Production and inventory planning have become crucial and challenging in nowadays competitive industrial and commercial sectors, especially when multiple plants or warehouses are involved. In this context, this paper addresses the complexity of uncapacitated multi-plant lot-sizing problems. We consider a multi-item uncapacitated multi-plant lot-sizing problem with fixed transfer costs and show that two of its very restricted special cases are already NP-hard. Namely, we show that the single-item uncapacitated multi-plant lot-sizing problem with a single period and the multi-item uncapacitated two-plant lot-sizing problem with fixed transfer costs are NP-hard. Furthermore, as a direct implication of the proven results, we also show that a two-echelon multi-item lot-sizing with joint setup costs on transportation is NP-hard.

在当今竞争激烈的工业和商业领域，尤其是涉及多个工厂或仓库时，生产和库存计划已变得至关重要且充满挑战。在这种情况下，本文解决了无能力的多工厂批量确定问题的复杂性。我们考虑了具有固定转移成本的多项目无能力的多工厂批量确定问题，并显示了其两个非常受限制的特殊情况已经是NP难的。即，我们表明具有固定期限的单项目单容量无能力多工厂批量问题和固定转移成本的多项目无容量两工厂批量问题都是NP难题。此外，作为已验证结果的直接暗示，我们还显示出带有联合设置运输成本的两级多项目批量估计是NP-hard的。