The heterogeneous Aquila optimization algorithm

Juan ZHAO; Zheng-Ming GAO; Juan ZHAO; Zheng-Ming GAO

doi:10.3934/mbe.2022275

Mathematical Biosciences and Engineering

2022, Volume 19, Issue 6: 5867-5904. doi: 10.3934/mbe.2022275

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Research article Special Issues

The heterogeneous Aquila optimization algorithm

Juan ZHAO ^1,,
Zheng-Ming GAO ^{2,3
,
,}

1.
School of electronics and information engineering, Jingchu University of Technology, Jingmen 448000, China
2.
School of computer engineering, Jingchu University of Technology, Jingmen 448000, China
3.
Institute of intelligent information technology, Hubei Jingmen industrial technology research institute, Jingmen 448000, China

Academic Editor: Runzhang Xu

Received: 10 February 2022 Revised: 21 March 2022 Accepted: 31 March 2022 Published: 08 April 2022

A new swarm-based optimization algorithm called the Aquila optimizer (AO) was just proposed recently with promising better performance. However, as reported by the proposer, it almost remains unchanged for almost half of the convergence curves at the latter iterations. Considering the better performance and the lazy latter convergence rates of the AO algorithm in optimization, the multiple updating principle is introduced and the heterogeneous AO called HAO is proposed in this paper. Simulation experiments were carried out on both unimodal and multimodal benchmark functions, and comparison with other capable algorithms were also made, most of the results confirmed the better performance with better intensification and diversification capabilities, fast convergence rate, low residual errors, strong scalabilities, and convinced verification results. Further application in optimizing three benchmark real-world engineering problems were also carried out, the overall better performance in optimizing was confirmed without any other equations introduced for improvement.

Keywords:

Citation: Juan ZHAO, Zheng-Ming GAO. The heterogeneous Aquila optimization algorithm[J]. Mathematical Biosciences and Engineering, 2022, 19(6): 5867-5904. doi: 10.3934/mbe.2022275

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Abstract

1. Introduction

Along with the development of modern science and technology, the details of problems in nature are considered so much as to every influential factor being considered. These factors would play the roles influencing both the apparent showing to human and each other. As a result, analytical solutions could be no longer obtained whenever the problems are a little complicated. To conquer such difficulties, the nature-inspired algorithms had been proposed several decades of years ago.

Genetic algorithm (GA) ^[1] might be the first candidate in literature, it soon became a hot spot to solve most of the problems human met. Convinced by the better performance, other optimization algorithms were soon proposed, such as the ant colony optimization (ACO) ^[2] algorithm, the particle swarm optimization (PSO) ^[3] algorithm and so on.

Talking about the PSO algorithm, scientists and engineers around the world were soon addicted to the beautiful and simple structure. Thorough details were contributed, the reason why it converged so fast, its stability together with the constraints of variables ${c}_{1}, {c}_{2}$ were confirmed ^[4]. Researches applied the PSO algorithm to solve every optimization problem they found, and they were also not satisfied with its current performance. Finding ways to improve the performance, in either convergence ratio or residual errors, soon became a hot spot at those years. Inertia weights ^[5], local unimodal sampling ^[6], regrouping ^[7], guaranteed convergence ^[8], and other improvements were soon proposed and better performance confirmed. Literally speaking, the improvements could be classified as: 1) improvements of variables, such as the chaotic mapping ^[9], Levy flight ^[10]. 2) re-constructions of updating equations, such as the guaranteed convergence, regrouping methods. 3) improvements of controlling parameters, for instance, the inertia weights. 4) Hybridizations, such as the hybridization of PSO with GA ^[11], PSO with ACO ^[12].

Among all of the improved PSO algorithms, the heterogeneous PSO (HPSO) improvement ^[13] was a most shining one, it might be an inspiration of most of the modern optimization algorithms proposed recently.

In the early years of computational optimization, all of the individuals in swarms would following a same style to update their positions. For example, in the GA, they would crossover, mutate, while in the PSO algorithm, all of the individuals would update their positions according to their current velocities, the global best candidate, and their current best trajectories. Individuals in swarms of the bat algorithm (BA) ^[14] would also follow a same equation to update their current positions. While in the HPSO algorithm, individuals in swarms would select a way randomly among the cognitive-only, social-only, barebones, and modified barebones operations. The HPSO was also confirmed better in optimization.

Heterogeneous improvements would give individuals in swarm multiple ways ^[15] or tunnels ^[16] to update their positions. In such conditions, individuals could seek help for more variable operations. Some of them could take larger steps for a better exploration capability, some of them could take smaller steps to exploit the local optima and confirm whether it is the global one or not, and they would not follow a same way, which could increase their capability in both exploration and exploitation procedure. We can see that individuals in the sine cosine algorithm (SCA) ^[17] have two choices to update their positions, the arithmetic optimization algorithm (AOA) ^[18] and Harris hawk optimization (HHO) algorithm have four choices. More ways to update their positions might not always achieve in better performance, individuals in swarms with such operations could indeed afford variable choices and multiple characteristics. Henceforth, the heterogeneous improvements could be induced to be a multiple updating principle.

Aquila optimizer (AO) ^[19] was just proposed recently, it was claimed and verified with fast convergence rate and better performance than most of the other algorithms, it have been applied in intrusion detection ^[20], production forecasting ^[21].

The individuals in swarms of the AO algorithm also have four ways to update their positions, however, they can only choose two of them during the first 2/3 full procedure in exploration, and two of them at the rest exploitation procedure. Although the overall results were quite promising as reported by the proposer, individuals in swarms could not always take further operations to obtain better results in the exploitation procedure, which could be easily found in figure 9 in referenced paper ^[19]. This paper would report a heterogeneous improvement of the Aquila optimization algorithm with abbreviation HAO. The constant value 2/3 which constraint the individuals to follow two ways each during their exploration and exploitation procedure would be changed to a probability value, and consequently, individuals in the HAO swarms could have four ways to update their positions.

The main contribution of this paper would be:

1) The heterogeneous improvement of the AO algorithm was proposed in this paper, which did not introduce any further equations and improve the computer complexity. The proposed HAO algorithm only reconstructed the strategies which were proposed by the proposer.

2) The four strategies were classified by two groups with two proportional values ${p}_{1}$ and ${p}_{2}$ , their influence of the convergence rate was simulated and a best set was chosen.

3) Qualitative analysis, intensification, diversification, and scalability capability of the proposed HAO algorithm were simulated and the acceleration rate were tested on either unimodal or multimodal benchmark functions.

4) Simulation experiments on unimodal, multimodal, together with three classical real-world engineering problems were carried out and comparison were made. Results confirmed the better performance of the propose HAO algorithm than the original AO algorithm together with some other well-known optimization algorithms.

The rest of this paper would be arranged as follows: In Section 2, the original AO algorithm was briefly reported, and the improved HAO was proposed. Simulation experiments would be carried out both on benchmark functions in Section 3 and real-world engineering problems in Section 4. Discussions would be made and conclusions would be drawn in Section 5.

2. The AO and proposed HAO algorithms

2.1. Strategies in the AO algorithm

There are four strategies for individuals in the AO algorithm:

Strategy 1: Expanded exploration.

${X}_{i}\left(t+1\right) = {X}_{best}\left(t\right)\times \left(1-\frac{t}{T}\right)+{X}_{M}\left(t\right)- \\ \qquad\qquad \quad\;{X}_{best}\left(t\right)*{r}_{1}$

(1)

where ${X}_{i}\left(t+1\right)$ , ${X}_{best}\left(t\right)$ , ${X}_{M}\left(t\right)$ represent the position of i-th individuals at iteration t + 1, the best location at the current iteration and the mean positions of all individuals at the current iteration respectively. ${X}_{M}\left(t\right)$ would be calculated as follows:

${X}_{M}\left(t\right) = \frac{1}{N}{\sum }_{i = 1}^{N}{X}_{i}\left(t\right)$

(2)

where ${X}_{i}\left(t\right)$ represents the position of i-th individuals at iteration t. N represents the number of individuals in swarms. ${r}_{1}$ is the random number in Gaussian distribution with the interval of 0 and 1.

Strategy 2: Narrowed exploration.

${X}_{i}\left(t+1\right) = {X}_{best}\left(t\right)\times Levy\left(D\right)+{X}_{R}\left(t\right)+\left(y-x\right)*{r}_{2}$

(3)

where $Levy\left(D\right)$ represents the Levy flights following equations:

$\mathrm{Levy}\left(\mathrm{D}\right) = \mathrm{s}\times \frac{{\rm{\mu }}\times {\rm{\sigma }}}{{\left|\nu \right|}^{\frac{1}{\beta }}}$

(4)

where $s = 0.01$ is a constant parameter, ${r}_{2}$ is another random number. $\mu$ , $\nu$ are random numbers between 0 and 1. $\sigma$ is calculated as follows:

$\sigma = \frac{\mathrm{\Gamma }\left(1+\beta \right)\times \mathrm{sin}\left(\frac{\pi \beta }{2}\right)}{\mathrm{\Gamma }\left(\frac{1+\beta }{2}\right)\times \beta \times {2}^{\frac{\beta -1}{2}}}$

(5)

where $\beta = 1.5$ is a constant value. ${X}_{R}\left(t\right)$ is a random selected candidate at the current iteration. $y$ and $x$ represent the spiral shape:

$y = r\times \mathrm{cos}\left(\theta \right)$

(6)

$x = r\times \mathrm{sin}\left(\theta \right)$

(7)

$r = {r}_{1}+U\times {D}_{1}$

(8)

$\theta = -\omega \times {D}_{1}+{\theta }_{1}$

(9)

${\theta }_{1} = \frac{3\pi }{2}$

(10)

where ${r}_{1}$ is a fixed number between 1 and 20. ${D}_{1}$ is integer numbers from 1 to the length of the problems. $\omega = 0.005$ is a fixed constant number.

Strategy 3: Expanded exploitation.

${X}_{i}\left(t+1\right) = \alpha \times \left[{X}_{best}\left(t\right)-{X}_{M}\left(t\right)\right] \\ \quad +\delta \times \left[\left(UB-LB\right)\times {r}_{3}+LB\right]$

(11)

where [LB, UB] is the definitional domain of the given problem. $\alpha$ and $\delta$ are two fixed small numbers. ${r}_{3}$ is the third random number in Gaussian distribution.

Strategy 4: Narrowed exploitation.

${X}_{i}\left(t+1\right) = QF\times {X}_{best}\left(t\right)-{G}_{1}\times {X}_{i}\left(t\right)\times {r}_{4} \\ \qquad\qquad -{G}_{2}\times Levy\left(D\right)+{r}_{5}\times G1$

(12)

where $QF$ denotes to a quality function used to equilibrium the search strategy, and calculated with the following equation:

$QF\left(t\right) = {t}^{\frac{2\times {r}_{6}-1}{{\left(1-T\right)}^{2}}}$

(13)

${G}_{1} = 2{r}_{7}-1$

(14)

${G}_{2} = 2\times \left(1-\frac{t}{T}\right)$

(15)

${r}_{4}$ , ${r}_{5}, {r}_{6}, {r}_{7}$ are the fourth to seventh random numbers involved in this algorithm.

2.2. Proposed HAO improved algorithm

Although individuals in the original AO swarm have four strategies and four ways to update their positions, however, according to the status of the prey and the Aquila, individuals could only choose the first two strategies in the early 2/3 iteration times called exploration procedure, while they would choose the latter two strategies at the rest of exploitation procedure. Apparently, the first two strategies would be more aggressive and could result in fast convergence, however, the latter two involve the re-initializing operations, which would give the individuals a chance to jump out of the local optima. Both of the first and latter two strategies have their own merits and result in better performance of the original AO algorithm.

However, we can easily find that the latter two strategies lack capability in approaching global optima, especially for the multimodal benchmark functions. Figure 9 in the referenced paper ^[19] showed that individuals would fail to obtain better positions any more in the later iterations for F5–F10, F12–F15, F17–23, who were some unimodal benchmark functions and most of the multimodal benchmark functions. To eliminate such defects, the choice of strategies could be a random way with different probabilities, let alone the solemn separation of exploration and exploitation. The individuals are recommended to choose a way with different probabilities to explore or exploit the whole definitional domain with their own willing. Therefore, we introduce the heterogeneous improvement for the original AO algorithm, and proposed an improvement called the heterogeneous Aquila Optimizer (HAO). For simplicity, a pseudo code of the proposed HAO algorithm would be shown in Table 1.

Table 1. Pseudo code of HAO algorithm.

Stage	Pseudo-code
Initializing	Maximum iteration number M
	population size N
	dimensionality D
	constraint variables
	probabilities ${p}_{1}, {p}_{2}, {p}_{3}$
Exploration and Exploitation	While iteration < M
	calculate the fitness values
	calculate ${\mathrm{X}}_{\mathrm{best}}, {X}_{M}$
	for i=1 to N
	update parameters
	random number ${\mathrm{r}}_{1}, {r}_{2}, {r}_{3}$
	if ${\mathrm{r}}_{1}$ < ${p}_{1}$
	if ${r}_{2} < {p}_{2}$
	strategy 1
	else
	strategy 2
	end if
	else
	if ${r}_{3} < {p}_{2}$
	strategy 3
	else
	strategy 4
	end if
	end if
	end for
	end while
output	the best candidate

| Show Table

DownLoad: CSV

The complexity would remain as $O(M\times N\times D)$ with a minimum change in proportional values.

3. Simulation experiments on benchmark functions

In this section, simulation experiments on benchmark functions would be carried out, simulation results would be discussed to find whether the proposed HAO algorithm would perform better than the original AO algorithm or not. And furthermore, considering the development of swarm intelligence and better performance, several other algorithms would also be involved in comparison, such as original AO, the antlion optimization (ALO) ^[22], African vultures optimization (AVOA) ^[23], equilibrium optimizer (EO) ^[24], grasshopper optimization algorithm (GOA) ^[25], the grey wolf optimizer (GWO) ^[26], Harris hawk optimization (HHO) ^[27], Moth-frame optimization (MFO) ^[28], mayfly optimization algorithm (MOA) ^[29], PSO, SCA and whale optimization algorithm (WOA) ^[30]. The parameters of all of the compared algorithms are shown in Table 2.

Table 2. Parameters setup.

Algorithm	values
AO	U = 0.00565; r₁ = 10; ω = 0.005; α = 0.1; δ = 0.1; G₁∈[−1, 1]; G₂∈[2,0]
ALO	r₁∈[0, 1]; I = 10^w; w = 2, 3, 4, 5, 6
AVOA	p₁ = 0.6; p₂ = 0.4; p₃ = 0.6; $\mathrm{\alpha }=0.8;\mathrm{\beta }=0.2;\mathrm{\gamma }=2.5$
EO	V = 1; a₁ = 2; a₂ = 1; GP = 0.5
GOA	cMax = 1; cMin = 4e-5
GWO	a₀ = 2; r₁, r₂∈[0, 1]
HHO	q∈[0, 1]; r∈[0, 1]; E0∈[−1, 1]; E1∈[2,0]; E∈[−2, 2];
MFO	b = 1
MOA	g = 0.8; gdamp = 0.8; a1 = 1.0; a2 = 1.5; a3 = 1.5; $\mathrm{\beta }=2;$ dance = 5; fl = 1; dance_damp = 0.8; fl_damp = 0.99;
PSO	c₁ = 1.5; c₂ = 1.5;
SCA	r₁∈[1,0]; r₂∈[0, 2 $\pi$ ]; r₃∈[0, 2]
WOA	r₁, r₂∈[0, 1]; a₀ = 2; b = 1;p∈[1,0]

| Show Table

DownLoad: CSV

3.1. Benchmark functions

Verifying the capability of an algorithm with benchmark functions is a classical way in optimization. In this paper, the improved HAO would be also applied in optimization benchmark functions. 5 unimodal, 5 unimodal with three-dimensional basin-like landscapes, and 11 multimodal benchmark functions would be involved ^[31], as shown in Table 3–5.

Table 3. unimodal scalable benchmark functions (D = 10).

No.	Names	Equations	domain[lb, ub]
F1	Ackley 1	$f\left(x\right)=-20{e}^{-0.02\sqrt{\frac{{\sum }_{i=1}^{d}{x}_{i}^{2}}{d}}}-{e}^{\frac{{\sum }_{i=1}^{d}\cos\left(2\pi {x}_{i}\right)}{d}}+20+e$	[−100,100]
F2	Exponential	$f\left(x\right)=1-{e}^{-0.5\sum_{i=1}^{d}{x}_{i}^{2}}$	[−3, 3]
F3	Powell Sum	$f\left(x\right)=\sum\limits_{i=1}^{d}{\left\|{x}_{i}\right\|}^{i+1}$	[−1, 1]
F4	Sargan	$f\left(x\right)=\sum\limits_{i=1}^{d}d\left({x}_{i}^{2}+0.4\sum\limits_{j=1, j\ne i}^{d}{x}_{i}{x}_{j}\right)$	[−100,100]
F5	Sphere	$f\left(x\right)=\sum\limits_{i=1}^{d}{x}_{i}^{2}$	[−100,100]

| Show Table

DownLoad: CSV

Table 4. unimodal scalable benchmark functions with three-dimensional basin-like- landscapes (D = 10).

No.	Names	Equations	domain[lb, ub]
F6	Chung Reynolds	$f\left(x\right)={\left(\sum\limits_{i=1}^{d}{x}_{i}^{2}\right)}^{2}$	[−100,100]
F7	Csendes	$f\left(x\right)=\sum\limits_{i=1}^{d}{x}_{i}^{6}\left(2+\mathrm{sin}\frac{1}{{x}_{i}}\right)$	[−100,100]
F8	Holzman's 2	$f\left(x\right)=\sum\limits_{i=1}^{d}i{x}_{i}^{4}$	[−100,100]
F9	Hyper-Ellipsoid	$f\left(x\right)=\sum\limits_{i=1}^{d}{i}^{2}{x}_{i}^{2}$	[−100,100]
F10	Schumer Steiglitz 2	$f\left(x\right)=\sum\limits_{i=1}^{d}{x}_{i}^{4}$	[−100,100]

| Show Table

DownLoad: CSV

Table 5. multimodal scalable benchmark functions (D = 10).

No.	Names	Equations	domain[lb, ub]
F11	Alpine 1	$f\left(x\right)=\sum\limits_{i=1}^{d}\left\|{x}_{isin}\left({x}_{i}\right)+0.1{x}_{i}\right\|$	[−100,100]
F12	Cosine Mixture	$f\left(x\right)=\frac{d}{10}+\sum\limits_{i=1}^{d}{x}_{i}^{2}-\frac{1}{10}\sum\limits_{i=1}^{d}\cos\left(5\pi {x}_{i}\right)$	[−100,100]
F13	Griewank	$f\left(x\right)=\sum\limits_{i=1}^{d}\frac{{x}_{i}^{2}}{4000}--\mathrm{cos}\prod \left(\frac{{x}_{i}}{\sqrt{i}}\right)+1$	[−100,100]
F14	Inverted Cosine-Wave	$\begin{array}{c}f\left(x\right)=d-1-{\sum }_{i=1}^{d}\left\{{e}^{\left[\frac{-\left({x}_{i}^{2}+{x}_{i+1}^{2}+0.5{x}_{i}{x}_{i+1}\right)}{8}\;\;\right]}cos\left(4\times \sqrt{{x}_{i}^{2}+{x}_{i+1}^{2}+0.5{x}_{i}{x}_{i+1}}\right)\right\}\end{array}$	[−100,100]
F15	Pathological	$f\left(x\right)=\sum\limits_{i=1}^{d-1}\left(0.5+\frac{{\mathrm{sin}}^{2}\sqrt{100{x}_{i}^{2}+{x}_{i+1}^{2}}-0.5}{1+0.001{\left({x}_{i}^{2}-2{x}_{i}{x}_{i+1}+{x}_{i+1}^{2}\right)}^{2}}\right)$	[−100,100]
F16 .	Rastrigin	$f\left(x\right)=\sum\limits_{i=1}^{d}\left[{x}_{i}^{2}-10\mathrm{cos}\left(2\pi {x}_{i}\right)+10\right]$	[−100,100]
F17	Salomon	$f\left(x\right)=1-\mathrm{cos}\left(2\pi \sqrt{\sum\limits_{i=1}^{d}{x}_{i}^{2}}\right)+0.1\sqrt{\sum \limits_{i=1}^{d}{x}_{i}^{2}}$	[−100,100]
F18	Shubert 6	$f\left(x\right)=\sum\limits_{i=1}^{d-1}0.5+\frac{{\mathrm{sin}}^{2}\sqrt{{x}_{i}^{2}+{x}_{i+1}^{2}}-0.5}{\left[1+0.001\left({x}_{i}^{2}+{x}_{i+1}^{2}\right)\right]}$	[−100,100]
F19	Streched V Sine Wave	$f\left(x\right)=\sum\limits_{i=1}^{d-1}{\left({x}_{i}^{2}+{x}_{i+1}^{2}\right)}^{0.25}\left\{{\mathrm{sin}}^{2}\left[50{\left({x}_{i}^{2}+{x}_{i+1}^{2}\right)}^{0.1}\right]+0.1\right\}$	[−100,100]
F20	Venter and Sobiezcczanski-Sobieski's	$f\left(x\right)=\sum\limits_{i=1}^{d}\left[{x}_{i}^{2}-100{\mathrm{cos}}^{2}\left({x}_{i}\right)-100\mathrm{cos}\left(\frac{{x}_{i}^{2}}{30}\right)\right]+400$	[−100,100]
F21	Xin-She YANG 6	$f\left(x\right)=1+\left\{\left[\sum\limits_{i=1}^{d}{\mathrm{sin}}^{2}\left({x}_{i}\right)\right]-{e}^{{\sum }_{i=1}^{d}{x}_{i}^{2}}\right\}\cdot {e}^{-{\sum }_{i=1}^{d}{\mathrm{sin}}^{2}\left(\sqrt{\left\|{x}_{i}\right\|}\right)}$	[−100,100]

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3.2. Simulation platform

Simulation experiments would be carried out with a HP server platform whose assembly is shown in Table 6.

Table 6. Hardware for simulation experiments.

Items	Component
Server	HPE ProLiant DL380 Gen10
CPU	Intel Xeon Bronze 3106×2
RAM	Kingston 32GB

| Show Table

DownLoad: CSV

The source code was compiled with Matlab 2021b. And for simplicity, the maximum iteration number would be fixed 200, and the population size would be 40. In order to reduce the influence of random numbers involved in the algorithms, Monte Carlo simulation methods would be introduced and all of the results, if needed, would be the averaged over 30 separated runs.

For fair comparison, all the algorithms involved in this paper would be set the same with the above setup.

3.3. Probability parameters

For the proposed HAO algorithm, there would be three probability parameters ${p}_{1}, {p}_{2}, {p}_{3}$ . According to the source code of the AO algorithm provided by the proposer, ${p}_{1}$ might be a constant number near 2/3, and ${p}_{2} = {p}_{3} = 0.5$ . However, we do not know exactly whether it is suitable for the HAO algorithm. As a probability value, they all in a definitional domain fallen into [0, 1], and therefore, we simulate both ${p}_{1}-{p}_{2}$ and ${p}_{1}-{p}_{3}$ with the mean least iteration number (MLIN) when the best fitness values are smaller than 5e–4, which is 0.5‰, a constant number frequently used in engineering. Reducing the influence or randomness involved in the algorithm, Monte Carlo method is introduced and the final results are the average over 30 separated independent runs. Results were shown in Figures 1–6.

Figure 1.

${p}_{1}-{p}_{2}$ relationship for different benchmark functions.

Fcn	Items	HAO	AO	ALO	EO	GWO	HHO	PSO	SCA	WOA
F1	best	−8.881E−16	−8.881E−16	0	0	0	−8.881E−16	0	0	0
	worst	0	0	4.9594886	1.09246E−13	0.5672173	0	0.6567802	0.6151628	2.1770439
	median	0	0	0	0	0	0	0	0	0
	mean	−1.33227E-16	–1.33227E-16	0.349809815	3.77476E−15	0.002836087	−1.33227E-16	0.05238038	0.004490143	0.016635884
	std	3.17939E−16	3.17939E−16	0.930088506	1.14419E−14	0.04010832	3.17939E−16	0.145110562	0.045383772	0.173740625
F2	best	0	0	0	0	0	0	0	0	0
	worst	0	0	2.41267E−08	1.11022E−16	2.22045E−16	0	1.07827E−08	2.57222E−06	1.11022E−16
	median	0	0	0	0	0	0	0	0	0
	mean	0	0	6.71821E−10	5.55112E−19	1.72085E−17	0	2.07952E−10	8.93685E−08	2.77556E−18
	std	0	0	2.55674E−09	7.85046E−18	4.32465E−17	0	1.00964E−09	3.84951E−07	1.73768E−17
F3	best	0	0	0	0	0	0	0	0	0
	worst	1.71345E−07	1.3404E−06	0.002712803	5.09874E−41	4.89302E−38	5.38875E−29	1.99386E−09	1.27563E−07	2.83673E−32
	median	0	0	0	0	0	0	0	0	0
	mean	8.59349E−10	1.32862E−08	0.000188208	6.06182E−43	7.02152E−40	2.76018E−31	5.25788E−11	8.6424E−10	4.08456E−34
	std	1.21158E−08	1.1398E−07	0.000528951	4.92415E−42	4.29465E−39	3.81024E−30	2.43252E−10	9.32274E−09	2.80826E−33
F4	best	0	0	0	0	0	0	0	0	0
	worst	1.6596E−123	2.68981E−59	0.008148981	8.83296E−23	1.47557E−19	2.22569E−39	5.58975E−06	0.94986434	4.73958E−08
	median	0	0	0	0	0	0	0	0	0
	mean	8.5616E−126	1.51816E−61	0.000121248	6.57998E−25	3.60452E−21	2.39605E−41	1.39242E−07	0.014161434	2.64588E−10
	std	1.1736E−124	1.91108E−60	0.0006726	6.47525E−24	1.7605E−20	2.19965E−40	5.45541E−07	0.092392958	3.35656E−09
F5	best	0	0	0	0	0	0	0	0	0
	worst	1.5031E−130	7.21475E−47	1.93144E−05	1.46308E−27	7.13202E−23	1.88131E−43	4.29984E−08	0.006006323	3.6791E−31
	median	0	0	0	0	0	0	0	0	0
	mean	1.6376E−132	3.60737E−49	6.89004E−07	2.55882E−29	7.41054E−25	9.67108E−46	7.31406E−10	0.000112339	2.52022E−33
	std	1.4821E−131	5.1016E−48	2.31881E−06	1.52145E−28	5.24025E−24	1.33026E−44	3.82728E−09	0.000660868	2.62669E−32
F6	best	0	0	0	0	0	0	0	0	0
	worst	4.6398E−240	4.6215E−103	5.46088E−11	7.90852E−56	8.05404E−45	1.92346E−82	6.57684E−16	0.000149337	2.58423E−59
	median	0	0	0	0	0	0	0	0	0
	mean	2.3199E−242	2.3108E−105	5.83399E−13	1.15805E−57	5.01269E−47	9.61742E−85	5.3483E−18	1.64084E−06	1.29331E−61
	std	0	3.2679E−104	4.19661E−12	7.23204E−57	5.72004E−46	1.36009E−83	4.83976E−17	1.30259E−05	1.82732E−60
F7	best	0	0	0	0	0	0	0	0	0
	worst	0	2.4556E−128	6.67468E−14	8.57806E−67	1.61691E−49	6.2895E−127	7.41427E−15	0.178898289	1.39788E−45
	median	0	0	0	0	0	0	0	0	0
	mean	0	1.2278E−130	6.32863E−16	8.40608E−69	9.51101E−52	3.1748E−129	4.02523E−17	0.000992967	8.0714E−48
	std	0	1.7364E−129	5.20692E−15	7.45379E−68	1.14851E−50	4.4473E−128	5.24803E−16	0.012706253	9.97234E−47
F8	best	0	0	0	0	0	0	0	0	0
	worst	1.7671E−253	6.1768E−100	8.61175E−10	1.23148E−46	3.58526E−38	2.11815E−84	5.42758E−12	0.014168893	1.41375E−46
	median	0	0	0	0	0	0	0	0	0
	mean	9.0354E−256	3.0884E−102	1.17105E−11	8.13848E−49	4.38731E−40	1.07815E−86	4.08357E−14	0.000180906	1.02521E−48
	std	0	4.3677E−101	7.01814E−11	8.95168E−48	3.35327E−39	1.4978E−85	4.06497E−13	0.001412517	1.06371E−47
F9	best	0	0	0	0	0	0	0	0	0
	worst	6.5633E−126	1.55672E−42	1902.684424	3.14469E−26	9.69724E−22	7.89724E−42	3.18004E−07	0.460748677	2.51842E−29
	median	0	0	0	0	0	0	0	0	0
	mean	3.2924E−128	7.78367E−45	85.65268113	3.44888E−28	1.70161E−23	6.59982E−44	9.55701E−09	0.005258053	2.9233E−31
	std	4.6409E−127	1.10077E−43	285.1771895	2.38927E−27	9.31255E−23	6.54555E−43	3.77459E−08	0.03811698	2.5359E−30
F10	best	0	0	0	0	0	0	0	0	0
	worst	5.3821E−243	2.1822E−116	5.07135E−10	1.35718E−47	2.7511E−38	1.39152E−81	1.69102E−13	0.029115168	2.47058E−46
	median	0	0	0	0	0	0	0	0	0
	mean	2.6911E−245	1.0939E−118	3.41545E−12	7.03595E−50	1.61158E−40	6.95764E−84	2.66075E−15	0.00027155	1.28746E−48
	std	0	1.543E−117	3.63283E−11	9.59781E−49	1.95278E−39	9.83952E−83	1.6196E−14	0.002562231	1.74734E−47

Algorithm	T_s(x₁)	T_h(x₂)	R(x₃)	L(x₄)	f(x)
GSA ^[32]	1.125	0.625	55.9886598	84.4542025	8538.8359
MVO ^[32]	0.8125	0.4375	42.090738	176.73869	6060.8066
WOA ^[30]	0.812500	0.437500	42.0982699	176.638998	6059.7410
BA ^[14]	0.812500	0.437500	42.098445	176.636595	6059.7143
GWO ^[26]	0.8125	0.4345	42.089181	176.758731	6051.5639
AOA ^[33]	0.8303737	0.4162057	42.75127	169.3454	6048.7844
AO ^[19]	1.0540	0.182806	59.6219	38.8050	5949.2258
HAO	0.810726461	0.400897167	42.16466765	175.8460143	5935.56831

Algorithm	${x}_{1}$	${x}_{2}$	$f\left(x\right)$
AVOA ^[23]	0.788680394972034	0.408233412073586	263.895843396802
GOA ^[25]	0.788897555578973	0.407619570115153	263.895881496069
MBA ^[34]	0.7885650	0.4085597	263.8958522
SSA ^[35]	0.788665414	0.408275784444547	263.895843
PSO-DE ^[36]	0.7886751	0.4082482	263.8958433
DEDS ^[36]	0.78867513	0.40824828	263.8958434
MFO ^[28]	0.788244771	0.409466905784741	263.8959797
MVO ^[32]	0.78860276	0.408453070000000	263.8958499
CS ^[37]	0.78867	0.40902	263.9716
Tsai ^[38]	0.788	0.408	263.68
AOA ^[33]	0.79369	0.39426	263.9154
AO ^[19]	0.7926	0.3966	263.8684
HAO	0.788609979	0.408362638	263.8916603

[1]	D. E. Goldberg, Genetic Algorithms in Search, Optimization, and Machine Learning. Addison-Wesley, 1989.
[2]	M. Dorigo, V. Maniezzo, A. Colorni, Ant system: optimization by a colony of cooperating agents, IEEE Trans. Syst. Man Cybern. B Cybern., 26 (1996), 29–41. https://doi.org/10.1109/3477.484436 doi: 10.1109/3477.484436
[3]	R. Eberhart, J. Kennedy, A new optimizer using particle swarm theory, in MHS'95. Proceedings of the Sixth International Symposium on Micro Machine and Human Science, (1995), 39–43. https://doi.org/10.1109/MHS.1995.494215
[4]	M. Clerc, J. Kennedy, The particle swarm - explosion, stability, and convergence in a multidimensional complex space, IEEE Trans. Evolut. Comput., 6 (2002), 58–73. https://doi.org/10.1109/4235.985692 doi: 10.1109/4235.985692
[5]	R. C. Eberhart, Y. Shi, Comparing inertia weights and constriction factors in particle swarm optimization, in Proceedings of the 2000 Congress on Evolutionary Computation. CEC00 (Cat. No.00TH8512), 1 (2000), 16–19. https://doi.org/10.1109/CEC.2000.870279
[6]	M. E. H. Pedersen, A. J. Chipperfield, Simplifying Particle Swarm Optimization, Appl. Soft Comput., 10 (2010), 618–628. https://doi.org/10.1016/j.asoc.2009.08.029 doi: 10.1016/j.asoc.2009.08.029
[7]	G. I. Evers, M. B. Ghalia, Regrouping particle swarm optimization: A new global optimization algorithm with improved performance consistency across benchmarks, in 2009 IEEE International Conference on Systems, Man and Cybernetics, (2009), 3901–3908. https://doi.org/10.1109/ICSMC.2009.5346625
[8]	F. v. d. Bergh, A. P. Engelbrecht, A new locally convergent particle swarm optimiser, in IEEE International Conference on Systems, Man and Cybernetics, 3 (2002). https://doi.org/10.1109/ICSMC.2002.1176018.
[9]	T. Xiang, X. Liao, K. W. Wong, An improved particle swarm optimization algorithm combined with piecewise linear chaotic map, Appl. Math. Comput., 190 (2007), 1637–1645. https://doi.org/10.1016/j.amc.2007.02.103 doi: 10.1016/j.amc.2007.02.103
[10]	H. Haklı, H. Uğuz, A novel particle swarm optimization algorithm with Levy flight, Appl. Soft Comput., 23 (2014), 333–345. http://dx.doi.org/10.1016/j.asoc.2014.06.034 doi: 10.1016/j.asoc.2014.06.034
[11]	H. Garg, A hybrid PSO-GA algorithm for constrained optimization problems, Appl. Math. Comput., 274 (2016), 292–305. https://doi.org/10.1016/j.amc.2015.11.001 doi: 10.1016/j.amc.2015.11.001
[12]	N. Holden, A. A. Freitas, A hybrid PSO/ACO algorithm for discovering classification rules in data mining, J.Artif. Evolut. Appl., (2008), 316145. https://doi.org/10.1155/2008/316145 doi: 10.1155/2008/316145
[13]	A. P. Engelbrecht, Heterogeneous particle swarm optimization, in Swarm Intelligence (eds. M. Dorigo et al.), Springer Berlin Heidelberg, (2010), 191–202.
[14]	X. S. Yang, A New Metaheuristic Bat-Inspired Algorithm, in Nature Inspired Cooperative Strategies for Optimization (NICSO 2010), Springer Berlin Heidelberg, (2010), 65–74.
[15]	Z. M. Gao, J. Zhao, X. R. Li, Y. R. Hu, An improved sine cosine algorithm with multiple updating ways for individuals, J. Phys. Conf. Ser., 1678 (2020), 012079. https://doi.org/10.1088/17426596/1678/1/012079 doi: 10.1088/17426596/1678/1/012079
[16]	J. Zhao, Z. M. Gao, An improved grey wolf optimization algorithm with multiple tunnels for updating, J. Phys. Conf. Ser., 1678 (2020), 012096. https://doi.org/10.1088/17426596/1678/1/012096 doi: 10.1088/17426596/1678/1/012096
[17]	S. Mirjalili, SCA: A Sine Cosine algorithm for solving optimization problems, Knowl. Based Syst., 96 (2016), 120–133. https://doi.org/10.1016/j.knosys.2015.12.022 doi: 10.1016/j.knosys.2015.12.022
[18]	L. Abualigah, A. Diabat, S. Mirjalili, M. Abd Elaziz, A. H. Gandomi, The arithmetic optimization algorithm, Comput. Methods Appl. Mech. Eng., 376 (2021), 113609. https://doi.org/10.1016/j.cma.2020.113609 doi: 10.1016/j.cma.2020.113609
[19]	L. Abualigaha, D. Yousrib, M. A. Elazizc, A. A. Eweesd, M. A. A. Al-qanesse, A. H. Gandomif, Aquila optimizer: a novel meta-heuristic optimization algorithm, Comput. Indust. Eng., 157 (2021), 107250. https://doi.org/10.1016/j.cie.2021.107250 doi: 10.1016/j.cie.2021.107250
[20]	A. Fatani, A. Dahou, M. A. A. Al-qaness, S. Lu, M. Abd Elaziz, Advanced feature extraction and selection approach using deep learning and Aquila optimizer for IoT intrusion detection system, Sensors, 22 (2022), 140. https://doi.org/10.3390/s22010140 doi: 10.3390/s22010140
[21]	A. M. AlRassas, M. A. A. Al-qaness, A. A. Ewees, S. Ren, M. Abd Elaziz, Optimized ANFIS model using Aquila optimizer for oil production forecasting, Processes, 9 (2021). https://doi.org/10.3390/pr9071194 doi: 10.3390/pr9071194
[22]	S. Mirjalili, The ant lion optimizer, Adv. Eng. Soft., 83 (2015), 80–98. http://dx.doi.org/10.1016/j.advengsoft.2015.01.010 doi: 10.1016/j.advengsoft.2015.01.010
[23]	B. Abdollahzadeh, F. S. Gharehchopogh, S. Mirjalili, African vultures optimization algorithm: A new nature-inspired metaheuristic algorithm for global optimization problems, Comput. Indust. Eng., 158 (2021), 107408. https://doi.org/10.1016/j.cie.2021.107408 doi: 10.1016/j.cie.2021.107408
[24]	A. Faramarzi, M. Heidarinejad, B. Stephens, S. Mirjalili, Equilibrium optimizer: A novel optimization algorithm, Knowl. Based Syst., (2019), 105190. https://doi.org/10.1016/j.knosys.2019.105190 doi: 10.1016/j.knosys.2019.105190
[25]	S. Saremi, S. Mirjalili, A. Lewis, Grasshopper optimisation algorithm: theory and application, Adv. Eng. Soft., 105 (2017), 30–47. https://doi.org/10.1016/j.advengsoft.2017.01.004 doi: 10.1016/j.advengsoft.2017.01.004
[26]	S. Mirjalili, S. M. Mirjalili, A. Lewis, Grey wolf optimizer, " Adv. Eng. Soft., 69 (2014), 46–61. http://dx.doi.org/10.1016/j.advengsoft.2013.12.007 doi: 10.1016/j.advengsoft.2013.12.007
[27]	A. A. Heidari, S. Mirjalili, H. Faris, I. Aljarah, M. Mafarja, H. Chen, Harris hawks optimization: Algorithm and applications, Future Gener. Comp. Syst., 2019. https://doi.org/10.1016/j.future.2019.02.028 doi: 10.1016/j.future.2019.02.028
[28]	S. Mirjalili, Moth-flame optimization algorithm: A novel nature-inspired heuristic paradigm, Knowl. based syst., 89 (2015), 228. https://doi.org/10.1016/j.knosys.2015.07.006 doi: 10.1016/j.knosys.2015.07.006
[29]	K. Zervoudakis, S. Tsafarakis, A mayfly optimization algorithm, Comput. Indust. Eng., 145 (2020), 106559. https://doi.org/10.1016/j.cie.2020.106559 doi: 10.1016/j.cie.2020.106559
[30]	S. Mirjalili, A. Lewis, The whale optimization algorithm, Adv. Eng. Softw., 95 (2016), 51–67. https://doi.org/10.1016/j.advengsoft.2016.01.008 doi: 10.1016/j.advengsoft.2016.01.008
[31]	Z. M. Gao, J. Zhao, Benchmark functions with Python, Golden Light Academic Publishing, (2020), 3–5.
[32]	S. Mirjalili, S. M. Mirjalili, A. Hatamlou, Multi-Verse Optimizer: a nature-inspired algorithm for global optimization, Neural Comput. Appl., 27 (2016), 495–513. https://doi.org/10.1007/s00521-015-1870-7 doi: 10.1007/s00521-015-1870-7
[33]	A. D. Laith Abualigah, S. Mirjalilid, M. Abd Elazizf, A. H. Gandomih, The Arithmetic Optimization Algorithm, Comput. Methods Appl. Mech. Eng., 376 (2021), 113609. https://doi.org/10.1016/j.cma.2020.113609 doi: 10.1016/j.cma.2020.113609
[34]	A. Sadollah, A. Bahreininejad, H. Eskandar, M. Hamdi, Mine blast algorithm: A new population based algorithm for solving constrained engineering optimization problems, Appl. Soft Comput., 13 (2013), 2592–2612. https://doi.org/10.1016/j.asoc.2012.11.026 doi: 10.1016/j.asoc.2012.11.026
[35]	S. Mirjalili, A. H. Gandomi, S. Z. Mirjalili, S. Saremi, H. Faris, S. M. Mirjalili, Salp swarm algorithm: A bio-inspired optimizer for engineering design problems, Adv. Eng. Softw., 114 (2017), 163–191. https://doi.org/10.1016/j.advengsoft.2017.07.002 doi: 10.1016/j.advengsoft.2017.07.002
[36]	M. Zhang, W. Luo, X. Wang, Differential evolution with dynamic stochastic selection for constrained optimization, Inform. Sci., 178 (2008), 3043–3074. https://doi.org/10.1016/j.ins.2008.02.014 doi: 10.1016/j.ins.2008.02.014
[37]	V. Bhargava, S. E. K. Fateen, A. Bonilla-Petriciolet, Cuckoo Search: A new nature-inspired optimization method for phase equilibrium calculations, Fluid Phase Equilibr., 337 (2013), 191–200. http://dx.doi.org/10.1016/j.fluid.2012.09.018 doi: 10.1016/j.fluid.2012.09.018
[38]	J. F. Tsai, Global optimization of nonlinear fractional programming problems in engineering design, Eng. Optimiz., 37 (2005), 399–409. https://doi.org/10.1080/03052150500066737 doi: 10.1080/03052150500066737
[39]	A. Faramarzi, M. Heidarinejad, S. Mirjalili, A. H.Gandomic, Marine predators algorithm: A nature-inspired metaheuristic, Expert Syst. Appl., 152 (2020), 113377. https://doi.org/10.1016/j.eswa.2020.113377 doi: 10.1016/j.eswa.2020.113377
[40]	J. M. Czerniak, H. Zarzycki, D. Ewald, AAO as a new strategy in modeling and simulation of constructional problems optimization, Simul. Model. Pract. Theory, 76 (2017), 22–33. https://doi.org/10.1016/j.simpat.2017.04.001 doi: 10.1016/j.simpat.2017.04.001
[41]	S. Li, H. Chen, M. Wang, A. A. Heidari, S. Mirjalili, Slime mould algorithm: A new method for stochastic optimization, Future Gener. Comput. Syst., 111 (2020), 300–323. https://doi.org/10.1016/j.future.2020.03.055 doi: 10.1016/j.future.2020.03.055

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Fcn	Items	HAO	AO	ALO	EO	GWO	HHO	PSO	SCA	WOA
F11	best	0	0	0	0	0	0	0	0	0
	worst	0.000102444	0.000212601	0.686498568	4.1191E−08	0.000466289	0	0.000971432	0.06394849	0.940399825
	median	0	0	0	0	0	0	0	0	0
	mean	5.12222E−07	1.40426E−06	0.035056268	2.34813E−10	2.64455E−05	0	1.42278E−05	0.000620885	0.03160676
	std	7.24392E−06	1.5517E−05	0.117127269	2.92632E−09	9.19686E−05	0	7.93889E−05	0.004755222	0.149338695
F12	best	0	0	0	0	0	0	0	0	0
	worst	0	0	0.591137009	0	3.33067E−16	0	0.147784252	8.23936E−06	3.33067E−16
	median	0	0	0	0	0	0	0	0	0
	mean	0	0	0.039901749	0	1.05471E−17	0	0.000738941	8.4255E−08	4.996E−18
	std	0	0	0.117323503	0	4.92719E−17	0	0.010449923	6.21508E−07	3.57161E−17
F13	best	–9	–9	–8.271743663	–9	–9	–9	–8.763739656	–8.999963768	–9
	worst	0	0	0	0	0	0	0	0	0
	median	0	0	0	0	0	0	0	0	0
	mean	–1.35	–1.35	–1.018132954	–1.332689027	–1.293365437	–1.35	–1.213239065	–1.24248688	–1.274377431
	std	3.22	3.22	2.465359483	3.180884043	3.091366535	3.22	2.898767185	2.991706983	3.072957522
F14	best	0	0	0	0	0	0	0	0	0
	worst	0	0	6.553034763	3.031861151	3.258463936	0	3.095329066	3.857462194	5.252117521
	median	0	0	0	0	0	0	0	0	0
	mean	0	0	0.619407638	0.234314999	0.234699015	0	0.315975027	0.135451624	0.098063259
	std	0	0	1.549603278	0.669642237	0.672165615	0	0.777237848	0.606556869	0.628334407
F15	best	0	0	0	0	0	0	0	0	0
	worst	0	0	3.999948399	2.552321449	2.647696583	0	3.501379209	3.745287122	3.999426852
	median	0	0	0	0	0	0	0	0	0
	mean	0	0	0.504369	0.282061586	0.30369492	0	0.401829418	0.376056186	0.217401183
	std	0	0	1.214378287	0.699697313	0.740864965	0	0.973607626	0.916560706	0.705258618
F16	best	0	0	0	0	0	0	0	0	0
	worst	0	0	55.71738174	4.974825684	8.97399354	0	19.36820549	35.47861084	36.42076237
	median	0	0	0	0	0	0	0	0	0
	mean	0	0	3.517170973	0.034827673	0.413835634	0	1.334837167	0.758800881	1.081133301
	std	0	0	9.2891052	0.36483464	1.481162818	0	3.446218849	3.868735337	5.511231645
F17	best	0	0	0	0	0	0	0	0	0
	worst	9.33509E−56	5.88658E−25	1.399873346	0.09987335	0.199873353	1.92798E−21	0.199874464	0.300008748	0.299873346
	median	0	0	0	0	0	0	0	0	0
	mean	4.67551E−58	3.3774E−27	0.059481002	0.014981002	0.016981002	1.79513E−23	0.021590163	0.020570275	0.018484168
	std	6.60086E−57	4.17777E−26	0.16893679	0.035751408	0.042619526	1.75014E−22	0.054933573	0.055290725	0.053082389
F18	best	0	0	0	0	0	0	0	0	0
	worst	0	0	0.881248897	0.451552733	1.110889767	0	0.475219164	1.343309798	1.14382478
	median	0	0	0	0	0	0	0	0	0
	mean	0	0	0.044907776	0.029224494	0.038940139	0	0.034195028	0.133216919	0.093117857

Fcn	HAO	AO	ALO	EO	GWO	HHO	PSO	SCA	WOA
F1	NA	NA	6.38644E−05	6.24874E−05	6.38644E−05	NA	6.38644E−05	6.38644E−05	5.93632E−05
F2	NA	NA	6.38644E−05	0.368120251	1.59379E−05	NA	6.38644E−05	6.38644E−05	0.368120251
F3	NA	0.001706249	0.000182672	0.212293836	0.472675594	0.427355314	0.000182672	0.00058284	0.427355314
F4	NA	0.000182672	0.000182672	0.000182672	0.000182672	0.000182672	0.000182672	0.000182672	0.000182672
F5	NA	0.000182672	0.000182672	0.000182672	0.000182672	0.000182672	0.000182672	0.000182672	0.000182672
F6	NA	0.000182672	0.000182672	0.000182672	0.000182672	0.000182672	0.000182672	0.000182672	0.000182672
F7	NA	6.38644E−05	6.38644E−05	6.38644E−05	6.38644E−05	6.38644E−05	6.38644E−05	6.38644E−05	6.38644E−05
F8	NA	0.000182672	0.000182672	0.000182672	0.000182672	0.000182672	0.000182672	0.000182672	0.000182672
F9	NA	0.000182672	0.000182672	0.000182672	0.000182672	0.000182672	0.000182672	0.000182672	0.000182672
F10	NA	0.000182672	0.000182672	0.000182672	0.000182672	0.000182672	0.000182672	0.000182672	0.000182672
F11	NA	0.584248553	8.74499E−05	0.001699468	0.00016688	0.368120251	0.000565815	8.74499E−05	0.000533651
F12	NA	NA	6.38644E−05	NA	NA	NA	6.38644E−05	6.38644E−05	0.168078319
F13	NA	NA	6.38644E−05	0.002212542	6.38644E−05	NA	6.38644E−05	6.38644E−05	0.000751179
F14	NA	NA	6.38644E−05	0.000231246	6.38644E−05	NA	6.38644E−05	6.38644E−05	0.03484304
F15	NA	NA	6.38644E−05	6.38644E−05	6.38644E−05	NA	6.38644E−05	6.38644E−05	0.002212542
F16	NA	NA	6.38644E−05	0.168078319	6.34029E−05	NA	6.38644E−05	6.38644E−05	0.005858055
F17	NA	0.000439639	0.000182672	0.000182672	0.000182672	0.000182672	0.000182672	0.000182672	0.000182672
F18	NA	NA	6.38644E−05	6.38644E−05	6.38644E−05	NA	6.38644E−05	6.38644E−05	6.38644E−05
F19	NA	0.000182672	0.025692445	0.000182672	0.000182672	0.000182672	0.000182672	0.000182672	0.000182672
F20	NA	NA	6.38644E−05	NA	0.368120251	NA	6.38644E−05	6.38644E−05	NA
F21	NA	0.368120251	6.38644E−05	6.38644E−05	6.38644E−05	NA	6.38644E−05	6.38644E−05	6.38644E−05

Algorithm	h	L	t	b	f(x)
GSA ^[32]	0.182129	3.856979	10.000	0.202376	1.87995
WOA ^[30]	0.205396	3.484293	9.037426	0.206276	1.730499
MVO ^[32]	0.205463	3.473193	9.044502	0.205695	1.72645
SSA ^[35]	0.2057	3.4714	9.0366	0.2057	1.72491
MPA ^[39]	0.205728	3.470509	9.036624	0.205730	1.724853
AVOA ^[23]	0.205730	3.470474	9.036621	0.205730	1.724852
AAO ^[40]	0.2057	3.4705	9.0366	0.2057	1.724
AOA ^[33]	0.194475	2.57092	10.000	0.201827	1.7164
SMA ^[41]	0.2054	3.2589	9.0384	0.2058	1.69604
AO ^[19]	0.1631	3.3652	9.0202	0.2067	1.6566
HAO	0.19952608	3.384869727	9.064048595	0.206681757	1.715727482

Mathematical Biosciences and Engineering

The heterogeneous Aquila optimization algorithm

Related Papers:

Abstract

1. Introduction

2. The AO and proposed HAO algorithms

2.1. Strategies in the AO algorithm

2.2. Proposed HAO improved algorithm

3. Simulation experiments on benchmark functions

3.1. Benchmark functions

3.2. Simulation platform

3.3. Probability parameters

3.4. Qualitative experiments

3.5. Intensification capability experiments

3.6. Diversification capability experiments

3.7. Acceleration convergence analysis

3.8. Scalability experiments

3.9. Wilcoxon rank sum test

4. Simulation experiments on real-world engineering problems

4.1. Pressure vessel design

4.2. Three-bar truss design

4.3. Welded beam design

5. Discussion and conclusions

Acknowledgements

Conflict of interest

References

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