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Linear Programming

Question
CBSEENMA12035860
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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: 

Machine Area occupied Labour force daily output ( in units )
A 1000 m2 12 men men 60
B 1200 m2 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 + 1200 y ≤ 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 0, 152 300  
B 94, 458 360 → maximum
C (6, 0) 360 → Maximum

 

The maximum value of Z is 360 units, which is attained at             B94, 458, and c( 6, 0 ).

Now the number of machines cannot be in fraction.

Thus, to maximise the daily output, 6 machines of type A and no machine of type B need to be bought.                  

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