Weighted power–weakness ratio for multi-criteria decision making

A new concept of higher-order similarity and the role of distance/similarity measures in local classification methods

“…This preliminary result fully justifies the introduction of the adjunct dissimilarity factor that improves the classification performance in about 50% of the analyzed cases. [49] for multi-criteria decision-making and ranking comparison.…”

“…wPWR is based on pairwise comparisons between objects and it has already been applied to address several issues, such as model comparison, hazardous compounds ranking, virtual screening and partial ordering [49]- [51]. The core of wPWR is the generation of the so-called tournament table (T), which summarizes the pairwise comparisons between the objects in a win-loss manner.…”

How to weight Hasse matrices and reduce incomparabilities

“…Finally, if the values are equal (x ik ≜ x jk ), the two objects "tie the comparison" for the kth criterion and half a credit is given to both. This weighting scheme was already introduced in our previous work [5], as it is a very simple and efficient way to weight criteria and compare objects. Note that all the values of T W range from 0 to 1…”

“…In this work, in particular, for each data set, we calculated the ranks from Copeland-like scores of (a) all the matrices of the H R family, (b) the original Hasse (HH), and (c) the weighted count matrix T W (Cop). For the sake of comparison, also wPWR ranks were taken into account since they derive from an eigenvector-eigenvalue decomposition of T W [5]. In this way, for each data set, each of the n objects was described by T + 3 ranks obtained by the approaches listed above, where T is the number of the H R family.…”

“…On the contrary, if at least one pair of criteria is in conflict, the objects are incomparable and the ordering is not possible. In this way, partial ordering techniques differ from total ordering techniques (e.g., simple average ranking [3], Copeland score [4], and weighted Power-Weakness Ratio [5]), which always set an ordering. Once a partial order is established, relations between alternatives are represented as graphs in the socalled Hasse diagrams, in which elements are vertices and ordering relationships are edges.…”