Incomplete pairwise comparisons in the analytic hierarchy process

The Analytic Hierarchy Process is a decision-analysis tool which was developed by T.L. Saaty in the 1970s and which has been applied to many different decision problems in corporate, governmental and other institutional settings. The most successful applications have come about in group decisionmaking sessions, where the group structures the problem in a hierarchical framework and pairwise comparisons are elicited from the group for each level of the hierarchy. However, the number of pairwise comparison necessary in a real problem often becomes overwhelming. For example, with 9 alternatives and 5 criteria, the group must answer 190 questions. This paper explores various methods for reducing the complexity of the preference eliciting process. The theory of a method based upon the graph-theoretic structure of the pairwise comparison matrix and the gradient of the right Perron vector is developed, and simulations of a series of random matrices are used to illustrate the properties of this approach.