Question

A cottage industry manufactures pedestal lamps and wooden shades, each requiring the use of a grinding/cutting machine and a sprayer. It takes 2 hours on grinding/cutting machine and 3 hours on the sprayer to manufacture a pedestal lamp. It takes 1 hour on the grinding/ cutting machine and 2 hours on the sprayer to manufacture a shade. On any day, the sprayer is available for at the most 20 hours and the grinding/cutting machine for at the most 12 hours. The profit from the sale of a lamp is Rs. 5 and that from a shade is Rs. 3. Assuming that the manufacturer can sell all the lamps and shades that he produces, how should he schedule his daily production in order to maximise his profit?

Answer 1

	Time on grinding machine	Time on sprayer	Profit
(i) pedestal lamp(x)	2hrs	3hrs	Rs 5
(ii) wooden shades(y)	1hr	2hrs	Rs. 3
(iii) Total	at most 12 hrs	at most 20 hrs

P = 5x + 3y subject to the constraints 

(i) 2x + y ≤ 12 

(ii) 3x + 2y ≤ 20 

(iii) x ≥0; y ≥0

ABCD is the solution region

A(0,0) P= 0 B(6,0) P = 5 x 6+3 x 0 = 30 Rs. 

C(4,4) P = 5 x 4+3 x 4 = 32 Rs. 

D(0,10) P = 5 x 0+3 x 10 = 30 Rs. 

At C (4, 4) p is maximised to 32 Rs. 

i.e. 4 pedestal lamps and 4 wooden shades.