## 4 Simulation Results
### Evaluation of Engineering Design Problems
#### 4.3.2 Engineering Design 2: REBD Problem
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Design** & **CBOA** & **BWOA** & **ATOA** & **AOSA** & **AEO** & **FAEO** & **MGO** & **Proposed** \\
**Parameters** & [62] & [62] & [62] & [62] & [62] & [62] & [62] & **IMGO** \\ \hline \(x_{1}\) & 125.721537 & 125 & 125.6450966 & 125.6954156 & 125.7166548 & 125.7190473 & 125.718792 & 125.719054 \\ \hline \(x_{2}\) & 21.41505373 & 20.20294186 & 21.40912385 & 21.3713786 & 21.42416992 & 21.42558213 & 21.4255085 & 21.4255819 \\ \hline \(x_{3}\) & 10.62467678 & 10.76935323 & 10.91688091 & 11.16793061 & 10.9060758 & 10.51638533 & 11.364617 & 10.7342173 \\ \hline \(x_{4}\) & 0.515 & 0.515 & 0.515 & 0.515 & 0.51500222 & 0.515001707 & 0.51500054 & 0.51500001 \\ \hline \(x_{5}\) & 0.515 & 0.515 & 0.6 & 0.534909785 & 0.516220193 & 0.558166915 & 0.51574389 & 0.51501414 \\ \hline \(x_{6}\) & 0.4 & 0.439875386 & 0.5 & 0.405654012 & 0.458634675 & 0.421633147 & 0.49524934 & 0.48689999 \\ \hline \(x_{7}\) & 0.657872851 & 0.6 & 0.7 & 0.614593561 & 0.634391445 & 0.660530794 & 0.69777003 & 0.6506474 \\ \hline \(x_{8}\) & 0.3 & 0.357539167 & 0.3 & 0.3 & 0.300016447 & 0.300000008 & 0.3000067 & 0.3000003 \\ \hline \(x_{9}\) & 0.03454596 & 0.095748315 & 0.028386125 & 0.05490016 & 0.07399689 & 0.04871366 & 0.03265539 & 0.09864157 \\ \hline \(x_{10}\) & 0.6 & 0.6 & 0.6 & 0.606110819 & 0.653987474 & 0.631641863 & 0.61206202 & 0.61419829 \\ \hline Load (\(x\)) & 85,475.01324 & 77,061.67537 & 85,411.68148 & 85,162.69751 & 85,535.83305 & 85,538.44468 & 85,547.295 & 85,549.152 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Optimal results of IMGO, MGO, and the other reported results for REBD problem. Figure 8: Best convergence curves of MGO and IMGO for the REBD problem. As demonstrated, the IMGO algorithm's mean load of 85,546.950 is significantly higher than those of MGO (85,530.002) and the other algorithms, which have considerably lower mean values, particularly BWOA, ATOA, and AOSA, where the mean values are extremely high (in the order of 63,995.023, 51,582.611, and 69,772.645). Also, IMGO outperforms all other methods in terms of the worst-case load, with a worst load of 85,535.607. The proposed IMGO has the lowest standard deviation at 3.416, indicating that its results are highly consistent across different runs. By contrast, MGO (9.767), FAEO (53.148), and the other algorithms show greater variation, with much higher standard deviations, especially compared to AOSA (16,605.259), BWOA (10,247.562), ATOA (8792.876), AEO (6019.295), and CBOA (1117.190), where the deviations are orders of magnitude larger. Alternatively, Table 9 presents a comparison of the maximum load achieved by various algorithms, including the IMGO, standard MGO, and the other reported algorithms used to solve the REBD problem. As shown in Table 9, the proposed IMGO achieves the highest load, outperforming all the other algorithms listed, including SDA (81,575.185), GA (81,841.511), MBO (81,843.686), Chaotic-JAYA (81,858.318), TLA (81,859.738), and MGO (185,547.295). The proposed IMGO demonstrates superior optimization performance compared to MGO and the other advanced algorithms, achieving the highest best load for the REBD problem. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Load** & **CBOA [62]** & **BWOA [62]** & **ATOA [62]** & **AOSA [62]** & **AEO [62]** & **FAEO [62]** & **MGO** & **Proposed IMGO** \\ \hline Best & 85,475.013 & 77,061.675 & 85,411.681 & 85,162.698 & 85,535.833 & 85,538.445 & 85,547.295 & 85,549.152 \\ \hline Mean & 83,176.664 & 63,995.023 & 51,582.611 & 69,772.645 & 82,641.699 & 85,489.774 & 85,530.002 & 85,546.950 \\ \hline Worst & 82,009.018 & 33,317.699 & 45,160.116 & 42,268.082 & 52,438.992 & 85,222.972 & 85,505.571 & 85,535.607 \\ \hline Standard deviation & 1117.190 & 10,247.562 & 8792.876 & 16,605.259 & 6019.295 & 53.148 & 9.767 & 3.416 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Statistical results of IMGO, MGO, and the other reported results for the REBD problem.