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.

The Heuristic Rating Estimation Method enables decision-makers to decide based on existing ranking data and expert comparisons. In this approach, the ranking values of selected alternatives are known in advance, while these values have to be calculated for the remaining ones. Their calculation can be performed using either an additive or a multiplicative method. Both methods assumed that the pairwise comparison sets involved in the computation were complete. In this paper, we show how these algorithms can be extended so that the experts do not need to compare all alternatives pairwise. Thanks to the shortening of the work of experts, the presented, improved methods will reduce the costs of the decision-making procedure and facilitate and shorten the stage of collecting decision-making data.

Pairwise comparison matrices are increasingly used in settings where some pairs are missing. However, there exist few inconsistency indices to analyse such incomplete data sets and even fewer measures have an associated threshold. This paper generalises the inconsistency index proposed by Saaty to incomplete pairwise comparison matrices. The extension is based on the approach of filling the missing elements to minimise the eigenvalue of the incomplete matrix. It means that the well-established values of the random index, a crucial component of the consistency ratio for which the famous threshold of 0.1 provides the condition for the acceptable level of inconsistency, cannot be directly adopted. The inconsistency of random matrices turns out to be the function of matrix size and the number of missing elements, with a nearly linear dependence in the case of the latter variable. Our results can be directly used by practitioners as a statistical criterion for accepting/rejecting an incomplete pairwise comparison matrix.

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.