## 4 Creation of Two-Head Self-Adaptive Metaheuristic Algorithms
The results of Experiment 2 obtained by applying the two-head self-adaptive algorithm \(SaNSDE(1,3)\), \(DE\)\((1)\) and \(DE\)\((3)\) to the test cases are summarized in Table 7. Figure 3 shows the results in a bar chart. The results show that the performance of \(SaNSDE(1,3)\) is either better than or the same as that of \(DE\)\((3)\). In particular, \(SaNSDE(1,3)\) significantly outperforms \(DE\)\((1)\) and \(DE\)\((3)\) for larger cases, Case 12, Case 13 and Case 14. The results show that the saying is valid for \(SaNSDE(1,3)\). \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Test Case**} & **Participant** & \multirow{2}{*}{_SaNSDE(1,2)_} & \multirow{2}{*}{_DE(1)_} & \multirow{2}{*}{_DE(2)_} \\ \cline{1-1}  & & & & \\ \hline
1 & 4/10 & 32.998/8.8 & 32.998/16.6 & 32.998/16.7 \\ \hline
2 & 5/11 & 63.615/16.6 & 63.615/32.2 & 50.8922/108.5 \\ \hline
3 & 5/12 & 41.715/16.7 & 41.715/39.7 & 41.715/26 \\ \hline
4 & 6/12 & 51.11/27.3333 & 51.11/43.8 & 50.9085/28.3 \\ \hline
5 & 7/13 & 30.063/16.5 & 30.063/31.8 & 26.1379/48.5 \\ \hline
6 & 8/14 & 73.328/37.7 & 72.328/48.3 & 68.6614/274 \\ \hline
7 & 9/15 & 89.03/34.3 & 89.03/101.4 & 86.9898/82.9 \\ \hline
8 & 10/16 & 54.02/31.1 & 54.02/61.3 & 53.7046/77.2 \\ \hline
9 & 11/17 & 74.05/31.8 & 74.05/59.7 & 65.5666/103.3 \\ \hline
10 & 12/18 & 50.9/94.4 & 50.0623/136.8 & 47.1623/121.1 \\ \hline
11 & 20/20 & 112.906/80.4 & 112.906/436.5 & 104.6357/14,180.2 \\ \hline
12 & 30/30 & 202.15/506 & 196.9089/5691.5 & 172.4707/9605.8 \\ \hline
13 & 40/40 & 204.767/977.8888 & 190.1664/15,185.8 & 161.4642/7370.7 \\ \hline
14 & 50/50 & 212.8168/9680.2 & 137.1625/17,131.7 & 135.1454/19,995.4 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: The results (average fitness values and average number of generations) obtained by applying \(SaNSDE(1,2)\), \(DE\)\((1)\) and \(DE\)\((2)\) to the test cases. \begin{table}
\begin{tabular}{c l l} \hline \hline \multicolumn{1}{c}{**Algorithm**} & **Parameters for Case 1 Through Case 10** & **Parameters for Case 11 Through Case 14** \\ \hline \(SaNSDE(s_{1},s_{2})\) & POP = 30, Gen = 1000, LP = 1000 & POP = 30, Gen = 50,000, LP = 1000 \\ \hline \multirow{2}{*}{\(DE(s_{1})\)} & POP = 30, Gen = 1000, \(CR\) = 0.5 & POP = 30, Gen = 50,000, \(CR\) = 0.5 \\  & \(F\): a value arbitrarily selected from uniform (0, 2) & \(F\): a value arbitrarily selected from uniform (0, 2) \\ \hline \multirow{2}{*}{\(DE(s_{2})\)} & POP = 30, Gen = 1000, \(CR\) = 0.5 & POP = 30, Gen = 50,000, \(CR\) = 0.5 \\  & \(F\): a value arbitrarily selected from uniform (0, 2) & \(F\): a value arbitrarily selected from uniform (0, 2) \\ \hline \multirow{2}{*}{\(PSO\)} & POP = 30, Gen = 1000, \(c_{1}\) = 0.4, \(c_{2}\) = 0.6, & POP = 30, Gen = 50,000, \(c_{1}\) = 0.4, \(c_{2}\) = 0.6, \\  & \(\omega\) = 0.4 & \(\omega\) = 0.4 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Parameters used in the algorithms \(SaNSDE(s_{1},s_{2})\), \(DE(s_{1})\), \(DE(s_{2})\) and \(PSO\). The results of Experiment 3 obtained by applying the two-head self-adaptive algorithm \(SaNSDE(1,4)\), \(DE\) (\(1\)) and \(DE\) (\(4\)) to the test cases are summarized in Table 8. Figure 4 shows the results in a bar chart. The results show that the performance of \(SaNSDE(1,4)\) is either better than or the same as that of \(DE\) (\(1\)). The results show that the performance of \(SaNSDE(1,4)\) is either better than or the same as that of \(DE\) (\(4\)). In particular, \(SaNSDE(1,4)\) significantly outperforms \(DE\) (\(1\)) and \(DE\) (\(3\)) for larger cases, Case 12, Case 13 and Case 14. The results show that the saying is valid for \(SaNSDE(1,4)\).