IDEAS home Printed from https://ideas.repec.org/a/inm/ormnsc/v25y1979i4p329-340.html
   My bibliography  Save this article

Cluster Analysis: An Application of Lagrangian Relaxation

Author

Listed:
  • John M. Mulvey

    (Princeton University)

  • Harlan P. Crowder

    (IBM T. J. Watson Research Center)

Abstract

This paper presents and tests an effective optimization algorithm for clustering homogeneous data. The algorithm iteratively employs a subgradient method for determining lower bounds and a simple search procedure for determining upper bounds. The overall objective is to assign n objects to m mutually exclusive "clusters" such that the sum of the distances from each object to a designated cluster median is minimum. The model represents a special case of the uncapacitated facility location and m-median problems. This technique has proven efficient for examples with n \le 200 (i.e., the number of 0-1 variables \le 40,000); computational experiences with 10 real-world clustering applications are provided. A comparison with a hierarchical agglomerative heuristic, the minimum squared error method, is included. It is shown that the optimization algorithm is an effective solution technique for the homogeneous clustering problem, and also a good method for providing tight lower bounds for evaluating the quality of solutions generated by other procedures.

Suggested Citation

  • John M. Mulvey & Harlan P. Crowder, 1979. "Cluster Analysis: An Application of Lagrangian Relaxation," Management Science, INFORMS, vol. 25(4), pages 329-340, April.
  • Handle: RePEc:inm:ormnsc:v:25:y:1979:i:4:p:329-340
    as

    Download full text from publisher

    File URL: http://dx.doi.org/10.1287/mnsc.25.4.329
    Download Restriction: no

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Kulkarni, Girish & Fathi, Yahya, 2007. "Integer programming models for the q-mode problem," European Journal of Operational Research, Elsevier, vol. 182(2), pages 612-625, October.
    2. Marshall L. Fisher, 2004. "The Lagrangian Relaxation Method for Solving Integer Programming Problems," Management Science, INFORMS, vol. 50(12_supple), pages 1861-1871, December.
    3. Chen, Ja-Shen & Heragu, Sunderesh S., 1999. "Stepwise decomposition approaches for large scale cell formation problems," European Journal of Operational Research, Elsevier, vol. 113(1), pages 64-79, February.
    4. Michael Brusco & Hans-Friedrich Köhn, 2009. "Exemplar-Based Clustering via Simulated Annealing," Psychometrika, Springer;The Psychometric Society, vol. 74(3), pages 457-475, September.
    5. Maravalle, Maurizio & Simeone, Bruno & Naldini, Rosella, 1997. "Clustering on trees," Computational Statistics & Data Analysis, Elsevier, vol. 24(2), pages 217-234, April.
    6. García, Sergio & Benati, Stefano, 2012. "A p-median problem with distance selection," DES - Working Papers. Statistics and Econometrics. WS ws121913, Universidad Carlos III de Madrid. Departamento de Estadística.
    7. Chen, Mu-Chen & Wu, Hsiao-Pin, 2005. "An association-based clustering approach to order batching considering customer demand patterns," Omega, Elsevier, vol. 33(4), pages 333-343, August.
    8. Lozano, S. & Guerrero, F. & Onieva, L. & Larraneta, J., 1998. "Kohonen maps for solving a class of location-allocation problems," European Journal of Operational Research, Elsevier, vol. 108(1), pages 106-117, July.
    9. Drexl, Andreas & Klose, Andreas, 2001. "Facility location models for distribution system design," Manuskripte aus den Instituten für Betriebswirtschaftslehre der Universität Kiel 546, Christian-Albrechts-Universität zu Kiel, Institut für Betriebswirtschaftslehre.
    10. Vakharia, Asoo J. & Mahajan, Jayashree, 2000. "Clustering of objects and attributes for manufacturing and marketing applications," European Journal of Operational Research, Elsevier, vol. 123(3), pages 640-651, June.
    11. Wayne DeSarbo & Vijay Mahajan, 1984. "Constrained classification: The use of a priori information in cluster analysis," Psychometrika, Springer;The Psychometric Society, vol. 49(2), pages 187-215, June.
    12. Michael Brusco & Douglas Steinley, 2011. "A Tabu-Search Heuristic for Deterministic Two-Mode Blockmodeling of Binary Network Matrices," Psychometrika, Springer;The Psychometric Society, vol. 76(4), pages 612-633, October.
    13. D'Alfonso, Thomas H. & Ventura, Jose A., 1995. "Assignment of tools to machines in a flexible manufacturing system," European Journal of Operational Research, Elsevier, vol. 81(1), pages 115-133, February.
    14. Mladenovic, Nenad & Brimberg, Jack & Hansen, Pierre & Moreno-Perez, Jose A., 2007. "The p-median problem: A survey of metaheuristic approaches," European Journal of Operational Research, Elsevier, vol. 179(3), pages 927-939, June.
    15. Mangiameli, Paul & Chen, Shaw K. & West, David, 1996. "A comparison of SOM neural network and hierarchical clustering methods," European Journal of Operational Research, Elsevier, vol. 93(2), pages 402-417, September.
    16. Lau, Kin-nam & Leung, Pui-lam & Tse, Ka-kit, 1999. "A mathematical programming approach to clusterwise regression model and its extensions," European Journal of Operational Research, Elsevier, vol. 116(3), pages 640-652, August.
    17. Klose, Andreas & Drexl, Andreas, 2005. "Facility location models for distribution system design," European Journal of Operational Research, Elsevier, vol. 162(1), pages 4-29, April.
    18. Morgan, Shona D. & Fathi, Yahya, 2008. "Algorithms for the q-model clustering problem with application in switching cabinet manufacturing," European Journal of Operational Research, Elsevier, vol. 189(3), pages 939-951, September.
    19. Chen, Albert Y. & Yu, Ting-Yi, 2016. "Network based temporary facility location for the Emergency Medical Services considering the disaster induced demand and the transportation infrastructure in disaster response," Transportation Research Part B: Methodological, Elsevier, vol. 91(C), pages 408-423.
    20. Michael Brusco & Douglas Steinley, 2015. "Affinity Propagation and Uncapacitated Facility Location Problems," Journal of Classification, Springer;The Classification Society, vol. 32(3), pages 443-480, October.
    21. Chiou, Yu-Chiun & Lan, Lawrence W., 2001. "Genetic clustering algorithms," European Journal of Operational Research, Elsevier, vol. 135(2), pages 413-427, December.
    22. Wojtek Krzanowski & Glenn Milligan & Stanley Wasserman & Joseph Galaskiewicz & Joel Levine & Elke Weber & Peter Fishburn & Theodore Crovello & Bernard Baum & Wayne DeSarbo, 1987. "Book reviews," Journal of Classification, Springer;The Classification Society, vol. 4(1), pages 111-141, March.
    23. Holmberg, Kaj, 1997. "Mean value cross decomposition applied to integer programming problems," European Journal of Operational Research, Elsevier, vol. 97(1), pages 124-138, February.
    24. Sáez-Aguado, Jesús & Trandafir, Paula Camelia, 2012. "Some heuristic methods for solving p-median problems with a coverage constraint," European Journal of Operational Research, Elsevier, vol. 220(2), pages 320-327.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:inm:ormnsc:v:25:y:1979:i:4:p:329-340. See general information about how to correct material in RePEc.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (Mirko Janc). General contact details of provider: http://edirc.repec.org/data/inforea.html .

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    We have no references for this item. You can help adding them by using this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service hosted by the Research Division of the Federal Reserve Bank of St. Louis . RePEc uses bibliographic data supplied by the respective publishers.