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Comprehensive learning Jaya algorithm for engineering design optimization problems

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Abstract

Jaya algorithm (JAYA) is a recently developed metaheuristic algorithm for global optimization problems. JAYA has a very simple structure and only needs the essential population size and terminal condition for solving optimization problems. However, JAYA is easy to get trapped in the local optimum for solving complex global optimization problems due to its single learning strategy. Motivated by this disadvantage of JAYA, this paper presents an improved JAYA, named comprehensive learning JAYA algorithm (CLJAYA), for solving engineering design optimization problems. The core idea of CLJAYA is the designed comprehensive learning mechanism by making full use of population information. The designed comprehensive learning mechanism consists of three different learning strategies to improve the global search ability of JAYA. To investigate the performance of CLJAYA, CLJAYA is first evaluated by the well-known CEC 2013 and CEC 2014 test suites, which include 50 multimodal test functions and eight unimodal test functions. Then CLJAYA is employed to solve five real-world engineering optimization problems. Experimental results demonstrate that CLJAYA can achieve better solutions for most test problems than JAYA and the other compared algorithms, which indicates the designed comprehensive learning mechanism is very effective. In addition, the source code of the proposed CLJAYA can be loaded from https://www.mathworks.com/matlabcentral/fileexchange/82134-the-source-code-for-cljaya.

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Abbreviations

\({\mathbf{X}}\) :

Population

\(N\) :

Population size

\(D\) :

Dimension

\({\mathbf{x}}_{i}\) :

The position of the ith individual

\({\mathbf{v}}_{i}\) :

The candidate position of the ith individual

\({\mathbf{x}}_{{{\text{BEST}},j}}\) :

The jth variable of the best individual

\({\mathbf{x}}_{{{\text{WORST}},j}}\) :

The jth variable of the worst individual

\({\mathbf{M}}\) :

The mean position of the population

\(p_{{{\text{switch}}}}\) :

Switch probability

\(p,q\) :

Integers between 1 and \(N\)

\(\kappa_{1} ,\kappa_{2} ,\varphi_{5} ,\varphi_{6} ,\chi\) :

Random numbers between 0 and 1 with uniform distribution

\(\varphi_{1} ,\varphi_{2} ,\varphi_{3} ,\varphi_{4}\) :

Random numbers with standard normal distribution

\({\mathbf{L}}\) :

The lower boundary of the variables

\({\mathbf{U}}\) :

The upper boundary of the variables

\(F_{{{\text{current}}}}\) :

The current number of function evaluations

\(F_{\max }\) :

The maximum number of function evaluations

\(R_{N}\) :

The number of independent runs

\({\mathbf{x}}^{*}\) :

The real optimal solution

\(h_{t}\) :

The tth equality constraint

\(g_{k}\) :

The kth inequality constraint

\(m\) :

The number of equality constraints

\(n\) :

The number of inequality constraints

\(P({\mathbf{x}})\) :

The total penalty function

\(H_{i} ({\mathbf{x}})\) :

The penalty function of the ith equality constraint

\(G_{j} ({\mathbf{x}})\) :

The penalty function of the jth inequality constraint

\(\eta_{i}\) :

The penalty factor of the penalty function of the ith equality constraint

\(\xi_{j}\) :

The penalty factor of the penalty function of the jth inequality constraint

MEAN:

Mean absolute error

STD:

Standard variance

NNA:

Neural network algorithm

GWO:

Grey wolf optimizer

WOA:

Whale optimization algorithm

SCA:

Sine cosine algorithm

JAYA:

Jaya algorithm

TLBO:

Teaching–learning-based optimization

CLJAYA:

Comprehensive learning Jaya algorithm

References

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Author information

Authors and Affiliations

  1. School of Electrical and Information Engineering, Tianjin University, Tianjin, 300072, People’s Republic of China

    Yiying Zhang & Zhigang Jin

Authors
  1. Yiying Zhang
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  2. Zhigang Jin
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Corresponding author

Correspondence to Zhigang Jin.

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Appendix A

Appendix A

A.1 welded beam design problem

$$\begin{array}{l} {\rm{Minimize }}f\left( x \right) = 1.10471x_1^2{x_2} + 0.04811{x_3}{x_4}\left( {14 + {x_2}} \right)\\ {\rm{Subject \, to:}}\\ {g_1}\left( x \right) = \tau \left( x \right) - {\tau _{\max }} \le 0\\ {g_2}\left( x \right) = \sigma \left( x \right) - {\sigma _{\max }} \le 0\\ {g_3}\left( x \right) = {x_1} - {x_4} \le 0\\ {g_4}\left( x \right) = 0.10471x_1^2 + 0.04811{x_3}{x_4}\left( {14 + {x_2}} \right) - 5 \le 0\\ {g_5}\left( x \right) = 0.125 - {x_1} \le 0\\ {g_6}\left( x \right) = \delta \left( x \right) - {\delta _{\max }} \le 0\\ {g_7}\left( x \right) = P - {P_c}\left( x \right) \le 0\\ 0.1 \le {x_i} \le 2{\rm{ }} {\rm{ }}i{\rm{ = 1,4}}\\ 0.1 \le {x_i} \le 10{\rm{ }} i{\rm{ = 2,3}}\\ {\rm{where,}}\\ \tau \left( x \right) = \sqrt {{{\left( {\tau '} \right)}^2} + 2\tau '\tau ''\frac{{{x_2}}}{{2R}} + {{\left( {\tau ''} \right)}^2}} ,\tau ' = \frac{P}{{\sqrt 2 {x_1}{x_2}}},\tau '' = \frac{{MR}}{J},M = P\left( {L + \frac{{{x_2}}}{2}} \right),R = \sqrt {{{\left( {\frac{{{x_2}}}{2}} \right)}^2} + {{\left( {\frac{{{x_1} + {x_3}}}{2}} \right)}^2}} ,\\ J = 2\left( {\sqrt 2 {x_1}{x_2}\left( {\frac{{x_2^2}}{{12}} + {{\left( {\frac{{{x_1} + {x_3}}}{2}} \right)}^2}} \right)} \right),\sigma \left( x \right) = \frac{{6PL}}{{{x_4}x_3^2}},\delta \left( x \right) = \frac{{4P{L^3}}}{{Ex_3^3{x_4}}},{P_c}\left( x \right) = \frac{{4.013E\sqrt {\frac{{x_3^2x_4^6}}{{36}}} }}{{{L^2}}}\left( {1 - \frac{{{x_3}}}{{2L}}\sqrt {\frac{E}{{4G}}} } \right)\\ P = 6000{\rm{lb}},L = 14{\rm{in}},E = 30 \times {10^6}{\rm{psi}},G = 12 \times {10^6}{\rm{psi}},{\tau _{\max }} = 13,600{\rm{psi}},{\rm{ }}{\sigma _{\max }} = 30,000{\rm{psi}},{\delta _{\max }} = 0.25{\rm{in}} \end{array}$$

A.2 Tension/compression spring design problem

$$\begin{array}{l} {\rm{Minimize }}f\left( x \right) = \left( {{x_3} + 2} \right){x_2}x_1^2\\ \\ {\rm{Subject \, to:}}\\ {g_1}\left( x \right) = 1 - \frac{{x_2^3{x_3}}}{{71,785x_1^4}} \le 0\\ {g_2}\left( x \right) = 4x_2^2 - \frac{{{x_1}{x_2}}}{{12.566\left( {{x_2}x_1^3 - x_1^4} \right)}} + \frac{1}{{5108x_1^2}} - 1 \le 0\\ {g_3}\left( x \right) = 1 - \frac{{140.45{x_1}}}{{x_2^2{x_3}}} \le 0\\ {g_4}\left( x \right) = {x_2} + \frac{{{x_1}}}{{1.5}} - 1 \le 0\\ {\rm{where}},\\ 0.05 \le {x_1} \le 2,{\rm{ }}0.25 \le {x_2} \le 1.30,{\rm{ }}2.00 \le {x_3} \le 15.00 \end{array}$$

A.3 Speed reducer design problem

$$\begin{array}{l} {\rm{Minimize }}f\left( x \right) = 0.7854{x_1}x_2^2\left( {3.3333x_3^2{\rm{ + }}14.9334{x_3} - 43.0934} \right) - 1.508{x_1}\left( {x_6^2 + x_7^2} \right) + 7.4777\left( {x_6^3 + x_7^3} \right) + \\ 0.7854\left( {{x_4}x_6^2 + {x_5}x_7^2} \right)\\ \\ {\rm{Subject \, to:}}\\ {g_1}\left( x \right) = \frac{{27}}{{{x_1}x_2^2{x_3}}} - 1 \le 0\\ {g_2}\left( x \right) = \frac{{397.5}}{{{x_1}x_2^2{x_3}}} - 1 \le 0\\ {g_3}\left( x \right) = \frac{{1.93x_4^3}}{{{x_2}x_6^4{x_3}}} - 1 \le 0\\ {g_4}\left( x \right) = \frac{{1.93x_5^3}}{{{x_2}x_7^4{x_3}}} - 1 \le 0\\ {g_5}\left( x \right) = \frac{{{{\left( {{{\left( {\frac{{745{x_4}}}{{{x_2}{x_3}}}} \right)}^2} + 16.9 \times {{10}^6}} \right)}^{1/2}}}}{{110x_6^3}} - 1 \le 0\\ {g_6}\left( x \right) = \frac{{{{\left( {{{\left( {\frac{{745{x_5}}}{{{x_2}{x_3}}}} \right)}^2} + 157.5 \times {{10}^6}} \right)}^{1/2}}}}{{85x_7^3}} - 1 \le 0\\ {g_7}\left( x \right) = \frac{{{x_2}{x_3}}}{{40}} - 1 \le 0\\ {g_8}\left( x \right) = \frac{{5{x_2}}}{{{x_1}}} - 1 \le 0\\ {g_9}\left( x \right) = \frac{{{x_1}}}{{12{x_2}}} - 1 \le 0\\ {g_{10}}\left( x \right) = \frac{{1.5{x_6} + 1.9}}{{{x_4}}} - 1 \le 0\\ {g_{11}}\left( x \right) = \frac{{1.1{x_7} + 1.9}}{{{x_5}}} - 1 \le 0\\ {\rm{where}},\\ 2.6 \le {x_1} \le 3.6,{\rm{ }}0.7 \le {x_2} \le 0.8,17 \le {x_3} \le 28,{\rm{ 7}}{\rm{.3}} \le {x_4} \le 8.3,{\rm{7}}{\rm{.3}} \le {x_5} \le 8.3,{\rm{ 2}}{\rm{.9}} \le {x_6} \le 3.9,{\rm{ 5}}{\rm{.0}} \le {x_7} \le 5.5 \end{array}$$

A.4 Three-bar truss design problem

$$\begin{array}{l} {\rm{Minimize }}f\left( x \right) = (2\sqrt 2 {x_1} + {x_2}) \times l\\ \\ {\rm{Subject \, to:}}\\ {g_1}\left( x \right) = \frac{{\sqrt 2 {x_1} + {x_2}}}{{\sqrt 2 x_1^2 + 2{x_1}{x_2}}}P - \sigma \le 0\\ {g_2}\left( x \right) = \frac{{{x_2}}}{{\sqrt 2 x_1^2 + 2{x_1}{x_2}}}P - \sigma \le 0\\ {g_3}\left( x \right) = \frac{1}{{\sqrt 2 x_1^2 + 2{x_1}{x_2}}}P - \sigma \le 0\\ {\rm{where}},\\ 0 \le {x_i} \le 1,{\rm{ }}i = 1,2\\ l = 100{\rm{cm, }}P{\rm{ = 2}}{{{\rm{k}}N} \mathord{\left/ {\vphantom {{{\rm{k}}N} {{\rm{c}}{{\rm{m}}^2},}}} \right. \kern-\nulldelimiterspace} {{\rm{c}}{{\rm{m}}^2},}}\sigma {\rm{ = 2}}{{{\rm{k}}N} \mathord{\left/ {\vphantom {{{\rm{k}}N} {{\rm{c}}{{\rm{m}}^2}}}} \right. \kern-\nulldelimiterspace} {{\rm{c}}{{\rm{m}}^2}}} \end{array}$$

A.5 Car impact design problem

$$\begin{array}{l} {\rm{Minimize }}f{\rm{(}}{\bf{x}}{\rm{) = 1}}{\rm{.98 + 4}}{\rm{.90}}{x_{\rm{1}}} + 6.67{x_2}{\rm{ + 6}}{\rm{.98}}{x_3} + 4.01{x_4}{\rm{ + 1}}{\rm{.78}}{x_5} + 2.73{x_7}\\ \\ {\rm{Subject \, to}}\\ {g_1}({\bf{x}}) = 1.16 - 0.3717{x_2}{x_4} - 0.00931{x_2}{x_{10}} - 0.484{x_3}{x_9} + 0.01343{x_6}{x_{10}} \le 1{\rm{KN}}\\ {g_2}({\bf{x}}) = 0.261 - 0.0159{x_1}{x_2} - 0.0188{x_1}{x_8} - 0.0191{x_2}{x_7} + 0.0144{x_3}{x_5} + 0.0008757{x_5}{x_{10}} + 0.08045{x_6}{x_9} + 0.00139{x_8}{x_{11}}\\ {\rm{ }} + 0.00001575{x_{10}}{x_{11}} \le 0.32{\rm{ m/s}}\\ {g_3}({\bf{x}}) = 0.214 + 0.00817{x_5} - 0.131{x_1}{x_8} - 0.0704{x_1}{x_9} + 0.03099{x_2}{x_6} - 0.018{x_2}{x_7} + 0.0208{x_3}{x_8} + 0.121{x_3}{x_9} - 0.00364{x_5}{x_6}\\ {\rm{ }} + 0.0007715{x_5}{x_{10}} - 0.0005354{x_6}{x_{10}} + 0.00121{x_8}{x_{11}} \le 0.32{\rm{ m/s}}\\ {g_4}({\bf{x}}) = 0.74 - 0.61{x_2} - 0.163{x_3}{x_8} + 0.001232{x_3}{x_{10}} - 0.166{x_7}{x_9} + 0.227x_2^2 \le 0.32{\rm{ m/s}}\\ {g_5}({\bf{x}}) = 28.98 + 3.818{x_3} - 4.2{x_1}{x_2} + 0.0207{x_5}{x_{10}} + 6.63{x_6}{x_9} - 7.7{x_7}{x_8} + 0.32{x_9}{x_{10}} \le 32{\rm{ mm}}\\ {g_6}({\bf{x}}) = 33.86 + 2.95{x_3} + 0.1792{x_{10}} - 5.057{x_1}{x_2} - 11.0{x_2}{x_8} - 0.0215{x_5}{x_{10}} - 9.98{x_7}{x_8} + 22.0{x_8}{x_9} \le 32{\rm{ mm}}\\ {g_7}({\bf{x}}) = 46.36 - 9.9{x_2} - 12.9{x_1}{x_8} - 5.057{x_1}{x_2} + 0.1107{x_3}{x_{10}} \le 32{\rm{ mm}}\\ {g_8}({\bf{x}}) = 4.72 - 0.5{x_4} - 0.19{x_2}{x_3} - 0.0122{x_4}{x_{10}} + 0.009325{x_6}{x_{10}} + 0.000191x_{11}^2 \le 4{\rm{KN}}\\ {g_9}({\bf{x}}) = 10.58 - 0.674{x_1}{x_2} - 1.95{x_2}{x_8} + 0.02054{x_3}{x_{10}} - 0.0198{x_4}{x_{10}} + 0.028{x_6}{x_{10}} \le 9.9{\rm{ mm/ms}}\\ {g_{10}}({\bf{x}}) = 16.45 - 0.489{x_3}{x_7} - 0.843{x_5}{x_6} + 0.0432{x_9}{x_{10}} - 0.0556{x_9}{x_{11}} - 0.000786x_{11}^2 \le 15.7{\rm{ mm/ms}}\\ {\rm{where }}0.5 \le {x_i} \le 1.5,{\rm{ }}i = 1,2,3,4,5,6,7;{\rm{ }}0.192 \le {x_i} \le 0.345,{\rm{ }}i = 8,9;{\rm{ }} - 30 \le {x_i} \le 30,{\rm{ }}i = 10,11.{\rm{ }} \end{array}$$

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Zhang, Y., Jin, Z. Comprehensive learning Jaya algorithm for engineering design optimization problems. J Intell Manuf 33, 1229–1253 (2022). https://doi.org/10.1007/s10845-020-01723-6

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