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# A factory owner purchases two types of machines, A and B, for his factory. The requirements and limitations for the machines are as follows :He has an area of 7600 sq.m available and 72 skilled men who can operate the machines. How many machines of each type should he buy to maximize the daily output?

Let required number of machine A and B are x and y respectively.

Since, products of each machine A and B are 60 and 40 units daily respectively. So, production by by x number of machine A and y number of machine B are 60x and 40y respectively.

Let Z denotes total output daily, so,

Z = 60x + 40y

Since, each machine of type A and B requires 1000sq. m and 1200 s. m area so, x machine of type A and y machine of type B require 1000x and 1200y sq. m area but,

Total available area for machine is 7600 sq. m. So,

1000x + 1200y 7600

or, 5x + 6y 38. {First Constraint}

Since each machine of type A and B requires 12 men and 8 men to work respectively. So, x machine of type A and y machine of type B require 12x and 8y men to work respectively.

But total men available for work are 72.

So,

12x + 8y 72

3x + 2y 18 {Second Constraint}

Hence mathematical formulation of the given LPP is,

Max Z = 50x + 40y

Subject to constraints,

5x + 6y 38

3x + 2y 18

x,y 0 [Since number of machines can not be less than zero]

Region 5x + 6y 38: line 5x + 6y = 38 meets the axes at A(,0), B(0,) respectively.

Region containing the origin represents 5x + 6y 38 as origin satisfies 5x + 6y 38

Region 3x + 2y 18: line 3x + 2y = 18 meets the axes at C(6,0), D(0,9) respectively.

Region containing the origin represents 3x + 2y 18 as origin satisfies 3x + 2y 18.

Region x,y 0: it represents the first quadrant.

Shaded region represents the feasible region.

The corner points are O(0,0), B(0,), E(4,3), C(6,0).

Thus the values of Z at these corner points are as follows:

The maximum value of Z is 360 which is attained at E(4,3), C(6,0).

Thus,the maximum output is Rs 360 obtained when 4 units of type A and 3 units of type B or 6 units of type A and 0 units of type B are manufactured.

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