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Archimedes optimization algorithm: a new metaheuristic algorithm for solving optimization problems

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Abstract

The difficulty and complexity of the real-world numerical optimization problems has grown manifold, which demands efficient optimization methods. To date, various metaheuristic approaches have been introduced, but only a few have earned recognition in research community. In this paper, a new metaheuristic algorithm called Archimedes optimization algorithm (AOA) is introduced to solve the optimization problems. AOA is devised with inspirations from an interesting law of physics Archimedes’ Principle. It imitates the principle of buoyant force exerted upward on an object, partially or fully immersed in fluid, is proportional to weight of the displaced fluid. To evaluate performance, the proposed AOA algorithm is tested on CEC’17 test suite and four engineering design problems. The solutions obtained with AOA have outperformed well-known state-of-the-art and recently introduced metaheuristic algorithms such genetic algorithms (GA), particle swarm optimization (PSO), differential evolution variants L-SHADE and LSHADE-EpSin, whale optimization algorithm (WOA), sine-cosine algorithm (SCA), Harris’ hawk optimization (HHO), and equilibrium optimizer (EO). The experimental results suggest that AOA is a high-performance optimization tool with respect to convergence speed and exploration-exploitation balance, as it is effectively applicable for solving complex problems. The source code is currently available for public from: https://www.mathworks.com/matlabcentral/fileexchange/79822-archimedes-optimization-algorithm

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Acknowledgments

The authors would like to thank Halwan University for supporting this research. This research is also partially supported by University of Electronic Science and Technology of China (UESTC) and National Natural Science Foundation of China (NSFC) under Grant No. 61772120.

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Authors and Affiliations

  1. Faculty of Engineering, Helwan University, Helwan, Egypt

    Fatma A. Hashim & Walid Al-Atabany

  2. Institute of Fundamental and Frontier Sciences, University of Electronic Science and Technology of China, Chengdu, Sichuan, China

    Kashif Hussain

  3. Faculty of Computers and Information, Minia University, Minia, Egypt

    Essam H. Houssein

  4. Faculty of Engineering, Misr University for Science and Technology, 6th of October, Egypt

    Mai S. Mabrouk

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  1. Fatma A. Hashim
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Correspondence to Essam H. Houssein.

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Appendices

Appendix A: Welded beam design problem

$$ \begin{array}{@{}rcl@{}} &&\text{Consider }\mathbf{x}=\left[ {{x}_{1}}{{x}_{2}}{{x}_{3}}{{x}_{4}} \right]=\left[ h l t b \right] \\ &&\text{Minimize } f\left( {\mathbf{x}} \right) = 1.10471{x_{1}^{2}}{{x}_{2}} + 0.04811{{x}_{3}}{{x}_{4}}\left( 14.0 + {{x}_{2}} \right) \\ &&\text{Subject to: } \\ &&{{g}_{1}}\left( {\mathbf{x}} \right)=\tau \left( {\mathbf{x}} \right)-13600\le 0 \\ &&{{g}_{2}}\left( {\mathbf{x}} \right)=\sigma \left( {\mathbf{x}} \right)-30000\le 0 \\ &&{{g}_{3}}\left( {\mathbf{x}} \right)={{x}_{1}}-{{x}_{4}}\le 0 \\ &&{{g}_{4}}\left( {\mathbf{x}} \right)=0.10471\left( {x_{1}^{2}} \right) + 0.04811{{x}_{3}}{{x}_{4}}\left( 14 + {{x}_{2}} \right) - 5.0\le 0 \\ &&{{g}_{6}}\left( {\mathbf{x}} \right)=\delta \left( {\mathbf{x}} \right)-0.25\le 0 \\ &&{{g}_{7}}\left( {\mathbf{x}} \right)=6000-{{p}_{c}}\left( {\mathbf{x}} \right)\le 0 \\ &&\text{where}\\ &&\tau \left( {\mathbf{x}} \right)=\sqrt{\left( {{\tau }^{\prime}} \right)+\left( 2{\tau }'{\tau }^{\prime\prime} \right)\frac{{{x}_{2}}}{2R}+{{\left( {{\tau }^{\prime\prime}} \right)}^{2}}} \\ &&{\tau }^{\prime}=\frac{6000}{\sqrt{2}{{x}_{1}}{{x}_{2}}} \end{array} $$
$$ \begin{array}{@{}rcl@{}} &&{\tau }^{\prime\prime}=\frac{MR}{J} \\ &&M=6000\left( 14+\frac{{{x}_{2}}}{2} \right) \\ &&R=\sqrt{\frac{{x_{2}^{2}}}{4}+{{\left( \frac{{{x}_{1}}+{{x}_{3}}}{2} \right)}^{2}}} \\ &&j=2\left\{ {{x}_{1}}{{x}_{2}}\sqrt{2}\left[ \frac{{x_{2}^{2}}}{12}+{{\left( \frac{{{x}_{1}}+{{x}_{3}}}{2} \right)}^{2}} \right] \right\} \\ &&\sigma \left( {\mathbf{x}} \right)=\frac{504000}{{{x}_{4}}{x_{3}^{2}}} \\ &&\delta \left( {\mathbf{x}} \right)=\frac{65856000}{\left( 30\times {{10}^{6}} \right){{x}_{4}}{x_{3}^{3}}} \\ &&{{p}_{c}}\left( {\mathbf{x}} \right)=\frac{4.013\left( 30\times {{10}^{6}} \right)\sqrt{\frac{{x_{3}^{2}}{x_{4}^{6}}}{36}}}{196}\left( 1-\frac{{{x}_{3}}\sqrt{\frac{30\times {{10}^{6}}}{4\left( 12\times {{10}^{6}} \right)}}}{28} \right) \\ &&\text{with } 0.1\le {{x}_{1}},{{x}_{4}}\le 2.0 and 0.1\le {{x}_{2}},{{x}_{3}}\le 10.0 \end{array} $$

Appendix B: Tension/compression spring design problem

$$ \begin{array}{@{}rcl@{}} &&\text{Consider:}\\ &&\mathbf{x}=\left[ {{x}_{1}}{{x}_{2}}{{x}_{3}} \right]=\left[ d D N \right] \\ &&\text{Min} f\left( {\mathbf{x}} \right)=\left( {{x}_{3}}+2 \right){{x}_{2}}{x_{1}^{2}} \\ &&\text{subject to: } \\ &&{{g}_{1}}\left( {\mathbf{x}} \right)=1-\frac{{x_{2}^{3}}{{x}_{3}}}{71785{x_{1}^{4}}}\le 0 \\ &&{{g}_{2}}\left( {\mathbf{x}} \right)=\frac{4{x_{2}^{2}}-{{x}_{1}}{{x}_{2}}}{12566\left( {{x}_{2}}{x_{1}^{3}}-{x_{1}^{4}} \right)}+\frac{1}{5108{x_{1}^{2}}}-1\le 0 \\ &&{{g}_{3}}\left( {\mathbf{x}} \right)=1-\frac{140.45{{x}_{1}}}{{x_{2}^{2}}{{x}_{3}}}\le 0 \\ &&{{g}_{4}}\left( {\mathbf{x}} \right)=\frac{{{x}_{1}}+{{x}_{2}}}{1.5}-1\le 0 \\ &&\text{with } 0.05\!\le\! {{x}_{1}} \!\le\! 2.0,0.25 \!\le\! {{x}_{2}} \!\le\! 1.3,and 2.0 \!\le\! {{x}_{3}} \!\le\! 15.0 \end{array} $$

Appendix C: Speed reducer design problem

$$ \begin{array}{@{}rcl@{}} &&\!\!\!\text{Min}f\left( {\mathbf{x}} \right) = 0.7854{{x}_{1}}{x_{2}^{2}}\left( 3.3333{x_{3}^{2}} + 14.9334{{x}_{3}} - 43.0934 \right)\\ &&-1.508{{x}_{1}} \left( {x_{6}^{2}}+{x_{7}^{2}} \right)+7.4777\left( {x_{6}^{3}}+{x_{7}^{3}} \right)+0.7854\\ &&\left( {{x}_{4}}{x_{6}^{2}}+{{x}_{5}}{x_{7}^{2}} \right) \\ &&\text{Subject to: } \\ &&{{g}_{1}}\left( {\mathbf{x}} \right)=\frac{27}{{{x}_{1}}{x_{2}^{2}}{{x}_{3}}}-1\le 0 \\ \end{array} $$
$$ \begin{array}{@{}rcl@{}} &&{{g}_{2}}\left( {\mathbf{x}} \right)=\frac{397.5}{{{x}_{1}}{x_{2}^{2}}{{x}_{3}}}-1\le 0 \\ &&{{g}_{3}}\left( {\mathbf{x}} \right)=\frac{1.93{x_{4}^{3}}}{{{x}_{2}}{{x}_{3}}{x_{6}^{4}}}-1 \\ &&{{g}_{4}}\left( {\mathbf{x}} \right)=\frac{1.93{x_{5}^{3}}}{{{x}_{2}}{{x}_{3}}{x_{7}^{4}}}-1\le 0 \\ &&{{g}_{5}}\left( {\mathbf{x}} \right)=\frac{1}{110{x_{6}^{3}}}\sqrt{{{\left( \frac{745{{x}_{4}}}{{{x}_{2}}{{x}_{3}}} \right)}^{2}}+16.9\times {{10}^{6}}}-1\le 0 \\ &&{{g}_{6}}\left( {\mathbf{x}} \right)=\frac{1}{85{x_{7}^{3}}}\sqrt{{{\left( \frac{745{{x}_{5}}}{{{x}_{2}}{{x}_{3}}} \right)}^{2}}+157.5\times {{10}^{6}}}-1\le 0 \\ &&{{g}_{7}}\left( {\mathbf{x}} \right)=\frac{{{x}_{2}}{{x}_{3}}}{40}-1\le 0 \\ &&{{g}_{8}}\left( {\mathbf{x}} \right)=\frac{5{{x}_{2}}}{{{x}_{1}}}-1\le 0 \\ &&{{g}_{9}}\left( {\mathbf{x}} \right)=\frac{{{x}_{1}}}{12{{x}_{2}}}-1\le 0 \\ &&{{g}_{10}}\left( {\mathbf{x}} \right)=\frac{1.5{{x}_{6}}+1.9}{x4}-1\le 0 \\ &&{{g}_{11}}\left( {\mathbf{x}} \right)=\frac{1.1{{x}_{7}}+1.9}{{{x}_{5}}}-1\le 0 \end{array} $$

with 2.6 ≤ x1 ≤ 3.6, 0.7 ≤ x2 ≤ 0.8, 17 ≤ x3 ≤ 28, 7.3 ≤ x4 ≤ 8.3, 7.8 ≤ x5 ≤ 8.3, 2.9 ≤ x6 ≤ 3.9,and5 ≤ x7 ≤ 5.5

Appendix D: Pressure vessel design problem

$$ \begin{array}{@{}rcl@{}} &&\text{Min}f(x)=0.6224{{x}_{1}}{{x}_{3}}{{x}_{4}}+1.7781{{x}_{2}}{x_{3}^{2}}+3.1661{x_{1}^{2}}{{x}_{4}}\\ &&+19.84{x_{1}^{2}}{{x}_{3}} \\ &&\text{Subject to: }\\ &&{{g}_{1}}(x)=-{{x}_{1}}+0.0193x \\ &&{{g}_{2}}(x)=-{{x}_{2}}+0.00954{{x}_{3}} \le 0 \\ &&{{g}_{3}}(x)=-\pi {x_{3}^{2}}{{x}_{4}}-(4/3)\pi {x_{3}^{3}}+1,296,000\le 0 \\ &&{{g}_{4}}(x)={{x}_{4}}-240\le 0 \\ &&0\le {{x}_{i}}\le 100, i=1,2 \\ &&10\le {{x}_{i}}\le 200, i=3,4 \end{array} $$

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Hashim, F.A., Hussain, K., Houssein, E.H. et al. Archimedes optimization algorithm: a new metaheuristic algorithm for solving optimization problems. Appl Intell 51, 1531–1551 (2021). https://doi.org/10.1007/s10489-020-01893-z

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