Maximise and minimise Z = x + 2y subject to the constraints x + 2 y ≥ 100, 2x – y ≤ 0, 2x+ y ≤ 200, x, y ≥ 0
Solve the above LPP graphically.

Our problem is to minimise and maximise
$\mathrm{Z}=\mathrm{x}+2\mathrm{y}.....\left(\mathrm{i}\right)$
Subject to constraints,
$\begin{array}{l}\mathrm{x}+2\mathrm{y}\ge 100 .......\left(\mathrm{ii}\right)\\ 2\mathrm{x}-\mathrm{y}\le 0 ..........\left(\mathrm{iii}\right)\\ 2\mathrm{x}+\mathrm{y}\le 200........\left(\mathrm{iv}\right)\\ \text{ and }\mathrm{x}\ge 0,\mathrm{y}\ge 0 ........\left(\mathrm{v}\right)\end{array}$
Table for line x+2y=,100 is
 

So, the half plane is towards Y-axis.
Table for line 2x+y=200 is 
 

So, the line 2 x+y=200 is passing through the points (0,200) and (100,0) .
On putting (0,0) in the inequality 2 x+y \leq 200, we get
$2\times 0+0\le 200$
$\Rightarrow 0\le 200$ (which is true)
So, the half plane is towards the origin.
$\text{ Also, }\mathrm{x},\mathrm{y}\ge 0$
$\mathrm{So}, \mathrm{the} \mathrm{region} \mathrm{lies} \mathrm{in} \mathrm{the} \mathrm{I} \mathrm{quadrant}.$

$\mathrm{Clearly}, \mathrm{feasible} \mathrm{region} \mathrm{is} \mathrm{ABCDA}.$
On solving equations $2\mathrm{x}-\mathrm{y}=0\text{ and }\mathrm{x}+2\mathrm{y}=100$, we get $3(20,40)$
Again, solving the equations $2\mathrm{x}-\mathrm{y}=0\text{ and }2\mathrm{x}+\mathrm{y}=200\text{, we get }\mathrm{C}(50,100)$
$\text{So, the line }\mathrm{x}+2\mathrm{y}=100 \mathrm{is} \mathrm{passing} \mathrm{through} \mathrm{the} \mathrm{points} (0,50)\text{ and }(100,0).$
$\text{ On putting }(0,0)\text{ in the inequality }x+2y\ge 100\text{, we get }$
$\begin{array}{l}0+2\times 0\ge 100\\ \Rightarrow 0\ge 100\text{ (which is false) }\end{array}$
So, the half plane is away from the origin.
$\text{Table for line }2x-y=0\text{ is }$
 

$\text{ So, the line }2x-y=0\text{ is passing through the points }(0,0)\text{ and }(10,20)\text{. }$
$\text{ On putting }(5,0)\text{ in the inequality }2\mathrm{x}-\mathrm{y}\le 0\text{, we get }$
$2\times 5-0\le 0$
$\Rightarrow 10\le 0\text{ (which is false) }$
The comer points of the feasible region are
$\mathrm{A}(0,50),\mathrm{B}(20,40),\mathrm{C}(50,100)\text{ and }\mathrm{D}(0,200)\text{. }$
The values of Z at corner points are given below:
 

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