Harmony Search Algorithms for Water and Environmental Systems - Music-Inspired Harmony Search Algorithm: Theory and Applications - page 113

2.1 Problem Formulation for Water Network Sizing Design
The objective function of the water network design is generally the total costs of pipes
and pumps that are functions of diameter sizes and pumping capacities. When certain
design constraints are violated, such as minimal nodal pressure [6], penalty is im-
posed to the objective function to improve feasibility.
There are three major design constraints considered: mass conservation, energy
conservation, and minimal nodal pressure. For every node, total inflows equal to total
outflows in order to conserve the mass. For every loop, the summation of energy loss
due to friction between fluid and metal is assumed to be zero (note an energy loss
may take a positive or negative sign depending on its flow direction). For every de-
mand node, the residual pressure should be greater than or equal to the minimal nodal
pressure. The energy loss is calculated using EPANET [7].
2.2 Two-Loop Benchmark Network Example
Alperovits and Shamir [8] first proposed the benchmark example of two-loop network
which has one reservoir, six demand nodes, and eight pipes, as shown in Figure 1.
The number of total candidate designs is 14 8 = 1.48 × 10 9 because the number of pipes
(= decision variables) is 8 and the number of candidate diameters for each pipe is 14.
Table 1 shows the computational results of various meta-heuristic algorithms such
as genetic algorithm (GA) [9], simulated annealing (SA) [10], shuffled frog leaping
(SFL) [11], cross entropy (CE) [12], scatter search (SS) [13], and HS [14].
Fig. 1. Schematic of two-loop benchmark network
Results show that HS reached the optimal design solution (cost = $419,000) with
the least number of function evaluations among competing algorithms, taking less
than one second to solve using a desktop computer with an Intel Celeron 1.8GHz
CPU. It should be also noted that GA and SA used less-strict hydraulic coefficients,
which makes it much easier for both algorithms to find the optimal solution.