The Heartland Distribution Company is a food warehouse and distributor that has a contract with a grocery store chain in several Midwest and Southeast cities. The company wants to construct new warehouses/distribution centers in some of the cities it services to serve the stores in those cities plus all the other stores in the other cities that don't have distribution centers. A distribution center can effectively service all stores within a 300-mile radius. The company also wants to limit its fixed annual costs to under $900,000. The company wants to build the minimum number of distribution centers possible. The following table shows the cities within 300 miles of every city and the projected fixed annual charge for a distribution center in each city:
City Annual Fixed Charge Cities Within 300 Miles ($1,000s)
1. Atlanta $276 Atlanta, Charlotte, Nashville
2. Charlotte $253 Atlanta, Charlotte, Richmond
3. Cincinnati $394 Cincinnati, Cleveland, Indianapolis, Louisville, Nashville, Pittsburgh
4. Cleveland $408 Cincinnati, Cleveland, Indianapolis, Pittsburgh
5. Indianapolis $282 Cincinnati, Cleveland, Indianapolis, Louisville, Nashville, St. Louis
6. Louisville $365 Cincinnati, Indianapolis, Louisville, Nashville, St. Louis
7. Nashville $268 Atlanta, Cincinnati, Indianapolis, Louisville, Nashville, St. Louis
8. Pittsburgh $323 Cincinnati, Cleveland, Pittsburgh, Richmond
9. Richmond $385 Charlotte, Pittsburgh, Richmond
10. St. Louis $298 Indianapolis, Louisville, Nashville, St. Louis
Formulate an Integer Programming Model for this problem.