LP objective function with unknown parameters

[Mark J.]

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

 

[lavinia]

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

I am having trouble reading this. can you redo it?

 

[fresh_42]

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.

 

[HallsofIvy]

No it is not possible to determine max and min without knowing $\lambda$. 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$\lambda$ so that knowing which is largest and which is smallest will depend upon $\lambda$.

 

[Mark J.]

So how to determine when problem has optimal solution. infinite or no solution depending on lambda??