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The Weiszfeld Algorithm: Proof, Amendments, and Extensions

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Foundations of Location Analysis

Part of the book series: International Series in Operations Research & Management Science ((ISOR,volume 155))

Abstract

Some time in the early seventeenth century, the following geometrical optimization problem was posed:

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References

  • Arriola R, Laporte G, Ortega FA (2005) The Weber problem in a multi-storey building. INFOR 43(3):157–169.

    Google Scholar 

  • Bajaj C (1988) The algebraic degree of geometric optimisation problems. Discrete and Comput Geom 3:177–191

    Google Scholar 

  • Balas E, Yu CS (1982) A note on the Weiszfeld-Kuhn algorithm for the general Fermat problem. Management Science Report 484:1–16, Graduate School of Industrial Administration, Carnegie Mellon University, Pittsburgh

    Google Scholar 

  • Brimberg J (2003) Further notes on convergence of the Weiszfeld algorithm. Yugoslav J Oper Res 13:199–206

    Google Scholar 

  • Brimberg J, Chen R, Chen D (1998) Accelerating convergence in the Fermat–Weber location problem. Oper Res Lett 22:151–157

    Google Scholar 

  • Brimberg J, Hansen P, Mladenović N (2006) Decomposition strategies for large-scale continuous location–allocation problems IMA J Manag Math 17:307–316

    Google Scholar 

  • Brimberg J, Hansen P, Mladenović N, Taillard ED (2000) Improvements and comparison of heuristics for solving the multisource Weber problem. Oper Res 48:444–460

    Google Scholar 

  • Brimberg J, Love RF (1993) Global convergence of a generalized iterative procedure for the minisum location problem with ℓ p distances. Oper Res 41:1153–1163

    Google Scholar 

  • Cánovas L, Canavate R, Marín A (2002) On the convergence of the Weiszfeld algorithm. Math Program 93:327–330

    Google Scholar 

  • Cera M, Ortega FA (2002) Locating the median hunter among n mobile prey on the plane. Inter J Ind Eng 9:6–15

    Google Scholar 

  • Chandrasekaran R, Tamir A (1989) Open questions concerning Weiszfeld’s algorithm for the Fermat-Weber location problem. Math Program 44:293–295

    Google Scholar 

  • Chen R (1991) An improved method for the solution of the problem of location on a inclined plane. RAIRO Rech Oper—Oper Res 25:45–53

    Google Scholar 

  • Chen R (2001) Optimal location of a single facility with circular demand areas. Comput Math Appl 41:1049–1061

    Google Scholar 

  • Cockayne EJ, Melzak ZA (1969) Euclidean constructibility in graph-minimization problems. Math Mag 42:206–208

    Google Scholar 

  • Cooper L (1963) Location-allocation problems. Oper Res 11:331–343

    Google Scholar 

  • Cooper L (1968) An extension of the generalized Weber problem. J Reg Sci 8:181–197

    Google Scholar 

  • Cooper L, Katz IN (1981) The Weber problem revisited. Comput Math Appl 7:225–234

    Google Scholar 

  • Dowling PD, Love RF (1986) Bounding methods for facilities location algorithms. Nav Res Logist Q 33:775–787

    Google Scholar 

  • Dowling PD, Love RF (1987) An evaluation of the dual as a lower bound in facilities location problems. Inst Ind Eng Trans 19:160–166

    Google Scholar 

  • Drezner Z (1984) The planar two center and two median problems. Transp Sci 18:351–361

    Google Scholar 

  • Drezner Z (1995) A note on accelerating the Weiszfeld procedure. Locat Sci 3:275–279

    Google Scholar 

  • Drezner Z (2009) On the convergence of the generalized Weiszfeld algorithm. Ann Oper Res 167:327–336

    Google Scholar 

  • Drezner T, Drezner Z (2004) Finding the optimal solution to the Huff based competitive location model. Comput Manag Sci 1:193–208

    Google Scholar 

  • Drezner Z, Klamroth K, Schöbel A, Wesolowsky GO (2002) The Weber problem. In Drezner Z, Hamacher H (eds) Facility Location: Applications and Theory. Springer, Berlin, pp. 1–136

    Google Scholar 

  • Drezner Z, Schaible S, Wesolowsky GO (1990) Queuing-location problems in the plane. Nav Res Logist Q 37:929–935

    Google Scholar 

  • Drezner Z, Wesolowsky GO (1978) Facility location on a sphere. J Oper Res Soc 29:997–1004

    Google Scholar 

  • Drezner Z, Wesolowsky GO (1980) Optimal location of a facility relative to area demands. Nav Res Logist Q 27:199–206

    Google Scholar 

  • Drezner Z, Wesolowsky GO (1991) The Weber problem on the plane with some negative weights. INFOR 29:87–99

    Google Scholar 

  • Eckhardt U (1980) Weber’s problem and Weiszfeld’s algorithm in general spaces. Math Program 18:186–196

    Google Scholar 

  • Eilon S, Watson-Gandy CDT, Christofides N (1971) Distribution management: Mathematical modelling and practical analysis. Charles Griffin & Co., Ltd., London

    Google Scholar 

  • Elzinga DJ, Hearn DW (1983) On stopping rules for facilities location algorithms. Inst of Ind Eng Trans 15:81–83

    Google Scholar 

  • Fagnano GF (1775) Problemata quaedam ad methodum maximorum et minimorum spectantia, Nova Acta Eruditorum 1775 Mensis Iunii (published in 1779), 281–303

    Google Scholar 

  • Fernández J, Pelegrín B, Plastria F, Tóth B (2007) Solving a Huff-like competitive location and design model for profit maximization in the plane. Eur J Oper Res 179:1274–1287

    Google Scholar 

  • Francis RL, McGinnis LF, White JA (1992) Facility layout and location: an analytical approach (2nd edition). Prentice Hall, Englewood Cliffs

    Google Scholar 

  • Franksen OI, Grattan-Guinness I (1989) The earliest contribution to location theory? Spatio-temporal equilibrium with Lamé and Clapeyron, 1829. Math Comput Simul 31:195–220

    Google Scholar 

  • Frenk JBG, Melo MT, Zhang S (1994) A Weiszfeld method for a generalized ℓ p distance minisum location model in continuous space. Locat Sci 2:111–127

    Google Scholar 

  • Gass SA (2004) In Memoriam, Andrew (Andy) Vazsonyi: 1916–2003, OR/MS Today, February 2004, http://www.lionhrtpub.com/orms/orms-2-04/frmemoriam.html, see also Ann Oper Res 167:2–5 (2009)

  • Gendreau M, Potvin J-Y (2008) Metaheuristics: a Canadian Perspective. INFOR 46:71–80

    Google Scholar 

  • Hoos HH, Stützle T (2005) Stochastic local search: foundations and applications. Elsevier, Amsterdam

    Google Scholar 

  • Hiriart-Urruty, J-B, Lemaréchal C (2001) Fundamentals of convex analysis. Springer, Berlin

    Google Scholar 

  • Illgen A (1979) Das Verhalten von Abstiegsverfahren an einer Singularität des Gradienten. Mathematik, Operationsforschung und Statistik, Ser Optim 10:39–55

    Google Scholar 

  • Juel H (1984) On a rational stopping rule for facilities location algorithms. Nav Res Logist Q 31:9–11

    Google Scholar 

  • Katz IN (1969) On the convergence of a numerical scheme for solving some locational equilibrium problems. SIAM J Appl Math 17:1224–1231

    Google Scholar 

  • Katz IN (1974) Local convergence in Fermat’s problem. Math Program 6:89–104

    Google Scholar 

  • Katz IN, Cooper L (1980) Optimal location on a sphere. Comput Math Appl 6:175–196

    Google Scholar 

  • Katz IN, Vogl SR (2010) A Weiszfeld algorithm for the solution of an asymmetric extension of the generalized Fermat location problem. Comput Math Appl 59:399–410

    Google Scholar 

  • Kuhn H (1967) On a pair of dual nonlinear programs. In: Abadie J (ed) Methods of nonlinear programming. North-Holland, Amsterdam, pp. 38–54

    Google Scholar 

  • Kuhn H (1973) A note on Fermat’s problem. Math Program 4:98–107

    Google Scholar 

  • Kuhn HW, Kuenne RE (1962) An efficient algorithm for the numerical solution of the generalized Weber problem in spatial economics. J Reg Sci 4:21–33

    Google Scholar 

  • Kupitz YS, Martini H (1997) Geometric aspects of the generalized Fermat-Torricelli problem. In: Intuitive geometry, Bolyai Society, Math Stud 6:55–127

    Google Scholar 

  • Lamé G, Clapeyron BPE (1829) Mémoire sur l’application de la statique à la solution des problèmes relatifs à la théorie des moindres distances. Journal des Voies de Communications 10: 26–49. (In French—Memoir on the application of statics to the solution of problems concerning the theory of least distances.) For a translation into English see Franksen & Grattan-Guinness (1989)

    Google Scholar 

  • Launhardt W (1885) Mathematische Begründung der Volkswirtschaftslehre, Wilhelm Engelmann, Leipzig

    Google Scholar 

  • Levin Y, Ben-Israel A. (2002) The Newton bracketing method for convex minimization. Comput Optim and Appl 21:213–229

    Google Scholar 

  • Li Y (1998) A Newton acceleration of the Weiszfeld algorithm for minimizing the sum of Euclidean distances. Comput Optim Appl 10:219–242

    Google Scholar 

  • Love RF, Dowling PD (1989) A new bounding method for single facility location models. Ann Oper Res 18:103–112

    Google Scholar 

  • Love RF, Morris JG, Wesolowsky GO (1988) Facilities location: models & methods. North-Holland, New York

    Google Scholar 

  • Love RF, Yeong WY (1981) A stopping rule for facilities location algorithms. Am Inst Ind Eng Trans 13:357–362

    Google Scholar 

  • Miehle W (1958) Link-length minimization in networks. Oper Res 6:232–243

    Google Scholar 

  • Morris JG, Verdini WA (1979) Minisum ℓ p distance location problems solved via a perturbed problem and Weiszfeld’s algorithm. Oper Res 27:1180–1188

    Google Scholar 

  • Ostresh LM Jr (1975) An efficient algorithm for solving the two-center location-allocation problem. J Reg Sci 15:209–216

    Google Scholar 

  • Ostresh LM Jr (1978a) Convergence and descent in the Fermat location problem. Transportation Science 12:153–164

    Google Scholar 

  • Ostresh LM Jr (1978b) On the convergence of a class of iterative methods for solving the Weber location problem. Oper Res 26:597–609

    Google Scholar 

  • Overton ML (1983) A quadratically convergent method for minimizing a sum of Euclidean norms. Math Program 27:34–63

    Google Scholar 

  • Plastria F (1987) Solving general continuous single facility location problems by cutting planes. Eur J Oper Res 29:98–110

    Google Scholar 

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

    Google Scholar 

  • Plastria F (1992b) When facilities coincide: exact optimality conditions in multifacility location. J Math Anal and Appl 169:476–498

    Google Scholar 

  • Plastria F (1993) On destination optimality in asymmetric distance Fermat-Weber problems. Ann Oper Res 40:355–369

    Google Scholar 

  • Plastria F, Elosmani M (2008) On the convergence of the Weiszfeld algorithm for continuous single facility location-allocation problems. TOP 16:388–406

    Google Scholar 

  • Rado F (1988) The Euclidean multifacility location problem. Oper Res 36:485–492

    Google Scholar 

  • Rautenbach D, Struzyna M, Szegedy C, Vygen J (2004) Weiszfeld’s algorithm revisited once again. Report 04946-OR, Research Institute for Discrete Mathematics, University of Bonn, Germany

    Google Scholar 

  • Rockafellar T (1970) Convex analysis. Princeton University Press, Princeton, NJ

    Google Scholar 

  • Rosen JB, Xue GL (1992) On the convergence of Miehle’s algorithm for the Euclidean multifacility location problem. Oper Res 40:188–191

    Google Scholar 

  • Rosen JB, Xue GL (1993) On the convergence of a hyperboloid approximation procedure for the perturbed Euclidean multifacility location problem. Oper Res 41:1164–1171

    Google Scholar 

  • Rosing K, Harris B (1992) Algorithmic and technical improvements: optimal solutions to the (generalized) multi-Weber problem. Pap in Reg Sci 71:331–352

    Google Scholar 

  • Schärlig A (1973) About the confusion between the center of gravity and Weber’s optimum. Reg Urban Econ 13:371–382

    Google Scholar 

  • Sturm R (1884) Über den Punkt kleinster Entfernungssumme von gegebenen Punkten. Journal für die reine und angewandte Mathematik 97:49–61. (In German: On the point of smallest distance sum from given points)

    Google Scholar 

  • Shi Y, Chang Q, Xu J (2007) Convergence of fixed point iteration for deblurring and denoising problem. Appl Math and Comput 189:1178–1185

    Google Scholar 

  • Valkonen T (2006) Convergence of a SOR-Weiszfeld Type Algorithm for Incomplete Data Sets. Numer Funct Anal and Optim 27;931–952. See also Errata, same journal, (2008), volume 29:1201–1203

    Google Scholar 

  • Vardi Y, Zhang C-H (2001) A modified Weiszfeld algorithm for the Fermat-Weber location problem. Math Program 90:559–566

    Google Scholar 

  • Vazsonyi A (2002) Which Door has the Cadillac. Writers Club Press, New York

    Google Scholar 

  • Vazsonyi A (2002) Pure mathematics and the Weiszfeld algorithm. Decision Line 33:12–13, http://www.decisionsciences.org/DecisionLine/Vol33/33_3/index.htm

    Google Scholar 

  • Weber A (1909) Über den Standort der Industrien. Tübingen, Germany. (English translation: Friedrich CJ (translator) (1929), Theory of the location of industries. University of Chicago Press, Chicago)

    Google Scholar 

  • Weida NC, Richardson R, Vazsonyi A (2001) Operations analysis using Excel. Duxbury, Pacific Grove

    Google Scholar 

  • Weiszfeld E (1937) Sur le point pour lequel la somme des distances de n points données est minimum. Tôhoku Math J (first series) 43:355–386

    Google Scholar 

  • Weiszfeld E, Plastria F (2009) On the point for which the sum of the distances to n given points is minimum. Ann Oper Res 167:7–41

    Google Scholar 

  • Wendell RE, Peterson EL (1984) A dual approach for obtaining lower bounds to the Weber problem. J Reg Sci 24:219–228

    Google Scholar 

  • Wesolowsky GO (1993) The Weber problem: history and perspectives. Locat Sci 1:5–23

    Google Scholar 

  • Wu S (1994) A polynomial time algorithm for solving the Fermat-Weber location problem with mixed norms. Optimization 30:227–234

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Operational Research, Statistics and Information Systems for Management, MOSI, Vrije Universiteit Brussel, Pleinlaan 2, B 1050, Brussels, Belgium

    Frank Plastria

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  1. Frank Plastria
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Correspondence to Frank Plastria .

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Editors and Affiliations

  1. Faculty of Business Administration, University of New Brunswick, Fredericton,, New Brunswick, Canada

    H. A. Eiselt

  2. Depto. Ingenieria Electrica, Pontificia Universidad Católica de Chile, Av. Vicuna Mackenna 4860, Macul, Santiago, Chile

    Vladimir Marianov

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Plastria, F. (2011). The Weiszfeld Algorithm: Proof, Amendments, and Extensions. In: Eiselt, H., Marianov, V. (eds) Foundations of Location Analysis. International Series in Operations Research & Management Science, vol 155. Springer, New York, NY. https://doi.org/10.1007/978-1-4419-7572-0_16

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