Threshold Boolean Form for Joint Probabilistic Constraints with Random Technology Matrix

Abstract

We develop a new modeling and exact solution method for stochastic programming problems thatinclude a joint probabilistic constraint in which the multi-row random technology matrix is discretely distributed. We binarize the probability distribution of the random variables in such a way that we can extract a threshold partially defined Boolean function (pdBf) representing the probabilistic constraint.We then construct a tight threshold Boolean minorant for the pdBf. Any separating structureof the tight threshold Boolean minorant defines sufficient conditions for the satisfaction of the probabilistic constraint and takes the form of a system of linear constraints. We use the separating structure to derive three new deterministic formulations equivalent to the studied stochastic problem. We derivea set of strengthening valid inequalities for the reformulated problems. A crucial feature ofthe new integer formulations is that the number of integer variables does not depend on the numberof scenarios used to represent uncertainty. The computational study, based on instances of thestochastic capital rationing problem, shows that the MIP reformulations are much easier and ordersof magnitude faster to solve than the MINLP formulation. The method integrating the derived valid inequalities in a branch-and-bound algorithm has the best performance.