Sparse and distributed Analytic Hierarchy Process

Abstract

The Analytic Hierarchy Process (AHP) is a de-facto standard technique in centralized decision making. Consider a situation where there is a need to rank a set of elements or alternatives, based on their value or utility, of which we just know pairwise relative information, i.e., the ratio of their values. AHP proved an effective tool to retrieve the value of each element, being able to handle also relative information affected by distortions, subjective biases and intransitivity. A downside of AHP, however, is that it requires complete information, i.e., knowledge on all pairs. In this paper, we extend the applicability of the AHP technique to the case of sparse information, i.e., when only a limited amount of information is available, and such an information corresponds to an undirected connected graph. We complement our sparse framework by developing novel criteria and metrics to evaluate the degree of consistency of the data at hand. Moreover, exploiting the proposed framework, we also provide a distributed formulation of AHP in which a set of agents, interacting through an undirected graph, are able to compute their own values (e.g., for ranking or leader election purposes), by only knowing the ratio of their values with respect to their neighbors. To this end, we develop a novel algorithm to let each agent i compute, the dominant eigenvalue and the ith component of the corresponding eigenvector of the sparse AHP matrix. We conclude the paper with a simulation campaign that numerically demonstrates the effectiveness of the proposed approach.