Abstract
The capacitated p-center problem is concerned with how to select p locations for facility centers and assign demand points to them such that the maximum distance between a demand point and its nearest center is minimized. This paper focuses on the capacitated p-center problem in an uncertain environment, in which demands and distances are regarded as uncertain variables. Consequently, two minimax models with uncertain parameters are formulated, and their crisp equivalences are investigated. Additionally, a hybrid algorithm based on the 99-method, a genetic algorithm and a tabu search algorithm is designed to solve the models. Finally, some numerical examples are presented to unveil the applications of the models and algorithm.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Albareda-Sambola, M., Martínez-Merino, L. I., & Rodríguez-Chía, A. M. (2019). The stratified \(p\)-center problem. Computers and Operations Research, 108, 213–225.
Barilan, J., Kortsarz, G., & Peleg, D. (1993). How to allocate network centers. Journal of Algorithms, 15(3), 385–415.
Du, B., Zhou, H., & Leus, R. (2020). A two-stage robust model for a reliable \(p\)-center facility location problem. Applied Mathematical Modelling, 77, 99–114.
Espejo, I., Marín, A., & Rodríguez-Chía, A. M. (2015). Capacitated \(p\)-center problem with failure foresight. European Journal of Operational Research, 247(1), 229–244.
Gao, Y. (2011). Shortest path problem with uncertain arc lengths. Computers and Mathematics with Applications, 62(6), 2591–2600.
Gao, Y. (2012). Uncertain models for single facility location problems on networks. Applied Mathematical Modelling, 36(6), 2592–2599.
Garfinkel, R. S., Neebe, A. W., & Rao, M. R. (1977). The \(m\)-center problem: Minimax facility location. Management Science, 23(10), 1133–1142.
Han, S., Peng, Z., & Wang, S. (2014). The maximum flow problem of uncertain network. Information Sciences, 265, 167–175.
Jaeger, M., & Goldberg, J. (1994). A polynomial algorithm for the equal capacity \(p\)-center problem on trees. Transportation Science, 28(2), 167–175.
Liu, B. (2007). Uncertainty theory (2nd ed.). Berlin: Springer.
Liu, B. (2009a). Theory and practice of uncertain programming (2nd ed.). Berlin: Springer.
Liu, B. (2009b). Some research problems in uncertainty theory. Journal of Uncertain Systems, 3(1), 3–10.
Liu, B. (2010). Uncertainty theory: A branch of mathematics for modeling human uncertainty. Berlin: Springer.
Majumder, S., Kar, M. B., Kar, S., & Pal, T. (2020). Uncertain programming models for multi-objective shortest path problem with uncertain parameters. Soft Computing, 24, 8975–8996.
Majumder, S., Kundu, P., Kar, S., & Pal, T. (2019). Uncertain multi-objective multi-item fixed charge solid transportation problem with budget constraint. Soft Computing, 23, 3279–3301.
Majumder, S., Saha, B., Anand, P., Kar, S., & Pal, T. (2018). Uncertainty based genetic algorithm with varying population for random fuzzy maximum flow problem. Expert Systems, 35(4), 12264.
ReVelle, C., & Hogan, K. (1989). The maximum reliability location problem and the \(\alpha \)-reliable \(p\)-center problem: Derivatives of the probabilistic location set covering problem. Annals of Operations Research, 18(1), 155–173.
Sylvester, J. J. (1857). A question in the geometry of situation. Quarterly Journal of Pure and Applied Mathematics, 1(5), 79.
Wang, K., & Yang, Q. (2014). Hierarchical facility location for the reverse logistics network design under uncertainty. Journal of Uncertain Systems, 8(4), 255–270.
Wen, M., Qin, Z., & Kang, R. (2014). The \(\alpha \)-cost minimization model for capacitated facility location–allocation problem with uncertain demands. Fuzzy Optimization and Decision Making, 13, 345–356.
Wu, X., Lan, Y., & Liu, H. (2014). Optimal revenue-sharing contract based on forecasting effort for uncertain agency problem. International Journal of Machine Learning and Cybernetics, 5, 971–979.
Yang, K., Zhao, R., & Lan, Y. (2016). Incentive contract design in project management with serial tasks and uncertain completion times. Engineering Optimization, 48(4), 629–651.
Zhang, B., Li, H., Li, S., & Peng, J. (2018). Sustainable multi-depot emergency facilities location-routing problem with uncertain information. Applied Mathematics and Computation, 333, 506–520.
Zhang, B., Peng, J., Li, S., & Chen, L. (2016). Fixed charge solid transportation problem in uncertain environment and its algorithm. Computers and Industrial Engineering, 102, 186–197.
Zhou, J., & Liu, B. (2003). New stochastic models for capacitated location–allocation problem. Computers and Industrial Engineering, 45, 111–125.
Acknowledgements
This work was supported by the National Natural Science Foundation of China (Grant Nos. 61873108, 61703438, and 11626234).
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Zhang, B., Peng, J. & Li, S. Minimax models for capacitated p-center problem in uncertain environment. Fuzzy Optim Decis Making 20, 273–292 (2021). https://doi.org/10.1007/s10700-020-09343-8
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10700-020-09343-8