A firm has to transport 1200 packages using large vans which can carry 200 packages 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.

Let the firm has $x$ number of large vans and $y$ number of small vans.

From the given information,

We have the following conditions as per the question, following corresponding constraint table.

Thus, we see that objective function for minimum cost is $Z = 400x + 200y$.

Subject to constraints
$200x + 80y \ge 1200$

[package constraint]
$\Rightarrow$ $5x + 2y \ge 30$

….(i)
and $400x + 200y \le 3000$ [cost constraint]
$\Rightarrow$ $2x + y \le 15$

….(ii)
and $x \le y$

[van constraint]

…..(iii)
and $x \ge 0,y \ge 0$

[non-negative constraints] ….(iv)

Thus, required LPP to minimise cost is minimise $Z = 400x + 200y$, subject to $5x + 2y \ge 30$.

$2x + y \le 15$

$x \le y$

$x \ge 0,y \ge 0$