Milan Hladík and Martin Černý. Weak and strong consistency of an interval comparison matrix. In V.-N. Huynh et al., editor, Integrated Uncertainty in Knowledge Modelling and Decision Making, LNAI, pp. 15–25, Springer, Cham, 2020.
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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.
@inCollection{HlaCer2020b,
author = "Milan Hlad\'{\i}k and Martin {\v{C}}ern\'{y}",
title = "Weak and strong consistency of an interval comparison matrix",
editor = "Huynh et al., V.-N.",
feditor = "Huynh, Van-Nam and Entani, Tomoe and Jeenanunta, Chawalit and Inuiguchi, Masahiro and Yenradee, Pisal",
booktitle = "Integrated Uncertainty in Knowledge Modelling and Decision Making",
publisher = "Springer",
address = "Cham",
series = "LNAI",
fseries = "Lecture Notes in Artificial Intelligence",
volume = "12482",
pages = "15-25",
year = "2020",
doi = "10.1007/978-3-030-62509-2_2",
isbn = "978-3-030-62509-2",
issn = "0302-9743",
url = "https://doi.org/10.1007/978-3-030-62509-2_2",
bib2html_dl_html = "https://link.springer.com/chapter/10.1007/978-3-030-62509-2_2",
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.",
keywords = "Consistency; Decision making; Pairwise comparison matrix; Interval analysis",
}
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