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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. - Mathematics

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Sum

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.

‚ÄčLet 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

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

 

 

 

Concept: Introduction of Linear Programming
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APPEARS IN

RD Sharma Class 12 Maths
Chapter 30 Linear programming
Exercise 30.1 | Q 8 | Page 16

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