Subcontracting and lot-sizing with constant capacities

In this paper, we consider the subcontracting and single-item lot-sizing problem with constant capacities, which is uncapacitated in subcontracting but capacitated in production. For a holistic understanding of the problem, an infinite-period model is proposed. Such a model provides a unified view of a capacitated lot-sizing problem. The usefulness of the infinite-period model is shown by the principle that the firm’s production schedule drives the subcontractor’s supply schedule. For efficient construction of the production schedule, an innovative technique–coined using the concept of the shadow period–is presented to designate the optimal marginal supply cost. The success of the infinite-period model is finally proved with the concept of the effective period, indicating the supply coverage over periods. This fills the gap between the uncapacitated techniques and the capacitated techniques, making it possible to apply the best techniques from the lot-sizing literature. With this novel approach, we improve existing algorithms and provide new ones for concave production and supply cost structures involving setups.