## 3 Proposed Methodology
### Phase 5: Application of the Opposition Nelder-Mead Algorithm
Since for \(D=20\), there was no observed difference in performance (\(1.16\times 10^{-01}>0.05\)), the focus is solely on showcasing the variations in performance among the algorithms for \(D=10\). In Table 7, the \(p\)-values obtained using [PERSON]'s test for \(D=10\) are presented, illustrating the pairwise comparisons of the performance of the algorithms. A value in bold signifies that the algorithm listed in the row outperformed the algorithm listed in the corresponding column. Conversely, a value not in bold indicates that there was no statistically significant difference in terms of performance between the compared algorithms. The values in bold provide a clear indication of the superior performers within the set of algorithms analyzed. +++ ==ERROR: No output for this page==
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c c} \hline \hline  & \multicolumn{2}{c}{\(F_{1}\)} & \multicolumn{2}{c}{\(F_{2}\)} & \multicolumn{2}{c}{\(F_{3}\)} & \multicolumn{2}{c}{\(F_{4}\)} & \multicolumn{2}{c}{\(F_{5}\)} & \multicolumn{2}{c}{\(F_{6}\)} & \multicolumn{2}{c}{\(F_{7}\)} & \multicolumn{2}{c}{\(F_{8}\)} & \multicolumn{2}{c}{\(F_{9}\)} & \multicolumn{2}{c}{\(F_{10}\)} & \multicolumn{2}{c}{\(F_{11}\)} & \multicolumn{2}{c}{\(F_{12}\)} & \multicolumn{2}{c}{\(F_{12}\)} & \multicolumn{2}{c}{\(F_{13}\)} & \multicolumn{2}{c}{\(F_{14}\)} & \multicolumn{2}{c}{\(F_{15}\)} & \multicolumn{2}{c}{\(F_{16}\)} & \multicolumn{2}{c}{\(F_{17}\)} & \multicolumn{2}{c}{\(F_{18}\)} & \multicolumn{2}{c}{\(F_{19}\)} & \multicolumn{2}{c}{\(F_{10}\)} \\ \cline{2-13} \cline{7-13} \cline{12-13} \cline{12-13} \cline{12-13} \cline{12-13} \cline{12-13} \cline{12-13} \cline{12-13} \cline{12-13} \cline{12-13} \cline{12-13} \cline{12-13} \cline{12-13} \cline{12-13} \cline{12-13} \cline{12-13} \cline{12-13} \cline{12-13} \cline{12-13} \cline{12-13} \cline{12-13} \cline{12-13} \cline{12-13} \cline{12-13} \cline{12-13} \cline{12-13} \cline{12-13} \cline{12-13} \cline{12-13} \clineine{12-13} \clineine{12-13} \clineine{12-13} \clineine{12-13} \clineineine{12-13} \clineineine{12-13} \clineineineine{12



## 5 Conclusions and Future Scope

In conclusion, this paper introduced a novel methodology that integrates the opposition Nelder-Mead algorithm into the selection phase of the genetic algorithm, aiming to improve its performance. Through a comprehensive comparative study, our methodology was rigorously evaluated against 11 highly regarded state-of-the-art algorithms known for their exceptional performance in the 2022 IEEE Congress on Evolutionary Computation (CEC 2022). The evaluation included [PERSON]'s post hoc test following a Friedman test. The results obtained were highly promising, showcasing the outstanding performance of our algorithm. In the majority of cases examined, our methodology demonstrated equal or superior performance compared to the competing algorithms. These findings affirm the effectiveness and competitiveness of our proposed approach for solving optimization problems. In future work, we plan to further explore and refine the integration of the opposition Nelder-Mead algorithm with other stages of the genetic algorithm. Additionally, we aim to conduct more extensive experiments on diverse benchmark problems to evaluate the robustness and generalizability of our methodology. Furthermore, investigating the scalability and efficiency of our approach for larger problem dimensions will be an important area of future research. Overall, we believe that our proposed methodology opens up promising avenues for advancements in evolutionary optimization techniques. Conceptualization, F.Z. and S.H. ; methodology, F.Z. and S.H. ; software, F.Z. and S.H. ; validation, F.Z. and S.H. ; formal analysis, F.Z. and S.H. ; writing--original draft preparation, F.Z. and S.H. ; writing--review and editing, F.Z. and S.H. All authors have read and agreed to the published version of the manuscript. This research received no external funding. Not applicable. The authors declare no conflict of interest.