## 4 Creation of Two-Head Self-Adaptive Metaheuristic Algorithms
In addition to comparing the two-head self-adaptive algorithms with the single-head DE algorithms, we also compared the two-head self-adaptive algorithms with the \(PSO\) algorithm. The results obtained by \(PSO\) are shown in Table 12 for comparing \(SaNSDE(1,2)\), \(SaNSDE(1,3),\)\(SaNSDE(1,4),\)\(SaNSDE(2,3),\)\(SaNSDE(2,4)\), \(SaNSDE(3,4)\) with \(PSO\). The results show that \(SaNSDE(1,2),\)\(SaNSDE(1,3),\)\(SaNSDE(1,4),\)\(SaNSDE(2,3),\)\(SaNSDE(2,4),\)\(SaNSDE(3,4)\) outperform \(PSO\) in terms of average fitness values for test Case 3 through test Case 14 and the average fitness values for test Case 1 and test Case 2 are the same for all of these algorithms. For convergence speed, \(SaNSDE(1,2),\)\(SaNSDE(1,3),\) \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Test Case**} & **Participant** & \multirow{2}{*}{_SaNSDE(3,4)_} & \multirow{2}{*}{_DE(3)_} & \multirow{2}{*}{_DE(4)_} \\  & **(D/P)** & & & \\ \hline
1 & 4/10 & 32.998/9.2 & 32.998/19.8 & 32.4747/11.9 \\ \hline
2 & 5/11 & 63.615/14 & 63.615/36.9 & 63.615/91.3 \\ \hline
3 & 5/12 & 41.715/18.1 & 41.715/47.6 & 41.715/33.9 \\ \hline
4 & 6/12 & 51.11/14.7777 & 51.11/50.3 & 51.11/34.3 \\ \hline
5 & 7/13 & 30.063/18 & 30.063/44.1 & 30.063/32.7 \\ \hline
6 & 8/14 & 73.328/19.3 & 72.328/67.8 & 72.328/40.3 \\ \hline
7 & 9/15 & 89.03/62.6 & 89.03/135 & 85.323/144.8 \\ \hline
8 & 10/16 & 54.02/25.5 & 54.02/78.6 & 53.2235/101.1 \\ \hline
9 & 11/17 & 74.05/45.3 & 74.05/65 & 73.1033/80.1 \\ \hline
10 & 12/18 & 50.9/89.1 & 50.9/146.3 & 40.9119/229.7 \\ \hline
11 & 20/20 & 112.906/59.3 & 112.906/817.1 & 112.1708/7149.9 \\ \hline
12 & 30/30 & 202.15/1395.6 & 200.1078/16,065.1 & 188.1118/7445.2 \\ \hline
13 & 40/40 & 204.2402/5388.3333 & 179.4224/27,574.4 & 194.7417/9897 \\ \hline
14 & 50/50 & 201.6048/7703.2 & 171.1756/22,745.3 & 161.954/14,146.2 \\ \hline \hline \end{tabular}
\end{table} 
Table 11: The results (average fitness values and average number of generations) obtained by applying \(SaNSDE(3,4),\)\(DE\) (3) and \(DE\) (4) to the test cases. Figure 6: Comparison of \(SaNSDE(2,4)\) with \(DE\) (2) and \(DE\) (4). \(SaNSDE(1,4)\), \(SaNSDE(2,3)\), \(SaNSDE(2,4)\), \(SaNSDE(3,4)\) outperform \(PSO\) in terms of the average number of generations for test Case 1 through test Case 12 and test Case 14. The only exception is test Case 13. For test Case 13, \(SaNSDE(1,2)\), \(SaNSDE(1,4)\), \(SaNSDE(2,4)\), \(SaNSDE(3,4)\) outperform \(PSO\) in terms of the average number of generations. The average numbers of generations of \(SaNSDE(1,3)\) and \(SaNSDE(2,3)\) are greater than that of \(PSO\) for test CASE 13. In spite of these exceptions, \(SaNSDE(1,2)\), \(SaNSDE(1,3)\), \(SaNSDE(1,4)\), \(SaNSDE(2,3)\)



deviation is calculated by the formula \(\sqrt{\frac{\sum\limits_{i=1}^{N}{(z_{i}-\overline{z})^{2}}}{N-1}}\), where \(\overline{z}\) is the mean of \(z_{1}\), \(z_{2}\), \(z_{3}\), \(\ldots\), \(z_{N}\). In this study, the outputs of experiments, \(z_{1}\), \(z_{2}\), \(z_{3}\), \(\ldots\), \(z_{N}\), in the above formula are the average fitness values of experiments for all test cases. To study the robustness property of the two-head self-adaptive algorithms, we calculated the average fitness value, standard deviation of average fitness values and normalized standard deviations of average fitness values of \(SaNSDE(1,2)\), \(SaNSDE(1,3)\), \(SaNSDE(1,4)\), \(SaNSDE(2,3)\), \(SaNSDE(2,4)\), \(SaNSDE(3,4)\). The results are shown in the three columns on the left in Table 13. It indicates that the normalized standard deviation of average fitness values of all of these two-head self-adaptive algorithms is zero for Case 1 through Case 12 and is very small for Case 13 and Case 14. To compare the robustness property of the two-head self-adaptive algorithms with \(DE\) (1), \(DE\) (2), \(DE\) (3) and \(DE\) (4), we also calculate the average fitness value, standard deviation of average fitness values and normalized standard deviation of average fitness values of \(DE\) (1), \(DE\) (2), \(DE\) (3) and \(DE\) (4).