Question

Maximise $Z=x+2y$
Subject to the constraints
$x+2y\ge 100$
$2x-y\le 0$
$2x+y\le 200$
$x,y\ge 0$.
Solve the given LPP graphically.

Answer 1

Given LPP is :
Objective function $Z=x+2y$
Subject to the constraints
$x+2y\ge 100$
$2x+y\le 200$
$2x-y\le 0$
$x\ge 0$ and $y\ge 0$



Consider $x+2y=100$

Table of solution is :

$\begin{array}{ccc}x& 0& 100\\ y& 50& 0\end{array}$

Consider $2x-y=0...\left(ii\right)$

Table of solution is :
$\begin{array}{ccc}x& 0& 50\\ y& 0& 100\end{array}$
Consider $2x+y=200...\left(iii\right)$

Table of solution is :

$\begin{array}{ccc}x& 100& 0\\ y& 0& 200\end{array}$

To solve the LPP, we draw the graph of the inequations and get the feasible solution shown (shaded) in the graph.

Corner points of the common shaded region are $A(0,200),B(0,50),C(20,40)\text{and}D(50,100)$. Value of $Z$ at each corner point is given as:

$\begin{array}{cc}\text{Corner Points}& \text{Value of the objective function}\\ & Z=x+2y\\ AtA(0,200)& Z=0+400=400\text{(Maximum)}\\ AtC(20,40)& Z=20+2\left(40\right)=100\\ AtB(0,50)& Z=0+2\left(50\right)=100\\ AtD(50,100)& Z=50+2\left(100\right)=250\end{array}$

Hence, $Z=400$ is the maximum value obtained by putting $x=0,y=200.$