Dantzig-Wolfe Decomposition for Solving Multistage Stochastic Capacity-Planning Problems

TLDR

The model uses a scenario tree to represent uncertainty, with a general mixed‑integer program defining operational submodels at each node and linking capacity‑expansion decisions across stages. The study proposes a multistage stochastic mixed‑integer programming model to plan capacity expansion of production facilities. The authors apply variable splitting to two model variants and use Dantzig‑Wolfe decomposition, defining at each scenario‑tree node a single‑period deterministic capacity‑planning subproblem and employing a dual‑stabilization method; the largest solved instance has six stages, 243 scenarios, and a quarter‑million binary variables. The Dantzig‑Wolfe master problem achieves a substantially stronger LP relaxation—over 700% stronger in one case—and solves readily, often yielding integer solutions and eliminating the need for branch‑and‑price, provided subproblems solve efficiently.

Abstract

We describe a multistage, stochastic, mixed-integer programming model for planning capacity expansion of production facilities. A scenario tree represents uncertainty in the model; a general mixed-integer program defines the operational submodel at each scenario-tree node, and capacity-expansion decisions link the stages. We apply “variable splitting” to two model variants, and solve those variants using Dantzig-Wolfe decomposition. The Dantzig-Wolfe master problem can have a much stronger linear programming relaxation than is possible without variable splitting, over 700% stronger in one case. The master problem solves easily and tends to yield integer solutions, obviating the need for a full branch-and-price solution procedure. For each scenario-tree node, the decomposition defines a subproblem that may be viewed as a single-period, deterministic, capacity-planning problem. An effective solution procedure results as long as the subproblems solve efficiently, and the procedure incorporates a good “duals stabilization method.” We present computational results for a model to plan the capacity expansion of an electricity distribution network in New Zealand, given uncertain future demand. The largest problem we solve to optimality has six stages and 243 scenarios, and corresponds to a deterministic equivalent with a quarter of a million binary variables.