Advertisement

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

New lower bounds for the facility location problem with clients’ preferences

Abstract

A bilevel facility location problem in which the clients choose suppliers based on their own preferences is studied. It is shown that the coopertative and anticooperative statements can be reduced to a particular case in which every client has a linear preference order on the set of facilities to be opened. For this case, various reductions of the bilevel problem to integer linear programs are considered. A new statement of the problem is proposed that is based on a family of valid inequalities that are related to the problem on a pair of matrices and the set packing problem. It is shown that this formulation is stronger than the other known formulations from the viewpoint of the linear relaxation and the integrality gap.

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

References

  1. 1.

    V. L. Beresnev, Discrete Location Problems and Polynomials of Boolean Variables (Institut matematiki, SO RAN, Novosibirsk, 2005) [in Russian].

  2. 2.

    L. E. Gorbachevskaya, Polynomially Solvable and NP-Hard Standardization Problems, Candidate’s Dissertation in Mathematics and Physics (IM SO RAN, Novosibirsk, 1998).

  3. 3.

    P. Hanjoul and D. Peeters, “A Facility Location Problem with Clients’ Preference Orderings,” Regional Sci. Urban Econom. 17, 451–473 (1987).

  4. 4.

    L. E. Gorbachevskaya, V. T. Dement’ev, and Yu. V. Shamardin, “Bilevel Standartization Problem with the Uniqueness of the Optimal Consumer Choice,” Diskretnyi Analiz Issl. Operatsii, Ser. 2, 6(2), 3–11 (1999).

  5. 5.

    G. Ausiello, P. Crescenzi, G. Gambosi, et al., Complexity and Approximation: Combinatorial Optimization Problems and Their Approximability Properties (Springer, Berlin, 1999).

  6. 6.

    P. Hansen, Y. Kochetov, and N. Mladenovic, “Lower Bounds for the Uncapacitated Facility Location Problem with User Preferences,” Technical Report, Les Cahiers du GERAD, G-2004-24 (2004).

  7. 7.

    P. Hansen, Y. Kochetov, and N. Mladenovic, “The Uncapacitated Facility Location Problem with User Preferences,” in Proc. DOM’2004 Workshop, Omsk-Irkutsk, 2004, pp. 50–55.

  8. 8.

    E. V. Alekseeva and Yu. A. Kochetov, “Genetic Local Search for the p-Median Problem with Client’s Preferences,” Diskretnyi Analiz Issl. Operatsii, Ser. 2, 14(1), 3–31 (2007).

  9. 9.

    L. Cánovas, S. García, M. Labbé, and A. Marín, “A Strengthened Formulation for the Simple Plant Location Problem with Order,” Operat. Res. Letts. 35(2), 141–150 (2007).

  10. 10.

    A. V. Kononov, Yu. A. Kochetov, and A. V. Plyasunov, “Competitive Facility Location Models,” Zh. Vychisl. Mat. Mat. Fiz. 49(6) (2009) [Comput. Math. Math. Phys. 49 (6), (2009)].

  11. 11.

    G. N. Nemhauser and L. A. Wolsey, Integer and Combinatiorial Optimization (Wiley-Interscience, Chichester, 1999).

  12. 12.

    M. W. Padberg, “On the Facial Structure of the Set Packing Polyhedra,” Math. Program. 5, 199–215 (1973).

  13. 13.

    Y. Pochet and L. A. Wolsey, Production Planning by Mixed Integer Programming (Springer, Berlin, 2006).

  14. 14.

    P. Avella and I. A. Vasil’ev, “A Computational Study of a Cutting Plane Algorithm for University Course Timetabling,” J. Scheduling 8, 497–514 (2005).

  15. 15.

    K. L. Hoffman and M. Padberg, “Solving Airline Crew Scheduling Problems by Branch-and-Cut,” Management Sci. 39, 657–682 (1993).

  16. 16.

    R. Borndorfer and R. Weismantel, “Set Packing Relaxations of Some Integer Programs,” Math. Program. 88, 425–450 (2000).

  17. 17.

    H. Waterer, E. L. Johnson, P. Nobili, and M. W. P. Savelsbergh, “The Relation of Time Indexed Formulations of Single Machine Scheduling Problems to the Node Packing Problem,” Math. Program. 93, 477–494 (2002).

  18. 18.

    E. Cheng and W. Y. Cunninghav, “Wheel Inequalities for Stable Set Polytopes,” Math. Program. 77, 389–421 (1997).

  19. 19.

    E. Cheng and S. Vries, “Antiweb-Wheel Inequalities and Their Separation Problems Over the Stable Set Polytopes,” Math. Program. 92, 153–175 (2002).

  20. 20.

    F. Rossi and S. Smriglio, “A Branch-and-Cut Algorithm for the Maximum Cardinality Stable Set Problem,” Operat. Res. Letts. 28, 63–74 (2001).

  21. 21.

    V. R. Khachaturov, V. E. Veselovskii, A. V. Zlotov, et al., Combinatorial Methods and Algorithms for Solving Large-Scale Discrete Optimization Problems (Nauka, Moscow, 2000) [in Russian].

Download references

Author information

Correspondence to I. L. Vasil’ev.

Additional information

Original Russian Text © I.L. Vasil’ev, K.B. Klimentova, Yu.A. Kochetov, 2009, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2009, Vol. 49, No. 6, pp. 1055–1066.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Vasil’ev, I.L., Klimentova, K.B. & Kochetov, Y.A. New lower bounds for the facility location problem with clients’ preferences. Comput. Math. and Math. Phys. 49, 1010–1020 (2009). https://doi.org/10.1134/S0965542509060098

Download citation

Keywords

  • bilevel facility location problems
  • bilevel programming
  • valid inequalities
  • lower bounds