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Math Help - Linear Algebra- Geometric Linear Programming

  1. #1
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    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?
    Last edited by Jesstess123; April 25th 2013 at 01:53 PM.
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  2. #2
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    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.
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    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.
    Last edited by Jesstess123; April 25th 2013 at 03:26 PM.
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  4. #4
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    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?
    Last edited by Jesstess123; April 25th 2013 at 05:35 PM.
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