Conceptos Básicos
An efficient orthogonalization method is introduced to find the closest consistent pairwise comparison matrix from an inconsistent one.
Resumen
The content presents a computationally efficient method for orthogonalizing pairwise comparison (PC) matrices. The key points are:
Orthogonalization is an important technique for approximating an inconsistent PC matrix with a consistent one, conforming to mathematical standards.
The authors introduce a generalized Frobenius inner product that allows for weighting the importance of different pairwise comparisons using a positive definite matrix W.
They derive a W-orthogonal basis of the subspace of additively consistent PC matrices (ln), which enables the orthogonal projection of any PC matrix onto the consistent and inconsistent subspaces (ln and hn,W).
A (non-orthogonal) basis of the inconsistent subspace hn is constructed using graph theory, avoiding the need for computations.
The proposed orthogonalization approach creates new opportunities for applications of pairwise comparison methods, which are widely used in decision-making processes.
Simple heuristics may also be useful for complex systems requiring immediate solutions, as demonstrated in prior work.