Problem 1: (Algorithmic)
Suppose that you are given a decision situation with three possible states of nature: S1, S2, and S3. The prior probabilities are P(S1) = 0.16, P(S2) = 0.57, and P(S3) = 0.27. With sample information P(I|S1) = 0.13, P(I|S2) = 0.07, and P(I|S3) = 0.19. Compute the revised or posterior probabilities: P(S1 | I), P(S2, I), and P(S3 | I). If required, round your answers to four decimal places.
State of Nature P(Sj|I)
S1 =
S2 =
S3 =
Problem 2: (Algorithmic)
Following is the payoff table for the Pittsburgh Development Corporation (PDC) Condominium Project. Amounts are in millions of dollars.
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State of Nature
|
|
Decision Alternative
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Strong Demand S1
|
Weak Demand S2
|
|
Small complex, d1
|
9
|
8
|
|
Medium complex, d2
|
14
|
4
|
|
Large complex, d3
|
19
|
-7
|
Suppose PDC is optimistic about the potential for the luxury high-rise condominium complex and that this optimism leads to an initial subjective probability assessment of 0.81 that demand will be strong (S1) and a corresponding probability of 0.19 that demand will be weak (S2). Assume the decision alternative to build the large condominium complex was found to be optimal using the expected value approach. Also, a sensitivity analysis was conducted for the payoffs associated with this decision alternative. It was found that the large complex remained optimal as long as the payoff for the strong demand was greater than or equal to $16.58 million and as long as the payoff for the weak demand was greater than or equal to -$17.32 million.
A) Consider the medium complex decision. How much could the payoff under strong demand increase and still keep decision alternative d3 the optimal solution? If required, round your answer to two decimal places.
The payoff for the medium complex under strong demand remains less than or equal to $ ____ million, the large complex remains the best decision.
B) Consider the small complex decision. How much could the payoff under strong demand increase and still keep decision alternative d3 the optimal solution? If required, round your answer to two decimal places.
Problem 3: Hemmingway, Inc., is considering a $5 million research and development (R&D) project. Profit projections appear promising, but Hemmingway's president is concerned because the probability that the R&D project will be successful is only 0.50. Furthermore, the president knows that even if the project is successful, it will require that the company build a new production facility at a cost of $20 million in order to manufacture the product. If the facility is built, uncertainty remains about the demand and thus uncertainty about the profit that will be realized. Another option is that if the R&D project is successful, the company could sell the rights to the product for an estimated $25 million. Under this option, the company would not build the $20 million production facility.
The decision tree is shown in Figure 4.16. The profit projection for each outcome is shown at the end of the branches. For example, the revenue projection for the high demand outcome is $59 million. However, the cost of the R&D project ($5 million) and the cost of the production facility ($20 million) show the profit of this outcome to be $59 - $5 - $20 = $34 million. Branch probabilities are also shown for the chance events.
A) Analyze the decision tree to determine whether the company should undertake the R&D project. If it does, and if the R&D project is successful, what should the company do?
B) What is the expected value of your strategy?
Expected value = $ ____ M
C) What must the selling price be for the company to consider selling the rights to the product?
Payoff for sell rights would have to be $ M or more. In order to recover the $5M R&D cost, the selling price would have to be $ M or more.
D) Develop a risk profile for the optimal strategy. If required, round your answers to two decimal places.
|
|
Associated Probability
|
|
Possible Profit
|
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$34M
|
|
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$20M
|
|
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$10M
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|
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-$5M
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|
Problem 4: (Algorithmic)
A real estate investor has the opportunity to purchase land currently zoned residential. If the county board approves a request to rezone the property as commercial within the next year, the investor will be able to lease the land to a large discount firm that wants to open a new store on the property. However, if the zoning change is not approved, the investor will have to sell the property at a loss. Profits (in thousands of dollars) are shown in the following payoff table:
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State of Nature
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|
Rezoning Approved
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Rezoning Not Approved
|
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Decision Alternative
|
S1
|
S2
|
|
Purchase, d1
|
550
|
-170
|
|
Do not purchase, d2
|
0
|
0
|
A) If the probability that the rezoning will be approved is 0.5, what decision is recommended?
Recommended decision:
What is the expected profit?
Expected profit: $
B) The investor can purchase an option to buy the land. Under the option, the investor maintains the rights to purchase the land anytime during the next three months while learning more about possible resistance to the rezoning proposal from area residents. Probabilities are as follows:
Let H = High resistance to rezoning
L = Low resistance to rezoning
P(H) = 0.53
P(S1 | H) = 0.15
P(S2 | H) = 0.85
P(L) = 0.47
P(S1 | L) = 0.88
P(S2 | L) = 0.12
C) What is the optimal decision strategy if the investor uses the option period to learn more about the resistance from area residents before making the purchase decision?
High resistance:
Low resistance:
D) If the option will cost the investor an additional $10,000, should the investor purchase the option?
Why or why not?
The input in the box below will not be graded, but may be reviewed and considered by your instructor.
What is the maximum that the investor should be willing to pay for the option?
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