Reciprocal distributions in the analytic hierarchy process

The paper analyses a special kind of probability distributions used to capture the existing uncertainty for the judgements in the Analytic Hierarchy Process (AHP). These distributions, called reciprocal, satisfy the reciprocity axiom of the AHP methodology. First, we define and characterise the reciprocal distributions and present some results and properties associated with them. We then study the relationship between the reciprocal and symmetrical random variables and the distribution of the priorities vector. Considering the eigenvector method (EGV), in the consistent case, and the row geometric mean method (RGM), as priorization methods, and assuming that the judgements follow reciprocal distributions, we prove that the elements of the priorities vector are distributed according to reciprocal random variables. The knowledge of these distributions will allow us to obtain the probability of two interesting events in practical selection problems: the probability that a given alternative would be ranked in the first position, and the probability of a given ranking (preference structure) for the whole set of alternatives. In particular, we find the analytic expressions for these probabilities when a reciprocal random variable, such as the lognormal, is used.