A manufacturer makes two types of toys A and B. Three machines are needed for this purpose and the time (in minutes) required for each toy on the machines is given below:
Each machine is available for a maximum of 6 hours per day. If the profit on each toy of type A is Rs 7.50 and that on each toy of type B is Rs 5, show that 15 toys of type A and 30 of type B should be manufactured in a day to get maximum profit.
Let x and y toys of type A and type B respectively be manufactured in a day.
The given problem can be formulated as follows.
Maximize $z=7.5 x+5 y$ (1)
subject to the constraints,
$2 x+y \leq 60$ (2)
$x \leq 20$ (3)
$2 x+3 y \leq 120$ (4)
$x, y \geq 0$ (5)
The feasible region determined by the constraints is as follows.
The corner points of the feasible region are A (20, 0), B (20, 20), C (15, 30), and D (0, 40).
The values of z at these corner points are as follows.
The maximum value of z is 262.5 at (15, 30).
Thus, the manufacturer should manufacture 15 toys of type A and 30 toys of type B to maximize the profit.
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