## 5 MGTO for Solving Engineering-Optimization Problems
### Tubular-Column Design  
According to Figure 11, a tubular column is given to decrease the cost of supporting the compressive load, \(p=2500\) kgf, and it consists of the yield stress, elastic modulus and density.  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline
**Algorithm** & **MGTO** & **GTO** & **AOA** & **SSA** & **WOA** & **ROLGWO** & **PSO** \\ \hline x1 & 0.5000 & 1.3249 & 0.50000 & 0.6770 & 1.1079 & 0.5008 & 0.5000 \\ x2 & 1.2294 & 0.8819 & 1.2827 & 1.1695 & 1.1202 & 1.2439 & 1.2222 \\ x3 & 0.5000 & 0.5000 & 0.6364 & 0.5000 & 0.6965 & 0.5000 & 1.5000 \\ x4 & 1.2006 & 1.2322 & 1.3636 & 1.1805 & 1.1535 & 1.1921 & 0.7412 \\ x5 & 0.5000 & 0.5000 & 0.50000 & 0.9549 & 0.6840 & 0.5037 & 0.5000 \\ x6 & 1.0917 & 1.0985 & 1.50000 & 0.9809 & 0.8071 & 1.2917 & 1.5000 \\ x7 & 0.5000 & 0.5000 & 0.50000 & 0.5000 & 0.9805 & 0.5012 & 0.5000 \\ x8 & 0.3450 & 0.34385 & 0.3450 & 0.3413 & 0.2748 & 0.3449 & 0.34500 \\ x9 & 0.3450 & 0.19200 & 0.3084 & 0.2229 & 0.2587 & 0.2536 & 0.34500 \\ x10 & 0.6436 & 4.4782 & 0.5795 & 3.5908 & 8.4836 & 3.1707 & \(-\)0.20845 \\ x11 & 0.3162 & 2.6609 & \(-\)9.62291 & \(-\)1.22041 & 17.8255 & 3.3177 & 3.2759 \\ Optimal weight & **23.1894** & 23.2046 & 25.8678 & 23.3121 & 28.9580 & 23.2527 & 23.1902 \\ \hline \end{tabular}
\end{table}
Table 12: The comparative optimization results of the car-crashworthiness problem.  
The functions, including the material and manufacturing costs, are considered. The formulas can be expressed as follows:  
Minimize:  
\[f(d,\ t)=9.8 dt+2d \tag{90}\]  
Constraints:  
\[g_{1}=\frac{P}{\pi d\sigma_{y}}-1\leq 0 \tag{91}\]  
\[g_{2}=\frac{8 PL^{2}}{\pi^{3}Edt(d^{2}+t^{2})}-1\leq 0 \tag{92}\]  
\[g_{3}=\frac{2.0}{d}-1\leq 0 \tag{93}\]  
\[g_{4}=\frac{d}{14}-1\leq 0 \tag{94}\]  
\[g_{5}=\frac{0.2}{t}-1\leq 0 \tag{95}\]  
Variable range:  
\[0.01\leq d,\ T\leq 100 \tag{96}\]  
The comparative optimization results of the tubular-column-design problem are shown in Table 13.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Algorithm**} & \multicolumn{2}{c}{**Optimal Values for Variables**} & \multirow{2}{*}{**Optimal Cost**} \\ \cline{2-3} \cline{5-5}  & \(d\) & \(T\) & \\ \hline
**MGTO** & **5.4511** & **0.2919** & **26.5313** \\ HSCAHS & 5.4170 & 0.3128 & 27.4745 \\ AOA & 7.5313 & 0.2223 & 31.5088 \\ WOA & 5.7562 & 0.2764 & 27.1415 \\ GWO & 5.4513 & 0.2919 & 26.5333 \\ ROLGWO & 5.4510 & 0.2919 & 26.5326 \\ DSCA & 5.5179 & 0.2899 & 26.7494 \\ \hline \hline \end{tabular}
\end{table}
Table 13: The comparative optimization results of the tubular-column-design problem.  
Figure 11: The schematic diagram of the tubular column.  
The optimal solution obtained by the MGTO is 26.5313 when the variable \(x\) is 5.4511 and 0.2919. It is obvious that the proposed algorithm has advantages in solving the tubular-column-design problem.  
In summary, the advantages of the proposed algorithm are shown in this section. Because the MGTO has great exploration and exploitation ability, it is superior to the traditional GTO and other existing algorithms in regard to performance. Therefore, the MGTO can be applied in practical engineering problems.