## 7 Application on Engineering Design and QoS-CSC Problem
### Pressure Vessel Design  
Cutting down budget requirements constitutes the purpose of pressure vessel engineering by optimizing four variables under four constraints. These four design variables are shell thickness (\(T\)s), head thickness (\(Th\)h), inner radius (\(R\)), and the length of the cylindrical part without head (\(L\)), as shown in Figure 6. Detailed below is the mathematical framework for the problem:  
\[\textit{Consider}:\vec{x}=[x_{1},x_{2},x_{3},x_{4}]=[T_{s},T_{h},R,L]\] \[\textit{Minimize}:f(\vec{x})=0.6224x_{1}x_{3}x_{4}+1.7781x_{2}x _{3}^{2}+3.1661x_{1}^{2}x_{4}+19.84x_{1}^{2}x_{3}\] \[\textit{Subject to}:g_{1}(\vec{x})=-x_{1}+0.0193x_{3}\leq 0\] \[g_{2}(\vec{x})=s_{3}+0.00954x_{3}\leq 0\] \[g_{3}(\vec{x})=-\tau x_{3}^{2}x_{4}-\frac{4}{3}\tau x_{3}^{3}+12 96000\leq 0\] \[g_{4}(\vec{x})=x_{4}-240\leq 0\] \[\textit{Variable ranges}:0\geq x_{1}\leq 99,\,0\geq x_{2}\leq 99,\,10 \geq x_{3}\leq 200,\,10\geq x_{4}\leq 200\]  
Figure 5: Convergence curves on benchmark functions.  
\[\begin{split}& Consider:\vec{x}=[x_{1},x_{2},x_{3}]=[d,D,N]\\ & Minimize:f(\vec{x})=(x_{3}+2)x_{2}x_{1}^{2}\\ & Subject to:g_{1}(\vec{x})=1-\frac{x_{2}^{3}x_{3}}{71785x_{1}^{4}} \leq 0\\ & g_{2}(\vec{x})=\frac{4x_{2}^{2}-x_{1}x_{2}}{12566(x_{2}x_{1}^{3} -x_{1}^{4})}+\frac{1}{5108x_{1}^{2}}\leq 0\\ & g_{3}(\vec{x})=1-\frac{140.45x_{1}}{x_{2}^{2}x_{3}}\leq 0\\ & g_{4}(\vec{x})=\frac{x_{1}+x_{2}}{1.5}-1\leq 0\\ & Variable ranges:0.05\leq x_{1}\leq 2.00,\,0.25\leq x_{2}\leq 1.30,\,2.0 0\leq x_{3}\leq 15.0\end{split}\]  
From the evidence found in Tables 12 and 13, MIARO significantly surpasses other algorithms in tackling the compression spring design challenge, resulting in the best solution.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Algorithm** & **Avg** & **Best** & **Std** \\ \hline MIARO & **0.01266731** & **0.01266523** & **1.07 \(\times\) 10\({}^{-7}\)** \\ ARO & 0.01480855 & 0.01272433 & 1.74 \(\times\) 10\({}^{-3}\) \\ HHO & 0.01334434 & 0.01266527 & 5.31 \(\times\) 10\({}^{-4}\) \\ SSA & 0.01309456 & 0.01272616 & 8.22 \(\times\) 10\({}^{-4}\) \\ GWO & 0.01274092 & 0.01267135 & 8.99 \(\times\) 10\({}^{-5}\) \\ WOA & 0.01381973 & 0.01270982 & 1.23 \(\times\) 10\({}^{-3}\) \\ MFO & 0.01262286 & 0.01266759 & 1.57 \(\times\) 10\({}^{-3}\) \\ SCA & 0.01307718 & 0.01279199 & 1.15 \(\times\) 10\({}^{-4}\) \\ PSO & 0.01293869 & 0.01266527 & 3.80 \(\times\) 10\({}^{-4}\) \\ \hline \hline \end{tabular}
\end{table}
Table 13: Best values and corresponding solutions for compression spring.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Algorithm** & **Avg** & **Best** & **Std** \\ \hline MIARO & **0.01266731** & **0.01266523** & **1.07 \(\times\) 10\({}^{-7}\)** \\ ARO & 0.01480855 & 0.01272433 & 1.74 \(\times\) 10\({}^{-3}\) \\ HHO & 0.01334434 & 0.01266527 & 5.31 \(\times\) 10\({}^{-4}\) \\ SSA & 0.01309456 & 0.01272616 & 8.22 \(\times\) 10\({}^{-4}\) \\ GWO & 0.01274092 & 0.01267135 & 8.99 \(\times\) 10\({}^{-5}\) \\ WOA & 0.01381973 & 0.01270982 & 1.23 \(\times\) 10\({}^{-3}\) \\ MFO & 0.0126286 & 0.01266579 & 1.57 \(\times\) 10\({}^{-3}\) \\ SCA & 0.01307718 & 0.01279199 & 1.15 \(\times\) 10\({}^{-4}\) \\ PSO & 0.01293869 & 0.01266527 & 3.80 \(\times\) 10\({}^{-4}\) \\ \hline \hline \end{tabular}
\end{table}
Table 12: Results of compression spring obtained by running 30 times.  
Figure 7: Compression spring.