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LP objective function with unknown parameters

  1. Nov 9, 2015 #1
    Hi All,

    I am struggling with minimization(maximization) problems which needs to be solved graphically but they have unknown parameter in objective function:

    For example:

    f = 2x_1 + \lambda x_2(min)

    for conditions:

    -x_1 + x_2 \leq 3
    x_1 + 2x_2 \leq 12
    3x_1 -x_2 \leq 15
    x_i \geq 0

    More than solution I need to understand the way so I can proceed for similar examples

    Regards
     
  2. jcsd
  3. Nov 9, 2015 #2

    lavinia

    User Avatar
    Science Advisor
    Gold Member

    I am having trouble reading this. can you redo it?
     
  4. Nov 10, 2015 #3

    fresh_42

    Staff: Mentor

    There are 4 straight lines limiting the search area and one as function f to minimize. Draw all 4 lines in cartesian coordinates, i.e a piece of paper with x_1 and x_2 axes. Then determine the areas defined by them (hatch them). Finally draw f and look whether you have to push it up or down to minimalize it's value. The searched point will be on the boundary you drew.
     
  5. Nov 10, 2015 #4

    HallsofIvy

    User Avatar
    Staff Emeritus
    Science Advisor

    No it is not possible to determine max and min without knowing [itex]\lambda[/itex]. The basic "rule" of linear programming is that max and min of a linear function on a convex polygon occurs at a vertex. It is fairly easy to determine the vertices of the given convex polygon but when you evaluate f at the vertices, the value will depend upon[itex]\lambda[/itex] so that knowing which is largest and which is smallest will depend upon [itex]\lambda[/itex].
     
  6. Nov 16, 2015 #5
    So how to determine when problem has optimal solution. infinite or no solution depending on lambda??
     
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