Extending the Vector Analytical Hierarchy Process Clustering Algorithm to Create More Coherent Subgroups

The Vector Analytical Hierarchy Process (VAHP) is an extension of the original Analytical Hierarchy Process that translates the decision-makers’ preferences into geometric resultant unit vectors. VAHP is beneficial in the creation of subgroups based on these vectors, but has some inherent problems that are magnified in relation to the diversity of the population being analyzed. Use of the original VAHP clustering algorithm can leave a large number of decision-makers out of the meaningful clusters and makes the interpretation of subgroup preference vectors more difficult. This paper proposes four extensions to the VAHP clustering algorithm that result in more coherent and meaningful subgroup creation. Track: Student Papers INTRODUCTION The Vector Analytical Hierarchy Process (VAHP) is an extension of the original Analytical Hierarchy Process that translates the decision-makers’ preferences into geometric resultant unit vectors. This allows for visualization of the results on an n-dimensional sphere, as well as allowing for the utilization of geometric calculations to determine relationships between vectors and the creation of coherent subgroups of preference vectors. VAHP is beneficial in the creation of these subgroups, but has some inherent problems that are magnified in relation to the diversity of the population being analyzed. As the relationships between the vectors become more pronounced, the use of the original VAHP clustering algorithm leaves a large number of decision-makers out of the meaningful clusters and makes the interpretation of the similarities and differences between preference vectors more difficult. This paper proposes four extensions to the VAHP clustering algorithm that result in more coherent and meaningful subgroup creation. ANALYTICAL HIERARCHY PROCESS The Analytical Hierarchy Process (AHP) [8] is a multi-criteria decision making approach that is used to elicit priorities from groups of decision makers. AHP allows for the hierarchical arrangement of the objectives, goals, and attributes of the decision. Priorities for the elements are derived through a series of pair-wise comparisons between each of the elements at one level of the hierarchy. The use of AHP is widespread, demonstrating usefulness in a variety of disciplines, for example: forecasting [4], politics [9], education [10] and environmental impact assessment [6]. Two of the most popular social choice axioms have been evaluated in relation to AHP [7], geometric mean method (GMM) and the weighted arithmetic mean method (WAMM). It was argued that the GMM will not always satisfy the Pareto optimality axiom. This claim was refuted [3] and it was argued that this was irrelevant if the proper aggregation approach was selected. If the group is assumed to act together, then aggregation of individual judgments (AIJ) should be used; while if the members of the group are acting as individuals, then aggregation of individual priorities (AIP) should be used. In AIJ, the members of the group are assumed to be acting in the best interests of the whole, subjugating their own preferences so the group behaves as one. AIP should be used when a group is comprised of members representing a variety of opinions and it is unlikely that they will be able to reach a consensus. When using AIJ only the geometric mean can be used, while AIP group preferences can be computed using either the arithmetic or geometric mean [1]. Creating homogeneous subgroups from a heterogeneous group, which can be implemented using either AIP or AIJ, better reflects some of the variability in the individual preferences and the subgroup preference results more accurately reflect the opinions of the decision makers within each subgroup as opposed to the creation of preferences only for the group as a whole. VECTOR ANALYTICAL HIERARCHY PROCESS The application of the Analytic Hierarchy Process chosen for this research involves an extension of the traditional AHP approach that translates the pairwise comparisons into Euclidean vector space, providing geometric meaning to the results [11]. This methodology, referred to as the vector analytical hierarchy process (VAHP), creates preference vectors in multi-dimensional space, where the number of dimensions equals the number of different alternatives being considered. A preference vector is created for each respondent in the population as a resultant unit vector normalized to unit length of one, i.e.