Answer :

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

Since, profits of each A and B are Rs. 40 and Rs. 50 respectively. So, profits on x number of type A and y number of type B are 40x and 50y respectively.

Let Z denotes total output daily, so,

Z = 40x + 50y

Since, each A and B requires 3 grams and 1 gram of silver respectively. So, x of type A and y of type B require 3x and y of silver respectively. But,

Total silver available is 9 grams. So,

3x + y 9 {First Constraint}

Since each A and B requires 1 gram and 2 grams of gold respectively. So, x of type A and y of type B require x and 2y respectively.

But total gold available is 8 grams.

So,

x + 2y 8 {Second Constraint}

Hence mathematical formulation of the given LPP is,

Max Z = 40x + 50y

Subject to constraints,

3x + y 9

x + 2y 8

x,y 0 [Since production of A and B can not be less than zero]

Region 3x + y 9: line 3x + y = 9 meets the axes at A(3,0), B(0,9) respectively.

Region containing the origin represents 3x + y 9

as origin satisfies 3x + y 9.

Region x + 2y 8: line x + 2y = 8 meets the axes at C(8,0), D(0,4) respectively.

Region containing the origin represents x + 2y 8 as origin satisfies x + 2y 8.

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

The corner points are O(0,0), D(0,4), E(2,3), A(3,0)

The values of Z at these corner points are as follows

The maximum value of Z is 230 which is attained at E(2,3).

Thus the maximum profit is of Rs 230 when 2 units of Type A 3 uniits of Type B are produced.

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