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

This paper mainly studied the problem of solving interval type-2 fuzzy relation equations $\tilde{A}\circ \tilde{X}=\tilde{B}$. 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 ${\tilde{A}}_{\mu }\circ {\tilde{X}}_{\mu }={\tilde{B}}_{\mu }$ and secondary fuzzy matrix equation ${\tilde{A}}_f\circ {\tilde{X}}_f={\tilde{B}}_f$. Since all elements of ${\tilde{X}}_f$ 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.