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Measuring consistency of interval-valued preference relations: comments and comparison

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The concepts of consistency definition and consistency index are usually used to measure the consistency of a preference relation. When interval numbers are used to express the preference information, the consistency of the derived interval-valued preference relations (IVPRs) is worth being investigated. In this study, a comment is provided for the ideas behind consistency definitions and consistency indexes of interval multiplicative reciprocal matrices (IMRMs) and interval additive reciprocal matrices (IARMs), respectively. A comparison is made by considering the two kinds of consistency definitions of IVPRs. It is found that the method of defining the consistency of IVPRs in terms of the imaginary intervals is equivalent to that of defining the approximate consistency. Numerical examples are reported to illustrate the differences of the two consistency definitions of IVPRs. The observations illustrate that the fundamental inconsistency of IVPRs is compatible with the underlying idea of fuzzy sets. It is revealed that a consistent preference relation is only a particular case with a fixed value of the defined consistency index. In general, the consistency index could be used to quantify the deviation degree from a consistent real-valued preference relation.

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The authors would like to thank the anonymous reviewers for the valuable comments and constructive suggestions improving the quality of the paper.

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Correspondence to Fang Liu.

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The work was supported by the National Natural Science Foundation of China (Nos. 71571054, 71871072), the Guangxi Natural Science Foundation for Distinguished Young Scholars (No. 2016GXNSFFA380004), 2017 Guangxi high school innovation team and outstanding scholars plan, and the Innovation Project of Guangxi Graduate Education (No. YCSW2019045).

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Liu, F., Huang, M., Huang, C. et al. Measuring consistency of interval-valued preference relations: comments and comparison. Oper Res Int J (2020). https://doi.org/10.1007/s12351-020-00551-z

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  • Interval-valued preference relation (IVPR)
  • Consistency definition
  • Consistency index
  • Deviation degree
  • Equivalence