The combined use of resilience and loop diameter uniformity as a good indirect measure of network reliability

The aim of this paper is to show that the combined use of the resilience index (Todini, 2000) with a loop based diameter uniformity index (here formulated) yields a good indirect reliability measure, which can be conveniently used within the optimization processes of the water distribution system design. The methodology adopted to show the advantages of the combined use of the two indexes consists of (a) a three-objective optimization performed in order to simultaneously minimize costs (first objective function) and maximize both the resilience and the loop diameter uniformity indexes (second and third objective functions respectively); (b) a retrospective assessment of performance indicators relative to critical operational scenarios on the solutions of the Pareto surface obtained at the end of the optimization process. Applications were performed considering a simple case study, which made it possible to easily compare the new approach, based on a three-objective optimization, with the two-objective optimization process based on the use of the resilience index alone and also with the two-objective optimization process based on the modified resilience index formulated by Prasad, Sung-Hoon, and Namsik (2003) (where the diameter uniformity is defined at nodal level and inserted as a weight in Todini's resilience index), being both indexes a surrogate to reliability. The comparison pointed out that using resilience and loop diameter uniformity as two separate objective functions in an optimization process leads to solutions which perform better during critical operational scenarios (particularly when dealing with segment isolation) than the equally expensive solutions obtained adopting the resilience index (independently of its formulation) alone as reliability related objective function. Since the proposed approach suggests that a three-objective optimization be utilized to perform an appropriate pipe-network optimal design, an improvement in the well-known NSGA-II algorithm (Deb, Pratap, Agarwal, & Meyarivan, 2002) is proposed as its original formulation proved to have some difficulties dealing with more than two objectives.