5. Conclusions and outlookThe PSO a

5. Conclusions and outlook
The PSO algorithm has been subject to many different modifications in order to adapt to problems with discrete
variables. The approach which was presented in this work has shown satisfactory performance when applied to water
supply systems, especially for the optimal design of the networks provided by the Hanoi and New York City water
distribution problems. In fact, the solutions we obtained with our method are among the best results compared to
the published results of the same benchmark examples, but required a significantly lower number of iterations for its
approximation.
Although the algorithm parameters have been used without any changes from previously published sources by
other authors, and for different problems, all our case studies of the PSO method have given feasible solutions for
the water supply problems under consideration. Presumably, a more elaborate analysis of the parameters for this case
with some fine-tuning will deliver more improvements. Such an analysis of the PSO algorithm for the problem at hand
would help to obtain better solutions in a more general framework and speed up the convergence of the procedure.
The application of PSO to problems related to the design of hydraulic networks seems to have a promising future,
not only for water distribution as shown in this work, but also for waste water, according to previous findings
by Montalvo et al. [27]. The PSO algorithm excels by its flexibility and adaptability in accommodating either
discrete or continuous types of optimization variables. Its simplicity allows for a straightforward implementation,
with relatively high execution speeds compared to other evolutionary algorithms, alongside a high convergence rate