Difference between revisions of "Facility location problems"

From optimization
Jump to: navigation, search
 
(125 intermediate revisions by 2 users not shown)
Line 2: Line 2:
 
Stewards: Dajun Yue and Fengqi You
 
Stewards: Dajun Yue and Fengqi You
  
[[File:johnny.jpg|thumb|right|350px|MILP [1]]]
+
[[File:weber.jpg|thumb|right|upright=1.2|German economist Alfred Weber (1868-1958).]]
  
  
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.
+
==Introduction==
Finding a factory location which minimizes total weighted distances from suppliers and customers where weights represent difficulty of transportation or amounts of material
+
  
 +
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.
  
==History==
+
==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:
 
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.''"  
+
"''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.''" [1]
  
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.
+
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. [2] This formulation is one the simplest continuous facility location models.
  
[[File:FermatPoints_700.gif|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:<br>
+
[[File:FermatPoints_700.gif|frame|none|alt=Alt text|The Fermat Point, or Torricelli Point, is the solution that minimizes distances from A, B, and C (the three corners of the black triangle).[3]]]
"''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.''"
+
  
 +
<br>
 +
 +
The Fermat-Weber problem is defined as: <br>
 +
Given finitely many distinct points <math>A_{1}, A_{2}, ..., A_{m}</math> and positive multipliers <math>w_{1}, w_{2}, ..., w_{m}</math> find a point <math>P \in \mathbb{R}^{n}</math> that minimizes
 +
 +
<math>f(P)= \sum_{i=1}^{m} w_{i} \left \| P-A_{i} \right \|</math>
 +
 +
where <math>\left \| X \right \| </math> denotes the Euclidean norm of <math>X \in \mathbb{R}^{n}</math> i.e. <math>\left \| (x_{1},..,x_{n}) \right \| = \sqrt{x_{1}^{2}+...+x_{n}^2}</math>
 +
 +
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:<br>
 +
"''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! [4]
  
 
==Description and Formulation==
 
==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.
  
==Examples and Applications==
+
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. [5]
  
 +
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: <br>
  
Suppose you are a manager at a company that builds
+
<br>A: The algorithm will stop after a given number of steps (e.g. number of customers and facilities)
Warehouse needs to be built in a central location so that the transportation costs are minimized
+
<br>B: There is an approximation ratio such that the calculated solution is within some small amount of the optimal solution
 +
[5]
  
 +
===Minisum and Minimax Location Problems===
  
Applications in physics, solution to Fermat-Weber problem allowing for "repulsion" or negative distances.
+
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: <br>
 +
<br> min <math>f(x) = \sum_{i=1}^{m} w_{i}d(X,P_{i})</math> <br>
 +
 
 +
where <br>
 +
<math>X</math> is the location of the new facility <br>
 +
<math>P</math> are the locations of existing facilities <br>
 +
<math>w_{i}</math> is the weight associated with traveling between the new facility and facility i <br>
 +
<math>d</math> is the distance between the new facility and facility i <br>
 +
 
 +
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. [6]
 +
 
 +
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.
 +
 
 +
For example:
 +
 
 +
 
 +
[[File:AppleModel.png|thumb|none|top|upright=3.6|]]
 +
 
 +
<br> Define decision variables <math>i=1,2,3,4 </math> such that <math>y_{i} = \begin{Bmatrix}
 +
1 \textrm{ if Apple store } i \textrm{ should be built }\\
 +
 
 +
0 \textrm{ if not}\end{Bmatrix}</math>. <br> Then the total expected benefit is: <math>10y_{1}+9y_{2}+6y_{3}+12y_{4}</math> <br> <br> and the total capital needed is: <math>8y_{1}+7y_{2}+5y_{3}+9y_{4}</math><br>
 +
 
 +
<br> <br> In total, the model can be given as: <br><br> max    <math>10y_{1}+9y_{2}+6y_{3}+12y_{4}</math><br><br>
 +
subject to    <math>8y_{1}+7y_{2}+5y_{3}+9y_{4} \leq 16</math> <br><br>
 +
where    <math>y_{1}, y_{2}, y_{3}, y_{4}</math> are binary variables. <br><br>
 +
 
 +
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. <br>
 +
 
 +
[[File:ChicagoApple.png|thumb|none|upright=1.0]]
 +
<br>
 +
 
 +
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==
 +
 
 +
[[File:OilandGasFacilities.png|thumb|left|upright=1.8|This map illustrates facility locations for natural gas processing, natural gas transmission, underground natural gas storage, LNG storage, and LNG import and export. The locations selected depend heavily on minimizing the distance to the nearest natural gas pipeline shown in grey.[7]]]
 +
 
 +
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).[8] 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. [5] 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 [9].
 +
 
 +
 
 +
[[File:Optimalbrazil.gif|thumb|none|upright=1.2|This figure gives the result after optimization for the location of steel production facilities in Brazil.[9]]]
  
 
==Conclusion==
 
==Conclusion==
 +
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.
 +
 +
  
 
==References==
 
==References==
Line 43: Line 124:
 
[1] Dorrie, H. (1965) 100 great problems of elementary mathematics: their history and solution. New York, NY: Dover Publications.
 
[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.
+
[2] Drezner, Z & Hamacher. H. W. (2004). Facility location: applications and theory. New York, NY: Springer.
 +
 
 +
[3] Weisstein, E. W. Fermat Points. Mathworld-- A wolfram web resource. http://mathworld.wolfram.com/FermatPoints.html
 +
 
 +
[4] Kuhn, H. W. and Kuenne, R. E. (1962). An efficient algorithm for the numerical solution of the generalized weber problem in spatial economics. Journal of Regional Science, 4.
 +
 
 +
[5] Vygen, J. (2005). Approximation algorithms for facility location problems. Bonn, Germany: Research Institute for Discrete Mathematics.
 +
 
 +
[6] Sliva F., Serra, D. (2007) A capacitated facility location problem with constrained backlogging probabilities. International Journal of Production Research, 45 (21), pp.5117-5134.
 +
 
 +
[7] United States Environmental Protection Agency. (2015, May 28). Petroleum and Natural Gas Systems. http://www.epa.gov/ghgreporting/ghgdata/reported/petroleum.html
 +
 
 +
[8] Korupolu, M. R. & Plaxton, G. C. & Rajaraman, R. (2000). Analysis of a local search heuristic for facility location problems.
 +
 
 +
[9] Conceição, S., Pedrosa, L., Neto, A., Vinagre, M., & Wolff, E. (2012). The facility location problem in the steel industry: A case study in Latin America. Production Planning & Control, 23(1), 26-46.

Latest revision as of 22:59, 7 June 2015

Author: Aaron Litoff
Stewards: Dajun Yue and Fengqi You

German economist Alfred Weber (1868-1958).


Contents

Introduction

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." [1]

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. [2] This formulation is one the simplest continuous facility location models.


Alt text
The Fermat Point, or Torricelli Point, is the solution that minimizes distances from A, B, and C (the three corners of the black triangle).[3]


The Fermat-Weber problem is defined as:
Given finitely many distinct points A_{1}, A_{2}, ..., A_{m} and positive multipliers w_{1}, w_{2}, ..., w_{m} find a point P \in \mathbb{R}^{n} that minimizes

f(P)= \sum_{i=1}^{m} w_{i} \left \| P-A_{i} \right \|

where \left \| X \right \| denotes the Euclidean norm of X \in \mathbb{R}^{n} i.e. \left \| (x_{1},..,x_{n}) \right \| = \sqrt{x_{1}^{2}+...+x_{n}^2}

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! [4]

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. [5]

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 [5]

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:

min f(x) = \sum_{i=1}^{m} w_{i}d(X,P_{i})

where
X is the location of the new facility
P are the locations of existing facilities
w_{i} is the weight associated with traveling between the new facility and facility i
d 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. [6]

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.

For example:


AppleModel.png


Define decision variables i=1,2,3,4 such that y_{i} = \begin{Bmatrix}
1 \textrm{ if Apple store } i \textrm{ should be built }\\ 

0 \textrm{ if not}\end{Bmatrix}.
Then the total expected benefit is: 10y_{1}+9y_{2}+6y_{3}+12y_{4}

and the total capital needed is: 8y_{1}+7y_{2}+5y_{3}+9y_{4}



In total, the model can be given as:

max 10y_{1}+9y_{2}+6y_{3}+12y_{4}

subject to 8y_{1}+7y_{2}+5y_{3}+9y_{4} \leq 16

where y_{1}, y_{2}, y_{3}, y_{4} 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.

ChicagoApple.png


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

This map illustrates facility locations for natural gas processing, natural gas transmission, underground natural gas storage, LNG storage, and LNG import and export. The locations selected depend heavily on minimizing the distance to the nearest natural gas pipeline shown in grey.[7]

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).[8] 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. [5] 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 [9].


This figure gives the result after optimization for the location of steel production facilities in Brazil.[9]

Conclusion

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.


References

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

[2] Drezner, Z & Hamacher. H. W. (2004). Facility location: applications and theory. New York, NY: Springer.

[3] Weisstein, E. W. Fermat Points. Mathworld-- A wolfram web resource. http://mathworld.wolfram.com/FermatPoints.html

[4] Kuhn, H. W. and Kuenne, R. E. (1962). An efficient algorithm for the numerical solution of the generalized weber problem in spatial economics. Journal of Regional Science, 4.

[5] Vygen, J. (2005). Approximation algorithms for facility location problems. Bonn, Germany: Research Institute for Discrete Mathematics.

[6] Sliva F., Serra, D. (2007) A capacitated facility location problem with constrained backlogging probabilities. International Journal of Production Research, 45 (21), pp.5117-5134.

[7] United States Environmental Protection Agency. (2015, May 28). Petroleum and Natural Gas Systems. http://www.epa.gov/ghgreporting/ghgdata/reported/petroleum.html

[8] Korupolu, M. R. & Plaxton, G. C. & Rajaraman, R. (2000). Analysis of a local search heuristic for facility location problems.

[9] Conceição, S., Pedrosa, L., Neto, A., Vinagre, M., & Wolff, E. (2012). The facility location problem in the steel industry: A case study in Latin America. Production Planning & Control, 23(1), 26-46.