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A minimax regret model for the leader–follower facility location problem

  • S.I.: Data-Driven OR in Transportation and Logistics
  • Published:
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Abstract

The leader–follower facility location problem consists of a leader and a follower who are competitors that locate new facilities sequentially. Traditional studies have generally assumed that the leader has partial or full advance information of the follower’s response when making a decision. However, this assumption might be invalid or impracticable in practice. In this paper, we consider that the leader needs to locate a predetermined number of new facilities without knowing anything about the follower’s response. By separating the scenarios in which the follower responds with different numbers of new facilities, a minimax regret model is proposed for the leader to minimise its maximum possible loss. Based on the structural characteristics of the proposed model, a set of solving procedures is provided that transforms the follower’s nonlinear (fraction) programming model into a linear model. In the numerical experiments, the proposed model is compared with two other location models, a deterministic model and a risk model, and the efficiency of the linearisation in decreasing the computation time is verified. The results show that the proposed model is more applicable to the leader when there is no information about the number or probability distribution of the follower’s new facilities.

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References

  • Aboolian, R., Berman, O., & Krass, D. (2007). Competitive facility location and design problem. European Journal of Operational Research, 182(1), 40–62.

    Article  Google Scholar 

  • Ahmadi-Javid, A., Seyedi, P., & Syam, S. S. (2016). A survey of healthcare facility location. Computers & Operations Research, 79, 223–263.

    Article  Google Scholar 

  • Ashtiani, M. G., Makui, A., & Ramezanian, R. (2013). A robust model for a leader–follower competitive facility location problem in a discrete space. Applied Mathematical Modelling, 37(1–2), 62–71.

    Article  Google Scholar 

  • Dilek, H., Karaer, O., & Nadar, E. (2017). Retail location competition under carbon penalty. European Journal of Operational Research, 269(1), 146–158.

    Article  Google Scholar 

  • Drezner, T., & Drezner, Z. (1998). Facility location in anticipation of future competition. Location Science, 6(1–4), 155–173.

    Article  Google Scholar 

  • Drezner, T., Drezner, Z., & Kalczynski, P. (2015). A leader–follower model for discrete competitive facility location. Computers & Operations Research, 64, 51–59.

    Article  Google Scholar 

  • Farahani, R. Z., Fallah, S., Ruiz, R., Hosseini, S., & Asgari, N. (2019). OR models in urban service facility location: A critical review of applications and future developments. European Journal of Operational Research, 276(1), 1–27.

    Article  Google Scholar 

  • Fernández, J., G.-Tóth, B., Redondo, J. L., & Ortigosa, P. M. (2019). The probabilistic customer’s choice rule with a threshold attraction value: Effect on the location of competitive facilities in the plane. Computers & Operations Research, 101, 234–249.

    Article  Google Scholar 

  • Fernández, J., G.-Tóth, B., Redondo, J. L., Ortigosa, P. M., & Arrondo, A. G. (2017). A planar single-facility competitive location and design problem under the multi-deterministic choice rule. Computers & Operations Research, 78, 305–315.

    Article  Google Scholar 

  • Fischer, K. (2002). Sequential discrete p-facility models for competitive location planning. Annals of Operations Research, 111(1–4), 253–270.

    Article  Google Scholar 

  • Gentile, J., Pessoa, A. A., Poss, M., & Roboredo, M. C. (2018). Integer programming formulations for three sequential discrete competitive location problems with foresight. European Journal of Operational Research, 265, 872–881.

    Article  Google Scholar 

  • Hakimi, S. L. (1983). On locating new facilities in a competitive environment. European Journal of Operational Research, 12(1), 29–35.

    Article  Google Scholar 

  • Hakimi, S. L. (1986). P-Median theorems for competitive locations. Annals of Operations Research, 6(4), 75–98.

    Article  Google Scholar 

  • Hotelling, H. (1929). Stability in competition. The Economic Journal, 39(153), 41.

    Article  Google Scholar 

  • Huff, D. L. (1964). Defining and estimating a trading area. Journal of Marketing, 28(3), 34–38.

    Article  Google Scholar 

  • Huff, D. L. (1966). A programmed solution for approximating an optimum retail location. Land Economics, 42(3), 293–303.

    Article  Google Scholar 

  • Kochetov, Y., Kochetova, N., & Plyasunov, A. (2013). A matheuristic for the leader–follower facility location and design problem. In Lau, H., Van Hentenryck, P., Raidl, G. (Eds.) Proceedings of the 10th metaheuristics international conference (MIC2013) (pp. 32/1–32/3). Singapore.

  • Kress, D., & Pesch, E. (2012). Sequential competitive location on networks. European Journal of Operational Research, 217(3), 483–499.

    Article  Google Scholar 

  • Kung, L., & Liao, W. (2018). An approximation algorithm for a competitive facility location problem with network effects. European Journal of Operational Research, 267(1), 176–186.

    Article  Google Scholar 

  • Lado-Sestayo, R., & Fernandez-Castro, A. S. (2019). The impact of tourist destination on hotel efficiency: A data envelopment analysis approach. European Journal of Operational Research, 272(2), 674–686.

    Article  Google Scholar 

  • Lopes, R. B., Ferreira, C., & Santos, B. S. (2016). A simple and effective evolutionary algorithm for the capacitated location-routing problem. Computers & Operations Research, 70, 155–162.

    Article  Google Scholar 

  • Moore, J. T., & Bard, J. F. (1990). The mixed integer linear bilevel programming problem. Operations Research, 38(5), 911–921.

    Article  Google Scholar 

  • Peréz, M. D. G., & Pelegrín, B. P. (2003). All stackelberg location equilibria in the hotelling’s duopoly model on a tree with parametric prices. Annals of Operations Research, 122(1), 177–192.

    Article  Google Scholar 

  • Plastria, F. (2001). Static competitive facility location: An overview of optimisation approaches. European Journal of Operational Research, 129(3), 461–470.

    Article  Google Scholar 

  • Plastria, F., & Vanhaverbeke, L. (2008). Discrete models for competitive location with foresight. Computers & Operations Research, 35, 683–700.

    Article  Google Scholar 

  • Qi, M., Xia, M., Zhang, Y., & Miao, L. (2017). Competitive facility location problem with foresight considering service distance limitations. Computers and Industrial Engineering, 112, 483–491.

    Article  Google Scholar 

  • Reilly, W. J. (1931). The law of retail gravitation. New York, NY: Knickerbocker Press.

    Google Scholar 

  • Saidani, N., Chu, F., & Chen, H. (2012). Competitive facility location and design with reactions of competitors already in the market. European Journal of Operational Research, 219(1), 9–17.

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Sedghi, N., Shavandi, H., & Abouee-Mehrizi, H. (2017). Joint pricing and location decisions in a heterogeneous market. European Journal of Operational Research, 261(1), 234–246.

    Article  Google Scholar 

  • Serra, D., & Revelle, C. (1994). Market capture by two competitors: The pre-emptive location problem. Journal of Regional Science, 34(4), 549–561.

    Article  Google Scholar 

  • Wang, X., & Ouyang, Y. (2013). A continuum approximation approach to competitive facility location design under facility disruption risks. Transportation Research Part B: Methodological, 50, 90–103.

    Article  Google Scholar 

  • Weber, A. (1909). Über den Standort der Industrien. 1. Teil: Reine Theorie des Standortes. TÜbingen. Translated as: On the location of industries. (p. 1929). Chicago, IL: University of Chicago Press.

  • Xia, Y., Chen, B., Jayaraman, V., & Munson, C. L. (2015). Competition and market segmentation of the call center service supply chain. European Journal of Operational Research, 247(2), 504–514.

    Article  Google Scholar 

  • Zhang, Y., & Atkins, D. (2019). Medical facility network design: User-choice and system-optimal models. European Journal of Operational Research, 273(1), 305–319.

    Article  Google Scholar 

Download references

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant Nos. 71722007, 71931001); the Funds for First-class Discipline Construction (XK18025); the China Postdoctoral Science Foundation (No. 2019M660426); and the Fundamental Research Funds for the Central Universities (buctrc201926).

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Authors and Affiliations

  1. School of Economics and Management, Beijing University of Chemical Technology, Beijing, 100029, China

    Xiang Li, Tianyu Zhang & Hongguang Ma

  2. China Europe International Business School, 699 Hongfeng Road, Pudong, Shanghai, 201206, China

    Liang Wang & Xiande Zhao

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  1. Xiang Li
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  2. Tianyu Zhang
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  3. Liang Wang
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  4. Hongguang Ma
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Correspondence to Hongguang Ma.

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Li, X., Zhang, T., Wang, L. et al. A minimax regret model for the leader–follower facility location problem. Ann Oper Res 309, 861–882 (2022). https://doi.org/10.1007/s10479-020-03826-y

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  • DOI: https://doi.org/10.1007/s10479-020-03826-y

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