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Linear Programming probelm

  1. Mar 5, 2009 #1
    1. The problem statement, all variables and given/known data

    Under current legislation, a company involved in a noxious chemical industry has a permit to pass into the atmosphere a maximum of 1500 units of sulfur dioxide, and a maximum of 1200 units of carbon monoxide each day. A new plant just installed releases 150 units of sulfur dioxide, and 80 units of carbon monoxide each hour it operates. The old plant releases 100 units of sulfur dioxide, and 120 units of carbon monoxide each hour it operates. The company must run each plant for at least 2 hours, but less than 8 hours to keep the plants operative.The new plant makes a profit of $1600 p/hour of operation, and the old plant makes a profit of $1000 per/hour of operation.

    A new environmental policy is proposed which will restrict the maximum amount of sulfur to be released to 1250 units per day, and the amount of carbon monoxide to 1000 units per day. How will this affect the companies profits


    3. The attempt at a solution

    Let x = hours of operation of Plant 1
    Let y = hours of operation of Plant 2

    With the above, I am not sure whether the x, y, should be units of sulfur and carbon monoxide though, or hours of operation of P1, and P2.

    This is why I am having so much trouble getting the constraints.

    The last part, in which the new environmental policy puts restrictions on the units of sulfur dioxide, and carbon monoxide allowed per day will become a new constraint that will have to be tested separately.

    In essence, once I can determine what x and y are and the constraints, I can then compare the constraints on levels of sulfur and carbon monoxide and how they affect profit.

    Please help. I really need to get this down-pat ASAP.
     
  2. jcsd
  3. Mar 5, 2009 #2

    HallsofIvy

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    Staff Emeritus
    Science Advisor

    First there are no "P1" or "P2" in the statement of the problem.! I assume that by P1 you mean the "old plant" and by P2 the "new plant". The units of sulfur and carbon monoxide are functions of the hours of operation. You can control the hours of operation, not the units of sulfur and carbond monoxide so the variables, x and y are the hours of operation. Let x be the number of hours of operation of P1 and y the number of hours of operation of P2.

    A new plant just installed releases 150 units of sulfur monoxide, and 80 units of carbon monoxide each hour it operates. The old plant releases 100 units of sulfur monoxide, and 120 units of carbon monoxide each hour it operates.

    So the sulfur monoxide production is 100x+ 150y and carbon monoxide production is 80x+ 120y

    The company must run each plant for at least 2 hours, but less than 8 hours to keep the plants operative.
    [itex]2\le x< 8[/itex] and [itex]2\le y< 8[/itex]

    The new plant makes a profit of $1600 p/hour of operation, and the old plant makes a profit of $1000 per/hour of operation.
    The object function is 1000x+ 1600y

    Under the old rules, the company could produce "a maximum of 1500 units of sulfur monoxide, and a maximum of 1200 units of carbon monoxide each day"
    [itex]100x+ 150y\le 1500[/itex] and carbon [itex]80x+ 120y\le 1200[/itex]

    Under the new rules, the company must "restrict the maximum amount of sulfur to be released to 1250 units per day, and the amount of carbon monoxide to 1000 units per day".
    [itex]100x+ 150y\le 1250[/itex] and [itex]80x+ 120y\le 1000[/itex]
     
  4. Mar 5, 2009 #3
    Thanks HallsofIvy.

    I worked this out on my own eventually as well.

    After analyzing effects on profit, there is a 16.5% reduction in profits with new regulations in place.
     
  5. Mar 5, 2009 #4
    Max Profit under current Legislation = $15,800

    Max Profit under new Legislation = $13, 200
     
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