Question

Solve the following linear programming problem graphically:

Maximise Z = 20x + 30y

Subject to the constraints:

x + y ≤ 80 

2x + 3y ≥ 100

x ≥ 14 

y ≥ 14

Answer 1

Maximise Z = 20x + 30y

Subject to constraints

x + y ≤ 80

2x + 3y ≥ 100 

x ≥ 14,

y ≥14 

Convert into equations

x + y = 80   ...(i)

x	0	80
y	80	0

2 + 3y = 100   ...(ii)

x	0	50
y	100/3	0

x = 14,    ...(iii)

y = 14    ...(iv)

Solving equations (i) and (iii),

x + y = 80

14 + y = 80

y = 66

x = 14

A (14, 66)

Equations (ii) and (iii),

2(14) + 3y = 100

3y = 100 − 28

3y = 72

y = `72/3`

y = 24

∴ x = 12, y = 24

B(14, 24)

Solving equations (ii) and (iv),

2x + 3(14) = 100

2x = 100 − 42

2x = 58

x = `58/2`

x = 29

∴ x = 29, y = 14

C(29, 14)

Solving equations (i) and (iv)

x + 14 = 80

x = 80 − 14

x = 66

D(66, 14)

Corner Points	Maximum Z = 20x + 30y
A(14, 66)	Z = 20 × 14 + 30 × 66 = 2260
B (14, 24)	Z = 20 × 14 + 30 × 24 = 1000
C(29, 14)	Z = 20 × 29 + 30 × 14 = 1000
D(66, 14)	Z = 20 × 66 + 30 × 14 = 1740

Maximum value of Z = 2260 at A(14, 66)