Question

The feasible region for an L.P.P. is shown in the adjoining figure:

Based on the given graph, answer the following questions.

1. Write the constraints for the L.P.P.
2. Find the coordinates of the point B.
3. Find the maximum value of the objective function Z = x + y.

Answer 1

a. From the figure:

Line AD passes through A(25, 0) and D(0, 50).

Intercept form:

`x/25 + y/50 = 1`

⇒ `50(x/25 + y/50) = 50`

⇒ 2x + y = 50

Line EC passes through E(40, 0) and C(0, 20).

Intercept form:

`x/40 + y/20 = 1`

⇒ `40(x/40 + y/20) = 40`

⇒ x + 2y = 40

Since the feasible region lies in the first quadrant and includes the origin,

2x + y ≤ 50

x + 2y ≤ 40

x ≥ 0, y ≥ 0

b. Point B is the intersection of:

2x + y = 50  ....(1)

x + 2y = 40  ....(2)

From (1):

y = 50 − 2x

Put in (2):

x + 2(50 − 2x) = 40

x + 100 − 4x = 40

−3x = −60

x = 20

y = 50 − 2(20) = 10

⇒ B(20, 10)

c. The corner points are O(0, 0), C(0, 20), B(20, 10) and A(25, 0).

Z = x + y

At O(0, 0), Z = 0

At C(0, 20), Z = 20

At B(20, 10), Z = 30

At A(25, 0), Z = 25

∴ Maximum value of Z is 30 at (20, 10).