Chapter 6.3, Problem 50E

In Problems 49-58, construct a mathematical model in the form of a linear programming problem. (the answers in the back of the book for these application problems indicate the model.) then solve the problem by applying the simplex method to the dual problem.

Mining. A mining company operates two mines, each producing three grades of ore. 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 in one hour of operation. The North Ridge mine can produce $2$ tons of high-grade ore, $1$ ton of medium-grade ore, and $2$ tons of high-grade ore in one hour of operation. To satisfy existing oredrs, the company needs to produce at least $100$ tons of low-grade ore, $60$ tons of medium-grade ore, and $80$ tons of high-grade ore. The cost of operating each mine varies, depending on conditions while extracting the ore. If it costs $\text{\$}400$ per hour to operate the West Summit mine and $\text{\$}600$ per hour to operate the North Ridge mine, how many hours should each mine operate to supply the required amounts of ore and minimize the cost of production? What is the minimum production cost?