Question

Minimize and maximize $Z=600x+400y$
Subject to the constraints
$x+2y\le 12$
$2x+y\le 12$
$4x+5y\le 20$
$x\ge 0;y\ge 0$ by graphical method

Answer 1

Draw the graph of $2x+y=12$ ..... $\left(i\right)$
We get intersection of $\left(i\right)$ with the coordinate axes at points $(0,12)$ and $(6,0)$
Draw the graph of $x+2y=12$ ..... $\left(ii\right)$

We get the intersection of $\left(ii\right)$ with the coordinate axes at points $\left(06\right)$ and $\left(120\right)$
Draw the graph of $4x+5y=20$..... $\left(iii\right)$
We get the intersection of $\left(iii\right)$ with the coordinate axes at points $(0,4)$ and $(5,0)$

Common shaded region is the feasible region with corner points $(5,0),(6,0),(4,4),(0,6),(0,4)$

Corner points	$Z=600x+400y$
$(0,4)$	$1600$ minimum
$\left(0.6\right)$	$2400$
$(4,4)$	$4000$ maximum
$(6,0)$	$3600$
$(5,0)$	$3000$

Minimum value of $z=1600$ at $(0,4)$
Maximum value of $z=4000$ at $(4,4)$