LP objective function with unknown parameters

Tags:
1. Nov 9, 2015

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

2. Nov 9, 2015

lavinia

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

3. Nov 10, 2015

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.

4. Nov 10, 2015

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 $\lambda$.

5. Nov 16, 2015

Mark J.

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