United aluminum company of cincinnati produces three grades (high, medium, and low) of aluminum at two mills. each mill has a different production capacity (in tons per day) for each grade, as follows: 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 2 $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 in order to meet the contract at the minimum cost. formulate a linear programming model for this problem.