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Math Help - Linear programming problem concerning constraints

  1. #1
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    Linear programming problem concerning constraints

    I got a LP problem that I'm stuck on.

    A furniture factory makes tables and chairs. It takes 2 hours to assemble a table and 30 minutes to assemble a chair. Assembly is carried out by four people on the basis of one eight-hour shift per day. Customers buy at four chairs with each table which means that the factory has to make at most four times as many chairs as tables. The selling price is 135 per table and 50 per chair.

    Formulate this as a linear programming problem to determine the daily production of tables and chairs which would maximise the total daily revenue to the factory and solve the problem using the simplex method.

    My progress:

    X1 = Amount of tables produced
    X2 = Amount of chairs produced

    Max Z = 135X1+50X2

    Subject to:
    120X1+30X2 ≤ 1920
    4X1 ≥ X2
    X1, X2 ≥ 0

    Standard form:

    Max Z = 135X1+50X2

    Subject to:
    4X1+X2+S1 = 64
    4X1-X2-S2 = 0
    X1, X2, S1, S2 ≥ 0

    Are my constraints correct?


    Thanks for your time.
    Last edited by fobster; August 20th 2006 at 01:49 AM.
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  2. #2
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    Quote Originally Posted by fobster
    I got a LP problem that I'm stuck on.

    A furniture factory makes tables and chairs. It takes 2 hours to assemble a table and 30 minutes to assemble a chair. Assembly is carried out by four people on the basis of one eight-hour shift per day. Customers buy at four chairs with each table which means that the factory has to make at most four times as many chairs as tables. The selling price is 135 per table and 50 per chair.

    Formulate this as a linear programming problem to determine the daily production of tables and chairs which would maximise the total daily revenue to the factory and solve the problem using the simplex method.

    My progress:

    X1 = Amount of tables produced
    X2 = Amount of chairs produced

    Max Z = 135X1+50X2

    Subject to:
    120X1+30X2 ? 1920
    4X1 ? X2
    X1, X2 ? 0

    Standard form:

    Max Z = 135X1+50X2

    Subject to:
    4X1+X2+S1 = 1920
    4X1-X2-S2 = 0
    X1, X2, S1, S2 ? 0

    Are my constraints correct?


    Thanks for your time.
    Looks good to me except the first constraint in the standard form has incorrect coefficients for X1 and X2.
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  3. #3
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    Hello, fobster!

    I simplified the language . . .


    A furniture factory makes tables and chairs.
    It takes 2 hours to assemble a table and 30 minutes to assemble a chair.
    Assembly is carried out by four people on the basis of one eight-hour shift per day.
    Customers buy at most four chairs with each table which means that
    the factory has to make at most four times as many chairs as tables.
    The selling price is 135 per table and 50 per chair.

    Formulate this as a linear programming problem to determine the daily production of tables and chairs
    which would maximise the total daily revenue to the factory
    and solve the problem using the simplex method.

    Those subscripts are confusing . . .

    Let T = number of tables produced: T \geq 0 [1]
    Let C = number of chairs produced: C \geq 0 [2]

    The T tables take a total of 2T hours to assemble.
    The C chairs take a total of \frac{C}{2} hours to assemble.
    So we have: . 2T + \frac{C}{2}\:\leq\:32\quad\Rightarrow\quad 4T + C \:\leq \:64 [3]

    There must be at most four chairs per table: . C \leq 4T\quad\Rightarrow\quad 4T + C \geq 0 [4]

    Now apply the Simplex Method to the four inequalties
    . . and maximize the revenue function: . R \:=\:135T + 50C

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