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.

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.

Considering only the naïve random mutation and one-point crossover, the best results after 1×105 function evaluations were achieved with a mutation rate equal to 0.05 and a crossover rate equal to 0.95. In this case the average objective function value was 8.8×105. Adding a proximity mutation does not improve the results, but considering only a proximity mutation and no random mutation has a significant influence on the outcomes. With only the proximity mutation, the best results after 1×105 function evaluations were achieved for a proximity mutation rate of 0.05. With this value, the best results were achieved for the mean as well as the best and worst values for the objective function. Figure 3a illustrates the influence of this variator on the computed objective function values.

Adding a flatiron mutation further improved the obtained results. The best results after 1×105 function evaluations, on average, were obtained for a combination of a crossover rate of 0.95, with a proximity mutation rate equal to 0.05 and a flatiron mutation rate equal to 0.8. The best result within one test was obtained for a slightly lower flatiron mutation rate, equal to 0.7. Figure 3b illustrates the influence of this flatiron mutation rate on the obtained objective function values.

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 ± standard deviation, and mean ±2 × standard deviation convergence curves for tests number 2 (RMNPC2) and 12 (FM4). The proximity and flatiron mutations lead to smoother curves and a faster convergence. The standard deviation between results of the different runs is also much lower, which means that the results are more stable.

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 6×105 was reached in around 700 generations, about 1.4 times faster than in LPM2, LPM5 and LMP6.

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 1×105 function evaluations (test numbers 13:FM5 and 14:FM6), i.e. the fastest convergence. All tested combinations which include either the flatiron or the proximity mutation exhibit a similar or worse performance. Albeit slower, particularly stable results were obtained with the proximity mutation (rate = 0.1) and no flatiron mutation. These runs show the smallest standard deviation in the results after 1×105 function evaluations.

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.