Analysis of Trajectories towards Pareto-Optimal Water Distribution Networks: A Dual Graph Approach

The multi-objective design of water distribution networks (WDNs) is a very challenging task and can be addressed with evolutionary algorithms. Especially for large WDNs, such a process is very computationally demanding, and it is difficult to assess if the obtained solutions could be further optimized. In other words, due to the stochastic nature of evolutionary algorithm, it is not straighforward to recognize whether the solutions are optimal or further generations need to be processed. Characteristics of the network graphs could reveal the progress of the optimization process. For optimal WDNs, the primal graph characteristics and network patterns have already been explored. However, literature is missing a dual graph approach to investigate the properties of optimal WDNs in a dual space. Such a dual graph approach is developed in this work. Therewith, the trajectories of the optimization process from random initialization to the (final) optimal generation are described and systematically investigated. With that the answer to the question is successfully addressed: When is an optimal stage at the optimization achieved, and how can that be assessed and predicted?