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Math Help - The simplex method

  1. #1
    Senior Member chella182's Avatar
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    The simplex method

    The question goes...

    Use the simplex method to solve the following linear programming problem. Maximise

    f(x_1, x_2, x_3)=3x_1+2x_2+4x_3

    subject to the constraints

    3x_1+x_2+4x_3\leq60
    x_1+2x_2+3x_3\leq30
    2x_1+2x_2+3x_3\leq600
    x_1, x_2, x_3\geq0
    Last edited by mr fantastic; August 4th 2009 at 07:20 PM. Reason: Linear methods \neq linear algebra
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  2. #2
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    Opalg's Avatar
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    This looks like a totally standard exercise in using the simplex method, so where are you having trouble with it?

    Just to get you started, you should introduce "slack variables" u,v,w, to convert the inequalities into equations, namely

    \begin{aligned}3x_1+x_2+4x_3+u\qquad\qquad &= 60,\\ x_1+2x_2+3x_3 \qquad+ v\qquad&= 30,\\ 2x_1+2x_2+3x_3\qquad\qquad+w &= 600.\end{aligned}

    The objective equation is M = 3x_1+2x_2+4x_3, which you write as -3x_1-2x_2-4x_3+M=0.

    Then you write down the simplex tableau, which is just the matrix of coefficients in these equations, and apply the simplex algorithm.

    If it's any help to you, I wrote down a systematic description of the simplex algorithm a few years ago when teaching this stuff. You can find a copy of it here (pdf file).
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  3. #3
    Senior Member chella182's Avatar
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    My problem is that I don't ever remember being taught this, and it's not in my notes anywhere, so I don't know what the simplex algorithm is or anything. I'll have a look at that .pdf, see if that helps.
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  4. #4
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    mr fantastic's Avatar
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    Quote Originally Posted by chella182 View Post
    My problem is that I don't ever remember being taught this, and it's not in my notes anywhere, so I don't know what the simplex algorithm is or anything.
    [snip]
    Then you should ask whoever gave it to you why you were given it.
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