## 4 Simulation Results
### Evaluation of Engineering Design Problems
#### 4.3.1 Engineering Design 1: WBD Problem  
The WBD problem focuses on minimizing the manufacturing cost of a beam, while ensuring constraints on bending stress, shear stress, buckling load, and end deflection are maintained [61]. Figure 4 illustrates the setup of a WB fixed to a substrate. As depicted, the beam supports a load of weight \(P\) positioned at a distance \(L\) from the substrate. The beam is secured with upper and lower welds of length \(W\) and thickness \(H\), creating a rectangular cross-section defined by width \(b\) and height \(t\). Constructed from steel, the beam's design in this study is optimized to minimize cost under the applied load. Specifically, the load weight is fixed at 6000 pounds, while the distance \(L\) is set at 14 inches. The design problem includes four continuous variables (\(x\)) as described in Equation (15) where \(H\), \(W\), \(t\), and \(b\) indicate the weld thickness, weld length, beam height, and beam width for minimizing the  
Figure 3: Visual comparison of key performance indicators between IMGO and MGO for the 23 benchmarks. manufacturing cost of a beam described in Equation (16), while ensuring constraints on bending stress, shear stress, buckling load, and end deflection are maintained [62]. \[x=[H,\ W,t,b] \tag{15}\]  
\[Cost(x)=\left(1.1047\times W\times H^{2}\right)+\left(0.0481\times b\times t \times(W+14)\right) \tag{16}\]  
The proposed IMGO and the standard MGO methods are applied to the WBD problem for 50 separate runs. The optimal settings and fabrication cost objectives are tabulated in Table 4. Also, the convergence curves of the compared algorithms are displayed in Figure 5. As shown, IMGO provides greater ability in exploring lower cost solutions compared to MGO through the generations, and finally, the proposed IMGO successfully attains a lower cost of 1.63443, while its MGO counterpart obtains a value of 1.63614 with a 0.104% reduction. Table 4 also tabulates the comparisons with the Artificial Ecosystem Optimization (AEO) [63], Fitness-Distance-Balance AEO (FAEO) [62], Chef-Based Optimization Algorithm (CBOA) [64], Beluga Whale Optimization Algorithm (BWOA) [65], Arithmetic-Trigonometric Optimization Algorithm (ATOA) [66], and Atomic Orbital Searching Algorithm (AOSA) [67; 68]. IMGO demonstrates significant improvements equivalent to 1.508%, 10.165%, 11.389%, 0.808%, 0.051%, and 0.023% compared to CBOA, BWOA, ATOA, AOSA, AEO, and FAEO, respectively. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Design Parameters** & **CBOA [62]** & **BWOA [62]** & **ATOA [62]** & **AOSA [62]** & **AEO [62]** & **FAEO [62]** & **MGO** & \
\begin{tabular}{c} **Proposed** \\ **IMGO** \\ \end{tabular} \\ \hline \(H\) & 0.176131 & 0.162265 & 0.181515 & 0.182966 & 0.185113 & 0.185216 & 0.18477 & 0.185409 \\ \hline \(W\) & 3.883728 & 4.411861 & 4.394343 & 3.716486 & 3.635691 & 3.632255 & 3.641876 & 3.627015 \\ \hline \(t\) & 9.475198 & 9.282771 & 10.27982 & 9.55396 & 9.51964 & 9.51898 & 9.522734 & 9.51893 \\ \hline \(b\) & 0.187231 & 0.205658 & 0.185175 & 0.185469 & 0.185423 & 0.185412 & 0.185439 & 0.18541 \\ \hline Cost (\(x\)) & 1.659462 & 1.819376 & 1.844511 & 1.647761 & 1.63528 & 1.63482 & 1.636145311 & 1.634443 \\ \hline
\begin{tabular}{c} Percentage \\ improvement \\ regarding IMGO \\ \end{tabular} & 1.508\% & 10.165\% & 11.389\% & 0.808\% & 0.051\% & 0.023\% & 0.104\% & - \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Optimal results of IMGO, MGO, and the other reported results for the WBD problem. Figure 4: WBD problem. Figure 6 displays the heatmap of the obtained cost using MGO and IMGO for the WBD problem. As shown, IMGO consistently achieves a lower cost compared to MGO across all runs, as seen by the near-constant values in the IMGO row. In contrast, the MGO results show significant variation, with costs fluctuating much more widely. This suggests that IMGO provides more stable and reliable solutions for this optimization problem.