## 5 Conclusions  
In this work, two ACO variations are applied to optimization problems in a dynamic environment. The TSP is used as the base problem to systematically generate dynamic test cases. Two types of dynamic changes are considered: (1) when nodes change and (2) when  
Figure 7: Box plots of \(\mathcal{MM}\)AS and P-ACO drawn between the first (\(Q_{0.25}\)) and third quartile (\(Q_{0.75}\)) of the distribution on pr1002 with node (**left**) and weight (**right**) changes.  
weights change. From the experiments, the follow concluding remarks can be drawn. First, P-ACO performs significantly better than \(\mathcal{MMAS}\) in most DTSPs with node changes. Second, \(\mathcal{MMAS}\) is significantly better than P-ACO in DPSP with weight changes. Third, P-ACO has better average performance than \(\mathcal{MMAS}\) in most DPSP with node changes. Fourth, \(\mathcal{MMAS}\) has a better peak performance in most DTSPs with weight changes.  
In general, complex problems arising in practical applications, such as training a neural network, require huge computational power. As a consequence, the computation power for optimizing complex problems as well as performing several independent executions (due to the stochastic nature of swarm and evolutionary computation algorithms) in parallel may not be available. Therefore, optimization algorithms with better average performances may be the best choice rather than optimization algorithms with better peak performance.  
For future work, it will be interesting to apply and evaluate ACO algorithms in more practical applications, such as the agile earth observation satellites' scheduling problem [36; 37]. In fact, this particular problem can be solved as a constrained TSP [38] and consists of several unexpected environmental changes that may affect the scheduled tasks [39; 40]. Hence, the adaptation capabilities and evaluation of the ACO demonstrated in this work may be suitable to address the aforementioned problem.  
Conceptualization, M.M.; methodology, M.M.; software, M.M.; validation, M.M. and M.N.A.; formal analysis, M.M. and M.N.A.; investigation, M.M.; resources, M.M.; data curation, M.M.; writing--original draft preparation, M.M.; writing--review and editing, M.M., M.N.A. and D.H.; visualization, M.M.; supervision, M.M.; funding acquisition, D.H. All authors have read and agreed to the published version of the manuscript.  
This research was partially funded by the 'EXCELSIOR' project (European Union's Horizon 2020 research and innovation programme), grant number \"857510\"; and the 'AI-OBSERVER' project (European Union's Horizon Europe framework programme), grant number \"101079468\".  
All data used in this research are provided in this article.  
The authors acknowledge the \"EXCELSIOR\": ERATOSTHENES: Excellence Research Centre for Earth Surveillance and Space-Based Monitoring of the Environment H2020 Widespread Teaming project (www.excelsior2020.eu, accessed on 26 November 2023). The \"EXCELSIOR\" project has received funding from the European Union's Horizon 2020 Research and Innovation Programme under Grant Agreement No 857510, from the Government of the Republic of Cyprus through the Directorate General for the European Programmes, Coordination and Development and the Cyprus University of Technology. The authors acknowledge the \"AI-OBSERVER\" project funded by the European Union's Horizon Europe Framework Programme HORIZON-WIDERA-2021-ACCESS-03 (Iwinning) under the Grant Agreement No. 101079468.  
The authors declare no conflict of interest.