A branch-and-cut algorithm for the stochastic uncapacitated lot-sizing problem

This paper addresses a multi-stage stochastic integer programming formulation of the uncapacitated lot-sizing problem under uncertainty. We show that the classical (`, S) inequalities for the deterministic lot-sizing polytope are also valid for the stochastic lot-sizing polytope. We then extend the (`, S) inequalities to a general class of valid inequalities, called the (Q, SQ) inequalities, and we establish necessary and sufficient conditions which guarantee that the (Q, SQ) inequalities are facet-defining. A separation heuristic for (Q, SQ) inequalities is developed and incorporated into a branch-and-cut algorithm. A computational study verifies the usefulness of the (Q, SQ) inequalities as cuts.