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Expensive
20,000
-5,000
Medium
10,000
2,000
Inexpensive
5,000
11,000
a.
Determine the expected value of each alternative and indicate which decision alternative is the
best.
b.
Determine the expected value with perfect information about the states of nature.
c.
Determine the expected value of perfect information.
8. An investor has a choice between four investments. The profitability of the investments depends upon
the market. The payoff table is given below for different market conditions.
States of Nature
Investments
Market
Increases
Market
Stays the Same
Market
Decreases
A
100,000
50,000
-40,000
B
70,000
30,000
-10,000
C
40,000
15,000
10,000
D
20,000
20,000
20,000
a.
A market economist has stated that there is a 25% chance that the market will stay the same, a
35% chance that the market will decrease, and a 40% chance that the market will increase.
Compute the expected value for each investment. Which investment is the best?
b.
Compute the expected value of perfect information.
9. A fashion designer wants to produce a new line of clothes. In the production of the clothes, expensive,
medium-priced, or inexpensive materials can be used. The profit associated with each type of material
depends upon economic conditions next year. Below you are given the payoff table.
States of Nature
Decisions
Economy
Improves
Economy
Stays the Same
Economy
Gets Worse
Expensive
80,000
40,000
10,000
Medium
40,000
60,000
70,000
Inexpensive
10,000
30,000
60,000
An economist believes that the probability that the economy will improve is 20%, the probability that
the economy will stay the same is 70%, and the probability that the economy will get worse is 10%.
a.
Compute the expected value for each investment. Which investment is the best?
b.
Compute the expected value of perfect information.
10. An automobile manufacturer must make an immediate decision on the car size which should account
for the majority of the firm's production two years from now. The firm perceives three possible states
of nature at that time: S1, gasoline will be rationed; S2, gasoline will be readily available at close to
current prices; and S3, gasoline will be readily available, but at much higher prices. The firm has
determined the following profit payoff table (in $l,000s).
States of Nature
Decision
Alternatives
S1
Gas Rationed
S2
Gas Readily Available at
Close to Current Prices
S3
Gas Readily Available
At Much Higher Prices
Make Mostly
Large Cars
-200
1,900
200
Make Mostly
Medium Cars
400
1,400
700
Make Mostly
Small Cars
900
800
1,400
a.
An economist at the auto company has advised the firm that the probabilities of the states of
nature are P(S1) = .2, P(S2) = .5, and P(S3) = .3. Find the expected value for the three decisions.
b.
Which decision should be chosen under the expected value criterion?
c.
Determine the expected value of perfect information.
11. Below you are given a payoff table involving two states of nature and two decision alternatives.
Decision
States of Nature
Alternatives
S1
S2
d1
20,000
60,000
d2
50,000
10,000
The probability of the occurrence of S1 is 0.3.
a.
Compute the expected value for each decision. Which decision is the best?
b.
Compute the expected value of perfect information.
12. The owner of a new gourmet kitchenware shop wishes to determine how many days and evenings to
keep the shop open. The various payoffs (in $ 1,000s) are indicated in the table below.
States of Nature
Decision
Alternatives
S1
High Demand
S2
Average Demand
S3
Low Demand
d1
30
20
10
d2
30
10
10
d3
20
20
40
d4
20
25
45
Assume the probabilities of the three states of nature are P(S1) = 0.60, P(S2) = 0.30, and P(S3) = 0.1.
a.
Determine the expected value of each alternative and indicate which decision alternative is the
best.
b.
Determine the expected value with perfect information about the states of nature.
c.
Determine the expected value of perfect information.
13. The Video Game Supply Company (VGS) is deciding whether to set production next year at 2,000,
2,500, or 3,000 games. Demand could be low, medium, or high. Using historical data, VGS estimates
the probabilities as 0.4 for low demand, 0.3 for medium demand, and 0.3 for high demand. The
following profit payoff table (in $100s) has been developed:
Production
Quantity Demanded
Target
Low
Medium
High
2,000
1,000
1,200
1,400
2,500
800
1,500
1,300
3,000
600
1,700
1,400
a.
Determine the expected value of each alternative and indicate what should be the production
target.
b.
Determine the expected value with perfect information about the states of nature.
c.
Determine the expected value of perfect information.
14. Shannon Lipscomb & Associates (SLA) are producers of a new brand of personal computers. SLA is
considering employing a market research firm to supply indicator information related to the demand
for their computers. The information would consist of forecasts of light demand (I1) or heavy demand
(I2) for SLA's computers. The following conditional probabilities reflect the accuracy of the market
research firm's forecasts:
P(I1 | S1) = 0.8
P(I1 | S2) = 0.6
P(I1 | S3) = 0.3
P(I2 | S1) = 0.2
P(I2 | S2) = 0.4
P(I2 | S3) = 0.7
a.
Compute the posterior probabilities.
b.
What decision should be taken if the market research firm forecasts light demand (I1)? Heavy
demand (I2)?
c.
Calculate the expected value of sample information.
d.
Compute the expected value of perfect information.
15. Assume you have a sum of money available which you would like to invest in one of the two available
investment plans: Stocks or bonds. The conditional payoffs of each plan under two possible economic
conditions are as follows:
States of Nature
Decision
Alternatives
Economic
Condition I
Economic
Condition II
Stocks
$40,000
$ -8,000
Bonds
$ 8,000
$12,000
a.
If the probability of Economic Condition I occurring is 0.8, where should you invest your
money? Use the expected value criterion and show your complete work.
b.
Compute the expected value of perfect information (EVPI).
c.
What kind of probabilities of Economic Conditions I and II should there be before you would
be indifferent between investing in stocks and bonds? (i.e., compute the probabilities for which
you will be indifferent between investing in stocks or bonds.)
16. Consider the following profit payoff table.
Decision
States of Nature
Alternatives
S1
S2
d1
8
6
d2
15
4
What should the probabilities of S1 and S2 be so that the expected values of the two decision
alternatives equal one another?
17. Super Cola is considering the introduction of a root beer drink. The company feels that the
probability of the new drink being successful is .6. The payoff table is as follows.
Decision
Alternatives
States of Nature
Success (S1)
Failure (S2)
Introduce
$250,000
$-300,000
Do Not Introduce
$-50,000
$-20,000
Super Cola has a choice of two research firms to obtain information for this new product. Stanton
Marketing has market indicators I1 and I2 for which P(I1|S1) = .7 and P(I1|S2) = .4. New World
Marketing has indicators J1 and J2 for which P(J1|S1) = .6 and P(J1|S2) = .3. (Be sure to compute
probabilities to the third decimal place.)
a.
What is the optimal decision if neither research firm is used?
b.
Compute the expected value of perfect information (EVPI).
c.
Find the EVSIs for Stanton and New World.
d.
If both research firms charge $5,000, which firm should be hired?
e.
If Stanton charges $10,000 and New World charges $5,000, which firm should Super
Cola hire?
18. Assume you have a sum of money available that you would like to invest in one of the three available
investment plans: stocks, bonds, or money market. The conditional payoffs of each plan under two
possible economic conditions are shown below. The probability of the occurrence of economic
condition I is 0.28.
Decision Alternative
Economic Condition I
Economic Condition II
Stocks
1000
3000
Bonds
2500
2000
Money Market
1800
4000
a.
Compute the expected value of the three investment options. Which investment option would
you select, based on the expected values?
b.
Compute the expected value with perfect information (i.e., expected value under certainty).
c.
Compute the expected value of perfect information (EVPI).
19. The following payoff table shows profits for two decision alternatives under three different states of
nature. It is known that the probability of the occurrence of state of nature 1 is 0.1.
Decision Alternative
state of nature 1
state of nature 2
state of nature 3
Decision 1
10
13
9
Decision 2
15
9
10
a.
What should the probabilities of states of nature 2 and 3 be so that the expected values of the
two decision alternatives equal one another?
b.
Determine the expected values.
20. Michael, Nancy, & Associates (MNA) produce color printers. The demand for their printers could be
light, medium, or high with the following probabilities.
Light Demand
Medium Demand
High Demand
Probability
0.4
0.3
0.3
The company has three production alternatives for the coming period. The payoffs (in millions of
dollars) associated with the three alternatives are shown below.
Light Demand
Medium Demand
High Demand
Alternative 1
18
28
20
Alternative 2
25
17
-5
Alternative 3
3
40
16
a.
Compute the expected value of the three alternatives. Which alternative would you select,
based on the expected values?
b.
Compute the expected value with perfect information (i.e., expected value under certainty).
c.
Compute the expected value of perfect information (EVPI).
21. You are given a decision situation with three possible states of nature S1, S2, and S3. The prior
probabilities of the three states are 0.20, 0.45, and 0.35. With sample information I, you are provided
with the following information.
P(I1S1) = 0.85
P(I1S2) = 0.70
P(I1S3) = 0.40
a.
Compute P(I).
b.
Compute the revised probabilities of P(S1I), P(S2I), and P(S3I).
22. Assume you are faced with the following decision alternatives and two states of nature. The payoff
table is shown below.
Decision Alternatives
State of Nature 1
State of Nature 2
Decision 1
26
32
Decision 2
40
18
Decision 3
28
25
Probability
0.42
0.58
a.
Determine the expected value of each alternative.
b.
Which decision is the optimal decision?
c.
Determine the expected value with perfect information.
d.
Compute the expected value of perfect information.
23.
An appliance dealer must decide how many (if any) new microwave ovens to order for next month.
The ovens cost $220 and sell for $300. Because the oven company is coming out with a new product
line in two months, any ovens not sold next month will have to be sold at the dealer's half price
clearance sale.
Additionally, the appliance dealer feels he suffers a loss of $25 for every oven demanded when he is
out of stock. On the basis of past months' sales data, the dealer estimates the probabilities of monthly
demand (D) for 0, 1, 2, or 3 ovens to be .3, .4, .2, and .1, respectively.
a. Construct a payoff table for this problem.
b. Determine the optimal decision using the expected value approach.
24. Cashman Co. will be leasing a new copier and is considering four plans. The company has
determined it will make 12,600, 14,400, 16,200, 18,000, 19,800, or 21,600 copies per month with
probabilities of .05, .10, .15, .25, .25, and .20 respectively.
Plan
Monthly Lease Cost
Unit Copy Cost
I
$100
$.020 for the first 10,000 copies; $.016 thereafter
II
$200
$.012 for all copies
III
$150
first 5,000 free; $.022 thereafter
IV
$300
$.005 for all copies
a. Construct a monthly payoff table for Cashman in terms of costs.
b. What is the optimal plan using the expected value approach? (Hint: This is a cost minimization
problem.)
25. A maintenance department replaces a malfunctioning machine with a standby machine if one is
available; otherwise, they repair the broken machine as soon as possible. When a standby machine is
available, production down time is greatly reduced. The department has reviewed its historical
maintenance records on machine breakdowns and found this pattern for the past four weeks:
Number of
Breakdowns
Occurrences
4
50
3
100
2
150
1
200
If a standby machine is not available when a breakdown occurs, the estimated cost is $400 due to lost
production time, overtime usage on the other machines, and emergency repair procedures. On the other
hand, weekly cost for machines not in use is estimated to be $200 due to storage and special handling
expenses. The department manager wants to use a payoff table to determine how many standby ma-
chines they should maintain.
a. Construct a table showing the cost associated with each decision alternative (number of computers
stocked) and state of nature (number of computers needed) combination.
b. Compute the probability of each state of nature.
c. How many standby computers should be stocked in order to minimize their expected costs?
26. An automobile manufacturer stocks an electric motor unit that is used in many of their production line
robots. As this is the major item to fail in a robot, it is important that enough of them are kept in stor-
age. Since these precision motors are very expensive (over $10,000 each) it is also very important not
to keep too many on the shelf. Long costs are $200 and short costs are $325 per unit. Data on monthly
breakdown experience is as follows:
Motor Units Demanded
Occurrences
5
10
10
15
15
18
20
9
a. Construct a table showing the cost associated with each decision alternative (number of motors
stocked) and state of nature (number of motors needed) combination.
b. Compute the probability of each state of nature.
c. How many standby motors should be stocked in order to minimize their expected costs?
