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

# A Firm Manufactures 3 Products A, B and C. the Profits Are Rs 3, Rs 2 and Rs 4 Respectively. the Firm 2 Machines and Below is Required Processing Time in Minutes for Each Machine on Each Product : - Mathematics

Sum

A firm manufactures 3 products AB and C. The profits are Rs 3, Rs 2 and Rs 4 respectively. The firm has 2 machines and below is the required processing time in minutes for each machine on each product :

 Machine Products A B C M1M2 4 3 5 2 2 4

Machines M1 and M2 have 2000 and 2500 machine minutes respectively. The firm must manufacture 100 A's, 200 B's and 50 C's but not more than 150 A's. Set up a LPP to maximize the profit.

#### Solution

Let the number of units of product A, B and C manufactured be  x, and z respectively.
Given, machine $M_1$ takes 4 minutes to manufacture 1 unit of product A, 3 minutes to manufacture one unit of product B and 5 minute to manufacture one unit of product C.

Machine $M_2$  takes 2 minutes to manufacture 1 unit of product A, 2 minutes to manufacture one unit of product B and 4 minute to manufacture one unit of product C.
The availability is 2000 minutes for
$M_1$  and 2500 minutes for  $M_2$
Thus,
$4x + 3y + 5z \leq 2000$
$2x + 2y + 4z \leq 2500$
Number of units of products cannot be negative.
So,
$x, y, z \geq 0$
Further, it is given that the firm should manufacture 100 A's, 200 B's and 50 C's but not more than 150 A's.
Then,
$100 \leq x \leq 150$
$y \geq 200$
$z \geq 50$
Let Z denotes the profit  $\therefore Z =$ 3x + 2y + 4z
Hence, the required LPP is as follows :
Maximize  Z =  3x + 2y + 4z
subject to
$4x + 3y + 5z \leq 2000$
$2x + 2y + 4z \leq 2500$

$100 \leq x \leq 150$
$y \geq 200$
$z \geq 50$
$x, y, z \geq 0$
Concept: Mathematical Formulation of Linear Programming Problem
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#### APPEARS IN

RD Sharma Class 12 Maths
Chapter 30 Linear programming
Exercise 30.1 | Q 3 | Page 14