# An Advanced Crow Search Algorithm for Solving Global Optimization Problem
## 4 Numerical Examples
### Engineering Problems
#### 4.2.2. Welded Beam Design Problem  
This problem posed here is to minimize the costs of welding and materials for the welding of two beams, as shown in Figure 10. \(h\) (welding height; \(x_{1}\)), \(l\) (welding length; \(x_{2}\)), \(t\) (thickness of beam 2; \(x_{3}\)), and \(b\) (width of beam 2; \(x_{4}\)) are design variables, and the range that the design variables can have is given by Equation (17). The welding cost minimization problem is expressed as an equation in Equation (18). Here, the load (\(P\)) applied to Beam 2 is 6000 lb, the length (\(L\)) of Beam 2 is 14.0 inches, the modulus of elasticity (\(E\)) is \(30\times 10^{6}\) psi, the modulus of shear elasticity (\(G\)) is \(12\times 10^{6}\) psi, the maximum shear stress (\(\tau_{max}\)) is 13, 600 psi, maximum stress (\(\sigma_{max}\)) is \(30,000\) psi, and the maximum displacement (\(\delta_{max}\)) is 0.25 inches. In addition, each design variable has the constraints provided by Equation (19).  
\[0.1\leq x_{1}\;or\;x_{4}\leq 2.0 \tag{17}\] \[0.1\leq x_{2}\;or\;x_{3}\leq 10.0\]  
\[min\;f(x)=1.10471x_{1}^{2}x_{2}+0.04811x_{3}x_{4}(14.0+x_{2}) \tag{18}\]  
\[\begin{split} Subject\;to\;:g_{1}(x)&=\tau(x)-\tau_{ max}\leq 0\\ g_{2}(x)&=\sigma(x)-\sigma_{max}\leq 0\\ g_{3}(x)&=x_{1}-x_{4}\leq 0\\ g_{4}(x)&=0.10471x_{1}^{2}+0.04811x_{3}x_{4}(14.0+x _{2})-5.0\leq 0\\ g_{5}(x)&=0.125-x_{1}\leq 0\\ g_{6}(x)&=\delta(x)-\delta_{max}\leq 0\\ g_{7}(x)&=P-P_{c}(x)\leq 0\end{split} \tag{19}\]  
Table 9 compares the results of the ACS algorithm with those of previous studies [24; 28; 29; 30]. The ACS algorithm derived the smallest cost of 1.7254 (design variables were 0.2057, 3.4747, 9.0365, and 0.2057), and all of the constraints were satisfied. The ACS algorithm reduced the cost by about 0.23% compared to the conventional CS algorithm and by 1.33% compared to a study by [PERSON] [24].  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
\begin{tabular}{c} **Design** \\ **Variables** \\ \end{tabular} & **[PERSON] [24]** & **[PERSON] [25]** & **[PERSON] \& **[PERSON]** & \multicolumn{2}{c}{**This Paper**} \\ \cline{3-6}  & & & **[PERSON] [26]** & [27] & **Conventional CS** & **ACS** \\ \hline \(x_{1}\) & 0.8125 & 0.9375 & 1.125 & 1.125 & 0.7783 & 0.7782 \\ \(x_{2}\) & 0.4375 & 0.5000 & 0.625 & 0.625 & 0.3860 & 0.3846 \\ \(x_{3}\) & 40.3239 & 48.3290 & 58.291 & 47.700 & 40.3211 & 40.3196 \\ \(x_{4}\) & 2000.0000 & 112.6790 & 43.690 & 117.701 & 200.0000 & 200.0000 \\ \(g_{1}(x)\) & \(-0.0343\) & \(-0.0048\) & 0.0000 & \(-0.2044\) & \(-7.2388\times 10^{-5}\) & \(-3.8296\times 10^{-9}\) \\ \(g_{2}(x)\) & \(-0.0528\) & \(-0.0389\) & \(-0.0689\) & \(-0.1699\) & \(-0.014\) & \(-6.4738\times 10^{-9}\) \\ \(g_{3}(x)\) & \(-27.1058\) & \(-3652.8768\) & \(-21.2201\) & 54.2260 & \(-1.0402\times 10^{2}\) & \(-3.4597\times 10^{-4}\) \\ \(g_{4}(x)\) & \(-40.0000\) & \(-127.3210\) & \(-196.3100\) & \(-122.2990\) & \(-40.0000\) & \(-40.0000\) \\ \hline \(F(x)\) & 6288.7445 & 6410.3811 & 7198.0428 & 8,129.1036 & 5890.288 & 5885.333 \\ \hline \hline \end{tabular}
\end{table}
Table 8: Results of PVD problem.  
Figure 10: Welded beam design problem.