  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.
