Solving interval type-2 fuzzy relation equations via semi-tensor product of interval matrices

Aidong Ge; Zhen Chang; Jun-e Feng; Aidong Ge; Zhen Chang; Jun-e Feng

doi:10.3934/mmc.2023027

Mathematical Modelling and Control

2023, Volume 3, Issue 4: 331-344. doi: 10.3934/mmc.2023027

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Solving interval type-2 fuzzy relation equations via semi-tensor product of interval matrices

1.
School of Mathematics, Shandong University, Ji'nan 250100, China
2.
School of Information and Automation, Qilu University of Technology (Shandong Academy of Sciences), Ji'nan 250353, China

Received: 31 January 2023 Revised: 14 May 2023 Accepted: 04 June 2023 Published: 14 December 2023

This paper mainly studied the problem of solving interval type-2 fuzzy relation equations $ \widetilde A \circ \widetilde X = \widetilde B $. First, to solve the interval type-2 fuzzy relation equations, we extend the semi-tensor product of matrices to interval matrices and give its specific definition. Second, the interval type-2 fuzzy relation equation was divided into two parts: primary fuzzy matrix equation $ {\widetilde A_\mu } \circ {\widetilde X_\mu }{\rm{ = }}{\widetilde B_\mu} $ and secondary fuzzy matrix equation $ {\widetilde A_f} \circ {\widetilde X_f} = {\widetilde B_f} $. Since all elements of $ {\widetilde X_f} $ equal to one, only the principal fuzzy matrix equation needs to be considered. Furthermore, it was proved that all solutions can be obtained from the parameter set solutions if the primary fuzzy matrix equation is solvable. Finally, with semi-tensor product of interval matrices, the primary fuzzy matrix equation was transformed into an algebraic equation and the specific algorithm for solving an interval type-2 fuzzy relation equation was proposed.
- interval type-2 fuzzy relation equations,
- semi-tensor product,
- semi-tensor product of interval matrices,
- primary fuzzy matrix equation
Citation: Aidong Ge, Zhen Chang, Jun-e Feng. Solving interval type-2 fuzzy relation equations via semi-tensor product of interval matrices[J]. Mathematical Modelling and Control, 2023, 3(4): 331-344. doi: 10.3934/mmc.2023027

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Abstract

This paper mainly studied the problem of solving interval type-2 fuzzy relation equations $ \widetilde A \circ \widetilde X = \widetilde B $. First, to solve the interval type-2 fuzzy relation equations, we extend the semi-tensor product of matrices to interval matrices and give its specific definition. Second, the interval type-2 fuzzy relation equation was divided into two parts: primary fuzzy matrix equation $ {\widetilde A_\mu } \circ {\widetilde X_\mu }{\rm{ = }}{\widetilde B_\mu} $ and secondary fuzzy matrix equation $ {\widetilde A_f} \circ {\widetilde X_f} = {\widetilde B_f} $. Since all elements of $ {\widetilde X_f} $ equal to one, only the principal fuzzy matrix equation needs to be considered. Furthermore, it was proved that all solutions can be obtained from the parameter set solutions if the primary fuzzy matrix equation is solvable. Finally, with semi-tensor product of interval matrices, the primary fuzzy matrix equation was transformed into an algebraic equation and the specific algorithm for solving an interval type-2 fuzzy relation equation was proposed.

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