Convergence Results and Approximations for Optimal (s, S) Policies

In this paper we consider the dynamic inventory model with a discrete demand and no discounting. We verify a conjecture of Iglehart about the asymptotic behaviour of the minimal total expected cost. To do this, we give for the denumerable state dynamic programming model a number of conditions under which the minimal total expected cost for the n-stage model minus n times the minimal average cost has a finite limit as n → ∞. For a positive demand distribution we establish a turnpike theorem which states that for all n sufficiently large the optimal n-stage policy (s_n, S_n) is average cost optimal. Further, we show that the computation of the (s_n, S_n) policies supplies monotonic upper and lower bounds on the minimal average cost. Also, the average cost of the (s_n, S_n) policy lies between the corresponding bounds. For a positive demand distribution these bounds converge as n → ∞ to the minimal average cost.