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6. 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?
Answers (1)
Let the cottage industry manufactures x pedestal lamps and y wooden shades. Thus, .
The given information can be represented in the table as :
| lamps | shades | availability | |
| machine (h) | 2 | 1 | 12 |
| sprayer (h) | 3 | 2 | 20 |
Profit on a lamp is Rs. 5 and on the shade is 3.
Therefore, constraint is
The feasible region determined by constraints is as follows:
The corner points of the feasible region are
The value of Z at corner points is as shown :
| Corner points |
|
|
| 30 | ||
| 32 | maximum | |
| 30 | ||
| 0 |
The maximum value of z is 32 at .
Thus, 4 shades and 4 pedestals lamps should be manufactured every day to get the maximum profit.
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