Lagrangian solution techniques and bounds for loosely coupled mixed-integer stochastic programs

Many production problems involve facility setups that lead to integer variables, production decisions that are continuous, and demands that are likely to be random. While these problems can be quite difficult to solve, we propose a model and an efficient solution technique for this basic class of stochastic mixed-integer programs. We use a set of scenarios to reflect uncertainty. The resulting mathematical model is solved using Lagrangian relaxation. We show that the duality gap of our relaxation is bounded above by a constant that depends on the cost function and the number of branching points in the scenario tree. We apply our technique to the problem of generating electric power. Numerical results indicate significant savings when the stochastic model is used instead of a deterministic one.