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

Nowadays it is universally accepted that optimal design of water supply and distribution networks is not solely a least cost problem, but involves a significant number of engineering issues, thus requiring a multiobjective optimisation approach. The problem of partial rehabilitation of an existing distribution network is harder than designing a new one. In this paper a model is presented for the optimal design rehabilitation of an existing network, involving selection of new pipes, cleaning and lining of old pipes, determining both the location and the size of new tanks (retaining the old ones) and possible addition of new pumps.

In all previously published work involving fuzziness and water network design, single objective models have been applied. To the best of the authors' knowledge, no other multi-objective fuzzy reasoning approach for water distribution network design optimisation exists in literature.

In this paper multiobjective optimisation is applied, i.e. minimizing costs and maximizing a benefit/quality function, the trade-off Pareto curve being produced by a genetic algorithm, whereas fuzzy reasoning is introduced to the evaluation of benefits for each potential solution. Multiobjective optimisation is carried out using a genetic (evolutionary) algorithm (GA), resulting in a trade-off curve consisting of non-inferior cost-benefit points, which evolve within the GA, as generations proceed. It is based on elitist Pareto optimality ranking.

A number of criteria relating to the performance of each potential solution are introduced for the benefit function, evaluating the performance of the system for any solution under multiple loadings. or each criterion and network component involved, the performance of the system is assessed independently, by estimating the partial membership function of each solution generated by the GA to the set of feasible solutions for the specific criterion, applying fuzzy aggregation when more than one system component or loading is involved. Classic fuzzy intersection (minimum operator) and all kinds of weighted generalized means may theoretically be used as total aggregators, but results and their impact on the GA will vary, as shown in the application included in the paper.

The approach adopted for criteria in the model is highly flexible: If, for any network, the engineer wishes to omit one or more criteria, whatever the reason, the weight assigned to it can be simply set to zero. On the other hand, should in the future more criteria be added, referring to other quality factors of the design (e.g. resiliency, risk reduction etc), the modular approach adopted by fuzzy reasoning allows for easy additions and modifications, without affecting the overall structure of the multi-objective model and algorithm.

By using fuzzy membership functions, tolerance to small constraint violations is simulated (as would any engineer do for real networks). On the other hand, by estimating benefit values through aggregators, the whole design algorithm moves away from strict mathematical functions, and resembles more a DSS model. There is no need for scaling; constraints, requirements and network properties are all being handled as a population of criteria (or factors), acting as decision makers. Each criterion assesses benefits separately, while in the end, all of them combine "opinions" in order to estimate the merits (benefits) of each solution.

Additionally the paper includes a novel approach for the simulation of tanks as network storage components within the genetic algorithm, reducing the number of decision variables needed, while tank shape parameters are entered as data for the GA, thus avoiding excesses for tank height or diameter in potential solutions.

The model is applied to a well-known example from the literature, the "Anytown" water distribution network [1], to benchmark the results. Comparison of results, included in the paper, shows that the model manages to find a better solution than any other previous approach in terms of cost, despite the multiple criteria applied for the benefit function being more extensive and stricter.

1

Centre for Water Systems, University of Exeter, "Anytown Water Distribution Network - Benchmark Data", www.ex.ac.uk/cws/bmanytown.htm, 2004.

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