New bounds and heuristics for (Q, r) policies

To clarify the impact of demand variability on single item stochastic inventory systems with setup costs, we subsume the distributional information of the lead time demand into its mean and variance and solve the resulting problem against the worst possible distribution in this class. For (Q, r) policies we obtain in closed form a distribution-free solution for Q and r, and upper bounds on the optimal long run average cost and on the optimal batch size. As a byproduct we develop a robust, distribution-free, batch size heuristic that causes a relative cost increase of no more than 6.07%. In addition, when the newsvendor cost is known, we obtain sharper lower and upper bounds on the long run average cost. These bounds clarify, in an exceedingly simple way, the cost impact of fixed setup costs, demand variability, and constraints on the batch size. We illustrate our bounds and heuristics on problems with Poisson and Compound Poisson demands.