A satisfactory consistency judging method of linguistic judgment matrix based on transitive closure

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Abstract

Language phrases are effective tools for depicting uncertain information and are highly consistent with the cognitive habits and expression needs of decision-makers when they evaluate things using natural language. During the decision-making process, decision-makers can compare the alternative options in pairs to obtain a linguistic judgment matrix. First, the comparative preference relationship among schemes is regarded as a binary relation on the set of decision schemes. The preference relation matrix derived from the linguistic judgment matrix is, in essence, the matrix-based expression of this set relation. From the perspective of relational transitivity, if the transitive closure of this set relation coincides with the preference relation matrix, meaning the relation possesses complete transitivity, the linguistic judgment matrix can be determined to meet the requirement of satisfactory consistency. Second, if decision-makers have logical conflicts in pairwise comparisons, the corresponding linguistic judgment matrix will inevitably fail to satisfy the satisfactory consistency condition. For the precise identification of such logical contradictions, the second-order submatrix derived from the preference relation matrix within the linguistic judgment matrix is amenable to application. This submatrix can represent illogical preference judgments formed by 3 or 4 decision schemes. Therefore, by examining whether there exists an illogical preference structure in the second-order submatrix, we could inversely infer whether the original linguistic judgment matrix possesses satisfactory consistency. We conducted analysis combined with decision cases, confirming the feasibility and application value of this theoretical framework from the perspective of case application.

Keywords

linguistic judgment matrix / loop matrix / satisfactory consistency / transitive closure / ranking matrix

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