Minimize and Maximize $Z = 5x + 10y$ subject to $x + 2y \le 120,x + y \ge 60,x - 2y \ge 0,x,y \ge 0.$

.: The system of constraints is :

$x + 2y \le 120$ …(1)

$x + y \ge 60$ …(2)

$x - 2y \ge 0$ …(3)

and $x,y \ge 0$ ….(4)

Let ${l_1}:x + 2y = 120$

${l_2}:x + y = 60$

${l_3}:x - 2y = 0$

It is observed that the feasible region CADE is bounded.

The co-ordinates of C, A, D, E are (60, 0), (120, 0), (60, 30), (40, 20).

Thus, we use Comer Point Method to determine the maximum and minimum values of Z.

We have : $Z = 5x + 10y$

Hence,${Z_{\min }} = 300$ at (60, 0) and

${Z_{\max }} = 600$ at all points on the line segment joining the points (120,0) and (60, 30).