Department of Pre-University Education, KarnatakaPUC Karnataka Science Class 12

# 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

Sum

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.

#### Solution

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 $M_2$
Also,
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

Thus,
$x + y \leq 400$

$2x + y \leq 600$

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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#### APPEARS IN

RD Sharma Class 12 Maths
Chapter 30 Linear programming
Exercise 30.1 | Q 4 | Page 15