# Multi-Strategy Hybrid Whale Optimization Algorithm Improvement
## 5 Practical Application
### Welded Beam Design  
The welded beam design problem is shown in Figure 12. The design of the beam bar includes the length, thickness, height, and welding thickness. Obviously, this is a nonlinear programming problem, and it is difficult to solve it in an exhaustive way.  
In practical scenarios, it is often necessary to minimize the cost of welding beams as much as possible. This is where the algorithm needs to strategically balance the variables.  
The problem can be formulated as follows:  
Minimize:  
\[f(\overline{x})=0.04811x_{3}x_{4}(x_{2}+14)+1.10471x_{1}^{2}x_{2} \tag{30}\]  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Algorithm** & **Best** & **Mean** & **Worst** &
\begin{tabular}{c} **Comparison to** \\ **the Best Results** \\ \end{tabular} \\ \hline MHWOA & 5304 & 5337 & 5386 & 0\% \\ GWO & 5921 & 5994 & 6004 & 10.96\% \\ AVOA & 5982 & 6056 & 6093 & 11.87\% \\ MFO & 6005 & 6181 & 6210 & 13.65\% \\ AO & 6998 & 7031 & 7088 & 24.09\% \\ SO & 7155 & 7304 & 7612 & 26.93\% \\ BWO & 21,579 & 24,680 & 28,437 & 78.38\% \\ WOA & 55,967 & 56,490 & 57,366 & 90.55\% \\ GTO & 7300 & 7319 & 7458 & 27.08\% \\ \hline \hline \end{tabular}
\end{table}
Table 10: Pressure vessel design cost comparison.  
Figure 12: A schematic diagram of the welded beam design problem.  
Subject to:  
\[\left\{\begin{array}{c}g_{1}(\overline{x})=x_{1}-x_{4}\leq 0\\ g_{2}(\overline{x})=\delta(\overline{x})-\delta_{max}\leq 0\\ g_{3}(\overline{x})=P\leq P_{c}(\overline{x})\\ g_{4}(\overline{x})=\tau_{max}\geq\tau(\overline{x})\\ g_{5}(\overline{x})=\sigma(\overline{x})-\sigma_{max}\leq 0\end{array}\right. \tag{31}\]  
where  
\[\left\{\begin{array}{c}\tau=\sqrt{\tau^{\prime 2}+\tau^{\prime \prime}{}^{2}+2\tau^{\prime}\tau^{\prime\prime}\frac{\overline{x_{2}}}{2R}},\ \tau^{\prime}=\frac{RM}{J},\ \tau^{ \prime}=\frac{P}{\sqrt{2x_{2}x_{1}}}\,\ M=p(\frac{\gamma_{2}}{2}+L)\\ R=\sqrt{\frac{\gamma_{2}^{2}}{4}+(\frac{\gamma_{1}+\gamma_{2}}{2})^{2}}\,\ J=2((\frac{\gamma_{2}^{2}}{4}+(\frac{\gamma_{1}+\gamma_{2}}{2})^{2}) \sqrt{2}x_{2}x_{1}\\ 4\end{array}\right. \tag{32}\] \[\sigma(\overline{x})=\frac{6 PL}{x_{4}x_{3}^{3}},\ \delta(\overline{x})=\frac{6 PL^{3}}{Ex_{4}x_{3}^{3}},\ P_{c}( \overline{x})=\frac{4.013 Ex_{3}x_{4}^{3}}{6L^{2}}(1-\frac{\gamma_{3}}{2L} \sqrt{\frac{E}{4C}})\] \[L=14\ \text{in},\ P=6000\ \text{lb},\ E=30.10^{6}\ \text{psi},\ \sigma_{max}=30,000\ \text{psi}\] \[\tau_{max}=13,600\ \text{psi},\ G=12.10^{6}\ \text{psi},\ \delta_{max}=0.25\ \text{in}\]  
With bounds:  
\[\left\{\begin{array}{c}0.125\leq x_{1}\leq 2\\ x_{2}\leq 10\\ x_{3}\geq 0.1\\ 0.1\leq x_{3}\leq 2\end{array}\right. \tag{33}\]  
By incorporating MHWOA into the problem and comparing it with standard swarm intelligence algorithms, Figure 13 shows that MHWOA achieves the highest computational accuracy among all the compared algorithms when dealing with this problem.  
Based on the comparison of the best cost, mean cost, and worst cost obtained from 30 runs, the cost savings rate of each algorithm can be statistically determined. Taking MHWOA as the benchmark, the cost savings rates compared with other algorithms are shown in Table 11.  
Figure 13: Convergence visibility of the welded beam design.  
MHWOA has a cost savings rate of 4.1% over GWO; 12.42% over AVOA; 13.13% over AVO; 34.8% over AO; 17.98% over GTO; 30.24% over SO; 23.35% over BWO; 52.54% over WOA.