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An exact method for the discrete \((r|p)\)-centroid problem

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Abstract

This paper provides a new exact iterative method for the following problem. Two decision makers, a leader and a follower, compete to attract customers from a given market. The leader opens \(p\) facilities, anticipating that the follower will react to the decision by opening \(r\) facilities. Each customer patronizes the closest opened facility. The goal is to find \(p\) facilities for the leader to maximize his market share. It is known that this problem is \(\Sigma ^P_2\)-hard and can be presented as an integer linear program with a large number of constraints. Based on this representation, we design the new iterative exact method. A local search algorithm is used at each iteration to find a feasible solution for a system of constraints. Computational results and comparison with other exact methods show that the new method can be considered as one of the alternative approaches among the most advanced exact methods for the problem.

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References

  1. Alekseeva, E., Beresnev, V., Kochetov, Y. et al.: Benchmark library: Discrete Location Problems (Russia), http://math.nsc.ru/AP/benchmarks/Competitive/p_me_comp_eng.html

  2. Alekseeva, E., Kochetov, Y.: Matheuristics and exact methods for the discrete (r|p)-centroid problem. In: Talbi, E.-G. (ed.) Metaheuristics for bi-level optimization. Studies in Computational Intelligence, vol. 482, pp. 189–219. Springer, Berlin (2013)

  3. Alekseeva, E., Kochetova, N., Kochetov, Yu., Plyasunov, A.: A heuristic and exact methods for the discrete (\(r|p\))-centroid problem. LNCS 6022, pp. 11–22 (2010)

  4. Banik, A., Das, S., Maheshwari, A., Smid, M.: The discrete Voronoi game in a simple polygon. LNCS 7936, pp. 197–207 (2013)

  5. Ben-Ayed, O.: Bilevel linear programming. Comput. Oper. Res. 20(5), 485–501 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  6. Benati, S., Laporte, G.: Tabu search algorithms for the (\(r|X_p\))-medianoid and (\(r|p\))-centroid problems. Locat. Sci. 2, 193–204 (1994)

    MATH  Google Scholar 

  7. Beresnev, V., Melnikov, A.: Approximate algorithms for the competitive facility location problem. J. Appl. Ind. Math. 5(2), 180–190 (2011)

    Google Scholar 

  8. Campos-Rodríguez, C.M., Moreno Pérez, J.A.: Multiple voting location problems. Eur. J. Oper. Res. 191, 437–453 (2008)

    Article  Google Scholar 

  9. Campos-Rodríguez, C.M., Moreno Pérez, J.A., Santos-Peñate, D.R.: Particle swarm optimization with two swarms for the discrete (\(r|p\))-centroid problem. LNCS 6927, pp. 432–439 (2012)

  10. Campos-Rodríguez, C.M., Santos-Peñate, D.R., Moreno-Pérez, J.A.: An exact procedure and LP formulations for the leader-follower location problem. TOP 18(1), 97–121 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  11. Davydov, I., Kochetov, Y., Carrizosa, E.: VNS heuristic for the (\(r|p\))-centroid problem on the plane. Electron. Notes Discrete Math. 39, 5–12 (2012)

    Article  MathSciNet  Google Scholar 

  12. Davydov I., Kochetov Y., Plyasunov A.: On the complexity of the (\(r|p\))-centroid problem in the plane. TOP (2013). doi:10.1007/s11750-013-0275-y

  13. Ghosh, A., Craig, C.: A Location allocation model for facility planning in a competitive environment. Geogr. Anal. 16, 39–51 (1984)

    Article  Google Scholar 

  14. Hakimi, S.L.: Locations with spatial interactions: competitive locations and games. In: Mirchandani, P.B., Francis, R.L. (eds.): Discrete Location Theory, pp. 439–478. Wiley-Interscience, New York (1990)

  15. Hotelling, H.: Stability in competition. Econ. J. 39, 41–57 (1929)

    Article  Google Scholar 

  16. Hansen, P., Labbé, M.: Algorithms for voting and competitive location on a network. Transp. Sci. 22(4), 278–288 (1988)

    Article  MATH  Google Scholar 

  17. Hansen, P., Mladenović, N.: Variable neighborhood search: principles and applications. Eur. J. Oper. Res. 130, 449–467 (2001)

    Article  MATH  Google Scholar 

  18. Küçükaudin, H., Aras, N., Altinel, I.K.: Competitive facility location problem with attractiveness adjustment of the follower: a bilevel programming model and its solution. Eur. J. Oper. Res. 208, 206–220 (2011)

    Article  Google Scholar 

  19. Kochetov, Y., Ivanenko, D.: Computationally difficult instances for the uncapacitated facility location problem. In: Ibaraki, T., et al. (eds.) Metaheuristics: Progress as Real Solvers, pp. 351–367. Springer, Berlin (2005)

    Chapter  Google Scholar 

  20. Kochetov, Y.A.: Facility location: discrete models and local search methods. In: Chvatal, V. (ed.) Combinatorial Optimization, Methods and Applications, pp. 97–134. IOS Press, Amsterdam (2011)

    Google Scholar 

  21. Kress, D., Pesch, E.: Sequential competitive location on networks. Eur. J. Oper. Res. 217(3), 483–499 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  22. Megiddo, N., Zemel, E., Hakimi, S.: The maximum coverage location problem. SIAM J. Algebraic Discrete Methods 4, 253–261 (1983)

    Article  MATH  MathSciNet  Google Scholar 

  23. Noltemeier, H., Spoerhase, J., Wirth, H.: Multiple voting location and single voting location on trees. Eur. J. Oper. Res. 181, 654–667 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  24. Plastria, F., Carrizosa, E.: Optimal location and design of a competitive facility. Math. Program. Ser. A. 100, 247–265 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  25. Roboredo, M.C., Pessoa, A.A.: A branch-and-cut algorithm for the discrete \((r|p)\)-centroid problem. Eur. J. Oper. Res. 224(1), 101–109 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  26. Resende, M.G.C., Werneck, R.F.: A hybrid multistart heuristic for the uncapacitated facility location problem. Eur. J. Oper. Res. 174(1), 54–68 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  27. Serra, D., ReVelle, C.: Market capture by two competitors: the pre-emptive capture problem. J. Reg. Sci. 34(4), 549–561 (1994)

    Article  Google Scholar 

  28. Spoerhase, J., Wirth, H.: (\(r, p\))-centroid problems on paths and trees. J. Theor. Comput. Sci. Arch. 410(47–49), 5128–5137 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  29. Saidani, N., Chu, F., Chen, H.: Competitive facility location and design with reactions of competitors already in the market. Eur. J. Oper. Res. 219, 9–17 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  30. Talbi, E.-G.: Metaheuristics: From Design to Implementation. Wiley, Chichester (2009)

    Book  Google Scholar 

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Acknowledgments

Many thanks to Alexander Kononov for the fruitful discussions.

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Authors and Affiliations

  1. Sobolev Institute of Mathematics, Novosibirsk State University, 4 pr. Akademika Koptyuga, Novosibirsk, Russia

    Ekaterina Alekseeva, Yury Kochetov & Alexandr Plyasunov

Authors
  1. Ekaterina Alekseeva
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  2. Yury Kochetov
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  3. Alexandr Plyasunov
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Corresponding author

Correspondence to Yury Kochetov.

Additional information

This research was partially supported by RFBR Grants 13-07-00016, 12-01-31090, 12-01-00077. The work of the first author was carried out during the tenure of an ERCIM “Alain Bensoussan” Fellowship Programme. The research leading to these results has received funding from the European Union Seventh Framework Programme (FP7/2007-2013) under grant agreement N246016.

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Alekseeva, E., Kochetov, Y. & Plyasunov, A. An exact method for the discrete \((r|p)\)-centroid problem. J Glob Optim 63, 445–460 (2015). https://doi.org/10.1007/s10898-013-0130-6

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