A factory owner purchases two types of machines, A and B, for his factory. The requirements and the limitations for the machines are as follows: Machine A: occupies $1000\ \text{m}^2$, needs 12 men, daily output 60 units. Machine B: occupies $1200\ \text{m}^2$, needs 8 men, daily output 40 units. He has a maximum area of $9000\ \text{m}^2$ available, and 72 skilled labourers who can operate both the machines. How many machines of each type should he buy to maximize the daily output?

CBSE Class 12 · 2008 · Linear ProgrammingLong Answer

A factory owner purchases two types of machines, A and B, for his factory. The requirements and the limitations for the machines are as follows:
Machine A: occupies

1000 m^{2}

, needs 12 men, daily output 60 units.
Machine B: occupies

1200 m^{2}

, needs 8 men, daily output 40 units.
He has a maximum area of

9000 m^{2}

available, and 72 skilled labourers who can operate both the machines. How many machines of each type should he buy to maximize the daily output?

Answer: Max output = 360 units at (6, 0): 6 machines of type A and none of type B

Solution

Let the owner buy

x

machines of type A and

y

of type B.
Daily output to maximise:

Z = 60 x + 40 y

.
Area:

1000 x + 1200 y \leq 9000 ⟹ 5 x + 6 y \leq 45

Labour:

12 x + 8 y \leq 72 ⟹ 3 x + 2 y \leq 18

x \geq 0, y \geq 0

Corner points:

(0, 0)

(6, 0)

(\frac{9}{4}, \frac{45}{8})

(intersection of the two lines) and

(0, \frac{15}{2})

.
Evaluate

Z = 60 x + 40 y

(0, 0)

Z = 0

(6, 0)

Z = 360

← maximum

(\frac{9}{4}, \frac{45}{8})

Z = 135 + 225 = 360

← also maximum

(0, \frac{15}{2})

Z = 300

The maximum

Z = 360

occurs at

(6, 0)

and at

(\frac{9}{4}, \frac{45}{8})

. Since the number of machines must be a whole number, only

(6, 0)

is admissible.

∴ Z_{m a x} = 360 at (6, 0)

The owner should buy 6 machines of type A (and none of type B).

Structure Feasible region for 5x + 6y = 45 and 3x + 2y = 18, with corner points (0, 0), (6, 0), (9/4, 45/8) and (0, 15/2). — Feasible region for 5x + 6y = 45 and 3x + 2y = 18, with corner points (0, 0), (6, 0), (9/4, 45/8) and (0, 15/2).

CBSE Class 12 Board · 2008 · Subjective

Practise more: all CBSE board PYQs on Linear Programming →

CBSE Class 12 · 2008 · Linear ProgrammingLong Answer

Solution

More board questions on Linear Programming