A probabilistic non-dominated sorting GA for optimization under uncertainty

Abstract

Purpose – 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. Design/methodology/approach – 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. Findings – 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. Originality/value – 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.

Genetic algorithms for condition-based maintenance optimization under uncertainty

“…This algorithm exploits the ranking criterion described above and assigns the same ranking position to all the solutions that are equivalent with respect to objective q. For example, if we have Finally, notice that the approach proposed in [49] to implement the NSGA-II algorithm also relies on the ranking positions of the solutions in a population. However, the definition of the rank in [49] is different from that given in this work.…”

“…For example, if we have Finally, notice that the approach proposed in [49] to implement the NSGA-II algorithm also relies on the ranking positions of the solutions in a population. However, the definition of the rank in [49] is different from that given in this work. Rather, it resembles the ranking method proposed in [44], which applies the Monte Carlo sampling method to estimate, for every solution X h , the probabilities of occupying the H positions in the ranking.…”

“…It is worth noticing that when the reference Dirac delta is positioned at zero, then the method proposed in [49] reduces to sorting the distributions based on their mean values. Then, also in this case the CDF sorting is based on points summarizing the CDFs, thus losing the information they encode.…”

“…Finally notice that a ranking technique has been proposed in [49] in support to GA, which establishes the order relation based on the comparison of the solutions with an ideal, pre-fixed Dirac delta distribution. It is worth noticing that when the reference Dirac delta is positioned at zero, then the method proposed in [49] reduces to sorting the distributions based on their mean values.…”

“…), has not received the necessary attention. In fact, as pointed out in [17] and in [49], few approaches have been propounded in the literature to effectively tackle such multiobjective optimization problems in the presence of uncertain objective functions. These works consider different frameworks for uncertainty representation: probability distributions in [17], [23], [39], [49], [52], [57] fuzzy sets in [31] and [56], and plausibility and belief functions in [14], [33].…”