Location Covering Models pp 177-201 | Cite as

# Continuous Space Coverage

## Abstract

An important distinction in location analysis and modeling has long been discrete versus continuous approaches. In previous chapters, for the most part, the reviewed coverage problems have been discrete in the sense that the places at which a facility may be sited are known and finite in number, and the demand locations to be served are also known and finite. This has enabled discrete integer programming formulations of models to be developed, allowing for efficient and exact solution in many cases. In some circumstances, however, neither potential facility sites nor demand locations are necessarily known and finite. Thus, one aspect of a continuous space location model is that facilities may be sited anywhere. An example is depicted in Fig. 8.1 where all locations within the region are feasible. The implication is that there are an infinite number of potential facility locations to be considered, in contrast to an assumed finite set of potential locations in discrete approaches (see Chap. 2). Another aspect of a continuous space problem is that demand too is not limited to a finite set of locations. Rather, demand is assumed to be continuously distributed across geographic space, varying over a study area. One example of this is shown in Fig. 8.2, where the height of the surface reflects demand for service. As is evident in the figure, some level of demand can be observed everywhere and this varies across space. Another example is given in Fig. 8.3. The Census unit color reflects the amount of demand in each area. Within a unit the demand varies in some manner, but given limited knowledge, a small geographic area and relative homogeneity, it is often thought that demand is uniformly distributed in the unit. Thus, Fig. 8.3 reflects discontinuities in demand variability, but it remains varying across space. Irrespective of representation, the implication is that demand for service is everywhere, and in some cases can possibly be defined/described by a mathematical function. From a practical standpoint, the study area could be a demand region, collection of areal units, set of line segments, or any group of spatial objects. This clearly makes dealing with demand and its geographic variability particularly challenging, especially when compared to a discrete representation view based on points. This contrasting view can be observed in Fig. 8.4 as centroids for each Census unit are identified, and demand is assumed to occur at these precise points in traditional modeling approaches.

## References

- Aly AA, White JA (1978) Probabilistic formulation of the emergency service location problem. J Oper Res Soc 29:1167–1179CrossRefGoogle Scholar
- Alexandris G, Giannikos I (2010) A new model for maximal coverage exploiting GIS capabilities. Eur J Oper Res 202:328–338CrossRefGoogle Scholar
- Benveniste R (1982) A note on the set covering problem. J Oper Res Soc 33:261–265CrossRefGoogle Scholar
- Berman O, Verter V, Kara BY (2007) Designing emergency response networks for hazardous materials transportation. Comput Oper Res 34:1374–1388CrossRefGoogle Scholar
- Berman O, Wang J (2011) The minmax regret gradual covering location problem on a network with incomplete information of demand weights. Eur J Oper Res 208(3):233–238CrossRefGoogle Scholar
- Brady SD, Rosenthal RE (1980) Interactive computer graphical solutions of constrained minimax location problems. AIIE Trans 12(3):241–248CrossRefGoogle Scholar
- Brady SD, Rosenthal RE, Young D (1983) Interactive graphical minimax location of multiple facilities with general constraints. AIIE Trans 15(3):242–254Google Scholar
- Capar I, Kuby M, Leon VJ, Tsai YJ (2013) An arc cover–path-cover formulation and strategic analysis of alternative-fuel station locations. Eur J Oper Res 227:142–151CrossRefGoogle Scholar
- Church RL (1984) The planar maximal covering location problem. J Reg Sci 24:185–201CrossRefGoogle Scholar
- Church RL, Meadows ME (1979) Location modeling utilizing maximum service distance criteria. Geogr Anal 11:358–373CrossRefGoogle Scholar
- Church RL, Revelle C (1974) The maximal covering location problem. Pap Reg Sci 32:101–118CrossRefGoogle Scholar
- Cromley RG, Lin J, Merwin DA (2012) Evaluating representation and scale error in the maximal covering location problem using GIS and intelligent areal interpolation. Int J Geogr Inf Sci 26(3):495–517CrossRefGoogle Scholar
- Current J, Schilling D (1990) Analysis of errors due to demand data aggregation in the set covering and maximal covering location problems. Geogr Anal 22:116–126CrossRefGoogle Scholar
- Daskin MS, Haghani AE, Khanal M, Malandraki C (1989) Aggregation effects in maximal covering models. Ann Oper Res 18:115–140CrossRefGoogle Scholar
- Drezner Z (1986) The p-cover problem. Eur J Oper Res 26:312–313CrossRefGoogle Scholar
- Elzinga J, Hearn DW (1972) Geometrical solutions for some minimax location problems. Transp Sci 6(4):379–394CrossRefGoogle Scholar
- Erdemir ET, Batta R, Rogerson PA, Blatt A, Flanigan M (2010) Joint ground and air emergency medical services coverage models: a greedy heuristic solution approach. Eur J Oper Res 207:736–749CrossRefGoogle Scholar
- Erdemir ET, Batta R, Spielman S, Rogerson PA, Blatt A, Flanigan M (2008) Location coverage models with demand originating from nodes and paths: application to cellular network design. Eur J Oper Res 190:610–632CrossRefGoogle Scholar
- Francis RL, McGinnis LF, White JA (1992) Facility layout and location: an analytical approach. Pearson College Division, New YorkGoogle Scholar
- Goodchild MF, Lee J (1989) Coverage problems and visibility regions on topographic surfaces. Ann Oper Res 18(1):175–186CrossRefGoogle Scholar
- Grubesic TH, Murray AT, Matisziw TC (2013) Putting a price on politics as usual: rural air transport in the United States. Transp Policy 30:117–124CrossRefGoogle Scholar
- Hodgson MJ (1990) A flow-capturing location-allocation model. Geogr Anal 22:270–279CrossRefGoogle Scholar
- Kershner R (1939) The number of circles covering a set. Am J Math 61:665–671CrossRefGoogle Scholar
- Kim K, Murray AT (2008) Enhancing spatial representation in primary and secondary coverage location modeling. J Reg Sci 48:745–768CrossRefGoogle Scholar
- Kuby M, Lim S (2005) The flow-refueling location problem for alternative-fuel vehicles. Socio Econ Plan Sci 39:125–145CrossRefGoogle Scholar
- Matisziw TC, Murray AT (2009a) Siting a facility in continuous space to maximize coverage of continuously distributed demand. Socio Econ Plan Sci 43:131–139CrossRefGoogle Scholar
- Matisziw TC, Murray AT (2009b) Area coverage maximization in service facility siting. J Geogr Syst 11:175–189CrossRefGoogle Scholar
- Mehrez A, Stulman A (1982) The maximal covering location problem with facility placement on the entire plane. J Reg Sci 22:361–365CrossRefGoogle Scholar
- Mehrez A, Stulman A (1984) An extended continuous maximal covering location problem with facility placement. Comput Oper Res 11:19–23CrossRefGoogle Scholar
- Murray AT (2005) Geography in coverage modeling: exploiting spatial structure to address complementary partial service of areas. Ann Assoc Am Geogr 95:761–772CrossRefGoogle Scholar
- Murray AT, Matisziw TC, Wei H, Tong D (2008b) A GeoComputational heuristic for coverage maximization in service facility siting. Trans GIS 12:757–773CrossRefGoogle Scholar
- Murray AT, O’Kelly ME (2002) Assessing representation error in point-based coverage modeling. J Geogr Syst 4:171–191CrossRefGoogle Scholar
- Murray AT, O’Kelly ME, Church RL (2008a) Regional service coverage modeling. Comput Oper Res 35:339–355CrossRefGoogle Scholar
- Murray AT, Tong D (2007) Coverage optimization in continuous space facility siting. Int J Geogr Inf Sci 21:757–776CrossRefGoogle Scholar
- Murray AT, Tong D, Kim K (2010) Enhancing classic coverage location models. Int Reg Sci Rev 33:115–133CrossRefGoogle Scholar
- Murray AT, Wei R (2013) A computational approach for eliminating error in the solution of the location set covering problem. Eur J Oper Res 224:52–64CrossRefGoogle Scholar
- Plastria F (2002) Continuous covering location problems. In: Drezner Z, Hamacher H (eds) Facility location: applications and theory. Springer, Berlin, pp 37–79CrossRefGoogle Scholar
- Poetranto DR, Hamacher HW, Horn S, Schöbel A (2009) Stop location design in public transportation networks: covering and accessibility objectives. TOP 17:335–346CrossRefGoogle Scholar
- ReVelle CS, Toregas C, Falkson L (1976) Applications of the location set covering problem. Geogr Anal 8:65–76CrossRefGoogle Scholar
- Schobel A, Hamacher HW, Liebers A, Wagner D (2009) The continuous stop location problem in public transportation networks. Asia Pac J Oper Res 26:13–30CrossRefGoogle Scholar
- Spaulding BD, Cromley RG (2007) Integrating the maximum capture problem into a GIS framework. J Geogr Syst 9(3):267–288CrossRefGoogle Scholar
- Suzuki A, Okabe A (1995) Using voronoi diagrams. In: Drezner Z (ed) Facility location: a survey of applications and methods. Springer, New York, pp 103–118CrossRefGoogle Scholar
- Suzuki A, Drezner Z (1996) On the p-center location problem in an area. Locat Sci 4:69–82CrossRefGoogle Scholar
- Tong D (2012) Regional coverage maximization: a new model to account implicitly for complementary coverage. Geogr Anal 44:1–14CrossRefGoogle Scholar
- Tong D, Church RL (2012) Aggregation in continuous space coverage modeling. Int J Geogr Inf Sci 26:795–816CrossRefGoogle Scholar
- Tong D, Murray AT (2009) Maximizing coverage of spatial demand for service. Pap Reg Sci 88:85–97CrossRefGoogle Scholar
- Toregas C, Swain R, ReVelle C, Bergman L (1971) The location of emergency service facilities. Oper Res 19:1363–1373CrossRefGoogle Scholar
- Toussaint GT (1983) Computing largest empty circles with location constraints. Int J Comp Inf Sci 12(5):347–358CrossRefGoogle Scholar
- Watson-Gandy CDT (1982) Heuristic procedures for the m-partial cover problem on a plane. Eur J Oper Res 11:149–157CrossRefGoogle Scholar
- Wei H, Murray AT, Xiao N (2006) Solving the continuous space p-center problem: Planning application issues. IMA J Manag Math 17:413–425CrossRefGoogle Scholar
- Wei R, Murray AT (2014) Evaluating polygon overlay to support spatial optimization coverage modeling. Geogr Anal 46(3):209–229CrossRefGoogle Scholar
- Wei R, Murray AT (2015) Continuous space maximal coverage: insights, advances and challenges. Comput Oper Res 62:325–336CrossRefGoogle Scholar
- Wei R, Murray AT (2016) A parallel algorithm for coverage optimization on multi-core architectures. Int J Geogr Inf Sci 30:432–450CrossRefGoogle Scholar
- Yao J, Murray AT (2013) Continuous surface representation and approximation: spatial analytical implications. Int J Geogr Inf Sci 27:883–897CrossRefGoogle Scholar
- Yin P, Mu L (2012) Modular capacitated maximal covering location problem for the optimal siting of emergency vehicles. Appl Geogr 34:247–254CrossRefGoogle Scholar