A mining company operates two mines, each of which produces three grades of ores. The West Summit mine can produce 2 tons of low-grade ore, 3 tons of medium-grade ore, and 1 ton of high-grade ore per hour of operation. The North Ridge mine can produce 2 tons of low-grade ore, 1 ton of medium-grade ore, and 2 tons of high-grade ore per hour of operation. To satisfy existing orders, the company needs at least 100 tons of low-grade ore, 60 tons of medium-grade ore, and 80 tons of high-grade ore. If it costs $400 per hour to operate the West Summit mine and $600 per hour to operate the North Ridge mine, how many hours should each mine be operated to supply the needed amounts of ore and, at the same time, minimize the cost of production?
Here is the mathematical model where x1 is the number of hours the West Summit mine operates and x2 is the number of hours the North Ridge Mine operates.
2x1 + 2x2 ≥ 100 tons of low-grade ore 3x1 + x2 ≥ 60 tons of medium-grade ore x1 + 2x2 ≥ 80 tons of high-grade ore x1, x2, x3 ≥ 0 Minimize C = 400x1 + 600x2