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Math Help - Linear Programming Problem

  1. #1
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    Linear Programming Problem

    Kunz manufactures two products that are used in the heavy equipment industry. Both products require maunfacturing operations in 2 different departments (A and B). The following are the production times (in hours) and profit contributions for the two products:

    Product Profit/Unit Dept A (in hours) Dept B (in hours)
    1 $25 6 12
    2 $20 8 10

    For the coming period, Kunz has a total of 900 hours of labour that can be allocated to either of the two departments. Formulate an LP to maximize the numbers of hours to allocate per department and total contribution to profit.

    So I created 4 variables:

    X1 - Hours of Product 1 in Dept A
    X2 - Hours of Product 1 in Dept B
    X3 - Hours of Product 2 in Dept A
    X4 - Hours of Product 2 in Dept B

    I formulated the following LP:

    Max Z = 25(X1 + X2) + 20(X3 + X4)
    s.t.
    6X1 + 12X2 + 8X3 + 10X4 <=900
    all variables => 0

    Is my solution correct?
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by statmajor View Post
    Kunz manufactures two products that are used in the heavy equipment industry. Both products require maunfacturing operations in 2 different departments (A and B). The following are the production times (in hours) and profit contributions for the two products:

    Product Profit/Unit Dept A (in hours) Dept B (in hours)
    1 $25 6 12
    2 $20 8 10

    For the coming period, Kunz has a total of 900 hours of labour that can be allocated to either of the two departments. Formulate an LP to maximize the numbers of hours to allocate per department and total contribution to profit.

    So I created 4 variables:

    X1 - Hours of Product 1 in Dept A
    X2 - Hours of Product 1 in Dept B
    X3 - Hours of Product 2 in Dept A
    X4 - Hours of Product 2 in Dept B

    I formulated the following LP:

    Max Z = 25(X1 + X2) + 20(X3 + X4)
    s.t.
    6X1 + 12X2 + 8X3 + 10X4 <=900
    all variables => 0

    Is my solution correct?
    That look like a correct formulation of the LP problem.

    CB
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  3. #3
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    Really? Because the objective function doesn't make sense to me. Would X1 = X2 in this case since product 1 has to spends 6X1 hours in Dept A and 12X2 hours in Dept B. Would X1 = X2 in this case?

    If they do, then X1 and X2 would each refer to the number of Products 1 produced.
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  4. #4
    Grand Panjandrum
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    Quote Originally Posted by statmajor View Post
    Really? Because the objective function doesn't make sense to me. Would X1 = X2 in this case since product 1 has to spends 6X1 hours in Dept A and 12X2 hours in Dept B. Would X1 = X2 in this case?

    If they do, then X1 and X2 would each refer to the number of Products 1 produced.
    As I read your question a unit of product 1 take 6 hours to produce in Dept A and 12 hours to produce in Dept B. If you think it means it takes 6 hours in DA and 12 hours in DB then X1=X2 (and similarly X3=X4), in fact there are only 2 variables Y1 the production of product 1 and Y2 the production of product 2.

    The latter interpretation makes better sense from one point of view, the profit is independently of department. Also under the other interpretation DA would dominate DB.

    The wording is still ambiguous, but I am coming round to preferring your new interpretation.

    CB

    CB
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  5. #5
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    I kinda think that the new objective function should be Max Z = 25X1 + 20X3. You should be able to replace X1 with X2, and X3 with X4.

    For example:

    If we wanted 3 units of Product 1, we would need (6*3)=18 hours in Dept A, and (12*3) = 36 hours in Dept B.

    We could reduce it down to two variables:

    Y1 - Product 1
    Y2 - Product 2

    So the new LP could be:

    Max Z = 25Y1 + 20Y2
    s.t
    18Y1 + 18Y2 <= 900
    all var => 0
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