# Linear Algebra- Geometric Linear Programming

• April 25th 2013, 02:20 PM
Jesstess123
Linear Algebra- Geometric Linear Programming
I've been stuck on this exercise for some time now.
A manufacturer produces sacks of chicken feed from two ingredients, A and B. Each sack is to contain atleast
10 ounces of nutrient N1, atleast 8 ounces of nutrient N2 and atleast 12 ounces of nutrient N3.
Ingredient A contains, per pound: 2 ounces of nutrient N1, 2 ounces of nutrient N2 and 6 ounces of nutrient N3
Ingredient B contains, per pound: 5 ounces of nutrient N1, 3 ounces of Nutrient N2 and 4 ounces of nutrient N3.
If ingredient A costs 8 cents per pound and Ingredient B costs 9 cents per pound, how much of ingredient should the manufacturer use in each sack of feed to minimize his cost.

I've called ingredient A for X1 and ingredient B for X2, thus the total cost Z will be:
Z = 8X1 + 9X1

Since it must contain atleast 10 ounce of N1:
2 X1 + 5X2 >= 10 (greater or equal to)

Since it must contain atleast 8 ounce of N2:
2X1 +3X2 >= 8

Since it must contain atleast 12 ounce of N3:

6X1 + 4X2 >= 12

Since each sack must totally contain 30 ounce all together (10+8+12)

X1+ X2 = 30

Have i set up the proper constraints?
If not, what have i done wrong.
If so, how do i solve this geometrically?
• April 25th 2013, 03:08 PM
HallsofIvy
Re: Linear Algebra- Geometric Linear Programming
Looks good to me. Now, since there are two variables, $X_1$, and $X_2$, graph each constraint on an " $X_1$", " $X_2$, graph. Each constraint, with "= " rather than " $\ge$", will be a straight line and the "feasible region", the region in which all constraints are satisfied is bounded by those lines. The maximum or minimum or a linear "target function" will lie on a vertex of that boundary. So find the vertices (which are always where two of the lines intersect) and evaluate the target function at each vertex.
• April 25th 2013, 03:32 PM
Jesstess123
Re: Linear Algebra- Geometric Linear Programming
Thanks! Good to hear.

I find the intersection by solving the equational system the lines provide right?

Edit: Hmm I'm also starting to doubt the X1 + X2 = 30 constraint, because i get some vertex points like (0,3) and (5,0) and those don't satisfy X1 +X2 = 30
Edit 2: Ugh, the ounces and pounds really make my head spin.
• April 25th 2013, 04:56 PM
Jesstess123
Re: Linear Algebra- Geometric Linear Programming
Should the X1 + X2 = 30 constraint be X1+X2 <= 30 instead?
I get the correct answer which is 0,4 pounds of ingredient A and 2,4 pounds of ingredient B.
But the constraint X1+ X2 = 30 cannot be correct right?