Minimize and Maximize $Z = x + 2y$ subject to $x + 2y \ge 100,2x - y \le 0,2x + y \le 200;x,y \ge 0$ .

.: The system of constraints is :

$x + 2y \ge 100$ ...(1)

$2x - y \le 0$ ...(2)

$2x + y \le 200$ ...(3)

and $x,y \ge 0$ ...(4)

Let ${l_1}:x + 2y = 100$

${l_2}:2x - y = 0$

${l_3}:2x + y = 200$

The shaded region in the adjoining figure is

the feasible region determined by the system of constraints (1) to (4).

It is observed that the feasible region ECDB is bounded.

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

We have :$Z = x + 2y$ ...(5)

The co-ordinates of E, C, D and B are
(20, 40) (on solving $x + 2y = 100$ and$2x - y = 0$)

(50, 100) (on solving $2x + y = 200$ and$2x - y = 0$ )

(0, 200) and (0, 50) respectively.

Hence, ${Z_{\max }} = 400$ at $(0,200)$ and ${Z_{\min }} = 100$

at all points on the line segment joining the points (0, 50) and (20, 40).