In many industrial engineering problems, we must select a design, select parameters of a process, or, in general, make a decision. Informally, this decision must be optimal, the best for the users. In traditional operations research, we assume that we know the objective function f(x) whose values describe the consequence of a decision x for the user. Optimization of well-defined functions is what started calculus in the first place: once we know the objective function f(x), we can use differentiation to find its maximum, e.g., as the point x at which the derivative of f with respect to x is equal to 0.
In real life, we often do not know the exact consequence f(x) of each possible decision x, we only know this consequence with uncertainty. The simplest case is when we have tolerance-type (interval) uncertainty, i.e., when all we know is that the deviation between the the actual (unknown) value f(x) and the approximate (known) value F(x) cannot exceed the (known) bound D(x). In precise terms, this means that f(x) belongs to the interval [F(x)-D(x),F(x)+D(x)]. In other situations, in addition to the interval that is guaranteed to contain f(x), experts can also provide us with narrower intervals that contain f(x) with certain degree of confidence alpha; such a nested family of intervals is equivalent to a more traditional definition of fuzzy set. To solve the corresponding optimization problems, in this paper, we extend differentiation formalisms to the cases of interval and fuzzy uncertainty.