## 6 Utilizing MRSO for Solving Engineering Design Issues
### Performance Comparison of MRSO with SCA, LCA, TSA, and DOA on Seven Engineering Design Problems  
In this section, we compare the performance of the MRSO with four other metaheuristic algorithms--the SCA, the LCA, the TSA, and the DOA--across seven engineering design problems. These include the Pressure Vessel Design, String Design, Three Bar Truss, Gear Train Design, Cantilever Beam, Welded Beam, and Tire Design. Table 10 provides a detailed breakdown of the average, standard deviation, and ranking for each algorithm on these problems. Our evaluation was conducted by running each algorithm 30 times to ensure reliable and accurate results. \begin{table}
\begin{tabular}{l c c c c} \hline \hline \multirow{2}{*}{**Engineering Application**} & \multicolumn{2}{c}{**MRSO**} & \multicolumn{2}{c}{**RSO**} \\ \cline{2-5}  & **Avg.** & **Std.** & **Avg.** & **Std.** \\ \hline Pressure Vessel Design & 6.956 \(\times\) 10\({}^{+3}\) & 4.479 \(\times\) 10\({}^{+2}\) & 1.72 \(\times\) 10\({}^{+4}\) & 1.031 \(\times\) 10\({}^{+4}\) \\ String Design & 1.417 \(\times\) 10\({}^{-2}\) & 9.721 \(\times\) 10\({}^{-4}\) & 3.954 \(\times\) 10\({}^{+8}\) & 4.217 \(\times\) 10\({}^{+8}\) \\ Three Bar Truss & 2.665 \(\times\) 10\({}^{+2}\) & 6.522 & 2.705 \(\times\) 10\({}^{+2}\) & 4.829 \\ Gear Train Design & 1.106 \(\times\) 10\({}^{-12}\) & 2.739 \(\times\) 10\({}^{-12}\) & 4.792 \(\times\) 10\({}^{-3}\) & 6.392 \(\times\) 10\({}^{-3}\) \\ Cantilever Beam & 1.343 & 2.932 \(\times\) 10\({}^{-3}\) & 2.382 & 8.673 \(\times\) 10\({}^{-1}\) \\ Welded Beam & 1.552 & 2.052 \(\times\) 10\({}^{-1}\) & 7.253 \(\times\) 10\({}^{+7}\) & 1.629 \(\times\) 10\({}^{+8}\) \\ Tire Design & 4.683 \(\times\) 10\({}^{-3}\) & 1.764 \(\times\) 10\({}^{-18}\) & 1.74 & 1.162 \\ \hline \hline \end{tabular}
\end{table} 
Table 9: The MRSO vs. standard RSO performance when applied to the seven engineering design problems: average, standard deviation, and statistical significance. In terms of average values, the MRSO showed superior performance in four out of the seven engineering design problems. In the Pressure Vessel Design, the MRSO secured second place with an average of 6.83 \(\times\) 10\({}^{+3}\), coming close to the DOA, which achieved the best performance with 5.98 \(\times\) 10\({}^{+3}\). However, in the Three Bar Truss, the MRSO outperformed all other algorithms, achieving the lowest average value of 2.51 \(\times\) 10\({}^{+2}\). Similarly, in the Cantilever Beam, the MRSO achieved the best average value of 1.34, proving its robustness in solving this particular problem. On the other hand, the MRSO ranked second in String Design, where the SCA led with an average of 1.31 \(\times\) 10\({}^{-2}\), compared to the MRSO's 1.38 \(\times\) 10\({}^{-2}\). In Gear Train Design, the MRSO was narrowly beaten by the DOA, placing second with an average of 1.75 \(\times\) 10\({}^{-13}\). In Tire Design, the MRSO also took first place, highlighting its strength across multiple problems. When considering standard deviation, the MRSO displayed strong consistency across most problems. In Pressure Vessel Design, the MRSO's standard deviation of 5.21 \(\times\) 10\({}^{+2}\) was lower than the SCA and the LCA, indicating more reliable performance. For the Three Bar Truss, the MRSO's standard deviation of 6.12 \(\times\) 10\({}^{+1}\) was considerably higher than the DOA and the TSA, suggesting some variability in its results, despite having the best average value. The MRSO also demonstrated stable results in the Cantilever Beam, with a lower standard deviation of 2.93 \(\times\) 10\({}^{-3}\), compared to the TSA's 1.13 \(\times\) 10\({}^{-2}\) and the LCA's 4.63 \(\times\) 10\({}^{-2}\). This highlights the reliability of the MRSO's performance in these cases. Regarding consistency, as represented by standard deviation, the MRSO exhibited reliable performance across most of the engineering problems.