An equivalent characterization for admissible orders on n-dimensional intervals generated by matrices

Abstract

The n-dimensional intervals are a generalization of intervals, and they are also the membership degrees of n-dimensional fuzzy sets. They can be used to represent high-dimensional data. This paper first defines a binary relation on n-dimensional intervals using a matrix. Then, we prove that this binary relation is an admissible order on n-dimensional intervals if and only if the determinant of the corresponding matrix is not equal to zero, and for each column of the matrix, the first non-zero element is greater than zero. Finally, we apply this type of admissible order to a specific multi-criteria group decision-making case.