This paper addresses a multi-stage stochastic integer programming formulation of the uncapacitated lot-sizing problem under uncertainty. We show that the classical (l,S) inequalities for the deterministic lot-sizing polytope are also valid for the stochastic lot-sizing polytope. We then extend the (l,S) inequalities to a general class of valid inequalities, called the (Q,S_Q) inequalities, and we establish necessary and sufficient conditions which guarantee that the (Q,S_Q) inequalities are facet-defining. A separation heuristic for (Q,S_Q) inequalities is developed and incorporated into a branch and cut algorithm. A computational study verifies the usefulness of the (Q,S_Q) inequalities as cuts.
Technical Report, School of Industrial & Systems Engineering, Georgia Institute of Technology, 2004.