Multi-Hierarchy Methods in AHP: Methodological Issues and Solutions

Maggie Yu Yue

Abstract

The Analytic Hierarchy Process (AHP) as a structured technique has been widely discussed and used in multi-criteria decision analysis. In many applications of AHP, benefit and cost are measured separately hence two vectors are associated with a given situation. Since decision maker has different directions of preference over benefit and cost, a single-hierarchy approach cannot directly apply in finding the best alternative. Thomas L. Saaty proposed a method by computing benefit/cost ratios to find the best alternative. Yet this method was criticized by Richard H. Bernhard and John R. Canada who instead proposed an incremental benefit/cost ratio analysis. This thesis reviews the two methods and looks for the justification of each method. Saaty’s benefit/cost ratio method is found inconsistent with the axioms of single-hierarchy approach with an unstated additional assumption on decision maker’s preference. Bernhard and Canada’s incremental benefit/cost ratio method does not have a presumption on decision maker’s utility function and is found consistent with utility theory. However, if the methodology of the incremental benefit/cost ratio analysis is extended in solving more complex multi-hierarchy problems, the practicality of this method reduces significantly. Hence this thesis raises an idea to find a way to reshape the problem in fitting the condition where single-hierarchy approach can apply then follow the standard single-hierarchy approach to arrive the best alternative. Two methods of transformation are considered in converting all attributes into the same direction of preference. In the process of assessing methods of transformation, the method of matrix transpose is rejected due to its potential in changing the decision. A linear transformation method is accepted which is simply the rescaling of normalized vector to preserve the attribute value space. This thesis concludes that the combination of linear transformation method and single-hierarchy approach can solve multi-hierarchy problems in general.