An O(T2) algorithm for the lot-sizing problem with limited inventory levels

The paper present an efficient O(T²) algorithm for the general single-item dynamic lot-sizing problem, where the time horizon spans over T periods, inventory levels are limited, and nonzero initial and safety stock levels are assumed to exist. The properties of the model were previously found by the author (1992) to transform the problem of finding an optimal, extreme solution into an equivalent shortest-path problem in a multigraph, which is a nontrivial generalization of the Wagner-Whitin (1958) model. Although the resulted multigraph contains O(T³) edges, this paper shows how to construct the shortest path solution in O(T²) time. For this type of the lot-sizing problems, the resulted algorithm has the lowest complexity over known methods