A Note on the Convexity of Service-Level Measures of the (r,q) System

A well-known method of inventory control is the reorder-point/order-quantity system, or (r, q) system, where an order of constant size q is placed whenever the inventory position (that is, stock on hand plus on order minus backorders) drops to a fixed reorder point r. Zipkin (1986) proves that, when stockouts are backordered, the average outstanding backorders, denoted by B(q, r), is a convex function of the two control variables q and r. He also shows (under Poisson demand and an additional mild assumption) that the average stockouts per unit time, denoted by A(q, r), is also convex in q and r. His proof is based on explicitly expressing B(q, r) and A(q, r) in terms of integrations involving probability distribution function of demand in a lead time, taking second partial derivatives and then checking for the nonnegative definiteness of the Hessian matrix. In this note, we present a much simpler proof that has a strong intuitive appeal. Let D and I be two random variables denoting demand in a lead time and the inventory position, respectively. Then the average outstanding backorders B(q, r) can by expressed as