The company has contracted with a manufacturing firm to supply at least $12$ tons of high$-$grade aluminum, $8$ tons of medium$-$grade aluminum, and $5$ tons of low$-$grade aluminum. It costs United $$6,000$ per day to operate mill $1$ and $$7,000$ per day to operate mill $2.$ The company wants to know the number of days to operate each mill to meet the contract at the minimum cost.

Formulate a linear programming model for this problem.