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Math Help - Linear programming

  1. #1
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    Linear programming

    Maximize f(x,y)= 2x+y

    subject to
    2x-y-8 \leq 0
    x+2y-14 \leq 0
    -x+y-4 \leq 0
    -x \leq 0
    -y \leq 0

    a)with Graphical method of Linear programming
    b)with Simplex method of Linear programming

    Thnak you in advance.
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  2. #2
    Member Last_Singularity's Avatar
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    Quote Originally Posted by abstar View Post
    Maximize f(x,y)= 2x+y

    subject to
    2x-y-8 \leq 0
    x+2y-14 \leq 0
    -x+y-4 \leq 0
    -x \leq 0
    -y \leq 0

    a)with Graphical method of Linear programming
    b)with Simplex method of Linear programming

    Thnak you in advance.
    Please show us your work and let us know where you get stuck; we are not here to do out the entire problem for you.

    To get you started, denote 2x+y as 2x_1 + x_2. Declare your objective function z=2x_1 + x_2.

    Then your constraints become
    2x_1-x_2-8 \leq 0
    x_1+2x_2-14 \leq 0
    -x_1+x_2-4 \leq 0

    The last two constraints can be ignored, as the Simplex Method itself requires all variables to be nonnegative. In other words, as you carry out your pivots correctly, the minimal ratio test will ensure that your variables say nonnegative.

    For these three constraints, introduce slack variables x_3,x_4,x_5 and use that as your initial basis for a feasible tableau:

    maximize z - 2x_1 - x_2 = 0
    2x_1-x_2 + x_3 = 8
    x_1+2x_2 + x_4 = 14
    -x_1+x_2 + x_5 = 4

    Your basis is currently [3,4,5] with an objective value of 0. Since both of them have positive reduced costs, choose one of x_1,x_2 as the entering variable k.

    Can you take it from there? Check your answers by comparing the graphical and the Simplex method - they should agree with each other.
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  3. #3
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    i've started to do this and got as far as the formulas part, i'm trying to work outthe new values in the new table

    where R3 is the pivot row

    row1 = (R1-R3)
    row2 = (r2-2(r3)
    row4 = (R4 + 2(r3)

    also is it always Rx - Pivot row, i get confused on the formula section, the rest i think im ok with. Could someone please explain what to do here?

    my pivot table is attached to this thread, could someone have a look and check for me please?
    Attached Files Attached Files
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  4. #4
    Senior Member DeMath's Avatar
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    Quote Originally Posted by abstar View Post
    Maximize f(x,y)= 2x+y

    subject to
    2x-y-8 \leq 0
    x+2y-14 \leq 0
    -x+y-4 \leq 0
    -x \leq 0
    -y \leq 0

    a)with Graphical method of Linear programming
    b)with Simplex method of Linear programming

    Thnak you in advance.
    a) with Graphical method of Linear programming



    \overrightarrow n is a normal-vector
    Last edited by DeMath; February 13th 2009 at 06:38 AM.
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