Continuous accounting of inventory costs with Brownian-motion and Poisson demand processes

We investigate a periodic review inventory system with demand that is either a Brownian motion or a Poisson process and with costs that accrue in continuous time over the period (and not at the end of the period). We find explicit expressions for the expected inventory costs and for the optimality equation. On the basis of these expressions, a simple search can be designed to obtain the optimal order-up-to level. In addition, we develop a new distribution function with a closed-form quantile function to approximate this level. The approximation can be adjusted to consider lead time. Numerical examples show that the suggested approximation produces substantially lower deviations from the optimal solution compared with approximations recommended in the literature.