Expert knowledge is needed for design under uncertainty: For p-boxes, backcalculation is, in general, NP-hard

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), Fmacrc(x)] on the cumulative distribution function F_c(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 F_c(x) for c = f(a,b) is within the given bounds for all possible values of the environmental variables b. The problem of computing such a is called backcalculation. For b, we also have ranges with different probabilities - i.e., also a p-box. Thus, we have backcalculation problem for p-boxes. For p-boxes, there exist efficient algorithms for finding a design a that satisfies the given constraints. The next natural question is to find a design that satisfies additional constraints: on the cost, on the efficiency, etc. In this paper, we prove that that without expert knowledge, the problem of finding such a design is computationally difficult (NP-hard). We show that this problem is NP-hard already in the simplest possible linearized case, when the dependence c = f(a, b) is linear. Thus, expert (fuzzy) knowledge is needed to solve design problems under uncertainty.