Exact accounting of inventory costs in stochastic periodic-review models

Most stochastic periodic-review inventory models calculate inventory costs according to the inventory level at the end of the review cycle. However, in most cases, actual inventory costs accrue at different time-points over the course of the cycle, suggesting that the outputs of these models might be suboptimal. This work proposes a means of overcoming this deficiency by attributing inventory costs to their actual timing. Formulas are presented for the expected profit and for its corresponding optimality equation in various multi-period models that follow complete or partial backlogging, with or without spoilage, stationary or varying inventory costs within cycle, and immediate or postponed revenues. We show that the optimality equation of all these model-variants is a kind of "newsvendor formula" with modified demand distribution and cost parameters. A simple approximation formula and bounds on the optimal order quantity, which are based only on the demand distribution over the entire cycle, are presented and combined together into an improved approximation formula. Numerical examples of a Brownian motion demand process demonstrate the benefit of the proposed approximation in comparison to the simple approximation formula.