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Linear programming problem

  1. Aug 25, 2010 #1
    1. The problem statement, all variables and given/known data
    Question (to be solved graphically)
    -------

    A coffee company sells coffee under a "Best blend" label and an "Economy blend" label. Both are blended from three basic grades of coffee:

    Best blend = 40% A + 40% B + 20 % C
    Economy = 20% A + 40% B + 40% C

    The market prices are
    - $1760/tonne for Best Blend
    - $1600/tone for Economy.

    The company is given the option of buying
    - up to 80 tonnes of grade A at $1600/tonne
    - up to 120 tonnes of grade B at $1000/tonne
    - up to 200 tonnes of grade C at $600 tonne.

    1. Calculate the profit per tonne of each blend of coffee.
    2. How much of each blend should the company produce to maximize its profit?
    3. What is the maximum profit?



    2. Relevant equations



    3. The attempt at a solution

    First find the cost per tonne

    Best blend : 0.4*1600 + 0.4*1000 + 0.2*600 = 1160
    Economy: 0.2*1600 + 0.4*1000 + 0.4*600 = 960

    Thus the total profit is: (1760 - 1160)(Best blend) + (1260 - 960)(Economy)

    Let x1 be the tonnes of Best blend produced, x2 the tonnes of economy produced.

    Maximum number of coffee to be produced is

    0.6*(80) + 0.8*(120) + 0.6*(200) (1)

    Max tonnes per blend is

    0.4*80 + 0.4*120 + 0.2*200 = 120 tonnes (2)
    0.2*80 + 0.4*120 + 0.4*200 = 144 tonnes (3)

    Problem is
    Max Z = 600*x1 + 300*x2
    st
    x1 + x2 <= 264
    x1 <= 120
    x2 <= 144

    The solution is then trivial, the optimal solution occurs at (144, 120) and the max profit is $122,400

    ---------

    I'm not sure if this solution is correct, can someone help...
     
  2. jcsd
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