Abstract
The sensor placement problem is modeled as a multi-objective optimization problem with Boolean decision variables. A new multi objective evolutionary algorithm (MOEA) is proposed for approximating and analyzing the set of Pareto optimal solutions. The evaluation of the objective functions requires the execution of a hydraulic simulation model of the network. To organize the simulation results a data structure is proposed which enables the dynamic representation of a sensor placement and its fitness as a heatmap. This allows the definition of information spaces, in which the fitness of a placement can be represented as a matrix or, in probabilistic terms as a histogram. The key element in the new algorithm is this probabilistic representation which is embedded in a space endowed with a metric based on a specific notion of distance. Among several distances between probability distributions the Wasserstein (WST) distance has been selected: WST has enabled to derive new genetic operators, indicators of the quality of the Pareto set and criteria to choose among the Pareto solutions. The new algorithm has been tested on a benchmark water distribution network with two objective functions showing an improvement over NSGA-II, in particular for low generation counts, making it a good candidate for expensive black-box multi-objective optimization
Highlights
Many real-world problems fit into this general framework among which water distribution networks (WDNs), where sensing spots are sensors deployed at specific locations, events are natural/intended contaminations and effects are spatio-temporal concentrations of the contaminant
This problem is usually known as optimal sensor placement (SP), where the goal is to optimize several objectives such as the amount of contaminated water, the number of inhabitants affected before detection, the detection time or the detection likelihood so that we faced with a multi-objective optimization problem (MOP)
The main result of this paper is the proposal of a new evolutionary algorithms The main for result of this paper is the proposal a new evolutionary optimal sensor placement in waterofdistribution networks.algorithms
Summary
Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. Distinguishing features of the new algorithm MOEA/WST are in particular the selection operator which drives a more effective diversification and exploration among candidate solutions and a problem specific crossover operator which generates two “feasible by design” children from two feasible parents This new algorithm, called multi-objective evolutionary algorithm-Wasserstein (MOEA-WST) has been tested on two benchmark and two real-world water distribution networks resulting in significant improvement over a standard algorithm both in terms of hypervolume improvement and coverage, in particular at low generation counts. The last section “Conclusions” elaborates on the advantages of the algorithm and on how the probabilistic representation has allowed substantial improvements in terms of hypervolume and coverage, requiring a significantly lower number of generations This makes MOEA/WST a good candidate for expensive multi-objective optimization problems
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