## 5 MGTO for Solving Engineering-Optimization Problems
### Car-Crashworthiness Design  
The car-crashworthiness-design problem proposed by [PERSON] et al. aims to improve the safety of the vehicle and reduce casualties. The details are shown in Figure 10 [62].  
In this case, eleven parameters are adopted to reduce the weight, namely the inside of the B-pillar, reinforcement of the B-pillar, the inside of the floor side, the crossbeam, the door beam, the door-strip-line reinforcement, the roof rail of the car (\(x_{1}\)-\(x_{7}\)), the materials of inside of B-pillar and reinforcement of B-pillar (\(x_{8}\), \(x_{9}\)), the thickness of barrier height and the impact position (\(x_{10}\), \(x_{11}\)). The mathematical formula is represented as follows:  
Minimize:  
\[f(\overrightarrow{x})=1.98+4.90x_{1}+6.67x_{2}+6.98x_{3}+4.01x_{4}+1.78x_{5} +2.73x_{7} \tag{78}\]  
Constraints:  
\[g_{1}(\overrightarrow{x})=1.16-0.3717x_{2}x_{4}-0.00931x_{2}x_{10}-0.484x_{3} x_{9}+0.01343x_{6}x_{10}\leq 1 \tag{79}\]  
\[\begin{array}{c}g_{2}(\overrightarrow{x})=0.261-0.0159x_{1}x_{2}-0.188x_{1 }x_{8}-0.019x_{2}x_{7}\\ +0.0144x_{3}x_{5}+0.0008757x_{5}x_{10}+0.080405x_{6}x_{9}\\ +0.00139x_{8}x_{11}+0.00001575x_{10}x_{11}\leq 0.32\end{array} \tag{80}\]  
Figure 10: Three-dimensional model diagram of car’s side crash.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Algorithm**} & \multicolumn{2}{c}{**Optimum Variables**} & \multirow{2}{*}{**Optimum Weight**} \\ \cline{2-3} \cline{5-5}  & \(x_{1}\) & & \(x_{2}\) & \\ \hline MGTO & 0.7884 & 0.4081 & **263.8523464** \\ AO & 0.7926 & 0.3966 & 263.8684 \\ HHO & 0.788662816 & 0.408283133832900 & 263.8958434 \\ SSA & 0.78866541 & 0.408275784 & 263.89584 \\ AOA & 0.79369 & 0.39426 & 263.9154 \\ MVO & 0.78860276 & 0.40845307000000 & 263.8958499 \\ MFO & 0.788244771 & 0.409466905784741 & 263.8959797 \\ GOA & 0.788897555578973 & 0.407619570115153 & 263.895881496069 \\ \hline \hline \end{tabular}
\end{table}
Table 11: The comparative optimization results of the three-bar-truss-design problem.  
\[\begin{split} g_{3}(\overrightarrow{x})&=0.214+0.00817x_{5 }-0.131x_{1}x_{8}-0.0704x_{1}x_{9}\\ &+0.03099x_{2}x_{6}-0.018x_{2}x_{7}+0.0208x_{3}x_{8}\\ &+0.121x_{3}x_{9}-0.00364x_{5}x_{6}+0.0007715x_{5}x_{10}\\ &-0.0005354x_{6}x_{10}+0.00121x_{8}x_{11}\leq 0.32\\ \\ g_{4}(\overrightarrow{x})&=0.074-0.061x_{2}-0.163x_{3} x_{8}+0.001232x_{3}x_{10}\\ &-0.166x_{7}x_{9}+0.227x_{2}^{2}\leq 0.32\\ \\ g_{5}(\overrightarrow{x})&=28.98+3.818x_{3}-4.2x_{1}x _{2}+0.0207x_{5}x_{10}\\ &+6.63x_{6}x_{9}-7.7x_{7}x_{8}+0.32x_{9}x_{10}\leq 32\\ \\ g_{6}(\overrightarrow{x})&=33.86+2.95x_{3}+0.1792x_{1 0}-5.057x_{1}x_{2}\\ -11.0x_{2}x_{8}-0.0215x_{5}x_{10}-9.98x_{7}x_{8}+22.0x_{8}x_{9} \leq 32\\ \\ g_{7}(\overrightarrow{x})&=46.36-9.9x_{2}-12.9x_{1 }x_{8}+0.1107x_{3}x_{10}\leq 32\\ \\ g_{8}(\overrightarrow{x})&=4.72-0.5x_{4}-0.19x_{2}x _{3}-0.0122x_{4}x_{10}\\ &+0.009325x_{6}x_{10}+0.000191x_{11}^{2}\leq 4\\ \\ g_{9}(\overrightarrow{x})&=10.58-0.674x_{1}x_{2}-1.95x _{2}x_{8}+0.02054x_{3}x_{10}\\ &-0.0198x_{4}x_{10}+0.028x_{6}x_{10}\leq 9.9\\ \\ g_{10}(\overrightarrow{x})&=16.45-0.489x_{3}x_{7}-0.843x _{5}x_{6}+0.0432x_{9}x_{10}\\ &-0.0556x_{9}x_{11}-0.000786x_{11}^{2}\leq 15.7\\ \end{split} \tag{81}\]  
Variable range:  
\[\left\{\begin{array}{llllllll}0.5\leq x_{1}-x_{7}\leq 1.5\\ x_{8},~{}x_{9}\in(0.192,~{}0.345)\\ -30\leq x_{10},~{}x_{11}\leq 30\\ \end{array}\right. \tag{82}\]  
The results of the car-crashworthiness-design problem are listed in Table 12. From the contents, we can see that the optimal weight obtained by the MGTO is 23.1894. Furthermore, it is certain that the MGTO is an efficient algorithm in solving the car-crashworthiness-design problem.