# Investigation of the iCC Framework Performance for Solving Constrained LSGO Problems +
## 4 Numerical Experiments and Results
### Effect of Mutation Strategy on the iCC Framework Performance  
It is well known that there are many different mutation strategies for DE algorithms. In this subsection, we investigate the iCC performance with various mutation operators. Many operators have been taken from [27]. The detailed description of each mutation strategy can be found in Table 8. The first column of Table 8 contains the original names of mutation schemes, the second column contains the formula for evaluating mutant vectors, and the third column contains the short notations. We have modified the traditional mutation operators (mut-1, mut-2, mut-3, mut-4, mut-5, and mut-6) by applying the tournament selection for choosing indexes. The tournament size has been set to 2. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**iCC vs.** & **+** & **–** & \(\approx\) \\ \hline CC (1) & 12 & 4 & 2 \\ CC (2) & 15 & 2 & 1 \\ CC (4) & 11 & 2 & 5 \\ CC (8) & 9 & 3 & 6 \\ CC (10) & 7 & 3 & 8 \\ Total amount & 54 & 14 & 22 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Mann–Whitney U test, pop_size is 100. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Population Size** & **CC (1)** & **CC (2)** & **CC (4)** & **CC (8)** & **CC (10)** & **iCC** \\ \hline
25 & 65 & 53 & 52 & 49 & 51 & 52 \\
50 & 72 & 55 & 48 & 46 & 54 & 53 \\
75 & 74 & 56 & 55 & 52 & 56 & 55 \\
100 & 69 & 55 & 48 & 48 & 54 & 54 \\ Total amount & 280 & 219 & 203 & 195 & 215 & 214 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: The Holm test, \(\varepsilon\)-CC-SHADE (m) vs. \(\varepsilon\)-iCC-SHADE p-value = 0.05. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Mutation Strategy** & **Formula** & **Reassignment** \\ \hline DE/rand/1 & \(x_{t_{1}}+F(x_{t_{2}}-x_{t_{3}})\) & mut-1 \\ DE/rand/2 & \(x_{t_{1}}+F(x_{t_{2}}-x_{t_{3}})+F(x_{t_{4}}-x_{t_{5}})\) & mut-2 \\ DE/best/1 & \(x_{best}+F(x_{t_{2}}-x_{t_{3}})\) & mut-3 \\ DE/best/2 & \(x_{best}+F(x_{t_{2}}-x_{t_{3}})+F(x_{t_{4}}-x_{t_{5}})\) & mut-4 \\ DE/cut-to-best/1 & \(x_{t}+F(x_{best}-x_{t_{3}})+F(x_{t_{2}}-x_{t_{3}})\) & mut-5 \\ DE/cur-to-\(p\)best/1 & \(x_{t}+F(x_{t_{p}}-x_{t_{3}})+F(x_{t_{2}}-x_{t_{3}})\) & mut-6 \\ DE/tour/1 & \(x_{t_{1}}+F(x_{t_{2}}-x_{t_{3}})\) & mut-7 \\ DE/tour/2 & \(x_{t_{1}}+F(x_{t_{2}}-x_{t_{3}})+F(x_{t_{4}}-x_{t_{5}})\) & mut-8 \\ DE/best/1(tour) & \(x_{best}+F(x_{t_{2}}-x_{t_{3}})\) & mut-9 \\ DE/best/2(tour) & \(x_{best}+F(x_{t_{2}}-x_{t_{3}})+F(x_{t_{4}}-x_{t_{5}})\) & mut-10 \\ DE/cut-to-best/1(tour) & \(x_{t}+F(x_{best}-x_{t})+F(x_{t_{2}}-x_{t_{3}})\) & mut-11 \\ DE/cur-to-\(p\)best/1(tour) & \(x_{t}+F(x_{t_{p}}-x_{t_{3}})+F(x_{t_{2}}-x_{t_{3}})\) & mut-12 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Mutation strategies. As we can see from Figure 13, mut-1, mut-6, mut-7, and mut-12 perform better on all cLSGO benchmark problems, including separable problems, and mut-6, mut-9, and mut-12 show better performance on the non-separable cLSGO benchmark problems. The mutation ranking results for each cLSGO problem are presented in Figures 14-17 (the lowest rank corresponds to the best algorithm) using parallel diagram plots. On the x-axis, we have problem numbers; on the y-axis, we have the rank for each mutation. Figure 13: The ranking of different mutation strategies on the cLSGO benchmark problems. Figure 16: The ranking of \(\varepsilon\)-iCC-SHADE with some mutation strategies for non-separable problems. Figure 14: The ranking of \(\varepsilon\)-iCC-SHADE with some mutation strategies for non-separable problems. Figure 15: The ranking of \(\varepsilon\)-iCC-SHADE with some mutation strategies for separable problems. Table 11 shows the results of the Holm post-hoc test (as Table 7). The first row of Table 11 presents the type of mutation schemes from 1 to 12. The second and the third rows present the total difference sum for 50 and 100 individuals, respectively. From the results of applying the Holm test, we can conclude that mutations under numbers 2, 4, 6, and 12 have the smallest performance difference.