Question

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.

1. Formulate a linear programming model for this problem.

2. Solve this model by using graphical analysis.

Answer 1

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