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A Different Perspective on a Scale for Pairwise Comparisons

  • J. Fülöp
  • W. W. Koczkodaj
  • S. J. Szarek
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6220)

Abstract

One of the major challenges for collective intelligence is inconsistency, which is unavoidable whenever subjective assessments are involved. Pairwise comparisons allow one to represent such subjective assessments and to process them by analyzing, quantifying and identifying the inconsistencies.

We propose using smaller scales for pairwise comparisons and provide mathematical and practical justifications for this change. Our postulate’s aim is to initiate a paradigm shift in the search for a better scale construction for pairwise comparisons. Beyond pairwise comparisons, the results presented may be relevant to other methods using subjective scales.

Keywords

Pairwise comparisons collective intelligence scale subjective assessment inaccuracy inconsistency 

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References

  1. 1.
    Anholcer, M., Babiy, V., Bozóki, S., Koczkodaj, W.W.: A simplified implementation of the least squares solution for pairwise comparisons matrices. Central European Journal of Operations Research (to appear)Google Scholar
  2. 2.
    Basile, L., D’Apuzzo, L., Marcarelli, G., Squillante, M.: Generalized Consistency and Representation of Preferences by Pairwise Comparisons. In: Panamerican Conference of Applied Mathematics, Huatulco, Mexico (2006)Google Scholar
  3. 3.
    Blankmeyer, E.: Approaches to consistency adjustments. Journal of Optimization Theory and Applications 54, 479–488 (1987)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Bozóki, S.: A method for solving LSM problems of small size in the AHP. Central European Journal of Operations Research 11, 17–33 (2003)zbMATHGoogle Scholar
  5. 5.
    Bozóki, S.: Solution of the least squares method problem of pairwise comparisons matrices. Central European Journal of Operations Research 16, 345–358 (2008)zbMATHCrossRefGoogle Scholar
  6. 6.
    Bozóki, S., Rapcsák, T.: On Saaty’s and Koczkodaj’s inconsistencies of pairwise comparison matrices. Journal of Global Optimization 42(2), 157–175 (2007)CrossRefGoogle Scholar
  7. 7.
    Brunelli, M., Fedrizzi, M.: Fair Consistency Evaluation in Fuzzy Preference Relations and in AHP. In: Apolloni, B., Howlett, R.J., Jain, L. (eds.) KES 2007, Part II. LNCS (LNAI), vol. 4693, pp. 612–618. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  8. 8.
    Bullions, P.: The Principles of English Grammar, 16th edn. Pratt, Woodford, & Co. (1846)Google Scholar
  9. 9.
    Cavallo, B., D’Apuzzo, L.: A general unified framework for pairwise comparison matrices in multicriterial methods. International Journal of Intelligent Systems 24(4), 377–398 (2009)zbMATHCrossRefGoogle Scholar
  10. 10.
    Chu, A.T.W., Kalaba, R.E., Spingarn, K.: A comparison of two methods for determining the weight belonging to fuzzy sets. Journal of Optimization Theory and Applications 4, 531–538 (1979)CrossRefMathSciNetGoogle Scholar
  11. 11.
    Choo, E.U., Wedley, W.C.: A common framework for deriving preference values from pairwise comparison matrices. Computers and Operations Research 31, 893–908 (2004)zbMATHCrossRefGoogle Scholar
  12. 12.
    Condorcet, M.: Essai sur l’Application de l’Analyse à la Probabilité des Décisions Rendues à la Pluralité des Voix, Paris (1785)Google Scholar
  13. 13.
    Crawford, G., Williams, C.: A note on the analysis of subjective judgment matrices. Journal of Mathematical Psychology 29, 387–405 (1985)zbMATHCrossRefGoogle Scholar
  14. 14.
    D’Apuzzo, L., Marcarelli, G., Squillante, M.: Generalized consistency and intensity vectors for comparison matrices. International Journal of Intelligent Systems 22(12), 1287–1300 (2007)zbMATHCrossRefGoogle Scholar
  15. 15.
    Debreu, G.: Topological methods in cardinal utility theory. In: Arrow, K.J., Karlin, S., Suppes, P. (eds.) Mathematical Methods in the Social Sciences, pp. 16–26. Stanford University Press, Stanford (1960)Google Scholar
  16. 16.
    De Jong, P.: A statistical approach to Saaty’s scaling method for priorities. Journal of Mathematical Psychology 28, 467–478 (1984)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Farkas, A., Lancaster, P., Rózsa, P.: Consistency adjustment for pairwise comparison matrices. Numer. Linear Algebra Applications 10, 689–700 (2003)zbMATHCrossRefGoogle Scholar
  18. 18.
    Fedrizzi, M., Fedrizzi, M., Marques Pereira, R.A.: On the issue of consistency in dynamical consensual aggregation. In: Bouchon Meunier, B., Gutierrez Rios, J., Magdalena, L., Yager, R.R. (eds.) Technologies for Constructing Intelligent Systems. Studies in Fuzziness and Soft Computing, vol. 1, 89, pp. 129–137. Springer, Heidelberg (2002)Google Scholar
  19. 19.
    Fedrizzi, M., Giove, S.: Incomplete pairwise comparison and consistency optimization. European Journal of Operational Research 183(1), 303–313 (2007)zbMATHCrossRefGoogle Scholar
  20. 20.
    Fülöp, J.: A method for approximating pairwise comparison matrices by consistent matrices. Journal of Global Optimization 42, 423–442 (2008)zbMATHCrossRefGoogle Scholar
  21. 21.
    Golany, B., Kress, M.: A multicriteria evaluation method for obtaining weights from ratio-scale matrices. European Journal of Operational Research 69, 210–220 (1993)zbMATHCrossRefGoogle Scholar
  22. 22.
    Holsztynski, W., Koczkodaj, W.W.: Convergence of inconsistency algorithms for the pairwise comparisons. Information Processing Letters 59(4), 197–202 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Hubbard, D.: How to measure anything. Wiley, Chichester (2007)Google Scholar
  24. 24.
    Jensen, R.E.: Comparison of eigenvector, least squares, chi squares and logarithmic least squares methods of scaling a reciprocal matrix, working paper 153, Trinity, University (1983)Google Scholar
  25. 25.
    Jensen, R.E.: Alternative scaling method for priorities in hierarchical structures. Journal of Mathematical Psychology 28, 317–332 (1984)CrossRefGoogle Scholar
  26. 26.
    Koczkodaj, W.W.: A new definition of consistency of pairwise comparisons. Mathematical and Computer Modelling 18(7), 79–84 (1993)zbMATHCrossRefGoogle Scholar
  27. 27.
    Koczkodaj, W.W., Szarek, S.J.: On distance-based inconsistency reduction algorithms for pairwise comparisons. Logic Journal of IGPL (2010) (advance access published January 17, 2010)Google Scholar
  28. 28.
    Luce, R.D., Edwards, W.: The derivation of subjective scales from just noticeable differences. Psychological Review 65(4), 222–237 (1958)CrossRefGoogle Scholar
  29. 29.
    Luce, R.D., Tukey, J.W.: Simultaneous conjoint measurement: a new scale type of fundamental measurement. Journal of Mathematical Psychology 1, 1–27 (1964)zbMATHCrossRefGoogle Scholar
  30. 30.
    Llull, R.: Artifitium electionis personarum (before 1283)Google Scholar
  31. 31.
    Mikhailov, L.: A fuzzy programming method for deriving priorities in the analytic hiarerchy process. Journal of the Operational Research Society 51, 341–349 (2000)zbMATHGoogle Scholar
  32. 32.
    Nagel, E.: Measurement. Erkenntnis 2(1), 313–335 (1931)CrossRefGoogle Scholar
  33. 33.
    Nguyen, N.T.: Advanced method in Inconsistency Knowledge Management, p. 356. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  34. 34.
    Saaty, T.L.: A scaling method for priorities in hierarchical structures. Journal of Mathematical Psychology 15, 234–281 (1977)zbMATHCrossRefMathSciNetGoogle Scholar
  35. 35.
    Saaty, T.L.: The Analytic Hierarchy Process. McGraw-Hill, New York (1980)zbMATHGoogle Scholar
  36. 36.
    Stevens, S.S.: On the theory of scales of measurement. Science 103, 677–680 (1946)CrossRefGoogle Scholar
  37. 37.
    Thurstone, L.L.: A law of comparative judgement. Psychological Review 34, 278–286 (1927)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • J. Fülöp
    • 1
  • W. W. Koczkodaj
    • 2
  • S. J. Szarek
    • 3
    • 4
  1. 1.Research Group of Operations Research and Decision Systems, Computer and Automation Research InstituteHungarian Academy of SciencesBudapestHungary
  2. 2.Computer ScienceLaurentian UniversitySudburyCanada
  3. 3.Department of MathematicsCase Western Reserve UniversityClevelandUSA
  4. 4.Université Pierre et Marie Curie-Paris 6ParisFrance

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