# Parameter Combination Framework for the Differential Evolution Algorithm
## 6 Experiments and Discussion
With jDE, PSCDE obtains better results for 24 functions and poorer results for three functions. Thus, PSCDE shows significant improvement in comparison to SHADE and jDE. Compared to SHADE and jDE simultaneously, PSCDE wins for 17 functions and fails for one function, f10. This effect results from the cooperation of the two strategies: they compensate for each other to obtain better result than each one alone. For 12 functions, PSCDE has the compromise effects between SHADE and jDE, or no significant improvement. Thus, with a combination algorithm, we can obtain a result exceeding all the sub-algorithms, or at least a result close to the best sub-algorithm. This robustness is attained by the combination algorithm. EPSDE, CoDE, and MPEDE are all multiple-mutation operations algorithms, and they concentrate on operation combination and use the unified parameter strategy. According to the rank-sum test results, PSCDE demonstrates better performance than these algorithms. The results of Wilcoxons rank-sum tests show that PSCDE is significantly better than EPSED, MPEDE, and CoDE on 29, 18, and 21 functions, respectively. It is significantly worse than EPSED, MPEDE, and CoDE on 0, 3, and 3 functions while there is no significant different between PSCDE and other comparative algorithms on 1, 9, and 6 functions, respectively. The test expresses that operation and parameter strategy combination is important, and that the combination of parameter strategies also plays an important role on the algorithm's effect. Furthermore, a multiple comparison test on the results of CEC2014 50D test is performed. The algorithms including the ones we proposed in this paper (PVCDE, L-PVCDE, PSCDE) and the comparison algorithms we used (SHADE with 200 individuals, L-SHADE with 100 individuals, jDE, EPSDE, MPEDE, CoDE). We used Friedman and Holm-Bonferrini test [PERSON] and Holm-Bonferrini (1998, 1999, 2000) to get the comparison result showed Table 7. The null hypothesis is set to: 'There was no significant differences between L-PVCDE and the j-th algorithm'. The null hypothesis is rejected in 3 cases out of 9. We use SPSS statistical software to perform further Friedman analysis. We get the uniform subsets of multiple algorithms, as shown in Table 8, the algorithms were divided into 4 subsets according to the similarity. The algorithms combined with strategy for NP parameter are superior than others, which means that it is very meaningful to control the population parameters and operational parameters simultaneously. In the algorithms without NP strategy, the PVCDE and PSCDE have similar performance, but superior than others. This also illustrates that further discussing the parameter control strategy is very important for differential evolution algorithm. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**j** & **Optimizer** & **Rank** & **zj** & **Pj** & **\&/j** & **Hypothesis** \\ \hline
1 & L-SHADE & 3.60 & 1.5606e+00 & 1.20e-01 & 5.00e-2 & Accepted \\
2 & PVCDE & 4.117 & 2.2258e+00 & 2.65e-02 & 2.50e-2 & Accepted \\
3 & PSCDE & 4.133 & 2.2386e+00 & 2.52e-02 & 1.67e-2 & Accepted \\
4 & SHADE200 & 5.817 & 4.4005e+00 & 1.12e-05 & 1.25e-2 & Rejected \\
5 & MPEDE & 5.917 & 4.5284e+00 & 6.16e-06 & 1.00e-2 & Rejected \\
6 & SHADE100 & 6.483 & 5.2447e+00 & 1.57e-07 & 8.33e-3 & Rejected \\
7 & CoDE & 6.867 & 5.7436e+00 & 9.70e-09 & 7.14e-3 & Rejected \\
8 & jDE & 7.050 & 5.9739e+00 & 2.37e-09 & 5.00e-2 & Rejected \\
9 & EPSDE & 8.633 & 7.9950e+00 & 1.33e-15 & 2.50e-2 & Rejected \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Holm–Bonferrini procedure ranking L-PVCDE against algorithms (reference: L-PVCDE, Rank = 2.383).