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Question
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.
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 \[x + 2y, 3x + y, 6x + 3y, 5x + 4y\] respectively.
The available capacities of these operations in hours for the given period are grinding 30, turning 60, assembling 200 and testing 200.
∴ \[x + 2y \leq 30, 3x + y \leq 60, 6x + 3y \leq 200, 5x + 4y \leq 200\]
Units of products cannot be negative.Therefore,
\[x, y \geq 0\]
Hence, the required LPP is as follows:
Maximize Z = 2x + 3y
subject to
\[x + 2y \leq 30, \]
\[3x + y \leq 60, \]
\[6x + 3y \leq 200, \]
\[5x + 4y \leq 200\]
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