Keywords: genetic algorithms, multiobjective optimisation, water networks, fuzzy sets.

The optimal design of water supply and distribution systems (WDS) is not solely a least-cost problem, but involves a significant number of engineering issues, thus requiring a multiobjective optimisation approach, resulting in Pareto trade-off non-inferior or non-dominated solution sets, typically using multiobjective genetic algorithms (MOGA). Generally all MOGA algorithms in literature present some common characteristics:

a)

They include rigorous sorting algorithms, based on non-domination, in order to define a Pareto front consisting of non-inferior solutions.

b)

They try to maintain diversity both in the genetic algorithm population and the Pareto-front solutions, by introducing mechanisms and metrics that detect solutions, which are very close to each other, and discourage (or limit) the selection of multiple quasi similar solutions for consequent generations.

In this paper a fuzzy hierarchical MOGA is applied for WDS optimisation, with two objective functions: For any solution i defined through the decision variable set (or string) X_i=[x₁,x₂,...,x_ND], the first objective function min C(X_i) stands for the sum of all costs, while the second objective function max B(X_i) represents the fuzzy benefit/quality/acceptance of the solution, while the MOGA structure is based on elitist Pareto optimality ranking (NSGA-II). The original NSGA-II applies tournament selection with the following criteria: solutions with lower domination rank are preferred, while for equal domination rank, solutions at less crowded regions (i.e. at a larger distance from each other) are selected, so as to maintain diversity in the population. However, in NSGA-II, as in all existing literature, the similarity or closeness of two solutions is determined solely according to the numerical values of the objective functions, with no engineering "meaning" assigned to either the variables or the solutions.

In this paper research is focused on the mechanism for detecting quasi-similar solutions, for which the innovative notion of "water network engineering similarity" (WNES) is introduced. Selecting different tank locations affects the overall structure and hydraulic behaviour of any WDS significantly more than the selection of alternative link diameters, which mainly affect the criteria relating to minimum pressure requirements. Accordingly, for two potential solutions sharing similar tank locations, there is a greater similarity in the hydraulic behaviour, if they also share similar tank levels, while sharing similar storage sizes can only be considered as a secondary factor. On the other hand, should two solutions have similar tanks (location, level and storage) they can be considered similar, if the objective function values are close, irrespective of the differences in link diameters and pump numbers.

Consequently, the similarity between two potential solutions i and j from the engineering point of view, i.e. the water network engineering similarity WNES(i,j) indicator, can be estimated using three consecutive indices: an index ITL(i,j) estimating similarity in tank locations, an index ITH(i,j) estimating similarity in tank levels and an index ITS(i,j) estimating similarity in storage sizes. The WNES(i,j) indicator takes a final binary form, with two possible outcomes: WNES(i,j)=0 (not similar) or WNES(i,j)=1 (similar). Thus, in the MOGA model presented in this paper, the numerical distance of two neighbouring strings will only be taken into account as the preference criterion in the selection process, if the solutions are similar, i.e. if WNES(i,i-1)=1. This new rule results in enhancing the GA search space, and maintains solutions, which seem to be "quite close" in terms of objective function values, although different from the engineering and hydraulic point of view.

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