Genetic algorithms can be a powerful tool for the automated design
of optimal drinking water distribution networks. Fast convergence of such
algorithms is a crucial factor for successful practical implementation at
the drinking water utility level. In this technical note, we therefore
investigate the performance of a suite of genetic variators that was
tailored to the optimization of a least-cost network design. Different
combinations of the variators are tested in terms of convergence rate and
the robustness of the results during optimization of the real-world drinking
water distribution network of Sittard, the Netherlands. The variator
configurations that reproducibly reach the furthest convergence after
10

Specific variators. Types include mutators (m) and crossover (c), classic (C) and heuristic (H).

Optimization techniques have been applied to the design (or more specifically, the dimensioning) of water networks for decades (see Bieupoude et al., 2012, and De Corte and Sörensen, 2013, for overviews). A widely applied approach is that of genetic algorithms (GAs) (Holland, 1975; Goldberg, 1989) and other members of the overarching family of evolutionary algorithms (EAs) (Maier et al., 2014). Though the classic genetic algorithm is very powerful, the various mechanisms of the genetic algorithm are commonly expanded, replaced or combined with heuristic tricks or complete heuristic algorithms to improve performance (as reviewed by e.g. El-Mihoub et al., 2006). The algorithms which include these are commonly referred to as hybrid genetic algorithms (HGAs) or memetic algorithms (MAs). An overview of these approaches is summarized below. Following one of the approaches, a selection of custom heuristic variators has been implemented in Gondwana, a generic optimization tool for drinking water networks (van Thienen and Vertommen, 2015). In this paper, these variators are described and it is demonstrated how they contribute to significantly faster convergence in an optimization problem case study.

A cornerstone of the HGA approach (Krasnogor and Smith, 2005; El-Mihoub et
al., 2006) is the observation that classic GAs are especially well suited for
quickly locating global optima in the solution space but subsequently have
difficulty converging to the optimum locally within a reasonable number of
iterations. To mitigate this, GAs are augmented with local search (LS)
methods. These are algorithms that iteratively modify a given solution
towards a predefined optimization criterion. LS methods find local optima
relatively quickly but are generally unable to escape this local optimum in
favour of a possibly different global optimum. The resulting HGA therefore
profits from the strengths of both techniques and yields better solutions.
El-Mihoub et al. (2006) identify the following general ways in which GA
capabilities can be expanded through hybridization:

A GA solution can be improved by running it through a LS method. This can be done to improve the final solution of the GA. Alternatively, the LS algorithm can be applied to refine intermediate solutions to promote the representation of different promising areas of the solution space within the population.

The number of iterations that are needed to achieve convergence can be reduced by replacing classic genetic operators with different ones to guide the search through the solution space.

Alternatively, system-specific knowledge can be used to modify the genetic operators in such a way that they only result in viable solutions. This does not guide the search but prevents time loss due to the evaluation of many illegal solutions, which may arise from random variations in heavily restricted GA problems.

The population size needed to achieve convergence can be reduced by dynamically controlling candidate selection with a LS method.

System-specific knowledge can be used to construct a model to quickly approximate the results of fitness functions that are expensive to calculate, speeding up the evaluation of the GA objectives.

Within the field of water network design optimization, algorithms that guide the GA to reduce the size of the search space is a specific challenge in current research (Maier et al., 2014). Table 1 lists a collection of genetic operators that was composed to tune a GA to the optimization of a least-cost design (Alperovitz and Shamir, 1977; Savic and Walters, 1997). This type of problem varies pipe diameters throughout the network in search of the minimum network costs while achieving a minimum pressure at each node. In addition to several classic GA variators (Holland, 1995; Goldberg, 1989), two heuristic variators are used that were constructed with the goal of a least-cost design in mind. In terms of the classification of hybrid metaheuristics by Talbi (2002), the resulting HGA is a low-level teamwork hybrid.

The heuristic flatiron mutator was custom-made to enhance convergence
according to approach 2 in the list above. It guides the search past a type
of artefact that commonly occurs in intermediate solutions for the least-cost design problem. This artefact occurs when classic mutation causes a
larger diameter pipe to be surrounded by smaller diameter pipes, which is
hydraulically insensible. These artefacts can take a long time to disappear
through random mutation only. The flatiron mutator speeds up convergence by
“smoothing out” these artefacts as follows:

For the mutating pipe, obtain the neighbour IDs from a lookup table with neighbouring pipes per pipe (it is worth noting that this lookup table is created at the start of the optimization, thereby limiting its impact on computation).

If the pipe connects to exactly 1 or 2 neighbouring pipes, compare the diameter of the mutating pipe to those of its neighbours.

If the mutating diameter is larger than the diameter of all neighbours, reduce it to the largest diameter among neighbours.

In order to evaluate the influence of the developed problem-specific variators, a series of tests was performed on a case study.

The case study consists in the design of part of the existing drinking water
distribution network of the Dutch village Sittard. The network has a total
length of 10.8 km and has 1000 connections, including connections to a
school, a residential building with 32 apartments and a care farm for mental
patients. The network is fed by a single reservoir and has a mean total
demand of 15 m

EPANET model of the drinking water distribution network of Sittard (Netherlands), consisting of 583 junctions, 491 pipes, 140 valves and 1 reservoir.

For the design of the network, the minimization of the product between pipe diameter and pipe length (surrogate for costs) was considered as the objective, constrained by a minimum pressure at each node equal to 34 m. The decision variables were the pipe diameters that could be chosen from the following: 0, 13.2, 21.2, 36, 42.6, 58.2, 66, 72.8, 87.3, 101.6, 130.8, 147.6, 163.6, 190, or 200 mm. A population of 100 individuals with an elitism rate of 15 % was used for each optimization. Selection between candidates was achieved through tournament selection with a tournament size of 2 and with the objective function – the product of pipe length and diameter – as the performance criterion.

A total of 16 tests were performed, wherein different rates for the specific
variators were considered in order to assess their influence on the network
design results. For each test, a total of

Problem-specific variator values considered in the different tests
and obtained results for 10 runs with

From the obtained results (Table 2) it is clear that the consideration of the heuristic flatiron mutation (FM) and proximity mutation (LPM) significantly improved the obtained results for the optimization problem. These results are graphically reported in Fig. 2.

Overview of the obtained results for the different tests.

Considering only the naïve random mutation and one-point crossover, the
best results after

Adding a flatiron mutation further improved the obtained results. The best
results after

The effect of the problem-specific variators can also clearly be seen on the
shape of the convergence curves. Figure 4 illustrates the mean, mean

Convergence curves (mean, first and second standard deviations of
10 runs) obtained for tests RMNPC2 and FM4.

The results presented in this paper clearly illustrate the value of applying
heuristic, non-classical variators in drinking water distribution system
design optimizations using genetic algorithms. While the difference between
the test with random mutation and the other tests is especially noticeable
in Fig. 2, it is worth noting that the smaller differences between the other
individual tests indicate a significant difference in convergence as well.
In Fig. 4a, for instance, it can be seen that, in FM4, the average objective
function value of

In the tests, the combination of a low rate for the proximity mutation with
a high rate of the flatiron mutation leads to the best results after

In future research and consulting projects with Gondwana, this combination of variators will be used in order to deal with the computational challenges of larger real-world networks.

The network model used here is the property of the Dutch utility WML and is therefore not made public.

KvL wrote the manuscript. IV carried out the comparison of optimization settings. All authors took part in discussing and interpreting the findings.

The authors declare that they have no conflict of interest.

The authors wish to thank Henk Vogelaar from Waterleiding Maatschappij Limburg (WML) for providing the Sittard network model used in the calculations. Edited by: Ran Shang Reviewed by: two anonymous referees