1. 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	Mill
1	2
High	6	2
Medium	2	2
Low	4	10

The company has contracted with a manufacturing firm to supply at least 12 tons of high-grade aluminum, and 5 tons of low-grade aluminum. It costs United $6000 per day to operate mill 1 and $7000 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 minimum cost.

Formulate a linear programming model for this problem in "Standard Form".

1. Solve the linear programming model formulated in problem 7 by using the computer and answer the following:
   1. Identify and explain the shadow prices for each of the aluminum grade contract requirements.
   2. Identify the sensitivity ranges for the objective function coefficients and the constraint quantity values.
   3. 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?