The General Model

We shall be concerned with the following stochastic dynamic inventory-production model (1) xk: stock on hand at the beginning of period k,(k=1,2,…,N+1) xk ɛ ℝ ∀ k, x1: initial stock; N: planning horizon (2) uk: production decision at the beginning of period k which results in a shipment in this period ukɛ Uk (k=1,2,…,N), where (1) Uk:=ℝ (cash balance case) (2) Uk:=ℝ+ (inventory case) (3) rk: stochastic demand in period k. All conditional expectations subsequently used are assumed to exist. {rk} is not to be influenced by the initial value x1.