Decision Making Under Uncertainty
Conditional value (or) Conditional Profit TableThe actual profit that would result following a given action, conditional on a given event occurring.
Opportunity lossAmount of profit forgone by not choosing the best act for each event. Thus, conditional opportunity lossthe relative loss following a given action, conditional on a given event occurring.
Expected monetary value (EMV)expected profit on average given uncertainty. Conditional values weighted by the probability of the events
occurring, and summed for each act.
Expected opportunity loss (EOL)Conditional opportunity losses weighted by the probability of events occurring, and summed
Optimum actGreatest EMV (or) smallest EOL
Expected profit with perfect predictionThe profit one could make on average if each event could be predicted in advance
Sum of (conditional profit of best acts given certain event occurring * probability of event)
Expected Value of Perfect InformationValue of obtaining a perfect prediction
= Expected profit with perfect prediction – EMV of the best act
Cost of additional information < additional profit of information
Also, note the following the following information:
EMV (of any act) + EOL (of same act) = Expected Profit with Perfect Prediction
-> EOL of the best act = expected value of perfect information in such a case
Decision nodes
Event nodes
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Discussion Problem #1
An analysis and forecast of next month’s sales results in the following probability distribution:
Event Demand
10 units
11 units
12 units
Probability
.10
.70
.20
1.00
The store stocks one of the three demand amounts. The unit is sold for $11. The cost of the product sold is
$6. If the product is not sold during the month, it is worthless (leftover units are of no value).
Required:
a. Construct a decision tree.
b. Determine the optimum act, compute its EMV.
c. Determine the expected profit with perfect prediction.
d. Determine the expected value of perfect information.
Decision Trees
Expected
Expected
Monetary Value Opportunity Loss
(EMV)
(EOL)
Values
1) Optimum act_______________ EMV of optimum act________________
2) Expected profit with perfect prediction________________
3) Expected value of perfect information_________________
2
Conditional
Losses
Discussion Problem #2
The Gorman Manufacturing Company must decide whether to purchase a component from a supplier or
manufacture the component at its Milan, Michigan plant. If demand is high, Gorman could profitably
manufacture the component. However, if demand is low, Gorman’s unit manufacturing cost would be high
due to underutilization of equipment. The following table shows the projected profit (in thousands of
dollars) for Gorman’s make-or-buy decision.
Decision Alternative
Manufacture component
Purchase component
Low
-20
10
Demand
Medium High
40
100
45
70
The probabilities are: P(low) = 0.35, P(medium)= 0.35, P(high)= 0.30.
a.
b.
c.
Use a decision tree to recommend a decision. What is its EMV?
What is expected profit with perfect prediction?
What is the expected value of perfect information?
Expected
Monetary Value
(EMV)
Conditional
Values
1) Optimum act_______________ EMV of optimum act________________
2) Expected profit with perfect prediction________________
3) Expected value of perfect information_________________
3
Discussion Problem #3
Campus Program sells programs at football games. The owner has collected the following data regarding
the pattern of sales
Quantity Sold (Cases)
400
600
700
Probability
.2
.5
.3
The owner is uncertain of the number of cases of programs to order. He must order one of the given
quantities. A case of programs sells for $300 and the purchase price is $100. Unsold programs are thrown
away.
a. Determine the number of cases of programs the firm should order. What is its EMV?
b. What is expected profit with perfect prediction?
c. What is the expected value of perfect information?
Expected
Monetary Value
(EMV)
Conditional
Values
1) Optimum act_______________ EMV of optimum act________________
2) Expected profit with perfect prediction________________
3) Expected value of perfect information_________________
4
Discussion Problem #4
Martin’s Service Station is considering investing in a heavy-duty snowplow this fall. Martin has analyzed
the situation carefully and believes that it would be a profitable investment if the snowfall is heavy. Martin
could still make a small profit if the snowfall is moderate, but he would lose money if the snowfall is light.
Specifically, Martin forecasts a profit of $7000 if the snowfall is heavy, $2000 if it is moderate, and a
$9000 loss if the snowfall is light. Based on the weather bureau’s long-range forecast, Martin estimates
that P(heavy snowfall) = 0.4, P(moderate snowfall) = 0.3, and P(light snowfall) = 0.3.
a.
b.
c.
Prepare a decision tree for Martin’s problem
What is the expected value at each state-of-nature node.
Would the expected value approach recommend that Martin invest in a snowplow?
Expected
Monetary Value
(EMV)
Conditional
Values
Optimum act_______________ EMV of optimum act________________
Suppose that Martin can purchase a blade to attach to his service truck that can also be used to plow
driveways and parking lots. This truck must also be available to start cars, so Martin will not be able to
generate as much revenue plowing snow if he chooses this alternative. But his loss will be smaller if the
snowfall is light. Under this alternative, Martin forecasts a profit of $3500 if the snow fall is heavy, $1000
if it is moderate, and a $1500 loss if the snowfall is light.
d.
e.
f.
Prepare a new decision tree showing all three alternatives.
What is the optimal decision using the expected value approach.
What is the expected value of perfect prediction?
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Discussion Problem #4 (cont’d)
Expected
Monetary Value
(EMV)
e)
f)
Conditional
Values
Optimum act_______________ EMV of optimum act________________
Expected profit with perfect prediction________________
Suppose that Martin decides to wait to check the September temperature pattern before making a final
decision. Estimates of the probabilities associated with an unseasonably cold September are
P(Unseasonably cold|heavy snowfall)= 0.30, P(Unseasonably cold|moderate snowfall)= 0.20,
P(Unseasonably cold|light snowfall)= 0.05.
g) If Martin observes an unseasonably cold September, what is the recommended decision?
h) If Martin does not observe an unseasonably cold September, what is the recommended decision?
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Discussion Problem #4 (cont’d)
Expected Value of Sample Information (EVSI)
_________________________________
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Discussion Problem #4 (cont’d)
September
Snowfall
Unseasonably cold
Not unseasonably cold
Marginals
Heavy
0.4
Moderate
0.3
Light
0.3
Marginals
Given:
P(heavy) =
0.4
P(unseasonably cold|heavy snowfall)=
0.3
P(moderate)=
0.3
P(unseasonably cold|moderate snowfall)=
0.2
P(light)=
0.3
P(unseasonably cold|light snowfall)=
0.05
Solve:
P(heavy snowfall & unseasonably cold)=
P(moderate snowfall & unseasonably cold)=
P(light snowfall & unseasonably cold)=
Need:
P(heavy snowfall|unseasonably cold)=
P(heavy snowfall| not unseasonably cold)=
P(moderate snowfall|unseasonably cold)=
P(moderate snowfall|not unseasonably cold)=
P(light snowfall|unseasonably cold)=
P(light snowfall| not unseasonably cold)=
Discussion Problem #5
Hale’s TV Productions is considering producing a pilot for a comedy series for a major television network.
The network may reject the pilot and the series, but it may also purchase the program for 1 or 2 years. Hale
may decide to produce the pilot or transfer the rights for the series to a competitor for $100,000. Hale’s
profits are summarized in the following profit (in thousands of dollars) payoff table.
Decision Alternative
Produce pilot
Sell to competitor
State of Nature
Reject 1 year 2 years
-100
50
150
100
100
100
The probabilities are: P(reject) = 0.2, P(1 year)= 0.3, P(2 years)= 0.5.
a.
b.
What should the company do?
What is the maximum that Hale should be willing to pay for inside information on what the
network will do?
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For a consulting fee of $2500, an agency will review the plans for the comedy series and indicate the
overall chances of a favorable network reaction to the series. If the special agency review results in a
favorable or an unfavorable evaluation, what should Hale’s decision strategy be? Assume that Hale
believes that the following conditional probabilities are realistic appraisals of the agnecy’s evaluation
accuracy.
P(favorable review|rejection)=
P(favorable review|1 year)=
P(favorable review|2 years)=
P(unfavorable review|rejection)=
P(unfavorable review|1 year)=
P(unfavorable review|2 years)=
c.
d.
e.
0.3
0.6
0.9
0.7
0.4
0.1
Show the decision tree for this problem.
What is the recommended decision strategy and the expected value, assuming that the agency
information is obtained?
What is the EVSI? Is the agency’s information worth the $2500 consulting fee? What is the
maximum that Hale should be willing to pay for the information?
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