Weight set decomposition for weighted rank and rating aggregation: An interpretable and visual decision support tool

Figure 1.  The set of convex weights $ \Lambda $ creates a triangle in $ \mathbb{R}^3 $ (left) that can be visualized in $ \mathbb{R}^2 $ (right). When the weights are equally weighted, i.e., $ \lambda = (\frac{1}{3}, \frac{1}{3}, \frac{1}{3}) $, the aggregated ranking is equidistant from the three corners, and hence, the three input rankings

Figure 2.  The triangle on the left shows the weight set $ \Lambda $ for Patient A, Anne, visualized in $ \mathbb{R}^2 $. Two sets of weights are shown. The weights $ (\frac{1}{2}, \frac{1}{2}, 0) $ is midway along the boundary between $ \bar r^1 $ and $ \bar r^2 $, and so it compromises between only two criteria (longevity and simplicity). The weight $ (\frac{5}{12}, \frac{5}{12}, \frac{1}{6}) $ is in the interior of the triangle, moved slightly toward the third criterion of cost. The triangle on the right shows colored axes that indicate the relative weight of a goal and guide users in movement toward or away from a particular goal

Figure 3.  (Left) Rank colormap for Patient A, Anne. (Right) Barchart displaying the percentage of $ \Lambda $ on the $ y $-axis per region with the corresponding ranking labeled on the $ x $-axis. A quick scan of this barchart shows the relative area associated with each ranking

Figure 4.  Rank colormap for Patient B, Bob. While Bob's colormap contains $ |A| = 18 $ regions, more than Anne's, several regions, and hence rankings, have insignificant areas

Figure 5.  Pairwise item analysis. (Left) Item 1 is ranked better than 5 in 96% of the weighted ranks, shown in blue. (Right) Item 2 is ranked better than 3 in 75% of the weighted ranks, shown in blue

Figure 6.  Illustrating the steps of the exact algorithm

Figure 7.  (left) The rating colormap shows $ |A| = 20 $ regions. When the data is converted to rankings as input, the ranking colormap (right) also has $ |A| = 14 $ regions. Clearly, the two colormaps differ. In short, our colormap work allows for input vectors $ \bar r^1 $, $ \bar r^2 $, and $ \bar r^3 $ that are either rating vectors or ranking vectors

Figure 8.  The map on the left shows the heatmap for Treatment $ T_1 $ Temozolomide. Lighter regions indicate that Treatment $ T_1 $ ranks better in the aggregated ranking associated with that region. Treatment $ T_1 $ scores poorly when quality of life is the most important consideration. The map on the right shows the heatmap for another treatment, $ T_3 $ Gliovac, which scores better on quality of life and not as well in the compromise area between quality of life and simplicity of the treatment regimen

Figure 9.  Sensitivity Map. The darker points near the center of a region are most robust, i.e., their ranking of treatments is least sensitive to small changes in the input weights $ \lambda_i $

Figure 10.  The $ j = 4 $ polytope in $ \mathbb{R}^3 $ (left). Planes through the polytope for fixed values of $ \lambda_4 $ (right). The largest plane is the $ \lambda_4 = 1 $ plane through the polytope, which is a face of the $ j = 4 $ polytope. The other planes are the when $ \lambda_4 = .75 $, $ \lambda_4 = .5 $, and $ \lambda_4 = .25 $. The origin corresponds to $ \lambda_4 = 0 $. There are infinitely many planes at fixed $ \lambda_4 $ through this polytope, each with color-coded points mapped to aggregated rankings

Figure 11.  (Left) Nonlinear function for weight $ \lambda_3 $ associated with cost. Cost of a treatment has little impact for small values of $ \lambda_3 $ but then increases rapidly. (Right) This nonlinear transformation affects the geometry of the indifference regions