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A manufacturer can produce two products, A and B, during a given time period. Each of these products requires four different manufacturing operations: grinding, turning, assembling and testing. The manufacturing requirements in hours per unit of products A and B are given below.

A B

Grinding 1 2

Turning 3 1

Assembling 6 3

Testing 5 4

The available capacities of these operations in hours for the given time period are: grinding 30; turning 60, assembling 200; testing 200. The contribution to profit is Rs 20 for each unit of A and Rs 30 for each unit of B. The firm can sell all that it produces at the prevailing market price. Determine the optimum amount of A and B to produce during the given time period. Formulate this as a LPP.

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Solution

Let x and y units of products A and B were manufactured respectively.
The contribution to profit is Rs 2 for each unit of A and Rs 3 for each unit of B.
Therefore for x units of A and y units of B,the contribution to profit would be Rs 2x and Rs 3y respectively.

Let Z denote the total profit

Then, Z = Rs (2x + 3y)

Total hours required for grinding, turning, assembling and testing are $x + 2 y, 3 x + y, 6 x + 3 y, 5 x + 4 y$ respectively.

The available capacities of these operations in hours for the given period are grinding 30, turning 60, assembling 200 and testing 200.

$∴$ $x + 2 y \leq 30, 3 x + y \leq 60, 6 x + 3 y \leq 200, 5 x + 4 y \leq 200$
Units of products cannot be negative.Therefore,
$x, y \geq 0$

Hence, the required LPP is as follows:

Maximize Z = 2x + 3y

subject to

$x + 2 y \leq 30, 3 x + y \leq 60, 6 x + 3 y \leq 200, 5 x + 4 y \leq 200$

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