Question

A company manufactures two types of novelty souvenirs made of plywood. Souvenirs of type A require 5 minutes each for cutting and 10 minutes each for assembling. Souvenirs of type B require 8 minutes each for cutting and 8 minutes each for assembling. There are 3 hours 20 minutes available for cutting and 4 hours of assembling. The profit is Rs 5 each for type A and Rs 6 each for type B souvenirs. How many souvenirs of each type should the company manufacture in order to maximize the profit?

Answer 1

Let the company manufacture x souvenirs of type A and y souvenirs of type B. Therefore,

x ≥ 0 and y ≥ 0

The given information can be complied in a table as follows

	Type A	Type B	Availability
Cutting (min)	5	8	3 × 60 + 20 =200
Assembling (min)	10	8	4 × 60 = 240

The profit on type A souvenirs is Rs 5 and on type B souvenirs is Rs 6. Therefore, the constraints are

`5x+8y<=200`

`10x+8y<=240` i.e.,`5x+4y<=120`

Total profit, Z = 5x + 6y

The mathematical formulation of the given problem is

Maximize Z = 5x + 6y … (1)

subject to the constraints,

`5x+8y<=200`… (2)

`5x+4y<=120` … (3)

x, y ≥ 0 … (4)

The feasible region determined by the system of constraints is as follows.

The corner points are A (24, 0), B (8, 20), and C (0, 25).

The values of Z at these corner points are as follows.

Corner point	Z = 5x + 6y
A(24, 0)	120
B(8, 20)	160	→ Maximum
C(0, 25)	150

The maximum value of Z is 200 at (8, 20).

Thus, 8 souvenirs of type A and 20 souvenirs of type B should be produced each day to get the maximum profit of Rs 160.