Skip to main content
SpringerLink
Log in

Location of Dimensional Facilities in a Continuous Space

  • Chapter
  • First Online:
Book cover Location Science

Abstract

In many cases, the facilities to be located cannot be represented by isolated points, but may be modeled as dimensional structures. Examples for one-dimensional facilities are straight lines, line segments, or circles while boxes, strips, or balls are two-dimensional facilities. The goal of this chapter is to review the location of lines and circles in the plane and the location of hyperplanes and hyperspheres in higher dimensional spaces. We also discuss the location of some other dimensional facilities. We formulate the resulting location problems, point out some of their important properties and review the basic solution techniques and algorithmic approaches. Our focus lies on presenting a unified understanding of the common characteristics these problems have, and on reviewing the new findings obtained in this field within the last 10 years.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 89.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 119.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  • Agarwal P, Efrat A, Sharir M, Toledo S (1993) Computing a segment center for a planar point set. J Algorithm 15:314–323

    Article  Google Scholar 

  • Agarwal P, Aronov B, Peled S, Sharir M (1999) Approximation and exact algorithms for minimum-width annuli and shells. In: Proceedings of the 15th ACM symposium on computational geometry, pp 380–389

    Google Scholar 

  • Agarwal P, Peled SH, Varadarajan K (2004) Approximation extent measures of points. J ACM 51:605–635

    Google Scholar 

  • Alonso J, Martini H, Spirova M (2012a) Minimal enclosing discs, circumcircles, and circumcenters in normed planes (part I). Comput Geom Theor Appl 45:258–274

    Article  Google Scholar 

  • Alonso J, Martini H, Spirova M (2012b) Minimal enclosing discs, circumcircles, and circumcenters in normed planes (part II). Comput Geom Theor Appl 45:350–369

    Article  Google Scholar 

  • Bennet K, Mangasarian O (1992) Robust linear programming discrimination of two linearly inseparable sets. Optim Method Softw 1:23–34

    Article  Google Scholar 

  • Bertsimas D, Shioda R (2007) Classification and regression via integer optimization. Oper Res 55:252–271

    Article  Google Scholar 

  • Blanquero R, Carrizosa E, Hansen P (2009) Locating objects in the plane using global optimization techniques. Math Oper Res 34:837–858

    Article  Google Scholar 

  • Blanquero R, Carrizosa E, Schöbel A, Scholz D (2011) Location of a line in the three-dimensional space. Eur J Oper Res 215:14–20

    Article  Google Scholar 

  • Brimberg J, Nickel S (2009) Constructing a DC decomposition for ordered median problems. J Global Optim 45:187–201

    Article  Google Scholar 

  • Brimberg J, Wesolowsky G (2000) Note: facility location with closest rectangular distances. Nav Res Log 47:77–84

    Article  Google Scholar 

  • Brimberg J, Juel H, Schöbel A (2002) Linear facility location in three dimensions - models and solution methods. Oper Res 50:1050–1057

    Article  Google Scholar 

  • Brimberg J, Juel H, Schöbel A (2003) Properties of 3-dimensional line location models. Ann Oper Res 122:71–85

    Article  Google Scholar 

  • Brimberg J, Juel H, Schöbel A (2009a) Locating a circle on the plane using the minimax criterion. Stud Locat Anal 17:46–60

    Google Scholar 

  • Brimberg J, Juel H, Schöbel A (2009b) Locating a minisum circle in the plane. Discrete Appl Math 157:901–912

    Article  Google Scholar 

  • Brimberg J, Juel H, Körner MC, Schöbel A (2011a) Locating a general minisum ‘circle’ on the plane. 4OR-Q J Oper Res 9:351–370

    Google Scholar 

  • Brimberg J, Juel H, Körner MC, Schöbel A (2011b) Locating an axis-parallel rectangle on a Manhattan plane. TOP 22:185–207

    Article  Google Scholar 

  • Brimberg J, Juel H, Körner MC, Schöbel A (2013a) On models for continuous facility location with partial coverage. J Oper Res Soc. doi:JORS.2013.142

    Google Scholar 

  • Brimberg J, Schieweck R, Schöbel A (2013b) Locating a median line with partial coverage distance. Preprint 32, Institut für Numerische und Angewandte Mathematik, Universität Göttingen. http://num.math.uni-goettingen.de/preprints/files/2013-32.pdf

  • Brodal GS, Jacob R (2002) Dynamic planar convex hull. In: Proceedings of the 43rd annual IEEE symposium on foundations of computer science, pp 617–626

    Google Scholar 

  • Carrizosa E, Plastria F (2008) Optimal expected-distance separating halfspace. Math Oper Res 33:662–677

    Article  Google Scholar 

  • Chan TM (2000) Approximating the diameter, width, smalat enclosing cylinder, and minimum-width annulus. In: Proceedings of the 16th annual symposium on computational geometry SCG ’00. ACM, New York, pp 300–309

    Google Scholar 

  • Cheng SW (1996) Widest empty L-shaped corridor. Inform Process Lett 58:277–283

    Article  Google Scholar 

  • Chernov N, Sapirstein P (2008) Fitting circles to data with correlated noise. Comput Stat Data Anal 52:5328–5337

    Article  Google Scholar 

  • Coope I (1993) Circle fitting by linear and nonlinear least squares. J Optim Theory Appl 76:381–388

    Article  Google Scholar 

  • Crawford J (1983) A non-iterative method for fitting circular arcs to measured points. Nucl Instrum Methods 211:223–225

    Article  Google Scholar 

  • Das G, Mukhopadhyay D, Nandy S (2009) Improved algorithm for the widest empty 1-corner corridor. Inform Process Lett 109:1060–1065

    Article  Google Scholar 

  • Deshpande A, Rademacher L, Vempala S, Wang G (2006) Matrix approximation and projective clustering via volume sampling. In: Proceedings of the 17th annual ACM-SIAM symposium on discrete algorithms. ACM, New York, pp 1117–1126

    Google Scholar 

  • Dey T (1998) Improved bounds for planar k-sets and related problems. Discrete Comput Geom 19:373–382

    Article  Google Scholar 

  • Díaz-Bánez JM, Mesa J, Schöbel A (2004) Continuous location of dimensional structures. Eur J Oper Res 152:22–44

    Article  Google Scholar 

  • Díaz-Bánez JM, López MA, Sellarès JA (2006a) Locating an obnoxious plane. Eur J Oper Res 173:556–564

    Article  Google Scholar 

  • Díaz-Bánez JM, López MA, Sellarès JA (2006b) On finding a widest empty 1-corner corridor. Inform Process Lett 98:199–205

    Article  Google Scholar 

  • Díaz-Bánez J, Korman M, Pérez-Lantero P, Ventura I (2013) The 1-median and 1-highway problem. Eur J Oper Res 225:552–557

    Article  Google Scholar 

  • Dicks DR (1985) Early Greek astronomy to aristotle (Aspects of Greek and Roman life series). Cornell University Press, Ithaca

    Google Scholar 

  • Drezner Z, Brimberg J (2014) Fitting concentric circles to measurements. Math Method Oper Res 29:119–133

    Article  Google Scholar 

  • Drezner T, Drezner Z (2007) Equity models in planar location. Comput Manag Sci 4:1–16

    Article  Google Scholar 

  • Drezner Z, Suzuki A (2004) The big triangle small triangle method for the solution of non-convex facility location problems. Oper Res 52:128–135

    Article  Google Scholar 

  • Drezner Z, Klamroth K, Schöbel A, Wesolowsky G (2001) The weber problem, chap 1. In: Drezner Z, Hamacher H (eds) Facility location - applications and theory. Springer, Berlin/Heidelberg, pp 1–36

    Google Scholar 

  • Drezner Z, Steiner S, Wesolowsky G (2002) On the circle closest to a set of points. Comput Oper Res 29:637–650

    Article  Google Scholar 

  • Ebara H, Fukuyama N, Nakano H, Nakanishi Y (1989) Roundness algorithms using the voronoi diagrams. In: Proceedings of the 1st Canadian conference on computational geometry, p 41

    Google Scholar 

  • Edelsbrunner H (1985) Finding transversals for sets of simple geometric figures. Theor Comput Sci 35:55–69

    Article  Google Scholar 

  • Efrat A, Sharir M (1996) A near-linear algorithm for the planar segment-center problem. Discrete Comput Geom 16:239–257

    Article  Google Scholar 

  • Espejo I, Rodríguez-Chía A (2011) Simultaneous location of a service facility and a rapid transit line. Comput Oper Res 38:525–538

    Article  Google Scholar 

  • Espejo I, Rodríguez-Chía A (2012) Simultaneous location of a service facility and a rapid transit line. Comput Oper Res 39:2899–2903

    Article  Google Scholar 

  • Farago F, Curtis M (1994) Handbook of dimensional measurement, 3rd edn. Industrial Press, New York

    Google Scholar 

  • Gander W, Golub G, Strebel R (1994) Least-squares fitting of circles and ellipses. BIT 34:558–578

    Article  Google Scholar 

  • García-López J, Ramos P, Snoeyink J (1998) Fitting a set of points by a circle. Discrete Comput Geom 20:389–402

    Article  Google Scholar 

  • Gluchshenko O (2008) Annulus and center location problems. Ph.D. thesis, Technische Universität Kaiserslautern

    Google Scholar 

  • Gluchshenko ON, Hamacher HW, Tamir A (2009) An optimal o(nlogn) algorithm for finding an enclosing planar rectilinear annulus of minimum width. Oper Res Lett 37:168–170

    Article  Google Scholar 

  • Golub G, van Loan C (1980) An analysis of the total least squares problem. SIAM J Numer Anal 17:883–893

    Article  Google Scholar 

  • Hamacher H, Nickel S (1995) Restricted planar location problems and applications. Nav Res Log 42:967–992

    Article  Google Scholar 

  • Har-Peled S, Varadarajan K (2002) Projective clustering in high dimensions using core-sets. In: Proceedings of the 18th annual symposium on computational geometry. ACM, New York, pp 312–318

    Google Scholar 

  • Helly E (1923) Über Mengen konvexer Körper mit gemeinschaftlichen Punkten. Jahrb Dtsch Math Ver 32:175–176

    Google Scholar 

  • Houle M, Toussaint G (1985) Computing the width of a set. In: Proceedings of the 1st ACM symposium on computational geometry, pp 1–7

    Google Scholar 

  • Imai H, Lee D, Yang CD (1992) 1-Segment center problems. ORSA J Comput 4:426–434

    Article  Google Scholar 

  • Janardan R, Preparata F (1996) Widest-corridos problems. Nord J Comput 1:231–245

    Google Scholar 

  • Kapelushnik L (2008) Computing the k-centrum and the ordered median hyperplane. Master’s thesis, School of Computer Science, Tel-Aviv University

    Google Scholar 

  • Kasa I (1976) A circle fitting procedure and its error analysis. IEEE T Instrum Meas 25:8–14

    Article  Google Scholar 

  • Kelachankuttu H, Batta R, Nagi R (2007) Contour line construction for a new rectangular facility in an existing layout with rectangular departments. Eur J Oper Res 180:149–162

    Article  Google Scholar 

  • Korneenko N, Martini H (1990) Approximating finite weighted point sets by hyperplanes. In: SWAT90. Lecture notes in computer science, vol 447. Springer, pp 276–286

    Google Scholar 

  • Korneenko N, Martini H (1993) Hyperplane approximation and related topics. In: Pach J (ed) New trends in discrete and computational geometry. Springer, New York, pp 135–162

    Chapter  Google Scholar 

  • Körner MC (2011) Minisum hyperspheres. Springer, New York

    Book  Google Scholar 

  • Körner MC, Brimberg J, Juel H, Schöbel A (2009) General circle location. In: Proceedings of the 21st Canadian conference on computational geometry, pp 111–114

    Google Scholar 

  • Körner MC, Brimberg J, Juel H, Schöbel A (2011) Geometric fit of a point set by generalized circles. J Global Optim 51:115–132

    Article  Google Scholar 

  • Körner MC, Martini H, Schöbel A (2012) Minisum hyperspheres in normed spaces. Discrete Appl Math 16:2221–2233

    Article  Google Scholar 

  • Krempasky T (2012) Locating median lines and hyperplanes with a restriction on the slope. Ph.D. thesis, Universität Göttingen

    Google Scholar 

  • Le V, Lee D (1991) Out-of-roundness problem revisited. IEEE Trans Pattern Anal 13:217–223

    Article  Google Scholar 

  • Lee D, Ching Y (1985) The power of geometric duality revisited. Inform Process Lett 21:117–122

    Article  Google Scholar 

  • Lozano AJ, Plastria F (2009) The ordered median Euclidean straight-line location problem. Stud Locat Anal 17:29–43

    Google Scholar 

  • Lozano AJ, Mesa J, Plastria F (2010) The k-centrum straight-line location problem. J Math Model Algorithms 9:1–17

    Article  Google Scholar 

  • Lozano AJ, Mesa J, Plastria F (2013) Location of weighted anti-ordered median straight lines with euclidean distances. Discrete Appl Math. doi:10.1016/j.dam.2013.04.016

    Google Scholar 

  • Mangasarian O (1999) Arbitrary-norm separating plane. Oper Res Lett 24:15–23

    Article  Google Scholar 

  • Martini H, Schöbel A (1998) Median hyperplanes in normed spaces—a survey. Discrete Appl Math 89:181–195

    Article  Google Scholar 

  • Martini H, Schöbel A (1999) A characterization of smooth norms. Geom Dedicata 77:173–183

    Article  Google Scholar 

  • Megiddo N (1984) Linear programming in linear time when the dimension is fixed. J ACM 31:114–127

    Article  Google Scholar 

  • Megiddo N, Tamir A (1982) On the complexity of locating linear facilities in the plane. Oper Res Lett 1:194–197

    Article  Google Scholar 

  • Megiddo N, Tamir A (1983) Finding least-distance lines. SIAM J Algebr Discrete Method 4:207–211

    Article  Google Scholar 

  • Morris J, Norback J (1980) A simple approach to linear facility location. Transp Sci 14:1–8

    Article  Google Scholar 

  • Morris J, Norback J (1983) Linear facility location - solving extensions of the basic problem. Eur J Oper Res 12:90–94

    Article  Google Scholar 

  • Moura L, Kitney R (1992) A direct method for least-squares circle fitting. Comput Phys Commun 64:57–63

    Article  Google Scholar 

  • Mukherjee J, Sinha Mahapatra PR, Karmakar A, Das S (2013) Minimum-width rectangular annulus. Theor Comput Sci 508:74–80

    Article  Google Scholar 

  • Narula SC, Wellington JF (1982) The minimum sum of absolute errors regression: a state of the art survey. Int Stat Rev 50:317–326

    Article  Google Scholar 

  • Nickel S, Puerto J (2005) Location theory: a unified approach. Springer, Berlin/Heidelberg

    Google Scholar 

  • Nievergelt Y (2002) A finite algorithm to fit geometrically all midrange lines, circles, planes, spheres, hyperplanes, and hyperspheres. Numer Math 91:257–303

    Article  Google Scholar 

  • Nievergelt Y (2004) Perturbation analysis for circles, spheres, and generalized hyperspheres fitted to data by geometric total least-squares. Math Comput 73:169–180

    Article  Google Scholar 

  • Nievergelt Y (2010) Median spheres: theory, algorithms, applications. Numer Math 114:573–606

    Article  Google Scholar 

  • Overmars MH, van Leeuwen J (1981) Maintenance of configurations in the plane. J Comput Syst Sci 23:166–204

    Article  Google Scholar 

  • Plastria F (1992) GBSSS: the generalized big square small square method for planar single-facility location. Eur J Oper Res 62:163–174

    Article  Google Scholar 

  • Plastria F (2001) Continuous covering location problems. In: Drezner Z, Hamacher H (eds) Facility location - applications and theory. Springer, Berlin/Heidelberg, pp 1–36

    Google Scholar 

  • Plastria F, Carrizosa E (2001) Gauge-distances and median hyperplanes. J Optim Theory Appl 110:173–182

    Article  Google Scholar 

  • Plastria F, Carrizosa E (2012) Minmax-distance approximation and separation problems: geometrical properties. Math Program 132:153–177

    Article  Google Scholar 

  • Rivlin T (1979) Approximation by circles. Computing 21:1–17

    Article  Google Scholar 

  • Robert JM (1991) Linear approximation and line transversals. Ph.D. thesis, School of Computer Sciences, McGill University, Montreal

    Google Scholar 

  • Robert JM, Toussaint G (1994) Linear approximation of simple objects. Comput Geom Theor Appl 4:27–52

    Article  Google Scholar 

  • Robins G, Shute C (1987) The Rhind mathematical papyrus. An ancient Egyptian text. British Museum

    Google Scholar 

  • Rockafellar R (1970) Convex analysis. Princeton Landmarks, Princeton

    Google Scholar 

  • Rorres C, Romano D (1997) Finding the center of a circular starting line in an Ancient Greek stadium. SIAM Rev 39:745–754

    Article  Google Scholar 

  • Sarkar A, Batta R, Nagi R (2007) Placing a finite size facility with a center objective on a rectangular plane with barriers. Eur J Oper Res 179:1160–1176

    Article  Google Scholar 

  • Savas S, Batta R, Nagi R (2002) Finite-size facility placement in the presence of barriers to rectilinear travel. Oper Res 50:1018–1031

    Article  Google Scholar 

  • Schieweck R (2013) Lower bounds for line location problems via demand regions. Technical report 28. Institut für Numerische und Angewandte Mathematik, Universität of Göttingen, Göttingen

    Google Scholar 

  • Schieweck R, Schöbel A (2012) Properties and algorithms for line location with extensions. In: Proceedings of the 28th European workshop on computational geometry, Assisi, Italy, pp 185–188

    Google Scholar 

  • Schöbel A (1996) Locating least-distant lines with block norms. Stud Locat Anal 10:139–150

    Google Scholar 

  • Schöbel A (1997) Locating line segments with vertical distances. Stud Locat Anal 11:143–158

    Google Scholar 

  • Schöbel A (1998) Locating least distant lines in the plane. Eur J Oper Res 106:152–159

    Article  Google Scholar 

  • Schöbel A (1999a) Locating lines and hyperplanes—theory and algorithms. Applied optimization series, vol 25. Kluwer, Dordrecht

    Google Scholar 

  • Schöbel A (1999b) Solving restricted line location problems via a dual interpretation. Discrete Appl Math 93:109–125

    Article  Google Scholar 

  • Schöbel A (2003) Anchored hyperplane location problems. Discrete Comput Geom 29:229–238

    Article  Google Scholar 

  • Schöbel A, Scholz D (2010) The Big Cube Small Cube solution method for multidimensional facility location problems. Comput Oper Res 37:115–122

    Article  Google Scholar 

  • Schömer E, Sellen J, Teichmann M, Yap C (2000) Smallest enclosing cylinders. Algorithmica 27:170–186

    Article  Google Scholar 

  • Späth H (1997) Least squares fitting of ellipses and hyperbolas. Comput Stat 12:329–341

    Google Scholar 

  • Späth H (1998) Least-squares fitting with spheres. J Optim Theory Appl 96:191–199

    Article  Google Scholar 

  • Sun T (2009) Applying particle swarm optimization algorithm to roundness measurement. Expert Syst Appl 36:3428–3438

    Article  Google Scholar 

  • Suzuki T (2005) Optimal location of orbital routes in a circular city. Presented at ISOLDE X—10th international symposium on locational decisions, Sevilla and Islantilla, 2–8 June 20005

    Google Scholar 

  • Swanson K, Lee DT, Wu V (1995) An optimal algorithm for roundness determination on convex polygons. Comput Geom Theor Appl 5:225–235

    Article  Google Scholar 

  • Ventura J, Yeralan S (1989) The minmax center estimation problem. Eur J Oper Res 41:64–72

    Article  Google Scholar 

  • Wang L, Gordon MD, Zhu J (2006) Regularized least absolute deviations regression and an efficient algorithm for parameter tuning. In: Proceedings of the 6th international conference on data mining. IEEE, New York, pp 690–700

    Google Scholar 

  • Wesolowsky G (1972) Rectangular distance location under the minimax optimality criterion. Transp Sci 6:103–113

    Article  Google Scholar 

  • Wesolowsky G (1975) Location of the median line for weighted points. Environ Plan A 7:163–170

    Article  Google Scholar 

  • Yamamoto P, Kato K, Imai K, Imai H (1988) Algorithms for vertical and orthogonal L 1 linear approximation of points. In: Proceedings of the 4th annual symposium on computational geometry, pp 352–361

    Google Scholar 

  • Yeralan S, Ventura J (1988) Computerized roundness inspection. Int J Prod Res 26:1921–1935

    Article  Google Scholar 

  • Zemel E (1984) An O(n) algorithm for the linear multiple choice knapsack problem and related problems. Inform Process Lett 18:123–128

    Article  Google Scholar 

Download references

Acknowledgements

I want to thank Robert Schieweck for providing useful hints on line and hyperplane location problems.

Author information

Authors and Affiliations

  1. University of Göttingen, Göttingen, Germany

    Anita Schöbel

Authors
  1. Anita Schöbel
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Anita Schöbel .

Editor information

Editors and Affiliations

  1. HEC Montréal, Montréal, Canada

    Gilbert Laporte

  2. Institute of Operations Research (IOR), Karlsruhe Institute of Technology, Karlsruhe, Germany

    Stefan Nickel

  3. DEIO-FCUL, University of Lisbon, Lisbon, Portugal

    Francisco Saldanha da Gama

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Schöbel, A. (2015). Location of Dimensional Facilities in a Continuous Space. In: Laporte, G., Nickel, S., Saldanha da Gama, F. (eds) Location Science. Springer, Cham. https://doi.org/10.1007/978-3-319-13111-5_7

Download citation

Publish with us

Policies and ethics

Search

Navigation