Exploring the interface between hierarchies, multiple objectives and fuzzy sets

Thomas Saaty
Joseph M. Katz Graduate School of Business
University of Pittsburgh
United States

Publication date: Jan, 0978

Journal: Fuzzy sets and systems
Vol.: 1- Issue: 1- Pages: 57-68

Abstract: The complexity of experience acquired through our senses and as interpreted by our mind, is fuzzy and must remain so as long as the meaning of things change as they are embedded in larger or different contexts to relate them to new ideas and new experiences. Here we give a method for measuring the relativity of fuzziness by structuring the functions of a system hierarchically in a multiple objective framework. The method is based on computing the principal eigenvector of a positive matrix with reciprocal entries (i.e., aji = 1/aij). The eigenvector provides an estimate for an (assumed) underlying ratio scale. For a single property the scale provides a measure of the grade of membership of elements in a fuzzy set according to that property and the departure of the eigenvalue from the dimension of the matrix serves as a measure of departure from consistency. For a number of properties, the principle of hierarchical composition enables us to compose the eigenvectors into a priority vector which measures the fuzziness of the elements in the lowest level of the hierarchy with respect to the relative dominance of the purposes or properties represented in the hierarchy.

Keywords: Hierarchies, Multiple objectives, Fuzzy sets