Probabilistic warm solutions-based multi-objective optimization algorithm, application in optimal design of water distribution networks

Single objective optimization of water distribution networks (WDNs) based on construction costs reduces system resiliency. Thus, researchers have developed various optimization frameworks to maximize system resilience while minimizing construction costs during the last decade. They generally utilized population-based meta -heuristic algorithms. These algorithms start with random populations (initial cold solutions), so they need high computational time to converge to the optimal solutions. This paper aims to reduce search space by introducing prebaked initial or warm solutions based on the hydraulic properties of WDNs which are discharge variance and dissipated power function. After generating initial warm solutions, several probabilistic models are used to bring them closer to the global optimal Pareto front and increase their diversity. The proposed approach is used to optimize the design of two well-known benchmarks, Alprovits and Hanoi, and a real-world large-scale WDN. Results show that using warm solutions as the initial population reduces time, the number of iterations, and fitness evaluations to reach a certain accuracy level, more than two times on average for the benchmarks. Also, results show that the proposed method reduces the average construction costs by 45% compared to the tradi-tional optimization method in the same execution time for real-world WDN.