# Distance Functions in Location Problems

• Marzie Zarinbal
Chapter
Part of the Contributions to Management Science book series (MANAGEMENT SC.)

## Abstract

Distance is a numerical description of how far apart objects are at any given moment in time. In physics or everyday discussion, distance may refer to a physical length, a period of time, or it is estimated based on other criteria.

While making location decisions, the distribution of travel distances among the service recipients (clients) is an important issue.

Most classical location studies focus on the minimization of the mean (or total) distance (the median concept) or the minimization of the maximum distance (the center concept) to the service facilities. (Ogryczak 2000) In these studies, the location modeling is divided into four broad categories:

Analytic models. These models are based on a large number of simplifying assumptions such as the fix cost of locating facility. The travel distances follow the Manhattan metric.

Continuous models. These models are the oldest location models, deal with geometrical representations of reality, and are based on the continuity of location area. The classic model in this area is the Weber problem. Distances in the Weber problem are often taken to be straight-line or Euclidean distances but almost all kind of the distance functions can be used here (Jiang and Xu 2006; Hamacher and Nickel 1998).

In the study of continuous location theory, it is generally assumed that the customers may be treated as points in space. This assumption is valid when the dimensions of the customers are small relative to the distances between the new facility and the customers. However, it is not always the case. Sometimes, we should not ignore the dimensions of the customers. Some researchers have treated the customers as demand regions representing the demand over a region.

Jiang and Xu (2006) discussed that some researchers such as Brimberg andWesolowsky in 1997, 2000 and 2002 and Nickel et al. in 2003 used the distance between the facility and the closest point of a demand region; and in the others, the distance between the facility and a demand region may be calculated as some form of expected or average travel distance.

Network models. Network models are composed of links and nodes. Absolute 1-median, un-weighted 2-center and q-criteria L-median on a tree models are some well-known models in this area. Distances are measured with respect to the shortest path.

Discrete models. In these models, there are a discrete set of candidate locations. Discrete N-median, un-capacitated facility location, and coverage models are some well-known models in this area. Like the distances in continuous models, all kind of the distance functions can be used here but sometimes it could be specified exogenously (Hamacher and Nickel 1998; Fouard and Malandain 2005).

Distances and norms are usually defined on the finite space E n and take real values. In discrete geometry, however, we sometimes need to have discrete distances defined on Z n with their values in Z. Since Z n is not a vector space, the notion of distances and norms had to be extended.

## Keywords

Distance Function Mahalanobis Distance Demand Point Hilbert Curve Levenshtein Distance
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## References

1. Aronov B, VanKreveld M, VanOostrum R, Varadarajan K (2005) Facility location on a polyhedral surface. Discrete Comput Geom 30:357–372
2. Chae A, Fromm H (2005) Supply chain management on demand. Springer, BerlinGoogle Scholar
3. Chew EP, Tang LC (1999) Travel time analysis for general item location assignment in a rectangular warehouse. Eur J Oper Res 112:582–597
4. Chung KL, Huang YL, Liu YW (2007) Efficient algorithms for coding Hilbert curve of arbitrary-sized image and application to window query. Inf Sci 177:2130–2151
5. Dearing PM, Klamroth K, Segars R Jr (2005) Planar location problems with block distance and barriers. Ann Oper Res 136:117–143
6. De Maesschalck R, Jouan-Rimbaud D, Massart DL (2000) The Mahalanobis distance. Chem Intell Lab Syst 50:1–18
7. Drezner Z (2008) Extensive experiments with hybrid genetic algorithms for the solution of the quadratic assignment problem. Comput Oper Res 35:717–736
8. Drezner T, Drezner Z (1998) Facility location in anticipation of future competition. Location Sci 6:155–173
9. Drezner T, Drezner Z (2007) Equity models in planar location. Comput Manage Sci 4:1–16
10. Drezner Z, Wesolowsky GO (2001) On the collection depots location problem. Eur J Oper Res 130:510–518
11. Fouard C, Malandain G (2005) 3-D chamfer distances and norms in anisotropic grids. Image Vision Comput 23:143–158
12. Hamacher HW, Nickel S (1998) Classification of location models. Location Sci 6:229–242
13. Hinojosa Y, Puerto J (2003) Single facility location problems with unbounded unit balls. Math Method Oper Res 58:87–104
14. Jiang J, Xu Y (2006) MiniSum location problem with farthest Euclidean distances. Math Methodol Oper Res 64:285–308
15. Munoz-Perez J, Saameno-Rodroguez JJ (1999) Location of an undesirable facility in a polygonal region with forbidden zones. Eur J Oper Res 114:372–379
16. Nickel S, Puerto J (2005) Location theory: A unified approach. Springer-Verlag, BerlinGoogle Scholar
17. Norman BA, Arapoglu R, Smith AE (2001) Integrated facilities design using a contour distance metric. IIE Trans 33:337–344Google Scholar
18. Ogryczak W (2000) Inequality measures and equitable approaches to location problems. Eur J Oper Res 122:347–391
19. Plastria F, Carrizosa E (2004) Optimal location and design of a competitive facility. Math Program 100:247–265
20. Schilling DA, Rosing KE, ReVelle CS (2000) Network distance characteristics that affect computational effort in p-median location problems. Eur J Oper Res 127:525–536
21. Song Z, Roussopoulos N (2002) Using Hilbert curve in image storing and retrieving. Inf Syst 27:523–536
22. Uster H, Love RF (2003) Formulation of confidence intervals for estimated actual distances. Eur J Oper Res 151:586–601
23. Yu J, Sarker BR (2003) Directional decomposition heuristic for a linear machine-cell location problem. Eur J Oper Res 149:142–184