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Math Help - Minimizing

  1. #1
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    Minimizing

    Someone please help me with this.

    Hard Math Question?

    Minimize Z=9x1 + 4x2 + 12x3

    The 1,2, and 3 are superscript.

    Subject to the constraints:

    3x1+x2+4x3 larger or equal to 24
    3x1 + x2 + 2x3 larger or equal to 18
    x1 >= 0, x2 >=0, x3 larger or equal to 0


    What is Z, x1, x2, and x3?

    All numbers after letter are superscript. Thanks!
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  2. #2
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    Are you sure you don't mean SUBscript?

    Did you set up a tableau?

    Can you find the intersections of the constraints?

    You may wish to hunt around x1 = 0 and x2 = 12, I suppose.
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  3. #3
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    x_{1}, x_{2}, x_{3} \geq 0

    3x_{1} +x_{2} +4x_{3} \geq 24
    3x_{1} + x_{2} + 2x_{3} \geq 18

    Subtract both inequalities to get: 2x_{3} \geq 6 \: \Rightarrow \: x_{3} \geq 3

    Since we're looking for the minimum values that'll satisfy both inequalities, we'll set x_{3} = 3 and find that: {\color{red}3x_{1} + x_{2} \geq 12}

    We want to somehow use this inequality in your equation for z to simplify your equation. You can do that by modifying the equation a bit (we also substitute x_{3} = 3 as that'll minimize the value of z):

    z = 9x_{1} + {\color{blue}4x_{2}} + 12x_{3}
    z = 9x_{1} + {\color{blue}3x_{2} + x_{2}} + 12(3)
    z = 3({\color{red}3x_{1} + x_{2}}) + x_{2} + 36 \: \: \geq \: \: 3({\color{red}12}) + x_{2} + 36
    z \geq 72 + x_{2}

    And since x_{2} \geq 0, you could probably figure out what value of x_{2} to choose and solve for x_{1}
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  4. #4
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    Hello, Greenbaumenom!

    This is a Linear Programming problem.
    . . Are you familiar with the techniques?
    I'll solve it by "graphing".


    Minimize: . P \:=\:9x + 4y + 12z
    Subject to the constraints:. \begin{array}{cccc}<br />
x &\geq& 0 & {\color{blue}[1]} \\<br />
y &\geq& 0 & {\color{blue}[2]} \\<br />
z & \geq & 0 & {\color{blue}[3]} \\<br />
3x+y+4z & \geq & 24 & {\color{blue}[4]} \\<br />
3x + y + 2z & \geq & 18 & {\color{blue}[5]} \end{array}
    Find x, y, z \text{ and }P.

    [1], [2], [3] places us in the first octant.

    [4] is a plane with intercepts: . (8,0,0),\;(0,24,0),\;(0,0,6)
    Graph the plane and shade the space above the plane.

    [5] is a plane with intercepts: . (6,0,0),\;(0,18,0),\;(0,0,9)
    Graph the plane and shade the space above the plane.


    The solution set is a polyhderon with vertices at:
    . . (6,0,0),\;(0,18,0),\;(0,0,9),\;(4,0,3),\;(0,12,3)

    Test these vertices in the P-function to find the minimum P.

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  5. #5
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    Grrr.... Am I maximizing again when I should be minimizing?!
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