Question

A firm manufactures 3 products A, B 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.

Answer 1

Let the number of units of product A, B and C manufactured be  x, y 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 

\[4x + 3y + 5z \leq 2000\]
\[2x + 2y + 4z \leq 2500\]

 

\[100 \leq x \leq 150\]
\[y \geq 200\]
\[z \geq 50\]
\[x, y, z \geq 0\]