Question

Solve the following linear programming problem graphically:

Minimise Z = x − 5y

subject to the constraints:

x − y ≥ 0

− x + 2y ≥ 2

x ≥ 3, y ≤ 4, y ≥ 0

Answer 1

Given LPP

Min Z = x − 5у 

Subject to constraints,

x − y ≥ 0

− x + 2y ≥ 2

x ≥ 3, y ≤ 4,

y ≥ 0

Let x − y = 0

x = y

x	0	1	4
y	0	1	4

and x + 2y = 2

x	0	2
y	1	0

x = 3 is a line parallel to the Y-axis at x = 3 

y = 4 is a line parallel to the X-axis at y = 4

For shading

Take point (1, 0) in

x − y ≥ 0 

1 ≥ 0 true

Shading towards (1, 0)

x + 2y ≥ 2

1 ≥ 2 false

x ≥ 3

1 ≥ 3 false

Shading away from (1, 0)

y ≤ 4 

0 ≤ 4 true

Towards (1, 0)

Intersection point x − y = 0 and x = 3 is (3,3)

(x − y) = 0 and y = 4 is (4, 4)

The region is unbounded with corner points

A(4, 4), B(3, 3), C(3, 0)

Corner point	Value of Z = x − 5y
A(4, 4)	Z = 4 − 20 = −16
B(3, 3)	Z = 3 − 15 = −12
C(3, 0)	Z = 3

From the table, 

Min Z = −16 at x = 4, y = 4