Arnoff Enterprises manufactures the central processing unit (CPU) for a line of personal computers. The CPUs are manufactured in Seattle, Columbus, New York, and Pittsburgh and shipped to warehouses in Denver, Los Angeles, and Washington, D.C., for further distribution. The following table shows the production capacity at each plant, the number of CPUs required by each warehouse, and unit shipping costs:

1. Formulate this problem to minimize the total shipping cost. [The explicit model formulation is fine; it is also fine if you prefer the generic model formulation. But please provide clear definitions of all the notations in the model. NO need to solve the model.]

2. (5 Points) There are a one-time set-up cost associated with each plant. The following table provides the fixed set-up cost for each plant:

Formulate the problem to minimize the total cost, including the total fixed cost and the total shipping cost. [The explicit model formulation is fine; it is also fine if you prefer the generic model formulation. But please provide clear definitions of all the notations in the model. NO need to solve the model.]

3. (5 Points) Please write down the LP relaxation of the integer programming model you developed in Question 2.

4. (10 Points) Please write constraints for the following requirements:

a. At least two plants should be selected.

b. The plant at New York or the plant at Pittsburgh must be selected, but not both.

c. If the plant at Pittsburgh is selected, then either the plant at Seattle or the plant at Columbus must be elected but not both.

Warehouse Plant Denver Los Angeles Seattle Columbus New York Pittsburgh CPUs Required $5 per unit $8 per unit $7 per unit $6 per unit 5,000 units $9 per unit $30 per unit $10 per unit $12 per unit 3,000 units Washington, D.C. $10 per unit $6 per unit $4 per unit $5 per unit 4,000 units Production Capacity 8,000 units 5,000 units 7,000 units 2,000 units Plant Seattle Columbus New York Pittsburgh Fixed Set-up Cost $300 $100 $300 $100