Page 470 -

450 CHAPTER 10 INVENTORY MODELS

Development of the Optimal Lot Size (Q*) Formula for the
Appendix 10.2
Production Lot Size Model

Given Equation (10.15) as the total annual cost for the production lot size model,

1 D D
TC ¼ 1 QC h þ C o (10:15)
2 P 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 D
¼ 1 C h þ C o ¼ 0
dQ 2 P Q 2
Solving for Q*, we have:

1 D D
1 C h ¼ C o
2 P Q 2

D
2
1 C h Q ¼ 2DC o
P
2
Q ¼ 2DC o
ð1 D=PÞC h
Hence,
s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

Q ¼ 2DC o (10:16)
ð1 D=PÞC h

The second derivative is:
2
d TC 2DC o
¼
dQ 2 Q 3
Because the value of the second derivative is greater than zero, Q*from equation
(10.16) is a minimum cost solution.

Copyright 2014 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has
deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.