# An Analysis of Differential Evolution Population Size
## 5 Experimental Setup
### Population Size in Relation to Problem Classes in DE  
To further investigate the correlations between population size and the fitness landscape of various problems (utilizing problem modality as the key characteristic of varied fitness landscapes), the mean average ranks were calculated for each population size across the unimodal, multimodal, hybrid, and composition problems in dimensions \(D=10\), \(D=30\), \(D=50\), and \(D=100\), as shown in Figure 3. A deeper understanding of the relationship between population size, problem modalities, and DE performance can be obtained by examining the mean average ranks. The trends observed in Figure 2 were further supported by the results presented in Figure 3. Similar trends were apparent for the unimodal, multimodal, and hybrid problems, as illustrated in Figure 3a, 3b, and 3c, respectively. It is evident from Figure 3a-c, that small population sizes are associated with high overall average ranks, indicating poorer performance. Conversely, larger population sizes improved the performance up to a certain threshold level, beyond which the performance either remained steady or even deteriorated. However, some exceptional observations were evident in the case of the multimodal problems with \(D=100\), as depicted in Figure 3b; specifically, increasing the population size beyond \(N_{P}=30\) did not lead to noticeable performance enhancements. Moreover, it was observed for the composition problems across all dimensions that the performance improved as the population size increased, as shown in Figure 3d. Notably, larger population sizes yielded better results for the composition problems. This investigation uncovers a unique observation not previously emphasized in the DE literature that larger population sizes benefit particular optimization problems, specifically, the composition problems. Thus, the findings resonate with the second objective of this paper and provide solid evidence about the relationship between problem characteristics and optimal population sizes. It is worth noting that the search space for the composition problems is more complex compared  
Figure 2: Relationship between mean average ranks and the population size for the DE algorithm on CEC 2014 problems for (**a**) \(D=10\), (**b**) \(D=30\), (**c**) \(D=50\), and (**d**) \(D=100\), where \(D\) is the problem dimension. to the unimodal problems, multimodal problems, and hybrid problems. More detailed information and plots regarding the CEC 2014 problems are available in [109; 112]. Figure 3: _Cont._  
As mentioned earlier, the Friedman-Shaffer statistical test was performed to verify the statistical significance of the results. For each problem per dimension and with 13 population sizes, a total of \(m=78\) comparisons were performed (\(m=K\times(K-1)/2\)), where \(K\) is the number of population sizes chosen for the experiment (i.e., 13). Therefore, a total of 9360 pair-wise comparisons were conducted throughout the experiment. It is important to note that the Friedman test calculates the average ranks of the population sizes and specifies whether these average ranks are statistically significantly different at \(\alpha=0.05\). In the case of multiple comparisons among all population sizes, the Shaffer procedure is needed to verify the significance of all possible pair-wise comparisons and to control the accumulated family pair-wise error rate (denoted as FWER and defined as the probability of making one or more false discoveries among all the hypotheses when performing multiple pair-wise tests). Accordingly, the statistical results in this research are interpreted as follows: considering all statistically significant pair-wise comparisons provided by the Shaffer post hoc procedure, the total number of wins is calculated for each population size. A win indicates that a particular population size performed significantly better (having a lower average rank compared to that of its competitor, and the test shows a statistically significant difference).