Abstract:
In a discrete review inventory process, when the demand
forms a stochastically convergent sequence of random variables,
it seems reasonable that the optimal stationary (s, S) inventory
policy will be a function of the limiting demand and cost structure
only. The intent of this paper is to provide a rigorous justification
of this conjecture under suitable restrictions. Assuming linear
costs and integer valued demand, the problem is essentially reduced
to showing the existence and finding an expression for the stationary
inventory distribution.
The stationary inventory distribution, with an (s,S) policy
in effect, is derived by applying renewal theory to the inventory
process with renewals defined as those periods in which a positive
amount is ordered. For this purpose a version of the key renewal
theorem for stochastically convergent sequences is proved and formulated in terms of integrals. The integral formulation is used
to derive the stationary distribution of the excess variable and the
stationary probability that a renewal will occur, or equivalently,
that an order will be placed.
