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

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.

Automated summary

This study provides a commentary on the ideas behind consistency definitions and indexes in interval multiplicative reciprocal matrices (IMRMs) and interval additive reciprocal matrices (IARMs). It compares two consistency definitions of interval-valued preference relations (IVPRs) and finds that defining the consistency in terms of imaginary intervals is equivalent to defining approximate consistency. Numerical examples illustrate differences between the two definitions.