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

Yunlong Zhang; Fengxia Jin; Jingfei Jiang; Juan LG Guirao; Huatao Chen; Yunlong Zhang; Fengxia Jin; Jingfei Jiang; Juan LG Guirao; Huatao Chen

doi:10.3934/era.2026283

Electronic Research Archive

2026, Volume 34, Issue 9: 6463-6482. doi: 10.3934/era.2026283

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

1.
Center for Dynamics and Intelligent Control Research, School of Mathematics and Statistics, Shandong University of Technology, ZiBo 255000, China
2.
School of Mathematical Sciences, Henan Institute of Science and Technology, Xinxiang 453003, China
3.
Department of Applied Mathematics and Statistics, Technical University of Cartagena, Hospital de Marina, Cartagena 30203, Spain
4.
Shandong Key Laboratory of Wind Power Equipment Testing Evaluation and Service Support Technology, Shandong University of Technology, ZiBo 255000, China

Received: 26 May 2026 Revised: 23 June 2026 Accepted: 14 July 2026 Published: 22 July 2026

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.
- linguistic judgment matrix,
- loop matrix,
- satisfactory consistency,
- transitive closure,
- ranking matrix
Citation: Yunlong Zhang, Fengxia Jin, Jingfei Jiang, Juan LG Guirao, Huatao Chen. A satisfactory consistency judging method of linguistic judgment matrix based on transitive closure[J]. Electronic Research Archive, 2026, 34(9): 6463-6482. doi: 10.3934/era.2026283

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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.

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