A New Approach to Understanding and Finding Remedy for Rank Reversals in the Additive Analytic Hierarchy Process

The Analytic Hierarchy Process (AHP) which was developed by Saaty (1977, 1980, 1995) has been a widely used, discussed and debated multiple criteria decision making (MCDM) technique during the last twenty five years or more. Belton and Gear (1983) showed that additive aggregation method allowed rank order of the alternatives to change when a new alternative is added or an existing one is deleted. In the MCDM arena, this observation initiated debates that are still continuing, often raising questions about the validity of the AHP itself (Dyer, 1990a, b; Harker and Vargas, 1990; Saaty 1990). Millet and Saaty (2000) noted that rank reversal was a perfectly valid phenomenon in many cases, but rank should be preserved in many others. They promoted two types of synthesis procedures (i.e., the distributive and ideal modes) to address both contingencies and proposed guidelines for the selection of one aggregation mode over the other. These guidelines are vague and difficult to follow in practice. Later it has been further noted (Saaty, 2006) that there are rank reversals with both the distributive and ideal modes of aggregation in special cases. When there is only one criterion, the ranks are always preserved (Harker and Vargas, 1990; Saaty 1990) if the ratio judgments in the pairwise comparison matrix (PCM) are fully consistent. But, Stam and Duarte (2003) pointed out that rank reversals can be caused solely by inconsistency of pairwise comparisons even in the case of single criterion. However, this type of rank reversal is natural and acceptable (Leskinen and Kangas, 2005). However, because of normalization, the relative priorities of the objects have an arbitrary unit of measure. Although the sums are all equal to one, what is often ignored is that “one here is not necessarily equal to one there”. This essentially causes rank reversals in AHP aggregation.