A small firm manufactures necklaces and bracelets. The total number of necklaces and bracelets that it can handle per day is at most24. It takes one hour to make a bracelet and half an hour to make a necklace. The maximum number of hours available per day is 16. If the profit on a necklace is ` 100 and that on a bracelet is ` 300. Formulate on L.P.P. for finding how many of each should be produced daily to maximize the profit? It is being given that at least one of each must be produced.

Let number of necklaces manufactured be x and y be the number of bracelets manufactured.

As the total number of items are at most 24;

∴ x + y ≤ 24 ……(i)

Necklaces takes half an hour and Bracelets takes 1 hour to manufacture.

⇒ x items take x/2 hours and y items take y hours to manufacture and the total time available is 16 hours.

∴ x + 2y ≤ 32 ……(ii)

The profit of one necklace is Rs. 100 and that of bracelet is Rs. 300.

Let the profit be Z

∴ Z = 100x + 300y ……(iii)

Equations (i), (ii) and (iii) forms the LPP.

(0,0),(0,16),(16,8) and (24,0) are the boundary points.

At (0,0) Z = 0

At (0,16) Z = 4800

At (16,8) Z = 4000

At (24,0) Z = 2400

Since at least one of each should be produced;

16 necklaces and 8 bracelets should be produced daily for maximum profit.