# Development of Practical Design Approaches for Water Distribution Systems
## 3 Model Formulation
### Study Network  
To verify the proposed practical WDSs design techniques, this study applies the three benchmark WDSs (i.e., Hanoi, Fossolo, and Cycling networks) considering single- and multi-objective frameworks. Among these applied networks, Hanoi and Fossolo networks consider a single-objective optimization for comparison with the existing technique [10], where the minimum construction cost is used as an objective function. For the Cycling network, the multi-objective optimization problems consider the minimum construction cost and maximum system resilience simultaneously. The applied WDSs design problems apply the pipe continuity constraint and hydraulic constraints such as nodal pressure. Figure 4 and Table 3 provide descriptions of the study networks.  
Hanoi networks are loop-type WDSs, which consist of one water source, 31 nodes, and 34 pipes. They were first introduced by [PERSON] and [PERSON] [22]. The Hazen-Williams (HW) roughness coefficient is 130 for all pipes. The minimum required pressure is 30 m. A total of six commercial pipe diameters ranging from 304.8 mm to 1016 mm are available. In the existing studies, the known global optimal solution for the Hanoi network cost $6,081,088, but it contained the areas that did not satisfy pipe size continuity [10]. To compare the computation times between the proposed algorithm and existing algorithm, we applied the proposed methodology in the Hanoi network mainly used in studies related to the optimal cost design of WDSs. The Fossolo network [23] consists of one water  
\begin{table}
\begin{tabular}{l} Input: _Objective functions_ (i.e., _Minimum Construction Cost_, _Maximum System Resilience_), _Algorithm parameters_ (i.e., _HMCR_, _PAR_, _HMS_, _BW_) \\ _Generate initial HM randomly_ \\ while _stopping criterion is not satisfied do_ \\ _Calculate Objective function values_ (_Cost and Resilience_) \\ _Constraints check_ (_Hydraulic and Pipe continuity_) \\ [_Hydraulic constraints_] \\ _if Hydraulic constraints condition is not satisfied_ \\ _Apply the Penalty function_ \\ end if \\ [_Continuity constraints_] \\ If _Flow(upstream) \textgreater{} Flow(downstream) and Diameter(upstream) \textless{} Diameter(downstream)_ \\ _Apply the Penalty function_ \\ else if _Flow(upstream) \textless{} Flow(downstream) and Diameter(upstream) \textgreater{} Diameter(downstream)_ \\ _Apply the Penalty function_ \\ end if \\ _Calculate the Pareto ranking of HM considering the non-dominated sorting method_ \\ _Apply the crowding-distance method_ \\ _if the ranking of new harmony is better than the worst solution in HM_ \\ _Update HM_ \\ end if \\ _Generate a new HM_ \\ if (rand \textless{} HMCR) \\ _choose an existing harmony randomly_ \\ _if (rand \textless{} PAR)_ \\ _adjust the pitch randomly within the limits_ \\ end if \\ else \\ _generate new harmony_ via _randomization1_ \\ end if \\ end while \\ \end{tabular}
\end{table}
Table 2: Pseudocode for the proposed practical water distribution systems (WDSs) design approach. HM: harmony memory, HMS: HM size, HMCR: harmony memory considering rate, PAR: pitch adjust rate, Bw: bandwidth.  
source, 36 nodes, and 58 pipes. The hydraulic constraints include the minimum nodal water pressure at 40 m and the boundary condition of pipe velocity approximately 0-1 m/s. The optimization was independently performed 50 times by applying the same constraint conditions of both the existing algorithm and the proposed algorithm in this study. The Cycling network is composed of nine closed loops, one reservoir, 165 nodes, and 242 pipes. The total length of the network is 65.38 km. A total of 10 commercial pipe diameters are used. The water pressure at each node in the pipeline denotes the hydraulic constraints. The constraints are 28 m for the minimum pressure at each node. A total of 500,000 iterations was set as the termination condition for each network.