Skip to main content
Log in

Studying a set of properties of inconsistency indices for pairwise comparisons

  • Original Paper
  • Published:
Annals of Operations Research Aims and scope Submit manuscript

Abstract

Pairwise comparisons between alternatives are a well-established tool to decompose decision problems into smaller and more easily tractable sub-problems. However, due to our limited rationality, the subjective preferences expressed by decision makers over pairs of alternatives can hardly ever be consistent. Therefore, several inconsistency indices have been proposed in the literature to quantify the extent of the deviation from complete consistency. Only recently, a set of properties has been proposed to define a family of functions representing inconsistency indices. The scope of this paper is twofold. Firstly, it expands the set of properties by adding and justifying a new one. Secondly, it continues the study of inconsistency indices to check whether or not they satisfy the above mentioned properties. Out of the four indices considered in this paper, in their present form, two fail to satisfy some properties. An adjusted version of one index is proposed so that it fulfills them.

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

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5

Similar content being viewed by others

References

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

    Article  Google Scholar 

  • Barzilai, J. (1997). Deriving weights from pairwise comparison matrices. The Journal of the Operational Research Society, 48(12), 1226–1232.

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Belton, V., & Gear, T. (1983). On a short-coming of Saaty’s method of analytic hierarchies. Omega, 11(3), 228–230.

    Article  Google Scholar 

  • Bozóki, S., & Rapcsák, T. (2008). On Saaty’s and Koczkodaj’s inconsistencies of pairwise comparison matrices. Journal of Global Optimization, 42(2), 157–175.

    Article  Google Scholar 

  • Bozóki, S., Dezső, L., Poesz, A., & Temesi, J. (2013). Analysis of pairwise comparison matrices: an empirical research. Annals of Operations Research, 211(1), 511–528.

    Article  Google Scholar 

  • Brunelli, M. (2011). A note on the article “Inconsistency of pair-wise comparison matrix with fuzzy elements based on geometric mean” [Fuzzy Sets and Systems 161 (2010) 1604–1613]. Fuzzy Sets and Systems, 176(1), 76–78.

    Article  Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

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

    Article  Google Scholar 

  • Brunelli, M., & Fedrizzi, M. (2015b). Boundary properties of the inconsistency of pairwise comparisons in group decisions. European Journal of Operational Research, 230(3), 765–773.

    Article  Google Scholar 

  • Cavallo, B., & D’Apuzzo, L. (2009). A general unified framework for pairwise comparison matrices in multicriterial methods. International Journal of Intelligent Systems, 24(4), 377–398.

    Article  Google Scholar 

  • Cavallo, B., & D’Apuzzo, L. (2012). Investigating properties of the \(\odot \)-consistency index. In Advances in Computational Intelligence. Communications in Computer and Information Science (Vol. 4, pp. 315–327).

  • Chen, K., Kou, G., Tarn, J. M., & Song, Y. (2015). Bridging the gap between missing and inconsistent values in eliciting preference from pairwise comparison matrices. Annals of Operations Research, 235(1), 155–175.

    Article  Google Scholar 

  • Cook, W. D., & Kress, M. (1988). Deriving weights from pairwise comparison ratio matrices: An axiomatic approach. European Journal of Operational Research, 37(3), 355–362.

    Article  Google Scholar 

  • Duszak, Z., & Koczkodaj, W. W. (1994). Generalization of a new definition of consistency for pairwise comparisons. Information Processing Letters, 52(5), 273–276.

    Article  Google Scholar 

  • Dyer, J. S. (1990a). Remarks on the analytic hierarchy process. Management Science, 36(3), 249–258.

    Article  Google Scholar 

  • Dyer, J. S. (1990b). A clarification of “Remarks on the analytic hierarchy process”. Management Science, 36(3), 274–275.

    Article  Google Scholar 

  • Ergu, D., Kou, G., Peng, Y., & Shi, Y. (2011). A simple method to improve the consistency ratio of the pair-wise comparison matrix in ANP. European Journal of Operational Research, 213(1), 246–259.

    Article  Google Scholar 

  • Fichtner, J. (1986). On deriving priority vectors from matrices of pairwise comparisons. Socio-Economic Planning Sciences, 20(6), 341–345.

    Article  Google Scholar 

  • Fishburn, P. C. (1968). Utility theory. Management Science, 14(5), 335–378.

    Article  Google Scholar 

  • Fishburn, P. C. (1999). Preference relations and their numerical representations. Theoretical Computer Science, 217(2), 359–383.

    Article  Google Scholar 

  • Gass, S. I. (2005). Model world: The great debate—MAUT versus AHP. Interfaces, 35(4), 308–312.

    Article  Google Scholar 

  • Hämäläinen, R. P., & Pöyhönen, M. (1996). On-line group decision support by preference programming in traffic planning. Group Decision and Negotiation, 5(4–6), 485–500.

    Article  Google Scholar 

  • Herman, M. W., & Koczkodaj, W. W. (1996). A Monte Carlo study of pairwise comparison. Information Processing Letters, 57(1), 25–29.

    Article  Google Scholar 

  • Irwin, F. W. (1958). An analysis of the concepts of discrimination and preference. The American Journal of Psychology, 71(1), 152–163.

    Article  Google Scholar 

  • Ishizaka, A., & Lusti, M. (2006). How to derive priorities in AHP: A comparative study. Central European Journal of Operations Research, 14(4), 387–400.

    Article  Google Scholar 

  • Kakiashvili, T., Koczkodaj, W. W., & Woodbury-Smith, M. (2012). Improving the medical scale predictability by the pairwise comparisons method: Evidence from a clinical data study. Computer Methods and Programs in Biomedicine, 105(3), 210–216.

    Article  Google Scholar 

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

    Google Scholar 

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

    Article  Google Scholar 

  • Koczkodaj, W. W., Herman, M. W., & Orlowski, M. (1999). Managing null entries in pairwise comparisons. Knowledge and Information Systems, 1(1), 119–125.

    Article  Google Scholar 

  • Koczkodaj, W. W., Kulakowski, K., & Ligeza, A. (2014). On the quality evaluation of scientific entities in Poland supported by consistency-driven pairwise comparisons method. Scientometrics, 99(3), 911–926.

    Article  Google Scholar 

  • Kou, G., & Lin, C. (2014). A cosine maximization method for the priority vector derivation in AHP. European Journal of Operational Research, 235(1), 225–232.

    Article  Google Scholar 

  • Kułakowski, K. (2015). Notes on order preservation and consistency in AHP. European Journal of Operational Research, 245(1), 333–337.

    Article  Google Scholar 

  • Lamata, M. T., & Peláez, J. I. (2002). A method for improving the consistency of judgements. International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, 10(6), 677–686.

    Article  Google Scholar 

  • Lin, C., Kou, G., & Ergu, D. (2013). An improved statistical approach for consistency test in AHP. Annals of Operations Research, 211(1), 289–299.

  • Luce, R. D., & Suppes, P. (1965). Preference, utility and subjective probability. In R. D. Luce, R. R. Bush, & E. H. Galanter (Eds.), Handbook of Mathematical Psychology (pp. 249–410). New York: Wiley.

  • Luce, R. D., & Raiffa, H. (1957). Games and decisions. New York: Wiley.

  • Maleki, H., & Zahir, S. (2013). A comprehensive literature review of the rank reversal phenomenon in the analytic hierarchy process. Journal of Multi-Criteria Decision Analysis, 20(3–4), 141–155.

    Article  Google Scholar 

  • Mustajoki, J., & Hämäläinen, R. P. (2000). Web-HIPRE: Global decision support by value tree and AHP analysis. INFOR Journal, 38(3), 208–220.

    Google Scholar 

  • Nikou, S., & Mezei, J. (2013). Evaluation of mobile services and substantial adoption factors with Analytic Hierarchy Process (AHP) analysis. Telecommunications Policy, 37(10), 915–929.

    Article  Google Scholar 

  • Nikou, S., Mezei, J., & Sarlin, P. (2015). A process view to evaluate and understand preference elicitation. Journal of Multi-Criteria Decision Analysis, 22(5–6), 305–329.

    Article  Google Scholar 

  • Pereira, V., & Costa, H. G. (2015). Nonlinear programming applied to the reduction of inconsistency in the AHP method. Annals of Operations Research, 229(1), 635–655.

    Article  Google Scholar 

  • Ramík, J., & Korviny, P. (2010). Inconsistency of pair-wise comparison matrix with fuzzy elements based on geometric mean. Fuzzy Sets and Systems, 161(11), 1604–1613.

    Article  Google Scholar 

  • Saaty, T. L. (1993). What is relative measurement? The ratio scale phantom. Mathematical and Computer Modelling, 17(4), 1–12.

    Article  Google Scholar 

  • Saaty, T. L. (2013). The modern science of multicriteria decision making and its practical applications: The AHP/ANP approach. Operations Research, 61(5), 1101–1118.

    Article  Google Scholar 

  • Salo, A. A., & Hämäläinen, R. P. (1995). Preference programming through approximate ratio comparisons. European Journal of Operational Research, 82(3), 458–475.

    Article  Google Scholar 

  • Salo, A. A., & Hämäläinen, R. P. (1997). On the measurement of preferences in the analytic hierarchy process. Journal of Multi-Criteria Decision Analysis, 6(6), 309–319.

    Article  Google Scholar 

  • Shiraishi, S., Obata, T., Daigo, M., & Nakajima, N. (1999). Assessment for an incomplete comparison matrix and improvement of an inconsistent comparison: computational experiments. In ISAHP 1999.

  • Stein, W. E., & Mizzi, P. J. (2007). The harmonic consistency index for the analytic hierarchy process. European Journal of Operational Research, 177(1), 488–497.

    Article  Google Scholar 

  • Tanino, T. (1984). Fuzzy preference orderings in group decision making. Fuzzy Sets and Systems, 12(2), 117–131.

    Article  Google Scholar 

  • Watson, S. R., & Freeling, A. N. S. (1982). Assessing attribute weights. Omega, 10(6), 582–583.

    Article  Google Scholar 

  • Watson, S. R., & Freeling, A. N. S. (1983). Comment on: assessing attribute weights by ratios. Omega, 11(1), 13.

    Google Scholar 

  • Wu, Z., & Xu, J. (2012). A consistency 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

Acknowledgments

The author is grateful to the reviewers and the Associate Editor for their precious comments. The manuscript benefited from the author’s discussions with Michele Fedrizzi and Ragnar Freij. A special mention goes to Sándor Bozóki who also had the intuition that a property might have been missing. This research has been financially supported by the Academy of Finland.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Matteo Brunelli.

Appendix: Proof of Theorem 1

Appendix: Proof of Theorem 1

To prove logical consistency, it is sufficient to find an instance of \(I:{\mathcal {A}}\rightarrow {\mathbb {R}}\) which satisfies all the properties P1–P6. One such instance is the following function

$$\begin{aligned} I^{*}({\mathbf {A}})=\sum _{i=1}^{n-2}\sum _{j=i+1}^{n-1}\sum _{k=j+1}^{n} \left( \frac{a_{ik}}{a_{ij}a_{jk}} + \frac{a_{ij}a_{jk}}{a_{ik}} - 2 \right) \end{aligned}$$
(14)

To prove the independence of P1–P6, it is sufficient to find a function satisfying all properties except one, for all the properties. The examples of inconsistency indices proposed by Brunelli and Fedrizzi (2015a) to prove the independence of the system P1–P5 are invariant under transposition. If follows that P1–P5 are logically independent within the system P1–P6. It remains to show that P6 does not depend on P1–P5. Consider that, if \({\mathbf {A}}\) has one row, say H, whose non-diagonal elements are all greater than one, i.e. \(a_{Hj}>1~\forall j\ne H\), then this property is shared by any matrix \({\mathbf {P}}{\mathbf {A}}{\mathbf {P}}^{T}\), where \({\mathbf {P}}\) is any permutation matrix, but not by its transpose \({\mathbf {A}}^{T}\). Taking into account the inconsistency index \(I^{*}\) in (14), and defining H as the row with the greatest non-diagonal element, then the function

$$\begin{aligned} I_{\lnot 6}({\mathbf {A}})=I^{*}({\mathbf {A}}) \cdot {\mathop {\underbrace{\left( 1 + \max \left\{ \min _{j\ne H} \{a_{Hj} - 1 \} ,0 \right\} \right) }}\limits _{M}} \end{aligned}$$
(15)

is invariant under row-column permutation but not under transposition. Hence, \(I_{\lnot 6}\) satisfies AP but not P6. To prove the independence of P6, it remains to show that (15) satisfies P1 and P3–P5. It is easy, and thus omitted, to show that P1, P3, and P5 are satisfied. To prove it for P4, we note that any \({\mathbf {A}}\in {\mathcal {A}}^{*}\) can be rewritten as

$$\begin{aligned} {\mathbf {A}}= \begin{pmatrix} 1 &{}\quad \! a_{12} &{}\quad \! a_{12}a_{23} &{}\quad \! \ldots &{}\quad \! a_{12} \cdot \ldots \cdot a_{n-1 \, n} \\ \frac{1}{a_{12}} &{}\quad \! 1 &{}\quad \! a_{23} &{}\quad \! \ldots &{}\quad \! a_{23} \cdot \ldots \cdot a_{n-1 \, n} \\ \ldots &{}\quad \! \ldots &{}\quad \! \ldots &{}\quad \! \ldots &{}\quad \! \ldots \\ \frac{1}{a_{12} \cdot \ldots \cdot a_{n-2 \, n-1}} &{}\quad \! \frac{1}{a_{23} \cdot \ldots \cdot a_{n-2 \, n-1}}&{}\quad \! \frac{1}{a_{34} \cdot \ldots \cdot a_{n-2 \, n-1}} &{}\quad \! \ldots &{}\quad \! a_{n-1 \, n} \\ \frac{1}{a_{12} \cdot \ldots \cdot a_{n-1 \, n}} &{}\quad \! \frac{1}{a_{23} \cdot \ldots \cdot a_{n-1 \, n}} &{}\quad \! \frac{1}{a_{34} \cdot \ldots \cdot a_{n-1 \, n}}&{}\quad \! \ldots &{}\quad \! 1 \end{pmatrix} \in {\mathcal {A}}^{*} \end{aligned}$$
(16)

Without loss of generality let us consider \(a_{1n}\) and its reciprocal \(a_{n1}\) and replace them with \(a_{1n}^{\delta }\) and \(a_{n1}^{\delta }\), respectively. Then, by calling \({\mathbf {A}}_{1n}^{\delta }\) the new matrix and bearing in mind that \({\mathbf {A}} \in {\mathcal {A}}^{*}\), we have

$$\begin{aligned} I^{*}({\mathbf {A}}_{1n}^{\delta })&= \sum _{j=2}^{n-1} \left( \frac{a_{1n}^{\delta }}{a_{1j}a_{jn}} + \frac{{a_{1j}a_{jn}}}{{a_{1n}^{\delta }}} - 2 \right) \\&= (n-2) \left( \frac{\left( a_{12} \cdot \ldots \cdot a_{n-1 \, n} \right) ^{\delta }}{a_{12} \cdot \ldots \cdot a_{n-1 \, n}} + \frac{a_{12} \cdot \ldots \cdot a_{n-1 \, n}}{\left( a_{12} \cdot \ldots \cdot a_{n-1 \, n} \right) ^{\delta }} -2 \right) \end{aligned}$$

If \(H \notin \{1,n \}\), then, in (15) M is constant and P4 holds in this case. Also if \(H \in \{1,n \}\) and \(\min _{j \ne H} \{a_{Hj} \} \ne a_{1n}\), then M is constant and P4 is satisfied. Finally, if \(H=1\) and \(\min _{j \ne H} \{a_{Hj} \} = a_{1n}\), it is

$$\begin{aligned} I_{\lnot 6}\left( {\mathbf {A}}_{1n}^{\delta }\right) = I^{*}\left( {\mathbf {A}}_{1n}^{\delta }\right) \cdot \left( 1 + a_{1n}^{\delta } - 1 \right) = I^{*}\left( {\mathbf {A}}_{1n}^{\delta }\right) \cdot a_{1n}^{\delta } \end{aligned}$$
(17)

which can be reduced to

$$\begin{aligned} I_{\lnot 6}({\mathbf {A}}_{1n}^{\delta })&={\mathop {\overbrace{(n-2) \left( \frac{\left( a_{12} \cdot \ldots \cdot a_{n-1 \, n} \right) ^{\delta }}{a_{12} \cdot \ldots \cdot a_{n-1 \, n}} + \frac{a_{12} \cdot \ldots \cdot a_{n-1 \, n}}{\left( a_{12} \cdot \ldots \cdot a_{n-1 \, n} \right) ^{\delta }} -2 \right) }}\limits ^{I^{*}({\mathbf {A}}^{\delta }_{1n})}}\\&\quad \cdot {\mathop {\overbrace{(a_{12} \ldots a_{n-1 \,n})^{\delta }}}\limits ^{a_{1n}^{\delta }}}\\&= (n-2) \left( \frac{a_{1n}^{2 \delta }}{a_{1n}} + a_{1n} - 2 a_{1n}^{\delta } \right) . \end{aligned}$$

Considering that, from \({\mathbf {A}}\in {\mathcal {A}}^{*}\) and \(H=1\), it follows that \(a_{1n} \ge 1\) and the partial derivative in \(\delta \) is

$$\begin{aligned} \frac{\partial I_{\lnot 6}({\mathbf {A}}_{1n}^{\delta })}{\partial \delta }&= (n-2) \left( 2a_{1n}^{2\delta - 1}\log (a_{1n}) - 2 a_{1n}^{\delta } \log (a_{1n}) \right) \\&={\mathop {\underbrace{(n-2)}}\limits _{>0}} {\mathop {\underbrace{\left( 2 a_{1n}^{\delta - 1}\right) }}\limits _{>0}} \left( a_{1n}^{\delta }-a_{1n} \right) {\mathop {\underbrace{\left( \log a_{1n} \right) }}\limits _{>0}}, \end{aligned}$$

which is always non-negative for \(\delta >1\) and non-positive for \( 0 < \delta < 1\). Thus, P4 is satisfied and the properties P1–P6 are logically independent. \(\square \)

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Brunelli, M. Studying a set of properties of inconsistency indices for pairwise comparisons. Ann Oper Res 248, 143–161 (2017). https://doi.org/10.1007/s10479-016-2166-8

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10479-016-2166-8

Keywords

Navigation