Question

Minimize and maximize $z=5x+10y$
subject to the constraints
$x+2y\le 120$
$x+y\ge 60$
$x-2y\ge 0$ and $x\ge 0,y\ge 0$ by graphical method.

Answer 1

$z=5x+10y$
subject to the constraints
$x+2y\le 120$
$x+y\ge 60$
$x-2y\ge 0$ and $x\ge 0,y\ge 0$

We take the given inequality to fing the region bounded by these inequalities.

$x+2y=120$

$x$	$0$	$60$	$120$
$y$	$60$	$30$	$0$

$x+y=60$

$x$	$0$	$40$	$60$
$y$	$60$	$20$	$0$

$x-2y=0$

$x$	$0$	$40$	$60$
$y$	$0$	$20$	$30$

The shaded region $BDEF$ determined by linear inequalities shows the easible region

Let us evaluate the objective function $Z$ at each point as shown below

At $B(120,0),$ $Z=5\times 120+10\times 0=600$

At $D(60,0),$ $Z=5\times 60+10\times 0=300$

At $E(40,20),$ $Z=5\times 40+10\times 20=400$

At $F(60,30),$ $Z=5\times 60+10\times 30=600$

Hence, Maximum value of $Z$ is $600$ at $F(60,30)$ and minimum value of $Z$ is $300$ at $D(60,0)$