On the use of group theory to generalize elements of pairwise comparisons matrix: A cautionary note

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Koczkodaj W. W., Liu F., Marek V. W., Mazurek J., Mazurek M., Mikhailov L., ...More

INTERNATIONAL JOURNAL OF APPROXIMATE REASONING, vol.124, pp.59-65, 2020 (SCI-Expanded) identifier identifier

  • Publication Type: Article / Article
  • Volume: 124
  • Publication Date: 2020
  • Doi Number: 10.1016/j.ijar.2020.05.008
  • Journal Name: INTERNATIONAL JOURNAL OF APPROXIMATE REASONING
  • Journal Indexes: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Academic Search Premier, PASCAL, Compendex, INSPEC, Linguistics & Language Behavior Abstracts, MathSciNet, zbMATH, DIALNET
  • Page Numbers: pp.59-65
  • Keywords: Pairwise comparisons, Inconsistency, Approximate reasoning, Group theory, UNIFIED FRAMEWORK, CONSISTENCY, ALGORITHMS, RANKING
  • Ankara University Affiliated: Yes

Abstract

This paper examines the constricted use of group theory in the study of pairwise comparisons. The presented approach is based on the application of the celebrated Levi Theorems of 1942 and 1943 for orderable groups. The theoretical foundation for multiplicative (ratio) pairwise comparisons is provided. Counterexamples are provided to support the theory. In our opinion, the scientific community must be made aware of the limitations of using the group theory in pairwise comparisons. Groups, which are not torsion free, cannot be used for ratios by Levi's theorems. (C) 2020 Elsevier Inc. All rights reserved.