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, 

$$4x+3y+5z\le 2000$$
$$2x+2y+4z\le 2500$$

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,

$$100\le x\le 150$$
$$y\ge 200$$
$$z\ge 50$$

 Let Z denotes the profit  $$\therefore Z=$$ 3x + 2y + 4z
Hence, the required LPP is as follows :
Maximize  Z =  3x + 2y + 4z 

subject to 

$$4x+3y+5z\le 2000$$
$$2x+2y+4z\le 2500$$

 

$$100\le x\le 150$$
$$y\ge 200$$
$$z\ge 50$$
$$x,y,z\ge 0$$