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Math Help - Linear Programming

  1. #1
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    Linear Programming

    I have a question on the most profitable mix. X1 = belts, X2 = watches

    Profit: 25X1 + 15X2

    I have 3X1 + 2X2 < or = 240 and 2x1 + X2 < or = 100.

    I then multiplied 2x1 + X2 < or = 100 by 1.5 to get a common term and got X2 < or = 180 and X1 < or = 40. We have to either do a graph or some other way find out what the most profitable mix of belts and watches are but I have no idea where to go from here. Any help would be appreciated, thanks.

    I think I have to draw it out on a graph and find the feasible region which is shaded and the most profitable area will be one of the 4 corners?
    Last edited by nycalex; May 5th 2008 at 05:52 AM.
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  2. #2
    A riddle wrapped in an enigma
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    You started out with "belts" and "watches" and ended up with "air conditioners" and "fans". I'm a little confused as well. Can you state the problem as it was given to you?
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  3. #3
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    sorry about that i just fixed it
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  4. #4
    A riddle wrapped in an enigma
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    I think you tried to find the intersection of your two linear functions, but you came up with the wrong ordered pair. I believe, if you check your work, you'll see:

    (1) 3x1 + 2x2 = 240
    (2) 2x1 + x2 = 100
    ---------------------
    Multiply (2) by -2 and add to (1)

    3x1 + 2x2 = 240
    -4x1 - 2x2 = 200
    -----------------
    -x1 = 40
    x1 = -40

    Substituting x1 = -40 into (1) yields x2 = 180

    But, that intersection is not part of your feasible region!

    Don't you need two more constraints to insure that x1 >= 0 and x2 >= 0?
    Even with these new constraints, your feasible region will only have 3 corners when graphed. Are there other constraints you haven't mentioned?
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  5. #5
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    I am very sorry about this I talked to the teacher and he said there was a mistake in the question. He assigned a new question which is

    6x1 + 2x2 <= 72
    6x1 + 4x2 <= 120

    I tried it out by graphing and my most profitable point on the feasible region came out to X1 being 4 and X2 being 24. I'm not sure if this is right?
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  6. #6
    A riddle wrapped in an enigma
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    Assuming you are using the same profit function 25x1 + 15x2, the vertex of (4, 24) would indeed yield the largest profit ($460).
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