# Extended Evolutionary Algorithms with Stagnation-Based Extinction Protocol
## 4 Experimental Design and Result
### Results and Discussion
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \multicolumn{4}{c}{**Extinction Probability (\(P_{e}\))**} \\  & **0.1** & **0.3** & **0.5** & **0.7** & **0.9** \\ \hline \(K\) & \(-4779.91\pm 178.69\) & \(-4830.64\pm 192.21\) & \(-4739.18\pm 187.37\) & \(-4870.65\pm 193.91\) & \(-4903.66\pm 155.85\) \\
10 & (478.55) & (514.75) & (501.78) & (519.30) & (417.37) \\ \hline \(100\) & \(-4812.89\pm 171.71\) & \(-4745.69\pm 218.27\) & \(-4801.74\pm 193.10\) & \(-4793.12\pm 186.45\) & \(-4820.04\pm 171.53\) \\ \(100\) & (459.86) & (584.53) & (517.14) & (499.33) & (459.37) \\ \hline \(200\) & \(-4878.61\pm 191.80\) & \(-4772.05\pm 180.67\) & \(-4669.98\pm 151.52\) & \(-4790.46\pm 252.18\) & \(-4685.11\pm 195.38\) \\ \(200\) & (513.66) & (483.85) & (405.77) & (675.35) & (523.23) \\ \hline \(300\) & \(-4639.88\pm 216.00\) & \(-4821.38\pm 205.53\) & \(-4814.87\pm 197.89\) & \(-4810.89\pm 156.37\) & \(-4740.43\pm 191.03\) \\ \(300\) & (578.46) & (504.41) & (529.96) & (418.7) & (511.59) \\ \hline \(400\) & \(-4849.50\pm 253.70\) & \(-4704.33\pm 179.80\) & \(-4629.31\pm 203.17\) & \(-4788.53\pm 250.73\) & \(-4766.24\pm 185.33\) \\ \(679.43\) & (481.50) & (544.11) & (671.47) & (496.31) \\ \hline \(500\) & \(-4822.09\pm 174.10\) & \(-4637.11\pm 179.41\) & \(-4803.43\pm 203.50\) & \(-4779.93\pm 180.51\) & \(-4713.39\pm 217.27\) \\ \(466.26\) & (480.48) & (544.99) & (483.41) & (581.86) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Fitness results of SDEP with random extinction on Schwefel (\(K\) is the stagnant counter); 95% confidence interval is presented after \(\pm\) sign, and standard deviation is presented inside parenthesis. Minimum results are in bold font. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \multicolumn{4}{c}{**Extinction Probability (\(P_{e}\))**} \\  & **0.1** & **0.3** & **0.5** & **0.7** & **0.9** \\ \hline \(K\) & \(6.82\pm 1.50\) & \(11.91\pm 1.83\) & \(16.04\pm 2.09\) & \(20.60\pm 2.51\) & \(33.39\pm 3.16\) \\ \(10\) & (4.01) & (4.90) & (5.58) & (6.74) & (8.46) \\ \hline \(100\) & \(3.31\pm 0.69\) & \(4.15\pm 0.67\) & \(6.82\pm 1.65\) & \(8.32\pm 1.55\) & \(19.93\pm 2.82\) \\ \(100\) & (1.86) & (1.80) & (4.43) & (4.15) & (7.56) \\ \hline \(200\) & \(2.92\pm 0.50\) & \(3.68\pm 0.72\) & \(4.46\pm 1.38\) & \(9.17\pm 2.24\) & \(14.31\pm 2.74\) \\ \(200\) & (1.34) & (1.94) & (3.70) & (5.99) & (7.34) \\ \hline \(300\) & \(\mathbf{2.82\pm 0.44}\) & \(3.72\pm 0.55\) & \(3.97\pm 0.72\) & \(4.75\pm 0.91\) & \(5.98\pm 0.90\) \\ \(300\) & (1.18) & (1.48) & (1.92) & (2.44) & (2.40) \\ \hline \(400\) & \(3.49\pm 0.58\) & \(3.16\pm 0.57\) & \(3.48\pm 0.61\) & \(4.11\pm 0.73\) & \(4.22\pm 0.63\) \\ \(400\) & (1.55) & (1.53) & (1.65) & (1.95) & (1.69) \\ \hline \(500\) & \(2.91\pm 0.45\) & \(3.35\pm 1.02\) & \(5.35\pm 1.64\) & \(6.94\pm 2.01\) & \(15.99\pm 3.33\) \\ \(500\) & (1.21) & (2.73) & (4.39) & (5.39) & (8.92) \\ \hline \end{tabular}
\end{table} 
Table 3: Fitness results of SDEP with random extinction on Rastrigin (\(K\) is the stagnant counter); 95% confidence interval is presented after \(\pm\) sign, and standard deviation is presented inside parenthesis. Minimum results are in bold font. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \multicolumn{4}{c}{**Extinction Probability (\(P_{e}\))**} \\  & **0.1** & **0.3** & **0.5** & **0.7** & **0.9** \\ \hline \(K\) & \(0.03\pm 0.01\) & \(0.04\pm 0.02\) & \(0.03\pm 0.01\) & \(0.03\pm 0.01\) & \(0.03\pm 0.02\) \\ \(10\) & (0.02) & (0.05) & (0.02) & (0.04) & (0.07) \\ \hline \(100\) & \(0.02\pm 0.01\) & \(0.05\pm 0.02\) & \(0.02\pm 0.01\) & \(0.03\pm 0.01\) & \(0.03\pm 0.01\) \\ \(  
To investigate further the statistical significance in comparing the algorithms, we perform a Friedman test [24] on our experimental results. Figure 7 shows a box and whisker plot with Friedman test results. To alleviate multiple comparison errors, \(p\)-values are adjusted using the Bonferroni method [25]. Note that a \(p\)-value \(<\) 0.05 means that the experimental result in Table 1 is statistically significant.