Paper Info
Title
The (Logarithmic) Least Squares Optimality Of The Arithmetic (Geometric) Mean Of Weight Vectors Calculated From All Spanning Trees For Incomplete Additive (Multiplicative) Pairwise Comparison Matrices
Abstract
Complete and incomplete additive/multiplicative pairwise comparison matrices are applied in preference modelling, multi-attribute decision making and ranking. The equivalence of two well known methods is proved in this paper. The arithmetic (geometric) mean of weight vectors, calculated from all spanning trees, is proved to be optimal to the (logarithmic) least squares problem, not only for complete, as it was recently shown in Lundy, M., Siraj, S., Greco, S. (2017): The mathematical equivalence of the spanning tree and row geometric mean preference vectors and its implications for preference analysis, European Journal of Operational Research 257(1) 197-208, but for incomplete matrices as well. Unlike the complete case, where an explicit formula, namely the row arithmetic/geometric mean of matrix elements, exists for the (logarithmic) least squares problem, the incomplete case requires a completely different and new proof. Finally, Kirchhoff's laws for the calculation of potentials in electric circuits is connected to our results.
Year
DOI
Venue
2019
10.1080/03081079.2019.1585432
INTERNATIONAL JOURNAL OF GENERAL SYSTEMS
Keywords
Field
DocType
Decision analysis, multi-criteria decision making, incomplete pairwise comparison matrix, additive, multiplicative, least squares, logarithmic least squares, Laplacian matrix, spanning tree
Least squares,Laplacian matrix,Discrete mathematics,Pairwise comparison,Arithmetic–geometric mean,Multiplicative function,Matrix (mathematics),Equivalence (measure theory),Spanning tree,Mathematics
Journal
Volume
Issue
ISSN
48
4
0308-1079
Citations 
PageRank 
References 
1
0.35
0
Authors
2
Name
Order
Citations
PageRank
Sándor Bozóki115810.98
Vitaliy Tsyganok281.87