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Covering Location Problems

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Abstract

When deciding where to locate facilities (e.g., emergency points where an ambulance will wait for a call) that provide a service, it happens quite often that a customer (e.g., a person) can receive this service only if she is located less than a certain distance from the nearest facility (e.g., the ambulance can arrive in less than 7 min at this person’s home). The problems that share this property receive the name of covering problems and have many applications. (analysis of markets, archaeology, crew scheduling, emergency services, metallurgy, nature reserve selection, etc.). This chapter surveys the most relevant problems in this field: the Set Covering Problem, the Maximal Covering Location Problem, and related problems, In addition, it is introduced a general model that has as particular cases the main covering location models. The most important theoretical results in this topic as well as exact and heuristic algorithms are reviewed. A Lagrangian approach to solve the general model is detailed, and, although the emphasis is on discrete models, some information on continuous covering is provided at the end of the chapter.

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References

  • Aguilera NE, Katz RD, Tolomei PB (2017) Vertex adjacencies in the set covering problem. Discrete Appl Math 218:40–56

    Article  MathSciNet  MATH  Google Scholar 

  • Al-Sultan KS, Hussain MF, Nizami JS (1996) A genetic algorithm for the set covering problem. J Oper Res Soc 47(5):702–709

    Article  MATH  Google Scholar 

  • Avella P, Boccia M, Vasilyev I (2009) Computational experience with general cutting planes for the set covering problem. Oper Res Lett 37(1):16–20

    Article  MathSciNet  MATH  Google Scholar 

  • Balas E (1980) Cutting planes from conditional bounds: a new approach to set covering. Math Program 12:19–36

    Article  MathSciNet  MATH  Google Scholar 

  • Balas E, Carrera MC (1996) A dynamic subradient-based branch-and-bound procedure for set covering. Oper Res 44(6):875–890

    Article  MathSciNet  MATH  Google Scholar 

  • Balas E, Ho A (1980) Set covering algorithms using cutting planes, heuristics and subgradient optimization: a computational study. Math Program 12:37–60

    Article  MathSciNet  MATH  Google Scholar 

  • Balas E, Ng SM (1989a) On the set covering polytope: I. All the facets with coefficients in {0,  1,  2}. Math Program 43:57–69

    Article  MathSciNet  MATH  Google Scholar 

  • Balas E, Ng SM (1989b) On the set covering polytope: II. Lifting the facets with coefficients in {0,  1,  2}. Math Program 45:1–20

    Article  MathSciNet  MATH  Google Scholar 

  • Balas E, Padberg MW (1976) Set partitioning: a survey. SIAM Rev 18(4):710–760

    Article  MathSciNet  MATH  Google Scholar 

  • Balinski ML (1965) Integer programming: methods, uses, computations. Manag Sci 12(3):253–313

    Article  MathSciNet  MATH  Google Scholar 

  • Batta R, Mannur NR (1990) Covering-location models for emergency situations that require multiple response units. Manag Sci 36(1):16–23

    Article  MathSciNet  MATH  Google Scholar 

  • Batta R, Dolan JM, Krishnamurthy NN (1989) The Maximal expected covering location problem: revisited. Transp Sci 23(4):277–287

    Article  MathSciNet  MATH  Google Scholar 

  • Beasley JE (1987) An algorithm for the set covering problem. Eur J Oper Res 31(1):85–93

    Article  MathSciNet  MATH  Google Scholar 

  • Beasley JE (1990) A Lagrangian heuristic for set-covering problems. Nav Res Log 37(1):151–164

    Article  MathSciNet  MATH  Google Scholar 

  • Beasley JE, Chu PC (1996) A genetic algorithm for the set covering problem. Eur J Oper Res 94(2):392–404

    Article  MATH  Google Scholar 

  • Beasley JE, Jørnsten K (1992) Enhancing an algorithm for set covering problems. Eur J Oper Res 58(2):293–300

    Article  MATH  Google Scholar 

  • Bell T, Church RL (1985) Location-allocation theory in archaeological settlement pattern research: some preliminary applications. World Archaeol 16(3):354–371

    Article  Google Scholar 

  • Bellmore M, Ratliff HD (1971) Set covering and involutory bases. Manag Sci 18(3):194–206

    Article  MathSciNet  MATH  Google Scholar 

  • Berge C (1957) Two theorems in graph theory. Proc Natl Acad Sci USA 43(9):842–844

    Article  MathSciNet  MATH  Google Scholar 

  • Berman O, Drezner Z, Krass D (2010) Generalized coverage: new developments in covering location models. Comput Oper Res 37:1675–1687

    Article  MathSciNet  MATH  Google Scholar 

  • Broin MW, Lowe TJ (1986) A dynamic programming algorithm for covering problems with (greedy) totally balanced constraint matrices. SIAM J Algebra Discr 7(3):348–357

    Article  MathSciNet  MATH  Google Scholar 

  • Brusco MJ, Jacobs LW, Thompson GM (1999) A morphing procedure to supplement a simulated annealing heuristic for cost- and coverage-correlated set-covering problems. Ann Oper Res 86:611–627

    Article  MathSciNet  MATH  Google Scholar 

  • Caprara A, Fischetti M, Toth P (1999) A heuristic method for the set covering problem. Oper Res 47(5):730–743

    Article  MathSciNet  MATH  Google Scholar 

  • Caprara A, Toth P, Fischetti M (2000) Algorithms for the set covering problem. Ann Oper Res 98:353–371

    Article  MathSciNet  MATH  Google Scholar 

  • Ceria S, Nobili P, Sassano A (1998) A Lagrangian-based heuristic for large-scale set covering problems. Math Program 81(2):215–228

    Article  MathSciNet  MATH  Google Scholar 

  • Chazelle BM, Lee DT (1986) On a circle placement problem. Computing 36:1–16

    Article  MathSciNet  MATH  Google Scholar 

  • Christofides N, Korman S (1975) A computational survey of methods for the set covering problem. Manag Sci 21(5):591–599

    Article  MathSciNet  MATH  Google Scholar 

  • Chung C (1986) Recent applications of the maximal covering location planning (M.C.L.P.) model. J Oper Res Soc 37(8):735–746

    Article  MATH  Google Scholar 

  • Church RL (1984) The planar maximal covering location problem. J Reg. Sci 24(2):185–201

    Article  MathSciNet  Google Scholar 

  • Church RL, Meadows ME (1979) Location modeling utilizing maximum service distance criteria. Geogr Anal 11(4):358–373

    Article  Google Scholar 

  • Church RL, ReVelle C (1974) The maximal covering location problem. Pap Reg Sci Assoc 32(1):101–118

    Article  Google Scholar 

  • Church RL, ReVelle C (1976) Theoretical and computational links between the p-median, location set-covering, and the maximal covering location problem. Geogr Anal 8(4):406–415

    Article  Google Scholar 

  • Church RL, Weaver JR (1986) Theoretical links between median and coverage location problems. Ann Oper Res 6:1–19

    Article  Google Scholar 

  • Church RL, Stoms DM, Davis FW (1996) Reserve selection as a maximal covering location problem. Biol Conserv 76(2):105–112

    Article  Google Scholar 

  • Chvátal V (1979) A greedy heuristic for the set-covering problem. Math Oper Res 4(3):233–235

    Article  MathSciNet  MATH  Google Scholar 

  • Cornuéjols G, Sassano A (1989) On the 0,1 facets of the set covering polytope. Math Program 43:45–55

    Article  MathSciNet  MATH  Google Scholar 

  • Cornuéjols G, Nemhauser GL, Wolsey LA (1980) A canonical representation of simple plant location problems and its applications. SIAM J Algebra Discrete Methods 1(3):261–272

    Article  MathSciNet  MATH  Google Scholar 

  • Current JR, Schilling DA (1990) Analysis of errors due to demand data aggregation in the set covering and maximal covering location problems. Geogr Anal 22(2):116–126

    Article  Google Scholar 

  • Current JR, Storbeck JE (1988) Capacitated covering problems. Environ Plann B 15(2):153–163

    Article  Google Scholar 

  • Curtin KM, Hayslett-McCall K, Qiu F (2010) Determining optimal police patrol areas with maximal covering and backup covering location models. Netw Span Econ 10:125–145

    Article  MathSciNet  MATH  Google Scholar 

  • Daskin MS (1983) A maximum expected covering location model: formulation, propeties and heuristic solution. Transp Sci 17(1):48–70

    Article  Google Scholar 

  • Daskin MS (2013) Covering problems. In: Networks and discrete location. Models, algorithms and applications, 2nd edn. Wiley, New York, pp 124–192

    Google Scholar 

  • Daskin MS, Stern EH (1981) A hierarchical objective set covering model for emergency medical service vehicle deployment. Transp Sci 15(2):137–152

    Article  MathSciNet  Google Scholar 

  • Daskin MS, Haghani AE, Khanal M, Malandraki C (1989) Aggregation effects in maximum covering problems. Ann Oper Res 18(1):115–140

    Article  MathSciNet  MATH  Google Scholar 

  • Downs BT, Camm JD (1996) An exact algorithm for the maximal covering problem. Nav Res Log 43(3):435–461

    Article  MathSciNet  MATH  Google Scholar 

  • Drezner Z (1981) On a modified one-center problem. Manag Sci 27(7):848–851

    Article  MATH  Google Scholar 

  • Drezner Z, Wesolowsky GO, Drezner T (2004) The gradual covering problem. Nav Res Log 51(6):841–855

    Article  MathSciNet  MATH  Google Scholar 

  • Dwyer FP, Evans JR (1981) A branch and bound algorithm for the list selection problem in direct mail advertising. Manag Sci 27(6):658–667

    Article  Google Scholar 

  • Eaton DJ, Sánchez HML, Lantigua RR, Morgan J (1986) Determining ambulance deployment in Santo Domingo, Dominican Republic. J Oper Res Soc 37(2):113–126

    Article  Google Scholar 

  • Elzinga D, Hearn D (1972) Geometric solutions for some minimax location problems. Transp Sci 6(4):379–394

    Article  Google Scholar 

  • Etcheberry J (1977) The set-covering problem: a new implicit enumeration algorithm. Oper Res 25(5):760–772

    Article  MathSciNet  MATH  Google Scholar 

  • Farahani RZ, Asgari N, Heidari N, Hosseininia M, Goh M (2012) Covering problems in facility location: a review. Comput Ind Eng 62(1):368–407

    Article  Google Scholar 

  • Feo TA, Resende MGC (1989) A probabilistic heuristic for a computationally difficult set covering problem. Oper Res Lett 8:67–71

    Article  MathSciNet  MATH  Google Scholar 

  • Fisher ML, Kedia P (1990) Optimal solutions of set covering/partitioning problems using dual heuristics. Manag Sci 36(6):674–688

    Article  MathSciNet  MATH  Google Scholar 

  • Galvão RD, ReVelle C (1996) A Lagrangean heuristic for the maximal covering location problem. Eur J Oper Res 88(1):114–123

    Article  MATH  Google Scholar 

  • Galvão RD, Espejo LGA, Boffey B (2000) A comparison of Lagrangean and surrogate relaxations for the maximal covering location problem. Eur J Oper Res 124(2):377–389

    Article  MathSciNet  MATH  Google Scholar 

  • Galvão RD, Chiyoshia FY, Morabito R (2005) Towards unified formulations and extensions of two classical probabilistic location models. Comput Oper Res 32(1):15–33

    Article  MathSciNet  MATH  Google Scholar 

  • García S, Labbé M, Marín A (2011) Solving large p-median problems with a radius formulation. INFORMS J Comput 23(4):546–556

    Article  MathSciNet  MATH  Google Scholar 

  • Garey MR, Johnson DS (1979) Computers and intractibility: a guide to the theory of NP-completeness. WH Freeman, New York

    MATH  Google Scholar 

  • Guignard M (2003) Lagrangean relaxation. TOP 11(2):151–200

    Article  MathSciNet  MATH  Google Scholar 

  • Haddadi S (2017) Benders decomposition for set covering problems almost satisfying the consecutive ones property. J Comb Optim 33(1):60–80

    Article  MathSciNet  MATH  Google Scholar 

  • Hakimi SL (1965) Optimum distribution of switching centers in a communication network and some related graph theoretic problems. Oper Res 13(3):462–475

    Article  MathSciNet  MATH  Google Scholar 

  • Hogan K, ReVelle C (1986) Concepts and applications of backup coverage. Manag Sci 32(11):1434–1444

    Article  Google Scholar 

  • Hohn F (1955) Mathematical aspects of switching. Am Math Mon 62:75–90

    Article  MathSciNet  MATH  Google Scholar 

  • Jacobs LW, Brusco MJ (1995) Note: a local-search heuristic for large set-covering problems. Nav Res Log 42(7):1129–1140

    Article  MathSciNet  MATH  Google Scholar 

  • Klastorin TD (1979) On the maximal covering location problem and the generalized asssignment problem. Manag Sci 25(1):107–112

    Article  MATH  Google Scholar 

  • Kolen A, Tamir A (1990) Covering problems. In: Mirchandani PB, Francis RL (eds) Discrete location theory. Wiley, New York, pp 263–304

    Google Scholar 

  • Krarup J, Pruzan PM (1983) The simple plant location problem: survey and synthesis. Eur J Oper Res 12(1):36–81

    Article  MathSciNet  MATH  Google Scholar 

  • Kuo YH, Leung JMY (2016) On the mixed set covering, packing and partitioning polytope. Discrete Optim 22:162–282

    Article  MathSciNet  MATH  Google Scholar 

  • Lemke CE, Salkin HM, Spielberg K (1971) Set covering by single branch enumeration with linear programming subproblems. Oper Res 19(4):998–1022

    Article  MathSciNet  MATH  Google Scholar 

  • Li XP, Ramshani M, Huang Y (2018) Cooperative maximal covering models for humanitarian relief chain management. Comput Ind Eng 119:301–308

    Article  Google Scholar 

  • Lorena LAN, Lopes FB (1994) A surrogate heuristic for set covering problems. Eur J Oper Res 79(1):138–150

    Article  MATH  Google Scholar 

  • Mannino C, Sassano A (1995) Solving hard set covering problems. Oper Res Lett 18(1):1–5

    Article  MathSciNet  MATH  Google Scholar 

  • Megiddo N, Zemel E, Hakimi SL (1983) The maximum coverage location problem. SIAM J Algebra Discrete Methods 4(2):253–261

    Article  MathSciNet  MATH  Google Scholar 

  • Murray AT (2016) Maximal coverage location problem: impacts, significance, and evolution. Int Reg Sci Rev 39(1):5–27

    Article  Google Scholar 

  • Neebe AW (1988) A procedure for locating emergency-service facilities for all possible response distances. J Oper Res Soc 39(8):743–748

    Article  MATH  Google Scholar 

  • Nobili P, Sassano A (1989) Facets and lifting procedures for the set covering polytope. Math Program 45:111–137

    Article  MathSciNet  MATH  Google Scholar 

  • Norman RZ, Rabin MO (1959) An algorithm for a minimum cover of a graph. Proc Am Math Soc 10:315–319

    Article  MathSciNet  MATH  Google Scholar 

  • Plane DR, Hendrick TE (1977) Mathematical programming and the location of fire companies for the Denver Fire. Oper Res 25(4):563–578

    Article  MATH  Google Scholar 

  • Plastria F (2002) Continuous covering location problems. In: Hamacher HW, Drezner Z (eds) Facility location: applications and theory. Springer, New York, pp. 37–79

    Chapter  MATH  Google Scholar 

  • ReVelle C (1989) Review, extension and prediction in emergency service siting models. Eur J Oper Res 40(1):58–69

    Article  MathSciNet  MATH  Google Scholar 

  • ReVelle C, Hogan K (1989a) The maximum reliability location problem and α-reliable p-center problem: derivatives of the probabilistic location set covering problem. Ann Oper Res 18:155–174

    Article  MathSciNet  MATH  Google Scholar 

  • ReVelle C, Hogan K (1989b) The maximum availability location problem. Transp Sci 23(3):192–200

    Article  MathSciNet  MATH  Google Scholar 

  • Roth R (1969) Computer solutions to minimum-cover problems. Oper Res 17(3):455–465

    Article  MATH  Google Scholar 

  • Sánchez-García M, Sobrón MI, Vitoriano B (1998) On the set covering polytope: facets with coefficients in {0,  1,  2,  3}. Ann Oper Res 81:343–356

    Article  MathSciNet  MATH  Google Scholar 

  • Sassano A (1989) On the facial structure of the set covering polytope. Math Program 44:181–202

    Article  MathSciNet  MATH  Google Scholar 

  • Schilling DA, Jayaraman V, Barkhi R (1993) A review of covering problems in facility location. Locat Sci 1(1):25–55

    MATH  Google Scholar 

  • Snyder LV (2011) Covering problems. In: Eiselt HA, Marianov V (eds). Foundations of location analysis. Springer, Berlin, pp 109–135

    Chapter  Google Scholar 

  • Storbeck JE (1988) The spatial structuring of central places. Geogr Anal 20(2):93–110

    Article  Google Scholar 

  • Storbeck JE, Vohra RV (1988) A simple trade-off model for maximal and multiple coverage. Geogr Anal 20(3):220–230

    Article  Google Scholar 

  • Toregas C, ReVelle C (1972) Optimal location under time or distance constraints. Pap Reg Sci Assoc 28(1):133–144

    Article  Google Scholar 

  • Toregas C, ReVelle C (1973) Binary logic solutions to a class of location problem. Geogr Anal 5(2):145–175

    Article  Google Scholar 

  • Toregas C, Swain A, ReVelle C, Bergman L (1971) The location of emergency service facilities. Oper Res 19(6):1363–1373

    Article  MATH  Google Scholar 

  • Vasko FJ, Wolf FE, Stott KL (1989) A set covering approach to metallurgical grade assignment. Eur J Oper Res 38(1):27–34

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The research of the authors has been partially supported by the research project 19320/PI/14 (Fundación Séneca, Región de Murcia, Spain). Alfredo Marín has also been supported by the research projects MTM2015-65915-R (MINECO, Spain) and “Cost-sensitive classification. A mathematical optimization approach” (Fundación BBVA).

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Authors and Affiliations

  1. School of Mathematics, University of Edinburgh, Edinburgh, UK

    Sergio García

  2. Departamento de Estadística e Investigación Operativa, Universidad de Murcia, Murcia, Spain

    Alfredo Marín

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  1. Sergio García
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  2. Alfredo Marín
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Correspondence to Sergio García .

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Editors and Affiliations

  1. HEC Montréal, Montréal, QC, Canada

    Gilbert Laporte

  2. Institute for Operations Research, Karlsruhe Institute of Technology, Karlsruhe, Germany

    Stefan Nickel

  3. Faculty of Science, University of Lisbon, Lisbon, Portugal

    Francisco Saldanha da Gama

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García, S., Marín, A. (2019). Covering Location Problems. In: Laporte, G., Nickel, S., Saldanha da Gama, F. (eds) Location Science. Springer, Cham. https://doi.org/10.1007/978-3-030-32177-2_5

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