Problem 1 (15 points): Problem 9.9 from the book Dwight Mooty is the manager of

Question

Problem 1 (15 points): Problem 9.9 from the book Dwight Mooty is the manager of a large farm with 1,000 acres of arable land. For greater efficiency, Dwight always devotes the form 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 season. For each of these crops, Dwight has obtained the following estimates of crop yields and net incomes per bushel under various weather conditions. Expected Yield: Bushels Acre Crop 2 15 20 30 Crop 3 30 25 25 Crop 4 40 40 40 Weather Crop 1 20 35 40 Moderate Net income per bushel $1.00 $1.50 $1.00 $0.50 After referring to the historical meteorological records, Dwight also has estimated the following prior probabilities for the weather during the growing season: Dry 0.3, Moderate 0.5, Damp 0.2. a) (5 points) Develop a decision analysis formulation of this problem by identifying the decision b) c) d) e) f) alternatives, the states of nature, and construct the payoff table. (2 points) Which alternative should be selected based on Maximax criterion? (2 points) Which alternative should be selected based on Maximin criterion? (2 points) Which alternative should be selected based on Maximum likelihood criterion? (2 points) Which alternative should be selected based on Baye's rule? (2 points) Dwight is able to consult a psychic for a fee of $5,000 to get the season's weather condition precisely. Should Dwight consult the psychic? (Hint: calculate the expected value of perfect information).

Explanation / Answer

1 a)

Different Decision alternatives are

Different state of nature are

pay off table is as follows

Payoff Table calculation example

For crop 1 alternative with dry weather state = 20x1000x$1.00=$20,0000.00

1 b)

The maximax criterion involves selecting the alternative that maximizes the maximum payoff available.

The highest possible payoff is $45000, therefore, Dwight should select alternative Crop 2.

1 c)

The maximin rule involves selecting the alternative that maximizes the minimum pay-off achievable.

Min possible pay off for crop 1: $ 20000.00

Min possible pay off for crop 2: $ 22500.00

Min possible pay off for crop 3: $ 25000.00

Min possible pay off for crop 4: $ 20000.00

Therefore, Dwight should go with Alternative Crop 3.

1 d)

The maximum likelihood criterion means selecting an alternative that maximizes payoff for a state with a maximum probability of occurring.

The state with the maximum likelihood of occurring: Moderate

Alternative with maximum pay off in the moderate state: Crop 1 with $ 35000.00

Therefore Dwight should go with alternative Crop 1

1e) As per Baye's rule we multiply payoff of every alternative with the probability of the corresponding state and sum all payoff of each state to a weighted payoff of that state

For calculation

for crop 1

The payoff in the Dry state: 0.3* $20000 = $6000

The payoff in the Moderate state: 0.5* $35000 = $17500

The payoff in the Damp state: 0.2* $40000 = $8000

Sum of all this is $31500

As per Baye's rule, Dwight should select alternative Crop 1.

1 e) If Dwight selects alternative on the basis of expected value, his expected payoff is $ 31500 as described above.

With perfect information from psychic, his payoff will be sum of maximum possible payoff for each state multiplied by the probability of that state that is

for the dry state: 0.3 x $30000=$9000

for moderate state: 0.5 x $35000=$17500

for the damp state: 0.2 x 45000=$9000

sum of alll above= $35500

expected value of perfect information = $35500-$31500 =$4000

Therefoore Dwight should not consult a psychic.

Weather Probability Expected yieeld in bushles/acres Crop 1 Crop 2 Crop 3 Crop 4 Dry 0.3 $ 20,000.00 $ 22,500.00 $ 30,000.00 $ 20,000.00 Moderate 0.5 $ 35,000.00 $ 30,000.00 $ 25,000.00 $ 20,000.00 Damp 0.2 $ 40,000.00 $ 45,000.00 $ 25,000.00 $ 20,000.00

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