A general formulation for some inconsistency indices of pairwise comparisons
We propose a unifying approach to the problem of measuring the inconsistency of judgments. More precisely, we define a general framework to allow several well-known inconsistency indices to be expressed as special cases of this new formulation. We consider inconsistency indices as aggregations of ‘local’, i.e. triple-based, inconsistencies. We show that few reasonable assumptions guarantee a set of good properties for the obtained general inconsistency index. Under this representation, we prove a property of Pareto efficiency and show that OWA functions and t-conorms are suitable aggregation functions of local inconsistencies. We argue that the flexibility of this proposal allows tuning of the index. For example, by using different types of OWA functions, the analyst can obtain the desired balance between an averaging behavior and a ‘largest inconsistency-focused’ behavior.
KeywordsPairwise comparisons Multiplicative preference relations Consistency Inconsistency index Analytic hierarchy process Aggregation OWA functions
The authors acknowledge the anonymous reviewers for their constructive comments which helped improve the original version of this manuscript.
- Beliakov, G., Pradera, A., & Calvo, T. (2007). Aggregation functions: A guide for practitioners, studies in fuzziness and soft computing (Vol. 221). Berlin: Springer.Google Scholar
- Chiclana, F., Herrera, F., & Herrera-Viedma, E. (2000). The ordered weighted geometric operator: Properties and application in MCDM problems. In Proceedings of 8th conference on information processing and management of uncertainty in knowledge-based systems (IPMU).Google Scholar
- Csató, L. (2017). Characterization of an inconsistency ranking for pairwise comparison matrices. Annals of Operations Research, 261(1–2), 155–165.Google Scholar
- Grabisch, M., Marichal, J. L., Mesiar, R., & Pap, E. (2009). Aggregation functions. Encyclopedia of mathematics and its applications (Vol. 127). Cambridge: Cambridge University Press.Google Scholar
- Keeney, R. L., & Raiffa, H. (1976). Decisions with multiple objectives: Preferences and value tradeoffs. New York: Wiley.Google Scholar
- Klir, G. J., & Yuan, B. (1995). Fuzzy sets and fuzzy logic: Theory and applications. Upper Saddle River: Pretience Hall.Google Scholar
- Koczkodaj, W. W., & Szwarc, R. (2014). On axiomatization of inconsistency indicators in pairwise comparisons. Fundamenta Informaticae, 132(4), 485–500.Google Scholar
- Obata, T., Shiraishi, S., Daigo, M., Nakajima, N. (1999). Assessment for an incomplete comparison matrix and improvement of an inconsistent comparison: Computational experiments. In ISAHP Google Scholar
- O’Hagan, M. (1988). Aggregating template or rule antecedents in real-time expert systems with fuzzy set logic. In Twenty-second asilomar conference on signals, systems and computers (pp. 681–689).Google Scholar