Specification of the AHP hierarchy and rank reversal
Date
2010
Authors
Journal Title
Journal ISSN
Volume Title
Publisher
University of Delaware
Abstract
Developed by Dr. Saaty in the early 1970’s, the Analytic Hierarchy
Process is a multi-criteria decision making tool. The initial step in applying AHP is to
accurately decompose a decision problem into a decision hierarchy, avoiding both the
over-specification (including irrelevant criteria/alternatives) and underspecification
(omitting relevant criteria/alternatives).
Aull-Hyde and Duke (2006) introduced the concept of a minimal possible
priority weight, the smallest priority weight for any alternative/criterion among n
alternatives/criteria. They suggested using the minimal priority weight to detect an
over-specified hierarchy. If the priority weight associated with a specific
alternative/criterion is within 10% of the corresponding minimal possible weight, the
alternative/criterion should be considered for omission from the decision hierarchy.
However, they assumed perfect consistency when determining the minimal possible
priority weight.
The first focus of the thesis is to extend their methodology for the case of
an inconsistent pairwise comparison matrix. For the case of a 3x3 pairwise comparison
matrix, the minimal possible priority weight is shown to be a unique function of the
consistency ratio. For higher dimension pairwise comparison matrices, the concept of
a consistency ratio set is used to group potential pairwise comparison matrixes according to their consistency ratios. Within each set, we propose a representative
matrix for that set and use its smallest priority weight as the minimal weight for the
entire set. Moreover, we numerically show that the minimal priority weight is a
decreasing function of the consistency ratio, indicating that higher levels of
inconsistency will generate smaller minimal priority weights.
The second focus of the thesis is to investigate any potential link between
over-specified hierarchies and the rank reversal phenomenon, via Monte Carlo
simulation. The analysis reveals that, as expected, the risk of rank reversal (in matrices
having an acceptable level of inconsistency and are at risk for over-specification)
increases dramatically as the number of decision alternatives increases. Given that a
pairwise comparison matrix, with an acceptable level of inconsistency, exhibits rank reversal,
the likelihood that the associated hierarchy is at risk for over-specification is
no more than 5%. This result indicates that no strong link exists between an over-specified
hierarchy and rank reversal phenomenon.