Compound improved Harris hawks optimization for global and engineering optimization

Abstract

Meta-heuristic algorithms, due to their high search speed and strong generalization ability, are frequently applied in programs mainly to discover the corresponding optimal strategy for any problem in view of their defined rules. After years of collision evolution, they have been continuously used to solve the complex, unordered and diverse optimization problems and improve efficiency. Aiming at the problems of low convergence accuracy and easy to fall into local optima of the traditional Harris hawks optimization algorithm, a compound improved Harris Hawks Optimization algorithm (CIHHO) is proposed. Firstly, the early circling exploration and later attack exploitation phase of the dynamic adjustment algorithm for environmental factors is introduced to regulate the energy of Harris hawks; Secondly, the concept of Versoria function is introduced to modify the random jump strength and raise the data grabbing ability of local space; Introducing the Levy flight function to adjust the factor and reduce the disturbance impact of Levy flight is beneficial for getting rid of the local space after entering the exploitation phase, and introducing random white noise to reduce step size and improve algorithm accuracy. Taking CEC 2017 test function suite set as the core, the performance of CIHHO algorithm is analyzed. Firstly, the performance of CIHHO algorithm is compared with HHO, HHO_JOS, LHHO, LMHHO and NCHHO. Secondly, the performance of unimodal function, multimodal function, mixed function and compound function is compared with other 7 improved algorithms. Finally, ablation experiments are carried out. The convergence value of the iterative curve obtained is more quantitative than the improved algorithm, The generality of the improved CIHHO algorithm in solving multiple optimization problems with different dimensions is verified. Further applying the CIHHO algorithm to three different engineering experiments, the minimum cost calculation results directly demonstrate that the CIHHO algorithm obtained has certain advantages in dealing with search space problems.

Appendices

Appendix 1

See Tables 7, 8, 9 and 10.

See Figs. 10, 11 and 12.

Appendix 2

Consider:

$${{\text{X}}}_{{\text{s}}}=[{{\text{x}}}_{\mathrm{0,1}},{{\text{x}}}_{\mathrm{0,2}},{{\text{x}}}_{\mathrm{0,3}},{{\text{x}}}_{\mathrm{0,4}}]=[{\text{H}},{\text{L}},{\text{T}},{\text{B}}]$$

Minimize:

$${\text{F}}=1.10471{{\text{x}}}_{\mathrm{0,1}}^{2}{{\text{x}}}_{\mathrm{0,2}}+0.04811{{\text{x}}}_{\mathrm{0,3}}{{\text{x}}}_{\mathrm{0,4}}(14.0+{{\text{x}}}_{\mathrm{0,2}})$$

Subject to:

$$\begin{gathered} g_{1} \left( x \right) = - \tau \left( {{\text{X}}_{{\text{s}}} } \right) + {\uptau }_{{{\text{max}}}} \le 0;\,g_{2} \left( x \right) = \sigma \left( {{\text{X}}_{{\text{s}}} } \right) - {\upsigma }_{{{\text{max}}}} \le 0;\,g_{3} \left( x \right) = - \delta \left( {{\text{X}}_{{\text{s}}} } \right) + {\updelta }_{{{\text{max}}}} \le 0; \hfill \\ g_{4} \left( x \right) = {\text{x}}_{0,1} - {\text{x}}_{0,4} \le 0;\,g_{5} \left( x \right) = P - {\text{P}}_{{\text{c}}} \left( {{\text{X}}_{{\text{s}}} } \right) \le 0;\,g_{6} \left( x \right) = 0.125 - {\text{x}}_{0,1} \le 0 \hfill \\ g_{7} \left( x \right) = 1.10471{\text{x}}_{0,1}^{2} + 0.04811{\text{x}}_{0,3} {\text{x}}_{0,4} \left( {14.0 + {\text{x}}_{0,2} } \right) - 5.0 \le 0;\,g_{4} \left( x \right) = {\text{x}}_{0,4} - 240 \le 0 \hfill \\ \end{gathered}$$

Where:

$$\tau \left( {{\text{X}}_{{\text{s}}} } \right) = \sqrt {\left( {\tau^{\prime}} \right)^{2} + 2\tau^{\prime}\tau^{\prime\prime}\frac{{x_{0,2} }}{2R} + \left( {\tau^{\prime\prime}} \right)^{2} } ;\,\,\tau^{\prime} = \frac{P}{{\sqrt 2 x_{0,1} x_{0,2} }};\,\,\tau^{\prime\prime} = \frac{MR}{J}$$

$$M = P\left( {L + \frac{{x_{0,2} }}{2}} \right);\,R = \sqrt {\frac{{x_{0,2}^{2} }}{4} + \left( {\frac{{x_{0,1} + x_{0,3} }}{2}} \right)^{2} } ;\,J = 2\left\{ {\sqrt 2 x_{0,1} x_{0,2} \left[ {\frac{{x_{0,2}^{2} }}{4} + \left( {\frac{{x_{0,1} + x_{0,3} }}{2}} \right)^{2} } \right]} \right\}$$

$$\sigma \left( X \right) = \frac{{6PL_{s} }}{{{\text{x}}_{0,4} {\text{x}}_{0,3}^{2} }};\,\delta \left( X \right) = \frac{{6PL_{s}^{3} }}{{E{\text{x}}_{0,2}^{2} {\text{x}}_{0,4} }}$$

$${\text{ P}}_{{\text{c}}} \left( {\text{X}} \right) = \frac{{4.013E\frac{{\sqrt {x_{0,3}^{2} x_{0,4}^{6} } }}{36}}}{{L^{2} }}\left( {1 - \frac{{x_{0,3} }}{{2L_{s} }}\sqrt{\frac{E}{4G}} } \right)$$

P = 6000LB; ${L}_{s}=14$; ${\text{E}}=30\times {1}^{6}\mathrm{ psi}$

$${\text{G}}=12\times {1}^{6}\mathrm{ psi }; {\delta }_{max}=0.25 ; {\tau }_{max}=13600 psi ; {\sigma }_{max}=3000 psi$$

With bounds:

0.1 ≤ ${{\text{x}}}_{\mathrm{0,1}}$,${{\text{x}}}_{\mathrm{0,4}}$≤2,0.1 ≤ ${{\text{x}}}_{\mathrm{0,2}}$,${{\text{x}}}_{\mathrm{0,3}}$≤10.

Appendix 3

Consider:

$${{\text{X}}}_{{\text{s}}}=[{{\text{x}}}_{\mathrm{1,1}},{{\text{x}}}_{\mathrm{1,2}},{{\text{x}}}_{\mathrm{1,3}},{{\text{x}}}_{\mathrm{1,4}}]=[{{\text{T}}}_{{\text{S}}},{{\text{T}}}_{{\text{h}}},{\text{R}},{\text{L}}]$$

Minimize:

$${\text{F}}={0.6224{\text{x}}}_{\mathrm{1,1}}{{\text{x}}}_{\mathrm{1,3}}{{\text{x}}}_{\mathrm{1,4}}+1.788{{\text{x}}}_{\mathrm{1,2}}{{\text{x}}}_{\mathrm{1,3}}^{2}+3.1661{{\text{x}}}_{\mathrm{1,1}}^{2}{{\text{x}}}_{\mathrm{1,4}}+19.84{{\text{x}}}_{\mathrm{1,1}}^{2}{{\text{x}}}_{\mathrm{1,3}}$$

Subject to:

$${g}_{1}(x)=-{{\text{x}}}_{\mathrm{1,1}}+0.0193{{\text{x}}}_{\mathrm{1,3}}\le 0$$

$${g}_{2}(x)=-{{\text{x}}}_{\mathrm{1,2}}+0.0954{{\text{x}}}_{\mathrm{1,3}}\le 0$$

$${g}_{3}(x)=-\uppi {{\text{x}}}_{\mathrm{1,3}}^{2}{{\text{x}}}_{\mathrm{1,4}}+\frac{4}{3}\uppi {{\text{x}}}_{\mathrm{1,3}}^{3}+1296000\le 0$$

$${g}_{4}(x)={{\text{x}}}_{\mathrm{1,4}}-240\le 0$$

With bounds:

0 ≤ ${{\text{x}}}_{\mathrm{1,1}}$,${{\text{x}}}_{\mathrm{1,2}}$≤ 100,10 ≤  ${{\text{x}}}_{\mathrm{1,3}}$, ${{\text{x}}}_{\mathrm{1,4}}$≤ 200.

Appendix 4

Consider:

$${{\text{X}}}_{{\text{s}}}=[{{\text{x}}}_{\mathrm{2,1}},{{\text{x}}}_{\mathrm{2,2}}]=[{{\text{A}}}_{1},{{\text{A}}}_{2}]$$

Minimize:

$${\text{F}}={{\text{l}}({\text{x}}}_{\mathrm{2,2}}+2\sqrt{2}{{\text{x}}}_{\mathrm{2,1}})$$

Subject to:

$${g}_{1}(x)=\frac{{{\text{x}}}_{\mathrm{2,2}}}{{2{\text{x}}}_{\mathrm{2,1}}{{\text{x}}}_{\mathrm{2,2}}+2\sqrt{2}{{\text{x}}}_{\mathrm{2,1}}^{2}}{\text{P}}-\upsigma \le 0$$

$${g}_{2}(x)=\frac{{{\text{x}}}_{\mathrm{2,2}}+\sqrt{2}{{\text{x}}}_{\mathrm{2,1}}}{2{{\text{x}}}_{\mathrm{2,1}}{{\text{x}}}_{\mathrm{2,2}}+2\sqrt{2}{{\text{x}}}_{\mathrm{2,1}}^{2}}{\text{P}}-\upsigma \le 0$$

$${g}_{3}(x)=\frac{1}{{{\text{x}}}_{\mathrm{2,1}}+2\sqrt{2}{{\text{x}}}_{\mathrm{2,2}}}{\text{P}}-\upsigma \le 0$$

where:

$${\text{l}}=100,{\text{P}}=2,\upsigma =2$$

With bounds:

0 ≤ ${{\text{x}}}_{\mathrm{2,1}}$,${{\text{x}}}_{\mathrm{2,2}}$≤100.