## 5 Results and Discussion
### Determination of Optimal Pipe Diameter Set  
The outcomes of the application of the HS algorithm provide a set of pipe diameters that is considered to be optimal based on the constraints imposed on the objective function. These constraints, as explained in the model construction section, guide the optimization process based on the limitations of head pressure on the nodes and the flow velocity in the pipes. For this reason, although most of the conduit sizes are reduced, some of them can be increased owing to hydraulic requirements. Table 4 shows the parameters used for the HS algorithm as well as the objective function constraints. In addition, before initializing the HS, it is necessary to set the upper and lower bounds between which the HS will find the solution for the optimization problem. In this case, both the upper and lower bounds were established as 80 mm and 900 mm, respectively. These values also correspond to the maximum and minimum pipe diameters that were initially set in the original model. However, if only the lower and upper bound are specified, the HS will blindly find continuous values within this range of sizes. Thus, a set of discrete commercial pipe diameters must be established to obtain real-world applicable solutions. Table 5 shows the set of commercial pipe diameters that are common in Korea for ductile iron, which was the material considered in this study. Although some of the conduits have increased in diameter owing to hydraulic requirements, a reduction in the average size of the pipes is achieved, providing a general reduction in the head pressure in the system, which was the main objective. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**NI** & **HMS** & **HMCR** & **PAR** & **Bw** & **H (m)** & **V (m/s)** \\ \hline
30,000 & 30 & 0.95 & 0.3 & 0.01 & 15–40 & 0.05–3 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Parameter settings for the HS algorithm and constraints on the objective function. \begin{table}
\begin{tabular}{c} \hline \hline
**Commercial Pipe Diameters (mm)** \\ \hline
80 100 125 150 200 250 300 350 400 450 500 600 700 800 900 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Set of commercial diameters for ductile iron pipes. Thus, an average reduction of the maximum pipe diameter from 900 mm to 800 mm is obtained along with a reduction of the average pipe diameter of approximately 25% (from 244.15 mm in the original model to 186.9 mm in the simulated one) using the HS algorithm. In the case of the hydraulic head, owing to the restrictions imposed as the constraints of the objective function, both the lower and upper boundaries were reduced, as expected. The minimum hydraulic head is reduced from 32.83 m in the original model to 17.46 m in the HS model. The maximum value of the hydraulic head decreases from 65.51 m in the original model to 39.06 m in the simulation. This corresponds to a reduction of approximately 50% in the lower bound and 40% in the upper bound for the hydraulic head range in the system. In this manner, the outcomes achieved by means of the HS algorithm along with the setting of the objective function have led to an optimization of the Cheongna WDN in terms of a reduction in the pressure in the system along with a reduction in the pipe diameter of the conduit network. Therefore, both results imply a reduction in the probability of damage due to high pressure as well as a reduction in the construction costs of the system. On the other hand, as the pipe diameter is generally reduced in the network, the water flow velocity consequently increases. In this case, the maximum flow velocity in the original model is 2.35 m/s and is 2.42 m/s in the HS model, representing an increase of approximately 3%. Furthermore, the average flow velocity also increases from 0.35 m/s in the original model to 0.54 m/s in the simulated model, which represents an increase of approximately 40%. Despite the increase in the flow velocity in the system, this parameter remains within the range of admissible values, as it does not exceed the 3 m/s upper bound.