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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 shade that he produces, how should he schedule his daily production in order to maximize his profit?



Answer -

Let the cottage industry manufacture x pedestal lamps and ywooden shades respectively

Hence,

x ≥ 0 and y ≥ 0

The given information can be compiled in a table is given below

Lamps

Shades

Availability

Grinding / Cutting Machine (h)

2

1

12

Sprayer (h)

3

2

20

The profit on a lamp is Rs 5 and on the shades is Rs 3. Hence,the constraints are

2x + y ≤ 12

3x + 2y ≤ 20

Total profit, Z = 5x + 3y …………….. (i)

Subject to the constraints,

2x + y ≤ 12 …………. (ii)

3x + 2y ≤ 20 ………… (iii)

x, y ≥ 0 …………. (iv)

The feasible region determined by the system of constraints isgiven below

A (6, 0), B (4, 4) and C (0, 10) are the corner points

The value of Z at these corner points are given below

Corner point

Z = 5x + 3y

A (6, 0)

30

B (4, 4)

32

Maximum

C (0, 10)

30

The maximum value of Z is 32 at point (4, 4)

Therefore, the manufacturer should produce 4 pedestal lamps and4 wooden shades to maximize his profits.

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