Dwight Moody is the manager of a large farm with 1,000 acres of arable land. For greater efficiency, Dwight 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, Dwight has obtained the following estimates of crop yields and net incomes per bushel under various weather conditions.

After referring to historical meteorological records, Dwight also 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) 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 re-solving when the prior probability of moderate weather is 0.2, 0.3, 0.4, and 0.6.

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Expected Yield, Bushels/Acre Weather Crop 1 Crop 2 15 20 30 Crop 3 Dry Moderate Damp 20 35 40 30 25 25 Crop 4 40 40 40 Net income per bushel $1.00 51.00 0.50