Question

A person makes two types of gift items A and B requiring the services of a cutter and a finisher.  Gift item A requires 4 hours of the cutter's time and 2 hours of finisher's time. Fifth item B requires 2 hours of the cutter's time and 4 hours of finisher's time. The cutter and finisher have 208 hours and 152 hours available time respectively every month. The profit on one gift item of type A is ₹ 75 and on one gift item of type, B is ₹ 125. Assuming that the person can sell all the gift items produced, determine how many gift items of each type should he make every month to obtain the best returns?

Answer 1

Let x: number of gift item A 

y: number of gift item B

As numbers of the items are never negative 

X ≥ 0; y ≥ 0 

	A (x)	B (y)	Max.time available
Cutter	4	2	208
Finisher	2	4	152
Profit	75	125

Total time required for the cutter = 4x + 2y 

Maximum available time 208 hours 

∴ 4x+ 2y ≤ 208 

Total time required for the finisher 2x +4y

Maximum available time 152 hours 

∴ 2x+ 4y ≤ 152

Total Profit is 75x + 125y 

∴ L.P.P. of the above problem is 

Minimize Z = 75x + 125y 

Subject to 4x+ 2y ≤ 208 

2x + 4y ≤ 152 

x ≥ 0 ; y ≥ 0

Graphical solution 

2x + y = 104
x	0	52
y	104	0
(0 , 104) (52 , 0)

 

x + 2y = 76
x	0	0
y	38	76
(0 , 38) (76 , 0)

Corner points

Now, Z at 

Z = (75x + 125y) 

0(0, 0) = 75 x 0 + 125 x 0 = 0

A(52, 0)  = 75 x 52 + 125 x 0 = 3900

B(44, 16) = 75 x 44 + 125 x 16 = 5300

C(O, 38) = 75 x 0 + 125 x 38 = 4750 

∴ A person should make 44 items of type A and 16 items of type Band his returns are ₹ 5,300.