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 can not exceed the number of small vans. Formulate this problem as a LPP given that the objective is to minimise cost.

Min $Z = 400x + 200y$ subject to $5x + 2y \geq 30, 2x + y \leq 15, x \leq y, x, y \geq 0$

The correct answer is Option (3) → Min $Z = 400x + 200y$ subject to $5x + 2y \geq 30, 2x + y \leq 15, x \leq y, x, y \geq 0$

Let x and y be the number of large and small vans respectively. From the given information, we construct the following corresponding constraints table;

Now the objective function for minimum cost is

$Z = 400x + 200y$

Subject to the constraints;

$200x + 80y ≥ 1200⇒ 5x+2y≥30$   ...(i)

$400x + 200y ≤ 3000⇒ 2x + y ≤ 15$   ...(ii)

$x≤y$   ...(iii)

and $x ≥0, y ≥ 0$ (non-negative constraints)

Hence, the required LPP is to minimise $Z = 400x + 200y$

Subject to the constraints $5x + 2y ≥ 30, 2x+ y ≤ 15$ and $x ≥0, y ≥ 0$