A company runs two mines P and Q. In a day, mine P gives 1 tonne of high-grade, 2 tonnes of medium-grade and 4 tonnes of low-grade ore, and mine Q gives 2 tonnes of high-grade, 2 tonnes of medium-grade and 1 tonne of low-grade ore. The company needs at least 12 tonnes of high-grade, 16 tonnes of medium-grade and 16 tonnes of low-grade ore. It costs Rs. 3000 a day to run mine P and Rs. 4000 a day to run mine Q. The least cost of meeting the needs is
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → Rs. 28000
Let mine P run for x days and mine Q for y days. Then
Minimise \(Z = 3000x + 4000y\)
subject to
\(x + 2y \ge 12\) (high grade)
\(2x + 2y \ge 16\), that is \(x + y \ge 8\) (medium grade)
\(4x + y \ge 16\) (low grade)
\(x \ge 0,\ y \ge 0\)
The lines \(x + 2y = 12\) and \(x + y = 8\) meet at (4, 4). The lines \(x + y = 8\) and \(4x + y = 16\) meet where \(3x = 8\), that is at
\(\displaystyle\left(\frac{8}{3}, \frac{16}{3}\right)\)
The region is unbounded, with corner points (12, 0), (4, 4), the point just found and (0, 16). Values of Z:
\(Z(12, 0) = 36000\)
\(Z(4, 4) = 12000 + 16000 = 28000\)
\(\displaystyle Z\left(\frac{8}{3}, \frac{16}{3}\right) = 8000 + \frac{64000}{3} \approx 29333\)
\(Z(0, 16) = 64000\)
The half-plane \(3000x + 4000y \lt 28000\) has no point in common with the region, so the least cost is Rs. 28000, with each mine run for 4 days.