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Question : 28
Total: 29
A factory owner purchases two types of machines, A and B for his factory. The requirements and the limitations for the machines are as follows:
He has maximum area of 9000 m2 available, and 72 skilled labourers who can operate both the machines. How many machines of each type should he buy to maximise the daily output?
| Machine | Area occupied | Labour force | Daily output (in units) |
|---|---|---|---|
| A | 1000 | 12 men | 60 |
| B | 1200 | 8 men | 40 |
He has maximum area of 9000 m2 available, and 72 skilled labourers who can operate both the machines. How many machines of each type should he buy to maximise the daily output?
Solution:
Let x and y respectively be the number of machines A and B, which the factory owner should buy.
Now, according to the given information, the linear programming problem is:
Maximise Z = 60x + 40y
Subject to the constraints
1000x + 1200y ≤ 9000
⇒ 5x+ 6y ≤ 45 ...(1)
12x + 8y ≤ 72
⇒ 3x + 2y ≤ 18 ...(2)
x ≥ 0, y ≥ 0 ...(3)
The inequalities (1), (2), and (3) can be graphed as:
The shaded portion OABC is the feasible region.
The value of Z at the corner points are given in the following table
The maximum value of Z is 360 units, which is attained at B(
,
)
Now, the number of machines cannot be in fraction.
Thus, to maximize the daily output, 6 machines of type A and no machine of type B need to be bought.
Now, according to the given information, the linear programming problem is:
Maximise Z = 60x + 40y
Subject to the constraints
1000x + 1200y ≤ 9000
⇒ 5x+ 6y ≤ 45 ...(1)
12x + 8y ≤ 72
⇒ 3x + 2y ≤ 18 ...(2)
x ≥ 0, y ≥ 0 ...(3)
The inequalities (1), (2), and (3) can be graphed as:
The shaded portion OABC is the feasible region.
The value of Z at the corner points are given in the following table
| Corner point | Z = 60x + 40y | |||||
|---|---|---|---|---|---|---|
| 0 (0,0) | 0 | |||||
| A | 300 | |||||
| B | 360 → | Maximum | ||||
| C (6,0) | 360 → | Maximum |
The maximum value of Z is 360 units, which is attained at B
Now, the number of machines cannot be in fraction.
Thus, to maximize the daily output, 6 machines of type A and no machine of type B need to be bought.
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