Two-Stage and Single-Stage Methods for the Optimization of Branched Water Distribution Networks

Branched pipe networks are critical for water distribution applications, such as agricultural irrigation, reclaimed water systems, and wastewater disposal. Most studies focus on optimizing pipe sizes for predefined network layouts, and less attention has been paid to optimizing the geographical configuration of the network itself. This paper presents optimization methods developed for designing minimum-cost branched water distribution networks, considering both network structure and demand patterns. Two groups of methods are proposed and were compared: two-stage and single-stage methods. Two-stage methods construct a spanning tree in the first stage, followed by Steiner point insertion in the second stage. Single-stage methods integrate tree construction and Steiner point placement into a single process. All methods utilize the Nelder–Mead optimization algorithm to determine Steiner point locations. The performance of these methods was evaluated on 50 randomly generated irrigation network design problems with 20, 40, 60, 80, and 100 nodes, as well as on a real-world irrigation project in Kemalpaşa, İzmir, Türkiye, demonstrating their practical applicability.