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  • A firm has to transport 1200 packages using large vans which can carry 200packages each and small vans which can take 80 packages each. The cost for engaging each large van is Rs 400 and each small van is Rs 200. Not more than Rs 3000 is to be spent on th

A firm has to transport 1200 packages using large vans which can carry 200packages each and small vans which can take 80 packages each. The cost for engaging each large van is Rs 400 and each small van is Rs 200. Not more than Rs 3000 is to be spent on the job and the number of large vans cannot exceed the number of small vans. Formulate this problem as a LPP given that the objective is to minimise cost.

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Let us consider that the firm has X number of large vans and Y number of small vans . We have made the following table:

\begin{array}{|l|l|l|l|} \hline & \text { Large Van (X) } & \text { Small Van (Y) } & \begin{array}{l} \text { Maximum/Minim } \\ \text { um } \end{array} \\ \hline \text { Packages } & 200 & 80 & 1200 \\ \hline \text { cost } & \text { Rs.400 } & \text { Rs.200 } & \text { Rs. 3000 } \\ \hline \end{array}

Looking at the table, we can see that the cost becomes Z=400x+200y.

Now when we minimize the cost, i.e. minimize Z=400x+200y

The subject to constraints are:

200x+80y \geq1200    

When we divide it by 40, we get:

\Rightarrow 5x+2y \geq 30 \ldots \text{(i)}

And 400 x+200 y \leq 3000

Now will divide throughout by 200, we get 

\Rightarrow 2 x+y \leq 15

Also given the number of large vans cannot exceed the number of small vans

\Rightarrow x \leq y

And x \geq 0, y \geq 0  [non-negative constraint]

So, minimize cost we have to minimize Z =400 \mathrm{x}+200 \mathrm{y} subject to

\begin{align*} 5x + 2y & \geq 30 \\ 2x + y & \leq 15 \\ x & \leq y \\ x & \geq 0, \quad y \geq 0 \end{align*}

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