Inventory Cost Rate Functions with Nonlinear Shortage Costs

This article considers five cost-rate models for inventory control, each summarizing the expected holding and shortage costs per period as a function of the inventory position. All models have linear holding costs and shortage cost coefficients of dimension [ $/unit/period], [ $/unit], and [ $/period]. The latter two coeficients may be the shadow costs of a fill-rate and a ready-rate service constraint, respectively. One of the cost-rate models is a new suggestion, intended to facilitate modeling of periodic-review inventory systems.If-and-only-if conditions on the demand process are presented for which the cost rate is quasi-convex in the inventory position. The typical sufficient condition requires that the cumulative demand distribution be logconcave, a condition that is met by most demand distributions commonly used in the inventory literature.The results simplify optimization and extend the known optimality of ( S,s) and ( nQ,r) policies to cost structures common in applications and to the presence of typical service constraints. As a prerequisite for the study, a series of new monotonicity results are derived for compound renewal processes.