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A variable neighborhood search algorithm for the \( (r{\mid }p) \) hub–centroid problem under the price war

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Abstract

This study considers the \( (r{\mid }p) \) hub–centroid problem under the price war, which was recently proposed in the literature. The objective is profit maximization by choosing the best hub and spoke topology, with the corresponding price structure, in a leader–follower setting. Because this bi–level optimization problem is NP–hard, the use of metaheuristics is a natural choice for solving real–size instances. A variable neighborhood search algorithm is designed as a solution approach for the leader. The characterization of optimal routes under the price equilibrium is given in order to simplify and improve the algorithm. When it comes to the follower, we have shown how to reformulate in a linear fashion the initial non–linear model. The computational experiments are conducted on the CAB instances. The results of these experiments are thoroughly discussed, highlighting the effects of different parameters and providing some interesting managerial insights.

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Acknowledgements

We are sincerely grateful to the two anonymous reviewers for their critical reading and constructive input which helped us to improve and clarify this manuscript.

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Authors and Affiliations

  1. Faculty of Natural Sciences and Mathematics, University of Banja Luka, Mladena Stojanovića 2, 78000, Banja Luka, Srpska Republic, Bosnia and Herzegovina

    Dimitrije D. Čvokić

  2. Sobolev Institute of Mathematics, pr. Akademika Koptyuga 4, Novosibirsk, 630090, Russia

    Yury A. Kochetov & Aleksandr V. Plyasunov

  3. Novosibirsk State University, 1, Pirogova str., Novosibirsk, 630090, Russia

    Yury A. Kochetov & Aleksandr V. Plyasunov

  4. Faculty of Mathematics, University of Belgrade, Studentski trg 16, 11000, Belgrade, Serbia

    Aleksandar Savić

Authors
  1. Dimitrije D. Čvokić
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  2. Yury A. Kochetov
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  3. Aleksandr V. Plyasunov
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  4. Aleksandar Savić
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Corresponding author

Correspondence to Dimitrije D. Čvokić.

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The research of the second author was supported by the Russian Foundation for Basic Research, Grant 18-07-00599. The research of the third author was supported by the program of fundamental scientific researches of the SB RAS. The research of the fourth author was partially supported by Serbian Ministry of Education, Science and Technological Development under the Grant No. 174010.

Appendix

Appendix

Tables 4, 5, and 6 present the computational results concerning the profit values, running times and number of AH iterations. In the first three columns, the instance values for the inter–hub discount factor \( \alpha \), sensitivity parameter \( \varTheta \), and hub backbone size (‘# of hubs’) are given, respectively. For every instance, the VNS algorithm was executed for 10 times. The best leader’s profit value found among all 10 executions is given in the column ‘best profit’. The mean value of best leader’s profits regarding these 10 executions is presented in the column ‘avg. profit’. Similarly, the mean value of times (in seconds) for which the VNS has found best solution is given in the column ‘avg. best time (s)’. The mean total time (in seconds) of VNS executions is presented in the column ‘avg. run time (s)’. In the last column ‘# of AH iterations’, the number of AH iterations is given.

Table 4 Computational results for \( |N|=15 \)
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Table 5 Computational results for \( |N|=20 \)
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Table 6 Computational results when \( |N|=25 \)
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Čvokić, D.D., Kochetov, Y.A., Plyasunov, A.V. et al. A variable neighborhood search algorithm for the \( (r{\mid }p) \) hub–centroid problem under the price war. J Glob Optim 83, 405–444 (2022). https://doi.org/10.1007/s10898-021-01036-9

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