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Linear Programming Problem Setup

  1. Sep 11, 2012 #1
    We were given this problem in a Linear Programming class and asked to define the constraints.

    1. The problem statement, all variables and given/known data

    max Z = max (xεS) {min {Z1, Z2...Zq}}

    where Zi=C1ix1 + C2ix2+...+Cnixn


    2. Relevant equations

    Constraints need to be defined to set up the problem.

    3. The attempt at a solution

    The first few Z equations would be:
    Z1=C11x1 + C21x2+...+Cn1xn

    Z2=C12x1 + C22x2+...+Cn2xn

    Zq=C1qx1 + C2qx2+...+Cnqxn

    I think the best way to ensure that the Z is at a minimum is to define the inequalities below:
    Z < Z1
    Z < Z2
    ...
    Z < Zq

    This ensures that we pick the minimum value of Z. But these constraints should also include some way to maximize x and I am confused as to how to include that.
     
  2. jcsd
  3. Sep 12, 2012 #2

    Ray Vickson

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    As written, your formulation will fail, but it can be modified slightly to work properly. Come back for additional hints when you have dealt with this issue.

    RGV
     
  4. Sep 12, 2012 #3
    Would it work to write the constraints as follows instead of saying Z<Zi. This way you are telling the program to choose the maximum values of x that lead to the minimum Z.

    Z ≤ c11x1+...+c1nxn

    Z ≤ c21x1+...+c2nxn

    ...

    Z ≤ cq1x1+...+cqnxn
     
  5. Sep 12, 2012 #4

    Ray Vickson

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    Why would you want to minimize Z? You were asked to maximize the minimum, not minimize it. Anyway, as written your problem would be unbounded, with Zmin = -∞.

    RGV
     
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