16. 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: Aluminum Grade High Medium Low 1 629 4 Mill 2 2 2 10 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 in order to meet the contract at the minimum cost. Formulate a linear programming model for this problem. 17. Solve the linear programming model formulated in Problem 16 for United Aluminum Company graphically. a. How much extra (i.e., surplus) high-, medium-, and low-grade aluminum does the company produce at the optimal solution? b. What would be the effect on the optimal solution if the cost of operating mill 1 increased from $6,000 to $7,500 per day? c. What would be the effect on the optimal solution if the company could supply only 10 tons of high-grade aluminum? 18. Solve the linear programming model formulated in Problem 16 for United Aluminum Company by using the computer. a. Identify and explain the shadow prices for each of the aluminum grade contract requirements. b. Identify the sensitivity ranges for the objective function coefficients and the constraint quantity values. c. Would the solution values change if the contract requirements for high-grade aluminum were increased from 12 tons to 20 tons? If yes, what would the new solution values be?

Fig: 1