Sustainability is one of the world’s fundamental objectives, and a wide variety of information types, parameters, and uncertainties need to be appraised and managed to assess it. In the present paper, Multi-Criteria Decision Analysis (MCDA) is used to prioritise the criteria of sustainable development based on regularly published indicators. In line with most approaches in the literature, the main criteria are Economy, Society and Environment. Complex criteria are decomposed into subcriteria until the performance with respect to them can be measured directly. Weights of importance are calculated by the Analytic Hierarchy Process (AHP), in decision support system PriEsT. The model is flexible to both the modification of criteria and re-weighting, and the PriEsT file is supplemented to the paper. Moreover, the results can also be applied in decisions on resource allocation. The proposed methodology has the potential of resulting in a new composite index to measure, compare or rank countries and regions regarding sustainable development or one of its subcriteria, as well as to track, year by year, the improvements or the impact of the policies introduced.

The method of pairwise comparisons is frequently applied for ranking purposes. This article aims to rank top women tennis players based on their win/lose ratios. Incomplete pairwise comparison matrices (PCMs) were constructed from data obtained from the WTA (Women’s Tennis Association) homepage. The database contains head-to-head results from the period between 1973 and 2022 for 28 players who had the position No. 1 in the official rankings of WTA. The weight vector was calculated from the incomplete PCM with the logarithmic least squares method and the eigenvector method. The results are not surprising: Serena Williams, Steffi Graf, and Martina Navratilova stand in the first three positions, and Martina Hingis, Kim Clijsters, and Justine Henin follow them. We also tested the frequently used probability-based Bradley-Terry method and found high rank-correlation values. Using graph representations, the results gave us a deeper insight into the properties of incomplete PCMs. Special attention was given to the nontransitive triads. A data modification was necessary to remove ties in order to apply the commonly used tests. The results indicate that ordinally nontransitive triads are not significant in the data we analysed.

A crucial, both from theoretical and practical points of view, problem in preference modelling is the number of questions to ask from the decision maker. We focus on incomplete pairwise comparison matrices based on graphs whose average degree is approximately 3 (or a bit more), i.e., each item is compared to three others in average. In the range of matrix sizes we considered, $$n=5,6,7,8,9,10$$n = 5 , 6 , 7 , 8 , 9 , 10 , this requires from 1.4n to 1.8n edges, resulting in completion ratios between 33% ( $$n=10$$n = 10 ) and 80% ( $$n=5$$n = 5 ). We analyze several types of union of two spanning trees (three of them building on additional ordinal information on the ranking), 2-edge-connected random graphs and 3-(quasi-)regular graphs with minimal diameter (the length of the maximal shortest path between any two vertices). The weight vectors are calculated from the natural extensions, to the incomplete case, of the two most popular weighting methods, the eigenvector method and the logarithmic least squares. These weight vectors are compared to the ones calculated from the complete matrix, and their distances (Euclidean, Chebyshev and Manhattan), rank correlations (Kendall and Spearman) and similarity (Garuti, cosine and dice indices) are computed in order to have cardinal, ordinal and proximity views during the comparisons. Surprisingly enough, only the union of two star graphs centered at the best and the second best items perform well among the graphs using additional ordinal information on the ranking. The union of two edge-disjoint spanning trees is almost always the best among the analyzed graphs.