Algorithms for Stochastic Lot-Sizing Problems with Backlogging

As a traditional model in the operations research and management science domain, deterministic lot-sizing problem is embedded in many application problems such as production and inventory planning and has been consistently drawing attentions from researchers. In this paper we consider basic versions of lot-sizing models in which problem parameters are stochastic and develop corresponding scenario tree based stochastic lot-sizing models. For these models, we develop production path properties and a general dynamic programming framework based on these properties. The dynamic programming framework allows us to show that the optimal value function is piecewise linear and continuous, which enables us to develop polynomial time algorithms for several different problems, including those with backlogging and varying capacities under certain conditions. Moreover, we develop polynomial time algorithms that run in O(n) and O(nT log n) times respectively for stochastic uncapacitated and constant capacitated lot-sizing problems with backlogging, regardless of the scenario tree structure.