Uncapacitated lot-sizing problem with production time windows, early productions, backlogs and lost sales

We consider the single item uncapacitated lot-sizing problem with production time windows, lost sales, early productions and backlogs over a planning horizon of T periods. In this context, a demand not processed within its time window can either be lost (lost sale), satisfied from a production that is processed before the release period of the demand (early production) or satisfied from a production that occurs after the demand due period (backlog). We present several properties of the optimal solution for different variants of the problem when production time windows are non-customer specific. We propose dynamic programming algorithms to solve the examined problems in O(T2).