Modeling unstructured decision problems — the theory of analytical hierarchies

Quantitative modeling of unstructured decision problems with social implications is new and challenging and has pressing needs. A new approach to scaling using largest eigenvalues and reciprocal matrices and the effect of inconsistent judgment are introduced and relevant theory discussed. In this approach inconsistency is accepted as a fact but measured to determine how bad it is. Since most decision problems are hierarchical in form as they fulfill higher and still higher objectives, the appropriate structure for representation is a hierarchy. A new formal definition of a hierarchy is given and the notion of measurement with eigenvalues is extended to hierarchies. Both the eigenvalue approach to measurement and the hierarchical approach are illustrated with examples. Finally, unstructured problems are illustrated through applications of forward-backward planning, a two-point boundary value problem.