On Convexity of Service-Level Measures of the Discrete (r,Q) Inventory System

In this paper, we first give the definition for convexity of real-valued function defined on the integer lattice and obtain a sufficient condition to convexity of real-value function defined on the integer lattice and its some properties. Furthermore, by using the results obtained in this paper, we directly prove, under the condition that demands are discrete, that the long-run average cost of outstanding backorders and the long-run average cost of inventories of the (r, Q) inventory system is a P-jointly-convex in (r, Q), which answers the long-standing conjecture about convexity of two often-used inventory service-level measures under the case that demands are discrete.