# Facility location problems

Author: Aaron Litoff
Stewards: Dajun Yue and Fengqi You

MILP [1]

The facility location problem deals with selecting the location of a facility (often from a list of integer possibilities) to best meet demand wether to the next line of customers. Goal here is to most efficiently serve the constraints demanded while doing so at the lowest cost. Finding a factory location which minimizes total weighted distances from suppliers and customers where weights represent difficulty of transportation or amounts of material

## History

The Fermat-Weber problem was one of the first facility location problems every proposed, and was done so as early as the 17th century. This "geometric median of three points" can be thought of as a version of the facility location problem where the assumption is made that transportation costs per distance are the same for all destinations. It was put forth by the French mathematician Pierre de Fermat to the Italian physicist Evangelista Torricelli as follows:

"Given three points in a plane, find a fourth point such that the sum of its distances to the three given points is as small as possible."

In 1909 Alfred Weber used a a three point version to determine the location for industry to minimize transportation costs with the weights being This is the simplest continous facility location model.

The Fermat Point, or Torricelli Point, is the solution that minimizes distances from A, B, and C the three corners of the black triangle.

A modern day engineering interpretation could be as follows:
"Find the best location for a refining plant between three cities in such a way that te sum of the connections between the power plant and the cities in minimal."

## Examples and Applications

Suppose you are a manager at a company that builds Warehouse needs to be built in a central location so that the transportation costs are minimized

Applications in physics, solution to Fermat-Weber problem allowing for "repulsion" or negative distances.

## References

[1] Dorrie, H. (1965) 100 great problems of elementary mathematics: their history and solution. New York, NY: Dover Publications.

Last, F. M. (Year Published) Book. City, State: Publisher.