# Decision Theory Problems

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```					                                      Decision Theory
Problem Set 1

DT1 B.F. Retread, a tire manufacturer, wants to select one of three feasible prototype
designs for a new tire; A, B, C.

Sales Event
Demand          Low            Medium           High
Probability     .30            .50              .20
A               120,000        255,000          390,000
B               130,000        295,000          460,000
C               100,000        300,000          480,000

a.   What is the best optimistic decision?
b.   What is the best cautious decision?
c.   What is the best decision in order to minimize losses?
d.   Determine the maximum expected payoff.
e.   Calculate the highest value you would pay for a highly reliable forecast.
f.   What is the minimum expected loss?

DT2 Venus Flycatcher Company sells exotic plants and is trying to decide which of two
hybrid plants to introduce into their product line.

Demand
Probabilities        .4              .3             .3
Hybrid/Demand        Low             Medium         High
Hybrid 1              -10,000        10,000         30,000
Hybrid 2              -15,000        10,000         35,000

a. If Venus wants to maximize expected profits, which Hybrid should be introduced?
b. What is the most that Venus would pay for a highly reliable demand forecast?

1
DT3 Management is uncomfortable stating probabilities for the states of nature for a
trade bill passing. Depending on its probability they will invest in one of three countries;
A, B, or C.
Let p denote the probability of the bill passing.
a. What does graphical sensitivity analysis tell management about location preferences?
b. After further review, management estimated the probability of the trade bill passing at
0.65. Based on the results in part a, which country should be selected and what is the
expected value associated with that decision?

Passes           Fails
A            320            160
B            260            210
C            370            0

DT4 The economy will either be in a state of prosperity or recession and the probability
of either is uncertain. A company wishes to invest in one of three ventures. Provide
some guidelines as to the probability of prosperity and each investment choice based on
the payoff table (number are stated in the millions) below.

Prosperity     Recession
A            5              -2
B            2              1
C            3              -1

DT5 The payoffs for venture C have been revised to 4 if a state of prosperity and 0 if a
recession. Provide some guidelines as to the probability of prosperity and each
investment choice based on the payoff table and also indicate at what probability of
prosperity you would be indifferent to investing in venture B versus venture C.

2
Decision Theory Solutions

DT1
Sales per 1000
Prob.        .30      .50       .20
Low      Med.      High         Maxi    Maxi      Minimax     Exp.        Exp.
max     min       Regret      Payoff      Losses
A            120       255        390        390     120       90          241,500     43,500
B            130       295        460        460     130       20          278,500     6,500
C            100       300        480        480     100       30          276,000     9,000
Max          .3(130)   .5(300)    .2(480)                                  EVwPI
Pay          39        150        96                                       285,000

Regrets 10             45         90
0              5          20
30             0          0

a.    Decision C
b.    Decision B
c.    Decision B
d.    \$278,500 Maximum expected payoff
e.    EVPI: 285,000-278,500 = \$6,500
f.    \$6,500 Minimum expected losses are always equal to EVPI

Loss Table          Sales Event
Low             Medium       High           Exp. Losses
.30             .50          .20
A                   10,000          45,000       90,000         43,500
B                   0               5,000        20,000         6,500
C                   30,000          0            0              9,000

DT2

Demand
Probabilities     .40             .30         .30
Plants            Low             Medium      High         Expected
Payoff
Hybrid 1           -10,000        10,000      30,000        8,000
Hybrid 2           -15,000         10,000     35,000        7,500
Max Payoff        .4(-10000)      .3(10000)   .3(35000)
-4,000          3,000      10,500        9,500 EVwPI

a. Hybrid 1
b. 9,500-8,000 = \$1,500 EVPI

3
DT3
EV (A) = p (320) + (1 - p)(160) = 160p +160
EV (B) = p (260) + (1 - p)(210) = 50p + 210
EV (C) = p (370) + (1 - p)(0) = 370p
C

370                                                 A

B

210

0                            1
.4545   .7619

To calculate each p, set each two lines that intersect equal to one another.
A and B
160p +160 = 210 + 50p
110p = 50
P=50/110 = .4545

A and C
160p +160 = 370p
160 = 210p
p =160/210 = .7619

a. What does graphical sensitivity analysis tell management about location
preferences?
If p < 0.4545 then B is the best decision.
If p > 0.4545 or p < 0.7619 then A is best.
If p > 0.7619 the C is the best decision.

b. After further review, management estimated the probability of the trade bill
passing at 0.65. Based on the results in part a, which country should be selected
and what is the expected value associated with that decision?
Select Location A since 0.65 > 0.4545 and < .7619
EV (A) = 0.65 (320) + 0.35 (160) = 264

4
DT4 The economy will either be in a state of prosperity or recession and the probability
of either is uncertain. A company wishes to invest in one of three ventures. Provide
some guidelines as to the probability of prosperity and each investment choice based on
the payoff table (number are stated in the millions) below.

Prosperity      Recession
A                    5               -2
B                    2               1
C                    3               -1

A

5
C

B

0                            1
.500

The only two lines that intersect at the greatest expected value are A and B.
C at all probabilities will have a lower expected value; hence, will never be a good
decision.
A and B
7p -2 = 1p +1
6p = 3
p =3/6 = .500

Below 50% B is the best decision and above 50% A is the best decision.
At 50% we are indifferent to the decision A or B.

5
DT5 The payoffs for venture C have been revised to 4 if a state of prosperity and 0 if a
recession. Provide some guidelines as to the probability of prosperity and each
investment choice based on the payoff table and also indicate at what probability of
prosperity you would be indifferent to investing in venture B versus venture C.

Prosperity        Recession
A                       5                 -2
B                       2                 1
C                       4                 0

5
A

C

B

0                             1

.33        .67

B and C
1p +1 = 4p
3p = 1
p =1/3 = .33
A and C
7p -2 = 4p
3p = 2
p =2/3 = .67

If p < 0.33 then B is the best decision.
If p > 0.33 or p < 0.67 then C is best.
If p > 0.67 then A is the best decision.

If prosperity had a probability of 33% we would be indifferent to venture B or C
because their expected values would be equal.

6

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