A novel chaotic Henry gas solubility optimization algorithm for solving real-world engineering problems

Abstract

The paper proposes a novel metaheuristic based on integrating chaotic maps into a Henry gas solubility optimization algorithm (HGSO). The new algorithm is named chaotic Henry gas solubility optimization (CHGSO). The hybridization is aimed at enhancement of the convergence rate of the original Henry gas solubility optimizer for solving real-life engineering optimization problems. This hybridization provides a problem-independent optimization algorithm. The CHGSO performance is evaluated using various conventional constrained optimization problems, e.g., a welded beam problem and a cantilever beam problem. The performance of the CHGSO is investigated using both the manufacturing and diaphragm spring design problems taken from the automotive industry. The results obtained from using CHGSO for solving the various constrained test problems are compared with a number of established and newly invented metaheuristics, including an artificial bee colony algorithm, an ant colony algorithm, a cuckoo search algorithm, a salp swarm optimization algorithm, a grasshopper optimization algorithm, a mine blast algorithm, an ant lion optimizer, a gravitational search algorithm, a multi-verse optimizer, a Harris hawks optimization algorithm, and the original Henry gas solubility optimization algorithm. The results indicate that with selecting an appropriate chaotic map, the CHGSO is a robust optimization approach for obtaining the optimal variables in mechanical design and manufacturing optimization problems.

Appendices

Appendix A

Cantilever problem

The problem can be formulated as follows:

Minimize:

$$f\left(X\right)=0.0624\left({x}_{1}+{x}_{2}+{x}_{3}+{x}_{4}+{x}_{5}\right).$$

Subject to:

$$\mathrm{g}\left(x\right)=\frac{61}{{x}_{1}^{3}}+\frac{37}{{x}_{2}^{3}}+\frac{19}{{x}_{3}^{3}}+\frac{7}{{x}_{4}^{3}}+\frac{1}{{x}_{5}^{3}}-1\le 0.$$

The design variables are the heights (or widths) of the different beam elements, and the thickness is held fixed (here $t$ = 2/3). The bound constraints are set as $0.01\le {x}_{j}\le 100$.

Welded beam design problem

The problem can be formulated as follows:

Minimize: $f\left(x\right)=1.10471{h}^{2}l+0.04811tb\left(14.0+l\right).$

Subject to:

$${g}_{1}\left(x\right)=\tau \left(x\right)-{\tau }_{max}\le 0, {g}_{2}\left(x\right)=\sigma \left(x\right)-{\sigma }_{max}\le 0,$$

$${g}_{3}\left(x\right)=h-b\le 0, {g}_{4}\left(x\right)=0.1047{h}^{2}+0.04811tb\left(14+l\right)-5\le 0,$$

$${g}_{5}\left(x\right)=0.125-h\le 0, {g}_{6}\left(x\right)=\delta \left(x\right)-{\delta }_{max}\le 0,$$

$${g}_{7}\left(x\right)=P-{P}_{c}(x)\le 0,$$

where $\tau = \sqrt {(\tau ^{\prime})^{2} {\text{ + }}\left( {\tau ^{\prime\prime}} \right)^{{\text{2}}} {\text{ + 2}}\tau ^{\prime}\tau ^{\prime\prime}\frac{{\text{l}}}{{{\text{2}}R}}} ,\;\tau ^{\prime}{\text{ = }}\frac{P}{{\sqrt {\text{2}} hl}},\tau ^{\prime\prime}{\text{ = }}\frac{{MR}}{J},M{\text{ = }}P\left( {L{\text{ + }}\frac{{\text{1}}}{{\text{2}}}} \right),R{\text{ = }}\sqrt {\frac{{{\text{l}}^{{\text{2}}} }}{{\text{4}}}{\text{ + }}\frac{{(h{\text{ + }}t)^{{\text{2}}} }}{{\text{4}}}} ,$

$$J = 2\left\{ {\sqrt 2 hl\left[ {\frac{{l^{2} }}{{12}} + \frac{{(h + t)^{2} }}{4}} \right]} \right\},\sigma = \frac{{6PL}}{{bt^{2} }},P_{c} = \frac{{4.013E\sqrt {t^{2} b^{6} /36} }}{{bt^{2} }}\left( {1 - \frac{t}{{2L}}\sqrt {\frac{E}{{4G}}} } \right),$$

$$P=6000lp, l=14in, E=30\times {10}^{6}psi, G=12\times {10}^{6}psi, {\tau }_{max}=13600 psi, {\delta }_{max}=0.25in, 0.1\le h,b\le 2,$$

and $1\le l,t\le 10.$