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SChoA: a newly fusion of sine and cosine with chimp optimization algorithm for HLS of datapaths in digital filters and engineering applications

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Abstract

The Chimp optimization algorithm (ChoA) inspired by the individual intelligence and sexual motivation of chimps in their group hunting, which is separate from the another social predators. Generally, it is developed for trapping in local optima on the complex functions and alleviate the slow convergence speed. This algorithm has been widely applied to find the best optima solutions of complex global optimization tasks due to its simplicity and inexpensive computational overhead. Nevertheless, premature convergence is easily trapped in the local optimum solution during search process and is ineffective in balancing exploitation and exploration. In this paper, we have developed a modified novel nature inspired optimizer algorithm based on the sine–cosine functions; it is called as sine–cosine chimp optimization algorithm (SChoA). During this research, the sine–cosine functions have been applied to update the equations of chimps during the search process for reducing the several drawbacks of the ChoA algorithm such as slow convergence rate, locating local minima rather than global minima, and low balance amid exploitation and exploration. Experimental solutions based on 23-standard benchmark and 06 engineering functions such as welded beam, tension/compression spring, pressure vessel, multiple disk clutch brake, planetary gear train and digital filters design, etc. demonstrate the robustness, effectiveness, efficiency, and convergence speed of the proposed algorithm in comparison with others.

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Authors and Affiliations

  1. Department of Electronics and Communication Engineering, Punjabi University, Patiala, India

    Mandeep Kaur & Ranjit Kaur

  2. Department of Mathematics, Punjabi University, Patiala, India

    Narinder Singh

  3. Government Bikram College of Commerce, Patiala, India

    Gaurav Dhiman

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  1. Mandeep Kaur
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  2. Ranjit Kaur
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Appendix

Appendix

See Table 15.

1.1 Welded beam design problem

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

1.2 Tension/compression spring design problem

Consider:

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

1.3 Pressure vessel design problem

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

1.4 Multiple disk clutch brake design problem

$$\begin{aligned}&\quad \text {Minimize}f(x)=\pi (x_{2}^{2}-x_{1}^{2})x_3(x_5+1)\rho \\&\text {Variable range} \\&x_1 \varepsilon \left\{ 60,61,...,79,80 \right\} \\&x_2 \varepsilon \left\{ 90,91,...,109,110 \right\} \\&x_3 \varepsilon \left\{ 1,1.5,...,3 \right\} \\&x_4 \varepsilon \left\{ 600,610,...,990,1000 \right\} \\&x_5 \varepsilon \left\{ 2,3,...,8,9 \right\} \\&\text {Subject to;} g_1(x)=x_2-x_1-\Delta r \le 0 \\&g_2(x)=L_{\max }-(x_5+1)(x_3+\delta ) \le 0 \\&g_3(x)=p_{\max }-p_{rz}\le 0 \\&g_4(x)=p_{\max }v_{\mathrm{sr},\, \max }-p_{rz}v_{sr}\le 0 \\&g_5(x)=v_{\mathrm{sr}, \max }-v_{sr}\le 0 \\&g_6(x)=M_h-sM_s \le 0 \\&g_7(x)=T \le 0 \\&g_8(x)=T_{\max }-T \le 0 \\&\Delta r =20\,(\mathrm{mm}), I_z = 55\,(\mathrm{kg},\,\mathrm{mm}^2), p_{\max }=1\,(\mathrm{MPa}), T_{max}=15 (\mathrm{s}), \mu =0.50 \\&s=1.50. M_s=40\,(\mathrm{Nm}), M_f=3\,(\mathrm{Nm}), n=250\,(\mathrm{rpm}),v_{\mathrm{sr},\, \mathrm{max}}=10(\frac{m}{s}), L_{\max }=30(mm). \end{aligned}$$

1.5 Planetary gear train design problem

$$\begin{aligned}&\quad \text {Maximize }f(x)=\max \left| i_k-i_{ok} \right| ; k=\left\{ 1,2,R \right\} \\&\mathrm{where} \\&i_1=N_6/N_4 \\&i_{01}=3.11 \\&i_2=\frac{N_6(N_1N_3+N_2N_4)}{N_1N_3(N_6-N_4)} \\&i_{02}=1.84 \\&i_R=-\frac{N_2N_6}{N_1N3} \\&i_{0R}=-3.11 \\&X= \left\{ N_1,N_2,N_3,N_4,N_5,p,m_1,m_2 \right\} , \\&\text {Subject to: }\\&g_1(X)=m_3(N_6+2.5)\le D_{\max } \\&g_2(X)=m_1(N_1+N_2)+m_1(N_2+2) \le D_{\max } \\&g_3(X)=m_3(N_4+N_5)+m_3(N_5+2) \le D_{\max } \\&g_4(X)=\left| m_1(N_1+N_2)-m_3(N_6-N_3) \right| \le m_1+m_3 \\&g_5(X)=(N_1+N_2)\sin \left( \frac{\pi }{p}\right) -N_2-2-\delta _{22} \ge 0 \\&g_6(X)=(N_6-N_3)\sin \left( \frac{\pi }{p}\right) -N_3-2-\delta _{33} \ge 0 \\&g_7(X)=(N_4+N_5)\sin \left( \frac{\pi }{p}\right) -N_5-2 -\delta _{55} \ge 0 \\&g_8(X)=(N_6-N_3)^2+(N_4+N_5)^2-(N_6-N_3)(N_4+N_5)cos(\frac{2\pi }{p}-\beta )\le (N+3+N_5+2+\delta _{35})^2 \\&\mathrm{where} \\&\beta = \frac{\cos ^{-1}(N_6-N_3)^2+(N_4+N-5)^2-(N_3+N_5)^2}{2(N_6-N_3)(N_4+N_5)} \\&g_9(X) = N_6-2N_3-N_4-4-2\delta _{34} \ge 0 \\&g_10(X)=N_6-N_4-2N_5-4-2\delta _{56} \ge 0 \\&h(X)=\frac{N_6-N_4}{p}\,=\, \mathrm{integer},\\&\mathrm{where} \\&D_{\max }=220, p=(3,4,5), m_1,m_3=(1.75,2.0,2.25,2.5,2.75,3.0), \delta _{22},\delta _{33},\delta _{55},\delta _{35},\delta _{56},=0.5\\&17\le N_1 \le 96, 14\le N_2 \le 54, 14\le N_3 \le 51, 17\le N_4 \le 46, \\&14\le N_5 \le 51, 48\le N_6 \le 124, N_i=integer. \end{aligned}$$
Table 15 23-standard benchmark test functions
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Kaur, M., Kaur, R., Singh, N. et al. SChoA: a newly fusion of sine and cosine with chimp optimization algorithm for HLS of datapaths in digital filters and engineering applications. Engineering with Computers 38 (Suppl 2), 975–1003 (2022). https://doi.org/10.1007/s00366-020-01233-2

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