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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

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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
M1
M2
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.

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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
  Is there an error in this question or solution?

APPEARS IN

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