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

To solve the problem graphically, we consider the constraints as equations and draw straight lines.

$\begin{aligned} x_1 + 4x_2 &= 20\qquad(1)\\ 2x_1 + x_2 &= 30\qquad(2)\\ x_1 + x_2 &= 8~~\qquad(3)\\ \end{aligned}$

The graph of the above equations is plotted below.

The region of feasibility is bounded by the extreme points OABC. Point B is the point of intersection of lines (1) and (3). Point B is obtained by solving equations (1) and (3).

The value of the objective at these extreme points is given in the following table

$\begin{array}{|c|c|} \hline \text{Extreme point} & \text{Value of~} z = 300x_1 + 700 x_2\\ \hline A(0,0) & 0\\ B(4,4) & 4000\\ C(8,0) & 2400\\ D(0,5) & 3500\\ \hline \end{array}$

From the above table, the maximum value is z = 4000 occurs at B(4,4).