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 $O(T^4)$ by adapting a well‐known algorithm in the literature. Restricting to non‐speculative costs, we propose an $O(T^3)$ 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 $\text{NP}$ ‐hard in the strong sense even if all the cost parameters and capacities of the instance are stationary and identical for each item.