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An improved teaching-learning-based optimization algorithm for numerical and engineering optimization problems

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Abstract

The teaching-learning-based optimization (TLBO) algorithm, one of the recently proposed population-based algorithms, simulates the teaching-learning process in the classroom. This study proposes an improved TLBO (ITLBO), in which a feedback phase, mutation crossover operation of differential evolution (DE) algorithms, and chaotic perturbation mechanism are incorporated to significantly improve the performance of the algorithm. The feedback phase is used to enhance the learning style of the students and to promote the exploration capacity of the TLBO. The mutation crossover operation of DE is introduced to increase population diversity and to prevent premature convergence. The chaotic perturbation mechanism is used to ensure that the algorithm can escape the local optimal. Simulation results based on ten unconstrained benchmark problems and five constrained engineering design problems show that the ITLBO algorithm is better than, or at least comparable to, other state-of-the-art algorithms.

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Acknowledgments

This research was supported by Major State Basic Research Development Program of China under Grant No. 2012CB720500, National Natural Science Foundation of China under Grant Nos. 61333010, 61222303, Fundamental Research Funds for the Central Universities, National High-Tech Research and Development Program of China under Grant No. 2013AA040701, National Key Scientific and Technical Project of China under Grant No. 2012BAF05B00, Shanghai R&D Platform Construction Program under Grant No. 13DZ2295300, and Open Research Fund of State Key Laboratory of Synthetical Automation for Process Industries.

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Authors and Affiliations

  1. Key Laboratory of Advanced Control and Optimization for Chemical Processes, Ministry of Education, East China University of Science and Technology, Shanghai , 200237, China

    Kunjie Yu & Zhenlei Wang

  2. Center of Electrical and Electronic Technology, Shanghai Jiao Tong University, Shanghai , 200240, China

    Xin Wang

Authors
  1. Kunjie Yu

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  2. Xin Wang

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  3. Zhenlei Wang

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Corresponding author

Correspondence toZhenlei Wang.

Appendix: Constrained engineering design problems

Appendix: Constrained engineering design problems

Problem 1: the welded beam design problem

Minimize:

$$\begin{aligned} f(x)=1.10471x_1^2 x_2 +0.04811x_3 x_4 (14.0+x_2 ) \end{aligned}$$

Subject to:

$$\begin{aligned} g_1 (x)&= \tau (x)-\tau _{\max } \le 0 \\ g_2 (x)&= \sigma (x)-\sigma _{\max } \le 0 \\ g_3 (x)&= x_1 -x_4 \le 0 \\ g_4 (x)&= 0.1047x_1^2 +0.04811x_3 x_4 (14.0+x_2 )-5.0\le 0 \\ g_5 (x)&= 0.125-x_1 \le 0 \\ g_6 (x)&= \delta (x)-\delta _{\max } \le 0 \\ g_7 (x)&= P-P_C (x)\le 0 \end{aligned}$$

where

$$\begin{aligned}&\tau (x)=\sqrt{(\tau ^{{\prime }})^{2}+2\tau ^{{\prime }}\tau ^{{\prime }{\prime }}\frac{x_2 }{2R}+(\tau ^{{\prime }{\prime }})^{2}} \\&\tau ^{{\prime }}=\frac{P}{\sqrt{2}x_1 x_2 },\tau ^{{\prime }{\prime }}=\frac{MR}{J}, M=P\left( L+\frac{x_2 }{2}\right) \\&R=\sqrt{\frac{x_2^2 }{4}+\left( \frac{x_1 +x_3 }{2}\right) ^{2}} \\&J=2\left\{ {\sqrt{2}x_1 x_2 \left[ \frac{x_2^2 }{12}+(\frac{x_1 +x_3 }{2})^{2}\right] } \right\} \\&\sigma (x)=\frac{6PL}{x_4 x_3^2 },\delta (x)=\frac{4PL^{3}}{Ex_3^3 x_4 } \\&P_c (x)=\frac{4.013E\sqrt{\frac{x_3^2 x_4^6 }{36}}}{L^{2}}\left( 1-\frac{x_3 }{2L}\sqrt{\frac{E}{4G}}\right) \\&P=6000\hbox {lb},L=14\hbox {in},E=30e6\hbox {psi},\\&G=12e6\hbox {psi,}\tau _{\max } =13600\hbox {psi}, \\&\sigma _{\max } =30000\hbox {psi,}\delta _{\max } =0.25\hbox {in} \\&0.1 \le x_1 \le 2.0,\quad 0.1\le x_2 \le 10.0 \\&0.1\le x_3 \le 10.0,\quad 0.1\le x_4 \le 2.0 \end{aligned}$$

Problem 2: the tension/compression spring design problem

Minimize:

$$\begin{aligned} f(x)=(x_3 +2)x_2 x_1^2 \end{aligned}$$

Subject to:

$$\begin{aligned} g_1 (x)&= 1-\frac{x_2 ^{3}x_3 }{71785x_1 ^{4}}\le 0 \\ g_2 (x)&= \frac{4x_2 ^{2}-x_1 x_2 }{12566(x_1 ^{3}x_2 -x_1 ^{4})}+\frac{1}{5108x_1 ^{2}}-1\le 0 \\ g_3 (x)&= 1-\frac{140.45x_1 }{x_2 ^{2}x_3 }\le 0 \\ g_4 (x)&= \frac{x_1 +x_2 }{1.5}-1\le 0 \end{aligned}$$

where

$$\begin{aligned} 0.05\le x_1 \le 2,0.25\le x_2 \le 1.3,2\le x_3 \le 15 \end{aligned}$$

Problem 3: the pressure vessel problem

Minimize:

$$\begin{aligned} f(x)&= 0.6224x_1 x_3 x_4 +1.7781x_2 x_3^2\\&\quad +\,3.1661x_1^2 x_4 +19.84x_1^2 x_3 \end{aligned}$$

Subject to:

$$\begin{aligned} g_1 (x)&= -x_1 +0.0193x_3 \le 0 \\ g_2 (x)&= -x_2 +0.00954x_3 \le 0 \\ g_3 (x)&= -\pi x_3^2 x_4 -\frac{4}{3}\pi x_3^3 +1296000\le 0 \\ g_4 (x)&= x_4 -240\le 0 \end{aligned}$$

where

$$\begin{aligned} 1\le x_1 \le 99,1\le x_2 \le 99,10\le x_3 \le 200,10\le x_4 \le 200 \end{aligned}$$

\(x_1 \) and\(x_2 \) are integer multiples of 0.0625.

Problem 4: the speed reducer design problem

Minimize:

$$\begin{aligned} f(x)&= 0.7854x_1 x_2^2 (3.3333x_3^2 +14.9334x_3 -43.0934) \\&\quad -\,1.508x_1 (x_6^2 +x_7^2 )+7.4777(x_6^3 +x_7^3 ) \\&\quad +\,0.7854(x_4 x_6^2 +x_5 x_7^2 ) \end{aligned}$$

Subject to:

$$\begin{aligned} g_1 (x)&= \frac{27}{x_1 x_2^2 x_3 }-1\le 0 \\ g_2 (x)&= \frac{397.5}{x_1 x_2^2 x_3^2 }-1\le 0 \\ g_3 (x)&= \frac{1.93x_4^3 }{x_2 x_3 x_6^4 }-1\le 0 \\ g_4 (x)&= \frac{1.93x_5^3 }{x_2 x_3 x_7^4 }-1\le 0 \\ g_5 (x)&= \frac{\sqrt{(\frac{745x_4 }{x_2 x_3 })^{2}+16.9e6}}{110x_6^3 }-1\le 0 \\ g_6 (x)&= \frac{\sqrt{(\frac{745x_5 }{x_2 x_3 })^{2}+157.5e6}}{85x_7^3 }-1\le 0 \\ g_7 (x)&= \frac{x_2 x_3 }{40}-1\le 0 \\ g_8 (x)&= \frac{5x_2 }{x_1 }-1\le 0 \\ g_9 (x)&= \frac{x_1 }{12x_2 }-1\le 0\\ g_{10} (x)&= \frac{1.56x_6 +1.9}{x_4 }-1\le 0 \\ g_{11} (x)&= \frac{1.1x_7 +1.9}{x_5 }-1\le 0 \end{aligned}$$

where

$$\begin{aligned}&2.6\le x_1 \le 3.6,0.7\le x_2 \le 0.8,17\le x_3 \le 28 \\&7.3\le x_4 \le 8.3,7.3\le x_5 \le 8.3,2.9\le x_6 \le 3.9 \\&5.0\le x_7 \le 5.5 \end{aligned}$$

Problem 5: the three-bar truss design problem

Minimize:

$$\begin{aligned} f(x)=(2\sqrt{2}x_1 +x_2 )l \end{aligned}$$

Subject to:

$$\begin{aligned} g_1 (x)&= \frac{\sqrt{2}x_1 +x_2 }{\sqrt{2}x_1^2 +2x_1 x_2 }p-\sigma \le 0 \\ g_2 (x)&= \frac{x_2 }{\sqrt{2}x_1^2 +2x_1 x_2 }p-\sigma \le 0 \\ g_3 (x)&= \frac{1}{\sqrt{2}x_2 +x_1 }p-\sigma \le 0 \end{aligned}$$

where

$$\begin{aligned}&0\le x_1 \le 1,0\le x_2 \le 1;l=100\hbox {cm, }p=2KN/\hbox {cm}^{2}, \\&\sigma =2 KN/\hbox {cm}^{2} \end{aligned}$$

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Yu, K., Wang, X. & Wang, Z. An improved teaching-learning-based optimization algorithm for numerical and engineering optimization problems.J Intell Manuf27, 831–843 (2016). https://doi.org/10.1007/s10845-014-0918-3

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