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A bi-level maximal covering location problem

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Abstract

In this research a bi-level maximal covering location problem is studied. The problem considers the following situation: a firm wants to enter a market, where other firms already operate, to maximize demand captured by locating p facilities. Customers are allowed to freely choose their allocation to open facilities. The problem is formulated as a bi-level mathematical programming problem where two decision levels are considered. In the upper level, facilities are located to maximize covered demand, and in the lower level, customers are allocated to facilities based on their preferences to maximize a utility function. In addition, two single-level reformulations of the problem are examined. The time required to solve large instances of the problem with the considered reformulations is very large, therefore, a heuristic is proposed to obtain lower bounds of the optimal solution. The proposed heuristic is a genetic algorithm with local search. After adjusting the parameters of the proposed algorithm, it is tested on a set of instances randomly generated based on procedures described in the literature. According to the obtained results, the proposed genetic algorithm with local search provides very good lower bounds requiring low computational time.

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Acknowledgements

The research of the first two authors has been partially supported by the Mexican National Council for Science and Technology (CONACYT) through grant SEP-CONACYTCB-2014-01-240814. Also, the second author thanks to the program of professional development of professors with the Grant PRODEP/511-6/17/7425 for research stays during his sabbatical year.

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Authors and Affiliations

  1. Facultad de Ciencias Físico Matemáticas, Universidad Autónoma de Nuevo León, Ave. Universidad s/n., 66455, San Nicolás de los Garza, Nuevo León, Mexico

    Martha-Selene Casas-Ramírez & José-Fernando Camacho-Vallejo

  2. Depto. de Actuaría, Física y Matemáticas, Universidad de las Américas Puebla, San Andrés Cholula, Mexico

    Juan A. Díaz

  3. Depto. de Ingeniería Industrial y Mecánica, Universidad de las Américas Puebla, San Andrés Cholula, Mexico

    Dolores E. Luna

Authors
  1. Martha-Selene Casas-Ramírez
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  2. José-Fernando Camacho-Vallejo
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  3. Juan A. Díaz
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  4. Dolores E. Luna
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Corresponding author

Correspondence to José-Fernando Camacho-Vallejo.

Appendix

Appendix

Figures that plot the objective function value against execution times for every sampled instance are shown. For each sampled instance, the six possible configurations are plotted. The label in each point indicates the corresponding configuration in the following order: (0.65, 0.01), (0.65, 0.03), (0.65, 0.05), (0.75, 0.01), (0.75, 0.03), and (0.75, 0.05), where the first number corresponds to crossover propabibility, and the second number corresponds to the probability of executing local search. In this analysis, if two configurations have the same value for one criterion, the one that has the worst value in the other criterion is considered as the dominated solution (Figs. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10).

Fig. 1
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Dominance among configurations for sampled instances 01–03

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Fig. 2
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Dominance among configurations for sampled instances 11–13

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Fig. 3
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Dominance among configurations for sampled instances 21–23

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Fig. 4
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Dominance among configurations for sampled instances 31–33

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Fig. 5
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Dominance among configurations for sampled instances 41–43

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Fig. 6
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Dominance among configurations for sampled instances 51–53

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Fig. 7
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Dominance among configurations for sampled instances 61–63

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Fig. 8
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Dominance among configurations for sampled instances 71–73

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Fig. 9
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Dominance among configurations for sampled instances 81–83

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Fig. 10
figure 10

Dominance among configurations for sampled instances 91–93

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Casas-Ramírez, MS., Camacho-Vallejo, JF., Díaz, J.A. et al. A bi-level maximal covering location problem. Oper Res Int J 20, 827–855 (2020). https://doi.org/10.1007/s12351-017-0357-y

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