Question

Solve the following problem graphically:
Minimize and Maximize $Z=3x+9y$
Subject to the constraints:
$x+3y\le 60$
$x+y\ge 10$
$x\le y$
$x\ge 0,y\ge 0$

Answer 1

Maximise and Minimise : $Z=3x+9y$
Subject to
$x+3y\le 60$, $x+y\ge 10$, $x\le y$
$x\ge 0,y\ge 0$

$x+3y=60$
$\begin{array}{ccc}x& 0& 60\\ y& 20& 0\end{array}$

$x+y=10$
$\begin{array}{ccc}x& 0& 10\\ y& 10& 0\end{array}$

$x=y$
$\begin{array}{ccc}x& 0& 20\\ y& 0& 20\end{array}$
$\begin{array}{cc}\text{Corner points}& \text{Value of}Z=3x+9y\\ (0,10)& 90\\ (5,5)& 60\\ (15,15)& 180\\ (0,20)& 180\end{array}$

$\therefore$ $Z=60$ is minimum at $(5,5)$
Also,
$Z$ is maximum at two points $(15,15)\&(0,20)$
$\therefore$ $Z=180$ is maximum at all points joining $(15,15)\&(0,20)$.