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TOP

, Volume 27, Issue 1, pp 70–93 | Cite as

A directional approach to gradual cover

  • Tammy Drezner
  • Zvi DreznerEmail author
  • Pawel Kalczynski
Original Paper
  • 51 Downloads

Abstract

The objective of classic cover location models is for facilities to cover demand within a given distance. Locating a given number of facilities to cover as much demand as possible is referred to as max-cover. Finding the minimum number of facilities required to cover all the demand is the set covering problem. The gradual (or partial) cover replaces abrupt drop from full cover to no cover by defining gradual decline in cover. If classic cover models consider 3 miles as the cover distance, then at 2.99 miles a demand point is fully covered while at 3.01 miles it is not covered at all. In gradual cover, a cover range is set. For example, up to 2 miles the demand is fully covered, beyond 4 miles it is not covered at all, and between 2 and 4 miles it is partially covered. In this paper, we propose, analyze, and test a new rule for calculating the joint cover of a demand point which is partially covered by several facilities. The algorithm is tested on a case study of locating cell phone towers in Orange County, California. The new approach provided better total cover than the cover obtained by existing procedures.

Keywords

Cover location models Partial cover Gradual cover 

Mathematics Subject Classification

90B80 90B85 90C27 

References

  1. Abramowitz M, Stegun I (1972) Handbook of mathematical functions. Dover Publications Inc., New York, NYGoogle Scholar
  2. Alkhalifa L, Brimberg J (2017) Locating a minisum annulus: a new partial coverage distance model. TOP 25(2):373–393CrossRefGoogle Scholar
  3. Altýnel I, Durmaz E, Aras N, Özkýsacýk K (2009) A location-allocation heuristic for the capacitated multi-facility Weber problem with probabilistic customer locations. Eur J Oper Res 198:790–799CrossRefGoogle Scholar
  4. Bagherinejad J, Bashiri M, Nikzad H (2018) General form of a cooperative gradual maximal covering location problem. J Ind Eng Int 14:241–253CrossRefGoogle Scholar
  5. Berman O, Drezner Z, Krass D (2010) Cooperative cover location problems: the planar case. IIE Trans 42:232–246CrossRefGoogle Scholar
  6. Berman O, Drezner Z, Krass D (2018) The multiple gradual cover location problem. J Oper Res Soc.  https://doi.org/10.1080/01605682.2018.1471376
  7. Berman O, Drezner Z, Wesolowsky GO (2009) The maximal covering problem with some negative weights. Geograph Anal 41:30–42CrossRefGoogle Scholar
  8. Berman O, Krass D (2002) The generalized maximal covering location problem. Comput Oper Res 29:563–591CrossRefGoogle Scholar
  9. Berman O, Krass D, Drezner Z (2003) The gradual covering decay location problem on a network. Eur J Oper Res 151:474–480CrossRefGoogle Scholar
  10. Berman O, Simchi-Levi D (1990) The conditional location problem on networks. Transp Sci 24:77–78CrossRefGoogle Scholar
  11. Beyer HW (1981) Standard mathematical tables. CRC Press, Boca Raton, FLGoogle Scholar
  12. Carrizosa E, Conde E, Muñoz-Marquez M, Puerto J (1995) The generalized weber problem with expected distances. RAIRO Oper Res 29:35–57CrossRefGoogle Scholar
  13. Chen R, Handler GY (1993) The conditional \(p\)-center in the plane. Naval Res Log 40:117–127CrossRefGoogle Scholar
  14. Church RL, ReVelle CS (1974) The maximal covering location problem. Pap Region Sci Assoc 32:101–118CrossRefGoogle Scholar
  15. Church RL, Roberts KL (1984) Generalized coverage models and public facility location. Pap Region Sci Assoc 53:117–135CrossRefGoogle Scholar
  16. Clenshaw CW, Curtis AR (1960) A method for numerical integration on an automatic computer. Numer Math 2:197–205CrossRefGoogle Scholar
  17. Cooper L (1963) Location-allocation problems. Oper Res 11:331–343CrossRefGoogle Scholar
  18. Cooper L (1964) Heuristic methods for location–allocation problems. SIAM Rev 6:37–53CrossRefGoogle Scholar
  19. Dennis J, Woods DJ (1987) Optimization on microcomputers: the Nelder-Mead simplex algorithm. In: Wouk A (ed) New computing environments: microcomputers in large-scale computing. SIAM Publications, Philadelphia, pp 116–122Google Scholar
  20. Diaz JA, Fernandez E (2006) Hybrid scatter search and path relinking for the capacitated \(p\)-median problem. Eur J Oper Res 169:570–585CrossRefGoogle Scholar
  21. Drezner T (2004) Location of casualty collection points. Environ Plan C Govern Policy 22:899–912CrossRefGoogle Scholar
  22. Drezner T, Drezner Z (1997) Replacing discrete demand with continuous demand in a competitive facility location problem. Naval Res Log 44:81–95CrossRefGoogle Scholar
  23. Drezner T, Drezner Z (2007) Equity models in planar location. Comput Manag Sci 4:1–16CrossRefGoogle Scholar
  24. Drezner T, Drezner Z (2014) The maximin gradual cover location problem. OR Spectr 36:903–921CrossRefGoogle Scholar
  25. Drezner T, Drezner Z, Goldstein Z (2010) A stochastic gradual cover location problem. Naval Res Log 57:367–372Google Scholar
  26. Drezner T, Drezner Z, Kalczynski P (2011) A cover-based competitive location model. J Oper Res Soc 62:100–113CrossRefGoogle Scholar
  27. Drezner T, Drezner Z, Kalczynski P (2012) Strategic competitive location: improving existing and establishing new facilities. J Oper Res Soc 63:1720–1730CrossRefGoogle Scholar
  28. Drezner T, Drezner Z, Kalczynski P (2015) A leader-follower model for discrete competitive facility location. Comput Oper Res 64:51–59CrossRefGoogle Scholar
  29. Drezner T, Drezner Z, Salhi S (2006) A multi-objective heuristic approach for the casualty collection points location problem. J Oper Res Soc 57:727–734CrossRefGoogle Scholar
  30. Drezner Z (1986) Location of regional facilities. Naval Res Log Q 33:523–529CrossRefGoogle Scholar
  31. Drezner Z (1995) On the conditional \(p\)-median problem. Comput Oper Res 22:525–530CrossRefGoogle Scholar
  32. Drezner Z (2015) The fortified Weiszfeld algorithm for solving the Weber problem. IMA J Manag Math 26:1–9CrossRefGoogle Scholar
  33. Drezner Z, Klamroth K, Schöbel A, Wesolowsky GO (2002) The Weber problem. In: Drezner Z, Hamacher HW (eds) Facility location: applications and theory. Springer, Berlin, pp 1–36CrossRefGoogle Scholar
  34. Drezner Z, Suzuki A (2004) The big triangle small triangle method for the solution of non-convex facility location problems. Oper Res 52:128–135CrossRefGoogle Scholar
  35. Drezner Z, Suzuki A (2010) Covering continuous demand in the plane. J Oper Res Soc 61:878–881CrossRefGoogle Scholar
  36. Drezner Z, Wesolowsky GO, Drezner T (2004) The gradual covering problem. Naval Res Log 51:841–855CrossRefGoogle Scholar
  37. Eiselt HA, Marianov V (2009) Gradual location set covering with service quality. Soc Econ Plan Sci 43:121–130CrossRefGoogle Scholar
  38. Fonseca MC, Captivo ME (1996) Location of semi obnoxious facilities with capacity constraints. Stud Loc Anal 9:51–52Google Scholar
  39. García S, Marín A (2015) Covering location problems. In: Laporte G, Nickel S, da Gama FS (eds) Location science. Springer, Heidelberg, pp 93–114Google Scholar
  40. Glover F, Laguna M (1997) Tabu search. Kluwer Academic Publishers, BostonCrossRefGoogle Scholar
  41. Goldberg DE (2006) Genetic algorithms. Pearson Education, DelhiGoogle Scholar
  42. Hansen P, Peeters D, Thisse J-F (1981) On the location of an obnoxious facility. Sistem Urban 3:299–317Google Scholar
  43. Holland JH (1975) Adaptation in natural and artificial systems. University of Michigan Press, Ann ArborGoogle Scholar
  44. Hosseininezhad SJ, Jabalameli MS, Naini SGJ (2013) A continuous covering location model with risk consideration. Appl Math Modell 37:9665–9676CrossRefGoogle Scholar
  45. Huff DL (1964) Defining and estimating a trade area. J Mark 28:34–38CrossRefGoogle Scholar
  46. Huff DL (1966) A programmed solution for approximating an optimum retail location. Land Econ 42:293–303CrossRefGoogle Scholar
  47. Karasakal O, Karasakal E (2004) A maximal covering location model in the presence of partial coverage. Comput Oper Res 31:15–26CrossRefGoogle Scholar
  48. Karatas M (2017) A multi-objective facility location problem in the presence of variable gradual coverage performance and cooperative cover. Eur J Oper Res 262:1040–1051CrossRefGoogle Scholar
  49. Kirkpatrick S, Gelat CD, Vecchi MP (1983) Optimization by simulated annealing. Science 220:671–680CrossRefGoogle Scholar
  50. Love RF (1972) A computational procedure for optimally locating a facility with respect to several rectangular regions. J Region Sci 12:233–242CrossRefGoogle Scholar
  51. Melo MT, Nickel S, Da Gama FS (2006) Dynamic multi-commodity capacitated facility location: a mathematical modeling framework for strategic supply chain planning. Comput Oper Res 33:181–208CrossRefGoogle Scholar
  52. Minieka E (1980) Conditional centers and medians on a graph. Networks 10:265–272CrossRefGoogle Scholar
  53. Miyagawa M (2017) Continuous location model of a rectangular barrier facility. TOP 25(1):95–110CrossRefGoogle Scholar
  54. Mladenović N, Hansen P (1997) Variable neighborhood search. Comput Oper Res 24:1097–1100CrossRefGoogle Scholar
  55. Morohosi H, Furuta T (2017) Two approaches to cooperative covering location problem and their application to ambulance deployment. In: Operations research proceedings 2015, Springer, New York, pp 361–366Google Scholar
  56. Nelder JA, Mead R (1965) A simplex method for function minimization. Comput J 7:308–313CrossRefGoogle Scholar
  57. Nickel S, Puerto J, Rodriguez-Chia AM (2003) An approach to location models involving sets as existing facilities. Math Oper Res 28:693–715CrossRefGoogle Scholar
  58. Ogryczak W, Zawadzki M (2002) Conditional median: a parametric solution concept for location problems. Ann Oper Res 110:167–181CrossRefGoogle Scholar
  59. Plastria F (2002) Continuous covering location problems. In: Drezner Z, Hamacher HW (eds) Facility location: applications and theory. Springer, New York, pp 39–83Google Scholar
  60. Puerto J, Ricca F, Scozzari A (2018) Extensive facility location problems on networks: an updated review. TOP 26(2):187—226Google Scholar
  61. Puerto J, Rodríguez-Chía AM (2011) On the structure of the solution set for the single facility location problem with average distances. Math Program 128:373–401CrossRefGoogle Scholar
  62. ReVelle C, Toregas C, Falkson L (1976) Applications of the location set covering problem. Geograph Anal 8:65–76CrossRefGoogle Scholar
  63. Snyder LV (2011) Covering problems. In: Eiselt HA, Marianov V (eds) Foundations of location analysis. Springer, New York, pp 109–135CrossRefGoogle Scholar
  64. Suzuki A, Drezner Z (1996) The \(p\)-center location problem in an area. Loc Sci 4:69–82CrossRefGoogle Scholar
  65. Taillard ÉD (1991) Robust tabu search for the quadratic assignment problem. Parall Comput 17:443–455CrossRefGoogle Scholar
  66. Wang S-C, Chen T-C (2017) Multi-objective competitive location problem with distance-based attractiveness and its best non-dominated solution. Appl Math Model 47:785–795CrossRefGoogle Scholar
  67. Weber A (1929) Über den Standort der Industrien, 1. Teil: Reine Theorie des Standortes. English Translation: on the Location of Industries. University of Chicago Press, Chicago, IL. Originally published in Tübingen, Germany in (1909)Google Scholar
  68. Wesolowsky GO, Love RF (1971) Location of facilities with rectangular distances among point and area destinations. Naval Res Log Q 18:83–90CrossRefGoogle Scholar

Copyright information

© Sociedad de Estadística e Investigación Operativa 2018

Authors and Affiliations

  1. 1.Steven G. Mihaylo College of Business and EconomicsCalifornia State University-FullertonFullertonUSA

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