## 3 The Theory of the Proposed OL-UPSGBO Algorithm
### Benchmarking of the OL-UPSGBO Algorithm
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Function Type** & **No.** & **Function** & **Optimum** \\ \hline \multirow{4}{*}{Unimodal} & \(F_{1}\) & Shifted and Rotated Bent Cigar Function & 100 \\ \cline{2-4}  & \(F_{2}\) & NA & NA \\ \cline{2-4}  & \(F_{3}\) & Shifted and Rotated Zakharov Function & 300 \\ \hline \multirow{4}{*}{Multimodal} & \(F_{4}\) & Shifted and Rotated Rosenbrock’s Function & 400 \\ \cline{2-4}  & \(F_{5}\) & Shifted and Rotated Rastrigin’s Function & 500 \\ \cline{2-4}  & \(F_{6}\) & Shifted and Rotated Expanded Scafter’s \(f_{6}\) Function & 600 \\ \cline{2-4}  & \(F_{7}\) & Shifted and Rotated Lunacek Bi\_Rastrigin Function & 700 \\ \cline{2-4}  & \(F_{8}\) & Shifted and Rotated Non-Continuous Rastrigin’s Function & 800 \\ \cline{2-4}  & \(F_{9}\) & Shifted and Rotated Levy Function & 900 \\ \cline{2-4}  & \(F_{10}\) & Shifted and Rotated Schwefel’s Function & 1000 \\ \hline \multirow{4}{*}{Hybrid} & \(F_{11}\) & Hybrid Function 1 (N = 3) & 1100 \\ \cline{2-4}  & \(F_{12}\) & Hybrid Function 2 (N = 3) & 1200 \\ \cline{2-4}  & \(F_{13}\) & Hybrid Function 3 (N = 3) & 1300 \\ \cline{2-4}  & \(F_{14}\) & Hybrid Function 4 (N = 4) & 1400 \\ \cline{2-4}  & \(F_{15}\) & Hybrid Function 5 (N = 4) & 1500 \\ \cline{2-4}  & \(F_{16}\) & Hybrid Function 6 (N = 4) & 1600 \\ \cline{2-4}  & \(F_{17}\) & Hybrid Function 6 (N = 5) & 1700 \\ \cline{2-4}  & \(F_{18}\) & Hybrid Function 6 (N = 5) & 1800 \\ \cline{2-4}  & \(F_{19}\) & Hybrid Function 6 (N = 5) & 1900 \\ \cline{2-4}  & \(F_{20}\) & Hybrid Function 6 (N = 6) & 2000 \\ \hline \multirow{4}{*}{Composite} & \(F_{21}\) & Composition Function 1 (N = 3) & 2100 \\ \cline{2-4}  & \(F_{22}\) & Composition Function 2 (N = 3) & 2200 \\ \cline{2-4}  & \(F_{23}\) & Composition Function 3 (N = 4) & 2300 \\ \cline{2-4}  & \(F_{24}\) & Composition Function 4 (N = 4) & 2400 \\ \cline{2-4}  & \(F_{25}\) & Composition Function 5 (N = 5) & 2500 \\ \cline{2-4}  & \(F_{26}\) & Composition Function 6 (N = 5) & 2600 \\ \cline{2-4}  & \(F_{27}\) & Composition Function 7 (N = 6) & 2700 \\ \cline{2-4}  & \(F_{28}\) & Composition Function 8 (N = 6) & 2800 \\ \cline{2-4}  & \(F_{29}\) & Composition Function 9 (N = 3) & 2900 \\ \cline{2-4}  & \(F_{30}\) & Composition Function 10 (N = 3) & 3000 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The CEC 2017 test suite [58; 59]. Moreover, the [PERSON] test and Wilcoxon test were conducted as two non-parametric statistical tests. The average ranking and total ranking based on the Friedman test can be observed in the last two rows of Table 3, which show that the proposed OLUPSGBO has the best rank among the tested algorithms. Furthermore, to further graphically illustrate the Friedman test ranks, the total ranks can be seen in Figure 5. The Wilcoxon test was also conducted and \(p\)-value results of the test can be tabulated in Table 4. It is clear that the OLUPSUBO algorithm is statistically significant to other algorithms corresponding to most studied functions with a \(p\)-value <0.05. Nevertheless, it is common to have statistical similarities between the algorithms for fewer cases. The overall results demonstrate that the proposed OL-UPSGBO algorithm has great performance when compared with the studied algorithms and the component algorithms for higher dimensions using the CEC 201  
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline
**Function** & **Property** & **GWO** & **WOA** & **MVO** & **SCA** & **GBO** & **AOA** & **EO** & **AO** & **OLPSGBO** \\ \hline \multirow{6}{*}{\(F_{1}\)} & Ave & \(1.1\times 10^{9}\) & \(2.6\times 10^{6}\) & \(3.7\times 10^{8}\) & \(1.6\times 10^{10}\) & \(1.9\times  
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline
**Function** & **Property** & **GWO** & **WOA** & **MVO** & **SCA** & **GBO** & **AOA** & **EO** & **AO** & **OLPSGBO** \\ \hline \multirow{4}{*}{\(F_{10}\)} & Ave & \(5.9\times 10^{3}\) & \(6.4\times 10^{3}\) & \(7.2\times 10