Answer to Question #292757 in Operations Research for Yihenew Awoke

Question

Solve the following LP problem by graphical method: Maximise Z = 300X1 + 700X2

Subject to the constraints: X1 + 4X2 ≤ 20 2X1 + X2 ≤ 30 X1 + X2 ≤ 8 And X1, X2 ≥ 0

Answer 1

Given, Maximize $Z = 300x_1 + 700x_2$

subject to the constraints:

$x_1 + 4x_2 \le 20~~~~~-(1)\\ 2x_1 + x_2 \le 30~~~~~-(2)\\ x_1 + x_2 \le 8 ~~~~~~~~~-(3)\\ ~\text{and}~~x_1, x_2 \ge 0$

We consider the constraints as equations to plot the graph.

$x_1 + 4x_2 = 20~~~~~-(1)\\ 2x_1 + x_2 = 30~~~~~-(2)\\ x_1 + x_2 = 8 ~~~~~~~~~-(3)\\$

From Equation 1, when $x_1 = 0$ , we get $x_2 = 5$ . When $x_2 = 0$ , we get $x_1 = 20$.

From Equation 2, when $x_1 = 0$ , we get $x_2 = 30$ . When $x_2 = 0$ , we get $x_1 = 15$.

From Equation 3, when $x_1 = 0$ , we get $x_2 = 8$ . When $x_2 = 0$ , we get $x_1 = 8$.

The following image shows the graph of the equation 1, 2, and 3.

The feasible region is bounded by the points OABC which is the region bounded by the lines given by equations 1 and 3. The values of the objective function at extreme points are given in the following table.

The above table shows the maximum value of the objective function z = 4000 occurs at the extreme point B (4,4). Hence, the optimal solution to the given linear programming problem is $x_1 = 4, x_2 = 4$  and maximum z = 4000.