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Math Help - Linear Programming Problem

  1. #1
    irishwisteria
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    Red face Linear Programming Problem

    Okay, I am not feeling really well, and this is frustrating me quite a bit.

    Here is the problem:
    A manufacturer of refrigerators must ship at least 100 refrigerators to its two West Coast warehouses. Warehouse A holds a maximum of 75 refrigerators and Warehous B holds a maximum of 80 refrigerators. It costs $12 to ship a refrigerator to warehouse A and $10 to ship one to warehouse B. Union rules require that at least 300 workers be hired. Shipping a refrigerator to Warehouse A requires 4 workers, while shipping a refrigerator to Warehouse B requires 2 workers. How many refrigerators should be shipped to each warehouse to minimize costs? What is the minimum cost?

    Okay, now here is my problem: Teacher says that there are three constraints and then the objective function. So, I have:

    Objective function: c=12x+10y

    Constraints: 1. 75x+80y >/= 100
    2. 4x + 2y >/= 300

    Now, could someone give me a hint as to what I am looking for in the third constraint? I mean I have a constraint for space and workers, what else is there? I have a feeling I am missing something obvious.
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  2. #2
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    Quote Originally Posted by irishwisteria
    Okay, I am not feeling really well, and this is frustrating me quite a bit.

    Here is the problem:
    A manufacturer of refrigerators must ship at least 100 refrigerators to its two West Coast warehouses. Warehouse A holds a maximum of 75 refrigerators and Warehous B holds a maximum of 80 refrigerators. It costs $12 to ship a refrigerator to warehouse A and $10 to ship one to warehouse B. Union rules require that at least 300 workers be hired. Shipping a refrigerator to Warehouse A requires 4 workers, while shipping a refrigerator to Warehouse B requires 2 workers. How many refrigerators should be shipped to each warehouse to minimize costs? What is the minimum cost?

    Okay, now here is my problem: Teacher says that there are three constraints and then the objective function. So, I have:

    Objective function: c=12x+10y

    Constraints: 1. 75x+80y >/= 100
    2. 4x + 2y >/= 300

    Now, could someone give me a hint as to what I am looking for in the third constraint? I mean I have a constraint for space and workers, what else is there? I have a feeling I am missing something obvious.
    The area in red is incorrect and should be split into 3 different equations...

    x\leq75 and y\leq80 these two show space constraints
    x+y\geq100 and this one shows minimum ref. sent
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  3. #3
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    x,y\geq 0 in the first quadrant
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  4. #4
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    Quote Originally Posted by irishwisteria
    Okay, I am not feeling really well, and this is frustrating me quite a bit.

    Here is the problem:
    A manufacturer of refrigerators must ship at least 100 refrigerators to its two West Coast warehouses. Warehouse A holds a maximum of 75 refrigerators and Warehous B holds a maximum of 80 refrigerators. It costs $12 to ship a refrigerator to warehouse A and $10 to ship one to warehouse B. Union rules require that at least 300 workers be hired. Shipping a refrigerator to Warehouse A requires 4 workers, while shipping a refrigerator to Warehouse B requires 2 workers. How many refrigerators should be shipped to each warehouse to minimize costs? What is the minimum cost?

    Okay, now here is my problem: Teacher says that there are three constraints and then the objective function. So, I have:

    Objective function: c=12x+10y

    Constraints: 1. 75x+80y >/= 100
    2. 4x + 2y >/= 300

    Now, could someone give me a hint as to what I am looking for in the third constraint? I mean I have a constraint for space and workers, what else is there? I have a feeling I am missing something obvious.
    Your teacher should have said there are four constraints:
    (You assumed x = number of refs delivered to Warehouse A, and y = number of refs to Warehouse B.)

    A manufacturer of refrigerators must ship at least 100 refrigerators to its two West Coast warehouses.
    x +y >= 100 ---------------------(1)

    Warehouse A holds a maximum of 75 refrigerators...
    x <= 75 ------------------(2)

    ...and Warehous B holds a maximum of 80 refrigerators.
    y <= 80 -------------(3)

    Union rules require that at least 300 workers be hired. Shipping a refrigerator to Warehouse A requires 4 workers, while shipping a refrigerator to Warehouse B requires 2 workers.
    4X +2y >= 300
    Or, in its lowest terms,
    2x +y >= 150 -------------(4)

    Those are the 4 constraints.

    --------------------------------------
    To solve the question.....

    If you plot those on the same x,y coordinates setup, and solve for the intersections of those four inequalities, you will find that the feasible region is a quadrilateral whose vertices are (35,80), (75,80), (75,25) and (50,50).

    Testing those 4 corner points against the objective function, you will find that corner (50,50) gives the lowest cost, so, to minimize cost, ship 50 refs to Warehouse A and also 50 refs to Warehouse B.
    And the minmum cost will be 50(12) +50(10) = $1100.
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  5. #5
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    Quote Originally Posted by ticbol
    Union rules require that at least 300 workers be hired. Shipping a refrigerator to Warehouse A requires 4 workers, while shipping a refrigerator to Warehouse B requires 2 workers.
    4X +2y >= 300
    Or, in its lowest terms,
    2x +y >= 150 -------------(4)
    I don't see how the union rules pose any constraint at all for this problem.

    If the shipping requirements for workers are less than 300, then some workers will be idle. There is nothing that says the total number of workers used must be at least 300.

    If the shipping requirements for workers exceed 300, the manufacturer can hire more. It says at least 300 must be hired. It does not say the manufacturer cannot hire more.

    So the union rules mean nothing here. Thus there are not 4 constraints, but 3 as the teacher says.
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  6. #6
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    Quote Originally Posted by JakeD
    I don't see how the union rules pose any constraint at all for this problem.

    If the shipping requirements for workers are less than 300, then some workers will be idle. There is nothing that says the total number of workers used must be at least 300.

    If the shipping requirements for workers exceed 300, the manufacturer can hire more. It says at least 300 must be hired. It does not say the manufacturer cannot hire more.

    So the union rules mean nothing here. Thus there are not 4 constraints, but 3 as the teacher says.
    "Union rules require that at least 300 workers be hired. Shipping a refrigerator to Warehouse A requires 4 workers, while shipping a refrigerator to Warehouse B requires 2 workers."

    Is there something not English in that Union requirement? Did it specifically say "Union rules require that at least 300 workers must be hired"?

    ----------------------------
    Then, you said :

    "If the shipping requirements for workers exceed 300, the manufacturer can hire more. It says at least 300 must be hired. It does not say the manufacturer cannot hire more."

    Duh.
    Do I understand English? Am I lost here?
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