Tighter bounds on the probability of failure than those provided by random set theory

• Published in: Computers & structures 2017-09, Vol.189, p.101-113
• Main Authors: Alvarez, Diego A., Hurtado, Jorge E., Ramírez, Juliana
• Format: Article
• Language: English
• Subjects: Boxes; Dempster-Shafer evidence theory; Design optimization; Distribution functions; Failure; Imprecise probabilities; Mathematical models; Monte Carlo simulation; Multivariate analysis; Numerical analysis; Probability distribution; Probability of failure; Random sets; Set theory; Stochastic subset optimization; Studies; Upper and lower probabilities
• ISSN: 0045-7949; 1879-2243
• DOI: 10.1016/j.compstruc.2017.04.006

•The bounds provided by random set theory are wider than necessary.•The bounds provided by the measurable selections in the random set are narrower.•Condition are given when both methods provide the same bounds on the probability.•Numerical examples illustrate the discussion. Random set theory is a generalization of Dempster-Shafer evidence theory, that employs an infinite number of focal elements. It can be used for the estimation of the bounds of the probability of failure of structural systems when there is both aleatory and epistemic uncertainty in the representation of the input variables. Indeed, this framework allows to model basic variables as cumulative distribution functions, distribution-free probability boxes, possibility distributions or families of intervals provided by experts, while representing the dependence of the implied variables by means of copulas. This paper reviews another method, which poses the calculation of the bounds of the probability of failure as a reliability-based-design-optimization problem. It is proved theoretically and by means of numerical experiments, that the latter method provides tighter bounds on the probability of failure than those estimated by random set theory. We also theoretically show some interesting relationships between the random set-based method and the optimization approach.