Skip to main content

Inconsistency Indices and Their Properties

  • Chapter
  • First Online:
Advances in Pairwise Comparisons

Part of the book series: Multiple Criteria Decision Making ((MCDM))

  • 125 Accesses

Abstract

This chapter provides a review of (cardinal) inconsistency indices, two systems of indices’ (desirable) properties, and an overview of satisfaction of these properties by selected indices. Further on, numerical studies on indices’ comparisons are presented and discussed as well.

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

Access this chapter

eBook
USD 16.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 139.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 139.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Aguarón, J., & Moreno-Jiménez, J. M. (2003). The geometric consistency index: Approximated threshold, European Journal of Operational Research, 147(1), 137–145.

    Article  Google Scholar 

  2. Aguarón, J., Escobar, M. T., Moreno-Jiménez, J. M., & Turón, A. (2020). The Triads Geometric Consistency Index in AHP-Pairwise Comparison Matrices. Mathematics, 8, 926. https://doi.org/10.3390/math8060926.

    Article  Google Scholar 

  3. Alonso, J. A., & Lamata M. T. (2006). Consistency in the analytic hierarchy process: a new approach. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 14(4), 445–459.

    Article  Google Scholar 

  4. Amenta, P., Lucadamo, A., & Marcarelli, G. (2020). On the transitivity and consistency approximated thresholds of some consistency indices for pairwise comparison matrices. Information Sciences, 507, 274–287. https://doi.org/10.1016/j.ins.2019.08.042.

    Article  Google Scholar 

  5. Barzilai, J. (1998). Consistency measures for pairwise comparison matrices. Journal of Multi-Criteria Decision Analysis, 7(3), 123–132.

    Article  Google Scholar 

  6. Bortot, S., Brunelli, M., Fedrizzi, M., & Marques Pereira, R. A. (2022). A novel perspective on the inconsistency indices of reciprocal relations and pairwise comparison matrices. Fuzzy Sets and Systems. https://doi.org/10.1016/j.fss.2022.04.020.

  7. Brunelli, M. (2017). Studying a set of properties of inconsistency indices for pairwise comparisons. Annals of Operations Research, 248(1,2), 143–161.

    Google Scholar 

  8. Brunelli, M. (2018). A survey of inconsistency indices for pairwise comparisons. International Journal of General Systems, 47(8), 751–771.

    Article  Google Scholar 

  9. Brunelli, M., & Fedrizzi M. (2015). Axiomatic properties of inconsistency indices for pairwise comparisons. Journal of the Operational Research Society, 66(1), 1–15.

    Article  Google Scholar 

  10. Brunelli, M., & Fedrizzi, M. (2015). Boundary properties of the inconsistency of pairwise comparisons in group decisions. European Journal of Operational Research, 240(3), 76—773. https://doi.org/10.1016/j.ejor.2014.07.045.

    Article  Google Scholar 

  11. Brunelli, M., & Fedrizzi, M. (2019). A general formulation for some inconsistency indices of pairwise comparisons. Annals of Operations Research, 274, 155–169. https://doi.org/10.1007/s10479-018-2936-6.

    Article  Google Scholar 

  12. Brunelli, M., Canal, L., & Fedrizzi, M. (2013). Inconsistency indices for pairwise comparison matrices: a numerical study. Annals of Operations Research, 211(1), 493–509.

    Article  Google Scholar 

  13. Brunelli, M., Critch, A., & Fedrizzi, M. (2013). A note on the proportionality between some consistency indices in the AHP. Applied Mathematics and Computation, 219(14), 7901–7906.

    Article  Google Scholar 

  14. Cavallo, B. (2017). Computing random consistency indices and assessing priority vectors reliability. Information Sciences, 420, 532–542. https://doi.org/10.1016/j.ins.2017.08.082.

    Article  Google Scholar 

  15. Cavallo, B. (2020). Functional relations and Spearman correlation between consistency indices. Journal of the Operational Research Society, 71(2), 301–311. https://doi.org/10.1080/01605682.2018.1516178.

    Article  Google Scholar 

  16. Cavallo, B., & D’Apuzzo, L. (2012). Investigating Properties of the ⊙–Consistency Index. IPMU, 4, 315–327.

    Google Scholar 

  17. Chu, A. T. W., Kalaba, R. E., & Spingarn, K. (1979). A comparison of two methods for determining the weights of belonging to fuzzy sets. Journal of Optimization Theory and Applications, 27(4), 531–538.

    Article  Google Scholar 

  18. Crawford, G., & Williams, C. (1985). A Note on the Analysis of Subjective Judgment Matrices. Journal of Mathematical Psychology, 29(4), 387–405.

    Article  Google Scholar 

  19. Csató, L. (2018). Characterization of an inconsistency measure for pairwise comparison matrices. Annals of Operations Research, 261(1–2), 155–165.

    Article  Google Scholar 

  20. Dixit, P. (2018). Entropy Production Rate as a Criterion for Inconsistency in Decision Theory. Journal of Statistical Mechanics: Theory and Experiment, 5, 053408.

    Article  Google Scholar 

  21. Duszak, Z., & Koczkodaj, W. W. (1994). Generalization of a new definition of consistency for pairwise comparisons. Information Processing Letters, 52, 273–276.

    Article  Google Scholar 

  22. Fedrizzi, M., & Ferrari, F. (2018). A Chi-Square-Based Inconsistency Index for PairwiseComparison Matrices. Journal of the Operational Research Society, 69(7), 1125–1134.

    Article  Google Scholar 

  23. Fedrizzi, M., & Giove, S. (2007). Incomplete pairwise comparison and consistency optimization. European Journal of Operational Research, 183, 303–313.

    Article  Google Scholar 

  24. Fedrizzi, M., Civolani, N., & Critch, A. (2020). Inconsistency evaluation in pairwise comparison using norm-based distances. Decisions in Economics and Finance, 43, 657–672. https://doi.org/10.1007/s10203-020-00304-9.

    Article  Google Scholar 

  25. Forman, E. H. (1990). Random indices for incomplete pairwise comparison matrices. European Journal of Operational Research, 48, 153–155.

    Article  Google Scholar 

  26. Gass, S. I., & Rapcsák, T. (2004). Singular Value Decomposition in AHP. European Journal of Operational Research, 154(3), 573–584.

    Article  Google Scholar 

  27. Golden, B., & Wang, Q. (1989). An alternate measure of consistency. In B. Golden, E. Wasil, & P. T. Harker (Eds.), The Analytic Hierarchy Process, Applications and Studies (pp. 68–81). Berlin: Springer.

    Chapter  Google Scholar 

  28. Grzybowski, A. Z. (2012). Note on a new optimization based approach for estimating priority weights and related consistency index. Expert Systems with Applications, 39(14), 11699–11708. https://doi.org/10.1016/j.eswa.2012.04.051.

    Article  Google Scholar 

  29. Grzybowski, A. Z. (2016). New results on inconsistency indices and their relationship withthe quality of priority vector estimation. Expert Systems with Applications, 43, 197–212.

    Article  Google Scholar 

  30. Karapetrovic, E. S., & Rosenbloom, A. (1999). A quality control approach to consistency paradoxes in AHP. European Journal of Operational Research, 119(3), 704–718. https://doi.org/10.1016/S0377-2217(98)00334-8.

    Article  Google Scholar 

  31. Koczkodaj, W. W. (1993). A new definition of consistency of pairwise comparisons. Mathematical and Computer Modeling, 18(7), 79–84.

    Article  Google Scholar 

  32. Koczkodaj, W. W., & Magnot, J.-P. (2017). Axiomatization of Inconsistency Indicators for Pairwise Comparisons. ArXiv:1509.03781v2.

    Google Scholar 

  33. Koczkodaj, W. W., & Szwarc, R. (2014). On axiomatization of inconsistency indicators for pairwise comparisons. Fundamenta Informaticae, 132(4), 485–500.

    Article  Google Scholar 

  34. Koczkodaj, W. W., & Urban, R. (2018). Axiomatization of inconsistency indicators for pairwise comparisons. International Journal of Approximate Reasoning, 94, 18–29. https://doi.org/10.1016/j.ijar.2017.12.001.

    Article  Google Scholar 

  35. Koczkodaj, W. W., Magnot, J.-P., Mazurek, J., Peters, J. F., Rakhshani, H., Soltys, M., Strzalka, D., Szybowski, J., & Tozzi, A. (2017). On normalization of inconsistency indicators in pairwise comparisons. International Journal of Approximate Reasoning, 86, 73–79.

    Article  Google Scholar 

  36. Kou, G., & Lin, C. (2014). A Cosine Maximization Method for the Priority VectorDerivation in AHP. European Journal of Operational Research, 235(1), 225–232.

    Article  Google Scholar 

  37. Kowal, B., Kuras, P. Strzalka, D., Mazurek, J., & Perzina, R. (2021). REDUCE: An online decision support tool for reduction of inconsistency in multiplicative pairwise comparisons. In Proceedings of the 3rd International conference on Decision making for Small and Medium-Sized Enterprises (pp. 294–300). Karvina: Silesian University in Opava, School of Business Administration in Karvina.

    Google Scholar 

  38. Kulakowski, K. (2015). Notes on Order Preservation and Consistency in AHP. European Journal of Operational Research, 245, 333–337.

    Article  Google Scholar 

  39. Kulakowski, K., & Szybowski, J. (2014). The new triad based inconsistency indices for pairwise comparisons. Procedia Computer Science, 35, 1132–1137.

    Article  Google Scholar 

  40. Lin, C., Kou, G., & Ergu, D. (2013). An improved statistical approach for consistency test in AHP. Annals of Operations Research, 211, 289–299. https://doi.org/10.1007/s10479-013-1413-5.

    Article  Google Scholar 

  41. Lin, C., Kou, G., & Ergu, D. (2014) A statistical approach to measure the consistency level of the pairwise comparison matrix. Journal of the Operational Research Society, 65(9), 1380–1386. https://doi.org/10.1057/jors.2013.92.

    Article  Google Scholar 

  42. Mazurek, J. (2018). Some notes on the properties of inconsistency indices in pairwise comparisons. Operations Research and Decisions, 1, 27–42.

    Google Scholar 

  43. Mazurek, J., Smalara, K., & Kowal, B. (2022). Percentile Tables for Selected Inconsistency Indices–Technical Paper. https://doi.org/10.13140/RG.2.2.23037.03044.

    Google Scholar 

  44. Mizuno, T. (2019). A Link Diagram for Pairwise Comparisons. In: I. Czarnowski, R. Howlett, L. Jain, & L. Vlacic (Eds.), Intelligent Decision Technologies 2018. KES-IDT 2018. Smart Innovation, Systems and Technologies (vol. 97). Cham: Springer. https://doi.org/10.1007/978-3-319-92028-3_19.

  45. Osei–Bryson, K.–M. (2006). An action learning approach for assessing the consistency of pairwise comparison data. European Journal of Operational Research, 174, 234–244.

    Google Scholar 

  46. Pant, S., Kumar, A., Ram, M., Klochkov, Y., & Sharma, H. K. (2022). Consistency Indices in Analytic Hierarchy Process: A Review. Mathematics, 10, 1206. https://doi.org/10.3390/math10081206.

    Article  Google Scholar 

  47. Peláez, J. I., & Lamata, M. T. (2003). A new measure of inconsistency for positive reciprocal matrices. Computers and Mathematics with Applications, 46(12), 1839–1845.

    Article  Google Scholar 

  48. Peláez, J. I., Martínez, E. A., & Vargas, L-G.(2018). Consistency in Positive Reciprocal Matrices: An Improvement in Measurement Methods. IEEE Access, 6, 25600–25609. https://doi.org/10.1109/ACCESS.2018.2829024.

    Article  Google Scholar 

  49. Saaty, T. L. (1980). Analytic Hierarchy Process. New York: McGraw-Hill.

    Google Scholar 

  50. Saaty, T. L. (2008). Decision making with the analytic hierarchy process, International Journal of Services Sciences, 1, 83–98.

    Article  Google Scholar 

  51. Salo, A. A., Hämäläinen, R. (1995). Preference Programming through Approximate Ratio Comparisons. European Journal of Operational Research, 82(3), 458–475.

    Article  Google Scholar 

  52. Sato, Y., & Tan, K. H. (2022). Inconsistency indices in pairwise comparisons: an improvement of the Consistency Index. Annals of Operations Research, 1–22. https://doi.org/10.1007/s10479-021-04431-3.

  53. Shiraishi, S., Obata, T., & Daigo, M. (1998). Properties of a Positive Reciprocal Matrix and their Application to AHP. Journal of the Operations Research Society of Japan, 41(3), 404–414.

    Article  Google Scholar 

  54. Siraj, S., Mikhailov, L., & Keane, J. A. (2015). PriEsT: an interactive decision support tool to estimate priorities from pairwise comparison judgments. International Transactions in Operational Research, 22, 217–235. https://doi.org/10.1111/itor.12054

    Article  Google Scholar 

  55. Stein, W. E., & Mizzi, P. J. (2007). The Harmonic Consistency Index for the AnalyticHierarchy Process. European Journal of Operational Research, 177(1), 488–497.

    Article  Google Scholar 

  56. Takeda, E. (1993). A note on consistent adjustments of pairwise comparison judgments. Mathematical and Computer Modelling, 17(7), 29–35.

    Article  Google Scholar 

  57. Tanino, T. (1984). Fuzzy preference orderings in group decision making. Fuzzy Sets and Systems, 12, 117–131.

    Article  Google Scholar 

  58. Vargas, L. G. (2008). The consistency index in reciprocal matrices: Comparison of deterministic and statistical approaches. European Journal of Operational Research, 191, 454–463.

    Article  Google Scholar 

  59. Wan, Z., Chen, M., & Zhang, L. (2013). New consistency index for comparison matrices and its properties. International Journal of Applied Mathematics and Statistics, 42(12), 206–218.

    Google Scholar 

  60. Wu, Z., & Xu, J. (2012). Inconsistency and consensus based decision support model for group decision making with multiplicative preference relations. Decision Support Systems, 52(3), 757–767.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

Copyright information

© 2023 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Mazurek, J. (2023). Inconsistency Indices and Their Properties. In: Advances in Pairwise Comparisons. Multiple Criteria Decision Making. Springer, Cham. https://doi.org/10.1007/978-3-031-23884-0_3

Download citation

Publish with us

Policies and ethics