We consider pairwise comparison matrices on a real divisible and continuous abelian linearly ordered group \(\mathcal{G}= (G, \odot, \leq)\), focusing on a proposed ⊙-consistency measure and its properties. We show that the proposed general ⊙-(in)consistency index satisfies some basic properties that can be considered naturally characterizing a consistency measure.


Multi-Criteria Decision Making Pairwise comparison matrices consistency index abelian linearly ordered group 


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  1. 1.
    Aguarón, J., Moreno-Jiménez, J.M.: The geometric consistency index: Approximated thresholds. European Journal of Operational Research 147(1), 137–145 (2003)zbMATHCrossRefGoogle Scholar
  2. 2.
    Barzilai, J.: Consistency measures for pairwise comparison matrices. Journal of Multi-Criteria Decision Analysis 7(3), 123–132 (1998)zbMATHCrossRefGoogle Scholar
  3. 3.
    Basile, L., D’Apuzzo, L.: Ranking and weak consistency in the a.h.p. context. Rivista di Matematica per le Scienze Economiche e Sociali 20(1), 99–110 (1997)zbMATHGoogle Scholar
  4. 4.
    Basile, L., D’Apuzzo, L.: Weak consistency and quasi-linear means imply the actual ranking. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 10(3), 227–239 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Basile, L., D’Apuzzo, L.: Transitive matrices, strict preference and intensity operators. Mathematical Methods in Economics and Finance 1, 21–36 (2006)MathSciNetGoogle Scholar
  6. 6.
    Basile, L., D’Apuzzo, L.: Transitive matrices, strict preference and ordinal evaluation operators. Soft Computing 10(10), 933–940 (2006)zbMATHCrossRefGoogle Scholar
  7. 7.
    Brunelli, M., Fedrizzi, M.: Characterizing properties for inconsistency indices in the ahp. In: ISAHP (2011)Google Scholar
  8. 8.
    Brunelli, M.: A note on the article inconsistency of pair-wise comparison matrix with fuzzy elements based on geometric mean. Fuzzy Sets and Systems 161, 1604–1613 (2010); Fuzzy Sets and Systems 176(1), 76–78 (2011)MathSciNetCrossRefGoogle 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.
    Cavallo, B., D’Apuzzo, L.: Characterizations of consistent pairwise comparison matrices over abelian linearly ordered groups. International Journal of Intelligent Systems 25(10), 1035–1059 (2010)zbMATHCrossRefGoogle Scholar
  11. 11.
    Cavallo, B., D’Apuzzo, L., Squillante, M.: Building Consistent Pairwise Comparison Matrices over Abelian Linearly Ordered Groups. In: Rossi, F., Tsoukias, A. (eds.) ADT 2009. LNCS (LNAI), vol. 5783, pp. 237–248. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  12. 12.
    Cavallo, B., D’Apuzzo, L., Squillante, M.: About a consistency index for pairwise comparison matrices over a divisible alo-group. International Journal of Intelligent Systems 27(2), 153–175 (2012)CrossRefGoogle Scholar
  13. 13.
    Cavallo, B., D’Apuzzo, L.: Deriving weights from a pairwise comparison matrix over an alo-group. Soft Computing - A Fusion of Foundations, Methodologies and Applications 16(2), 353–366 (2012)Google Scholar
  14. 14.
    Crawford, G., Williams, C.: A note on the analysis of subjective judgment matrices. Journal of Mathematical Psychology 29(4), 387–405 (1985)zbMATHCrossRefGoogle Scholar
  15. 15.
    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
  16. 16.
    Golden, B.L., Wang, Q.: An alternate measure of consistency. In: Golden, B.L., Wasil, E.A., Harker, P.T. (eds.) The Analythic Hierarchy Process, Applications and Studies, pp. 68–81. Springer, Heidelberg (1989)CrossRefGoogle Scholar
  17. 17.
    Herrera-Viedma, E., Herrera, F., Chiclana, F., Luque, M.: Some issue on consistency of fuzzy preferences relations. European Journal of Operational Research 154, 98–109 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Peláez, J.I., Lamata, M.T.: A new measure of consistency for positive reciprocal matrices. Computers & Mathematics with Applications 46(12), 1839–1845 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Ramík, J., Korviny, P.: Inconsistency of pair-wise comparison matrix with fuzzy elements based on geometric mean. Fuzzy Sets Syst. 161, 1604–1613 (2010)zbMATHCrossRefGoogle Scholar
  20. 20.
    Saaty, T.L.: A scaling method for priorities in hierarchical structures. J. Math. Psychology 15, 234–281 (1977)MathSciNetzbMATHCrossRefGoogle Scholar
  21. 21.
    Saaty, T.L.: The Analytic Hierarchy Process. McGraw-Hill, New York (1980)zbMATHGoogle Scholar
  22. 22.
    Saaty, T.L.: Axiomatic foundation of the analytic hierarchy process. Management Science 32(7), 841–855 (1986)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Bice Cavallo
    • 1
  • Livia D’Apuzzo
    • 1
  1. 1.University Federico IINaplesItaly

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