A company manufactures two types of screws $A$ and $B.$ All the screws have to pass through a threading machine and a slotting machine. A box of Type $A$ screws requires $2\, minutes$ on the threading machine and $3\, minutes$ on the slotting machine. A box of type $B$ screws requires $8\, minutes$ of threading on the threading machine and $2\, minutes$ on the slotting machine. In a week, each machine is available for $60\, hours.$ On selling these screws, the company gets a profit of $Rs.\, 100$ per box on type $A$ screws and $Rs.\, 170$ per box on type $B$ screws. Formulate this problem as a $LPP$ given that the objective is to maximise profit.
A company manufactures two types of screws $A$ and $B.$ All the screws have to pass through a threading machine and a slotting machine. A box of Type $A$ screws requires $2\, minutes$ on the threading machine and $3\, minutes$ on the slotting machine. A box of type $B$ screws requires $8\, minutes$ of threading on the threading machine and $2\, minutes$ on the slotting machine. In a week, each machine is available for $60\, hours.$ On selling these screws, the company gets a profit of $Rs.\, 100$ per box on type $A$ screws and $Rs.\, 170$ per box on type $B$ screws. Formulate this problem as a $LPP$ given that the objective is to maximise profit.
Let the company manufactures $x$ boxes of type $A$ screws and $y$ boxes of type $B$ screws.
From the given information, we have following corresponding constraint table.
| Type $A \,(x)$ | Type $B \,(y)$ | Maximum time available on each machine in a week | |
| Time required for screws on threading machine | $2$ | $8$ | $60 \times 60(\min )$ |
|
Time required for screws on slotting machine |
$3$ | $2$ | $60 \times 60(\min )$ |
| Profit | $Rs.\,100$ | $Rs.\,170$ |
Thus, the objective function for maximum profit is $Z=100 x+170 y$ Subject to constraints
$2 x+8 y \leq 60 \times 60$ [time constraint for threading machine]
$\Rightarrow \quad x+4 y \leq 1800$
and $3 x+2 y \leq 60 \times 60$ [time constraint for slotting machine]
$\Rightarrow \quad 3 x+2 y \leq 3600$
Also, $x \geq 0, y \geq 0$ [non-negative constraints]
$\therefore$ Required $LLP$ is,
Maximize $Z=100 x+170 y,$ subject to constraints
$x+4 y \leq 1800,3 x+2 y \leq 3600, x \geq 0, y \geq 0$
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