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DEVELOPMENT OF THE OPTIMAL ORDER QUANTITY (Q) FORMULA FOR THE EOQ MODEL 449

Managerial Report unit in order to cover the daily absences? These
extra daily needs will be filled by the additional
Develop a report that will enable the Fire Chief to
firefighters and, when necessary, the more
determine the necessary numbers for the Fire
expensive use of overtime by off-duty firefighters.
Department. Include, at a minimum, the following
3 On a given day, what is the probability that Kelley-
items in your report.
day firefighters will be called in to work overtime?
1 Assuming no daily absences and taking into
4 Based on the three-unit organization, how
account the need to staff Kelley days, determine the
many firefighters should be assigned to each
base number of firefighters needed by each unit.
unit? What is the total number of full-time
2 Using a minimum cost criterion, how many firefighters required for the River City Fire
additional firefighters should be added to each Department?

Development of the Optimal Order Quantity (Q) Formula for
Appendix 10.1
the EOQ Model

You will need to be familiar with differential calculus for this Appendix 10.1 and
10.2. Given Equation (10.4) as the total annual cost for the EOQ model,

1 D
TC ¼ QC h þ C o (10:4)
2 Q

we can find the order quantity Q that minimizes the total cost by setting the
derivative, dTC/dQ, equal to zero and solving for Q*.
dTC 1 D
¼ C h C o ¼ 0
dQ 2 Q 2
1 D
C h ¼ C o
2 Q 2
2
C h Q ¼ 2DC o
2
Q ¼ 2DC o
C h
Hence,

s ffiffiffiffiffiffiffiffiffiffiffiffi

Q ¼ 2DC o (10:5)
C h

The second derivative is:

2
d TC 2D
¼ C o
dQ 2 Q 3
Because the value of the second derivative is greater than zero, Q*from equation
(10.5) is the minimum cost solution.

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