Automation and Remote Control

, Volume 75, Issue 7, pp 1221–1230 | Cite as

Solving a maximin location problem on the plane with given accuracy

  • G. G. ZabudskiiEmail author
  • A. A. Koval’
Topical Issue


We consider the optimal placement problem in a bounded region on a plane with fixed objects. We specify minimal admissible distances between placed and fixed objects. The optimization criterion is to maximize the minimal weighted distances from placed objects to fixed ones. We propose a quasipolynomial combinatorial algorithm to solve this problem with a given accuracy. We show the results of a computational experiment with the integer programming model and the IBM ILOG CPLEX suite.


Remote Control Integer Linear Programming Corner Point Recognition Problem Vertical Edge 
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.


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  1. 1.
    Beresnev, V.L., Diskretnye zadachi razmeshcheniya i polinomy ot bulevykh peremennykh (Discrete Location Problems and Polynomials of Boolean Variables), Novosibirsk: Inst. Mat. Sib. Otd. Ross. Akad. Nauk, 2005.Google Scholar
  2. 2.
    Panyukov, A.V. and Pelzwerger, B.V., Optimal Location of a Tree in a Finite Set, Zh. Vychisl. Mat. Mat. Fiz., 1988, vol. 28, pp. 618–620.zbMATHGoogle Scholar
  3. 3.
    Panyukov, A.V. and Pelzwerger, B.V., Polynomial Algorithms to Finite Veber Problem for a Tree Network, J. Comput. Appl. Math., 1991, vol. 35, pp. 291–296.CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Ageev, A.A., Gimadi, E.Kh., and Kurochkin, A.A., A Polynomial Algorithm for the Location Problem on a Chain with Identical Industrial Capacities of Plants, Diskret. Anal. Issled. Oper., 2009, vol. 16, no. 5, pp. 3–18.MathSciNetGoogle Scholar
  5. 5.
    Zabudskii, G.G., Model Building and Location Problem Solving in a Plane with Forbidden Gaps, Autom. Remote Control, 2006, vol. 67, no. 12, pp. 1986–1990.CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Zabudskii, G.G. and Amzin, I.V., Search Region Contraction of the Weber Problem Solution on the Plane with Rectangular Forbidden Zones, Autom. Remote Control, 2012, vol. 73, no. 5, pp. 821–830.CrossRefzbMATHGoogle Scholar
  7. 7.
    Kochetov, Yu.A., Pashchenko, M.G., and Plyasunov, A.V., On the Complexity of Local Search for the p-Median Problem, Diskret. Anal. Issled. Oper., Ser. 2, 2005, vol. 12, no. 2, pp. 44–71.zbMATHMathSciNetGoogle Scholar
  8. 8.
    Farahani, R.Z. and Hekmatfar, M., Facility Location: Concepts, Models, Algorithms and Case Studies, Heidelberg: Physica-Verlag Heidelberg, 2009.CrossRefGoogle Scholar
  9. 9.
    Nickel, S. and Puerto, J., Location Theory, Heidelberg: Springer-Verlag, 2005.zbMATHGoogle Scholar
  10. 10.
    Zabudskii, G.G. and Koval’, A.A., Object Location on a Plane with Maximin Criterion and Minimal Admissible Distances, Proc. XIV All-Russian Conf. “Mathematical Programming and Applications,” Yekaterinburg: Inst. Mat. Mekh. Ural. Otd. Ross. Akad. Nauk, 2011, p. 88.Google Scholar
  11. 11.
    Zabudskii, G.G. and Markhotskaya, N.V., Solving the Maximin Location Problem on a Plane with Minimal Admissible Distances, Proc. XIV Baikal Int. School-Seminar “Optimization Methods and Their Applications,” Irkutsk: Inst. Sist. Energ. Sib. Otd. Ross. Akad. Nauk, 2008, vol. 1, pp. 380–387.Google Scholar
  12. 12.
    Brimberg, J. and Mehrez, A., Multi-Facility Location Using a Maximin Criterion and Rectangular Distances, Location Sci., 1994, vol. 2, no. 1, pp. 11–19.zbMATHGoogle Scholar
  13. 13.
    Katz, M.J., Kedem, K., and Segal, M., Improved Algorithms for Placing Undesirable Facilities, Comput. Oper. Res., 2002, no. 29, pp. 1859–1872.Google Scholar
  14. 14.
    Tamir, A., Locating Two Obnoxious Facilities Using the Weighted Maximin Criterion, Oper. Res. Lett., 2006, no. 34, pp. 97–105.Google Scholar
  15. 15.
    Levin, G.M. and Tanaev, V.S., Dekompozitsionnye metody optimizatsii proektnykh reshenii (Decomposition Optimization Methods for Design Solutions), Minsk: Nauka i Tekhnika, 1978.Google Scholar
  16. 16.
    Tanaev, V.S., Dekompozitsiya i agregirovanie v zadachakh matematicheskogo programmirovaniya (Decomposition and Aggregation in Mathematical Programming Problems), Minsk: Nauka i Tekhnika, 1987.Google Scholar
  17. 17.
    Preparata, F.P. and Shamos, M.G., Computational Geometry. An Introduction, New York: Springer, 1985. Translated under the title Vychislitel’naya geometriya. Vvedenie, Moscow: Mir, 1988.CrossRefGoogle Scholar
  18. 18.
    Dearing, P.M. and Francis, R.L., A Network Flow Solution to a Multifacility Minimax Location Problem Involving Rectilinear Distances, Transportat. Sci., 1974, vol. 8, pp. 126–141.CrossRefMathSciNetGoogle Scholar

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© Pleiades Publishing, Ltd. 2014

Authors and Affiliations

  1. 1.Omsk Branch of Sobolev Institute of Mathematics, Siberian BranchRussian Academy of SciencesOmskRussia

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