# An Analysis of Differential Evolution Population Size
## 5 Experimental Setup
### Statistical Analysis and Extended Discussion on Population Size Implications  
To simplify and consolidate the results of the statistical analysis into a single performance measure, a metric referred to as the'score' is applied. The score for each population size is computed as the difference between the total number of wins and losses. A higher score indicates better statistically significant performance associated with a specific population size. The statistical results, represented by the scores, are summarized in Tables A1-A4. The overall scores for each population size and problem modality are reported at the bottom of each table, with the best scores achieved highlighted in boldface. It is worth mentioning that the statistically insignificant results were disregarded in the analysis. Referring to the overall total scores at the bottom of Table A1, it is observed that the best overall score was obtained with population sizes of \(N_{P}=20\) and \(N_{P}=30\). Conversely, the population size of \(N_{P}=10\) exhibited comparatively lower scores across all instances except for the multimodal problems. Moderate scores were achieved with population sizes of \(N_{P}=40\), 50, 70, and 100. A noteworthy observation inferred from Table A1 is that the lowest overall score was obtained with the largest population size. More precisely, the population size of \(N_{P}=1000\) resulted in the worst overall score across all cases. Figure 3: Relationship between overall average ranks and population size for 10-, 30-, 50-, and 100-dimensional unimodal, multimodal, hybrid, and composition problems from CEC 2014. Furthermore, a prominent trend observed in Tables A2-A4 reveals that the highest overall scores were attained with population sizes of \(N_{P}\) = 20 and \(N_{P}\) = 30. Moderate population sizes of \(N_{P}\) = 40, 50, 70, and 100 achieved moderate scores. The worst scores were obtained with \(N_{P}\) = 1000. However, an exception was noticed for the composition problems in Tables A2-A4. In Table A2, the population size of \(N_{P}\) = 1000 performed the worst across all cases except for the composition functions, which reported the lowest overall score with \(N_{P}\) = 10. Moreover, in Table A3, it is notable that population sizes of \(N_{P}\) = 400 and \(N_{P}\) = 500 produced the highest scores for the composition problems. Finally, focusing on Table A4, the composition problems demonstrated the best performance with population sizes of \(N_{P}\) = 400 and \(N_{P}\) = 1000. Concerning the characteristics of the problem modality, the following observations were made from Tables A1-A4: for the unimodal problems, the highest overall score was achieved with population size \(N_{P}\) = 20 for all dimensions, except for \(D\) = 10 where both \(N_{P}\) = 20 and \(N_{P}\) = 30 exhibited equal performance. For the multimodal problems, the highest overall scores were reported with population sizes of \(N_{P}\) = 10, 20, 20, and 30 for dimensions 10, 30, 50, and 100, respectively. Regarding the hybrid problems, the population size of \(N_{P}\) = 30 achieved the highest scores for \(D\) = 10, \(D\) = 30, and \(D\) = 50, while \(N_{P}\) = 20 yielded the highest overall score for \(D\) = 100. In the case of the composition problems, the population sizes of \(N_{P}\) = 20 and \(N_{P}\) = 30 achieved the highest scores for \(D\) = 10 and \(D\) = 30, respectively. However, for higher dimensions of \(D\) = 50 and \(D\) = 100, the population sizes which achieved the highest overall score were \(N_{P}\) = 400 and \(N_{P}\) = 500 in dimension \(D\) = 50, whereas for \(D\) = 100, the population sizes of \(N_{P}\) = 400 and \(N_{P}\) = 1000 produced the highest scores. Furthermore, it is evident from Tables A1-A4 that, collectively, increasing the population size can be beneficial and generally leads to improved performance, but only up to a specific threshold, beyond which the performance gradually deteriorates.