Group teaching optimization algorithm with information sharing for numerical optimization and engineering optimization

Abstract

Most of the reported metaheuristic methods need the control parameters except the essential population size and terminal condition. When these methods are used for solving an unknown problem, how to set the most suitable values for their control parameters to achieve the optimal solution is a great challenge. Group teaching optimization algorithm (GTOA) is a newly presented metaheuristic method, whose remarkable feature is that it only relies on the essential population size and terminal condition for optimization. However, GTOA may get trapped in the local optimal solutions for solving complex optimization problems due to the lack of communication between outstanding group and average group. In order to improve the performance of GTOA, this paper proposes a new variant of GTOA, namely group teaching optimization algorithm with information sharing (ISGTOA). Like GTOA, ISGTOA doesn’t introduce any other control parameters, which enhances the communication between outstanding group and average group by reusing the individuals in the built two archives. The performance of ISGTOA is investigated by CEC 2014 test suite, CEC 2015 test suite, and four challenging constrained engineering design problems. Experimental results prove the superiority of ISGTOA for expensive optimization problems with multimodal properties by comparing with GTOA and other powerful methods. The source codes of the proposed ISGTOA can be found in https://ww2.mathworks.cn/matlabcentral/fileexchange/98629-the-source-code-of-isgtoa and https://github.com/jsuzyy/The-source-code-of-ISGTOA-for-global-optimization.

Appendices

Appendix A: The download links of the applied algorithms

• 1. ISGTOA: https://ww2.mathworks.cn/matlabcentral/fileexchange/98629-the-source-code-of-isgtoa.

• 2. JAYA: https://ww2.mathworks.cn/matlabcentral/fileexchange/74004-jaya-a-simple-and-new-optimization-algorithm?s_tid=srchtitle.

• 3. TSO: https://ww2.mathworks.cn/matlabcentral/fileexchange/82640-transient-search-algorithm-tso?s_tid=srchtitle.

• 4. AOA: https://ww2.mathworks.cn/matlabcentral/fileexchange/84742-the-arithmetic-optimization-algorithm-aoa?s_tid=srchtitle.

• 5. SOA: https://ww2.mathworks.cn/matlabcentral/fileexchange/75180-seagull-optimization-algorithm-soa?s_tid=srchtitle.

• 6. JSO: https://ww2.mathworks.cn/matlabcentral/fileexchange/78961-jellyfish-search-optimizer-js?s_tid=srchtitle.

• 7. TLBO: https://yarpiz.com/83/ypea111-teaching-learning-based-optimization.

Appendix B: Mathematical formulas of engineering design problems

Appendix B.1: Pressure vessel design problem

$$ \begin{array}{l} {\text{Minimize }}f\left( {\mathbf{x}} \right) = 0.6224x_{1} x_{3} x_{4} + 1.7781x_{2} x_{3}^{2} + 3.1661x_{1}^{2} x_{4} + 19.84x_{1}^{2} x_{3} \hfill \\ {\text{Subject to:}} \hfill \\ g_{1} \left( {\mathbf{x}} \right) = - x_{1} + 0.0193x_{3} \le 0 \hfill \\ g_{2} \left( {\mathbf{x}} \right) = - x_{2} + 0.00954x_{3} \le 0 \hfill \\ g_{3} \left( {\mathbf{x}} \right) = - \pi x_{3}^{2} x_{4} - \frac{4}{3}\pi x_{3}^{3} + 1296,000 \le 0 \hfill \\ g_{4} \left( {\mathbf{x}} \right) = x_{4} - 240 \le 0 \hfill \\ \end{array} $$

where $0 \le x_{i} \le 100, \, i = 1,2; \, 10 \le x_{i} \le 200, \, i = 3,4.$

Appendix B.2: Tension/compression spring design problem

$$ \begin{array}{l} {\text{Minimize }}f\left( {\mathbf{x}} \right) = \left( {x_{3} + 2} \right)x_{2} x_{1}^{2} \hfill \\ {\text{Subject to:}} \hfill \\ g_{1} \left( {\mathbf{x}} \right) = 1 - \frac{{x_{2}^{3} x_{3} }}{{71,785x_{1}^{4} }} \le 0 \hfill \\ g_{2} \left( {\mathbf{x}} \right) = 4x_{2}^{2} - \frac{{x_{1} x_{2} }}{{12.566\left( {x_{2} x_{1}^{3} - x_{1}^{4} } \right)}} + \frac{1}{{5108x_{1}^{2} }} - 1 \le 0 \hfill \\ g_{3} \left( {\mathbf{x}} \right) = 1 - \frac{{140.45x_{1} }}{{x_{2}^{2} x_{3} }} \le 0 \hfill \\ g_{4} \left( {\mathbf{x}} \right) = x_{2} + \frac{{x_{1} }}{1.5} - 1 \le 0 \hfill \\ \end{array} $$

where $0.05 \le x_{1} \le 2, \, 0.25 \le x_{2} \le 1.30, \, 2.00 \le x_{3} \le 15.00.$

Appendix B.3: Car side impact design problem

$$ \begin{array}{l} {\text{Minimize }}f{(}{\mathbf{x}}{) = 1}{\text{.98 + 4}}{.90}x_{{1}} + 6.67x_{2} { + 6}{\text{.98}}x_{3} + 4.01x_{4} { + 1}{\text{.78}}x_{5} + 2.73x_{7} \hfill \\ {\text{Subject to}} \hfill \\ g_{1} ({\mathbf{x}}) = 1.16 - 0.3717x_{2} x_{4} - 0.00931x_{2} x_{10} - 0.484x_{3} x_{9} + 0.01343x_{6} x_{10} \le 1{\text{KN}} \hfill \\ g_{2} ({\mathbf{x}}) = 0.261 - 0.0159x_{1} x_{2} - 0.0188x_{1} x_{8} - 0.0191x_{2} x_{7} + 0.0144x_{3} x_{5} + 0.0008757x_{5} x_{10} + 0.08045x_{6} x_{9} + 0.00139x_{8} x_{11} \hfill \\ \, + 0.00001575x_{10} x_{11} \le 0.32{\text{ m/s}} \hfill \\ g_{3} ({\mathbf{x}}) = 0.214 + 0.00817x_{5} - 0.131x_{1} x_{8} - 0.0704x_{1} x_{9} + 0.03099x_{2} x_{6} - 0.018x_{2} x_{7} + 0.0208x_{3} x_{8} + 0.121x_{3} x_{9} - 0.00364x_{5} x_{6} \hfill \\ \, + 0.0007715x_{5} x_{10} - 0.0005354x_{6} x_{10} + 0.00121x_{8} x_{11} \le 0.32{\text{ m/s}} \hfill \\ g_{4} ({\mathbf{x}}) = 0.74 - 0.61x_{2} - 0.163x_{3} x_{8} + 0.001232x_{3} x_{10} - 0.166x_{7} x_{9} + 0.227x_{2}^{2} \le 0.32{\text{ m/s}} \hfill \\ g_{5} ({\mathbf{x}}) = 28.98 + 3.818x_{3} - 4.2x_{1} x_{2} + 0.0207x_{5} x_{10} + 6.63x_{6} x_{9} - 7.7x_{7} x_{8} + 0.32x_{9} x_{10} \le 32{\text{ mm}} \hfill \\ g_{6} ({\mathbf{x}}) = 33.86 + 2.95x_{3} + 0.1792x_{10} - 5.057x_{1} x_{2} - 11.0x_{2} x_{8} - 0.0215x_{5} x_{10} - 9.98x_{7} x_{8} + 22.0x_{8} x_{9} \le 32{\text{ mm}} \hfill \\ g_{7} ({\mathbf{x}}) = 46.36 - 9.9x_{2} - 12.9x_{1} x_{8} - 5.057x_{1} x_{2} + 0.1107x_{3} x_{10} \le 32{\text{ mm}} \hfill \\ g_{8} ({\mathbf{x}}) = 4.72 - 0.5x_{4} - 0.19x_{2} x_{3} - 0.0122x_{4} x_{10} + 0.009325x_{6} x_{10} + 0.000191x_{11}^{2} \le 4{\text{KN}} \hfill \\ g_{9} ({\mathbf{x}}) = 10.58 - 0.674x_{1} x_{2} - 1.95x_{2} x_{8} + 0.02054x_{3} x_{10} - 0.0198x_{4} x_{10} + 0.028x_{6} x_{10} \le 9.9{\text{ mm/ms}} \hfill \\ g_{10} ({\mathbf{x}}) = 16.45 - 0.489x_{3} x_{7} - 0.843x_{5} x_{6} + 0.0432x_{9} x_{10} - 0.0556x_{9} x_{11} - 0.000786x_{11}^{2} \le 15.7{\text{ mm/ms}} \hfill \\ \end{array} $$

where $0.5 \le x_{i} \le 1.5, \, i = 1,2,3,4,5,6,7; \, 0.192 \le x_{i} \le 0.345, \, i = 8,9; \, - 30 \le x_{i} \le 30, \, i = 10,11.$

Appendix B.4: Rolling element bearing design problem

$$ \begin{array}{l} {\text{M}}aximum \, f{(}{\mathbf{x}}{)} = \left\{ \begin{gathered} f_{{\text{c}}} Z^{2/3} D_{{\text{b}}}^{1.8} ,{\text{ if }}D \le 25.4{\text{mm}} \hfill \\ 3.647f_{{\text{c}}} Z^{2/3} D_{{\text{b}}}^{1.4} ,{\text{ if }}D > 25.4{\text{mm}} \hfill \\ \end{gathered} \right. \hfill \\ {\text{Subject to}} \hfill \\ g_{1} \left( {\mathbf{x}} \right) = \frac{{\phi_{0} }}{{2\sin^{ - 1} \left( {D_{{\text{b}}} /D_{{\text{m}}} } \right)}} - Z + 1 \ge 0 \hfill \\ g_{2} \left( {\mathbf{x}} \right) = 2D_{{\text{b}}} - K_{{{\text{D}}\min }} \left( {D - d} \right) \ge 0 \hfill \\ g_{3} \left( {\mathbf{x}} \right) = K_{{{\text{D}}\max }} \left( {D - d} \right) - 2D_{{\text{b}}} \ge 0 \hfill \\ g_{4} \left( {\mathbf{x}} \right) = \zeta B_{{\text{w}}} - D_{{\text{b}}} \le 0 \hfill \\ g_{5} \left( {\mathbf{x}} \right) = D_{{\text{m}}} - 0.5\left( {D + d} \right) \ge 0 \hfill \\ g_{6} \left( {\mathbf{x}} \right) = \left( {0.5 + e} \right)\left( {D + d} \right) - D_{{\text{m}}} \ge 0 \hfill \\ g_{7} \left( {\mathbf{x}} \right) = 0.5\left( {D - D_{{\text{m}}} - D_{{\text{b}}} } \right) - \varepsilon D_{{\text{b}}} \ge 0 \hfill \\ g_{8} \left( {\mathbf{x}} \right) = f_{{\text{i}}} \ge 0.515 \hfill \\ g_{9} \left( {\mathbf{x}} \right) = f_{{\text{o}}} \ge 0.515 \hfill \\ \gamma = \frac{{D_{{\text{b}}} }}{{D_{{\text{m}}} }}, \, f_{{\text{i}}} = \frac{{r_{{\text{i}}} }}{{D_{{\text{b}}} }}, \, f_{{\text{o}}} = \frac{{r_{{\text{o}}} }}{{D_{{\text{b}}} }}, \, T = D - d - 2D_{{\text{b}}} , \, D = 160, \, d = 90, \, B_{{\text{w}}} = 30, \, D = 160, \, r_{{\text{i}}} = r_{{\text{o}}} = 11.033 \hfill \\ 0.5\left( {D + d} \right) \le D_{{\text{m}}} \le 0.6\left( {D + d} \right), \, 0.15\left( {D - d} \right) \le D_{{\text{b}}} \le 0.45\left( {D - d} \right),{ 4} \le Z \le {50, 0}{\text{.515}} \le f_{{\text{i}}} \le 0.6,{ 0}{\text{.515}} \le f_{{\text{o}}} \le 0.6 \hfill \\ 0.4 \le K_{{{\text{D}}\min }} \le 0.5, \, 0.6 \le K_{{{\text{D}}\max }} \le 0.7, \, 0.3 \le e \le 0.4, \, 0.02 \le \varepsilon \le 0.1, \, 0.6 \le \zeta \le 0.85 \hfill \\ f_{{\text{c}}} = 37.91\left[ {1 + \left\{ {1.04\left( {\frac{1 - \gamma }{{1 + \gamma }}} \right)^{1.72} \left( {\frac{{f_{{\text{i}}} \left( {2f_{{\text{o}}} - 1} \right)}}{{f_{{\text{o}}} \left( {2f_{{\text{i}}} - 1} \right)}}} \right)^{0.41} } \right\}^{10/3} } \right]^{ - 0.3} \times \left( {\frac{{\gamma^{0.3} \left( {1 - \gamma } \right)^{1.39} }}{{\left( {1 + \gamma } \right)^{1./3} }}} \right)\left[ {\frac{{2f_{{\text{i}}} }}{{2f_{{\text{i}}} - 1}}} \right]^{0.41} \hfill \\ \phi_{0} =2\pi - 2\cos^{ - 1} \frac{{\left[ {\left\{ {(D - d)/2 - 3(T/4)} \right\}^{2} + \left\{ {D/2 - (T/4) - D_{{\text{b}}} } \right\}^{2} - \left\{ {d/2 + (T/4)} \right\}^{2} } \right]}}{{2\left\{ {(D - d)/2 - 3(T/4)} \right\}\left\{ {D/2 - (T/4) - D_{{\text{b}}} } \right\}}} \hfill \\ \end{array} $$