# An Analysis of Differential Evolution Population Size
## 5 Experimental Setup
### Statistical Analysis and Extended Discussion on Population Size Implications
The scale factor \(F\) and the crossover rate \(CR\) were sampled in increments of 0.1 in the ranges [0.1, 1], producing a total of 1300 parameter configurations per problem and collectively producing 39,000 simulations for 30 problems, each of which comprised 51 independent runs. The algorithm was terminated as per the specified number of fitness evaluations, specifically, \(10,000\times D\) evaluations. The fANOVA approach was used to analyze the importance of the population size and highlight the mutual interactions between the population size and the other DE control parameters. As a powerful statistical approach, the fANOVA allowed for the assessment of the proportion of variance in fitness associated with each control parameter and, hence, to predict the individual importance of each control parameter. The results of the analysis indicating the individual importance of each control parameter are presented in Table 1.  
Table 1 presents the individual influence of each control parameter on the overall fitness variation. The results in Table 1 indicate that the crossover rate \(CR\) exhibits the highest contribution to the overall average fitness variation across almost all problems. On average, \(CR\) alone accounted for 43.6% (or 0.436) of the total variance, making it the most influential control parameter that requires careful tuning. Following the scale factor \(F\), the population size \(N_{P}\) emerged as the second most influential control parameter, contributing to 21.9% (or 0.219) of the overall average fitness variation. In contrast, the scale factor, \(F\), was found to be the least important influential control parameter, accounting for only 3% (or 0.030) of the overall average fitness variance. Figure 4: Results of the statistical analysis represented by score for all population sizes with 10-dimensional, 30-dimensional, 50-dimensional, and 100-dimensional problems. (**a**) \(D=\) 10, (**b**) \(D=\) 30, (**c**) \(D=\) 50, and (**d**) \(D=\) 100, where \(D\) is the problem dimension. Recall that \(F6\) is a multimodal problem, while \(F29\) is a composition problem. The different shapes of the plots in Figure 5a,b, for different problems further support the findings of Experiment 1, indicating that the optimal population size is problem-dependent. Experimental evidence collected indicates that different behaviors were observed for population sizes across different problem modalities. To further investigate the mutual interactions and interrelationships between the DE control parameters, pairwise marginals were obtained using the fANOVA technique. Considering all possible pairs of control parameters, the pairwise marginals provide insight  
Figure 5: Average fitness, with respect to \(N_{P}\), on (**a**) \(F6\) and (**b**) \(F29\) problems. The solid line through the middle indicates the average fitness, and the red area shows the standard deviation. into how the variations in one control parameter affect the impact of another control parameter on the algorithm's performance. The pairwise marginal results are presented in Table 2.  
It is evident that different pairwise marginals were obtained for different problems, which suggests that the interactions between DE control parameters are problem-specific and that parameter tuning approaches need to be problem-tailored. Moreover, the results reveal that, on average, the combined influence of the population size \(N_{P}\) and the crossover rate \(CR\) was found to be marginally the most important across the CEC 2014 problems. Specifically, the combined effect of \(N_{P}\) and \(CR\) on the DE performance exhibited the highest impact on DE performance for twenty problems, while the combined effect of \(F\) and \(CR\) had the highest impact on DE performance for ten problems. Notably, the combination of \(N_{P}\) and \(F\) was not found to have a high impact for any of the problems.