Inventory Cost Rate Functions with Nonlinear Shortage Costs

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Abstract
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.