A relationship between probability interval and random sets and its application to linear optimization with uncertainties

Random sets and probability intervals could be represented as uncertain parameters in a linear program. However, they are not the same. This paper presents a method by which an intersection of random sets provides an approximation of a given probability interval. Two specific random sets generating this approximation are used to construct probability density mass functions that provide the smallest and the largest expected values with respect to the probability interval uncertainty. We show that our construction is computationally more efficient than computing directly from probability intervals. These results are applied to linear programs with random set and probability interval uncertainties to obtain optimal solutions with respect to pessimistic, optimistic, and minimax regret approaches.