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INVENTORY MODEL WITH PLANNED SHORTAGES 423

So, the average inventory is expressed in terms of two inventory decisions: how much
we will order (Q) and the maximum number of backorders (S).
The formula for the annual number of orders placed using this model is identical
to that for the EOQ model. With D representing the annual demand, we have:

D
Annual number of orders ¼ (10:21)
Q

The next step is to develop an expression for the average backorder level.
Because we know the maximum for backorders is S, we can use the same logic we
used to establish average inventory in finding the average number of backorders. We
have an average number of backorders during the period t 2 of ½ the maximum
number of backorders or ½S. We do not have any backorders during the t 1 days we
have inventory, therefore we can calculate the average backorders in a manner
similar to Equation (10.17). Using this approach, we have:

0t 1 þðS=2Þt 2 ðS=2Þt 2
Average backorders ¼ ¼ (10:22)
T T

When we let the maximum number of backorders reach an amount S at a daily rate
of d, the length of the backorder portion of the inventory cycle is:

S
t 2 ¼ (10:23)
d

Using Equations (10.23) and (10.19) in Equation (10.22), we have:

ðS=2ÞðS=dÞ S 2
Average backorders ¼ ¼ (10:24)
Q=d 2Q

Let

C h ¼ cost to hold one unit in inventory for one year
C o ¼ cost per order
C b ¼ cost to maintain one unit on backorder for one year

The total annual cost (TC) for the inventory model with backorders becomes:

ðQ SÞ 2 D S 2
TC ¼ C h þ C o þ C b (10:25)
2Q Q 2Q

Given C h , C o and C b and the annual demand D, differential calculus can be used
to show that the minimum cost values for the order quantity Q* and the planned
backorders S* are as follows:

s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Q ¼ 2DC o C h þ C b (10:26)
C h C b

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