Question

A Set Covering Example American Parcel Service (APS) has determined that it needs to add several new package distri- bution hubs to service cities east of the Mississippi River. The company wants to construct the minimum set of new hubs in the following 12 cities so that there is a hub within t each city (i.e. every city is covered by a hub): counse neds needs Tailor eBooks to your City Cities Within 300 Miles 1. Atlanta 2. Boston 3. Charlotte 4. Cincinnati 5. Detroit Atlanta. Charlotte. Nashville Boston, New York Atlanta. Charlotte. Richmond Cincinnati. Detroit. Indianapolis. Nashville. Pittsburgh Cincinnati, Detroit, Indianapolis, Milwaukee, Pittsburgh Cincinnati. Detroit, Indianapolis, Milwaukee. Nashville. St. Louis Detroit. Indianapolis, Milwaukee Atlanta. Cincinnati. Indianapolis, Nashville, St. Louis Boston. New York, Richmond Cincinnati. Detroit. Pittsburgh. Richmond Charlotte. New York, Pittsburgh. Richmond Indianapolis. Nashville, St. Louis 6. Indianapolis 7. Milwaukee 8. Nashville 9. New York 10. Pittsburgh I1. Richmond 12. St. Louis Write an appropriate Mathematical Model this LP.

Answer 1

Here decision variable are available cities –

x _i = city i

i = 1 to 12

Where if city i is not selected as a hub then x _i = 0

And if city i is selected as a hub then x _i = 1

The objective function (Z) is to minimize the number of hubs:

Minimize Z = x1 + x2 +x3 + x4 + x5 + x6 + x7+ x8 + x9 + x10 + x11 + x12

So, appropriate mathematical model is –

Minimize Z = x1 + x2 +x3 + x4 + x5 + x6 + x7+ x8 + x9 + x10 + x11 + x12

Subject to

Atlanta: x1 + x3 = x8 ≥ 1

Boston: x2 + x9 ≥ 1

Charlotte: x1 + x3 + x11 ≥ 1

Cincinnati: x4 + x5 + x6 + x8 + x10 ≥ 1

Detroit: x4 + x5 + x6 + x7 + x10 ≥ 1

Indianapolis: x4 + x5 + x6 + x7 + x8 + x12 ≥ 1

Milwaukee: x5 + x6 + x7 ≥ 1

Nashville: x1 + x4 + x6 + x8 + x12 ≥ 1

New York: x2 + x9 + x11 ≥ 1

Pittsburgh: x4 + x5+ x10 + x11 ≥ 1

Richmond: x3 + x9 + x10 + x11 ≥ 1

St. Louis: x6 + x8 + x12 ≥ 1 x i = 0 or 1