The Separation, and Separation-Deviation Methodology

In a generic group decision scenario, the decision makers review alternatives and then provide their own individual ranking. The aggregate ranking problem is to obtain a ranking that is fair and representative of the individual rankings. We argue here that using cardinal pairwise comparisons provides several advantages over scorewise models. The aggregate group ranking problem is then formalized as the separation model and separation-deviation model. The premise of the models is to use the implied or explicit pairwise comparisons in the reviewers’ input and assign an aggregate ranking that minimized the penalty of not agreeing with the scores, and the penalty for not agreeing on the intensity of the pairwise preference. The latter permits to the incorporation of confidence levels in the input provided by reviewers on specific pairwise comparisons, as well as on specific scores. Both separation and separation-deviation models have been shown to be solved efficiently, for convex penalties. This tutorial presents several group ranking scenarios where the pairwise comparisons are the input, such as sports competitions. We show that using cardinal, rather than ordinal, pairwise comparisons, and the proposed separation model, is advantageous. In group ranking contexts, where pairwise comparisons are not inherently available, there is also an advantage of using implied pairwise comparisons. The latter contexts include, e.g., the National Science Foundation review panels, choosing winning projects, determining countries’ credit risk, and customer segmentation. We unify the group decision problem with the problem of webpages rankings and ranking academic papers in terms of citations. We compare and contrast the separation approach with PageRank and the principal eigenvector methods. The problem of aggregating rankings “optimally” with pairwise comparisons is shown to be linked to a problem we call the inverse equal-paths problem. The graph representation provides insights and enables the introduction of a specific performance measure for the quality of the aggregate ranking as per its deviations from the individual rankings observations. We show that for convex penalties of deviating from the reviewers’ inputs the problem is polynomial-time solvable, by combinatorial and polynomial-time algorithms related to network flows. As such, the approach is very efficient. We demonstrate further how graph properties are related to the quality of the resulting aggregate ranking. Our graph representation paradigm provides a unifying framework for problems of aggregate ranking, group decision making, and data mining.