# A manufacturer produces two products A and B. Both the products are processed on two different machines. The available capacity of the first machine is 12 hours and that of the second machine is 9 hours per day. Each unit of product A requires 3 hours on both machines, and each unit of product B requires 2 hours on the first machine and 1 hour on the second machine. Each unit of product A is sold at ` 7 profit and that of B at a profit of ` 4. Find the production level per day for maximum profit graphically.

Given: A manufacturer produces two products A and B

To find: the production level of A and B per day for maximum profit graphically

Let the number of units of products A and B to be produced each day be x and y respectively Let z be a total profit of manufacture

Each unit of product A is sold at Rs. 7 profit and that of B at a profit of Rs. 4

z = 7x + 4y

We need to maximize the cost

Hence, the mathematical formulation of LPP is

Maximize z = 7x + 4y

subject to the constraints,

3x + 2y 12

3x + y 9

x, y 0

The feasible region determined by the system of constraints is as follows: The corner points of the enclosed region are A(0, 6) , B(2, 3), C(3, 0)

The value of z at these corners points is as follows:

Case 1: A(0, 6)

z = 7x + 4y

z = 7(0) + 4(6)

z = 0 + 24

z = 24

Case 2: B(2, 3)

z = 7x + 4y

z = 7(2) + 4(3)

z = 14 + 12

z = 26

Case 3: C(3, 0)

z = 7x + 4y

z = 7(3) + 4(0)

z = 21 + 0

z = 21

The value of z is maximum in the second case at point B(2, 3)

Hence, the unit of products A and B to be produced are 2 and 3 respectively, and the total maximum profit is Rs. 26

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