A farmer has 5(X) acres of land and wishes to determine the acreage allocated to the following three crops: wheat, corn and soybeans. The man-days required, preparation cost, and profit per acre of the three crops are summarized. Suppose that the maximum number of man-days available are 5,000, and that the farmer has $80,000 for preparation.
|
Crop
|
Man-Days
|
Preparation Costs ($)
|
Profits ($/acre)
|
|
Wheat
|
6
|
100
|
60
|
|
Corn
|
8
|
150
|
100
|
|
Soy Beans
|
10
|
120
|
80
|
The linear program for this problem P4 is given below along with the optimal solution for the dual problem D4. Note that D4 has three variables (yi, Y2, y3) and three slack variables (y4, y5, y6).
max 60x1 + 100x2 + 80x3
P4 s.t. x1 + X2 + X3 ≤ 500 (Acreage)
6x1 + 8x2 + 10x3 ≤ 5, 000 (Labor)
100x1 + 150x2 + 120x3 ≤ 80, 000 (Money)
X1, X2, X3 ≥ 0
Y* = [Yi Y2, Y3, Y4, Y5, Y6]T = [100, 0, 0, 40, 0, 20]T
(a) How many acres of each crop should the farmer plant?
(b) The farmer is worried that the price of corn has been very volitile lately. How sensitive is this solution to that price? That is, how much can the price rise or drop before the current solution is no longer the optimal one?
(c) Assuming an 8-hour work day, would it be profitable to the farmer to acquire an additional 15 workers at $3 per hour? Why or why not?
(d) The farmer's distant relative dies and leaves him $2,500. Which of the following will be more profitable for the farmer (considering the current year only)
i. Store the money under his mattress
ii. Use the money to cover preparation costs
iii. Use the money to rent land from a neighboring farmer at $80 per acre
iv. Put the money into a savings account with an annual yield of 14% (i.e. he earns $350 in interest for the year)
Explain why you chose the option you did and show how much profit the farmer will make from his $2,500.