The single-product lot-sizing problem with constant parameters and backlogging: exact results, a new solution, and all parameter stability regions

We consider the single-product lot-sizing problem over a finite planning horizon. Demand at each period is constant, and excess demand is completely backlogged. Holding and backlogging costs are proportional to the amount of inventory stocked or backlogged, while ordering cost is fixed, independent of the quantity ordered. The optimal policy targets to minimize the total relevant costs over the planning horizon. The key results of this paper are: (1) an explicit formula for the optimal total cost as a function of the model parameters and the number of cycles of the policy; (2) a new, polynomial-time algorithm which determines the overall optimal policy; and (3) stability regions for any solution considering simultaneous variations on all cost and demand parameters. The proposed algorithm is easy to implement and therefore is suitable for practical use.