Lotsizing with backlogging and start-ups: the case of Wagner–Whitin costs

Abstract

We examine the uncapacitated single-item lotsizing problem with backlogging and start-up costs where Wagner–Whitin costs are assumed. We generalize some theoretical results obtained in [5] for the polyhedral description of the convex hull of feasible solutions for models that can be viewed as particular cases of the one treated in this paper (models without start-up costs and models where backlog is not allowed). In the presence of Wagner–Whitin costs (which satisfy p t+ h ̃ t +−p t+1⩾0, for 0⩽ t⩽ n−1, and p t+1+ h ̃ t −−p t⩾0 , for 1⩽ t⩽ n, where p t, h ̃ t + and h ̃ t − are the unit production, storage and backlogging costs; in the case that unit production costs are constant over time, the Wagner–Whitin assumption corresponds to non-negative holding costs and backlogging costs) we present a linear extended formulation with O( n) variables and O( n 2) constraints. By projection, we obtain a linear formulation in the original space of variables with an exponential number of constraints.