Pure and modified base-stock policies for the lost sales inventory system with negligible set-up costs and constant lead times

Abstract

The studied inventory system with continuous review has an easily computed optimal ( S−1, S) policy when unsatisfied demands are backlogged. We assume that unsatisfied demands are lost and then it is also easy to compute the best ( S−1, S) policy. But, as demonstrated by Roger Hill at the ISIR Symposium in 1996, this pure base-stock policy can never be optimal if S⩾2. Our focus is on periodic review. We use Erlang's loss formula to derive approximate expressions for the stockout probability and the average cost. These expressions are used to approximate the average cost and to compute a good base-stock. We formulate and implement a Markov decision model to find the optimal replenishment policy. The model is solved by a policy-iteration algorithm. Because the optimal policy is often rather complicated, we introduce modified base-stock policies. They are specified by a pair ( S, t) where S is the base-stock and t is a lower bound for the number of review periods between review epochs in which placing a replenishment order is permitted. A simple one has S equal to the base-stock computed from Erlang's formula and fixes t as the largest integer which is less than or equal to the ratio of the number of review periods per delivery period and S. Our numerical examples show that the simple modified base-stock policy provides most of the cost reduction which can be obtained by replacing the best pure base-stock policy by the optimal policy.