A farmer can grow three crops on 1,000 acres of land: corn, soybeans, and wheat. In producing these three crops, the farmer has to: (1) plow the land (pl), (2) plant corn (pc), (3) plant soybeans (ps), (4) plant wheat (pw), (5) harvest corn (hc), (6) harvest soybeans (hs), (7) harvest wheat (hw), (8) sell the corn after harvest (sc), (9) sell the soybeans after harvest (ss), and (10) sell the wheat after harvest (sw). The farmer must plow the land prior to planting and must plant the crops prior to harvesting the crops. In addition to the land endowment of 1,000 acres, the farmer has a total of 1,200 hours available to perform all of the above production operations. The farmer does not want to plant more than 400 acres to corn for soil conservation reasons. The farmer's expec- tations regarding the labor requirements (hours per acre) and variable costs for each operation, as well as expected price and yield (bushels per acre) at harvest for the three crops are presented below:
|
Operation
|
Labor Requirement (hours/acre)
|
Variable Cost ($/acre)
|
Crop Yields and Prices
|
|
Plowing (pl)
|
0.40
|
4.00
|
|
|
Plant Corn (pc)
|
0.39
|
114.00
|
|
|
Plant Soybeans (ps)
|
0.30
|
80.00
|
|
|
Plant Wheat (pw)
|
0.30
|
78.00
|
|
|
Harvest Corn (hc)
|
0.60
|
48.00
|
120
|
|
Harvest Soybeans (hs)
|
0.30
|
17.00
|
40
|
|
Harvest Wheat (hw)
|
0.28
|
10.00
|
70
|
|
Corn Price ($/bushel)
|
|
2.90
|
|
|
Soybean Price ($/bushel)
|
|
5.75
|
|
|
Wheat Price ($/bushel)
|
|
3.00
|
|
Assume that the farmer's objective is to maximize net revenue from corn, soybeans, and wheat production.
a. Write the LP tableau for this problem.
b. There are many assumptions regarding the decision process this farmer follows in this model. List three of these assumptions.
c. Solve this exercise using Solver. Summarize the optimal solution and SPs. List three factors (other than relative crop prices) that influence the optimal crop mix in this exercise.