Exact approaches for competitive facility location with discrete attractiveness

Abstract

We study a variant of the competitive facility location problem, in which a company is to locate new facilities in a market where competitor’s facilities already exist. We consider the scenario where only a limited number of possible attractiveness levels is available, and the company has to select exactly one level for each open facility. The goal is to decide the facilities’ locations and attractiveness levels that maximize the profit. We apply the gravity-based rule to model the behavior of the customers and formulate a multi-ratio linear fractional 0–1 program. Our main contributions are the exact solution approaches for the problem. These approaches allow for easy implementations without the need for designing complicated algorithms and are “friendly” to the users without a solid mathematical background. We conduct computational experiments on the randomly generated datasets to assess their computational performance. The results suggest that the mixed-integer quadratic conic approach outperforms the others in terms of computational time. Besides that, it is also the most straightforward one that only requires the users to be familiar with the general form of a conic quadratic inequality. Therefore, we recommend it as the primary choice for such a problem.

This is a preview of subscription content, access via your institution.

Fig. 1

Notes

  1. 1.

    https://www.gurobi.com/pdfs/WebSeminar-Gurobi-6.5.pdf.

References

  1. 1.

    Adjiman, C.S., Dallwig, S., Floudas, C.A., Neumaier, A.: A global optimization method, \(\alpha \)bb, for general twice-differentiable constrained NLPS–I. Theoretical advances. Comput. Chem. Eng. 22(9), 1137–1158 (1998)

    Article  Google Scholar 

  2. 2.

    Atamtürk, A., Berenguer, G., Shen, Z.J.: A conic integer programming approach to stochastic joint location-inventory problems. Oper. Res. 60(2), 366–381 (2012)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Atamtürk, A., Narayanan, V.: Lifting for conic mixed-integer programming. Math. Program. 126(2), 351–363 (2011)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Ben-Tal, A., Nemirovski, A.: Lectures on Modern Convex Optimization: Analysis, Algorithms, and Engineering Applications, vol. 2. SIAM, Philadelphia (2001)

    Google Scholar 

  5. 5.

    Benati, S., Hansen, P.: The maximum capture problem with random utilities: problem formulation and algorithms. Eur. J. Oper. Res. 143(3), 518–530 (2002)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Borrero, J.S., Gillen, C., Prokopyev, O.A.: Fractional 0–1 programming: applications and algorithms. J. Glob. Optim. 69(1), 255–282 (2017)

    MathSciNet  Article  Google Scholar 

  7. 7.

    Boyd, S., Vandenberghe, L.: Convex Optimization. Cambridge University Press, Cambridge (2004)

    Google Scholar 

  8. 8.

    Çezik, M.T., Iyengar, G.: Cuts for mixed 0–1 conic programming. Math. Program. 104(1), 179–202 (2005)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Diamond, S., Boyd, S.: CVXPY: a Python-embedded modeling language for convex optimization. J. Mach. Learn. Res. 17(83), 1–5 (2016)

    MathSciNet  MATH  Google Scholar 

  10. 10.

    Drezner, T., Drezner, Z., Kalczynski, P.: A leader–follower model for discrete competitive facility location. Comput. Oper. Res. 64, 51–59 (2015)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Drezner, T., Drezner, Z., Salhi, S.: Solving the multiple competitive facilities location problem. Eur. J. Oper. Res. 142(1), 138–151 (2002)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Drezner, T., Drezner, Z., Zerom, D.: Competitive facility location with random attractiveness. Oper. Res. Lett. 46(3), 312–317 (2018)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Elhedhli, S.: Exact solution of a class of nonlinear knapsack problems. Oper. Res. Lett. 33(6), 615–624 (2005)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Fernández, J., Pelegrı, B., Plastria, F., Tóth, B.: Solving a huff-like competitive location and design model for profit maximization in the plane. Eur. J. Oper. Res. 179(3), 1274–1287 (2007)

    Article  Google Scholar 

  15. 15.

    Grant, M., Boyd, S.: Graph implementations for nonsmooth convex programs. In: Blondel, V.D., Boyd, S.P., Kimura, H. (eds.) Recent Advances in Learning and Control, pp. 95–110. Springer, Berlin (2008)

    Google Scholar 

  16. 16.

    Grant, M., Boyd, S., Ye, Y.: Disciplined convex programming. In: Liberti, L., Maculan, N. (eds.) Global Optimization, pp. 155–210. Springer, Berlin (2006)

    Google Scholar 

  17. 17.

    Haase, K., Müller, S.: A comparison of linear reformulations for multinomial logit choice probabilities in facility location models. Eur. J. Oper. Res. 232(3), 689–691 (2014)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Küçükaydın, H., Aras, N., Altınel, İ.: A discrete competitive facility location model with variable attractiveness. J. Oper. Res. Soc. 62(9), 1726–1741 (2011)

    Article  Google Scholar 

  19. 19.

    Küçükaydin, H., Aras, N., Altınel, I.K.: Competitive facility location problem with attractiveness adjustment of the follower: a bilevel programming model and its solution. Eur. J. Oper. Res. 208(3), 206–220 (2011)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Küçükaydın, H., Aras, N., Altınel, İ.K.: A leader–follower game in competitive facility location. Comput. Oper. Res. 39(2), 437–448 (2012)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Lančinskas, A., Fernández, P., Pelegín, B., Žilinskas, J.: Improving solution of discrete competitive facility location problems. Optim. Lett. 11(2), 259–270 (2017)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Ljubić, I., Moreno, E.: Outer approximation and submodular cuts for maximum capture facility location problems with random utilities. Eur. J. Oper. Res. 266(1), 46–56 (2018)

    MathSciNet  Article  Google Scholar 

  23. 23.

    McCormick, G.P.: Computability of global solutions to factorable nonconvex programs: Part I—convex underestimating problems. Math. program. 10(1), 147–175 (1976)

    Article  Google Scholar 

  24. 24.

    Méndez-Díaz, I., Bront, J.J., Vulcano, G., Zabala, P.: A branch-and-cut algorithm for the latent-class logit assortment problem. Discrete Appl. Math. 164, 246–263 (2014)

    MathSciNet  Article  Google Scholar 

  25. 25.

    Qi, M., Xia, M., Zhang, Y., Miao, L.: Competitive facility location problem with foresight considering service distance limitations. Comput. Ind. Eng. 112, 483–491 (2017)

    Article  Google Scholar 

  26. 26.

    Sáiz, M.E., Hendrix, E.M., Fernández, J., Pelegrín, B.: On a branch-and-bound approach for a huff-like stackelberg location problem. OR Spectrum 31(3), 679–705 (2009)

    MathSciNet  Article  Google Scholar 

  27. 27.

    Şen, A., Atamtürk, A., Kaminsky, P.: A conic integer optimization approach to the constrained assortment problem under the mixed multinomial logit model. Oper. Res. 66(4), 994–1003 (2018)

    MathSciNet  Article  Google Scholar 

  28. 28.

    Vielma, J.P., Ahmed, S., Nemhauser, G.L.: A lifted linear programming branch-and-bound algorithm for mixed-integer conic quadratic programs. INFORMS J. Comput. 20(3), 438–450 (2008)

    MathSciNet  Article  Google Scholar 

  29. 29.

    Vielma, J.P., Dunning, I., Huchette, J., Lubin, M.: Extended formulations in mixed integer conic quadratic programming. Math. Program. Comput. 9(3), 369–418 (2017)

    MathSciNet  Article  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Yun Hui Lin.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

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

Lin, Y.H., Tian, Q. Exact approaches for competitive facility location with discrete attractiveness. Optim Lett 15, 377–389 (2021). https://doi.org/10.1007/s11590-020-01596-x

Download citation

Keywords

  • Competitive facility location
  • Gravity model
  • Conic programming
  • Outer approximation
  • Mixed-integer linear programming