# Personalized-Template-Guided Intelligent Evolutionary Algorithm
## 3 Results
### Engineering Problem Experiment
#### 3.4.2 Welded Beam Structure Problem  
The structural design of a welded beam involves four parameters, including weld thickness \(h\), bar clamping length \(l\), bar height \(t\), and bar thickness \(b\). The objective function is to obtain the minimum manufacturing cost of a welded beam under the premise of satisfying the shear stress \(\tau\), the bending stress in the beam \(\sigma\), the bending load of the beam \begin{table}
\begin{tabular}{l c c c c c c} \hline \hline
**Algorithm** & \(\mathbf{d}\) & \(\mathbf{D}\) & \(\mathbf{P}\) & \(\mathbf{Best}\) & \(\mathbf{Mean}\) & \(\mathbf{Std}\) \\ \hline PTG & \(5.17\times 10^{-2}\) & \(3.57\times 10^{-1}\) & \(1.13\times 10^{1}\) & \(1.266506\times 10^{-2}\) & \(1.27\times 10^{-2}\) & \(6.57\times 10^{-5}\) \\ WOA & \(5.16\times 10^{-2}\) & \(3.54\times 10^{-1}\) & \(1.14\times 10^{1}\) & \(1.266523\times 10^{-2}\) & \(1.35\times 10^{-2}\) & \(1.08\times 10^{-3}\) \\ HHO & \(5.27\times 10^{-2}\) & \(3.81\times 10^{-1}\) & \(1.00\times 10^{1}\) & \(1.268195\times 10^{-2}\) & \(1.37\times 10^{-2}\) & \(1.04\times 10^{-3}\) \\ COA & \(5.13\times 10^{-2}\) & \(3.47\times 10^{-1}\) & \(1.19\times 10^{1}\) & \(1.268248\times 10^{-2}\) & \(3.83\times 10^{-1}\) & \(2.02\) \\ TSA & \(5.21\times 10^{-2}\) & \(3.66\times 10^{-1}\) & \(1.08\times 10^{1}\) & \(1.267502\times 10^{-2}\) & \(1.29\times 10^{-2}\) & \(2.19\times 10^{-4}\) \\ RSA & \(5.00\times 10^{-2}\) & \(3.10\times 10^{-1}\) & \(1.50\times 10^{1}\) & \(1.319337\times 10^{-2}\) & \(2.26\times 10^{-2}\) & \(2.46\times 10^{-2}\) \\ FATA & \(5.13\times 10^{-2}\) & \(3.48\times 10^{-1}\) & \(1.18\times 10^{1}\) & \(1.266776\times 10^{-2}\) & \(1.29\times 10^{-2}\) & \(3.20\times 10^{-4}\) \\ MFO & \(5.17\times 10^{-2}\) & \(3.58\times 10^{-1}\) & \(1.12\times 10^{1}\) & \(1.266508\times 10^{-2}\) & \(1.32\times 10^{-2}\) & \(6.45\times 10^{-4}\) \\ ECO & \(5.20\times 10^{-2}\) & \(3.65\times 10^{-1}\) & \(1.08\times 10^{1}\) & \(1.266712\times 10^{-2}\) & \(1.33\times 10^{-2}\) & \(7.13\times 10^{-4}\) \\ HO & \(5.17\times 10^{-2}\) & \(3.58\times 10^{-1}\) & \(1.12\times 10^{1}\) & \(1.268031\times 10^{-2}\) & \(1.30\times 10^{-2}\) & \(4.97\times 10^{-4}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Comparison results of PTG and advanced algorithm in compression spring design. Figure 7: Schematic diagram of design problems of compression spring. \(P_{c}\), and the deflection of the beam end \(\delta\). The specific problem is illustrated in Figure 8. For specific definitions, see Equations (28)-(30). Definition of the problem:  
\[X=[x_{1},x_{2},x_{3},x_{4}]=[h,l,t,b] \tag{28}\]  
The objective function:  
\[f(x)=1.10471x_{1}^{2}x_{2}+0.04811x_{3}x_{4}(14+x_{2}) \tag{29}\]  
Constraint setting:  
\[ \begin{split}& g_{1}(x)=\tau(x)-\tau_{\max}\leq 0,\quad g_{2}(x)= \sigma(x)-\sigma_{\max}\leq 0,\quad g_{3}(x)=\delta(x)-\delta_{\max}\leq 0,\\ & g_{4}(x)=x_{1}-x_{4}\leq 0,\quad g_{5}(x)=P-P_{c}(x)\leq 0,\quad g _{6}(x)=0.125-x_{1}\leq 0,\\ & g\tau(x)=1.10471x_{1}^{2}x_{2}+0.04811x_{3}x_{4}(14+x_{2})-5 \leq 0,\\ & 0.1\leq x_{1}\leq 2,\quad 0.1\leq x_{2}\leq 10,\quad 0.1\leq x_{3} \leq 10,\quad 0.1\leq x_{4}\leq 2,\\ &\tau(x)=\sqrt{(\tau^{\prime})^{2}+2\tau^{\prime}\tau^{\prime} \frac{x_{2}}{2R}+(\tau^{\prime\prime})^{2}},\quad\tau^{\prime}=\frac{P}{ \sqrt{2}x_{1}x_{2}},\quad\tau^{\prime\prime}=\frac{MR}{J},\quad M=P\Big{(}L+ \frac{x_{2}}{2}\Big{)},\\ & R=\sqrt{\frac{x_{2}^{2}}{4}+\bigg{(}\frac{x_{1}+x_{3}}{2} \bigg{)}^{2}},\quad J=2\sqrt{2}x_{1}x_{2}\bigg{(}\frac{x_{2}^{2}}{12}+\bigg{(} \frac{x_{1}+x_{3}}{2}\bigg{)}^{2}\bigg{)},\quad\sigma(x)=\frac{6 PL}{x_{4}x_{3}^ {2}},\\ &\delta(x)=\frac{4 PL^{3}}{Ex_{3}^{3}x_{4}},\quad P_{c}(X)=\frac{4.013E}{L^{2}}\sqrt{\frac{x_{3}^{2}x_{4}^{6}}{36}}\bigg{(}1-\frac{x_{3}}{2L} \sqrt{\frac{E}{4G}}\bigg{)},\quad P=6000,\quad L=14,\\ &\delta_{\max}=0.25,\quad E=30\times 10^{6},\quad G=12\times 10^{6 },\quad\tau_{\max}=13600,\quad\sigma_{\max}=30000\end{split}