Decision Sciences Journal
Volume 28, Number 3
Summer 1997
Stochastic Judgments in the AHP: The Measurement of Rank
Reversal Probabilities
Antonie Stam
Terry College of Business, The University of Georgia, Athens, GA
30602, U.S.A., and International Institute for Applied Systems
Analysis, A-2361 Laxenburg, Austria
A. Pedro Duarte Silva
Universidade Catolica Portuguesa, Faculdade de Cincias Economicas
e Empresariais, Centro Regional do Porto, Rua Diogo Botelho 1327,
4150 Porto, Portugal
ABSTRACT
This paper presents a methodology for analyzing Analytic Hierarchy
Process (AHP) rankings if the pairwise preference judgments are
uncertain (stochastic). If the relative preference statements are
represented by judgment intervals, rather than single values, then
the rankings resulting from a traditional (deterministic) AHP
analysis based on single judgment values may be reversed, and
therefore incorrect. In the presence of stochastic judgments, the
traditional AHP rankings may be stable or unstable, depending on
the nature of the uncertainty.
We develop multivariate statistical techniques to obtain both point
estimates and confidence intervals of the rank reversal
probabilities, and show how simulation experiments can be used as
an effective and accurate tool for analyzing the stability of the
preference rankings under uncertainty. If the rank reversal
probability is low, then the rankings are stable and the decision
maker can be confident that the AHP ranking is correct. However, if
the likelihood of rank reversal is high, then the decision maker
should interpret the AHP rankings cautiously, as there is a
subtantial probability that these rankings are incorrect. High rank
reversal probabilities indicate a need for exploring alternative
problem formulations and methods of analysis.
The information about the extent to which the ranking of the
alternatives is sensitive to the stochastic nature of the pairwise
judgments should be valuable information into the decision-making
process, much like variability and confidence intervals are crucial
tools for statistical inference. We provide simulation experiments
and numerical examples to evaluate our method.
Our analysis of rank reversal due to stochastic judgments is not
related to previous research on rank reversal that focuses on
mathematical properties inherent to the AHP methodology, for
instance, the occurrence of rank reversal if a new alternative is
added or an existing one is deleted.
Subject Areas: Analytic Hierarchy Process, Decision
Analysis, Judgments, Multicriteria Decision Making, and
Uncertainty.
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