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,
$$\begin{gathered} 4x+3y+5z\le 2000 \\ 2x+2y+4z\le 2500 \end{gathered}$$

Number of units of products cannot be negative.
So, $x,y,z\ge 0$

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,
$$\begin{gathered} 100\le x\le 150 \\ y\ge 200 \\ z\ge 50 \end{gathered}$$

Let Z denotes the profit

$\therefore \mathrm{Z}=$3x + 2y + 4z

​Hence, the required LPP is as follows :

Maximize $\mathrm{Z}=$3x + 2y + 4z

subject to

$$\begin{gathered} 4x+3y+5z\le 2000 \\ 2x+2y+4z\le 2500 \end{gathered}$$

$$\begin{gathered} 100\le x\le 150 \\ y\ge 200 \\ z\ge 50 \\ x,y,z\ge 0 \end{gathered}$$