Decision Making Under Uncertainty

Conditional value (or) Conditional Profit Table-

The actual profit that would result following a given action, conditional on a given event occurring.

Opportunity loss-

Amount of profit forgone by not choosing the best act for each event. Thus, conditional opportunity loss- the 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 act-

Greatest EMV (or) smallest EOL

Expected profit with perfect prediction-

The 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 Information-

Value 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

Discussion Problem #1

An analysis and forecast of next month’s sales results in the following probability distribution:

Event DemandProbability

10 units.10

11 units.70

12 units.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:

  1. Construct a decision tree.
  2. Determine the optimum act, compute its EMV.
  3. Determine the expected profit with perfect prediction.

d. Determine the expected value of perfect information.

Decision Trees

Expected Expected Monetary Value Opportunity Loss Conditional

(EMV) (EOL) Values Losses

1)Optimum act______EMV of optimum act______

2)Expected profit with perfect prediction______

3)Expected value of perfect information______

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 AlternativeDemand

LowMediumHigh

Manufacture component-20 40100

Purchase component 10 45 70

The probabilities are: P(low) = 0.35, P(medium)= 0.35, P(high)= 0.30.

  1. Use a decision tree to recommend a decision. What is its EMV?
  2. What is expected profit with perfect prediction?
  3. What is the expected value of perfect information?

Expected

Monetary Value Conditional

(EMV) Values

1)Optimum act______EMV of optimum act______

2)Expected profit with perfect prediction______

3)Expected value of perfect information______

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)Probability

400.2

600 .5

700 .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 Conditional

(EMV) Values

1) Optimum act______EMV of optimum act______

2) Expected profit with perfect prediction______

3) Expected value of perfect information______

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.

  1. Prepare a decision tree for Martin’s problem
  2. What is the expected value at each state-of-nature node.
  3. Would the expected value approach recommend that Martin invest in a snowplow?

Expected

Monetary Value Conditional

(EMV) 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.

  1. Prepare a new decision tree showing all three alternatives.
  2. What is the optimal decision using the expected value approach.
  3. What is the expected value of perfect prediction?

Discussion Problem #4 (cont’d)

Expected

Monetary Value Conditional

(EMV) Values

e)Optimum act______EMV of optimum act______

f)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

SnowfallUnseasonably coldNot unseasonably coldMarginals

Heavy0.4

Moderate0.3

Light0.3

Marginals

Given:

P(heavy) = 0.4P(unseasonably cold|heavy snowfall)= 0.3

P(moderate)= 0.3P(unseasonably cold|moderate snowfall)= 0.2

P(light)= 0.3P(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 State of Nature

Reject1 year2 years

Produce pilot-100 50150

Sell to competitor 100 100100

The probabilities are: P(reject) = 0.2, P(1 year)= 0.3, P(2 years)= 0.5.

  1. What should the company do?
  2. What is the maximum that Hale should be willing to pay for inside information on what the network will do?

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)= 0.3

P(favorable review|1 year)= 0.6

P(favorable review|2 years)= 0.9

P(unfavorable review|rejection)= 0.7

P(unfavorable review|1 year)= 0.4

P(unfavorable review|2 years)= 0.1

  1. Show the decision tree for this problem.
  2. What is the recommended decision strategy and the expected value, assuming that the agency information is obtained?
  3. 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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