## 6 Utilizing MRSO for Solving Engineering Design Issues
### Performance Comparison of MRSO with SCA, LCA, TSA, and DOA on Seven Engineering Design Problems
For instance, in Pressure Vessel Design, the MRSO had a significantly lower standard deviation (5.21 \(\times\) 10\({}^{+2}\)) than the SCA and the LCA, indicating more stable results across iterations. Similarly, the MRSO demonstrated strong consistency in the Cantilever Beam, where its standard deviation was 2.93 \(\times\) 10\({}^{-3}\), considerably lower than the TSA (1.13 \(\times\) 10\({}^{-2}\)) and the LCA (4.63 \(\times\) 10\({}^{-2}\)). In Gear Train Design, the MRSO maintained a lower standard deviation (2.08 \(\times\) 10\({}^{-13}\) \begin{table}
\begin{tabular}{l c c c c c c} \hline \hline \multirow{2}{*}{**Engineering Problems**} & \multirow{2}{*}{**Metrics**} & \multicolumn{6}{c}{**Algorithms**} \\ \cline{3-6}  & & **MRSO** & **SCA** & **LCA** & **TSA** & **DOA** \\ \hline \multirow{3}{*}{Pressure Vessel Design} & Avg. & 6.83 \(\times\) 10\({}^{+3}\) & 7.28 \(\times\) 10\({}^{+3}\) & 1.64 \(\times\) 10\({}^{+4}\) & 7.24 \(\times\) 10\({}^{+3}\) & 5.98 \(\times\) 10\({}^{+3}\) \\  & Std. & 5.21 \(\times\) 10\({}^{+2}\) & 7.80 \(\times\) 10\({}^{+2}\) & 2.02 \(\times\) 10\({}^{+3}\) & 7.18 \(\times\) 10\({}^{+2}\) & 3.32 \(\times\) 10\({}^{+3}\) \\  & Rank & 2 & 4 & 5 & 3 & 1 \\ \hline \multirow{3}{*}{String Design} & Avg. & 1.38 \(\times\) 10\({}^{-2}\) & 1.31 \(\times\) 10\({}^{-2}\) & 2.86 \(\times\) 10\({}^{+8}\) & 1.45 \(\times\) 10\({}^{-2}\) & 6.37 \(\times\) 10\({}^{+7}\) \\  & Std. & 8.98 \(\times\) 10\({}^{-4}\) & 2.42 \(\times\) 10\({}^{-4}\) & 3.19 \(\times\) 10\({}^{+8}\) & 1.72 \(\times\) 10\({}^{-3}\) & 1.65 \(\times\) 10\({}^{+8}\) \\  & Rank & 2 & 1 & 5 & 3 & 4 \\ \hline \multirow{3}{*}{Three Bar Truss} & Avg. & 2.51 \(\times\) 10\({}^{+2}\) & 2.67 \(\times\) 10\({}^{+2}\) & 2.79 \(\times\) 10\({}^{+2}\) & 2.64 \(\times\) 10\({}^{+2}\) & 2.64 \(\times\) 10\({}^{+2}\) \\  & Std. & 6.12 \(\times\) 10\({}^{+1}\) & 6.44 & 1.13 \(\times\) 10\({}^{+1}\) & 3.53 \(\times\) 10\({}^{-1}\) & 3.59 \(\times\) 10\({}^{-1}\) \\  & Rank & 1 & 4 & 5 & 3 & 2 \\ \hline \multirow{3}{*}{Gear Train Design} & Avg. & 1.75 \(\times\) 10\({}^{-13}\) & 2.01 \(\times\) 10\({}^{-9}\) & 5.28 \(\times\) 10\({}^{-3}\) & 5.44 \(\times\) 10\({}^{-10}\) & 1.27 \(\times\) 10\({}^{-13}\) \\  & Std. & 2.08 \(\times\) 10\({}^{-13}\) & 4.63 \(\times\) 10\({}^{-9}\) & 7.29 \(\times\) 10\({}^{-3}\) & 9.02 \(\times\) 10\({}^{-10}\) & 5.87 \(\times\) 10\({}^{-13}\) \\  & Rank & 2 & 4 & 5 & 3 & 1 \\ \hline \multirow{3}{*}{Cantilever Beam} & Avg. & 1.34 & 1.41 & 1.57 & 1.36 & 1.4 \\  & Std. & 2.93 \(\times\) 10\({}^{-3}\) & 2.75 \(\times\) 10\({}^{-2}\) & 4.63 \(\times\) 10\({}^{-2}\) & 1.13 \(\times\) 10\({}^{-2}\) & 9.62 \(\times\) 10\({}^{-2}\) \\  & Rank & 1 & 4 & 5 & 2 & 3 \\ \hline \multirow{3}{*}{Welded Beam} & Avg. & 1.55 & 1.59 & 4.94 \(\times\) 10\({}^{-7}\) & 1.56 & 1.95 \\  & Std. & 2.05 \(\times\) 10\({}^{-1}\) & 4 \(\times\) 10\({}^{-2}\) & 1.36 \(\times\) 10\({}^{-8}\) & 3.64 \(\times\) 10\({}^{-2}\) & 6.13 \(\times\) 10\({}^{-1}\) \\  & Rank & 1 & 3 & 5 & 2 & 4 \\ \hline \multirow{3}{*}{Tire Design} & Avg. & 4.68 \(\times\) 10\({}^{-3}\) & 4.68 \(\times\) 10\({}^{-3}\) & 2.15 & 4.68 \(\times\) 10\({}^{-3}\) & 8.24 \(\times\) 10\({}^{-3}\) \\  & Std. & 1.76 \(\times\) 10\({}^{-18}\) & 1.11 \(\times\) 10\({}^{-17}\) & 1.31 & 8.82 \(\times\) 10\({}^{-19}\) & 1.36 \(\times\) 10\({}^{-2}\) \\ \cline{1-1}  & Rank & 1 & 2 & 5 & 3 & 4 \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Performance comparison of MRSO, SCA, LCA, TSA, and DOA on engineering design problems. compared to the SCA and the LCA, showcasing its reliability in delivering consistent results. However, in the Three Bar Truss, the MRSO exhibited a higher standard deviation of \(6.12\times 10^{4}\), indicating some variability in its performance despite having the best average value. When analyzing the ranking based on performance across the seven engineering problems, the MRSO secured top positions in four out of seven problems. In the Three Bar Truss and the Cantilever Beam, the MRSO ranked first, outperforming all other algorithms in these tasks.