The analytic hierarchy process: can wash criteria be ignored?

We define a wash criterion as one where the decision-maker is indifferent among the alternatives when they are compared on that criterion. In view of the Belton-Gear example and other such anomalies associated with the analytic hierarchy process (AHP), we ask whether eliminating a wash criterion will affect the overall ranking of objects. In the case where there is only one level of criteria, the rank-order of objects is unaffected by leaving out a wash criterion. However, in the case where the wash criterion is a subcriterion, the rank order may be affected by leaving it out.