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

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.