Question

A manufacturer has three machines I, II and III installed in his factory. Machine I and II are capable of being operated for atmost 12 hours whereas machine III must be operated for atleast 5 hours a day. He produces only two items M and N each requiring the use of all the three machines.

The number of hours required for producing 1 unit of M and N on three machines are given in the following table:

Items	Number of hours required on machines
	I	II	III
M	1	2	1
N	2	1	1.25

He makes a profit of ₹ 600 and ₹ 400 on one unit of items M and N respectively. How many units of each item should he produce so as to maximize his profit assuming that he can sell all the items that he produced. What will be the maximum profit?

Answer 1

Let manufacturer produces x units of Product M and y units of product N.

	Product M (x)	Product N (y)	Time
Machine I	1	2	12
Machine II	2	1	12
Machine III	1	1.25	5

Subject to constraints

x + 2y ≤ 12

2x + y ≤ 12

x + 1.25y ≥ 5

or, 4x + 5y ≥ 20, x ≥ 0, y ≥ 0

Maximize Z = 600x + 400y

L₁: x + 2y = 12

x	0	12	4
y	6	0	4

L₂: 2x + y = 12

x	0	6	4
y	12	0	4

L₃: 4x + 5y = 20

x	0	5
y	4	0

Vertices of feasible region are (0, 4), (5, 0), (6, 0), (4, 4) and (0, 6)

(x, y)	Z = 600x + 400y
(0, 4)	Z = 1600
(5, 0)	Z = 3000
(6, 0)	Z = 3600
(4, 4)	Z = 4000 (Max.)
(0, 6)	Z = 2400

Maximum Value of Z = 4000 at x = 4 and y = 4