# 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.

## Keywords

Bi-level programming Maximal covering Customer preferences Facility location## Mathematics Subject Classification

90B80 90C59 90C99## Notes

### 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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