# A bi-level maximal covering location problem

- 147 Downloads

## 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.

## References

- Alekseeva EV, Kochetov TA (2007) Genetic local search for the p-median problem with client’s preferences. Diskret Anal Issled Oper 14:3–31Google Scholar
- Arroyo JM, Fernández FJ (2013) A genetic algorithm for power system vulnerability analysis under multiple contingencies. In: Talbi EG (ed) Metaheuristics for bi-level optimization. Springer, Berlin, pp 41–68CrossRefGoogle Scholar
- Atta S, Sinha PR, Mukhopadhyay A (2017) Solving maximal covering location problem using genetic algorithm with local refinement. Soft Comput. doi: 10.1007/s00500-017-2598-3 Google Scholar
- Belotti P, Labbé M, Maffioli F, Ndiaye M (2007) A branch-and-cut method for the obnoxious p-median problem. 4OR 5(4):299–314CrossRefGoogle Scholar
- Bhadury J, Jaramillo J, Batta R (2002) On the use of genetic algorithms for location problems. Comput Oper Res 29:761–779CrossRefGoogle Scholar
- Calvete H, Galé C, Mateo PM (2008) A new approach for solving linear bi-level problems using genetic algorithms. Eur J Oper Res 188:14–28CrossRefGoogle Scholar
- Camacho-Vallejo JF, Cordero-Franco AE, González-Ramírez RG (2014) Solving the bi-level facility location problem under preferences by a Stackelberg-evolutionary algorithm. Math Probl Eng 2014:14CrossRefGoogle Scholar
- Camacho-Vallejo JF, Mar-Ortiz J, López-Ramos F, Pedraza R (2015) A genetic algorithm for the bi-level topological design of local area networks. PLoS ONE. doi: 10.1371/journal.pone.0128067 Google Scholar
- Cánovas L, García S, Labbé M, Marín A (2007) A strengthened formulation for the simple plant location problem with order. Oper Res Lett 35(2):141–150CrossRefGoogle Scholar
- Channakrishnaraju M (2014) Design of memetic algorithm to enhance coverage in wireless sensor networks with minimum number of sensors. Sensors 3(6):7272–7278Google Scholar
- Church RL, ReVelle C (1974) The maximal covering location problem. Pap Reg Sci Assoc 32(1):101–118CrossRefGoogle Scholar
- Colombo F, Cordone R, Lulli G (2016) The multimode covering location problem. Comput Oper Res 67:25–33CrossRefGoogle Scholar
- Corrêa FA, Chaves AA, Nogueira LA (2008) Hybrid heuristics for the probabilistic maximal covering location-allocation problem. Oper Res Int J 7(3):323–344CrossRefGoogle Scholar
- Davari S, Fazel MH, Hemmati A (2011) Maximal covering location problem (MCLP) with fuzzy travel times. Expert Syst Appl 38:14535–14541CrossRefGoogle Scholar
- Davari S, Fazel MH, Turksen B (2013) A greedy variable neighborhood search heuristic for the maximal covering location problem with fuzzy coverage radii. Knowl Based Syst 41:68–76CrossRefGoogle Scholar
- Díaz JA, Luna E, Camacho-Vallejo JF, Casas-Ramírez MS (2017) GRASP and hybrid GRASP-Tabu heuristics to solve a maximal covering location problem with customer preference ordering. Expert Syst Appl 82:67–76CrossRefGoogle Scholar
- Eiselt HA, Laporte G, Thisse JF (1993) Competitive location models: a framework and bibliography. Transp Sci 17:451–473Google Scholar
- ElKady SK, Abdelsalam HM (1996) A modified particle swarm optimization algorithm for solving capacitated maximal covering location problem in healthcare systems. In: Hassanien A-E et al (eds) Applications of intelligent optimization in biology and medicine. Springer, Berlin, pp 117–133Google Scholar
- Farahani RZ, Asgari N, Heidari N, Hosseininia M, Goh M (2012) Covering problems in facility location: a review. Comput Ind Eng 62:368–407CrossRefGoogle Scholar
- Farahani RZ, Hassani A, Mousavi SM, Baygi MB (2014) A hybrid artificial bee colony for disruption in a hierarchical maximal covering location problem. Comput Ind Eng 75:129–141CrossRefGoogle Scholar
- Fazel MH, Davari S, Hadda SA (2011) The large scale maximal covering location problem. Sci Iran E 18(6):1564–1570CrossRefGoogle Scholar
- Fisher K (2002) Sequential discrete \(p\)-facility models for competitive location planning. Ann Oper Res 111:253–270CrossRefGoogle Scholar
- García S, Marín A (2015) Covering location problems. In: Laporte G, Nickel S, Saldanha da Gama F (eds) Location science. Springer International Publishing, New York, pp 93–114Google Scholar
- Hakimi SL (1964) Optimum locations of switching centers and the absolute centers and medians of a graph. Oper Res 12(3):450–459CrossRefGoogle Scholar
- Hakimi SL (1983) On locating new facilities in a competitive environment. Eur J Oper Res 12:29–35CrossRefGoogle Scholar
- Hanjoul P, Peeters D (1987) A facility location problem with clients’ preference orderings. Reg Sci Urban Econ 27:44–54Google Scholar
- Hansen P, Kochetov Y, Mladenovic N (2004) Lower bounds for the uncapacitated facility location problem with user preferences, Preprint G-2004-24, Mart 2004. GERAD-HEC, Montreal, CanadaGoogle Scholar
- Hejazi SR, Memariani A, Jahanshahloo G, Sepehri MM (2002) Linear bi-level programming solution by genetic algorithm. Comput Oper Res 29:1913–1925CrossRefGoogle Scholar
- Kress D, Pesch E (2012) Sequential competitive location on networks. Eur J Oper Res 217:483–499CrossRefGoogle Scholar
- Land MWS, Belew RK (1998) Evolutionary algorithms with local search for combinatorial optimization. Ph.D. thesisGoogle Scholar
- Larranaga P, Kuijpers CMH, Murga RH, Inza I, Dizdarevic S (1999) Genetic algorithms for the travelling salesman problem: a review of representations and operators. Artif Intell Rev 13(2):129–170CrossRefGoogle Scholar
- Lee JM, Lee YH (2012) Facility location and scale decision problem with customer preference. Comput Ind Eng 63:184–191CrossRefGoogle Scholar
- Li H, Wang Y (2007) A genetic algorithm for solving a special class of nonlinear bi-level programming problems. Lect Notes Comput Sci 4490:1159–1162CrossRefGoogle Scholar
- Maldonado-Pinto S, Casas-Ramírez MS, Camacho-Vallejo JF (2016) Analyzing the performance of a hybrid heuristic for solving a bi-level location problem under different approaches to tackle the lower level. Math Probl Eng 2016:10CrossRefGoogle Scholar
- Mathieu R, Pittard L, Anandalingam G (1994) Genetic algorithm based approach to bi-level lineal programming. RAIRO Oper Res 28:1–21CrossRefGoogle Scholar
- Marić M, Stanimirović Z, Milenković N (2012) Metaheuristic methods for solving the bi-level uncapacitated facility location problem with clients’ preferences. Electron Notes Discret Math 39:43–50CrossRefGoogle Scholar
- Marić M, Stanimirović Z, Milenković N, Djenić A (2015) Metaheuristic approaches to solving large-scale bi-level uncapacitated facility location problem with clients’ preferences. Yugosl J Oper Res 25(2):361–378Google Scholar
- Oduguwa V, Roy R (2002) Bi-level optimisation using genetic algorithm. In: IEEE international conference proceedings of artificial intelligence systems (ICAIS02), pp 32Google Scholar
- Ognjanović I, Gaševi D, Bagheri E (2013) A stratified framework for handling conditional preferences: an extension of the analytic hierarchy process. Expert Syst Appl 40(4):1094–1115CrossRefGoogle Scholar
- Paul NR, Lunday BJ, Nurre SG (2017) A multiobjective, maximal conditional covering location problem applied to the relocation of hierarchical emergency response facilities. Omega 66:147–158CrossRefGoogle Scholar
- Plastria F (2001) Static competitive facility location: an overview of optimisation approaches. Eur J Oper Res 129:461–470CrossRefGoogle Scholar
- Plastria F, Vanhaverbeke L (2009) Maximal covering location problem with price decision for revenue maximization in a competitive environment. OR Spectr 31:555–571CrossRefGoogle Scholar
- Resende M (1998) Computing approximate solutions of the maximum covering problem with GRASP. J Heuristics 4:161–177CrossRefGoogle Scholar
- Sinha A, Malo P, Deb K (2014) An improved bilevel evolutionary algorithm based on quadratic approximations. In: IEEE congress on evolutionary computation. IEEE, pp 1870–1877Google Scholar
- Schilling DA, Jayaraman V, Barkhi R (1993) A review of covering problem in facility location. Locat Sci 1:25–55Google Scholar
- Vasilyev IL, Klimentova KB (2010) The branch and cut method for the facility location problem with client’s preferences. J Appl Ind Math 4(3):441–454CrossRefGoogle Scholar
- Vasil’ev IL, Klimentova KB, Kochetov YA (2009) New lower bounds for the facility location problem with clients’ preferences. Comput Math Math Phys 49(6):1010–1020CrossRefGoogle Scholar
- Wang Y, Fan K, Horng J (1997) Genetic-based search for error-correcting graph isomorphism. IEEE Trans Syst Man Cybern Part B 27(4):588–597CrossRefGoogle Scholar
- Wang G, Wan Z, Wang X, Lv Y (2008) Genetic algorithm based on simplex method for solving linear-quadratic bilevel programming problem. Comput Math Appl 56:2550–2555CrossRefGoogle Scholar
- Zhang B, Peng J, Li S (2017) Covering location problem of emergency service facilities in an uncertain environment. Appl. Math. Mod. 51:429–447CrossRefGoogle Scholar