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  • 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 s

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.

 

Answers (1)

Let’s assume that the company manufactures X boxes of type A screws and Y boxes of type B screws. Look at the following table:

\begin{array}{|l|l|l|l|} \hline & \begin{array}{l} \text { Type A Screws } \\ (\mathrm{x} \text { boxes }) \end{array} & \begin{array}{l} \text { Type B Screws } \\ (\mathrm{y} \text { boxes }) \end{array} & \begin{array}{l} \text { Maximum time } \\ \text { available on } \\ \text { each machine } \\ \text { in a week } \end{array} \\ \hline \begin{array}{l} \text { Time } \\ \text { required for } \\ \text { screws on } \\ \text { threading } \\ \text { machine } \end{array} & 2 & 8 & \begin{array}{l} 60 \text { hours } \\ =60 \times 60 \mathrm{~min} \\ =3600 \mathrm{~min} \end{array} \\ \hline \begin{array}{l} \text { Time } \\ \text { required for } \\ \text { screws on } \\ \text { slotting } \\ \text { machine } \end{array} & 3 & 2 & \begin{array}{l} 60 \text { hours } \\ =60 \times 60 \mathrm{~min} \\ =3600 \mathrm{~min} \end{array} \\ \hline \text { Profit } & \text { Rs } 100 &\text { Rs } 170 & \\ \hline \end{array}

According to the table, we can see that profit becomes \mathrm{Z}=100 \mathrm{x}+170 \mathrm{y}

Now, we have to maximize the profit, i.e., maximize \mathrm{Z}=100 \mathrm{x}+170 \mathrm{y}
The constraints so obtained, i.e., subject to the constraints,
2 x+8 y \leq 3600[time constraints for threading machine]
Now will divide throughout by 2, we get
\Rightarrow x+4 y \leq 1800

And 3 x+2 y \leq 3600 [time constraints for slotting machine]
\Rightarrow 3 x+2 y \leq 3600 \ldots \ldots \ldots \ldots . .(i i)
And x \geq 0,y\geq 0  [non-negative constraint]
So, to maximize profit we have to maximize \mathrm{Z}=100 \mathrm{x}+170 \mathrm{y} subject to
\begin{align*} x + 4y & \leq 1800 \\ 3x + 2y & \leq 3600 \\ x & \geq 0, \quad y \geq 0 \end{align*}

 

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