Pairwise comparison

Quantum mechanics

Scale (ratio)

Discrete mathematics

Statistics

Materials science

Applied mathematics

Accounting

Eigenvalues and eigenvectors

Economics

Market economy

Data mining

Computer science

Scaling

Statistical physics

Business

Matrix (chemical analysis)

Physics

Hierarchy

Variance (accounting)

Mathematics

Politics

Political science

Multidimensional scaling

Geometry

Composite material

Representation (politics)

Measure (data warehouse)

Consistency (knowledge bases)

Journal of Mathematical Psychology

The purpose of this paper is to investigate a method of scaling ratios using the principal eigenvector of a positive pairwise comparison matrix. Consistency of the matrix data is defined and measured by an expression involving the average of the nonprincipal eigenvalues. We show that λmax = n is a necessary and sufficient condition for consistency. We also show that twice this measure is the variance in judgmental errors. A scale of numbers from 1 to 9 is introduced together with a discussion of how it compares with other scales. To illustrate the theory, it is then applied to some examples for which the answer is known, offering the opportunity for validating the approach. The discussion is then extended to multiple criterion decision making by formally introducing the notion of a hierarchy, investigating some properties of hierarchies, and applying the eigenvalue approach to scaling complex problems structured hierarchically to obtain a unidimensional composite vector for scaling the elements falling in any single level of the hierarchy. A brief discussion is also included regarding how the hierarchy serves as a useful tool for decomposing a large-scale problem, in order to make measurement possible despite the now-classical observation that the mind is limited to 7 ± 2 factors for simultaneous comparison.

A scaling method for priorities in hierarchical structures