Deterministic Inventory Lot Size Models—A General Root Law

Abstract

The article presents a continuous time inventory model with known time varying demand. The relevant costs are the carrying and replenishment costs; backlogs are prohibited, replenishments are instantaneous, and the planning horizon is finite and known, The problem is to find the optimal schedule of replenishments, i.e., their number and the schedule of time intervals between consecutive orders. A complete solution is given for demand functions of the type b(t) = ktr, k > 0, r > −2, where t stands for time, and the asymptotic properties of the solution when the planning horizon tends to infinity are investigated. For r = 0 the solution reduces to the results obtained by Carr and Howe for the constant demand case with finite horizons, and to the classical “square root law” for infinite horizons. For r = 1 it yields the “cubic root law” given by Resh, Friedman, and Barbosa for time proportional demand rate. More generally, for r integer the solution can be expressed in an “(r + 2) root law.”