## 4 Simulation Results
### Evaluation of CEC 2017 Benchmark Models  
When evaluating the efficacy of optimization efforts, benchmark functions are essential. In this study, fifteen mathematical functions from the CEC 2017 competition set are used to compare the performance of the regular MGO with the suggested IMGO approaches. These functions, which come in unimodal, multimodal, mixed, and composite varieties [60], offer a variety of testing environments to evaluate the algorithms' flexibility and resilience. Table 1 lists the specifics of these unrestricted benchmark functions. Each function runs in a 30-dimensional space with control variables that are bounded between [\(-\)100, 100]. As a result, IMGO is used in conjunction with MGO, considering 30 search material agents and 500 generations, and Table 2 summarizes the comparison of their outcomes.  
\begin{table}
\begin{tabular}{c c c} \hline
**No.** & **Function** & **Optimal** \\ \hline \(Fn_{1}\) & Shifted and rotated bent cigar & 100.000 \\ \(Fn_{2}\) & Shifted and rotated & 400.000 \\ \(Fn_{3}\) & Hybrid 1 (\(N=3\)) & 1100.000 \\ \(Fn_{4}\) & Hybrid 2 (\(N=3\)) & 1200.000 \\ \(Fn_{5}\) & Hybrid 3 (\(N=3\)) & 1300.000 \\ \(Fn_{6}\) & Hybrid 4 (\(N=4\)) & 1400.000 \\ \(Fn_{7}\) & Hybrid 5 (\(N=4\)) & 1500.000 \\ \(Fn_{8}\) & Hybrid 6 (\(N=4\)) & 1600.000 \\ \(Fn_{9}\) & Hybrid 6 (\(N=5\)) & 1800.000 \\ \(Fn_{10}\) & Hybrid 6 (\(N=5\)) & 1900.000 \\ \(Fn_{11}\) & Hybrid 6 (\(N=6\)) & 2000.000 \\ \(Fn_{12}\) & Composition 1 (\(N=3\)) & 2100.000 \\ \(Fn_{13}\) & Composition 5 (\(N=5\)) & 2500.000 \\ \(Fn_{14}\) & Composition 7 (\(N=6\)) & 2700.000 \\ \(Fn_{15}\) & Composition 9 (\(N=3\)) & 2900.000 \\ \hline \end{tabular}
\end{table}
Table 1: Benchmarks under investigation regarding the CEC 2017 suite.  
The results presented in Table 2 compare the performance of the proposed IMGO technique with the standard MGO method across various benchmark functions from the CEC 2017 suite. IMGO consistently demonstrates significant improvements in certain benchmarks, as indicated by the improvement percentages across the \"Best Score\", \"Mean Score\", and \"Worst Score\" columns. As shown, the suggested IMGO successfully achieves lower fitness scores equivalent to 93.33%, 86.66%, and 86.66% of the investigated benchmarks regarding the best, mean and worst scores, respectively. The suggested IMGO shows a particularly strong improvement in benchmarks such as \(Fn_{5}\) (39.784%), \(Fn_{13}\) (8.410%), and \(Fn_{14}\) (7.149%) in terms of best performance scores. Also, IMGO exhibits substantial improvements in mean performance across many functions. For example, in \(Fn_{5}\), IMGO achieves a 52.619% improvement in mean scores, indicating more consistent performance across multiple runs. Other notable improvements in mean scores include \(Fn_{10}\) (4.906%) and \(Fn_{14}\) (4.253%), showing that IMGO performs more robustly across iterations and is less likely to become trapped in local optima. The improvement in the worst score is a critical indicator of the algorithm's reliability in avoiding poor solutions. In \(Fn_{5}\), IMGO improves the worst-case performance by 81.060%, highlighting its ability to maintain solution quality even in challenging scenarios. Improvements in the worst score are also notable in \(Fn_{8}\) (6.906%) and \(Fn_{10}\) (5.886%), underscoring IMGO's robustness. These improvements reflect IMGO's effectiveness in achieving better optimization accuracy, consistency, and resilience against suboptimal results, making it a promising method for complex optimization tasks.