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The geometry of optimal partitions in location problems

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Abstract

Given the position of some facilities, we study the shape of optimal partitions of the customers’ area in a general planar demand region minimizing total average cost that depends on a set up cost plus some function of the travelling distances. By taking into account different norms, according to the considered situation of the location problem, we characterize optimal consumers’ partitions and describe their geometry. The case of dimensional facilities is also investigated.

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References

  1. Ambrosio, L.: Lecture Notes on Optimal Transport Problems. In: Colli, P., Rodrigues, J.F. (eds.) Ambrosio, L., et al.: LNM 1812. Springer, Berlin, pp. 1–52 (2003)

  2. Ambrosio, L., Kirchheim, B., Pratelli, A.: Existence of optimal transport maps for crystalline norms. Duke Math. J. 125(2), 207–241 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  3. Ambrosio, L.B., Pratelli, A.: Existence and stability results in the \(L^1\) theory of optimal transportation. In: Caffarelli, L.A., Salsa, S. (Eds.) Ambrosio, L. et al.: LNM 1813, pp. 123–160. Springer, Berlin (2003)

  4. Borwein, J.M., Lewis, A.S.: Partially finite convex programming, part II: explicit lattice models. Math. Program. 57(1), 4983 (1992)

    Google Scholar 

  5. Bouchitté, G., Juimenez, C., Mahadevan, R.: Asymptotic analysis of a class of optimal location problems. J. Math. Pures Appl. 95, 382–419 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  6. Brenier, Y.: Décomposition polaire et réarrangement monotone des champs de vecteurs. C. R. Acad. Sci. Paris Sér. I Math. 305(19), 805–808 (1987)

    MathSciNet  MATH  Google Scholar 

  7. Brimberg, J., Juel, H., Korner, M.C., Schobel, A.: Locating a general minisum circle on the plane. 4OR J Oper. Res. 9, 351–370 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  8. Carrizosa, E., Conde, E., Munoz-Marquez, M., Puerto, J.: The generalized Weber problem with expected distances. RAIRO-Oper. Res. 29(1), 35–57 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  9. Carrizosa, E., Munoz-Marquez, M., Puerto, J.: The Weber problem with regional demand. Eur. J. Oper. Res. 104(2), 358–365 (1998)

    Article  MATH  Google Scholar 

  10. Crippa, G., Jimenez, C., Pratelli, A.: Optimum and equilibrium in a transport problem with queue penalization effect. Adv. Calc. Var. 2, 207–246 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  11. Diaz-Banez, J.M., Mesa, J.A., Schobel, A.: Continuous location of dimensional structures. Eur. J. Oper. Res. 152, 22–44 (2004)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  13. Icking, C., Klein, R., Ma, L., Nickel, S., Weißler, A.: On bisectors for different distance functions. Discret. Appl. Math. 109, 139–161 (2001)

  14. Kalcsics, J.: Districting problems. In: Laporte, G., Nickel, S., Saldanha da Gama, F. (eds.) Location Science. Springer, Berlin (2015)

    Google Scholar 

  15. Kantorovich, L.V.: On the transfer of masses. Dokl. Akad. Nauk. 37, 227–229 (1942)

    Google Scholar 

  16. Knott, M., Smith, C.S.: On the optimal mapping of distributions. J. Optim. Theory Appl. 43(1), 39–49 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  17. Lowe, T.J., Hurter Jr., A.P.: The generalized market area problem. Manag. Sci. 22, 11051115 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  18. Love, R.F., Morris, J.G.: Modelling inter-city road distances by mathematical functions. Oper. Res. Q. 23(1), 61–71 (1972)

    Article  MATH  Google Scholar 

  19. Mallozzi, L., D’Amato, E., Daniele, E.: A planar location–allocation problem with waiting time costs. In: Rassias, T.M., Toth, L. (eds.) Topics in Mathematical and Applications, Springer Optimization and Its Applications, vol. 94, Ch. 23, pp. 549–562. Springer, Berlin (2014)

  20. Mallozzi, L., Passarelli di Napoli, A.: Optimal transport and a bilevel location–allocation problem. J. Glob. Optim (2015). doi:10.1007/s10898-015-0347-7

  21. Marín, A., Nickel, S., Puerto, J., Velten, S.: A flexible model and efficient solution strategies for discrete location problems. Discret. Appl. Math. 157, 1128–1145 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  22. Monge, G.: Memoire sur la Theorie des Dèblais et des Remblais. Histoire de L’Acad. de Sciences de Paris (1781)

  23. Nickel, S., Puerto, J.: Facility Location—A Unified Approach. Springer, Berlin (2005)

    MATH  Google Scholar 

  24. Nickel, S., Puerto, J., Rodríguez-Chía, A.M.: An approach to location models involving sets as existing facilities. Math. Oper. Res. 28(4), 693–715 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  25. Okabe, A., Boots, B., Sugihara, K.: Spatial Tessellations: Concepts and Applications of Voronoi Diagrams. Wiley, New York (1992)

    MATH  Google Scholar 

  26. Puerto, J., Rodríguez-Chía, A.M.: On the structure of the solution set for the single facility location problem with average distances. Math. Program. 128, 373–401 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  27. Puerto, J., Tamir, A., Perea, F.: A cooperative location game based on the 1-center location problem. Eur. J. Oper. Res. 214, 317–330 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  28. Silva, A., Tembine, H., Altman, E., Debbah, M.: Optimum and equilibrium in assignment problems with congestion: mobile terminals association to base station. IEEE Trans. Autom. Control 58(8), 2018–2031 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  29. Todd, M.J.: Solving the generalized market area problem. Manag. Sci. 24(14), 15491554 (1978)

    Article  MathSciNet  Google Scholar 

  30. Villani, C.: Optimal Transport, Old and New. Fundamental Principles of Mathematical Sciences, vol. 338. Springer, Berlin (2009)

    Google Scholar 

  31. Ward, J.E., Wendell, R.E.: Using block norms for location modeling. Oper. Res. 33, 1074–1090 (1985)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The first author was supported by STAR 2014—linea 1 (project: Variational Analysis and Equilibrium Models in Physical and Social Economic Phenomena), University of Naples Federico II, Italy and by GNAMPA 2016 (project: Analisi Variazionale per Modelli Competitivi con Incertezza e Applicazioni). The second author was supported by Spanish Ministry of Economy and Competitiveness through Grants MTM2013-46962-C02-01 and MTM2016-74983-C02-01 (MINECO/FEDER).

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Authors and Affiliations

  1. University of Naples Federico II, Naples, Italy

    Lina Mallozzi

  2. Universidad de Sevilla, Sevilla, Spain

    Justo Puerto

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  1. Lina Mallozzi
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  2. Justo Puerto
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Correspondence to Lina Mallozzi.

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Mallozzi, L., Puerto, J. The geometry of optimal partitions in location problems. Optim Lett 12, 203–220 (2018). https://doi.org/10.1007/s11590-017-1156-3

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