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 (sn,Sn) is average cost optimal. Further, we show that the computation of the (sn,Sn) policies supplies monotonic upper and lower bounds on the minimal average cost. Also, the average cost of the (sn,Sn) policy lies between the corresponding bounds. For a positive demand distribution these bounds converge as n → ∞ to the minimal average cost.