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 Rs5 and that from a shade is Rs3.Assuming that the manufacturer can sell all the lamps and shades that he produces,how should he schedule his daily production in order to maximize his profit?

Answer 1

Let the cottage manufacture industry x pedestal lamps and y wooden shades.

Therefore, x≥0 and y≥0 The given information can be compiled in a table as follows.

| Lamps| Shades| Availability
---|---|---|---
**Grinding/Cutting Machine(h) **| 2| 1| 12
Sprayer(h)| 3| 2| 20

Therefore, the constraints are 2x+y≤12 3x+2y≤20 Total profit,Z=5x+3y

The mathematical formulation of the given problem is
Maximize Z=5x+3y...(1)

Subject to the constraints,
2x+y≤12...(2)
3x+2y≤20....(3)
x,y≥0....(4)

The feasible region determined by the system of constraints is as follows.

The corner points are A(6,0),B(4,4)and C(0,10).
The value of Z at these corner points are as follows.

Corner points| Z=5x+3y|
---|---|---
A(6,0)| 30|
B(4,4)| 32| →Maximum
C(0,10)| 30|

Thus, the manufacturer should produce 4 pedestal lamps and 4 wooden shades to maximize his profits.