基于信噪比的学习型哈里斯鹰优化算法

张林; 沈佳颖; 胡传陆; 朱东林

doi:10.13700/j.bh.1001-5965.2023.0433

基于信噪比的学习型哈里斯鹰优化算法

doi: 10.13700/j.bh.1001-5965.2023.0433

1.
江西理工大学软件工程学院，南昌 330000
2.
浙江师范大学计算机科学与技术学院，金华 321004

基金项目:

国家自然科学基金(62272418, 62002046)

详细信息

通讯作者:
E-mail：2078570674@qq.com

中图分类号: TP301.6
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- 被引次数: 0
出版历程
- 收稿日期: 2023-07-03
- 录用日期: 2023-11-27
- 网络出版日期: 2024-02-04
- 整期出版日期: 2025-07-31

Learning Harris Hawks optimization algorithm with signal-to-noise ratio

1.
School of Software Engineering，Jiangxi University of Science and Technology，Nanchang 330000，China
2.
School of Computer Science and Technology，Zhejiang Normal University，Jinhua 321004，China

Funds:

National Natural Science Foundation of China (62272418, 62002046)

More Information

Corresponding author: E-mail：2078570674@qq.com

摘要

摘要:
针对哈里斯鹰优化（HHO）算法存在种群学习性与适应性不足的问题，提出一种基于信噪比的学习型哈里斯鹰优化（SLHHO）算法。该算法通过引入信噪比的概念来判断个体的位置信息，设计了一种协调学习策略，可以更合理地更新种群内个体的位置，进而对逃逸距离重新设计，提升了算法的适应与寻优能力。以12个基准函数为标准，将所提算法与哈里斯鹰算法的变体及其他算法进行性能测试，并在时间复杂度、多样性、探索与开发等评价指标中进行对比分析，结果显示，SLHHO算法具有较强的竞争力与可行性，在压力容器设计问题中，验证了SLHHO算法的实用性。
- 哈里斯鹰优化 /
- 信噪比 /
- 协调学习 /
- 逃逸距离 /
- 基准函数
Abstract:
Aiming at the problem of insufficient population learning and adaptability of the Harris hawks optimization (HHO) algorithm, this paper proposes a learning Harris hawks optimization based on the signal-to-noise ratio（SLHHO）algorithm. By using the signal-to-noise ratio as a metric to assess individual position information, the algorithm creates a coordinated learning strategy that can more realistically update the positions of individuals within the population. It then reworks the escape distance to enhance the algorithm’s capacity for adaptation and optimization seeking. With 12 benchmark functions as the standard, the proposed algorithm was tested for its performance with the variants of the Harris hawk algorithm and other algorithms, and compared and analyzed in the evaluation indexes such as time complexity, diversity, exploration and development, etc. The results show that SLHHO algorithm is highly competitive and feasible, and finally, its practicability is verified in the pressure vessel design problem.
- Harris hawks optimization /
- signal to noise ratio /
- coordinated learning /
- escape distance /
- benchmark function

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图 1 HHO算法中迭代变化的J

Figure 1. Iterative change of J in HHO algorithm

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图 2 SLHHO算法中迭代变化的J

Figure 2. Iterative change of J in SLHHO algorithm

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图 3 协调学习原理

Figure 3. Coordinated learning schematic

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图 4 各算法平均收敛图

Figure 4. Average convergence plot of each algorithm

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图 5 HHO算法与SLHHO算法的探索与开发分布

Figure 5. Distribution of exploration and development of HHO algorithm and SLHHO algorithm

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图 6 HHO算法与SLHHO算法的多样性分布

Figure 6. Diversity distribution of HHO algorithm and SLHHO algorithm

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表 1 各函数信息

Table 1. Information for each function

函数	区间	最小值
$ {f}_{1}\left(x\right)=\displaystyle\sum_{i=1}^{n}{x}_{i}^{2} $	[−100,100]	0
$ {f}_{2}\left(x\right)=\displaystyle\sum_{i=1}^{n}\left\|{x}_{i}\right\|+\prod _{i=1}^{n}\left\|{x}_{i}\right\| $	[−10,10]	0
$ {f}_{3}\left(x\right)=\displaystyle\sum_{i=1}^{n}\left(\displaystyle\sum_{j=1}^{i}{x}_{j}\right)^{2} $	[−100,100]	0
${f}_{4}\left(x\right)={\rm{max} }_{i}\left\{\left\|{x}_{i}\right\|,1 \leqslant i \leqslant n\right\}$	[−100,100]	0
$ {f}_{5}\left(x\right)=\displaystyle\sum _{i=1}^{n-1}[100{\left({x}_{i+1}-{x}_{i}^{2}\right)}^{2}+{({x}_{i}-1)}^{2}] $	[−30,30]	0
$ {f}_{6}\left(x\right)=\displaystyle\sum _{i=1}^{n}{\left( {x}_{i}+0.5 \right)}^{2} $	[−100,100]	0
${f}_{7}\left(x\right)=\displaystyle\sum _{i=1}^{n}{ix}_{i}^{4}+{\rm{random}}\left[\mathrm{0,1}\right)$	[−1.28,1.28]	0
$ {f}_{8}\left(x\right)=\displaystyle\sum _{i=1}^{n}-{x}_{i}\mathrm{s}\mathrm{i}\mathrm{n}\left(\sqrt{\left\|{x}_{i}\right\|}\right) $	[−500,500]	−418.9829n
${f}_{9}\left(x\right)=\displaystyle\sum _{i=1}^{n}({x}_{i}^{2}-10\mathrm{cos}\left(2 \text{π} {x}_{i}\right)+10)$	[−5.12,5.12]	0
${f}_{10}\left(x\right)=-20\mathrm{exp}\left(-0.2\sqrt{\dfrac{1}{n}\sum\limits_{i = 1}^n {x}_{i}^{2} }\right)-\mathrm{e}\mathrm{x}\mathrm{p}\left(\dfrac{1}{n}\sum\limits_{i = 1}^n \mathrm{cos}\left(2 \text{π} {x}_{i}\right)\right)+20+{\rm{e}}$	[−32,32]	0
$ {f}_{11}\left(x\right)=\dfrac{1}{4\;000}\displaystyle\sum _{i=1}^{n}{x}^{2}-\prod _{i=1}^{n}\mathrm{c}\mathrm{o}\mathrm{s}\left(\dfrac{{x}_{i}}{\sqrt{i}}\right) $	[−600,600]	0
$\left\{\begin{gathered}{f}_{12}\left(x\right)=\dfrac{ \text{π} }{n}\left[10\mathrm{sin}\left( \text{π} {y}_{1}\right)+\displaystyle\sum _{i=1}^{n-1}{\left({y}_{i}-1\right)}^{2}\left(1+10{\mathrm{sin} }^{2}\left( \text{π} {y}_{i+1}\right)\right)+{\left({y}_{n}-1\right)}^{2}\right]+\displaystyle\sum _{i=1}^{n}u\left({x}_{i},10, \mathrm{100,4}\right) \\{y}_{i}=1+\dfrac{{x}_{i}+1}{4} \\ u\left({x}_{i},a,k,m\right)=\left\{\begin{array}{*{20}{l}}k{\left({x}_{i}-a\right)}^{m} & {x}_{i} > a\\ 0 & -a < {x}_{i} < a\\ k{\left(-{x}_{i}-a\right)}^{m} & {x}_{i} < -a\end{array}\right. \\\end{gathered}\right. $	$ [-\mathrm{50,50}] $	0

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表 2 各算法需设置的内部参数

Table 2. Internal parameters to be set for each algorithm

算法	参数
ATOA^[26]	f_min=0.2，f_max=1.0，α=5，$ \mu $=0.499，$ \varepsilon $=2.2204$ \times {10}^{-16} $
MFPA^[30]	n=50, P=0.8, γ₁=1, γ₂=3, λ=1.5, cloning array = [9,8,7,6,5,4,3,2,1,1,1,1,1,1]
PSOGSA^[31]	w=rand(1)，c₁=0.5，c₂=1.5，G₀=1，α=20
TACPSO^[32]	W_max=0.9，W_min=0.4，c₁=c₂=2

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表 3 SLHHO算法与其他算法的变体

Table 3. SLHHO algorithm and other algorithm variants

算法	f_min
算法	f₁	f₂	f₃	f₄	f₅	f₆	f₇	f₈	f₉	f₁₀	f₁₁	f₁₂
SLHHO	0	8.60×10⁻²³¹	4.89×10⁻¹⁰⁰	1.25×10⁻⁵⁹	0	0	2.82×10⁻⁶	−1.26×10⁴	0	8.88×10⁻¹⁶	0	3.58×10⁻¹⁹
ATOA^[26]	5.59×10⁻¹²	1.18×10⁻³⁹	4.12×10⁻³	2.79×10⁻³	2.88×10⁺¹	5.56×10⁻¹	6.13×10⁻⁴	−6.67×10³	0	3.97×10⁻¹⁰	0	2.53×10⁻¹
DAOA^[27]	4.21×10⁴	0	0	7.57×10¹	7.36×10³	4.61×10⁴	3.49×10⁻²	−4.03×10³	3.57×10¹	8.88×10⁻¹⁶	9.19×10¹	1.85×10⁸
EJAYA^[28]	6.17×10⁻⁶	1.52×10⁻³	1.57	1.27×10⁻¹	2.31×10¹	1.37×10⁻⁵	6.34×10⁻³	−7.30×10³	8.05	3.06×10⁻³	1.48×10⁻⁵	3.39×10⁻⁷
IGWO^[29]	6.14×10⁻³⁴	1.85×10⁻²¹	3.52×10⁻⁵	8.41×10⁻⁹	2.51×10¹	3.93×10⁻²	6.78×10⁻⁴	−7.55×10³	5.68	1.15×10⁻¹⁴	0	1.12×10⁻³
MFPA^[30]	1.27	9.02×10⁻⁴	3.84×10²	8.28	2.20×10¹	2.69	7.71×10⁻²	−1.08×10⁴	2.15×10¹	2.57	8.57×10⁻¹	7.38×10⁻¹
PSOGSA^[31]	4.88×10⁻¹⁹	3.05×10⁻⁹	1.08×10³	1.85×10¹	1.87×10¹	4.84×10⁻¹⁹	1.89×10⁻²	−9.76×10³	7.86×10¹	5.80×10⁻¹⁰	7.94×10⁻³	3.10×10⁻²
TACPSO^[32]	1.59×10⁻⁸	8.43×10⁻⁵	2.02	6.52×10⁻¹	2.78	2.33×10⁻⁹	5.95×10⁻³	−1.04×10⁴	2.89×10¹	1.04×10⁻⁴	4.90×10⁻⁹	4.05×10⁻⁸
HHO^[9]	3.64×10⁻⁴⁴	3.02×10⁻¹⁹	3.15×10⁻³²	2.08×10⁻²⁰	1.14×10⁻⁵	4.97×10⁻⁹	2.58×10⁻⁶	−1.26×10⁴	0	8.88×10⁻¹⁶	0	1.65×10⁻⁸

算法	f_ave
算法	f₁	f₂	f₃	f₄	f₅	f₆	f₇	f₈	f₉	f₁₀	f₁₁	f₁₂
SLHHO	0	8.85×10⁻⁵⁸	5.45×10⁻⁸²	1.95×10⁻⁵⁰	1.66×10⁻⁷	6.05×10⁻¹²	4.34×10⁻⁵	−1.26×10⁴	0	8.88×10⁻¹⁶	0	1.48×10⁻⁸
ATOA^[26]	7.56×10⁻⁴	7.02×10⁻⁴	4.00×10²	3.47×10⁻²	2.92×10¹	8.05×10⁻¹	5.62×10⁻³	−5.34×10³	0	9.30×10⁻³	5.17×10⁻⁷	9.47×10⁻¹
DAOA^[27]	6.12×10⁴	4.23×10¹	7.82×10⁴	8.40×10¹	1.06×10⁴	5.95×10⁴	3.11×10¹	−2.75×10³	1.13×10²	1.08×10¹	5.13×10²	4.69×10⁸
EJAYA^[28]	7.52×10⁻⁵	6.34×10⁻³	9.76	4.80×10⁻¹	2.41×10⁺¹	9.62×10⁻⁵	1.48×10⁻²	−6.37×10³	5.00×10¹	1.64	1.70×10⁻²	1.03×10⁻¹
IGWO^[29]	1.04×10⁻³¹	8.36×10⁻²¹	4.15×10⁻³	4.58×10⁻⁷	2.55×10¹	8.72×10⁻²	2.14×10⁻³	−5.88×10³	6.45×10¹	1.76×10⁻¹⁴	6.55×10⁻³	3.80×10⁻³
MFPA^[30]	1.12×10¹	3.29×10⁻²	1.63×10³	1.56×10¹	2.99×10¹	6.86	1.54×10⁻¹	−8.20×10³	5.66×10¹	1.50×10¹	1.07	4.24
PSOGSA^[31]	2.02×10⁻¹	1.87×10⁻²	6.94×10³	3.56×10¹	2.44×10¹	3.37×10²	5.89×10⁻²	−8.32×10³	1.23×10²	1.02×10¹	9.21	4.50
TACPSO^[32]	6.55×10⁻⁵	4.87×10⁻³	4.29×10¹	2.02	3.33×10¹	8.81×10⁻⁵	1.88×10⁻²	−9.27×10³	5.82×10¹	7.01×10⁻¹	1.32×10⁻²	2.07×10⁻¹
HHO^[9]	1.44×10⁻²⁷	2.18×10⁻¹⁵	3.00×10⁻¹⁷	1.83×10⁻¹⁵	8.10×10⁻⁴	1.08×10⁻⁵	1.12×10⁻⁴	−1.26×10⁴	0	2.79×10⁻¹⁴	0	7.97×10⁻⁷

算法	f_std
算法	f₁	f₂	f₃	f₄	f₅	f₆	f₇	f₈	f₉	f₁₀	f₁₁	f₁₂
SLHHO	0	4.85×10⁻⁵⁷	2.22×10⁻⁸¹	9.12×10⁻⁵⁰	9.07×10⁻⁷	2.63×10⁻¹¹	4.05×10⁻⁵	2.48×10⁻²	0	1.00×10⁻³¹	0	7.42×10⁻⁸
ATOA^[26]	9.91×10⁻⁴	3.83×10⁻³	2.18×10⁺³	1.83×10⁻²	1.16	1.83×10⁻¹	4.12×10⁻³	5.50×10²	0	1.23×10⁻²	2.83×10⁻⁶	5.89×10⁻¹
DAOA^[27]	6.62×10³	1.35×10¹	3.41×10⁴	3.79	1.37×10³	5.42×10³	2.55×10¹	4.73×10²	3.60×10¹	9.62	1.03×10²	1.08×10⁸
EJAYA^[28]	9.26×10⁻⁵	4.82×10⁻³	9.07	2.16×10⁻¹	5.34×10⁻¹	7.18×10⁻⁵	4.96×10⁻³	6.31×10²	3.22×10¹	6.13×10⁻¹	3.75×10⁻²	1.41×10⁻¹
IGWO^[29]	1.84×10⁻³¹	7.11×10⁻²¹	9.53×10⁻³	3.44×10⁻⁷	1.97×10⁻¹	7.33×10⁻²	1.22×10⁻³	6.19×10²	4.94×10¹	3.51×10⁻¹⁵	1.30×10⁻²	2.34×10⁻³
MFPA^[30]	1.93×10¹	8.68×10⁻²	7.72×10²	4.10	7.96	4.40	4.36×10⁻²	1.12×10³	1.81×10¹	6.80	7.77×10⁻²	2.31
PSOGSA^[31]	5.74×10⁻¹	5.64×10⁻²	3.89×10³	1.96×10¹	1.92	1.84×10³	1.62×10⁻²	7.08×10²	2.83×10¹	6.78	2.75×10¹	3.25
TACPSO^[32]	3.28×10⁻⁴	4.94×10⁻³	4.02×10¹	1.32	2.33×10¹	4.33×10⁻⁴	7.82×10⁻³	6.19×10²	1.90×10¹	7.05×10⁻¹	1.44×10⁻²	2.79×10⁻¹
HHO^[9]	6.83×10⁻²⁷	4.05×10⁻¹⁵	1.64×10⁻¹⁶	4.48×10⁻¹⁵	1.13×10⁻³	1.82×10⁻⁵	1.45×10⁻⁴	1.75×10⁻¹	0	1.31×10⁻¹³	0	8.28×10⁻⁷

算法	f_cm
算法	f₁	f₂	f₃	f₄	f₅	f₆	f₇	f₈	f₉	f₁₀	f₁₁	f₁₂
SLHHO
ATOA^[26]	+	+	+	+	+	+	+	+	=	+	+	+
DAOA^[27]	+	+	+	+	+	+	+	+	+	+	+	+
EJAYA^[28]	+	+	+	+	+	+	+	+	+	+	+	+
IGWO^[29]	+	+	+	+	+	+	+	+	+	+	+	+
MFPA^[30]	+	+	+	+	+	+	+	+	+	+	+	+
PSOGSA^[31]	+	+	+	+	+	+	+	+	+	+	+	+
TACPSO^[32]	+	+	+	+	+	+	+	+	+	+	+	+
HHO^[9]	+	+	+	+	+	+	+	+	=	+	=	+

算法	f_rank
算法	f₁	f₂	f₃	f₄	f₅	f₆	f₇	f₈	f₉	f₁₀	f₁₁	f₁₂
SLHHO	1	1	1	1	1	1	1	1	1	1	1	1
ATOA^[26]	6	4	6	4	6	6	4	8	1	4	2	6
DAOA^[27]	9	9	9	9	9	9	9	9	6	8	8	9
EJAYA^[28]	5	6	4	5	3	4	5	6	2	6	5	4
IGWO^[29]	2	2	3	3	5	5	3	7	5	2	3	3
MFPA^[30]	8	8	7	7	7	7	8	5	3	9	6	7
PSOGSA^[31]	7	7	8	8	4	8	7	4	7	7	7	8
TACPSO^[32]	4	5	5	6	8	3	6	3	4	5	4	5
HHO^[9]	3	3	2	2	2	2	2	2	1	3	1	2

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表 4 SLHHO算法与HHO算法的变体（维度为30）

Table 4. SLHHO algorithm and HHO algorithm variants (dim=30)

算法	f_min
算法	f₁	f₂	f₃	f₄	f₅	f₆	f₇	f₈	f₉	f₁₀	f₁₁	f₁₂
SLHHO	0	0	4.89×10⁻¹⁰⁰	1.25×10⁻⁵⁹	0	0	2.82×10⁻⁶	−8.60	0	8.88×10⁻¹⁶	0	3.58×10⁻¹⁹
NCHHO^[33]	3.93×10⁻²⁴⁶	3.93×10⁻²⁴⁶	1.75×10⁻²²⁰	1.73×10⁻¹¹⁵	2.88×10¹	1.62	4.24×10⁻⁶	−6.80	0	8.88×10⁻¹⁶	0	4.56×10⁻²
LMHHO^[34]	3.07×10⁻⁴⁴	3.07×10⁻⁴⁴	3.01×10⁻⁴⁴	1.31×10⁻²⁵	7.35×10⁻⁹	1.33×10⁻¹⁰	5.00×10⁻⁷	−8.60	0	8.88×10⁻¹⁶	0	5.54×10⁻¹⁰
LHHO^[35]	3.14×10⁻⁷⁵	3.14×10⁻⁷⁵	3.09×10⁻⁷⁰	4.16×10⁻³⁹	2.27×10⁻⁶	1.10×10⁻¹⁰	1.23×10⁻⁵	−8.60	0	8.88×10⁻¹⁶	0	3.61×10⁻¹⁰
THHO^[10]	1.28×10⁻²⁴⁶	1.28×10⁻²⁴⁶	4.75×10⁻²⁴⁶	2.62×10⁻¹²⁷	7.20×10⁻⁸	2.95×10⁻¹³	1.48×10⁻⁶	−8.60	0	8.88×10⁻¹⁶	0	9.20×10⁻¹²
DAHHO^[20]	2.03×10⁻⁶⁰	2.03×10⁻⁶⁰	1.30×10⁻⁵⁰	1.03×10⁻³⁰	2.54×10⁻⁶	4.11×10⁻⁸	2.37×10⁻⁶	−8.60	0	8.88×10⁻¹⁶	0	6.93×10⁻⁸
JOS-HHO^[36]	1.50×10⁻¹⁶	1.50×10⁻¹⁶	1.39×10⁻¹⁵	1.11×10⁻⁹	8.06×10⁻⁵	5.24×10⁻⁵	6.60×10⁻⁶	−8.60	0	8.77×10⁻¹¹	0	9.23×10⁻⁷
HHO-1	8.19×10⁻⁴²	8.19×10⁻⁴²	2.68×10⁻³⁰	3.25×10⁻¹⁸	0	1.51×10⁻⁸	4.06×10⁻⁵	−8.60	0	8.88×10⁻¹⁶	0	3.74×10⁻¹⁰
HHO-2	7.82×10⁻¹²³	7.82×10⁻¹²³	4.38×10⁻¹⁰⁴	1.48×10⁻⁵⁹	5.19×10⁻⁸	5.48×10⁻⁷	2.88×10⁻⁶	−8.60	0	8.88×10⁻¹⁶	0	5.90×10⁻²⁴

算法	f_ave
算法	f₁	f₂	f₃	f₄	f₅	f₆	f₇	f₈	f₉	f₁₀	f₁₁	f₁₂
SLHHO	0	8.85×10⁻⁵⁸	5.45×10⁻⁸²	1.95×10⁻⁵⁰	1.66×10⁻⁷	6.05×10⁻¹²	4.34×10⁻⁵	−8.60	0	8.88×10⁻¹⁶	0	1.48×10⁻⁸
NCHHO^[33]	3.61×10⁻²²²	9.70×10⁻¹¹⁵	3.26×10⁻²⁰²	2.25×10⁻¹⁰⁶	2.88×10¹	2.36	6.59×10⁻⁵	−6.16×10¹	0	8.88×10⁻¹⁶	0	1.02×10⁻¹
LMHHO^[34]	1.33×10⁻²⁹	6.45×10⁻¹⁶	7.63×10⁻²⁸	3.78×10⁻¹⁶	1.70×10⁻⁴	5.98×10⁻⁷	6.22×10⁻⁵	−8.60	0	1.36×10⁻¹⁵	0	2.11×10⁻⁷
LHHO^[35]	5.71×10⁻⁶⁴	4.29×10⁻³²	2.86×10⁻⁵⁴	8.66×10⁻³³	3.85×10⁻³	2.21×10⁻⁶	9.27×10⁻⁵	−8.60	0	8.88×10⁻¹⁶	0	1.03×10⁻⁶
THHO^[10]	3.67×10⁻²³⁴	9.42×10⁻¹¹⁷	3.74×10⁻²²⁶	9.85×10⁻¹¹⁷	1.46×10⁻³	6.16×10⁻⁷	7.59×10⁻⁵	−8.60	0	8.88×10⁻¹⁶	0	2.46×10⁻⁶
DAHHO^[20]	1.36×10⁻⁴⁷	2.04×10⁻²⁶	2.55×10⁻³⁵	4.94×10⁻²⁶	5.28×10⁻³	4.96×10⁻⁵	8.77×10⁻⁵	−8.60	0	8.88×10⁻¹⁶	0	3.76×10⁻⁶
JOS-HHO^[36]	6.37×10⁻¹⁰	6.25×10⁻⁶	2.59×10⁻⁷	6.87×10⁻⁶	1.51×10⁻²	1.47×10⁻³	3.38×10⁻⁴	−8.60	4.15×10⁻¹⁰	1.86×10⁻⁶	3.34×10⁻⁹	7.92×10⁻⁵
HHO-1	5.49×10⁻²⁴	1.68×10⁻¹²	3.18×10⁻¹²	6.57×10⁻¹³	1.53×10⁻²¹	4.75×10⁻⁵	2.45×10⁻⁴	−8.60	0	1.30×10⁻¹²	0	2.63×10⁻⁶
HHO-2	9.46×10⁻¹⁰²	2.31×10⁻⁵⁵	3.35×10⁻⁸⁹	7.67×10⁻⁵³	1.84×10⁻⁴	3.42×10⁻⁵	5.91×10⁻⁵	−8.60	0	8.88×10⁻¹⁶	0	8.22×10⁻¹⁴

算法	f_std
算法	f₁	f₂	f₃	f₄	f₅	f₆	f₇	f₈	f₉	f₁₀	f₁₁	f₁₂
SLHHO	0	4.85×10⁻⁵⁷	2.22×10⁻⁸¹	9.12×10⁻⁵⁰	9.07×10⁻⁷	2.63×10⁻¹¹	4.05×10⁻⁵	2.48×10⁻²	0	1.00×10⁻³¹	0	7.42×10⁻⁸
NCHHO^[33]	0	5.21×10⁻¹¹⁴	0	1.18×10⁻¹⁰⁵	2.11×10⁻²	4.51×10⁻¹	7.15×10⁻⁵	1.74×10⁺³	0	1.00×10⁻³¹	0	3.90×10⁻²
LMHHO^[34]	5.41×10⁻²⁹	1.59×10⁻¹⁵	3.94×10⁻²⁷	1.78×10⁻¹⁵	4.21×10⁻⁴	9.84×10⁻⁷	5.70×10⁻⁵	3.49×10⁻²	0	2.59×10⁻¹⁵	0	4.59×10⁻⁷
LHHO^[35]	1.68×10⁻⁶³	1.71×10⁻³¹	8.21×10⁻⁵⁴	2.98×10⁻³²	7.80×10⁻³	3.19×10⁻⁶	5.07×10⁻⁵	1.82×10⁻²	0	1.00×10⁻³¹	0	1.77×10⁻⁶
THHO^[10]	0	5.10×10⁻¹¹⁶	0	5.10×10⁻¹¹⁶	5.77×10⁻³	1.92×10⁻⁶	8.68×10⁻⁵	1.16	0	1.00×10⁻³¹	0	7.00×10⁻⁶
DAHHO^[20]	7.25×10⁻⁴⁷	8.76×10⁻²⁶	1.09×10⁻³⁴	1.25×10⁻²⁵	5.54×10⁻³	7.85×10⁻⁵	1.04×10⁻⁴	5.42×10⁻¹	0	1.00×10⁻³¹	0	5.35×10⁻⁶
JOS-HHO^[36]	1.97×10⁻⁹	8.74×10⁻⁶	7.26×10⁻⁷	1.02×10⁻⁵	2.11×10⁻²	1.77×10⁻³	2.93×10⁻⁴	1.35	8.53×10⁻¹⁰	2.66×10⁻⁶	1.45×10⁻⁸	7.29×10⁻⁵
HHO-1	2.82×10⁻²³	4.49×10⁻¹²	1.22×10⁻¹¹	1.90×10⁻¹²	8.40×10⁻²¹	6.01×10⁻⁵	2.08×10⁻⁴	5.75×10⁻²	0	5.98×10⁻¹²	0	3.55×10⁻⁶
HHO-2	5.18×10⁻¹⁰¹	8.26×10⁻⁵⁵	1.83×10⁻⁸⁸	2.63×10⁻⁵²	1.96×10⁻⁴	4.72×10⁻⁵	5.16×10⁻⁵	3.45×10⁻²	0	1.00×10⁻³¹	0	4.06×10⁻¹³

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表 5 SLHHO算法与HHO算法的变体(维度为100)

Table 5. SLHHO algorithm and HHO algorithm variants (dim=100)

算法	f_min
算法	f₁	f₂	f₃	f₄	f₅	f₆	f₇	f₈	f₉	f₁₀	f₁₁	f₁₂
SLHHO	1.10×10⁻¹¹²	2.48×10⁻⁶⁸	2.05×10⁻¹⁰⁵	2.70×10⁻⁶⁰	0	5.56×10⁻⁸	1.85×10⁻⁶	−37.9	0	8.88×10⁻¹⁶	0	1.54×10⁻²⁰
NCHHO^[33]	1.25×10⁻²³⁹	6.64×10⁻¹²²	3.12×10⁻²¹⁴	1.28×10⁻¹¹⁶	9.82×10¹	5.87	2.51×10⁻⁶	−36.6	0	8.88×10⁻¹⁶	0	1.15×10⁻¹
LMHHO^[34]	2.28×10⁻⁴²	5.70×10⁻²³	9.74×10⁻³⁸	2.61×10⁻²³	2.51×10⁻⁷	5.25×10⁻¹¹	4.95×10⁻⁶	−37.9	0	8.88×10⁻¹⁶	0	2.49×10⁻¹²
LHHO^[35]	6.78×10⁻⁸⁰	9.25×10⁻³⁸	8.52×10⁻⁷¹	6.23×10⁻³⁹	1.22×10⁻⁵	3.99×10⁻⁸	9.72×10⁻⁶	−37.9	0	8.88×10⁻¹⁶	0	7.51×10⁻¹¹
THHO^[10]	7.71×10⁻²⁵⁴	9.11×10⁻¹²⁵	2.86×10⁻²⁴⁸	1.53×10⁻¹²⁶	1.27×10⁻⁷	1.66×10⁻¹¹	3.42×10⁻⁷	−37.9	0	8.88×10⁻¹⁶	0	8.16×10⁻¹³
DAHHO^[20]	6.55×10⁻⁵⁸	2.58×10⁻³²	3.92×10⁻⁵⁰	3.62×10⁻³²	0	1.10×10⁻⁶	1.41×10⁻⁵	−37.9	0	8.88×10⁻¹⁶	0	5.71×10⁻⁹
JOS-HHO^[36]	2.34×10⁻¹⁷	2.57×10⁻⁸	1.81×10⁻¹¹	2.17×10⁻¹²	7.43×10⁻³	2.72×10⁻⁴	1.72×10⁻⁵	−37.9	0	1.46×10⁻¹⁰	1.89×10⁻¹⁵	7.14×10⁻⁷
HHO-1	9.94×10⁻²¹	2.09×10⁻¹¹	2.41×10⁻²⁰	2.00×10⁻¹⁰	2.11×10⁻⁵	4.57×10⁻⁵	3.40×10⁻⁶	−37.9	0	2.30×10⁻¹¹	0	6.70×10⁻⁷
HHO-2	9.26×10⁻⁷⁷	4.64×10⁻⁴¹	5.63×10⁻⁶³	6.69×10⁻³⁹	7.85×10⁻⁵	1.53×10⁻⁸	1.24×10⁻⁷	−37.9	0	8.88×10⁻¹⁶	0	1.32×10⁻⁸

算法	f_ave
算法	f₁	f₂	f₃	f₄	f₅	f₆	f₇	f₈	f₉	f₁₀	f₁₁	f₁₂
SLHHO	5.33×10⁻⁹⁹	3.46×10⁻⁵⁴	1.73×10⁻⁷⁷	2.41×10⁻⁵⁰	0	2.01×10⁻⁴	4.90×10⁻⁵	−37.9	0	8.88×10⁻¹⁶	0	2.89×10⁻⁹
NCHHO^[33]	5.23×10⁻²²⁰	2.09×10⁻¹¹⁵	2.42×10⁻¹⁹⁸	3.53×10⁻¹⁰⁷	9.83×10¹	7.95	8.19×10⁻⁵	−21.2	0	8.88×10⁻¹⁶	0	1.87×10⁻¹
LMHHO^[34]	1.16×10⁻³⁰	2.17×10⁻¹⁴	1.01×10⁻²⁵	3.41×10⁻¹⁶	5.20×10⁻⁴	1.74×10⁻⁶	7.11×10⁻⁵	−37.9	0	1.72×10⁻¹⁵	0	1.86×10⁻⁸
LHHO^[35]	1.56×10⁻⁶¹	3.36×10⁻³¹	6.89×10⁻⁴⁸	5.71×10⁻³²	1.21×10⁻²	1.97×10⁻⁵	1.05×10⁻⁴	−37.9	0	8.88×10⁻¹⁶	0	8.80×10⁻⁷
THHO^[10]	5.65×10⁻²²⁹	2.20×10⁻¹¹⁷	2.12×10⁻²²⁸	2.10×10⁻¹¹⁷	2.26×10⁻³	8.72×10⁻⁶	4.88×10⁻⁵	−37.9	0	8.88×10⁻¹⁶	0	1.94×10⁻⁸
DAHHO^[20]	1.07×10⁻⁴⁶	1.42×10⁻²⁶	1.20×10⁻³⁷	7.80×10⁻²⁶	3.32×10⁻⁶	2.26×10⁻³	3.04×10⁻⁴	−37.9	0	8.88×10⁻¹⁶	0	9.55×10⁻⁶
JOS-HHO^[36]	4.38×10⁻⁹	3.01×10⁻⁵	6.64×10⁻⁵	5.43×10⁻⁶	1.76×10⁻¹	8.32×10⁻³	2.71×10⁻⁴	−37.9	7.18×10⁻⁹	4.93×10⁻⁶	5.14×10⁻⁹	5.67×10⁻⁵
HHO-1	2.98×10⁻¹⁰	3.02×10⁻⁷	3.06×10⁻⁸	5.27×10⁻⁷	5.05×10⁻²	5.54×10⁻³	1.22×10⁻⁴	−37.9	3.16×10⁻¹²	3.93×10⁻⁸	2.18×10⁻¹²	1.50×10⁻⁵
HHO-2	5.80×10⁻⁶²	2.03×10⁻³⁴	5.49×10⁻⁴⁷	4.17×10⁻³²	1.74×10⁻³	4.57×10⁻⁵	3.27×10⁻⁵	−37.9	0	8.88×10⁻¹⁶	0	4.15×10⁻⁷

算法	f_std
算法	f₁	f₂	f₃	f₄	f₅	f₆	f₇	f₈	f₉	f₁₀	f₁₁	f₁₂
SLHHO	2.88×10⁻⁹⁸	1.84×10⁻⁵³	8.16×10⁻⁷⁷	7.89×10⁻⁵⁰	0	3.14×10⁻⁴	3.69×10⁻⁵	2.41×10⁻¹	0	1.00×10⁻³¹	0	6.78×10⁻⁹
NCHHO^[33]	0	8.89×10⁻¹¹⁵	0	1.77×10⁻¹⁰⁶	6.68×10⁻²	1.95	7.59×10⁻⁵	8.50×10³	0	1.00×10⁻³¹	0	6.79×10⁻²
LMHHO^[34]	4.72×10⁻³⁰	8.97×10⁻¹⁴	5.52×10⁻²⁵	1.27×10⁻¹⁵	1.25×10⁻³	3.20×10⁻⁶	6.49×10⁻⁵	1.14	0	3.32×10⁻¹⁵	0	3.66×10⁻⁸
LHHO^[35]	8.32×10⁻⁶¹	1.75×10⁻³⁰	3.13×10⁻⁴⁷	2.76×10⁻³¹	2.22×10⁻²	3.56×10⁻⁵	6.56×10⁻⁵	3.48×10⁻²	0	1.00×10⁻³¹	0	1.28×10⁻⁶
THHO^[10]	0	1.08×10⁻¹¹⁶	0	7.70×10⁻¹¹⁷	8.19×10⁻³	2.88×10⁻⁵	4.07×10⁻⁵	3.57	0	1.00×10⁻³¹	0	5.66×10⁻⁸
DAHHO^[20]	5.66×10⁻⁴⁶	4.52×10⁻²⁶	4.68×10⁻³⁷	1.99×10⁻²⁵	1.82×10⁻⁵	3.11×10⁻³	3.33×10⁻⁴	3.05	0	1.00×10⁻³¹	0	2.15×10⁻⁵
JOS-HHO^[36]	1.07×10⁻⁸	4.91×10⁻⁵	2.04×10⁻⁴	1.09×10⁻⁵	2.07×10⁻¹	8.01×10⁻³	2.48×10⁻⁴	2.68×10¹	3.64×10⁻⁸	1.20×10⁻⁵	1.19×10⁻⁸	6.68×10⁻⁵
HHO-1	1.62×10⁻⁹	6.43×10⁻⁷	6.42×10⁻⁸	1.47×10⁻⁶	6.57×10⁻²	4.61×10⁻³	1.76×10⁻⁴	7.88	1.14×10⁻¹¹	7.55×10⁻⁸	6.06×10⁻¹²	2.02×10⁻⁵
HHO-2	3.15×10⁻⁶¹	9.48×10⁻³⁴	2.23×10⁻⁴⁶	1.51×10⁻³¹	2.60×10⁻³	6.19×10⁻⁵	2.90×10⁻⁵	3.59×10⁻¹	0	0	0	5.59×10⁻⁷

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表 6 不同维度下SLHHO算法与HHO算法的变体排序值

Table 6. Variant sorting values of SLHHO algorithm and HHO algorithm in different dimensions

算法	f_rank（维度为30）	f_rank（维度为100）
SLHHO	2.9167	3.1667
NCHHO^[33]	5.1250	5.0417
LMHHO^[34]	4.9583	4.7917
LHHO^[35]	4.8333	4.5833
THHO^[10]	3.3333	2.6667
DAHHO^[20]	5.6667	5.2500
JOS-HHO^[36]	8.3750	8.2917
HHO-1	6.2083	7.3750
HHO-2	3.5833	3.8333

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表 7 压力容器设计问题优化结果

Table 7. Pressure vessel design problem optimization results

函数	f_best	f_mean	f_worst	T_s/mm	T_h/mm	R/mm	L/mm
SLHHO	5887.5364	5974.2155	7225.6231	0.7786	0.3853	40.3403	199.7153
HHO^[9]	5981.9440	6909.8281	7654.0635	1.1752	0.5786	59.8934	37.9625
PSO^[1]	5951.9270	6200.6677	7217.0457	0.8153	0.4030	42.2448	174.8309
GWO^[3]	5981.9440	6909.8281	7654.0635	1.1752	0.5786	59.8934	37.9625
WOA^[4]	6103.9794	7578.9269	10436.7044	0.8442	0.4089	42.7487	168.7424

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