## 5 Comparison of Results
### Experiments and Results for the Benchmark Control Problem
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Method** & **DE without Noise FLC** & **DE with Noise FLC** & **FDE without Noise FLC** & **GFDE without Noise FLC** & **GFDE with Noise FLC** \\ \hline Minimum & \(1.23\times 10^{-1}\) & \(1.42\times 10^{-1}\) & \(4.28\times 10^{-3}\) & \(4.28\times 10^{-3}\) & \(2.31\times 10^{-3}\) & \(3.95\times 10^{-4}\) \\ Maximum & \(4.57\times 10^{0}\) & \(5.06\times 10^{0}\) & \(1.32\times 10^{0}\) & \(1.05\times 10^{0}\) & \(1.94\times 10^{0}\) & \(1.50\times 10^{0}\) \\ Average & \(1.08\times 10^{0}\) & \(1.35\times 10^{0}\) & \(3.83\times 10^{-1}\) & \(2.17\times 10^{-1}\) & \(2.50\times 10^{-1}\) & \(8.78\times 10^{-2}\) \\ Std. & \(1.14\times 10^{0}\) & \(1.06\times 10^{0}\) & \(3.53\times 10^{-1}\) & \(2.59\times 10^{-1}\) & \(4.02\times 10^{-1}\) & \(2.68\times 10^{-1}\) \\ Execution time & \(19\),\(372.6\) s & \(23\),\(489.3\) s & \(37\),\(471.16\) s & \(78\),\(320.8\) s & \(155\),\(640.5\) s & \(179\),\(568.2\) s \\ \hline \hline \end{tabular}
\end{table} 
Table 15: Results of the OFDE method applied to robot mobile controller. To corroborate the results from both algorithms, Table 17 is presented, which includes some results using different algorithms and different ways of optimizing applied to this control problem. Table 17 contains the minimum, maximum, average, standard deviation, and the number of experiments for other metaheuristics that were applied to the same problem. In order to corroborate all the experimentation carried out with our proposal, three methods are chosen from Table 17 to perform statistical tests, and these methods are the only ones that meet the characteristics to perform a Z test. The following explains how the statistical tests are applied:  
Table 18 shows the OFDE results with noise which represents our \(\mu_{1}\) and we perform a statistical test for each of the three methods (GAs optimization FLC Type-2, GA optimization Type-1, and GA optimization FLC Type-2), which is represented with \(\mu_{2}\) in each of the statistical tests. Figure 11: Best simulation OFDE (**a**) without noise and (**b**) with noise. Figure 12: Best simulation OFHS (**a**) without noise and (**b**) with noise. The same dynamics is applied in Table 19, where three statistical tests are performed, but in a comparison with the OFHS results with noise. The claim is that the OFDE or OFHS method has lower (better) values compared to the GA optimization FLC Type-2 or GA optimization Type-1 or GA optimization FLC Type-2 method. The parameters of the statistical test are as follows: Alpha 0.05, confidence level 95%, sample size is 30 and the critical value is \(-1645\), where \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Method** & **OfHS with Noise FLC** & \begin{tabular}{c} **Resulted GAs** \\ **Optimization FLC** \\ **Type-2 [47]** \\ \end{tabular} & \
\begin{tabular}{c} **GA Optimization** \\ **Type-1 [48]** \\ \end{tabular} &
\begin{tabular}{c} **GA Optimization** \\ **FLC Type-2 [48]** \\ \end{tabular} \\ \hline Minimum & \(3.95\times 10^{-4}\) & \(3.99\times 10^{-1}\) & \(4.08\times 10^{-1}\) & \(3.99\times 10^{-1}\) \\ Maximum & \(1.50\times 10^{0}\) & \(4.11\times 10^{-1}\) & \(5.70\times 10^{-1}\) & \(4.11\times 10^{-1}\) \\ Average & \(8.78\times 10^{-2}\) & \(4.01\times 10^{-1}\) & \(4.40\times 10^{-1}\) & \(4.01\times 10^{-1}\) \\ Std. & \(2.68\times 10^{-1}\) & \(3.25\times 10^{-3}\) & \(4.89\times 10^{-2}\) & \(3.40\times 10^{-3}\) \\ Experiments & 30 & 30 & 37 & 37 \\ Evidence & & Significance & Significance & Significance \\ \hline \hline \end{tabular}
\end{table} 
Table 19: Statistical test results for OFHS.