Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Solution methods for a min–max facility location problem with regional customers considering closest Euclidean distances

  • 185 Accesses

Abstract

We study a facility location problem where a single facility serves multiple customers each represented by a (possibly non-convex) region in the plane. The aim of the problem is to locate a single facility in the plane so that the maximum of the closest Euclidean distances between the facility and the customer regions is minimized. Assuming that each customer region is mixed-integer second order cone representable, we firstly give a mixed-integer second order cone programming formulation of the problem. Secondly, we consider a solution method based on the Minkowski sums of sets. Both of these solution methods are extended to the constrained case in which the facility is to be located on a (possibly non-convex) subset of the plane. Finally, these two methods are compared in terms of solution quality and time with extensive computational experiments.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

References

  1. 1.

    Alizadeh, F., Goldfarb, D.: Second order cone programming. Math. Program. 95, 3–51 (2003). https://doi.org/10.1007/s10107-002-0339-5

  2. 2.

    Aly, A.A., Marucheck, A.S.: Generalized Weber problem with rectangular regions. J. Oper. Res. Soc. 33(11), 983–989 (1982). https://doi.org/10.1057/jors.1982.209

  3. 3.

    Avigad, J., Donnelly, K.: Formalizing O notation in Isabelle/HOL. In: International Joint Conference on Automated Reasoning, pp. 357–371. Springer (2004). https://doi.org/10.1007/978-3-540-25984-8_27

  4. 4.

    Bennett, C.D., Mirakhor, A.: Optimal facility location with respect to several regions. J. Reg. Sci. 14(1), 131–136 (1974). https://doi.org/10.1111/j.1467-9787.1974.tb00435.x

  5. 5.

    Benson, H., Saglam, U.: Mixed-integer second order cone programming: a survey. Tutor. Oper. Res. 34, 11 (2005). https://doi.org/10.1287/educ.1053.0000

  6. 6.

    Bentley, J.L., Ottmann, T.A.: Algorithms for reporting and counting geometric intersections. IEEE Trans. Comput. 9(9), 643–647 (1979). https://doi.org/10.1109/TC.1979.1675432

  7. 7.

    Berger, A., Grigoriev, A., Winokurow, A.: An efficient algorithm for the single facility location problem with polyhedral norms and disk-shaped demand regions. Comput. Optim. Appl. 68(3), 661–669 (2017). https://doi.org/10.1007/s10589-017-9935-4

  8. 8.

    Blanco, V.: Ordered p-median problems with neighbourhoods. Comput. Optim. Appl. (2019). https://doi.org/10.1007/s10589-019-00077-x

  9. 9.

    Blanco, V., Fernández, E., Puerto, J.: Minimum spanning trees with neighborhoods: Mathematical programming formulations and solution methods. Eur. J. Oper. Res. 262(3), 863–878 (2016). https://doi.org/10.1016/j.ejor.2017.04.023

  10. 10.

    Blanco, V., Puerto, J., Ben-Ali, S.E.H.: Revisiting several problems and algorithms in continuous location with \(\ell _p\) norms. Comput. Optim. Appl. 58(3), 563–595 (2014). https://doi.org/10.1007/s10589-014-9638-z

  11. 11.

    Blanco, V., Puerto, J., Ben-Ali, S.E.H.: Continuous multifacility ordered median location problems. Eur. J. Oper. Res. 250(1), 56–64 (2016). https://doi.org/10.1016/j.ejor.2015.10.065

  12. 12.

    Brimberg, J., Wesolowsky, G.: Note: facility location with closest rectangular distances. Naval Res. Logist. 47(1), 77–84 (2000). https://doi.org/10.1002/(SICI)1520-6750(200002)47:1<77::AID-NAV5>3.0.CO;2-#

  13. 13.

    Brimberg, J., Wesolowsky, G.O.: Locating facilities by minimax relative to closest points of demand areas. Comput. Oper. Res. 29(6), 625–636 (2002). https://doi.org/10.1016/S0305-0548(00)00106-4

  14. 14.

    Brimberg, J., Wesolowsky, G.O.: Minisum location with closest Euclidean distances. Ann. Oper. Res. 111(1–4), 151–165 (2002). https://doi.org/10.1023/A:1020901719463

  15. 15.

    Carrizosa, E., Conde, E., Muñoz Marquez, M., Puerto, J.: The generalized Weber problem with expected distances. RAIRO Oper. Res. 29(1), 35–57 (1995). https://doi.org/10.1051/ro/1995290100351

  16. 16.

    Carrizosa, E., Muñoz-Márquez, M., Puerto, J.: Location and shape of a rectangular facility in \({\mathbb{R}}^{n}\). Convexity properties. Math. program. 83(1–3), 277–290 (1998). https://doi.org/10.1007/BF02680563

  17. 17.

    Cooper, L.: Bounds on the Weber problem solution under conditions of uncertainty. J. Reg. Sci. 18(1), 87–92 (1978). https://doi.org/10.1111/j.1467-9787.1978.tb00530.x

  18. 18.

    De Berg, M., Van Kreveld, M., Overmars, M., Schwarzkopf, O.: Springer. Computational Geometry (1997). https://doi.org/10.1007/978-3-662-03427-9_1

  19. 19.

    Dinler, D., Tural, M.K.: A minisum location problem with regional demand considering farthest Euclidean distances. Optim. Meth. Softw 31(3), 446–470 (2016). https://doi.org/10.1080/10556788.2015.1121486

  20. 20.

    Dinler, D., Tural, M.K., Iyigun, C.: Heuristics for a continuous multi-facility location problem with demand regions. Comput. Oper. Res. 62, 237–256 (2015). https://doi.org/10.1016/j.cor.2014.09.001

  21. 21.

    Drezner, Z., Wesolowsky, G.O.: Location models with groups of demand points. Inf. Syst. Oper. Res. 38(4), 359–372 (2000). https://doi.org/10.1080/07408170108936860

  22. 22.

    Fogel, E., Halperin, D., Weibel, C.: On the exact maximum complexity of Minkowski sums of polytopes. Discret. Comput. Geom. 42(4), 654 (2009). https://doi.org/10.1007/s00454-009-9159-1

  23. 23.

    Gugat, M., Pfeiffer, B.: Weber problems with mixed distances and regional demand. Math. Methods Oper. Res. 66(3), 419–449 (2007). https://doi.org/10.1007/s00186-007-0165-x

  24. 24.

    Jeroslow, R.G., Lowe, J.K.: Modelling with integer variables. In: Korte, B.K., Klaus, R. (eds.) Mathematical Programming at Oberwolfach II, pp. 167–184. Springer, Berlin (1984). https://doi.org/10.1007/BFb0121015

  25. 25.

    Jiang, J., Yuan, X.: A Barzilai-Borwein-based heuristic algorithm for locating multiple facilities with regional demand. Comput. Optim. Appl. 51(3), 1275–1295 (2012). https://doi.org/10.1007/s10589-010-9392-9

  26. 26.

    Jiang, Jl, Xu, Y.: Minisum location problem with farthest Euclidean distances. Math. Methods Oper. Res. 64(2), 285–308 (2006). https://doi.org/10.1007/s00186-006-0084-2

  27. 27.

    Juel, H.: Bounds in the generalized Weber problem under locational uncertainty. Oper. Res. 29(6), 1219–1227 (1981). https://doi.org/10.1287/opre.29.6.1219

  28. 28.

    Lobo, M., Vandenberghe, L., Boyd, S., Lebret, H.: Applications of second-order cone programming. Linear Algebra Appl. 284, 193–228 (1998). https://doi.org/10.1016/S0024-3795(98)10032-0

  29. 29.

    Love, R.F.: A computational procedure for optimally locating a facility with respect to several rectangular regions. J. Reg.Sci. 12(2), 233–242 (1972). https://doi.org/10.1111/j.1467-9787.1972.tb00345.x

  30. 30.

    Megiddo, N.: Linear-time algorithms for linear programming in \({\mathbb{R}}^3\) and related problems. SIAM J. Comput. 12(4), 759–776 (1983). https://doi.org/10.1137/0212052

  31. 31.

    Megiddo, N.: The weighted Euclidean 1-center problem. Math. Oper. Res. 8(4), 498–504 (1983). https://doi.org/10.1287/moor.8.4.498

  32. 32.

    Megiddo, N., Tamir, A.: New results on the complexity of p-centre problems. SIAM J. Comput. 12(4), 751–758 (1983). https://doi.org/10.1137/0212051

  33. 33.

    Mesadi, F., Erdil, E., Cetin, M., Tasdizen, T.: Image segmentation using disjunctive normal Bayesian shape and appearance models. IEEE Trans. Med. Imaging 37(1), 293–305 (2017). https://doi.org/10.1109/TMI.2017.2756929

  34. 34.

    Nickel, S., Puerto, J., Rodriguez-Chia, A.M.: An approach to location models involving sets as existing facilities. Math. Oper. Res. 28(4), 693–715 (2003). https://doi.org/10.1287/moor.28.4.693.20521

  35. 35.

    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(1–2), 373–401 (2011). https://doi.org/10.1007/s10107-009-0308-3

  36. 36.

    Shamos, M.I., Hoey, D.: Geometric intersection problems. In: 17th Annual Symposium on Foundations of Computer Science (sfcs 1976), pp. 208–215. IEEE (1976). https://doi.org/10.1109/SFCS.1976.16

  37. 37.

    Sylvester, J.J.: A question in the geometry of situation. Q. J. Pure Appl. Math. 1(1), 79–80 (1857)

  38. 38.

    Vielma, J.P.: Mixed integer linear programming formulation techniques. Siam Rev. 57(1), 3–57 (2015). https://doi.org/10.1137/130915303

  39. 39.

    Welzl, E.: Smallest enclosing disks (balls and ellipsoids). In: Maurer, H. (ed.) New Results and New Trends in Computer Science, pp. 359–370. Springer, Berlin (1991)

  40. 40.

    Wesolowsky, G.O., Love, R.: Location of facilities with rectangular distances among point and area destinations. Naval Res. Logist. Q. 18(1), 83–90 (1971). https://doi.org/10.1002/nav.3800180107

Download references

Author information

Correspondence to Nazlı Dolu.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Dolu, N., Hastürk, U. & Tural, M.K. Solution methods for a min–max facility location problem with regional customers considering closest Euclidean distances. Comput Optim Appl 75, 537–560 (2020). https://doi.org/10.1007/s10589-019-00163-0

Download citation

Keywords

  • Facility location
  • Min–max problem
  • Second order cone programming
  • Minkowski sum
  • Regional customer