# Facility location problem

Authors: Liz Cantlebary, Lawrence Li (CHEME 6800 Fall 2020)

Stewards: Allen Yang, Fengqi You

## Introduction

The Facility Location Problem (FLP) is a classic optimization problem that determines the best location for a factory or warehouse to be placed based on geographical demands, facility costs, and transportation distances. These problems generally aim to maximize the supplier's profit based on the given customer demand and location. FLP can be further broken down into capacitated and uncapacitated problems, depending on whether the facilities in question have a maximum capacity or not.

## Theory and Formulation

### Weber Problem

The Weber Problem is a simple FLP that consists of locating the geometric median between three points with different weights. The geometric median is a point between three given points in space such that the sum of the distances between the median and the other three points is minimized. It is based on the premise of minimizing transportation costs from one point to various destinations, where each destination has a different associated cost per unit distance.

Given ${\displaystyle N}$ points ${\displaystyle (a_{1},b_{1})...(a_{N},b_{N})}$ on a plane with associated weights ${\displaystyle w_{1}...w_{N}}$, the 2-dimensional Weber problem to find the geometric median ${\displaystyle (x,y)}$ is formulated as(1)

${\displaystyle {\underset {x,y}{min}}\{W(x,y)=\sum _{i=1}^{N}w_{i}d_{i}(x,y,a_{i},b_{i})\}}$

where

${\displaystyle d_{i}(x,y,a_{i},b_{i})={\sqrt {(x-a_{i})^{2}+(y-b_{i})^{2}}}}$

### Uncapacitated and Capacitated FLPs

In an uncapacitated facility problem, the amount of product each facility can produce and transport is assumed to be unlimited, and the optimal solution results in customers being supplied by the lowest-cost, and usually the nearest, facility.

A capacitated facility problem applies constraints to the production and transportation capacity of each facility. As a result, customers may not be supplied by the most immediate facility, since this facility may not be able to satisfy the given customer demand.

In a problem with ${\displaystyle N}$ given facilities, the capacitated formulation defines a binary variable ${\displaystyle x_{i}}$ for each facility ${\displaystyle i}$, where ${\displaystyle x_{i}=1}$ if facility ${\displaystyle i}$ is open, and ${\displaystyle x_{i}=0}$ otherwise, and a variable ${\displaystyle y_{ij}}$ for each facility ${\displaystyle i}$ and each customer ${\displaystyle j}$, which represents the fraction of the demand ${\displaystyle d_{j}}$ that facility ${\displaystyle i}$ has satisfied. The capacitated FLP is defined as

${\textstyle {\begin{array}{rl}\min &\displaystyle \sum _{i=1}^{N}\sum _{j=1}^{m}c_{ij}d_{j}y_{ij}+\sum _{i=1}^{n}f_{i}x_{i}\\{\text{s.t.}}&\displaystyle \sum _{i=1}^{n}y_{ij}=1{\text{ for all }}j=1,\dots ,m\\&\displaystyle \sum _{j=1}^{m}d_{j}y_{ij}\leqslant u_{i}x_{i}{\text{ for all }}i=1\dots ,n\\&y_{ij}\geqslant 0{\text{ for all }}i=1,\dots ,n{\text{ and }}j=1,\dots ,m\\&x_{i}\in \{0,1\}{\text{ for all }}i=1,\dots ,n\end{array}}}$

## References

1. http://www.pitt.edu/~lol11/ie1079/notes/ie2079-weber-slides.pdf