# Estimation of Distribution Algorithms with Fuzzy Sampling for Stochastic Programming Problems
## 4 Numerical Experiments
### Simulation-Based Optimization Results  
In this section, we give more details about the experimental results of the comparison between the proposed FSEDA and ASEDA algorithms. Then, the results of the best method are compared with other benchmark methods. Table 6 shows the best and average errors obtained by the two methods using the seven test functions in Set C. The FSEDA method could obtain better solutions than the other  
Figure 2: The performance of Fuzzy Sampling Random Search (FSRS) and SPRA with different sample sizes. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Criterion** & **Methods** & \(R^{+}\) & \(R^{-}\) & \(p\)**-Value** & **Best Method** \\ \hline \multirow{2}{*}{Average Errors} & DEDA, SS & 30.5 & 74.5 & 0.1982 & – \\  & DEDA, DSS & 21.5 & 83.5 & 0.2063 & – \\ \hline \multirow{2}{*}{Success Rates} & DEDA, SS & 54.5 & 50.5 & 0.6995 & – \\  & DEDA, DSS & 54.5 & 50.5 & 0.6995 & – \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Wilcoxon rank-sum test for the results of Table 4.  
method for six out of seven test functions, and its average solutions are better in five out of seven test functions. Therefore, we used the FSEDA results to compare with the other benchmark methods. Two main comparison experiments are presented to test the FSEDA performance against some benchmark methods. The first experiment used Test Set C to make the comparisons with methods in [10; 68], while the other experiment used Test Set D with methods in [12; 13]. Table 7 shows the best and the average errors obtained by the proposed FSEDA method and the following evolutionary-based methods:  
* DESSP: Directed evolution strategies for stochastic programming [10]. * DSSSP: Directed scatter search for stochastic programming [10; 68]. The results were obtained over 25 independent runs with maximum function evaluations equal to 500,000. Moreover, Table 8 shows the statistical measures of the results compared in Table 7. These statistical measures reveal that there is no significant difference between the proposed method and the other two methods in terms of the best solution found or the average errors. However, for high dimensional function \(f_{7}\), the proposed method demonstrated the best performance. \begin{table}
\begin{tabular}{l l l l l l l} \hline \hline  & \multicolumn{3}{c}{**ASEDA**} & \multicolumn{3}{c}{**FSEDA**} \\ \hline \(f\) & **Best** & **Av.** & **Std.** & **Best** & **Av.** & **Std.** \\ \hline \(f_{1}\) & \(1.05\times 10^{-2}\) & \(1.76\times 10^{-1}\) & \(3.88\times 10^{-2}\) & \(1.30\times 10^{-2}\) & \(7.15\times 10^{-2}\) & \(2.00\times 10^{-1}\) \\ \(f_{2}\) & \(3.67\) & \(5.93\) & \(2.50\) & \(3.05\) & \(5.90\) & \(8.12\times 10^{-1}\) \\ \(f_{3}\) & \(9.50\times 10^{-1}\) & \(1.86\) & \(1.50\) & \(3.68\times 10^{-1}\) & \(2.66\) & \(5.43\times 10^{-1}\) \\ \(f_{4}\) & \(3.79\) & \(6.98\) & \(3.57\) & \(2.81\) & \(6.21\) & \(2.12\) \\ \(f_{5}\) & \(4.60\times 10^{-1}\) & \(1.26\) & \(6.86\times 10^{-1}\) & \(2.45\times 10^{-1}\) & \(9.99\times 10^{-1}\) & \(5.12\times 10^{-1}\) \\ \(f_{6}\) & \(7.71\times 10^{-1}\) & \(1.08\) & \(5.02\times 10^{-1}\) & \(3.43\times 10^{-1}\) & \(1.07\) & \(2.77\times 10^{-1}\) \\ \(f_{7}\) & \(1.67\) & \(2.21\) & \(3.76\times 10^{-1}\) & \(1.39\) & \(1.80\) & \(2.80\times 10^{-1}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 6: The best and average errors of the the Average Sampling Estimation of Distribution Algorithm (ASEDA) and FSDEA methods using Test Set C.