I am learning a little bit of Linear Programming and came across this question which I am unable to solve or even begin solving.
The question is: Construct an LP with a feasible region containing four corner points at which the objective function assumes values z = 0, z = 2 and z = 8.
I managed to work out that these points must lie on either axis and z = 0 being at (0,0). Beyond that, I am stuck.
All help is appreciated.
One of the main ideas is to build a symmetrical domain (with respect ot the exhange $x \leftrightarrow y$ for feasible solutions.
Here is one.
Consider, for $x \geq 0, y geq 0$, the constraints:
$$\begin{cases} \ \ 2x-y & \leq & 4\\-x+2y &\leq &4\end{cases}$$
and maximize function $x+y$ on this domain.
You will see that the corner points are $(0,0)$,$(0,2)$, $(2,0)$ and $(4,4)$ with values of the objective function $0,2,2,8$ resp. as desired.
Have a look at the picture below where you see how the linear function to be maximized "sweeps" the domain.
