Question

Minimise and Maximise $Z=x+2y$
subject to $x+2y\ge 100,2x-y\le 0,2x+y\le 200;x,y\ge 0$

Answer 1

The feasible region determined by the constraints $x+2y\ge 100,2x-y\le 0,2x+y\le 200;x\ge 0,y\ge 0$ is as shown.
The corner points of the feasible region are $A(0,50),B(20,40),C(50,100)$ and $D(0,200)$
The values of $Z$ at these corner points are as follows.

Corner point	$Z=x+2y$
$A(0,50)$	$100$	$\to$ Minimum
$B(20,40)$	$100$	$\to$ Minimum
$C(50,100)$	$250$
$D(0,200)$	$400$	$\to$ Maximum

The maximum value of $Z$ is $400$ at $(0,200)$ and the minimum value of $Z$ is $100$ at all the points on the line segment joining the points $(0,50)$ and $(20,40)$.