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Parallel chaotic local search enhanced harmony search algorithm for engineering design optimization

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Abstract

In this paper, we present a parallel chaotic local search enhanced harmony search algorithm (MHS–PCLS) for solving engineering design optimization problems. The concept of chaos has been previously successfully applied in metaheuristics. However, chaos sequences are sensitive to their initial conditions and cause unstable performance in algorithms. The proposed parallel chaotic local search method searches from several different initial points and diminishes the sensitivity of the initial condition, thereby increasing the robustness of the harmony search method. Numerical benchmark problems are tested to validate the effectiveness of MHS–PCLS. The simulation results confirm that MHS–PCLS obtains superior results for mathematical examples compared to other harmony search variants. Several well-known constrained engineering design problems are also tested using the new approach. The computational results demonstrate that the proposed MHS–PCLS algorithm requires a smaller number of function evaluations and in the majority of cases delivers improved and more robust results compare to other algorithms.

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Acknowledgments

The authors would like to thank the cloud system in HUST for providing us the computing services. This research work is supported by the National Natural Science Foundation of China (NSFC) under Grant Nos. 51435009, 61232008 and 51421062, and Youth Science & Technology Chenguang Program of Wuhan under Grant no. 2015070404010187.

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Authors and Affiliations

  1. The State Key Laboratory of Digital Manufacturing Equipment and Technology, School of Mechanical Science and Technology, Huazhong University of Science and Technology, Wuhan, 430074, People’s Republic of China

    Jin Yi, Xinyu Li & Liang Gao

  2. Department of Industrial Engineering and Engineering Management, National Tsing-Hua University, Hsinchu, Taiwan

    Chih-Hsing Chu

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  1. Jin Yi
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  2. Xinyu Li
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  3. Chih-Hsing Chu
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  4. Liang Gao
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Corresponding author

Correspondence to Xinyu Li.

Appendix: Mathematical model of the design problems

Appendix: Mathematical model of the design problems

1. Design of tension/compression spring

$$\begin{aligned}&Minimize: \quad f({{{\varvec{x}}}})=(x_3+2)x_2x^2_1, \end{aligned}$$
(12)
$$\begin{aligned}&Subject \quad to: \left\{ \begin{array}{rcl} &{}&{} {g_1({{{\varvec{x}}}})=1-\frac{x^3_2x_3}{71785x^4_1}\le 0} \\ &{}&{} {g_2({{{\varvec{x}}}})= \frac{4x^2_2-x_1x_2}{12566(x_2x^3_1-x^4_1)}+\frac{1}{5108x^2_1}-1\le 0} \\ &{}&{} {g_3({{{\varvec{x}}}})= 1-\frac{140.45x_1}{x^2_2x_3}\le 0} \\ &{}&{} {g_4({{{\varvec{x}}}})= \frac{x_1+x_2}{1.5}-1\le 0} \\ \end{array} \right. \nonumber \\ \end{aligned}$$
(13)

where \(0.05\le x_1 \le 2, 0.25\le x_2\le 1.3, 2\le x_3\le 15\).

2. Design of welded beam

$$\begin{aligned}&Minimize: \quad f({{{\varvec{x}}}})=1.10471x^2_1x_2\nonumber \\&\qquad \qquad \qquad \qquad +\,0.04811x_3x_4(14.0+x_2), \end{aligned}$$
(14)
$$\begin{aligned}&Subject \quad to: \left\{ \begin{array}{rcl} &{} &{} {g_1({{{\varvec{x}}}})=\tau ({{{\varvec{x}}}})-13000\le 0} \\ &{} &{} {g_2({{{\varvec{x}}}})= \sigma ({{{\varvec{x}}}})-30000\le 0} \\ &{} &{} {g_3({{{\varvec{x}}}})= x_1-x_4\le 0} \\ &{} &{} g_4({{{\varvec{x}}}})= 0.1047x^2_1\\ &{}&{}\qquad \qquad +\,0.04811x_3x_4(14.0+x_2)-5.0 \le 0 \\ &{} &{} {g_5({{{\varvec{x}}}})= 0.125-x_1 \le 0} \\ &{} &{} {g_6({{{\varvec{x}}}})= \delta ({{{\varvec{x}}}})-0.25 \le 0} \\ &{} &{} {g_7({{{\varvec{x}}}})= 6000-P_c({{{\varvec{x}}}}) \le 0} \end{array} \right. \nonumber \\ \end{aligned}$$
(15)

where \(\tau ({{{\varvec{x}}}})=\sqrt{(\tau ^{'})^2+2\tau ^{'}\tau ^{''} \frac{x_2}{2R}+(\tau ^{''})^2}, \tau ^{'}=\frac{6000}{\sqrt{2}x_1x_2}\)

\(\tau ^{''}=\frac{MR}{J}\),\(M=6000(14+\frac{x_2}{2})\),\(R=\sqrt{\frac{x^2_2}{4}+(\frac{x_1+x_3}{2})^2}\)

\(J=2\{\sqrt{2}x_1x_2[\frac{x^2_2}{12}+(\frac{x_1+x_3}{2})^2]\}\), \(\sigma ({{\varvec{x}}})=\frac{504000}{x_4x^2_3}\)

\(\delta ({{\varvec{x}}})=\frac{2.1952}{x^2_3x_4}\),\(P_c({{\varvec{x}}})=64746.022(1-0.0282346x_3)x_3x^3_4\)

\(0.1\le x_1\le 2\),\(0.1\le x_2\le 10\),\(0.1\le x_3\le 10\),\(0.1\le x_4\le 2\).

3. Design of pressure vessel

$$\begin{aligned} Minimize: \quad f({{\varvec{x}}})= & {} 0.6224x_1x_3x_4+1.7781x_2x^2_3\nonumber \\&+\,3.1661x^2_1x_4+19.84x^2_1x_3, \end{aligned}$$
(16)
$$\begin{aligned}&Subject \quad to: \left\{ \begin{array}{rcl} &{} &{} {g_1({{{\varvec{x}}}})=-x_1+0.0193x_3 \le 0} \\ &{} &{} {g_2({{{\varvec{x}}}})= -x_2+0.00954x_3\le 0} \\ &{} &{} {g_3({{{\varvec{x}}}})= -\pi x^2_3x_4-\frac{4}{3}\pi x^3_3+1296000\le 0} \\ &{} &{} {g_4({{{\varvec{x}}}})= x_4-240 \le 0} \end{array} \right. \nonumber \\ \end{aligned}$$
(17)

where \(0\le x_1 \le 99, 0\le x_2\le 99, 10\le x_3\le 200, 10\le x_4\le 200\).

4. Design of speed reducer

$$\begin{aligned}&Minimize:\nonumber \\&\quad f({{{\varvec{x}}}}) = 0.7854x_1x^2_2(3.3333x^2_3+14.9334x_3-43.0934) \nonumber \\&\qquad -\, 1.508x_1(x^2_6+x^2_7)+7.4777(x^3_6+x^3_7) \nonumber \\&\qquad +\, 0.7854(x_4x^2_6+x_5x^2_7)\end{aligned}$$
(18)
$$\begin{aligned}&Subject \quad to: \left\{ \begin{array}{rcl} &{}&{} {g_1({{{\varvec{x}}}})=\frac{27}{x_1x^2_2x_3}-1 \le 0} \\ &{}&{} {g_2({{{\varvec{x}}}})= \frac{397.5}{x_1x^2_2x^2_3}-1\le 0} \\ &{}&{} {g_3({{{\varvec{x}}}})= \frac{1.93x^3_4}{x_2x^4_6x_3}-1\le 0 } \\ &{}&{} {g_4({{{\varvec{x}}}})= \frac{1.93x^3_5}{x_2x^4_7x_3}-1 \le 0 } \\ &{}&{} {g_5({{{\varvec{x}}}})= \frac{[745(x_4/x_2x_3)^2+16.9\times 10^6]^{1/2}}{110x^3_6}-1\le 0 } \\ &{}&{} {g_6({{{\varvec{x}}}})= \frac{[745(x_5/x_2x_3)^2+157.5\times 10^6]^{1/2}}{85x^3_7}-1\le 0 } \\ &{}&{} {g_7({{{\varvec{x}}}})= \frac{x_2x_3}{40}-1 \le 0 } \\ &{}&{} {g_8({{{\varvec{x}}}})= \frac{5x_2}{x_1}-1 \le 0 } \\ &{}&{} {g_9({{{\varvec{x}}}})= \frac{x_1}{12x_2}-1 \le 0 } \\ &{}&{} {g_{10}({{{\varvec{x}}}})= \frac{1.5x_6+1.9}{x_4}-1 \le 0 } \\ &{}&{} {g_{11}({{{\varvec{x}}}})= \frac{1.1x_6+1.9}{x_5}-1 \le 0 } \\ \end{array} \right. \nonumber \\ \end{aligned}$$
(19)

where \(2.6\le x_1 \le 3.6, 0.7\le x_2\le 0.8, 17\le x_3\le 28, 7.3\le x_4\le 8.3, 7.3\le x_5\le 8.3, 2.9\le x_4\le 3.9, 5.0\le x_4\le 5.5\).

5. Car side impact design

$$\begin{aligned}&Minimize: \quad f({{{\varvec{x}}}}) = 1.98+4.90x_1+6.67x_2+6.98x_3+4.01x_4+1.78x_5+2.73x_7 - 1.508x_1(x^2_6+x^2_7)\nonumber \\&\qquad \qquad \qquad \qquad \qquad \qquad +\,7.4777(x^3_6+x^3_7) + 0.7854(x_4x^2_6+x_5x^2_7) \end{aligned}$$
(20)
$$\begin{aligned}&Subject \quad to: \left\{ \begin{array}{rcl} &{}&{} g_1=F_a=1.16-0.3717x_2x_4-0.00931x_2x_{10}-0.484x_3x_9+0.01343x_6x_{10}-1 \le 0 \\ &{}&{} g_2= VC_u= 0.261-0.0159x_1x_2-0.188x_1x_8-0.019x_2x_7+0.0144x_3x_5+0.0008757x_5x_{10} \\ &{}&{} \qquad \quad +0.080405x_6x_9+0.00139x_8x_{11}+0.00001575x_{10}x_{11}-0.32 \le 0 \\ &{}&{} g_3= VC_m= 0.214+0.00817x_5-0.131x_1x_8-0.0704x_1x_9+0.03099x_2x_6-0.018x_2x_7 \\ &{}&{} \qquad \quad +0.0208x_3x_8+0.121x_3x_9-0.00364x_5x_6+0.0007715x_5x_{10}-0.0005354x_6x_{10} \\ &{}&{} \qquad \quad +0.00121x_8x_{11}-0.32 \le 0 \\ &{}&{} g_4=VC_l=0.074-0.061x_2-0.163x_3x_8+0.001232x_3x_{10}-0.166x_7x_9+0.27x^2_2-0.32 \le 0 \\ &{}&{} g_5= \Delta _{ur}=28.98+3.818x_3-4.2x_1x_2+0.0207x_5x_{10}+6.63x_6x_9-7.7x_7x_8+0.32x_9x_{10}-32 \le 0 \\ &{}&{} g_6=\Delta _{mr}=33.86+2.95x_3+0.1792x_{10}-5.057x_1x_2-11.0x_2x_8-0.0215x_5x_{10} \\ &{}&{} \qquad \quad -9.98x_7x_8+22.0x_8x_9-32 \le 0 \\ &{}&{} g_7=\Delta _{lr}= 46.36-9.9x_2-12.9x_1x_8+0.1107x_3x_{10}-32 \le 0 \\ &{}&{} g_8= F_p=4.72-0.5x_4-0.19x_2x_3-0.0122x_4x_{10}+0.009325x_6x_{10}+0.000191x^2_{11}-4 \le 0 \\ &{}&{} g_9= V_{MBP}=10.58-0.674x_1x_2-1.95x_2x_8+0.02054x_3x_{10}-0.0198x_4x_{10}+0.028x_6x_{10}-9.9 \le 0 \\ &{}&{} g_10=V_{FD}=16.45-0.489x_3x_7-0.843x_5x_6+0.0432x_9x_{10}-0.0556x_9x_{11}-0.000786x^2_{11}-15.7 \le 0 \end{array} \right. \nonumber \\ \end{aligned}$$
(21)

where \(0.5 \le x_1 \sim x_7 \le 1.5, x_8,x_9 \in (0.192,0.345), -30\le x_{10},x_{11}\le 30\).

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Yi, J., Li, X., Chu, CH. et al. Parallel chaotic local search enhanced harmony search algorithm for engineering design optimization. J Intell Manuf 30, 405–428 (2019). https://doi.org/10.1007/s10845-016-1255-5

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