The Martin-Beck Company operates a plant in St. Louis with an annual capacity of 30,000 units. Product is shipped to regional distribution centers located in Boston, Atlanta, and Houston. Because of an anticipated increase in demand, Martin-Beck plans to increase capacity by constructing a new plant in one or more of the following cities: Detroit, Toledo, Denver, or Kansas City. The estimated annual fixed cost and the annual capacity for the four proposed plants are as follows: Proposed Plant Annual Fixed Cost Annual Capacity
Detroit $175,000 10,000
Toledo $300,000 20,000
Denver $375,000 30,000
Kansas City $500,000 40,000

The company's long-range planning group developed forecasts of the anticipated annual demand at the distribution centers as follows:

Distribution Center Annual Demand
Boston 30,000
Atlanta 20,000
Houston 20,000

The shipping cost per unit from each plant to each distribution center is as follows:

Plant Site Boston Atlanta Houstan
Detroit 5 2 3
Toledo 4 3 4
Denver 9 7 5
Kansas City 10 4 2
St. Louis 8 4 3

Required:
a. Formulate a mixed-integer programming model that could be used to help Martin-Beck determine which new plant or plants to open in order to satisfy anticipated demand.
b. Solve the model you formulated in part a. What is the optimal cost? What is the optimal set of plants to open?
c. Find a second-best solution. What is the increase in cost versus the best solution from part b?