A factory manufactures two types of screws, $A$ and $B$ . Each type of screw requires the use of two machines, an automatic and a hand-operated. It takes $4$ minutes on the automatic and $6$ minutes on hand-operated machines to manufacture a package of screws $A$ , while it takes $6$ minutes on automatic and $3$ minutes on the hand-operated machines to manufacture a package of screws $B$ . Each machine is available for at the most $4$ hours on any day. The manufacturer can sell a package of screws $A$ at a profit of $R s . 7$ and screws $B$ at a profit of $R s . 10$ . Assuming that he can sell all the screws he manufactures, how many packages of each type should the factory owner produce in a day in order to maximize his profit? Determine the maximum profit.

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Solution

Let number of package of screws A to be made is $X$ and number of package of screws B to be made is $Y$

Since, screws A package requires $4$ minutes and screws B package requires $6$ minutes of automatic machine work time. Also, there is maximum $4$ hours (240 minutes) of automatic machine time available.
$∴ 4 X + 6 Y \leq 240$

$\Rightarrow 2 X + 3 Y \leq 120 . . . (1)$

Since, screws A package requires $6$ minutes and screws B package requires $3$ minutes of hand operated machine work time. Also, there is maximum $4$ hours (240 minutes) of hand operated machine time available.
$∴ 6 X + 3 Y \leq 240$

$\Rightarrow 2 X + Y \leq 80 . . . (2)$

Since, count of package can't be negative.
$∴ X \geq 0, Y \geq 0 . . . (3)$

We have to maximize profit of the factory.
Here, profit on screws A package is $7 R s$ and on screws B package is $10 R s$

So, objective function is $Z = 7 X + 10 Y$

Plotting all the constraints given by equation (1), (2) and (3), we got the feasible region as shown in the image.

Corner points Value of $Z = 7 X + 10 Y$
A $(0, 40)$ $400$
B $(30, 20)$ $410$ (maximum)
C $(40, 0)$ $280$
Hence, factory owner should produce 30 packages of screws A and 20 packages of screws B in a day to maximise his profit. Also, maximum profit will be $410 R s$

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