A polynomial time algorithm for the single-item lot sizing problem with capacities, minimum order quantities and dynamic time windows

This paper deals with the single-item capacitated lot sizing problem with concave production and storage costs, considering minimum order quantity and dynamic time windows. This problem models a lot sizing where the production lots are constrained in amount and frequency. In this problem, a demand must be satisfied at each period t over a planning horizon of T periods. This demand can be satisfied from the stock or by a production at the same period. When a production is made at period t, the produced quantity must be greater than a minimum order quantity (L) and lower than the production capacity (U). The frequency constraints on the production lots are modeled by dynamic time windows. Between two consecutive production lots, there is at least Q periods and at most R periods. An optimal algorithms in O(T 9) is given. The complexity of the algorithm is reduced to O(T 7) when all the demands are strictly positive.