## 5 Experimental Results
### The Compared Experimental Results
#### 5.4.1 Comparisons on Solution Accuracy  
The accuracy results of the uni-modal, multimodal, and composition functions are given in Tables 8-10, respectively. The accuracy results are in terms of the mean optimum solution and the standard deviation of the solutions obtained by each algorithm over 300,000 function evaluation times (FES) or 10,000 maximum generations. In all experiments, the dimensions of all problems are 30. The best \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Func.** & **PTBO** & **PSO** & **DE** & **BA** & **CS** & **BSO** & **WWO** & **WCCA** & **SMS** \\ \hline F01 & \(4.31\times 10^{-28}\) & \(3.56\times 10^{1}\) & \(2.65\times 10^{3}\) & \(1.05\times 10^{-3}\) & \(4.46\times 10^{1}\) & \(3.71\times 10^{-3}\) & \(2.62\times 10^{-28}\) & \(\mathbf{1.58\times 10^{-28}}\) & \(4.63\times 10^{4}\) + \\  & \(5.96\times 10^{-28}\) & \(2.65\times 10^{3}\) & \(8.98\times 10^{-1}\) & \(1.29\times 10^{-4}\) & \(3.10\times 10^{2}\) & \(5.71\times 10^{-3}\) & \(\mathbf{1.30\times 10^{-28}}\) & \(1.45\times 10^{-28}\) & \(4.24\times 10^{3}\) \\ \hline F02 & \(8.63\times 10^{5}\) & \(1.12\times 10^{7}\) & \(1.56\times 10^{7}\) & \(4.82\times 10^{4}\) & \(4.21\times 10^{7}\) & \(1.17\times 10^{6}\) & \(1.20\times 10^{6}\) & \(\mathbf{7.53\times 10^{6}}\) & \(4.91\times 10^{8}\) + \\  & \(6.60\times 10^{5}\) & \(1.56\times 10^{7}\) & \(\mathbf{2.05\times 10^{8}}\) & \(1.81\times 10^{4}\) & \(4.04\times 10^{6}\) & \(3.35\times 10^{5}\) & \(7.19\times 10^{5}\) & \(7.54\times 10^{6}\) & \(8.49\times 10^{7}\) \\ \hline F03 & \(\mathbf{3.84\times 10^{7}}\) & \(4.98\times 10^{10}\) & \(4.72\times 10^{10}\) & \(8.96\times 10^{8}\) & \(1.72\times 10^{10}\) & \(1.66\times 10^{8}\) + & \(3.98\times 10^{8}\) + & \(2.94\times 10^{9}\) + & \(1.00\times 10^{10}\) + \\  & \(5.22\times 10^{7}\) & \(4.72\times 10^{10}\) & \(3.57\times 10^{7}\) & \(3.48\times 10^{9}\) & \(2.11\times 10^{9}\) & \(2.22\times 10^{8}\) & \(6.71\times 10^{8}\) & \(4.30\times 10^{9}\) & **0.00** \\ \hline F04 & \(8.26\times 10^{3}\) & \(5.64\times 10^{3}\) & \(8.41\times 10^{3}\) & \(2.42\times 10^{4}\) & \(6.05\times 10^{4}\) & \(5.44\times 10^{3}\) + & \(5.57\times 10^{4}\) + & \(\mathbf{6.55\times 10^{4}}\) + & \(8.14\times 10^{4}\) + \\  & \(3.71\times 10^{5}\) & \(8.41\times 10^{3}\) & \(1.27\times 10^{3}\) & \(3.91\times 10^{2}\) & \(6.61\times 10^{2}\) & \(2.22\times 10^{10}\) & \(1.05\times 10^{2}\) & \(\mathbf{2.07\times 10^{2}}\) & \(9.79\times 10^{3}\) \\ \hline F05 & \(\mathbf{2.64\times 10^{-18}}\) & \(1.58\times 10^{4}\) & \(1.25\times 10^{3}\) & \(1.37\times 10^{-2}\) + & \(1.09\times 10^{3}\) + & \(4.12\times 10^{-2}\) & \(5.53\times 10^{-2}\) + & \(2.95\times 10^{-12}\) & \(1.38\times 10^{4}\) + \\  & \(\mathbf{4.44\times 10^{-18}}\) & \(1.25\times 10^{3}\) & \(1.84\times 10^{1}\) & \(2.08\times 10^{-3}\) & \(6.70\times 10^{3}\) & \(1.74\times 10^{-2}\) & \(2.90\times 10^{-1}\) & \(7.89\times 10^{-12}\) & \(2.66\times 10^{3}\) \\ \hline \multicolumn{7}{c}{\(\dagger/L/\cdots\)} & \multicolumn{1}{c}{\(4/1/0\)} & \multicolumn{1}{c}{\(4/1/0\)} & \multicolumn{1}{c}{\(4/1/0\)} & \multicolumn{1}{c}{\(5/0/0\)} & \multicolumn{1}{c}{\(4/1/0\)} & \multicolumn{1}{c}{\(2/3/0\)} & \multicolumn{1}{c}{\(5/0/0\)} \\ \hline \multicolumn{7}{c}{\(\ddagger\)+, \(\ddagger\), and \"\(\sim\)\" denote that the performance of PTBO is better than, worse than, and similar to that of the corresponding algorithm, respectively, Bold text is the best result obtained by the compared algorithms.} \\ \hline \hline \end{tabular}
\end{table} 
Table 8: The results of solution accuracy for the uni-modal functions. **(2) The accuracy results of multimodal functions**  
From Table 9, it can be observed that the mean value and the standard deviation value of PSO displays the best results for the function F15. DE obtains the best results on F07, F09, and F12, and BA has the best result for the function F06. BSO has the best result for the function F16. WCA obtains the best results on F8 and F10.