## 5 D MMAEO for Engineering Application Problems
### Results and Analysis of DMMAEO on Engineering Application Problems  
For examining the practicability of DMMAEO, the seven engineering application problems (EA1-EA7) mentioned above are solved by the DMMAEO in this subsection, which are also solved by several recent promising algorithms with three categories mentioned in the previous section (EO [11], PSO [8], GWO [5], HHO [6], SMA [7], MPSO [36], MELGWO [37], LWMEO [20], SASS [52], and SaCHBA_PDN [53]) for comparison. Among the compared algorithms, SASS is the best algorithm for addressing engineering optimization problem, and SaCHBA_PDN is the best algorithm for addressing the path planning problem. Similar to the previous section, the compared algorithms adopt identical experimental settings to ensure the fairness of optimization performance comparison in solving engineering application problems. In this section, the best fitness value (Best), average fitness value (Ave.), worst fitness value (Worst), and standard deviation of fitness value (Std.) are applied in assessing the algorithm performance. Tables 15-21 show the results for the compared algorithms, which are obtained by implementing each algorithm 30 times independently on each engineering application problem. In Tables 15-21, the best results and the top-performing algorithm are highlighted in bold. As per the results of the three-bar truss design (EA1) from Table 15, the DMMAEO exhibits a superior performance compared to the other algorithms in minimizing the volume of three-bar truss. Regarding the spring design (EA2), the DMMAEO achieves the strongest performance in minimizing the spring weight compared to other algorithms based on the results from Table 16. Regarding the pressure vessel design (EA3), the \begin{table}
\begin{tabular}{c c c c c} \hline
**Algorithm** & **Best** & **Ave.** & **Worst** & **Std.** \\ \hline EO & 26.53133 & 26.53136 & 26.53178 & 7.2532 \(\times 10^{-5}\) \\ PSO & 26.53133 & 26.69082 & 29.22785 & 5.1219 \(\times 10^{-1}\) \\ GWO & 26.53133 & 26.72259 & 27.35287 & 2.0327 \(\times 10^{-5}\) \\ HHO & 26.53133 & 27.05176 & 31.16788 & 1.1008 \(\times 10^{9}\) \\ SMA & 26.53133 & 26.54724 & 26.72936 & 4.5165 \(\times 10^{-2}\) \\ MPSO & 26.53133 & 26.53173 & 26.54194 & 1.9825 \(\times 10^{-3}\) \\ MELGWO & 26.53133 & 26.53138 & 26.53252 & 1.8041 \(\times 10^{-4}\) \\ LWMEO & 26.53133 & 26.53133 & 26.53133 & 2.6632 \(\times 10^{-11}\) \\ SASS & 26.48636 & 26.48636 & 26.48637 & 2.2160 \(\times 10^{-6}\) \\ DMMAEO & **26.48636** & **26.48636** & **26.48636** & **1.1349 \(\times\) 10\({}^{-15}\)** \\ \hline \end{tabular}
\end{table} 
Table 18: Results for compared algorithms across tubular column design (EA4). \begin{table}
\begin{tabular}{c c c c} \hline
**Algorithm** & **Best** & **Ave.** & **Worst** & **Std.** \\ \hline EO & 26.53133 & 26.53136 & 26.53178 & 7.2532 \(\times 10^{-5}\) \\ PSO & 26.53133 & 26.69082 & 29.22785 & 5.1219 \(\times 10^{-1}\) \\ GWO & 26.53133 & 26.72259 & 27.35287 & 2.0327 \(\times 10^{-5}\) \\ HHO & 26.53133 & 27.05176 & 31.16788 & 1.1008 \(\times 10^{9}\) \\ SMA & 26.53133 & 26.54724 & 26.72936 & 4.5165 \(\times 10^{-2}\) \\ MPSO & 26.53133 & 26.53133 & 26.54194 & 1.9825 \(\times 10^{-3}\) \\ MELGWO & 26.53133 & 26.53138 & 26.53252 & 1.8041 \(\times 10^{-4}\) \\ LWMEO & 26.53133 & 26.53133 & 26.53133 & 2.6632 \(\times 10^{-11}\) \\ SASS & 26.48636 & 26.48636 & 26.48637 & 2.2160 \(\times 10^{-6}\) \\ DMMAEO & **26.48636** & **26.48636** & **26.48636** & **1.1349 \(\times\) 10\({}^{-15}\)** \\ \hline \end{tabular}
\end{table} 
Table 19: Results for compared algorithms across piston lever design (EA5).