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Prove nonempty, unbounded feasible region has no optimal solution

  1. Feb 9, 2012 #1
    1. The problem statement, all variables and given/known data

    Let C be the set of all points (x,y) in the plane satisfying x≥0, y≥0, -x-2y≥-8.

    a. Show that C is nonempty and unbounded.
    b. Prove that the LP problem: Max M=2x+3y subject to the constraint that (x,y) lie in C has no feasible, optimal solution.
    c. Show that the LP problem: Max M=-3x-6y subject to the constraint that (x,y) lie in C does have a feasible, optimal solution.


    2. Relevant equations



    3. The attempt at a solution

    a. I graphed the constraints and showed that the feasible region is the entire first quadrant, and therefore C is nonempty and unbounded (provided attachment of my work - is this enough?)

    b. I could "show" this but I have no idea how to "prove" it. Does it involve the simplex method?

    c. Same question as above
     

    Attached Files:

  2. jcsd
  3. Feb 9, 2012 #2

    Ray Vickson

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    There is something wrong with the question: what it is asking you is false if you have copied down the constraints correctly. The constraints are x ≥ 0, y ≥ 0 and -x-2y ≥ -8, which is the *same as* x + 2y ≤ 8. Here the feasible region is nonempty and bounded, and so the problems max 2x + 3y and max -3x - 6y (which is the same as min 3x + 6y) both have feasible, optimal solutions.

    RGV
     
  4. Feb 9, 2012 #3
    Sorry here is the problem, corrected:
    Let C be the set of all points (x,y) in the plane satisfying x≥0, y≥0, -x-2y≤-8.

    a. Show that C is nonempty and unbounded.
    b. Prove that the LP problem: Max M=2x+3y subject to the constraint that (x,y) lie in C has no feasible, optimal solution.
    c. Show that the LP problem: Max M=-3x-6y subject to the constraint that (x,y) lie in C does have a feasible, optimal solution.
     
  5. Feb 9, 2012 #4
    someone please help :(
     
  6. Feb 9, 2012 #5

    Ray Vickson

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    I have already helped, but you have ignored my assistance.

    RGV
     
  7. Feb 9, 2012 #6
    No, I had the problem written down wrong. You helped me with the wrong problem
     
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