Specification of the AHP hierarchy and rank reversal

Xiao, Guang
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University of Delaware
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.