Dwight and Hattie have run the family farm for over thirty years. They are curre
ID: 3043444 • Letter: D
Question
Dwight and Hattie have run the family farm for over thirty years. They are currently planning the mix of crops to plant on their 120-acrea farm for the upcoming season. The table below gives the labor hours and fertilizer required per acre, as well as the total expected profit per acre for each of the potential crops under consideration. Dwight, Hattie, and their children can work at most 6,500 total hours during the upcoming season. They have 200 tons of fertilizer available. Define the Variable. Formulate an OPL.
Crop
Labor Required
Fertilizer Required
Expected Profit
Oats
50
1.5
$500
Wheat
60
2
$600
Corn
105
4
$950
Family Farm
Oats
Wheat
Corn
Profit (per acre)
$500
$600
$950
Total
Total
Resources
Resources Used per Acre
Used
Available
Land(acres)
1
1
1
120
<=
120
Labor(acres)
50
60
105
6400
<=
6500
Fertilizer(tons)
1.5
2
4
200
<=
200
Total Profit
Acres Planted
80
40
0
$64,000
Oats should be planted on 80 acres and wheat on 40 acres while not planting any corn will result in a total profit of $64,000.
Crop
Labor Required
Fertilizer Required
Expected Profit
Oats
50
1.5
$500
Wheat
60
2
$600
Corn
105
4
$950
Explanation / Answer
let the number of acres used on oats , wheat and corn are
x ,y, and z respectively
Maximize
Z = 500x + 600y + 950 z
x+y+ z <= 120
50x+60y + 105z <= 6500
1.5 x + 2y + 4z <= 200
Tableau #1
x y z s1 s2 s3 z
1 1 1 1 0 0 0 120
50 60 105 0 1 0 0 6500
1.5 2 4 0 0 1 0 200
-500 -600 -950 0 0 0 1 0
Tableau #2
x y z s1 s2 s3 z
0.625 0.5 0 1 0 -0.25 0 70
10.625 7.5 0 0 1 -26.25 0 1250
0.375 0.5 1 0 0 0.25 0 50
-143.75 -125 0 0 0 237.5 1 47500
Tableau #3
x y z s1 s2 s3 z
1 0.8 0 1.6 0 -0.4 0 112
0 -1 0 -17 1 -22 0 60
0 0.2 1 -0.6 0 0.4 0 8
0 -10 0 230 0 180 1 63600
Tableau #4
x y z s1 s2 s3 z
1 0 -4 4 0 -2 0 80
0 0 5 -20 1 -20 0 100
0 1 5 -3 0 2 0 40
0 0 50 200 0 200 1 64000
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