IDEAS home Printed from https://ideas.repec.org/a/spr/operea/v22y2022i1d10.1007_s12351-020-00547-9.html
   My bibliography  Save this article

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
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s12351-020-00547-9
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s12351-020-00547-9?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    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.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Bice Cavallo, 2019. "Coherent weights for pairwise comparison matrices and a mixed-integer linear programming problem," Journal of Global Optimization, Springer, vol. 75(1), pages 143-161, September.
    2. Bice Cavallo & Livia D’Apuzzo, 2020. "Relations between coherence conditions and row orders in pairwise comparison matrices," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 43(2), pages 637-656, December.
    3. Liang, Fuqi & Brunelli, Matteo & Rezaei, Jafar, 2020. "Consistency issues in the best worst method: Measurements and thresholds," Omega, Elsevier, vol. 96(C).
    4. D'ora Gr'eta Petr'oczy & L'aszl'o Csat'o, 2019. "Revenue allocation in Formula One: a pairwise comparison approach," Papers 1909.12931, arXiv.org, revised Dec 2020.
    5. Csató, László & Tóth, Csaba, 2020. "University rankings from the revealed preferences of the applicants," European Journal of Operational Research, Elsevier, vol. 286(1), pages 309-320.
    6. L'aszl'o Csat'o & Csaba T'oth, 2018. "University rankings from the revealed preferences of the applicants," Papers 1810.04087, arXiv.org, revised Feb 2020.
    7. Zsombor Szádoczki & Sándor Bozóki & Patrik Juhász & Sergii V. Kadenko & Vitaliy Tsyganok, 2023. "Incomplete pairwise comparison matrices based on graphs with average degree approximately 3," Annals of Operations Research, Springer, vol. 326(2), pages 783-807, July.
    8. Aguarón, Juan & Escobar, María Teresa & Moreno-Jiménez, José María, 2021. "Reducing inconsistency measured by the geometric consistency index in the analytic hierarchy process," European Journal of Operational Research, Elsevier, vol. 288(2), pages 576-583.
    9. Bice Cavallo & Alessio Ishizaka, 2023. "Evaluating scales for pairwise comparisons," Annals of Operations Research, Springer, vol. 325(2), pages 951-965, June.
    10. Salvatore Greco & Sajid Siraj & Michele Lundy, 2021. "Supporting decisions by unleashing multiple mindsets using pairwise comparisons method," Papers 2107.01731, arXiv.org.
    11. László Csató, 2018. "Characterization of the Row Geometric Mean Ranking with a Group Consensus Axiom," Group Decision and Negotiation, Springer, vol. 27(6), pages 1011-1027, December.
    12. Csató, László & Petróczy, Dóra Gréta, 2021. "On the monotonicity of the eigenvector method," European Journal of Operational Research, Elsevier, vol. 292(1), pages 230-237.
    13. Bozóki, Sándor & Fülöp, János, 2018. "Efficient weight vectors from pairwise comparison matrices," European Journal of Operational Research, Elsevier, vol. 264(2), pages 419-427.
    14. Csató, László, 2019. "A characterization of the Logarithmic Least Squares Method," European Journal of Operational Research, Elsevier, vol. 276(1), pages 212-216.
    15. Vladimír Bureš & Jiří Cabal & Pavel Čech & Karel Mls & Daniela Ponce, 2020. "The Influence of Criteria Selection Method on Consistency of Pairwise Comparison," Mathematics, MDPI, vol. 8(12), pages 1-13, December.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:operea:v:22:y:2022:i:1:d:10.1007_s12351-020-00547-9. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.