2865. Tensor Basis for Pairwise Comparisons
Invited abstract in session WA-10: Pairwise comparisons and preference relations 2, stream Multiple Criteria Decision Aiding.
Wednesday, 8:30-10:00Room: Clarendon SR 1.06
Authors (first author is the speaker)
| 1. | Konrad Kułakowski
|
| Applied Computer Science, AGH University of Science and Technology | |
| 2. | Ryszard Smarzewski
|
| Military University of Technology |
Abstract
In the presented study, we construct the first orthogonal basis for the space of additively consistent matrices in the context of pairwise comparisons. This achievement resolves a long-standing problem concerning the logarithmic least-squares method by Koczkodaj and Orłowski (1997). Our approach relies on a tensor-based representation of the best additively consistent approximations of skew-symmetric matrices. The proposed orthogonal basis enables a novel logarithmic consistent projection and provides a new perspective on the orthogonal windowing of pairwise comparison matrices. We compare our approach with the established windowing methods of Saaty and singular value decomposition (SVD), deriving new composite formulae for logarithmic, Saaty, and SVD projections. These results are supported by theoretical analysis and numerical examples. Furthermore, we discuss the implications of different matrix norms in approximation quality and propose directions for future research on weighted inner products and alternative consistency measures in pairwise comparisons.
Keywords
- Decision Theory
- Decision Support Systems
- Decision Analysis
Status: accepted
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