Uncapacitated lot-sizing problem with production time windows, early production, backlog and lost sale

We consider the single item uncapacitated lot-sizing problem with production time windows, lost sale, early production and backlog 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 optimally the examined problems in O(T ).