Multi-item uncapacitated lot sizing problem with inventory bounds

We study a multi-item lot sizing problem with inventory bounds, where the production of the items is uncapacitated but a storage capacity is considered, limiting at each period the amount of products that can be held in stock. We prove that the problem is strongly NP-hard even with no holding cost and stationary setup costs. For non-speculative costs, this lot-sizing problem remains NP-hard even when restricted to only 2 periods. However, in this case, we show that it can be polynomially solved for any fixed number of items.