A cottage industry manufactures pedestal lamps and wooden shades, each requiring the use of grinding/cutting machine and a sprayer. It takes 2 hours on the grinding/cutting machine and 3 hours on the sprayer to manufacture a pedestal lamp while 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 most 20 hours and the grinding/cutting machine for at most 12 hours. The profit from the sale of a lamp is Rs 5.00 and a shade is Rs 3.00. 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?

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Solution

Let the cottage industry manufacture x pedestal lamps and y wooden shades.
Number of lamps and wooden shades cannot be negative.
Therefore, x ≥ 0 and y ≥ 0

The given information can be compiled in a table as follows:

Lamps

Shades

Availability

Grinding/Cutting Machine (hr)

2

1

12

Sprayer (hr)

3

2

20

Therefore, the constraints are

The profit on a lamp is Rs 5 and on the shades is Rs 3.Therefore, profit on x pedestal lamps and y wooden shades is Rs 5x and Rs 3y respectively.

Total profit, Z = 5x + 3y

The mathematical formulation of the given problem is

Maximize Z = 5x + 3y

subject to the constraints,

x, y ≥ 0

First we will convert inequations into equations as follows:
2x + y =12, 3x + 2y = 20, x = 0 and y = 0

Region represented by 2x + y ≤ 12:
The line 2x + y = 12 meets the coordinate axes at A(6, 0) and B(0, 12) respectively. By joining these points we obtain the line
2x + y = 12. Clearly (0,0) satisfies the inequation 2x + y ≤ 12. So,the region which contains the origin represents the solution set of the inequation 2x + y ≤ 12.

Region represented by 3x + 2y ≤ 20:
The line 3x + 2y = 20 meets the coordinate axes at $C (\frac{20}{3}, 0)$ and D(0, 10) respectively. By joining these points we obtain the line
3x + 2y = 20. Clearly (0,0) satisfies the inequation 3x + 2y ≤ 20. So,the region which contains the origin represents the solution set of the inequation 3x + 2y ≤ 20.

Region represented by x ≥ 0 and y ≥ 0:
Since, every point in the first quadrant satisfies these inequations. So, the first quadrant is the region represented by the inequations x ≥ 0, and y ≥ 0.

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

The corner points are O(0, 0), A(6, 0), E(4, 4), and D(0, 10).

The values of Z at these corner points are as follows

Corner point Z = 5x + 3y

O 0

A 30

E 32

D 30

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

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

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