Question 1:
Lee is the manager of a large farm with 1,000 acres or arable land. For greater efficiency, Lee always devotes the farm to growing one crop at a time. He now needs to make a decision on which one of four crops to grow during the upcoming growing season. For each of these crops, Lee has obtained the following estimates of crop yields and net incomes per bushel under various weather conditions.
|
Weather
|
Expected Yield, Bushels/Acre
|
|
Crop 1
|
Crop 2
|
Crop 3
|
Crop 4
|
|
Dry
|
20
|
15
|
30
|
40
|
|
Moderate
|
35
|
20
|
25
|
40
|
|
Damp
|
40
|
30
|
25
|
40
|
|
Net income per bushel
|
$1.00
|
$1.50
|
$1.00
|
$0.50
|
After referring to historical meteorological records, Lee also has estimated the following prior probabilities for the weather during the growing season:
|
Dry
|
0.3
|
|
Moderate
|
0.5
|
|
Damp
|
0.2
|
a. Develop a decision analysis formulation of this problem by identifying the decision alternatives, the states of nature, and the payoff table.
b. Construct a decision tree for this problem and use Bayes' decision rule to determine which crop to grow.
c. Using Bayes' decision rule, do sensitivity analysis with respect to the prior probabilities of moderate weather and damp weather (without changing the prior probability of dry weather) by resolving when the prior probability of moderate weather is 0.2, 0.3, 0.4 and 0.6.
Question 2:
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 items 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.
b. Assuming the single item is not inspected in advance, use Bayes' decision rule to determine which decision alternative should be chosen.
c. Find the expected value of perfect information. Does this answer indicate that consideration should be given to inspecting the single item in advance?
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.
e. Construct and solve the decision tree for this entire problem.
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 os using the laboratory be and will be worthwhile?