4.3 Article

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

期刊

INTERNATIONAL JOURNAL OF GENERAL SYSTEMS
卷 48, 期 4, 页码 362-381

出版社

TAYLOR & FRANCIS LTD
DOI: 10.1080/03081079.2019.1585432

关键词

Decision analysis; multi-criteria decision making; incomplete pairwise comparison matrix; additive; multiplicative; least squares; logarithmic least squares; Laplacian matrix; spanning tree

资金

  1. Janos Bolyai Research Fellowship of the Hungarian Academy of Sciences [BO/00154/16/3]
  2. Hungarian Scientific Research Fund (OTKA) [K111797]
  3. Bolyai+ New National Excellence Program of the Ministry of Human Capacities, Hungary [UNKP-18-4-BCE-90]

向作者/读者索取更多资源

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.

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