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ORDER-QUANTITY, REORDER POINT MODEL WITH PROBABILISTIC DEMAND 433

expensive cost of overestimating demand and larger order quantity provides a lower
having a surplus will tend to be avoided. probability of a stock-out in an attempt to avoid
Finally, whenever c u > c o ,alarger order the more expensive cost of underestimating
quantity will be recommended. In this case, the demand and experiencing a stock-out.

Order-Quantity, Reorder Point Model with Probabilistic
10.7
Demand

In the previous section we considered a single-period inventory model with
probabilistic demand. In this section we extend our discussion to a multiperiod
order-quantity, reorder point inventory model with probabilistic demand. In the
The inventory model in multiperiod model, the inventory system operates continuously with many
this section is based on repeating periods or cycles; inventory can be carried from one period to the
the assumptions of the
EOQ model shown in next. Whenever the inventory position reaches the reorder point, an order for Q
Table 10.3 with the units is placed. Because demand is probabilistic, the time the reorder point will
exception that demand is be reached, the time between orders and the time the order of Q units will arrive
probabilistic rather than in inventory cannot be determined in advance.
deterministic. With
probabilistic demand, The inventory pattern for the order-quantity, reorder point model with proba-
occasional shortages bilistic demand will have the general appearance shown in Figure 10.10. Note that
may occur. the increases or jumps in the inventory occur whenever an order of Q units arrives.
The inventory decreases at a nonconstant rate based on the probabilistic demand. A
new order is placed whenever the reorder point is reached. At times, the order
quantity of Q units will arrive before inventory reaches zero. However, at other
times, higher demand will cause a stock-out before a new order is received. As with
other order-quantity, reorder point models, the manager must determine the order
quantity Q and the reorder point r for the inventory system.
The exact mathematical formulation of an order-quantity, reorder point inventory
model with probabilistic demand is beyond the scope of this text. However, we
present a procedure that can be used to obtain good, workable order quantity and

Figure 10.10 Inventory Pattern for an Order-Quantity, Reorder Point Model with
Probabilistic Demand

Order Quantity Probabilistic Demand
of Size Q Reduces Inventory
Arrives

Q
Inventory Q

Reorder
Order Order Order Stock-Out Point
Placed Placed Placed
0
Time

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