Stochastic Inventory Models with Continuous and Poisson Demands and Discounted and Average Costs

It has been more than ninety years since the classical square-root EOQ formula was given by Harris (1913). Yet there is no continuous-review stochastic inventory model published in the literature with a general enough demand, whose optimal policy would reduce to the square-root formula in the absence of the stochastic components of the underlying demand. Why? In this paper, we surmise the reasons , develop a model with continuous and Poisson demands for the first time, and prove the optimality of an (s, S)-policy. We also verify that the policy reduces to the EOQ formula, as it must, when the intensity of the Poisson process goes to zero. In the process, we develop a new, unified approach of dealing with both the average cost and the discounted cost criteria. We introduce new average discounted-cost formulas along with intuitive interpretations. We do not require the surplus cost function to be convex or quasi-convex as has been assumed in the literature. We show that while the optimal ordering level is unique, there may be more than one optimal order-up-to levels.