Construction of hierarchical fuzzy relational models from input-output data

Changle Sun; Haitao Li; Wenxiu Zhao; Linlin Guo; Yalu Li; Changle Sun; Haitao Li; Wenxiu Zhao; Linlin Guo; Yalu Li

doi:10.3934/math.2026787

AIMS Mathematics

2026, Volume 11, Issue 7: 19379-19399. doi: 10.3934/math.2026787

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Research article

Construction of hierarchical fuzzy relational models from input-output data

1.
School of Artificial Intelligence, Shandong Women's University, Jinan 250399, China
2.
School of Mathematics and Statistics, Shandong Normal University, Jinan 250014, China
3.
School of Information Science and Engineering, University of Jinan, Jinan, 250022, China
4.
School of Mathematics and Statistics, Qingdao University, Qingdao, 266071, China

Received: 12 May 2026 Revised: 22 June 2026 Accepted: 26 June 2026 Published: 01 July 2026
MSC : 03B25, 15B15

As an important branch of fuzzy systems, hierarchical fuzzy systems (HFSs) are widely used in information science and engineering. Based on the semi-tensor product method, this paper investigates the hierarchical fuzzy relational models (HFRMs) of HFSs, and develops the corresponding construction algorithms. First, the construction methods of fuzzy relation matrices of fuzzy logic units (FLUs) are proposed by means of input-output data, based on which, the algebraic formulation is presented. After that, HFRMs of HFSs are explored via the fuzzy relation matrices based on the input-output data, and some effective algorithms are proposed to construct HFRMs with respect to the different hierarchical structures. Finally, the effectiveness of obtained results is verified by the on-ramp metering of freeway.
- {hierarchical fuzzy systems,
- hierarchical fuzzy relational models,
- fuzzy relation matrices,
- algebraic formulation,
- semi-tensor product of matrices}
Citation: Changle Sun, Haitao Li, Wenxiu Zhao, Linlin Guo, Yalu Li. Construction of hierarchical fuzzy relational models from input-output data[J]. AIMS Mathematics, 2026, 11(7): 19379-19399. doi: 10.3934/math.2026787

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Abstract

As an important branch of fuzzy systems, hierarchical fuzzy systems (HFSs) are widely used in information science and engineering. Based on the semi-tensor product method, this paper investigates the hierarchical fuzzy relational models (HFRMs) of HFSs, and develops the corresponding construction algorithms. First, the construction methods of fuzzy relation matrices of fuzzy logic units (FLUs) are proposed by means of input-output data, based on which, the algebraic formulation is presented. After that, HFRMs of HFSs are explored via the fuzzy relation matrices based on the input-output data, and some effective algorithms are proposed to construct HFRMs with respect to the different hierarchical structures. Finally, the effectiveness of obtained results is verified by the on-ramp metering of freeway.

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