Stochastic Judgments in the AHP: The Measurement of Rank Reversal Probabilities

Stam, A. & Duarte Silva, A.P. (1994). Stochastic Judgments in the AHP: The Measurement of Rank Reversal Probabilities. IIASA Working Paper. IIASA, Laxenburg, Austria: WP-94-101

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Abstract

Recently, the issue of rank reversal of alternatives in the Analytic Hierarchy Process (AHP) has captured the attention of a number of researchers. Most of the research on rank reversal has addressed the case where the pairwise comparisons of the alternatives are represented by single values, focusing on mathematical properties inherent to the AHP methodology that can lead to rank reversal if a new alternative is added or an existing one is deleted. A second situation, completely unrelated to the mathematical foundations of the AHP, in which rank reversal can occur is the case where the pairwise judgments are stochastic, rather than single values.

If the relative preference ratings are uncertain, one has judgment intervals, and as a consequence there is a possibility that the rankings resulting from an AHP analysis are reversed, i.e., incorrect. It is important for modeler and decision maker alike to be aware of the likelihood that this situation of rank reversal will occur. In this paper, we introduce methods for assessing the relative preference of the alternatives in terms of their rankings, if the pairwise comparisons of the alternatives are stochastic.

We develop multivariate statistical techniques to obtain 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. This 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. Although the focus of our analysis is on stochastic preference judgments, our sampling method for estimating rank reversal probabilities can be extended to the case of non-stochastic imprecise fuzzy judgments.

We provide simulation experiments and numerical examples comparing our method with that proposed previously by Saaty and Vargas (1987) for imprecise interval judgments.

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