Introduction to Management Science

1.The 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 grad, as follows.

                       Mill
Aluminum
Grade             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, 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.

2. Solve the linear programming model formulated in problem 1 for the United Aluminum Company 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 quanity 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?

3. The manager of a Burger Doodle franchise wants to determine how many sausage biscuits and ham biscuits to prepare each morning for breakfast customers. Each type of biscuit requires the following resources.

Biscuit        Labor(hr)         Sausage(lb)        Ham(lb)           Flour(lb)

Sausage     0.010               0.10                    ?                    0.04
Ham           0.024                 ?                     0.15                0.04

The franchise has 6 hours of labor available each morning. The manager has a contract with a local grocer for 30 pounds of sausage and 30 pounds of ham each morning. The manager also purchases 16 pounds of flour. The profit for a sausage biscuit is $0.60; the profit for a ham biscuit is $0.50. The manager wants to know the number of each type of biscuit to prepare each morning in order to maximize profit.

Formulate a linear programming model for this problem.

4. Solve the linear programming model developed in problem 3 for the Burger Doodle restaurant usng the computer.

a. Identify and explain the shadow prices for each of the resource constraints.
b. Which of the resources constrain profit the most?
c. Identify the sensitivity ranges for the profit of a sausage biscuit and the amount of sausage available. Explain these sensitivity ranges.