STS-TransUNet: Semi-supervised Tooth Segmentation Transformer U-Net for dental panoramic image

Duolin Sun; Jianqing Wang; Zhaoyu Zuo; Yixiong Jia; Yimou Wang; Duolin Sun; Jianqing Wang; Zhaoyu Zuo; Yixiong Jia; Yimou Wang

doi:10.3934/mbe.2024104

Mathematical Biosciences and Engineering

2024, Volume 21, Issue 2: 2366-2384. doi: 10.3934/mbe.2024104

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Research article Special Issues

STS-TransUNet: Semi-supervised Tooth Segmentation Transformer U-Net for dental panoramic image

1.
University of Science and Technology of China, Hefei, China
2.
Hefei Institutes of Physical Science, Chinese Academy of Sciences, Hefei, China
3.
Hangzhou Sai Future Technology Co., Ltd, Hangzhou, China
4.
Department of Electrical and Electronic Engineering, The University of Hong Kong, Hong Kong, China

Academic Editor: Junxin Chen
^† Co-First Authors: These two authors contributed equally.

Received: 16 November 2023 Revised: 22 December 2023 Accepted: 23 December 2023 Published: 15 January 2024

In this paper, we introduce a novel deep learning method for dental panoramic image segmentation, which is crucial in oral medicine and orthodontics for accurate diagnosis and treatment planning. Traditional methods often fail to effectively combine global and local context, and struggle with unlabeled data, limiting performance in varied clinical settings. We address these issues with an advanced TransUNet architecture, enhancing feature retention and utilization by connecting the input and output layers directly. Our architecture further employs spatial and channel attention mechanisms in the decoder segments for targeted region focus, and deep supervision techniques to overcome the vanishing gradient problem for more efficient training. Additionally, our network includes a self-learning algorithm using unlabeled data, boosting generalization capabilities. Named the Semi-supervised Tooth Segmentation Transformer U-Net (STS-TransUNet), our method demonstrated superior performance on the MICCAI STS-2D dataset, proving its effectiveness and robustness in tooth segmentation tasks.

Keywords:

Citation: Duolin Sun, Jianqing Wang, Zhaoyu Zuo, Yixiong Jia, Yimou Wang. STS-TransUNet: Semi-supervised Tooth Segmentation Transformer U-Net for dental panoramic image[J]. Mathematical Biosciences and Engineering, 2024, 21(2): 2366-2384. doi: 10.3934/mbe.2024104

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Abstract

1. Introduction

With the development of science and technology in human society, traditional technology can no longer not be used to solve complex practical problems. Therefore, the meta-heuristic algorithm came into being. Meta-heuristic algorithms are widely used in these complex practical problems because of their simple concept and flexible parameters. At present, the proposed algorithms are mainly divided into four categories ^[1], The first category is swarm-based algorithms, such as Artificial Bee Colony (ABC) ^[2]. The second category is human-based algorithms, such as Teaching based learning algorithm (TBLA) ^[3]. The third category is physics-based algorithms, such as Fireworks Algorithm (FWA) ^[4]. The last category is evolutionary algorithms, such as Tree Growth Algorithm (TGA) ^[5]. The classification of the meta-heuristic algorithm is shown in Table 1.

Table 1. Classification of meta-heuristic algorithm.

Classes	Algorithm
Evolutionary	Tree Growth Algorithm (TGA) ^[5]
	Arithmetic Optimization Algorithm (AOA) ^[6]
	Genetic Algorithm (GA) ^[7]
	Differential evolution (DE) ^[8]
	Genetic Programming (GP) ^[9]
	Evolutionary Strategies (ES) ^[10]
Human-based	Teaching based learning algorithm (TBLA) ^[3]
	Harmony Search (HS) ^[11]
	Imperialist Competitive Algorithm (ICA) ^[12]
	Fireworks Algorithm (FWA) ^[4]
	Collective Decision Optimization (CSO) ^[13]
	Socio Evolution & Learning Optimization Algorithm (SELOA) ^[14]
	Political Optimizer (PO) ^[15]
Physics-based	Henry Gas Solubility Optimization (HGSO) ^[16]
	Big Bang–Big Crunch (BBBC) ^[17]
	Multi-verse Optimizer (MVO) ^[18]
	Electromagnetic Field Optimization (EFO) ^[19]
	Gravitational Search Algorithm (GSA) ^[20]
	Thermal Exchange Optimization (TEO) ^[21]
	Central Force Optimization (CFO) ^[22]
Swarm-based	Artificial Bee Colony (ABC) ^[23]
	Particle Swarm Optimization (PSO) ^[24]
	Moth Flame Optimization (MFO) ^[25]
	Ant Colony Optimization (ACO) ^[26]
	Marine Predators Algorithm (MPA) ^[27]
	Seagull optimization algorithm (SOA) ^[28]
	Sooty Tern Optimization Algorithm (STOA) ^[29]
	Aquila Optimizer (AO) ^[30]
	Grey Wolf Optimizer (GWO) ^[31]
	Salp Swarm Algorithm (SSA) ^[32]
	Harris Hawks Optimization (HHO) ^[33]
	Monarch Butterfly Optimization (MBO) ^[34]
	Slime Mould Algorithm (SMA) ^[35]
	Moth Search Algorithm (MSA) ^[36]
	Hunger Games Search (HGS) ^[37]
	Runge Kutta Method (RUN) ^[38]
	Colony Predation Algorithm (CPA) ^[39]
	Welghted mean of vectors (INFO) ^[40]
	Fick's Law Algorithm(FLA) ^[41]
	Ant Lion Optimization ^[42]

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However, not all problems can be solved ^[43] by these algorithms, especially the multidisciplinary problems that have become increasingly complex in recent years. Therefore, we need to find better algorithms for solving multidisciplinary and complex problems.

The improvement for AO is less because the time to propose the algorithm is not long, so far there is not enough research on AO algorithms. Wang et al. ^[44] proposed a hybrid improvement of the AO algorithm and the HHO algorithm and added hunting strategies and adversarial-based learning rules to them. It has been verified by experiments that the improved algorithm has a very good performance. Elaziz et al ^[45] proposed a Mobile Net V3 AO hybrid model. He uses the AO algorithm to reduce the dimensionality of the image, thereby improving the accuracy of the data. Zhang et al. ^[46] proposed the hybridization of the AO algorithm and the AOA algorithm. Zhao et al. ^[47] applied the JAYA algorithm to the parameter extraction of photovoltaic models and added a chaotic adaptive strategy. Zhang et al. ^[48] used the enhanced adaptive comprehensive learning hybrid algorithm of Rao-1 and JAYA algorithm for photovoltaic model parameter extraction. Zhou et al. proposed ^[49] metaphor-free dynamic spherical evolution for parameter estimation of photovoltaic modules, it helps the research on the application of swarm intelligence algorithm in photovoltaic modules. We believe that the enhanced AO algorithm can be used in photovoltaic model parameter extraction. In the future, we will also conduct relevant research. Simrandeep Singh et al. ^[50] proposed Arithmetic optimization based image segmentation. Hussien et al. ^[51] proposed comparative research and application of Harris Eagle optimization research progress. Wang et al. ^[52] proposed an enhanced remora optimization algorithm. Hashim et al. ^[53] proposed a snake optimizer. Zheng et al. ^[54] proposed an improved wild horse optimizer. Hussien et al. ^[55] proposed an enhanced COOT optimization algorithm for dimensionality reduction. Yu et al. ^[56] proposed an enhanced aquila optimizer for global optimization and constrained engineering problems. Yang et al. ^[57] proposed an efficient DBSCAN optimized by the arithmetic optimization algorithm. Cui et al. ^[58] proposed a modified slime mold algorithm via levy flight. Yang et al. proposed ^[59] an opposition learning and spiral modelling based arithmetic optimization algorithm. Hussien et al. ^[60] proposed boosting whale optimization with an evolution strategy and Gaussian random walks, and used it to image segmentation. Hussien et al. ^[61] proposed a comprehensive review of moth-flame optimization: variants, hybrids, and applications. Also, swarm intelligence algorithms are used to solve real-world engineering problems. Yu et al. ^[62] proposed laplace crossover and random replacement strategy boosted Harris hawks optimization. Qi et al. ^[63] proposed directional mutation and crossover for the immature performance of whale algorithm with application to engineering optimization. Zhao et al. ^[64] proposed opposition-based ant colony optimization with an all-dimension neighborhood search for engineering design. Zhou et al. ^[65] proposed advanced orthogonal learning and Gaussian barebone hunger games for engineering design.

Considering the powerful global exploration capability of the AO and early convergence ^[44] caused by insufficient exploration and exploitation in the later phases, an enhanced AO with velocity-aided global search mechanism and adaptive opposition-based learning, called VAIAO. The proposed VAIAO has better search ability than the original algorithm, and it is got better performance easily. This article was inspired by the velocity-aided global search mechanism ^[66], and this paper applies the velocity-aided global search Mechanism to the AO. On the other land, the adaptive opposition-based learning rule is introduced to avoid being trapped in the local optimum.

The performance of the VAIAO is tested by using 27 classical benchmark functions, the Wilcoxon statistical test, and five engineering optimization problems. In order to compare the various experimental results, IAO ^[67], AO ^[30] and some well-known algorithms including Arithmetic AOA ^[6], HHO ^[33], STOA ^[29], ChOA ^[68], SOA ^[28], PSO ^[24], DE ^[8] are introduced. The results of the experiment show that the proposed VAIAO is better than another algorithm.

The basic AO and improved strategies will be introduced later. And carry out simulation experiments and engineering optimization problems using VAIAO and comparison algorithms to solve. Finally, the conclusion is given at the end of the article.

2. Aquila optimizer

When the Aquila is preying, it can switch four strategies in different situations when facing the prey in different situations to escape the eagle's predation strategy, which can be represented by the following four strategies.

2.1. Strategy 1: Flying high for searching prey

In this strategy, Aquila flies through high-altitude hunting areas to find the best spot for prey, and once it finds the best spot, Aquila swoops vertically at prey. This behavior can be expressed by the formal.

$X\left(t+1\right) = {X}_{best}\left(t\right)\times \left(1-\frac{t}{T}\right)+\left({X}_{M}\left(t\right)-{X}_{best}\left(t\right)\times rand\right)$

(1)

where, ${X}_{best}\left(t\right)$ represent the global optimal position, ${X}_{M}\left(t\right)$ represents the current mean position, and $\mathrm{t}$ and $\mathrm{T}$ represent the current number of iterations and maximum number of iterations.

2.2. Short glide attack and fight

In this strategy, Aquila will switch from flying at high altitude to hovering on the head of the prey, preparing for Aquila.

$X\left(t+1\right) = {X}_{best}\left(t\right)\times LF\left(D\right)+{X}_{R}\left(t\right)+\left(\mathrm{y}-\mathrm{x}\right)\times \mathrm{r}\mathrm{a}\mathrm{n}\mathrm{d}$

(2)

where, the Aquila's random position is ${X}_{R}\left(t\right)$ , and $D$ is the size of dimension. $LF$ is Levy flight function. The $\mathrm{y}$ and $\mathrm{x}$ represents the shape of the search, which can be expressed by the formula:

$\left\{\begin{array}{c}x = \left({r}_{1}+0.00565\times {D}_{1}\right)\times \mathit{sin}\left(-\omega \times {D}_{1}+\frac{3\times \mathit{\Pi } }{2}\right)\\ y = \left({r}_{1}+0.00565\times {D}_{1}\right)\times \mathit{cos}\left(-\omega \times {D}_{1}+\frac{3\times \mathit{\Pi } }{2}\right)\end{array}\right.$

(3)

$LF\left(x\right) = 0.01\times \frac{\mu \times \sigma }{{\left|v\right|}^{\frac{1}{\beta }}}{, }\sigma = {\left(\frac{r\left(1+\beta \right)\times \mathit{sin}\left(\frac{\mathit{\Pi } \beta }{2}\right)}{r\left(\frac{1+\beta }{2}\right)\times \beta \times {2}^{\left(\frac{\beta -1}{2}\right)}}\right)}^{\frac{1}{\beta }}$

(4)

where, ${\mathrm{r}}_{1}$ is Aquila's search cycles which number between $\left[\mathrm{1, 20}\right]$ , ${\mathrm{D}}_{1}$ is a random integer from one to D dimensions, and ${\rm{ \mathsf{ ω} }}$ is a constant of 0.005.

2.3. Approaching prey and attack

In this strategy, Aquila first finds and determines the approximate location of the prey through strategy 2. After finding the prey, Aquila descends vertically for initial hunting. This behavior can be expressed by the formula:

$X\left(t+1\right) = {(X}_{best}\left(t\right)-{X}_{M}\left(t\right)\times \alpha -rand+\left(\left(UB-LB\right)rand+LB\right)\times \delta$

(5)

where ${\rm{ \mathsf{ α} }}$ and ${\rm{ \mathsf{ δ} }}$ are adjustment parameters during the development process, fixed at 0.1, $\mathrm{U}\mathrm{B}$ and $\mathrm{L}\mathrm{B}$ are the upper and lower bounds of the search space, respectively.

2.4. Catching prey on land

In this strategy, Aquila will follow the prey's escape track to land and chase it, then attack it. This behavior be expressed by the formula:

$\left(t+1\right) = QF\times {X}_{best}\left(t\right)-\left({G}_{1}\times X\left(t\right)\times rand\right)-{G}_{2}\times LF\left(D\right)+rand{\times G}_{1}$

(6)

$QF\left(t\right) = {t}^{\frac{2\times rand-1}{{\left(1-T\right)}^{2}}}$

(7)

$\left\{\begin{array}{c}{G}_{1} = 2\times rand-1\\ {G}_{2} = 2\times \left(1-\frac{t}{T}\right)\end{array}\right.$

(8)

where, ${G}_{1}$ is the random number which between [1, −1], and ${G}_{2}$ is the flight slope when Aquila tracking the prey between [2,0].

3. The simplified IAO

The IAO ^[67] is a simplification of the AO ^[69]. Although the AO algorithm has better convergence speed and exploration ability, the AO algorithm has poor convergence performance in the later stage. For solving this problem, the IAO algorithm selects only strategies 1 and 2 based on the Aquila algorithm. In this way, IAO algorithm can avoid the situation that the convergence speed decreased in the later period. The simplified improved Aquila will be easier than ever.

4. The proposed VAIAO algorithm

During exploration, individuals perform fast flights and hunts of Aquila in the search space. Even though the simplified algorithm speeds up the convergence and improves the search ability. Individuals can still easily fall into local optima. And the speed of convergence and exploration ability of AO can be strengthened That's why we should improve and overcome these shortcomings.

4.1. Velocity-Aided global search mechanism

When Aquila individuals move in the search space without considering the two features of velocity and acceleration, these movements do not always occur continuously or smoothly. The movement of search agents in the search space may break down, then leading to potential drift. Due to this reason, search agents may drift when executing search strategies. Considering the velocity term can avoid premature convergence by helping the search agent maintain its unique trajectory, enhancing the search agent's exploration capabilities, and balancing the exploration and development phases of the optimization process. In VAIAO, an initial random velocity is defined for each Aquila search agent as it performs a search policy move, and an initial random position is defined for each search agent in the search space. Therefore, each search agent has a velocity and position in each dimension of the optimization problem. This idea be expressed by this formula:

${V}_{j}^{t+1} = k\times \left(sgn\left({A}_{j}^{t}\right)\times \left|{V}_{j}^{t}\right|\right)+{A}_{j}^{t}\times {D}_{j}^{t}$

(9)

where, ${V}_{j}^{t}$ represent the velocity of the Aquila search agent in the $j$ th dimension. $sgn$ is the sign function. ${A}_{j}^{t}$ represents the acceleration terms of the search agents, and ${A}_{j}^{t}$ be expressed by this formula:

${A}_{j}^{t} = \left(2\times {r}_{1}-1\right)\times {a}^{2}$

(10)

where, ${r}_{1}$ represents a random number between 1 and 0, Furthermore, a is a linearly decreasing parameter determined as follows:

$a = {a}_{max}-\left({a}_{max}-{a}_{min}\right)\times \frac{t-1}{{t}_{max}\ \ -1}$

(11)

${D}_{j}^{t}$ a represents the modified distance between the focused search agent and the leading Aquila. These distances in each dimension can be calculated as follows:

${D}_{j}^{t} = \left|{C}_{j}^{t}\times {X}_{1, j}^{t}-{X}_{i, j}^{t}\right|$

(12)

${C}_{j}^{t} = 1+\left(2\times {r}_{2}-1\right)\times {c}^{2}$

(13)

$c = {c}_{max}-\left({c}_{max}-{c}_{min}\right)\times \frac{t-1}{{t}_{max}\ \ -1}$

(14)

where, ${C}_{j}^{t}$ can take these uncertainties into account and help the algorithm better perform the exploration phase. Especially in the early iterations of the optimization process. $c$ is an adaptively determined parameter as follows. ${r}_{2}$ represents a random number between 0 and 1.

$k$ acts as a tuning parameter, acting as an inertia weight, while facilitating a proper and reliable transition from exploration to exploitation, and is iterated through iterative computation as follows:

$k = {k}_{max}-\left({k}_{max}-{k}_{min}\right)\times \frac{t-1}{{t}_{max}\ \ -1}$

(15)

Finally, the next position of the search agent can be expressed by this formula:

${X}_{1, j}^{t+1} = {X}_{j}^{t+1}-{V}_{j}^{t+1}$

(16)

The new position of search agent is calculated by this formula:

${X}_{i, j}^{t+1} = \frac{\left({X}_{1, j}^{t}-{X}_{1, j}^{t+1}\right)}{2}$

(17)

where, is the position of the ith search agent (Aquila) in the jth dimension in the (t+1)th iteration.

4.2. Adaptive opposition-based learning

The starting point of introducing reverse learning into IAO is to get rid of the precocious dilemma when it falls into local convergence through reverse learning. Introducing the reverse learning probability of reverse learning requires a large amount of experimental data to determine its value in a hybrid algorithm designed by the original reverse learning and other algorithms. Hussien et al. ^[70] proposed opposition-based learning and chaotic local search strategy are used in Harris Hawks optimization algorithm for global optimization and feature selection. Hussien et al. ^[71] An enhanced opposition-based Salp Swarm Algorithm. Therefore, we find that the adaptive opposition-based learning rules can help the algorithm jump out of the local optimum. That's why we introduce adaptive opposition-based learning rules to improve the IAO algorithm. It can be expressed by this formula:

${X}_{jOBL}\left(t+1\right) = {LB}_{j}+{UB}_{j}-{X}_{j}$

(18)

where, ${\mathrm{L}\mathrm{B}}_{\mathrm{j}}$ is the lower bound of the current problem in the jth dimension, and ${\mathrm{U}\mathrm{B}}_{\mathrm{j}}$ is the upper bound of the current problem in the jth dimension.

4.3. Overview of the proposed VAIAO algorithm

The flowchart of the VAIAO algorithm is given in . The pseudo-code for the VAIAO algorithm is shown in . The computational time complexity is O(T $\times$ D $\times$ N).

Figure 1. The flowchart of the VAIAO algorithm.

DownLoad: Full-Size Img PowerPoint

Table 2. The pseudo-code of the VAIAO algorithm.

Initialization of population size N and the positions of Individuals ${\mathrm{X}}_{\mathrm{i}}(\mathrm{i}=\mathrm{1, 2}, ······, \mathrm{n})$
While( $\mathrm{t}\le \mathrm{T}$ )
For $\mathrm{i}=1:\mathrm{N}$
Updata $\mathrm{X}\left(\mathrm{t}\right)$
Updata the parameters
If rand < 0.5
Updata $\mathrm{X}(\mathrm{t}+1)$ with Eq.(1)
Else
Updata $\mathrm{X}(\mathrm{t}+1)$ with Eq.(2)
End
For $\mathrm{i}=1:\mathrm{N}$
Updata $\mathrm{X}(\mathrm{t}+1)$ with Eq.(9)-Eq.(17)
End For
For $\mathrm{i}=1:\mathrm{N}$
Updata $\mathrm{X}(\mathrm{t}+1)$ with Eq.(18)
End For
For $\mathrm{i}=1:\mathrm{N}$
Check boundaries
Calculate fitness of $\mathrm{X}\left(\mathrm{t}\right)$
Updata ${\mathrm{X}}_{\mathrm{b}\mathrm{e}\mathrm{s}\mathrm{t}}\left(\mathrm{t}\right)$
End For
t+t+1
End While
Return ${\mathrm{X}}_{\mathrm{b}\mathrm{e}\mathrm{s}\mathrm{t}}\left(\mathrm{t}\right)$

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DownLoad: CSV

5. Experiments and results

In this section, the 27 classic functions, the Wilcoxon test and Friedman is used to test the performance of VAIAO algorithm. The results of this experimental show that the VAIAO algorithm has better performance than the original AO algorithm.

5.1. Experimental setup

In this section, some well-known algorithms are selected to verify the performance of the proposed VAIAO algorithm, such as IAO ^[67], AO ^[30], AOA ^[6], HHO ^[33], STOA ^[29], ChOA ^[68], SOA ^[28], PSO ^[24], DE ^[8]. To ensure the fairness of the experimental, set the population data to 30, and set the number of iterations is 500. The worst value, average value, optimal value, median value, and standard deviation of the output are compared to set the parameters of the algorithm according to the original version. The parameter settings for the comparative algorithms are shown in Table 3.

Table 3. Para settings for the comparative algorithm.

Algorithm	Parameters
VAIAO	$a\in \left[\sqrt{2}, 0\right]; c\in \left[\mathrm{1, 0}\right]; k=\mathrm{0.9, 0.4}$
IAO	$r\in \left[\mathrm{1, 0}\right]$
AO	$U=0.00565;{r}_{1}=10;\omega =0.005;\alpha =0.1;\delta =0.1;G1\in \left[-\mathrm{1, 1}\right]; G2\in \left[\mathrm{2, 0}\right]$
AOA	${r}_{1}\in \left[\mathrm{1, 0}\right]; {r}_{2}\in \left[\mathrm{1, 0}\right]; {r}_{3}\in \left[\mathrm{1, 0}\right]$
HHO	$q\in \left[\mathrm{0, 1}\right]; r\in \left[\mathrm{0, 1}\right]; E0\in \left[-\mathrm{1, 1}\right]; E1\in \left[\mathrm{2, 0}\right]; E\in [-\mathrm{2, 2}]$
STOA	$Sa\in \left[\mathrm{2, 0}\right]; r1\in \left[\mathrm{0.5, 0}\right]; r2\in \left[\mathrm{0.5, 0}\right]; b=1$
ChOA	$f\in \left[\mathrm{2, 0}\right]; r1\in \left[\mathrm{0, 1}\right]; r2\in \left[\mathrm{0, 1}\right]$
SOA	$Fc\in \left[\mathrm{2, 0}\right]; r1\in \left[\mathrm{1, 0}\right]; r2\in \left[\mathrm{1, 0}\right]; b=1$
PSO	${C}_{1}=1.49445, {C}_{2}=1.49445, v\in \left[-\mathrm{0.5, 0.5}\right]$
DE	${F}_{0}=0.5, {C}_{R}=0.7$

| Show Table

DownLoad: CSV

5.2. Test function

In this section, there are 27 typical benchmark selection functions, including 9 single-peak functions (shown in Table 4), 9 two-dimensional multi-channel functions (shown in Table 5), and 9 multidimensional multi-peak functions (shown in Table 6). These benchmark functions are used to test the basic performance of the proposed OLAO algorithm. All simulation experiments will be conducted on Intel(R) Core (TM) i5-10300H CPU @2.50GHz 3106×2 core and MATLAB 2017b software will be used.

Table 4. Unimodal benchmark functions.

Functions	Description	Dimensions	Range	fmin
F1	$f\left(x\right)=\sum\limits_{i=1}^{D}{x}_{i}^{2}$	30, 60,100,300,500	[−100,100]	0
F2	$f\left(x\right)=\sum\limits_{i=1}^{D}\left\|{x}_{i}\right\|$	30, 60,100,300,500	[−100,100]	0
F3	$f\left(x\right)={10}^{6\frac{i-1}{D-1}}\times \sum\limits_{i=1}^{D}{x}_{i}^{2}$	30, 60,100,300,500	[−100,100]	0
F4	$f\left(x\right)=\sum\limits_{i=1}^{D}\left({x}_{i+1}^{2}\mathrm{*}{x}_{i}^{2}\right)$	30, 60,100,300,500	[−100,100]	0
F5	$f\left(x\right)=max\left\{\left\|{x}_{i}\right\|, 1\le i\le D\right\}$	30, 60,100,300,500	[−100,100]	0
F6	$f\left(x\right)=\sum\limits_{i=1}^{D}\left\|{x}_{i}\right\|+\prod\limits_{i=1}^{D}\left\|{x}_{i}\right\|$	30, 60,100,300,500	[−100,100]	0
F7	$f\left(x\right)=\sum\limits_{i=1}^{D}{ix}_{i}^{2}$	30, 60,100,300,500	[−100,100]	0
F8	$f\left(x\right)=f\left(x\right)={\sum\limits_{i=1}^{D}(\sum\limits_{j=1}^{i}i{x}_{i}^{2})}^{2}$	30, 60,100,300,500	[−100,100]	0
F9	$f\left(x\right)=\sum\limits_{i=1}^{D}{ix}_{i}^{4}$	30, 60,100,300,500	[−100,100]	0

| Show Table

DownLoad: CSV

Table 5. Two-dimensional multimodal functions.

Functions	Descriptions	Dimensions	Ranges	fmin
F10	$f\left(x\right)=7{x}_{1}^{2}-6\sqrt{3}{x}_{1}{x}_{2}+13{x}_{2}^{2}$	2	[−100,100]	0
F11	$f\left(x\right)=195.6316-{200e}^{\left(-0.02\sqrt{{x}_{1}^{2}+{x}_{2}^{2}}\right)}+5{e}^{(cos3{x}_{1}+sin3{x}_{2})}$	2	[−100,100]	0
F12	$f\left(x\right)={({x}_{1}^{2}+{x}_{2}^{2})}^{0.25}\times \left({{1+\mathrm{s}\mathrm{i}\mathrm{n}\left(50\right(3{x}_{1}^{2}+{x}_{2}^{2})}^{0.1})}^{2}\right)$	2	[−100,100]	0
F13	$f\left(x\right)=0.5+\frac{(sin{\left(\sqrt{{x}_{1}^{2}+{x}_{2}^{2}}\right)}^{2}-0.5)}{{(1+0.001({x}_{1}^{2}+{x}_{2}^{2}\left)\right)}^{2}}$	2	[−5, 5]	0
F14	$f\left(x\right)=0.1+{\mathrm{s}\mathrm{i}\mathrm{n}\left({x}_{1}\right)}^{2}+{\mathrm{s}\mathrm{i}\mathrm{n}\left({x}_{2}\right)}^{2}-0.1{e}^{(-{x}_{1}^{2}-{x}_{2}^{2})}$	2	[−10, 10]	0
F15	$f\left(x\right)=\left(sin4-\frac{\mathrm{sin}\left(2\sqrt{{x}_{1}^{2}+{x}_{2}^{2}}\right)}{2}+\frac{\mathrm{sin}\left(3\sqrt{{x}_{1}^{2}+{x}_{2}^{2}}\right)}{3}-\frac{\mathrm{sin}\left(4\sqrt{{x}_{1}^{2}+{x}_{2}^{2}}\right)}{4}+4\right)\times \mathrm{cos}\left(argtan\frac{{x}_{2}}{{x}_{1}}-\pi \right)+2$	2	[−100,100]	0
F16	$f\left(x\right)=2{x}_{1}^{2}-1.05{x}_{1}^{4}+\frac{{x}_{1}^{6}}{6}+{x}_{1}{x}_{2}+{x}_{2}^{2}$	2	[−100,100]	0
F17	$f\left(x\right)=0.26\left({x}_{1}^{2}+{x}_{2}^{2}\right)-0.48{x}_{1}{x}_{2}$	2	[−100,100]	0
F18	$f\left(x\right)=0.5+\frac{\left(\left(sin{\left(\sqrt{{x}_{1}^{2}+{x}_{2}^{2}}\right)}^{2}-0.5\right)\right)}{{(1+0.001({x}_{1}^{2}+{x}_{2}^{2}\left)\right)}^{2}}$	2	[−20, 20]	0

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Table 6. Multi-dimensional multimodal functions.

Functions	Descriptions	Dimensions	Ranges	fmin
F19	$f\left(x\right)=1+(\sum\limits_{i=1}^{D}{\mathrm{s}\mathrm{i}\mathrm{n}\left({x}_{i}\right)}^{2}-{e}^{(-\sum\nolimits_{i=1}^{D}{x}_{i}^{2})}\times {e}^{(-{\sum }_{i=1}^{D}\mathrm{s}\mathrm{i}\mathrm{n}({\sqrt{\left\|{x}_{i}\right\|})}^{2})}$	30, 60,100,300,500	[−100,100]	0
F20	$f\left(x\right)=\sum\limits_{i=1}^{D}{x}_{i}^{6}(2+\mathrm{s}\mathrm{i}\mathrm{n}(\frac{1}{{x}_{i}}\left)\right)$	30, 60,100,300,500	[−100,100]	0
F21	$f\left(x\right)=\sum\limits_{i=1}^{D}[100{\left({x}_{i+1}-{x}_{i}^{2}\right)}^{2}+{({x}_{i}-1)}^{2}]$	30, 60,100,300,500	[−100,100]	0
F22	$f\left(x\right)=\sum\limits_{i=1}^{D}\left\|{x}_{i}\mathrm{sin}\left({x}_{i}\right)+0.1{x}_{i}\right\|$	30, 60,100,300,500	[−100,100]	0
F23	$f\left(x\right)=\frac{D}{10}+\sum\limits_{i=1}^{D}{x}_{i}^{2}-\frac{{\sum }_{i=1}^{D}\mathrm{c}\mathrm{o}\mathrm{s}\left(5\pi {x}_{i}\right)}{10}$	30, 60,100,300,500	[−100,100]	0
F24	$f\left(x\right)=1+\frac{{\sum }_{i=1}^{D}{x}_{i}^{2}}{4000}-\prod\limits_{i=1}^{D}\mathrm{c}\mathrm{o}\mathrm{s}\left(\frac{{x}_{i}}{\sqrt{i}}\right)$	30, 60,100,300,500	[−100,100]	0
F25	$f\left(x\right)=1-\mathrm{c}\mathrm{o}\mathrm{s}(2\pi \sqrt{\sum\limits_{i=1}^{D}{x}_{i}^{2}}+0.1\sqrt{\sum\limits_{i=1}^{D}{x}_{i}^{2}})$	30, 60,100,300,500	[−100,100]	0
F26	$f\left(x\right)=10D+\sum\limits_{i=1}^{D}[{x}_{i}^{2}-10\mathrm{c}\mathrm{o}\mathrm{s}(2\pi {x}_{i}\left)\right]$	30, 60,100,300,500	[−100,100]	0
F27	$f\left(x\right)=-20{e}^{\left(-2\sqrt{\frac{{\sum }_{i=1}^{D}{x}_{i}^{2}}{D}}\right)}-{e}^{\left(\frac{{\sum }_{i=1}^{D}\mathrm{cos}\left(2\pi {x}_{i}\right)}{D}\right)}+20+e$	30, 60,100,300,500	[−100,100]	0

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5.3. Experimental series 1: Intensification capability experiments

In this section, the test function contains an unimodal test function and a multimodal test function. The algorithm tests the single peak function to verify the algorithm's exploration ability is effective, because the unimodal test function only has one global optimal value. The results of the experiment are shown in Table 7. In most instances, the VAIAO algorithm has the best convergence performance in almost all case and has better performance on unimodal functions than other algorithm.

Table 7. Test results of Unimodal function (F1–F9), the dimension is fixed to 30.

Fun	Items	SOIAO	IAO	AO	AOA	HHO	STOA	CHIMP	SOA	PSO	DE
F1	Average	0.0000	1.3374 × 10⁻²⁹¹	5.2757 × 10⁻¹⁵²	2.4485 × 10⁻⁹³	3.6079 × 10⁻¹⁰²	1.9011 × 10⁻⁶	1.3008 × 10⁻⁵	2.8797 × 10⁻¹²	4.1938 × 10⁻⁴	4.5181 × 10⁻⁴
	Std	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
F2	Average	3.7644 × 10⁻¹⁴⁷	9.7308 × 10⁻¹⁴⁶	6.5298 × 10⁻⁵⁹	6.3692 × 10⁻⁵²	6.4248 × 10⁻⁵¹	1.7544 × 10⁻⁴	2.1821 × 10⁻⁴	2.7349 × 10⁻⁷	2.0394 × 10⁻²	9.6032 × 10²
	Std	8.4104 × 10⁻¹⁴⁷	1.2976 × 10⁻¹⁴⁵	1.4601 × 10⁻⁵⁸	1.4242 × 10⁻⁵¹	1.2502 × 10⁻⁵⁰	1.1095 × 10⁻⁴	2.4623 × 10⁻⁴	1.9075 × 10⁻⁷	8.9718 × 10⁻³	38.142
F3	Average	1.1656 × 10⁻³⁰⁰	4.2723 × 10⁻²⁹⁶	1.4097 × 10⁻¹⁴¹	3.6188 × 10⁻³	1.5452 × 10⁻⁹⁴	74.685	1.4667 × 10²	6.1053 × 10⁻⁶	2.1579 × 10³	6.7414 × 10¹¹
	Std	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
F4	Average	0.0000	0.0000	1.2913 × 10⁻²⁹⁶	0.0000	2.3320 × 10⁻¹⁸⁹	6.1574 × 10⁻¹⁸	3.5583 × 10⁻¹⁶	5.9353 × 10⁻²⁸	2.4914 × 10⁻⁸	5.1324 × 10⁷
	Std	0.0000	0.0000	0.0000	0.0000	0.0000	1.0005 × 10⁻¹⁷	5.0628 × 10⁻¹⁶	9.0818 × 10⁻²⁸	5.3972 × 10⁻⁸	7.9633 × 10⁶
F5	Average	3.7482 × 10⁻¹⁴⁵	3.1301 × 10⁻¹⁵⁰	1.1888 × 10⁻⁵²	2.7044 × 10⁻²	2.0140 × 10⁻⁴⁹	4.8732 × 10⁻²	0.34245	1.2416 × 10⁻²	1.7174	74.286
	Std	8.3764 × 10⁻¹⁴⁵	4.4009 × 10⁻¹⁵⁰	2.6583 × 10⁻⁵²	2.5004 × 10⁻²	2.8430 × 10⁻⁴⁹	6.8444 × 10⁻²	0.35991	2.5710 × 10⁻²	0.40172	4.1678
F6	Average	3.6797 × 10⁻¹⁴³	3.8905 × 10⁻¹⁴²	4.3251 × 10⁻⁷¹	0.0000	2.3336 × 10⁻⁴⁹	5.5061 × 10⁻⁶	6.6415 × 10⁻⁵	2.6600 × 10⁻⁸	1.9753 × 10⁻²	5.4108 × 10⁸
	Std	8.2280 × 10⁻¹⁴³	8.6993 × 10⁻¹⁴²	9.6703 × 10⁻⁷¹	0.0000	5.0420 × 10⁻⁴⁹	4.7519 × 10⁻⁶	1.1828 × 10⁻⁴	8.7207 × 10⁻⁹	1.1563 × 10⁻²	7.3281 × 10⁸
F7	Average	8.8651 × 10⁻³⁰⁸	9.0167 × 10⁻²⁶³	6.9932 × 10⁻¹¹⁰	3.5958 × 10⁻³⁰	1.0178 × 10⁻⁹⁴	6.0971 × 10⁻⁶	8.9469 × 10⁻⁵	2.2386 × 10⁻¹⁰	8.4813 × 10⁻⁴	6.4148 × 10⁵
	Std	0.0000	0.0000	1.5637 × 10⁻¹⁰⁹	8.0405 × 10⁻³⁰	2.2759 × 10⁻⁹⁴	4.7054 × 10⁻⁶	1.3921 × 10⁻⁴	3.3203 × 10⁻¹⁰	8.9580 × 10⁻⁴	6.9981 × 10⁴
F8	Average	0.0000	0.0000	3.9573 × 10⁻²⁷⁶	4.4146 × 10⁻³⁶	3.9112 × 10⁻¹⁹³	1.5903 × 10⁻¹⁰	2.1632 × 10⁻⁶	2.0882 × 10⁻¹⁸	5.0434 × 10⁻⁶	1.7330 × 10⁸
	Std	0.0000	0.0000	0.0000	9.8714 × 10⁻³⁶	0.0000	2.5138 × 10⁻¹⁰	4.8222 × 10⁻⁶	4.0422 × 10⁻¹⁸	4.4984 × 10⁻⁶	2.7526 × 10⁷
F9	Average	0.0000	0.0000	2.2060 × 10⁻²⁸²	9.5734 × 10⁻¹¹⁹	6.6071 × 10⁻²⁰⁷	4.0249 × 10⁻⁹	2.2641 × 10⁻⁵	1.2280 × 10⁻¹⁹	1.4946 × 10⁻⁴	2.3440 × 10⁹
	Std	0.0000	0.0000	0.0000	2.1407 × 10⁻¹¹⁸	0.0000	6.8554 × 10⁻⁹	4.7015 × 10⁻⁵	1.2419 × 10⁻¹⁹	2.5801 × 10⁻⁴	2.6115 × 10⁸

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The multimodal test functions include many local optimal solutions and global optimal solutions. With the increase of the dimension, the algorithm with poor performance can easily fall into local optimum. Therefore, it is very effective to use these functions to test the diversity of algorithms. The results of the multimodal two-dimensional in this section are shown in Tables 8 and 9. In the low-dimensional diversification experiment, the VAIAO algorithm proposed in this paper can converge to 0 many times, and obtain the optimal value through the data in Tables 8 and 9.

Table 8. Test results of multimodal two-dimensional function (F10–F18), the dimension is fixed to 30.

Fun	Items	SOIAO	IAO	AO	AOA	HHO	STOA	CHIMP	SOA	PSO	DE
F10	Average	0.0000	0.0000	6.1912 × 10^-2	10.000	0.0000	10.000	10.000	10.000	10.000	10.000
F10	Std	0.0000	0.0000	5.7285 × 10^-2	6.1740 × 10^-10	0.0000	1.6385 × 10^-11	1.9981 × 10^-09	1.5984 × 10^-10	6.4737 × 10^-16	1.2519 × 10^-08
F11	Average	0.0000	0.0000	0.0000	2.0226 × 10^-50	3.0654 × 10^-286	1.9485 × 10^-11	1.4912 × 10^-08	2.6768 × 10^-23	1.1258 × 10^-4	1.4997 × 10⁺¹²
F11	Std	0.0000	0.0000	0.0000	4.5226 × 10-50	0.0000	3.6242 × 10^-11	3.3336 × 10^-8	3.0150 × 10^-23	2.1042 × 10^-4	4.1024 × 10⁺¹¹
F12	Average	2.5155 × 10^-2	4.3679 × 10^-1	6.5901 × 10^-3	71.927	6.2121 × 10^-3	73.546	89.006	73.509	75.565	11.242
F12	Std	5.4232 × 10^-2	9.5290 × 10^-1	1.0058 × 10^-2	4.9991 ×10^-1	1.0974 ×10^-2	4.1366 ×10^-1	1.2286 ×10^-1	4.2066 ×10^-1	15.983	6.4118 ×10⁸
F13	Average	3.3299 × 10^-38	9.8432 × 10^-124	9.7428 ×10^-5	6.6943 × 10^-24	2.8332 × 10^-48	2.3757 × 10^-1	1.0823 × 10^-1	2.1251 ×10^-3	6.3621 ×10^-2	5.1495 ×10²
F13	Std	7.4458 × 10^-38	2.2010 × 10^-123	2.1786 × 10^-04	1.4969 × 10^-23	6.3024 × 10^-48	4.0764 × 10^-1	2.0948 × 10^-1	1.4765 × 10^-3	9.5643 × 10^-2	25.793
F14	Average	0.0000	0.0000	0.0000	0.0000	0.0000	2.9961 × 10^-2	4.4001 × 10^-6	1.2201 × 10^-11	4.7933 × 10^-1	4.4415 × 10⁺⁴
F14	Std	0.0000	0.0000	0.0000	0.0000	0.0000	6.6979 × 10^-2	3.7691 × 10^-6	1.7271 × 10^-11	3.5923 × 10^-1	3.7541 × 10⁺³
F15	Average	0.0000	0.0000	0.0000	0.0000	0.0000	2.2648 × 10^-2	3.6068 × 10^-2	8.5477 × 10^-3	3.9600 × 10^-3	12.304
F15	Std	0.0000	0.0000	0.0000	0.0000	0.0000	2.4951 × 10^-2	2.3192 × 10^-2	1.9113 × 10^-2	5.3962 × 10^-3	2.0706
F16	Average	2.1276 × 10^-139	1.3235 × 10^-129	4.9230 × 10^-37	9.9873 × 10^-2	1.8757 × 10^-50	2.5987 × 10^-1	1.5301 × 10^-1	1.3987 ×10^-1	4.6473 × 10^-1	22.130
F16	Std	4.7576 × 10^-139	2.9594 × 10^-129	1.1007 × 10^-36	1.5245 × 10^-8	3.0802 × 10^-50	5.4772 × 10^-2	5.0490 × 10^-2	5.4772 × 10^-2	8.5849 × 10^-2	1.0707
F17	Average	0.0000	0.0000	0.0000	3.4987 × 10^-3	0.0000	22.912	17.773	84.342	6.5628 × 10⁺¹	4.8665 × 10⁺⁴
F17	Std	0.0000	0.0000	0.0000	7.8233 × 10^-3	0.0000	10.347	14.383	7.9822	43.406	2.4056 × 10⁺³
F18	Average	8.8818 × 10^-16	8.8818 × 10^-16	8.8818 × 10^-16	8.8818 × 10^-16	8.8818 × 10^-16	20.000	20.000	20.000	12.575	21.266
F18	Std	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	10.815	3.2831 × 10^-2

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Table 9. Test results of Multi-peak multidimensional function (F19–F27), the dimension is fixed to 30.

Fun	Items	SOIAO	IAO	AO	AOA	HHO	STOA	CHIMP	SOA	PSO	DE
F19	Average	3.7921 × 10^-284	4.2388 × 10^-152	0.0000	7.5115 × 10^-123	5.3123 × 10^-60	8.4732 × 10^-109	1.2217 × 10^-94	5.7790 × 10^-43	1.1176 × 10⁺²	3.7921 × 10^-284
F19	Std	0.0000	6.8746 × 10^-152	0.0000	1.4946 × 10^-122	1.1458 × 10^-59	1.7112 × 10^-108	2.7296 × 10^-94	1.2922 × 10^-42	1.5591 × 10⁺²	0.0000
F20	Average	3.1780 × 10^-5	6.1035 × 10^-4	3.6168 × 10^-3	2.6850 × 10^-3	2.5718 × 10^-3	2.5765 × 10^-3	2.6067 × 10^-3	2.5818 × 10^-3	2.5717 × 10^-3	12.733
F20	Std	9.3107 × 10^-5	3.0590 × 10^-4	4.8087 × 10^-4	1.1191 × 10^-4	1.1817 × 10^-7	2.0037 × 10^-6	2.8916 × 10^-5	6.9897 × 10^-6	1.4845 × 10^-15	6.6951
F21	Average	7.4397 × 10^-67	1.0670 × 10^-54	8.0752 × 10^-37	0.0000	2.0674 × 10^-28	1.5729 × 10^-16	1.1379 × 10^-32	6.3436 × 10^-27	8.1781 × 10^-15	2.5053
F21	Std	1.6082 × 10^-66	2.3860 × 10^-54	1.6215 × 10^-36	0.0000	2.9078 × 10^-28	1.9113 × 10^-16	2.5444 × 10^-32	7.8475 × 10^-27	1.1609 × 10^-14	8.1600 × 10^-1
F22	Average	0.0000	0.0000	0.0000	0.0000	0.0000	1.9432 × 10^-3	6.3022 × 10^-3	3.8864 × 10^-3	1.9432 × 10^-3	9.7478 × 10^-3
F22	Std	0.0000	0.0000	0.0000	0.0000	0.0000	4.3451 × 10^-3	4.7119 × 10^-3	5.3216 × 10^-3	4.3452 × 10^-3	5.5634 × 10^-5
F23	Average	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	8.0236 × 10^-2	2.0024 × 10^-2	9.9192 × 10^-2	8.7557 × 10^-2
F23	Std	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	4.4853 × 10^-2	4.4775 × 10^-2	1.7146 × 10^-1	3.0053 × 10^-2
F24	Average	6.1118 × 10^-145	1.7857 × 10^-138	3.2553 × 10^-53	0.0000	6.3114 × 10^-54	2.7565	4.5736	1.8520	4.6233	4.6436
F24	Std	1.3661 × 10^-144	3.9697 × 10^-138	7.2790 × 10^-53	0.0000	1.1169 × 10^-53	2.5178	1.6283 × 10^-1	2.5359	5.9274 × 10^-2	6.9033 × 10^-2
F25	Average	0.0000	0.0000	1.7368 × 10^-156	0.0000	2.6981 × 10^-146	1.2028 × 10^-54	9.2557 × 10^-69	5.9487 × 10^-84	4.0707 × 10^-49	7.2193 × 10^-1
F25	Std	0.0000	0.0000	1.2975 × 10^-139	0.0000	1.4173 × 10^-107	9.7993 × 10^-64	2.4634 × 10^-140	7.0872 × 10^-100	1.2175 × 10^-56	33.863
F26	Average	0.0000	0.0000	1.7368 × 10^-156	0.0000	2.6981 × 10^-146	1.2028 × 10^-54	9.2557 × 10^-69	5.9487 × 10^-84	4.0707 × 10^-49	7.2193 × 10^-1
F26	Std	0.0000	0.0000	1.2975 × 10^-139	0.0000	1.4173 × 10^-107	9.7993 × 10^-64	2.4634 × 10^-140	7.0872 × 10^-100	1.2175 × 10^-56	33.863
F27	Average	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	2.0085
F27	Std	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	5.8149 × 10^-1

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5.4. Experiment series 3: Accelerated convergence experiments

In order to understand the performance of the algorithm, this section will perform accelerated convergence analysis of these algorithms on 27 classic benchmark functions to compare the performance of the algorithm, the results are shown in Figures 2–5. The results in Figure 2 show the VAIAO algorithm has faster convergence s peed, higher convergence accuracy, and more stable convergence curve.

Figure 2. Convergence curves of 27 benchmark functions (F1–F6).

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Figure 3. Convergence curves of 27 benchmark functions (F7–F14).

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Figure 4. Convergence curves of 27 benchmark functions (F15–F22).

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Figure 5. Convergence curves of 27 benchmark functions (F23–F27).

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5.5. Experiment series 4: Scalability experiments

One of the problems which the algorithm must solve is the high-dimensional problem. The dimensions are selected as 60,100,300,500 to solve this problem, and the results are shown in Tables 10–13.

Table 10. Test results of benchmark function (F1–F9, F19–F27), the dimension is fixed to 60.

Fun	Item	VAIAO	IAO	AO	AOA	HHO	STOA	CHIMP	SOA	PSO	DE
F1	Average	4.9791 × 10^-285	7.9894 × 10^-284	1.6265 × 10^-101	7.9491 × 10^-83	1.2280 × 10^-97	1.2062 × 10^-6	3.0010 × 10^-6	2.2287 × 10^-12	1.2383 × 10^-4	4.5836 × 10⁺⁴
F1	Std	0.0000	0.0000	3.6369 × 10^-101	1.4446 × 10^-82	2.7459 × 10^-97	1.8624 × 10^-6	3.2504 × 10^-6	2.0211 × 10^-12	1.0144 × 10^-4	3.3363 × 10⁺³
F2	Average	1.0754 × 10^-144	1.0762 × 10^-142	4.4908 × 10^-62	8.6497 × 10^-43	4.7376 × 10^-50	7.2859 × 10^-5	5.4243 × 10^-4	2.9238 × 10^-7	2.8354 × 10^-2	9.3463 × 10⁺²
F2	Std	2.4046 × 10^-144	2.4064 × 10^-142	1.0042 × 10^-61	1.9341 × 10^-42	9.9176 × 10^-50	9.0603 × 10^-5	6.9310 × 10^-4	2.7641 × 10^-7	1.7438 × 10^-2	85.440
F3	Average	0.0000	6.4756 × 10^-295	4.0162 × 10^-130	4.2837 × 10^-127	7.4047 × 10^-96	1.1059 × 10^-1	7.9538	1.6480 × 10^-4	2.0313 × 10⁺⁴	6.7097 × 10⁺¹¹
F3	Std	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
F4	Average	0.0000	0.0000	1.6672 × 10^-227	0.0000	2.9404 × 10^-204	9.4603 × 10^-18	7.5156 × 10^-16	8.3905 × 10^-28	1.8487 × 10^-9	5.9847 × 10⁺⁷
F4	Std	0.0000	0.0000	0.0000	0.0000	0.0000	1.2690 × 10^-17	1.1246 × 10^-15	1.4975 × 10^-27	3.3947 × 10^-9	4.5947 × 10⁺⁶
F5	Average	5.3277 × 10^-150	5.8938 × 10^-150	2.8218 × 10^-67	3.3871 × 10^-2	3.2133 × 10^-50	6.5490 × 10^-2	4.1592 × 10^-1	1.0360 × 10^-3	1.2916	76.030
F5	Std	1.2621 × 10^-149	1.3179 × 10^-149	6.2674 × 10^-67	1.8970 × 10^-2	4.2164 × 10^-50	8.9912 × 10^-2	4.3917 × 10^-1	7.9839 × 10^-4	4.3288 × 10^-1	2.0916 ×
F6	Average	3.3763 × 10^-149	1.1833 × 10^-145	5.5194 × 10^-51	0.0000	5.3515 × 10^-52	1.4184 × 10^-5	6.6697 × 10^-5	1.0693 × 10^-8	1.9238 × 10^-2	1.9707 × 10⁺⁸
F6	Std	7.5442 × 10^-149	2.6406 × 10^-145	1.2342 × 10^-50	0.0000	9.5332 × 10^-52	1.4988 ×10^-5	7.7537 ×10^-5	4.9822 × 10^-9	3.6831 × 10^-3	2.2429 × 10⁺⁸
F7	Average	5.7989 × 10^-283	2.7115 × 10^-281	9.1577 × 10^-135	4.6576 × 10^-87	8.4403 × 10^-98	5.6128 ×10^-6	1.2090 ×10^-5	1.6166 × 10^-10	9.5755 × 10^-4	6.5491 × 10⁺⁵
F7	Std	0.0000	0.0000	2.0477 × 10^-134	1.0268 × 10^-86	1.5796 × 10^-97	6.8124 ×10^-6	2.0955 ×10^-5	1.4812 × 10^-10	9.0545 × 10^-4	4.1739 × 10⁺⁴
F8	Average	0.0000	0.0000	6.1451 × 10^-295	1.1543 × 10^-99	1.8381 × 10^-186	3.4122 × 10^-11	1.0365 × 10^-7	2.7425 × 10^-19	2.1931 × 10^-5	1.7272 × 10^-8
F8	Std	0.0000	0.0000	0.0000	2.5810 × 10^-99	0.0000	3.4613 × 10^-11	1.0793 × 10^-7	4.6732 × 10^-19	3.1971 × 10^-5	4.2488 × 10⁺⁷
F9	Average	0.0000	0.0000	1.2212 × 10^-255	6.2093 × 10^-110	1.2294 × 10^-200	3.1520 × 10^-8	8.8864 × 10^-7	1.5077 × 10^-18	6.5741 × 10^-4	2.0827 × 10⁺⁹
F9	Std	0.0000	0.0000	0.0000	1.3884 × 10^-109	0.0000	6.5054 × 10^-8	1.4509 × 10^-6	3.2314 × 10^-18	1.1546 × 10^-3	1.8815 × 10⁺⁸
F10	Average	0.0000	0.0000	0.0000	1.0000	0.0000	1.0000	1.0000	1.0000	1.0000	1.0000
F10	Std	0.0000	0.0000	0.0000	1.1073 × 10^-9	0.0000	2.4964 × 10^-11	4.8343 × 10^-9	4.7418 × 10^-11	2.1617 × 10^-8	1.5491 × 10^-8
F11	Average	0.0000	0.0000	0.0000	1.1151 × 10^-9	3.6025 × 10^-293	4.7543 × 10^-12	5.2735 × 10^-5	3.6730 × 10^-25	1.0178 × 10^-2	1.8080 × 10⁺¹²
F11	Std	0.0000	0.0000	0.0000	2.4935 × 10^-9	0.0000	6.9713 × 10^-12	1.1788 × 10^-4	8.1060 × 10^-25	1.3883 × 10^-2	3.1086 × 10⁺¹¹
F12	Average	1.1053	1.5749 × 10^-2	2.9848 × 10^-3	7.1238	1.6186 × 10^-3	7.7015	8.7626	7.7813	5.6736	1.0216 × 10⁺⁰⁹
F12	Std	2.4646	1.4847 × 10^-2	2.8920 × 10^-3	4.5150 × 10^-1	1.9301 × 10^-3	5.3349 × 10^-1	3.8776 × 10^-1	7.2438 × 10^-1	4.4997	6.1546 × 10⁺⁰⁸
F13	Average	5.1562 × 10^-139	6.6950 × 10^-145	2.5938 × 10^-41	2.2555 × 10^-36	9.1534 × 10^-52	3.3102 × 10^-1	7.5026 × 10^-3	2.2971 × 10^-2	1.6973 × 10^-1	5.2031 × 10⁺⁰²
F13	Std	1.1530 × 10^-138	1.4667 × 10^-144	5.8000 × 10^-41	5.0434 × 10^-36	1.4544 × 10^-51	4.7732 × 10^-1	4.6404 × 10^-3	4.6946 × 10^-2	1.4654 × 10^-1	47.448
F14	Average	0.0000	0.0000	0.0000	0.0000	0.0000	8.2893 × 10^-6	8.4925 × 10^-5	7.4010 × 10^-11	4.7624 × 10^-1	4.5788 × 10⁺⁴
F14	Std	0.0000	0.0000	0.0000	0.0000	0.0000	1.4189 × 10^-5	1.3325 × 10^-4	9.5731 × 10^-11	3.1832 × 10^-1	5.6324 × 10⁺³
F15	Average	0.0000	0.0000	0.0000	0.0000	0.0000	3.5722 × 10^-2	1.4915 × 10^-2	3.0534 × 10^-3	3.9471 × 10^-3	12.181
F15	Std	0.0000	0.0000	0.0000	0.0000	0.0000	3.7178 × 10^-2	2.0853 × 10^-2	6.8275 × 10^-3	8.8018 × 10^-3	1.5118
F16	Average	1.0131 × 10^-96	3.9346 × 10^-135	6.6523 × 10^-53	9.9873 × 10^-2	4.6541 × 10^-51	2.3987 × 10^-1	1.9987 × 10^-1	1.5987 × 10^-1	4.8059 × 10^-1	22.228
F16	Std	2.2654 × 10^-96	8.7981 × 10^-135	1.4875 × 10^-52	6.9961 × 10^-08	1.0002 × 10^-50	5.4772 × 10^-2	4.4412 × 10^-7	5.4772 × 10^-2	8.3927 × 10^-2	7.9026 × 10^-1
F17	Average	0.0000	0.0000	0.0000	0.0000	0.0000	22.710	24.708	5.4079	53.609	4.8653 × 10⁺⁴
F17	Std	0.0000	0.0000	0.0000	0.0000	0.0000	14.530	35.191	5.0580	5.4390	4.5805 × 10⁺³
F18	Average	8.8818 × 10^-16	8.8818 × 10^-16	8.8818 × 10^-16	1.5987 × 10^-15	8.8818 × 10^-16	20.000	20.000	20.000	21.217	21.211
F18	Std	0.0000	0.0000	0.0000	1.5888 × 10^-15	0.0000	0.0000	1.6313 × 10^-9	0.0000	7.6156 × 10^-2	1.1811 × 10^-1

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Table 11. Test results of benchmark function (F1–F9, F19–F27), the dimension is fixed to 100.

Fun	Item	VAIAO	IAO	AO	AOA	HHO	STOA	CHIMP	SOA	PSO	DE
F1	Average	2.0856 × 10^-288	1.3525 × 10^-271	1.9209 × 10^-144	5.8406 × 10^-22	6.0336 × 10^-99	1.0055 × 10^-6	1.4164 × 10^-6	1.3739 × 10^-11	2.2488 × 10^-4	4.7702 × 10⁺⁴
F1	Std	0.0000	0.0000	4.0786 × 10^-144	1.3060 × 10^-21	1.3491 × 10^-98	5.6687 × 10^-7	1.5866 × 10^-6	2.2845 × 10^-11	1.5536 × 10^-4	8.9312 × 10⁺³
F2	Average	5.5074 × 10^-148	4.1899 × 10^-147	9.4978 × 10^-65	7.8562 × 10^-31	4.5422 × 10^-51	1.1767 × 10^-4	1.1669 × 10^-4	1.7749 × 10^-7	2.2129 × 10^-2	9.3520 × 10-²
F2	Std	1.2075 × 10^-148	9.3243 × 10^-147	2.1238 × 10^-64	1.7567 × 10^-30	9.7386 × 10^-51	8.6044 × 10^-5	1.4799 × 10^-4	6.9353 × 10^-8	1.8698 × 10^-2	30.797
F3	Average	0.0000	0.0000	2.4261 × 10^-144	4.9514 × 10^-82	1.3037 × 10^-101	1.9101 × 10^-1	4.7998 × 10^-1	1.0582 × 10^-5	8.5986 × 10⁺³	7.5167 × 10⁺¹¹
F3	Std	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
F4	Average	0.0000	0.0000	1.8263 × 10^-290	0.0000	3.5375 × 10^-204	5.7413 × 10^-18	1.1299 × 10^-15	6.5828 × 10^-27	2.6545 × 10^-10	4.6675 × 10⁺⁷
F4	Std	0.0000	0.0000	0.0000	0.0000	0.0000	6.6767 × 10^-18	2.5264 × 10^-15	1.1839 × 10^-26	3.3196 × 10^-10	1.0026 × 10⁺⁷
F5	Average	1.0731 × 10^-147	1.1287 × 10^-147	7.7323 × 10^-75	2.6114 × 10^-2	1.3253 × 10^-50	2.5638 × 10^-2	2.7609 × 10^-1	1.0539 × 10^-2	1.5166	75.742
F5	Std	2.4467 × 10^-147	2.5238 × 10^-147	7.1508 × 10^-75	2.3856 × 10^-2	2.9470 × 10^-50	4.3785 × 10^-3	1.7849 × 10^-1	1.4339 × 10^-2	3.3536 × 10^-1	2.7620
F6	Average	2.7824 × 10^-154	9.8495 × 10^-149	1.9648 × 10^-64	0.0000	1.6351 × 10^-50	1.3057 × 10^-5	4.3568 × 10^-5	1.2533 × 10^-8	2.4265 × 10^-2	2.9054 × 10⁺⁸
F6	Std	3.9894 × 10^-154	2.2024 × 10^-148	4.3933 × 10^-64	0.0000	3.2917 × 10^-50	1.6997 × 10^-5	5.0613 × 10^-5	6.3306 × 10^-9	1.4935 × 10^-2	3.3341 × 10⁺⁸
F7	Average	3.4587 × 10^-286	7.4886 × 10^-262	1.3214 × 10^-115	2.7430 × 10^-30	2.4926 × 10^-102	4.5084 × 10^-6	1.3141 × 10^-4	4.0396 × 10^-11	7.3701 × 10⁴	6.9182 × 10⁺⁵
F7	Std	0.0000	0.0000	2.9548 × 10^-115	6.1334 × 10^-30	2.7141 × 10^-102	4.7226 × 10^-6	2.4179 × 10^-4	4.5370 × 10^-11	4.7857 × 10⁴	3.9992 × 10⁺⁴
F8	Average	0.0000	0.0000	2.3053 × 10^-297	1.4862 × 10^-173	9.8413 × 10^-193	5.5288 × 10^-11	6.0337 × 10^-10	6.6808 × 10^-19	2.9822 × 10^-4	1.8184 × 10⁺⁸
F8	Std	0.0000	0.0000	0.0000	0.0000	0.0000	5.9165 × 10^-11	4.7341 × 10^-10	1.4725 × 10^-18	5.2236 × 10^-4	9.5505 × 10⁺⁶
F9	Average	0.0000	0.0000	3.4185 × 10^-197	2.4835 × 10^-125	4.3701 × 10^-191	6.5793 × 10^-10	5.2847 × 10^-06	1.0413 × 10^-18	3.3135 × 10^-4	2.5114 × 10⁺⁹
F9	Std	0.0000	0.0000	0.0000	5.5533 × 10^-125	0.0000	4.2526 × 10^-10	1.1817 × 10^-05	1.3940 × 10^-18	4.2915 × 10^-4	3.2430 × 10⁺⁸
F10	Average	0.0000	0.0000	6.1061 × 10^-2	1.0000	0.0000	1.0000	1.0000	1.0000	1.0000	1.0000
F10	Std	0.0000	0.0000	1.3458 × 10^-1	1.4541 × 10^-9	0.0000	1.7788 × 10^-11	1.8392 × 10^-9	5.4131 × 10^-11	9.9584 × 10^-9	1.7382 × 10^-8
F11	Average	0.0000	0.0000	0.0000	2.3162 × 10^-40	6.9359 × 10^-291	2.5794 × 10^-12	8.2743 × 10^-10	3.4449 × 10^-25	1.5298 × 10^-4	1.3530 × 10⁺¹²
F11	Std	0.0000	0.0000	0.0000	5.1792 × 10^-40	0.0000	4.9553 × 10^-12	1.2241 × 10^-9	7.6965 × 10^-25	3.1283 × 10^-4	1.4707 × 10⁺¹¹
F12	Average	2.3293 × 10^-2	2.6402 × 10-^-2	8.3243 × 10^-3	7.5737	5.7798 × 10^-3	7.3658	8.9861	7.7392	7.5901	7.9727 × 10⁺⁸
F12	Std	2.6047 × 10^-2	4.2889 × 10^-2	1.3581 × 10^-2	3.4765 × 10^-1	6.7518 × 10^-3	3.9510 × 10^-1	1.1386 × 10^-2	4.8710 × 10^-1	4.9830	3.0547 × 10⁺⁸
F13	Average	2.7741 × 10^-134	9.5046 × 10^-131	2.0574 × 10^-45	2.9736 × 10^-16	3.6069 × 10^-51	1.1388	1.0799 × 10^-2	9.7007 × 10^-1	1.4721 × 10^-1	5.5924 × 10⁺²
F13	Std	6.2031 × 10^-134	2.1253 × 10^-130	4.6005 × 10^-45	6.6491 × 10^-16	7.9918 × 10^-51	1.9680	1.0189 × 10^-2	2.1645	1.4404 × 10^-1	7.7151
F14	Average	0.0000	0.0000	0.0000	0.0000	0.0000	2.9766 × 10^-2	1.3756 × 10^-5	1.4978 × 10^-11	7.7284 × 10^-1	4.7685 × 10⁺⁴
F14	Std	0.0000	0.0000	0.0000	0.0000	0.0000	6.6554 × 10^-2	5.5149 × 10^-6	1.6901 × 10^-11	1.0194 × 10^-1	3.1623 × 10⁺³
F15	Average	0.0000	0.0000	0.0000	0.0000	0.0000	2.6034 × 10^-2	1.6210 × 10^-2	2.1853 × 10^-12	1.4857 × 10^-3	11.962
F15	Std	0.0000	0.0000	0.0000	0.0000	0.0000	3.6158 × 10^-2	2.3634 × 10^-2	4.5888 × 10^-12	3.3054 × 10^-3	20.858
F16	Average	2.4308 × 10^-132	1.4778 × 10^-123	2.0964 × 10^-56	9.9873 × 10^-2	8.6246 × 10^-51	2.3987 × 10^-1	1.7988 × 10^-1	1.3987 × 10^-1	5.0070 × 10^-1	22.327
F16	Std	5.4354 × 10^-132	3.3046 × 10^-123	4.6877 × 10^-56	1.2637 × 10^-8	1.6438 × 10^-50	5.4772 × 10^-2	4.4713 × 10^-1=2	5.4772 × 10^-2	1.2189 × 10^-1	5.5834 × 10^-1
F17	Average	0.0000	0.0000	0.0000	0.0000	0.0000	21.667	23.760	5.1447	51.996	4.5831 × 10⁺⁴
F17	Std	0.0000	0.0000	0.0000	0.0000	0.0000	22.730	21.422	4.6792	14.896	5.0338 × 10⁺³
F18	Average	8.8818 × 10^-16	8.8818 × 10^-16	8.8818 × 10^-16	2.0365 × 10^-9	8.8818 × 10^-16	20.000	20.000	20.000	12.009	21.208
F18	Std	0.0000	0.0000	0.0000	4.5537 × 10^-9	0.0000	3.6343 × 10^-9	0.0000	5.8102 × 10^-9	10.943	8.3420 × 10^-2

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Table 12. Test results of benchmark function (F1-F9, F19-F27), the dimension is fixed to 300.

Fun	Item	VAIAO	IAO	AO	AOA	HHO	STOA	CHIMP	SOA	PSO	DE
F1	Average	8.5034 × 10^-270	2.3079 × 10^-261	3.3665 × 10^-142	5.1609 × 10^-52	4.3526 × 10^-103	1.8230 × 10^-7	2.7129 × 10^-7	5.2054 × 10^-12	5.1871 × 10^-5	4.6751 × 10⁺⁴
F1	Std	0.0000	0.0000	7.1672 × 10^-142	1.1540 × 10^-51	5.1572 × 10^-103	1.6248 × 10^-7	3.2089 × 10^-7	7.8125 × 10^-12	3.9173 × 10^-5	2.9835 × 10⁺³
F2	Average	3.7912 × 10^-145	7.2883 × 10^-119	2.4149 × 10^-63	1.3091 × 10^-30	8.5396 × 10^-50	2.1334 × 10^-4	2.9038 × 10^-4	2.8768 × 10^-7	2.0031 × 10^-2	9.2501 × 10⁺²
F2	Std	8.4775 × 10^-145	1.6297 × 10^-118	5.1948 × 10^-63	2.9273 × 10^-30	1.8982 × 10^-49	3.7048 × 10^-4	3.1308 × 10^-4	3.0723 × 10^-7	1.1255 × 10^-2	53.869
F3	Average	7.3174 × 10^-303	7.7808 × 10^-294	2.8079 × 10^-141	1.8273 × 10^-47	1.6119 × 10^-99	9.4355	18.046	2.1366 × 10^-5	2.7337 × 10⁺³	6.5635 × 10⁺¹¹
F3	Std	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
F4	Average	0.0000	0.0000	1.0180 × 10^-214	0.0000	1.9773 × 10^-218	2.8943 × 10^-17	5.9752 × 10^-19	4.6456 × 10^-29	1.1970 × 10^-9	5.9402 × 10⁺⁷
F4	Std	0.0000	0.0000	0.0000	0.0000	0.0000	5.6868 × 10^-17	1.3144 × 10^-18	8.3512 × 10^-29	2.5358 × 10^-9	1.3212 × 10⁺⁷
F5	Average	3.5737 × 10^-147	1.5840 × 10^-146	3.9392 × 10^-55	2.5504 × 10^-2	1.8021 × 10^-47	4.1602 × 10^-2	1.3958 × 10^-1	2.4564 × 10^-3	1.3620	75.163
F5	Std	7.9911 × 10^-147	2.4424 × 10^-146	8.8083 × 10^-55	2.3298 × 10^-2	3.8369 × 10^-47	3.5228 × 10^-2	1.3822 × 10^-1	1.9369 × 10^-3	2.4597 × 10^-1	1.1950
F6	Average	8.6006 × 10^-154	4.8798 × 10^-138	4.5346 × 10^-59	0.0000	1.4295 × 10^-48	7.3639 × 10^-6	3.8962 × 10^-5	2.4319 × 10^-8	2.7580 × 10^-2	8.9581 × 10⁺⁷
F6	Best	3.4887 × 10^-169	2.0631 × 10^-163	1.6957 × 10^-75	0.0000	1.3140 × 10^-57	3.9492 × 10^-6	8.9013 × 10^-6	7.6014 × 10^-9	8.2748 × 10-^-3	4.1897 × 10⁺⁷
F7	Middle	8.6568 × 10^-155	1.4478 × 10^-162	2.0853 × 10^-73	0.0000	1.0650 × 10^-53	4.9859 × 10^-6	4.3948 × 10^-5	1.7779 × 10^-8	1.0212 × 10^-2	8.2422 × 10⁺⁷
F7	Std	1.8039 × 10^-153	1.0912 × 10^-137	1.0140 × 10^-58	0.0000	3.1961 × 10^-48	4.2023 × 10^-6	2.3412 × 10^-5	1.9170 × 10^-8	3.1222 × 10^-2	4.1768 × 10⁺⁷
F8	Average	2.6834 × 10^-290	2.3487 × 10^-291	5.0829 × 10^-107	2.9621 × 10^-22	1.3730 × 10^-95	3.1965 × 10^-6	7.6805 × 10^-5	1.4931 × 10^-10	6.8479 × 10^-3	6.3921 × 10⁺⁵
F8	Std	0.0000	0.0000	1.1366 × 10^-106	6.6234 × 10^-22	3.0614 × 10^-95	3.1031 × 10^-6	7.3720 × 10^-5	2.5066 × 10^-10	5.3827 × 10^-3	5.9461 × 10⁺⁴
F9	Average	0.0000	0.0000	8.6268 × 10^-297	1.0809 × 10^-85	9.3269 × 10^-199	2.4077 × 10^-10	1.8286 × 10^-8	8.6589 × 10^-18	6.6912 × 10^-6	1.5498 × 10⁺⁸
F9	Std	0.0000	0.0000	0.0000	2.4170 × 10^-85	0.0000	3.8745 × 10^-10	2.1243 × 10^-8	1.7220 × 10^-17	6.5714 × 10^-6	2.3799 × 10⁺⁷
F10	Average	0.0000	0.0000	2.1467 × 10^-278	1.5224 × 10^-84	7.5095 × 10^-192	1.6258 × 10^-10	8.2123 × 10^-8	1.0161 × 10-^-17	1.6788 × 10^-3	2.3457 × 10⁺⁹
F10	Std	0.0000	0.0000	0.0000	3.4043 × 10^-84	0.0000	2.5954 × 10^-10	1.7044 × 10^-7	1.9271 × 10^-17	3.5113 × 10^-3	5.2772 × 10⁺⁸
F11	Average	0.0000	0.0000	0.0000	1.0000	0.0000	1.0000	1.0000	1.0000	1.0000	1.0000
F11	Std	0.0000	0.0000	0.0000	5.5017 × 10^-9	0.0000	1.9167 × 10^-11	7.2876 × 10^-9	4.9847 × 10^-11	5.3948 × 10^-14	1.2245 × 10^-8
F12	Average	0.0000	0.0000	0.0000	1.4957 × 10^-187	1.8815 × 10^-284	3.1720 × 10^-12	1.0875 × 10^-8	1.5745 × 10^-23	4.3439 × 10^-3	1.7600 × 10⁺¹²
F12	Std	0.0000	0.0000	0.0000	0.0000	0.0000	2.8874 × 10^-12	1.5643 × 10^-8	3.2855 × 10^-23	9.0737 × 10^-3	3.1814 × 10⁺¹¹
F13	Average	9.4565 × 10^-3	2.0277 × 10^-2	4.8216 × 10^-2	7.1695	9.6104 × 10^-3	7.6772	8.9345	7.8181	34.758	7.9339 × 10⁺⁸
F13	Std	6.9496 × 10^-3	2.9629 × 10^-2	6.3765 × 10^-2	5.1244 × 10^-1	1.9708 × 10^-2	4.8319 × 10^-1	9.7361 × 10^-2	3.9107 × 10^-1	66.473	3.7199 × 10⁺⁸
F14	Average	1.3961 × 10^-115	1.5967 × 10^-19	1.8333 × 10^-54	3.0679 × 10^-61	3.6483 × 10^-33	2.1743 × 10^-1	1.1566 × 10^-1	1.9336 × 10^-3	23.391	5.2977 × 10⁺²
F14	Std	3.1217 × 10^-115	3.5703 × 10^-19	4.0993 × 10^-54	6.8601 × 10^-61	8.1579 × 10^-33	4.1511 × 10^-1	2.4825 × 10^-1	2.3224 × 10^-3	52.087	34.131
F15	Average	0.0000	0.0000	0.0000	0.0000	0.0000	1.3587 × 10^-5	3.1783 × 10^-5	1.7297 × 10-10	7.4304 × 10^-1	4.5561 × 10⁺⁴
F15	Std	0.0000	0.0000	0.0000	0.0000	0.0000	2.5788 × 10^-5	4.4060 × 10^-5	3.5587 × 10^-10	3.4849 × 10^-1	2.4965 × 10⁺³
F16	Average	0.0000	0.0000	0.0000	0.0000	0.0000	4.5692 × 10^-2	1.4218 × 10^-2	2.0649 × 10^-2	9.8611 × 10^-3	12.818
F16	Std	0.0000	0.0000	0.0000	0.0000	0.0000	5.1199 × 10^-2	1.9480 × 10^-2	2.8551 × 10^-2	6.7337 × 10^-3	8.1075 × 10^-1
F17	Average	5.0522 × 10^-31	2.4786 × 10^-138	1.0165 × 10^-3	9.9873 × 10^-2	1.3072 × 10^-50	2.3987 × 10^-1	1.7987 × 10^-1	1.3987 × 10^-1	4.8019 × 10^-1	22.103
F17	Std	1.1297 × 10^-30	4.0631 × 10^-138	2.2730 × 10^-3	1.7264 × 10^-8	2.8236 × 10^-50	5.4772 × 10^-2	4.4721 × 10^-2	5.4772 × 10^-2	4.4542 × 10^-2	1.0527
F18	Average	0.0000	0.0000	0.0000	0.0000	0.0000	20.896	10.683	5.5791	58.920	4.2836 × 10⁺⁴
F18	Std	0.0000	0.0000	0.0000	0.0000	0.0000	6.5017	6.6700	6.4969	18.029	8.4056 × 10⁺³

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Table 13. Test results of benchmark function (F1–F9, F19–F27), the dimension is fixed to 500.

Fun	Item	VAIAO	IAO	AO	AOA	HHO	STOA	CHIMP	SOA	PSO	DE
F1	Average	6.6313 × 10^-248	1.0171 × 10^-241	8.4862 × 10^-144	7.1006 × 10^-48	8.8334 × 10^-95	3.8707 × 10^-7	3.3576 × 10^-6	4.3111 × 10^-12	6.6836 × 10^-5	4.5996 × 10⁺⁴
F1	Std	0.0000	0.0000	1.7980 × 10^-143	1.4166 × 10^-47	1.7142 × 10^-94	3.7231 × 10^-7	2.7747 × 10^-6	1.7571 × 10^-12	5.5540 × 10^-5	5.1543 × 10⁺³
F2	Average	1.7819 × 10^-145	6.9639 × 10^-136	1.5808 × 10^-69	8.6734 × 10^-53	5.6451 × 10^-49	1.2085 × 10^-4	1.6912 × 10^-4	3.6835 × 10^-7	1.1760 × 10^-1	9.5069 × 10⁺²
F2	Std	3.9837 × 10^-145	1.5564 × 10^-135	3.4327 × 10^-69	1.3821 × 10^-52	1.1327 × 10^-48	4.7589 × 10^-5	1.2149 × 10^-4	3.3018 × 10^-7	1.3612 × 10^-1	39.409
F3	Average	0.0000	1.0874 × 10^-291	8.6120 × 10^-109	1.6761 × 10^-93	1.5190 × 10^-97	1.2145	8.2760 × 10^-2	1.4082 × 10^-6	9.1394 × 10⁺²	6.3988 × 10⁺¹¹
F3	Std	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000
F4	Average	0.0000	0.0000	4.5259 × 10^-295	0.0000	1.7267 × 10^-197	2.4355 × 10^-15	1.1832 × 10^-16	1.9752 × 10^-27	2.5327 × 10^-9	4.5378 × 10⁺⁷
F4	Std	0.0000	0.0000	0.0000	0.0000	0.0000	5.4241 × 10^-15	2.6449 × 10^-16	4.1972 × 10^-27	5.3850 × 10^-9	8.1540 × 10⁺⁶
F5	Average	1.1538 × 10^-141	2.3397 × 10^-129	5.8828 × 10^-58	1.9148 × 10^-2	4.9412 × 10^-50	2.7255 × 10^-2	5.4022 × 10^-1	1.4371 × 10^-3	1.4054	74.030
F5	Std	2.5801 × 10^-141	5.2316 × 10^-129	1.3154 × 10^-57	1.6284 × 10^-2	1.0977 × 10^-49	1.1281 × 10^-2	5.1437 × 10^-1	1.2835 × 10^-3	2.3314 × 10^-1	1.5767
F6	Average	5.1864 × 10^-148	5.6589 × 10^-142	3.2096 × 10^-68	0.0000	5.1573 × 10^-53	5.5248 × 10^-6	2.1736 × 10^-5	1.7515 × 10^-8	1.3136 × 10^-2	7.1716 × 10⁺⁸
F6	Std	1.1597 × 10^-147	8.7834 × 10^-142	7.1768 × 10^-68	0.0000	7.0852 × 10^-53	3.5567 × 10^-6	2.4572 × 10^-5	1.0438 × 10^-8	1.0137 × 10^-2	1.4532 × 10⁺⁹
F7	Average	3.0218 × 10^-299	4.5202 × 10^-280	1.5705 × 10^-146	2.4540 × 10^-28	5.9460 × 10^-96	4.2891 × 10^-6	8.1426 × 10^-5	1.3674 × 10^-10	2.9373 × 10^-3	6.2494 × 10⁺⁵
F7	Std	0.0000	0.0000	3.5076 × 10^-146	5.4872 × 10^-28	1.3267 × 10^-95	2.7558 × 10^-6	1.2553 × 10^-4	1.6128 × 10^-10	3.0603 × 10^-3	9.8639 × 10⁺⁴
F8	Average	0.0000	0.0000	4.9884 × 10^-283	1.7473 × 10^-89	4.7178 × 10^-184	1.0944 × 10^-10	1.1862 × 10^-6	2.4482 × 10^-19	1.3297 × 10^-4	1.5343 × 10⁺⁸
F8	Std	0.0000	0.0000	0.0000	3.9071 × 10^-89	0.0000	2.0330 × 10^-10	2.2994 × 10^-6	3.0112 × 10^-19	2.1119 × 10^-4	2.5921 × 10⁺⁷
F9	Average	0.0000	0.0000	1.3983 × 10^-289	8.5594 × 10^-120	5.0159 × 10^-202	2.9436 × 10^-10	6.4955 × 10^-7	2.6418 × 10^-17	1.0147 × 10^-3	2.2796 × 10⁺⁹
F9	Std	0.0000	0.0000	0.0000	1.9139 × 10^-119	0.0000	3.4713 × 10^-10	9.5399 × 10^-7	5.8755 × 10^-17	1.0989 × 10^-3	3.6372 × 10⁺⁸
F10	Average	0.0000	0.0000	1.1662 × 10^-2	1.0000	0.0000	1.0000	1.0000	1.0000	1.0000	1.0000
F10	Std	0.0000	0.0000	2.6076 × 10^-2	1.3383 × 10^-9	0.0000	2.6928 × 10^-11	7.6557 × 10^-10	2.5327 × 10^-11	1.9230 × 10^-16	1.8846 × 10^-8
F11	Average	0.0000	0.0000	8.3800 × 10^-294	2.2833 × 10^-131	1.3353 × 10^-295	1.5095 × 10^-10	6.2053 × 10^-9	7.3865 × 10^-24	6.6536 × 10^-6	1.4321 × 10⁺¹²
F11	Std	0.0000	0.0000	0.0000	5.1056 × 10^-131	0.0000	2.7488 × 10^-10	9.7885 × 10^-9	1.6436 × 10^-23	7.2883 × 10^-6	3.1233 × 10⁺¹¹
F12	Average	6.5458 × 10^-3	8.8714 × 10^-2	9.5595 × 10^-3	7.5066	9.7117 × 10^-3	7.3632	8.9502	7.3032	6.0514	9.4655 × 10⁺⁸
F12	Std	6.5556 × 10^-3	1.7098 × 10^-1	1.0124 × 10^-2	4.4457 × 10^-1	1.0480 × 10^-2	3.7827 × 10^-1	1.0031 × 10^-1	2.3831 × 10^-1	2.9727	3.9936 × 10⁺⁸
F13	Average	1.0427 × 10^-123	1.0306 × 10^-121	5.0944 × 10^-20	2.7879 × 10^-44	8.2408 × 10^-52	7.3789 × 10^-1	3.4293 × 10^-1	9.3585 × 10^-4	6.9881	5.2362 × 10⁺²
F13	Std	2.3314 × 10^-123	2.1964 × 10^-121	1.1391 × 10^-19	5.7506 × 10^-44	1.5881 × 10^-51	1.5253	6.0003 × 10^-1	7.5831 × 10^-4	1.4919 × 10⁺¹	4.1865 × 10⁺¹
F14	Average	0.0000	0.0000	0.0000	0.0000	0.0000	6.0417 × 10^-6	4.5754 × 10^-5	2.5965 × 10^-11	6.9428 × 10^-1	4.6528 × 10⁺⁴
F14	Std	0.0000	0.0000	0.0000	0.0000	0.0000	7.1292 × 10^-6	9.8098 × 10^-6=	2.6781 × 10^-11	3.3943 × 10^-1	2.3086 × 10⁺³
F15	Average	0.0000	0.0000	0.0000	8.9007 × 10^-9	0.0000	3.6310 × 10^-8	6.0050 × 10^-2	1.8778 × 10^-8	6.4113 × 10^-3	12.866
F15	Std	0.0000	0.0000	0.0000	1.9903 × 10^-8	0.0000	4.4207 × 10^-8	7.9379 × 10^-2	4.1980 × 10^-8	9.7735 × 10^-3	9.4698 × 10^-1
F16	Average	4.7865 × 10^-115	1.2149 × 10^-26	9.0592 × 10^-64	9.9873 × 10^-2	4.6927 × 10^-50	2.1987 × 10^-1	1.7988 × 10^-1	1.7987 × 10^-1	4.8939 × 10^-1	21.397 ×
F16	Std	1.0703 × 10^-114	2.7166 × 10^-26	2.0257 × 10^-63	1.2063 × 10^-8	1.0476 × 10^-49	4.4721 × 10^-2	4.4723 × 10^-2	4.4722 × 10^-2	7.5154 × 10^-2	1.7984
F17	Average	0.0000	0.0000	3.3621 × 10^-2	0.0000	0.0000	12.486	20.821	4.1970	43.587	4.8691 × 10⁺⁴
F17	Std	0.0000	0.0000	7.5178 × 10^-2	0.0000	0.0000	10.150	27.344	2.8825	9.4970	3.4889 × 10⁺³
F18	Average	8.8818 × 10^-16	8.8818 × 10^-16	8.8818 × 10^-16	8.8818 × 10^-16	8.8818 × 10^-16	20.000	20.000	20.000	20.688	21.197
F18	Std	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	0.0000	6.1862 × 10^-1	1.7209 × 10^-1

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5.6. Wilcoxon statistical sign-rank test and Friedman test on the 27 benchmark Functions

The Wilcoxon statistical tests are often introduced to verify the performance of the proposed algorithm. If the proposed algorithm value is smaller than another algorithm, it's proved the proposed algorithm is superior. This experiment is carried out 30 times, and the results of the Wilcoxon statistical test on Table 14. Table 14 shows that the Wilcoxon statistical test results for 27 functions are less than 0.05 in most cases. The results of the Wilcoxon sign-rank test and Friedman test on 27 functions are shown in Tables 15–18. That also means the VAIAO algorithm is better than the others.

Table 14. The results of the Wilcoxon statistical test on 27 functions.

Fun	VAIAO Vs. IAO	VAIAO Vs. AO	VAIAO Vs. AOA	VAIAO Vs. HHO	VAIAO Vs. STOA	VAIAO Vs. ChOA	VAIAO Vs. SOA
F1	3.1600 × 10⁻²	1.7300 × 10⁻⁶	1.7300 × 10⁻⁶	1.7300 × 10⁻⁶	1.7300 × 10⁻⁶	1.7300 × 10⁻⁶	1.7300 × 10⁻⁶
F2	3.8811 × 10⁻⁴	1.7300 × 10⁻⁶	1.7300 × 10⁻⁶	1.7300 × 10⁻⁶	1.7300 × 10⁻⁶	1.7300 × 10⁻⁶	1.7300 × 10⁻⁶
F3	2.5600 × 10⁻²	1.7300 × 10⁻⁶	1.7300 × 10⁻⁶	1.7300 × 10⁻⁶	1.7300 × 10⁻⁶	1.7300 × 10⁻⁶	1.7300 × 10⁻⁶
F4	NAN	1.7300 × 10⁻⁶	NAN	1.7300 × 10⁻⁶	1.7300 × 10⁻⁶	1.7300 × 10⁻⁶	1.7300 × 10⁻⁶
F5	2.0000 × 10⁻³	1.7300 × 10⁻⁶	1.7300 × 10⁻⁶	1.7300 × 10⁻⁶	1.7300 × 10⁻⁶	1.7300 × 10⁻⁶	1.7300 × 10⁻⁶
F6	1.4000 × 10⁻²	1.7300 × 10⁻⁶	3.7900 × 10⁻⁶	1.7300 × 10⁻⁶	1.7300 × 10⁻⁶	1.7300 × 10⁻⁶	1.7300 × 10⁻⁶
F7	4.0400 × 10⁻²	1.7300 × 10⁻⁶	1.7300 × 10⁻⁶	1.7300 × 10⁻⁶	1.7300 × 10⁻⁶	1.7300 × 10⁻⁶	1.7300 × 10⁻⁶
F8	NAN	1.7300 × 10⁻⁶	1.7300 × 10⁻⁶	1.7300 × 10⁻⁶	1.7300 × 10⁻⁶	1.7300 × 10⁻⁶	1.7300 × 10⁻⁶
F9	NAN	1.7300 × 10⁻⁶	1.7300 × 10⁻⁶	1.7300 × 10⁻⁶	1.7300 × 10⁻⁶	1.7300 × 10⁻⁶	1.7300 × 10⁻⁶
F10	1.7000 × 10⁻³	7.8100 × 10⁻³	1.7300 × 10⁻⁶	1.7300 × 10⁻⁶	1.7300 × 10⁻⁶	1.7300 × 10⁻⁶	1.7300 × 10⁻⁶
F11	3.5152 × 10⁻⁶	NAN	1.2300 × 10⁻⁵	1.9200 × 10⁻⁶	3.1800 × 10⁻⁶	3.6100 × 10⁻³	5.2200 × 10⁻⁶
F12	2.3000 × 10⁻²	1.4000 × 10⁻²	1.7300 × 10⁻⁶	1.7300 × 10⁻⁶	1.7300 × 10⁻⁶	1.7300 × 10⁻⁶	1.7300 × 10⁻⁶
F13	NAN	9.3200 × 10⁻⁶	1.7300 × 10⁻⁶	NAN	4.8800 × 10⁻⁴	8.3000 × 10⁻⁶	1.9500 × 10⁻³
F14	NAN	NAN	NAN	NAN	7.8100 × 10⁻³	1.8200 × 10⁻⁵	3.9100 × 10⁻³
F15	4.0700 × 10⁻²	NAN	NAN	1.7300 × 10⁻⁶	1.7300 × 10⁻⁶	1.7300 × 10⁻⁶	1.7300 × 10⁻⁶
F16	9.8000 × 10⁻³	1.7300 × 10⁻⁶	1.7300 × 10⁻⁶	1.7300 × 10⁻⁶	1.7300 × 10⁻⁶	1.7300 × 10⁻⁶	1.7300 × 10⁻⁶
F17	1.9600 × 10⁻²	0.50000	0.25000	1.7300 × 10⁻⁶	1.7300 × 10⁻⁶	1.7300 × 10⁻⁶	1.7300 × 10⁻⁶
F18	NAN	NAN	NAN	NAN	NAN	NAN	NAN
F19	NAN	1.7300 × 10⁻⁶	6.1000 × 10⁻⁵	NAN	1.7300 × 10⁻⁶	1.7300 × 10⁻⁶	1.7300 × 10⁻⁶
F20	NAN	3.8500 × 10⁻³	1.7500 × 10⁻²	1.8200 × 10⁻⁵	1.7300 × 10⁻⁶	1.7300 × 10⁻⁶	1.7300 × 10⁻⁶
F21	4.9500 × 10⁻²	1.7300 × 10⁻⁶	9.7700 × 10⁻⁴	1.2900 × 10⁻³	1.7300 × 10⁻⁶	1.7300 × 10⁻⁶	1.7300 × 10⁻⁶
F22	2.3000 × 10⁻²	NAN	NAN	1.4000 × 10⁻²	1.7300 × 10⁻⁶	1.7300 × 10⁻⁶	3.1800 × 10⁻⁶
F23	NAN	NAN	NAN	NAN	1.7300 × 10⁻⁶	1.7300 × 10⁻⁶	1.7300 × 10⁻⁶
F24	NAN	1.7300 × 10⁻⁶	1.2200 × 10⁻⁴	NAN	1.7300 × 10⁻⁶	1.7300 × 10⁻⁶	1.7300 × 10⁻⁶
F25	2.3000 × 10⁻³	1.7300 × 10⁻⁶	5.9600 × 10⁻⁵	3.5900 × 10⁻⁴	1.7300 × 10⁻⁶	1.7300 × 10⁻⁶	1.7300 × 10⁻⁶
F26	NAN	1.7300 × 10⁻⁶	1.2200 × 10⁻⁴	NAN	1.7300 × 10⁻⁶	1.7300 × 10⁻⁶	1.7300 × 10⁻⁶
F27	NAN	NAN	NAN	NAN	6.0500 × 10⁻⁷	4.3200 × 10⁻⁸	1.4500 × 10⁻⁷

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Table 15. The results of the Wilcoxon sign-rank test between VAIAO and the compared algorithms (IAO, AO, AOA).

No.	VAIAO vs IAO			VAIAO vs AO			VAIAO vs AOA
No.	R⁺	R^-	sign	R⁺	R^-	sign	R⁺	R^-	sign
F1	465.0	0.0	+	0.0	465.0	−	0.0	465.0	−
F2	0.0	465.0	−	465.0	0.0	+	465.0	0.0	+
F3	465.0	0.0	+	465.0	0.0	+	465.0	0.0	+
F4	232.5	232.5	≈	465.0	0.0	+	232.5	232.5	≈
F5	465.0	0.0	+	465.0	0.0	+	465.0	0.0	+
F6	465.0	0.0	+	465.0	0.0	+	0.0	465.0	−
F7	465.0	0.0	+	465.0	0.0	+	465.0	0.0	+
F8	232.5	232.5	≈	465.0	0.0	+	465.0	0.0	+
F9	232.5	232.5	≈	232.5	232.5	≈	232.5	232.5	≈
F10	232.5	232.5	≈	232.5	232.5	≈	465.0	0.0	+
F11	232.5	232.5	≈	232.5	232.5	≈	232.5	232.5	≈
F12	0.0	465.0	−	0.0	465.0	−	465.0	0.0	+
F13	465.0	0.0	+	465.0	0.0	+	465.0	0.0	+
F14	232.5	232.5	≈	232.5	232.5	≈	232.5	232.5	≈
F15	232.5	232.5	≈	232.5	232.5	≈	232.5	232.5	≈
F16	465.0	0.0	+	465.0	0.0	+	465.0	0.0	+
F17	232.5	232.5	≈	232.5	232.5	≈	232.5	232.5	≈
F18	232.5	232.5	≈	232.5	232.5	≈	232.5	232.5	≈
F19	232.5	232.5	≈	465.0	0.0	+	232.5	232.5	≈
F20	465.0	0.0	+	0.0	465.0	−	0.0	465.0	−
F21	232.5	232.5	≈	232.5	232.5	≈	232.5	232.5	≈
F22	232.5	232.5	≈	232.5	232.5	≈	232.5	232.5	≈
F23	232.5	232.5	≈	232.5	232.5	≈	232.5	232.5	≈
F24	0.0	465.0	−	465.0	0.0	+	0.0	465.0	−
F25	0.0	465.0	−	465.0	0.0	+	0.0	465.0	−
F26	232.5	232.5	≈	232.5	232.5	≈	232.5	232.5	≈
F27	232.5	232.5	≈	465.0	0.0	+	232.5	232.5	≈
+-≈	8/4/15			13/3/11			10/5/12

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Table 16. The results of the Wilcoxon sign-rank test between VAIAO and the compared algorithms (HHO, STOA, ChOA).

No.	VAIAO vs HHO			VAIAO vs STOA			VAIAO vs ChOA
No.	R+	R-	sign	R+	R-	sign	R+	R-	sign
F1	0.0	465.0	−	0.0	465.0	−	0.0	465.0	−
F2	465.0	0.0	+	465.0	0.0	+	465.0	0.0	+
F3	465.0	0.0	+	465.0	0.0	+	465.0	0.0	+
F4	465.0	0.0	+	465.0	0.0	+	232.5	232.5	+
F5	465.0	0.0	+	465.0	0.0	+	465.0	0.0	+
F6	465.0	0.0	+	465.0	0.0	+	465.0	0.0	+
F7	465.0	0.0	+	465.0	0.0	+	465.0	0.0	+
F8	465.0	0.0	+	465.0	0.0	+	465.0	0.0	+
F9	232.5	232.5	≈	232.5	232.5	≈	232.5	232.5	≈
F10	232.5	232.5	≈	465.0	0.0	+	465.0	0.0	+
F11	465.0	0.0	+	465.0	0.0	+	465.0	0.0	+
F12	0.0	465.0	−	465.0	0.0	+	465.0	0.0	+
F13	465.0	0.0	+	465.0	0.0	+	465.0	0.0	+
F14	232.5	232.5	≈	465.0	0.0	+	465.0	0.0	+
F15	232.5	232.5	≈	232.5	232.5	≈	232.5	232.5	≈
F16	465.0	0.0	+	465.0	0.0	+	465.0	0.0	+
F17	232.5	232.5	≈	232.5	232.5	≈	232.5	232.5	≈
F18	232.5	232.5	≈	232.5	232.5	≈	232.5	232.5	≈
F19	465.0	0.0	+	465.0	0.0	+	465.0	0.0	+
F20	0.0	465.0	−	0.0	465.0	−	0.0	465.0	−
F21	232.5	232.5	≈	232.5	232.5	≈	232.5	232.5	≈
F22	232.5	232.5	≈	465.0	0.0	+	465.0	0.0	+
F23	232.5	232.5	≈	232.5	232.5	≈	232.5	232.5	≈
F24	465.0	0.0	+	465.0	0.0	+	465.0	0.0	+
F25	465.0	0.0	+	465.0	0.0	+	465.0	0.0	+
F26	232.5	232.5	≈	232.5	232.5	≈	232.5	232.5	≈
F27	232.5	232.5	≈	465.0	0.0	+	232.5	232.5	≈
+-≈	13/3/11			18/2/7			17/2/8

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Table 17. The results of the Wilcoxon sign-rank test between VAIAO and the compared algorithms (SOA, PSO, DE).

No.	VAIAO vs SOA			VAIAO vs PSO			VAIAO vs DE
	R+	R-	sign	R+	R-	sign	R+	R-	sign
F1	0.0	465.0	−	0.0	465.0	−	0.0	465.0	−
F2	465.0	0.0	+	465.0	0.0	+	465.0	0.0	+
F3	465.0	0.0	+	465.0	0.0	+	465.0	0.0	+
F4	465.0	0.0	+	465.0	0.0	+	232.5	232.5	+
F5	465.0	0.0	+	465.0	0.0	+	465.0	0.0	+
F6	465.0	0.0	+	465.0	0.0	+	465.0	0.0	+
F7	465.0	0.0	+	465.0	0.0	+	465.0	0.0	+
F8	465.0	0.0	+	465.0	0.0	+	465.0	0.0	+
F9	232.5	232.5	≈	232.5	232.5	≈	232.5	232.5	≈
F10	465.0	0.0	+	465.0	0.0	+	465.0	0.0	+
F11	465.0	0.0	+	465.0	0.0	+	465.0	0.0	+
F12	465.0	0.0	+	465.0	0.0	+	465.0	0.0	+
F13	465.0	0.0	+	465.0	0.0	+	465.0	0.0	+
F14	465.0	0.0	+	465.0	0.0	+	465.0	0.0	+
F15	232.5	232.5	≈	232.5	232.5	≈	232.5	232.5	≈
F16	465.0	0.0	+	465.0	0.0	+	465.0	0.0	+
F17	0.0	465.0	−	0.0	465.0	−	0.0	465.0	−
F18	232.5	232.5	≈	232.5	232.5	≈	232.5	232.5	≈
F19	465.0	0.0	+	465.0	0.0	+	465.0	0.0	+
F20	0.0	465.0	−	465.0	0.0	+	0.0	465.0	−
F21	232.5	232.5	≈	232.5	232.5	≈	232.5	232.5	≈
F22	465.0	0.0	+	465.0	0.0	+	232.5	232.5	≈
F23	465.0	0.0	+	465.0	0.0	+	232.5	232.5	≈
F24	465.0	0.0	+	465.0	0.0	+	465.0	0.0	+
F25	465.0	0.0	+	465.0	0.0	+	465.0	0.0	+
F26	232.5	232.5	≈	232.5	232.5	≈	232.5	232.5	≈
F27	232.5	232.5	≈	465.0	0.0	+	232.5	232.5	≈
+-≈	18/3/6			20/2/5			16/3/9

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Table 18. The results of the Friedman test on 27 Functions.

Algorithm	Final-Rank
VAIAO	3.48
IAO	3.64
AO	3.75
AOA	4.31
HHO	3.75
STOA	6.57
ChOA	6.53
SOA	5.57
PSO	9.87
DE	7.48

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5.7. Ablation experiments

Since the proposed VAIAO introduced two improvement strategies, velocity-aided global search mechanism and adaptive opposition-based learning. Therefore, it is necessary to improve these strategies by ablation experiments.

In VAIAO, there are two variants. The first variant is VAIAO-2 which is with only velocity-aided global search mechanism. The second variant is VAIAO-3 which with adaptive opposition-based learning. Three single-peak functions (F1, F2, F3), three two-dimensional multi-channel functions (F12, F13, F16), and 3 multidimensional multi-peak functions (F24, F25, F26) are tested. The results of the ablation experiments are shown in Table 19.

Table 19. the results of the ablation experiments on 27 Functions.

Functions	Item	VAIAO	VAIAO-2	VAIAO-3
F1	Average	1.6451 × 10⁻²⁹³	5.9584 × 10⁻²⁷⁵	1.1838 × 10⁻²⁴⁵
	Std	0.0000	0.0000	0.0000
F2	Average	2.5965 × 10⁻¹⁴⁸	5.0994 × 10⁻¹³⁴	1.1056 × 10⁻¹³²
	Std	4.5876 × 10⁻¹⁴⁸	1.1403 × 10⁻¹³³	2.4723 × 10⁻¹³²
F3	Average	3.9676 × 10⁻²⁷⁶	3.2373 × 10⁻²⁵⁹	3.2917 × 10⁻²⁸⁹
	Std	0.0000	0.0000	0.0000
F12	Average	3.6658 × 10⁻³	0.11316	3.1808 × 10⁻²
	Std	1.5875 × 10⁻³	0.24508	3.6911 × 10⁻²
F13	Average	2.6766 × 10⁻¹²⁹	4.3596 × 10⁻¹²³	1.5940 × 10⁻¹²⁴
	Std	5.9850 × 10⁻¹²⁹	9.7483 × 10⁻¹²³	2.9863 × 10⁻¹²⁴
F16	Average	3.0590 × 10⁻¹⁴⁶	6.4337 × 10⁻¹³⁴	6.9369 × 10⁻¹²²
	Std	0.0000	0.0000	0.0000
F24	Average	8.5225 × 10⁻¹³⁸	2.5222 × 10⁻¹³⁵	1.1379 × 10⁻¹²⁸
	Std	1.9057 × 10⁻¹³⁷	5.6294 × 10⁻¹³⁵	2.5445 × 10⁻¹²⁸
F25	Average	7.4699 × 10⁻³	3.5276 × 10⁻²	1.9608 × 10⁻²
	Std	0.0000	0.0000	0.0000
F26	Average	1.4884 × 10⁻³⁰¹	4.3692 × 10⁻²⁹³	1.4242 × 10⁻²⁷²
	Std	0.0000	0.0000	0.0000

| Show Table

DownLoad: CSV

6. Engineering design problem

Engineering design problems are used to test the performance of algorithm, the results of the engineering design problem in real world can reflect the advantages and disadvantages of an algorithms. It is important to use the traditional engineering design problem to test proposed VAIAO algorithm.

6.1. Three-bar truss design problem

The three-bar truss is a classic optimization problem which is an engineering design problem. The schematic model of three-bar truss is shown in . The pressure of each Truss member is $\sigma$ . the cross-sectional area is ${A}_{1}( = {x}_{1})$ and ${A}_{2}( = {x}_{2})$ . The two parameters are optimized to minimize the total weight in the case of satisfying three limiting conditions. The three constraints are as follows:

$\stackrel{⃑}{x} = \left[{x}_{1}{x}_{2}\right] = \left[{A}_{1}{A}_{2}\right]$

Figure 6. The schematic model of three-bar truss ^[72].

DownLoad: Full-Size Img PowerPoint

Objective:

$f\left(\overrightarrow{x}\right) = \left(2\sqrt{2}{x}_{1}+{x}_{2}\right)\mathrm{*}l$

Subject to:

${g}_{1}\left(\overrightarrow{x}\right) = \frac{\sqrt{2}{x}_{1}+{x}_{2}}{\sqrt{2}{x}_{1}^{2}+2{x}_{1}{x}_{2}}p-\sigma \le 0$

${g}_{2}\left(\overrightarrow{x}\right) = \frac{{x}_{2}}{\sqrt{2}{x}_{2}^{2}+2{x}_{1}{x}_{2}}p-\sigma \le 0$

${g}_{1}\left(\overrightarrow{x}\right) = \frac{1}{\sqrt{2}{x}_{2}+{x}_{1}}p-\sigma \le 0$

Variable range:

$0\le {x}_{1}, {x}_{2}\le 1$

Wheel $l = 100cm, P = 2KN/c{m}^{2}, \sigma = 2KN/c{m}^{2}$ , Table 20 shows the results of three bar truss design problem. T It can be seen from the data results in the table 20 that the best value of VAIAO smaller than that of other algorithms. From this reason, it can be concluded that VAIAO can better solve the engineering problem.

Table 20. The results of three bar truss design problem.

Algorithm	X₁	X₂	Best
VAIAO	0.78975	0.40516	263.8926
IAO	0.80211	0.38269	265.1395
AO	0.77612	0.44565	264.0855
AOA	0.809	0.35403	264.2239
HHO	0.76304	0.48595	264.4203
STOA	0.79306	0.39626	263.9367
ChOA	0.77768	0.44029	263.9912
SOA	0.79261	0.39721	263.9045
PSO	0.78867	0.40818	264.5890
DE	0.84867	0.71405	318.1953

| Show Table

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6.2. Compression spring design problem

The compression spring design problem is one of the classic engineering optimization problems in mechanical engineering. The schematic model of compression spring design problem is shown in Figure 7. The essence of the compression spring design problem is to minimize the weight of the tension and compression spring by optimizing three parameters (wire diameter (d), average coil diameter (D) and effective coil number (N)).

Figure 7. The schematic model of compression spring design problem ^[72].

DownLoad: Full-Size Img PowerPoint

Consider:

$\overrightarrow{{x}_{1}} = \left[{x}_{1}, {x}_{2}, {x}_{3}\right] = \left[d\ D\ N\right]$

Objective:

$f\left(\overrightarrow{x}\right) = \left({x}_{3}+2\right){x}_{2}{x}_{1}^{2}$

Subject to:

${g}_{1}\left(\overrightarrow{x}\right) = 1-\frac{{x}_{2}^{3}{x}_{3}}{71785{x}_{1}^{4}}\le 0$

${g}_{2}\left(\overrightarrow{x}\right) = \frac{4{x}_{2}^{2}-{x}_{1}{x}_{2}}{12566({x}_{2}{x}_{1}^{3}-{x}_{1}^{4})}+\frac{1}{5108{x}_{1}^{2}}-1\le 0$

${g}_{3}\left(\overrightarrow{x}\right) = 1-\frac{140.45{x}_{3}}{{x}_{2}^{2}{x}_{3}}\le 0$

${g}_{4}\left(\overrightarrow{x}\right) = \frac{{x}_{1}+{x}_{2}}{1.5}-1\le 0$

Variable ranges:

$0.05\le {x}_{1}\le 2.00$

$0.25\le {x}_{2}\le 1.30$

$2.00\le {x}_{3}\le 15.00$

The experimental results are shown in Table 21. The schematic model of the Compression Spring Design Problem is shown in Figure 7. Table 21 shows the final results of VAIAO and comparative algorithms after solving this engineering problem. As shown in the table 16, VAIAO achieves the minimum weight of the compression spring. IAO and AO are in all cases applied to this engineering problem the algorithm presents the worst objective value.

Table 21. The results of compression spring design problem.

Algorithm	X₁	X₂	X₃	Best
VAIAO	0.05	0.316923	14.1159	0.012769
IAO	0.075746	0.592546	11.6982	0.04657
AO	0.07433	1.0468	2.4959	0.026002
AOA	0.05	0.311514	15	0.013239
HHO	0.057138	0.50254	6.0288	0.013173
STOA	0.051846	0.359471	11.3453	0.012895
ChOA	0.05	0.315274	14.4887	0.012996
SOA	0.05014	0.320512	13.997	0.01289
PSO	0.05886	0.55504	5.039	0.014581
DE	0.05000	0.32349	14.0266	0.0129611

| Show Table

DownLoad: CSV

6.3. Pressure vessel design problem

The pressure vessel design is also an engineering problem which needs to optimize cost, including material cost, molding cost and welding cost, to minimize the total cost. This problem has four issues that need to be optimized (the thickness of the shell (X1), the thickness of the head (X2), the inner radius (X3), the length of the vessel section (X4). There are 4 constraints in this problem.

Consider:

$\overrightarrow{x} = \left[{x}_{1}, {x}_{2}, {x}_{3}, {x}_{4}\right]$

Objective:

$f\left(\overrightarrow{x}\right) = 0.6224{x}_{1}{x}_{3}{x}_{4}+1.7781{x}_{2}{x}_{3}^{2}+3.1661{x}_{1}^{2}{x}_{4}+19.84{x}_{1}^{2}{x}_{3}$

Subject to:

${g}_{1}\left(\overrightarrow{x}\right) = -{x}_{1}+0.0193{x}_{3}\le 0$

${g}_{2}\left(\overrightarrow{x}\right) = -{x}_{3}+0.0954{x}_{3}\le 0$

${g}_{3}\left(\overrightarrow{x}\right) = -\pi {x}_{3}^{2}{x}_{4}-\frac{4}{3}\pi {x}_{3}^{3}+1296000\le 0$

${g}_{4}\left(\overrightarrow{x}\right) = {x}_{4}-240\le 0$

Variable ranges:

$0\le {x}_{1}\le 99$

$0\le {x}_{2}\le 99$

$10\le {x}_{3}\le 200$

$10\le {x}_{4}\le 200$

The schematic model of pressure vessel is shown in Figure 8. Table 22 shows that the best value of VAIAO is the smallest, which means that the optimized solution is obtained by VAIAO.

Figure 8. The schematic model of pressure vessel ^[72].

DownLoad: Full-Size Img PowerPoint

Table 22. The results of pressure vessel.

Algorithm	X₁	X₂	X₃	X₄	Best
VAIAO	1.00954	0.496241	52.2862	90.309	6.73 × 10³
IAO	1.33753	1.42724	53.4433	86.9307	1.35 × 10⁴
AO	1.04199	0.558116	54.073	70.3731	6.78 × 10³
AOA	1.03106	1.867994	44.41967	200	1.39 × 10⁴
HHO	1.1552	0.5725	58.5983	42.0081	6.99 × 10³
STOA	1.23586	0.598644	63.3739	18.6267	7.23 × 10³
ChOA	1.27118	0.658186	65.6949	10	7.73 × 10³
SOA	0.779165	0	40.32663	200	1.53 × 10⁵
PSO	0.983267	0.486026	51.0843	89.9709	6.34 × 10³
DE	76.03506	27.32045	159.8693	18.81918	3.4 × 10⁵

| Show Table

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6.4. Speeds reducer design problem

The speeds reducer design problem needs to optimize 7 variables to get the minimize the weight of the reducer. Face width parameter is b ( = ${x}_{1}$ ), Tooth modulus is m ( = ${x}_{2}$ ), the number of teeth in the pinion is z ( = ${x}_{3}$ ). Between bearings, the length of the first shaft is l1 ( = ${x}_{4}$ ) and the length of the second shaft l2 ( = ${x}_{5}$ ). the diameter of first shafts is d1 ( = ${x}_{6}$ ), and the diameter of second shafts is d2 ( = ${x}_{7}$ ). At the same time, the speed reducer design problem has 11 constraints:

Consider:

$\overrightarrow{x} = \left[{x}_{1}, {x}_{2}, {x}_{3}, {x}_{4}, {x}_{5}, {x}_{6}, {x}_{7}\right]$

Objective:

$\begin{array}{c}f\left(x\right) = 0.7854{x}_{1}{x}_{2}^{2}\left(3.3333{x}_{3}^{2}+14.9334{x}_{3}-43.0934\right)-1.508{x}_{1}\left({x}_{6}^{3}+{x}_{7}^{3}\right)+0.7854\left({x}_{4}{x}_{6}^{2}+{x}_{5}{x}_{7}^{2}\right)\end{array}$

Subject to:

${g}_{1}\left(\overrightarrow{x}\right) = \frac{27}{{x}_{1}{x}_{2}^{2}{x}_{3}}-1\le 0$

${g}_{2}\left(\overrightarrow{x}\right) = \frac{397.5}{{x}_{1}{x}_{2}^{2}{x}_{3}^{2}}-1\le 0$

${g}_{3}\left(\overrightarrow{x}\right) = \frac{1.934{x}_{4}^{3}}{{x}_{2}{x}_{3}{x}_{6}^{4}}-1\le 0$

${g}_{4}\left(\overrightarrow{x}\right) = \frac{1.93{x}_{5}^{3}}{{x}_{2}{x}_{3}{x}_{7}^{4}}-1\le 0$

${g}_{5}\left(\overrightarrow{x}\right) = \frac{\sqrt{{\left(\frac{745{x}_{4}}{{x}_{2}{x}_{3}}\right)}^{2}+16.9\times {10}^{6}}}{110{x}_{6}^{3}}-1\le 0$

${g}_{6}\left(\overrightarrow{x}\right) = \frac{\sqrt{{\left(\frac{745{x}_{4}}{{x}_{2}{x}_{3}}\right)}^{2}+157.5\times {10}^{6}}}{85{x}_{7}^{3}}-1\le 0$

${g}_{7}\left(\overrightarrow{x}\right) = \frac{{x}_{2}{x}_{3}}{40}-1\le 0$

${g}_{8}\left(\overrightarrow{x}\right) = \frac{{5x}_{2}}{{x}_{1}}-1\le 0$

${g}_{9}\left(\overrightarrow{x}\right) = \frac{{x}_{1}}{{12x}_{2}}-1\le 0$

${g}_{10}\left(\overrightarrow{x}\right) = \frac{1.5{x}_{6}+1.9}{{x}_{4}}-1\le 0$

${g}_{11}\left(\overrightarrow{x}\right) = \frac{{1.1x}_{1}+1.9}{{x}_{5}}-1\le 0$

Variable ranges:

$2.6\le {x}_{1}\le 3.6$

$0.7\le {x}_{2}\le 0.8$

$17\le {x}_{3}\le 28$

$7.3\le {x}_{4}\le 8.3$

$7.8\le {x}_{5}\le 8.3$

$2.9\le {x}_{6}\le 3.9$

$5.0\le {x}_{7}\le 5.5$

The schematic model of speeds reducer design problem is shown in Figure 9. The experimental results are shown in Table 23. Table 23 shows the final results of the VAIAO and comparison algorithms after solving the speed reducer design problem. As shown in Table 21, VAAIO has the smallest best value compared to another algorithm.

Figure 9. The schematic model of speeds reducer design problem ^[72].

DownLoad: Full-Size Img PowerPoint

Table 23. The results of speeds reducer design problem.

Algorithm	X₁	X₂	X₃	X₄	X₅	X₆	X₇	Best
VAIAO	3.48616	0.7	17	7.42066	8.01342	3.35348	5.31633	3032.273
IAO	3.6	0.7	17	8.3	8.3	3.9	5.5	3363.876
AO	3.49415	0.7	17.0532	8.26393	7.80686	3.43739	5.32147	3060.208
AOA	3.6	0.7	17	7.3	8.3	3.48962	5.29573	3089.259
HHO	3.49436	0.7	17	7.95063	7.8	3.76166	5.28586	3121.865
STOA	3.55498	0.7	17	7.98411	7.89628	3.38054	5.29571	3039.726
ChOA	3.54913	0.7	17	8.3	8.3	3.51904	5.32825	3108.260
SOA	3.49757	0.7	17	8.3	7.8	3.47918	5.28433	3039.816
PSO	3.6	0.7	17	8.3	7.8	3.35006	5.2857	3035.2958
DE	2.70401	0.752413	21.7262	7.61006	8.3	3.89253	5.43619	7120.8812

| Show Table

DownLoad: CSV

6.5. Gear design problem

The gear design problem is also a classic engineering optimization problem. The problem is to achieve the minimum transmission ratio cost by optimizing the number of teeth. The four gears are A( ${x}_{1}$ ), B( ${x}_{2}$ ), C( ${x}_{3}$ ), A( ${x}_{4}$ ). This engineering optimization problems has no constraints.

Consider:

$\overrightarrow{x} = \left[{x}_{1}, {x}_{2}, {x}_{3}, {x}_{4}\right]$

Objective:

$f\left(\overrightarrow{x}\right) = {(\frac{1}{6.913}-\frac{{x}_{2}{x}_{3}}{{x}_{1}{x}_{4}})}^{2}$

Variable ranges:

$12\le {x}_{1}, {x}_{2}, {x}_{3}, {x}_{4}\le 60$

The schematic model of gear design problem is shown in Figure 10. The results of gear design problem are shown in Table 24 The results of VAIAO algorithm are also the best compared with other algorithms. As shown in Table 24, Compared with IAO, AO and other objective algorithms, the most competitive results.

Figure 10. The schematic model of gear design problem ^[72].

DownLoad: Full-Size Img PowerPoint

Table 24. The results of gear design problem.

Algorithm	X₁	X₂	X₃	X₄	Best
VAIAO	44	20	16	49	2.70 × 10⁻¹²
IAO	52	14	14	27	4.78 × 10⁻⁷
AO	49	23	18	55	1.17 × 10⁻¹⁰
AOA	60	13	12	19	2.73 × 10⁻⁸
HHO	44	17	19	50	2.70 × 10⁻¹²
STOA	47	12	14	24	9.92 × 10⁻¹⁰
ChOA	48	13	12	23	1.36 × 10⁻⁹
SOA	50	14	17	34	2.36 × 10⁻⁹
PSO	23	12	13	47	9.92 × 10⁻¹⁰
DE	24	23	18	60	6.74 × 10⁻³

| Show Table

DownLoad: CSV

7. Conclusions and future works

In this paper, an enhanced AO algorithm is proposed to improve the exploration ability and convergence speed of IAO. Inspired by the velocity-Aided Global Search Mechanism, the velocity parameters and acceleration parameters are introduced into the AO algorithm to help the search agent update the position and prevent a number of good positions from being missed during the optimization process. In addition, introduced the adaptive opposition-based learning rule is introduced to improve the local optimum. The performance proposed VAIAO algorithm and the comparison algorithms are tested by 27 classical benchmark functions, and five engineering optimization problems. The results of the experiment show that the VAIAO algorithm is easier to obtain better global exploration and exploitation capabilities, faster convergence speed and higher convergence accuracy than any other algorithm. Because the effect of the proposed VAIAO algorithm is better than the original algorithm in terms of optimization. In future work, the proposed VAIAO will be used to solve complex multidisciplinary problems and real-meaning engineering problems, such as Photovoltaic module model parameter extraction, image segmentation and engineering problems of multi-objective optimization.

Acknowledgments

This work was supported by the Innovation Fund for Industry-University-Research in Chinese Universities with grant number 2021ZYA11012, the open project of Hubei Engineering Research Center for Specialty Flowers Biological Breeding with grant number 2022ZD006, the educational teaching research project of Jingchu university of technology with grant number JX 2022-016, the science research project of Jingchu university of technology with grant number YY202203 and the general funded science and technology research project of Jingmen with grant number 2022YFYB152.

Conflict of interest

All authors declare no conflicts of interest in this paper.

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Mathematical Biosciences and Engineering

STS-TransUNet: Semi-supervised Tooth Segmentation Transformer U-Net for dental panoramic image

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Abstract

1. Introduction

2. Aquila optimizer

2.1. Strategy 1: Flying high for searching prey

2.2. Short glide attack and fight

2.3. Approaching prey and attack

2.4. Catching prey on land

3. The simplified IAO

4. The proposed VAIAO algorithm

4.1. Velocity-Aided global search mechanism

4.2. Adaptive opposition-based learning

4.3. Overview of the proposed VAIAO algorithm

5. Experiments and results

5.1. Experimental setup

5.2. Test function

5.3. Experimental series 1: Intensification capability experiments

5.4. Experiment series 3: Accelerated convergence experiments

5.5. Experiment series 4: Scalability experiments

5.6. Wilcoxon statistical sign-rank test and Friedman test on the 27 benchmark Functions

5.7. Ablation experiments

6. Engineering design problem

6.1. Three-bar truss design problem

6.2. Compression spring design problem

6.3. Pressure vessel design problem

6.4. Speeds reducer design problem

6.5. Gear design problem

7. Conclusions and future works

Acknowledgments

Conflict of interest

References

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Mathematical Biosciences and Engineering

STS-TransUNet: Semi-supervised Tooth Segmentation Transformer U-Net for dental panoramic image

Related Papers:

Abstract

1. Introduction

2. Aquila optimizer

2.1. Strategy 1: Flying high for searching prey

2.2. Short glide attack and fight

2.3. Approaching prey and attack

2.4. Catching prey on land

3. The simplified IAO

4. The proposed VAIAO algorithm

4.1. Velocity-Aided global search mechanism

4.2. Adaptive opposition-based learning

4.3. Overview of the proposed VAIAO algorithm

5. Experiments and results

5.1. Experimental setup

5.2. Test function

5.3. Experimental series 1: Intensification capability experiments

5.4. Experiment series 3: Accelerated convergence experiments

5.5. Experiment series 4: Scalability experiments

5.6. Wilcoxon statistical sign-rank test and Friedman test on the 27 benchmark Functions

5.7. Ablation experiments

6. Engineering design problem

6.1. Three-bar truss design problem

6.2. Compression spring design problem

6.3. Pressure vessel design problem

6.4. Speeds reducer design problem

6.5. Gear design problem

7. Conclusions and future works

Acknowledgments

Conflict of interest

References

This article has been cited by:

Reader Comments

通讯作者: 陈斌, bchen63@163.com

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