Intricate interrelation between robustness and probability in the context of structural optimization

In this study, we deal with the problem of structural optimization under uncertainty. In previous studies, either of three philosophies were adopted: (a) probabilistic methodology, (b) fuzzy-sets-based design, or (c) nonprobabilistic approach in the form of given bounds of variation of uncertain quantities. In these works, authors are postulating knowledge of either involved probability densities, membership functions, or bounds in the form of boxes or ellipsoids, where the uncertainty is assumed to vary. Here, we consider the problem in its apparently pristine setting, when the initial raw data are available and the uncertainty model in the form of bounds must be constructed. We treat the often-encountered case when scarce data are available and the unknown-but-bounded uncertainty is dealt with. We show that the probability concepts ought to be invoked for predicting the worst-and best-possible designs. The Chebyshev inequality, applied to the raw data, is superimposed with the study of the robustness of the associated deterministic optimal design. We demonstrate that there is an intricate relationship between robustness and probability.