# An Analysis of Differential Evolution Population Size
## 5 Experimental Setup
### Statistical Analysis and Extended Discussion on Population Size Implications
Another observation is that the population size of \(N_{P}\) =1000 yielded the lowest scores across almost all problem modalities except for the composition problems with high dimensions (i.e., \(D\) = 100). Moreover, it can be further observed that the population sizes of 10 \(\times\)\(D\) did not achieve the highest scores except for the composition problems in high dimensions (i.e., \(D\) = 50, \(D\) = 100). Consequently, the assumption that a population size of 10 \(\times\)\(D\) represents an optimal choice is negated by the evidence, as the population size of 10 \(\times\)\(D\) did not demonstrate superior performance over other alternatives. To examine the implications of the observed trends in more detail, the results of statistical analysis represented by the overall score are shown in Figure 4. One important finding in Figure 4 is the noticeable decline in the overall score for population sizes equal to or larger than \(N_{P}\) = 200. The certain degradation of performance after a specific value suggests the presence of a threshold value, beyond which increasing the population size does not contribute to further performance improvement. Notably, excessively large population sizes may even hamper the algorithm's performance. Moreover, it is also observed that relatively small population sizes, such as \(N_{P}\) = 20 and \(N_{P}\) = 30, consistently outperformed larger population sizes with statistical significance. For the majority of the cases, population sizes smaller than or equal to D were more beneficial. For the unimodal problems, a relatively smaller population size of \(N_{P}\) = 20 consistently exhibited the best performance across all dimensions. Compared to the unimodal problems, the multimodal problems with larger population sizes were more beneficial. For the multimodal problems, the population sizes that yielded the best performance increased as the dimension increased, benefiting from populations of \(N_{P}\) = 10, \(N_{P}\) = 20, \(N_{P}\) = 20, and \(N_{P}\) = 30 individuals for the dimensions of \(D\) = 10, \(D\) = 30, \(D\) = 50, and \(D\) = 100, respectively. For the hybrid problems, the population size of \(N_{P}\) = 30 produced the best performance across dimensions of \(D\) = 10, \(D\) = 30, and \(D\) = 50; a lower population size of \(N_{P}\) = 20 yielded the The previous discussion, supported by the results and statistical evidence, highlights the significance of carefully considering the population size selection guidelines and avoiding excessively large population sizes. Optimal population sizes are not uniformly determined by simple formulas such as \(10\times D\), and larger population sizes can lead to diminishing outcomes. The results above also show that the choice of population size is definitely problem-dependent. To determine a population size that is best suited for a specific problem, it is necessary to consider the characteristics of the problem landscape. Consequently, it is evident that a more thoughtful approach is needed when determining the population size for various optimization problems in various dimensions to mitigate the unnecessary computational burden and achieve better performance. Figure 4: _Cont._  
7 Experiment 2: The Importance of Population Size and Interrelation with Other DE Control Parameters  
The primary objective of this experiment is to investigate the importance of the population size as a control parameter within the framework of the DE algorithm. Additionally, this experiment aims to examine the mutual interactions and interrelationships between the population size and other DE control parameters, namely, the scale factor, \(F\), and the crossover rate, \(CR\). The experiment was performed on a 20-dimensional CEC 2014 benchmark suite. A total of 13 different population sizes were employed, specifically, 10, 20, 30, 40, 50, 70, 100, 200, 300, 400, 500, 700, and 1000 individuals.