  We study a two-level uncapacitated lot-sizing problem with inventory bounds
that occurs in a supply chain composed of a supplier and a retailer. The first
level with the demands is the retailer level and the second one is the supplier
level. The aim is to minimize the cost of the supply chain so as to satisfy the
demands when the quantity of item that can be held in inventory at each period
is limited. The inventory bounds can be imposed at the retailer level, at the
supplier level or at both levels. We propose a polynomial dynamic programming
algorithm to solve this problem when the inventory bounds are set on the
retailer level. When the inventory bounds are set on the supplier level, we
show that the problem is NP-hard. We give a pseudo-polynomial algorithm which
solves this problem when there are inventory bounds on both levels. In the case
where demand lot-splitting is not allowed, i.e. each demand has to be satisfied
by a single order, we prove that the uncapacitated lot-sizing problem with
inventory bounds is strongly NP-hard. This implies that the two-level
lot-sizing problems with inventory bounds are also strongly NP-hard when demand
lot-splitting is considered.
