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.

在本文中，我们考虑具有恒定生产能力的分包和单项批量问题，该问题在分包中没有能力，但在生产中却没有能力。为了全面了解该问题，提出了一个无限周期模型。这样的模型提供了一个容量很大的批量问题的统一视图。无限期模型的有用性由公司的生产进度表驱动分包商的供应进度表的原理证明。为了有效地构建生产进度计划，提出了一种创新技术，该技术采用了“影子期”的概念来指定最佳边际供应成本。最后，通过有效期的概念证明了无限期模型的成功，该有效期表明了各个时期的供应范围。这填补了能力丧失的技术与能力丧失的技术之间的空白，从而有可能应用批量研究文献中的最佳技术。通过这种新颖的方法，我们改进了现有算法，并为涉及设置的凹面生产和供应成本结构提供了新算法。