Location Covering Models pp 229-253 | Cite as

# Coverage of Network-Based Structures: Paths, Tours and Trees

## Abstract

Generally speaking, location analysis and modeling is often characterized as a network based problem. Classic location books have reinforced this characterization, such as that by Handler and Mirchandani (1979) titled *Location on Networks* and that of Daskin (1995) titled *Network and Discrete Location*, as well as many other books and papers (the seminal work of Hakimi 1965, likely influenced this too). In fact, location modeling tools in ArcGIS, as an example, are part of the Network Analyst toolbox and strictly require an underlying network in order to structure and solve associated location (coverage) models. However, as reflected in the much of this book, there is no inherent need or assumption that location decision making be restricted to a network, even if attribute data is derived from a corresponding network. This chapter deviates from much of the book in this respect, and assumes the underlying representation of an analysis region, including demand and potential facility locations, is based on a network. Most network location problems involve the positioning of one or more facilities at nodes or along arcs in order to optimize a level of service or access. Often, however, site selection is restricted to the nodes of the network. In the early 1980s Morgan and Slater (1980), Current (1981), and Slater (1982) broke new ground within location science when they independently suggested that facilities may be represented by some type of network structure. Examples include a path, a tree, a tour, or even the network itself. Slater (1982) suggested that facilities “… can be of an extended nature, rather than occupying a single point of a network.” They suggested, for example, that facilities may be modeled as network paths representing railroad lines, pipelines, or transit routes. Slater (1982) proposed that four classes of locational problems should be considered within the context of network location: point-serves-point, point-serves-structure, structure-serves-point, and structure-serves-structure (see also Slater 1981, 1983). In this chapter we address the problem of designing a structure to serve points. One of the first examples in the literature is that of Morgan and Slater (1980) who defined the “core” of a tree network as the shortest path connecting two endpoints of the tree, minimizing the sum of distances to all other nodes of the tree.

## References

- Arkin EM, Hassin R (1994) Approximation algorithms for the geometric covering salesman problem. Discret Appl Math 55:197–218CrossRefGoogle Scholar
- Arkin EM, Haldorsson NM, Hassin R (1993) Approximating tree and tour covers of a graph. Inf Process Lett 47:275–282CrossRefGoogle Scholar
- Boffey B, Narula SC (1998) Models for multi-path covering-routing problems. Ann Oper Res 82:331–342CrossRefGoogle Scholar
- Church R, Current J (1993) Maximal covering tree problems. Nav Res Logist 40:129–142CrossRefGoogle Scholar
- Church R, Noronha, V, Lei T, Corrigan W, Burbidge S, Marston J (2005) Spatial and temporal utility modeling to increase transit ridership: final report to Caltrans. VITAL, University of California, Santa Barbara, CAGoogle Scholar
- Cohon JL (1978) Multiobjective programming and planning. Academic, New YorkGoogle Scholar
- Current JR (1981) Multiobjective design of transportation networks. PhD dissertation, The Johns Hopkins University, Baltimore, MDGoogle Scholar
- Current JR, Schilling DA (1989) The covering salesman problem. Transp Sci 23:208–213CrossRefGoogle Scholar
- Current JR, Schilling DA (1994) The median tour and maximal covering tour problems: formulations and heuristics. Eur J Oper Res 73:114–126CrossRefGoogle Scholar
- Current J, ReVelle C, Cohon J (1984) The shortest covering path problem: an application of locational constraints to network design. J Reg Sci 24:161–184CrossRefGoogle Scholar
- Current JR, ReVelle CS, Cohon JL (1985) The maximum covering/shortest path problem: a multiobjective network design and routing formulation. Eur J Oper Res 21:189–199CrossRefGoogle Scholar
- Current J, Pirkul H, Rolland E (1994) Efficient algorithms for solving the shortest covering path problem. Transp Sci 28(4):317–327CrossRefGoogle Scholar
- Curtin KM, Biba S (2011) The transit route arc-node service maximization problem. Eur J Oper Res 208:46–56CrossRefGoogle Scholar
- Dantzig G, Fulkerson R, Johnson S (1954) Solution of a large-scale traveling-salesman problem. J Oper Res Soc Am 2:393–410Google Scholar
- Daskin MS (1995) Network and discrete location: models, algorithms, and applications. Wiley, New YorkCrossRefGoogle Scholar
- Finke GA, Claus A, Gunn E (1983) A two commodity network flow approach to the travelling salesman problem. In Proceedings of the 14th South Eastern conference on combinatorics, graph theory and computing, Atlantic University, FLGoogle Scholar
- Fischetti M, Toth P (1988) An additive approach for the optimal solution the prize-collecting salesman problem. In: Golden BL, Assad AA (eds) Vehicle routing: methods and studies. North Holland, AmsterdamGoogle Scholar
- Gavish B, Graves SC (1978) The travelling salesman problem and related problems. Working Paper OR-078-78. Operations Research Center, MIT Press, Cambridge, MAGoogle Scholar
- Gendreau M, Laporte G, Semet F (1997) The covering tour problem. Oper Res 45:568–576CrossRefGoogle Scholar
- Golden B, Naji-Azimi Z, Raghavan S, Salari M, Toth P (2012) The generalized covering salesman problem. INFORMS J Comput 24:534–553CrossRefGoogle Scholar
- Hakimi SL (1965) Optimum distribution of switching centers in a communication network and some related graph theoretic problems. Oper Res 13(3):462–475CrossRefGoogle Scholar
- Handler GY, Mirchandani PB (1979) Location on networks: theory and algorithms. MIT Press, Cambridge, MAGoogle Scholar
- Hutson VA, ReVelle C (1989) Maximal direct covering tree problems. Transp Sci 23:288–289CrossRefGoogle Scholar
- Hutson VA, ReVelle C (1993) Indirect covering tree problems on spanning tree networks. Eur J Oper Res 65:20–32CrossRefGoogle Scholar
- Kruskal JB Jr (1956) On the shortest spanning subtree of a graph and the travelling salesman problem. Proc Am Math Soc 7:48–50CrossRefGoogle Scholar
- Laporte G, Martello S (1990) The selective travelling salesman problem. Discret Appl Math 26:193–207CrossRefGoogle Scholar
- Matisziw TC, Murray AT, Kim C (2006) Strategic route extension in transit networks. Eur J Oper Res 171:661–673CrossRefGoogle Scholar
- Medrano FA, Church RL (2014) Corridor location for infrastructure development: a fast bi-objective shortest path method for approximating the Pareto frontier. Int Reg Sci Rev 37(2):129–148CrossRefGoogle Scholar
- Medrano FA, Church RL (2015) A parallel computing framework for finding the supported solutions to a biobjective network optimization problem. J Multicrit Decis Anal 22(5–6):244–259CrossRefGoogle Scholar
- Miller CE, Tucker AW, Zemlin RA (1960) Integer programming formulation of travelling salesman problems. J ACM 3:326–329CrossRefGoogle Scholar
- Morgan CA, Slater PJ (1980) A linear algorithm for a core of a tree. J Algorithm 1:247–258CrossRefGoogle Scholar
- Murawski L, Church RL (2009) Improving accessibility to rural health services: the maximal covering network improvement problem. Socio Econ Plan Sci 43:102–110CrossRefGoogle Scholar
- Murray AT, Wu X (2003) Accessibility tradeoffs in public transit planning. J Geogr Syst 5:93–107CrossRefGoogle Scholar
- Niblett TJ (2016) On the development of a new class of covering-path models. PhD dissertation, UCSB, Santa Barbara, CAGoogle Scholar
- Niblett TJ, Church RL (2016) The shortest covering path problem a new perspective and model. Int Reg Sci Rev 39:131–151CrossRefGoogle Scholar
- Oppong JR (1996) Accommodating the rainy season in third world location-allocation applications. Socio Econ Plan Sci 30:121–137CrossRefGoogle Scholar
- Orman AJ, Williams HP (2004) A survey of different integer programming formulations of the travelling salesman problem. Working Paper No. LSEOR 04.67. London School of Economics and Political Science, LondonGoogle Scholar
- ReVelle CS, Laporte G (1993) New directions in plant location. Stud Locational Anal 5:32–58Google Scholar
- Slater PJ (1981) On locating a facility to service areas within a network. Oper Res 29:523–531CrossRefGoogle Scholar
- Slater PJ (1982) Locating central paths in a graph. Transp Sci 16:1–18CrossRefGoogle Scholar
- Slater PJ (1983) Some definitions of central structures. Lect Notes Math 1073:169–178CrossRefGoogle Scholar
- Vajda S (1961) Mathematical programming. Addison-Wesley, LondonGoogle Scholar
- Wu C, Murray AT (2005) Optimizing public transit quality and system access: the multiple-route, maximal covering/shortest-path problem. Environ Plann B Plann Des 32:163–178CrossRefGoogle Scholar