A GOLD ANY The Impala Gold Company operated a gold mine in the Orange Free State, South Africa. The mining operation consisted of mining underground, at a depth of 4.000 feet, gold- bearing rock. The rock was transported up the mine shafts to a mill that crushed the rock and extracted the gold. The Impala mine had three shafts. Information on these shafts is given in Table 2-20. Note that the rock mined in each shaft area has a different gold content as well as different costs. Rock mined from all three shafts was sent to the mill to be crushed and refined. The mill capacity depended on how fine the rock was ground. If the rock was ground fine, mill capac- ity was 240,000 tons per month, and 95 percent of the gold was recovered in the operation. Rock from each shaft could tocgroind separately. The cost of milling a ton of rock ground fine was 1.12 rand per ton. If the rock was ground coarse, mill capacity was 250,000 tons per month, but gold recovery dropped to 90 percent. The cost of milling a ton of rock ground coarse was 0.85 rand. The mine could sell all the gold it produced at a price of 0.80 rand per gram. The mine manager was concerned about how much rock he should mine in each shaft area. He noted that the mill capacity was not sufficient to handle all three shafts operating at full capacity. The problem was further complicated by the legal requirements that a mine could not mine "above the average grade" of the ore reserves. In the Impala mine, this TABLE 2-20 Hoist capacity of shaft (tons per month) Ore grade (grams of gold per ton of rock) Variable cost of mining rock (rands per ton) Shaft Shaft Shaft 3 2 85,000 90,000 95,000 25 20 15 6 5 4 Pare I Modeling and Optimization average grade was 20 grams per ton. Thus, there was the legal restriction that the mix of rock from the three shafts could not exceed an average of 20 grams per ton in ore grade. Formulate a linear programming model to maximize profit from operating the nine.