A new stopping criterion for multi-objective evolutionary algorithms: application in the calibration of a hydrologic model
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Multi-objective genetic algorithms have been successfully applied in a wide variety of problems. Although widely used, there are few theoretical guidelines for determining when to stop the search. Many users commonly use rules like stopping when there is no significant improvement during the last generations or when a certain number of generations are reached. In this paper, we propose a new stopping criterion approach and evaluate its performance with three widely used evolutionary algorithms in the calibration of a hydrologic model. The stopping criterion is based on the minimum number of generations required to achieve a determined number of non-dominated solutions in Pareto Front. The new stopping criterion was tested in the lumped hydrologic model IPH-II calibration, using the genetic algorithms NSGA-II, NSGA-III, and SPEA-II and two objective functions. The generational distance, spacing, and maximum spread metrics were used to assess the performance of the proposed stopping criterion in comparison to the standard criterion. Results show no significant loss in goodness of fit associated with the proposed stopping criterion, both in calibration and validation periods. Performance metrics have shown similar values when the standard and the proposed stopping criteria were compared. However, the average computational time to complete the optimization process was reduced up to 38.2% when the proposed stopping criterion was used. Thus, it can be concluded that the new stopping criterion reduces the iteration workload without compromising the accuracy of solution sets.
KeywordsMulti-objective evolutionary algorithm Lumped hydrologic model Stopping criterion NSGA-II NSGA-III SPEA-II
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We thank the anonymous reviewers and editor whose comments/suggestions helped to improve and clarify this manuscript.
The Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq: www.cnpq.br) of Brazil gave the scholarships awarded to JCTG and DSA, process numbers 141181/2015-0 and 141448/2015-6. This work was also supported by CNPq’s project number 443834/2014-8.
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