## 4 Results
### Case 1 Results: Benchmark Functions  
Table 6 shows the parameters used for the benchmark functions. Column 1 of Table 7 shows the number that represents the particular function; column 2 shows the minimum of the function; columns 3 and 4 show the averages and standard deviations respectively obtained with the WDO; finally, columns 5 and 6 shows the results of DSO. Each of these functions was evaluated 30 times with the same parameters to obtain the averages and standard deviations. The parameters used in the WDO, DSO, and FA methods are presented in Table 8. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Population** & **Iterations** & **Dimensions** \\ \hline
30 & 500 & 30 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Parameters used in the experiments. \begin{table}
\begin{tabular}{c c c} \hline \hline \multicolumn{2}{c}{**WDO**} & \multicolumn{2}{c}{**DSO**} \\ \hline
**Function** & \(f\)**min** & **Average** & \
\begin{tabular}{c} **Standard** \\ **Deviation** \\ \end{tabular} & **Average** &
\begin{tabular}{c} **Standard** \\ **Deviation** \\ \end{tabular} \\ \hline \(f1\) & 0 & \(3.66\times 10^{-27}\) & \(3.87\times 10^{-27}\) & \(4.95\times 10^{-09}\) & \(1.39\times 10^{-08}\) \\ \(f2\) & 0 & \(3.26\times 10^{-14}\) & \(2.70\times 10^{-14}\) & \(8.94\times 10^{-09}\) & \(2.68\times 10^{-08}\) \\ \(f3\) & 0 & \(2.42\times 10^{-20}\) & \(2.02\times 10^{-20}\) & \(1.16\times 10^{-03}\) & \(3.77\times 10^{-03}\) \\ \(f4\) & 0 & \(5.90\times 10^{-13}\) & \(6.18\times 10^{-13}\) & \(3.00\times 10^{-03}\) & \(6.76\times 10^{-03}\) \\ \(f5\) & 0 & \(28.5696\) & \(7.65\times 10^{-02}\) & \(9.82\times 10^{+00}\) & \(1.15\times 10^{-01}\) \\ \(f6\) & 0 & \(1.74\times 10^{-02}\) & \(4.52\times 10^{-03}\) & \(2.71\times 10^{-03}\) & \(2.14\times 10^{-05}\) \\ \(f7\) & 0 & \(1.12\times 10^{-02}\) & \(3.09\times 10^{-02}\) & \(2.71\times 10^{-03}\) & \(1.64\times 10^{-03}\) \\ \(f9\) & 0 & \(-118.27\) & 0 & \(5.58\times 10^{+00}\) & \(9.24\times 10^{+00}\) \\ \(f10\) & 0 & \(9.53\times 10^{-15}\) & \(1.23\times 10^{-14}\) & \(3.47\times 10^{-07}\) & \(7.22\times 10^{-07}\) \\ \(f11\) & 0 & 0 & \(0\) & \(3.66\times 10^{-11}\) & \(1.41\times 10^{-10}\) \\ \(f15\) & 0.00030 & \(3.07\times 10^{-04}\) & \(2.18\times 10^{-07}\) & \(3.07\times 10^{-04}\) & \(1.39\times 10^{-15}\) \\ \(f16\) & \(-1.0316\) & 1.0316 & \(6.78\times 10^{-16}\) & \(-1.03\times 10^{+00}\) & \(4.59\times 10^{-16}\) \\ \(f17\) & 0.398 & 0.3979 & \(6.78\times 10^{-05}\) & 0 & \\ \(f18\) & 3 & 7.7827 & \(3.61\times 10^{-15}\) & 3 & \(3.26\times 10^{-15}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Results for 30 dimensions. \begin{table}
\begin{tabular}{c} \hline \hline If then Rules \\ \hline
1. If (ev is N) and (ew is N) then (T1 is N)(T2 is N) (1) \\
2. If (ev is N) and (ew is Z) then (T1 is N)(T2 is Z) (1) \\
3. If (ev is N) and (ew is P) then (T1 is N)(T2 is P) (1) \\
4. If (ev is Z) and (ew is N) then (T1 is Z)(T2 is N) (1) \\
5. If (ev is Z) and (ew is Z) then (T1 is Z)(T2 is Z) (1) \\
6. If (ev is Z) and (ew is P) then (T1 is Z)(T2 is P) (1) \\
7. If (ev is P) and (ew is N) then (T1 is P)(T2 is N) (1) \\
8. If (ev is P) and (ew is Z) then (T1 is P)(T2 is Z) (1) \\
9. If (ev is P) and (ew is P) then (T1 is P)(T2 is P) (1) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: If-then fuzzy rules.N: negative, P: positive, Z: zero. The WDO average and standard deviation results with 64 dimensions can be compared with the results obtained with 30 dimensions. Table 9 above shows the optimization results of benchmark functions where the minimum of functions f15, f16, f17, and f18 is different from zero. With the evaluation of functions with different minima, the performance of the optimization in the functions with the WDO and DSO can be better observed. Table 10 shows the parameters used in the above-mentioned methods. These parameters are the same for each of these algorithms as a fair form of competition is aimed.