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HSSAHHO: a novel hybrid Salp Swarm-Harris Hawks optimization algorithm for complex engineering problems

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Abstract

With Metaheuristic algorithms hybridization is used typically to improve the performances of original algorithm, so it became a recent trend of research is to hybridize two and several number of variants to find out better quality of solution of practical and to tackle with recent real applications in the field of global optimization problems. Harris Hawks Optimization (HHO) is powerful optimization tool that is robust, has smooth transitions between exploration and exploitation, that permits to provide competitive results in complex problems. Also, the Salp Swarm Algorithm (SSA) has been widely used to solve complex global optimization tasks due to its implementation simplicity and inexpensive computational overhead. However, SSA has premature convergence, is easily trapped in the local optimum solution and is ineffective in balancing exploration and exploitation. In this study, to overcome the shortcomings of HHO and SSA, a hybrid Salp Swarm Algorithm with Harris Hawks Optimization (HHO) called (HSSAHHO) can be considered as a good choice for this purpose to solve these issues. Based on this consideration, hybrid SSAHHO is combination of SSA used for exploitation phase and HHO for exploration phase in uncertain environment. The movement directions and speed of the Salp’ is improved using position update equations of HHO. To evaluate the effectiveness in solving the global optimization problems, HSSAHHO has been tested on 29-standard IEEE CEC 2017, 10-standard IEEE CEC 2020 benchmark test problems and also tested on six engineering design optimization problems. The numerical and statistical solutions obtained with HSSAHHO approach is compared with other meta-heuristics approaches such as Salp Swarm Optimizer (SSA), Particle Swarm Optimization (PSO), Moth-flame Optimization (MFO), Sine Cosine Algorithm (SCA), Enhanced grey wolf optimisation (EGWO), Augmented Grey Wolf Optimizer (AGWO) Hybrid Particle Swarm Optimization-Grey Wolf Optimizer (PSOGWO), Harris Hawks Optimization (HHO) and Coot algorithm. The conducted experimental statistical and convergence prove that the proposed hybrid variant can highly be effective in solving benchmark and real life global optimization problems.

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Authors and Affiliations

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

    Narinder Singh & S. B. Singh

  2. Faculty of Computers and Information, Minia University, El Minya, Egypt

    Essam H. Houssein

  3. Department of Computer Science, Government Bikram College of Commerce, Patiala, Punjab, India

    Gaurav Dhiman

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  1. Narinder Singh
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  2. Essam H. Houssein
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  3. S. B. Singh
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  4. Gaurav Dhiman
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Correspondence to Gaurav Dhiman.

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Appendices

Appendix 1: Welded beam design problem

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

with \(0.1\le {{x}_{1}},{{x}_{4}}\le 2.0\,\mathrm{and}\,0.1\le {{x}_{2}},{{x}_{3}}\le 10.0\)

Appendix 2: Tension/compression spring design problem

$$\begin{aligned}&\qquad \text {Consider:}\\&\vec {x}=\left[ {{x}_{1}}{{x}_{2}}{{x}_{3}} \right] =\left[ d\,D\,N \right] \\&\text {Minimize } f\left( {\vec {x}} \right) =\left( {{x}_{3}}+2 \right) {{x}_{2}}x_{1}^{2} \\&\text {subject to: } \\&{{g}_{1}}\left( {\vec {x}} \right) =1-\frac{x_{2}^{3}{{x}_{3}}}{71785x_{1}^{4}}\le 0 \\&{{g}_{2}}\left( {\vec {x}} \right) =\frac{4x_{2}^{2}-{{x}_{1}}{{x}_{2}}}{12566\left( {{x}_{2}}x_{1}^{3}-x_{1}^{4} \right) }+\frac{1}{5108x_{1}^{2}}-1\le 0 \\&{{g}_{3}}\left( {\vec {x}} \right) =1-\frac{140.45{{x}_{1}}}{x_{2}^{2}{{x}_{3}}}\le 0 \\&{{g}_{4}}\left( {\vec {x}} \right) =\frac{{{x}_{1}}+{{x}_{2}}}{1.5}-1\le 0 \end{aligned}$$

with \(0.05\le {{x}_{1}}\le 2.0,0.25\le {{x}_{2}}\le 1.3,and\,2.0\le {{x}_{3}}\le 15.0\)

Appendix 3: Rolling element bearing design problem

$$\begin{aligned}&\qquad \text {Maximum}[C_d(X)]={\left\{ \begin{array}{ll}max\left( -f_cz^{2/3}D_b^{1.8}\right) &{} if D_b =\le 25.5mm 0\\ max\left( -3.647f_cz^{2/3}D_b^{1.4}\right) &{} if D_b =\le 25.5mm \end{array}\right. } \\&\text {Subject to: }\\&g_1(x)=\frac{\phi _0}{2\sin ^{-1}(D_b/D_m)}-Z+1\ge 0, \\&g_2(x)=2D_b-K_{D_{min}}(D-d)\ge 0,\\&g_3(x)=K_{D_{max}}(D-d)-2D_b\ge 0,\\&g_4(x)=\zeta B_w-D_b\ge 0,\\&g_5(x)=D_m-0.5(D+d)\ge 0,\\&g_6(x)=(0.5+e)(D+d)-D_m\ge 0,\\&g_7(x)=0.5\left( D-D_m-D_b\right) -\zeta D_b\ge 0,\\&g_8(x)=f_1\ge 0.515,\\&g_9(x)=f_0\ge 0.515,\\&where \\&f_c=37.91\left[ 1+\left[ 1.04\left( \frac{1-\gamma }{1+\gamma }\right) ^{1.72}\right] \right. \\&\qquad \left. \left. \left( \frac{f_i(2f_0-1)}{f_i-1}\right) ^{0.41}\right] ^{10/3}\right] ^{-0.3}\\&\gamma = \frac{D_b\cos \alpha }{D_m},\\&f_1=\frac{r_1}{D_b}\\&\phi _0=2\pi -20\cos ^{-1}\\&\qquad \left[ \frac{(D-d)/2-3(T/4)^2+\left[ D/2-(T/4-D_b)^2\right] -\left[ d/2+(T/4)]^2\right] }{\left[ 2(D-d/2-3(T/4)\right] \left[ (T/4)-D_b\right] }\right] \\&T=D-d-2D_b,\\&D=160, d=90, B_w=30,\\&0.5(D+d)\le D_m\le 0.6(D+d),\\&0.15(D-d)\le D_b\le 0.45(D-d),\\&4\le Z\le 50, 0.515 \le f_1\le 0.6, 0.515\le f_0 \le 0.6,\\&0.4 \le K_{D_{min}}\le 0.5, 0.6 \le K_{D_{max}}\le 0.7,\\&0.3\le \epsilon \le 0.4, 0.02\le e \le 0.1, 0.6\le \xi \le 0.85 \end{aligned}$$

Appendix 4: Multiple disk clutch brake design problem

$$\begin{aligned}&\quad {Minimize } f(x) = \pi \left( r_0^2-r_i^2\right) (Z+1)\rho t\\&s.t ;\\&g_1(x)=r_0-r_i-\Delta r\geqslant 0 \\&g_2(x)=l_max-(Z+1)(t+\delta )\geqslant 0 \\&g_3(x)=P_max-P_{rz}\geqslant 0\\&g_4(x)=P_max v_{vrmax}-P_{rz}v_{sr}\geqslant 0 \\&g_5(x)=v_{srmax}-v_{sr}\geqslant 0 \\&g_6(x)=T_max-T\geqslant 0 \\&g_7(x)=M_h-sM_s\geqslant 0 \\&g_8(x)=T\geqslant 0\\&where \\&M_h=\frac{2}{3}\mu FZ\frac{r_0^3-r_i^3}{r_0^2-r_i^2}\\&P_{rz}=\frac{2}{3}\frac{F}{\pi \left( r_0^2-r_i^2\right) }\\&v_{rz}=\frac{2\pi n \left( r_0^3-r_i^3\right) }{90\left( r_0^3-r_i^3\right) }\\&T = \frac{I_z\pi n}{30\left( M_h-M_f\right) }\\&\Delta r=20\,mm, I_z=55\,kgm^2, P_{max}=1\,MPa,\\&F_{max}=1000 N, T_{max}=15 s,\mu =0.5\\&s=1.5, M_s=40\,Nm, M_f=3 Nm, N = 250\,r/min,\\&v_{srmax}=10\,m/s, l_{max}=30\,mm \\&60 mm\le r_i\le 80\,mm, 90\,mm\le r_0\le , 110\,mm, 1.5\,mm\le t \le 3mm,\\&600N\le F\le 1000 N, 2 \le Z \le 9 \end{aligned}$$

Appendix 5: Speed reducer design problem

$$\begin{aligned}&\quad {Minimize } (f(x))=0.7854x_1x_2^2\left( 3.3333x_3^2+14.9334x_3\right. \\&\left. \qquad \qquad -43.0934\right) -1.508x_1\left( x_6^2+x_7^2\right) +7.4777\left( x_6^3+x_7^3\right) \\&\qquad \qquad +0.7854\left( x_4x_6^2+x_5x_7^2\right) \\&s.t;\\&g_1(x)=\frac{27}{x_1x_2^2x_3}-1\le 0\\&g_2(x)=\frac{397.5}{x_1x_2^2x_3^2}-1\le 0\\&g_3(x)=\frac{1.93x_4^3}{x_2x_6^4x_3}-1\le 0\\&g_4(x)=\frac{1.93x_5^3}{x_2x_7^4x_3}-1\le 0\\&g_5(x)=\frac{\left[ \left( 745x_4/x_2x_3\right) ^2+16.9 \times 10^6\right] ^0.5}{110x_6^3}-1\le 0\\&g_6(x)=\frac{\left[ \left( 745x_5/x_2x_3\right) ^2+157.5 \times 10^6\right] ^0.5}{85x_7^3}-1\le 0\\&g_7(x)=x_2x_3/40-1\le 0\\&g_8(x)=\frac{5x_2}{x_1}-1\le 0\\&g_9(x)=\frac{x_1}{12x_2}-1\le 0\\&g_10(x)=\frac{1.5x_6+1.9}{x_4}-1\le 0\\&g_11(x)=\frac{1.1x_7+1.9}{x_5}-1\le 0\\&where\\&2.6\le x_1\le , 0.7\le x_2\le 0.8, 17\le x_3 \le 28, 7.3\le \\&\qquad \quad x_4 \le 8.3, 7.3\le x_5\le 8.3 \\&2.9\le x_6\le 3.9, 5.0\le x_7\le 5.5 \end{aligned}$$

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Singh, N., Houssein, E.H., Singh, S.B. et al. HSSAHHO: a novel hybrid Salp Swarm-Harris Hawks optimization algorithm for complex engineering problems. J Ambient Intell Human Comput 14, 11569–11605 (2023). https://doi.org/10.1007/s12652-022-03724-0

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