Abstract
The p-dispersion facility location problem first emerged in the 1970s as a byproduct of covering and center location problems and the dual relationship between the continuous 1-center solution on a tree network and the two most distant points on the tree. Building on this insight, Shier (Transpor Sci 11(3):243–252, 1977) was the first to recognize the dispersion problem as worthy of study in its own right, with useful real-world applications such as locating oil storage tanks and retail franchises. In the 1980s, two milestones were achieved by researchers who were notably motivated by trying to solve other problems. First, Chandrasekaran and Daughety (Math Oper Res 6(1):50–57, 1981) introduced the “dispersion” nomenclature for the problem of maximizing the diversity of location of points on a tree network, motivated by the need to diversify simulation experiments for fitting a railroad cost surface (Daughety and Turnquist (Oper Res 29(3):485–500, 1981)). Second, Kuby (Geograph Analy 19(4):315–329, 1987) developed the first mixed-integer programming formulation of the discrete dispersion problem in order to generate a regularly spaced solution for his optimization model of central place theory, a geographical theory of retail and services in an urban hierarchy. Finally, Erkut and Neuman (Eur J Oper Res 40(3):275–291, 1989) and Erkut (Eur J Oper Res 46(1):48–60, 1990) popularized dispersion for locating “mutually obnoxious” types of undesirable facilities and developed faster solution methods for the discrete diversity problem.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
I would like to correct my youthful self at the bottom of p. 321 where I suggested that the maxisum dispersion problem is related to the maximin p-dispersion problem in the same way the minisum p-median problem is related to the minimax p-center problem. I was probably looking at both the median and maxisum-dispersion objective functions and seeing double summations. This analogy does not hold up, however, because the median objective counts only the allocation to the nearest facility, while the maxisum dispersion model counts all of the separation distances from all facilities to all other facilities. Ignoring the number of summations in the objective function, the p-median problem is a closer analogue to the p-defense problem, in that both objectives sum the distances to only the nearest facility.
References
Bahrenberg, G.: Providing an adequate social infrastructure in rural areas: an application of a maximal supply dispersion model to elementary school planning in Rotenburg/Wümme (FRG). Environ. Plann. A 13(12), 1515–1527 (1981)
Beaumont J.R.: Location-allocation models and central place theory. In: Ghosh, A., Rushton, G. (eds.) Spatial Analysis and Location-Allocation Models. Van Nostrand Reinhold, New York (1987)
Chandrasekaran, R., Daughety, A.: Location on tree networks: p-centre and n-dispersion problems. Math. Oper. Res. 6(1), 50–57 (1981)
Chandrasekaran, R., Tamir, A.: Polynomially bounded algorithms for locating p-centers on a tree. Math. Program. 22(1), 304–315 (1982)
Chaudhry, S., Moon, I.: Analytical models for locating obnoxious facilities. In: Chatterji M. (ed.) Material Disposal: Siting and, Management, pp. 275–282. Gower, Brookfield, VT (1987)
Christaller, W.: Central Places in Southern Germany. Baskin, C. W. Translator. Prentice-Hall, Englewood Cliffs (1966)
Church, R.L., Garfinkel, R.S.: Locating an obnoxious facility on a network. Transport. Sci. 12(2), 107–118 (1978)
Curtin K.M.: Models for multiple-type discrete dispersion. PhD Thesis, University of California, Santa Barbara (2002)
Curtin, K.M., Church, R.L.: A family of location models for multiple-type discrete dispersion. Geograph. Analy. 38(3), 248–270 (2006)
Curtin, K.M., Church, R.L.: Optimal dispersion and central places. J. Geograph. Syst. 9(2), 167–187 (2007)
Daughety, A.F., Turnquist, M.A.: Budget constrained optimization of simulation models via estimation of their response surfaces. Oper. Res. 29(3), 485–500 (1981)
Dearing, P.M., Francis, R.L.: A minimax location problem on a network. Transport. Sci. 8(4), 333–343 (1974)
Erkut, E.: The discrete p-dispersion problem. Eur. J. Oper. Res. 46(1), 48–60 (1990)
Erkut, E., Neuman, S.: Analytical models for locating undesirable facilities. Eur. J. Oper. Res. 40(3), 275–291 (1989)
Fischer, K.: Central places: the theories of von Thünen, Christaller, and Lösch. In: Eiselt, H.A., Marianov, V. (eds.) Foundations of Location Analysis, pp. 471–505. Springer, New York (2011)
Goldman, A.: Minimax location of a facility in a network. Transport. Sci. 6(4), 407–418 (1972)
Goldman, A.: Optimal facility-location. J. Res. Natl Instit. Stand. Technol. 111(2), 97–101 (2006)
Goldman, A., Dearing, P.: Concepts of optimal location for partially noxious facilities. Bull. Oper. Res. Soc. Amer. 23(1), B31–85 (1975)
Hakimi, S.L.: Optimum locations of switching centers and the absolute centers and medians of a graph. Oper. Res. 12(3), 450–459 (1964)
Hakimi, S.L.: Optimum distribution of switching centers in a communication network and some related graph theoretic problems. Oper. Res. 13(3), 462–475 (1965)
Handler, G.Y.: Minimax location of a facility in an undirected tree graph. Transport. Sci. 7(3), 287–293 (1973)
Handler, G.Y.: Minimax network location: theory and algorithms. PhD Thesis, Cambridge, Mass.: Massachusetts Institute of Technology (1974)
Hsu, W.T., Zou, X.: Central place theory and the power law for cities. In: The Mathematics of Urban Morphology, pp 55–75. Springer, Berlin (2019)
Kohsaka, H.: A central-place model as a two-level location-allocation system. Environ. Plann. A Econ. Space 15(1), 5–14 (1983)
Kuby, M.: A location-allocation model of lösch’s central place theory: testing on a uniform lattice network. Geograph. Analy. 21(4), 316–337 (1989)
Kuby, M.J.: Programming models for facility dispersion: the p-dispersion and maxisum dispersion problems. Geograph. Analy. 19(4), 315–329 (1987)
Kulshrestha, D.: Duality with distant point and median of a graph. In: Third International Symposium on Locational Decisions, Boston, MA, pp. 7–12 (1984)
Lösch, A.: Economics of Location. Yale University Press, New Haven, CT (1954)
Martí, R., Martínez-Gavara, A., Pérez-Peló, S., Sánchez-Oro, J.: A review on discrete diversity and dispersion maximization from an OR perspective. Eur. J. Oper. Res. 299(3), 795–813 (2022)
Meir, A., Moon, J.: Relations between packing and covering numbers of a tree. Pacific J. Math. 61(1), 225–233 (1975)
Minieka, E.: The m-center problem. Siam Rev. 12(1), 138–139 (1970)
Moon, I.D., Chaudhry, S.S.: An analysis of network location problems with distance constraints. Manag. Sci. 30(3), 290–307 (1984)
Narula, S.C.: Hierarchical location-allocation problems: a classification scheme. Eur. J. Oper. Res. 15(1), 93–99 (1984)
Puryear, D.: A programming model of central place theory. J. Reg. Sci. 15(3), 307–316 (1975)
Ratick, S.J., White, A.L.: A risk-sharing model for locating noxious facilities. Environ. Plann. B Plann. Design 15(2), 165–179 (1988)
ReVelle, C.: Facility siting and integer-friendly programming. Eur. J. Oper. Res. 65(2), 147–158 (1993)
ReVelle, C.S., Swain, R.W.: Central facilities location. Geograph. Analy. 2(1), 30–42 (1970)
Shier, D.R.: A min-max theorem for p-center problems on a tree. Transport. Sci. 11(3), 243–252 (1977)
Storbeck, J.E.: The spatial structuring of central places. Geograph. Analy. 20(2), 93–110 (1988)
Tansel, B.C., Francis, R.L., Lowe, T.J., Chen, M.L.: Duality and distance constraints for the nonlinear p-center problem and covering problem on a tree network. Oper. Res. 30(4), 725–744 (1982)
Tansel, B.C., Francis, R.L., Lowe, T.J.: State of the art—location on networks: a survey. Part I: the p-center and p-median problems. Manag. Sci. 29(4), 482–497 (1983)
Teitz, M.B.: Toward a theory of urban public facility location. Papers Reg. Sci. 21(1), 35–51 (1968)
von Thünen, J.H.: von Thünen’s ‘Isolated State’. An English Edition of ‘Der Isolierte Staat’ Edited with an Introduction by Peter Hall. Pergamon Press, New York (1966)
Weber, A.: Alfred Weber’s Theory of the Location of Industries. University of Chicago Press, Chicago (1929)
Witzgall C.: Mathematical methods of site selection for Electronic Message Systems (EMS). Technical Report NBSIR 75-737. National Bureau of Standards, Washington, D.C. (1975)
Witzgall, C., Goldman, A.: Optimal location of a central facility: Mathematical models and concepts. Technical Report NBS Technical Report No. 8388. National Bureau of Standards, Washington, D.C. (1964)
Acknowledgements
In the course of writing this chapter, I had the unique privilege of chatting either by Zoom or email with Douglas Shier, Ramaswamy Chandrasekaran, and Andrew Daughety about their foundational work on the p-dispersion problem. I thank them for sharing their recollections and insights, which I hope I have captured accurately here. I would also like to thank my old friend Susan Neuman for detailed copy editing of my first draft. Finally, I am grateful to co-editors Rafa Martí, for initially inviting me to contribute a chapter, and Anna Martínez-Gavara, for her very generous and much-needed assistance with formatting in Latex.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2023 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this chapter
Cite this chapter
Kuby, M. (2023). The Origins of Discrete Diversity. In: Martí, R., Martínez-Gavara, A. (eds) Discrete Diversity and Dispersion Maximization. Springer Optimization and Its Applications, vol 204. Springer, Cham. https://doi.org/10.1007/978-3-031-38310-6_2
Download citation
DOI: https://doi.org/10.1007/978-3-031-38310-6_2
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-38309-0
Online ISBN: 978-3-031-38310-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)