# A Modified Cloud Particles Differential Evolution Algorithm for Real-Parameter Optimization
## 4 Experiments and Discussion
### Comparison with Nine State-of-the-Art Intelligent Algorithms on 10 and 30 Dimension
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \(\mathbf{F}\) & **PSO** & **PSO** & **PSOcf** & **TLBO** & **DE** & **JADE** & **CoDE** & **jDE** & **CMA-ES** & **CPDE** & **MCPDE** \\ \hline \(F_{\text{max}}\) & 333 & 312 & 219 & 20.7 & 226 & 200 & 260 & 574 & 211 & 200 \\ \(f_{\text{58}}\) & 5D & 61 & 842 & 50 & 27.5 & 54.2 & \(5.20\times 10^{3}\) & 86.9 & 504 & 35.1 & \(22.6\times 10^{4}\) \\ \(f_{\text{38}}\) & 373 & 391 & 352 & 316 & 345 & 200 & 389 & \(1.87\times 10^{3}\) & 317 & 200 \\ Min & 200 & 200 & 200 & 200 & 200 & 200 & 200 & 132 & 200 & 200 \\ \(Comp\)Rank\(-\) & \(-/8\) & \(-/5\) & \(-/3\) & \(-/6\) & \(-/2\) & \(-/7\) & \(-/10\) & \(-/4\) & \(\surd\) \\ \hline \(F_{\text{sum}}\) & 966 & \(1.04\times 10^{3}\) & 820 & 363 & 691 & 962 & \(1.11\times 10^{4}\) & 595 & 416 & 300 \\ \(f_{\text{77}}\) & 30 & 9.5 & 75.9 & \(85.5\times 10\) & 85.4 & 228 & 153 & 32.7 & 123 & 109 & \(1.19\times 10^{3}\) \\ Min & \(1.10\times 10^{3}\) & \(1.20\times 10^{3}\) & 961 & 513 & \(1.00\times 10^{3}\) & \(1.17\times 10^{3}\) & 1.17\times 10^{3}\) & 799 & 617 & 300 \\ Min & \(7^{\circ}\) & 861 & 660 & 300 & 309 & 659 & \(1.04\times 10^{3}\) & 387 & 300 & 300 \\ \(Comp\)Rank\(-\) & \(-/7\) & \(-/9\) & \(-/6\) & \(-/2\) & \(-/5\) & \(-/8\) & \(-/10\) & \(-/4\) & \(-/3\) & \(\surd\)1 \\ \hline \(F_{\text{sum}}\) & 385 & \(2.13\times 10^{3}\) & 514 & 300 & 300 & 300 & 300 & 300 & 300 & 300 & 300 \\ \(f_{\text{58}}\) & 325 & 298 & 639 & \(2.27\times 10^{3}\) & 0 & 6.78\times 10^{3}\) & 0 & 3.75 \(\times 10^{3}\) & \(1.84\times 10^{3}\) & 2.65 \(\times 10^{3}\) \\ Max & \(1.63\times 10^{3}\) & \(2.84\times 10^{3}\) & \(2.69\times 10^{3}\) & 300 & 300 & 300 & 300 & \(1.34\times 10^{4}\) & 300 & 300 \\ Min & 300 & \(1.67\times 10^{3}\) & 100 & 300 & 300 & 300 & 300 & 300 & 100 & 300 & 300 \\ \(Comp\)Rank\(-\) & \(-/7\) & \(-/10\) & \(-/8\) & \(\surd\) & \(\surd\) & \(-/1\) & \(-/6\) & \(\surd\) & \(-/9\) & \(\surd\) & \(\surd\)1 \\ \hline \(-/\surd\)/\(\ast\) & 7/1/0 & 7/0/1 & 8/0/0 & 7/1/0 & 5/1/2 & 8/0/0 & 5/2/1 & 8/0/0 & 7/1/0 & \(\surd\) \\ Avg-Rank & 6.00 & 8.13 & 6.25 & 4.25 & 3.63 & 6.88 & 5.25 & 7.63 & 3.75 & 1.50 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: _Cont._ \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline \(\mathbf{D}\) & & **PSO** & **PSOcf** & **TLBO** & **DE** & **JADE** & **CoDE** & **jDE** & **CMA-ES** & **CPDE** & **MCPDE** \\ \hline \multirow{3}{*}{10} & \(-/\!\!\approx\!/+\) & 26/2/0 & 25/1/2 & 26/2/0 & 23/5/0 & 9/9/10 & 24/2/2 & 19/7/2 & 22/3/3 & 18/9/1 & \(\surd\) \\  & Avg-rank & 6.39 & 7.43 & 6.21 & 5.07 & 2.46 & 5.50 & 4.50 & 7.46 & 4.00 & 1.71 \\ \hline \multirow{3}{*}{30} & \(-/\!\!\approx\!/+\) & 23/5/0 & 22/3/3 & 26/2/0 & 24/4/0 & 12/5/11 & 27/1/0 & 14/5/9 & 24/0/4 & 24/3/1 & \(\surd\) \\  & Avg-rank & 6.14 & 7.54 & 6.61 & 5.43 & 2.79 & 6.39 & 4.07 & 6.61 & 4.64 & 2.00 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Comparison of MCPDE with PSO, PSOcf, TLBO, DE, JADE, CoDE, jDE, CMA-ES and CPDE on the CEC2013 benchmarks (\(D=10\) and 30 dimensions). Figure 2: Evolution of the mean function error values derived from PSO, PSOG, TIBO, DE, JADE, CODE, JDE, CMA-ES, CPDE and MCPDE versus the number of _FES_ on sixteen test problems with D = 10. (a) \(f_{2}\); (b) \(f_{3}\) (\(6\)) \(f_{5}\) (\(6\)) \(f_{7}\); (c) \(f_{9}\) (\(6\)) \(f_{10}\); (d) \(f_{12}\) (\(6\)) \(f_{14}\); (f) \(f_{15}\) (\(6\)) \(f_{16}\) (\(6\)) \(f_{23}\) (\(6\)) \(f_{24}\) (\(6\)) \(f_{25}\) (\(6\)) \(f_{26}\). Figure 3: Evolution of the mean function error values derived from PSO, PSO, TIBO, DE, JADE, CoDE, JDE, CMA-ES, CPDE and MCPDE versus the number of _FES_ on sixteen test problems with D = 30. (a) \(f_{23}\); (b) \(f_{35}\) (\(6\)) \(f_{36}\); (d) \(f_{72}\); (e) \(f_{37}\); (f) \(f_{38}\); (f) \(f_{39}\) (f) \(f_{37}\); (f) \(f_{39}\) (f) \(f_{39}\) (f) \(f_{39}\) (f) \(f_{39}\) (f) \(f_{39}\).