Measuring Additive Consistency of Linguistic Preference Relations in a Personalized-Individual-Semantics Context: A Systematic Investigation With Axiomatic Design

Consistency is usually associated with transitive properties, among which the additive transitivity is one of the most popular methods. Although various linguistic additive consistency measurements have been proposed in the literature, the existing additive consistency measurements are usually directly proposed without a general definition with axiomatic design, and thus there is a lack of axiomatic comparisons among them. Moreover, the existing additive consistency measurements seldom consider the general case of personalized individual semantics (PIS) where a word has different numerical meanings to different people. In this study, we aim to develop a comparative study for the linguistic additive consistency measurements in the PIS context from an axiomatic perspective. Specifically, we first revisit the most widely used additive consistency measurements of simple linguistic preference relations (SLPRs) and hesitant fuzzy linguistic preference relations (HFLPRs) in the PIS context. Next, we present several rational axioms that the additive consistency measurements with PIS in SLPRs and HFLPRs should satisfy. Finally, we compare these additive consistency measurements by verifying the satisfaction of the proposed axioms. The results in this article show that the existing linguistic additive consistency measurements behave differently from the axiomatic perspective: they perform well in SLPRs but cannot satisfy all axioms in HFLPRs.