The Kalo Fertilizer Company makes a fertilizer using two chemicals that provide nitrogen, phosphate, and potassium. A pound of ingredient 1 contributes 10 ounces of nitrogen and 6 ounces of phosphate, whereas a pound of ingredient 2 contributes 2 ounces of nitrogen, 6 ounces of phosphate, and 1 ounce of potassium. Ingredient 1 costs $3 per pound, and ingredient 2 costs $5 per pound. The company wants to know how many pounds of each chemical ingredient to put into a bag of fertilizer to meet the minimum requirements of 20 ounces of nitrogen, 36 ounces of phosphate, and 2 ounces of potassium while minimizing cost.
Formulate a linear programming model for this problem.
Solve this model by using graphical analysis.
make a table
nitrogen | phosphate | potassium | cost | |
x=Ingredient 1 | 10 | 6 | 0 | $3 |
y=Ingredient 2 | 2 | 6 | 1 | $5 |
minimum requirements | 20 | 36 | 2 |
system is
subject to
.
The value of the objective function at each of these extreme points is as follows:
Extreme Point Coordinates (x,y) |
Objective function value z=3x+5y |
A(0,10) | 3(0)+5(10)=50 |
B(1,5) | 3(1)+5(5)=28 |
C(4,2) | 3(4)+5(2)=22 |
The minimum value of the objective function z=22 occurs at the
extreme point (4,2).
Hence, the optimal solution to the given LP problem is: x=4,y=2,
and min z=22
.
4 pound of ingredient 1
2 pound of ingredient 2
minimum cost is $22
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