# An Analysis of Differential Evolution Population Size
## 5 Experimental Setup
### Statistical Analysis and Extended Discussion on Population Size Implications
To further investigate the interaction effects of the population size with respect to other control parameters, the pairwise marginal plots between \(N_{P}\) and \(F\), and between \(N_{P}\) and \(CR\) are depicted in Figures 6 and 7, respectively. Figure 6 further supports the fANOVA results, demonstrating that the combined effect of \(N_{P}\) and \(F\) was less influential on overall DE performance. Conversely, the plots in Figure 7 indicate that the influence of the population size \(N_{P}\) was significantly higher for smaller values of the \(CR\) compared to larger values. Such interaction effects could  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Functions** & \(N_{P}\)**-\(CR\)** & \(F\)**-\(CR\)** & \(N_{P}\)**-\(F\)** \\ \hline \(F1\) & 0.105 & **0.124** & 0.030 \\ \(F2\) & 0.061 & **0.118** & 0.019 \\ \(F3\) & **0.067** & 0.038 & 0.014 \\ \(F4\) & 0.099 & **0.124** & 0.024 \\ \(F5\) & **0.175** & 0.052 & 0.023 \\ \(F6\) & **0.035** & 0.002 & 0.031 \\ \(F7\) & 0.067 & **0.130** & 0.017 \\ \(F8\) & **0.077** & 0.045 & 0.009 \\ \(F9\) & **0.070** & 0.044 & 0.011 \\ \(F10\) & **0.085** & 0.024 & 0.007 \\ \(F11\) & **0.139** & 0.033 & 0.014 \\ \(F12\) & **0.264** & 0.042 & 0.020 \\ \(F13\) & 0.089 & **0.111** & 0.019 \\ \(F14\) & 0.067 & **0.131** & 0.020 \\ \(F15\) & **0.125** & 0.118 & 0.047 \\ \(F16\) & **0.313** & 0.055 & 0.020 \\ \(F17\) & **0.236** & 0.043 & 0.023 \\ \(Fnot be demonstrated by the single marginal plots and should be considered when control parameters are configured. Two significant findings arise from the aforementioned analysis: Firstly, the evidence collected demonstrates that the behavior of the population size varies across different problem modalities, and there are notable interaction effects between \(N_{p}\), \(CR\), and \(F\). As a result, it can be reasonably claimed that tuning of control parameters individually without considering the inter-dependencies with other control parameters may not result in an optimal parameter configuration and hence may not lead to optimized performance. Secondly, the research findings from Experiment 2 further highlight the necessity of incorporating the characteristics of the search space when control parameters are tuned. Different problems may exhibit different complexities, requiring tailored parameter configurations to attain optimal performance. Figure 6: Plots for pairwise marginal importance of \(N_{p}\) and \(F\) for problems (**a**) \(F6\) and (**b**) \(F29\). In summary, based on the analysis conducted in the second experiment on the CEC 2014 problems, the population size as a control parameter has demonstrated critical significance, in alignment with the findings of the past studies on the DE algorithm. The results provide empirical evidence that the population size was the second most influential control parameter with respect to overall fitness. Moreover, the effect of the interaction between \(N_{P}\) and \(CR\) was found to be the most influential on DE performance. The observed interaction results revealed that \(N_{P}\) as a control parameter is of critical importance for small values of \(CR\) compared to large values. Figure 7: Plots for pairwise marginal importance of \(NP\) and \(CR\) for problems (**a**) \(F6\) and (**b**) \(F29\).