## 4 Simulation Results
### Evaluation of Engineering Design Problems
#### 4.3.3 Engineering Design 3: PVD Problem  
Figure 10 presents an engineering optimization problem in which the primary goal of the Pressure Vessel Design (PVD) is to reduce total fabrication costs. These costs encompass expenses related to materials, forming, and welding. The objective of this problem is to achieve the lowest possible production cost for the design depicted in Figure 10[77]. This optimization involves adjusting four variables, represented as \(x\), which are specified in Equation (19) and aim to minimize the fabrication costs detailed in Equation (20). Additionally, the problem includes four constraints that must be adhered to for a feasible solution. \[x=[Ts,\ Th,R,\ L] \tag{19}\]  
\[Cost(x)=(0.6224\times Ts\times R\times L)+\left(1.7781\times Th\times R^{2} \right)+\left(3.1661\times Ts^{2}\times L\right)+\left(19.84\times Ts^{2}\times R\right) \tag{20}\]  
The proposed IMGO and the standard MGO methods are applied for the PVD problem, and the optimal settings and fabrication cost objectives are tabulated in Table 10. Also, the convergence curves of the compared algorithms are displayed in Figure 11. As shown, the proposed IMGO successfully attains a lower cost of 5885.3329 while its MGO counterpart is equivalent to 5885.5584. Table 10 also tabulates comparisons with AEO, FAEO, CBOA, BWOA, ATOA, and AOSA [62]. As shown, the proposed IMGA achieves the lowest cost compared to all others. \begin{table}
\begin{tabular}{c c} \hline \hline
**Algorithm** & **Best Load** \\ \hline Supply–Demand-based Algorithm (SDA) [73] & 81,575.185 \\ GA [72] & 81,841.511 \\ Mine Blast Optimizer (MBO) [74] & 81,843.686 \\ Chaotic-JAYA [75] & 81,858.318 \\ Teaching Learning-based Algorithm (TLA) [76] & 81,859.738 \\ MGO & 85,547.295 \\ Proposed IMGO & 85,549.152 \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Comparisons of the best load between IMGO, MGO, and the other reported algorithms for the REBD problem. Figure 10: PVD problem. Figure 12 shows that IMGO consistently achieves lower costs compared to MGO for the PVD problem, with near-constant values in the IMGO row. IMGO provides more stable, reliable solutions, with results close to approximately 5885 across most runs indicating its effectiveness. Moreover, Table 11 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 PVD problem. As demonstrated, IMGO achieves the lowest mean cost value at 5894.55, which outperforms all other algorithms, including MGO (5889.51) and the other reported algorithms (CBOA, BWOA, ATOA, AOSA, AEO, and FAEO) which attain higher costs (20,603.45, 11,436.81, 27,135.55, 6825.11, 6338.51, and 5959.46). MGO and IMGO also outperform the other methods in terms of the standard deviations, which are equivalent to 20.4 and 21.8, respectively. The other algorithms exhibit very high variation, with much higher standard deviations, especially for CBOA, BWOA, and ATOA, where the deviations are significantly larger with values equivalent to 13,257.38, 3385.68, and 25,920.65, respectively. For tackling the PVD problem, Table 12 compares the best prices attained by the several strategies reported. As demonstrated, the suggested IMGO outperforms all other algorithms provided and attains the lowest best cost of 5885.33. Therefore, the suggested IMGO achieves the lowest optimal cost for the PVD issue and exhibits better optimization performance compared to MGO and the other sophisticated algorithms.