation reconstruction mechanism, is shown in Figure 4. The time complexity analysis of EAPO is provided in Appendix A.1, and its pseudocode is as follows:

Algorithm 1 Pseudocode for EAPO

Input:$\mathrm{Population} \mathrm{size} N$$, \mathrm{maximum} \mathrm{evaluations} {Max}_{FES}$$, \mathrm{bounds} lb$$\& ub$$, \mathrm{dimension} dim$$, \mathrm{objective} \mathrm{function} objfun$

Output:$\mathrm{Best} \mathrm{solution} X_{best}$$, \mathrm{best} \mathrm{fitness} Best\_Score$

Set Parameters: $\lambda = 0.5,$ ${\alpha }_{max}=0.9$, ${\alpha }_{min}=0.1,$ $\omega =0.2$, ${pf}_{max}=0.1$, $np=1,$ $k=5$

1: Initialize population $X$ using Latin hypercube sampling and elite strategy. 2: Initialize behavior counter $S$, $C$.

$3:\mathbf{while} FES \le {Max}_{FES}$ do

$4:\hspace{1em} \mathrm{Calculate} {\omega }_r$$, {\omega }_{oi}$ using Equations (38) and (39).

$5:\hspace{1em} \mathbf{for} \mathrm{each} \mathrm{action} a$ = 1 to 4 do

$6:\hspace{1em}\hspace{1em} \mathbf{if} c\left[a\right] > 0$ then

$7:\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{Calculate} SR\left(a\right),CF\left(a\right)$$, PF\left(a\right)$$, p\left(a\right)$ using Equations (34), (36), (37) and (42).

8:   else

$9:\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{Calculate} p\left(a\right)$ using Equation (43).

10:   end if

11:    end for

$12:\hspace{1em} \mathrm{Calculate} \widehat{p}\left(a\right)$$\mathrm{using} \mathrm{Equation} \left(44\right). \mathrm{Sort} \mathrm{the} \mathrm{population} \mathrm{by} \mathrm{fitness}. r_i\leftarrow$$\mathrm{Randomly} \mathrm{select} ⌈ps·p_{max}·rand\left(\right)⌉$ individuals.

$13:\hspace{1em} \mathbf{for} i$$= 1 \mathrm{to} ps$ do

$14:\hspace{1em}\hspace{1em} \mathbf{if} i\in r_i$ then

$15:\hspace{1em}\hspace{1em}\hspace{1em} \mathrm{Select} \mathrm{behavior} (a_3, a_4$) using Equation (46).

$16:\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathbf{if} a_3$ then

$17:\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{update} X_i^{new}$ using Equation (52).

18:     else

$19:\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{update} X_i^{new}$ using Equation (29).

20:     end if

21:   else

$22:\hspace{1em}\hspace{1em}\hspace{1em} \mathrm{Calculate} f$$, \nabla f\left(X_i\right)$$, \alpha \left(t\right)$, f using Equations (19), (49) and (52).

$23:\hspace{1em}\hspace{1em}\hspace{1em} \mathrm{Select} \mathrm{behavior} (a_1, a_2$) using Equation (45).

$24:\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathbf{if} a_1$ then

$25\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{update} X_i^{new}$ using Equation (50).

26:     else

$27:\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{update} X_i^{new}$ using Equation (51).

28:     end if

$29:\hspace{1em}\hspace{1em}\hspace{1em} \mathrm{Update} S$$, C$$. \mathrm{Update} X_{best},$ $Best\_Score$ if better solution found.

30:  end for

$31:\hspace{1em}\mathrm{Update} S$$, C$ using Equations (47) and (48).

32:   end while

$33: \mathrm{Return} X_{best},$$Best\_Score$.

5. Parameter Sensitivity Analysis of EAPO

This section presents a parameter sensitivity analysis of the proposed Enhanced Artificial Protozoa Optimizer (EAPO). The main purpose of this analysis is to determine suitable values for key parameters used in the algorithm.

Two parameters are investigated. The first is the neighborhood size $K$ in the sensory field strategy. This parameter determines how many neighboring individuals are used to estimate the search direction. The second parameter is the decay factor $\lambda$ in the historical success mechanism. This parameter controls how strongly past information influences behavior selection.

In the experiments, different values of $K$ and $\lambda$ are tested within predefined ranges, while all other parameters are kept unchanged. The results are compared to identify parameter values that lead to stable convergence and good solution quality. Based on these results, appropriate values for $K$ and $\lambda$ are selected for the proposed algorithm.

The experiments are conducted on the CEC2022 benchmark suite with problem dimensions of 10 and 20. For each parameter setting, the best value, mean value, and standard deviation are recorded. In addition, the Friedman test is used to evaluate the overall performance, and the average rankings are calculated. To enhance readability, detailed results are presented in

Table A4. Sensitivity analysis results for parameter $\lambda$ on CEC2022 (dim = 10).

Function	Item Name	$\mathit{\lambda }$
		0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9
F1	Best	3.640 × 103	2.979 × 103	2.246 × 103	2.452 × 103	1.899 × 103	2.187 × 103	2.588 × 103	1.003 × 103	2.973 × 103
	Mean	5.365 × 103	5.419 × 103	4.344 × 103	4.599 × 103	5.576 × 103	4.738 × 103	5.743 × 103	5.379 × 103	5.271 × 103
	Std	1.611 × 103	1.624 × 103	1.367 × 103	1.736 × 103	2.252 × 103	1.927 × 103	1.744 × 103	2.384 × 103	1.828 × 103
	Rank	2	3	1	8	7	5	9	6	4
F2	Best	4.070 × 102	4.069 × 102	4.071 × 102	4.068 × 102	4.071 × 102	4.072 × 102	4.071 × 102	4.070 × 102	4.070 × 102
	Mean	4.085 × 102	4.083 × 102	4.083 × 102	4.084 × 102	4.094 × 102	4.113 × 102	4.090 × 102	4.087 × 102	4.109 × 102
	Std	1.453 × 100	1.248 × 100	9.623 × 10−1	1.151 × 100	2.393 × 100	1.018 × 101	1.272 × 100	1.125 × 100	7.269 × 100
	Rank	8	2	4	3	6	1	7	5	9
F3	Best	6.000 × 102	6.000 × 102	6.000 × 102	6.000 × 102	6.000 × 102	6.000 × 102	6.000 × 102	6.000 × 102	6.000 × 102
	Mean	6.002 × 102	6.002 × 102	6.002 × 102	6.001 × 102	6.002 × 102	6.002 × 102	6.003 × 102	6.004 × 102	6.005 × 102
	Std	1.751 × 10−1	1.833 × 10−1	2.374 × 10−1	1.000 × 10−1	1.840 × 10−1	2.197 × 10−1	3.171 × 10−1	3.620 × 10−1	4.053 × 10−1
	Rank	2	1	7	5	3	4	6	9	8
F4	Best	8.088 × 102	8.116 × 102	8.110 × 102	8.083 × 102	8.112 × 102	8.108 × 102	8.116 × 102	8.137 × 102	8.076 × 102
	Mean	8.160 × 102	8.174 × 102	8.181 × 102	8.154 × 102	8.160 × 102	8.166 × 102	8.175 × 102	8.179 × 102	8.171 × 102
	Std	4.376 × 100	3.710 × 100	3.838 × 100	4.165 × 100	2.917 × 100	3.642 × 100	3.905 × 100	2.496 × 100	3.622 × 100
	Rank	4	1	9	7	2	3	5	8	6
F5	Best	9.010 × 102	9.009 × 102	9.011 × 102	9.003 × 102	9.005 × 102	9.008 × 102	9.018 × 102	9.009 × 102	9.020 × 102
	Mean	9.056 × 102	9.042 × 102	9.042 × 102	9.036 × 102	9.041 × 102	9.046 × 102	9.054 × 102	9.057 × 102	9.069 × 102
	Std	3.460 × 100	2.349 × 100	3.087 × 100	2.705 × 100	2.268 × 100	2.661 × 100	3.416 × 100	3.560 × 100	3.977 × 100
	Rank	3	2	1	4	5	7	6	8	9
F6	Best	2.177 × 103	1.998 × 103	1.942 × 103	1.963 × 103	2.032 × 103	2.192 × 103	2.066 × 103	2.353 × 103	2.018 × 103
	Mean	3.054 × 103	2.717 × 103	2.796 × 103	3.019 × 103	3.412 × 103	3.139 × 103	3.173 × 103	3.383 × 103	3.343 × 103
	Std	8.711 × 102	6.923 × 102	6.287 × 102	8.537 × 102	1.334 × 103	9.945 × 102	7.738 × 102	7.866 × 102	9.896 × 102
	Rank	6	3	2	1	7	4	5	9	8
F7	Best	2.025 × 103	2.023 × 103	2.025 × 103	2.015 × 103	2.017 × 103	2.025 × 103	2.025 × 103	2.023 × 103	2.026 × 103
	Mean	2.030 × 103	2.031 × 103	2.031 × 103	2.029 × 103	2.032 × 103	2.032 × 103	2.031 × 103	2.032 × 103	2.031 × 103
	Std	3.768 × 100	4.223 × 100	3.444 × 100	4.787 × 100	4.728 × 100	4.068 × 100	2.963 × 100	3.152 × 100	3.339 × 100
	Rank	6	2	3	4	8	1	5	9	7
F8	Best	2.224 × 103	2.213 × 103	2.215 × 103	2.217 × 103	2.215 × 103	2.217 × 103	2.215 × 103	2.217 × 103	2.221 × 103
	Mean	2.226 × 103	2.224 × 103	2.225 × 103	2.223 × 103	2.224 × 103	2.224 × 103	2.224 × 103	2.224 × 103	2.225 × 103
	Std	1.022 × 100	3.216 × 100	2.679 × 100	2.950 × 100	2.519 × 100	2.673 × 100	2.730 × 100	2.670 × 100	1.655 × 100
	Rank	2	1	6	7	5	9	4	3	8
F9	Best	2.529 × 103	2.529 × 103	2.529 × 103	2.529 × 103	2.529 × 103	2.529 × 103	2.529 × 103	2.529 × 103	2.529 × 103
	Mean	2.530 × 103	2.530 × 103	2.530 × 103	2.530 × 103	2.530 × 103	2.530 × 103	2.530 × 103	2.531 × 103	2.530 × 103
	Std	3.514 × 10−1	1.227 × 100	1.063 × 100	6.917 × 10−1	4.754 × 10−1	1.342 × 100	1.370 × 100	2.200 × 100	4.480 × 10−1
	Rank	2	4	3	6	1	5	7	8	9
F10	Best	2.500 × 103	2.500 × 103	2.500 × 103	2.500 × 103	2.500 × 103	2.500 × 103	2.500 × 103	2.500 × 103	2.500 × 103
	Mean	2.502 × 103	2.501 × 103	2.501 × 103	2.501 × 103	2.500 × 103	2.506 × 103	2.500 × 103	2.500 × 103	2.501 × 103
	Std	4.413 × 100	6.506 × 10−1	1.578 × 100	4.345 × 100	7.847 × 10−2	2.566 × 101	5.978 × 10−2	1.105 × 10−1	1.522 × 100
	Rank	8	3	9	4	5	7	2	6	1
F11	Best	2.716 × 103	2.696 × 103	2.660 × 103	2.683 × 103	2.647 × 103	2.722 × 103	2.666 × 103	2.655 × 103	2.675 × 103
	Mean	2.738 × 103	2.739 × 103	2.739 × 103	2.730 × 103	2.727 × 103	2.739 × 103	2.734 × 103	2.735 × 103	2.736 × 103
	Std	1.215 × 101	1.457 × 101	2.229 × 101	2.270 × 101	3.000 × 101	9.953 × 100	1.942 × 101	2.292 × 101	2.045 × 101
	Rank	4	3	9	5	1	7	2	6	8
F12	Best	2.860 × 103	2.862 × 103	2.860 × 103	2.862 × 103	2.860 × 103	2.862 × 103	2.859 × 103	2.862 × 103	2.862 × 103
	Mean	2.864 × 103	2.864 × 103	2.864 × 103	2.863 × 103	2.864 × 103	2.864 × 103	2.863 × 103	2.864 × 103	2.863 × 103
	Std	1.027 × 100	9.228 × 10−1	1.201 × 100	7.583 × 10−1	1.329 × 100	6.742 × 10−1	1.362 × 100	6.663 × 10−1	5.828 × 10−1
	Rank	4	1	6	8	7	9	2	5	3
Mean Rank		4.25	2.17	5	5.17	4.75	5.17	5	6.83	6.67

,

Table A5. Sensitivity analysis results for parameter $\lambda$ on CEC2022 (dim = 10).

Function	Item Name	$\mathit{\lambda }$
		0.1	0.2	0.3	0.4	0.5	0.6	0.7	0.8	0.9
F1	Best	1.454 × 104	1.155 × 104	1.258 × 104	1.440 × 104	1.574 × 104	1.179 × 104	1.535 × 104	1.626 × 104	1.531 × 104
	Mean	2.262 × 104	2.233 × 104	1.819 × 104	2.092 × 104	1.904 × 104	1.568 × 104	2.250 × 104	2.572 × 104	2.545 × 104
	Std	8.043 × 103	9.517 × 103	2.546 × 103	5.583 × 103	3.031 × 103	2.439 × 103	5.362 × 103	5.489 × 103	4.472 × 103
	Rank	6	5	2	4	3	1	7	9	8
F2	Best	4.449 × 102	4.491 × 102	4.491 × 102	4.453 × 102	4.491 × 102	4.423 × 102	4.491 × 102	4.491 × 102	4.491 × 102
	Mean	4.517 × 102	4.555 × 102	4.508 × 102	4.537 × 102	4.516 × 102	4.497 × 102	4.540 × 102	4.527 × 102	4.518 × 102
	Std	6.395 × 100	1.024 × 101	2.767 × 100	8.569 × 100	7.301 × 100	4.496 × 100	7.587 × 100	7.812 × 100	7.323 × 100
	Rank	1	8	5	7	3	2	9	6	4
F3	Best	6.001 × 102	6.001 × 102	6.001 × 102	6.001 × 102	6.001 × 102	6.001 × 102	6.001 × 102	6.001 × 102	6.001 × 102
	Mean	6.002 × 102	6.002 × 102	6.002 × 102	6.002 × 102	6.002 × 102	6.002 × 102	6.003 × 102	6.002 × 102	6.003 × 102
	Std	1.077 × 10−1	7.721 × 10−2	8.023 × 10−2	8.878 × 10−2	1.337 × 10−1	7.131 × 10−2	2.319 × 10−1	7.130 × 10−2	1.528 × 10−1
	Rank	7	3	4	2	6	1	8	5	9
F4	Best	8.392 × 102	8.460 × 102	8.542 × 102	8.469 × 102	8.451 × 102	8.463 × 102	8.457 × 102	8.468 × 102	8.462 × 102
	Mean	8.528 × 102	8.556 × 102	8.573 × 102	8.545 × 102	8.573 × 102	8.546 × 102	8.569 × 102	8.532 × 102	8.550 × 102
	Std	8.538 × 100	6.799 × 100	2.475 × 100	8.171 × 100	6.573 × 100	6.112 × 100	5.521 × 100	5.115 × 100	7.660 × 100
	Rank	5	6	9	2	7	4	8	1	3
F5	Best	9.000 × 102	9.001 × 102	9.005 × 102	9.000 × 102	9.003 × 102	9.000 × 102	9.006 × 102	9.000 × 102	9.000 × 102
	Mean	9.017 × 102	9.009 × 102	9.028 × 102	9.053 × 102	9.019 × 102	9.019 × 102	9.032 × 102	9.016 × 102	9.012 × 102
	Std	2.414 × 100	7.856 × 10−1	2.713 × 100	6.173 × 100	1.541 × 100	2.965 × 100	4.780 × 100	1.454 × 100	1.051 × 100
	Rank	4	1	9	7	6	3	8	5	2
F6	Best	5.252 × 103	3.185 × 103	3.493 × 103	2.360 × 103	5.564 × 103	3.745 × 103	6.716 × 103	3.330 × 103	4.982 × 103
	Mean	9.018 × 103	7.781 × 103	1.126 × 104	8.838 × 103	1.319 × 104	9.031 × 103	2.244 × 104	2.035 × 104	1.823 × 104
	Std	3.432 × 103	4.294 × 103	7.965 × 103	5.657 × 103	5.972 × 103	3.573 × 103	2.047 × 104	1.327 × 104	1.009 × 104
	Rank	3	1	4	2	6	5	9	7	8
F7	Best	2.048 × 103	2.052 × 103	2.047 × 103	2.055 × 103	2.042 × 103	2.046 × 103	2.054 × 103	2.044 × 103	2.051 × 103
	Mean	2.065 × 103	2.072 × 103	2.067 × 103	2.070 × 103	2.067 × 103	2.062 × 103	2.067 × 103	2.068 × 103	2.063 × 103
	Std	8.660 × 100	1.134 × 101	1.009 × 101	1.018 × 101	1.183 × 101	1.031 × 101	9.601 × 100	9.835 × 100	7.442 × 100
	Rank	3	9	6	8	5	1	4	7	2
F8	Best	2.228 × 103	2.227 × 103	2.227 × 103	2.229 × 103	2.227 × 103	2.228 × 103	2.227 × 103	2.227 × 103	2.228 × 103
	Mean	2.230 × 103	2.230 × 103	2.230 × 103	2.230 × 103	2.230 × 103	2.230 × 103	2.229 × 103	2.230 × 103	2.230 × 103
	Std	1.030 × 100	1.176 × 100	1.721 × 100	8.783 × 10−1	2.234 × 100	1.214 × 100	1.600 × 100	1.729 × 100	1.498 × 100
	Rank	9	1	6	3	5	4	2	7	8
F9	Best	2.481 × 103	2.481 × 103	2.481 × 103	2.481 × 103	2.481 × 103	2.481 × 103	2.481 × 103	2.481 × 103	2.481 × 103
	Mean	2.482 × 103	2.481 × 103	2.481 × 103	2.481 × 103	2.481 × 103	2.481 × 103	2.481 × 103	2.482 × 103	2.481 × 103
	Std	7.970 × 10−1	3.344 × 10−1	1.347 × 10−1	3.013 × 10−1	6.915 × 10−1	3.124 × 10−1	6.724 × 10−1	1.116 × 100	6.122 × 10−1
	Rank	9	5	1	3	8	2	4	6	7
F10	Best	2.500 × 103	2.500 × 103	2.500 × 103	2.500 × 103	2.500 × 103	2.500 × 103	2.501 × 103	2.500 × 103	2.500 × 103
	Mean	2.521 × 103	2.501 × 103	2.501 × 103	2.750 × 103	2.504 × 103	2.518 × 103	2.534 × 103	2.501 × 103	2.511 × 103
	Std	5.197 × 101	7.140 × 10−2	5.893 × 10−2	5.266 × 102	1.064 × 101	5.544 × 101	6.802 × 101	7.363 × 10−2	3.372 × 101
	Rank	6	1	5	4	7	2	9	3	8
F11	Best	2.901 × 103	2.901 × 103	2.902 × 103	2.901 × 103	2.738 × 103	2.901 × 103	2.901 × 103	2.902 × 103	2.901 × 103
	Mean	2.902 × 103	2.902 × 103	2.942 × 103	2.902 × 103	2.886 × 103	2.938 × 103	2.902 × 103	2.942 × 103	2.973 × 103
	Std	3.703 × 10−1	2.481 × 10−1	1.246 × 102	7.237 × 10−1	5.203 × 101	1.134 × 102	4.294 × 10−1	1.248 × 102	1.516 × 102
	Rank	1	2	7	5	4	6	3	8	9
F12	Best	2.941 × 103	2.939 × 103	2.936 × 103	2.939 × 103	2.939 × 103	2.940 × 103	2.938 × 103	2.940 × 103	2.941 × 103
	Mean	2.945 × 103	2.944 × 103	2.946 × 103	2.945 × 103	2.945 × 103	2.943 × 103	2.944 × 103	2.945 × 103	2.945 × 103
	Std	3.393 × 100	2.125 × 100	7.639 × 100	3.704 × 100	3.551 × 100	2.440 × 100	3.718 × 100	4.017 × 100	2.629 × 100
	Rank	3	4	7	8	9	1	2	5	6
Mean Rank		4.75	3.83	5.42	4.58	5.75	2.67	6.08	5.75	6.17

,

Table A6. Sensitivity analysis results for parameter $k$ on CEC2022 (dim = 10).

Function	Item Name	$\mathit{k}$
		1	2	3	4	5	6	7	8	9	10
F1	Best	3.128 × 103	2.559 × 103	3.674 × 103	2.096 × 103	4.632 × 103	3.356 × 103	3.700 × 103	3.433 × 103	3.954 × 103	2.599 × 103
	Mean	5.458 × 103	3.985 × 103	5.114 × 103	4.894 × 103	6.058 × 103	6.273 × 103	5.865 × 103	5.272 × 103	6.151 × 103	4.610 × 103
	Std	1.946 × 103	1.025 × 103	1.541 × 103	1.696 × 103	1.177 × 103	2.071 × 103	1.248 × 103	1.002 × 103	1.779 × 103	1.707 × 103
	Rank	6	1	4	3	7	10	8	5	9	2
F2	Best	4.073 × 102	4.072 × 102	4.075 × 102	4.074 × 102	4.056 × 102	4.073 × 102	4.072 × 102	4.072 × 102	4.072 × 102	4.075 × 102
	Mean	4.084 × 102	4.080 × 102	4.096 × 102	4.100 × 102	4.094 × 102	4.089 × 102	4.087 × 102	4.088 × 102	4.083 × 102	4.091 × 102
	Std	9.251 × 10−1	1.006 × 100	2.952 × 100	3.695 × 100	3.798 × 100	9.312 × 10−1	9.228 × 10−1	9.202 × 10−1	9.169 × 10−1	7.495 × 10−1
	Rank	4	1	7	10	5	8	3	6	2	9
F3	Best	6.002 × 102	6.001 × 102	6.001 × 102	6.001 × 102	6.000 × 102	6.000 × 102	6.001 × 102	6.000 × 102	6.000 × 102	6.000 × 102
	Mean	6.005 × 102	6.004 × 102	6.004 × 102	6.003 × 102	6.003 × 102	6.003 × 102	6.003 × 102	6.004 × 102	6.003 × 102	6.001 × 102
	Std	2.410 × 10−1	3.504 × 10−1	2.801 × 10−1	1.990 × 10−1	1.952 × 10−1	2.656 × 10−1	3.714 × 10−1	3.445 × 10−1	2.673 × 10−1	1.850 × 10−1
	Rank	10	9	8	4	6	2	5	7	3	1
F4	Best	8.114 × 102	8.089 × 102	8.097 × 102	8.154 × 102	8.121 × 102	8.137 × 102	8.103 × 102	8.101 × 102	8.099 × 102	8.128 × 102
	Mean	8.166 × 102	8.151 × 102	8.155 × 102	8.167 × 102	8.155 × 102	8.182 × 102	8.171 × 102	8.159 × 102	8.158 × 102	8.176 × 102
	Std	4.633 × 100	4.299 × 100	3.858 × 100	1.611 × 100	2.125 × 100	3.027 × 100	4.182 × 100	4.285 × 100	3.761 × 100	3.324 × 100
	Rank	4	1	5	7	3	10	8	2	6	9
F5	Best	9.008 × 102	9.009 × 102	9.025 × 102	9.009 × 102	9.011 × 102	9.009 × 102	9.018 × 102	9.011 × 102	9.010 × 102	9.014 × 102
	Mean	9.065 × 102	9.057 × 102	9.056 × 102	9.052 × 102	9.036 × 102	9.036 × 102	9.033 × 102	9.047 × 102	9.024 × 102	9.059 × 102
	Std	4.760 × 100	4.671 × 100	2.922 × 100	2.764 × 100	2.367 × 100	2.145 × 100	1.223 × 100	3.220 × 100	9.747 × 10−1	4.160 × 100
	Rank	9	5	10	8	4	2	3	6	1	7
F6	Best	2.659 × 103	2.115 × 103	2.139 × 103	1.959 × 103	2.094 × 103	2.166 × 103	1.922 × 103	2.346 × 103	2.066 × 103	2.541 × 103
	Mean	3.968 × 103	3.039 × 103	3.421 × 103	3.060 × 103	2.905 × 103	3.168 × 103	2.713 × 103	3.557 × 103	2.970 × 103	3.799 × 103
	Std	1.630 × 103	1.174 × 103	1.006 × 103	7.081 × 102	8.531 × 102	9.538 × 102	5.070 × 102	9.450 × 102	7.538 × 102	9.708 × 102
	Rank	9	2	7	6	3	5	1	8	4	10
F7	Best	2.025 × 103	2.027 × 103	2.025 × 103	2.026 × 103	2.026 × 103	2.027 × 103	2.024 × 103	2.027 × 103	2.029 × 103	2.022 × 103
	Mean	2.031 × 103	2.031 × 103	2.029 × 103	2.032 × 103	2.030 × 103	2.033 × 103	2.031 × 103	2.030 × 103	2.032 × 103	2.031 × 103
	Std	3.198 × 100	2.697 × 100	3.383 × 100	3.674 × 100	3.378 × 100	3.968 × 100	4.169 × 100	2.515 × 100	2.828 × 100	4.330 × 100
	Rank	4	6	1	8	3	10	5	2	9	7
F8	Best	2.221 × 103	2.220 × 103	2.221 × 103	2.209 × 103	2.216 × 103	2.211 × 103	2.220 × 103	2.219 × 103	2.220 × 103	2.219 × 103
	Mean	2.224 × 103	2.224 × 103	2.225 × 103	2.222 × 103	2.221 × 103	2.224 × 103	2.225 × 103	2.225 × 103	2.225 × 103	2.224 × 103
	Std	1.609 × 100	2.604 × 100	1.605 × 100	5.887 × 100	3.501 × 100	4.970 × 100	1.773 × 100	2.305 × 100	1.928 × 100	2.718 × 100
	Rank	2	5	6	3	1	9	7	4	10	8
F9	Best	2.529 × 103	2.529 × 103	2.529 × 103	2.529 × 103	2.529 × 103	2.529 × 103	2.529 × 103	2.529 × 103	2.529 × 103	2.529 × 103
	Mean	2.530 × 103	2.531 × 103	2.530 × 103	2.530 × 103	2.530 × 103	2.530 × 103	2.530 × 103	2.530 × 103	2.530 × 103	2.531 × 103
	Std	8.413 × 10−1	2.076 × 100	4.889 × 10−1	2.068 × 100	7.222 × 10−1	1.814 × 100	1.379 × 100	8.030 × 10−1	3.129 × 10−1	3.065 × 100
	Rank	5	10	6	7	3	9	2	1	8	4
F10	Best	2.500 × 103	2.500 × 103	2.500 × 103	2.500 × 103	2.500 × 103	2.500 × 103	2.500 × 103	2.500 × 103	2.500 × 103	2.500 × 103
	Mean	2.500 × 103	2.500 × 103	2.501 × 103	2.500 × 103	2.501 × 103	2.501 × 103	2.501 × 103	2.500 × 103	2.501 × 103	2.500 × 103
	Std	1.133 × 10−1	2.319 × 10−2	1.465 × 10−1	6.388 × 10−2	7.403 × 10−1	7.931 × 10−2	1.817 × 10−1	6.871 × 10−2	2.892 × 10−1	5.213 × 10−2
	Rank	2	3	5	1	8	9	6	7	10	4
F11	Best	2.679 × 103	2.701 × 103	2.713 × 103	2.656 × 103	2.716 × 103	2.688 × 103	2.706 × 103	2.697 × 103	2.645 × 103	2.646 × 103
	Mean	2.737 × 103	2.735 × 103	2.738 × 103	2.729 × 103	2.738 × 103	2.732 × 103	2.743 × 103	2.736 × 103	2.736 × 103	2.734 × 103
	Std	2.195 × 101	1.743 × 101	1.347 × 101	2.651 × 101	1.098 × 101	1.905 × 101	1.447 × 101	1.762 × 101	3.272 × 101	3.212 × 101
	Rank	8	4	6	1	2	3	10	5	9	7
F12	Best	2.862 × 103	2.863 × 1003	2.863 × 103	2.862 × 103	2.860 × 103	2.861 × 103	2.860 × 103	2.861 × 103	2.862 × 103	2.863 × 103
	Mean	2.863 × 103	2.864 × 1003	2.864 × 103	2.864 × 103	2.863 × 103	2.863 × 103	2.863 × 103	2.863 × 103	2.864 × 103	2.863 × 103
	Std	6.712 × 10−1	5.493 × 10−1	6.343 × 10−1	1.183 × 100	1.210 × 100	1.279 × 100	1.166 × 100	9.406 × 10−1	6.436 × 10−1	3.016 × 10−1
	Rank	2	8	10	7	6	4	5	1	9	3
Mean Rank		5.42	4.58	6.25	5.42	4.25	6.75	5.25	4.5	6.67	5.92

and

Table A7. Sensitivity analysis results for parameter $k$ on CEC2022 (dim = 20).

Function	Item Name	$\mathit{k}$
		1	2	3	4	5	6	7	8	9	10
F1	Best	1.313 × 104	8.988 × 103	1.140 × 104	1.152 × 104	1.288 × 104	1.321 × 104	1.429 × 104	1.224 × 104	1.258 × 104	1.333 × 104
	Mean	2.256 × 104	1.836 × 104	2.359 × 104	2.027 × 104	1.882 × 104	2.015 × 104	2.009 × 104	1.764 × 104	2.433 × 104	2.060 × 104
	Std	6.043 × 103	5.429 × 103	7.038 × 103	8.987 × 103	4.098 × 103	6.461 × 103	3.846 × 103	3.764 × 103	1.072 × 104	5.498 × 103
	Rank	9	2	10	3	4	5	6	1	8	7
F2	Best	4.491 × 102	4.492 × 102	4.491 × 102	4.491 × 102	4.491 × 102	4.491 × 102	4.491 × 102	4.491 × 102	4.491 × 102	4.491 × 102
	Mean	4.541 × 102	4.541 × 102	4.564 × 102	4.557 × 102	4.530 × 102	4.541 × 102	4.521 × 102	4.567 × 102	4.552 × 102	4.527 × 102
	Std	8.072 × 100	8.240 × 100	1.212 × 101	1.020 × 101	7.360 × 100	1.052 × 101	7.448 × 100	1.082 × 101	9.286 × 100	8.937 × 100
	Rank	9	10	6	8	7	3	1	4	5	2
F3	Best	6.002 × 102	6.001 × 102	6.001 × 102	6.001 × 102	6.001 × 102	6.001 × 102	6.001 × 102	6.001 × 102	6.001 × 102	6.001 × 102
	Mean	6.003 × 102	6.003 × 102	6.002 × 102	6.003 × 102	6.003 × 102	6.002 × 102	6.002 × 102	6.002 × 102	6.002 × 102	6.002 × 102
	Std	1.149 × 10−1	1.008 × 10−1	8.454 × 10−2	9.944 × 10−2	2.539 × 10−1	1.741 × 10−1	9.909 × 10−2	1.044 × 10−1	9.300 × 10−2	1.290 × 10−1
	Rank	10	8	5	9	4	1	3	6	7	2
F4	Best	8.465 × 102	8.453 × 102	8.453 × 102	8.512 × 102	8.365 × 102	8.273 × 102	8.423 × 102	8.460 × 102	8.501 × 1002	8.431 × 102
	Mean	8.562 × 102	8.530 × 102	8.570 × 102	8.571 × 102	8.535 × 102	8.530 × 102	8.540 × 102	8.549 × 102	8.581 × 102	8.556 × 102
	Std	6.388 × 100	6.182 × 100	5.873 × 100	3.494 × 100	6.710 × 100	1.033 × 101	7.164 × 100	5.527 × 100	5.408 × 100	7.629 × 100
	Rank	7	2	6	8	3	5	1	4	10	9
F5	Best	9.000 × 102	9.001 × 102	9.000 × 102	9.001 × 102	9.001 × 102	9.002 × 102	9.001 × 102	9.011 × 102	9.001 × 102	9.004 × 102
	Mean	9.049 × 102	9.031 × 102	9.029 × 102	9.019 × 102	9.027 × 102	9.017 × 102	9.016 × 102	9.043 × 102	9.013 × 102	9.017 × 102
	Std	7.111 × 100	5.835 × 100	3.551 × 100	2.130 × 100	2.247 × 100	1.689 × 100	1.169 × 100	4.876 × 100	1.452 × 100	1.007 × 100
	Rank	8	2	4	6	9	5	3	10	1	7
F6	Best	5.371 × 103	2.468 × 103	6.923 × 103	4.078 × 103	2.898 × 103	3.524 × 103	3.488 × 103	2.682 × 103	3.643 × 103	4.973 × 103
	Mean	1.662 × 104	1.591 × 104	1.400 × 104	1.283 × 104	1.183 × 104	1.089 × 104	1.285 × 104	9.385 × 103	1.237 × 104	1.030 × 104
	Std	7.920 × 103	1.244 × 104	5.953 × 103	7.449 × 103	8.854 × 103	1.011 × 104	6.398 × 103	4.497 × 103	8.187 × 103	4.175 × 103
	Rank	10	8	9	5	3	2	7	1	6	4
F7	Best	2.053 × 103	2.053 × 103	2.044 × 103	2.059 × 103	2.052 × 103	2.054 × 103	2.060 × 103	2.052 × 103	2.035 × 103	2.043 × 103
	Mean	2.069 × 103	2.062 × 103	2.063 × 103	2.070 × 103	2.068 × 103	2.063 × 103	2.069 × 103	2.066 × 103	2.065 × 103	2.061 × 103
	Std	7.043 × 100	7.263 × 100	1.284 × 101	6.545 × 100	1.464 × 101	6.634 × 100	5.752 × 100	1.156 × 101	1.550 × 101	9.566 × 100
	Rank	10	3	4	9	7	1	8	6	5	2
F8	Best	2.227 × 103	2.228 × 103	2.229 × 103	2.226 × 103	2.229 × 103	2.226 × 103	2.229 × 103	2.228 × 103	2.227 × 103	2.226 × 103
	Mean	2.230 × 103	2.230 × 103	2.230 × 103	2.229 × 103	2.230 × 103	2.229 × 103	2.230 × 103	2.230 × 103	2.229 × 103	2.229 × 103
	Std	1.474 × 100	1.182 × 100	8.012 × 10−1	1.305 × 100	1.093 × 100	1.409 × 100	7.423 × 10−1	1.018 × 100	1.504 × 100	2.361 × 100
	Rank	9	7	5	1	8	3	6	10	4	2
F9	Best	2.481 × 103	2.481 × 103	2.481 × 103	2.481 × 103	2.481 × 103	2.481 × 103	2.481 × 103	2.481 × 103	2.481 × 103	2.481 × 103
	Mean	2.482 × 103	2.481 × 103	2.481 × 103	2.482 × 103	2.482 × 103	2.481 × 103	2.482 × 103	2.481 × 103	2.481 × 103	2.482 × 103
	Std	5.243 × 10−1	3.630 × 10−1	4.709 × 10−1	5.124 × 10−1	6.593 × 10−1	5.065 × 10−1	1.153 × 100	6.032 × 10−1	3.866 × 10−1	9.217 × 10−1
	Rank	9	4	5	10	8	7	1	3	2	6
F10	Best	2.501 × 103	2.500 × 103	2.500 × 103	2.500 × 103	2.500 × 103	2.500 × 103	2.500 × 103	2.500 × 103	2.500 × 103	2.500 × 103
	Mean	2.516 × 103	2.512 × 103	2.543 × 103	2.518 × 103	2.508 × 103	2.537 × 103	2.623 × 103	2.513 × 103	2.501 × 103	2.510 × 103
	Std	3.072 × 101	3.539 × 101	1.040 × 102	3.559 × 101	1.564 × 101	9.630 × 101	3.824 × 102	2.740 × 101	7.912 × 10−2	2.270 × 101
	Rank	10	5	2	8	6	9	4	1	3	7
F11	Best	2.905 × 103	2.902 × 103	2.902 × 103	2.902 × 103	2.902 × 103	2.902 × 103	2.901 × 103	2.901 × 103	2.901 × 103	2.901 × 103
	Mean	2.988 × 103	3.021 × 103	2.902 × 103	2.926 × 103	2.902 × 103	2.941 × 103	2.943 × 103	2.941 × 103	2.955 × 103	2.952 × 103
	Std	1.700 × 102	1.912 × 102	4.893 × 10−1	7.579 × 101	5.263 × 10−1	1.239 × 102	1.311 × 102	1.227 × 102	1.232 × 102	1.167 × 102
	Rank	10	9	6	3	7	4	2	8	1	5
F12	Best	2.937 × 103	2.941 × 103	2.938 × 103	2.940 × 103	2.940 × 103	2.941 × 103	2.939 × 103	2.940 × 103	2.935 × 103	2.939 × 103
	Mean	2.944 × 103	2.946 × 103	2.944 × 103	2.945 × 103	2.945 × 103	2.946 × 103	2.946 × 103	2.943 × 103	2.944 × 103	2.943 × 103
	Std	3.589 × 100	2.610 × 100	3.001 × 100	3.427 × 100	4.687 × 100	3.069 × 100	4.654 × 100	2.995 × 100	5.198 × 100	2.867 × 100
	Rank	6	9	3	5	7	10	8	1	4	2
Mean Rank		8.92	5.75	5.42	6.25	6.08	4.58	4.17	4.58	4.67	4.58

within Appendix B. The Friedman average rankings over the 12 CEC2022 functions for the two dimensions are shown in Figure 5 and Figure 6, respectively.

From the results, it can be observed that the parameter combination λ = 0.4 and k = 8 achieves the lowest average Friedman ranking in both dimensions. Therefore, these values are selected as the default parameter settings for EAPO in all subsequent experiments.

6. Algorithm Performance Testing and Analysis

This section evaluates the performance of EAPO through a series of benchmark tests and practical engineering applications. The evaluation consists of two main parts: standard benchmark function testing and real-world engineering problem validation.

First, ablation experiments are conducted to assess the effectiveness of each proposed strategy. Quantitative analysis is used to examine the algorithm’s ability to maintain a dynamic balance between exploration and exploitation. Subsequently, EAPO is systematically evaluated using the CEC2022 and CEC2020 benchmark suites, which contain 12 and 10 test functions, respectively. These benchmarks are designed to assess convergence accuracy, convergence speed, robustness, and adaptability across different function types. Finally, EAPO is applied to ten real-world engineering optimization problems to demonstrate its practical applicability and effectiveness in solving complex constrained optimization tasks. The detailed definitions of the CEC2022 and CEC2020 functions are provided in

Table A1. CEC2022 benchmark functions.

Type	No.	Functions	${\mathit{F}}_{\mathit{i}}^{\mathbf{\ast }}={\mathit{F}}_{\mathit{i}}\left({\mathit{X}}^{\mathbf{\ast }}\right)$
Unimodal Functions	1	Shifted and full Rotated Zakharov Function	300
Basic Functions	2	Shifted and full Rotated Rosenbrock’s Function	400
	3	Shifted and full Rotated Expanded Schaffer’s f6 Function	600
	4	Shifted and full Rotated Non-Continuous Rastrigin’s Function	800
	5	Shifted and full Rotated Levy Function	900
Hybrid Functions	6	Hybrid Function 1 (N = 3)	1800
	7	Hybrid Function 2 (N = 3)	2000
	8	Hybrid Function 3 (N = 3)	2200
Composition Functions	9	Composition Function 1 (N = 3)	2300
	10	Composition Function 2 (N = 4)	2400
	11	Composition Function 3 (N = 5)	2600
	12	Composition Function 4 (N = 6)	2700
$\mathrm{Search} \mathrm{Range}: {[-100, 100]}^D$

and

Table A2. CEC2020 benchmark functions.

Type	No.	Functions	${\mathit{F}}_{\mathit{i}}^{\mathbf{\ast }}={\mathit{F}}_{\mathit{i}}\left({\mathit{X}}^{\mathbf{\ast }}\right)$
Unimodal Functions	1	Shifted and Rotated Bent Cigar Function	100
Basic Functions	2	Shifted and Rotated Schwefel’s Function	1100
	3	Shifted and Rotated Rastrigin’s Function	700
	4	Shifted and Rotated Lunacek bi-Rastrigin Function	1900
Hybrid Functions	5	Hybrid Function 1 (N = 3)	1700
	6	Hybrid Function 2 (N = 3)	1600
	7	Hybrid Function 3 (N = 3)	2100
Composition Functions	8	Composition Function 1 (N = 3)	2200
	9	Composition Function 2 (N = 4)	2400
	10	Composition Function 3 (N = 5)	2500
$\mathrm{Search} \mathrm{Range}: {[-100, 100]}^D$

of Appendix A.2.

We selected 13 representative optimization algorithms across four categories for comparison: (1) classic high-citation algorithms: PSO, DE, WOA, HHO, CMA-ES (Matrix Adaptation—Evolution strategies, 2016) [32]; (2) highly cited algorithms proposed within the last three years: DBO, AOA, HO (Hippopotamus Optimization Algorithm, 2024) [33], HOA (Hiking Optimization Algorithm, 2024) [34], TGCOA (The tetragonula carbonaria Optimization Algorithm, 2025) [35]; (3) mature algorithm variants: NCHHO (Nonlinear Chaotic Harris Hawks Optimization, 2021) [36], CL-PSO (Comprehensive Learning Particle Swarm Optimization, 2009) [37]; (4) the fundamental artificial protozoa optimization algorithm: APO.

6.1. Experimental Configuration

The population size for all algorithms was uniformly set to 100. Other parameters were configured according to their original references or widely adopted standard values, as detailed in

Table 1. Parameters setting.

Algorithms	Parameters Setting
PSO	$C_1=1,C_2=1,v_{min}=-10, v_{max}=10, \omega =[0.9, 0.4]$
DE	$F=0.5,$ $CR=0.5$
CMA-ES	$\sigma =0.75$
WOA	$a=[2, 0]$
HHO	$a=5, \beta =1.5$
CL-PSO	$C=1.2,$ $F_{max}=7, {\omega }_{max}=0.9$$, {\omega }_{min}=0.4$
NCHHO	$a=4$
DBO	$k=0.1, b=0.3, S=0.5$
HO	Parameter free
HOA	$\theta =[0, 50]$$, SF=[1, 2]$
AOA	$\mu =0.499,$ $\alpha =5$
TGCOA	$lT=10, uT=40, k=0.03, \alpha =50, a=0, b=10$
APO	${pf}_{max}=0.1$$, np=1$
EAPO	$\lambda =0.4,$${\alpha }_{max}=0.9$$, {\alpha }_{min}=0.1,$ $\omega =0.5$$, {pf}_{max}=0.1$$, np=1,$ $k=8$

. The maximum number of function evaluations in the CEC2022 and CEC2020 tests were set to 1000 × dim, while the maximum evaluations for real-world engineering problems were set to 20,000. All experiments were programmed and implemented in the MATLAB R2021b environment. To mitigate randomness, each algorithm was independently run 30 times on each test problem. The minimum, average, and standard deviation values from each run were recorded as performance metrics. For rigorous statistical analysis of experimental results, both the Wilcoxon rank-sum test and Friedman test were employed.

6.2. Ablation Experiment

To verify the effectiveness of the proposed improvement strategies, an ablation study was conducted on the CEC2022 benchmark suite with a problem dimension of 10. Four enhanced variants of the original APO algorithm were constructed by individually integrating each strategy: APO_L (APO with Latin Hypercube Sampling initialization), APO_H (APO with the success-history-based adaptive parameter adjustment), APO_S (APO with sensory-field-based environmental structure modeling strategy), and APO_D (APO with the adaptive dormancy–reconstruction mechanism).

The experimental results are summarized in

Table 2. Ablation experiment results.

Function	Item Name	APO	APO_L	APO_H	APO_S	APO_D	EAPO
F1	Best	3.441 × 103	3.829 × 103	1.729 × 103	2.098 × 103	2.094 × 103	2.464 × 103
	Mean	7.274 × 103	7.249 × 103	6.752 × 103	7.078 × 103	6.201 × 103	5.198 × 103
	Std	1.671 × 103	2.228 × 103	2.236 × 103	2.347 × 103	2.271 × 103	1.903 × 103
	Rank	6	5	3	4	2	1
F2	Best	4.080 × 102	4.078 × 102	4.076 × 102	4.076 × 102	4.073 × 102	4.071 × 102
	Mean	4.107 × 102	4.102 × 102	4.126 × 102	4.098 × 102	4.088 × 102	4.086 × 102
	Std	2.409 × 100	2.082 × 100	6.368 × 100	1.962 × 100	1.059 × 100	8.597 × 10−1
	Rank	6	4	5	3	2	1
F3	Best	6.009 × 102	6.008 × 102	6.004 × 102	6.009 × 102	6.008 × 102	6.000 × 102
	Mean	6.016 × 102	6.016 × 102	6.010 × 102	6.019 × 102	6.012 × 102	6.002 × 102
	Std	3.579 × 10−1	3.740 × 10−1	2.703 × 10−1	4.945 × 10−1	2.811 × 10−1	2.659 × 10−1
	Rank	4	5	2	6	3	1
F4	Best	8.080 × 102	8.102 × 102	8.111 × 102	8.119 × 102	8.064 × 102	8.072 × 102
	Mean	8.172 × 102	8.154 × 102	8.170 × 102	8.169 × 102	8.140 × 102	8.165 × 102
	Std	3.619 × 100	2.937 × 100	3.260 × 100	3.122 × 100	3.801 × 100	3.506 × 100
	Rank	6	2	5	4	1	3
F5	Best	9.053 × 102	9.136 × 102	9.080 × 102	9.086 × 102	9.029 × 102	9.004 × 102
	Mean	9.217 × 102	9.215 × 102	9.165 × 102	9.163 × 102	9.104 × 102	9.048 × 102
	Std	7.598 × 100	7.471 × 100	5.562 × 100	5.999 × 100	4.161 × 100	2.820 × 100
	Rank	6	5	4	3	2	1
F6	Best	2.449 × 103	2.437 × 103	1.909 × 103	1.993 × 103	2.058 × 103	1.904 × 103
	Mean	4.302 × 103	4.389 × 103	3.669 × 103	4.199 × 103	4.352 × 103	3.513 × 103
	Std	1.703 × 103	1.600 × 103	1.661 × 103	1.383 × 103	1.458 × 103	1.162 × 103
	Rank	3	6	2	4	5	1
F7	Best	2.021 × 103	2.012 × 103	2.018 × 103	2.016 × 103	2.017 × 103	2.021 × 103
	Mean	2.029 × 103	2.027 × 103	2.028 × 103	2.030 × 103	2.027 × 103	2.031 × 103
	Std	3.696 × 100	4.167 × 100	3.824 × 100	4.671 × 100	3.720 × 100	4.225 × 100
	Rank	4	2	3	5	1	6
F8	Best	2.213 × 103	2.214 × 103	2.217 × 103	2.221 × 103	2.216 × 103	2.218 × 103
	Mean	2.222 × 103	2.223 × 103	2.223 × 103	2.225 × 103	2.223 × 103	2.225 × 103
	Std	3.209 × 100	2.653 × 100	1.995 × 100	1.213 × 100	2.361 × 100	2.015 × 100
	Rank	1	3	4	6	2	5
F9	Best	2.530 × 103	2.529 × 103	2.530 × 103	2.529 × 103	2.529 × 103	2.529 × 103
	Mean	2.534 × 103	2.532 × 103	2.532 × 103	2.531 × 103	2.531 × 103	2.530 × 103
	Std	6.907 × 100	1.757 × 100	2.232 × 100	1.541 × 100	1.534 × 100	1.471 × 100
	Rank	6	5	4	3	2	1
F10	Best	2.500 × 103	2.500 × 103	2.500 × 103	2.500 × 103	2.500 × 103	2.500 × 103
	Mean	2.501 × 103	2.501 × 103	2.501 × 103	2.505 × 103	2.501 × 103	2.501 × 103
	Std	1.421 × 100	8.305 × 10−1	2.753 × 100	2.113 × 101	2.036 × 100	4.090 × 10−1
	Rank	5	4	3	6	2	1
F11	Best	2.716 × 103	2.741 × 103	2.738 × 103	2.733 × 103	2.720 × 103	2.665 × 103
	Mean	2.747 × 103	2.747 × 103	2.749 × 103	2.746 × 103	2.744 × 103	2.735 × 103
	Std	7.346 × 100	2.849 × 100	2.994 × 100	4.653 × 100	7.199 × 100	1.858 × 101
	Rank	5	4	6	3	2	1
F12	Best	2.862 × 103	2.861 × 103	2.863 × 103	2.861 × 103	2.861 × 103	2.861 × 103
	Mean	2.864 × 103	2.864 × 103	2.864 × 103	2.864 × 103	2.864 × 103	2.863 × 103
	Std	7.076 × 10−1	7.757 × 10−1	7.868 × 10−1	8.428 × 10−1	7.667 × 10−1	9.569 × 10−1
	Rank	2	4	5	6	3	1
Mean Rank		4.5	4.08	4.08	4.17	2.25	1.92
Final Ranking		6	3	3	5	2	1

, including the optimal values, mean values, standard deviations, and Friedman rankings obtained from testing. The best results in the table are highlighted in bold and underlined.. The results demonstrate that the fully enhanced EAPO algorithm, which integrates all four strategies, achieves the best overall performance and attains the highest number of optimal performance indicators across the test functions, ranking first overall. In addition, the single-strategy variants also exhibit competitive performance, as each of them achieves the best results on certain individual metrics or specific test functions, and all ranking higher than the original APO. These findings indicate that the proposed strategies are not only individually effective in improving the search performance of APO, but also complementary in nature, and their combination leads to a more robust and efficient optimization capability.

6.3. Exploration and Exploitation Capability Analysis

To quantitatively evaluate the global optimization performance of the algorithm, this section assesses the exploration and exploitation capabilities of EAPO using the population dimensional diversity metric [38,39]. This metric measures diversity changes during the search process by calculating the average deviation of population individuals from the median across each dimension. Its formula is as follows:

$$\begin{array}{c}Div={\displaystyle \frac{1}{D}}{\displaystyle \sum _{j=1}^D}{\displaystyle \frac{1}{N}}\left({\displaystyle \sum _{i=1}^N}\left|median\left(X_j\right)-X_{i,j}\right|\right)\end{array}$$

(54)

On this basis, the exploration and exploitation ratios in each iteration of the algorithm can be defined as follows:

$$\begin{array}{c} Exploration\%={\displaystyle \frac{Div}{{Div}_{max}}}\end{array}$$

(55)

$$\begin{array}{c} Exploitation\%={\displaystyle \frac{\left|Div-{Div}_{max}\right|}{{Div}_{max}}}\end{array}$$

(56)

where $Div$ represents the diversity metric of the current population; ${Div}_{max}$ denotes the maximum diversity value recorded throughout the entire iteration process; $X_{i,j}$ indicates the position of the $i$th individual in the $j$th dimension; and $median\left(X_j\right)$ signifies the median value of all individuals in the $j$th dimensional variable.

Six functions from CEC2022 were selected for analysis, and the results are shown in Figure 7. The exploration and exploitation ratios of EAPO exhibit clear changes during the iteration process. In the early stage, the algorithm maintains a high exploration ratio. This allows extensive searching of the solution space to locate promising regions. As the iteration proceeds, the exploration ratio gradually decreases. At the same time, the exploitation ratio steadily increases.

This behavior indicates a gradual transition from global exploration to local refinement. It shows that EAPO can effectively maintain a dynamic balance between exploration and exploitation.

6.4. Testing on the CEC2022 Benchmark Suite

Test results obtained on the 10-dimensional and 20-dimensional CEC2022 benchmarks are shown in

Table 3. Test results of EAPO and other algorithms on CEC2022 (dim = 10).

Function	Item Name	PSO	DE	CMA-ES	WOA	HHO	CL-PSO	NC-HHO	DBO	HO	HOA	AOA	TG-COA	APO	EAPO
F1	Best	4.192 × 103	7.024 × 103	2.971 × 103	7.056 × 103	4.694 × 103	5.551 × 103	5.076 × 103	5.348 × 102	1.659 × 103	9.697 × 102	1.727 × 104	1.563 × 103	2.769 × 103	2.739 × 103
	Mean	7.337 × 103	2.080 × 104	1.848 × 104	2.583 × 104	8.749 × 103	1.744 × 104	9.386 × 103	3.226 × 103	6.559 × 103	3.516 × 103	1.207 × 105	5.136 × 103	7.999 × 103	5.136 × 103
	Std	2.401 × 103	5.527 × 103	1.054 × 104	1.341 × 104	1.474 × 103	4.335 × 103	1.471 × 103	1.997 × 103	2.535 × 103	1.272 × 103	3.379 × 105	2.226 × 103	2.156 × 103	1.987 × 103
	p	1.709 × 10−3	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.127 × 10−5	1.734 × 10−6	3.515 × 10−6	2.831 × 10−4	2.067 × 10−2	1.197 × 10−0	1.734 × 10−6	7.655 × 10−1	6.320 × 10−5	~
	Rank	6	12	10	13	8	11	9	1	5	2	14	3	7	4
F2	Best	4.138 × 102	4.295 × 102	4.950 × 102	4.016 × 102	4.028 × 102	5.135 × 102	4.418 × 102	4.006 × 102	4.197 × 102	4.122 × 102	7.812 × 102	4.019 × 102	4.081 × 102	4.069 × 102
	Mean	4.563 × 102	4.623 × 102	6.021 × 102	4.451 × 102	5.171 × 102	6.894 × 102	5.789 × 102	4.278 × 102	4.997 × 102	5.099 × 102	2.298 × 103	4.770 × 102	4.107 × 102	4.090 × 102
	Std	7.060 × 101	1.703 × 101	6.205 × 101	7.397 × 101	1.065 × 102	1.498 × 102	9.076 × 101	3.058 × 101	6.966 × 101	7.573 × 101	1.029 × 103	4.194 × 101	2.991 × 100	2.243 × 100
	p	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	4.390 × 10−3	4.286 × 10−6	1.734 × 10−6	1.734 × 10−6	1.245 × 10−2	1.734 × 10−6	1.921 × 10−6	1.734 × 10−6	2.127 × 10−6	5.287 × 10−4	~
	Rank	5	6	12	4	8	13	11	3	9	10	14	7	2	1
F3	Best	6.114 × 102	6.084 × 102	6.000 × 102	6.175 × 102	6.237 × 102	6.229 × 102	6.217 × 102	6.000 × 102	6.144 × 102	6.112 × 102	6.502 × 102	6.173 × 102	6.005 × 102	6.000 × 102
	Mean	6.215 × 102	6.229 × 102	6.156 × 102	6.365 × 102	6.405 × 102	6.427 × 102	6.475 × 102	6.018 × 102	6.318 × 102	6.214 × 102	6.865 × 102	6.463 × 102	6.015 × 102	6.002 × 102
	Std	5.720 × 100	4.489 × 100	1.923 × 101	1.509 × 101	1.167 × 101	9.533 × 100	1.266 × 101	3.465 × 100	1.244 × 101	6.113 × 100	1.405 × 101	1.439 × 101	4.004 × 10−1	2.276 × 10−1
	p	1.734 × 10−6	1.734 × 10−6	4.492 × 10−2	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	2.415 × 10−3	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	~
	Rank	5	7	4	9	10	11	13	2	8	6	14	12	3	1
F4	Best	8.371 × 102	8.496 × 102	8.111 × 102	8.150 × 102	8.187 × 102	8.490 × 102	8.227 × 102	8.070 × 102	8.127 × 102	8.113 × 102	8.858 × 102	8.145 × 102	8.102 × 102	8.091 × 102
	Mean	8.576 × 102	8.678 × 102	8.255 × 102	8.382 × 102	8.361 × 102	8.660 × 102	8.408 × 102	8.316 × 102	8.303 × 102	8.253 × 102	9.148 × 102	8.387 × 102	8.159 × 102	8.168 × 102
	Std	7.439 × 100	7.699 × 100	7.242 × 100	1.449 × 101	1.060 × 101	7.449 × 100	1.031 × 101	1.623 × 101	8.938 × 100	8.850 × 100	1.365 × 101	1.474 × 101	3.133 × 100	3.471 × 100
	p	1.734 × 10−6	1.734 × 10−6	5.792 × 10−5	2.353 × 10−6	1.921 × 10−6	1.734 × 10−6	1.734 × 10−6	1.742 × 10−4	6.984 × 10−6	1.036 × 10−3	1.734 × 10−6	1.734 × 10−6	5.577 × 10−1	~
	Rank	11	13	4	7	8	12	10	5	6	3	14	9	1	2
F5	Best	1.012 × 103	1.394 × 103	9.000 × 102	9.242 × 102	1.047 × 103	1.266 × 103	1.258 × 103	9.005 × 102	9.882 × 102	9.241 × 102	2.429 × 103	1.191 × 103	9.116 × 102	9.017 × 102
	Mean	1.155 × 103	1.971 × 103	9.000 × 102	1.473 × 103	1.385 × 103	1.901 × 103	1.469 × 103	9.384 × 102	1.193 × 103	1.031 × 103	4.235 × 103	1.665 × 103	9.222 × 102	9.046 × 102
	Std	8.676 × 101	3.356 × 102	2.404 × 10−5	3.646 × 102	1.716 × 102	3.669 × 102	1.507 × 102	7.870 × 101	1.752 × 102	8.543 × 101	8.750 × 102	3.818 × 102	7.559 × 100	2.573 × 100
	p	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	2.105 × 10−3	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	~
	Rank	6	13	1	9	8	12	10	3	7	5	14	11	4	2
F6	Best	6.735 × 104	1.566 × 105	3.017 × 105	2.142 × 103	1.845 × 103	2.490 × 104	1.908 × 103	1.882 × 103	2.299 × 103	2.227 × 103	8.483 × 107	1.869 × 103	1.936 × 103	1.905 × 103
	Mean	2.179 × 106	2.489 × 106	5.242 × 106	7.117 × 103	4.163 × 103	3.550 × 106	4.117 × 103	6.077 × 103	5.559 × 104	4.015 × 106	8.333 × 108	3.458 × 103	3.590 × 103	3.172 × 103
	Std	1.478 × 106	2.093 × 106	6.734 × 106	7.599 × 103	6.861 × 103	5.157 × 106	3.370 × 103	2.322 × 103	1.168 × 105	7.733 × 106	4.633 × 108	1.868 × 103	1.268 × 103	1.018 × 103
	p	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	4.196 × 10−4	2.989 × 10−1	1.734 × 10−6	2.536 × 10−1	2.843 × 10−5	8.944 × 10−4	9.316 × 10−6	1.734 × 10−6	9.426 × 10−1	2.989 × 10−1	~
	Rank	12	11	13	6	1	10	5	8	7	9	14	2	4	3
F7	Best	2.040 × 103	2.034 × 103	2.026 × 103	2.029 × 103	2.044 × 103	2.049 × 103	2.036 × 103	2.000 × 103	2.028 × 103	2.028 × 103	2.090 × 103	2.046 × 103	2.018 × 103	2.025 × 103
	Mean	2.062 × 103	2.046 × 103	2.092 × 103	2.083 × 103	2.088 × 103	2.086 × 103	2.108 × 103	2.024 × 103	2.062 × 103	2.053 × 103	2.221 × 103	2.100 × 103	2.027 × 103	2.031 × 103
	Std	1.298 × 101	6.873 × 100	4.101 × 101	3.178 × 101	3.695 × 101	1.948 × 101	3.482 × 101	7.764 × 100	2.357 × 101	1.326 × 101	5.565 × 101	3.455 × 101	2.997 × 100	3.136 × 100
	p	1.734 × 10−6	1.734 × 10−6	6.339 × 10−6	1.921 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	5.792 × 10−5	2.603 × 10−6	1.921 × 10−6	1.734 × 10−6	1.734 × 10−6	5.307 × 10−5	~
	Rank	7	4	9	8	10	11	13	1	6	5	14	12	2	3
F8	Best	2.230 × 103	2.226 × 103	2.229 × 103	2.224 × 103	2.225 × 103	2.226 × 103	2.214 × 103	2.221 × 103	2.224 × 103	2.217 × 103	2.255 × 103	2.225 × 103	2.216 × 103	2.217 × 103
	Mean	2.242 × 103	2.230 × 103	2.241 × 103	2.239 × 103	2.240 × 103	2.234 × 103	2.247 × 103	2.226 × 103	2.252 × 103	2.227 × 103	2.503 × 103	2.263 × 103	2.223 × 103	2.224 × 103
	Std	2.247 × 101	2.250 × 100	8.718 × 100	1.329 × 101	1.490 × 101	4.525 × 100	2.094 × 101	3.771 × 100	4.589 × 101	3.649 × 100	1.720 × 102	5.795 × 101	2.460 × 100	3.264 × 100
	p	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	2.879 × 10−6	1.734 × 10−6	1.921 × 10−6	2.603 × 10−6	8.730 × 10−3	2.603 × 10−6	5.287 × 10−4	1.734 × 10−6	1.921 × 10−6	1.306 × 10−1	~
	Rank	10	5	12	9	8	6	13	3	7	4	14	11	1	2
F9	Best	2.531 × 103	2.534 × 103	2.531 × 103	2.530 × 103	2.532 × 103	2.576 × 103	2.540 × 103	2.529 × 103	2.550 × 103	2.618 × 103	2.727 × 103	2.548 × 103	2.529 × 103	2.529 × 103
	Mean	2.555 × 103	2.547 × 103	2.557 × 103	2.586 × 103	2.655 × 103	2.653 × 103	2.639 × 103	2.533 × 103	2.650 × 103	2.660 × 103	2.953 × 103	2.617 × 103	2.531 × 103	2.530 × 103
	Std	3.365 × 101	1.006 × 101	3.500 × 101	5.412 × 101	5.141 × 101	3.709 × 101	5.798 × 101	7.776 × 100	5.002 × 101	1.667 × 101	1.421 × 102	3.440 × 101	1.629 × 100	2.956 × 10−1
	p	1.921 × 10−6	1.734 × 10−6	2.127 × 10−6	2.879 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	9.918 × 10−1	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	~
	Rank	4	5	6	7	11	12	9	2	10	13	14	8	3	1
F10	Best	2.502 × 103	2.510 × 103	2.513 × 103	2.500 × 103	2.501 × 103	2.516 × 103	2.501 × 103	2.500 × 103	2.501 × 103	2.501 × 103	2.540 × 103	2.501 × 103	2.500 × 103	2.500 × 103
	Mean	2.576 × 103	2.520 × 103	2.771 × 103	2.563 × 103	2.609 × 103	2.550 × 103	2.627 × 103	2.544 × 103	2.580 × 103	2.570 × 103	3.192 × 103	2.564 × 103	2.501 × 103	2.500 × 103
	Std	8.326 × 101	6.368 × 10	3.486 × 102	1.625 × 102	3.051 × 102	4.279 × 101	1.398 × 102	6.332 × 101	7.441 × 101	6.330 × 101	6.623 × 102	1.163 × 102	2.448 × 10	9.281 × 10−2
	p	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	2.127 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	2.353 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	8.221 × 10−2	~
	Rank	10	7	13	4	6	11	12	3	9	8	14	5	2	1
F11	Best	2.783 × 103	2.791 × 103	2.806 × 103	2.635 × 103	2.717 × 103	2.809 × 103	2.738 × 103	2.600 × 103	2.736 × 103	2.702 × 103	2.966 × 103	2.732 × 103	2.716 × 103	2.633 × 103
	Mean	2.991 × 103	2.832 × 103	2.915 × 103	2.804 × 103	2.947 × 103	3.108 × 103	3.132 × 103	2.752 × 103	2.941 × 103	2.857 × 103	4.402 × 103	2.865 × 103	2.746 × 103	2.736 × 103
	Std	8.339 × 102	2.604 × 102	1.327 × 103	4.846 × 102	1.422 × 103	9.098 × 102	1.250 × 103	8.121 × 102	7.938 × 102	7.909 × 102	2.010 × 103	9.381 × 102	2.256 × 101	7.474 × 101
	p	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	3.160 × 10−2	4.729 × 10−6	1.734 × 10−6	2.127 × 10−6	4.284 × 10−1	3.515 × 10−6	2.597 × 10−5	1.734 × 10−6	4.729 × 10−6	3.501 × 10−2	~
	Rank	10	9	12	4	8	13	11	3	7	5	14	6	2	1
F12	Best	2.860 × 103	2.866 × 103	2.870 × 103	2.864 × 103	2.867 × 103	2.866 × 103	2.871 × 103	2.861 × 103	2.866 × 103	2.910 × 103	2.975 × 103	2.868 × 103	2.861 × 103	2.862 × 103
	Mean	2.870 × 103	2.867 × 103	2.875 × 103	2.893 × 103	2.923 × 103	2.895 × 103	2.937 × 103	2.864 × 103	2.892 × 103	2.954 × 103	3.109 × 103	2.919 × 103	2.864 × 103	2.863 × 103
	Std	5.579 × 100	1.143 × 100	2.742 × 100	3.055 × 101	5.743 × 101	1.724 × 101	4.529 × 101	1.504 × 100	3.364 × 101	2.685 × 101	7.643 × 101	2.899 × 101	7.257 × 10−1	8.084 × 10−1
	p	4.286 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	4.277 × 10−2	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	8.590 × 10−2	~
	Rank	5	4	6	8	10	9	12	3	7	13	14	11	2	1
Mean Rank		7.58	8	8.5	7.33	8	10.92	10.67	3.08	7.33	6.92	14	8.08	2.75	1.83
Final Ranking		7	8	11	6	8	13	12	3	5	4	14	10	2	1
+/=/−		12/0/0	12/0/0	11/0/1	12/0/0	11/1/0	12/0/0	11/1/0	8/1/3	12/0/0	11/1/0	12/0/0	10/0/0	7/3/2	~

and

Table 4. Test results of EAPO and other algorithms on CEC2022 (dim = 20).

Function	Item Name	PSO	DE	CMA-ES	WOA	HHO	CL-PSO	NC-HHO	DBO	HO	HOA	AOA	TG-COA	APO	EAPO
F1	Best	2.941 × 104	2.869 × 104	3.745 × 104	1.272 × 104	1.819 × 104	3.662 × 104	1.803 × 104	1.603 × 104	1.880 × 104	9.905 × 103	8.485 × 104	1.360 × 104	1.821 × 104	1.346 × 104
	Mean	5.077 × 104	6.218 × 104	6.962 × 104	2.168 × 104	3.946 × 104	5.997 × 104	3.257 × 104	3.347 × 104	2.974 × 104	1.979 × 104	2.827 × 107	2.870 × 104	2.783 × 104	1.995 × 104
	Std	1.569 × 104	9.490 × 103	1.957 × 104	7.843 × 103	1.489 × 104	1.053 × 104	9.899 × 103	1.166 × 104	6.187 × 103	4.989 × 103	1.466 × 108	8.135 × 103	4.366 × 103	4.297 × 103
	p	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	4.284 × 10−1	5.216 × 10−6	1.734 × 10−6	6.984 × 10−6	2.843 × 10−5	4.286 × 10−6	9.918 × 10−1	1.734 × 10−6	6.892 × 10−5	2.163 × 10−5	~
	Rank	10	12	13	3	9	11	8	7	6	2	14	5	4	1
F2	Best	5.611 × 102	6.176 × 102	6.497 × 102	4.564 × 102	5.133 × 102	1.072 × 103	5.733 × 102	4.067 × 102	5.727 × 102	6.573 × 102	2.978 × 103	5.545 × 102	4.461 × 102	4.449 × 102
	Mean	7.674 × 102	8.061 × 102	1.293 × 103	5.366 × 102	9.269 × 102	2.268 × 103	9.636 × 102	4.897 × 102	7.982 × 102	8.664 × 102	7.456 × 103	8.880 × 102	4.523 × 102	4.536 × 102
	Std	1.813 × 102	8.522 × 101	3.483 × 102	5.630 × 101	3.234 × 102	6.481 × 102	2.338 × 102	5.268 × 101	1.436 × 102	1.393 × 102	2.185 × 103	2.478 × 102	6.009 × 100	9.448 × 100
	p	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	2.225 × 10−4	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.254 × 10−1	~
	Rank	5	7	12	4	10	13	11	3	6	9	14	8	2	1
F3	Best	6.302 × 102	6.275 × 102	6.406 × 102	6.333 × 102	6.404 × 102	6.504 × 102	6.527 × 102	6.048 × 102	6.388 × 102	6.285 × 102	6.876 × 102	6.346 × 102	6.004 × 102	6.001 × 102
	Mean	6.443 × 102	6.326 × 102	6.637 × 102	6.620 × 102	6.675 × 102	6.609 × 102	6.752 × 102	6.170 × 102	6.615 × 102	6.425 × 102	7.182 × 102	6.687 × 102	6.006 × 102	6.002 × 102
	Std	6.435 × 100	2.914 × 100	1.042 × 101	1.208 × 101	1.307 × 101	5.290 × 100	1.074 × 101	7.650 × 100	1.083 × 101	8.660 × 100	1.299 × 101	1.224 × 101	1.698 × 10−1	8.296 × 10−2
	p	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.921 × 10−6	~
	Rank	6	4	10	8	11	7	13	3	9	5	14	12	2	1
F4	Best	9.437 × 102	9.590 × 102	8.830 × 102	8.739 × 102	8.810 × 102	9.504 × 102	9.041 × 102	8.560 × 102	8.549 × 102	8.563 × 102	1.055 × 103	9.066 × 102	8.317 × 102	8.308 × 102
	Mean	9.677 × 102	9.936 × 102	9.794 × 102	9.246 × 102	9.183 × 102	9.920 × 102	9.465 × 102	9.025 × 102	9.084 × 102	9.011 × 102	1.104 × 103	9.365 × 102	8.446 × 102	8.535 × 102
	Std	1.357 × 101	1.428 × 101	3.826 × 101	2.562 × 101	2.699 × 101	1.841 × 101	2.163 × 101	3.196 × 101	1.910 × 101	1.864 × 101	2.109 × 101	2.360 × 101	7.077 × 100	8.997 × 100
	p	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.965 × 10−3	~
	Rank	10	13	11	7	6	12	9	4	5	3	14	8	1	2
F5	Best	1.646 × 103	4.266 × 103	9.000 × 102	1.852 × 103	1.944 × 103	3.204 × 103	2.213 × 103	1.058 × 103	1.770 × 103	1.523 × 103	8.116 × 103	2.189 × 103	9.018 × 102	9.000 × 102
	Mean	4.010 × 103	6.840 × 103	1.151 × 103	3.655 × 103	3.019 × 103	5.616 × 103	3.036 × 103	1.891 × 103	2.825 × 103	2.162 × 103	1.246 × 104	3.195 × 103	9.061 × 102	9.027 × 102
	Std	1.057 × 103	1.145 × 103	9.812 × 102	1.388 × 103	5.897 × 102	1.128 × 103	4.659 × 102	8.300 × 102	4.354 × 102	4.901 × 102	1.814 × 103	7.338 × 102	4.191 × 100	3.071 × 100
	p	1.734 × 10−6	1.734 × 10−6	4.534 × 10−4	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.036 × 10−3	~
	Rank	11	13	1	10	8	12	7	4	6	5	14	9	3	2
F6	Best	2.999 × 107	4.160 × 107	3.832 × 107	4.905 × 104	1.155 × 104	2.440 × 107	1.134 × 107	2.032 × 103	4.433 × 105	3.783 × 106	1.939 × 109	1.280 × 104	1.360 × 104	2.375 × 103
	Mean	1.454 × 108	7.068 × 107	5.967 × 108	5.363 × 105	7.310 × 107	4.382 × 108	8.449 × 107	1.148 × 106	2.585 × 107	1.047 × 108	5.571 × 109	2.017 × 107	9.664 × 104	1.271 × 104
	Std	8.499 × 107	1.878 × 107	3.849 × 108	5.391 × 105	1.052 × 108	3.395 × 108	8.887 × 107	5.279 × 106	2.989 × 107	9.671 × 107	2.202 × 109	6.931 × 107	5.277 × 104	1.017 × 104
	p	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	6.424 × 10−3	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	2.353 × 10−6	~
	Rank	11	10	13	4	7	12	8	2	6	9	14	5	3	1
F7	Best	2.099 × 103	2.093 × 103	2.137 × 103	2.100 × 103	2.100 × 103	2.115 × 103	2.161 × 103	2.036 × 103	2.068 × 103	2.092 × 103	2.226 × 103	2.081 × 103	2.045 × 103	2.047 × 103
	Mean	2.159 × 103	2.132 × 103	2.212 × 103	2.222 × 103	2.217 × 103	2.172 × 103	2.237 × 103	2.099 × 103	2.166 × 103	2.143 × 103	2.504 × 103	2.195 × 103	2.060 × 103	2.069 × 103
	Std	3.439 × 101	2.323 × 101	2.914 × 101	7.585 × 101	6.176 × 101	3.753 × 101	5.173 × 101	3.749 × 101	4.657 × 101	3.770 × 101	1.179 × 102	6.184 × 101	7.695 × 100	1.086 × 101
	p	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	2.412 × 10−4	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	3.609 × 10−3	~
	Rank	7	4	12	10	11	8	13	3	6	5	14	9	1	2
F8	Best	2.250 × 103	2.234 × 103	2.247 × 103	2.236 × 103	2.242 × 103	2.243 × 103	2.240 × 103	2.222 × 103	2.230 × 103	2.230 × 103	2.672 × 103	2.239 × 103	2.225 × 103	2.227 × 103
	Mean	2.302 × 103	2.259 × 103	2.384 × 103	2.282 × 103	2.282 × 103	2.273 × 103	2.310 × 103	2.277 × 103	2.298 × 103	2.281 × 103	5.128 × 103	2.451 × 103	2.227 × 103	2.230 × 103
	Std	4.045 × 101	1.349 × 101	9.697 × 101	6.334 × 101	5.555 × 101	2.397 × 101	7.782 × 101	5.858 × 101	6.218 × 101	5.801 × 101	4.190 × 103	1.193 × 102	8.364 × 10−1	1.050 × 100
	p	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	9.711 × 10−5	1.921 × 10−6	4.286 × 10−6	1.734 × 10−6	1.734 × 10−6	2.353 × 10−6	~
	Rank	11	5	12	6	7	8	10	3	9	4	14	13	1	2
F9	Best	2.511 × 103	2.496 × 103	2.633 × 103	2.491 × 103	2.522 × 103	2.598 × 103	2.554 × 103	2.481 × 103	2.549 × 103	2.685 × 103	3.048 × 103	2.524 × 103	2.481 × 103	2.481 × 103
	Mean	2.648 × 103	2.510 × 103	2.830 × 103	2.538 × 103	2.633 × 103	2.803 × 103	2.703 × 103	2.498 × 103	2.664 × 103	2.802 × 103	3.937 × 103	2.685 × 103	2.482 × 103	2.481 × 103
	Std	7.640 × 101	8.349 × 100	1.224 × 102	4.245 × 101	6.514 × 101	9.317 × 101	7.475 × 101	2.427 × 101	5.318 × 101	7.774 × 101	4.471 × 102	6.228 × 101	9.008 × 10−1	6.883 × 10−1
	p	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	2.613 × 10−4	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.742 × 10−4	~
	Rank	7	4	13	5	6	11	10	3	8	12	14	9	2	1
F10	Best	2.517 × 103	2.532 × 103	2.609 × 103	2.501 × 103	2.528 × 103	2.575 × 103	2.557 × 103	2.501 × 103	2.502 × 103	2.521 × 103	2.695 × 103	2.503 × 103	2.500 × 103	2.500 × 103
	Mean	5.059 × 103	2.669 × 103	6.094 × 103	4.539 × 103	4.993 × 103	3.880 × 103	5.623 × 103	3.310 × 103	4.521 × 103	3.739 × 103	7.466 × 103	4.393 × 103	2.534 × 103	2.506 × 103
	Std	1.665 × 103	2.392 × 102	1.402 × 103	1.323 × 103	1.546 × 103	1.285 × 103	1.121 × 103	9.327 × 102	1.845 × 103	1.304 × 103	1.684 × 103	1.689 × 103	7.810 × 101	2.087 × 101
	p	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	5.792 × 10−5	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	4.286 × 10−6	7.655 × 10−1	~
	Rank	10	4	13	6	11	7	12	3	9	5	14	8	2	1
F11	Best	3.456 × 103	3.833 × 103	3.879 × 103	3.025 × 103	4.285 × 103	5.079 × 103	4.299 × 103	2.850 × 103	3.749 × 103	4.701 × 103	9.720 × 103	4.213 × 103	2.953 × 103	2.901 × 103
	Mean	4.862 × 103	4.227 × 103	7.302 × 103	3.359 × 103	6.230 × 103	7.301 × 103	6.055 × 103	3.109 × 103	4.770 × 103	5.855 × 103	1.299 × 104	5.697 × 103	2.987 × 103	2.921 × 103
	Std	7.499 × 102	2.275 × 102	1.107 × 103	6.378 × 102	1.359 × 103	1.090 × 103	1.069 × 103	1.829 × 102	7.110 × 102	7.680 × 102	1.746 × 103	9.836 × 102	1.975 × 101	8.469 × 101
	p	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	5.216 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	3.112 × 10−5	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	3.405 × 10−5	~
	Rank	7	5	12	4	11	13	10	3	6	9	14	8	2	1
F12	Best	2.955 × 103	2.951 × 103	3.002 × 103	2.963 × 103	3.012 × 103	3.056 × 103	3.003 × 103	2.941 × 103	2.960 × 103	3.166 × 103	3.494 × 103	3.102 × 103	2.941 × 103	2.938 × 103
	Mean	3.010 × 103	2.962 × 103	3.052 × 103	3.066 × 103	3.310 × 103	3.143 × 103	3.277 × 103	2.967 × 103	3.106 × 103	3.486 × 103	4.130 × 103	3.391 × 103	2.945 × 103	2.945 × 103
	Std	3.870 × 101	5.283 × 100	2.164 × 101	7.354 × 101	2.323 × 102	5.165 × 101	1.949 × 102	2.332 × 101	8.774 × 101	1.302 × 102	2.882 × 102	1.622 × 102	3.114 × 100	3.806 × 100
	p	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	3.182 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	7.036 × 10−1	~
	Rank	5	3	6	7	11	9	10	4	8	13	14	12	1	2
Mean Rank		8.33	7	10.67	6.17	9	10.25	10.08	3.5	7	6.75	14	8.83	2	1.41
Final Ranking		8	6	13	4	10	12	11	3	6	5	14	9	2	1
+/=/−		12/0/0	12/0/0	12/0/0	11/1/0	12/0/0	12/0/0	12/0/0	12/0/0	12/0/0	11/1/0	12/0/0	12/0/0	9/2/1	~

, including optimal values, averages, variances, p-values from the Wilcoxon rank-sum test, and Friedman rankings. The best results in the table are highlighted in bold and underlined. The symbols “+”, “=”, and “−” in the tables represent that EAPO’s performance is superior, equal, or inferior to other comparison algorithms, respectively, based on the Wilcoxon rank-sum test results. Iteration curves and boxplots for the 10-dimensional and 20-dimensional problems are shown in Figure 8, Figure 9, Figure 10 and Figure 11.

Based on experimental results, the EAPO algorithm demonstrates superior performance in both 10-dimensional and 20-dimensional problems. Specifically, in the 10-dimensional test, EAPO achieved the best results in 14 out of 36 metrics (covering 12 functions’ optimal values, means, and standard deviations), with an average ranking of 1.83; In the 20-dimensional test, EAPO performed even more outstandingly, achieving the best results in 20 metrics, with its average ranking further improving to 1.41. Based on the comprehensive evaluation of the above metrics, EAPO ranked first among all comparison algorithms, fully demonstrating its effectiveness and stability in solving optimization problems of different dimensions, with superior accuracy compared to the original APO algorithm.

6.5. Testing on the CEC2020 Benchmark Suite

The results obtained in the 10-dimensional and 20-dimensional tests of the CEC2020 benchmark suite are shown in

Table 5. Test results of EAPO and other algorithms on CEC2020 (dim = 10).

Function	Item Name	PSO	DE	CMA-ES	WOA	HHO	CL-PSO	NC-HHO	DBO	HO	HOA	AOA	TG-COA	APO	EAPO
F1	Best	1.419 × 108	3.522 × 108	1.292 × 109	9.222 × 105	1.848 × 107	1.349 × 109	1.990 × 108	1.326 × 102	1.493 × 108	1.932 × 108	7.592 × 109	8.608 × 106	3.190 × 106	1.259 × 103
	Mean	7.644 × 108	8.076 × 108	3.401 × 109	1.768 × 107	8.225 × 108	5.160 × 109	2.096 × 109	4.980 × 106	7.235 × 108	1.805 × 109	1.433 × 1010	2.112 × 108	5.919 × 106	3.338 × 103
	Std	6.282 × 108	2.580 × 108	1.165 × 109	1.519 × 107	1.448 × 109	2.136 × 109	1.799 × 109	1.777 × 107	4.869 × 108	9.251 × 108	4.255 × 109	3.870 × 108	1.650 × 106	2.237 × 103
	p	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	2.765 × 10−3	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	~
	Rank	7	9	12	4	6	13	10	2	8	11	14	5	3	1
F2	Best	1.978 × 103	1.846 × 103	2.237 × 103	1.597 × 103	1.507 × 103	2.088 × 103	1.534 × 103	1.262 × 103	1.731 × 103	1.412 × 103	2.839 × 103	1.428 × 103	1.599 × 103	1.283 × 103
	Mean	2.538 × 103	2.302 × 103	2.697 × 103	2.323 × 103	2.414 × 103	2.429 × 103	2.340 × 103	1.831 × 103	2.409 × 103	1.911 × 103	3.475 × 103	2.177 × 103	1.783 × 103	1.776 × 103
	Std	2.520 × 102	1.554 × 102	1.893 × 102	3.462 × 102	3.427 × 102	1.374 × 102	4.034 × 102	2.621 × 102	3.012 × 102	2.659 × 102	2.713 × 102	3.825 × 102	1.217 × 102	1.651 × 102
	p	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	8.466 × 10−6	2.879 × 10−6	1.734 × 10−6	9.316 × 10−6	2.895 × 10−1	3.882 × 10−6	1.319 × 10−2	1.734 × 10−6	6.320 × 10−5	8.612 × 10−1	~
	Rank	12	6	13	7	9	10	8	3	11	4	14	5	2	1
F3	Best	7.579 × 102	7.724 × 102	7.273 × 102	7.410 × 102	7.618 × 102	7.974 × 102	7.428 × 102	7.107 × 102	7.422 × 102	7.296 × 102	8.835 × 102	7.439 × 102	7.219 × 102	7.173 × 102
	Mean	8.168 × 102	8.050 × 102	7.398 × 102	7.876 × 102	8.010 × 102	8.579 × 102	8.015 × 102	7.336 × 102	7.769 × 102	7.515 × 102	1.109 × 103	7.878 × 102	7.266 × 102	7.310 × 102
	Std	1.877 × 101	1.293 × 101	4.855 × 100	2.381 × 101	2.470 × 101	3.465 × 101	2.349 × 101	1.259 × 101	1.868 × 101	1.313 × 101	6.833 × 101	2.543 × 101	3.182 × 100	4.386 × 100
	p	1.734 × 10−6	1.734 × 10−6	4.286 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	4.528 × 10−1	1.734 × 10−6	4.286 × 10−6	1.734 × 10−6	1.734 × 10−6	6.156 × 10−4	~
	Rank	12	11	4	7	9	13	10	3	6	5	14	8	1	2
F4	Best	1.907 × 103	1.914 × 103	1.951 × 103	1.902 × 103	1.903 × 103	2.327 × 103	1.906 × 103	1.901 × 103	1.903 × 103	1.907 × 103	1.836 × 104	1.904 × 103	1.901 × 103	1.901 × 103
	Mean	1.928 × 103	1.931 × 103	2.783 × 103	1.908 × 103	2.108 × 103	1.189 × 104	2.822 × 103	1.904 × 103	2.405 × 103	2.826 × 103	9.525 × 105	1.957 × 103	1.902 × 103	1.902 × 103
	Std	4.423 × 101	1.653 × 101	7.388 × 102	5.425 × 100	4.800 × 102	1.806 × 104	2.390 × 103	3.778 × 100	1.542 × 103	1.338 × 103	8.239 × 105	6.971 × 101	3.186 × 10−1	4.323 × 10−1
	p	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.204 × 10−1	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	3.872 × 10−2	~
	Rank	5	8	12	4	9	13	10	3	7	11	14	6	1	2
F5	Best	1.163 × 104	9.216 × 104	1.218 × 104	7.560 × 103	9.704 × 103	3.651 × 103	4.026 × 103	2.520 × 103	3.445 × 103	2.786 × 103	4.384 × 105	2.626 × 103	2.560 × 103	3.282 × 103
	Mean	9.556 × 104	7.830 × 105	1.708 × 105	4.179 × 105	3.588 × 105	9.797 × 104	1.144 × 105	4.429 × 104	7.965 × 104	2.032 × 105	1.006 × 107	2.060 × 104	1.145 × 104	9.740 × 103
	Std	7.971 × 104	6.335 × 105	1.949 × 105	6.434 × 105	2.483 × 105	8.431 × 104	1.515 × 105	7.554 × 104	1.520 × 105	1.529 × 105	1.159 × 107	3.270 × 104	6.384 × 103	5.862 × 103
	p	1.734 × 10−6	1.734 × 10−6	2.127 × 10−6	3.515 × 10−6	1.734 × 10−6	3.882 × 10−6	1.127 × 10−5	5.667 × 10−3	2.849 × 10−2	3.515 × 10−6	1.734 × 10−6	8.290 × 10−1	2.452 × 10−1	~
	Rank	8	13	10	11	12	7	6	4	5	9	14	1	3	2
F6	Best	1.622 × 103	1.667 × 103	1.875 × 103	1.629 × 103	1.604 × 103	1.666 × 103	1.646 × 103	1.601 × 103	1.642 × 103	1.639 × 103	2.113 × 103	1.724 × 103	1.604 × 103	1.610 × 103
	Mean	1.787 × 103	1.830 × 103	1.991 × 103	1.836 × 103	1.881 × 103	1.879 × 103	1.969 × 103	1.764 × 103	1.884 × 103	1.851 × 103	2.724 × 103	1.978 × 103	1.629 × 103	1.640 × 103
	Std	1.014 × 102	7.346 × 101	6.859 × 101	9.580 × 101	1.671 × 102	1.116 × 102	2.008 × 102	1.373 × 102	1.450 × 102	1.289 × 102	2.477 × 102	1.262 × 102	1.663 × 101	2.224 × 101
	p	1.921 × 10−6	1.734 × 10−6	1.734 × 10−6	1.921 × 10−6	1.921 × 10−6	1.734 × 10−6	1.921 × 10−6	2.831 × 10−4	1.921 × 10−6	1.921 × 10−6	1.734 × 10−6	1.734 × 10−6	2.703 × 10−2	~
	Rank	4	6	13	5	8	10	11	3	9	7	14	12	1	2
F7	Best	4.155 × 103	2.024 × 104	5.643 × 103	3.626 × 103	4.122 × 103	3.653 × 103	6.712 × 103	2.381 × 103	2.330 × 103	2.830 × 103	5.623 × 104	2.676 × 103	2.292 × 103	2.288 × 103
	Mean	1.735 × 104	2.769 × 105	3.260 × 105	4.441 × 105	5.679 × 105	1.217 × 104	6.326 × 105	8.418 × 103	1.190 × 104	6.809 × 104	4.980 × 106	8.061 × 103	2.836 × 103	2.740 × 103
	Std	9.366 × 103	1.783 × 105	5.082 × 105	9.759 × 105	1.057 × 106	8.113 × 103	1.271 × 106	8.456 × 103	1.083 × 104	1.341 × 105	7.005 × 106	5.316 × 103	3.142 × 102	3.144 × 102
	p	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.025 × 10−5	2.127 × 10−6	1.734 × 10−6	1.734 × 10−6	1.921 × 10−6	2.289 × 10−1	~
	Rank	7	13	9	12	11	6	10	3	5	8	14	4	2	1
F8	Best	2.314 × 103	2.317 × 103	2.300 × 103	2.268 × 103	2.302 × 103	2.323 × 103	2.280 × 103	2.213 × 103	2.252 × 103	2.261 × 103	3.146 × 103	2.277 × 103	2.291 × 103	2.285 × 103
	Mean	2.394 × 103	2.569 × 103	2.619 × 103	2.317 × 103	2.459 × 103	2.763 × 103	2.583 × 103	2.301 × 103	2.361 × 103	2.385 × 103	3.876 × 103	2.356 × 103	2.306 × 103	2.300 × 103
	Std	4.532 × 101	1.115 × 1002	5.178 × 102	1.598 × 101	1.312 × 102	2.167 × 102	2.400 × 102	2.644 × 101	5.474 × 101	5.418 × 101	4.149 × 102	4.495 × 101	3.329 × 100	2.865 × 100
	p	1.734 × 10−6	1.734 × 10−6	6.435 × 10−1	2.052 × 10−4	1.734 × 10−6	1.734 × 10−6	1.921 × 10−6	2.765 × 10−3	2.597 × 10−5	2.603 × 10−6	1.734 × 10−6	4.286 × 10−6	2.370 × 10−5	~
	Rank	9	12	5	4	10	13	11	2	7	8	14	6	3	1
F9	Best	2.591 × 103	2.675 × 103	2.791 × 103	2.538 × 103	2.544 × 103	2.671 × 103	2.573 × 103	2.512 × 103	2.544 × 103	2.508 × 103	2.849 × 103	2.522 × 103	2.581 × 103	2.598 × 103
	Mean	2.763 × 103	2.763 × 103	2.809 × 103	2.768 × 103	2.791 × 103	2.788 × 103	2.803 × 103	2.740 × 103	2.760 × 103	2.775 × 103	2.958 × 103	2.781 × 103	2.733 × 103	2.733 × 103
	Std	5.035 × 101	3.334 × 101	8.519 × 100	6.581 × 101	8.131 × 101	5.123 × 101	9.564 × 101	7.332 × 101	7.312 × 101	7.508 × 101	6.457 × 101	9.228 × 101	4.037 × 101	3.922 × 101
	p	1.593 × 10−3	8.217 × 10−3	1.734 × 10−6	3.589 × 10−4	4.390 × 10−3	4.534 × 10−4	4.390 × 10−3	1.319 × 10−2	2.765 × 10−3	1.833 × 10−3	1.734 × 10−6	1.833 × 10−3	5.038 × 10−1	~
	Rank	5	4	13	7	10	9	12	3	6	8	14	11	2	1
F10	Best	2.946 × 103	2.976 × 103	2.936 × 103	2.872 × 103	2.927 × 103	2.967 × 103	2.966 × 103	2.898 × 103	2.949 × 103	2.926 × 103	3.195 × 103	2.746 × 103	2.906 × 103	2.901 × 103
	Mean	2.994 × 103	3.011 × 103	3.091 × 103	2.953 × 103	3.054 × 103	3.177 × 103	3.066 × 103	2.938 × 103	2.994 × 103	2.992 × 103	4.211 × 103	2.994 × 103	2.941 × 103	2.935 × 103
	Std	2.928 × 101	1.993 × 101	7.328 × 101	3.128 × 101	1.210 × 102	1.160 × 102	8.135 × 101	2.818 × 101	4.673 × 101	4.779 × 101	5.076 × 102	9.466 × 101	9.829 × 100	1.621 × 101
	p	1.921 × 10−6	1.734 × 10−6	1.921 × 10−6	1.108 × 10−2	2.127 × 10−6	1.734 × 10−6	1.734 × 10−6	8.936 × 10−1	1.734 × 10−6	2.127 × 10−6	1.734 × 10−6	4.534 × 10−4	1.254 × 10−1	~
	Rank	8	10	12	4	9	13	11	3	7	6	14	5	2	1
Mean Rank		7.5	9.3	10.2	6.5	9.4	10.6	10.1	2.9	6.7	8.4	14	6	2	1.4
Final Ranking		7	9	12	5	10	13	11	3	6	8	14	4	2	1
+/=/−		10/0/0	10/0/0	10/1/0	10/0/0	10/0/0	10/0/0	10/0/0	5/4/1	10/0/0	10/0/0	10/0/0	9/1/0	4/6/1	~

and

Table 6. Test results of EAPO and other algorithms on CEC2020 (dim = 20).

Function	Item Name	PSO	DE	CMA-ES	WOA	HHO	CL-PSO	NC-HHO	DBO	HO	HOA	AOA	TG-COA	APO	EAPO
F1	Best	2.632 × 109	2.397 × 109	8.724 × 109	2.643 × 107	7.027 × 108	1.255 × 1010	5.643 × 109	1.020 × 103	2.552 × 109	5.216 × 109	2.975 × 1010	1.317 × 109	2.450 × 105	6.582 × 103
	Mean	6.571 × 109	3.968 × 109	1.744 × 1010	7.331 × 107	8.219 × 109	2.171 × 1010	1.091 × 1010	3.373 × 106	7.716 × 109	1.028 × 1010	5.106 × 1010	7.270 × 109	5.500 × 105	1.460 × 104
	Std	1.771 × 109	8.734 × 108	5.121 × 109	3.515 × 107	4.500 × 109	5.131 × 109	3.607 × 109	5.604 × 106	3.094 × 109	3.266 × 109	9.958 × 109	3.034 × 109	2.535 × 105	6.298 × 103
	p	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	4.897 × 10−4	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	~
	Rank	6	5	12	4	9	13	11	2	8	10	14	7	3	1
F2	Best	4.315 × 103	3.195 × 103	4.917 × 103	3.080 × 103	3.110 × 103	3.944 × 103	3.779 × 103	2.043 × 103	3.641 × 103	3.059 × 103	6.326 × 103	3.694 × 103	1.995 × 103	2.454 × 103
	Mean	5.108 × 103	3.660 × 103	5.629 × 103	3.915 × 103	4.521 × 103	4.529 × 103	4.785 × 103	2.937 × 103	4.732 × 103	3.961 × 103	6.848 × 103	4.634 × 103	2.617 × 103	2.880 × 103
	Std	4.130 × 102	2.002 × 102	2.743 × 102	5.294 × 102	5.172 × 102	2.645 × 102	5.263 × 102	5.088 × 102	5.072 × 102	4.745 × 102	2.627 × 102	4.499 × 102	2.547 × 102	2.373 × 102
	p	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.921 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	6.435 × 10−1	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.965 × 10−3	~
	Rank	12	4	13	5	7	8	11	3	10	6	14	9	1	2
F3	Best	1.030 × 103	9.732 × 102	8.036 × 102	8.245 × 102	9.237 × 102	9.984 × 102	9.553 × 102	7.403 × 102	8.901 × 102	8.331 × 102	1.817 × 103	8.114 × 102	7.426 × 102	7.592 × 102
	Mean	1.154 × 103	1.067 × 103	8.185 × 102	9.458 × 102	1.001 × 103	1.172 × 103	9.995 × 102	7.853 × 102	9.408 × 102	8.743 × 102	2.112 × 103	9.554 × 102	7.548 × 102	7.768 × 102
	Std	7.951 × 101	4.047 × 101	5.706 × 100	5.474 × 101	3.539 × 101	9.090 × 101	2.549 × 101	3.405 × 101	2.765 × 101	2.982 × 101	1.571 × 102	5.832 × 101	6.289 × 100	7.849 × 100
	p	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	3.933 × 10−1	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	~
	Rank	12	11	4	7	10	13	9	3	6	5	14	8	1	2
F4	Best	2.000 × 103	2.115 × 103	4.075 × 103	1.916 × 103	2.272 × 103	4.361 × 104	2.049 × 103	1.904 × 103	2.225 × 103	2.614 × 103	1.259 × 106	2.071 × 103	1.903 × 103	1.904 × 103
	Mean	6.563 × 103	4.053 × 103	1.306 × 105	1.982 × 103	3.421 × 104	3.881 × 105	3.771 × 104	1.916 × 103	8.683 × 103	2.810 × 104	6.543 × 106	1.934 × 104	1.904 × 103	1.906 × 103
	Std	6.587 × 103	9.507 × 102	9.504 × 104	9.119 × 101	3.859 × 104	3.382 × 105	5.154 × 104	1.224 × 101	7.711 × 103	2.500 × 104	3.256 × 106	3.479 × 104	6.551 × 10−1	7.098 × 10−1
	p	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	4.449 × 10−5	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	2.603 × 10−6	~
	Rank	6	5	12	4	10	13	9	3	7	11	14	8	1	2
F5	Best	7.484 × 105	1.185 × 106	1.286 × 106	9.891 × 104	1.651 × 105	6.724 × 105	2.584 × 105	3.771 × 104	2.774 × 105	7.996 × 104	1.604 × 107	7.094 × 104	3.756 × 104	2.718 × 104
	Mean	2.824 × 106	4.837 × 106	9.443 × 106	1.748 × 106	2.123 × 106	3.397 × 106	2.022 × 106	6.159 × 105	1.978 × 106	1.272 × 106	8.384 × 107	7.290 × 105	1.060 × 105	8.271 × 104
	Std	1.533 × 106	2.432 × 106	6.965 × 106	1.427 × 106	1.595 × 106	2.106 × 106	1.388 × 106	6.742 × 105	1.170 × 106	7.694 × 105	5.529 × 107	5.763 × 105	3.577 × 104	3.598 × 104
	p	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	2.353 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	2.353 × 10−6	3.160 × 10−2	~
	Rank	10	12	13	6	8	11	7	3	9	5	14	4	2	1
F6	Best	2.061 × 103	1.689 × 103	2.339 × 103	1.789 × 103	2.052 × 103	2.249 × 103	2.292 × 103	1.623 × 103	2.032 × 103	1.840 × 103	3.405 × 103	2.392 × 103	1.645 × 103	1.666 × 103
	Mean	2.394 × 103	1.800 × 103	2.761 × 103	2.482 × 103	2.721 × 103	2.588 × 103	2.866 × 103	1.983 × 103	2.540 × 103	2.367 × 103	4.277 × 103	2.853 × 103	1.696 × 103	1.719 × 103
	Std	1.949 × 102	6.006 × 101	1.895 × 102	3.223 × 102	3.749 × 102	1.656 × 102	3.243 × 102	1.785 × 102	2.824 × 102	2.680 × 102	3.995 × 102	3.218 × 102	3.377 × 101	2.865 × 101
	p	1.734 × 10−6	5.216 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	4.286 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.114 × 10−3	~
	Rank	6	3	11	7	10	9	12	4	8	5	14	13	1	2
F7	Best	1.815 × 105	4.964 × 105	3.600 × 105	8.786 × 104	6.482 × 104	9.171 × 104	1.301 × 105	9.613 × 103	1.271 × 105	7.606 × 104	6.979 × 106	1.716 × 104	4.283 × 103	4.901 × 103
	Mean	8.156 × 105	1.835 × 106	4.209 × 106	1.141 × 106	5.698 × 105	7.369 × 105	1.203 × 106	2.384 × 105	7.795 × 105	5.922 × 105	4.887 × 107	2.358 × 105	2.246 × 104	1.569 × 104
	Std	4.977 × 105	9.568 × 105	3.407 × 106	1.001 × 106	4.257 × 105	5.173 × 105	1.243 × 106	2.979 × 105	6.737 × 105	4.792 × 105	3.804 × 107	2.343 × 105	2.143 × 104	1.281 × 104
	p	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	2.353 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.921 × 10−6	3.286 × 10−1	~
	Rank	9	12	13	10	5	8	11	3	7	6	14	4	2	1
F8	Best	2.736 × 103	3.869 × 103	3.745 × 103	2.319 × 103	2.802 × 103	3.513 × 103	3.427 × 103	2.300 × 103	2.470 × 103	2.795 × 103	6.215 × 103	2.594 × 103	2.309 × 103	2.300 × 103
	Mean	5.002 × 103	4.603 × 103	6.617 × 103	3.876 × 103	5.264 × 103	5.508 × 103	5.157 × 103	2.656 × 103	2.953 × 103	4.036 × 103	8.159 × 103	4.620 × 103	2.312 × 103	2.302 × 103
	Std	1.834 × 103	4.044 × 102	7.734 × 102	1.857 × 103	1.135 × 103	6.683 × 102	1.056 × 103	7.745 × 102	4.155 × 102	1.239 × 103	5.592 × 102	1.555 × 103	1.547 × 100	1.209 × 100
	p	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	3.182 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	~
	Rank	9	8	13	5	11	12	10	3	4	6	14	7	2	1
F9	Best	2.914 × 103	2.922 × 103	2.983 × 103	2.915 × 103	2.935 × 103	2.990 × 103	3.029 × 103	2.851 × 103	2.947 × 103	3.063 × 103	3.245 × 103	3.006 × 103	2.822 × 103	2.835 × 103
	Mean	2.952 × 103	2.951 × 103	3.026 × 103	2.984 × 103	3.118 × 103	3.087 × 103	3.286 × 103	2.899 × 103	3.062 × 103	3.194 × 103	3.604 × 103	3.189 × 103	2.841 × 103	2.848 × 103
	Std	2.943 × 101	1.064 × 101	2.039 × 101	5.029 × 101	1.038 × 102	4.947 × 101	1.238 × 102	2.145 × 101	7.490 × 101	7.417 × 101	1.602 × 102	1.091 × 102	7.777 × 100	6.755 × 100
	p	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	2.415 × 10−3	~
	Rank	4	5	7	6	10	9	13	3	8	12	14	11	1	2
F10	Best	3.073 × 103	3.153 × 103	3.629 × 103	2.928 × 103	3.163 × 103	4.006 × 103	3.141 × 103	2.914 × 103	3.065 × 103	3.114 × 103	5.833 × 103	3.046 × 103	2.914 × 103	2.914 × 103
	Mean	3.293 × 103	3.438 × 103	4.287 × 103	3.025 × 103	3.696 × 103	4.902 × 103	3.624 × 103	2.961 × 103	3.326 × 103	3.514 × 103	1.067 × 104	3.429 × 103	2.947 × 103	2.945 × 103
	Std	1.705 × 102	1.438 × 102	4.686 × 102	4.541 × 101	3.981 × 102	5.709 × 102	3.066 × 102	5.800 × 101	1.956 × 102	1.919 × 102	2.319 × 103	2.425 × 102	3.088 × 101	3.088 × 101
	p	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	4.286 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	4.405 × 10−1	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	1.734 × 10−6	9.426 × 10−1	~
	Rank	5	8	12	4	11	13	10	2	6	9	14	7	3	1
Mean Rank		7.9	7.3	11	5.8	9.1	10.9	10.3	2.9	7.3	7.5	14	7.8	1.7	1.5
Final Ranking		9	5	13	4	10	12	11	3	5	7	14	8	2	1
+/=/−		10/0/0	10/0/0	10/0/0	10/0/0	10/0/0	10/0/0	10/0/0	7/3/0	10/0/0	10/0/0	10/0/0	9/1/0	4/2/4	~

. The meanings of the metrics in these tables are the same as those described in Section 6.4. The best results in the table are highlighted in bold and underlined. The average fitness iteration curves and box plots across different dimensions are shown in Figure 12, Figure 13, Figure 14 and Figure 15.

Based on experimental results, the EAPO algorithm also demonstrated strong performance on the 10-dimensional and 20-dimensional problems in CEC2020, particularly excelling in mean and standard deviation metrics, proving its robust capabilities. Specifically, in the 10-dimensional test, EAPO achieved the best performance in 13 out of 30 metrics (covering the optimal values, mean, and standard deviation of 10 functions), with an average ranking of 1.83. In the 20-dimensional test, EAPO achieved the best results in 14 metrics, with an average ranking of 1.41. Consequently, EAPO ranked first among all comparison algorithms.

6.6. Testing and Analysis of Engineering Optimization Problems

To validate the performance of the proposed algorithm in complex engineering optimization scenarios, this section selects 10 typical real-world engineering problems from the CEC2020 Real-World Constrained Optimization Problem Set (CEC2020-RW) as test benchmarks. CEC2020-RW encompasses optimization problems across multiple typical engineering application domains. Unlike conventional numerical optimization problems, these datasets exhibit highly nonlinear and nonconvex characteristics, incorporating diverse equality and inequality constraints that closely mirror real-world engineering optimization requirements [40,41]. From the industrial chemical processes, process synthesis and design, mechanical engineering, and livestock feed ration optimization domains within this test suite, several representative problems were selected from each field to form a total of 10 problems. These were solved using EAPO alongside 13 other algorithms. Definitions for these 10 typical engineering problems are provided in

Table A3. Definitions of 10 Typical Real-World Engineering Problems.

No.	Name	$\mathit{D}$	$\mathit{g}$	$\mathit{h}$	${\mathit{f}}^{\mathbf{\ast }}$
Industrial Chemical Processes
RW01	Heat Exchanger Network Design 2	11	0	9	7.049 × 103
RW02	Blending-Pooling-Separation problem	38	0	32	1.864 × 100
RW03	Propane, Isobutane, n-Butane Non-sharp Separation	48	0	38	2.116 × 100
Process Synthesis and Design Problems
RW04	Process synthesis problem 2	7	9	0	2.925 × 100
RW05	Multi-product batch plant	10	10	0	5.364 × 104
Mechanical Engineering Problems
RW06	Optimal Design of Industrial refrigeration System	14	15	0	3.221 × 10−2
RW07	Pressure vessel design	4	4	0	5.885 × 103
RW08	Step-cone pulley problem	5	8	3	1.607 × 101
Livestock Feed Ration Optimization
RW09	Beef Cattle (case 1)	59	14	1	4.551 × 103
RW10	Beef Cattle (case 2)	59	14	1	3.349 × 103

of Appendix A.2, where $D$, $g$, $h$, and $f^{\ast }$ denote problem dimension, inequality constraints, equality constraints, and known feasible optimal objective function values, respectively.

The test results are shown in

Table 7. Test results of EAPO and other algorithms on 10 real-world engineering problems.

Function	Item Name	PSO	DE	CMA-ES	WOA	HHO	CL-PSO	NC-HHO	DBO	HO	HOA	AOA	TG-COA	APO	EAPO
RW01	Best	5.199 × 1015	7.031 × 1013	9.257 × 1021	2.354 × 1013	4.887 × 1010	9.257 × 1021	7.779 × 1013	7.049 × 103	2.896 × 1014	2.176 × 1016	1.137 × 1017	1.131 × 1015	4.699 × 109	5.036 × 107
	Mean	3.636 × 1016	2.487 × 1014	9.257 × 1021	4.408 × 1014	1.083 × 1015	9.270 × 1021	1.201 × 1015	3.834 × 1014	1.167 × 1016	5.948 × 1016	4.777 × 1017	2.309 × 1016	6.585 × 1010	3.224 × 109
	Std	2.852 × 1016	1.209 × 1014	7.466 × 1006	2.938 × 1014	3.288 × 1015	1.880 × 1019	1.419 × 1015	6.695 × 1014	1.280 × 1016	2.446 × 1016	1.726 × 1017	2.192 × 1016	4.587 × 1010	5.103 × 109
	p	3.020 × 10−11	3.020 × 10−11	1.212 × 10−12	3.020 × 10−11	3.020 × 10−11	2.982 × 10−11	3.020 × 10−11	6.607 × 10−01	3.020 × 10−11	3.020 × 10−11	3.020 × 10−11	3.020 × 10−11	8.993 × 10−11	3.020 × 10−11
	Rank	10	5	13	6	4	14	7	3	8	11	12	9	2	1
RW02	Best	2.514 × 100	2.644 × 100	7.883 × 100	2.828 × 100	3.243 × 100	5.102 × 102	3.243 × 100	1.102 × 100	3.035 × 100	9.474 × 100	1.243 × 102	9.245 × 100	1.920 × 100	1.405 × 100
	Mean	8.391 × 100	3.177 × 100	1.027 × 101	5.049 × 100	7.466 × 101	2.541 × 104	1.662 × 101	2.373 × 100	5.819 × 100	2.168 × 101	2.192 × 102	2.653 × 101	2.579 × 100	1.926 × 100
	Std	5.649 × 100	2.961 × 10−1	4.875 × 100	4.556 × 100	6.273 × 101	2.576 × 104	1.006 × 101	9.085 × 10−1	4.238 × 100	7.489 × 100	4.877 × 101	1.101 × 101	2.510 × 10−1	2.706 × 10−1
	p	3.020 × 10−11	3.020 × 10−11	6.387 × 10−12	3.020 × 10−11	2.521 × 10−11	3.020 × 10−11	3.020 × 10−11	1.911 × 10−02	3.012 × 10−11	3.020 × 10−11	3.020 × 10−11	3.020 × 10−11	1.070 × 10−9	3.020 × 10−11
	Rank	7	4	8	5	12	14	9	2	6	10	13	11	3	1
RW03	Best	2.324 × 100	2.669 × 100	6.356 × 101	2.067 × 100	2.672 × 100	2.627 × 102	3.143 × 100	1.282 × 100	2.663 × 100	9.996 × 100	3.348 × 101	5.703 × 100	2.052 × 100	1.345 × 100
	Mean	6.313 × 100	3.333 × 100	7.183 × 101	2.929 × 100	9.258 × 100	1.623 × 103	9.347 × 100	2.419 × 100	6.088 × 100	1.529 × 101	4.998 × 101	1.170 × 101	2.476 × 100	1.615 × 100
	Std	3.428 × 100	3.401 × 10−1	5.558 × 100	4.773 × 10−1	6.093 × 100	1.123 × 103	4.455 × 100	4.628 × 10−1	1.852 × 100	2.989 × 100	8.998 × 100	3.332 × 100	2.033 × 10−1	2.041 × 10−1
	p	3.020 × 10−11	3.020 × 10−11	2.932 × 10−11	3.685 × 10−11	3.020 × 10−11	3.020 × 10−11	3.020 × 10−11	6.518 × 10−9	3.020 × 10−11	3.020 × 10−11	3.020 × 10−11	3.020 × 10−11	4.077 × 10−11	3.020 × 10−11
	Rank	6	5	13	4	8	14	9	2	7	11	12	10	3	1
RW04	Best	3.907 × 100	3.010 × 100	1.117 × 101	2.951 × 100	4.762 × 100	8.280 × 100	2.926 × 100	2.925 × 100	2.925 × 100	2.928 × 100	1.412 × 109	4.618 × 100	2.925 × 100	2.926 × 100
	Mean	6.534 × 100	6.412 × 100	1.615 × 101	5.419 × 100	7.964 × 100	1.178 × 101	3.782 × 100	4.040 × 100	3.327 × 100	3.903 × 100	2.492 × 1012	4.799 × 100	2.947 × 100	2.937 × 100
	Std	1.709 × 100	3.113 × 100	1.313 × 100	2.015 × 100	2.035 × 100	2.717 × 100	8.537 × 10−1	9.609 × 10−1	5.337 × 10−1	6.146 × 10−1	6.830 × 1012	2.023 × 10−1	3.861 × 10−2	1.323 × 10−2
	p	3.020 × 10−11	3.012 × 10−11	2.088 × 10−11	6.066 × 10−11	3.020 × 10−11	3.009 × 10−11	9.260 × 10−9	1.269 × 10−8	3.010 × 10−7	3.197 × 10−9	3.020 × 10−11	3.020 × 10−11	9.234 × 10−01	3.020 × 10−11
	Rank	10	9	13	8	11	12	4	5	3	6	14	7	2	1
RW05	Best	5.507 × 104	5.551 × 104	7.663 × 1011	6.260 × 104	6.504 × 104	7.663 × 1011	5.747 × 104	5.380 × 104	6.414 × 104	6.567 × 104	1.261 × 105	6.739 × 104	5.407 × 104	5.368 × 104
	Mean	6.789 × 104	5.892 × 104	7.663 × 1011	8.824 × 104	9.873 × 104	7.663 × 1011	8.248 × 104	6.388 × 104	7.502 × 104	8.515 × 104	2.044 × 106	9.069 × 104	5.528 × 104	5.472 × 104
	Std	9.398 × 103	1.731 × 103	3.725 × 10−4	1.484 × 104	2.619 × 104	1.885 × 106	1.823 × 104	1.284 × 104	7.852 × 103	1.034 × 104	2.850 × 106	1.062 × 104	1.401 × 103	1.243 × 103
	p	2.154 × 10−10	4.200 × 10−10	1.212 × 10−12	3.020 × 10−11	3.020 × 10−11	2.232 × 10−11	3.690 × 10−11	1.385 × 10−6	3.020 × 10−11	3.020 × 10−11	3.020 × 10−11	3.020 × 10−11	2.709 × 10−2	2.154 × 10−10
	Rank	5	3	13	8	10	14	7	4	6	9	12	11	2	1
RW06	Best	5.833 × 10−2	4.474 × 10−1	5.925 × 103	7.620 × 10−2	1.371 × 100	1.619 × 103	1.753 × 100	4.424 × 10−2	5.685 × 100	2.090 × 100	6.628 × 106	6.435 × 103	5.173 × 10−2	3.663 × 10−2
	Mean	1.220 × 105	4.346 × 100	6.748 × 104	1.513 × 105	1.617 × 106	6.079 × 105	5.300 × 104	8.748 × 104	1.195 × 106	1.180 × 103	5.973 × 107	1.159 × 106	3.062 × 104	1.872 × 104
	Std	4.733 × 105	4.329 × 100	1.950 × 105	2.419 × 105	3.726 × 106	5.654 × 105	1.470 × 105	9.512 × 104	1.740 × 106	2.240 × 103	4.988 × 107	1.328 × 106	6.899 × 104	5.713 × 104
	p	5.462 × 10−6	1.868 × 10−5	8.131 × 10−8	8.352 × 10−8	6.010 × 10−8	4.200 × 10−10	1.359 × 10−7	1.111 × 10−3	1.174 × 10−9	1.596 × 10−7	3.020 × 10−11	5.573 × 10−10	1.004 × 10−3	5.462 × 10−6
	Rank	4	2	10	8	9	12	7	5	11	6	14	13	3	1
RW07	Best	5.762 × 103	5.745 × 103	6.183 × 103	5.921 × 103	6.322 × 103	6.181 × 103	6.100 × 103	5.743 × 103	5.918 × 103	6.161 × 103	9.732 × 103	5.746 × 103	5.743 × 103	5.746 × 103
	Mean	6.314 × 103	6.012 × 103	6.420 × 103	7.285 × 103	1.005 × 104	6.352 × 103	8.443 × 103	5.926 × 103	6.846 × 103	6.719 × 103	2.152 × 104	6.783 × 103	5.923 × 103	5.888 × 103
	Std	5.552 × 102	1.262 × 102	2.962 × 102	1.289 × 103	3.381 × 103	2.007 × 102	3.291 × 103	2.855 × 102	9.953 × 102	3.610 × 102	7.230 × 103	1.094 × 103	1.623 × 102	1.023 × 102
	p	2.126 × 10−4	2.891 × 10−3	1.174 × 10−9	2.439 × 10−9	3.690 × 10−11	8.101 × 10−10	5.494 × 10−11	1.180 × 10−1	5.462 × 10−9	9.919 × 10−11	3.020 × 10−11	1.385 × 10−6	7.845 × 10−1	2.126 × 10−4
	Rank	5	4	8	11	13	6	12	2	9	10	14	7	3	1
RW08	Best	1.821 × 101	1.611 × 101	1.010 × 1010	1.619 × 101	1.655 × 101	1.010 × 1010	1.652 × 101	1.604 × 101	1.633 × 101	1.640 × 101	9.152 × 102	1.606 × 101	1.606 × 101	1.605 × 101
	Mean	2.145 × 101	1.675 × 101	1.010 × 1010	1.684 × 101	1.068 × 102	1.010 × 1010	3.900 × 101	1.711 × 101	1.791 × 101	1.998 × 101	2.447 × 109	3.502 × 101	1.636 × 101	1.616 × 101
	Std	4.588 × 100	3.519 × 10−1	1.185 × 102	2.642 × 10−1	3.133 × 102	2.899 × 101	2.668 × 101	1.994 × 100	1.967 × 100	9.937 × 100	2.196 × 109	2.943 × 101	2.289 × 10−1	1.642 × 10−1
	p	3.020 × 10−11	4.573 × 10−9	3.020 × 10−11	1.613 × 10−10	3.690 × 10−11	3.020 × 10−11	5.494 × 10−11	3.912 × 10−2	1.464 × 10−10	8.993 × 10−11	3.020 × 10−11	7.773 × 10−9	1.430 × 10−5	3.020 × 10−11
	Rank	10	4	14	5	11	13	9	3	6	7	12	8	2	1
RW09	Best	5.313 × 106	7.108 × 104	4.422 × 106	3.539 × 104	1.926 × 105	1.016 × 1010	4.538 × 104	4.525 × 103	4.291 × 104	1.653 × 106	2.757 × 1010	8.635 × 104	2.493 × 105	1.934 × 104
	Mean	1.345 × 109	2.362 × 105	5.089 × 107	8.291 × 105	6.760 × 106	1.406 × 1010	6.792 × 105	1.555 × 105	4.936 × 105	3.040 × 106	3.375 × 1010	1.347 × 106	9.880 × 105	4.434 × 104
	Std	7.180 × 108	1.380 × 105	1.924 × 107	4.172 × 105	4.956 × 106	2.018 × 109	3.451 × 105	2.646 × 105	4.274 × 105	8.519 × 105	2.691 × 109	1.337 × 106	4.279 × 105	2.822 × 104
	p	3.020 × 10−11	9.919 × 10−11	3.020 × 10−11	2.154 × 10−10	3.020 × 10−11	3.020 × 10−11	1.206 × 10−10	1.171 × 10−2	1.547 × 10−9	3.020 × 10−11	3.020 × 10−11	4.077 × 10−11	3.020 × 10−11	3.020 × 10−11
	Rank	12	3	11	6	9	13	5	2	4	10	14	7	8	1
RW10	Best	4.840 × 108	4.882 × 103	1.972 × 107	7.763 × 103	3.392 × 106	2.433 × 109	5.272 × 103	4.082 × 103	5.220 × 103	3.250 × 105	2.894 × 1010	2.787 × 104	4.618 × 103	4.136 × 103
	Mean	3.708 × 109	7.854 × 103	6.825 × 107	1.942 × 105	8.833 × 106	6.574 × 109	3.572 × 104	6.643 × 103	1.804 × 104	1.557 × 106	3.532 × 1010	4.819 × 106	6.430 × 103	4.878 × 103
	Std	2.608 × 109	3.580 × 103	1.791 × 107	4.086 × 105	4.771 × 106	1.722 × 109	2.817 × 104	3.235 × 103	6.969 × 103	8.407 × 105	3.637 × 109	8.760 × 106	2.331 × 103	3.970 × 102
	p	3.020 × 10−11	3.020 × 10−11	3.020 × 10−11	3.020 × 10−11	3.020 × 10−11	3.020 × 10−11	3.020 × 10−11	6.414 × 10−01	3.020 × 10−11	3.020 × 10−11	3.020 × 10−11	3.020 × 10−11	3.020 × 10−11	3.020 × 10−11
	Rank	12	4	11	7	10	13	6	2	5	8	14	9	3	1
Mean Rank		8.1	4.3	11.4	6.8	9.7	12.5	7.5	3	6.5	8.8	13	9.2	3.1	1
Final Ranking		8	4	12	6	11	13	7	2	5	9	14	10	3	1
+/=/−		10/0/0	10/0/0	10/0/0	10/0/0	10/0/0	10/0/0	10/0/0	7/3/0	10/0/0	10/0/0	10/0/0	10/0/0	8/2/0	~

, including optimal values, mean values, variance, Wilcoxon rank-sum test p-values, and Friedman rankings. The best results in the table are highlighted in bold and underlined. The experimental results demonstrate that EAPO achieved the best average values across all problems, the best standard deviation for all problems except RW02, RW03, and RW06, and ranked first in the Friedman ranking. This fully validates the algorithm’s advantages in solution accuracy and stability compared to other algorithms. Across multiple engineering problems with complex constraints, EAPO consistently converged stably to high-quality solutions, demonstrating its exceptional constraint handling capability and global optimization performance.

7. UAV Path Planning Simulation Experiment

7.1. Experimental Configuration for UAV Path Planning

Four map scenarios with different sizes and levels of complexity were constructed, as shown in Figure 16. In the figures, red spheres represent artilleries, and blue spheres represent radars. Blue cylinders denote no-fly zones. The square and the five-pointed star indicate the starting point and the target point, respectively.

Seven algorithms that showed strong performance on the CEC2022 and CEC2020 benchmark tests, together with the proposed EAPO, were selected for comparative UAV path planning experiments. For all algorithms, the population size was set to 100. The maximum number of function evaluations was set to 10,000. The remaining parameters were configured according to Table 1.

The constraint parameters of the unmanned aerial vehicle dynamics are listed in

Table 8. Setting of constraint parameters for UAV dynamics.

Constraint Parameter	Symbol	Numerical Value
Minimum flight speed	$v_{min}$	$20 \mathrm{m}/\mathrm{s}$
Maximum flight speed	$v_{max}$	$60 \mathrm{m}/\mathrm{s}$
Maximum turn angle	${\theta }_{max}$	$30°$
Maximum climb angle	${\psi }_{climb}$	$45°$
Maximum descent angle	${\psi }_{descent}$	$-45°$
Minimum turn radius	$R_{min}$	$30 \mathrm{m}$
Maximum climb rate	${ROC}_{max}$	$8 \mathrm{m}/\mathrm{s}$
Maximum descent rate	${ROD}_{max}$	$6 \mathrm{m}/\mathrm{s}$
Minimum flight segment length	$L_{min}$	$10 \mathrm{m}$
Minimum flight altitude	$h_{min}$	$150 \mathrm{m}$
Maximum flight altitude	$h_{max}$	$350 \mathrm{m}$
Safety margin coefficient	$M_{safty}$	$1.2$
Minimum terrain clearance	$d_{terrain}$	$10 \mathrm{m}$
Maximum acceleration	$a_{max}$	$3 \mathrm{m}/{\mathrm{s}}^2$
Maximum deceleration	$a_{dec}$	$4 \mathrm{m}/{\mathrm{s}}^2$

to simulate realistic flight conditions. In the simulation environment, the number of waypoints was set to 10. The coordinates of the starting point and the target point, the number of intermediate waypoints, and the parameters of the threat zones and terrain model are also provided in

Table 9. Parameter Configuration for the UAV Path Planning Simulation Environment.

Map	Starting Point	Target Point	Threat				No-Fly Zone
			Radar		Artillery
			Center	R	Center	R	Center	R
1	(80, 80, 200)	(900, 800, 250)	(400, 500, 190) (600, 200, 220)	100 100	(300, 250, 200)	100	(700, 500)	100
2	(80, 80, 200)	(900, 800, 250)	(380, 420, 200) (550, 200, 215) (720, 650, 185) (250, 650, 195)	85 75 70 80	(150, 280, 215) (650, 450, 205) (820, 320, 180)	95 85 90	(540, 620) (850, 500) (350, 100)	75 70 65
3	(80, 80, 150)	(380, 380, 200)	(150, 380, 90)	50	(150, 125, 100)	50	(350, 250)	60
4	(80, 80, 150)	(380, 380, 200)	(200, 200, 105) (150, 75, 110) (410, 150, 85)	45 40 40	(85, 160, 110) (310, 250, 120)	50 45	(325, 135) (400, 300) (200, 330)	40 45 45

.

To reduce randomness, each algorithm was independently executed 30 times. Performance was evaluated using the best, mean, and variance of fitness values, alongside Friedman’s test for ranking. In addition, path quality evaluation metrics were employed to conduct comparative assessments of the paths generated by EAPO and APO planning. The comparison metrics included path length, path smoothness, flight altitude variation, actual maximum climb angle, actual maximum descent angle, energy consumption, threat distance, and computation time, all averaged over 30 runs. These comparisons further validate the performance advantages of EAPO over APO in UAV path planning tasks.

7.2. Path Quality Evaluation Metrics

To comprehensively evaluate the quality of the planned UAV trajectories, multiple path evaluation metrics are adopted from geometric, kinematic, safety, and energy perspectives. These metrics provide a detailed assessment of path feasibility, smoothness, safety levels, and energy expenditure.

7.2.1. Path Length

The total path length $L$ is defined as the cumulative Euclidean distance between consecutive waypoints:

$$\begin{array}{c} L={\displaystyle \sum _{i=1}^{n-1}}\sqrt{{\left(x_{i+1}-x_i\right)}^2+{\left(y_{i+1}-y\right)}^2+{\left(z_{i+1}-z_i\right)}^2}\end{array}$$

(57)

where $n$ is the total number of waypoints; $(x_i,y_i,z_i)$ denotes the 3D coordinates of the $i$th waypoint.

7.2.2. Path Smoothness

Path smoothness reflects the continuity of direction changes along the trajectory. First, the turning angle between consecutive segments is calculated as:

$$\begin{array}{c}{\theta }_i=\arccos \left({\displaystyle \frac{{\overrightarrow{v}}_i·{\overrightarrow{v}}_{i+1}}{\left|{\overrightarrow{v}}_i\right|\left|{\overrightarrow{v}}_{i+1}\right|}}\right)\end{array}$$

(58)

where ${\theta }_i$ is the turning angle between two adjacent segments; ${\overrightarrow{v}}_i$ and ${\overrightarrow{v}}_{i+1}$ are consecutive path segment vectors.

Based on the accumulated turning angles, the smoothness index $S$ is defined as:

$$\begin{array}{c} S={\displaystyle \frac{1}{1+\sum {\theta }_i}}\end{array}$$

(59)

7.2.3. Altitude Variation

Altitude variation is used to evaluate vertical motion stability and is quantified by Equation (60).

$$\begin{array}{c}{ ∆H}_{total}=\sum \left|z_{i+1}-z_i\right|\end{array}$$

(60)

The altitude range is defined as:

$$\begin{array}{c} H_{range}=z_{max}-z_{min}\end{array}$$

(61)

where ${∆H}_{total}$ and ${∆H}_{total}$ measure altitude fluctuation.

7.2.4. Climb and Descent Angles

The climb or descent angle ${\psi }_i$ at each path segment is computed as:

$$\begin{array}{c} {\psi }_i=\arctan \left({\displaystyle \frac{z_{i+1}-z_i}{\sqrt{{\left(x_{i+1}-x_i\right)}^2+{\left(y_{i+1}-y_i\right)}^2}}}\right)\end{array}$$

(62)

7.2.5. Threat Distance

The threat distance is defined as the minimum clearance between the planned path and a threat region. Each threat region is modeled as a sphere, and the distance is computed as the shortest distance from a path segment to the surface of the sphere. If the path intersects the threat region, the threat distance is set to zero. The threat distance $d_{threat}$ is calculated as:

$$\begin{array}{c} d_{threat}=\max \left(0,‖P_{closest}-O‖\right)-R\end{array}$$

(63)

where $P_{closest}$ denotes the closest point on the path segment to the threat center; $O$ is the center of the spherical threat region; $R$ is the radius of the threat sphere including the safety margin.

7.2.6. Energy Consumption

The total energy consumption $E$ is estimated as:

$$\begin{array}{c}E=k_{1}L+k_2{∆H}_{climb}+k_3\sum {\theta }_i\end{array}$$

(64)

where $k_1$, $k_2$ and $k_3$ represent the coefficients for horizontal flight, climbing, and turning energy costs, respectively; ${∆H}_{climb}$ is the accumulated climbing altitude.

7.3. Analysis of Experimental Results

The optimal UAV path planning results and the corresponding average convergence curves in four simulated environments are shown in Figure 17, Figure 18, Figure 19 and Figure 20. The results indicate that EAPO successfully avoids all threat zones and no-fly zones and generates feasible paths from the starting point to the target point. The planned paths satisfy the imposed constraints and maintain good smoothness.

Table 10. Path planning fitness value results for EAPO and other algorithms.

Map	Item Name	PSO	DE	WOA	DBO	HO	HOA	APO	EAPO
1	Best	5.997 × 10−1	6.431 × 10−1	5.984 × 10−1	5.635 × 10−1	5.809 × 10−1	1.180 × 10−1	9.865 × 10−2	8.854 × 10−2
	Mean	6.409 × 10−1	6.699 × 10−1	6.883 × 10−1	5.897 × 10−1	6.050 × 10−1	5.468 × 10−1	4.524 × 10−1	3.451 × 10−1
	Std	2.433 × 10−2	1.360 × 10−2	8.804 × 10−2	1.660 × 10−2	1.466 × 10−2	1.131 × 10−1	2.245 × 10−1	2.449 × 10−1
	Rank	6	8	7	4	5	2	3	1
2	Best	6.323 × 10−1	6.716 × 10−1	5.949 × 10−1	5.850 × 10−1	6.213 × 10−1	6.206 × 10−1	6.249 × 10−1	6.325 × 10−1
	Mean	1.662 × 100	1.140 × 100	2.118 × 100	8.932 × 10−1	7.848 × 10−1	6.498 × 10−1	6.483 × 10−1	6.453 × 10−1
	Std	1.538 × 100	7.829 × 10−1	2.905 × 100	6.840 × 10−1	7.260 × 10−1	1.754 × 10−2	1.003 × 10−2	9.010 × 10−3
	Rank	7	8	6	1	4	5	3	2
3	Best	6.266 × 10−1	6.398 × 10−1	6.007 × 10−1	6.041 × 10−1	6.400 × 10−1	6.494 × 10−1	6.200 × 10−1	6.130 × 10−1
	Mean	6.423 × 10−1	6.496 × 10−1	6.636 × 10−1	6.199 × 10−1	6.545 × 10−1	6.684 × 10−1	6.237 × 10−1	6.194 × 10−1
	Std	8.482 × 10−3	4.902 × 10−3	4.046 × 10−2	9.723 × 10−3	1.145 × 10−2	1.244 × 10−2	2.320 × 10−3	3.072 × 10−3
	Rank	4	5	6	2	7	8	3	1
4	Best	6.237 × 10−1	6.466 × 10−1	6.282 × 10−1	6.250 × 10−1	6.481 × 10−1	6.567 × 10−1	6.238 × 10−1	6.174 × 10−1
	Mean	6.428 × 10−1	6.594 × 10−1	7.119 × 10−1	6.343 × 10−1	6.774 × 10−1	6.823 × 10−1	6.318 × 10−1	6.291 × 10−1
	Std	1.119 × 10−2	6.859 × 10−3	5.142 × 10−2	4.532 × 10−3	2.043 × 10−2	1.394 × 10−2	4.231 × 10−3	5.481 × 10−3
	Rank	4	5	8	3	6	7	2	1
Mean Rank		5.25	6.5	6.75	2.5	5.5	5.5	2.75	1.25
Final Ranking		4	7	8	2	5	5	3	1

summarizes the fitness values obtained by EAPO and seven comparison algorithms across the four maps.

Table 11. Comparison of Average Results for UAV Path Planning Between EAPO and APO (MAP 1).

Item Name	APO	EAPO	Percentage Improvement
Path length	1.419 × 103	1.422 × 103	−0.21%
Path smoothness index	1.926 × 10−1	2.201 × 10−1	14.28%
Flight altitude variation	1.902 × 102	1.549 × 102	18.56%
Actual maximum climb angle	29.66	28.28	4.65%
Actual maximum descent angle	−21.29	−13.77	35.18%
Energy consumption	1.722 × 103	1.680 × 103	2.44%
Threat distance	6.330 × 101	6.599 × 101	4.25%
Computation time	1.436	1.618	−12.67%

,

Table 12. Comparison of Average Results for UAV Path Planning Between EAPO and APO (MAP 2).

Item Name	APO	EAPO	Percentage Improvement
Path length	1.521 × 103	1.515 × 103	0.39%
Path smoothness index	1.250 × 10−1	1.312 × 10−1	4.96%
Flight altitude variation	3.774 × 102	3.579 × 102	5.17%
Actual maximum climb angle	42.91	40.32	6.04%
Actual maximum descent angle	−34.20	−34.93	−2.13%
Energy consumption	2.059 × 103	2.028 × 103	15.06%
Threat distance	4.604 × 100	5.040 × 100	9.47%
Computation time	2.108	2.365	−12.19%

,

Table 13. Comparison of Average Results for UAV Path Planning Between EAPO and APO (MAP 3).

Item Name	APO	EAPO	Percentage Improvement
Path length	6.227 × 102	6.189 × 102	0.61%
Path smoothness index	1.560 × 10−1	1.638 × 10−1	5.00%
Flight altitude variation	1.965 × 102	1.838 × 102	6.46%
Actual maximum climb angle	44.43	44.55	−0.27%
Actual maximum descent angle	−37.83	−37.32	1.35%
Energy consumption	9.336 × 102	9.137 × 102	2.13%
Threat distance	3.542 × 101	3.517 × 101	0.71%
Computation time	1.443	1.605	−11.22%

and

Table 14. Comparison of Average Results for UAV Path Planning Between EAPO and APO (MAP 4).

Item Name	APO	EAPO	Percentage Improvement
Path length	6.385 × 102	6.412 × 102	−0.42%
Path smoothness index	1.814 × 10−2	1.797 × 10−2	0.94%
Flight altitude variation	2.139 × 102	2.074 × 102	3.04%
Actual maximum climb angle	43.80	44.08	−0.64%
Actual maximum descent angle	−39.11	−38.10	2.58%
Energy consumption	9.716 × 102	9.660 × 102	0.58%
Threat distance	2.808 × 101	2.812 × 101	0.14%
Computation time	1.845	2.044	−10.78%

report the comparative results of path quality evaluation metrics for EAPO and APO. The optimum value for each indicator in the table is highlighted in bold and and underlined.

As shown in Table 10, EAPO achieves the best average fitness values on all four maps. Compared with the original APO algorithm, the improvements are 14.0%, 4.5%, 8.7%, and 31.42%, respectively. The composite average ranking of EAPO is 1.25, which consistently ranks first among all algorithms. On Maps 1 and 4, EAPO obtains the best results in terms of both the best value and the mean value. On Map 2, EAPO achieves the lowest mean fitness value among all algorithms. On Map 3, it provides the best mean value and the lowest standard deviation, indicating stable convergence behavior. Although HOA and DBO slightly outperform EAPO in terms of the best value on Maps 2 and 3, EAPO still shows clear overall advantages over the original APO algorithm.

The results in Table 11, Table 12, Table 13 and Table 14 further show that EAPO outperforms APO on most path quality metrics. On Maps 2 and 3, EAPO produces paths that are shorter, smoother, more energy-efficient, and safer. On Maps 1 and 4, although the path length is not shorter than that of APO, EAPO achieves better performance in terms of smoothness, elevation variation, and safety.

These results indicate that the proposed enhancement strategies improve both global exploration and local exploitation abilities. Although EAPO requires more computational time than APO, the performance improvement compensates for this increase in runtime.

8. Conclusions

To address the complex constraints and multimodal optimization challenges in three-dimensional path planning for UAVs, this paper proposes an Enhanced Artificial Protozoa Optimization (EAPO) algorithm. Building upon the original APO, this algorithm implements systematic enhancements across four dimensions: initialization strategy, behavioral decision mechanism, environmental perception capability, and population diversity maintenance. First, a hybrid initialization method combining Latin hypercube sampling enhances the uniformity and quality of the initial population within the search space, establishing a solid foundation for global exploration. Second, an adaptive behavior selection mechanism based on historical success is introduced, enabling the algorithm to dynamically choose search behaviors and effectively balance exploration and exploitation needs across different optimization phases. Additionally, the algorithm constructs an environmental structure model based on a perceptual field and designs an adaptive hibernation and reconstruction mechanism accordingly. This mechanism allows the algorithm to promptly escape from local optima when population activity declines, enhancing its global escape capability.

To address the complex constraints and multimodal optimization challenges in three-dimensional UAV path planning, this paper proposes an Enhanced Artificial Protozoa Optimizer (EAPO). The proposed algorithm is developed based on the original APO framework and introduces several targeted improvements. These improvements focus on population initialization, behavior selection, environmental perception, and population diversity maintenance.

First, a hybrid initialization strategy based on Latin hypercube sampling is adopted. This strategy improves the uniformity and quality of the initial population. It provides a more reliable starting point for global exploration. Second, a behaviorally adaptive selection mechanism based on historical success is introduced. This mechanism dynamically adjusts behavior selection probabilities during the optimization process. It helps maintain a balance between exploration and exploitation at different search stages. In addition, a sensory field–based environmental structure modeling strategy is incorporated. This strategy provides perceptual environmental information to guide the search direction. An adaptive hibernation–reconstruction mechanism is also designed. It restores population diversity when stagnation occurs and helps the algorithm escape from local optima.

The effectiveness of the proposed strategies is verified through ablation experiments. The exploration–exploitation balance is further analyzed using dedicated experiments. Extensive tests on the CEC2022 benchmark suite, the CEC2020 benchmark suite, and ten real-world engineering optimization problems demonstrate that EAPO outperforms the original APO and other advanced optimization algorithms in terms of convergence accuracy, stability, and robustness.

In four simulated UAV path planning scenarios with different sizes and complexities, EAPO consistently generates feasible paths that satisfy all constraints. The results in Table 11, Table 12, Table 13 and Table 14 further show that EAPO outperforms APO on most path quality evaluation metrics. On Maps 2 and 3, EAPO produces paths that are shorter, smoother, more energy-efficient, and safer. On Maps 1 and 4, although the path length is not shorter than that of APO, EAPO achieves better performance in terms of smoothness, altitude variation, and safety. Moreover, compared to the other seven algorithms, EAPO achieved the most favorable average fitness value, with Friedman ranking first overall. This indicates EAPO’s path planning capability surpasses that of the other algorithms.

Although EAPO demonstrated superior performance in experiments, future research can be expanded in several directions: first, extending EAPO to a multi-objective optimization framework to simultaneously optimize conflicting goals such as path length, energy consumption, and safety. Second, integrating deep learning or reinforcement learning methods to enhance the algorithm’s adaptive and real-time planning capabilities in dynamic threat environments. Third, further exploring the algorithm’s application potential in broader engineering scenarios such as power system dispatch, hyperparameter optimization, and image processing.