Dantzig-Wolfe Decomposition for Solving Multistage Stochastic Capacity-Planning Problems
Singh, Kavinesh J.
Philpott, Andy B.
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We describe a multistage, stochastic, mixed-integer programming model for planning capacity expansion of production facilities. A scenario tree represents uncertainty in the mode; a general mixed-integer program defines the operation submodel at each scenario-tree mode, 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-Wolf 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 soltuions, obviating the need for a full branch-and-price soltuion 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.
Operations Research, 57, pp. 1271-1286.
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