# Facility Location Problem

### Objective and Prerequisites

In this example, we will solve a facility location problem where we want to build warehouses to supply a certain number of supermarkets. We will construct a mixed-integer programming (MIP) model of this problem, implement this model in the Gurobi Python interface, and compute an optimal solution.

This modeling example is at the beginner level, where we assume that you know Python and that you have some knowledge about building mathematical optimization models.

### Motivation

The study of facility location problems -also known as location analysis- is a branch of operations research and computational geometry concerned with the optimal placement of facilities to minimize transportation costs while considering factors like avoiding placing hazardous materials near housing, and the location of competitors‘ facilities.

The Fermat-Weber problem, formulated in the 17’th century, was one of the first facility location problems ever proposed.

The Fermat-Weber problem can be described 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 minimal. This problem can be interpreted as a version of the facility location problem, where the assumption is made that the transportation costs per distance are the same for all destinations.

Facility location problems have applications in a wide variety of industries. For supply chain management and logistics, this problem can be used to find the optimal location for stores, factories, warehouses, etc. Other applications range from public policy (e.g. positioning police officers in a city), telecommunications (e.g. cell towers in a network), and even particle physics (e.g. separation distance between repulsive charges). Another application of the facility location problem is to determine the locations for natural gas transmission equipment. Finally, facility location problems can be applied to cluster analysis.

### Problem Description

A large supermarket chain in the UK needs to build warehouses for a set of supermarkets it is opening in Northern England. The locations of the supermarkets have been decided, but the locations of the warehouses have yet to be determined.

Several good candidate locations for the warehouses have been identified, but decisions must be made regarding how many warehouses to open and at which candidate locations to build them.

Opening many warehouses would be advantageous as this would reduce the average distance a truck has to drive from the warehouse to the supermarket, and hence reduce the delivery cost. However, opening a warehouse has a fixed cost associated with it.

In this example, our goal is to find the optimal tradeoff between delivery cost and the cost of building new facilities.

### Proposed Solution

A mixed-integer programming (MIP) formulation for the facility location problem.

Modeling examples are coded using the Gurobi Python API in Jupyter Notebook. In order to use the Jupyter Notebooks, you must have a Gurobi License. If you do not have a license, you can request an Evaluation License as a Commercial User or download a free license as an Academic User.

#### Access the Jupyter Notebook Modeling Example

Click on the button below to be directed to the GitHub HTML page, where you can download the repository for the Facility Location Jupyter Notebook modeling example.

Facility Location Problem