A probabilistic non-dominated sorting GA for optimization under uncertainty

Giovanni Petrone (Mechanical Engineering Department, Stanford University, Stanford, California, USA)
John Axerio-Cilies (Mechanical Engineering Department, Stanford University, Stanford, California, USA)
Domenico Quagliarella (Fluid Mechanics Department, CIRA – Italian Centre for Aerospace Research, Capua, Italy)
Gianluca Iaccarino (Mechanical Engineering Department, Stanford University, Stanford, California, USA)

Engineering Computations

ISSN: 0264-4401

Publication date: 11 November 2013



A probabilistic non-dominated sorting genetic algorithm (P-NSGA) for multi-objective optimization under uncertainty is presented. The purpose of this algorithm is to create a tight coupling between the optimization and uncertainty procedures, use all of the possible probabilistic information to drive the optimizer, and leverage high-performance parallel computing.


This algorithm is a generalization of a classical genetic algorithm for multi-objective optimization (NSGA-II) by Deb et al. The proposed algorithm relies on the use of all possible information in the probabilistic domain summarized by the cumulative distribution functions (CDFs) of the objective functions. Several analytic test functions are used to benchmark this algorithm, but only the results of the Fonseca-Fleming test function are shown. An industrial application is presented to show that P-NSGA can be used for multi-objective shape optimization of a Formula 1 tire brake duct, taking into account the geometrical uncertainties associated with the rotating rubber tire and uncertain inflow conditions.


This algorithm is shown to have deterministic consistency (i.e. it turns back to the original NSGA-II) when the objective functions are deterministic. When the quality of the CDF is increased (either using more points or higher fidelity resolution), the convergence behavior improves. Since all the information regarding uncertainty quantification is preserved, all the different types of Pareto fronts that exist in the probabilistic framework (e.g. mean value Pareto, mean value penalty Pareto, etc.) are shown to be generated a posteriori. An adaptive sampling approach and parallel computing (in both the uncertainty and optimization algorithms) are shown to have several fold speed-up in selecting optimal solutions under uncertainty.


There are no existing algorithms that use the full probabilistic distribution to guide the optimizer. The method presented herein bases its sorting on real function evaluations, not merely measures (i.e. mean of the probabilistic distribution) that potentially do not exist.



Petrone, G., Axerio-Cilies, J., Quagliarella, D. and Iaccarino, G. (2013), "A probabilistic non-dominated sorting GA for optimization under uncertainty", Engineering Computations, Vol. 30 No. 8, pp. 1054-1085. https://doi.org/10.1108/EC-05-2012-0110

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