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Weak and Strong Consistency of an Interval Comparison Matrix

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Integrated Uncertainty in Knowledge Modelling and Decision Making (IUKM 2020)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 12482))

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Abstract

We consider interval-valued pairwise comparison matrices and two types of consistency – weak (consistency for at least one realization) and strong (acceptable consistency for all realizations). Regarding weak consistency, we comment on the paper [Y. Dong and E. Herrera-Viedma, Consistency-Driven Automatic Methodology to Set Interval Numerical Scales of 2-Tuple Linguistic Term Sets and Its Use in the Linguistic GDM With Preference Relation, IEEE Trans. Cybern., 45(4):780–792, 2015], where, among other results, a characterization of weak consistency was proposed. We show by a counterexample that in general the presented condition is not sufficient for weak consistency. It provides a full characterization only for matrices up to size of three. We also show that the problem of having a closed form expression for weak consistency is closely related with P-completeness theory and that an optimization version of the problem is indeed P-complete. Regarding strong consistency, we present a sufficient condition and a necessary condition, supplemented by a small numerical study on their efficiency. We leave a complete characterization as an open problem.

With a correction on paper DOI 10.1109/TCYB.2014.2336808.

Supported by the Czech Science Foundation Grant P403-18-04735S.

The counterexample and the correction provided by this paper were originally submitted to IEEE Trans. Cybern. as a technical note, but the journal editors resigned to accept this note which points to incorrect statements published there.

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Notes

  1. 1.

    Sometimes the notion of “acceptable consistency” is used, but due to ambiguity we do not use it in this sense.

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Authors and Affiliations

  1. Faculty of Mathematics and Physics, Department of Applied Mathematics, Charles University, Malostranské nám. 25, 118 00, Prague, Czech Republic

    Milan Hladík & Martin Černý

Authors
  1. Milan Hladík
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  2. Martin Černý
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Correspondence to Milan Hladík .

Editor information

Editors and Affiliations

  1. Japan Advanced Institute of Science and Technology, Nomi, Ishikawa, Japan

    Van-Nam Huynh

  2. University of Hyogo, Kobe, Japan

    Tomoe Entani

  3. Sirindhorn International Institute of Technology, Thammasat University, Pathum Thani, Thailand

    Chawalit Jeenanunta

  4. Graduate School of Engineering Science, Osaka University, Toyonaka, Osaka, Japan

    Masahiro Inuiguchi

  5. Sirindhorn International Institute of Technology, Thammasat University, Pathum Thani, Thailand

    Pisal Yenradee

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Hladík, M., Černý, M. (2020). Weak and Strong Consistency of an Interval Comparison Matrix. In: Huynh, VN., Entani, T., Jeenanunta, C., Inuiguchi, M., Yenradee, P. (eds) Integrated Uncertainty in Knowledge Modelling and Decision Making. IUKM 2020. Lecture Notes in Computer Science(), vol 12482. Springer, Cham. https://doi.org/10.1007/978-3-030-62509-2_2

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  • DOI: https://doi.org/10.1007/978-3-030-62509-2_2

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