Facility location problems
Author: Aaron Litoff
Stewards: Dajun Yue and Fengqi You
Facility location problems deal with selecting the placement of a facility (often from a list of integer possibilities) to best meet the demanded constraints. The problem often consists of selecting a factory location that minimizes total weighted distances from suppliers and customers, where weights are representative of the difficulty of transporting materials. The solution to this problem gives the highest profit choice that most efficiently serves the needs of all consumers.
Background and 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 three point version to model possible industrial locations in order to minimize transportation costs from two sources of materials to a single customer or market.  This formulation is one the simplest continuous facility location models.
The Fermat-Weber problem is defined as:
Given finitely many distinct points and positive multipliers find a point that minimizes
where denotes the Euclidean norm of i.e.
The simplest version of this problem with all w=1 and n=2 gives the minimum distance in a flat plane.
A modern day engineering interpretation of Fermat's formulation could be as follows:
"Find the best location for a refining plant between three cities in such a way that the sum of the connections between the plant and the cities in minimal."
Although this problem was first solved geometrically by Torricelli in 1645 it did not have a direct numerical solution until Kuhn and Kuenne's iterative method was published over 300 years later in 1962! 
Description and Formulation
In the basic formulation of the facility location problem, the number of possible locations is now finite as we have discrete choices of where to build. These choices are governed by binary decision variables y.
The problem is often defined as a set of customers D, a set of facilities F, a fixed cost for opening each facility, and a variable cost for each facility. We are looking for the subset S of facilities that we should open and an assignment of S to D such that all customers will be serviced by a facility and such that the sum of fixed costs, variable costs, and transportation costs (modeled by distance) are minimized. 
These problems have been studied extensively in the literature and are often solved using an approximation algorithm. The approximation algorithm looks for a feasible solution where:
A: The algorithm will stop after a given number of steps (e.g. number of customers and facilities)
B: There is an approximation ratio such that the calculated solution is within some small amount of the optimal solution 
Minisum and Minimax Location Problems
Facility location problems are often formulated in one of two ways, minisum and minimax.
A minisum FLP looks to place a new facility in the location that minimizes the sum of the weighted distances between the new facility and the already existing facilities. The minisum location problem is as follows:
is the location of the new facility
are the locations of existing facilities
is the weight associated with traveling between the new facility and facility i
is the distance between the new facility and facility i
The minimax FLP, by contrast, looks for the optimal location to place a facility with the goal of minimizing the maximum distance between the newly placed facility and all existing facilities.
Capacitated vs. Uncapacitated Facility Location Problems
An additional variation on the facility location problem is whether a problem is capacitated or uncapacitated. In a capacitated model the capacity of each facility is known and accounted for. This means that even if a facility is located at the least cost source for a demand node it may not be able to fully serve the demand at that node because its production is limited by its own capacity. 
For a uncapacitated model however, the assumption is made that each facility can produce and ship unlimited quantities of the commodity under consideration. A concern when designing a manufacturing and transportation system, for example, is the rate of capacity that each factory can produce or can ship its product.
Continous vs. Discrete Facility Location Problems
Finally, in a continuous FLP the selection for the new facility can be any location within the space, whereas for a discrete FLP there are given set of choices for the facility's location.
The example that follows in the next section falls into the category of a discrete, uncapacitated facility location problem because there are a given set of four possible locations and it is assumed that each store has no limit on the amount it can sell.
A Real World Example
Suppose you are a manager at Apple and you are identifying locations to sell computers and iPhones in a new location in Chicago. You would start by identifying potential sites in a variety of neighborhoods throughout the city and finding the demand for Apple products in each neighborhood. Then to construct your model you could create a table of neighborhoods, the cost of building a new Apple store in each one, and the expected profit. Your goal now is to select which facilities should be built in order to maximize profit.
Define decision variables such that .
Then the total expected benefit is:
and the total capital needed is:
In total, the model can be given as:
where are binary variables.
Additional constrains are that the total available capital is $16M, and the number of new stores should not be more than two. Solving this model using an MILP solver in GAMS gives us the maximum when y2=y4=1. This indicates that the Logan Square and Hyde Park Apple stores should both be built in order to maximize profit.
Note that in another formulation of this problem you could minimize distance for customers to travel to the store regardless of profit.
Applications in Industry
Facility location problems have applications in a wide variety of fields and projects. As examined it can be used to find the optimal location for a store, plant, warehouse, etc. but these formulation methods can also be used is less obvious ways. Applications range from public policy (e.g. locating police officers in a city), telecommunications (e.g. cell towers in a network), and even particle physics (e.g. separation distance between repulsive charges). All of these problems have in common that discrete locations must be chosen and the objective is to meet the demand of consumers or users in the most efficient way.  The figure at the right demonstrates an example of how the facility location problem may be used in industry to determine the locations for natural gas transmission equipment. Another case study that has been analyzed in the literature was the determination of quantity and location of distribution centers of steel facilities in Latin America .
Facility location problems seek to optimize the placement of facilities such that the demands of consumers can be met at the lowest cost and/or shortest distance. The minisum formulation minimizes the sum of the weighted distances between facilities while the maxisum formulation minimizes the overall maximum distance between facilities. The study of these problems has numerous applications in the fields of mathematics, economics, physics, and engineering.
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