Department of Pre-University Education, KarnatakaPUC Karnataka Science Class 12
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A Firm Manufactures Two Types of Products a and B and Sells Them at a Profit of Rs 2 on Type a and Rs 3 on Type B. Each Product is Processed on Two Machines M1 and M2 - Mathematics

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A firm manufactures two types of products A and B and sells them at a profit of Rs 2 on type A and Rs 3 on type B. Each product is processed on two machines M1 and M2. Type A requires one minute of processing time on M1 and two minutes of M2; type B requires one minute on M1 and one minute on M2. The machine M1 is available for not more than 6 hours 40 minutes while machine M2 is available for 10 hours during any working day. Formulate the problem as a LPP.

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Let the firm produces units of product A and y units of product B.
Since, each unit of product A requires one minute on machine \[M_1\] and two minutes on machine  \[M_2\] Therefore, x units of product A will require product x minutes on machine ​  \[M_1\] and 2x  minutes on machine \[M_2\]
Since each unit of product B requires one minute on machine \[M_1\]  and one minute on machine  \[M_2\] Therefore, y  units of product A will require product y minutes on machine ​ \[M_1\] and y  minutes on machine \[M_2\] 
It is given that the machine \[M_1\] is available for  \[6 \text{ hours and 40 minutes} \] i.e. 400 minutes  and machine \[M_2\] is available for 10 hours i.e. 600 minutes

\[x + y \leq 400\]

\[2x + y \leq 600\]

  Since,units of the products cannot be negative,so
\[x, y \geq\] 0  Let Z denotes the total profit
\[\therefore Z = 2x + 3y\] which is to be maximised
Hence, the required LPP is as follows:
Maximize Z = 2x + 3y
subject to  \[x + y \leq 400\]

\[2x + y \leq 600\]

\[x, y \geq\] 0
Concept: Introduction of Linear Programming
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RD Sharma Class 12 Maths
Chapter 30 Linear programming
Exercise 30.1 | Q 4 | Page 15

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