Question

A small firm manufactures gold rings and chains. The total number of rings and chains manufactured per day is at most 24. It takes 1 hour to make a ring and 30 minutes to make a chain. The maximum number of hours available per day is 16. If the profit on a ring is Rs. 300 and that on a chain is Rs 190, find the number of rings and chains that should be manufactured per day, so as to earn the maximum profit. Make it as an L.P.P. and solve it graphically.

Answer 1

Total no. of rings & chain manufactured per day = 24. 

Time taken in manufacturing ring = 1 hour 

Time taken in manufacturing chain = 30 minutes 

One time available per day = 16 

Maximum profit on ring = Rs 300 

Maximum profit on chain = Rs 190 

Let gold rings manufactured per day = x 

Chains manufactured per day = y 

L.P.P. is maximize Z = 300x + 190y

Subject to x ≥ 0, 

y ≥ 0, 

x + y ≤ 24 and x + 1/2y ≤ 16 

On plotting graph of above constraints or inequalities, we get shaded region. 

Possible points for maximum Z are (16, 0), (8, 16) and (0, 24). 

At (16, 0), Z = 4800 + 0 = 4800 

At (8, 16), Z = 2400 + 3040 = 5440 ← Maximum 

At (0, 24), Z = 0 + 4560 = 4560 

Z is maximum at (8, 16). 

∴ 8 gold rings & 16 chains must be manufactured per day.