# MSCI432 Chapter Notes - Chapter 5: Spyker Cars, Probability Distribution, Soup Kitchen

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16 Oct 2011

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Chapter 05 - Inventory Control Subject to Uncertain Demand

5-1

Inventory Control Subject to Uncertain Demand

Solutions To Problems From Chapter 5

5.7 A newsboy keeps careful records of the number of papers he sells each day and the

various costs that are relevant to his decision regarding the optimal number of newspapers

to purchase. For what reason might his results be inaccurate? What would he need to do

in order to accurately measure the daily demand for newspapers?

5.7 If he only keeps track of the number of sales, he has no way to accurately estimate the

demand since demand = sales + lost sales. He would need some way to gauge the lost

sales. One method would be to increase his supply for a period of time so that he would

be able to meet all demand.

5.8 Billy’s Bakery bakes fresh bagels each morning. The daily demand for bagels is a random

variable with a distribution estimated from prior experience given by

Number of bagels

sold in one day Probability

0 0.5

5 .1

10 .1

15 .2

20 .25

25 .15

30 .1

35 .05

The bagels cost Billy’s 8 cents to make, and they are sold for 35 cents each. Bagels unsold

at the end of the day are purchase by a nearby charity soup kitchen for 3 cents each.

a. Based on the given discrete distribution, how many bagels should Billy’s bake at the start

of each day? (Your answer should be a multiple of 5)

b. If you were to approximate the discrete distribution with a normal distribution, would you

expect the resulting solution to be close to the answer that you obtained in part (a)? Why

or why not?

c. Determine the optimal number of bagels to bake each day using a normal approximation.

(Hint: you must compute the mean µ and the variance σ

2

of the demand from the given

discrete distribution)

5.8 a) c

0

= .08 - .03 = .05

c

u

= .35 - .08 = .27

Critical ratio =

.

27

.

05

+

.

27

= .84375

From the given distribution, we have:

Chapter 05 - Inventory Control Subject to Uncertain Demand

5-2

Q f(Q) F(Q)

0 .05 .05

5 .10 .15

10 .10 .25

15 .20 .45

20 .25 .70

< - - - - .84375

25 .15 .85

30 .10 .95

35 .05 1.00

Since the critical ratio falls between 20 and 25 the optimal is Q = 25 bagels.

b) The answers should be close since the given distribution appears to be close to the

normal.

c) µ =

∑

xf(x) = (0)(.05) + (5)(.10) +...+(35)(.05) = 18

σ

2

=

∑

x

2

f(x) - µ

2

= 402.5 - (18)

2

= 78.5

σ =

(2)(32)(1032)

.36

= 8.86

The z value corresponding to a critical ratio of .84375 is 1.01.

Hence,

Q* = σz + µ = (8.86)(1.01) + 18 = 26.95 ~ 27.

5.10 Happy Henry’s car dealership sells Spyker cars. Once every three months, a shipment of

the cars is made. Emergency shipments can be made between these three-month intervals

to resupply the cars when inventory falls short of demand. The emergency shipments

require two weeks and buyers are willing to wait this long for the cars, but will generally

go elsewhere before the next three-month shipment is due.

From experience, it appears that the demand for the cares over a three-month interval is

normally distributed with a mean of 60 and a variance of 36. The cost of holding a car for

one year is $500. Emergency shipments cost $250 per car over and above normal shipping

costs.

a. How many cars should Happy Henry’s be purchasing every three months?

b. Repeat the calculations, assuming that excess demands are back-ordered from one three-

month period to the next. Assume a loss-of-goodwill cost of $100 for customers having to

wait until the next three-month period and a cost of $50 per customer for bookkeeping

expenses.

c. Repeat calculations, assuming that when Happy Henry’s is out of stock, the customer will

purchase the car elsewhere. In this case, assume that the cars cost Henry an average of

$10,000 and sell for an average of $13,500. Ignore loss-of-goodwill costs for this