## 4 Simulation Results
### Evaluation of Engineering Design Problems
#### 4.3.2 Engineering Design 2: REBD Problem  
Figure 7 illustrates the engineering problem where the REBD focuses on optimizing the bearing's load-carrying capacity [72]. In this context, ten design variables likely encompass factors such as the diameter, length, material characteristics, and shape configurations of the rolling elements, in addition to parameters related to the bearing raceways and clearances. Each design variable significantly influences the bearing's load capacity, durability, and operational reliability. This design challenge involves ten continuous variables (\(x\)), as defined in Equation (17), aiming to maximize the bearing's load capacity, as outlined in Equation (18). Furthermore, ten constraints are applied to regulate these variables. These constraints may account for material strength, geometric restrictions, stress and wear limitations, temperature tolerances, and other engineering criteria essential to ensuring its safe and effective performance within the bearing's intended application [62]. \[x=[D,\;Db,Z,\;f_{i},\;f_{o},\;K_{D,min},\;K_{D,max},\;\epsilon,\xi] \tag{17}\]  
\[Load(x)=Maximize\Bigg{\{} \begin{matrix}f_{c}\times Z^{0.67}\times Db^{1.80}& \text{if }Db\leq 25.4\;\text{mm}\\ 6.647\times f_{c}\times Z^{0.67}\times Db^{1.40}&\text{if }Db>25.4\;\text{mm} \end{matrix}  \tag{18}\]  
The proposed IMGO and the standard MGO methods are applied for the REBD problem and the optimal settings and loading objective are tabulated in Table 7. Also, the convergence curves of the compared algorithms are displayed in Figure 8.  
In this figure, both techniques seek to minimize the negative objective which represents the maximization of the load-carrying capacity. As shown, IMGO demonstrates a greater ability in exploring lower objective solutions compared to MGO through the generations, and finally, the proposed IMGO successfully attains the lower objective of \(-85,\)549.152, while its MGO counterpart value is equivalent to \(-85,\)547.295. Table 7 also tabulates comparisons with the other reported settings-based algorithms of AEO, FAEO, CBOA, BWOA, ATOA, and AOSA [62]. The proposed IMGO achieves the highest load performance of 85,549.152, slightly outperforming MGO (85,547.295) and all the other algorithms. CBOA, BWOA, ATOA, AOSA, AEO, and FAEO produce lower load values (i.e., 85,475.013 for CBOA, 77,061.675 for BWOA, 85,411.681 for ATOA, 85,162.698 for AOSA, 85,535.833 for AEO, and 85,538.445 for FAEO) than IMGO and MGO. This indicates that IMGO provides  
Figure 7: REBD problem. a more optimal solution compared to the other reported algorithms, confirming IMGO's competitive edge in precision and performance. Figure 9 displays the heatmap of the obtained cost using MGO and IMGO for the REBD problem. The contrast in the heatmap effectively visualizes the performance gap between IMGO and MGO, underscoring the superiority of the proposed IMGO method in maximizing load values consistently. As shown, IMGO consistently produces higher load values (approximately 85,549) across all runs, as visualized by the bottom row, showing a more saturated color band in the heatmap. This uniformity signifies IMGO's ability to achieve stable and optimized load results, outperforming MGO in every instance. On the other hand, the MGO results indicate a less stable performance compared to IMGO. Moreover, Table 8 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 REBD problem.