A firm has to transport at least 1200 packages daily using large vans which carry 200 packages each and small vans which can take 80 packages each. The cost of engaging each large van is ₹400 and each small van is ₹200. Not more than ₹3000 is to be spent daily 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 minimize cost.

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Solution

Let the number of large vans and small vans used for transporting the packages be x and y, respectively.

It is given that the cost of engaging each large van is ₹400 and each small van is ₹200.

Cost of engaging x large vans = ₹400x

Cost of engaging y small vans = ₹200y

Let Z be the total cost of engaging x large vans and y small vans.

∴ Z = ₹(400x + 200y)

The firm has to transport at least 1200 packages daily using large vans which carry 200 packages each and small vans which can take 80 packages each.

∴ Number of packages transported by x large vehicles + Number of packages transported by y small vehicles ≥ 1200

⇒ 200x + 80y ≥ 1200

Not more than ₹3000 is to be spent daily on the transportation.

∴ 400x + 200y ≤ 3000

Also, the number of large vans cannot exceed the number of small vans.

∴ x ≤ y

Thus, the linear programming problem of the given problem is

Minimise Z = ₹(400x + 200y)

Subject to constraints

200x + 80y ≥ 1200

400x + 200y ≤ 3000

x ≤ y

x ≥ 0, y ≥ 0

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