# An Analysis of Differential Evolution Population Size
## 9 Conclusions  
The current approaches to set the population size in the differential evolution (DE) algorithm based on problem dimensionality or as a subjective fixed value lack the reliability and precision necessary for optimizing DE performance across diverse problems complexities. This study meticulously examined the efficacy of existing guidelines for setting the population size within the DE algorithm across various problem types and modalities utilizing the CEC 2014 benchmark suite. Moreover, the relative importance and interrelationships between DE control parameters were investigated using the functional analysis of variance (fANOVA) approach. The analysis revealed that conventional guidelines may often lead to over-estimations of initial populations and hence highlights the need for a reevaluation of how population sizes are determined in the DE algorithms. Furthermore, a link between population size and fitness landscape characteristics was found, revealing that different problems favor various population sizes. Specifically complex composition problems were found to benefit from larger population sizes, a finding not previously emphasized in the DE literature. Additionally, the investigation highlighted the paramount importance of the population size as a critical control parameter in the DE algorithm, ranking its significance second only to the crossover rate \((CR)\) and preceding the scale factor \((F)\) in terms of impact. Notably, the crossover rate contributed 43.6% to the total fitness variation, making it the most influential control parameter, followed by the population size at 21.9%. In contrast, the scale factor accounted for only 3% of the fitness variation. These results highlight the critical need to carefully tune DE control parameters for optimal DE performance. This study also explores the interrelationships between DE control parameters and identifies the crossover rate as the most influential factor affecting fitness variance in the DE algorithm. Moreover, the interaction between population size and the crossover rate shows a significant influence on the DE performance, with larger population sizes proving particularly advantageous at lower crossover rates to optimize DE performance. An important and potential direction for future research involves conducting experiments on real-world problems. Currently, the focus of the study is on the CEC 2014 benchmark problems, which may not fully capture the diversity of real-world problem modalities. Moreover, a promising future work is the integration of machine learning techniques (such as regression trees and random forests) to predict optimal population size preemptively. Another direction for future work could also focus on developing adaptive DE strategies to dynamically adjust the population size based on fitness landscape characteristics of problems and convergence. This involves investigating the relationship between the population diversity and convergence speed across problems of various fitness landscape characteristics, particularly incorporating additional fitness landscape characteristics, such as ruggedness, deception, and searchability. Additionally, a potential avenue for future work is to further investigate the interaction between DE control parameters considering a broader spectrum of problems of various complexities including real-world problems. This investigation is not only needed to gain a better understanding of DE control parameters optimization but also to facilitate the development of more sophisticated, problem-specific guidelines for setting population sizes in the DE algorithms. Conceptualization, A.S. and A.P.E. ; methodology, A.S., A.P.E. and S.A.K. ; software, A.S.; validation, A.S., A.P.E. and S.A.K. ; formal analysis, A.S., A.P.E. and S.A.K. ; investigation, A.S. and A.P.E. ; resources, A.S.; data curation, A.S. and A.P.E. ; writing--original draft preparation, A.S., A.P.E. and S.A.K. ; writing--review and editing, A.S., A.P.E. and S.A.K. ; visualization, A.S. and A.P.E. ; supervision, A.P.E. and S.A.K.