Capacitated lot sizing problems with inventory bounds

In this paper, we study the single-item and the multi-item capacitated lot sizing problem in presence of inventory bounds (CLSP-IB). That is, in any period, both the quantity produced and the stock on hand are limited. For the single-item CLSP-IB with a stationary production capacity, time-dependent inventory bounds and concave costs, we show that the problem can be solved in time by adapting a well-known algorithm in the literature. Restricting to non-speculative costs, we propose an time dynamic programming algorithm. For the multi-item CLSP-IB, we consider the lot-sizing problem with item dedicated machines and a common capacitated storage space shared by all the items. We show that this problem is -hard in the strong sense even if all the cost parameters and capacities of the instance are stationary and identical for each item.