## 4 Simulation Results
### Evaluation of Engineering Design Problems
#### 4.3.1 Engineering Design 1: WBD Problem
IMGO produces results that are consistently close to approximately 1.6344 across all runs, while MGO frequently yields higher costs, often above 1.65 and even reaching values around 1.70. This indicates that IMGO is more effective in achieving lower-cost solutions within the given constraints. Figure 5: Best convergence curves of MGO and IMGO for the WBD problem. Figure 6: Heatmap of the obtained cost using MGO and IMGO for the WBD problem. Moreover, Table 5 provides a statistical comparison of the IMGO, standard MGO, and various other algorithms (CBOA, BWOA, ATOA, AOSA, AEO, FAEO) in terms of their performance on the WBD problem. The comparison includes the following key statistics: the best, mean, worst, and standard deviation of the costs obtained by each algorithm. As demonstrated, IMGO achieves the lowest best cost value of 1.634443, which outperforms all other algorithms, including MGO (1.636145) and the other reported algorithms (CBOA, BWOA, ATOA, AOSA, AEO, and FAEO). The IMGO algorithm's mean cost of 1.634564 is significantly lower than those of MGO (1.666251) and the other algorithms, which have considerably higher mean values, particularly ATOA and CBOA, where the mean values are extremely high (in the order of 10 \(\times\) 10\({}^{19}\) and 10 \(\times\) 10\({}^{21}\)). IMGO also outperforms other methods in terms of the worst-case cost, with a worst cost of 1.638706, while MGO has a higher worst-case cost of 1.713294. The other algorithms exhibit very high worst costs, indicating poor reliability in finding low-cost solutions in some runs. The proposed IMGO has the lowest standard deviation of 0.000606, indicating that its results are highly consistent across different runs. In contrast, MGO and the other algorithms show greater variation, with much higher standard deviations, especially for CBOA, BWOA, and ATOA, where the deviations are orders of magnitude larger. This low standard deviation further emphasizes IMGO's stability and reliability in consistently achieving low-cost solutions. Alternatively, Table 6 presents a comparison of the best costs achieved by various algorithms, including the IMGO, standard MGO, and the other reported algorithms, for solving the WBD problem. As shown in Table 6, the proposed IMGO achieves the lowest best cost of 1.634443, outperforming all other algorithms listed, including WCMFOA (1.732704), ESO (1.728783), HSOGA (1.727341), HOA (1.722109), ADEA (1.7248523085), HMOA (1.695247), and MGO (1.636145). The proposed IMGO shows significant improvements equivalent to 5.67%, 5.46%, 5.38%, 5.24%, 5.09%, 3.59%, and 0.10% compared to WCMFOA, ESO, HSOGA, HOA, ADEA, HMOA, and MGO, respectively. Thus, the proposed IMGO demonstrates superior optimization performance compared to MGO and the other advanced algorithms, \begin{table}
\begin{tabular}{c c} \hline \hline
**Algorithm** & **Best Cost** \\ \hline Water Cycle Moth Flame Optimization Algorithm (WCMFOA) [69] & 1.732704 \\ Evolutionary Strategy Optimization (ESO) [70] & 1.728783 \\ Hybrid Self-adaptive Orthogonal Genetic Algorithm (HSOGA) [70] & 1.727341 \\ Adaptive Differential Evolution Algorithm (ADEA) [69] & 1.7248523085 \\ Harris Hawks Optimization Algorithm (HOA) [71] & 1.722109 \\ Hybrid Multi-population Optimization Algorithm (HMOA) [71] & 1.695247 \\ MGO & 1.636145 \\ Proposed IMGO & 1.634443 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Comparisons of the best cost between IMGO, MGO, and the other reported algorithms for WBD.