Skip to main content
Log in

Aggregation error for location models: survey and analysis

  • Published:
Annals of Operations Research Aims and scope Submit manuscript

Abstract

Location problems occurring in urban or regional settings may involve many tens of thousands of “demand points,” usually individual private residences. In modeling such problems it is common to aggregate demand points to obtain tractable models. We survey aggregation approaches to a large class of location models, consider and compare various aggregation error measures, identify some effective (and ineffective) aggregation error measures, and discuss some open research areas.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  • Agarwal, P. K., & Procopiuc, C. M. (2002). Exact and approximation algorithms for clustering. Algorithmica, 33, 201–226.

    Article  Google Scholar 

  • Agarwal, P. K., & Varadarajan, K. R. (1999). Approximation algorithms for bipartite and nonbipartite matchings in the plane, In 10th ACM-SIAM symposium on discrete algorithms (SODA), pp. 805–814.

  • Agarwal, P. K., Efrat, A., & Sharir, M. (1999). Vertical decomposition of shallow levels in 3-dimensional arrangements and its applications. SIAM Journal on Computing, 29, 912–953.

    Article  Google Scholar 

  • Agarwal, P. K., Procopiuc, C. M., & Varadarajan, K. R. (2002). Approximation algorithms for k-line center, In Proceedings 10-th annual European symposium on algorithms (ESA 2002), pp. 54–63.

  • Agarwal, P. K., Har-Peled, S., & Varadarajan, K. R. (2005). Geometric approximation via coresets. In J. E. Goodman, J. Pach, & E. Welzl (Eds.), Combinatorial and computational geometry. New York: Cambridge University Press.

    Google Scholar 

  • Ahuja, R. K., Magnanti, T. L., & Orlin, J. B. (1993). Network flows: theory, algorithms, and applications. Englewood Cliffs: Prentice–Hall (Exercise 12.23 on page 505 describes an O(n2.5 log n) algorithm for the bottleneck assignment problem.)

    Google Scholar 

  • Andersson, G., Francis, R. L., Normark, T., & Rayco, M. B. (1998). Aggregation method experimentation for large-scale network location problems. Location Science, 6, 25–39.

    Article  Google Scholar 

  • Bach, L. (1981). The problem of aggregation and distance for analysis of accessibility and access opportunity in location-allocation models. Environment and Planning A, 13, 955–978.

    Article  Google Scholar 

  • Ballou, R. H. (1994). Measuring transport costing error in customer aggregation for facility location. Transportation Journal, 33, 49–54.

    Google Scholar 

  • Bender, T., Hennes, H., Kalcsics, J., Melo, T., & Nickel, S. (2001). Location software and interface with GIS and supply chain management. In Z. Drezner & H. Hamacher (Eds.), Facility location: applications and theory. Berlin: Springer.

    Google Scholar 

  • Bowerman, R. L., Calamai, P. H., & Hall, B. (1999). The demand partitioning method for reducing aggregation errors in p-median problems. Computers and Operations Research, 26, 1097–1111.

    Article  Google Scholar 

  • Carrizosa, E., Hamacher, H. W., Nickel, S., & Klein, R. (2000). Solving nonconvex planar location problems by finite dominating sets. Journal of Global Optimization, 18, 195–210.

    Article  Google Scholar 

  • Casillas, P. A. (1987). Data aggregation and the p-median problem in continuous space. In A. Ghosh & G. Rushton (Eds.), Spatial analysis and location-allocation models (pp. 227–244). New York: Van Nostrand Reinhold Publishers.

    Google Scholar 

  • Chelst, K. R., Schultz, J. P., & Sanghvi, N. (1988). Issues and decision aids for designing branch networks. Journal of Retail Banking X, 2, 5–17.

    Google Scholar 

  • Cooper, L. (1967). Solutions of generalized location equilibrium models. Journal of Regional Science, 1, 334–336.

    Google Scholar 

  • Cornuejols, G., Fisher, M. L., & Nemhauser, G. L. (1977). Location of bank accounts to optimize float: an analytical study of exact and approximate algorithms. Management Science, 23, 789–810.

    Article  Google Scholar 

  • Current, J. R., & Schilling, D. A. (1987). Elimination of source A and B errors in p-median problems. Geographical Analysis, 19, 95–110.

    Google Scholar 

  • Current, J. R., & Schilling, D. A. (1990). Analysis of errors due to demand data aggregation in the set covering and maximal covering location problems. Geographical Analysis, 22, 116–126.

    Google Scholar 

  • Daskin, M. S. (1995). Network and discrete location: models, algorithms, and applications. New York: Wiley.

    Google Scholar 

  • Daskin, M. S., Haghani, A. E., Khanal, M., & Malandraki, C. (1989). Aggregation effects in maximum covering models. Annals of Operations Research, 18, 115–139.

    Article  Google Scholar 

  • Dekle, J., Lavieri, M., Martin, E., Emir-Farinas, H., & Francis, R. L. (2005). A Florida county locates disaster recovery centers. Interfaces, 35(2), 133–139.

    Article  Google Scholar 

  • Domich, P. D., Hoffman, K. L., Jackson, R. H. F., & McClain, M. A. (1991). Locating tax facilities: a graphics-based microcomputer optimization model. Management Science, 37, 960–979.

    Article  Google Scholar 

  • Drezner, Z. (Ed.). (1995a). Facility location: a survey of applications and methods. Berlin: Springer.

    Google Scholar 

  • Drezner, Z. (1995b). Replacing discrete demand with continuous demand. In Z. Drezner (Ed.), Facility location: a survey of applications and methods. Berlin: Springer.

    Google Scholar 

  • Drezner, T., & Drezner, Z. (1997). Replacing discrete demand with continuous demand in a competitive facility location problem. Naval Research Logistics, 44, 81–95.

    Article  Google Scholar 

  • Drezner, Z., & Hamacher, H. W. (Eds.). (2002). Facility location: theory and algorithms. Berlin: Springer.

    Google Scholar 

  • Efrat, A., Itai, A., & Katz, M. J. (2001). Geometry helps in bottleneck matching and related problems. Algorithmica, 31, 1–28.

    Article  Google Scholar 

  • Emir-Farinas, H. (2002). Aggregation of demand points for the planar covering location problem. Ph. D. Dissertation, Department of Industrial and Systems Engineering, University of Florida, Gainesville, FL.

  • Emir-Farinas, H., & Francis, R. L. (2005). Demand point aggregation for planar covering location models. Annals of Operations Research, 136, 175–192.

    Article  Google Scholar 

  • Erkut, E., & Bozkaya, B. (1999). Analysis of aggregation errors for the p-median problem. Computers and Operations Research, 26, 1075–1096.

    Article  Google Scholar 

  • Erkut, E., & Neuman, S. (1989). Analytical models for locating undesirable facilities. European Journal of Operational Research, 40, 275–291.

    Article  Google Scholar 

  • Ernst, A., Hamacher, H. W., Jiang, H. W., Krishnamorthy, M., & Woeginger, G. (2002a). Uncapacitated single and multiple allocation p-hub center problems. Report CSIRO, Melbourne, Australia.

  • Ernst, A., Hamacher, H. W., Jiang, H. W., Krishnamorthy, M., & Woeginger, G. (2002b). Heuristic algorithms for the uncapacitated hub center single allocation problem. Report CSIRO, Melbourne, Australia.

  • Fortney, J., Rost, K., & Warren, J. (2000). Comparing alternative methods of measuring geographic access to health services. Health Services and Outcomes Research Methodology, 1, 173–184.

    Article  Google Scholar 

  • Fotheringham, A. S., Densham, P., & Curtis, A. (1995). The zone definition problem in location-allocation modeling. Geographical Analysis, 27, 60–77.

    Google Scholar 

  • Francis, R. L., & Lowe, T. J. (1992). On worst-case aggregation analysis for network location problems. Annals of Operations Research, 40, 229–246.

    Article  Google Scholar 

  • Francis, R. L., & Rayco, M. B. (1996). Asymptotically optimal aggregation for some unweighted p-center problems with rectilinear distances. Studies in Locational Analysis, 10, 25–36.

    Google Scholar 

  • Francis, R. L., & White, J. A. (1974). Facility layout and location: an analytical approach. Englewood Cliffs: Prentice–Hall (Homework problem 7.25, p. 324).

    Google Scholar 

  • Francis, R. L., McGinnis, L. F., & White, J. A. (1992). Facility layout and location: an analytical approach (2nd ed.). Englewood Cliffs: Prentice–Hall.

    Google Scholar 

  • Francis, R. L., Lowe, T. J., & Rayco, M. B. (1996). Row-column aggregation for rectilinear p-median problems. Transportation Science, 30, 160–174.

    Article  Google Scholar 

  • Francis, R. L., Lowe, T. J., Rushton, G., & Rayco, M. B. (1999). A synthesis of aggregation methods for multi-facility location problems: strategies for containing error. Geographical Analysis, 31, 67–87.

    Google Scholar 

  • Francis, R. L., Lowe, T. J., & Tamir, A. (2000). On aggregation error bounds for a class of location models. Operations Research, 48, 294–307.

    Article  Google Scholar 

  • Francis, R. L., Lowe, T. J., & Tamir, A. (2002a). Worst-case incremental analysis for a class of p-facility location problems. Networks, 39, 139–143.

    Article  Google Scholar 

  • Francis, R. L., Lowe, T. J., & Tamir, A. (2002b). Demand point aggregation of location models. In Z. Drezner & H. Hamacher (Eds.), Facility location: applications and theory. Berlin: Springer.

    Google Scholar 

  • Francis, R. L., Lowe, T. J., Rayco, M. B., & Tamir, A. (2003). Exploiting self-canceling demand point aggregation error for some planar rectilinear median problems. Naval Research Logistics, 50, 614–637.

    Article  Google Scholar 

  • Francis, R. L., Lowe, T. J., & Tamir, A. (2004a). Demand point aggregation analysis for a class of constrained location models: a penalty function approach. IIE Transactions, 36, 601–609.

    Article  Google Scholar 

  • Francis, R. L., Lowe, T. J., Tamir, A., & Emir-Farinas, H. (2004b). Aggregation decomposition and aggregation guidelines for a class of minimax and covering location models. Geographical Analysis, 36, 332–349.

    Article  Google Scholar 

  • Francis, R. L., Lowe, T. J., Tamir, A., & Emir-Farinas, H. (2004c). A framework for demand point and solution space aggregation analysis for location models. European Journal of Operational Research, 159, 574–585.

    Article  Google Scholar 

  • Frieze, A. M. (1980). Probabilistic analysis of some Euclidean clustering problems. Discrete Applied Mathematics, 10, 295–309.

    Article  Google Scholar 

  • Gavriliouk, E. O. (2003). Aggregation in hub location models. M. Sc. Thesis, Department of Mathematics, Clemson University, Clemson, SC.

  • Geoffrion, A. (1977). Objective function approximations in mathematical programming. Mathematical Programming, 13, 23–37.

    Article  Google Scholar 

  • Goldberg, R. (1976). Methods of real analysis (2nd ed.). New York: Wiley.

    Google Scholar 

  • Goodchild, M. F. (1979). The aggregation problem in location-allocation. Geographical Analysis, 11, 240–255.

    Google Scholar 

  • Hakimi, S. L. (1965). Optimum location of switching centers and the absolute centers and medians of a graph. Operations Research, 12, 450–459.

    Article  Google Scholar 

  • Hakimi, S. L., Labbe’, M., & Schmeichel, E. (1992). The Voronoi partitioning of a network and its implications in network location theory. ORSA Journal on Computing, 4, 412–417.

    Google Scholar 

  • Hale, T., & Hale, L. (2000). An aggregation technique for location problems with one-dimensional forbidden regions. International Journal of Industrial Engineering, 7, 133–139.

    Google Scholar 

  • Hale, T., & Moberg, C. (2003). Location science research: a review. Annals of Operations Research, 123, 21–35.

    Article  Google Scholar 

  • Hale, T., Wysk, R., & Smith, D. (2000). A gross aggregation technique for the p-median location problem. International Journal of Operations and Quantitative Management, 6, 129–135.

    Google Scholar 

  • Handler, G. Y., & Mirchandani, P. B. (1979). Location on networks: theory and algorithms. Cambridge: MIT Press.

    Google Scholar 

  • Hansen, P., Jaumard, B., & Tuy, H. (1995). Global optimization in location. In Z. Drezner (Ed.), Facility location: a survey of applications and methods. Berlin: Springer.

    Google Scholar 

  • Har-Peled, S. (2004a). Clustering motion. Discrete Computational Geometry, 31, 545–565.

    Article  Google Scholar 

  • Har-Peled, S. (2004b). No coreset, no cry. In Proceedings 24-th conf. found. soft. tech. theoretical computer science, pp. 324–335.

  • Har-Peled, S., & Kushal, A. (2007). Smaller coresets for k-median and k-means clustering. Discrete Computational Geometry, 37, 3–19.

    Article  Google Scholar 

  • Har-Peled, S., & Mazumdar, S. (2004). Coresets for k-means and k-median clustering and their applications. In Proceedings 36-th annual ACM symposium on theory of computing, pp. 291–300.

  • Hassin, R., & Tamir, A. (1991). Improved complexity bounds for location problems on the real line. O.R. Letters, 10, 395–402.

    Article  Google Scholar 

  • Hillsman, E. L., & Rhoda, R. (1978). Errors in measuring distances from populations to service centers. Annals of Regional Science, 12, 74–88.

    Article  Google Scholar 

  • Hodgson, M. J. (2002). Data surrogation error in p-median models. Annals of Operations Research, 110, 153–165.

    Article  Google Scholar 

  • Hodgson, M. J., & Hewko, J. (2003). Aggregation and surrogation error in the p-median model. Annals of Operations Research, 123, 53–66.

    Article  Google Scholar 

  • Hodgson, M. J., & Neuman, S. (1993). A GIS approach to eliminating source C aggregation error in p-median models. Location Science, 1, 155–170.

    Google Scholar 

  • Hodgson, M. J., Shmulevitz, F., & Körkel, M. (1997). Aggregation error effects on the discrete-space p-median model: the case of Edmonton, Canada. The Canadian Geographer, 41, 415–428.

    Article  Google Scholar 

  • Hooker, J. N., Garfinkel, R. S., & Chen, C. K. (1991). Finite dominating sets for network location problems. Operations Research, 39, 100–118.

    Article  Google Scholar 

  • Kariv, O., & Hakimi, S. L. (1979). An algorithmic approach to network location problems: part 1, the p-centers; part 2, the p-medians. SIAM Journal of Applied Mathematics, 37, 513–560.

    Article  Google Scholar 

  • Kolen, A., & Tamir, A. (1990). Covering problems. In P. B. Mirchandani & R. L. Francis (Eds.), Discrete location theory (pp. 263–304). New York: Wiley–Interscience.

    Google Scholar 

  • Love, R., Morris, J., & Wesolowsky, G. (1988). Facility location: models and methods. Amsterdam: North-Holland.

    Google Scholar 

  • Marchetti-Spaccamela, A., & Talamo, M. (1983). Probabilistic analysis of two Euclidean location problems. R.A.I.R.O. Informatique Theorique/Theoretical Informatics, 17, 387–395.

    Google Scholar 

  • Megiddo, N., & Supowit, K. J. (1984). On the complexity of some common geometric location problems. SIAM Journal on Computing, 13, 182–196.

    Article  Google Scholar 

  • Mirchandani, P. B., & Francis, R. L. (Eds.). (1990). Discrete location theory. New York: Wiley–Interscience.

    Google Scholar 

  • Mirchandani, P. B., & Reilly, J. M. (1986). Spatial nodes in discrete location problems. Annals of Operations Research, 10, 329–350.

    Google Scholar 

  • Murray, A. T., & Gottsegen, J. M. (1997). The influence of data aggregation on the stability of p-median location model solutions. Geographical Analysis, 29, 200–213.

    Google Scholar 

  • Nickel, S., & Puerto, J. (1999). A unified approach to network location problems. Networks, 34, 283–290.

    Article  Google Scholar 

  • Nickel, S., & Puerto, J. (2005). Location theory: a unified approach. Berlin: Springer.

    Google Scholar 

  • Ohsawa, Y., Koshizuka, T., & Kurita, O. (1991). Errors caused by rounded data in two simple facility location problems. Geographical Analysis, 23, 56–73.

    Google Scholar 

  • Pardalos, P. M., & Resende, M. (Eds.). (2002). Handbook of applied optimization. Oxford: Oxford University Press.

    Google Scholar 

  • Plastria, F. (1992). GBSSS: the generalized big square small square method for planar single-facility location. European Journal of Operational Research, 62, 163–174.

    Article  Google Scholar 

  • Plastria, F. (2000). New error bounds in continuous minisum location for aggregation at the gravity centre. Studies in Locational Analysis, 14, 101–119.

    Google Scholar 

  • Plastria, F. (2001). On the choice of aggregation points for continuous p-median problems: a case for the gravity center. TOP, 9, 217–242.

    Article  Google Scholar 

  • Puerto, J., & Fernandez, F. R. (2000). Geometrical properties of the symmetrical single facility location problem. Journal of Nonlinear Convex Analysis, 1, 321–342.

    Google Scholar 

  • Rayco, M. B. (1996). Algorithmic approaches to demand point aggregation for location models. Ph. D. Dissertation, Department of Industrial and Systems Engineering, University of Florida, Gainesville, FL.

  • Rayco, M. B., Francis, R. L., & Lowe, T. J. (1997). Error-bound driven demand point aggregation for the rectilinear distance p-center model. Location Science, 4, 213–235.

    Article  Google Scholar 

  • Rayco, M. B., Francis, R. L., & Tamir, A. (1999). A p-center grid-positioning aggregation procedure. Computers and Operations Research, 26, 1113–1124.

    Article  Google Scholar 

  • Reeves, C. (1993). Modern heuristic techniques for combinatorial problems. Oxford: Blackwell Scientific Press.

    Google Scholar 

  • Resende, M. G. C., & de Sousa, J. P. (Eds.). (2004). Metaheuristics: computer decision-making. Boston: Kluwer Academic.

    Google Scholar 

  • Rockafellar, R. T. (1970). Convex analysis. Princeton: Princeton University Press.

    Google Scholar 

  • Rodriguez-Bachiller, A. (1983). Errors in the measurement of spatial distances between discrete regions. Environment and Planning A, 15, 781–799.

    Article  Google Scholar 

  • Rodriguez-Chia, A. M., Nickel, S., Puerto, J., & Fernandez, F. R. (2000). A flexible approach to location problems. Mathematical Methods of Operations Research, 51, 69–89.

    Article  Google Scholar 

  • Rogers, D. F., Plante, R. D., Wong, R. T., & Evans, J. R. (1991). Aggregation and disaggregation techniques and methodology in optimization. Operations Research, 39, 553–582.

    Article  Google Scholar 

  • Romero-Morales, D., Carrizosa, E., & Conde, E. (1997). Semi-obnoxious location models: a global optimization approach. European Journal of Operational Research, 102, 295–301.

    Article  Google Scholar 

  • Rosenbaum, R. A. (1950). Sub-additive functions. Duke Mathematical Journal, 17, 227–247.

    Article  Google Scholar 

  • Rushton, G. (1989). Applications of location models. Annals of Operations Research, 18, 25–42.

    Article  Google Scholar 

  • Sheffi, Y. (1985). Urban transportation networks: equilibrium analysis with mathematical programming models (pp. 14–16). Englewood Cliffs: Prentice–Hall.

    Google Scholar 

  • Shier, D. R., & Dearing, P. M. (1983). Optimal locations for a class of nonlinear, single-facility location problems on a network. Operations Research, 31, 292–303.

    Article  Google Scholar 

  • Varadarajan, K. R. (1998). A divide and conquer algorithm for min-cost perfect matching in the plane. In Proceedings 38-th annual IEEE symposium on foundations of computer sciences, pp. 320–331.

  • Webber, M. J. (1980). A theoretical analysis of aggregation in spatial interaction models. Geographical Analysis, 12, 129–141.

    Google Scholar 

  • Zemel, E. (1985). Probabilistic analysis of geometric location problems. SIAM J. Algebraic and Discrete Methods, 6, 189–200.

    Article  Google Scholar 

  • Zhao, P. (1996). Analysis of aggregation effects in location problems. Ph. D. Dissertation, Dept. of Industrial Engineering, University at Buffalo (SUNY), Buffalo, NY.

  • Zhao, P., & Batta, R. (1999). Analysis of centroid aggregation for the Euclidean distance p-median problem. European Journal of Operational Research, 113, 147–168.

    Article  Google Scholar 

  • Zhao, P., & Batta, R. (2000). An aggregation approach to solving the network p-median problem with link demands. Networks, 36, 233–241.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to R. L. Francis.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Francis, R.L., Lowe, T.J., Rayco, M.B. et al. Aggregation error for location models: survey and analysis. Ann Oper Res 167, 171–208 (2009). https://doi.org/10.1007/s10479-008-0344-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10479-008-0344-z

Keywords

Navigation