Question

A company manufactures two types of novelty souvenirs made of polywood.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 building.3 hours 20 minutes are 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 types of 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, $x≥0,y≥0$
The given information can be compiled 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 value of Z at these corner points is as follows.

Corner point	Z=5x+6y
A(24,0)	120
B(8,20)	160 (Max)
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.