## 5 MGTO for Solving Engineering-Optimization Problems
### Welded-Beam Design  
The welded-beam-design problem is a common engineering-optimization problem that was developed by [PERSON] et al. [52]. Figure 5 offers a schematic diagram of a welded beam. In this problem, the constraints may be divided into four categories: shear stress, bending stress, buckling load and deflection of beam. The cost of welding materials is minimized as much as possible under the four constraints. Furthermore, there are four variables in the welded-beam-design problem: The thickness of the weld (\(h\)), the length of the weld (\(l\)), the height of the welded beam (\(t\)) and the thickness of the bar (\(b\)).  
The mathematical model of the above problem is presented as follows:  
Consider:  
\[\overset{\rightarrow}{x}=[x_{1},\ x_{2},\ x_{3},\ x_{4}]=[h,\ l,\ t,\ b] \tag{32}\]  
Minimize:  
\[f(\overset{\rightarrow}{x})=1.10471x_{1}^{2}x_{2}+0.04811x_{3}x_{4}(14.0+x_{ 2}) \tag{33}\]  
Constraints:  
\[\left\{\begin{array}{l}g_{1}(\overset{\rightarrow}{x})=\tau(\overset{ \rightarrow}{x})-\tau_{max}\\ g_{2}(\overset{\rightarrow}{x})=\sigma(\overset{\rightarrow}{x})-\sigma_{ max}\\ g_{3}(\overset{\rightarrow}{x})=\delta(\overset{\rightarrow}{x})-\delta_{ max}\\ g_{4}(\overset{\rightarrow}{x})=x_{1}-x_{4}\leq 0\\ g_{5}(\overset{\rightarrow}{x})=P-P_{C}(\overset{\rightarrow}{x})\leq 0\\ g_{6}(\overset{\rightarrow}{x})=0.125-x_{1}\leq 0\\ g_{7}(\overset{\rightarrow}{x})=1.10471x_{1}^{2}+0.04811x_{3}x_{4}(14.0+x_{ 2})-5\leq 0\end{array}\right. \tag{34}\]  
where we have the following:  
\[\tau(\overset{\rightarrow}{x})=\sqrt{\left(\tau^{\prime}\right)^{2}+2\tau^{ \prime}\tau^{\prime\prime}\frac{x_{2}}{2R}+\left(\tau^{\prime\prime}\right)^{2}} \tag{35}\]  
\[\tau^{\prime}=\frac{P}{\sqrt{2}x_{1}x_{2}},\ \tau^{\prime\prime}=\frac{MR}{J},\ M =P\Big{(}L+\frac{x_{2}}{2}\Big{)} \tag{36}\]  
\[R=\sqrt{\frac{x_{2}^{2}}{4}+\left(\frac{x_{1}+x_{3}}{2}\right)^{2}} \tag{37}\]  
\[J=2\Bigg{\{}\sqrt{2}x_{1}x_{2}\Bigg{[}\frac{x_{2}^{2}}{4}+\left(\frac{x_{1}+x _{3}}{2}\right)^{2}\Bigg{]}\Bigg{\}} \tag{38}\]  
Figure 5: Welded-beam-design problem: three-dimensional model diagram (**left**) and the structural parameter (**right**).  
\[\sigma(\stackrel{{\rightarrow}}{{x}})=\frac{6 PL}{x_{3}^{2}x_{4}},\, \delta(\stackrel{{\rightarrow}}{{x}})=\frac{6 PL^{3}}{Ex_{3}^{3}x_{4}} \tag{39}\]  
\[P_{\text{C}}(\stackrel{{\rightarrow}}{{x}})=\frac{4.013E\sqrt{ \frac{\frac{2}{3}x_{4}^{2}}{36}}}{L^{2}}\left(1-\frac{x_{3}}{2L}\sqrt{\frac{E} {4G}}\right) \tag{40}\]  
\[\tau(\stackrel{{\rightarrow}}{{x}})=\sqrt{\left(\tau^{\prime} \right)^{2}+2\tau^{\prime}\tau^{\prime\prime}\frac{x_{2}}{2R}+\left(\tau^{ \prime\prime}\right)^{2}} \tag{41}\]  
Variable range:  
\[P=6000\text{ lb, }L=14\text{ in, }E=30\times 10^{6}\text{ psi, }G=12\times 10^{6}\text{ psi} \tag{42}\]  
\[\tau_{max}=13600\text{ psi, }\sigma_{max}=30000\text{ psi, }\delta_{max}=0.25\text{ in} \tag{43}\]  
The results of the MGTO and other comparative optimization algorithms are listed in Table 7. According to Table 7, it can be concluded that the MGTO was able to obtain the effective solution for welded-beam-design problems compared with other algorithms.