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Math Help - Linear programming

  1. #1
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    Linear programming

    Hi
    I have one problem
    Trucks production company must produce 1000 trucks. Company has four factories. The cost of truck production (raw materials, work) in each separate factory is shown in table:
    [HTML]
    -------------------------------------------------
    |Factory |Cost x 10^3 |Work |Raw materials |
    -------------------------------------------------
    |1 |15 |2 |3 |
    -------------------------------------------------
    |2 |10 |3 |4 |
    ------------------------------------------------
    |3 |9 |4 |5 |
    ------------------------------------------------
    |4 |7 |5 |6 |
    ------------------------------------------------[/HTML]

    Automaker trade union requires that at least 400 trucks should be made in 3rd factory. It was given 3300 work hours and 4000 pcs. raw materials to each factory for order lead time. Make linear programming problem which minimize the cost of 1000 trucks production.

    Can anyone help me to express the objective function and the constraints??
    Last edited by figo; January 10th 2009 at 02:45 PM.
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  2. #2
    Member Last_Singularity's Avatar
    Joined
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    Quote Originally Posted by figo View Post
    Hi
    I have one problem
    Trucks production company must produce 1000 trucks. Company has four factories. The cost of truck production (raw materials, work) in each separate factory is shown in table:
    [html]
    -------------------------------------------------
    |Factory |Cost x 10^3 |Work |Raw materials |
    -------------------------------------------------
    |1 |15 |2 |3 |
    -------------------------------------------------
    |2 |10 |3 |4 |
    ------------------------------------------------
    |3 |9 |4 |5 |
    ------------------------------------------------
    |4 |7 |5 |6 |
    ------------------------------------------------[/html]Automaker trade union requires that at least 400 trucks should be made in 3rd factory. It was given 3300 work hours and 4000 pcs. raw materials to each factory for order lead time. Make linear programming problem which minimize the cost of 1000 trucks production.

    Can anyone help me to express the objective function and the constraints??
    Denote x_1, x_2, x_3, x_4 as your variables, representing the amount that each factory produces. So your objective is:
    z = 15x_1 + 10x_2 + 9x_3 + 7x_4

    Because it needs to produce 1,000 trucks, one constraint is
    x_1 + x_2 + x_3 + x_4 = 1000

    At least 400 trucks need to be made in the 3rd factory, so
    x_3 \geq 400

    Subject to work hours, we have:
    2 x_1 + 3 x_2 + 4 x_3 +5 x_4 \leq 3300

    Subject to raw materials, we have:
    3 x_1 + 4 x_2 + 5 x_3 +6 x_4 \leq 4000

    However, I am unsure about whether the previous two constraints are correct or not. They would be if the integers that you provided me with were in units of work hours per truck made and raw materials per truck made. If it is the other way around, you need to flip the numbers around.

    Don't forget to implement:
    x_1,x_2,x_3,x_4 \geq 0 - this is NOT a trivial constraint because if you solve it with linear programming software, it could give you negative production.
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  3. #3
    Senior Member DeMath's Avatar
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    MS Excel has shown such a result:
    z_{\min}=11600 if x_1=400, x_2=200, x_3=400, x_4=0.
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