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Generalized p-median problem for the optimal distribution of facilities


The generalized p-median problem is related with the task to find an optimal distribution of p facilities for given heterogeneous population distribution. In the real world, the meaning of what is optimal strongly depends on the type and the purpose of the facility: the optimal distribution of fire stations may not be the same as the ones for hospitals, coffee shops, and schools. We start from the variational approach in Gastner and Newman (Phys Rev E 74:016117, 2006) and generalize the objective function in such a way that the accessibility of a facility is an algebraic function of distance with an exponent \(\beta\). Applying the second-order variational approach, we find that the extremum of the objective function corresponds to either a minimum or a maximum, depending on the value of the exponent \(\beta\). We find that the optimal distribution of public (private) facilities minimizes (maximizes) the objective function. We also perform simulations for the real population distribution of South Korea to validate our results from the variational approach.

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  1. S.M. Gusein-Zade, Geogr. Anal. 14, 246 (1982)

    Article  Google Scholar 

  2. N. Megiddo, K.J. Supowit, SIAM J. Comput. 13, 182 (1984)

    MathSciNet  Article  Google Scholar 

  3. J. Mycielski, W. Trzeciakowski, J. Reg. Sci. 5, 59 (1963)

    Article  Google Scholar 

  4. M.T. Gastner, Phys. Rev. E 84, 036112 (2011)

    ADS  Article  Google Scholar 

  5. M.J. Lee, K. Kim, J. Son, D.-S. Lee, Sci. Rep. 10, 1 (2020)

    ADS  Article  Google Scholar 

  6. A. Murat, V. Verter, G. Laporte, Comput. Oper. Res. 37, 123 (2010)

    MathSciNet  Article  Google Scholar 

  7. L. Cooper, Oper. Res. 11, 331 (1963)

    Article  Google Scholar 

  8. S.L. Hakimi, Oper. Res. 13, 462 (1965)

    Article  Google Scholar 

  9. C.S. ReVelle, R.W. Swain, Geogr. Anal. 2, 30 (1970)

    Article  Google Scholar 

  10. G. Laporte, S. Nickel, F.S. da Gama, Location Science (Springer, 2015)

    MATH  Google Scholar 

  11. K.E. Rosing, Eur. J. Oper. Res. 58, 414 (1992)

    ADS  Article  Google Scholar 

  12. E. Carrizosa, E. Conde, M. Munoz-Marquez, J. Puerto, RAIRO-Oper. Res. 29, 35 (1995)

    Article  Google Scholar 

  13. Z. Drezner, H.W. Hamacher, Facility Location: Applications and Theory (Springer, 2002)

    Book  Google Scholar 

  14. M.T. Gastner, M.E. Newman, Phys. Rev. E 74, 016117 (2006)

    ADS  Article  Google Scholar 

  15. J. Um, S.-W. Son, S.-I. Lee, H. Jeong, B.J. Kim, Proc. Natl. Acad. Sci. USA. 106, 14236 (2009)

    ADS  Article  Google Scholar 

  16. D. Kim, S.-W. Son, H. Jeong, J. Korean Phys. Soc. 60, 585 (2012)

    ADS  Article  Google Scholar 

  17. O. Kwon, Physica A 503, 209 (2018)

    ADS  Article  Google Scholar 

  18. A. Gomez-Lievano, H. Youn, L.M. Bettencourt, PLoS One 7, e40393 (2012)

    ADS  Article  Google Scholar 

  19. V.C. Yang, A.V. Papachristos, D.M. Abrams, Phys. Rev. E 100, 032306 (2019)

    ADS  Article  Google Scholar 

  20. A. Okabe, A. Suzuki, Eur. J. Oper. Res. 98, 445 (1997)

    Article  Google Scholar 

  21. J.-H. Qian, C.-H. Yang, D.-D. Han, Y.-G. Ma, Physica A 391, 5146 (2012)

    ADS  Article  Google Scholar 

  22. J.C. Leitao, J.M. Miotto, M. Gerlach, E.G. Altmann, R. Soc. Open Sci. 3, 150649 (2016)

    ADS  MathSciNet  Article  Google Scholar 

  23. Population data are downloaded from Korean Statistical Information Service (, and the areas of counties from Ministry of Land, Infrastructure, and Transport ( We also use the boundary map data for each county downloaded from Statistical Geographic Information Service of Korea (

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This research was supported by a National Research Foundation of Korea (NRF) grant funded by the Korea government’s Ministry of Science and ICT (MSIT) Grant No. 2019R1A2C2089463.

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Correspondence to Beom Jun Kim.

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Park, J.I., Kim, B.J. Generalized p-median problem for the optimal distribution of facilities. J. Korean Phys. Soc. 80, 352–358 (2022).

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  • Facility location
  • Location-allocation problem
  • Variational method