An experimental approach: The graph of graphs

One of the essential issues in decision problems and preference modeling is the number of comparisons and their pattern to ask from the decision maker. We focus on the optimal patterns of pairwise comparisons and the sequence including the most (close to) optimal cases based on the results of a color selection experiment. In the test, six colors (red, green, blue, magenta, turquoise, yellow) were evaluated with pairwise comparisons as well as in a direct manner, on color-calibrated tablets in ISO standardized sensory test booths of a sensory laboratory. All the possible patterns of comparisons resulting in a connected representing graph were evaluated against the complete data based on 301 individual's pairwise comparison matrices (PCMs) using the logarithmic least squares weight calculation technique. It is shown that the empirical results, i.e., the empirical distributions of the elements of PCMs, are quite similar to the former simulated outcomes from the literature. The obtained empirically optimal patterns of comparisons were the best or the second best in the former simulations as well, while the sequence of comparisons that contains the most (close to) optimal patterns is exactly the same. In order to enhance the applicability of the results, besides the presentation of graph of graphs, and the representing graphs of the patterns that describe the proposed sequence of comparisons themselves, the recommendations are also detailed in a table format as well as in a Java application.

An application of incomplete pairwise comparison matrices for ranking top tennis players

Pairwise comparison is an important tool in multi-attribute decision making. Pairwise comparison matrices (PCM) have been applied for ranking criteria and for scoring alternatives according to a given criterion. Our paper presents a special application of incomplete PCMs: ranking of professional tennis players based on their results against each other. The selected 25 players have been on the top of the ATP rankings for a shorter or longer period in the last 40 years. Some of them have never met on the court. One of the aims of the paper is to provide ranking of the selected players, however, the analysis of incomplete pairwise comparison matrices is also in the focus. The eigenvector method and the logarithmic least squares method were used to calculate weights from incomplete PCMs. In our results the top three players of four decades were Nadal, Federer and Sampras. Some questions have been raised on the properties of incomplete PCMs and remains open for further investigation.