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438 CHAPTER 10 INVENTORY MODELS

where

Q ¼ the order quantity
M ¼ the replenishment level
H ¼ the inventory on hand at the review period

Because the demand is probabilistic, the inventory on hand at the review period, H,
will vary. Thus, the order quantity that must be sufficient to bring the inventory
position back to its maximum or replenishment level M can be expected to vary each
period. For example, if the replenishment level for a particular product is 50 units,
and the inventory on hand at the review period is H ¼ 12 units, an order of Q ¼ M
H ¼ 50 12 ¼ 38 units should be made. Thus, under the periodic review model,
enough units are ordered each review period to bring the inventory position back up
to the replenishment level.
A typical inventory pattern for a periodic review system with probabilistic demand
is shown in Figure 10.13. Note that the time between periodic reviews is predeter-
mined and fixed. The order quantity Q at each review period can vary and is shown
to be the difference between the replenishment level and the inventory on hand.
Finally, as with other probabilistic models, an unusually high demand can result in
an occasional stock-out.
The decision variable in the periodic review model is the replenishment level
M.Todetermine M, we could begin by developing a total cost model, including
holding, ordering and stock-out costs. Instead, we describe an approach that
is oftenusedinpractice. In this approach, the objectiveistodetermine a
replenishment level that will meet a desired performance level, such as a rea-
sonably low probability of stock-out or a reasonably low number of stock-outs
per year.
In the Dollar Discounts problem, we assume that management’s objective is to
determine the replenishment level with only a 1 per cent chance of a stock-out.

Figure 10.13 Inventory Pattern for Periodic Review Model with Probabilistic Demand

M Replenishment
Level

Q

Q
Inventory Q

Lead Time

Review Review
Period Period
Stock-Out
Time

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