Narrow Results

We extend previous work regarding the transitivity of fuzzy matrices and propose the concept of completely transitive matrices. We further investigate their properties, indices, periods and relations with canonical fuzzy matrices. Our research shows that the completely transitive matrix type is a novel type, which contains previously defined transitive matrix types. Based on the proposed theories, we illustrate that completely transitive matrices can be applied in fuzzy orderings.

Fuzzy mathematics is a generalization in which fuzzy numbers replace real numbers and fuzzy arithmetic replaces real arithmetic. It is an excellent scope for modeling vague and uncertain aspects of the actual environments. In this important area, Dubois and Prade¹ defined a fuzzy matrix as a rectangular array of fuzzy numbers. They have also defined the LR type fuzzy numbers with some useful approximate arithmetic operators. The aim of this paper is to extend some useful aspects of linear algebra e.g. determinant, norm and eigenvalue for fuzzy matrices with LR fuzzy number entries by the use of fuzzy arithmetic. Finally, applications in fuzzy analytical hierarchy process (AHP) are investigated.

In this paper, the fuzzy analytic network process (FANP) is proposed. For achieving this purpose, two problems are highlighted and overcome in this paper. First, the postulate of the reciprocal matrix should be released, because this property is not satisfied in the fuzzy comparison matrix. Second, the convergent problem for raising the fuzzy supermatrix to limiting power should be appropriately handled. In this paper, we directly fuzzify Cogger and Yu's method for obtaining the fuzzy local vectors, because their method releases the postulate of the reciprocal matrix in the analytic hierarchy process (AHP). Then, we derive the particular matrix problem for obtaining the fuzzy global weight vector so that the convergent problem in a fuzzy limiting supermatrix can be overcome.

Eigenvalues and eigenvectors are widely used in various applications. Particularly, these concepts underlie analysis of consistency of a decision maker’s (DMs) preference knowledge. In real-world problems, DMs knowledge is inherently associated with imprecision and partial reliability. This involves combination of fuzzy and probabilistic information. The concept of a Z-number is a formal construct to describe such kind of information. In this study, we formulate the concepts of Z-number valued eigenvalue and eigenvector for matrices components of which are Z-numbers. A formal statement of the problem and a solution method for computation of Z-number valued eigensolutions are proposed. Numerical examples and an application devoted to foreign market selection problem are provided to show the usefulness of the proposed approach.