Harmony Search Algorithms for Water and Environmental Systems - Music-Inspired Harmony Search Algorithm: Theory and Applications - page 117

The total number of candidate designs is 16 21 = 1.93 × 10 25 , where 21 is the num-
ber of parallel pipes and 16 is the number of candidate diameters for each parallel
pipe.
Table 4 shows the computational results of various meta-heuristic algorithms such
as GA [21], ACO [22], SFL [11], and HS [14]. Results show that HS reached the
least-cost design solution (cost = $ 38.64 million) with the least number of function
evaluations, taking three seconds on a desktop computer with Intel Celeron 1.8GHz
CPU.

Table 4. Design Results of New York City Network

Algorithms Minimal Cost Number of Evaluations

Genetic Algorithm
Ant Colony Optimization
Shuffled Frog Leaping
Harmony Search

$ 38.64 M
$ 38.64 M
$ 38.80 M
$ 38.64 M

800,000
7,014
21,569
3,373

2.6 Other Issues

In the previous sections, how the HS algorithm was applied to the design of water dis-
tribution networks has been shown. Recently another study showed how HS can be
applied to the design of a pump-included water network [23]. In the study, HS found
the identical least-cost solution with less number of function evaluations when com-
pared with SA.
HS was also applied to the layout design of tree-type water networks [24, 25]. For
the layout problem of 64-node benchmark network reported in [24] that consists of
1.26 × 10 25 possible combinations in total, HS reached the global optimum after only
1,500 evaluations while GA [26] and evolutionary algorithm (EA) [27] reached near-
optima after 3,200 evaluations. For the layout problem of 100-node real-world network
in [25] that consists of 3.65 × 10 54 possible combinations, HS reached a near-optimum
within 0.4% of the optimal objective value, while EA [27] reached another near-
optimum within 1.0% and ACO [28] within 11.4%.
Recently, HS has been further improved by incorporating particle swarm technique
[14]. Results showed that particle-swarm HS converges much faster than the original
HS especially for small or medium sized networks such as two-loop network and
Hanoi network.

3 Scheduling of Multiple Dams

A dam is a barrier structure retaining water for the purpose of irrigation, urban water
supply, navigation, industrial uses, hydroelectric power generation, recreation uses,
wildlife habitat creation, and flood control [29].
For the optimal scheduling of multiple-unit, multiple-purpose dam systems, tradi-
tionally researchers have used dynamic programming techniques [30] to solve it. The
dynamic programming approach suffers from the so-called curse of dimensionality,