A manufacturer considers that men and women workers are equally efficient and so he pays them at the same rate. He has 30 and 17 units of workers (male and female) and capital respectively, which he uses to produce two types of goods A and B. To produce one unit of A, 2 workers and 3 units of capital are required while 3 workers and 1 unit of capital is required to produce one unit of B. If A and B are priced at Rs 100 and Rs 120 per unit respectively, how should he use his resources to maximize the total revenue? Form the above as an LPP and solve graphically. Do you agree with this view of the manufacturer that men and women workers are equally efficient and so should be paid at the same rate?

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Solution

Let the number of units of A and B that are produced be x and y respectively. If A and B are priced at Rs 100 and Rs 120 per unit respectively.
Therefore, total revenue R = Rs (100x + 120y)
Clearly, x, y ≥ 0
To produce one unit of A, 2 workers while 3 workers are required to produce one unit of B.
Therefore, total number of workers used in the production of given units of A and B is

To produce one unit of A, 3 units of capital while 1 unit of capital is required to produce one unit of B.
Therefore, total capital used in the production of given units of A and B is 3x + y.
As per the information given in the question, the following must hold true:

$2 x + 3 y \leq 30 and 3 x + y \leq 17$

We have to maximize Z = 100x + 120y
subject to

$2 x + 3 y \leq 30 3 x + y \leq 17 x, y \geq 0$

First we will convert inequations into equations as follows:
2x + 3y = 30, 3x + y = 17, x = 0 and y = 0

Region represented by 2x + 3y ≤ 30:
The line 2x + 3y = 30 meets the coordinate axes at A₁(15, 0) and B₁(0, 10) respectively. By joining these points we obtain the line 2x + 3y = 30. Clearly (0,0) satisfies the 2x + 3y = 30. So, the region which contains the origin represents the solution set of the inequation 2x + 3y ≤ 30.

Region represented by 3x + y ≤ 17:
The line 3x + y = 17 meets the coordinate axes at $C_{1} (\frac{17}{3}, 0)$ and D₁(0, 17) respectively. By joining these points we obtain the line 3x + y = 17. Clearly (0,0) satisfies the inequation 3x + y ≤ 17. So,the region which contains the origin represents the solution set of the inequation 3x + y ≤ 17.

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 2x + 3y ≤ 30, 3x + y ≤ 17, x ≥ 0 and y ≥ 0 are as follows.

The coordinates of the corner points are O(0, 0), $C_{1} (\frac{17}{3}, 0)$ , E₁(3, 8) and B₁(0, 10).
The value of the objective function at the corner points

Corner points Z = 100x + 120y

O(0, 0) 0

$C_{1} (\frac{17}{3}, 0)$ $\frac{1700}{3}$

E₁(3, 8) 1260

B₁(0, 10) 1200

Therefore, maximum revenue would be obtained when 3 units of A and 8 units of B are produced. In doing so, 30 workers and 17 units of capital must be used.

Yes, men and women workers are equally efficient and so should be paid at the same rate.

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30

workers (male and female) and

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units capital; which he uses to produce two types of goods

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and

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Do you agree with this view of the manufacturer that men and women workers are equally efficient and so should be paid at the same rate?

A manufacturer considers that men and women workers are equally efficient and so he pays them at the same rate. He has 30 and 17 units of workers (male and female) and capital respectively, which he uses to produce two types of goods A and B. To produce one unit of A, 2 workers and 3 units of capital are required while 3 workers and 1 unit of capital is required to produce one unit of B. If A and B are priced at ₹100 and ₹120 per unit respectively, how should he use his resources to maximise the total revenue? Form the above as an LPP and solve graphically. Do you agree with this view of the manufacturer that men and women workers are equally efficient and so should be paid at the same rate?

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