## 3 Proposed Methodology
### Phase 5: Application of the Opposition Nelder-Mead Algorithm
Each of these algorithms represents a significant approach in the field of optimization. The comparison was conducted by measuring and reporting the average and standard deviation values for each algorithm. This comprehensive evaluation allowed for a comprehensive understanding of the proposed methodology's performance in relation to other well-established algorithms. By considering a diverse range of state-of-the-art algorithms, we were able to gain valuable insights into the strengths, weaknesses, and comparative performance of the proposed methodology. To enhance clarity, the algorithms originally labeled Co-PPSO are renamed \(A_{2}\), and the algorithms originally labeled EA4 eigN100-10 are renamed \(A_{3}\). This renaming convention was also used for the remaining algorithms. By utilizing the new nomenclature (\(A_{2},A_{3},\ldots,A_{12}\)), the presentation and interpretation of the results are more straightforward and unambiguous. Finally, our methodology is renamed \(A_{1}\). Table 3 presents the diversity measurements (\(\Delta\)) calculated using Equation (35) during the initialization of the initial population. The diversity values were examined for two different dimensions, \(D=10\) and \(D=20\). For \(D=10\), it was observed that the highest diversity value was achieved when employing the chaotic map outlined in Equation (13) in conjunction with the opposition-based learning technique provided in Equation (23). On the other hand, for \(D=20\), the maximum diversity value was obtained when utilizing the chaotic map described in Equation (11) and the opposition-based learning technique provided in Equation (20). Consequently, these specific parameters were chosen to conduct the comparative study, as they demonstrated superior diversity in the initial population. \[\Delta=\frac{1}{D}\sum_{j=1}^{D}\left(\frac{1}{N}\sum_{i=1}^{N}\|\mathbf{\bar {x}}-\mathbf{x}_{i}\|\right) \tag{35}\]

\[\mathbf{\bar{x}}=\frac{\mathbf{x}_{1}+\ldots+\mathbf{x}_{N}}{N}\]Table 4 presents the initial values selected for the various parameters of the proposed methodology. This table serves as a comprehensive reference for the parameter configurations utilized during the initial stages of the study. Additionally, it is important to note that the initialization of the parameters for the algorithms employed in the comparative study was derived from their respective papers. By incorporating the parameters outlined in the original research papers, we ensure consistency and comparability between our study and previous works. This approach allows for a fair evaluation and unbiased comparison of the performance and effectiveness of the proposed methodology against existing algorithms. The proposed methodology was executed 30 times in order to facilitate the application of the Friedman statistical test. The Cayley-Menger determinant is a determinant that provides the volume of a simplex in \(D\) dimensions. If \(S\) is a \(D\)-simplex in \(\mathds{R}^{D}\) with vertices \(\{\mathbf{x}_{1},\dots,\mathbf{x}_{D+1}\}\) and \(B=(\beta_{ij})\) denotes the \((D+1)\times(D+1)\) matrix modeled in Equation (36), then the volume of the simplex \(S\), denoted by \(V(S)\), is computed using Equation (37). \[\beta_{ij}=\|\mathbf{x}_{i}-\mathbf{x}_{j}\|^{2} \tag{36}\]

\[V(S)=\sqrt{\frac{(-1)^{D+1}}{2^{D}(D!)^{2}}det(\hat{B})} \tag{37}\]

where \(\hat{B}\) is the \((D+2)\times(D+2)\) matrix obtained from \(B\) by bordering \(B\) with a top vector \([0,1,...,1]\) and a left column \([0,1,...,1]^{T}\). Here, the vector L2-norms \(|\mathbf{x}_{i}-\mathbf{x}_{j}|_{2}\) are the edge lengths, and the determinant in Equation (37) is the Cayley-Menger determinant [89; 90].