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16. 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 ?.5 and that from a shade is ?.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 maximize his profit?

0 3 Views | Posted 4 months ago
Asked by Shiksha User

  • 1 Answer

  • V

    Answered by

    Vishal Baghel | Contributor-Level 10

    4 months ago

    Let the cottage industry manufacture 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

    The profit on a lamp is Rs 5 and on the shades is Rs 3. 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 values of Z at these corner points are as follows

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

    Thus, the manufacturer should produce 4 p

    ...more

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