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Polyad inconsistency measure for pairwise comparisons matrices: max-plus algebraic approach

Author

Listed:
  • Hiroyuki Goto

    (Hosei University)

  • Shaohua Wang

    (University of Illinois at Urbana-Champaign)

Abstract

A max-algebraic approach is applied in this study to assess the inconsistency of pairwise comparisons (PC) matrices. An input PC matrix is flexible: It can be nonreciprocal, inconsistent, and incomplete. Contrary to previous studies, inconsistency is examined for all polyads with varying cycle lengths, while typical assessment methods are triad based. Central is max-plus algebra, also known as tropical geometry. An input PC matrix is converted by a logarithmic mapping into linear space, an eigenvalue problem in max-plus algebra is then solved to obtain the most significant inconsistent cycle across the associated graph. The max-algebraic eigenvalue reflects the extent of inconsistency, and its exponential inversion signifies the maximum geometric mean of cycle inconsistencies. The new measure can thus be comprehended in both geometric mean and polyad contexts. First-time users of max-plus algebra can compute an approximate value with an off-the-shelf computational environment that provides matrix functionalities. The resulting measure passes all property requirements stipulated in three related categories of literature. Analytical results highlight the significance of our method.

Suggested Citation

  • Hiroyuki Goto & Shaohua Wang, 2022. "Polyad inconsistency measure for pairwise comparisons matrices: max-plus algebraic approach," Operational Research, Springer, vol. 22(1), pages 401-422, March.
    • Handle: RePEc:spr:operea:v:22:y:2022:i:1:d:10.1007_s12351-020-00547-9
    DOI: 10.1007/s12351-020-00547-9
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    References listed on IDEAS

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    1. Bice Cavallo & Alessio Ishizaka & Maria Grazia Olivieri & Massimo Squillante, 2019. "Comparing inconsistency of pairwise comparison matrices depending on entries," Journal of the Operational Research Society, Taylor & Francis Journals, vol. 70(5), pages 842-850, May.
    2. Lundy, Michele & Siraj, Sajid & Greco, Salvatore, 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, Elsevier, vol. 257(1), pages 197-208.
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