Dynamics of Two Classes of Continuous-Review Inventory Systems

Abstract

The classes of inventory systems studied in this paper involve random, captive demand and assume continuous review of inventory and replenishment of stock in lots of size Q. The replenishment doctrine is to initiate an order for Q items whenever the sum of stock on hand and stock on order falls below a pre-determined level R. Possible examples are inventories of spare parts for maintenance, of military supplies, or of merchandising stock, so costs of purchase, of storage, and of backordenng are included in the measures of effectiveness. A general relation, between cost of back-orders and minimal-cost probability of stock-out, is derived, similar to that reported earlier [Morse, P. M. 1959. Solution of a class of discrete-time inventory problems. Opns Res. 7 67–78.] for discrete-time systems. For the first class of system studied, demands arrive at random, with a stationary probability distribution of arbitrary form, and all replenishment times are of the same length. For the second class, demand arrivals are Poisson and replenishment times are distributed exponentially. Exact solutions, together with expressions for expected values of operating cost, probability of stock-out, Pout, of stock on hand, etc., are obtained for both classes of system. For the first class, specific formulas are given for the frequently-encountered cases of Poisson and stuttering Poisson demand arrivals. An asymptotic form for these solutions is obtained, valid for both classes of system over the range of parameters of practical interest. In terms of this asymptotic formulation, equations, tables, and graphs are given, from which the re-order level R and replenishment lot size Q can be determined for minimal cost and/or for a pre-set value of Pout, or to satisfy other managerial requirements. For Poisson demand, comparison between the two systems shows that when Q is small, increase in variance of replenishment time makes very little difference in the requirements for R, but when Q is large an increase in variance of replenishment time requires an increase in the value of R, the re-order level, to maintain the optimality of the solution.