The method of judging satisfactory consistency of linguistic judgment matrix based on adjacency matrix and 3-loop matrix

Fengxia Jin; Feng Wang; Kun Zhao; Huatao Chen; Juan L.G. Guirao; Fengxia Jin; Feng Wang; Kun Zhao; Huatao Chen; Juan L.G. Guirao

doi:10.3934/math.2024922

AIMS Mathematics

2024, Volume 9, Issue 7: 18944-18967. doi: 10.3934/math.2024922

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The method of judging satisfactory consistency of linguistic judgment matrix based on adjacency matrix and 3-loop matrix

1.
School of Mathematical Sciences, Henan Engineering and Technology Research Center of Digital Agriculture, Henan Institute of Science and Technology, Xinxiang 453003, China
2.
Zibo Hospital of Traditional Chinese Medicine, Zibo, 255046, China
3.
Beijing Electro-Mechanical Engineering Institute, Beijing 100074, China
4.
Center for Dynamics and Intelligent Control Research, School of Mathematics and Statistics, Shandong University of Technology, Zibo 255000, China
5.
Department of Applied Mathematics and Statistics, Technical University of Cartagena, Hospital de Marina, 30203 Cartagena, Spain

Received: 10 January 2024 Revised: 29 April 2024 Accepted: 13 May 2024 Published: 05 June 2024
MSC : 15, 94D05

Language phrases are an effective way to express uncertain pieces of information, and easily conforms to the language habits of decision makers to describe the evaluation of things. The consistency judgment of a linguistic judgment matrices is the key to analytic hierarchy process (AHP). If a linguistic judgment matrix has a satisfactory consistency, then the rank of the decision schemes can be determined. In this study, the comparison relation between the decision schemes is first represented by a directed graph. The preference relation matrix of the linguistic judgment matrix is the adjacency matrix of the directed graph. We can use the $ n - 1 $ st power of the preference relation to judge the linguistic judgment matrix whether has a satisfactory consistency. The method is utilized if there is one and only one element in the $ n - 1 $ st power of the preference relation, and the element 1 is not on the main diagonal. Then the linguistic judgment matrix has a satisfactory consistency. If there are illogical judgments, the decision schemes that form a 3-loop can be identified and expressed through the second-order sub-matrix of the preference relation matrix. The feasibility of this theory can be verified through examples. The corresponding schemes for illogical judgments are represented in spatial coordinate system.
- adjacency matrix,
- linguistic judgment matrix,
- satisfactory consistency,
- 3-loop matrix
Citation: Fengxia Jin, Feng Wang, Kun Zhao, Huatao Chen, Juan L.G. Guirao. The method of judging satisfactory consistency of linguistic judgment matrix based on adjacency matrix and 3-loop matrix[J]. AIMS Mathematics, 2024, 9(7): 18944-18967. doi: 10.3934/math.2024922

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Abstract

Language phrases are an effective way to express uncertain pieces of information, and easily conforms to the language habits of decision makers to describe the evaluation of things. The consistency judgment of a linguistic judgment matrices is the key to analytic hierarchy process (AHP). If a linguistic judgment matrix has a satisfactory consistency, then the rank of the decision schemes can be determined. In this study, the comparison relation between the decision schemes is first represented by a directed graph. The preference relation matrix of the linguistic judgment matrix is the adjacency matrix of the directed graph. We can use the $ n - 1 $ st power of the preference relation to judge the linguistic judgment matrix whether has a satisfactory consistency. The method is utilized if there is one and only one element in the $ n - 1 $ st power of the preference relation, and the element 1 is not on the main diagonal. Then the linguistic judgment matrix has a satisfactory consistency. If there are illogical judgments, the decision schemes that form a 3-loop can be identified and expressed through the second-order sub-matrix of the preference relation matrix. The feasibility of this theory can be verified through examples. The corresponding schemes for illogical judgments are represented in spatial coordinate system.

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