# Estimation of Distribution Algorithms with Fuzzy Sampling for Stochastic Programming Problems
## 4 Numerical Experiments
### Simulation-Based Optimization Results
\begin{table}
\begin{tabular}{l l l l l l l l l l} \hline \hline  & \multicolumn{3}{c}{**DESSP**} & \multicolumn{3}{c}{**DSSSP**} & \multicolumn{3}{c}{**FSDEA**} \\ \hline \(f\) & **Best** & **Average** & **Std.** & **Best** & **Average** & **Std.** & **Best** & **Average** & **Std.** \\ \hline \(f_{1}\) & \(\mathbf{5.02\times 10^{-4}}\) & \(2.33\times 10^{-1}\) & \(1.06\) & \(1.46\times 10^{-2}\) & \(2.94\times 10^{-1}\) & \(2.97\times 10^{-1}\) & \(1.30\times 10^{-2}\) & \(\mathbf{7.15\times 10^{-2}}\) & \(2.00\times 10^{-1}\) \\ \(f_{2}\) & \(8.05\times 10^{-1}\) & \(\mathbf{3.55}\) & \(1.37\) & \(\mathbf{4.08\times 10^{-1}}\) & \(6.56\) & \(3.74\) & \(3.05\) & \(5.90\) & \(8.12\times 10^{-1}\) \\ \(f_{3}\) & \(\mathbf{1.05\times 10^{-3}}\) & \(\mathbf{3.31\times 10^{-1}}\) & \(4.98\times 10^{-1}\) & \(5.90\times 10^{-1}\) & \(1.04\) & \(2.92\times 10^{-1}\) & \(3.68\times 10^{-1}\) & \(2.66\) & \(5.43\times 10^{-1}\) \\ \(f_{4}\) & \(\mathbf{1.44\times 10^{-1}}\) & \(\mathbf{3.75}\) & \(5.96\) & \(2.75\) & \(6.71\) & \(3.97\) & \(2.81\) & \(6.21\) & \(2.12\) \\ \(f_{5}\) & \(\mathbf{4.13\times 10^{-4}}\) & \(\mathbf{3.87\times 10^{-1}}\) & \(\mathbf{5.99\times 10^{-1}}\) & \(2.82\times 10^{-1}\) & \(1.91\) & \(6.50\times 10^{-1}\) & \(2.45\times 10^{-1}\) & \(9.99\times 10^{-1}\) & \(5.12\times 10^{-1}\) \\ \(f_{6}\) & \(\mathbf{1.24\times 10^{-5}}\) & \(\mathbf{2.13\times 10^{-2}}\) & \(7.68\times 10^{-2}\) & \(3.85\times 10^{-4}\) & \(9.21\times 10^{-2}\) & \(9.31\times 10^{-2}\) & \(3.43\times 10^{-1}\) & \(1.07\) & \(2.77\times 10^{-1}\) \\ \(f_{7}\) & \(2.79\times 10\) & \(3.96\times 10\) & \(5.67\) & \(8.41\) & \(1.24\times 10\) & \(2.64\) & \(\mathbf{1.39}\) & \(\mathbf{1.80}\) & \(2.80\times 10^{-1}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 7: The best and average errors of the FSDEA method compared with the DESSP and DSSSP methods. \begin{table}
\begin{tabular}{l l l l l l l l} \hline \hline \multicolumn{2}{c}{**Criterion**} & \multicolumn{1}{c}{**Methods**} & \multicolumn{1}{c}{\(\mathbf{R}^{+}\)} & \multicolumn{1}{c}{\(\mathbf{R}^{-}\)} & \multicolumn{1}{c}{\(\boldsymbol{p}\)**-Value**} & \multicolumn{1}{c}{**Best Method**} \\ \hline \multirow{2}{*}{Best Solutions} & FSDEA, DESSP & 21 & 7 & 0.1282 & – \\  & FSDEA, DSSSP & 14 & 14 & 0.9015 & – \\ \hline \multirow{2}{*}{Average Errors} & FSDEA, DESSP & 20 & 8 & 0.6200 & – \\  & FSDEA, DSSSP & 11 & 17 & 0.6200 & – \\ \hline \hline \end{tabular}
\end{table} 
Table 8: [PERSON] rank-sum test for the results of Table 7.  
The other comparison experiment compared the FSEDA results with those of the following benchmark methods [12; 13] using Test Set D with dimensions 30 and 100. * EDA-MMSS: An EDA-based method with a modified sampling search mechanism called min-max sampling [13]. * DE/rand/1: A modified version of the standard differential evolution of DE/rand/1/bin algorithm [12]. * jDE: An adaptive differential evolution method [69]. * GADS: Genetic algorithm with duration sizing [70]. * DERSETS: Differential evolution with randomized scale factor and threshold-based selection [71]. * OBDE: Opposition-based differential evolution [72]. * NADE: Noise analysis differential evolution [73]. * MUDE: Memetic differential evolution for noisy optimization [74]. * MDE-DS: Modified differential evolution with a distance-based selection [12]. * NRDE: Noise resilient differential evolution [75]. The average errors over 30 independent runs are reported in the following tables, with computational budgets of 100,000 and 300,000 function evaluations for dimensions 30 and 100, respectively. Table 9 displays the average errors for test functions with \(n=30\), and the statistical measures of these results are presented in Table 10. The FSEDA outperforms seven out of nine methods in terms of obtaining better average solutions.