Engineering Design under Imprecise Probabilities: Computational Complexity

In engineering design problems, we want to make sure that a certain quantity c of the designed system lies within given bounds – or at least that the probability of this quantity to be outside these bounds does not exceed a given threshold. We may have several such requirements – thus the requirement can be formulated as bounds [F c(x), F c(x)] on the cumulative distribution function Fc(x) of the quantity c; such bounds are known as a p-box. The value of the desired quantity c depends on the design parameters a and the parameters b characterizing the environment: c = f(a, b). To achieve the design goal, we need to find the design parameters a for which the distribution Fc(x) for c = f(a, b) is within the given bounds for all possible values of the environmental variables b. The problem of com-puting such a is called backcalculation. For b, we also have ranges with different probabilities – i.e., also a p-box. Thus, we have backcalculatio