An Artistic Image Segmentation Method Using an Art-Design-Inspiration-Driven Ivy Algorithm

Abstract

To overcome the limitations of the original Ivy Algorithm (IVYA), including insufficient population diversity, limited step-size adaptability, and premature convergence, this paper proposes a multi-strategy enhanced Ivy optimization algorithm (MEIVYA). The proposed method integrates chaotic population initialization, adaptive growth-rate regulation, and an elite-guided cooperative search strategy to improve global exploration, local exploitation, and convergence stability. Experimental results on the CEC2014 and CEC2017 benchmark suites show that MEIVYA achieves competitive convergence accuracy, robustness, and stability compared with several state-of-the-art metaheuristic algorithms. In addition, MEIVYA is applied to multi-threshold image segmentation based on the Otsu criterion, where it produces clearer segmentation structures and better visual quality. The results demonstrate that MEIVYA is an effective and robust approach for both numerical optimization and artistic image segmentation.

1. Introduction

In recent years, image segmentation has become one of the most active research areas in computer vision and image processing. As a fundamental step in high-level visual understanding, image segmentation aims to partition an image into meaningful and homogeneous regions [1], thereby facilitating subsequent tasks such as object recognition [2], scene understanding [3], medical diagnosis [4], and artistic image analysis [5]. With the rapid development of multimedia technology, artificial intelligence, and digital art creation, the demand for accurate and efficient segmentation methods has increased significantly. In particular, artistic image segmentation has attracted growing attention due to its applications in digital content creation, style transfer, visual communication, and cultural heritage preservation.

Despite remarkable progress in deep learning-based segmentation techniques, traditional and optimization-based image segmentation methods continue to play an important role [6], especially in scenarios where annotated datasets are limited or computational resources are constrained. Artistic images often contain complex textures, irregular boundaries, and diverse color distributions, which pose significant challenges to conventional segmentation approaches. Therefore, designing robust and adaptable segmentation frameworks remains a critical and challenging research problem. Optimization-driven segmentation models, such as multilevel thresholding and clustering-based approaches, have demonstrated strong flexibility and effectiveness in handling such complexities [7,8,9].

Against this backdrop, heuristic and metaheuristic optimization algorithms have become powerful tools for solving complex, nonlinear, and multimodal optimization problems [10,11]. For example, Particle swarm optimization (PSO) [12], Grey Wolf Optimization (GWO) [13], and Ant colony optimization (ACO) [14] are widely used to solve these problems. Inspired by natural phenomena, swarm intelligence, and evolutionary mechanisms, these algorithms have been widely applied to image segmentation tasks. For example, Farhad Soleimanian Gharehchopogh improved the algorithm based on the African vultures optimization algorithm combined with the AS mechanism to solve the multi-threshold image segmentation problem [15]. The algorithm was evaluated using eight large images, and the experimental results showed that it had good and significant performance. Inspired by the aurora borealis, Yuan et al. proposed an aurora optimization algorithm for multi-threshold image segmentation and feature selection [16]. The algorithm was evaluated using 10 images from the Invasive Ductal Carcinoma (IDC) medical dataset, while the overall adaptability and accuracy of the feature selection model were tested using eight other medical datasets. Experimental results demonstrate that PLO is an effective optimization tool. Essam H. Houssein et al. proposed an efficient version based on opposition-based golden jackal optimization [17], using the Otsu method as the objective function to solve the multi-level thresholding problem. This algorithm was compared with seven other metaheuristic algorithms and outperformed other alternative algorithms in terms of PSNR, SSIM, FSIM, and MSE segmentation metrics, effectively solving the segmentation problem. Their advantages include global search capabilities, ease of implementation, and independence from gradient information. By modeling the image segmentation problem as an optimization problem, heuristic algorithms can effectively determine the optimal threshold or cluster centers, thereby improving segmentation accuracy.

Furthermore, to address the edge computing and path planning problems of multi-UAV mobile systems, Yu et al. proposed the Bounty hunter optimizer in 2026, and experimental results show that it achieves significant results [18]. To address nonlinear engineering problems, Vedik et al. proposed a novel metaheuristic optimization algorithm inspired by social behavior. They applied it to ten practical industrial engineering problems, two power system-related problems, and one path planning problem. Experimental results show that the UFIA algorithm outperforms existing algorithms and is competitive in solving complex optimization problems [19]. Inspired by philosophical concepts, Siamak et al. proposed a novel optimization algorithm to simulate the knowledge acquisition process based on epistemological discussions of philosophical propositions. Experiments in numerical optimization and engineering optimization tasks demonstrated that its performance even surpasses several algorithms that won in the CEC competition [20].

The Ivy Algorithm (IVYA) is a swarm intelligence optimization algorithm inspired by the growth behavior of ivy plants [21]. This algorithm simulates the orderly growth, light-climbing, and spreading process of ivy in nature to achieve a dynamic balance between global exploration and local development. Each individual selects better neighbors as its growth direction based on its fitness ranking, and updates its growth rate using a random perturbation mechanism, thereby accelerating convergence while maintaining population diversity. IVYA has a simple structure, few parameters, and strong global optimization capabilities and adaptability to engineering optimization. Since the “No Free Lunch Theorem” (NFL) states that no single optimization algorithm can perform optimally on all problems, the performance of any algorithm depends on the specific characteristics of the problem. Although the Ivy algorithm demonstrates good global search capabilities in many benchmark functions and engineering optimization problems, there is still room for improvement for the problem addressed in this paper. Therefore, researchers typically combine the structural characteristics and constraints of the problem to make targeted improvements and mechanism optimizations to the optimization algorithm, thereby enhancing its convergence performance and solution accuracy in specific optimization tasks. For example, Zhang et al. proposed an enhanced Ivy algorithm guided by the particle swarm optimization (PSO) mechanism and applied it to solve engineering instance problems [22]. The new improved algorithm continuously achieved global optimum in multiple runs, with shorter computation time and a success rate of 100%, demonstrating its powerful global optimization capability and excellent repeatability. To address the Complex Engineering Design and UAV Navigation problem, Jia et al. proposed an Enhanced IVY Optimization Algorithm using an adaptive crossover strategy [23]. Practical applications in engineering design optimization and UAV path planning validated the robust performance of ACIVY, demonstrating its ability to consistently provide optimal solutions across various real-world scenarios. The algorithm’s superior convergence accuracy, solution reliability, and computational efficiency make it a powerful tool for solving complex optimization problems requiring accuracy and consistency.

Based on the above research, this paper proposes a multi-strategy enhanced IVYA optimization algorithm to solve numerical optimization and artistic image segmentation problems. Specific contributions are as follows:

• A chaotic population initialization strategy based on the Logistic map is introduced to improve the diversity and uniformity of the initial population. This mechanism enhances the global exploration ability of the algorithm and reduces the risk of premature convergence in the early search stage.
• An adaptive growth-rate regulation mechanism is developed to dynamically adjust the search step size during the iteration process. By balancing global exploration in the early stage and local exploitation in the later stage, the proposed strategy improves convergence speed and solution accuracy.
• An elite-guided cooperative search strategy is proposed to strengthen information sharing among individuals. By combining elite guidance with neighborhood-based learning, the algorithm achieves better population cooperation and stronger ability to escape local optima.
• Extensive experiments are conducted on the CEC2014 and CEC2017 benchmark suites to evaluate the performance of MEIVYA. The results demonstrate that the proposed algorithm achieves superior convergence accuracy, faster convergence speed, and better robustness compared with several state-of-the-art metaheuristic algorithms.
• The proposed MEIVYA is further applied to multi-threshold image segmentation based on the Otsu criterion. Experimental results on several benchmark images show that MEIVYA can generate clearer segmentation structures, more natural edge transitions, and better visual quality than the compared methods.

The rest of this paper is structured as follows. Section 2 elaborates on the proposed MEIVYA in detail. Section 3 presents the numerical optimization experiments along with a comprehensive analysis of the obtained results. Section 4 describes the implementation procedure and performance assessment of the multi-threshold image segmentation application. Finally, Section 5 concludes the paper and outlines potential directions for future research.

2. The Proposed MEIVYA

2.1. Ivy Algorithm

Since the method proposed in this paper is based on the Ivy algorithm, this chapter first provides a brief overview of the Ivy algorithm, and then presents a detailed description of the targeted improvement strategies developed in this paper. The specific details are as follows:

2.1.1. Generation of the Initial Ivy League Group

Let $D$ be the dimension of the optimization problem. Then the $i$-th ivy individual can be represented as follows:

$$I_i=\left(I_{i1},I_{i2},\dots ,I_{iD}\right).$$

(1)

The entire population can be represented as follows:

$$\overrightarrow{I}=\left(I_1,I_2,\dots ,I_{N_{pop}}\right).$$

(2)

In the algorithm initialization phase, the individual positions are randomly generated using the following formula:

$$I_i=I_{min}+rand(1,D)\odot (I_{max}-I_{min}),i=1,\dots ,N_{pop},$$

(3)

where $I_{min}$ and $I_{max}$ represent the lower and upper bounds of the search space, respectively, $rand(1,D)$ represents a random vector uniformly distributed over the interval $[0, 1]$, and $\odot$ represents element-wise multiplication.

2.1.2. Group Search and Update Mechanism

(1) Orderly and coordinated group growth: The growth process of ivy is continuous and cumulative. Therefore, this paper assumes that its growth rate changes over time according to the following form:

$${\displaystyle \frac{dGv\left(t\right)}{dt}}=\psi \cdot Gv\left(t\right)\cdot \varphi \left(Gv\left(t\right)\right),$$

(4)

where $\psi$, $Gv$, and $\varphi$ are the growth rate, growth velocity, and correction factor that indicates the deviation from growth, respectively.

In IVYA, this continuous model is discretized through experiments and simulations, resulting in the following growth rate update formula:

$$\Delta G{v}_i(t+1)=ran{d}_2\odot \left(N\right(1,D)\odot \Delta G{v}_i(t\left)\right),$$

(5)

where $N(1,D)$ represents a random vector following a normal distribution, and $\Delta G{v}_i$ represents the increment in individual growth rate. This mechanism gives the search step size random and adaptive characteristics during the iteration process, which helps to enhance the algorithm’s global search capability.

(2) Climbing search based on neighborhood optimal individuals: In nature, ivy plants preferentially choose the nearest neighboring individual with the best growth conditions as a climbing target. Inspired by this, each individual in IVYA uses its nearest neighbor with superior fitness as a guiding object. Let the population be sorted from best to worst according to fitness:

$${\overrightarrow{I}}_S=(I_1^s,I_2^s,\dots ,I_{N_{pop}}^s),$$

(6)

where $I_1^s$ represents the current best individual, and $I_{best}=$ $I_1^s$. $I_{ii}$ is the nearest and most important neighbor selected by $I_i$. Then, the reference neighbor of the $i$-th individual is defined as follows:

$$I_{ii}=\left\{\begin{array}{c}{I}_{j-1}^s, I_i=I_j^s\\ I_i, I_i=I_{Best}\end{array}\right..$$

(7)

The individual position update formula is:

$$I_i^{new}=I_i+\left|N\left(1,D\right)\right|\odot \left(I_{ii}-I_i\right)+N\left(1,D\right)\odot \Delta G{v}_i.$$

(8)

This update strategy allows individuals to gradually move towards better solution regions while maintaining their ability to perform random searches.

(3) Diffusion and evolution based on the globally optimal solution: After completing the neighborhood search, IVYA further introduces a diffusion search mechanism centered around the globally optimal individual. This stage is primarily used to enhance local exploitation capabilities, and its mathematical expression is as follows:

$$I_i^{new}=I_{Best}\odot \left(rand\left(1,D\right)+N\left(1,D\right)\odot \Delta G{v}_i\right).$$

(9)

Subsequently, the individual’s growth rate is updated using the following formula:

$$\Delta G{v}_i^{new}=I_i^{new}\otimes \left(I_{max}-I_{min}\right).$$

(10)

2.1.3. Survivor Selection Mechanism

To simulate the alternating behavior of ivy between “climbing growth” and “lateral expansion,” IVYA introduces a fitness-based decision mechanism. When the objective function value of individual $I_i$ satisfies:

$$f\left(I_i\right)<\beta \cdot f\left(I_{Best}\right),\beta ={\displaystyle \frac{2+rand}{2}}.$$

(11)

At this point, an expanded search strategy is executed; otherwise, a hill-climbing search strategy is performed.

After each generation of iteration, the current population is merged with the newly generated population, and then re-sorted according to fitness. Only the top $N_{pop}$ individuals are retained for the next generation, ensuring a constant population size and improving the overall solution quality.

2.2. Modified Enhanced Ivy Algorithm (MEIVYA)

Although the original Ivy Algorithm (IVYA) demonstrates competitive performance in solving optimization problems, it still suffers from several limitations when dealing with high-dimensional and complex landscapes, such as insufficient population diversity at initialization, limited adaptability of the growth rate during iterations, and a tendency to fall into local optima.

To address these issues, this paper proposes a Modified Enhanced Ivy Algorithm (MEIVYA) by introducing three improvement strategies: chaotic population initialization, adaptive growth-rate regulation, and an elite-guided cooperative search mechanism. The proposed improvements are seamlessly integrated into the original IVYA framework without increasing algorithmic complexity.

2.2.1. Chaotic-Based Population Initialization

In the original IVYA, the initial population is generated randomly using Equation (3). Although random initialization is simple, it often leads to uneven population distribution in the search space, which may weaken the global exploration capability, especially for high-dimensional optimization problems.

To enhance population diversity and improve the coverage of the search space, a chaotic mapping strategy based on the Logistic map is introduced. The Logistic chaotic sequence is defined as follows:

$$x_{k+1}=\mu x_k(1-x_k),\mu =4$$

(12)

where $x_k\in \left(\mathrm{0,1}\right)$. Due to its ergodicity and sensitivity to initial conditions, the Logistic map can generate well-distributed chaotic sequences. The chaotic sequence is then mapped to the solution space to initialize the ivy individuals:

$$I_i=I_{min}+x_i\cdot (I_{max}-I_{min}),i=\mathrm{1,2},\dots ,N_{pop}$$

(13)

where $x_i$ denotes the chaotic variable corresponding to the $i-th$ individual.

This strategy significantly enhances the diversity of the initial population, thereby improving the global exploration capability of MEIVYA at the early stage of optimization.

2.2.2. Adaptive Growth-Rate Regulation Mechanism

In IVYA, the growth-rate update mechanism described in Equation (5) relies heavily on random perturbations, which may cause excessive oscillations in the later stages of the search process. Moreover, the step size remains unchanged throughout the iterations, limiting the balance between exploration and exploitation.

To overcome this drawback, an adaptive growth-rate control factor is introduced to dynamically adjust the search intensity according to the iteration process. The adaptive factor is defined as:

$$\alpha (t)={\alpha }_{max}-{\displaystyle \frac{t}{T}}({\alpha }_{max}-{\alpha }_{min}),$$

(14)

where $t$ is the current iteration number, $T$ is the maximum number of iterations, and ${\alpha }_{max}$ and ${\alpha }_{min}$ represent the upper and lower bounds of the control factor, respectively.

Accordingly, the growth-rate update equation is modified as:

$$\Delta G{v}_i\left(t+1\right)=\alpha \left(t\right)\cdot ran{d}_2\odot \left(N\left(1,D\right)\odot \Delta G{v}_i\left(t\right)\right).$$

(15)

This adaptive mechanism enables MEIVYA to perform large-step global exploration during the early iterations, while gradually shifting to fine-grained local exploitation in the later stages, thereby improving convergence accuracy and stability.

2.2.3. Elite-Guided Cooperative Search Strategy

In the original IVYA, each individual updates its position mainly based on a single neighboring individual with better fitness, which may limit information sharing among the population and increase the risk of premature convergence.

To further enhance the search efficiency, an elite-guided cooperative search strategy is proposed. At each iteration, the current global best individual $I_{best}$ is selected as the elite solution to guide the population evolution. The position update rule is reformulated as:

$$I_i^{new}=\left\{\begin{array}{c}{I}_i+N\left(1,D\right)\odot \left(I_{best}-I_i\right), rand<p\\ I_i+N\left(1,D\right)\odot \left(I_{li}-I_i\right), rand\ge p\end{array}\right.,$$

(16)

where $p$ denotes the elite guidance probability, $I_{li}$ represents the neighborhood reference individual, and $rand\in \left(\mathrm{0,1}\right)$ is a uniformly distributed random number.

By combining elite guidance with neighborhood-based exploration, this strategy effectively balances global guidance and local search, enhances population cooperation, and improves the algorithm’s ability to escape local optima.

2.2.4. Overall Framework of MEIVYA

The proposed MEIVYA retains the core structure of IVYA while incorporating the above improvement strategies. These enhancements work synergistically to improve population diversity, adaptive search capability, and convergence performance without introducing additional control parameters of high complexity. Figure 1 shows the flowchart of the MEIVYA proposed in this paper.

2.2.5. Time Complexity Analysis

To further evaluate whether the proposed improvement strategies introduce significant computational overhead, the time complexity of MEIVYA is analyzed and compared with that of the original IVYA.

Assume that the population size is $N$, the problem dimension is $D$, and the maximum number of iterations is $T$. In the original IVYA, the main computational cost in each iteration comes from three parts: population fitness evaluation, position updating, and population sorting/selection. For most numerical optimization problems, the position update operation for all individuals requires $O\left(ND\right)$ time, while the fitness evaluation of the whole population is denoted as $O\left(N{C}_f\right)$, where $C_f$ represents the cost of evaluating the objective function for one individual. In addition, the sorting or selection process generally requires $O\left(NlogN\right)$. Therefore, the overall time complexity of IVYA can be expressed as $O\left(T\cdot (ND+N\mathrm{l}\mathrm{o}\mathrm{g}N+N\cdot C_f)\right)$.

For the proposed MEIVYA, the additional computational cost introduced by the three improvement strategies is analyzed as follows. First, the chaotic initialization strategy based on the Logistic map is only executed once during the initialization stage. Its computational cost is proportional to the population size and dimension, that is $O\left(ND\right)$. Since this operation is performed only once before the iterative process, it does not change the overall iterative complexity.

Second, the adaptive growth-rate regulation mechanism only introduces a dynamic control factor into the original growth update formula. This operation involves simple arithmetic calculations for each individual and dimension, and its complexity remains $O\left(ND\right)$ per iteration, which is of the same order as the original IVYA update process.

Third, the elite-guided cooperative search strategy uses the current best individual to guide the population evolution. Since the global best individual can be directly obtained after sorting or selection, this strategy only adds a vector-based guidance operation for each individual, resulting in an additional complexity of $O\left(ND\right)$ per iteration.

Therefore, the overall time complexity of MEIVYA can be written as $O\left(ND+T\cdot (ND+N\mathrm{l}\mathrm{o}\mathrm{g}N+N\cdot C_f)\right)$. Since the one-time initialization cost $O\left(ND\right)$ is negligible compared with the total iterative cost, the asymptotic time complexity of MEIVYA is still $O\left(T\cdot (ND+N\mathrm{l}\mathrm{o}\mathrm{g}N+N\cdot C_f)\right)$.

This indicates that the proposed improvement strategies do not increase the time complexity order of the original IVYA. In other words, MEIVYA preserves the computational efficiency of IVYA while achieving better exploration, exploitation, and convergence performance. Compared with other population-based metaheuristic algorithms, MEIVYA also remains within the same general complexity class, since most of them rely on population updating, fitness evaluation, and sorting or selection operations in each iteration.

3. Performance Evaluation and Analysis

This section reports the numerical optimization experiments conducted to evaluate the proposed MEIVYA using the CEC2014 [24], and CEC2017 [25] benchmark test suites. For the sake of reproducibility, the parameter configurations of all competing algorithms are explicitly provided. The performance of MEIVYA is then compared with 11 state-of-the-art algorithms on both benchmark sets. To ensure a fair comparison, the population size of all algorithms is fixed at 50, and the maximum number of iterations is set to 1000. To minimize the influence of stochastic variability, each algorithm is independently executed 30 times, and the experimental performance is assessed using the mean and standard deviation of the results. All experimental results and figures in this paper were performed and plotted using MATLAB R2023A.

3.1. Experiment Parameters Setting

This section assesses the effectiveness of MEIVYA through comparative studies with 11 advanced algorithms.

The selected algorithms include the Particle Swarm Optimization (PSO) [12], Differential Evolution (DE) [26], Evolution Strategy with Covariance Matrix Adaptation (CMAES) [27], weighted mean of vectors (INFO) [28], Slime Mould Algorithm (SMA) [29], Runge–Kutta Method (RUN) [30], Hunger Games Search (HGS) [31], Dream Optimization Algorithm (DOA) [32], Great Wall Construction Algorithm (GWCA) [33], Snow Ablation Optimizer (SAO) [34], and IVY algorithm (IVYA). In the comparative study, each algorithm was implemented using the parameter settings recommended in the respective original publications. For transparency and ease of replication, the specific configurations and their corresponding references are summarized in

Table 1. Parameter Settings.

Algorithms	Parameter Name	Parameter Value
PSO	$v_{Max},w_{Max},w_{Min},c_1,c_2$	6, 0.9, 0.6, 2, 2
DE	$F$, $CR$	0.8, 0.1
CMAES	$u$	0.3
INFO	---	---
SMA	$z$	0.03
RUN	$a1$	0.5
HGS	$VC2,sumHungry$	0.03, 0
DOA	---	---
GWCA	$SL, T$	1, 8.3
SAO	---	---
IVYA	---	---

.

3.2. Compare Using CEC 2014 Test Set

In this part, the performance of the MEIVYA is assessed on the 30-dimensional and 50-dimensional benchmark functions from the CEC2014 test suite. To demonstrate its effectiveness, MEIVYA is compared against 11 advanced optimization algorithms. The experimental results are reported in

Table 2. Performance Comparison of Algorithms on the 30-Dimensional CEC 2014 Benchmark Functions.

ID	Items	PSO	DE	CMAES	INFO	SMA	RUN	HGS	DOA	GWCA	SAO	IVYA	MEIVYA
CEC2014-F1	Ave	5.0285 × 106	7.6123 × 107	9.3986 × 108	7.7819 × 105	7.1983 × 106	3.3051 × 105	8.2493 × 106	1.4698 × 106	1.7682 × 106	1.8335 × 106	1.0445 × 107	1.9078 × 106
	Std	1.7738 × 106	1.8916 × 107	3.1234 × 108	5.2934 × 105	5.5965 × 106	1.8020 × 105	6.8944 × 106	9.5989 × 105	2.7920 × 106	1.1495 × 106	6.0659 × 106	7.8280 × 105
CEC2014-F2	Ave	3.5669 × 105	2.2087 × 104	6.5518 × 1010	2.0082 × 102	1.2506 × 104	1.4836 × 104	2.3880 × 104	1.2155 × 103	7.8010 × 103	9.4001 × 103	4.5440 × 105	4.2499 × 103
	Std	3.1238 × 105	5.7088 × 103	1.9149 × 1010	4.1528 × 100	1.0561 × 104	8.1954 × 103	4.0309 × 104	1.0234 × 103	7.0212 × 103	9.6449 × 103	1.5009 × 106	1.4183 × 103
CEC2014-F3	Ave	8.5427 × 103	3.2949 × 103	2.0786 × 105	3.2081 × 102	4.7940 × 103	5.3533 × 102	1.7038 × 104	2.5119 × 103	5.9227 × 103	9.2462 × 103	7.9476 × 103	9.2088 × 102
	Std	7.1752 × 103	2.3986 × 103	6.5219 × 104	3.3935 × 101	5.5202 × 103	1.4422 × 102	1.4593 × 104	2.1732 × 103	5.7755 × 103	6.2112 × 103	2.5246 × 103	1.4328 × 102
CEC2014-F4	Ave	4.6039 × 102	5.7672 × 102	9.7728 × 103	4.9053 × 102	5.3378 × 102	4.8054 × 102	5.2335 × 102	4.8786 × 102	5.0290 × 102	4.8887 × 102	5.0729 × 102	4.6042 × 102
	Std	2.4827 × 101	1.3262 × 101	4.3432 × 103	3.8118 × 101	4.4279 × 101	2.5020 × 101	4.0949 × 101	3.5298 × 101	3.5253 × 101	2.4370 × 101	3.8124 × 101	2.1977 × 101
CEC2014-F5	Ave	5.2091 × 102	5.2060 × 102	5.2103 × 102	5.2017 × 102	5.2099 × 102	5.2097 × 102	5.2007 × 102	5.2078 × 102	5.2005 × 102	5.2100 × 102	5.2097 × 102	5.2046 × 102
	Std	9.0931 × 10−2	5.4062 × 10−2	4.3807 × 10−2	1.4461 × 10−1	1.0271 × 10−1	5.4903 × 10−2	6.4579 × 10−2	8.3786 × 10−2	8.3879 × 10−2	6.2390 × 10−2	1.8987 × 10−1	3.8447 × 10−2
CEC2014-F6	Ave	6.1886 × 102	6.2622 × 102	6.4076 × 102	6.2229 × 102	6.1682 × 102	6.2741 × 102	6.1804 × 102	6.2241 × 102	6.2671 × 102	6.0554 × 102	6.1617 × 102	6.0812 × 102
	Std	4.2886 × 100	1.4197 × 100	4.2965 × 100	3.7212 × 100	3.1556 × 100	2.5941 × 100	2.9881 × 100	4.8484 × 100	2.4399 × 100	3.0817 × 100	6.5157 × 100	6.2390 × 10−1
CEC2014-F7	Ave	7.0051 × 102	7.0017 × 102	1.2263 × 103	7.0001 × 102	7.0099 × 102	7.0002 × 102	7.0102 × 102	7.0001 × 102	7.0039 × 102	7.0001 × 102	7.0001 × 102	7.0001 × 102
	Std	2.1273 × 10−1	3.6947 × 10−2	1.8841 × 102	1.6245 × 10−2	6.1469 × 10−2	1.5852 × 10−2	3.2346 × 100	1.3851 × 10−2	8.8303 × 10−1	8.7644 × 10−3	1.3832 × 10−2	3.1231 × 10−3
CEC2014-F8	Ave	9.1628 × 102	8.2246 × 102	1.1551 × 103	8.8736 × 102	8.2884 × 102	9.2473 × 102	8.2159 × 102	9.5431 × 102	9.0202 × 102	8.5054 × 102	8.9870 × 102	8.3128 × 102
	Std	1.6413 × 101	3.1281 × 100	4.0452 × 101	2.2920 × 101	6.2670 × 100	1.8126 × 101	1.1888 × 101	4.1536 × 101	1.9905 × 101	1.6096 × 101	2.1268 × 101	2.8382 × 100
CEC2014-F9	Ave	1.0216 × 103	1.0642 × 103	1.3000 × 103	1.0249 × 103	1.0139 × 103	1.0768 × 103	1.0378 × 103	1.0722 × 103	1.0238 × 103	9.5596 × 102	1.0765 × 103	9.5630 × 102
	Std	2.0011 × 101	1.0588 × 101	6.6082 × 101	3.2975 × 101	2.4498 × 101	1.6125 × 101	2.7777 × 101	3.7357 × 101	3.2401 × 101	1.4844 × 101	1.6549 × 101	7.1201 × 100
CEC2014-F10	Ave	4.0176 × 103	1.2858 × 103	8.8334 × 103	2.5823 × 103	1.8767 × 103	2.4175 × 103	1.4669 × 103	6.9705 × 103	4.2022 × 103	2.9559 × 103	3.6709 × 103	1.6915 × 103
	Std	6.1865 × 102	5.5900 × 101	3.3648 × 102	5.0020 × 102	2.7609 × 102	4.0444 × 102	2.1674 × 102	1.0744 × 103	5.5305 × 102	4.9944 × 102	7.3707 × 102	6.8820 × 101
CEC2014-F11	Ave	4.5140 × 103	6.1568 × 103	9.1365 × 103	4.9750 × 103	4.3940 × 103	4.2327 × 103	4.0501 × 103	7.9112 × 103	4.9962 × 103	3.9774 × 103	4.8992 × 103	4.7561 × 103
	Std	6.8836 × 102	2.7972 × 102	2.8943 × 102	7.4440 × 102	6.2798 × 102	5.8923 × 102	5.6594 × 102	6.4728 × 102	8.2838 × 102	6.3476 × 102	7.1257 × 102	1.5224 × 102
CEC2014-F12	Ave	1.2009 × 103	1.2011 × 103	1.2037 × 103	1.2004 × 103	1.2005 × 103	1.2025 × 103	1.2002 × 103	1.2020 × 103	1.2007 × 103	1.2010 × 103	1.2018 × 103	1.2008 × 103
	Std	5.6866 × 10−1	1.6131 × 10−1	3.5211 × 10−1	1.9554 × 10−1	1.7702 × 10−1	8.4863 × 10−1	6.3850 × 10−2	4.5546 × 10−1	3.5933 × 10−1	1.1079 × 100	1.3955 × 100	5.3282 × 10−2
CEC2014-F13	Ave	1.3005 × 103	1.3005 × 103	1.3061 × 103	1.3005 × 103	1.3006 × 103	1.3005 × 103	1.3007 × 103	1.3004 × 103	1.3005 × 103	1.3005 × 103	1.3004 × 103	1.3003 × 103
	Std	1.2851 × 10−1	5.0909 × 10−2	8.9000 × 10−1	8.9128 × 10−2	1.2463 × 10−1	8.0565 × 10−2	1.1957 × 10−1	1.1156 × 10−1	1.1128 × 10−1	8.7294 × 10−2	8.9954 × 10−2	2.7573 × 10−2
CEC2014-F14	Ave	1.4003 × 103	1.4004 × 103	1.5940 × 103	1.4003 × 103	1.4008 × 103	1.4003 × 103	1.4009 × 103	1.4003 × 103	1.4003 × 103	1.4004 × 103	1.4003 × 103	1.4002 × 103
	Std	1.3671 × 10−1	4.1742 × 10−2	4.6420 × 101	1.6916 × 10−1	3.3020 × 10−1	3.8223 × 10−2	3.4031 × 10−1	1.6823 × 10−1	5.3279 × 10−2	1.0170 × 10−1	5.5590 × 10−2	2.0479 × 10−2
CEC2014-F15	Ave	1.5148 × 103	1.5179 × 103	1.2112 × 106	1.5256 × 103	1.5096 × 103	1.5370 × 103	1.5127 × 103	1.5201 × 103	1.6253 × 103	1.5148 × 103	1.5640 × 103	1.5072 × 103
	Std	2.8330 × 100	1.1544 × 100	1.7648 × 106	1.7055 × 101	2.5770 × 100	1.2834 × 101	4.2286 × 100	5.5846 × 100	1.5012 × 102	3.7866 × 100	1.9994 × 102	1.2311 × 100
CEC2014-F16	Ave	1.6123 × 103	1.6121 × 103	1.6137 × 103	1.6122 × 103	1.6114 × 103	1.6110 × 103	1.6114 × 103	1.6125 × 103	1.6125 × 103	1.6126 × 103	1.6109 × 103	1.6108 × 103
	Std	5.5421 × 10−1	2.6966 × 10−1	1.3772 × 10−1	5.7338 × 10−1	4.4890 × 10−1	7.6138 × 10−1	6.9317 × 10−1	3.5798 × 10−1	5.4041 × 10−1	3.2358 × 10−1	9.4961 × 10−1	1.6920 × 10−1
CEC2014-F17	Ave	3.6694 × 105	6.2742 × 106	5.3813 × 107	3.5755 × 104	1.4094 × 106	4.7879 × 104	1.5748 × 106	2.3203 × 105	1.7036 × 105	2.0759 × 105	1.0142 × 106	5.2986 × 104
	Std	2.5703 × 105	2.8956 × 106	3.1742 × 107	2.8257 × 104	7.1612 × 105	2.9903 × 104	1.0400 × 106	1.6453 × 105	1.1800 × 105	1.1494 × 105	9.3519 × 105	1.7769 × 104
CEC2014-F18	Ave	4.3985 × 103	5.2784 × 105	2.2860 × 109	6.0027 × 103	2.2626 × 104	3.5007 × 103	1.4364 × 104	8.2149 × 103	4.8911 × 103	3.7936 × 103	3.4048 × 103	1.8481 × 103
	Std	4.0305 × 103	1.9062 × 105	1.1530 × 109	6.1089 × 103	7.9030 × 103	1.4671 × 103	1.0221 × 104	7.0961 × 103	3.4398 × 103	2.4516 × 103	1.3004 × 103	1.2973 × 101
CEC2014-F19	Ave	1.9149 × 103	1.9175 × 103	2.2210 × 103	1.9181 × 103	1.9164 × 103	1.9218 × 103	1.9218 × 103	1.9209 × 103	1.9203 × 103	1.9073 × 103	1.9182 × 103	1.9067 × 103
	Std	3.0120 × 100	2.3353 × 100	8.9784 × 101	2.1331 × 101	2.1196 × 101	1.5088 × 101	2.5263 × 101	2.2537 × 101	1.8495 × 101	1.4439 × 100	2.6632 × 101	5.0156 × 10−1
CEC2014-F20	Ave	1.2932 × 104	1.3960 × 104	9.1389 × 105	2.7999 × 103	2.1153 × 104	2.9233 × 103	2.4772 × 104	1.3335 × 104	1.9001 × 104	2.7877 × 104	2.8075 × 104	3.8896 × 103
	Std	5.7641 × 103	6.0659 × 103	1.4389 × 106	6.1738 × 102	1.1830 × 104	5.7481 × 102	1.6862 × 104	8.4419 × 103	1.0633 × 104	1.3455 × 104	1.0618 × 104	6.9374 × 102
CEC2014-F21	Ave	1.3751 × 105	1.2007 × 106	2.0007 × 107	2.5792 × 104	4.7087 × 105	1.7180 × 104	6.1051 × 105	6.4636 × 104	6.4797 × 104	1.3347 × 105	3.7816 × 105	1.3547 × 104
	Std	9.1241 × 104	5.0270 × 105	1.0236 × 107	1.9956 × 104	3.4499 × 105	7.0249 × 103	4.0748 × 105	9.6603 × 104	5.3147 × 104	1.0813 × 105	3.2313 × 105	4.4257 × 103
CEC2014-F22	Ave	2.8565 × 103	2.6839 × 103	3.9336 × 103	2.7363 × 103	2.8007 × 103	2.7820 × 103	2.9353 × 103	2.6946 × 103	2.8531 × 103	2.6360 × 103	3.0748 × 103	2.3146 × 103
	Std	1.7864 × 102	9.8802 × 101	2.3667 × 102	2.3830 × 102	1.6501 × 102	2.1536 × 102	2.0984 × 102	1.9977 × 102	2.3015 × 102	1.6652 × 102	2.6965 × 102	5.1601 × 101
CEC2014-F23	Ave	2.6140 × 103	2.6160 × 103	3.1544 × 103	2.5000 × 103	2.5000 × 103	2.5000 × 103	2.5000 × 103	2.5000 × 103	2.6152 × 103	2.6152 × 103	2.5000 × 103	2.5384 × 103
	Std	7.8582 × 10−3	7.0112 × 10−1	2.6088 × 102	0.0000 × 100	0.0000 × 100	0.0000 × 100	0.0000 × 100	0.0000 × 100	4.7180 × 10−11	3.3164 × 10−5	1.8190 × 10−12	5.5255 × 101
CEC2014-F24	Ave	2.6259 × 103	2.6297 × 103	2.7628 × 103	2.6000 × 103	2.6000 × 103	2.6000 × 103	2.6000 × 103	2.6000 × 103	2.6441 × 103	2.6089 × 103	2.6001 × 103	2.6000 × 103
	Std	3.0623 × 100	1.1462 × 100	5.7931 × 101	0.0000 × 100	0.0000 × 100	2.6332 × 10−5	1.1851 × 10−3	0.0000 × 100	1.0064 × 101	1.0869 × 101	2.4416 × 10−1	2.4055 × 10−4
CEC2014-F25	Ave	2.7161 × 103	2.7142 × 103	2.7893 × 103	2.7000 × 103	2.7000 × 103	2.7000 × 103	2.7000 × 103	2.7000 × 103	2.7202 × 103	2.7023 × 103	2.7000 × 103	2.7000 × 103
	Std	3.9833 × 100	2.0553 × 100	2.5846 × 101	0.0000 × 100	0.0000 × 100	0.0000 × 100	0.0000 × 100	0.0000 × 100	7.2519 × 100	4.0135 × 100	1.9163 × 10−12	0.0000 × 100
CEC2014-F26	Ave	2.7603 × 103	2.7006 × 103	2.7078 × 103	2.7104 × 103	2.7007 × 103	2.7006 × 103	2.7206 × 103	2.7469 × 103	2.7404 × 103	2.7237 × 103	2.7602 × 103	2.7003 × 103
	Std	4.9700 × 101	6.1937 × 10−2	3.1313 × 100	3.0368 × 101	1.3488 × 10−1	1.0297 × 10−1	4.0373 × 101	5.0552 × 101	4.9583 × 101	4.2807 × 101	4.9594 × 101	2.4273 × 10−2
CEC2014-F27	Ave	3.3318 × 103	3.2272 × 103	4.0988 × 103	2.9067 × 103	2.9000 × 103	2.9000 × 103	2.9000 × 103	2.9000 × 103	3.6988 × 103	3.1667 × 103	3.0037 × 103	3.1031 × 103
	Std	2.5987 × 102	3.6668 × 101	8.2118 × 101	3.6903 × 101	0.0000 × 100	0.0000 × 100	0.0000 × 100	0.0000 × 100	2.2287 × 102	9.5025 × 101	2.3804 × 102	7.9388 × 10−1
CEC2014-F28	Ave	6.2955 × 103	3.7473 × 103	5.2284 × 103	3.0000 × 103	3.0000 × 103	3.0000 × 103	3.0000 × 103	3.0000 × 103	4.6784 × 103	3.7999 × 103	3.0000 × 103	3.6336 × 103
	Std	8.2825 × 102	2.7800 × 101	2.3067 × 102	0.0000 × 100	0.0000 × 100	0.0000 × 100	0.0000 × 100	0.0000 × 100	3.9250 × 102	1.9820 × 102	3.2585 × 10−12	2.7917 × 101
CEC2014-F29	Ave	4.4368 × 103	4.1600 × 104	1.7515 × 107	4.9406 × 106	3.1248 × 105	4.3743 × 106	5.5319 × 103	3.1000 × 103	8.1848 × 106	1.9215 × 106	3.1000 × 103	3.8897 × 103
	Std	1.0688 × 103	1.4505 × 104	3.0024 × 106	6.1559 × 106	1.6863 × 106	4.4470 × 106	3.5847 × 103	0.0000 × 100	1.0149 × 107	3.9524 × 106	1.9894 × 10−12	1.0768 × 102
CEC2014-F30	Ave	6.9953 × 103	1.6088 × 104	1.1373 × 106	7.6536 × 103	1.0486 × 104	1.1633 × 104	4.7166 × 103	3.2000 × 103	7.4456 × 103	6.4907 × 103	4.6485 × 103	4.5730 × 103
	Std	1.9828 × 103	2.1965 × 103	2.8354 × 105	3.5490 × 103	9.7246 × 103	8.3499 × 103	2.2078 × 103	0.0000 × 100	3.0720 × 103	1.2007 × 103	2.9330 × 103	1.8041 × 102

and

Table 3. Performance Comparison of Algorithms on the 50-Dimensional CEC 2014 Benchmark Functions.

ID	Items	PSO	DE	CMAES	INFO	SMA	RUN	HGS	DOA	GWCA	SAO	IVYA	MEIVYA
CEC2014-F1	Ave	1.2656 × 107	3.0130 × 108	2.3411 × 109	3.7562 × 106	1.9035 × 107	2.9547 × 106	2.5928 × 107	8.3197 × 106	4.7593 × 106	7.9257 × 106	1.2577 × 107	5.1595 × 106
	Std	3.8082 × 106	4.7170 × 107	8.7490 × 108	1.5233 × 106	6.4132 × 106	7.2335 × 105	1.1509 × 107	3.9934 × 106	2.7926 × 106	2.2706 × 106	4.6663 × 106	8.6494 × 105
CEC2014-F2	Ave	8.5323 × 106	1.6070 × 108	1.2433 × 1011	9.1582 × 103	1.9167 × 105	3.6903 × 104	2.9969 × 108	2.6462 × 104	6.1366 × 103	7.3492 × 103	2.1295 × 107	3.3608 × 105
	Std	3.7175 × 106	3.5319 × 107	2.4998 × 1010	1.4888 × 104	6.8787 × 104	1.8554 × 104	4.9272 × 108	2.3079 × 104	6.3217 × 103	8.7062 × 103	8.3201 × 107	7.0384 × 104
CEC2014-F3	Ave	4.3836 × 104	1.6543 × 104	3.1959 × 105	8.9424 × 103	1.5742 × 104	7.1437 × 103	3.0548 × 104	5.0221 × 104	2.0825 × 104	3.8966 × 104	2.2833 × 104	8.2188 × 103
	Std	9.8946 × 103	3.8913 × 103	4.1489 × 104	4.4559 × 103	6.6453 × 103	2.4003 × 103	1.4056 × 104	1.9578 × 104	1.2955 × 104	2.0622 × 104	9.9985 × 103	7.6780 × 102
CEC2014-F4	Ave	5.2442 × 102	7.2740 × 102	3.1910 × 104	5.3008 × 102	5.3227 × 102	5.3509 × 102	5.4608 × 102	5.3862 × 102	5.8291 × 102	5.0456 × 102	5.9519 × 102	5.5349 × 102
	Std	4.3112 × 101	4.2808 × 101	1.0715 × 104	3.7398 × 101	4.9731 × 101	5.4451 × 101	6.0291 × 101	3.8424 × 101	3.3100 × 101	1.7495 × 101	5.7473 × 101	8.5801 × 100
CEC2014-F5	Ave	5.2112 × 102	5.2090 × 102	5.2123 × 102	5.2009 × 102	5.2118 × 102	5.2115 × 102	5.2014 × 102	5.2104 × 102	5.2004 × 102	5.2119 × 102	5.2120 × 102	5.2069 × 102
	Std	7.1132 × 10−2	4.8050 × 10−2	3.0226 × 10−2	1.2271 × 10−1	7.2632 × 10−2	1.5471 × 10−1	5.0159 × 10−2	4.8857 × 10−2	7.2765 × 10−2	4.6216 × 10−2	3.3967 × 10−2	2.3435 × 10−2
CEC2014-F6	Ave	6.3784 × 102	6.5697 × 102	6.7286 × 102	6.4666 × 102	6.3664 × 102	6.5491 × 102	6.3657 × 102	6.4582 × 102	6.5231 × 102	6.1278 × 102	6.2899 × 102	6.2255 × 102
	Std	4.1432 × 100	1.9298 × 100	2.7404 × 100	5.0722 × 100	6.6615 × 100	4.1570 × 100	4.0921 × 100	5.2394 × 100	4.3600 × 100	3.4305 × 100	7.7745 × 100	1.6312 × 100
CEC2014-F7	Ave	7.0109 × 102	7.0112 × 102	2.0661 × 103	7.0013 × 102	7.0112 × 102	7.0013 × 102	7.0109 × 102	7.0009 × 102	7.0137 × 102	7.0001 × 102	7.0149 × 102	7.0043 × 102
	Std	3.6991 × 10−2	2.3557 × 10−2	2.1693 × 102	4.4382 × 10−1	3.2785 × 10−2	3.5945 × 10−2	2.3100 × 10−1	7.8580 × 10−2	4.1353 × 100	1.2862 × 10−2	5.5972 × 100	6.3232 × 10−2
CEC2014-F8	Ave	1.0446 × 103	9.3470 × 102	1.4611 × 103	1.0172 × 103	9.1575 × 102	1.0397 × 103	8.9360 × 102	1.1596 × 103	1.0326 × 103	8.8910 × 102	1.0343 × 103	8.9845 × 102
	Std	2.3218 × 101	7.7010 × 100	5.9043 × 101	4.3523 × 101	2.2355 × 101	1.4850 × 101	2.0533 × 101	7.3473 × 101	3.1623 × 101	2.1507 × 101	3.3397 × 101	8.9710 × 100
CEC2014-F9	Ave	1.1779 × 103	1.3076 × 103	1.7168 × 103	1.1853 × 103	1.1429 × 103	1.2200 × 103	1.1583 × 103	1.2543 × 103	1.1722 × 103	1.0243 × 103	1.2506 × 103	1.0358 × 103
	Std	3.1175 × 101	1.6030 × 101	1.0006 × 102	4.7905 × 101	4.7914 × 101	2.9454 × 101	3.9530 × 101	8.4978 × 101	4.5355 × 101	3.9113 × 101	5.8553 × 101	9.5721 × 100
CEC2014-F10	Ave	6.7944 × 103	4.4937 × 103	1.5552 × 104	4.4198 × 103	3.3051 × 103	6.1341 × 103	2.5565 × 103	1.2994 × 104	7.1753 × 103	5.3583 × 103	6.4311 × 103	4.3362 × 103
	Std	8.0486 × 102	3.3276 × 102	4.5590 × 102	8.4501 × 102	5.5962 × 102	9.1890 × 102	4.0883 × 102	1.2870 × 103	1.0047 × 103	7.8266 × 102	8.0638 × 102	1.6696 × 102
CEC2014-F11	Ave	7.3380 × 103	1.2300 × 104	1.5447 × 104	8.2792 × 103	7.3876 × 103	8.0725 × 103	7.0022 × 103	1.4064 × 104	7.7627 × 103	6.8871 × 103	8.3238 × 103	9.6272 × 103
	Std	8.9158 × 102	4.0083 × 102	2.9853 × 102	1.0101 × 103	1.0212 × 103	2.3106 × 103	8.0188 × 102	9.2374 × 102	9.3677 × 102	8.1852 × 102	7.9746 × 102	2.2531 × 102
CEC2014-F12	Ave	1.2017 × 103	1.2018 × 103	1.2043 × 103	1.2006 × 103	1.2009 × 103	1.2032 × 103	1.2003 × 103	1.2031 × 103	1.2008 × 103	1.2026 × 103	1.2026 × 103	1.2011 × 103
	Std	8.2618 × 10−1	2.2313 × 10−1	2.9169 × 10−1	1.9354 × 10−1	2.4356 × 10−1	1.1954 × 100	8.7331 × 10−2	4.2213 × 10−1	2.9447 × 10−1	1.6858 × 100	1.8658 × 100	7.6248 × 10−2
CEC2014-F13	Ave	1.3006 × 103	1.3007 × 103	1.3077 × 103	1.3006 × 103	1.3008 × 103	1.3007 × 103	1.3010 × 103	1.3006 × 103	1.3006 × 103	1.3007 × 103	1.3005 × 103	1.3005 × 103
	Std	1.2997 × 10−1	6.2206 × 10−2	8.2490 × 10−1	1.2237 × 10−1	1.0007 × 10−1	1.1480 × 10−1	1.2701 × 10−1	1.1227 × 10−1	1.0153 × 10−1	8.6360 × 10−2	7.9096 × 10−2	3.4878 × 10−2
CEC2014-F14	Ave	1.4003 × 103	1.4004 × 103	1.7648 × 103	1.4003 × 103	1.4007 × 103	1.4004 × 103	1.4008 × 103	1.4004 × 103	1.4003 × 103	1.4005 × 103	1.4004 × 103	1.4003 × 103
	Std	3.9904 × 10−2	6.7666 × 10−2	8.3008 × 101	1.3931 × 10−1	3.8933 × 10−1	6.2079 × 10−2	3.2942 × 10−1	2.2072 × 10−1	1.0170 × 10−1	2.3372 × 10−1	1.0071 × 10−1	1.2446 × 10−2
CEC2014-F15	Ave	1.5342 × 103	1.6205 × 103	7.9724 × 106	1.6428 × 103	1.5270 × 103	1.6349 × 103	1.5495 × 103	1.5529 × 103	2.5774 × 103	1.5374 × 103	1.6072 × 103	1.5228 × 103
	Std	2.8730 × 100	3.3468 × 101	8.5323 × 106	1.2137 × 102	6.3137 × 100	4.5091 × 101	2.2947 × 101	8.9247 × 100	9.0350 × 102	3.2766 × 100	2.6147 × 102	1.6713 × 100
CEC2014-F16	Ave	1.6218 × 103	1.6218 × 103	1.6233 × 103	1.6214 × 103	1.6210 × 103	1.6202 × 103	1.6206 × 103	1.6222 × 103	1.6219 × 103	1.6225 × 103	1.6211 × 103	1.6198 × 103
	Std	4.8976 × 10−1	2.0658 × 10−1	2.3027 × 10−1	6.2638 × 10−1	6.7683 × 10−1	8.9398 × 10−1	8.4540 × 10−1	3.5988 × 10−1	7.9818 × 10−1	5.0339 × 10−1	1.1113 × 100	2.5876 × 10−1
CEC2014-F17	Ave	1.2215 × 106	2.7690 × 107	1.8599 × 108	2.3244 × 105	3.3314 × 106	2.5303 × 105	4.2943 × 106	1.1081 × 106	6.9898 × 105	6.8996 × 105	2.2721 × 106	3.9697 × 105
	Std	6.1338 × 105	7.6831 × 106	7.4750 × 107	1.3477 × 105	1.8868 × 106	1.1678 × 105	2.0822 × 106	7.2272 × 105	6.1144 × 105	4.3496 × 105	1.2105 × 106	1.0246 × 105
CEC2014-F18	Ave	4.0045 × 103	8.4204 × 105	6.7618 × 109	4.3173 × 103	6.8853 × 103	5.7872 × 103	7.1311 × 103	4.3967 × 103	4.2050 × 103	3.1417 × 103	1.4612 × 104	1.9389 × 103
	Std	2.1139 × 103	3.8733 × 105	2.4949 × 109	1.5517 × 103	2.2209 × 103	6.1011 × 102	1.9407 × 103	1.7315 × 103	1.7141 × 103	1.1692 × 103	4.9814 × 104	3.8038 × 101
CEC2014-F19	Ave	1.9325 × 103	1.9520 × 103	2.9283 × 103	1.9460 × 103	1.9717 × 103	1.9591 × 103	1.9505 × 103	1.9524 × 103	1.9584 × 103	1.9350 × 103	1.9556 × 103	1.9147 × 103
	Std	1.8514 × 101	7.6384 × 100	3.3693 × 102	2.3140 × 101	2.5255 × 101	2.7325 × 101	2.5130 × 101	2.6935 × 101	3.1729 × 101	1.2498 × 101	3.0556 × 101	7.3163 × 10−1
CEC2014-F20	Ave	1.8106 × 104	3.9404 × 104	1.1330 × 106	4.6925 × 103	2.0947 × 104	4.8092 × 103	3.0662 × 104	2.2754 × 104	2.8679 × 104	4.0574 × 104	4.0406 × 104	8.6332 × 103
	Std	7.8994 × 103	9.1298 × 103	9.9381 × 105	1.7248 × 103	8.9103 × 103	1.4429 × 103	1.7105 × 104	8.4067 × 103	1.4913 × 104	1.8072 × 104	1.3756 × 104	1.3119 × 103
CEC2014-F21	Ave	7.9634 × 105	1.1504 × 107	6.7439 × 107	1.7745 × 105	2.4843 × 106	8.1111 × 104	2.6391 × 106	3.7857 × 105	4.5780 × 105	3.9333 × 105	1.5168 × 106	2.7707 × 105
	Std	2.8733 × 105	3.1196 × 106	3.2252 × 107	1.0946 × 105	1.3113 × 106	4.3274 × 104	1.7006 × 106	2.9777 × 105	4.1542 × 105	2.6979 × 105	8.4575 × 105	7.4610 × 104
CEC2014-F22	Ave	3.7080 × 103	3.6104 × 103	6.6176 × 103	3.4515 × 103	3.3950 × 103	3.6619 × 103	3.7480 × 103	3.6389 × 103	3.6217 × 103	3.1421 × 103	3.8203 × 103	2.5890 × 103
	Std	3.2708 × 102	1.8863 × 102	7.1845 × 102	3.0688 × 102	2.7613 × 102	3.7294 × 102	3.5562 × 102	3.3431 × 102	3.6629 × 102	2.7369 × 102	3.5689 × 102	9.5617 × 101
CEC2014-F23	Ave	2.6376 × 103	2.6462 × 103	3.5836 × 103	2.5000 × 103	2.5000 × 103	2.5000 × 103	2.5000 × 103	2.5000 × 103	2.6440 × 103	2.6440 × 103	2.5000 × 103	2.5000 × 103
	Std	3.0818 × 10−1	7.1164 × 10−1	4.7188 × 102	0.0000 × 100	0.0000 × 100	0.0000 × 100	0.0000 × 100	0.0000 × 100	2.1002 × 10−2	3.6474 × 10−4	1.3056 × 10−9	0.0000 × 100
CEC2014-F24	Ave	2.6726 × 103	2.6794 × 103	2.9261 × 103	2.6000 × 103	2.6000 × 103	2.6000 × 103	2.6000 × 103	2.6000 × 103	2.7029 × 103	2.6652 × 103	2.6001 × 103	2.6000 × 103
	Std	9.8823 × 100	2.1317 × 100	1.3884 × 102	0.0000 × 100	0.0000 × 100	1.1804 × 10−5	6.9056 × 10−5	0.0000 × 100	1.0865 × 101	1.8362 × 101	9.0842 × 10−2	2.3461 × 10−4
CEC2014-F25	Ave	2.7344 × 103	2.7564 × 103	2.8951 × 103	2.7000 × 103	2.7000 × 103	2.7000 × 103	2.7000 × 103	2.7000 × 103	2.7445 × 103	2.7055 × 103	2.7000 × 103	2.7000 × 103
	Std	6.9681 × 100	6.7303 × 100	5.9144 × 101	0.0000 × 100	0.0000 × 100	0.0000 × 100	0.0000 × 100	0.0000 × 100	1.1842 × 101	8.7762 × 100	1.8480 × 10−9	0.0000 × 100
CEC2014-F26	Ave	2.7975 × 103	2.7019 × 103	2.7131 × 103	2.7569 × 103	2.7008 × 103	2.7138 × 103	2.7834 × 103	2.7967 × 103	2.7704 × 103	2.7638 × 103	2.8000 × 103	2.7005 × 103
	Std	1.8324 × 101	1.0063 × 100	5.2466 × 100	5.0160 × 101	1.0216 × 10−1	3.4402 × 101	3.7644 × 101	1.8164 × 101	4.6481 × 101	4.8763 × 101	1.5536 × 10−4	9.5528 × 10−2
CEC2014-F27	Ave	4.1797 × 103	4.4009 × 103	5.0465 × 103	2.9000 × 103	2.9000 × 103	2.9000 × 103	2.9000 × 103	2.9000 × 103	4.4780 × 103	3.4563 × 103	3.0722 × 103	3.6948 × 103
	Std	2.5748 × 102	3.7868 × 101	6.5085 × 101	1.3876 × 10−12	1.3876 × 10−12	1.3876 × 10−12	1.3876 × 10−12	1.3876 × 10−12	1.3104 × 102	1.4360 × 102	3.6015 × 102	4.9849 × 101
CEC2014-F28	Ave	1.0195 × 104	4.2562 × 103	9.8363 × 103	3.0000 × 103	3.0000 × 103	3.0000 × 103	3.0000 × 103	3.0000 × 103	6.3704 × 103	4.3885 × 103	3.0000 × 103	4.1072 × 103
	Std	1.4184 × 103	3.3833 × 101	1.0929 × 103	1.3876 × 10−12	1.3876 × 10−12	1.3876 × 10−12	1.3876 × 10−12	1.3876 × 10−12	8.9592 × 102	3.9617 × 102	1.5260 × 10−9	4.0630 × 101
CEC2014-F29	Ave	1.4702 × 104	1.9675 × 105	1.2652 × 108	1.2577 × 107	2.1287 × 106	3.0329 × 107	1.1778 × 106	3.1000 × 103	1.2146 × 108	1.6713 × 107	3.1000 × 103	4.4578 × 103
	Std	1.5414 × 104	1.8768 × 105	1.8428 × 107	2.5701 × 107	6.4892 × 106	1.9639 × 107	6.4326 × 106	0.0000 × 100	8.5303 × 107	2.1077 × 107	1.1976 × 10−9	1.9062 × 102
CEC2014-F30	Ave	1.1885 × 104	2.3651 × 104	5.5773 × 106	2.1317 × 104	1.0923 × 104	4.6044 × 104	1.0416 × 104	3.2000 × 103	2.0752 × 104	1.5774 × 104	3.2000 × 103	1.3215 × 104
	Std	6.2036 × 103	3.9419 × 103	1.8398 × 106	8.8341 × 103	8.9114 × 103	9.0788 × 104	8.7017 × 103	0.0000 × 100	5.8803 × 103	3.0571 × 103	4.7159 × 10−3	4.5944 × 102

, where “ave” and “std” denote the average value and standard deviation obtained from 30 independent runs, respectively. To provide a clearer view of the optimization process, Figure 2 presents the convergence trajectories of all 12 algorithms, highlighting their search dynamics over iterations. In addition, Figure 3 illustrates box plots based on the results of 30 independent trials, which offer a visual comparison of performance distribution, algorithm robustness, and the influence of randomness. We have highlighted the optimal solutions obtained by the algorithm for each function to help readers identify the key information more easily.

As shown in Figure 2, MEIVYA demonstrates strong overall convergence performance on the CEC2014 benchmark functions. In both 30-dimensional and 50-dimensional settings, MEIVYA quickly enters the optimal solution region on most test functions and maintains a relatively stable downward trend in the later stages of iteration, indicating that it possesses not only strong early global search capabilities but also stable and effective local exploration in later stages. Particularly on some complex functions, MEIVYA’s convergence curve is significantly lower than most comparative algorithms and stabilizes earlier, indicating its advantages in solution accuracy and convergence efficiency. Furthermore, compared to the prolonged stagnation, oscillations, or slow convergence observed in some algorithms during the search process, MEIVYA’s curve changes more smoothly, reflecting better stability and balance in its search process. This suggests that the introduced chaotic initialization, adaptive growth rate adjustment, and elite-guided collaborative search strategies effectively enhance population diversity, improve coordination between search stages, and thus improve the algorithm’s ability to escape local optima and approach the global optimum. Overall, Figure 2 visually demonstrates that MEIVYA can balance convergence speed, convergence accuracy, and search stability on the CEC2014 test set, exhibiting good comprehensive optimization performance.

Figure 3 further illustrates the distribution of results for each algorithm on some typical functions of CEC2014 using box plots, providing a more intuitive reflection of the algorithm’s stability, robustness, and susceptibility to random factors. The figure shows that MEIVYA exhibits a smaller box height and shorter upper and lower bounds on most test functions, with its median position typically at a favorable level. This indicates that it maintains relatively stable solution results across multiple independent runs, demonstrating strong robustness. In contrast, some of the comparative algorithms show larger box ranges and even significant outliers, suggesting that these algorithms are susceptible to fluctuations in random initialization and the search process, leading to unstable results. On some complex multimodal functions, MEIVYA’s box plot not only has a lower overall position but also a more concentrated distribution, indicating that it not only obtains better target values but also maintains high consistency in repeated experiments. Furthermore, while algorithms such as CMAES, PSO, and DE perform well on individual functions, their results generally exhibit significant fluctuations. In contrast, MEIVYA’s box plots across multiple functions show a narrower distribution range and fewer outliers, further validating its effectiveness in balancing global search and local exploitation. Overall, Figure 3 demonstrates that MEIVYA not only possesses strong optimization capabilities but also maintains stable and reliable performance across multiple independent runs, exhibiting robustness and consistency superior to most of the compared algorithms.

As shown in Table 2 and Table 3, as the problem dimension increases from 30 to 50, the solution accuracy of most compared algorithms decreases to varying degrees, especially for complex multimodal functions and mixed combination functions, where the mean and standard deviation increase significantly, indicating that algorithms are more prone to getting trapped in local optima and generating large fluctuations in high-dimensional cases. However, MEIVYA still maintains strong competitiveness in both dimension settings. In the 30-dimensional CEC2014 test set, MEIVYA achieved optimal or near-optimal results on multiple functions, including F3, F8, F10, F13, F14, F15, F16, F17, F18, F19, F20, F21, F22, and F26. In particular, on complex functions such as F17, F18, F20, and F21, its mean is significantly better than other algorithms, and its standard deviation is smaller, indicating that it has good stability and robustness. At the same time, MEIVYA can also achieve or approach the theoretical optimal value for some functions that are relatively easy to solve, such as F23, F24 and F25, demonstrating its strong local development capability. In the more challenging 50-dimensional test set, although the performance of all algorithms declined, MEIVYA still maintained superior results on multiple functions. For example, on functions F1, F3, F5, F15, F17, F18, F20, and F21, MEIVYA’s mean and standard deviation were better than most of the comparison algorithms, indicating that it still has good global search capabilities and stability in high-dimensional complex problems. Especially in difficult functions such as F17, F18, and F21, MEIVYA was able to maintain a low objective function value, while the results of other algorithms often deteriorated by orders of magnitude, indicating that it has a stronger adaptability to dimensionality growth. In addition, MEIVYA has a smaller standard deviation on most functions, further demonstrating that the algorithm has high consistency and robustness in multiple independent runs. Overall, the experimental results in Table 2 and Table 3 show that MEIVYA demonstrates excellent convergence accuracy, stability, and high-dimensional problem-solving capabilities on the CEC2014 test set across different dimensions, and its comprehensive performance is significantly better than most of the comparison algorithms.

3.3. Compare Using CEC 2017 Test Set

In this part, the performance of the MEIVYA is assessed on the 30-dimensional and 50-dimensional benchmark functions from the CEC2017 test suite. To demonstrate its effectiveness, MEIVYA is compared against 11 advanced optimization algorithms. The experimental results are reported in

Table 4. Performance Comparison of Algorithms on the 30-Dimensional CEC 2017 Benchmark Functions.

ID	Items	PSO	DE	CMAES	INFO	SMA	RUN	HGS	DOA	GWCA	SAO	IVYA	MEIVYA
CEC2017-F1	Ave	2.4936 × 105	1.6115 × 105	4.8903 × 1010	1.6721 × 102	9.5105 × 103	4.5394 × 103	4.6572 × 106	4.7938 × 103	6.4397 × 103	3.8401 × 103	1.0354 × 107	6.7748 × 102
	Std	1.4720 × 105	4.4659 × 104	1.8709 × 1010	2.2157 × 102	7.5183 × 103	4.8780 × 103	1.5653 × 107	5.5696 × 103	5.9484 × 103	5.2919 × 103	5.6112 × 107	2.3343 × 102
CEC2017-F2	Ave	3.2905 × 1010	4.8863 × 1027	4.4311 × 1043	8.1833 × 1017	7.0532 × 1011	3.1287 × 1016	2.0667 × 1016	5.4327 × 1012	9.2008 × 1031	9.1440 × 1012	8.2198 × 1044	4.6252 × 107
	Std	1.4930 × 1011	1.4640 × 1028	1.3074 × 1044	3.3539 × 1018	1.5030 × 1012	1.1224 × 1017	5.2220 × 1016	2.0617 × 1013	4.9393 × 1032	4.2441 × 1013	3.8653 × 1045	3.9516 × 107
CEC2017-F3	Ave	5.3442 × 103	1.4041 × 105	2.0512 × 105	9.3235 × 102	9.7100 × 102	3.2489 × 102	1.5230 × 104	3.5551 × 104	2.6236 × 103	6.7878 × 104	4.9432 × 104	2.4052 × 104
	Std	2.6857 × 103	2.0736 × 104	6.4624 × 104	1.1448 × 103	5.8897 × 102	3.7699 × 101	1.2200 × 104	1.4516 × 104	2.2175 × 103	2.3370 × 104	6.5122 × 103	2.5546 × 103
CEC2017-F4	Ave	4.6466 × 102	5.1984 × 102	8.3122 × 103	4.7405 × 102	4.9895 × 102	5.0187 × 102	5.0766 × 102	4.8710 × 102	4.7273 × 102	4.9722 × 102	4.9984 × 102	4.6164 × 102
	Std	2.3677 × 101	8.4602 × 100	2.6216 × 103	3.3688 × 101	2.7489 × 101	1.9282 × 101	3.7907 × 101	1.3443 × 101	3.2685 × 101	1.6839 × 101	1.9507 × 101	2.1683 × 101
CEC2017-F5	Ave	6.6346 × 102	6.6005 × 102	9.2263 × 102	6.3768 × 102	6.0763 × 102	7.0089 × 102	6.4685 × 102	6.9456 × 102	6.5199 × 102	5.5032 × 102	7.0294 × 102	5.5069 × 102
	Std	2.8412 × 101	1.1087 × 101	5.0202 × 101	3.1747 × 101	2.6293 × 101	3.6562 × 101	3.6132 × 101	6.4347 × 101	4.1530 × 101	1.2880 × 101	3.8668 × 101	5.7030 × 100
CEC2017-F6	Ave	6.4265 × 102	6.0020 × 102	6.8468 × 102	6.1800 × 102	6.0468 × 102	6.4141 × 102	6.0400 × 102	6.3414 × 102	6.3307 × 102	6.0002 × 102	6.0512 × 102	6.0002 × 102
	Std	6.7544 × 100	3.3332 × 10−2	1.2512 × 101	9.5387 × 100	3.3494 × 100	8.7573 × 100	2.6164 × 100	1.6120 × 101	7.8882 × 100	5.0463 × 10−2	1.1076 × 101	3.9405 × 10−3
CEC2017-F7	Ave	8.5336 × 102	8.9775 × 102	1.7035 × 103	9.3498 × 102	8.5201 × 102	1.0128 × 103	9.1083 × 102	1.0500 × 103	9.9828 × 102	8.4267 × 102	1.1533 × 103	7.7927 × 102
	Std	3.4396 × 101	1.2952 × 101	4.2565 × 102	5.7884 × 101	2.9480 × 101	6.2956 × 101	5.0413 × 101	8.6404 × 101	7.0075 × 101	6.4846 × 101	1.0611 × 102	6.0627 × 100
CEC2017-F8	Ave	9.2537 × 102	9.6325 × 102	1.1843 × 103	9.1301 × 102	9.0777 × 102	9.4550 × 102	9.3607 × 102	9.4631 × 102	9.0059 × 102	8.5591 × 102	9.4752 × 102	8.5067 × 102
	Std	2.0046 × 101	1.3117 × 101	5.9633 × 101	2.6529 × 101	3.1938 × 101	2.0088 × 101	2.7754 × 101	3.8017 × 101	2.0196 × 101	1.3911 × 101	3.3370 × 101	4.8853 × 100
CEC2017-F9	Ave	4.9499 × 103	2.9543 × 103	1.3736 × 104	2.5615 × 103	3.2582 × 103	3.5865 × 103	4.0118 × 103	2.9491 × 103	3.0616 × 103	9.1705 × 102	5.1922 × 103	9.1117 × 102
	Std	1.6756 × 103	4.7352 × 102	5.0031 × 103	6.1259 × 102	1.4596 × 103	6.3661 × 102	1.0385 × 103	1.9540 × 103	7.0184 × 102	5.1164 × 101	6.0269 × 102	3.5361 × 100
CEC2017-F10	Ave	4.6286 × 103	6.2787 × 103	8.7155 × 103	4.8841 × 103	4.2509 × 103	4.3686 × 103	4.4399 × 103	7.6164 × 103	5.0268 × 103	3.7501 × 103	5.1280 × 103	4.9902 × 103
	Std	5.3494 × 102	2.9869 × 102	3.2496 × 102	5.7151 × 102	7.1573 × 102	7.6318 × 102	4.2108 × 102	7.9836 × 102	6.9321 × 102	6.2513 × 102	7.2890 × 102	2.0310 × 102
CEC2017-F11	Ave	1.2023 × 103	2.1324 × 103	1.7500 × 104	1.2416 × 103	1.2697 × 103	1.1914 × 103	1.2320 × 103	1.2616 × 103	1.2968 × 103	1.1832 × 103	1.2050 × 103	1.1264 × 103
	Std	2.6720 × 101	5.8778 × 102	5.3388 × 103	5.4792 × 101	5.6311 × 101	2.2896 × 101	3.7530 × 101	6.2969 × 101	6.9534 × 101	4.4560 × 101	3.9367 × 101	4.7943 × 100
CEC2017-F12	Ave	1.4174 × 106	2.0326 × 107	8.2361 × 109	7.0403 × 104	2.8228 × 106	1.1903 × 106	3.2113 × 106	1.6032 × 105	1.1411 × 105	3.0352 × 105	2.1027 × 106	1.1091 × 105
	Std	9.0145 × 105	4.9822 × 106	2.5248 × 109	7.1271 × 104	2.0811 × 106	7.2505 × 105	2.4369 × 106	2.7781 × 105	1.4956 × 105	3.4748 × 105	1.2041 × 106	3.3399 × 104
CEC2017-F13	Ave	1.1673 × 104	2.7536 × 106	4.6224 × 109	1.5184 × 104	3.6319 × 104	3.1953 × 104	4.0360 × 104	2.2127 × 104	2.4929 × 104	1.9262 × 104	3.8921 × 104	1.7330 × 103
	Std	1.2461 × 104	1.2525 × 106	2.0861 × 109	1.5849 × 104	2.6247 × 104	1.3189 × 104	2.5888 × 104	2.4790 × 104	2.1338 × 104	1.5516 × 104	3.5663 × 104	2.6174 × 102
CEC2017-F14	Ave	2.3257 × 104	2.9279 × 105	4.2491 × 106	2.1860 × 103	9.2217 × 104	2.9715 × 103	9.7975 × 104	6.6545 × 103	1.2844 × 104	2.5579 × 104	5.9268 × 105	1.9434 × 103
	Std	2.5675 × 104	1.8285 × 105	2.8654 × 106	1.1218 × 103	6.4206 × 104	1.4294 × 103	1.0101 × 105	3.8896 × 103	1.9262 × 104	2.6324 × 104	4.9138 × 105	1.7636 × 102
CEC2017-F15	Ave	8.8699 × 103	5.9493 × 105	6.4747 × 108	5.9671 × 103	2.6885 × 104	1.4702 × 104	2.0736 × 104	1.4002 × 104	1.1723 × 104	6.0723 × 103	1.0316 × 104	1.5805 × 103
	Std	7.5002 × 103	3.5850 × 105	4.1663 × 108	5.1516 × 103	1.3097 × 104	2.3899 × 103	1.6214 × 104	1.2855 × 104	9.8326 × 103	6.4591 × 103	7.7011 × 103	2.8358 × 101
CEC2017-F16	Ave	2.5638 × 103	2.7569 × 103	5.0281 × 103	2.6121 × 103	2.5816 × 103	2.6515 × 103	2.7837 × 103	2.7998 × 103	2.7772 × 103	2.3321 × 103	2.8290 × 103	1.8298 × 103
	Std	2.9538 × 102	1.6566 × 102	3.9693 × 102	3.6101 × 102	2.8046 × 102	3.1730 × 102	3.9475 × 102	4.2604 × 102	4.1684 × 102	2.6960 × 102	3.4391 × 102	6.7495 × 101
CEC2017-F17	Ave	2.2383 × 103	2.1120 × 103	3.6755 × 103	2.1825 × 103	2.2806 × 103	2.2054 × 103	2.3780 × 103	2.1960 × 103	2.3431 × 103	2.0249 × 103	2.5131 × 103	1.7361 × 103
	Std	2.1736 × 102	8.0566 × 101	3.7540 × 102	2.5423 × 102	1.9734 × 102	2.1112 × 102	2.2025 × 102	1.9711 × 102	2.2654 × 102	1.6548 × 102	2.8430 × 102	7.1878 × 100
CEC2017-F18	Ave	3.3736 × 105	1.5004 × 106	5.9111 × 107	4.2719 × 104	1.2108 × 106	4.2890 × 104	1.3048 × 106	1.4157 × 105	1.6230 × 105	3.5633 × 105	9.7626 × 105	4.5015 × 104
	Std	2.8180 × 105	7.5808 × 105	2.7747 × 107	2.6496 × 104	1.0214 × 106	1.5062 × 104	1.2124 × 106	1.1756 × 105	1.7135 × 105	3.5166 × 105	7.7947 × 105	8.1985 × 103
CEC2017-F19	Ave	9.3421 × 103	3.7088 × 105	9.3614 × 108	2.9800 × 103	3.0316 × 104	8.1345 × 103	2.2091 × 104	1.0634 × 104	1.3009 × 104	6.7259 × 103	1.4295 × 104	2.3819 × 103
	Std	7.6001 × 103	1.9634 × 105	5.3082 × 108	1.8074 × 103	2.1166 × 104	3.3370 × 103	2.1039 × 104	1.3085 × 104	1.4250 × 104	4.8956 × 103	1.7560 × 104	2.4763 × 102
CEC2017-F20	Ave	2.6314 × 103	2.3992 × 103	3.1845 × 103	2.5061 × 103	2.4874 × 103	2.4059 × 103	2.5657 × 103	2.6359 × 103	2.5337 × 103	2.3036 × 103	2.7911 × 103	2.0372 × 103
	Std	1.6910 × 102	8.0345 × 101	1.4656 × 102	1.9254 × 102	2.0422 × 102	1.3673 × 102	2.1435 × 102	1.4343 × 102	2.2724 × 102	1.6230 × 102	2.5998 × 102	1.0239 × 101
CEC2017-F21	Ave	2.4657 × 103	2.4601 × 103	2.7018 × 103	2.4108 × 103	2.4128 × 103	2.4252 × 103	2.4305 × 103	2.4553 × 103	2.4142 × 103	2.3556 × 103	2.3920 × 103	2.3383 × 103
	Std	3.6138 × 101	1.3582 × 101	5.0995 × 101	3.4243 × 101	2.9907 × 101	2.9626 × 101	3.9639 × 101	6.4532 × 101	3.0564 × 101	1.8390 × 101	3.6661 × 101	4.9507 × 100
CEC2017-F22	Ave	4.2793 × 103	4.5646 × 103	8.8582 × 103	5.2014 × 103	5.9817 × 103	3.5680 × 103	4.8199 × 103	5.0138 × 103	4.7745 × 103	2.5768 × 103	3.4006 × 103	2.3001 × 103
	Std	2.3321 × 103	1.7922 × 103	2.1255 × 103	2.0406 × 103	9.2543 × 102	2.0239 × 103	1.4523 × 103	3.3967 × 103	2.3163 × 103	8.4695 × 102	2.0666 × 103	1.7034 × 10−2
CEC2017-F23	Ave	3.1005 × 103	2.8023 × 103	3.1768 × 103	2.7989 × 103	2.7599 × 103	2.7740 × 103	2.7796 × 103	2.8328 × 103	2.8274 × 103	2.7032 × 103	2.7724 × 103	2.6878 × 103
	Std	1.2837 × 102	1.1996 × 101	5.7420 × 101	4.5631 × 101	3.1447 × 101	3.2499 × 101	2.8019 × 101	8.5767 × 101	5.7959 × 101	1.3899 × 101	5.9955 × 101	6.2762 × 100
CEC2017-F24	Ave	3.2253 × 103	3.0185 × 103	3.3020 × 103	2.9601 × 103	2.9378 × 103	2.9311 × 103	3.0021 × 103	2.9982 × 103	3.0005 × 103	2.8780 × 103	2.9362 × 103	2.8715 × 103
	Std	9.8767 × 101	1.3999 × 101	4.1354 × 101	4.1460 × 101	3.1663 × 101	3.7250 × 101	5.6470 × 101	7.7371 × 101	6.2220 × 101	1.4231 × 101	6.3384 × 101	6.5233 × 100
CEC2017-F25	Ave	2.8927 × 103	2.9108 × 103	6.0741 × 103	2.8991 × 103	2.8936 × 103	2.9044 × 103	2.8925 × 103	2.8882 × 103	2.8993 × 103	2.8892 × 103	2.9051 × 103	2.8843 × 103
	Std	2.1223 × 101	7.5830 × 100	1.6980 × 103	1.8412 × 101	1.3948 × 101	1.8464 × 101	1.2368 × 101	5.2201 × 100	1.8445 × 101	9.5636 × 100	2.4060 × 101	4.1499 × 10−1
CEC2017-F26	Ave	5.3736 × 103	5.1805 × 103	9.3521 × 103	5.3910 × 103	4.9308 × 103	5.5109 × 103	5.0364 × 103	5.0107 × 103	5.0633 × 103	4.1771 × 103	5.6038 × 103	2.8004 × 103
	Std	2.1103 × 103	1.5784 × 102	6.8797 × 102	1.1285 × 103	2.7695 × 102	1.4968 × 103	4.8820 × 102	1.1286 × 103	1.2557 × 103	2.8482 × 102	2.0515 × 103	1.6392 × 10−1
CEC2017-F27	Ave	3.3302 × 103	3.2221 × 103	3.6192 × 103	3.2572 × 103	3.2198 × 103	3.2709 × 103	3.2274 × 103	3.2780 × 103	3.2823 × 103	3.2194 × 103	3.2692 × 103	3.2074 × 103
	Std	1.8184 × 102	2.9744 × 100	1.0274 × 102	2.8237 × 101	1.1957 × 101	2.5004 × 101	1.4308 × 101	6.0081 × 101	4.5551 × 101	1.2094 × 101	3.4493 × 101	5.2105 × 100
CEC2017-F28	Ave	3.2193 × 103	3.2994 × 103	6.9586 × 103	3.1987 × 103	3.2467 × 103	3.1952 × 103	3.2464 × 103	3.1995 × 103	3.2201 × 103	3.2207 × 103	3.2305 × 103	3.2064 × 103
	Std	2.2889 × 101	1.3664 × 101	1.1750 × 103	3.3840 × 101	4.3856 × 101	1.8957 × 101	2.5552 × 101	3.5108 × 101	2.4178 × 101	2.9314 × 101	2.1502 × 101	2.8615 × 100
CEC2017-F29	Ave	4.0811 × 103	3.8139 × 103	5.8521 × 103	4.0707 × 103	3.9303 × 103	4.0748 × 103	3.8321 × 103	4.0343 × 103	4.1535 × 103	3.5952 × 103	4.3224 × 103	3.3417 × 103
	Std	2.9124 × 102	1.2463 × 102	3.1995 × 102	2.5816 × 102	2.4981 × 102	2.5450 × 102	2.2434 × 102	2.4002 × 102	2.6170 × 102	1.5194 × 102	2.4104 × 102	2.1221 × 101
CEC2017-F30	Ave	3.2676 × 104	2.3290 × 105	7.6592 × 108	1.0712 × 104	2.6943 × 104	4.0812 × 104	1.2930 × 105	1.1650 × 104	1.0200 × 104	8.2822 × 103	4.7599 × 104	6.1557 × 103
	Std	1.0538 × 104	9.9758 × 104	3.3649 × 108	4.0500 × 103	1.1573 × 104	1.5089 × 104	1.5075 × 105	4.3549 × 103	3.7701 × 103	3.5580 × 103	7.0351 × 104	2.3694 × 102

and

Table 5. Performance Comparison of Algorithms on the 50-Dimensional CEC 2017 Benchmark Functions.

ID	Items	PSO	DE	CMAES	INFO	SMA	RUN	HGS	DOA	GWCA	SAO	IVYA	MEIVYA
CEC2017-F1	Ave	8.9984 × 106	2.9543 × 108	1.0849 × 1011	6.3721 × 103	1.7869 × 105	1.8958 × 104	3.7784 × 107	4.8277 × 104	5.7542 × 103	3.6434 × 103	6.8340 × 108	3.4827 × 105
	Std	3.4965 × 106	6.8153 × 107	3.5737 × 1010	7.9193 × 103	6.8704 × 104	6.7792 × 103	1.1827 × 108	8.2938 × 104	7.6591 × 103	4.9119 × 103	1.6236 × 109	6.1778 × 104
CEC2017-F2	Ave	2.4421 × 1023	1.9197 × 1059	6.7291 × 1076	1.9384 × 1046	1.0838 × 1026	4.4081 × 1044	3.6962 × 1041	4.3962 × 1034	5.2229 × 1060	6.0302 × 1046	6.7709 × 1084	2.7623 × 1021
	Std	6.7860 × 1023	2.7897 × 1059	2.0018 × 1077	9.3931 × 1046	4.8145 × 1026	1.8543 × 1045	1.6770 × 1042	2.3969 × 1035	2.7616 × 1061	3.1865 × 1047	2.1048 × 1085	2.2033 × 1021
CEC2017-F3	Ave	7.5568 × 104	3.1062 × 105	3.6272 × 105	2.6060 × 104	3.3878 × 104	1.0137 × 104	7.1517 × 104	1.5969 × 105	6.1465 × 104	2.3383 × 105	2.0273 × 105	1.0208 × 105
	Std	1.6207 × 104	3.3580 × 104	7.4296 × 104	8.6544 × 103	1.5416 × 104	3.1403 × 103	2.2889 × 104	5.0240 × 104	2.8255 × 104	5.1726 × 104	6.0221 × 104	8.1244 × 103
CEC2017-F4	Ave	5.3757 × 102	7.8582 × 102	2.3643 × 104	5.4275 × 102	5.7893 × 102	5.1637 × 102	6.3012 × 102	5.3909 × 102	5.8739 × 102	5.4922 × 102	5.6188 × 102	4.7981 × 102
	Std	4.9575 × 101	3.7430 × 101	7.4373 × 103	5.5391 × 101	5.4426 × 101	5.8654 × 101	5.9058 × 101	5.7311 × 101	4.7514 × 101	5.0319 × 101	1.5588 × 102	4.6427 × 100
CEC2017-F5	Ave	7.8344 × 102	8.9931 × 102	1.2723 × 103	8.0087 × 102	7.5574 × 102	8.3378 × 102	7.9170 × 102	8.5063 × 102	7.7265 × 102	6.2738 × 102	8.4996 × 102	6.5100 × 102
	Std	3.5683 × 101	1.3974 × 101	7.2215 × 101	5.5366 × 101	4.5242 × 101	3.3300 × 101	3.7276 × 101	5.7278 × 101	4.6633 × 101	6.7431 × 101	3.3698 × 101	1.4314 × 101
CEC2017-F6	Ave	6.5249 × 102	6.0413 × 102	6.9565 × 102	6.3388 × 102	6.3045 × 102	6.5531 × 102	6.1601 × 102	6.5677 × 102	6.4544 × 102	6.0049 × 102	6.2127 × 102	6.0024 × 102
	Std	6.7721 × 100	4.5898 × 10−1	9.4430 × 100	8.0421 × 100	1.5056 × 101	4.3858 × 100	6.1314 × 100	1.2552 × 101	8.6961 × 100	3.2123 × 10−1	1.9512 × 101	2.8140 × 10−2
CEC2017-F7	Ave	1.0918 × 103	1.1877 × 103	1.9948 × 103	1.2858 × 103	1.0572 × 103	1.4065 × 103	1.2050 × 103	1.5277 × 103	1.4258 × 103	1.1679 × 103	1.5733 × 103	8.9068 × 102
	Std	7.8698 × 101	2.2120 × 101	4.2690 × 102	1.3331 × 102	6.3059 × 101	1.2166 × 102	1.6826 × 102	1.3182 × 102	1.4927 × 102	5.0014 × 101	1.6425 × 102	1.1980 × 101
CEC2017-F8	Ave	1.0895 × 103	1.1977 × 103	1.5607 × 103	1.0954 × 103	1.0503 × 103	1.1290 × 103	1.0583 × 103	1.1576 × 103	1.0782 × 103	9.1809 × 102	1.1745 × 103	9.5834 × 102
	Std	2.8345 × 101	1.5351 × 101	9.2648 × 101	5.4973 × 101	4.2418 × 101	3.5931 × 101	4.7676 × 101	6.7012 × 101	6.0036 × 101	2.7068 × 101	3.3831 × 101	1.1839 × 101
CEC2017-F9	Ave	2.1769 × 104	1.4101 × 104	4.0023 × 104	7.1040 × 103	1.2141 × 104	1.0317 × 104	1.2581 × 104	1.5807 × 104	9.2882 × 103	1.2255 × 103	1.2856 × 104	1.0270 × 103
	Std	4.6985 × 103	1.9748 × 103	8.0958 × 103	1.9936 × 103	4.2147 × 103	1.6064 × 103	1.9298 × 103	6.1410 × 103	1.8850 × 103	8.7165 × 102	8.1432 × 102	2.6445 × 101
CEC2017-F10	Ave	7.4316 × 103	1.2015 × 104	1.5465 × 104	8.4236 × 103	7.4296 × 103	7.4792 × 103	6.8925 × 103	1.4285 × 104	8.2107 × 103	6.7148 × 103	7.8806 × 103	9.1430 × 103
	Std	9.5097 × 102	4.2809 × 102	2.6300 × 102	1.0121 × 103	8.9927 × 102	1.1821 × 103	7.2537 × 102	7.0350 × 102	1.1526 × 103	1.2369 × 103	9.4104 × 102	2.2766 × 102
CEC2017-F11	Ave	1.3233 × 103	7.4337 × 103	4.6404 × 104	1.3226 × 103	1.3880 × 103	1.2623 × 103	1.3794 × 103	1.4367 × 103	1.4341 × 103	1.3829 × 103	1.9123 × 103	1.2303 × 103
	Std	4.3633 × 101	2.2378 × 103	1.2002 × 104	4.8502 × 101	6.7231 × 101	3.0172 × 101	7.7350 × 101	1.2369 × 102	8.9409 × 101	1.2015 × 102	3.7591 × 102	1.8956 × 101
CEC2017-F12	Ave	1.2563 × 107	3.8404 × 108	4.2854 × 1010	1.5131 × 106	2.0884 × 107	7.2915 × 106	3.9622 × 107	2.5833 × 106	2.1796 × 106	2.8155 × 106	6.6278 × 106	1.3490 × 106
	Std	6.3162 × 106	7.7068 × 107	9.0261 × 109	8.4278 × 105	7.5676 × 106	2.5387 × 106	2.8618 × 107	1.5778 × 106	2.6816 × 106	1.7576 × 106	2.7046 × 106	2.8637 × 105
CEC2017-F13	Ave	2.9159 × 104	1.2130 × 107	1.7976 × 1010	1.1038 × 104	6.9630 × 104	2.6591 × 104	3.3740 × 104	1.2304 × 104	1.1461 × 104	5.7210 × 103	2.1216 × 104	2.3204 × 103
	Std	1.1574 × 104	4.6787 × 106	5.4850 × 109	9.2785 × 103	2.0750 × 104	7.3534 × 103	1.5296 × 104	1.0596 × 104	8.9029 × 103	6.6140 × 103	8.2431 × 103	2.5103 × 102
CEC2017-F14	Ave	1.4449 × 105	2.3607 × 106	2.0093 × 107	1.6638 × 104	4.5941 × 105	2.0721 × 104	6.9501 × 105	7.8916 × 104	1.0451 × 105	8.6554 × 104	1.3287 × 106	1.2609 × 104
	Std	1.2191 × 105	1.0908 × 106	1.1109 × 107	1.4714 × 104	2.6268 × 105	1.5330 × 104	4.1285 × 105	7.0498 × 104	1.0220 × 105	7.4793 × 104	7.7277 × 105	3.9023 × 103
CEC2017-F15	Ave	7.6415 × 103	1.4580 × 106	4.6456 × 109	1.3080 × 104	2.6185 × 104	2.3937 × 104	1.9091 × 104	1.5067 × 104	8.3177 × 103	1.2752 × 104	2.2320 × 107	2.1881 × 103
	Std	4.7806 × 103	7.9927 × 105	1.8192 × 109	7.5906 × 103	1.1003 × 104	4.0775 × 103	8.5867 × 103	7.6895 × 103	7.6517 × 103	5.9353 × 103	1.2215 × 108	4.5840 × 102
CEC2017-F16	Ave	3.2603 × 103	4.1009 × 103	7.5955 × 103	3.4889 × 103	3.3632 × 103	3.3192 × 103	3.9561 × 103	3.6860 × 103	3.3770 × 103	3.0794 × 103	3.4369 × 103	2.2904 × 103
	Std	3.4543 × 102	1.8754 × 102	5.3193 × 102	4.1179 × 102	3.9218 × 102	3.4512 × 102	4.7828 × 102	5.3948 × 102	3.9821 × 102	5.0816 × 102	4.4560 × 102	9.7071 × 101
CEC2017-F17	Ave	3.3167 × 103	3.3494 × 103	8.4638 × 103	3.2500 × 103	3.3443 × 103	3.1962 × 103	3.4019 × 103	3.3316 × 103	3.3684 × 103	2.8641 × 103	3.4922 × 103	2.1682 × 103
	Std	3.4393 × 102	1.6786 × 102	5.2872 × 103	3.4157 × 102	4.0012 × 102	3.6608 × 102	3.1672 × 102	2.9662 × 102	3.5024 × 102	4.0751 × 102	3.5093 × 102	9.8894 × 101
CEC2017-F18	Ave	1.5329 × 106	8.5829 × 106	1.4644 × 108	1.4648 × 105	3.2071 × 106	9.0158 × 104	3.9299 × 106	4.3374 × 105	2.6936 × 105	1.0521 × 106	2.6610 × 106	1.7602 × 105
	Std	1.1606 × 106	3.7927 × 106	7.2975 × 107	7.1832 × 104	2.2277 × 106	5.2015 × 104	2.6937 × 106	3.9587 × 105	2.1786 × 105	7.7898 × 105	1.2558 × 106	6.3034 × 104
CEC2017-F19	Ave	1.7191 × 104	4.3339 × 105	1.9233 × 109	2.0315 × 104	1.9770 × 104	7.8100 × 104	2.5210 × 104	1.9997 × 104	1.4383 × 104	2.1494 × 104	2.2084 × 104	7.6682 × 103
	Std	8.1128 × 103	2.0717 × 105	1.1659 × 109	1.0053 × 104	1.7100 × 104	3.6743 × 104	1.7125 × 104	1.3117 × 104	9.0261 × 103	1.1885 × 104	1.4037 × 104	1.7757 × 103
CEC2017-F20	Ave	3.2297 × 103	3.3133 × 103	4.3741 × 103	3.3197 × 103	3.1329 × 103	3.0831 × 103	3.1625 × 103	3.5423 × 103	3.5201 × 103	2.7686 × 103	3.3427 × 103	2.1891 × 103
	Std	2.4324 × 102	1.4494 × 102	2.5506 × 102	2.7970 × 102	2.8936 × 102	2.7662 × 102	3.6140 × 102	3.7406 × 102	3.9347 × 102	3.8141 × 102	2.8537 × 102	4.2389 × 101
CEC2017-F21	Ave	2.6647 × 103	2.7092 × 103	3.0750 × 103	2.5727 × 103	2.5357 × 103	2.5936 × 103	2.5694 × 103	2.6401 × 103	2.5816 × 103	2.4247 × 103	2.5327 × 103	2.4175 × 103
	Std	5.4818 × 101	1.5113 × 101	4.7494 × 101	6.3327 × 101	5.6604 × 101	5.2670 × 101	4.7706 × 101	1.0339 × 102	4.4368 × 101	2.4289 × 101	7.4168 × 101	7.1099 × 100
CEC2017-F22	Ave	9.8517 × 103	1.3667 × 104	1.7119 × 104	9.6250 × 103	8.8785 × 103	9.5162 × 103	8.7779 × 103	1.5183 × 104	9.7262 × 103	7.8068 × 103	1.0243 × 104	2.3052 × 103
	Std	1.1223 × 103	6.9575 × 102	4.0474 × 102	9.9468 × 102	8.0924 × 102	1.5198 × 103	6.2053 × 102	2.7277 × 103	8.3917 × 102	2.2021 × 103	9.0975 × 102	9.2478 × 10−1
CEC2017-F23	Ave	3.6968 × 103	3.1170 × 103	3.7754 × 103	3.0803 × 103	2.9567 × 103	3.1007 × 103	3.0052 × 103	3.1639 × 103	3.2137 × 103	2.8497 × 103	2.9551 × 103	2.8407 × 103
	Std	1.8550 × 102	1.6172 × 101	1.1566 × 102	6.8373 × 101	4.5273 × 101	7.2691 × 101	5.2899 × 101	8.9106 × 101	1.0996 × 102	2.8760 × 101	8.5255 × 101	1.1595 × 101
CEC2017-F24	Ave	3.5284 × 103	3.3517 × 103	3.9029 × 103	3.2308 × 103	3.1218 × 103	3.1578 × 103	3.3114 × 103	3.3100 × 103	3.3582 × 103	3.0269 × 103	3.1240 × 103	3.0415 × 103
	Std	1.4587 × 102	1.7691 × 101	8.2390 × 101	9.1082 × 101	4.7547 × 101	6.5756 × 101	1.1453 × 102	9.7309 × 101	9.9111 × 101	6.0363 × 101	9.6533 × 101	1.2604 × 101
CEC2017-F25	Ave	3.0029 × 103	3.2542 × 103	1.5433 × 104	3.0731 × 103	3.0510 × 103	3.1033 × 103	3.0805 × 103	3.0619 × 103	3.1054 × 103	3.0549 × 103	3.1078 × 103	3.0944 × 103
	Std	3.8120 × 101	3.2068 × 101	8.5713 × 103	2.8341 × 101	2.9756 × 101	2.0121 × 101	2.7714 × 101	2.3651 × 101	4.1471 × 101	3.0091 × 101	2.5744 × 101	1.4670 × 101
CEC2017-F26	Ave	7.2694 × 103	7.7066 × 103	1.4788 × 104	8.3675 × 103	5.1381 × 103	1.0962 × 104	5.7548 × 103	5.3413 × 103	8.5667 × 103	4.8385 × 103	9.6538 × 103	2.9740 × 103
	Std	3.6478 × 103	2.0636 × 102	1.1829 × 103	2.0577 × 103	1.6678 × 103	1.8956 × 103	1.8039 × 103	3.5908 × 103	1.4171 × 103	4.5454 × 102	2.3402 × 103	2.2250 × 101
CEC2017-F27	Ave	3.9228 × 103	3.4795 × 103	4.8319 × 103	3.6336 × 103	3.4366 × 103	3.7048 × 103	3.4762 × 103	3.5761 × 103	3.7323 × 103	3.3631 × 103	3.7055 × 103	3.3419 × 103
	Std	7.8472 × 102	2.7277 × 101	1.9108 × 102	1.5058 × 102	7.1314 × 101	1.2206 × 102	1.0088 × 102	1.5753 × 102	1.6882 × 102	7.9869 × 101	1.8419 × 102	1.4553 × 101
CEC2017-F28	Ave	3.2998 × 103	4.7483 × 103	1.2148 × 104	3.3228 × 103	3.3357 × 103	3.3435 × 103	3.3871 × 103	3.3358 × 103	3.3831 × 103	3.2916 × 103	3.3515 × 103	3.3577 × 103
	Std	3.2989 × 101	4.6069 × 102	2.1705 × 103	2.7016 × 101	2.5056 × 101	3.5342 × 101	5.8657 × 101	3.4901 × 101	5.1507 × 101	2.1536 × 101	4.5743 × 101	2.0320 × 101
CEC2017-F29	Ave	5.0415 × 103	4.7103 × 103	1.5821 × 104	5.1679 × 103	4.6073 × 103	5.1878 × 103	4.4403 × 103	4.9308 × 103	5.0274 × 103	3.9418 × 103	4.8221 × 103	3.4953 × 103
	Std	3.5704 × 102	2.0684 × 102	4.4297 × 103	4.9179 × 102	3.9560 × 102	4.1225 × 102	3.1466 × 102	4.5425 × 102	4.3740 × 102	2.8709 × 102	4.7151 × 102	7.0603 × 101
CEC2017-F30	Ave	4.7210 × 106	7.7823 × 106	3.4226 × 109	8.7590 × 105	3.1455 × 106	7.2455 × 106	2.1176 × 106	1.6488 × 106	1.1577 × 106	1.0564 × 106	1.8579 × 106	7.2261 × 105
	Std	1.7439 × 106	2.1405 × 106	1.2985 × 109	1.4373 × 105	1.3100 × 106	1.7740 × 106	9.0991 × 105	7.8624 × 105	4.1352 × 105	3.8581 × 105	6.5891 × 105	3.4857 × 104

, where “ave” and “std” denote the average value and standard deviation obtained from 30 independent runs, respectively. To provide a clearer view of the optimization process, Figure 4 presents the convergence trajectories of all 12 algorithms, highlighting their search dynamics over iterations. In addition, Figure 5 illustrates box plots based on the results of 30 independent trials, which offer a visual comparison of performance distribution, algorithm robustness, and the influence of randomness. We have highlighted the optimal solutions obtained by the algorithm for each function to help readers identify the key information more easily.

Figure 4 illustrates the convergence behavior of each algorithm on the CEC2017 test functions. Overall, MEIVYA exhibits a faster descent rate and lower final convergence value on most test functions, indicating its ability to quickly approach the optimal region with fewer iterations and maintain stable local optimization capabilities in the later stages. Compared to algorithms such as PSO, DE, CMAES, and SMA, MEIVYA’s convergence curve typically reaches a stable phase earlier and has higher final convergence accuracy, demonstrating a more effective balance between global and local search. Particularly on complex multimodal and mixed combinatorial functions, where some algorithms exhibit premature convergence, prolonged stagnation, or late-stage oscillations, MEIVYA continues to descent and obtain better solutions, indicating a strong ability to escape local optima. Furthermore, MEIVYA’s convergence curve is generally smooth without drastic fluctuations, reflecting a more stable search process and less susceptibility to random factors. This is mainly due to the fact that chaotic initialization improves the diversity of the initial population, the adaptive growth rate adjustment mechanism enhances the search capability at different stages, and the elite-guided collaborative search strategy further improves the efficiency of information sharing among the population. Overall, Figure 4 fully demonstrates that MEIVYA has strong convergence speed, high solution accuracy, and excellent stability on the CEC2017 test set.

Figure 5 uses box plots to further illustrate the distribution of results for each algorithm on the CEC2017 benchmark function. It is clear from the figure that MEIVYA has a narrower box and shorter upper and lower bands on most test functions, and its median is usually in a better position, indicating that it can obtain more stable and consistent optimization results in multiple independent runs. In contrast, some of the comparison algorithms have larger box ranges and more outliers, indicating that these algorithms are susceptible to fluctuations in random initialization and the search process, leading to unstable results. On some complex multimodal functions and mixed combination functions, MEIVYA’s box plot is not only lower overall but also more concentrated, indicating that it not only obtains better target values but also has stronger robustness and resistance to random disturbances. Furthermore, while traditional algorithms such as PSO, DE, and CMAES can achieve good results on some functions, their box plots are generally wider, indicating larger performance fluctuations; while MEIVYA maintains a smaller standard deviation and higher consistency on most functions. Overall, Figure 5 shows that MEIVYA not only has excellent solving capabilities on the CEC2017 test set, but also exhibits strong stability and robustness, and its overall performance is better than most of the comparison algorithms.

As shown in Table 4 and Table 5, the solution difficulty of all algorithms increases with the problem dimension from 30 to 50. Most algorithms exhibit increased average value, higher standard deviation, and decreased stability in high-dimensional cases, especially with complex multimodal functions, mixed functions, and combined functions. However, MEIVYA demonstrates a strong competitive advantage in both dimension settings. In the 30-dimensional CEC2017 test set, MEIVYA achieved optimal or near-optimal results for most functions. Particularly for some complex functions, its average value is significantly lower than other algorithms, and its standard deviation is smaller, indicating that it not only has high solution accuracy but also maintains stable and consistent performance in multiple independent runs. At the same time, on some simple unimodal functions, MEIVYA can also quickly converge to the vicinity of the theoretical optimum, indicating its strong local exploitation capability.

In the more challenging 50-dimensional test set, although the performance of most algorithms showed significant degradation, MEIVYA still maintained a low objective function value and a small standard deviation, indicating its strong adaptability to dimensionality growth. Especially on high-dimensional complex multimodal functions and mixed combinatorial functions, MEIVYA maintained better convergence accuracy and more stable results compared to algorithms such as PSO, DE, CMAES, and SMA, while some of the compared algorithms showed significant performance degradation, even orders of magnitude difference. Furthermore, MEIVYA generally had a smaller standard deviation for most functions, further verifying its consistency and robustness in repeated experiments. Overall, the results in Table 4 and Table 5 show that MEIVYA not only obtains better solutions in low-dimensional cases but also exhibits good search ability, stability, and generalization performance in high-dimensional complex problems, demonstrating strong comprehensive optimization capabilities.

3.4. Friedman Mean Rank Test

The Friedman mean rank test is a non-parametric statistical method widely used to compare the performance of multiple algorithms over multiple benchmark problems. Unlike parametric tests, it does not assume normal distribution of the data, making it particularly suitable for optimization experiments where performance results may vary irregularly across different test functions. In this test, each algorithm is ranked on every problem according to its performance, and the average rank across all problems is then calculated. A lower mean rank indicates better overall performance. The Friedman test helps determine whether there are statistically significant differences among the compared algorithms and provides a reliable basis for comprehensive performance evaluation [35,36]. The experimental results of the Friedman mean rank test are illustrated in Figure 6. The “*” in the figure indicate the optimal values obtained by all algorithms across each dimension.

Figure 6 shows the average ranking of each algorithm based on the Friedman Mean Rank Test. It can be seen that MEIVYA achieved the best or near-best average ranking among all compared algorithms, indicating that it maintains stable and excellent overall performance under different test functions and dimensional conditions. Since the Friedman Mean Rank Test ranks algorithms based on overall performance across all test problems, a lower average ranking means that the algorithm not only performs well on individual functions but also maintains high solution quality and stability on most functions. In contrast, while some traditional algorithms achieve good results on certain simple functions, they are prone to performance degradation on complex multimodal functions and high-dimensional combined functions, resulting in relatively lower overall rankings. MEIVYA’s superior average ranking demonstrates that the proposed chaotic initialization, adaptive growth rate adjustment, and elite-guided collaborative search strategies effectively enhance the algorithm’s adaptability to different problems, achieving a good balance between global search capability, local exploitation capability, and stability. Overall, the Friedman average ranking results in Figure 6 further validate that MEIVYA outperforms most of the comparison algorithms in terms of comprehensive performance on the CEC2014 and CEC2017 test sets, demonstrating strong robustness and universality.

3.5. Wilcoxon Signed-Rank Test

The Wilcoxon signed-rank test is an important statistical tool for evaluating performance differences between metaheuristic algorithms. Since the results of heuristic and evolutionary algorithms typically fluctuate across multiple independent runs and often do not satisfy the assumption of a normal distribution, traditional parametric tests may not be suitable for reliable comparisons. In contrast, the Wilcoxon signed-rank test is a non-parametric method that can effectively determine whether the performance difference between two algorithms is statistically significant based on paired experimental results. Therefore, it provides a more rigorous and convincing basis for performance comparisons and helps avoid biased conclusions based solely on averages or single-run results. For this reason, the Wilcoxon signed-rank test has been widely used in the statistical analysis of optimization algorithms. In this section, we performed the Wilcoxon signed-rank test on MEIVYA, and the experimental results are shown below.

Table 6. Experimental results of the comparison algorithms.

Algorithm	CEC2014 (Dim = 30)	CEC2014 (Dim = 50)	CEC2017 (Dim = 30)	CEC2017 (Dim = 50)
PSO (W/T/L)	23/7/0	23/7/0	25/5/0	26/4/0
DE (W/T/L)	24/6/0	24/3/3	28/2/0	29/0/1
CMAES (W/T/L)	29/1/0	29/0/1	30/0/0	30/0/0
INFO (W/T/L)	25/0/5	21/9/0	26/4/0	25/5/0
SMA (W/T/L)	21/9/0	23/7/0	25/5/0	22/8/0
RUN (W/T/L)	26/4/0	25/5/0	27/3/0	29/1/0
HGS (W/T/L)	22/0/8	24/6/0	22/8/0	23/7/0
DOA (W/T/L)	23/0/7	21/9/0	23/7/0	22/8/0
GWCA (+/=/−)	26/4/0	23/7/0	22/8/0	23/7/0
SAO (W/T/L)	21/9/0	22/8/0	21/9/0	19/11/0
IVYA (W/T/L)	22/8/0	22/8/0	25/5/0	25/5/0

presents the statistical analysis results based on the Wilcoxon signed-rank test, used to examine whether the performance difference between MEIVYA and other comparative algorithms is statistically significant. The results show that MEIVYA obtained significantly more “W” results and fewer “L” results in most comparisons, indicating its superiority over other algorithms on most test functions. Particularly when compared with traditional or classic algorithms such as PSO, DE, CMAES, SMA, and HGS, MEIVYA exhibits a more significant advantage, demonstrating its strong competitiveness in terms of solution accuracy and stability. Furthermore, most p-values in the table are less than 0.05, and some even reach a more stringent significance level, indicating that MEIVYA’s performance improvement over other algorithms is not due to random factors but has a statistically significant advantage. Although MEIVYA still exhibits “T” or “L” results on a few functions when compared with some high-performing algorithms, its overall W/T/L results remain superior, indicating that its comprehensive performance is more stable and reliable. Overall, the Wilcoxon signed-rank test results in Table 6 further validate MEIVYA’s superiority and consistency across different test functions, and also demonstrate that the proposed multi-strategy improvement mechanism can effectively enhance the algorithm’s global optimization capability and robustness.

As shown in

Table A1. P_holm-values for various algorithms on the CEC 2014 (dim = 30).

Item	PSO	DE	CMAES	INFO	SMA	RUN	HGS	DOA	GWCA	SAO	IVYA
F1	2.5697 × 10−2	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.0000 × 100	1.9078 × 10−5	1.0000 × 100	1.9078 × 10−5	1.9411 × 10−5	1.9078 × 10−5	4.0772 × 10−1
F2	1.9078 × 10−5	1.0511 × 10−1	1.9078 × 10−5	1.9078 × 10−5	1.0511 × 10−1	1.0511 × 10−1	8.1302 × 10−1	1.9078 × 10−5	1.3237 × 10−3	1.0511 × 10−1	2.4089 × 10−1
F3	1.4897 × 10−4	1.1048 × 10−1	1.9078 × 10−5	1.9078 × 10−5	1.1048 × 10−1	1.9078 × 10−5	8.1607 × 10−5	1.4704 × 10−1	8.1972 × 10−5	1.9078 × 10−5	3.3104 × 10−5
F4	1.0139 × 10−4	1.9078 × 10−5	1.9078 × 10−5	7.7688 × 10−4	6.1766 × 10−1	9.5163 × 10−4	1.0000 × 100	3.3196 × 10−3	2.7343 × 10−2	1.0454 × 10−3	1.0000 × 100
F5	1.9078 × 10−5	1.1748 × 10−2	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	7.4485 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5
F6	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9422 × 10−4	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9422 × 10−4
F7	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	2.2102 × 10−1	1.9078 × 10−5	1.7889 × 10−3	1.7061 × 10−4	3.1599 × 10−4	1.6782 × 10−3	1.9078 × 10−5	1.6782 × 10−3
F8	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	7.4485 × 10−5	1.9078 × 10−5	4.4807 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.3059 × 10−1	1.9078 × 10−5
F9	1.9411 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.5921 × 10−4	1.1413 × 10−3	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.1413 × 10−3	7.1954 × 10−5	1.9078 × 10−5
F10	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.7377 × 10−4	9.7539 × 10−1	1.7377 × 10−4	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5
F11	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	2.4563 × 10−4	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	2.4563 × 10−4	1.9078 × 10−5	8.7297 × 10−3
F12	1.6586 × 10−3	1.5043 × 10−1	1.9078 × 10−5	1.9078 × 10−5	2.7175 × 10−5	6.1478 × 10−5	1.9078 × 10−5	1.9078 × 10−5	2.4626 × 10−3	2.6230 × 10−1	3.8599 × 10−3
F13	1.0454 × 10−3	1.9078 × 10−5	1.9078 × 10−5	5.5281 × 10−3	1.9078 × 10−5	1.6586 × 10−3	1.9078 × 10−5	6.3207 × 10−2	6.7654 × 10−3	3.0000 × 10−5	6.3207 × 10−2
F14	1.1617 × 10−1	1.9078 × 10−5	1.9078 × 10−5	6.2263 × 10−2	1.9078 × 10−5	4.2942 × 10−3	1.9078 × 10−5	1.5286 × 10−1	1.1617 × 10−1	1.9078 × 10−5	6.2263 × 10−2
F15	5.0729 × 10−1	1.1787 × 10−2	1.9078 × 10−5	1.2074 × 10−2	3.9175 × 10−3	1.9078 × 10−5	9.0993 × 10−1	5.2777 × 10−2	1.9078 × 10−5	1.7129 × 10−1	1.0387 × 10−2
F16	2.0358 × 10−4	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.0000 × 100	1.0000 × 100	1.0000 × 100	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.0000 × 100
F17	9.0993 × 10−1	1.9078 × 10−5	1.9078 × 10−5	6.1527 × 10−5	6.1527 × 10−5	1.9078 × 10−5	7.4285 × 10−5	4.0939 × 10−4	1.9554 × 10−1	4.1924 × 10−2	1.3269 × 10−3
F18	1.0000 × 100	1.9078 × 10−5	1.9078 × 10−5	2.9918 × 10−1	6.7729 × 10−5	8.5343 × 10−1	3.4940 × 10−5	1.3531 × 10−2	7.2498 × 10−3	1.0000 × 100	1.0000 × 100
F19	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.5325 × 10−3	2.2551 × 10−3	1.9078 × 10−5	1.5325 × 10−3	1.9078 × 10−5	1.9078 × 10−5	1.5325 × 10−3	1.5325 × 10−3
F20	9.4549 × 10−4	3.1854 × 10−3	1.9078 × 10−5	1.9078 × 10−5	9.4549 × 10−4	1.9078 × 10−5	3.8035 × 10−5	3.8723 × 10−2	9.4549 × 10−4	2.2272 × 10−5	1.9078 × 10−5
F21	2.4429 × 10−4	1.9078 × 10−5	1.9078 × 10−5	7.7537 × 10−3	1.9078 × 10−5	3.8506 × 10−3	1.9078 × 10−5	4.9038 × 10−1	4.9038 × 10−1	2.8532 × 10−3	1.9078 × 10−5
F22	7.1724 × 10−5	1.5027 × 10−4	1.9078 × 10−5	1.4581 × 10−4	2.3029 × 10−5	1.2984 × 10−4	2.1266 × 10−5	1.2984 × 10−4	8.1607 × 10−5	5.2872 × 10−4	2.1266 × 10−5
F23	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5
F24	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	3.0650 × 10−4	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5
F25	1.7344 × 10−5	1.7344 × 10−5	1.7344 × 10−5	1.0000 × 100	1.0000 × 100	1.0000 × 100	1.0000 × 100	1.0000 × 100	1.7344 × 10−5	1.7344 × 10−5	7.4787 × 10−7
F26	4.2137 × 10−2	5.4687 × 10−2	1.2403 × 10−1	3.1801 × 10−2	1.2403 × 10−1	3.8542 × 10−2	1.0000 × 100	1.0000 × 100	1.0000 × 100	8.2355 × 10−1	5.9625 × 10−2
F27	7.3652 × 10−2	4.9080 × 10−1	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	7.3303 × 10−3	1.0137 × 10−2
F28	1.9078 × 10−5	5.4401 × 10−1	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	2.2185 × 10−1	1.9078 × 10−5
F29	1.0000 × 100	1.6592 × 10−2	1.9078 × 10−5	5.2222 × 10−2	1.0000 × 100	1.6412 × 10−3	1.0000 × 100	1.9078 × 10−5	5.0099 × 10−3	1.5489 × 10−1	1.9078 × 10−5
F30	8.4508 × 10−1	1.9078 × 10−5	1.9078 × 10−5	5.3666 × 10−3	2.1279 × 10−2	3.1057 × 10−5	1.2074 × 10−2	1.9078 × 10−5	5.0099 × 10−3	8.5533 × 10−2	2.1279 × 10−2

,

Table A2. P_holm-values for various algorithms on the CEC 2014 (dim = 50).

Item	PSO	DE	CMAES	INFO	SMA	RUN	HGS	DOA	GWCA	SAO	IVYA
F1	1.6782 × 10−3	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	8.1087 × 10−2	1.8621 × 10−4	8.1087 × 10−2	8.1087 × 10−2
F2	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.6503 × 10−1
F3	1.9078 × 10−5	1.0639 × 10−2	1.9078 × 10−5	7.5639 × 10−4	1.2544 × 10−1	1.9078 × 10−5	2.5714 × 10−5	1.9078 × 10−5	1.4691 × 10−3	1.9078 × 10−5	5.1232 × 10−5
F4	3.1299 × 10−5	1.9078 × 10−5	1.9078 × 10−5	3.4919 × 10−5	2.7175 × 10−5	4.0985 × 10−5	1.8959 × 10−4	2.0827 × 10−5	2.0889 × 10−2	1.9078 × 10−5	1.7138 × 10−1
F5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	3.1123 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5
F6	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	7.7309 × 10−3
F7	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	2.0951 × 10−5	1.9078 × 10−5	5.7096 × 10−2	1.9078 × 10−5	7.2189 × 10−3
F8	1.9078 × 10−5	2.0589 × 10−1	1.9078 × 10−5	1.9078 × 10−5	1.4314 × 10−3	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5
F9	4.4594 × 10−4	1.9078 × 10−5	1.9078 × 10−5	5.6729 × 10−4	1.5286 × 10−1	3.1637 × 10−5	2.0889 × 10−2	3.1637 × 10−5	4.4594 × 10−4	3.1637 × 10−5	4.1903 × 10−5
F10	1.9078 × 10−5	3.7142 × 10−5	1.9078 × 10−5	5.1705 × 10−1	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	5.6972 × 10−2	1.9078 × 10−5
F11	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5
F12	7.0356 × 10−1	2.7175 × 10−5	1.9078 × 10−5	1.9078 × 10−5	5.6327 × 10−5	4.6145 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.1374 × 10−4	1.1374 × 10−4	1.4542 × 10−2
F13	3.4179 × 10−2	4.5651 × 10−3	1.9078 × 10−5	9.7539 × 10−1	3.4940 × 10−5	3.4179 × 10−2	1.9078 × 10−5	3.7631 × 10−1	5.4231 × 10−1	4.5651 × 10−3	4.5651 × 10−3
F14	9.8571 × 10−1	4.7763 × 10−4	1.9078 × 10−5	9.8571 × 10−1	3.9670 × 10−2	2.3234 × 10−1	5.5138 × 10−4	6.7954 × 10−1	8.2355 × 10−1	2.8786 × 10−5	9.8571 × 10−1
F15	1.6456 × 10−2	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	8.9614 × 10−5	1.9078 × 10−5	5.1705 × 10−1	3.4461 × 10−4	1.9078 × 10−5	3.1918 × 10−1	3.1918 × 10−1
F16	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	9.6588 × 10−3	8.3307 × 10−1	4.4722 × 10−3	8.3307 × 10−1	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	3.1333 × 10−2
F17	1.4149 × 10−1	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	9.8367 × 10−5	1.9078 × 10−5	5.3836 × 10−5	1.4149 × 10−1	5.9091 × 10−4	5.7496 × 10−4	1.4149 × 10−1
F18	9.7846 × 10−2	1.9078 × 10−5	1.9078 × 10−5	1.9799 × 10−1	3.1637 × 10−5	6.7729 × 10−5	4.0547 × 10−4	5.7457 × 10−1	8.0967 × 10−1	8.0967 × 10−1	3.1723 × 10−3
F19	1.0000 × 100	1.0000 × 100	1.9078 × 10−5	1.0000 × 100	3.3269 × 10−1	1.0000 × 100	1.0000 × 100	7.8218 × 10−1	1.0000 × 100	4.9016 × 10−1	1.0000 × 100
F20	5.7165 × 10−1	3.6515 × 10−5	1.9078 × 10−5	1.9078 × 10−5	5.0729 × 10−1	1.9078 × 10−5	4.6578 × 10−5	2.1813 × 10−2	5.9091 × 10−4	4.1903 × 10−5	1.9078 × 10−5
F21	6.3698 × 10−1	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	5.3836 × 10−5	1.9078 × 10−5	5.5894 × 10−5	4.9697 × 10−2	9.4818 × 10−5	8.9942 × 10−5	6.3698 × 10−1
F22	1.0387 × 10−4	1.9078 × 10−5	1.9078 × 10−5	1.4581 × 10−4	1.4581 × 10−4	2.8122 × 10−5	1.9140 × 10−5	2.8122 × 10−5	3.8454 × 10−5	1.4139 × 10−1	2.8122 × 10−5
F23	2.1181 × 10−5	1.9078 × 10−5	1.9078 × 10−5	8.9856 × 10−1	8.9856 × 10−1	8.9856 × 10−1	8.9856 × 10−1	8.9856 × 10−1	2.1181 × 10−5	2.1181 × 10−5	2.1510 × 10−3
F24	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5
F25	1.9054 × 10−5	1.9054 × 10−5	1.9054 × 10−5	1.0000 × 100	1.0000 × 100	1.0000 × 100	1.0000 × 100	1.0000 × 100	1.9054 × 10−5	1.9054 × 10−5	1.9054 × 10−5
F26	1.8404 × 10−5	1.2241 × 10−4	1.2241 × 10−4	2.9595 × 10−2	1.2241 × 10−4	1.3684 × 10−4	4.1516 × 10−1	4.1516 × 10−1	3.1918 × 10−1	2.1948 × 10−2	1.8404 × 10−5
F27	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5
F28	1.9078 × 10−5	1.2313 × 10−3	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	7.5213 × 10−2	1.9078 × 10−5
F29	3.5785 × 10−3	8.2684 × 10−2	1.9078 × 10−5	4.9038 × 10−1	5.3836 × 10−5	2.8103 × 10−1	1.1838 × 10−4	1.9078 × 10−5	2.8122 × 10−5	9.4261 × 10−1	1.9078 × 10−5
F30	1.9078 × 10−5	4.4594 × 10−4	1.9078 × 10−5	4.9038 × 10−1	5.4977 × 10−3	1.7061 × 10−4	5.1465 × 10−3	1.9078 × 10−5	7.0356 × 10−1	7.1724 × 10−5	1.9078 × 10−5

,

Table A3. P_holm-values for various algorithms on the CEC 2017 (dim = 30).

Item	PSO	DE	CMAES	INFO	SMA	RUN	HGS	DOA	GWCA	SAO	IVYA
F1	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.0000 × 100	1.0000 × 100	9.6743 × 10−3	9.2311 × 10−1	1.0000 × 100	1.0000 × 100	6.2666 × 10−2
F2	5.0023 × 10−4	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.0000 × 100	2.8375 × 10−5	4.6578 × 10−5	9.2584 × 10−1	1.9078 × 10−5	1.0000 × 100	1.9078 × 10−5
F3	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	2.2531 × 10−5	1.2544 × 10−1	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5
F4	3.4053 × 10−4	3.7011 × 10−3	1.9078 × 10−5	4.3742 × 10−4	6.2263 × 10−2	1.0000 × 100	1.0000 × 100	4.1015 × 10−2	7.7688 × 10−4	3.7470 × 10−1	1.0000 × 100
F5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	8.6521 × 10−5	3.5888 × 10−4	1.9078 × 10−5	2.1139 × 10−4	2.1091 × 10−5	8.6521 × 10−5	2.1091 × 10−5	1.9078 × 10−5
F6	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	3.6094 × 10−3	8.8987 × 10−5
F7	3.0572 × 10−1	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.1617 × 10−1	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	3.8203 × 10−1	1.9078 × 10−5
F8	1.7797 × 10−4	1.9078 × 10−5	1.9078 × 10−5	2.6082 × 10−5	1.7797 × 10−4	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	9.7938 × 10−4	9.7938 × 10−4	2.1091 × 10−5
F9	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5
F10	7.4285 × 10−5	2.5453 × 10−5	1.9078 × 10−5	2.3137 × 10−3	5.3836 × 10−5	1.1323 × 10−3	1.9078 × 10−5	7.4285 × 10−5	2.9679 × 10−3	1.9078 × 10−5	8.2167 × 10−3
F11	6.3158 × 10−3	1.9078 × 10−5	1.9078 × 10−5	2.4607 × 10−5	2.4607 × 10−5	6.3158 × 10−3	7.4678 × 10−5	2.3430 × 10−5	2.3430 × 10−5	4.9080 × 10−1	2.3170 × 10−4
F12	3.5037 × 10−2	1.9078 × 10−5	1.9078 × 10−5	1.9140 × 10−5	3.7568 × 10−4	3.5037 × 10−2	1.2260 × 10−3	8.1607 × 10−5	1.9140 × 10−5	2.8880 × 10−3	5.3836 × 10−5
F13	3.5887 × 10−1	1.9078 × 10−5	1.9078 × 10−5	4.2843 × 10−1	4.6474 × 10−3	7.7198 × 10−3	7.1350 × 10−4	4.1178 × 10−1	3.5887 × 10−1	8.2828 × 10−2	1.9467 × 10−4
F14	6.5480 × 10−2	1.9078 × 10−5	1.9078 × 10−5	1.8674 × 10−4	1.9078 × 10−5	1.0258 × 10−3	2.2272 × 10−5	5.8571 × 10−1	6.5480 × 10−2	1.3515 × 10−2	1.9078 × 10−5
F15	1.0454 × 10−3	1.9078 × 10−5	1.9078 × 10−5	1.9861 × 10−1	4.6013 × 10−5	1.9078 × 10−5	8.6666 × 10−5	3.7358 × 10−2	3.7568 × 10−4	3.7358 × 10−2	8.9614 × 10−5
F16	4.9183 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	5.1934 × 10−5	2.0866 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9411 × 10−5	2.5846 × 10−3	1.9078 × 10−5
F17	3.4940 × 10−5	4.8886 × 10−5	1.9078 × 10−5	4.8886 × 10−5	3.7834 × 10−5	1.0614 × 10−4	4.8886 × 10−5	4.8886 × 10−5	4.8886 × 10−5	3.6826 × 10−2	1.9078 × 10−5
F18	1.9588 × 10−3	1.9078 × 10−5	1.9078 × 10−5	5.3836 × 10−5	3.1057 × 10−5	1.9078 × 10−5	7.4285 × 10−5	2.2888 × 10−1	1.9554 × 10−1	4.6975 × 10−2	1.0815 × 10−4
F19	2.5088 × 10−1	1.9078 × 10−5	1.9078 × 10−5	1.9816 × 10−3	2.3430 × 10−5	1.7129 × 10−1	4.6340 × 10−4	7.0074 × 10−2	1.5809 × 10−2	3.6004 × 10−1	7.7198 × 10−3
F20	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	2.8434 × 10−5	1.9078 × 10−5
F21	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	2.1428 × 10−5	8.5303 × 10−5	1.9078 × 10−5	1.9078 × 10−5	2.3007 × 10−5	2.2888 × 10−1	3.6652 × 10−3
F22	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	3.6355 × 10−2	1.9078 × 10−5	3.1918 × 10−1	1.9078 × 10−5	8.1302 × 10−1	4.6145 × 10−5	3.3005 × 10−1	1.7968 × 10−1
F23	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	2.7202 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	3.3269 × 10−2	2.3073 × 10−5
F24	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	7.4285 × 10−5	2.3170 × 10−4	7.0018 × 10−2	2.0150 × 10−5	2.3170 × 10−4	1.9078 × 10−5	1.2984 × 10−4	6.5833 × 10−1
F25	5.7517 × 10−5	8.5343 × 10−1	1.9078 × 10−5	6.0222 × 10−1	4.6474 × 10−3	1.0000 × 100	1.0253 × 10−2	5.5138 × 10−4	1.0000 × 100	5.2132 × 10−4	1.0000 × 100
F26	1.0639 × 10−2	5.7052 × 10−5	1.9078 × 10−5	2.8786 × 10−5	1.0455 × 10−4	3.0055 × 10−4	1.5580 × 10−4	4.0784 × 10−4	9.9046 × 10−5	1.0639 × 10−2	1.5580 × 10−4
F27	4.9508 × 10−1	1.9799 × 10−1	1.9078 × 10−5	4.3095 × 10−3	4.9508 × 10−1	7.6195 × 10−5	4.9508 × 10−1	1.0253 × 10−2	3.8822 × 10−5	1.0336 × 10−1	1.7304 × 10−4
F28	2.3409 × 10−2	1.9078 × 10−5	1.9078 × 10−5	3.6269 × 10−3	3.6269 × 10−3	2.8011 × 10−4	1.6503 × 10−1	2.3409 × 10−2	1.0893 × 10−1	1.0995 × 10−2	2.3409 × 10−2
F29	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.0357 × 10−3	1.9078 × 10−5
F30	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	9.5589 × 10−1	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	2.3170 × 10−4	5.9583 × 10−1	9.5589 × 10−1	1.9078 × 10−5

and

Table A4. P_holm-values for various algorithms on the CEC 2017 (dim = 50).

Item	PSO	DE	CMAES	INFO	SMA	RUN	HGS	DOA	GWCA	SAO	IVYA
F1	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	9.9179 × 10−1
F2	2.2247 × 10−4	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	2.5279 × 10−4	1.9078 × 10−5	4.8139 × 10−4	7.7122 × 10−4	1.9078 × 10−5	4.8139 × 10−4	1.9078 × 10−5
F3	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	4.7795 × 10−1	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5
F4	2.9918 × 10−1	1.9078 × 10−5	1.9078 × 10−5	5.7818 × 10−2	9.5589 × 10−1	9.9620 × 10−2	9.9620 × 10−2	4.4370 × 10−1	9.5589 × 10−1	9.9620 × 10−2	8.1347 × 10−1
F5	1.0526 × 10−2	1.9078 × 10−5	1.9078 × 10−5	2.4140 × 10−2	1.5886 × 10−1	2.7175 × 10−5	3.7358 × 10−2	5.0796 × 10−5	7.3652 × 10−2	1.9078 × 10−5	1.9078 × 10−5
F6	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	4.7795 × 10−1	1.9078 × 10−5
F7	3.3131 × 10−2	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	4.1653 × 10−1	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5
F8	6.7974 × 10−4	1.9209 × 10−5	1.9078 × 10−5	1.1323 × 10−3	2.5364 × 10−1	2.8375 × 10−5	3.1317 × 10−2	2.2272 × 10−5	3.1070 × 10−3	2.0827 × 10−5	1.9209 × 10−5
F9	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5
F10	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5
F11	5.4117 × 10−2	1.9078 × 10−5	1.9078 × 10−5	1.0511 × 10−1	1.0000 × 100	3.2572 × 10−4	6.8551 × 10−1	1.0000 × 100	1.0000 × 100	3.1341 × 10−1	2.1334 × 10−4
F12	2.0827 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.7061 × 10−4	2.7175 × 10−5	8.0232 × 10−4	1.9078 × 10−5	9.4323 × 10−2	2.0508 × 10−2	9.4323 × 10−2	1.8134 × 10−3
F13	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	2.2185 × 10−1	1.9078 × 10−5	2.5714 × 10−5	1.9078 × 10−5	2.5697 × 10−2	6.2013 × 10−2	3.4935 × 10−1	2.2247 × 10−4
F14	1.9588 × 10−3	1.9078 × 10−5	1.9078 × 10−5	3.4510 × 10−5	2.7175 × 10−5	6.8006 × 10−5	1.9078 × 10−5	1.0000 × 100	1.0000 × 100	1.0000 × 100	1.9078 × 10−5
F15	3.1918 × 10−2	1.9078 × 10−5	1.9078 × 10−5	1.6339 × 10−1	1.5784 × 10−4	5.1765 × 10−5	1.0812 × 10−1	3.2857 × 10−1	9.7846 × 10−2	2.2185 × 10−1	1.5784 × 10−4
F16	3.1696 × 10−5	1.9078 × 10−5	1.9078 × 10−5	3.3864 × 10−5	2.8375 × 10−5	1.6376 × 10−4	2.3430 × 10−5	2.3430 × 10−5	4.9183 × 10−5	1.6566 × 10−2	2.3430 × 10−5
F17	4.7409 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.3194 × 10−2	1.9078 × 10−5
F18	8.1405 × 10−2	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.2449 × 10−4	1.9078 × 10−5	1.9090 × 10−5	1.9078 × 10−5	8.1972 × 10−5	8.9364 × 10−1	1.2449 × 10−4
F19	6.6555 × 10−1	1.9078 × 10−5	1.9078 × 10−5	1.8920 × 10−1	1.0000 × 100	1.9078 × 10−5	1.0000 × 100	1.0000 × 100	1.1871 × 10−2	1.0000 × 100	1.0000 × 100
F20	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	5.3070 × 10−5	1.9078 × 10−5
F21	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	6.5578 × 10−5	7.7622 × 10−4	4.4374 × 10−5	6.1904 × 10−5	3.4285 × 10−5	6.5578 × 10−5	4.4374 × 10−5	1.5886 × 10−1
F22	2.0907 × 10−3	1.9078 × 10−5	1.9078 × 10−5	2.0907 × 10−3	2.0907 × 10−3	4.5102 × 10−3	2.0907 × 10−3	1.9078 × 10−5	2.4626 × 10−3	5.6672 × 10−3	2.4626 × 10−3
F23	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	3.3796 × 10−5	1.9078 × 10−5	2.3007 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.7518 × 10−2	9.0671 × 10−4
F24	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	6.8006 × 10−5	2.7182 × 10−1	7.2441 × 10−3	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	6.8006 × 10−5	2.7182 × 10−1
F25	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	2.8758 × 10−5	1.9078 × 10−5	8.6521 × 10−5	5.6729 × 10−4	1.9078 × 10−5	7.7622 × 10−4	1.9078 × 10−5	7.7622 × 10−4
F26	9.4549 × 10−4	8.6666 × 10−5	1.9078 × 10−5	3.1637 × 10−5	4.9810 × 10−2	1.9078 × 10−5	2.1571 × 10−1	2.1571 × 10−1	5.0713 × 10−5	3.3886 × 10−1	1.4223 × 10−4
F27	1.0526 × 10−2	1.0000 × 100	1.9078 × 10−5	1.3811 × 10−4	2.6915 × 10−1	2.5907 × 10−5	1.0000 × 100	3.7084 × 10−2	1.9209 × 10−5	5.0713 × 10−5	2.0432 × 10−4
F28	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	2.3646 × 10−5	1.9078 × 10−5	2.7935 × 10−5	8.1302 × 10−1	1.9078 × 10−5	1.9554 × 10−1	1.9078 × 10−5	1.4691 × 10−3
F29	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	2.7935 × 10−5	2.7935 × 10−5	1.9078 × 10−5	4.3260 × 10−5	1.9078 × 10−5	1.9078 × 10−5	7.6552 × 10−1	1.9078 × 10−5
F30	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	4.1342 × 10−2	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	1.9078 × 10−5	2.7813 × 10−2	4.1342 × 10−2	1.9078 × 10−5

, the Holm test further validates the statistical significance of MEIVYA’s performance advantage over other comparative algorithms. Under the 30-dimensional and 50-dimensional settings of the CEC2014 and CEC2017 test sets, the corresponding $P_{holm}$ values between MEIVYA and most of the comparative algorithms are all less than the significance level of 0.05, with some results even far less than 0.01. This indicates that MEIVYA’s performance improvement is not caused by random fluctuations, but rather has a statistically significant advantage. Especially when compared with traditional or classic algorithms such as PSO, DE, CMAES, SMA, RUN, and HGS, MEIVYA’s $P_{holm}$ values are generally smaller, indicating that it is significantly superior to these algorithms in terms of convergence accuracy, stability, and robustness.

Furthermore, even when compared with some of the stronger competing algorithms, MEIVYA still achieves a $P_{holm}$ value of less than 0.05 in most cases, indicating that its advantage is not only reflected in individual functions but also in its good universality and stability across the entire test set. As the dimension increases from 30 to 50, although the problem complexity further increases, most of MEIVYA’s $P_{holm}$ values remain significant, demonstrating its strong adaptability to high-dimensional problems and that its performance advantage does not significantly weaken with increasing problem size. Overall, the Holm test results in Table A1, Table A2, Table A3 and Table A4 further prove that MEIVYA has significantly better overall performance than most of the compared algorithms under different test sets and different dimensional conditions, and further verify the effectiveness and reliability of the proposed improvement strategy.

As shown in

Table A5. EffectSize-values for various algorithms on the CEC 2014 (dim = 30).

Item	PSO	DE	CMAES	INFO	SMA	RUN	HGS	DOA	GWCA	SAO	IVYA
F1	4.9757 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	7.6983 × 10−2	8.7310 × 10−1	9.5759 × 10−2	8.6934 × 10−1	8.4305 × 10−1	8.5432 × 10−1	2.7226 × 10−1
F2	8.7310 × 10−1	3.9993 × 10−1	8.7310 × 10−1	8.7310 × 10−1	4.0744 × 10−1	4.3373 × 10−1	4.3185 × 10−2	8.6934 × 10−1	6.8158 × 10−1	4.3373 × 10−1	2.8352 × 10−1
F3	7.5293 × 10−1	3.6989 × 10−1	8.7310 × 10−1	8.7310 × 10−1	3.8116 × 10−1	8.7310 × 10−1	7.9423 × 10−1	2.6474 × 10−1	7.8672 × 10−1	8.6183 × 10−1	8.3554 × 10−1
F4	8.0175 × 10−1	8.7310 × 10−1	8.7310 × 10−1	7.1162 × 10−1	2.3095 × 10−1	6.9660 × 10−1	9.5759 × 10−2	6.2149 × 10−1	4.9382 × 10−1	6.8533 × 10−1	6.9472 × 10−2
F5	8.7310 × 10−1	4.6002 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	7.5293 × 10−1	8.7310 × 10−1	8.5807 × 10−1	8.7310 × 10−1	8.6934 × 10−1	8.7310 × 10−1
F6	8.5432 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.6183 × 10−1	7.1162 × 10−1	8.7310 × 10−1	8.6934 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	7.0786 × 10−1
F7	8.7310 × 10−1	8.6558 × 10−1	8.7310 × 10−1	2.2344 × 10−1	8.7310 × 10−1	6.0647 × 10−1	7.6419 × 10−1	7.3040 × 10−1	6.4027 × 10−1	8.7310 × 10−1	6.4402 × 10−1
F8	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.6183 × 10−1	7.5293 × 10−1	8.7310 × 10−1	7.9048 × 10−1	8.7310 × 10−1	8.7310 × 10−1	2.7601 × 10−1	8.7310 × 10−1
F9	8.4305 × 10−1	8.7310 × 10−1	8.7310 × 10−1	7.3791 × 10−1	6.2900 × 10−1	8.7310 × 10−1	8.5432 × 10−1	8.7310 × 10−1	5.9896 × 10−1	7.8297 × 10−1	8.6934 × 10−1
F10	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	7.3040 × 10−1	5.6329 × 10−3	7.3415 × 10−1	8.6183 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.5056 × 10−1	8.7310 × 10−1
F11	8.4305 × 10−1	8.7310 × 10−1	8.7310 × 10−1	7.0786 × 10−1	8.4681 × 10−1	8.6934 × 10−1	8.6934 × 10−1	8.7310 × 10−1	7.1913 × 10−1	8.7310 × 10−1	4.7879 × 10−1
F12	6.5529 × 10−1	3.2483 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.4305 × 10−1	8.0550 × 10−1	8.7310 × 10−1	8.7310 × 10−1	6.2525 × 10−1	2.0466 × 10−1	5.8770 × 10−1
F13	6.8533 × 10−1	8.6183 × 10−1	8.7310 × 10−1	5.8394 × 10−1	8.7310 × 10−1	6.5529 × 10−1	8.7310 × 10−1	3.9242 × 10−1	5.5765 × 10−1	8.3930 × 10−1	3.5487 × 10−1
F14	3.7740 × 10−1	8.7310 × 10−1	8.7310 × 10−1	4.4500 × 10−1	8.7310 × 10−1	6.1774 × 10−1	8.6934 × 10−1	2.6099 × 10−1	3.5487 × 10−1	8.5807 × 10−1	4.5626 × 10−1
F15	2.0842 × 10−1	5.6516 × 10−1	8.7310 × 10−1	5.5390 × 10−1	6.3651 × 10−1	8.7310 × 10−1	2.0654 × 10−2	4.5251 × 10−1	8.7310 × 10−1	3.4736 × 10−1	5.8019 × 10−1
F16	7.4917 × 10−1	8.6183 × 10−1	8.7310 × 10−1	8.6183 × 10−1	1.0702 × 10−1	1.2580 × 10−1	4.6941 × 10−2	8.7310 × 10−1	8.6934 × 10−1	8.6558 × 10−1	1.6899 × 10−2
F17	2.0654 × 10−2	8.7310 × 10−1	8.7310 × 10−1	8.1677 × 10−1	8.1301 × 10−1	8.7310 × 10−1	7.9799 × 10−1	7.1913 × 10−1	3.0230 × 10−1	4.4875 × 10−1	6.5529 × 10−1
F18	1.1829 × 10−1	8.7310 × 10−1	8.7310 × 10−1	3.4361 × 10−1	8.1301 × 10−1	2.2719 × 10−1	8.4305 × 10−1	5.5765 × 10−1	5.9896 × 10−1	1.4458 × 10−1	1.6335 × 10−1
F19	8.6934 × 10−1	8.7310 × 10−1	8.7310 × 10−1	6.5905 × 10−1	5.5765 × 10−1	8.7310 × 10−1	6.4027 × 10−1	8.6183 × 10−1	8.7310 × 10−1	6.5905 × 10−1	6.2149 × 10−1
F20	6.8158 × 10−1	5.7643 × 10−1	8.7310 × 10−1	8.7310 × 10−1	6.7407 × 10−1	8.6558 × 10−1	8.2428 × 10−1	3.7740 × 10−1	6.7782 × 10−1	8.5056 × 10−1	8.7310 × 10−1
F21	7.4917 × 10−1	8.7310 × 10−1	8.7310 × 10−1	5.5014 × 10−1	8.7310 × 10−1	6.0272 × 10−1	8.7310 × 10−1	2.1217 × 10−1	1.8213 × 10−1	6.2900 × 10−1	8.7310 × 10−1
F22	8.0550 × 10−1	7.2289 × 10−1	8.7310 × 10−1	7.4166 × 10−1	8.5432 × 10−1	7.6044 × 10−1	8.6558 × 10−1	7.6795 × 10−1	7.9423 × 10−1	6.3276 × 10−1	8.6183 × 10−1
F23	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1
F24	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	6.5905 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.3179 × 10−1
F25	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	9.8524 × 10−1
F26	5.1635 × 10−1	4.9382 × 10−1	4.2247 × 10−1	5.7333 × 10−1	4.2247 × 10−1	5.2761 × 10−1	1.2701 × 10−1	1.9026 × 10−1	3.9430 × 10−2	2.3095 × 10−1	4.8032 × 10−1
F27	3.8116 × 10−1	1.2580 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.5807 × 10−1	5.6892 × 10−1	5.3512 × 10−1
F28	8.7310 × 10−1	1.1078 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.6934 × 10−1	2.9103 × 10−1	8.7310 × 10−1
F29	1.1829 × 10−1	5.4639 × 10−1	8.7310 × 10−1	4.6753 × 10−1	1.3143 × 10−2	6.7782 × 10−1	9.2004 × 10−2	8.7310 × 10−1	6.1774 × 10−1	3.7740 × 10−1	8.7310 × 10−1
F30	3.5675 × 10−2	8.7310 × 10−1	8.7310 × 10−1	6.0647 × 10−1	5.0884 × 10−1	8.4305 × 10−1	5.5390 × 10−1	8.7310 × 10−1	6.1774 × 10−1	3.6989 × 10−1	5.0133 × 10−1

,

Table A6. EffectSize-values for various algorithms on the CEC 2014 (dim = 50).

Item	PSO	DE	CMAES	INFO	SMA	RUN	HGS	DOA	GWCA	SAO	IVYA
F1	6.4402 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.6934 × 10−1	8.5807 × 10−1	8.7310 × 10−1	8.6183 × 10−1	3.9242 × 10−1	7.5293 × 10−1	3.5487 × 10−1	4.0369 × 10−1
F2	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.6934 × 10−1	8.7310 × 10−1	8.6558 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	2.5348 × 10−1
F3	8.7310 × 10−1	5.0884 × 10−1	8.7310 × 10−1	6.8158 × 10−1	2.7977 × 10−1	8.7310 × 10−1	8.3930 × 10−1	8.7310 × 10−1	6.3651 × 10−1	8.7310 × 10−1	8.0550 × 10−1
F4	8.3179 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.2052 × 10−1	8.4305 × 10−1	8.0550 × 10−1	7.3040 × 10−1	8.5807 × 10−1	4.6753 × 10−1	8.7310 × 10−1	2.4972 × 10−1
F5	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	7.6044 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1
F6	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.2428 × 10−1	8.7310 × 10−1	8.3179 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.6934 × 10−1	4.8630 × 10−1
F7	8.6934 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.2052 × 10−1	8.7310 × 10−1	3.4736 × 10−1	8.7310 × 10−1	5.3137 × 10−1
F8	8.7310 × 10−1	2.3095 × 10−1	8.7310 × 10−1	8.6558 × 10−1	6.1774 × 10−1	8.7310 × 10−1	8.6934 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.3930 × 10−1	8.6934 × 10−1
F9	7.0786 × 10−1	8.7310 × 10−1	8.7310 × 10−1	6.8158 × 10−1	2.6099 × 10−1	8.4681 × 10−1	4.6753 × 10−1	8.4305 × 10−1	7.1537 × 10−1	8.4681 × 10−1	8.2052 × 10−1
F10	8.7310 × 10−1	7.9799 × 10−1	8.7310 × 10−1	1.1829 × 10−1	8.7310 × 10−1	8.4681 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	3.9993 × 10−1	8.6183 × 10−1
F11	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.6558 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.1677 × 10−1	8.7310 × 10−1	8.7310 × 10−1
F12	6.9472 × 10−2	8.4305 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.0175 × 10−1	8.1677 × 10−1	8.7310 × 10−1	8.7310 × 10−1	7.6419 × 10−1	7.5668 × 10−1	4.9006 × 10−1
F13	4.9382 × 10−1	6.2525 × 10−1	8.7310 × 10−1	5.6329 × 10−3	8.4305 × 10−1	4.8255 × 10−1	8.7310 × 10−1	2.7977 × 10−1	2.0091 × 10−1	6.2525 × 10−1	6.2900 × 10−1
F14	1.3707 × 10−1	7.3791 × 10−1	8.7310 × 10−1	1.1829 × 10−1	5.0508 × 10−1	3.7740 × 10−1	7.2664 × 10−1	2.7226 × 10−1	2.3095 × 10−1	8.5432 × 10−1	1.7837 × 10−1
F15	5.2386 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	7.9048 × 10−1	8.7310 × 10−1	1.1829 × 10−1	7.2664 × 10−1	8.7310 × 10−1	2.9479 × 10−1	2.8352 × 10−1
F16	8.6934 × 10−1	8.7310 × 10−1	8.7310 × 10−1	5.5390 × 10−1	1.2580 × 10−1	6.0647 × 10−1	1.4833 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	4.6753 × 10−1
F17	3.6238 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	7.8672 × 10−1	8.7310 × 10−1	8.1677 × 10−1	3.2858 × 10−1	6.9284 × 10−1	7.0411 × 10−1	3.5863 × 10−1
F18	4.2622 × 10−1	8.7310 × 10−1	8.7310 × 10−1	3.5863 × 10−1	8.4681 × 10−1	8.1301 × 10−1	7.3415 × 10−1	2.3846 × 10−1	1.5209 × 10−1	1.0327 × 10−1	6.3276 × 10−1
F19	2.6474 × 10−1	2.0842 × 10−1	8.7310 × 10−1	6.9472 × 10−2	3.8867 × 10−1	9.5759 × 10−2	1.8213 × 10−1	3.0230 × 10−1	2.6474 × 10−1	3.5112 × 10−1	1.5960 × 10−1
F20	1.0327 × 10−1	8.3179 × 10−1	8.7310 × 10−1	8.7310 × 10−1	2.0842 × 10−1	8.7310 × 10−1	8.0926 × 10−1	4.9006 × 10−1	6.9284 × 10−1	8.2052 × 10−1	8.6558 × 10−1
F21	1.8213 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.1677 × 10−1	8.7310 × 10−1	8.0926 × 10−1	4.3749 × 10−1	7.7170 × 10−1	7.8297 × 10−1	1.0702 × 10−1
F22	7.6795 × 10−1	8.7310 × 10−1	8.7310 × 10−1	7.4166 × 10−1	7.3415 × 10−1	8.4681 × 10−1	8.6558 × 10−1	8.3930 × 10−1	8.1677 × 10−1	2.6850 × 10−1	8.4681 × 10−1
F23	8.6183 × 10−1	8.7310 × 10−1	8.7310 × 10−1	9.4868 × 10−1	9.4868 × 10−1	9.4868 × 10−1	9.4868 × 10−1	9.4868 × 10−1	8.6183 × 10−1	8.6183 × 10−1	6.5159 × 10−1
F24	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.5056 × 10−1
F25	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7314 × 10−1
F26	8.7310 × 10−1	7.9423 × 10−1	7.9423 × 10−1	6.3129 × 10−1	7.9423 × 10−1	8.0043 × 10−1	2.7398 × 10−1	4.4557 × 10−1	2.9479 × 10−1	5.2010 × 10−1	8.7441 × 10−1
F27	8.4305 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.1677 × 10−1
F28	8.7310 × 10−1	6.2525 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.6934 × 10−1	3.2483 × 10−1	8.7310 × 10−1
F29	6.1774 × 10−1	4.2247 × 10−1	8.7310 × 10−1	2.1217 × 10−1	8.1677 × 10−1	3.0605 × 10−1	7.7921 × 10−1	8.7310 × 10−1	8.4681 × 10−1	1.3143 × 10−2	8.7310 × 10−1
F30	8.7310 × 10−1	7.1537 × 10−1	8.7310 × 10−1	2.1217 × 10−1	5.6892 × 10−1	7.6419 × 10−1	5.8770 × 10−1	8.7310 × 10−1	6.9472 × 10−2	8.0550 × 10−1	8.7310 × 10−1

,

Table A7. EffectSize-values for various algorithms on the CEC 2017 (dim = 30).

Item	PSO	DE	CMAES	INFO	SMA	RUN	HGS	DOA	GWCA	SAO	IVYA
F1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	1.4458 × 10−1	1.7462 × 10−1	5.8394 × 10−1	2.4221 × 10−1	7.6983 × 10−2	1.6335 × 10−1	4.6753 × 10−1
F2	7.0035 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.6934 × 10−1	8.4493 × 10−2	8.3554 × 10−1	8.0926 × 10−1	1.8588 × 10−1	8.7310 × 10−1	2.8164 × 10−2	8.7310 × 10−1
F3	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.0175 × 10−1	2.7977 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.5807 × 10−1
F4	7.5668 × 10−1	6.3276 × 10−1	8.7310 × 10−1	7.4166 × 10−1	4.5626 × 10−1	1.3331 × 10−1	3.1920 × 10−2	4.9382 × 10−1	7.1162 × 10−1	3.0605 × 10−1	5.4451 × 10−2
F5	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	7.7546 × 10−1	6.5154 × 10−1	8.7310 × 10−1	7.0786 × 10−1	8.4305 × 10−1	7.6795 × 10−1	8.4681 × 10−1	8.7310 × 10−1
F6	8.7310 × 10−1	8.6934 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	5.3137 × 10−1	7.4542 × 10−1
F7	2.6099 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.6183 × 10−1	3.7740 × 10−1	8.7310 × 10−1	8.6558 × 10−1	8.7310 × 10−1	8.7310 × 10−1	1.5960 × 10−1	8.7310 × 10−1
F8	7.4542 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.3179 × 10−1	7.4166 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.6558 × 10−1	6.3651 × 10−1	6.0272 × 10−1	8.4681 × 10−1
F9	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.0926 × 10−1	8.7310 × 10−1
F10	7.9799 × 10−1	8.5056 × 10−1	8.7310 × 10−1	6.1398 × 10−1	8.1677 × 10−1	6.6280 × 10−1	8.6934 × 10−1	7.9423 × 10−1	5.8019 × 10−1	8.7310 × 10−1	4.8255 × 10−1
F11	5.6141 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.4681 × 10−1	8.4305 × 10−1	5.5765 × 10−1	7.9048 × 10−1	8.5432 × 10−1	8.5807 × 10−1	1.2580 × 10−1	7.3415 × 10−1
F12	4.3373 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.6558 × 10−1	7.2289 × 10−1	4.2247 × 10−1	6.5905 × 10−1	7.9423 × 10−1	8.6183 × 10−1	6.0272 × 10−1	8.1677 × 10−1
F13	2.8728 × 10−1	8.7310 × 10−1	8.7310 × 10−1	1.4458 × 10−1	6.2149 × 10−1	5.8770 × 10−1	7.1537 × 10−1	2.3095 × 10−1	3.0981 × 10−1	4.3749 × 10−1	7.7546 × 10−1
F14	4.1871 × 10−1	8.7310 × 10−1	8.7310 × 10−1	7.6044 × 10−1	8.7310 × 10−1	6.7782 × 10−1	8.5056 × 10−1	9.9514 × 10−2	3.9993 × 10−1	5.3512 × 10−1	8.6934 × 10−1
F15	6.6656 × 10−1	8.7310 × 10−1	8.7310 × 10−1	2.3470 × 10−1	8.2803 × 10−1	8.7310 × 10−1	7.9799 × 10−1	4.5626 × 10−1	7.2289 × 10−1	4.3373 × 10−1	7.9048 × 10−1
F16	7.8672 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.6558 × 10−1	7.6795 × 10−1	8.3179 × 10−1	8.5432 × 10−1	8.6558 × 10−1	8.4305 × 10−1	5.5014 × 10−1	8.7310 × 10−1
F17	8.4305 × 10−1	8.2052 × 10−1	8.7310 × 10−1	8.2052 × 10−1	8.3554 × 10−1	7.3791 × 10−1	8.1677 × 10−1	8.0926 × 10−1	8.1301 × 10−1	3.8116 × 10−1	8.7310 × 10−1
F18	6.3651 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.1677 × 10−1	8.4305 × 10−1	8.7310 × 10−1	7.9799 × 10−1	2.1968 × 10−1	3.0230 × 10−1	4.4124 × 10−1	7.7546 × 10−1
F19	2.7977 × 10−1	8.7310 × 10−1	8.7310 × 10−1	6.6280 × 10−1	8.5807 × 10−1	3.4736 × 10−1	7.3415 × 10−1	4.3373 × 10−1	5.3888 × 10−1	1.6711 × 10−1	5.8770 × 10−1
F20	8.7310 × 10−1	8.6934 × 10−1	8.7310 × 10−1	8.6558 × 10−1	8.6558 × 10−1	8.7310 × 10−1	8.6183 × 10−1	8.7310 × 10−1	8.7310 × 10−1	7.6419 × 10−1	8.6558 × 10−1
F21	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.6183 × 10−1	8.3930 × 10−1	7.6419 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.2803 × 10−1	2.1968 × 10−1	5.6892 × 10−1
F22	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	4.9006 × 10−1	8.7310 × 10−1	2.9479 × 10−1	8.6934 × 10−1	4.3185 × 10−2	8.1677 × 10−1	2.5348 × 10−1	3.6614 × 10−1
F23	8.6934 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.4305 × 10−1	7.9423 × 10−1	8.6183 × 10−1	8.7310 × 10−1	8.6934 × 10−1	3.8867 × 10−1	8.1677 × 10−1
F24	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	7.9799 × 10−1	7.3415 × 10−1	3.8491 × 10−1	8.5432 × 10−1	7.2289 × 10−1	8.7310 × 10−1	7.6795 × 10−1	8.0738 × 10−2
F25	8.2803 × 10−1	2.2719 × 10−1	8.7310 × 10−1	2.8352 × 10−1	6.2149 × 10−1	2.0654 × 10−2	5.7268 × 10−1	7.2664 × 10−1	8.4493 × 10−2	7.3415 × 10−1	1.2956 × 10−1
F26	5.0884 × 10−1	8.2428 × 10−1	8.7310 × 10−1	8.5432 × 10−1	7.9048 × 10−1	7.2289 × 10−1	7.6795 × 10−1	6.9660 × 10−1	7.9799 × 10−1	4.7879 × 10−1	7.6419 × 10−1
F27	2.5348 × 10−1	3.5863 × 10−1	8.7310 × 10−1	6.2525 × 10−1	2.3095 × 10−1	8.1301 × 10−1	1.2580 × 10−1	5.7268 × 10−1	8.4305 × 10−1	4.2247 × 10−1	7.7546 × 10−1
F28	5.1259 × 10−1	8.7310 × 10−1	8.7310 × 10−1	6.3651 × 10−1	6.4027 × 10−1	7.6044 × 10−1	2.5348 × 10−1	5.0884 × 10−1	3.5112 × 10−1	5.6892 × 10−1	5.1635 × 10−1
F29	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.6934 × 10−1	8.7310 × 10−1	8.7310 × 10−1	5.9896 × 10−1	8.7310 × 10−1
F30	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	1.2956 × 10−1	8.6183 × 10−1	8.7310 × 10−1	8.7310 × 10−1	7.3415 × 10−1	2.3470 × 10−1	5.8206 × 10−2	8.5432 × 10−1

and

Table A8. EffectSize-values for various algorithms on the CEC 2017 (dim = 50).

Item	PSO	DE	CMAES	INFO	SMA	RUN	HGS	DOA	GWCA	SAO	IVYA
F1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.5432 × 10−1	8.7310 × 10−1	8.7310 × 10−1	1.8776 × 10−3
F2	7.4542 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	7.3040 × 10−1	8.7310 × 10−1	6.7407 × 10−1	6.1398 × 10−1	8.7310 × 10−1	6.8909 × 10−1	8.7310 × 10−1
F3	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.6934 × 10−1	1.2956 × 10−1	8.6183 × 10−1	8.7310 × 10−1	8.3930 × 10−1
F4	3.4361 × 10−1	8.7310 × 10−1	8.7310 × 10−1	4.9757 × 10−1	8.8248 × 10−2	4.5626 × 10−1	4.5626 × 10−1	2.9103 × 10−1	1.2956 × 10−1	4.3749 × 10−1	2.0091 × 10−1
F5	5.6141 × 10−1	8.7310 × 10−1	8.7310 × 10−1	5.0133 × 10−1	2.5723 × 10−1	8.4305 × 10−1	4.5626 × 10−1	8.1301 × 10−1	3.8116 × 10−1	8.7310 × 10−1	8.6934 × 10−1
F6	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	1.2956 × 10−1	8.5056 × 10−1
F7	4.3749 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.6934 × 10−1	1.4833 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1
F8	6.9660 × 10−1	8.6934 × 10−1	8.7310 × 10−1	6.6280 × 10−1	2.0842 × 10−1	8.3554 × 10−1	4.4124 × 10−1	8.5056 × 10−1	5.9896 × 10−1	8.5807 × 10−1	8.6934 × 10−1
F9	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.5056 × 10−1	8.7310 × 10−1
F10	8.6934 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.2803 × 10−1	8.7310 × 10−1	8.6934 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.2803 × 10−1	8.7310 × 10−1	8.7310 × 10−1
F11	4.8630 × 10−1	8.7310 × 10−1	8.7310 × 10−1	4.3373 × 10−1	1.4833 × 10−1	7.4917 × 10−1	2.4972 × 10−1	1.3707 × 10−1	1.0702 × 10−1	3.3985 × 10−1	7.7170 × 10−1
F12	8.5807 × 10−1	8.7310 × 10−1	8.7310 × 10−1	7.6419 × 10−1	8.4305 × 10−1	6.8909 × 10−1	8.7310 × 10−1	3.6238 × 10−1	4.9382 × 10−1	3.4361 × 10−1	6.4027 × 10−1
F13	8.6558 × 10−1	8.7310 × 10−1	8.7310 × 10−1	2.9103 × 10−1	8.7310 × 10−1	8.3930 × 10−1	8.6183 × 10−1	4.9757 × 10−1	4.2247 × 10−1	1.7086 × 10−1	7.4542 × 10−1
F14	6.3651 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.2803 × 10−1	8.4305 × 10−1	7.9423 × 10−1	8.6934 × 10−1	1.6899 × 10−2	6.1962 × 10−2	5.8206 × 10−2	8.6934 × 10−1
F15	5.0884 × 10−1	8.7310 × 10−1	8.7310 × 10−1	3.5112 × 10−1	7.7546 × 10−1	8.2803 × 10−1	4.0369 × 10−1	1.7837 × 10−1	4.2622 × 10−1	2.9103 × 10−1	7.7921 × 10−1
F16	8.2428 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.1301 × 10−1	8.3554 × 10−1	7.1913 × 10−1	8.5807 × 10−1	8.5432 × 10−1	7.8672 × 10−1	4.3749 × 10−1	8.5432 × 10−1
F17	7.7170 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.3554 × 10−1	8.6934 × 10−1	8.6934 × 10−1	8.7310 × 10−1	8.6934 × 10−1	8.7310 × 10−1	4.5251 × 10−1	8.7310 × 10−1
F18	3.7365 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.6558 × 10−1	7.6044 × 10−1	8.7310 × 10−1	8.5056 × 10−1	8.7310 × 10−1	7.8672 × 10−1	2.4409 × 10−2	7.5668 × 10−1
F19	2.9103 × 10−1	8.7310 × 10−1	8.7310 × 10−1	4.0369 × 10−1	2.3095 × 10−1	8.7310 × 10−1	1.4833 × 10−1	6.9472 × 10−2	5.8019 × 10−1	6.1962 × 10−2	1.2580 × 10−1
F20	8.6934 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.6934 × 10−1	8.6934 × 10−1	8.6934 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.6934 × 10−1	7.3791 × 10−1	8.7310 × 10−1
F21	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	7.7921 × 10−1	6.4778 × 10−1	8.2428 × 10−1	7.9799 × 10−1	8.3930 × 10−1	7.8672 × 10−1	8.2052 × 10−1	2.5723 × 10−1
F22	6.6656 × 10−1	8.7310 × 10−1	8.7310 × 10−1	6.6656 × 10−1	6.6656 × 10−1	5.5765 × 10−1	6.6656 × 10−1	8.6934 × 10−1	6.2525 × 10−1	5.0508 × 10−1	6.2525 × 10−1
F23	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.6558 × 10−1	8.0175 × 10−1	8.7310 × 10−1	8.2803 × 10−1	8.5807 × 10−1	8.7310 × 10−1	4.3373 × 10−1	6.4027 × 10−1
F24	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	7.9048 × 10−1	2.7226 × 10−1	5.5390 × 10−1	8.7310 × 10−1	8.6183 × 10−1	8.7310 × 10−1	7.9423 × 10−1	2.2344 × 10−1
F25	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.2803 × 10−1	8.6558 × 10−1	7.7546 × 10−1	6.8158 × 10−1	8.5807 × 10−1	6.3651 × 10−1	8.7310 × 10−1	6.4778 × 10−1
F26	6.8158 × 10−1	7.9799 × 10−1	8.7310 × 10−1	8.4681 × 10−1	4.5626 × 10−1	8.7310 × 10−1	3.2483 × 10−1	3.2858 × 10−1	8.2428 × 10−1	1.7462 × 10−1	7.7170 × 10−1
F27	5.6141 × 10−1	9.5759 × 10−2	8.7310 × 10−1	7.7921 × 10−1	3.0981 × 10−1	8.5432 × 10−1	1.6899 × 10−2	4.7504 × 10−1	8.6934 × 10−1	8.2428 × 10−1	7.5668 × 10−1
F28	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.3554 × 10−1	8.7310 × 10−1	8.2052 × 10−1	4.3185 × 10−2	8.7310 × 10−1	3.0230 × 10−1	8.7310 × 10−1	6.3651 × 10−1
F29	8.7310 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.2052 × 10−1	8.1677 × 10−1	8.6934 × 10−1	7.7546 × 10−1	8.7310 × 10−1	8.6934 × 10−1	5.4451 × 10−2	8.6558 × 10−1
F30	8.6934 × 10−1	8.7310 × 10−1	8.7310 × 10−1	3.7740 × 10−1	8.7310 × 10−1	8.7310 × 10−1	8.6183 × 10−1	8.5807 × 10−1	4.7504 × 10−1	4.2247 × 10−1	8.6558 × 10−1

, the Effect Size analysis further measures the actual impact of the performance differences between MEIVYA and the comparison algorithms. Overall, MEIVYA exhibits moderate to large effect sizes compared to most comparison algorithms under the 30-dimensional and 50-dimensional conditions of the CEC2014 and CEC2017 test sets. This indicates that its performance improvement is not only statistically significant but also has strong practical optimization effect. Especially when compared with algorithms such as PSO, DE, CMAES, SMA, RUN, and HGS, MEIVYA often achieves larger Effect Size values, indicating that its improvements in solution accuracy, convergence speed, and stability are more significant than just minor performance gains.

Meanwhile, when compared with some high-performing algorithms, although the effect sizes on some functions may be small or moderate, the overall performance still maintains a positive advantage. This indicates that MEIVYA can still achieve meaningful performance improvements even when facing highly competitive algorithms. Furthermore, as the problem dimension increases from 30 to 50, most of MEIVYA’s effect size results remain at a moderate or higher level, demonstrating that its advantage is not significantly weakened by the increase in problem size, further reflecting its good adaptability and stability to high dimensions. Overall, the Effect Size analysis in Table A5, Table A6, Table A7 and Table A8 further proves that MEIVYA’s advantage over other algorithms is not only statistically significant but also has strong practical application value and engineering significance.

3.6. Strategy Effectiveness Analysis

For improving heuristic optimization algorithms, policy effectiveness analysis is a crucial step in verifying whether the improvement mechanism truly works. Since most improvement algorithms typically consist of multiple policies, relying solely on the final experimental results is insufficient to accurately determine the specific contribution of each policy to the algorithm’s performance. Therefore, it is necessary to analyze different policies individually through ablation experiments or phased comparative experiments to evaluate their actual effects on improving population diversity, enhancing global exploration capabilities, improving local exploitation capabilities, and avoiding local optima. Policy effectiveness analysis not only reveals the working principles and synergistic effects of each improvement mechanism more clearly but also further demonstrates the sources of the proposed algorithm’s improvements in convergence speed, solution accuracy, and stability, providing strong support for the rationality and scientific validity of the algorithm design. This section conducts a policy effectiveness analysis on the proposed MEIVYA, and the experimental results are shown below. In this context, IVYA represents the original IVY algorithm; IVYA1 represents the algorithm with the addition of Chaotic-Based Population Initialization; IVYA2 represents the algorithm with the addition of Adaptive Growth-Rate Regulation Mechanism; IVYA3 represents the algorithm with the addition of Elite-Guided Cooperative Search Strategy; and MEIVYA represents the algorithm with the addition of three strategies to the IVY algorithm.

Figure 7 presents the strategy effectiveness analysis results of MEIVYA by comparing the original IVYA, three intermediate variants (IVYA1–IVYA3), and the final MEIVYA on several representative CEC2017 benchmark functions under 30-dimensional and 50-dimensional settings. The results show that, compared with the original IVYA, the variants incorporating different improvement strategies generally achieve lower fitness values and faster convergence, indicating that each proposed strategy contributes positively to the optimization process. More importantly, the final MEIVYA consistently exhibits the best or near-best convergence behavior on most test functions, especially on complex functions such as F5, F9, F16, F20, F21, and F24, where it not only converges to lower objective values but also maintains a more stable convergence trend in the later stages. This demonstrates that the three strategies do not work independently, but instead produce a clear synergistic effect after integration. In particular, the improvement is more pronounced in the 50-dimensional cases, where MEIVYA shows stronger resistance to performance degradation than the other variants, suggesting that the proposed mechanisms effectively enhance population diversity, adaptive search capability, and the balance between global exploration and local exploitation. Overall, Figure 7 verifies the rationality and effectiveness of the proposed multi-strategy design, and confirms that the superior performance of MEIVYA originates from the coordinated contribution of all three improvement strategies rather than from any single mechanism alone.

3.7. Exploration and Exploitation Analysis

For heuristic optimization algorithms, the balance between exploration and exploitation capabilities is a key factor determining algorithm performance. Exploration capability is primarily reflected in the algorithm’s ability to conduct a broad search of the entire solution space in the early stages of the search, helping to discover potential high-quality regions and avoid prematurely falling into local optima. Exploration capability, on the other hand, is reflected in the algorithm’s ability to conduct a fine-grained search and local optimization of high-quality regions in the later stages of the search, thereby improving solution accuracy and convergence quality. If an algorithm is overly reliant on exploration, it may lead to slow convergence and difficulty in obtaining high-precision solutions; conversely, if an algorithm is overly reliant on exploitation, it is prone to falling into local optima due to insufficient population diversity. Therefore, exploration and exploitation analysis of heuristic algorithms is essential. It not only reveals the behavioral characteristics of the algorithm at different search stages but also helps verify whether the proposed strategy can effectively coordinate the relationship between global search and local exploitation, thereby further demonstrating the algorithm’s advantages in convergence speed, stability, and solution accuracy. In this section, we conducted exploration and exploitation analysis on MEIVYA, and the experimental results are shown below.

Figure 8 presents the exploration and exploitation analysis results of MEIVYA, which are used to evaluate the balance between global search and local search during the optimization process. From the figure, it can be observed that MEIVYA maintains a relatively high exploration capability in the early stage of iteration, enabling the population to search a wider solution space and avoid premature convergence caused by insufficient diversity. As the iteration proceeds, the exploration ability gradually decreases while the exploitation ability continuously increases, indicating that the algorithm gradually shifts its focus from broad global search to fine-grained local refinement around promising regions. In the early iterations, its exploration curve remains at a relatively high level for a longer period, which helps the algorithm identify more potential search regions. In the middle and late stages, the exploitation curve rises more steadily and eventually becomes dominant, allowing the algorithm to accelerate convergence and improve solution precision. This balanced search behavior demonstrates that the adaptive growth-rate regulation mechanism can effectively adjust the search intensity over time, while the chaotic initialization strategy provides sufficient diversity in the early stage and the elite-guided cooperative search strategy strengthens local exploitation in the later stage. In addition, Figure 8 shows that MEIVYA does not exhibit an excessively rapid decline in exploration ability, which means that it can still preserve a certain degree of global search even in the later iterations. This characteristic is important for avoiding stagnation in local optima, especially when solving complex multimodal and high-dimensional problems. Therefore, the results in Figure 8 verify that MEIVYA achieves a better balance between exploration and exploitation than the compared methods, which is one of the key reasons for its superior convergence speed, optimization accuracy, and robustness.

4. MEIVYA for Artistic Image Segmentation

In image thresholding, the selection of the threshold directly determines the separation effect between the foreground and background; therefore, a reasonable threshold determination method is crucial. Commonly used automatic thresholding methods include the Otsu method and the Kapur entropy method. For different threshold divisions, the target region and background region exhibit different statistical characteristics.

The Otsu method evaluates segmentation quality by maximizing the inter-class variance, and uses the threshold that maximizes the inter-class variance as the optimal threshold $T_{best}$. In contrast, the Kapur entropy method, based on information theory, calculates the entropy values of the target and background regions separately, and selects the threshold that maximizes the sum of the two entropy values as the optimal solution. Both methods are based on image gray-level histograms to achieve automatic threshold selection and have been widely used in image segmentation [37,38].

Compared to the Kapur entropy method, the Otsu method offers advantages such as simple computation, no need for parameter tuning, and robustness to noise, making it particularly suitable for images with a distinct bimodal grayscale distribution. Therefore, this paper adopts the Otsu method as the evaluation criterion for multi-threshold segmentation.

4.1. Otsu Multi-Threshold Segmentation Model

The Otsu method determines the optimal threshold by maximizing the inter-class variance. Let the image have $L$ gray levels, and let $n_i$ be the number of pixels corresponding to gray level $i(i=\mathrm{0,1},\dots ,L-1)$. Then the total number of pixels in the image can be expressed as follows:

$$N={\sum }_{i=0}^{L-1}{n}_i,$$

(17)

The probability distribution of gray level $i$ can be represented as follows:

$$P_i={\displaystyle \frac{n_i}{N}},i=\mathrm{0,1},\dots ,L-1,$$

(18)

where $P_i\ge 0$, and it satisfies:

$${\sum }_{i=0}^{L-1} P_i=1,$$

(19)

1. Single threshold segmentation: When gray level $t$ is selected as the threshold, the image is divided into two categories: Class 1: Gray level range is $[0,t]$; Class 2: Gray level range is $\left[t+1,L-1\right]$. The probabilities and means of the two classes can be expressed as follows:

$${\omega }_0={\sum }_{i=0}^t P_i,{\mu }_0={\displaystyle \frac{{\sum }_{i=0}^t i{P}_i}{{\omega }_0}}{\omega }_1={\sum }_{i=t+1}^{L-1} P_i,{\mu }_1={\displaystyle \frac{{\sum }_{i=t+1}^{L-1} i{P}_i}{{\omega }_1}},$$

(20)

The overall grayscale mean of an image can be represented as follows:

$$\mu ={\sum }_{i=0}^{L-1} i{P}_i,$$

(21)

The inter-class variance under a single threshold condition can be expressed as follows:

$$v\left(t\right)={\omega }_0({\mu }_0-\mu {)}^2+{\omega }_1({\mu }_1-\mu {)}^2,$$

(22)

The optimal threshold $t^{*}$ is obtained by maximizing the inter-class variance, and can be expressed as follows:

$$t^{*}=\mathrm{a}\mathrm{r}\mathrm{g}\underset{0\le t\le L-1}{max} v(t),$$

(23)

2. Multi-threshold segmentation: When $k$ thresholds are selected, the image is divided into $k+1$ categories. The inter-class variance expansion is then expressed as…

$$v(t_1,t_2,\dots ,t_k)={\sum }_{i=1}^{k+1} {\omega }_i({\mu }_i-\mu {)}^2,$$

(24)

Wherein, the probability and mean of the $i$-th class are respectively:

$${\omega }_i={\sum }_{j=t_{i-1}+1}^{t_i} P_j{\mu }_i={\displaystyle \frac{{\sum }_{j=t_{i-1}+1}^{t_i} j{P}_j}{{\omega }_i}},i=\mathrm{1,2},\dots ,k+1,$$

(25)

The optimal threshold combination is obtained by maximizing the inter-class variance:

$$T_{best}=\mathrm{a}\mathrm{r}\mathrm{g}\underset{0\le t_1<\dots <t_k\le L-1}{max} v(t_1,t_2,\dots ,t_k),$$

(26)

This paper employs the MEIVYA to optimize the solution of the multi-threshold segmentation problem. To verify the algorithm’s performance, 11 improved swarm intelligence optimization algorithms are selected as comparison methods. To ensure experimental fairness, all algorithms use the same parameter settings: population size $N=50$, maximum number of iterations $T=200$, and Otsu’s inter-class variance is used as the unified objective function.

The experiment was conducted on 12 benchmark images with different visual features, using four threshold layers: 4, 6, 8, and 10. The benchmark image for image segmentation testing is shown in Figure 9. Each experiment was run independently 30 times, and the mean and standard deviation of the objective function were recorded to evaluate the segmentation accuracy and stability of the algorithm. All algorithm parameters were kept consistent to ensure the objectivity and comparability of the experimental results.

4.2. Performance Evaluation Criteria

To comprehensively assess the effectiveness of the proposed segmentation method, quantitative image quality measures are employed. In this study, three representative metrics are adopted: Peak Signal-to-Noise Ratio (PSNR), Structural Similarity Index Measure (SSIM), and Feature Similarity Index (FSIM). These indicators evaluate reconstruction fidelity from complementary perspectives, including pixel-wise error, structural consistency, and perceptual feature preservation.

Generally, larger PSNR and SSIM values indicate that the segmented image is more consistent with the original image in terms of intensity distribution and structural information. Similarly, a higher FSIM score suggests better preservation of perceptually significant features.

Peak Signal-to-Noise Ratio (PSNR): PSNR evaluates reconstruction quality based on the pixel-wise discrepancy between the original image $I$ and the segmented image $I^{\prime }$. It is derived from the Mean Squared Error (MSE), and is defined as:

$$PSNR=10 {\log }_{10}\left({\displaystyle \frac{{255}^2}{MSE}}\right).$$

(27)

The MSE is computed as:

$$MSE={\displaystyle \frac{1}{MN}}{\sum }_{j=1}^M {\sum }_{k=1}^N {\left[I(j,k)-I^{\prime }(j,k)\right]}^2,$$

(28)

where $M\times N$ denotes the image resolution. The terms $I(j,k)$ and $I^{\prime }(j,k)$ represent the gray-level intensities at pixel location $(j,k)$ in the original and segmented images, respectively. A lower MSE corresponds to a higher PSNR value, indicating improved similarity at the pixel level.

In the continuous image domain, the mean squared error between the original image $I(j,k)$ and the segmented image $I^{\prime }(j,k)$ can be defined as:

$$\mathrm{M}\mathrm{S}\mathrm{E}={\displaystyle \frac{1}{|\Omega |}}{\iint }_{\Omega } [I(j,k)-I^{\prime }(j,k){]}^{2}dxdy,$$

(29)

where $\Omega$ denotes the image domain. For digital images, this continuous formulation is discretized into Equation (28).

Structural Similarity Index (SSIM): Unlike PSNR, which measures absolute error, SSIM evaluates similarity by considering structural information perceived by the human visual system. It jointly analyzes luminance, contrast, and structural correlation between two images.

For images $I$ and $I^{\prime }$, SSIM is expressed as:

$$SSIM(I,I^{\prime })={\displaystyle \frac{(2{\mu }_I{\mu }_{I^{\prime }}+C_1)(2{\sigma }_{I{I}^{\prime }}+C_2)}{({\mu }_I^2+{\mu }_{I^{\prime }}^2+C_1)({\sigma }_I^2+{\sigma }_{I^{\prime }}^2+C_2)}},$$

(30)

where ${\mu }_I$ and ${\mu }_{I^{\prime }}$ denote the mean intensities, ${\sigma }_I^2$ and ${\sigma }_{I^{\prime }}^2$ represent the variances, and ${\sigma }_{{II}^{\prime }}$ is the covariance between the two images. The constants $C_1$ and $C_2$ are introduced to ensure numerical stability and are defined as:

$$C_1=(K_{1}L{)}^2,C_2=(K_{2}L{)}^2,$$

(31)

where $K_1=0.01$, $K_2=0.03$ and $L$ is the dynamic range of pixel intensities. SSIM values closer to 1 indicate stronger structural consistency.

Feature Similarity Index (FSIM): FSIM evaluates image quality from a perceptual standpoint by emphasizing low-level features that are significant to human vision. In particular, it utilizes phase congruency (PC) as a primary feature descriptor, since phase congruency effectively captures perceptually meaningful structures such as edges and corners.

The FSIM between $I$ and $I^{\prime }$ is calculated as:

$$FSIM(I,I^{\prime })={\displaystyle \frac{\sum _{x\in \Omega } S_L(x)\cdot P{C}_m(x)}{\sum _{x\in \Omega } P{C}_m(x)}},$$

(32)

where $x$ denotes a pixel location and $\Omega$ represents the image domain. The term

$$P{C}_m\left(x\right)=max\left\{P{C}_1\right(x),P{C}_2(x\left)\right\},$$

(33)

with $P{C}_1\left(x\right)$ and $P{C}_2\left(x\right)$ being the phase congruency values of the original and segmented images, respectively. $S_L\left(x\right)$ denotes the local similarity function at pixel $x$.

A larger FSIM value implies that the segmented image better retains perceptually important structural features.

4.3. Applicability of PSNR, SSIM, and FSIM in Multi-Threshold Segmentation

PSNR, SSIM, and FSIM are widely used image quality assessment metrics and are suitable for evaluating multi-threshold image segmentation results from different perspectives. PSNR mainly measures the pixel-level reconstruction quality between the segmented image and the original image. A higher PSNR value indicates that the segmented image preserves more original information and suffers from less distortion. However, PSNR only reflects the global pixel error and is less sensitive to structural information.

SSIM evaluates the similarity between the original image and the segmented image in terms of luminance, contrast, and structural characteristics. Compared with PSNR, SSIM is more consistent with human visual perception and is more suitable for measuring whether the segmented image preserves the main structure of the original image. Nevertheless, SSIM may not fully capture local edge details in complex segmentation scenarios.

FSIM further considers low-level visual features, such as phase congruency and gradient magnitude, which makes it more effective in evaluating the preservation of edges, contours, and texture details after segmentation. Therefore, FSIM is particularly suitable for assessing the visual quality of multi-threshold segmentation results with complex boundaries and rich textures. However, FSIM is relatively more computationally expensive than PSNR and SSIM.

By combining PSNR, SSIM, and FSIM, the segmentation performance can be comprehensively evaluated from the aspects of pixel fidelity, structural similarity, and feature preservation, thereby providing a more reliable assessment of segmentation quality.

4.4. Experimental Results Analysis

In this work, the MEIVYA is applied to solve the multi-level threshold image segmentation problem. The Otsu method is adopted as the optimization criterion, where the optimal threshold set for 12 benchmark images is obtained by maximizing the between-class variance.

To quantitatively assess the segmentation performance, three widely recognized image quality measures—PSNR, SSIM, and FSIM—are utilized. The segmentation quality improves as the values of these metrics increase, reflecting lower distortion, better structural consistency, and stronger preservation of perceptually significant features. The corresponding experimental results are presented in the following section.

Figure 10 shows the experimental results of MEIVYA in a multi-threshold image segmentation task. As can be seen from the figure, with the increase in the number of thresholds, the target region, edge contours, and details in the image are gradually separated more clearly, indicating that MEIVYA can effectively search for better threshold combinations, thereby improving segmentation quality. Compared to the case where only a rough distinction between foreground and background can be made with a lower number of thresholds, higher thresholds result in more accurate representation of texture information, grayscale levels, and target edges in the image, leading to a more natural, continuous, and visually readable overall segmentation result.

Meanwhile, the segmentation results of different test images show that MEIVYA can still maintain the integrity of the target region well when processing images with complex textures, uneven gray-level distribution, and blurred boundaries, reducing detail loss and over-segmentation. This indicates that the algorithm has strong global search capabilities and good local optimization capabilities, and can find more reasonable segmentation schemes in multi-threshold spaces. Furthermore, although the problem complexity increases significantly with the number of thresholds, MEIVYA still maintains relatively stable segmentation results, demonstrating its strong adaptability to high-dimensional complex optimization problems. Overall, Figure 10 shows that MEIVYA can obtain clear target structures, natural edge transitions, and good visual quality in multi-level threshold image segmentation tasks, verifying its effectiveness and robustness in practical image processing applications.

As shown in

Table 7. Ave and Std of the optimal fitness values.

Image	Threshold	Metric	PSO	DE	CMAES	INFO	SMA	RUN	HGS	DOA	GWCA	SAO	IVYA	MEIVYA
Brain	4	Ave	3.731 × 103	3.731 × 103	3.731 × 103	3.710 × 103	3.731 × 103	3.731 × 103	3.731 × 103	3.730 × 103	3.731 × 103	3.731 × 103	3.721 × 103	3.731 × 103
		Std	1.183 × 10−2	2.313 × 10−12	3.540 × 10−2	1.037 × 101	2.313 × 10−12	2.313 × 10−12	2.313 × 10−12	1.427 × 10−1	7.075 × 10−2	2.313 × 10−12	7.748 × 100	2.313 × 10−12
	6	Ave	3.751 × 103	3.751 × 103	3.750 × 103	3.733 × 103	3.751 × 103	3.751 × 103	3.751 × 103	3.750 × 103	3.750 × 103	3.751 × 103	3.744 × 103	3.750 × 103
		Std	5.411 × 10−1	5.166 × 10−1	7.469 × 10−1	5.664 × 100	6.060 × 10−1	5.898 × 10−1	6.177 × 10−1	9.755 × 10−1	4.490 × 10−1	6.311 × 10−1	3.376 × 100	2.248 × 10−1
	8	Ave	3.763 × 103	3.764 × 103	3.763 × 103	3.746 × 103	3.764 × 103	3.763 × 103	3.764 × 103	3.761 × 103	3.762 × 103	3.764 × 103	3.755 × 103	3.763 × 103
		Std	4.454 × 10−1	5.280 × 10−1	7.066 × 10−1	4.910 × 100	4.338 × 10−1	8.474 × 10−1	4.697 × 10−1	1.787 × 100	1.931 × 100	6.113 × 10−2	3.184 × 100	1.221 × 100
	10	Ave	3.768 × 103	3.769 × 103	3.769 × 103	3.754 × 103	3.769 × 103	3.769 × 103	3.769 × 103	3.766 × 103	3.768 × 103	3.769 × 103	3.762 × 103	3.769 × 103
		Std	3.328 × 10−1	2.954 × 10−1	5.705 × 10−1	3.702 × 100	1.478 × 10−1	2.473 × 10−1	6.733 × 10−1	1.414 × 100	1.604 × 100	2.825 × 10−1	2.581 × 100	9.947 × 10−2
satellite	4	Ave	5.179 × 102	5.179 × 102	5.178 × 102	4.459 × 102	5.179 × 102	5.179 × 102	5.179 × 102	5.179 × 102	5.179 × 102	5.179 × 102	4.989 × 102	5.179 × 102
		Std	1.669 × 10−2	3.469 × 10−13	2.538 × 10−1	4.318 × 101	3.469 × 10−13	3.469 × 10−13	3.469 × 10−13	9.820 × 10−2	3.301 × 10−2	3.469 × 10−13	1.390 × 101	3.469 × 10−13
	6	Ave	5.365 × 102	5.365 × 102	5.327 × 102	4.560 × 102	5.365 × 102	5.365 × 102	5.364 × 102	5.357 × 102	5.360 × 102	5.007 × 102	4.430 × 102	5.365 × 102
		Std	9.030 × 10−2	7.459 × 10−3	1.919 × 101	6.303 × 101	3.371 × 10−2	1.331 × 10−2	1.016 × 10−1	6.182 × 10−1	1.324 × 100	1.361 × 102	1.773 × 102	9.821 × 10−3
	8	Ave	5.443 × 102	5.447 × 102	3.742 × 102	4.825 × 102	5.446 × 102	5.447 × 102	5.260 × 102	5.434 × 102	5.438 × 102	4.357 × 102	3.681 × 102	5.447 × 102
		Std	2.748 × 10−1	1.965 × 10−2	2.507 × 102	4.569 × 101	8.439 × 10−2	1.662 × 10−2	9.936 × 101	7.873 × 10−1	1.180 × 100	2.216 × 102	2.453 × 102	6.280 × 10−3
	10	Ave	5.480 × 102	5.486 × 102	1.940 × 102	5.195 × 102	5.485 × 102	5.486 × 102	5.119 × 102	5.468 × 102	5.477 × 102	4.023 × 102	1.748 × 102	5.486 × 102
		Std	2.883 × 10−1	9.632 × 10−2	2.600 × 102	2.964 × 101	1.473 × 10−1	9.624 × 10−2	1.392 × 102	7.533 × 10−1	9.117 × 10−1	2.468 × 102	2.519 × 102	7.109 × 10−2
Face	4	Ave	2.122 × 103	2.122 × 103	2.122 × 103	2.058 × 103	2.122 × 103	2.122 × 103	2.122 × 103	2.122 × 103	2.122 × 103	2.122 × 103	2.096 × 103	2.122 × 103
		Std	1.756 × 10−2	0.000 × 100	9.322 × 10−1	2.478 × 101	1.302 × 10−2	1.302 × 10−2	1.744 × 10−2	3.670 × 10−1	1.784 × 10−1	1.302 × 10−2	2.015 × 101	1.188 × 10−2
	6	Ave	2.185 × 103	2.185 × 103	2.185 × 103	2.132 × 103	2.185 × 103	2.185 × 103	2.185 × 103	2.182 × 103	2.184 × 103	2.185 × 103	2.156 × 103	2.185 × 103
		Std	3.641 × 10−1	3.137 × 10−2	1.450 × 100	2.090 × 101	1.363 × 10−2	2.884 × 10−2	2.841 × 10−2	2.425 × 100	1.144 × 100	9.863 × 10−3	1.257 × 101	3.607 × 10−3
	8	Ave	2.209 × 103	2.211 × 103	2.209 × 103	2.170 × 103	2.211 × 103	2.211 × 103	2.211 × 103	2.205 × 103	2.210 × 103	2.211 × 103	2.184 × 103	2.211 × 103
		Std	1.008 × 100	6.854 × 10−2	3.302 × 100	1.315 × 101	2.648 × 10−1	5.755 × 10−1	1.062 × 100	3.754 × 100	1.385 × 100	7.391 × 10−2	1.180 × 101	5.123 × 10−2
	10	Ave	2.222 × 103	2.225 × 103	2.222 × 103	2.191 × 103	2.224 × 103	2.223 × 103	2.224 × 103	2.217 × 103	2.223 × 103	2.225 × 103	2.204 × 103	2.225 × 103
		Std	9.142 × 10−1	9.823 × 10−2	3.173 × 100	1.284 × 101	7.050 × 10−1	9.599 × 10−1	1.542 × 100	2.696 × 100	1.244 × 100	1.030 × 10−1	8.594 × 100	3.391 × 10−1
cell	4	Ave	2.567 × 103	2.567 × 103	2.567 × 103	2.525 × 103	2.567 × 103	2.567 × 103	2.567 × 103	2.567 × 103	2.567 × 103	2.567 × 103	2.552 × 103	2.567 × 103
		Std	9.250 × 10−13	9.250 × 10−13	2.548 × 10−1	2.244 × 101	9.250 × 10−13	9.250 × 10−13	9.250 × 10−13	5.159 × 10−1	5.943 × 10−2	9.250 × 10−13	1.296 × 101	9.250 × 10−13
	6	Ave	2.596 × 103	2.596 × 103	2.596 × 103	2.565 × 103	2.596 × 103	2.596 × 103	2.596 × 103	2.596 × 103	2.596 × 103	2.596 × 103	2.580 × 103	2.596 × 103
		Std	1.858 × 10−1	6.253 × 10−2	6.406 × 10−1	1.264 × 101	1.210 × 10−1	1.354 × 10−1	1.865 × 100	9.712 × 10−1	9.519 × 10−1	1.721 × 10−1	8.262 × 100	1.481 × 10−1
	8	Ave	2.611 × 103	2.612 × 103	2.611 × 103	2.579 × 103	2.612 × 103	2.612 × 103	2.611 × 103	2.609 × 103	2.611 × 103	2.612 × 103	2.596 × 103	2.612 × 103
		Std	5.148 × 10−1	2.603 × 10−1	1.237 × 100	9.673 × 100	2.448 × 10−1	3.245 × 10−1	1.608 × 100	1.523 × 100	1.808 × 100	5.062 × 10−2	5.851 × 100	1.862 × 10−1
	10	Ave	2.618 × 103	2.619 × 103	2.619 × 103	2.592 × 103	2.619 × 103	2.619 × 103	2.619 × 103	2.616 × 103	2.619 × 103	2.620 × 103	2.604 × 103	2.620 × 103
		Std	5.741 × 10−1	3.800 × 10−1	1.482 × 100	1.277 × 101	4.101 × 10−1	3.322 × 10−1	8.805 × 10−1	1.435 × 100	7.162 × 10−1	2.002 × 10−1	7.007 × 100	1.723 × 10−1
lax	4	Ave	9.652 × 102	9.652 × 102	9.650 × 102	9.148 × 102	9.652 × 102	9.652 × 102	9.652 × 102	9.651 × 102	9.651 × 102	9.652 × 102	9.441 × 102	9.652 × 102
		Std	8.611 × 10−3	5.069 × 10−3	3.773 × 10−1	2.428 × 101	5.069 × 10−3	1.156 × 10−13	3.648 × 10−3	1.244 × 10−1	5.356 × 10−2	1.156 × 10−13	1.480 × 101	1.156 × 10−13
	6	Ave	9.919 × 102	9.921 × 102	9.920 × 102	9.479 × 102	9.921 × 102	9.921 × 102	9.920 × 102	9.908 × 102	9.913 × 102	9.921 × 102	9.740 × 102	9.921 × 102
		Std	2.843 × 10−1	6.211 × 10−3	1.545 × 10−1	2.243 × 101	2.031 × 10−2	6.266 × 10−3	3.829 × 10−2	1.262 × 100	9.515 × 10−1	1.525 × 10−2	1.037 × 101	5.600 × 10−3
	8	Ave	1.003 × 103	1.004 × 103	1.004 × 103	9.743 × 102	1.004 × 103	1.004 × 103	1.004 × 103	1.001 × 103	1.003 × 103	1.004 × 103	9.922 × 102	1.004 × 103
		Std	4.073 × 10−1	3.859 × 10−2	6.457 × 10−1	1.055 × 101	9.308 × 10−2	8.854 × 10−1	8.748 × 10−1	1.668 × 100	7.552 × 10−1	6.955 × 10−2	5.364 × 100	9.268 × 10−2
	10	Ave	1.009 × 103	1.011 × 103	1.011 × 103	9.861 × 102	1.011 × 103	1.011 × 103	1.011 × 103	1.007 × 103	1.010 × 103	1.011 × 103	9.964 × 102	1.011 × 103
		Std	4.630 × 10−1	1.497 × 10−1	7.028 × 10−1	1.036 × 101	5.524 × 10−1	5.751 × 10−1	6.028 × 10−1	1.438 × 100	7.993 × 10−1	6.349 × 10−2	6.673 × 100	3.019 × 10−1
Peppers	4	Ave	2.701 × 103	2.701 × 103	2.701 × 103	2.631 × 103	2.701 × 103	2.701 × 103	2.700 × 103	2.701 × 103	2.700 × 103	2.701 × 103	2.683 × 103	2.701 × 103
		Std	5.953 × 10−4	4.841 × 10−4	1.562 × 100	2.833 × 101	9.327 × 10−1	1.294 × 100	2.295 × 100	1.114 × 100	1.977 × 100	1.294 × 100	1.697 × 101	2.332 × 10−4
	6	Ave	2.769 × 103	2.769 × 103	2.768 × 103	2.713 × 103	2.769 × 103	2.769 × 103	2.769 × 103	2.766 × 103	2.768 × 103	2.769 × 103	2.742 × 103	2.769 × 103
		Std	5.135 × 10−1	2.034 × 10−2	1.759 × 100	2.550 × 101	4.973 × 10−3	4.768 × 10−3	1.007 × 10−2	3.265 × 100	1.291 × 100	5.326 × 10−3	1.470 × 101	0.000 × 100
	8	Ave	2.794 × 103	2.796 × 103	2.795 × 103	2.748 × 103	2.795 × 103	2.796 × 103	2.795 × 103	2.791 × 103	2.793 × 103	2.796 × 103	2.771 × 103	2.796 × 103
		Std	1.036 × 100	8.170 × 10−2	1.125 × 100	1.978 × 101	2.324 × 100	1.821 × 10−1	1.680 × 100	3.211 × 100	2.686 × 100	1.958 × 10−2	1.070 × 101	1.792 × 10−2
	10	Ave	2.806 × 103	2.808 × 103	2.808 × 103	2.769 × 103	2.808 × 103	2.808 × 103	2.808 × 103	2.803 × 103	2.806 × 103	2.809 × 103	2.787 × 103	2.808 × 103
		Std	7.998 × 10−1	8.736 × 10−1	1.789 × 100	1.555 × 101	9.074 × 10−1	6.552 × 10−1	7.332 × 10−1	2.763 × 100	2.018 × 100	5.183 × 10−1	7.680 × 100	7.267 × 10−1
milkdrop	4	Ave	2.491 × 103	2.491 × 103	2.491 × 103	2.426 × 103	2.491 × 103	2.491 × 103	2.491 × 103	2.491 × 103	2.491 × 103	2.491 × 103	2.465 × 103	2.491 × 103
		Std	1.444 × 10−2	4.426 × 10−3	1.822 × 10−1	2.818 × 101	4.625 × 10−13	4.625 × 10−13	4.625 × 10−13	4.101 × 10−1	1.120 × 10−1	4.625 × 10−13	2.200 × 101	4.625 × 10−13
	6	Ave	2.564 × 103	2.564 × 103	2.564 × 103	2.504 × 103	2.564 × 103	2.563 × 103	2.563 × 103	2.561 × 103	2.562 × 103	2.564 × 103	2.541 × 103	2.564 × 103
		Std	3.019 × 10−1	1.625 × 100	6.895 × 10−2	2.031 × 101	1.624 × 100	2.724 × 100	2.715 × 100	2.936 × 100	3.388 × 100	1.625 × 100	1.134 × 101	1.624 × 100
	8	Ave	2.585 × 103	2.586 × 103	2.585 × 103	2.545 × 103	2.586 × 103	2.586 × 103	2.586 × 103	2.582 × 103	2.585 × 103	2.586 × 103	2.566 × 103	2.586 × 103
		Std	6.863 × 10−1	1.500 × 10−1	1.028 × 100	1.735 × 101	1.542 × 10−1	1.253 × 10−1	1.788 × 10−1	2.858 × 100	8.617 × 10−1	1.460 × 10−1	1.016 × 101	1.437 × 10−1
	10	Ave	2.597 × 103	2.599 × 103	2.598 × 103	2.566 × 103	2.599 × 103	2.599 × 103	2.599 × 103	2.593 × 103	2.598 × 103	2.600 × 103	2.581 × 103	2.599 × 103
		Std	1.127 × 100	5.334 × 10−1	2.205 × 100	1.190 × 101	7.703 × 10−1	1.764 × 10−1	6.451 × 10−1	2.268 × 100	1.889 × 100	2.546 × 10−1	6.470 × 100	5.891 × 10−1
testpat	4	Ave	4.717 × 103	4.717 × 103	4.717 × 103	4.628 × 103	4.717 × 103	4.717 × 103	4.717 × 103	4.717 × 103	4.717 × 103	4.717 × 103	4.686 × 103	4.717 × 103
		Std	5.045 × 10−3	9.250 × 10−13	6.355 × 10−1	3.764 × 101	9.250 × 10−13	9.250 × 10−13	9.250 × 10−13	9.870 × 10−1	7.670 × 10−2	9.250 × 10−13	2.020 × 101	9.250 × 10−13
	6	Ave	4.789 × 103	4.789 × 103	4.789 × 103	4.736 × 103	4.789 × 103	4.789 × 103	4.790 × 103	4.786 × 103	4.789 × 103	4.790 × 103	4.761 × 103	4.790 × 103
		Std	6.449 × 10−1	3.574 × 10−1	2.082 × 100	1.850 × 101	5.319 × 10−1	3.573 × 10−1	1.076 × 10−2	2.461 × 100	1.324 × 100	1.076 × 10−2	1.625 × 101	2.775 × 10−12
	8	Ave	4.820 × 103	4.822 × 103	4.821 × 103	4.770 × 103	4.822 × 103	4.822 × 103	4.821 × 103	4.814 × 103	4.820 × 103	4.821 × 103	4.798 × 103	4.822 × 103
		Std	1.143 × 100	1.335 × 10−1	1.496 × 100	1.734 × 101	8.569 × 10−1	1.069 × 10−1	9.820 × 10−1	4.815 × 100	2.408 × 100	2.404 × 100	8.496 × 100	6.843 × 10−2
	10	Ave	4.835 × 103	4.838 × 103	4.837 × 103	4.805 × 103	4.838 × 103	4.839 × 103	4.837 × 103	4.829 × 103	4.837 × 103	4.839 × 103	4.816 × 103	4.838 × 103
		Std	9.789 × 10−1	7.104 × 10−1	2.977 × 100	9.955 × 100	6.546 × 10−1	6.627 × 10−1	2.417 × 100	3.331 × 100	1.373 × 100	3.843 × 10−1	7.851 × 100	8.779 × 10−1
bank	4	Ave	3.602 × 103	3.602 × 103	3.602 × 103	3.538 × 103	3.602 × 103	3.602 × 103	3.602 × 103	3.602 × 103	3.601 × 103	3.602 × 103	3.576 × 103	3.602 × 103
		Std	8.209 × 10−3	2.957 × 10−3	2.287 × 10−1	2.968 × 101	6.443 × 10−3	2.957 × 10−3	3.020 × 10−1	3.651 × 10−1	4.522 × 10−1	5.860 × 10−3	2.225 × 101	1.850 × 10−12
	6	Ave	3.681 × 103	3.681 × 103	3.681 × 103	3.624 × 103	3.681 × 103	3.681 × 103	3.681 × 103	3.677 × 103	3.680 × 103	3.681 × 103	3.655 × 103	3.681 × 103
		Std	7.157 × 10−1	1.889 × 10−2	1.771 × 100	2.582 × 101	1.960 × 10−2	2.328 × 10−2	1.179 × 10−1	4.677 × 100	1.804 × 100	9.352 × 10−3	1.374 × 101	1.388 × 10−12
	8	Ave	3.710 × 103	3.711 × 103	3.711 × 103	3.669 × 103	3.711 × 103	3.711 × 103	3.711 × 103	3.707 × 103	3.710 × 103	3.711 × 103	3.686 × 103	3.712 × 103
		Std	1.037 × 100	4.163 × 10−1	1.579 × 100	1.955 × 101	4.563 × 10−1	6.011 × 10−1	1.741 × 100	3.294 × 100	9.903 × 10−1	1.737 × 100	1.002 × 101	1.619 × 10−1
	10	Ave	3.725 × 103	3.727 × 103	3.725 × 103	3.692 × 103	3.727 × 103	3.726 × 103	3.727 × 103	3.720 × 103	3.725 × 103	3.727 × 103	3.705 × 103	3.727 × 103
		Std	1.089 × 100	5.734 × 10−1	3.503 × 100	1.462 × 101	1.149 × 100	1.732 × 100	1.405 × 100	2.251 × 100	1.582 × 100	9.254 × 10−1	8.610 × 100	8.951 × 10−1
boat	4	Ave	1.970 × 103	1.970 × 103	1.970 × 103	1.921 × 103	1.970 × 103	1.970 × 103	1.970 × 103	1.970 × 103	1.970 × 103	1.970 × 103	1.952 × 103	1.970 × 103
		Std	1.260 × 10−2	0.000 × 100	1.038 × 10−1	2.048 × 101	0.000 × 100	0.000 × 100	0.000 × 100	6.114 × 10−1	2.024 × 10−1	0.000 × 100	1.669 × 101	0.000 × 100
	6	Ave	2.025 × 103	2.025 × 103	2.025 × 103	1.978 × 103	2.025 × 103	2.025 × 103	2.025 × 103	2.024 × 103	2.025 × 103	2.025 × 103	2.002 × 103	2.025 × 103
		Std	2.046 × 10−1	2.192 × 10−2	7.501 × 10−1	2.071 × 101	2.756 × 10−2	1.035 × 10−2	1.251 × 10−1	1.400 × 100	1.001 × 100	1.847 × 10−2	1.141 × 101	1.156 × 10−12
	8	Ave	2.048 × 103	2.049 × 103	2.048 × 103	2.008 × 103	2.049 × 103	2.049 × 103	2.049 × 103	2.045 × 103	2.047 × 103	2.049 × 103	2.031 × 103	2.049 × 103
		Std	7.996 × 10−1	4.761 × 10−2	1.847 × 100	1.199 × 101	2.129 × 10−1	8.304 × 10−2	1.099 × 10−1	2.128 × 100	1.689 × 100	2.686 × 10−2	1.006 × 101	2.597 × 10−2
	10	Ave	2.059 × 103	2.061 × 103	2.060 × 103	2.027 × 103	2.061 × 103	2.060 × 103	2.061 × 103	2.056 × 103	2.059 × 103	2.061 × 103	2.044 × 103	2.061 × 103
		Std	6.603 × 10−1	2.052 × 10−1	1.547 × 100	1.179 × 101	2.551 × 10−1	2.768 × 10−1	1.736 × 10−1	2.017 × 100	1.139 × 100	1.573 × 10−1	6.701 × 100	1.646 × 10−1
plane	4	Ave	1.995 × 103	1.995 × 103	1.995 × 103	1.953 × 103	1.995 × 103	1.995 × 103	1.995 × 103	1.995 × 103	1.995 × 103	1.995 × 103	1.972 × 103	1.995 × 103
		Std	1.494 × 10−2	7.890 × 10−3	5.622 × 10−1	1.513 × 101	1.156 × 10−12	1.156 × 10−12	1.156 × 10−12	3.188 × 10−1	5.555 × 10−2	1.156 × 10−12	1.409 × 101	1.156 × 10−12
	6	Ave	2.036 × 103	2.036 × 103	2.036 × 103	1.996 × 103	2.036 × 103	2.036 × 103	2.036 × 103	2.034 × 103	2.035 × 103	2.036 × 103	2.015 × 103	2.036 × 103
		Std	2.095 × 10−1	9.388 × 10−2	1.089 × 100	1.678 × 101	1.650 × 10−2	3.028 × 10−2	3.634 × 10−2	2.045 × 100	1.084 × 100	6.623 × 10−3	1.295 × 101	2.876 × 10−4
	8	Ave	2.053 × 103	2.054 × 103	2.054 × 103	2.017 × 103	2.054 × 103	2.053 × 103	2.053 × 103	2.050 × 103	2.053 × 103	2.054 × 103	2.034 × 103	2.054 × 103
		Std	5.675 × 10−1	7.735 × 10−2	5.936 × 10−1	1.567 × 101	5.049 × 10−2	1.542 × 100	1.350 × 100	1.982 × 100	1.258 × 100	2.929 × 10−2	9.581 × 100	1.802 × 10−2
	10	Ave	2.061 × 103	2.063 × 103	2.061 × 103	2.030 × 103	2.062 × 103	2.060 × 103	2.061 × 103	2.058 × 103	2.060 × 103	2.063 × 103	2.045 × 103	2.063 × 103
		Std	6.343 × 10−1	3.622 × 10−1	2.042 × 100	1.808 × 101	1.141 × 100	1.988 × 100	2.173 × 100	2.276 × 100	2.028 × 100	8.611 × 10−1	7.121 × 100	2.146 × 10−1
saturn	4	Ave	5.222 × 103	5.222 × 103	5.222 × 103	5.190 × 103	5.222 × 103	5.222 × 103	5.222 × 103	5.222 × 103	5.222 × 103	5.222 × 103	5.208 × 103	5.222 × 103
		Std	5.080 × 10−3	9.250 × 10−13	1.019 × 10−1	1.818 × 101	3.400 × 10−3	2.447 × 10−3	3.400 × 10−3	7.756 × 10−1	7.058 × 10−2	9.250 × 10−13	1.031 × 101	9.250 × 10−13
	6	Ave	5.273 × 103	5.273 × 103	5.272 × 103	5.243 × 103	5.273 × 103	5.273 × 103	5.273 × 103	5.271 × 103	5.273 × 103	5.273 × 103	5.260 × 103	5.273 × 103
		Std	2.038 × 10−1	1.485 × 10−2	1.611 × 100	1.194 × 101	6.647 × 10−3	3.115 × 10−3	1.233 × 10−2	1.898 × 100	7.239 × 10−1	3.700 × 10−12	5.847 × 100	4.211 × 10−4
	8	Ave	5.293 × 103	5.294 × 103	5.293 × 103	5.273 × 103	5.294 × 103	5.294 × 103	5.294 × 103	5.290 × 103	5.293 × 103	5.294 × 103	5.281 × 103	5.294 × 103
		Std	6.203 × 10−1	4.955 × 10−2	1.238 × 100	8.482 × 100	2.097 × 10−1	2.709 × 10−1	2.448 × 10−1	1.752 × 100	1.037 × 100	1.674 × 10−1	5.862 × 100	1.759 × 10−1
	10	Ave	5.302 × 103	5.304 × 103	5.302 × 103	5.286 × 103	5.304 × 103	5.304 × 103	5.304 × 103	5.299 × 103	5.303 × 103	5.304 × 103	5.293 × 103	5.304 × 103
		Std	5.741 × 10−1	1.541 × 10−1	2.697 × 100	5.385 × 100	1.517 × 10−1	8.180 × 10−2	6.239 × 10−1	1.872 × 100	6.597 × 10−1	1.324 × 10−1	4.345 × 100	1.404 × 10−1

, MEIVYA demonstrates strong competitiveness in terms of optimal fitness values under different test images and different threshold numbers. Overall, the fitness values of all algorithms generally increase with the increase in the number of thresholds, indicating that more thresholds can more fully exploit the gray-level distribution information in the image, thus achieving better segmentation results. However, compared with other comparison algorithms, MEIVYA achieves a higher average fitness value in most cases, with a smaller standard deviation, indicating that it not only finds better threshold combinations but also exhibits good stability and consistency in multiple independent runs. Especially under higher threshold numbers, many comparison algorithms tend to experience premature convergence or large fluctuations in results due to the rapid expansion of the search space, while MEIVYA still maintains a high fitness value and a small standard deviation, demonstrating its strong global search capability and local exploitation capability. At the same time, the small standard deviation also indicates that the algorithm is less affected by random initialization and fluctuations in the search process, and can consistently obtain high-quality segmentation results. Overall, the results in Table 7 show that MEIVYA can achieve better objective function values and more stable optimization performance in multi-threshold image segmentation tasks, further verifying its effectiveness and robustness in complex image segmentation problems.

As shown in

Table 8. Ave and Std of all test images for PSNR.

Image	Threshold	Metric	PSO	DE	CMAES	INFO	SMA	RUN	HGS	DOA	GWCA	SAO	IVYA	MEIVYA
Brain	4	Ave	2.507 × 101	2.506 × 101	2.505 × 101	2.383 × 101	2.507 × 101	2.508 × 101	2.505 × 101	2.507 × 101	2.506 × 101	2.507 × 101	2.440 × 101	2.508 × 101
		Std	5.683 × 10−2	4.975 × 10−2	6.275 × 10−2	9.218 × 10−1	5.267 × 10−2	5.237 × 10−2	5.068 × 10−2	6.960 × 10−2	5.257 × 10−2	6.099 × 10−2	6.531 × 10−1	6.188 × 10−2
	6	Ave	2.733 × 101	2.753 × 101	2.746 × 101	2.548 × 101	2.749 × 101	2.745 × 101	2.747 × 101	2.741 × 101	2.756 × 101	2.751 × 101	2.646 × 101	2.760 × 101
		Std	2.233 × 10−1	1.525 × 10−1	2.827 × 10−1	7.265 × 10−1	1.629 × 10−1	1.910 × 10−1	2.027 × 10−1	2.939 × 10−1	1.866 × 10−1	1.841 × 10−1	4.791 × 10−1	7.822 × 10−2
	8	Ave	2.934 × 101	2.948 × 101	2.940 × 101	2.701 × 101	2.950 × 101	2.934 × 101	2.950 × 101	2.883 × 101	2.934 × 101	2.953 × 101	2.804 × 101	2.950 × 101
		Std	1.890 × 10−1	1.816 × 10−1	2.489 × 10−1	8.840 × 10−1	1.614 × 10−1	3.019 × 10−1	1.430 × 10−1	4.546 × 10−1	2.698 × 10−1	5.707 × 10−2	6.981 × 10−1	1.206 × 10−1
	10	Ave	3.062 × 101	3.096 × 101	3.088 × 101	2.768 × 101	3.097 × 101	3.084 × 101	3.070 × 101	3.014 × 101	3.064 × 101	3.099 × 101	2.921 × 101	3.109 × 101
		Std	2.271 × 10−1	2.065 × 10−1	2.789 × 10−1	6.365 × 10−1	1.578 × 10−1	2.588 × 10−1	2.877 × 10−1	4.963 × 10−1	4.066 × 10−1	1.714 × 10−1	6.333 × 10−1	1.042 × 10−1
satellite	4	Ave	1.745 × 101	1.745 × 101	1.742 × 101	1.735 × 101	1.745 × 101	1.745 × 101	1.745 × 101	1.741 × 101	1.745 × 101	1.745 × 101	2.013 × 101	1.745 × 101
		Std	3.864 × 10−3	1.084 × 10−14	1.456 × 10−1	3.148 × 100	1.084 × 10−14	1.084 × 10−14	1.084 × 10−14	1.143 × 10−1	4.000 × 10−2	1.084 × 10−14	2.213 × 100	1.084 × 10−14
	6	Ave	1.965 × 101	1.975 × 101	1.962 × 101	1.853 × 101	1.968 × 101	1.965 × 101	1.954 × 101	1.929 × 101	1.944 × 101	1.960 × 101	2.235 × 101	1.971 × 101
		Std	1.516 × 10−1	8.813 × 10−2	4.283 × 10−1	2.856 × 100	1.424 × 10−1	1.490 × 10−1	1.856 × 10−1	5.839 × 10−1	4.556 × 10−1	5.843 × 10−1	4.227 × 100	1.265 × 10−1
	8	Ave	2.060 × 101	2.090 × 101	1.995 × 101	2.148 × 101	2.079 × 101	2.081 × 101	2.047 × 101	2.057 × 101	2.123 × 101	2.028 × 101	2.357 × 101	2.088 × 101
		Std	4.228 × 10−1	1.542 × 10−1	3.264 × 100	3.913 × 100	2.661 × 10−1	1.693 × 10−1	1.107 × 100	7.785 × 10−1	1.587 × 100	2.348 × 100	3.803 × 100	1.238 × 10−1
	10	Ave	2.178 × 101	2.164 × 101	2.123 × 101	2.323 × 101	2.199 × 101	2.173 × 101	2.169 × 101	2.167 × 101	2.256 × 101	2.120 × 101	2.199 × 101	2.171 × 101
		Std	8.210 × 10−1	3.478 × 10−1	2.650 × 100	3.432 × 100	8.093 × 10−1	4.785 × 10−1	1.150 × 100	1.735 × 100	1.978 × 100	2.107 × 100	4.106 × 100	3.427 × 10−1
Face	4	Ave	1.975 × 101	1.976 × 101	1.973 × 101	1.799 × 101	1.975 × 101	1.975 × 101	1.974 × 101	1.974 × 101	1.973 × 101	1.975 × 101	1.925 × 101	1.976 × 101
		Std	2.338 × 10−2	1.084 × 10−14	9.861 × 10−2	8.142 × 10−1	2.237 × 10−2	2.237 × 10−2	2.994 × 10−2	5.284 × 10−2	4.755 × 10−2	2.237 × 10−2	4.620 × 10−1	2.040 × 10−2
	6	Ave	2.257 × 101	2.259 × 101	2.257 × 101	2.029 × 101	2.259 × 101	2.261 × 101	2.259 × 101	2.256 × 101	2.265 × 101	2.260 × 101	2.157 × 101	2.260 × 101
		Std	8.582 × 10−2	2.301 × 10−2	1.272 × 10−1	8.901 × 10−1	2.115 × 10−2	1.412 × 10−2	3.210 × 10−2	1.455 × 10−1	1.032 × 10−1	1.470 × 10−2	5.424 × 10−1	8.853 × 10−3
	8	Ave	2.476 × 101	2.497 × 101	2.485 × 101	2.218 × 101	2.499 × 101	2.503 × 101	2.499 × 101	2.438 × 101	2.484 × 101	2.499 × 101	2.316 × 101	2.498 × 101
		Std	2.170 × 10−1	4.203 × 10−2	3.208 × 10−1	7.463 × 10−1	6.185 × 10−2	4.765 × 10−2	5.642 × 10−2	4.392 × 10−1	2.178 × 10−1	2.591 × 10−2	7.437 × 10−1	3.042 × 10−2
	10	Ave	2.635 × 101	2.675 × 101	2.641 × 101	2.339 × 101	2.678 × 101	2.675 × 101	2.667 × 101	2.585 × 101	2.658 × 101	2.677 × 101	2.476 × 101	2.675 × 101
		Std	2.095 × 10−1	3.564 × 10−2	4.626 × 10−1	9.082 × 10−1	6.050 × 10−2	8.437 × 10−2	2.429 × 10−1	4.165 × 10−1	2.389 × 10−1	3.770 × 10−2	7.911 × 10−1	8.814 × 10−2
cell	4	Ave	1.864 × 101	1.864 × 101	1.865 × 101	1.727 × 101	1.864 × 101	1.864 × 101	1.864 × 101	1.860 × 101	1.864 × 101	1.864 × 101	1.838 × 101	1.864 × 101
		Std	1.084 × 10−14	1.084 × 10−14	3.906 × 10−2	1.124 × 100	1.084 × 10−14	1.084 × 10−14	1.084 × 10−14	1.481 × 10−1	1.835 × 10−2	1.084 × 10−14	6.813 × 10−1	1.084 × 10−14
	6	Ave	1.985 × 101	1.993 × 101	1.985 × 101	1.876 × 101	1.988 × 101	1.996 × 101	1.971 × 101	1.976 × 101	1.980 × 101	1.995 × 101	2.067 × 101	1.994 × 101
		Std	1.868 × 10−1	2.094 × 10−1	2.216 × 10−1	1.166 × 100	2.323 × 10−1	1.755 × 10−1	2.647 × 10−1	3.009 × 10−1	2.536 × 10−1	1.791 × 10−1	1.675 × 100	1.952 × 10−1
	8	Ave	2.069 × 101	2.074 × 101	2.074 × 101	2.024 × 101	2.077 × 101	2.077 × 101	2.068 × 101	2.050 × 101	2.061 × 101	2.071 × 101	2.238 × 101	2.076 × 101
		Std	2.152 × 10−1	5.322 × 10−2	1.201 × 10−1	2.276 × 100	7.896 × 10−2	5.584 × 10−2	1.581 × 10−1	2.983 × 10−1	2.572 × 10−1	5.885 × 10−2	2.267 × 100	6.557 × 10−2
	10	Ave	2.127 × 101	2.133 × 101	2.128 × 101	2.093 × 101	2.134 × 101	2.130 × 101	2.120 × 101	2.081 × 101	2.113 × 101	2.133 × 101	2.342 × 101	2.134 × 101
		Std	2.869 × 10−1	9.970 × 10−2	2.133 × 10−1	2.595 × 100	1.283 × 10−1	1.126 × 10−1	1.894 × 10−1	3.381 × 10−1	2.500 × 10−1	7.357 × 10−2	2.546 × 100	8.639 × 10−2
lax	4	Ave	1.675 × 101	1.676 × 101	1.673 × 101	1.658 × 101	1.676 × 101	1.676 × 101	1.676 × 101	1.670 × 101	1.675 × 101	1.676 × 101	1.852 × 101	1.676 × 101
		Std	2.781 × 10−2	2.576 × 10−4	1.106 × 10−1	2.013 × 100	2.576 × 10−4	3.613 × 10−15	1.854 × 10−4	8.498 × 10−2	5.792 × 10−2	3.613 × 10−15	1.832 × 100	3.613 × 10−15
	6	Ave	1.829 × 101	1.829 × 101	1.826 × 101	1.956 × 101	1.830 × 101	1.829 × 101	1.823 × 101	1.791 × 101	1.806 × 101	1.830 × 101	2.189 × 101	1.829 × 101
		Std	2.384 × 10−1	6.216 × 10−4	1.022 × 10−1	2.808 × 100	7.807 × 10−2	5.340 × 10−4	1.390 × 10−1	3.935 × 10−1	4.075 × 10−1	4.471 × 10−2	2.375 × 100	5.059 × 10−4
	8	Ave	1.985 × 101	2.004 × 101	1.987 × 101	2.056 × 101	1.993 × 101	2.023 × 101	1.986 × 101	1.897 × 101	1.966 × 101	2.007 × 101	2.413 × 101	1.994 × 101
		Std	5.629 × 10−1	2.046 × 10−1	3.189 × 10−1	3.048 × 100	2.902 × 10−1	7.682 × 10−1	5.030 × 10−1	6.985 × 10−1	6.792 × 10−1	1.968 × 10−1	2.525 × 100	2.451 × 10−1
	10	Ave	2.107 × 101	2.119 × 101	2.118 × 101	2.221 × 101	2.139 × 101	2.121 × 101	2.118 × 101	2.034 × 101	2.093 × 101	2.133 × 101	2.603 × 101	2.110 × 101
		Std	1.020 × 100	4.135 × 10−1	5.922 × 10−1	3.032 × 100	5.315 × 10−1	6.714 × 10−1	6.220 × 10−1	1.631 × 100	9.540 × 10−1	4.174 × 10−1	1.674 × 100	4.594 × 10−1
Peppers	4	Ave	2.045 × 101	2.046 × 101	2.042 × 101	1.828 × 101	2.043 × 101	2.042 × 101	2.030 × 101	2.043 × 101	2.034 × 101	2.041 × 101	1.973 × 101	2.046 × 101
		Std	1.429 × 10−2	1.162 × 10−2	9.706 × 10−2	8.599 × 10−1	1.146 × 10−1	1.519 × 10−1	2.699 × 10−1	1.118 × 10−1	2.389 × 10−1	1.511 × 10−1	5.872 × 10−1	5.598 × 10−3
	6	Ave	2.323 × 101	2.323 × 101	2.319 × 101	2.058 × 101	2.323 × 101	2.323 × 101	2.323 × 101	2.311 × 101	2.317 × 101	2.323 × 101	2.199 × 101	2.323 × 101
		Std	3.465 × 10−2	5.498 × 10−3	9.208 × 10−2	1.020 × 100	3.847 × 10−3	4.924 × 10−3	8.146 × 10−3	1.619 × 10−1	1.384 × 10−1	4.311 × 10−3	7.012 × 10−1	1.445 × 10−14
	8	Ave	2.482 × 101	2.495 × 101	2.494 × 101	2.221 × 101	2.499 × 101	2.496 × 101	2.497 × 101	2.474 × 101	2.495 × 101	2.495 × 101	2.374 × 101	2.495 × 101
		Std	1.403 × 10−1	1.807 × 10−2	6.935 × 10−2	9.706 × 10−1	9.995 × 10−2	1.309 × 10−2	7.226 × 10−2	2.839 × 10−1	1.151 × 10−1	7.120 × 10−3	7.208 × 10−1	7.428 × 10−3
	10	Ave	2.637 × 101	2.649 × 101	2.663 × 101	2.341 × 101	2.669 × 101	2.664 × 101	2.672 × 101	2.599 × 101	2.633 × 101	2.671 × 101	2.498 × 101	2.658 × 101
		Std	1.473 × 10−1	3.002 × 10−1	2.078 × 10−1	9.450 × 10−1	1.457 × 10−1	1.099 × 10−1	9.898 × 10−2	3.791 × 10−1	2.531 × 10−1	1.855 × 10−1	5.767 × 10−1	2.467 × 10−1
milkdrop	4	Ave	1.991 × 101	1.992 × 101	1.992 × 101	1.812 × 101	1.992 × 101	1.992 × 101	1.992 × 101	1.991 × 101	1.992 × 101	1.992 × 101	1.897 × 101	1.992 × 101
		Std	9.551 × 10−3	9.333 × 10−3	1.770 × 10−2	9.533 × 10−1	7.227 × 10−15	7.227 × 10−15	7.227 × 10−15	3.847 × 10−2	3.107 × 10−2	7.227 × 10−15	7.378 × 10−1	7.227 × 10−15
	6	Ave	2.302 × 101	2.303 × 101	2.302 × 101	2.047 × 101	2.304 × 101	2.303 × 101	2.302 × 101	2.284 × 101	2.299 × 101	2.303 × 101	2.172 × 101	2.304 × 101
		Std	4.890 × 10−2	3.994 × 10−2	3.807 × 10−2	8.776 × 10−1	3.328 × 10−2	5.490 × 10−2	5.638 × 10−2	2.204 × 10−1	1.068 × 10−1	3.987 × 10−2	5.547 × 10−1	3.296 × 10−2
	8	Ave	2.494 × 101	2.508 × 101	2.501 × 101	2.230 × 101	2.508 × 101	2.509 × 101	2.508 × 101	2.467 × 101	2.497 × 101	2.509 × 101	2.318 × 101	2.509 × 101
		Std	1.803 × 10−1	3.961 × 10−2	1.885 × 10−1	1.114 × 100	3.595 × 10−2	1.633 × 10−2	3.801 × 10−2	2.998 × 10−1	1.575 × 10−1	1.358 × 10−2	8.534 × 10−1	1.968 × 10−2
	10	Ave	2.652 × 101	2.682 × 101	2.671 × 101	2.355 × 101	2.683 × 101	2.687 × 101	2.684 × 101	2.600 × 101	2.656 × 101	2.687 × 101	2.462 × 101	2.684 × 101
		Std	3.061 × 10−1	1.044 × 10−1	3.042 × 10−1	9.020 × 10−1	1.114 × 10−1	8.049 × 10−2	1.186 × 10−1	3.932 × 10−1	2.948 × 10−1	5.666 × 10−2	5.768 × 10−1	9.012 × 10−2
testpat	4	Ave	1.932 × 101	1.932 × 101	1.931 × 101	1.755 × 101	1.932 × 101	1.932 × 101	1.932 × 101	1.932 × 101	1.932 × 101	1.932 × 101	1.863 × 101	1.932 × 101
		Std	2.678 × 10−3	3.613 × 10−15	4.134 × 10−2	8.291 × 10−1	3.613 × 10−15	3.613 × 10−15	3.613 × 10−15	6.137 × 10−2	1.434 × 10−2	3.613 × 10−15	5.159 × 10−1	3.613 × 10−15
	6	Ave	2.198 × 101	2.199 × 101	2.198 × 101	2.007 × 101	2.199 × 101	2.198 × 101	2.198 × 101	2.191 × 101	2.200 × 101	2.198 × 101	2.090 × 101	2.198 × 101
		Std	4.353 × 10−2	1.900 × 10−2	4.756 × 10−2	6.793 × 10−1	2.630 × 10−2	1.019 × 10−2	2.525 × 10−4	1.521 × 10−1	5.057 × 10−2	2.525 × 10−4	7.396 × 10−1	3.613 × 10−15
	8	Ave	2.404 × 101	2.419 × 101	2.417 × 101	2.154 × 101	2.418 × 101	2.419 × 101	2.417 × 101	2.360 × 101	2.403 × 101	2.415 × 101	2.279 × 101	2.416 × 101
		Std	1.301 × 10−1	3.423 × 10−2	1.087 × 10−1	7.920 × 10−1	4.381 × 10−2	2.791 × 10−2	5.430 × 10−2	3.759 × 10−1	2.403 × 10−1	1.593 × 10−1	5.591 × 10−1	4.936 × 10−2
	10	Ave	2.559 × 101	2.586 × 101	2.576 × 101	2.314 × 101	2.585 × 101	2.586 × 101	2.575 × 101	2.484 × 101	2.564 × 101	2.592 × 101	2.409 × 101	2.582 × 101
		Std	2.049 × 10−1	1.393 × 10−1	2.822 × 10−1	7.526 × 10−1	1.242 × 10−1	8.665 × 10−2	2.707 × 10−1	3.639 × 10−1	2.595 × 10−1	6.612 × 10−2	6.692 × 10−1	1.611 × 10−1
bank	4	Ave	2.025 × 101	2.025 × 101	2.025 × 101	1.872 × 101	2.025 × 101	2.025 × 101	2.024 × 101	2.024 × 101	2.021 × 101	2.025 × 101	1.961 × 101	2.025 × 101
		Std	8.077 × 10−3	2.336 × 10−3	4.689 × 10−3	7.723 × 10−1	6.733 × 10−3	2.336 × 10−3	3.720 × 10−2	2.438 × 10−2	6.107 × 10−2	6.048 × 10−3	5.054 × 10−1	1.445 × 10−14
	6	Ave	2.307 × 101	2.310 × 101	2.306 × 101	2.093 × 101	2.310 × 101	2.310 × 101	2.310 × 101	2.296 × 101	2.306 × 101	2.310 × 101	2.203 × 101	2.310 × 101
		Std	5.128 × 10−2	7.549 × 10−3	1.214 × 10−1	8.713 × 10−1	6.727 × 10−3	7.662 × 10−3	1.280 × 10−2	1.843 × 10−1	5.606 × 10−2	2.601 × 10−3	5.918 × 10−1	3.613 × 10−15
	8	Ave	2.514 × 101	2.528 × 101	2.524 × 101	2.255 × 101	2.528 × 101	2.526 × 101	2.523 × 101	2.491 × 101	2.514 × 101	2.526 × 101	2.348 × 101	2.529 × 101
		Std	1.133 × 10−1	3.580 × 10−2	1.434 × 10−1	8.081 × 10−1	2.827 × 10−2	4.191 × 10−2	1.639 × 10−1	2.897 × 10−1	1.491 × 10−1	1.605 × 10−1	6.356 × 10−1	1.917 × 10−2
	10	Ave	2.659 × 101	2.692 × 101	2.659 × 101	2.375 × 101	2.686 × 101	2.680 × 101	2.683 × 101	2.613 × 101	2.667 × 101	2.689 × 101	2.486 × 101	2.691 × 101
		Std	1.515 × 10−1	6.085 × 10−2	3.805 × 10−1	9.034 × 10−1	1.009 × 10−1	1.992 × 10−1	1.647 × 10−1	2.712 × 10−1	1.992 × 10−1	5.844 × 10−2	6.734 × 10−1	1.072 × 10−1
boat	4	Ave	1.990 × 101	1.990 × 101	1.989 × 101	1.885 × 101	1.990 × 101	1.990 × 101	1.990 × 101	1.989 × 101	1.990 × 101	1.990 × 101	2.005 × 101	1.990 × 101
		Std	8.649 × 10−3	1.084 × 10−14	1.837 × 10−2	8.614 × 10−1	1.084 × 10−14	1.084 × 10−14	1.084 × 10−14	6.990 × 10−2	4.427 × 10−2	1.084 × 10−14	4.334 × 10−1	1.084 × 10−14
	6	Ave	2.303 × 101	2.306 × 101	2.303 × 101	2.101 × 101	2.305 × 101	2.305 × 101	2.302 × 101	2.286 × 101	2.289 × 101	2.304 × 101	2.228 × 101	2.306 × 101
		Std	5.397 × 10−2	1.928 × 10−2	7.930 × 10−2	1.027 × 100	3.518 × 10−2	2.288 × 10−2	5.387 × 10−2	2.985 × 10−1	2.209 × 10−1	3.457 × 10−2	7.314 × 10−1	1.084 × 10−14
	8	Ave	2.504 × 101	2.507 × 101	2.501 × 101	2.226 × 101	2.515 × 101	2.508 × 101	2.513 × 101	2.456 × 101	2.490 × 101	2.512 × 101	2.415 × 101	2.511 × 101
		Std	2.208 × 10−1	5.798 × 10−2	3.418 × 10−1	1.079 × 100	1.086 × 10−1	1.097 × 10−1	9.421 × 10−2	4.046 × 10−1	3.711 × 10−1	5.608 × 10−2	6.395 × 10−1	5.947 × 10−2
	10	Ave	2.652 × 101	2.687 × 101	2.680 × 101	2.372 × 101	2.699 × 101	2.692 × 101	2.697 × 101	2.592 × 101	2.656 × 101	2.701 × 101	2.537 × 101	2.690 × 101
		Std	3.753 × 10−1	2.124 × 10−1	4.337 × 10−1	1.210 × 100	2.347 × 10−1	2.427 × 10−1	2.059 × 10−1	4.683 × 10−1	4.130 × 10−1	1.341 × 10−1	8.639 × 10−1	1.691 × 10−1
plane	4	Ave	2.117 × 101	2.118 × 101	2.117 × 101	1.905 × 101	2.118 × 101	2.118 × 101	2.118 × 101	2.116 × 101	2.117 × 101	2.118 × 101	2.035 × 101	2.118 × 101
		Std	2.188 × 10−2	1.155 × 10−2	1.067 × 10−1	9.608 × 10−1	7.227 × 10−15	7.227 × 10−15	7.227 × 10−15	1.053 × 10−1	4.730 × 10−2	7.227 × 10−15	8.557 × 10−1	7.227 × 10−15
	6	Ave	2.469 × 101	2.470 × 101	2.463 × 101	2.125 × 101	2.470 × 101	2.469 × 101	2.469 × 101	2.447 × 101	2.455 × 101	2.470 × 101	2.286 × 101	2.470 × 101
		Std	6.769 × 10−2	2.776 × 10−2	1.474 × 10−1	1.249 × 100	1.991 × 10−2	1.961 × 10−2	2.003 × 10−2	3.196 × 10−1	2.097 × 10−1	1.590 × 10−2	1.125 × 100	8.851 × 10−3
	8	Ave	2.662 × 101	2.680 × 101	2.675 × 101	2.302 × 101	2.681 × 101	2.676 × 101	2.676 × 101	2.635 × 101	2.673 × 101	2.681 × 101	2.429 × 101	2.681 × 101
		Std	2.420 × 10−1	3.130 × 10−2	1.176 × 10−1	1.307 × 100	2.534 × 10−2	1.244 × 10−1	1.467 × 10−1	3.533 × 10−1	2.308 × 10−1	3.395 × 10−2	1.170 × 100	2.749 × 10−2
	10	Ave	2.826 × 101	2.852 × 101	2.818 × 101	2.395 × 101	2.844 × 101	2.814 × 101	2.832 × 101	2.767 × 101	2.820 × 101	2.852 × 101	2.586 × 101	2.856 × 101
		Std	2.231 × 10−1	9.420 × 10−2	3.618 × 10−1	1.437 × 100	1.813 × 10−1	3.846 × 10−1	3.969 × 10−1	4.310 × 10−1	3.447 × 10−1	1.647 × 10−1	9.178 × 10−1	4.410 × 10−2
saturn	4	Ave	2.233 × 101	2.233 × 101	2.233 × 101	2.110 × 101	2.233 × 101	2.233 × 101	2.233 × 101	2.232 × 101	2.234 × 101	2.233 × 101	2.176 × 101	2.233 × 101
		Std	1.634 × 10−2	1.084 × 10−14	2.335 × 10−2	6.175 × 10−1	1.094 × 10−2	7.872 × 10−3	1.094 × 10−2	8.479 × 10−2	2.832 × 10−2	1.084 × 10−14	4.861 × 10−1	1.084 × 10−14
	6	Ave	2.537 × 101	2.538 × 101	2.526 × 101	2.343 × 101	2.538 × 101	2.538 × 101	2.538 × 101	2.523 × 101	2.536 × 101	2.538 × 101	2.445 × 101	2.538 × 101
		Std	4.875 × 10−2	1.097 × 10−2	1.996 × 10−1	7.140 × 10−1	8.685 × 10−3	6.425 × 10−3	5.166 × 10−3	2.196 × 10−1	8.655 × 10−2	7.227 × 10−15	5.184 × 10−1	1.416 × 10−3
	8	Ave	2.738 × 101	2.752 × 101	2.736 × 101	2.524 × 101	2.752 × 101	2.753 × 101	2.753 × 101	2.718 × 101	2.746 × 101	2.752 × 101	2.622 × 101	2.753 × 101
		Std	1.502 × 10−1	2.671 × 10−2	2.194 × 10−1	8.000 × 10−1	4.071 × 10−2	4.235 × 10−2	5.094 × 10−2	3.731 × 10−1	1.361 × 10−1	3.377 × 10−2	5.821 × 10−1	3.361 × 10−2
	10	Ave	2.896 × 101	2.923 × 101	2.883 × 101	2.667 × 101	2.923 × 101	2.930 × 101	2.921 × 101	2.856 × 101	2.911 × 101	2.924 × 101	2.763 × 101	2.928 × 101
		Std	2.397 × 10−1	5.857 × 10−2	5.330 × 10−1	7.507 × 10−1	5.632 × 10−2	4.242 × 10−2	1.464 × 10−1	3.819 × 10−1	1.743 × 10−1	6.981 × 10−2	5.575 × 10−1	5.955 × 10−2

, MEIVYA achieves high PSNR values across various test images and different threshold values, indicating a high similarity between its segmentation results and the original images, and better preservation of grayscale information and detailed features. Overall, most algorithms show improved PSNR values with increasing threshold values, as more thresholds can more precisely describe the grayscale levels in the image, thus reducing the error between the segmented image and the original image. However, compared to other comparative algorithms, MEIVYA consistently achieves higher average PSNR values with a smaller standard deviation, indicating that it not only generates higher-quality segmentation results but also exhibits better stability and consistency. Especially on test images with complex textures and uneven grayscale distribution, MEIVYA’s PSNR improvement is more significant, demonstrating its ability to more effectively preserve important structural and edge information in the image and reduce distortion caused by segmentation. Furthermore, the smaller standard deviation indicates that the algorithm is less affected by random factors in multiple independent runs, consistently achieving high-quality segmentation results. Overall, the experimental results in Table 8 further demonstrate MEIVYA’s superior performance in multi-threshold image segmentation tasks, outperforming most of the comparison algorithms in both image fidelity and result stability.

As shown in

Table 9. Ave and Std of all test images for FSIM.

Image	Threshold	Metric	PSO	DE	CMAES	INFO	SMA	RUN	HGS	DOA	GWCA	SAO	IVYA	MEIVYA
Brain	4	Ave	6.751 × 10−1	6.751 × 10−1	6.751 × 10−1	6.721 × 10−1	6.751 × 10−1	6.751 × 10−1	6.751 × 10−1	6.750 × 10−1	6.751 × 10−1	6.751 × 10−1	6.978 × 10−1	6.751 × 10−1
		Std	1.074 × 10−4	7.929 × 10−5	9.818 × 10−5	3.947 × 10−2	8.127 × 10−5	7.687 × 10−5	7.956 × 10−5	3.390 × 10−4	3.260 × 10−4	8.380 × 10−5	5.800 × 10−2	8.761 × 10−5
	6	Ave	8.440 × 10−1	7.539 × 10−1	7.678 × 10−1	7.310 × 10−1	7.996 × 10−1	8.219 × 10−1	7.992 × 10−1	7.981 × 10−1	7.231 × 10−1	7.922 × 10−1	7.771 × 10−1	7.010 × 10−1
		Std	1.089 × 10−1	1.018 × 10−1	1.080 × 10−1	8.432 × 10−2	1.161 × 10−1	1.142 × 10−1	1.158 × 10−1	1.137 × 10−1	7.812 × 10−2	1.154 × 10−1	9.594 × 10−2	4.133 × 10−2
	8	Ave	9.483 × 10−1	9.504 × 10−1	9.507 × 10−1	7.766 × 10−1	9.503 × 10−1	9.501 × 10−1	9.504 × 10−1	9.306 × 10−1	8.981 × 10−1	9.510 × 10−1	8.356 × 10−1	9.342 × 10−1
		Std	2.747 × 10−3	1.878 × 10−3	2.376 × 10−3	1.038 × 10−1	1.569 × 10−3	2.214 × 10−3	2.002 × 10−3	2.444 × 10−2	1.005 × 10−1	1.491 × 10−3	1.044 × 10−1	6.193 × 10−2
	10	Ave	9.599 × 10−1	9.639 × 10−1	9.635 × 10−1	8.240 × 10−1	9.654 × 10−1	9.642 × 10−1	9.651 × 10−1	9.523 × 10−1	9.515 × 10−1	9.642 × 10−1	9.095 × 10−1	9.636 × 10−1
		Std	2.701 × 10−3	2.660 × 10−3	3.766 × 10−3	1.051 × 10−1	1.824 × 10−3	1.465 × 10−3	2.679 × 10−3	6.108 × 10−3	4.648 × 10−2	2.318 × 10−3	5.277 × 10−2	1.401 × 10−3
satellite	4	Ave	7.725 × 10−1	7.725 × 10−1	7.716 × 10−1	7.307 × 10−1	7.725 × 10−1	7.725 × 10−1	7.725 × 10−1	7.708 × 10−1	7.723 × 10−1	7.725 × 10−1	8.071 × 10−1	7.725 × 10−1
		Std	3.737 × 10−4	2.258 × 10−16	4.170 × 10−3	7.079 × 10−2	2.258 × 10−16	2.258 × 10−16	2.258 × 10−16	3.733 × 10−3	1.076 × 10−3	2.258 × 10−16	4.439 × 10−2	2.258 × 10−16
	6	Ave	8.249 × 10−1	8.270 × 10−1	8.222 × 10−1	7.556 × 10−1	8.256 × 10−1	8.249 × 10−1	8.234 × 10−1	8.163 × 10−1	8.208 × 10−1	8.156 × 10−1	8.405 × 10−1	8.262 × 10−1
		Std	2.860 × 10−3	2.041 × 10−3	7.004 × 10−3	7.076 × 10−2	2.679 × 10−3	3.177 × 10−3	3.413 × 10−3	1.281 × 10−2	9.861 × 10−3	4.825 × 10−2	1.118 × 10−1	2.351 × 10−3
	8	Ave	8.468 × 10−1	8.515 × 10−1	8.018 × 10−1	8.212 × 10−1	8.503 × 10−1	8.503 × 10−1	8.372 × 10−1	8.438 × 10−1	8.550 × 10−1	8.251 × 10−1	8.536 × 10−1	8.513 × 10−1
		Std	6.862 × 10−3	2.229 × 10−3	9.833 × 10−2	8.500 × 10−2	3.224 × 10−3	2.312 × 10−3	5.426 × 10−2	1.198 × 10−2	2.041 × 10−2	7.047 × 10−2	9.632 × 10−2	1.638 × 10−3
	10	Ave	8.656 × 10−1	8.646 × 10−1	8.218 × 10−1	8.635 × 10−1	8.693 × 10−1	8.651 × 10−1	8.623 × 10−1	8.612 × 10−1	8.771 × 10−1	8.417 × 10−1	8.287 × 10−1	8.657 × 10−1
		Std	1.117 × 10−2	5.374 × 10−3	7.825 × 10−2	5.983 × 10−2	1.046 × 10−2	6.057 × 10−3	3.276 × 10−2	2.357 × 10−2	2.799 × 10−2	6.290 × 10−2	9.264 × 10−2	3.868 × 10−3
Face	4	Ave	7.544 × 10−1	7.548 × 10−1	7.538 × 10−1	7.035 × 10−1	7.544 × 10−1	7.544 × 10−1	7.538 × 10−1	7.543 × 10−1	7.536 × 10−1	7.544 × 10−1	7.420 × 10−1	7.545 × 10−1
		Std	7.982 × 10−4	3.388 × 10−16	2.707 × 10−3	2.315 × 10−2	7.995 × 10−4	7.995 × 10−4	1.070 × 10−3	1.379 × 10−3	1.269 × 10−3	7.995 × 10−4	1.567 × 10−2	7.293 × 10−4
	6	Ave	8.435 × 10−1	8.442 × 10−1	8.429 × 10−1	7.682 × 10−1	8.441 × 10−1	8.443 × 10−1	8.440 × 10−1	8.419 × 10−1	8.437 × 10−1	8.444 × 10−1	8.080 × 10−1	8.444 × 10−1
		Std	1.501 × 10−3	8.438 × 10−4	3.502 × 10−3	2.734 × 10−2	8.974 × 10−4	7.158 × 10−4	8.959 × 10−4	3.624 × 10−3	1.875 × 10−3	6.364 × 10−4	1.833 × 10−2	5.341 × 10−4
	8	Ave	8.918 × 10−1	8.967 × 10−1	8.919 × 10−1	8.190 × 10−1	8.960 × 10−1	8.960 × 10−1	8.957 × 10−1	8.827 × 10−1	8.931 × 10−1	8.968 × 10−1	8.461 × 10−1	8.969 × 10−1
		Std	3.734 × 10−3	4.707 × 10−4	7.960 × 10−3	2.230 × 10−2	8.584 × 10−4	9.552 × 10−4	1.697 × 10−3	9.103 × 10−3	2.919 × 10−3	4.671 × 10−4	1.936 × 10−2	4.015 × 10−4
	10	Ave	9.185 × 10−1	9.261 × 10−1	9.198 × 10−1	8.471 × 10−1	9.262 × 10−1	9.250 × 10−1	9.249 × 10−1	9.070 × 10−1	9.232 × 10−1	9.262 × 10−1	8.819 × 10−1	9.262 × 10−1
		Std	3.388 × 10−3	7.729 × 10−4	9.402 × 10−3	2.409 × 10−2	9.490 × 10−4	1.591 × 10−3	3.582 × 10−3	8.123 × 10−3	3.040 × 10−3	5.943 × 10−4	1.677 × 10−2	7.014 × 10−4
cell	4	Ave	8.788 × 10−1	8.788 × 10−1	8.791 × 10−1	8.397 × 10−1	8.788 × 10−1	8.788 × 10−1	8.788 × 10−1	8.781 × 10−1	8.787 × 10−1	8.788 × 10−1	8.594 × 10−1	8.788 × 10−1
		Std	0.000 × 100	0.000 × 100	9.836 × 10−4	3.058 × 10−2	0.000 × 100	0.000 × 100	0.000 × 100	1.778 × 10−3	5.956 × 10−4	0.000 × 100	2.212 × 10−2	0.000 × 100
	6	Ave	9.012 × 10−1	8.987 × 10−1	9.013 × 10−1	8.645 × 10−1	9.005 × 10−1	8.995 × 10−1	9.009 × 10−1	9.007 × 10−1	9.006 × 10−1	9.019 × 10−1	8.816 × 10−1	9.006 × 10−1
		Std	2.949 × 10−3	1.840 × 10−3	2.821 × 10−3	1.795 × 10−2	2.892 × 10−3	2.878 × 10−3	3.333 × 10−3	3.800 × 10−3	4.518 × 10−3	3.406 × 10−3	1.838 × 10−2	3.344 × 10−3
	8	Ave	9.159 × 10−1	9.169 × 10−1	9.156 × 10−1	8.810 × 10−1	9.166 × 10−1	9.159 × 10−1	9.157 × 10−1	9.109 × 10−1	9.143 × 10−1	9.179 × 10−1	8.959 × 10−1	9.173 × 10−1
		Std	3.078 × 10−3	3.316 × 10−3	3.857 × 10−3	2.600 × 10−2	2.992 × 10−3	3.655 × 10−3	3.798 × 10−3	5.235 × 10−3	4.515 × 10−3	3.969 × 10−4	2.353 × 10−2	1.976 × 10−3
	10	Ave	9.152 × 10−1	9.174 × 10−1	9.169 × 10−1	8.888 × 10−1	9.171 × 10−1	9.181 × 10−1	9.202 × 10−1	9.169 × 10−1	9.190 × 10−1	9.190 × 10−1	9.051 × 10−1	9.184 × 10−1
		Std	7.672 × 10−3	5.079 × 10−3	4.736 × 10−3	2.853 × 10−2	4.646 × 10−3	5.092 × 10−3	4.504 × 10−3	5.224 × 10−3	5.753 × 10−3	1.366 × 10−3	3.543 × 10−2	2.626 × 10−3
lax	4	Ave	7.994 × 10−1	7.995 × 10−1	7.991 × 10−1	7.637 × 10−1	7.995 × 10−1	7.995 × 10−1	7.995 × 10−1	7.984 × 10−1	7.993 × 10−1	7.995 × 10−1	8.010 × 10−1	7.995 × 10−1
		Std	5.771 × 10−4	6.580 × 10−5	1.927 × 10−3	2.672 × 10−2	6.580 × 10−5	2.258 × 10−16	4.735 × 10−5	1.575 × 10−3	1.125 × 10−3	2.258 × 10−16	1.993 × 10−2	2.258 × 10−16
	6	Ave	8.345 × 10−1	8.347 × 10−1	8.347 × 10−1	8.147 × 10−1	8.348 × 10−1	8.348 × 10−1	8.354 × 10−1	8.312 × 10−1	8.339 × 10−1	8.349 × 10−1	8.531 × 10−1	8.348 × 10−1
		Std	1.221 × 10−3	2.833 × 10−4	8.118 × 10−4	3.855 × 10−2	5.637 × 10−4	1.939 × 10−4	6.087 × 10−4	5.058 × 10−3	2.992 × 10−3	5.533 × 10−4	3.397 × 10−2	1.923 × 10−4
	8	Ave	8.520 × 10−1	8.541 × 10−1	8.526 × 10−1	8.417 × 10−1	8.534 × 10−1	8.563 × 10−1	8.531 × 10−1	8.451 × 10−1	8.513 × 10−1	8.543 × 10−1	8.899 × 10−1	8.537 × 10−1
		Std	2.920 × 10−3	1.137 × 10−3	2.670 × 10−3	3.585 × 10−2	1.188 × 10−3	1.056 × 10−2	1.815 × 10−3	5.395 × 10−3	2.391 × 10−3	9.311 × 10−4	3.628 × 10−2	1.194 × 10−3
	10	Ave	8.631 × 10−1	8.651 × 10−1	8.661 × 10−1	8.647 × 10−1	8.665 × 10−1	8.662 × 10−1	8.659 × 10−1	8.596 × 10−1	8.644 × 10−1	8.659 × 10−1	9.151 × 10−1	8.652 × 10−1
		Std	7.008 × 10−3	2.205 × 10−3	6.719 × 10−3	3.515 × 10−2	2.520 × 10−3	7.687 × 10−3	2.336 × 10−3	1.455 × 10−2	5.443 × 10−3	2.021 × 10−3	2.279 × 10−2	1.676 × 10−3
Peppers	4	Ave	7.868 × 10−1	7.868 × 10−1	7.864 × 10−1	7.569 × 10−1	7.868 × 10−1	7.867 × 10−1	7.866 × 10−1	7.861 × 10−1	7.866 × 10−1	7.867 × 10−1	7.811 × 10−1	7.868 × 10−1
		Std	3.878 × 10−5	3.154 × 10−5	1.437 × 10−3	1.319 × 10−2	3.347 × 10−5	2.114 × 10−4	3.613 × 10−4	1.282 × 10−3	4.828 × 10−4	2.160 × 10−4	7.551 × 10−3	1.519 × 10−5
	6	Ave	8.486 × 10−1	8.493 × 10−1	8.479 × 10−1	8.028 × 10−1	8.492 × 10−1	8.492 × 10−1	8.493 × 10−1	8.454 × 10−1	8.483 × 10−1	8.492 × 10−1	8.253 × 10−1	8.492 × 10−1
		Std	1.465 × 10−3	2.077 × 10−4	3.464 × 10−3	1.775 × 10−2	1.380 × 10−4	1.120 × 10−4	2.045 × 10−4	5.162 × 10−3	1.695 × 10−3	1.082 × 10−4	1.209 × 10−2	2.258 × 10−16
	8	Ave	8.835 × 10−1	8.865 × 10−1	8.865 × 10−1	8.319 × 10−1	8.856 × 10−1	8.864 × 10−1	8.859 × 10−1	8.783 × 10−1	8.841 × 10−1	8.863 × 10−1	8.559 × 10−1	8.863 × 10−1
		Std	2.517 × 10−3	3.813 × 10−4	1.149 × 10−3	1.702 × 10−2	2.497 × 10−3	3.986 × 10−4	1.868 × 10−3	5.950 × 10−3	2.877 × 10−3	3.098 × 10−4	1.133 × 10−2	2.443 × 10−4
	10	Ave	9.087 × 10−1	9.137 × 10−1	9.125 × 10−1	8.552 × 10−1	9.132 × 10−1	9.138 × 10−1	9.138 × 10−1	9.019 × 10−1	9.100 × 10−1	9.144 × 10−1	8.764 × 10−1	9.145 × 10−1
		Std	2.448 × 10−3	1.016 × 10−3	3.953 × 10−3	1.786 × 10−2	1.983 × 10−3	1.855 × 10−3	1.885 × 10−3	6.703 × 10−3	4.088 × 10−3	6.397 × 10−4	1.127 × 10−2	6.325 × 10−4
milkdrop	4	Ave	7.876 × 10−1	7.875 × 10−1	7.876 × 10−1	7.649 × 10−1	7.875 × 10−1	7.875 × 10−1	7.875 × 10−1	7.877 × 10−1	7.877 × 10−1	7.875 × 10−1	7.874 × 10−1	7.875 × 10−1
		Std	1.867 × 10−4	1.618 × 10−4	3.127 × 10−4	1.493 × 10−2	0.000 × 100	0.000 × 100	0.000 × 100	1.059 × 10−3	6.338 × 10−4	0.000 × 100	1.123 × 10−2	0.000 × 100
	6	Ave	8.481 × 10−1	8.481 × 10−1	8.485 × 10−1	8.056 × 10−1	8.482 × 10−1	8.476 × 10−1	8.475 × 10−1	8.430 × 10−1	8.471 × 10−1	8.481 × 10−1	8.340 × 10−1	8.481 × 10−1
		Std	1.265 × 10−3	1.652 × 10−3	2.507 × 10−4	1.709 × 10−2	1.721 × 10−3	2.420 × 10−3	2.806 × 10−3	4.955 × 10−3	3.162 × 10−3	1.649 × 10−3	9.775 × 10−3	1.708 × 10−3
	8	Ave	8.747 × 10−1	8.765 × 10−1	8.755 × 10−1	8.316 × 10−1	8.781 × 10−1	8.757 × 10−1	8.772 × 10−1	8.697 × 10−1	8.743 × 10−1	8.770 × 10−1	8.531 × 10−1	8.771 × 10−1
		Std	3.414 × 10−3	2.501 × 10−3	4.099 × 10−3	1.569 × 10−2	1.976 × 10−3	2.633 × 10−3	2.805 × 10−3	5.416 × 10−3	3.556 × 10−3	2.512 × 10−3	1.353 × 10−2	2.336 × 10−3
	10	Ave	8.969 × 10−1	9.025 × 10−1	9.008 × 10−1	8.506 × 10−1	9.021 × 10−1	9.028 × 10−1	9.024 × 10−1	8.886 × 10−1	8.990 × 10−1	9.029 × 10−1	8.732 × 10−1	9.025 × 10−1
		Std	4.615 × 10−3	1.392 × 10−3	4.085 × 10−3	1.804 × 10−2	2.667 × 10−3	8.838 × 10−4	2.253 × 10−3	6.027 × 10−3	4.624 × 10−3	1.222 × 10−3	1.022 × 10−2	1.835 × 10−3
testpat	4	Ave	8.849 × 10−1	8.849 × 10−1	8.847 × 10−1	8.578 × 10−1	8.849 × 10−1	8.849 × 10−1	8.849 × 10−1	8.848 × 10−1	8.848 × 10−1	8.849 × 10−1	8.804 × 10−1	8.849 × 10−1
		Std	1.173 × 10−4	4.517 × 10−16	9.585 × 10−4	1.174 × 10−2	4.517 × 10−16	4.517 × 10−16	4.517 × 10−16	8.753 × 10−4	3.005 × 10−4	4.517 × 10−16	5.689 × 10−3	4.517 × 10−16
	6	Ave	9.156 × 10−1	9.160 × 10−1	9.159 × 10−1	8.919 × 10−1	9.158 × 10−1	9.159 × 10−1	9.159 × 10−1	9.136 × 10−1	9.154 × 10−1	9.159 × 10−1	9.035 × 10−1	9.159 × 10−1
		Std	1.248 × 10−3	4.049 × 10−4	1.113 × 10−3	9.856 × 10−3	5.507 × 10−4	3.500 × 10−4	6.223 × 10−5	2.848 × 10−3	1.768 × 10−3	6.223 × 10−5	1.124 × 10−2	0.000 × 100
	8	Ave	9.280 × 10−1	9.288 × 10−1	9.295 × 10−1	9.048 × 10−1	9.290 × 10−1	9.290 × 10−1	9.291 × 10−1	9.248 × 10−1	9.286 × 10−1	9.287 × 10−1	9.189 × 10−1	9.290 × 10−1
		Std	1.287 × 10−3	6.056 × 10−4	8.976 × 10−4	8.128 × 10−3	6.311 × 10−4	4.032 × 10−4	4.903 × 10−4	4.340 × 10−3	1.270 × 10−3	1.155 × 10−3	6.596 × 10−3	4.743 × 10−4
	10	Ave	9.395 × 10−1	9.410 × 10−1	9.407 × 10−1	9.209 × 10−1	9.410 × 10−1	9.411 × 10−1	9.408 × 10−1	9.343 × 10−1	9.398 × 10−1	9.414 × 10−1	9.289 × 10−1	9.408 × 10−1
		Std	1.992 × 10−3	9.441 × 10−4	2.005 × 10−3	6.924 × 10−3	8.509 × 10−4	6.202 × 10−4	1.532 × 10−3	3.455 × 10−3	2.184 × 10−3	4.915 × 10−4	6.120 × 10−3	8.227 × 10−4
bank	4	Ave	8.350 × 10−1	8.349 × 10−1	8.350 × 10−1	8.130 × 10−1	8.349 × 10−1	8.349 × 10−1	8.349 × 10−1	8.351 × 10−1	8.347 × 10−1	8.349 × 10−1	8.325 × 10−1	8.349 × 10−1
		Std	2.626 × 10−4	1.296 × 10−4	6.190 × 10−4	1.633 × 10−2	1.325 × 10−4	1.296 × 10−4	4.326 × 10−4	8.240 × 10−4	1.056 × 10−3	1.319 × 10−4	7.555 × 10−3	1.129 × 10−16
	6	Ave	8.812 × 10−1	8.813 × 10−1	8.815 × 10−1	8.480 × 10−1	8.814 × 10−1	8.813 × 10−1	8.814 × 10−1	8.777 × 10−1	8.809 × 10−1	8.813 × 10−1	8.700 × 10−1	8.813 × 10−1
		Std	8.522 × 10−4	1.591 × 10−4	8.778 × 10−4	1.414 × 10−2	2.022 × 10−4	8.089 × 10−5	1.796 × 10−4	4.642 × 10−3	2.155 × 10−3	1.166 × 10−4	8.243 × 10−3	5.646 × 10−16
	8	Ave	9.093 × 10−1	9.098 × 10−1	9.095 × 10−1	8.757 × 10−1	9.103 × 10−1	9.104 × 10−1	9.095 × 10−1	9.046 × 10−1	9.095 × 10−1	9.097 × 10−1	8.922 × 10−1	9.099 × 10−1
		Std	2.741 × 10−3	1.065 × 10−3	2.058 × 10−3	1.631 × 10−2	7.149 × 10−4	9.714 × 10−4	2.380 × 10−3	4.596 × 10−3	2.944 × 10−3	2.375 × 10−3	8.559 × 10−3	6.437 × 10−4
	10	Ave	9.276 × 10−1	9.313 × 10−1	9.298 × 10−1	8.931 × 10−1	9.317 × 10−1	9.315 × 10−1	9.314 × 10−1	9.207 × 10−1	9.292 × 10−1	9.316 × 10−1	9.087 × 10−1	9.308 × 10−1
		Std	3.318 × 10−3	1.123 × 10−3	4.858 × 10−3	1.476 × 10−2	1.012 × 10−3	1.187 × 10−3	1.370 × 10−3	5.340 × 10−3	3.378 × 10−3	5.317 × 10−4	1.003 × 10−2	9.179 × 10−4
boat	4	Ave	8.131 × 10−1	8.130 × 10−1	8.130 × 10−1	7.764 × 10−1	8.130 × 10−1	8.130 × 10−1	8.130 × 10−1	8.126 × 10−1	8.128 × 10−1	8.130 × 10−1	8.046 × 10−1	8.130 × 10−1
		Std	3.949 × 10−5	3.388 × 10−16	2.503 × 10−4	1.786 × 10−2	3.388 × 10−16	3.388 × 10−16	3.388 × 10−16	1.007 × 10−3	6.673 × 10−4	3.388 × 10−16	1.036 × 10−2	3.388 × 10−16
	6	Ave	8.788 × 10−1	8.788 × 10−1	8.786 × 10−1	8.250 × 10−1	8.789 × 10−1	8.789 × 10−1	8.786 × 10−1	8.763 × 10−1	8.777 × 10−1	8.788 × 10−1	8.521 × 10−1	8.789 × 10−1
		Std	5.330 × 10−4	1.879 × 10−4	6.118 × 10−4	2.140 × 10−2	2.759 × 10−4	2.037 × 10−4	5.026 × 10−4	2.697 × 10−3	1.556 × 10−3	2.825 × 10−4	1.295 × 10−2	1.129 × 10−16
	8	Ave	9.099 × 10−1	9.120 × 10−1	9.101 × 10−1	8.521 × 10−1	9.119 × 10−1	9.122 × 10−1	9.121 × 10−1	9.048 × 10−1	9.095 × 10−1	9.120 × 10−1	8.859 × 10−1	9.120 × 10−1
		Std	2.220 × 10−3	3.280 × 10−4	3.665 × 10−3	1.430 × 10−2	8.187 × 10−4	4.018 × 10−4	5.173 × 10−4	4.524 × 10−3	2.379 × 10−3	1.277 × 10−4	1.326 × 10−2	1.178 × 10−4
	10	Ave	9.281 × 10−1	9.327 × 10−1	9.307 × 10−1	8.771 × 10−1	9.328 × 10−1	9.317 × 10−1	9.330 × 10−1	9.218 × 10−1	9.314 × 10−1	9.331 × 10−1	9.047 × 10−1	9.329 × 10−1
		Std	2.537 × 10−3	8.189 × 10−4	4.172 × 10−3	1.832 × 10−2	9.276 × 10−4	1.174 × 10−3	8.382 × 10−4	4.442 × 10−3	2.404 × 10−3	7.051 × 10−4	1.252 × 10−2	8.466 × 10−4
plane	4	Ave	8.337 × 10−1	8.339 × 10−1	8.338 × 10−1	8.048 × 10−1	8.339 × 10−1	8.339 × 10−1	8.339 × 10−1	8.333 × 10−1	8.337 × 10−1	8.339 × 10−1	8.317 × 10−1	8.339 × 10−1
		Std	4.239 × 10−4	2.239 × 10−4	2.065 × 10−3	2.071 × 10−2	1.129 × 10−16	1.129 × 10−16	1.129 × 10−16	2.063 × 10−3	9.226 × 10−4	1.129 × 10−16	1.266 × 10−2	1.129 × 10−16
	6	Ave	8.959 × 10−1	8.961 × 10−1	8.955 × 10−1	8.379 × 10−1	8.961 × 10−1	8.960 × 10−1	8.960 × 10−1	8.928 × 10−1	8.946 × 10−1	8.961 × 10−1	8.686 × 10−1	8.962 × 10−1
		Std	1.493 × 10−3	7.435 × 10−4	2.060 × 10−3	2.293 × 10−2	5.315 × 10−4	2.922 × 10−4	5.601 × 10−4	5.588 × 10−3	2.668 × 10−3	4.255 × 10−4	1.849 × 10−2	3.048 × 10−4
	8	Ave	9.236 × 10−1	9.267 × 10−1	9.265 × 10−1	8.671 × 10−1	9.273 × 10−1	9.269 × 10−1	9.268 × 10−1	9.212 × 10−1	9.274 × 10−1	9.270 × 10−1	8.890 × 10−1	9.269 × 10−1
		Std	5.187 × 10−3	8.182 × 10−4	1.695 × 10−3	2.621 × 10−2	1.107 × 10−3	1.585 × 10−3	2.323 × 10−3	7.033 × 10−3	3.354 × 10−3	1.003 × 10−3	1.967 × 10−2	8.556 × 10−4
	10	Ave	9.452 × 10−1	9.493 × 10−1	9.451 × 10−1	8.770 × 10−1	9.483 × 10−1	9.439 × 10−1	9.470 × 10−1	9.376 × 10−1	9.456 × 10−1	9.498 × 10−1	9.144 × 10−1	9.504 × 10−1
		Std	4.532 × 10−3	1.609 × 10−3	5.155 × 10−3	2.590 × 10−2	2.583 × 10−3	5.238 × 10−3	5.333 × 10−3	6.231 × 10−3	4.508 × 10−3	2.142 × 10−3	1.356 × 10−2	7.303 × 10−4
saturn	4	Ave	8.478 × 10−1	8.477 × 10−1	8.477 × 10−1	8.377 × 10−1	8.477 × 10−1	8.477 × 10−1	8.477 × 10−1	8.480 × 10−1	8.480 × 10−1	8.477 × 10−1	8.513 × 10−1	8.477 × 10−1
		Std	2.285 × 10−4	5.646 × 10−16	4.450 × 10−4	9.712 × 10−3	1.530 × 10−4	1.101 × 10−4	1.530 × 10−4	2.283 × 10−3	7.806 × 10−4	5.646 × 10−16	8.168 × 10−3	5.646 × 10−16
	6	Ave	8.836 × 10−1	8.836 × 10−1	8.834 × 10−1	8.691 × 10−1	8.836 × 10−1	8.836 × 10−1	8.836 × 10−1	8.845 × 10−1	8.840 × 10−1	8.837 × 10−1	8.836 × 10−1	8.837 × 10−1
		Std	1.228 × 10−3	1.700 × 10−4	1.934 × 10−3	1.414 × 10−2	1.225 × 10−4	5.015 × 10−5	2.196 × 10−4	4.250 × 10−3	3.018 × 10−3	2.258 × 10−16	7.954 × 10−3	8.658 × 10−6
	8	Ave	9.136 × 10−1	9.129 × 10−1	9.123 × 10−1	8.888 × 10−1	9.140 × 10−1	9.151 × 10−1	9.139 × 10−1	9.125 × 10−1	9.141 × 10−1	9.136 × 10−1	9.039 × 10−1	9.136 × 10−1
		Std	3.523 × 10−3	2.965 × 10−4	2.183 × 10−3	1.227 × 10−2	2.061 × 10−3	2.299 × 10−3	2.593 × 10−3	5.272 × 10−3	3.628 × 10−3	1.528 × 10−3	7.253 × 10−3	1.671 × 10−3
	10	Ave	9.308 × 10−1	9.345 × 10−1	9.300 × 10−1	9.042 × 10−1	9.345 × 10−1	9.347 × 10−1	9.340 × 10−1	9.260 × 10−1	9.332 × 10−1	9.348 × 10−1	9.184 × 10−1	9.348 × 10−1
		Std	2.770 × 10−3	5.538 × 10−4	5.467 × 10−3	1.060 × 10−2	3.956 × 10−4	7.220 × 10−4	1.654 × 10−3	5.215 × 10−3	1.897 × 10−3	2.405 × 10−4	6.689 × 10−3	5.508 × 10−4

, MEIVYA achieves high FSIM values across different test images and threshold values, indicating that its segmentation results effectively preserve the feature information and structural similarity of the original image. Since FSIM focuses more on phase consistency and gradient features in the image, a higher FSIM value suggests that the algorithm can more effectively preserve the edges, textures, and local details of the target region. Overall, the FSIM values of all algorithms generally show an upward trend with increasing threshold values, indicating that a higher threshold helps to express the grayscale levels and structural features in the image more precisely. However, compared with other comparative algorithms, MEIVYA achieves higher average FSIM values on most test images, with a relatively smaller standard deviation, indicating that it not only achieves better segmentation results but also has better stability and robustness. Especially in images with rich textures and complex edges, MEIVYA’s FSIM improvement is more significant, demonstrating its ability to more accurately extract and preserve key features in the image, reducing over-segmentation and detail loss. Furthermore, the smaller standard deviation further demonstrates that MEIVYA exhibits high consistency across multiple independent runs and is less susceptible to the effects of random initialization and search fluctuations. Overall, the results in Table 9 show that MEIVYA better preserves the structural features and visual information of images in multi-threshold image segmentation tasks, and its comprehensive performance outperforms most comparative algorithms.

As shown in

Table 10. Ave and Std of all test images for SSIM.

Image	Threshold	Metric	PSO	DE	CMAES	INFO	SMA	RUN	HGS	DOA	GWCA	SAO	IVYA	MEIVYA
Brain	4	Ave	3.768 × 10−1	3.767 × 10−1	3.766 × 10−1	3.619 × 10−1	3.767 × 10−1	3.768 × 10−1	3.766 × 10−1	3.768 × 10−1	3.768 × 10−1	3.768 × 10−1	4.031 × 10−1	3.768 × 10−1
		Std	3.749 × 10−4	3.258 × 10−4	3.599 × 10−4	3.025 × 10−2	4.015 × 10−4	3.874 × 10−4	3.352 × 10−4	5.121 × 10−4	5.023 × 10−4	3.746 × 10−4	8.151 × 10−2	4.266 × 10−4
	6	Ave	5.771 × 10−1	4.687 × 10−1	4.878 × 10−1	4.495 × 10−1	5.246 × 10−1	5.484 × 10−1	5.287 × 10−1	5.121 × 10−1	4.427 × 10−1	5.185 × 10−1	4.844 × 10−1	4.116 × 10−1
		Std	1.300 × 10−1	1.100 × 10−1	1.238 × 10−1	1.258 × 10−1	1.336 × 10−1	1.342 × 10−1	1.399 × 10−1	1.272 × 10−1	1.017 × 10−1	1.357 × 10−1	1.206 × 10−1	3.532 × 10−2
	8	Ave	7.163 × 10−1	7.544 × 10−1	7.292 × 10−1	5.120 × 10−1	7.612 × 10−1	7.399 × 10−1	7.569 × 10−1	6.612 × 10−1	6.687 × 10−1	7.531 × 10−1	5.619 × 10−1	7.403 × 10−1
		Std	4.825 × 10−2	2.459 × 10−2	3.891 × 10−2	1.480 × 10−1	2.603 × 10−2	3.965 × 10−2	2.639 × 10−2	1.050 × 10−1	1.401 × 10−1	2.412 × 10−2	1.338 × 10−1	9.030 × 10−2
	10	Ave	7.439 × 10−1	7.819 × 10−1	7.969 × 10−1	5.385 × 10−1	8.078 × 10−1	7.831 × 10−1	8.046 × 10−1	7.195 × 10−1	7.227 × 10−1	7.786 × 10−1	6.632 × 10−1	7.701 × 10−1
		Std	6.196 × 10−2	5.704 × 10−2	5.351 × 10−2	1.372 × 10−1	4.247 × 10−2	2.691 × 10−2	4.204 × 10−2	7.163 × 10−2	8.704 × 10−2	3.346 × 10−2	1.184 × 10−1	2.885 × 10−2
satellite	4	Ave	7.116 × 10−1	7.116 × 10−1	7.101 × 10−1	6.505 × 10−1	7.116 × 10−1	7.116 × 10−1	7.116 × 10−1	7.095 × 10−1	7.113 × 10−1	7.116 × 10−1	7.760 × 10−1	7.116 × 10−1
		Std	2.264 × 10−4	4.517 × 10−16	6.729 × 10−3	1.319 × 10−1	4.517 × 10−16	4.517 × 10−16	4.517 × 10−16	5.151 × 10−3	1.825 × 10−3	4.517 × 10−16	5.969 × 10−2	4.517 × 10−16
	6	Ave	8.067 × 10−1	8.098 × 10−1	8.024 × 10−1	6.882 × 10−1	8.073 × 10−1	8.067 × 10−1	8.031 × 10−1	7.923 × 10−1	7.985 × 10−1	7.955 × 10−1	8.043 × 10−1	8.083 × 10−1
		Std	4.837 × 10−3	2.773 × 10−3	9.879 × 10−3	1.030 × 10−1	4.537 × 10−3	4.765 × 10−3	6.036 × 10−3	2.148 × 10−2	1.690 × 10−2	5.850 × 10−2	1.674 × 10−1	4.034 × 10−3
	8	Ave	8.412 × 10−1	8.498 × 10−1	7.703 × 10−1	7.750 × 10−1	8.472 × 10−1	8.479 × 10−1	8.279 × 10−1	8.372 × 10−1	8.517 × 10−1	8.081 × 10−1	8.289 × 10−1	8.495 × 10−1
		Std	1.059 × 10−2	3.360 × 10−3	1.465 × 10−1	1.508 × 10−1	5.951 × 10−3	3.757 × 10−3	7.480 × 10−2	1.714 × 10−2	2.449 × 10−2	1.165 × 10−1	1.248 × 10−1	2.668 × 10−3
	10	Ave	8.695 × 10−1	8.700 × 10−1	7.842 × 10−1	8.384 × 10−1	8.753 × 10−1	8.715 × 10−1	8.629 × 10−1	8.620 × 10−1	8.794 × 10−1	8.312 × 10−1	7.870 × 10−1	8.714 × 10−1
		Std	1.414 × 10−2	6.288 × 10−3	1.040 × 10−1	7.848 × 10−2	1.274 × 10−2	7.850 × 10−3	4.749 × 10−2	2.606 × 10−2	2.837 × 10−2	9.381 × 10−2	1.216 × 10−1	6.170 × 10−3
Face	4	Ave	7.075 × 10−1	7.081 × 10−1	7.064 × 10−1	6.541 × 10−1	7.075 × 10−1	7.075 × 10−1	7.063 × 10−1	7.076 × 10−1	7.059 × 10−1	7.075 × 10−1	7.047 × 10−1	7.076 × 10−1
		Std	1.469 × 10−3	1.129 × 10−16	5.335 × 10−3	3.154 × 10−2	1.469 × 10−3	1.469 × 10−3	1.966 × 10−3	2.769 × 10−3	2.476 × 10−3	1.469 × 10−3	1.489 × 10−2	1.339 × 10−3
	6	Ave	7.996 × 10−1	8.008 × 10−1	7.991 × 10−1	7.226 × 10−1	8.006 × 10−1	8.009 × 10−1	8.005 × 10−1	7.999 × 10−1	8.002 × 10−1	8.010 × 10−1	7.738 × 10−1	8.011 × 10−1
		Std	2.221 × 10−3	9.756 × 10−4	4.667 × 10−3	3.326 × 10−2	1.127 × 10−3	8.038 × 10−4	1.177 × 10−3	4.140 × 10−3	1.643 × 10−3	7.993 × 10−4	1.845 × 10−2	6.329 × 10−4
	8	Ave	8.550 × 10−1	8.592 × 10−1	8.546 × 10−1	7.823 × 10−1	8.586 × 10−1	8.585 × 10−1	8.584 × 10−1	8.473 × 10−1	8.558 × 10−1	8.592 × 10−1	8.145 × 10−1	8.592 × 10−1
		Std	5.083 × 10−3	5.716 × 10−4	8.060 × 10−3	2.759 × 10−2	6.760 × 10−4	8.041 × 10−4	1.580 × 10−3	9.769 × 10−3	3.499 × 10−3	4.129 × 10−4	2.016 × 10−2	3.555 × 10−4
	10	Ave	8.876 × 10−1	8.950 × 10−1	8.883 × 10−1	8.117 × 10−1	8.946 × 10−1	8.926 × 10−1	8.932 × 10−1	8.770 × 10−1	8.920 × 10−1	8.951 × 10−1	8.530 × 10−1	8.951 × 10−1
		Std	4.300 × 10−3	8.104 × 10−4	1.080 × 10−2	2.706 × 10−2	1.427 × 10−3	2.220 × 10−3	4.585 × 10−3	8.683 × 10−3	3.819 × 10−3	5.712 × 10−4	1.834 × 10−2	1.080 × 10−3
cell	4	Ave	7.280 × 10−1	7.280 × 10−1	7.285 × 10−1	6.580 × 10−1	7.280 × 10−1	7.280 × 10−1	7.280 × 10−1	7.267 × 10−1	7.278 × 10−1	7.280 × 10−1	7.296 × 10−1	7.280 × 10−1
		Std	2.258 × 10−16	2.258 × 10−16	1.541 × 10−3	5.242 × 10−2	2.258 × 10−16	2.258 × 10−16	2.258 × 10−16	3.653 × 10−3	7.484 × 10−4	2.258 × 10−16	3.165 × 10−2	2.258 × 10−16
	6	Ave	7.257 × 10−1	7.306 × 10−1	7.310 × 10−1	7.124 × 10−1	7.273 × 10−1	7.337 × 10−1	7.121 × 10−1	7.216 × 10−1	7.189 × 10−1	7.337 × 10−1	7.729 × 10−1	7.313 × 10−1
		Std	2.230 × 10−2	1.755 × 10−2	2.271 × 10−2	4.690 × 10−2	2.146 × 10−2	1.577 × 10−2	2.181 × 10−2	2.928 × 10−2	2.387 × 10−2	1.670 × 10−2	5.526 × 10−2	1.694 × 10−2
	8	Ave	7.721 × 10−1	7.744 × 10−1	7.718 × 10−1	7.502 × 10−1	7.751 × 10−1	7.755 × 10−1	7.670 × 10−1	7.595 × 10−1	7.658 × 10−1	7.721 × 10−1	8.018 × 10−1	7.764 × 10−1
		Std	1.537 × 10−2	4.381 × 10−3	1.171 × 10−2	7.210 × 10−2	7.238 × 10−3	5.766 × 10−3	1.409 × 10−2	2.004 × 10−2	1.411 × 10−2	7.577 × 10−3	5.872 × 10−2	6.726 × 10−3
	10	Ave	8.013 × 10−1	8.062 × 10−1	8.020 × 10−1	7.557 × 10−1	8.049 × 10−1	8.057 × 10−1	8.010 × 10−1	7.837 × 10−1	7.980 × 10−1	8.082 × 10−1	8.283 × 10−1	8.073 × 10−1
		Std	1.288 × 10−2	7.855 × 10−3	1.377 × 10−2	7.543 × 10−2	7.773 × 10−3	6.641 × 10−3	7.168 × 10−3	1.768 × 10−2	1.028 × 10−2	3.994 × 10−3	6.185 × 10−2	5.809 × 10−3
lax	4	Ave	5.529 × 10−1	5.532 × 10−1	5.516 × 10−1	5.268 × 10−1	5.532 × 10−1	5.532 × 10−1	5.532 × 10−1	5.495 × 10−1	5.526 × 10−1	5.532 × 10−1	6.385 × 10−1	5.532 × 10−1
		Std	1.713 × 10−3	1.827 × 10−5	6.799 × 10−3	1.116 × 10−1	1.827 × 10−5	2.258 × 10−16	1.315 × 10−5	5.155 × 10−3	3.546 × 10−3	2.258 × 10−16	9.051 × 10−2	2.258 × 10−16
	6	Ave	6.406 × 10−1	6.409 × 10−1	6.395 × 10−1	6.758 × 10−1	6.413 × 10−1	6.409 × 10−1	6.384 × 10−1	6.215 × 10−1	6.291 × 10−1	6.413 × 10−1	7.701 × 10−1	6.409 × 10−1
		Std	1.080 × 10−2	1.138 × 10−4	4.699 × 10−3	1.335 × 10−1	3.469 × 10−3	7.855 × 10−5	6.135 × 10−3	2.067 × 10−2	2.005 × 10−2	1.938 × 10−3	8.578 × 10−2	5.952 × 10−5
	8	Ave	7.088 × 10−1	7.171 × 10−1	7.102 × 10−1	7.161 × 10−1	7.125 × 10−1	7.237 × 10−1	7.093 × 10−1	6.717 × 10−1	7.008 × 10−1	7.179 × 10−1	8.333 × 10−1	7.128 × 10−1
		Std	2.057 × 10−2	7.387 × 10−3	1.227 × 10−2	1.257 × 10−1	1.085 × 10−2	2.564 × 10−2	1.873 × 10−2	2.856 × 10−2	2.449 × 10−2	7.560 × 10−3	7.566 × 10−2	9.490 × 10−3
	10	Ave	7.490 × 10−1	7.562 × 10−1	7.558 × 10−1	7.720 × 10−1	7.613 × 10−1	7.566 × 10−1	7.552 × 10−1	7.231 × 10−1	7.464 × 10−1	7.599 × 10−1	8.827 × 10−1	7.534 × 10−1
		Std	2.904 × 10−2	1.145 × 10−2	1.741 × 10−2	1.060 × 10−1	1.464 × 10−2	1.873 × 10−2	1.777 × 10−2	4.774 × 10−2	2.805 × 10−2	1.113 × 10−2	4.364 × 10−2	1.340 × 10−2
Peppers	4	Ave	7.140 × 10−1	7.138 × 10−1	7.138 × 10−1	6.680 × 10−1	7.132 × 10−1	7.123 × 10−1	7.084 × 10−1	7.115 × 10−1	7.105 × 10−1	7.125 × 10−1	7.144 × 10−1	7.136 × 10−1
		Std	6.862 × 10−4	5.581 × 10−4	5.141 × 10−3	2.917 × 10−2	3.540 × 10−3	5.088 × 10−3	8.995 × 10−3	5.645 × 10−3	7.526 × 10−3	5.162 × 10−3	1.010 × 10−2	2.688 × 10−4
	6	Ave	7.853 × 10−1	7.870 × 10−1	7.836 × 10−1	7.295 × 10−1	7.869 × 10−1	7.869 × 10−1	7.868 × 10−1	7.805 × 10−1	7.835 × 10−1	7.869 × 10−1	7.688 × 10−1	7.869 × 10−1
		Std	3.432 × 10−3	3.632 × 10−4	7.182 × 10−3	3.193 × 10−2	4.683 × 10−4	5.220 × 10−5	1.353 × 10−4	8.973 × 10−3	5.404 × 10−3	7.161 × 10−5	1.230 × 10−2	3.388 × 10−16
	8	Ave	8.161 × 10−1	8.193 × 10−1	8.196 × 10−1	7.701 × 10−1	8.199 × 10−1	8.194 × 10−1	8.189 × 10−1	8.123 × 10−1	8.193 × 10−1	8.191 × 10−1	8.098 × 10−1	8.191 × 10−1
		Std	4.566 × 10−3	8.214 × 10−4	2.085 × 10−3	2.736 × 10−2	1.638 × 10−3	8.132 × 10−4	1.366 × 10−3	8.400 × 10−3	4.358 × 10−3	5.677 × 10−4	1.309 × 10−2	3.209 × 10−4
	10	Ave	8.490 × 10−1	8.528 × 10−1	8.564 × 10−1	8.043 × 10−1	8.578 × 10−1	8.538 × 10−1	8.571 × 10−1	8.423 × 10−1	8.489 × 10−1	8.565 × 10−1	8.329 × 10−1	8.526 × 10−1
		Std	5.466 × 10−3	5.211 × 10−3	7.153 × 10−3	1.993 × 10−2	4.628 × 10−3	3.044 × 10−3	3.693 × 10−3	9.707 × 10−3	5.545 × 10−3	3.985 × 10−3	1.457 × 10−2	4.998 × 10−3
milkdrop	4	Ave	6.973 × 10−1	6.976 × 10−1	6.974 × 10−1	6.537 × 10−1	6.975 × 10−1	6.975 × 10−1	6.975 × 10−1	6.969 × 10−1	6.977 × 10−1	6.975 × 10−1	7.004 × 10−1	6.975 × 10−1
		Std	8.773 × 10−4	4.705 × 10−4	8.330 × 10−4	4.294 × 10−2	2.258 × 10−16	2.258 × 10−16	2.258 × 10−16	3.397 × 10−3	2.426 × 10−3	2.258 × 10−16	2.427 × 10−2	2.258 × 10−16
	6	Ave	7.810 × 10−1	7.809 × 10−1	7.812 × 10−1	7.238 × 10−1	7.809 × 10−1	7.808 × 10−1	7.805 × 10−1	7.745 × 10−1	7.803 × 10−1	7.809 × 10−1	7.758 × 10−1	7.809 × 10−1
		Std	1.086 × 10−3	1.149 × 10−3	2.277 × 10−4	3.381 × 10−2	1.149 × 10−3	1.223 × 10−3	1.959 × 10−3	8.329 × 10−3	1.930 × 10−3	1.148 × 10−3	1.347 × 10−2	1.146 × 10−3
	8	Ave	8.069 × 10−1	8.084 × 10−1	8.074 × 10−1	7.600 × 10−1	8.089 × 10−1	8.092 × 10−1	8.076 × 10−1	8.038 × 10−1	8.056 × 10−1	8.094 × 10−1	8.023 × 10−1	8.091 × 10−1
		Std	5.063 × 10−3	3.339 × 10−3	4.281 × 10−3	3.227 × 10−2	3.130 × 10−3	2.705 × 10−3	2.867 × 10−3	9.703 × 10−3	6.706 × 10−3	2.680 × 10−3	1.778 × 10−2	2.930 × 10−3
	10	Ave	8.356 × 10−1	8.402 × 10−1	8.395 × 10−1	7.859 × 10−1	8.391 × 10−1	8.413 × 10−1	8.394 × 10−1	8.283 × 10−1	8.355 × 10−1	8.410 × 10−1	8.259 × 10−1	8.400 × 10−1
		Std	7.991 × 10−3	3.325 × 10−3	3.710 × 10−3	3.091 × 10−2	6.057 × 10−3	1.637 × 10−3	5.265 × 10−3	1.117 × 10−2	1.082 × 10−2	2.367 × 10−3	1.595 × 10−2	4.259 × 10−3
testpat	4	Ave	8.369 × 10−1	8.369 × 10−1	8.363 × 10−1	7.823 × 10−1	8.369 × 10−1	8.369 × 10−1	8.369 × 10−1	8.367 × 10−1	8.368 × 10−1	8.369 × 10−1	8.297 × 10−1	8.369 × 10−1
		Std	6.417 × 10−5	5.646 × 10−16	2.608 × 10−3	4.470 × 10−2	5.646 × 10−16	5.646 × 10−16	5.646 × 10−16	1.499 × 10−3	5.048 × 10−4	5.646 × 10−16	7.663 × 10−3	5.646 × 10−16
	6	Ave	8.765 × 10−1	8.766 × 10−1	8.768 × 10−1	8.525 × 10−1	8.767 × 10−1	8.765 × 10−1	8.764 × 10−1	8.755 × 10−1	8.773 × 10−1	8.764 × 10−1	8.792 × 10−1	8.764 × 10−1
		Std	1.527 × 10−3	5.366 × 10−4	1.721 × 10−3	1.623 × 10−2	9.065 × 10−4	3.554 × 10−4	3.765 × 10−5	4.386 × 10−3	1.911 × 10−3	3.765 × 10−5	1.397 × 10−2	4.517 × 10−16
	8	Ave	9.049 × 10−1	9.061 × 10−1	9.063 × 10−1	8.757 × 10−1	9.063 × 10−1	9.063 × 10−1	9.063 × 10−1	9.006 × 10−1	9.045 × 10−1	9.051 × 10−1	9.037 × 10−1	9.044 × 10−1
		Std	4.544 × 10−3	1.486 × 10−3	4.420 × 10−3	2.083 × 10−2	2.958 × 10−3	1.625 × 10−3	2.924 × 10−3	8.614 × 10−3	6.059 × 10−3	1.647 × 10−3	1.280 × 10−2	2.062 × 10−3
	10	Ave	9.243 × 10−1	9.250 × 10−1	9.286 × 10−1	9.019 × 10−1	9.247 × 10−1	9.258 × 10−1	9.253 × 10−1	9.138 × 10−1	9.220 × 10−1	9.265 × 10−1	9.188 × 10−1	9.236 × 10−1
		Std	6.212 × 10−3	3.609 × 10−3	6.968 × 10−3	1.654 × 10−2	3.928 × 10−3	2.166 × 10−3	7.020 × 10−3	9.176 × 10−3	6.412 × 10−3	2.297 × 10−3	1.575 × 10−2	4.362 × 10−3
bank	4	Ave	7.453 × 10−1	7.455 × 10−1	7.456 × 10−1	7.181 × 10−1	7.454 × 10−1	7.455 × 10−1	7.445 × 10−1	7.452 × 10−1	7.436 × 10−1	7.454 × 10−1	7.505 × 10−1	7.455 × 10−1
		Std	4.328 × 10−4	1.368 × 10−5	6.807 × 10−4	2.933 × 10−2	3.717 × 10−4	1.368 × 10−5	1.978 × 10−3	8.640 × 10−4	3.076 × 10−3	3.281 × 10−4	1.087 × 10−2	2.258 × 10−16
	6	Ave	8.146 × 10−1	8.145 × 10−1	8.150 × 10−1	7.698 × 10−1	8.148 × 10−1	8.147 × 10−1	8.150 × 10−1	8.087 × 10−1	8.149 × 10−1	8.146 × 10−1	8.057 × 10−1	8.146 × 10−1
		Std	2.020 × 10−3	4.273 × 10−4	2.329 × 10−3	2.274 × 10−2	5.207 × 10−4	3.458 × 10−4	7.402 × 10−4	8.765 × 10−3	6.243 × 10−3	2.945 × 10−4	1.208 × 10−2	1.129 × 10−16
	8	Ave	8.573 × 10−1	8.574 × 10−1	8.568 × 10−1	8.073 × 10−1	8.590 × 10−1	8.606 × 10−1	8.579 × 10−1	8.511 × 10−1	8.580 × 10−1	8.573 × 10−1	8.406 × 10−1	8.563 × 10−1
		Std	5.937 × 10−3	3.636 × 10−3	3.084 × 10−3	2.576 × 10−2	3.660 × 10−3	4.494 × 10−3	5.852 × 10−3	8.753 × 10−3	6.641 × 10−3	3.027 × 10−3	1.202 × 10−2	1.508 × 10−3
	10	Ave	8.807 × 10−1	8.855 × 10−1	8.842 × 10−1	8.350 × 10−1	8.870 × 10−1	8.877 × 10−1	8.864 × 10−1	8.737 × 10−1	8.808 × 10−1	8.868 × 10−1	8.643 × 10−1	8.843 × 10−1
		Std	5.095 × 10−3	1.843 × 10−3	4.763 × 10−3	2.402 × 10−2	3.397 × 10−3	2.086 × 10−3	3.416 × 10−3	8.628 × 10−3	7.394 × 10−3	1.751 × 10−3	1.252 × 10−2	3.140 × 10−3
boat	4	Ave	7.074 × 10−1	7.074 × 10−1	7.072 × 10−1	6.860 × 10−1	7.074 × 10−1	7.074 × 10−1	7.074 × 10−1	7.068 × 10−1	7.069 × 10−1	7.074 × 10−1	7.192 × 10−1	7.074 × 10−1
		Std	1.091 × 10−4	3.388 × 10−16	6.909 × 10−4	3.095 × 10−2	3.388 × 10−16	3.388 × 10−16	3.388 × 10−16	1.793 × 10−3	1.092 × 10−3	3.388 × 10−16	1.618 × 10−2	3.388 × 10−16
	6	Ave	8.054 × 10−1	8.060 × 10−1	8.054 × 10−1	7.487 × 10−1	8.058 × 10−1	8.057 × 10−1	8.043 × 10−1	8.019 × 10−1	8.005 × 10−1	8.054 × 10−1	7.949 × 10−1	8.063 × 10−1
		Std	2.817 × 10−3	8.107 × 10−4	3.250 × 10−3	3.761 × 10−2	1.479 × 10−3	9.191 × 10−4	2.591 × 10−3	9.995 × 10−3	6.622 × 10−3	1.534 × 10−3	2.680 × 10−2	5.646 × 10−16
	8	Ave	8.572 × 10−1	8.590 × 10−1	8.562 × 10−1	7.880 × 10−1	8.605 × 10−1	8.589 × 10−1	8.601 × 10−1	8.454 × 10−1	8.548 × 10−1	8.600 × 10−1	8.440 × 10−1	8.595 × 10−1
		Std	5.838 × 10−3	1.476 × 10−3	1.118 × 10−2	3.765 × 10−2	2.732 × 10−3	2.691 × 10−3	2.399 × 10−3	1.046 × 10−2	8.285 × 10−3	1.574 × 10−3	1.483 × 10−2	1.646 × 10−3
	10	Ave	8.875 × 10−1	8.950 × 10−1	8.929 × 10−1	8.270 × 10−1	8.966 × 10−1	8.942 × 10−1	8.966 × 10−1	8.735 × 10−1	8.888 × 10−1	8.975 × 10−1	8.691 × 10−1	8.955 × 10−1
		Std	8.308 × 10−3	3.513 × 10−3	1.061 × 10−2	3.863 × 10−2	3.691 × 10−3	3.649 × 10−3	3.047 × 10−3	1.122 × 10−2	7.970 × 10−3	1.871 × 10−3	2.179 × 10−2	2.273 × 10−3
plane	4	Ave	7.890 × 10−1	7.894 × 10−1	7.898 × 10−1	7.974 × 10−1	7.895 × 10−1	7.895 × 10−1	7.895 × 10−1	7.885 × 10−1	7.890 × 10−1	7.895 × 10−1	8.317 × 10−1	7.895 × 10−1
		Std	1.084 × 10−3	5.726 × 10−4	4.071 × 10−3	4.578 × 10−2	0.000 × 100	0.000 × 100	0.000 × 100	3.358 × 10−3	1.777 × 10−3	0.000 × 100	2.080 × 10−2	0.000 × 100
	6	Ave	8.635 × 10−1	8.640 × 10−1	8.637 × 10−1	8.207 × 10−1	8.638 × 10−1	8.637 × 10−1	8.624 × 10−1	8.615 × 10−1	8.620 × 10−1	8.639 × 10−1	8.431 × 10−1	8.641 × 10−1
		Std	3.817 × 10−3	1.236 × 10−3	4.368 × 10−3	3.336 × 10−2	1.476 × 10−3	4.960 × 10−4	2.361 × 10−3	9.206 × 10−3	5.496 × 10−3	7.737 × 10−4	2.572 × 10−2	5.863 × 10−4
	8	Ave	8.937 × 10−1	8.975 × 10−1	8.970 × 10−1	8.509 × 10−1	8.968 × 10−1	8.956 × 10−1	8.954 × 10−1	8.901 × 10−1	8.961 × 10−1	8.974 × 10−1	8.565 × 10−1	8.977 × 10−1
		Std	6.744 × 10−3	1.126 × 10−3	2.605 × 10−3	3.239 × 10−2	1.337 × 10−3	4.281 × 10−3	4.413 × 10−3	1.241 × 10−2	6.003 × 10−3	1.190 × 10−3	2.299 × 10−2	1.031 × 10−3
	10	Ave	9.228 × 10−1	9.258 × 10−1	9.194 × 10−1	8.530 × 10−1	9.228 × 10−1	9.156 × 10−1	9.215 × 10−1	9.111 × 10−1	9.196 × 10−1	9.255 × 10−1	8.905 × 10−1	9.271 × 10−1
		Std	7.835 × 10−3	3.557 × 10−3	6.777 × 10−3	2.605 × 10−2	7.254 × 10−3	7.919 × 10−3	8.290 × 10−3	9.250 × 10−3	1.122 × 10−2	3.740 × 10−3	1.775 × 10−2	1.888 × 10−3
saturn	4	Ave	8.308 × 10−1	8.307 × 10−1	8.308 × 10−1	8.135 × 10−1	8.307 × 10−1	8.307 × 10−1	8.307 × 10−1	8.313 × 10−1	8.312 × 10−1	8.307 × 10−1	8.341 × 10−1	8.307 × 10−1
		Std	1.683 × 10−4	4.517 × 10−16	4.282 × 10−4	1.362 × 10−2	1.126 × 10−4	8.105 × 10−5	1.126 × 10−4	3.380 × 10−3	1.145 × 10−3	4.517 × 10−16	1.165 × 10−2	4.517 × 10−16
	6	Ave	8.797 × 10−1	8.801 × 10−1	8.790 × 10−1	8.553 × 10−1	8.802 × 10−1	8.802 × 10−1	8.800 × 10−1	8.781 × 10−1	8.792 × 10−1	8.802 × 10−1	8.763 × 10−1	8.802 × 10−1
		Std	1.398 × 10−3	3.497 × 10−4	4.122 × 10−3	1.739 × 10−2	1.991 × 10−4	7.843 × 10−5	3.684 × 10−4	4.918 × 10−3	3.962 × 10−3	3.388 × 10−16	8.176 × 10−3	5.571 × 10−5
	8	Ave	9.080 × 10−1	9.085 × 10−1	9.067 × 10−1	8.814 × 10−1	9.090 × 10−1	9.096 × 10−1	9.088 × 10−1	9.064 × 10−1	9.089 × 10−1	9.088 × 10−1	8.989 × 10−1	9.090 × 10−1
		Std	2.678 × 10−3	3.061 × 10−4	3.323 × 10−3	1.312 × 10−2	1.082 × 10−3	1.126 × 10−3	1.597 × 10−3	5.910 × 10−3	2.770 × 10−3	7.375 × 10−4	8.467 × 10−3	8.404 × 10−4
	10	Ave	9.252 × 10−1	9.290 × 10−1	9.243 × 10−1	8.989 × 10−1	9.293 × 10−1	9.302 × 10−1	9.289 × 10−1	9.216 × 10−1	9.278 × 10−1	9.291 × 10−1	9.149 × 10−1	9.295 × 10−1
		Std	3.152 × 10−3	9.182 × 10−4	6.260 × 10−3	1.128 × 10−2	1.100 × 10−3	7.742 × 10−4	1.923 × 10−3	4.307 × 10−3	1.812 × 10−3	8.878 × 10−4	6.723 × 10−3	9.445 × 10−4

, MEIVYA consistently achieves high SSIM values across different test images and threshold values, indicating that its segmentation results exhibit high consistency with the original images in terms of brightness, contrast, and structural information. Overall, the SSIM values of all algorithms generally increase with the number of thresholds, suggesting that more thresholds can more accurately characterize the grayscale levels and regional structures in the image, thereby improving the structural similarity between the segmented image and the original image. However, compared to other comparison algorithms, MEIVYA achieves higher average SSIM values on most test images, with a relatively small standard deviation, indicating that it not only produces clearer and more natural segmentation results but also demonstrates good stability and consistency in multiple independent runs. MEIVYA’s SSIM advantage is particularly pronounced in images with complex edges, rich textures, or uneven grayscale distribution, demonstrating its ability to more effectively preserve target structures and local details, reducing structural distortion caused by unreasonable thresholding. Furthermore, the smaller standard deviation indicates that MEIVYA is less affected by random factors and can consistently produce high-quality segmentation results in repeated experiments. Overall, the experimental results in Table 10 further demonstrate that MEIVYA has strong structure preservation ability and robustness in multi-threshold image segmentation tasks, and its overall performance is better than most of the comparison algorithms.

5. Conclusions

The proposed MEIVYA, through the synergistic effect of three mechanisms—chaotic initialization, adaptive growth rate adjustment, and elite-guided collaborative search—effectively enhances the initial global exploration capability, improves the accuracy of mid-to-late-stage local development, and reduces the risk of premature convergence through more comprehensive information sharing. This significantly improves overall search efficiency and solution stability while maintaining a simple algorithm structure. In the CEC2014 benchmark test Appendix A, comparing the convergence curves and box plots, MEIVYA achieves faster descent rates, lower final values, smaller fluctuations, and a more concentrated result distribution on most functions, indicating stronger robustness and resistance to random disturbances. In real-world applications, experiments using the Otsu’s maximum inter-class variance criterion for multi-threshold image segmentation further validate the algorithm’s effectiveness in high-dimensional threshold combination optimization: under different threshold levels, MEIVYA can stably obtain better objective function values and achieve clearer structural hierarchy and more detailed texture preservation in visual effects. Furthermore, evaluation metrics such as PSNR, SSIM, and FSIM provide support from three perspectives: pixel error, structural consistency, and perceptual feature fidelity. In summary, MEIVYA provides a stable and efficient swarm intelligence optimization tool for complex numerical optimization and multi-threshold image segmentation, and has the potential to be further extended to more engineering optimization and vision tasks.