Variable Neighborhood Search Algorithms for a Competitive Location Problem with Elastic Demand


Under consideration is the situation in a competitive market when a new Company plans to make profit from opening its own facilities that offer goods or services. The Company have to take it into account that there are several projects for opening each facility, and similar facilities of the Competitor are already placed on the market. Moreover, customers themselves choose the places to meet their demand in dependence on where and which facilities are located. The Company’s goal is to choose locations and projects for opening new facilities in order to attract the largest share of all customer demand. The special type of demand leads to nonlinearity of the objective function and to additional difficulties in finding an optimal solution. In this article we construct some variants of variable neighborhood search algorithms, perform their experimental analysis by using the upper estimates, obtain a posteriori accuracy estimates, and discuss the results.

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

Fig. 1.
Fig. 2.


  1. 1

    T. V. Levanova and A. S. Fedorenko, “Variable Neighborhood Search for the Two-Stage Facility Location Problem,” Diskret. Anal. Issled. Oper. 15 (3), 43–57 (2008).

    MATH  Google Scholar 

  2. 2

    Discrete Location Theory (Wiley, New York, 1990).

  3. 3

    Yu. A. Kochetov, E. V. Alekseeva, T. V. Levanova, and M. A. Loresh, “Large Neighborhood Local Search for the \(p\)-Median Problem,” Yugoslav J. Oper. Res. 15 (1), 53–63 (2005).

    MathSciNet  Article  Google Scholar 

  4. 4

    R. Sridharan, “The Capacitated Plant Location Problem,” European J. Oper. Res. 87 (1), 203–213 (1995).

    Article  Google Scholar 

  5. 5

    H. Hotelling, “Stability in Competition,” Econ. J. 39, 41–57 (1929).

    Article  Google Scholar 

  6. 6

    W. J. Reilly, The Law of Retail Gravitation (Knickerbocker Press, New York, 1931).

    Google Scholar 

  7. 7

    D. L. Huff, “Defining and Estimating a Trade Area,” J. Mark. 28, 34–38 (1964).

    Article  Google Scholar 

  8. 8

    D. L. Huff, “A Programmed Solution for Approximating an Optimum Retail Location,” Land Econ. 42, 293–303 (1966).

    Article  Google Scholar 

  9. 9

    A. G. Wilson, “Retailers’ Profits and Consumers’ Welfare in a Spatial Interaction Shopping Mode,” in Theory and Practice in Regional Science (Pion, London, 1976), pp. 42–59.

  10. 10

    O. Berman and D. Krass, “Locating Multiple Competitive Facilities: Spatial Interaction Models with Variable Expenditures,” Ann. Oper. Res. 111, 197–225 (2002).

    MathSciNet  Article  Google Scholar 

  11. 11

    R. Aboolian, O. Berman, and D. Krass, “Competitive Facility Location Model with Concave Demand,” European J. Oper. Res. 181 (2), 598–619 (2007).

    MathSciNet  Article  Google Scholar 

  12. 12

    R. Aboolian, O. Berman, and D. Krass, “Competitive Facility Location and Design Problem,” European J. Oper. Res. 182 (1), 40–62 (2007).

    MathSciNet  Article  Google Scholar 

  13. 13

    T. Drezner and Z. Drezner, “Lost Demand in a Competitive Environment,” J. Oper. Res. Soc. 59, 362–371 (2008).

    Article  Google Scholar 

  14. 14

    Z. Drezner and C. H. Scott, “Optimizing the Location of a Production Firm,” Network. Spat. Econ. 10, 411–425 (2010).

    MathSciNet  Article  Google Scholar 

  15. 15

    T. Drezner, Z. Drezner, and D. Zerom, “Competitive Facility Location with Random Attractiveness,” Oper. Res. Lett. 46, 312–317 (2018).

    MathSciNet  Article  Google Scholar 

  16. 16

    H. Küukaydin, N. Necati Aras, and I. K. Altinel, “A Leader–Follower Game in Competitive Facility Location,” Comput. Oper. Res. 39 (2), 437–448 (2012).

    MathSciNet  Article  Google Scholar 

  17. 17

    T. V. Levanova and A. Yu. Gnusarev, “Variable Neighborhood Search Approach for the Location and Design Problem,” in Discrete Optimization and Operations Research: Proceedings of 9th International Conference DOOR-2016 (Vladivostok, Russia, September 19–23, 2016) (Springer, Heidelberg, 2016), pp. 159–170 [Lecture Notes in Computer Science, Vol. 9869)].

  18. 18

    T. V. Levanova and A. Yu. Gnusarev, “Ant Colony Optimization for Competitive Facility Location Problem with Elastic Demand,” in Optimization Problems and Their Applications: Proceedings of School-Seminar OPTA-SCL-2018 (Omsk, Russia, July 8–14, 2018) (RWTH Aachen Univ., Aachen, 2018), pp. 239–248 [CEUR Workshop Proceedings, Vol. 2098]. Available at (accessed July 2, 2020).

  19. 19

    T. V. Levanova and A. Yu. Gnusarev, “Development of Threshold Algorithms for a Location Problem with Elastic Demand,” in Large-Scale Scientific Computing: Revised Selected Papers of 11th International Conference LSSC-2017 (Sozopol, Bulgaria, June 5–9, 2017) (Springer, Cham, 2017), pp. 382–389 [Lecture Notes in Computer Science, Vol. 10665].

  20. 20

    T. V. Levanova and A. Yu. Gnusarev, “Simulated Annealing for Competitive \(p \)-Median Facility Location Problem,” J. Phys. Conf. Ser. 1050, 012044.1–5 (2018). Available at 10.1088/1742-6596/1050/1/012044/pdf (accessed July 2, 2020).

  21. 21

    T. V. Levanova and A. Yu. Gnusarev, “Development of Ant Colony Optimization Algorithm for Competitive \(P\)-Median Facility Location Problem with Elastic Demand,” in Mathematical Optimization Theory and Operations Research: Revised Selected Papers of 18th International Conference MOTOR-2019 (Ekaterinburg, Russia, July 8–12, 2019) (Springer, Cham, 2019), pp. 68–78 [Communications in Computer and Information Science, Vol. 1090].

  22. 22

    R. G. McGarvey and T. M. Cavalier, “Determining the Location and Capacity of Competitive Facilities,” Internat. J. Math. Oper. Res. 2 (6), 694–723 (2010).

    MathSciNet  Article  Google Scholar 

  23. 23

    J. Perl and P. Ho, “Public Facilities Location under Elastic Demand,” Transp. Sci. 24 (2), 117–136 (1990).

    MathSciNet  Article  Google Scholar 

  24. 24

    A. Pahlavani and M. Saidi-Mehrabad, “A Competitive Facility Location Model with Elastic Demand and Patronising Behavior Sensitive to Location, Price, and Waiting Time,” Internat. J. Logist. Syst. Managem. 10 (3), 293–312 (2011).

    Google Scholar 

  25. 25

    X. Wang, “Location and Design Decisions of Facilities in a Distribution System with Elastic Customer Demand,” J. Shanghai Jiaotong Univ. (Sci.) 14 (5), 606–612 (2009).

    Article  Google Scholar 

  26. 26

    M. G. Ashtiani, “Competitive Location: A State-of-Art Review,” Internat. J. Ind. Eng. Comput. 7, 1–18 (2016).

    Google Scholar 

  27. 27

    O. Berman, T. Drezner, Z. Drezner, and D. Krass, “Modeling Competitive Facility Location Problems: New Approaches and Results,” in Decision Technologies and Applications (Inst. Oper. Res. Manage. Sci., Catonsville, 2009), pp. 156–181.

  28. 28

    T. Drezner, “Gravity Models in Competitive Facility Location,” in Contributions to Location Analysis (Springer, Cham, 2019), pp. 253–275.

  29. 29

    H. A. Eiselt, V. Marianov, and T. Drezner, “Competitive Location Models,” in Location Science (Springer, Cham, 2015), pp. 365–398.

  30. 30

    A. Karakitsiou, Modeling Discrete Competitive Facility Location (Springer, Cham, 2015).

    Google Scholar 

  31. 31

    D. Kress and E. Pesch, “Sequential Competitive Location on Networks,” European J. Oper. Res. 217 (3), 483–499 (2012).

    MathSciNet  Article  Google Scholar 

  32. 32

    A. E. Bakhtin, A. A. Kolokolov, and Z. V. Korobkova, Discrete Problems of Production and Transport Type (Nauka, Novosibirsk, 1978) [in Russian].

    Google Scholar 

  33. 33

    R. Aboolian, O. Berman, and D. Krass, “Capturing Market Share: Facility Location and Design Problem,” in Proceedings of Internatinal Conference “Discrete Optimization and Operations Research” (Novosibirsk, Russia, June 24–28, 2013) (Inst. Mat., Novosibirsk, 2013), pp. 7–11.

  34. 34

    T. A. J. Nicholson, “A Sequential Method for Discrete Optimization Problems and Its Application to the Assignment, Traveling Salesman, and Tree Scheduling Problems,” J. Inst. Math. Appl. 13, 362–375 (1965).

    MATH  Google Scholar 

  35. 35

    Yu. A. Kochetov, N. Mladenović and P. Hansen, “Local Search with Alternating Neighborhoods,” Diskret. Anal. Issled. Oper. Ser. 2, 10 (1), 11–43 (2003).

    MathSciNet  MATH  Google Scholar 

  36. 36

    P. Hansen and N. Mladenović, “Variable Neighborhood Search: Principles and Applications,” European J. Oper. Res. 130 (3), 449–467 (2001).

    MathSciNet  Article  Google Scholar 

  37. 37

    A. Anokić, Z. Stanimirović, T. Davidović and D. J. Stakić, “Variable Neighborhood Search Based Approaches to a Vehicle Scheduling Problem in Agriculture,” Internat. Trans. Oper. Res. 27 (1), 26–56 (2020).

    MathSciNet  Article  Google Scholar 

  38. 38

    Handbook of Metaheuristics (Springer, New York, 2010)

  39. 39

    P. A. Kononova and Yu. A. Kochetov, “The Variable Neighborhood Search for the Two Machine Flow Shop Problem with a Passive Prefetch,” J. Appl. Ind. Math. 7 (1), 54–67 (2013).

    MathSciNet  Article  Google Scholar 

  40. 40

    J. Pei, N. Mladenović, D. UroŢević, J. Brimberg, and X. Liu, “Solving the Traveling Repairman Problem with Profits: A Novel Variable Neighborhood Search Approach,” Inform. Sci. 507, 108–123 (2020).

    MathSciNet  Article  Google Scholar 

  41. 41

    J. Dréo, A. Pétrowski, P. Siarry, and E. Taillard, Metaheuristics for Hard Optimization (Springer, Heidelberg, 2006).

    Google Scholar 

  42. 42

    T. V. Levanova and S. E. Belan, “Application of Upper Bounds for Analysis of Approximate Algorithms for a Competitive Location Problem with Elastic Demand,” Vestnik Omsk. Gos. Univ. No. 4, 4–10 (2017).

  43. 43

    The General Algebraic Modeling System (GAMS Development, Fairfax, 2020). Available at http://www. (accessed July 2, 2020).

  44. 44

    LocalSolver (LocalSolver, Paris, 2020). Available at (accessed July 2, 2020).

Download references


The authors are grateful to N. Mladenović and Yu.A.Kochetov for helpful advice.


Sections 1 and 2 are performed by T.V. Levanova with the support by the Program for Fundamental Scientific Research of the State Academies of Sciences for 2013–2020 No. I.5 (project no. 0314–2019–0019). Section 3 is performed by A.Yu. Gnusarev with the support by the Russian Foundation for Basic Research (project no. 18–07–00599).

Author information



Corresponding authors

Correspondence to T. V. Levanova or A. Yu. Gnusarev.

Additional information

Translated by L.B. Vertgeim

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Levanova, T.V., Gnusarev, A.Y. Variable Neighborhood Search Algorithms for a Competitive Location Problem with Elastic Demand. J. Appl. Ind. Math. 14, 693–705 (2020).

Download citation


  • location problem
  • competition
  • elastic demand
  • heuristic
  • variable neighborhood search