Computing a Stationary Base-Stock Policy for a Finite Horizon Stochastic Inventory Problem with Non-linear Shortage Costs

Abstract

We consider a periodic-review stochastic inventory problem in which demands for a single product in each of a finite number of periods are independent and identically distributed random variables. We analyze the case where shortages (stockouts) are penalized via fixed and proportional costs simultaneously. For this problem, due to the finiteness of the planning horizon and non-linearity of the shortage costs, computing the optimal inventory policy requires a substantial effort as noted in the previous literature. Hence, our paper is aimed at reducing this computational burden. As a resolution, we propose to compute “the best stationary policy.” To this end, we restrict our attention to the class of stationary base-stock policies, and show that the multi-period, stochastic, dynamic problem at hand can be reduced to a deterministic, static equivalent. Using this important result, we introduce a model for computing an optimal stationary base-stock policy for the finite horizon problem under consideration. Fundamental analytic conclusions, some numerical examples, and related research findings are also discussed.

Key Words:

• Stochastic inventory problem
• Periodic review inventory system

Acknowledgment

This research was supported in part by NSF Grant CAREER/ DMI-0093654 and Natural Sciences and Engineering Research Council of Canada under Grant A5872.

Notes

^aWe note that the results obtained in this paper are applicable for the case where there is a delay of, say, ν periods between ordering goods and receiving them. In this case, z _n is not available to satisfy demand until period ν + n. The formulation given here can be modified in a straightforward fashion to handle the positive lead-time case as described in Ref.13, pp. 75–79.