Pairing inequalities and stochastic lot-sizing problems: A study in integer programming
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Based on the recent successes in stochastic linear programming and mixed integer programming, in this thesis we combine these two important areas of mathematical programming; specifically we study stochastic integer programming. We first study a simple and important stochastic integer programming problem, called stochastic uncapacitated lot-sizing (SLS), which is motivated by production planning under uncertainty. We describe a multi-stage stochastic integer programming formulation of the problem and develop a family of valid inequalities, called the (Q, S) inequalities. We establish facet-defining conditions and show that these inequalities are sufficient to describe the convex hull of integral solutions for two-period instances. A separation heuristic for (Q, S) inequalities is developed and incorporated into a branch-and-cut algorithm. A computational study verifies the usefulness of the inequalities as cuts. Then, motivated by the polyhedral study of (Q, S) inequalities for SLS, we analyze the underlying integer programming scheme for general stochastic integer programming problems. We present a scheme for generating new valid inequalities for mixed integer programs by taking pair-wise combinations of existing valid inequalities. The scheme is in general sequence-dependent and therefore leads to an exponential number of inequalities. For some special cases, we identify combination sequences that lead to a manageable set of all non-dominated inequalities. For the general scenario tree case, we identify combination sequences that lead to non-dominated inequalities. We also analyze the conditions such that the inequalities generated by our approach are facet-defining and describe the convex hull of integral solutions. We illustrate the framework for some deterministic and stochastic integer programs and we present computational results which show the efficiency of adding the new generated inequalities as cuts.