Solutions of a Class of Discrete-Time Inventory Problems

In solving problems of inventory control, where fluctuation in demand and/or supply must be taken into account, two classes of operational procedure lead to two different techniques for solution. If operating decisions (orders for replacement, for example) are made whenever the inventory reaches certain preassigned levels, the queuing theory model may be used. But if operating decisions are made at the ends of finite periods of time, on the basis of inventory counts made then, the techniques of the theory of Markov processes are usually applicable. Several general situations of the latter class are analyzed in the present paper those where the size of the replenishment order is equal to the number of demands arriving during the last period, or where the order is “quantized” in multiples of some lot-size q , those where the replenishment order is delivered within the next period, and those where sometimes delivery is delayed one or more periods. Techniques of solution are displayed for finding the steady-state probabilities that inventories of various sizes are present at the beginning of a period and various measures of effectiveness of the operation are computed from these probabilities. Finally, it is shown how asymptotic formulas for large demand rates can be obtained for these measures, in terms of the standard deviation of the interarrival times between demands.