Relations between coherence conditions and row orders in pairwise comparison matrices

Abstract

Coherence of preferences has been a long standing issue in decision analysis. This paper focuses on preferences expressed by means of pairwise comparison matrices. In particular, by following the idea proposed in previous papers concerning the relation between some coherence conditions and row orders of the matrix, we provide similar relations for further coherence conditions that are the restricted max–max transitivity, the index exchange ability and the quasi-consistency.

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

Fig. 1

References

  1. Bana e Costa, C .A., Vansnick, J.-C.: A critical analysis of the eigenvalue method used to derive priorities in AHP. Eur. J. Oper. Res. 187(3), 1422–1428 (2008)

    Article  Google Scholar 

  2. Barzilai, J., Cook, W., Golany, B.: Consistent weights for judgements matrices of the relative importance of alternatives. Oper. Res. Lett. 6(3), 131–134 (1987)

    Article  Google Scholar 

  3. Barzilai, J., Golany, B.: Deriving weights from pairwise comparison matrices: the additive case. Oper. Res. Lett. 9(6), 407–410 (1990)

    Article  Google Scholar 

  4. Bortot, S., Pereira, R.A.M.: Algebraic representations of the weighted mean. Fuzzy Sets Syst. 308, 85–105 (2017)

    Article  Google Scholar 

  5. Brunelli, M., Cavallo, B.: Distance-based measures of incoherence for pairwise comparisons. Knowl.-Based Syst. 187, 104808 (2020)

    Article  Google Scholar 

  6. Capuano, N., Chiclana, F., Fujita, H., Herrera-Viedma, E., Loia, V.: Fuzzy group decision making with incomplete information guided by social influence. IEEE Trans. Fuzzy Syst. 26(3), 1704–1718 (2018)

    Article  Google Scholar 

  7. Cavallo, B.: A further discussion of “A semiring-based studyof judgment matrices: properties and models”  [Information Sciences 181 (2011) 2166–2176]. Inf. Sci. 287, 61–67 (2014)

    Article  Google Scholar 

  8. Cavallo, B.: Computing random consistency indices and assessing priority vectors reliability. Inf. Sci. 420, 532–542 (2017)

    Article  Google Scholar 

  9. Cavallo, B.: Coherent weights for pairwise comparison matrices and a mixed-integer linear programming problem. J. Glob.Optim. 75(1), 143–161 (2019)

    Article  Google Scholar 

  10. Cavallo, B.: Functional relations and Spearman correlation between consistency indices. J. Oper. Res. Soc. 71, 301–311 (2020)

    Article  Google Scholar 

  11. Cavallo, B., Brunelli, M.: A general unified framework for interval pairwise comparison matrices. Int. J. Approx. Reason. 93, 178–198 (2018)

    Article  Google Scholar 

  12. Cavallo, B., D’Apuzzo, L.: A general unified framework for pairwise comparison matrices in multicriterial methods. Int. J. Intell. Syst. 24(4), 377–398 (2009)

    Article  Google Scholar 

  13. Cavallo, B., D’Apuzzo, L.: Characterizations of consistent pairwise comparison matrices over abelian linearly ordered groups. Int. J. Intell. Syst. 25(10), 1035–1059 (2010)

    Article  Google Scholar 

  14. Cavallo, B., D’Apuzzo, L.: Deriving weights from a pairwise comparison matrix over an alo-group. Soft. Comput. 16(2), 353–366 (2012)

    Article  Google Scholar 

  15. Cavallo, B., D’Apuzzo, L.: Reciprocal transitive matrices over abelian linearly ordered groups: Characterizations and application to multi-criteria decision problems. Fuzzy Sets Syst. 266, 33–46 (2015)

    Article  Google Scholar 

  16. Cavallo, B., D’Apuzzo, L.: Ensuring reliability of the weighting vector: Weak consistent pairwise comparison matrices. Fuzzy Sets Syst. 296, 21–34 (2016)

    Article  Google Scholar 

  17. Cavallo, B., D’Apuzzo, L.: Preservation of preferences intensity of an inconsistent pairwise comparison matrix. Int. J. Approx. Reason. 116, 33–42 (2020)

    Article  Google Scholar 

  18. Cavallo, B., D’Apuzzo, L., Basile, L.: Weak consistency for ensuring priority vectors reliability. J. Multi-Criteria Decis. Anal. 23(3–4), 126–138 (2016)

    Article  Google Scholar 

  19. Cavallo, B., D’Apuzzo, L., Vitale, G.: Reformulating Arrow’s conditions in terms of cardinal pairwise comparison matrices defined over a general framework. Group Decis. Negot. 27(1), 107–127 (2018)

    Article  Google Scholar 

  20. Cavallo, B., Ishizaka, A., Olivieri, M.G., Squillante, M.: Comparing inconsistency of pairwise comparison matrices depending on entries. J. Oper. Res. Soc. 70(5), 842–850 (2019)

    Article  Google Scholar 

  21. Chiclana, F., Herrera-Viedma, E., Alonso, S., Herrera, F.: Cardinal consistency of reciprocal preference relations: a characterization of multiplicative transitivity. IEEE Trans. Fuzzy Syst. 17(1), 14–23 (2009)

    Article  Google Scholar 

  22. Crawford, G., Williams, C.: A note on the analysis of subjective judgment matrices. J. Math. Psychol. 29(4), 387–405 (1985)

    Article  Google Scholar 

  23. D’Apuzzo, L., Marcarelli, G., Squillante, M.: Generalized consistency and intensity vectors for comparison matrices. Int. J. Intell. Syst. 22(12), 1287–1300 (2007)

    Article  Google Scholar 

  24. De Graan, J. G.: Extensions of the multiple criteria analysis method of T. L. Saaty, (1980)

  25. Fishburn, P.C.: Utility Theory for Decision Making. Wiley, Hoboken (1970)

    Google Scholar 

  26. Fraleigh, J.B.: A First Course in Abstract Algebra, 7th edn. Pearson, London (2002)

    Google Scholar 

  27. Hou, F.: A semiring-based study of judgment matrices: properties and models. Inf. Sci. 181(11), 2166–2176 (2011)

    Article  Google Scholar 

  28. Hou, F.: A multiplicative alo-group based hierarchical decision model and application. Commun. Stat. Simul. Comput. 45(8), 2846–2862 (2016)

    Article  Google Scholar 

  29. Keeney, R.L., Raiffa, H.: Decisions with Multiple Objectives: Preferences and Value Tradeoffs. Wiley, New York (1976)

    Google Scholar 

  30. Koczkodaj, W.W., Szybowski, J., Wajch, E.: Inconsistency indicator maps on groups for pairwise comparisons. Int. J. Approx. Reason. 69, 81–90 (2016)

    Article  Google Scholar 

  31. Kou, G., Ergu, D., Lin, C., Chen, Y.: Pairwise comparison matrix in multiple criteria decision making. Technol. Econ. Dev. Econ. 22(5), 738–765 (2016)

    Article  Google Scholar 

  32. Kulakowski, K., Mazurek, J., Ramík, J., Soltys, M.: When is the condition of order preservation met? Eur. J. Oper. Res. 277(1), 248–254 (2019)

    Article  Google Scholar 

  33. Lundy, M., Siraj, S., Greco, S.: The mathematical equivalence of the “spanning tree”and row geometric mean preference vectors and its implications for preference analysis. Eur. J. Oper. Res. 257(1), 197–208 (2017)

    Article  Google Scholar 

  34. Nida-Rümelin, J.: Rationality, Rules, and Structure, chapter Rationality: coherence and structure, pp. 1–16. Springer Netherlands, (2000)

  35. Rabinowitz, G.: Some comments on measuring world influence. J Peace Sci. 2(1), 49–55 (1976)

    Article  Google Scholar 

  36. Ramík, J.: Isomorphisms between fuzzy pairwise comparison matrices. Fuzzy Optim. Decis. Making 14(2), 199–209 (2015)

    Article  Google Scholar 

  37. Roberts, F.S.: Measurement Theory with Applications to Decisionmaking, Utility, and the Social Sciences. Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge (1985)

    Google Scholar 

  38. Saaty, T.L.: A scaling method for priorities in hierarchical structures. J. Math. Psychol. 15(3), 234–281 (1977)

    Article  Google Scholar 

  39. Saaty, T.L., Vargas, L.G.: The possibility of group choice: pairwise comparisons and merging functions. Soc. Choice Welf. 38(3), 481–496 (2012)

    Article  Google Scholar 

  40. Tanino, T.: Fuzzy preference orderings in group decision making. Fuzzy Sets Syst. 12(2), 117–131 (1984)

    Article  Google Scholar 

  41. Xia, M., Chen, J.: Consistency and consensus improving methods for pairwise comparison matrices based on Abelian linearly ordered group. Fuzzy Sets Syst. 266, 1–32 (2015)

    Article  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Bice Cavallo.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Cavallo, B., D’Apuzzo, L. Relations between coherence conditions and row orders in pairwise comparison matrices. Decisions Econ Finan (2020). https://doi.org/10.1007/s10203-020-00289-5

Download citation

Keywords

  • Multi-criteria decision making
  • Coherence
  • Order relations
  • Preferences representation

JEL Classification

  • C440
  • C650