Hostname: page-component-745bb68f8f-v2bm5 Total loading time: 0 Render date: 2025-02-10T14:13:03.759Z Has data issue: false hasContentIssue false
  • English
  • Français

Matrix Reorganization and Dynamic Programming: Applications to Paired Comparisons and Unidimensional Seriation

Published online by Cambridge University Press:  01 January 2025

L. J. Hubert  and
R. G. Golledge
L. J. Hubert*
Affiliation:
The University of California, Santa Barbara
R. G. Golledge*
Affiliation:
The University of California, Santa Barbara
*
Requests for reprints should be sent to either L. J. Hubert, Graduate School of Education, University of California, Santa Barbara, California 93106 or R. G. Golledge, Department of Geography, University of California, Santa Barbara, California 93106.
Requests for reprints should be sent to either L. J. Hubert, Graduate School of Education, University of California, Santa Barbara, California 93106 or R. G. Golledge, Department of Geography, University of California, Santa Barbara, California 93106.
Get access Rights & Permissions [Opens in a new window]

Abstract

A recursive dynamic programming strategy is discussed for optimally reorganizing the rows and simultaneously the columns of an n × n proximity matrix when the objective function measuring the adequacy of a reorganization has a fairly simple additive structure. A number of possible objective functions are mentioned along with several numerical examples using Thurstone's paired comparison data on the relative seriousness of crime. Finally, the optimization tasks we propose to attack with dynamic programming are placed in a broader theoretical context of what is typically referred to as the quadratic assignment problem and its extension to cubic assignment.

Keywords

combinatorial optimizationquadratic assignmentcubic assignment
Type
Original Paper
Information
Psychometrika , Volume 46 , Issue 4 , December 1981 , pp. 429 - 441
Copyright
Copyright © 1981 The Psychometric Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

Partial support for this research was provided by NIJ Grant 80-IJ-CX-0061.

References

Adelson, R. M., Norman, J. M., & Laporte, G. A dynamic programming formulation with diverse applications. Operational Research Quarterly, 1976, 27, 119–121.CrossRefGoogle Scholar
Arabie, P. & Carroll, J. D. MAPCLUS: A mathematical programming approach to fitting the ADCLUS model. Psychometrika, 1980, 45, 211–235.CrossRefGoogle Scholar
Baker, F. B. & Hubert, L. J. Applications of combinatorial programming to data analysis: Seriation using asymmetric proximity measures. British Journal of Mathematical and Statistical Psychology, 1977, 30, 154–164.CrossRefGoogle Scholar
Blin, J. M. On tournament patterns and rational group decisions. Proceedings of IEEE 1973 Conference on Cybernetics and Society, 1973, 55–59.Google Scholar
Defays, D. A short note on a method of seriation. British Journal of Mathematical and Statistical Psychology, 1978, 31, 49–53.CrossRefGoogle Scholar
Dreyfus, S. E. An appraisal of some shortest-path algorithms. Operation Research, 1969, 17, 395–412.CrossRefGoogle Scholar
Elmaghraby, S. E. The sequencing of “related” jobs. Naval Research Logistics Quarterly, 1968, 15, 23–32.CrossRefGoogle Scholar
Flueck, J. A. & Korsh, J. F. A branch search algorithm for maximum likelihood paired comparison ranking. Biometrika, 1974, 61, 621–626.CrossRefGoogle Scholar
Greenberg, M. G. A method of successive cumulations for scaling of pair-comparison judgements. Psychometrika, 1965, 30, 441–448.CrossRefGoogle Scholar
Hanan, M. & Kurtzberg, J. M. Placement techniques. In Breuer, M. A. (Eds.), Design Automation of Digital Systems, 1972, Englewood Cliffs, New Jersey: Prentice-Hall.Google Scholar
Hartigan, J. A. Clustering Algorithms, 1975, New York: Wiley.Google Scholar
Holman, E. W. Completely nonmetric multidimensional scaling. Journal of Mathematical Psychology, 1978, 18, 39–51.CrossRefGoogle Scholar
Hubert, L. J. Some applications of graph theory and related nonmetric techniques to problems of approximate seriation: The case of symmetric proximity measures. British Journal of Mathematical and Statistical Psychology, 1974, 27, 133–153.CrossRefGoogle Scholar
Hubert, L. J. Problems of seriation using a subject by item response matrix. Psychological Bulletin, 1974, 81, 976–983.CrossRefGoogle Scholar
Hubert, L. J. Seriation using asymmetric proximity measures. British Journal of Mathematical and Statistical Psychology, 1976, 29, 32–52.CrossRefGoogle Scholar
Hubert, L. J. & Baker, F. B. Applications of combinatorial programming to data analysis: The traveling salesman and related problems. Psychometrika, 1978, 43, 81–91.CrossRefGoogle Scholar
Johnson, S. C. Hierarchical clustering schemes. Psychometrika, 1967, 32, 241–254.CrossRefGoogle ScholarPubMed
Karp, R. M. Reducibility among combinatorial problems. In Miller, R. E. & Thatcher, J. W. (Eds.), Complexity of Computer Computations. New York: Plenum Press. 1972, 85–104.CrossRefGoogle Scholar
Korte, B. & Oberhofer, W. Triangularizing input-output matrices and the structures of production. European Economic Review, 1971, 2, 493–522.CrossRefGoogle Scholar
Lawler, E. L. A comment on minimum feedback arc sets. IEEE Transactions on Circuit Theory, 1964, 11, 296–297.CrossRefGoogle Scholar
Lenstra, J. K. & Rinnooy Kan, A. H. G. Some simple applications of the traveling salesman problem. Operational Research Quarterly, 1975, 26, 717–733.CrossRefGoogle Scholar
Mirkin, B. G. Group choice, 1979, New York: Wiley.Google Scholar
Reinhold, E. M., Nievergelt, J. & Deo, N. Combinatorial algorithms: Theory and practice, 1977, Englewood Cliffs, N. J.: Prentice-Hall.Google Scholar
Roberts, F. S. Graph theory and its applications to problems of society, 1978, Philadelphia: Society for Industrial and Applied Mathemetics.CrossRefGoogle Scholar
Shepard, R. N. The analysis of proximities: Multidimensional scaling with an unknown distance function I, II. Psychometrika, 1962, 17, 125–140.CrossRefGoogle Scholar
Shepard, R. N. A taxonomy of some principal types of data and of multidimensional methods for their analysis. In Shepard, R. N. et al. (Eds.), Multidimensional scaling: Theory and applications in the behavioral sciences, 1972, New York: Seminar Press.Google Scholar
Szczotka, F. A. On a method of ordering and clustering of objects. Zastosowania Mathematyki, 1972, 13, 23–33.Google Scholar
Thurstone, L. L. The method of paired comparisons for social values. Journal of Abnormal and Social Psychology, 1927, 31, 384–400.CrossRefGoogle Scholar