A firm has to transport 900 packages using large vans, which carry 150 packages each, and small vans, which carry 75 packages each. Hiring a large van costs Rs. 500 and hiring a small van costs Rs. 300. The number of large vans cannot be more than the number of small vans. The least cost of the job is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → Rs. 3200
Let x large vans and y small vans be hired. Then
Minimise \(Z = 500x + 300y\)
subject to
\(150x + 75y \ge 900\), that is \(2x + y \ge 12\) (packages)
\(x \le y\) (large vans not more than small vans)
\(x \ge 0,\ y \ge 0\)
The lines \(2x + y = 12\) and \(y = x\) meet where
\(3x = 12\)
\(x = 4,\ y = 4\)
The feasible region is unbounded, with corner points (0, 12) and (4, 4). Values of Z:
\(Z(0, 12) = 3600\)
\(Z(4, 4) = 2000 + 1200 = 3200\)
The half-plane \(500x + 300y \lt 3200\), that is \(5x + 3y \lt 32\), has no point in common with the feasible region, so the least cost is Rs. 3200 (4 large and 4 small vans).