Help with Optimization and Decision Support Modeling for Business HW4
Module 7 Practice Problems OSCM 471/571 Optimization and Decision Support Modeling for Business
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Study Materials: Lecture Note 7: Decision Making under Uncertainty
1. You are given the following decision tree, where the numbers in parentheses are probabilities and the
numbers on the right are payoffs at these terminal points.
a. Analyze this decision tree to obtain the optimal policy. b. Use Excel TreePlan Add-in to construct and solve the same decision tree. Answer:
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2. Using Bayes’ decision rule, consider the decision analysis problem having the following payoff table (in units of thousands of dollars).
a. Which alternative should be chosen? What is the resulting expected payoff? Answer: Alternative A3 has the highest expected payoff of $35,000.
b. You are offered the opportunity to obtain information that will tell you with certainty whether
the first state of nature S1 will occur. What is the maximum amount you should pay for the information? Assuming you will obtain the information, how should this information be used to choose an alternative? What is the resulting expected payoff (excluding the payment)? Answer:
If S1 occurs for certain then choose alternative A3 (payoff is $10,000). If S1 does not occur for certain then the chance of S2 occurring is 3/8 and the chance of S3 occurring is 5/8. So choose A1 (expected payoff is $66,250). A1: (3/8)(10) + (5/8)(100) = 66.25 A2: (3/8)(20) + (5/8)(50) = 38.75 A3: (3/8)(10) + (5/8)(60) = 41.25 EP(with information) = (0.2)(10) + (0.8)(66.25) = 55 EVI = EP (with information) – EP (without more information) = 55 – 35 = $20,000 The maximum amount you should pay for the information is $20,000. The decision with this information would be to choose A3 if S1 will occur. Otherwise choose A1. The expected payoff is $55,000 (excluding the payment for information).
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A B C D E F Payoff Table ($thousands) Expected
State of Nature Payoff Alternative S1 S2 S3 ($thousands)
A1 -100 10 100 33 A2 -10 20 50 29 A3 10 10 60 35
Prior Probability 0.2 0.3 0.5
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c. Now repeat part b if the information offered concerns S2 instead of S1. Answer:
If S2 occurs for certain then choose alternative A2 (payoff is $20,000). If S2 does not occur for certain then the chance of S1 occurring is 2/7 and the chance of S3 occurring is 5/7. So choose A3 (expected payoff is $45,714). A1: (2/7)(–100) + (5/7)(100) = 42.857 A2: (2/7)(–10) + (5/7)(50) = 32.857 A3: (2/7)(10) + (5/7)(60) = 45.714 EP(with information) = (0.3)(20) + (0.7)(42.857) = 38 EVI = EP (with information) – EP (without more information) = 38 – 35 = $3,000 The maximum amount you should pay for the information is $3,000. The decision with this information would be to choose A2 if S2 will occur. Otherwise choose A3. The expected payoff is $38,000 (excluding the payment for information).
d. Now repeat part b if the information offered concerns S3 instead of S1.
Answer: If S3 occurs for certain then choose alternative A1 (payoff is $100,000).
If S3 does not occur for certain then the chance of S1 occurring is 2/5 and the chance of S2 occurring is 3/5. So choose A3 (expected payoff is $10,000). A1: (2/5)(–100) + (3/5)(10) = –34 A2: (2/5)(–10) + (3/5)(20) = 8 A3: (2/5)(10) + (3/5)(10) = 10 EP(with information) = (0.5)(100) + (0.5)(10) = 55 EVI = EP (with information) – EP (without more information) = 55 – 35 = $20,000 The maximum amount you should pay for the information is $20,000. The decision with this information would be to choose A1 if S3 will occur. Otherwise choose A3. The expected payoff is $55,000 (excluding the payment for information).
e. Now suppose that the opportunity is offered to provide information that will tell you with
certainty which state of nature will occur (perfect information). What is the maximum amount you should pay for the information? Assuming you will obtain the information, how should this information be used to choose an alternative? What is the resulting expected payoff (excluding the payment)? Answer:
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With perfect information, choose A3 for when the state is S1, A2 when the state is S2, and A1 when the state is S3. EP(with perfect information) = (0.2)(10) + (0.3)(20) + (0.5)(100) = $58,000 EVPI = EP(with perfect information) – EP (without more information) = 58 – 35 = $23,000 A maximum of $23,000 should be paid for the information. With perfect information, choose A3 for when the state is S1, A2 when the state is S2, and A1 when the state is S3. The resulting expected payoff is $58,000.
f. If you have the opportunity to do some testing that will give you partial additional information
(not perfect information) about the state of nature, what is the maximum amount you should consider paying for this information? Answer:
The maximum amount you should ever pay for testing is $23,000.
3. The Hit-and-Miss Manufacturing Company produces items that have a probability p of being defective. These items are produced in lots of 150. Past experience indicates that p for an entire lot is either 0.05 or 0.25. Furthermore, in 80 percent of the lots produced, p equals 0.05 (so p equals 0.25 in 20 percent of the lots). These items are then used in an assembly, and ultimately their quality is determined before the final assembly leaves the plant. Initially the company can either screen each item in a lot at a cost of $10 per item and replace defective items or use the items directly without screening. If the latter action is chosen, the cost of rework is ultimately $100 per defective item. Because screening requires scheduling of inspectors and equipment, the decision to screen or not screen must be made two days before the screening is to take place. However, one item can be taken from the lot and sent to a laboratory for inspection, and its quality (defective or nondefective) can be reported before the screen/no-screen decision must be made. The cost of this initial inspection is $125.
a. Develop a decision analysis formulation of this problem by identifying the decision alternatives, the states of nature, and the payoff table if the single item is not inspected in advance. Answer:
State of Nature Alternative p=0.05 p=0.25
Screen –$1,500 –$1,500 Don’t screen –$750 –$3,750
Prior Probabilities 0.8 0.2
b. Assuming the single item is not inspected in advance, use Bayes’ decision rule to determine which decision alternative should be chosen.
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Answer: Choosing not to screen maximizes the expected payoff. The expected cost is $1,350.
c. Find the expected value of perfect information. Does this answer indicate that consideration should be given to inspecting the single item in advance? Answer:
With perfect information, they would screen if p = 0.25, and don’t screen if p = 0.05. EP(with perfect information) = (0.8)(–$750) + (0.2)(–$1,500) = –$900 EVPI = EP(with perfect information) – EP(without more information) = (–$900) – (–$1,350) = $450. This indicates that consideration should be given to inspecting the single item.
d. Assume now that the single item is inspected in advance. Find the posterior probabilities of
the respective states of nature for each of the two possible outcomes of this inspection. Answer:
e. Construct and solve the decision tree for this entire problem. Answer: The optimal policy is not to pre-screen or screen.
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A B C D E Payoff Table Expected
Alternative p = 0.05 p = 0.25 Payoff Screen -$1,500 -$1,500 -$1,500
Don't Screen -$750 -$3,750 -$1,350
Prior Probability 0.8 0.2
State of Nature
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B C D E F G H Data:
State of Prior Nature Probability Defective Nondefective
p = 0.05 0.8 0.05 0.95 p = 0.25 0.2 0.25 0.75
Posterior Probabilities:
Finding P(Finding) p = 0.05 p = 0.25 Defective 0.09 0.444 0.556
Nondefective 0.91 0.835 0.165
P(State | Finding) State of Nature
P(Finding | State) Finding
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f. Find the expected value of sample information. If the cost of using the laboratory to inspect the single item in advance is open to negotiation, how large can the cost of using the laboratory be and still be worthwhile? Answer:
EVSI = EP(with information ignoring cost of information) – EP(without information) = (–1392.95 + 125) – (–1350) = (–1267.95) – (–1350) = $82.95. If the prescreening inspection cost is less than $82.96, then it would be worthwhile to use.
4. Silicon Dynamics has developed a new computer chip that will enable it to begin producing and marketing a personal computer if it so desires. Alternatively, it can sell the rights to the computer chip for $15 million. If the company chooses to build computers, the profitability of the venture depends on the company’s ability to market the computer during the first year. It has sufficient access to retail outlets that it can guarantee sales of 10,000 computers. On the other hand, if this computer catches on, the company can sell 100,000 machines. For analysis purposes, these two levels of sales are taken to be the two possible outcomes of marketing the computer, but it is unclear what their prior
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probabilities are. The cost of setting up the assembly line is $6 million. The difference between the selling price and the variable cost of each computer is $600.
State of Nature Alternative Sell 10,000 Sell 100,000
Build Computers $0 $54 million Sell Rights $15 million $15 million
Dynamics now is considering doing full-fledged market research at an estimated cost of $1 million
to predict which of the two levels of demand is likely to occur. Previous experience indicates that such market research is correct two-thirds of the time.
a. Find the expected value of perfect information for this problem.
Answer: With perfect information, they should build computers if they will sell 100,000 of them, and
sell the rights if they could only sell 10,000 computers. EP(with perfect information) = (0.5)(54) + (0.5)(15) = $34.5 million EVPI = EP(with perfect information) – EP without more information) = 34.5 – 27 = $7.5 million.
b. Does the answer in part a indicate that it might be worthwhile to perform this market research?
Answer: Since the market research will cost $1 million it might be worthwhile to perform it.
c. Develop a probability tree diagram to obtain the posterior probabilities of the two levels of
demand for each of the two possible outcomes of the market research. Answer:
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d. Use the corresponding Excel template to check your answers in part c. Answer:
5. You are given the following payoff table (in units of dollars).
You have the option of paying $100 to have research done to better predict which state of nature will occur. When the true state of nature is S1, the research will accurately predict S1 60 percent of the time (but will inaccurately predict S2 40 percent of the time). When the true state of nature is S2, the research will accurately predict S 2 80 percent of the time (but will inaccurately predict S 1 20 percent of the time).
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B C D E F G H Data:
State of Prior Nature Probability Predict Sell 10,000 Predict Sell 100,000
Sell 10,000 0.5 0.667 0.333 Sell 100,000 0.5 0.333 0.667
Posterior Probabilities:
Finding P(Finding) Sell 10,000 Sell 100,000 Predict Sell 10,000 0.5 0.667 0.333
Predict Sell 100,000 0.5 0.333 0.667
P(State | Finding) State of Nature
P(Finding | State) Finding
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a. Given that the research is not done, use Bayes’ decision rule to determine which decision alternative should be chosen. Answer:
Alternative A1 maximizes the expected payoff ($100).
b. Use a decision tree to help find the expected value of perfect information. Does this answer indicate that it might be worthwhile to do the research? Answer:
EVPI = EP (with perfect info) – EP (without more info) = $220 – $100 = $120 This indicates that it might be worthwhile to do the research.
c. Given that the research is done, find the joint probability of each of the following pairs of outcomes: (i) the state of nature is S 1 and the research predicts S 1 , (ii) the state of nature is S 1 and the research predicts S 2 , (iii) the state of nature is S 2 and the research predicts S 1 , and (iv) the state of nature is S2 and the research predicts S2. Answer:
P(state and finding) = P(state) P(finding | state) i) P(Predict S1 and Actual S1) = (0.4)(0.6) = 0.24 ii) P(Predict S1 and Actual S2) = (0.4)(0.4) = 0.16 iii) P(Predict S2 and Actual S1) = (0.6)(0.2) = 0.12 iv) P(Predict S2 and Actual S2) = (0.6)(0.8) = 0.48
d. Find the unconditional probability that the research predicts S1. Also find the unconditional
probability that the research predicts S2. Answer: P(Predict S1) = 0.24 + 0.12 = 0.36 P(Predict S2) = 0.16 + 0.48 = 0.64
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A B C D E Payoff Table Expected
Alternative S1 S2 Payoff A1 $400 -$100 $100 A2 $0 $100 $60
Prior Probability 0.4 0.6
State of Nature
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e. Given that the research is done, use your answers in parts c and d to determine the posterior probabilities of the states of nature for each of the two possible predictions of the research. Answer:
P(state | finding) = P(state and finding) / P(finding) P(Actual S1 | Predict S1) = 0.24 / 0.36 = 0.667 P(Actual S1 | Predict S2) = 0.16 / 0.64 = 0.250 P(Actual S2 | Predict S1) = 0.12 / 0.36 = 0.333 P(Actual S2 | Predict S2) = 0.48 / 0.64 = 0.750
f. Use the corresponding Excel template to obtain the answers for part e.
Answer:
g. Given that the research predicts S1, use Bayes’ decision rule to determine which decision alternative should be chosen and the resulting expected payoff. Answer:
If S1 is predicted, then choosing alternative A1 maximizes the expected payoff ($233.33).
h. Repeat part g when the research predicts S2. Answer:
If S2 is predicted, then choosing alternative A2 maximizes the expected payoff ($75).
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B C D E F G H Data:
State of Prior Nature Probability Predict S1 Predict S2
Actual S1 0.4 0.6 0.4 Actual S2 0.6 0.2 0.8
Posterior Probabilities:
Finding P(Finding) Actual S1 Actual S2 Predict S1 0.36 0.667 0.333 Predict S2 0.64 0.250 0.750
P(State | Finding) State of Nature
P(Finding | State) Finding
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A B C D E Payoff Table Expected
Alternative S1 S2 Payoff A1 $400 -$100 $233.33 A1 $0 $100 $33.33
Prior Probability 0.6667 0.3333
State of Nature
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A B C D E Payoff Table Expected
Alternative S1 S2 Payoff A1 $400 -$100 $25 A1 $0 $100 $75
Prior Probability 0.25 0.75
State of Nature
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i. Given that research is done, what is the expected payoff when using Bayes’ decision rule?
Answer: Expected payoff given research is (0.36)($233.33) + (0.64)($75) – $100 = $32.
j. Use the preceding results to determine the optimal policy regarding whether to do the
research and the choice of the decision alternative. Answer:
The optimal policy is to do no research and simply choose A1.
k. Construct and solve the decision tree to show the analysis for the entire problem. (Using
TreePlan is optional.) Answer:
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6. You are given the following payoff table. Assume that your utility function for the payoffs is 𝑈𝑈(𝑥𝑥) = √𝑥𝑥. Plot the expected utility of each decision alternative versus the value of p on the same graph. For each decision alternative, find the range of values of p over which this alternative maximizes the expected utility. Answer:
Expected utility of A1 = pU(25) + (1 – p)U(36) = 5p + 6(1 – p) = 6 – p.
Expected utility of A2 = pU(100) + (1 – p)U(0) = 10p + 0 = 10p. Expected utility of A3 = pU(0) + (1 – p)U(49) = 7(1 – p) = 7 – 7p. A1 and A3 cross when 6 – p = 7 – 7p, or p = 1/6. A1 and A2 cross when 6 – p =10p, or p = 6/11. Thus, A3 is best when p ≤ 1/6, A1 is best when 1/6 < p ≤ 6/11, and A2 is best when p > 6/11.