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A gradient method for inconsistency reduction of pairwise comparisons matrices

Author

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  • Jean-Pierre Magnot

    (LAREMA - Laboratoire Angevin de Recherche en Mathématiques - UA - Université d'Angers - CNRS - Centre National de la Recherche Scientifique)

  • Jiří Mazurek
  • Viera Cernanova

    (Trnava University)

Abstract

We investigate an application of a mathematically robust minimization methodthe gradient method-to the consistencization problem of a pairwise comparisons (PC) matrix. Our approach sheds new light on the notion of a priority vector and leads naturally to the definition of instant priority vectors. We describe a sample family of inconsistency indicators based on various ways of taking an average value, which extends the inconsistency indicator based on the "sup"-norm. We apply this family of inconsistency indicators both for additive and multiplicative PC matrices to show that the choice of various inconsistency indicators lead to non-equivalent consistencization procedures.

Suggested Citation

  • Jean-Pierre Magnot & Jiří Mazurek & Viera Cernanova, 2021. "A gradient method for inconsistency reduction of pairwise comparisons matrices," Working Papers hal-03313878, HAL.
    • Handle: RePEc:hal:wpaper:hal-03313878
    Note: View the original document on HAL open archive server: https://hal.science/hal-03313878
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    References listed on IDEAS

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    1. Jean-Pierre Magnot, 2018. "A Mathematical Bridge between Discretized Gauge Theories in Quantum Physics and Approximate Reasoning in Pairwise Comparisons," Advances in Mathematical Physics, Hindawi, vol. 2018, pages 1-5, January.
    2. JosÉ Figueira & Salvatore Greco & Matthias Ehrogott, 2005. "Multiple Criteria Decision Analysis: State of the Art Surveys," International Series in Operations Research and Management Science, Springer, number 978-0-387-23081-8, September.
    3. Jean-Pierre Magnot, 2018. "A Mathematical Bridge between Discretized Gauge Theories in Quantum Physics and Approximate Reasoning in Pairwise Comparisons," Post-Print hal-01831631, HAL.
    4. Sándor Bozóki & János Fülöp & Attila Poesz, 2015. "On reducing inconsistency of pairwise comparison matrices below an acceptance threshold," Central European Journal of Operations Research, Springer;Slovak Society for Operations Research;Hungarian Operational Research Society;Czech Society for Operations Research;Österr. Gesellschaft für Operations Research (ÖGOR);Slovenian Society Informatika - Section for Operational Research;Croatian Operational Research Society, vol. 23(4), pages 849-866, December.
    5. Ergu, Daji & Kou, Gang & Peng, Yi & Shi, Yong, 2011. "A simple method to improve the consistency ratio of the pair-wise comparison matrix in ANP," European Journal of Operational Research, Elsevier, vol. 213(1), pages 246-259, August.
    6. Jean-Pierre Magnot, 2019. "On Mathematical Structures On Pairwise Comparisons Matrices With Coefficients In A Group Arising From Quantum Gravity," Post-Print hal-01835958, HAL.
    7. Kang Xu & Jiuping Xu, 2020. "A direct consistency test and improvement method for the analytic hierarchy process," Fuzzy Optimization and Decision Making, Springer, vol. 19(3), pages 359-388, September.
    8. Valdecy Pereira & Helder Costa, 2015. "Nonlinear programming applied to the reduction of inconsistency in the AHP method," Annals of Operations Research, Springer, vol. 229(1), pages 635-655, June.
    9. Sándor Bozóki & János Fülöp & Attila Poesz, 2011. "On pairwise comparison matrices that can be made consistent by the modification of a few elements," Central European Journal of Operations Research, Springer;Slovak Society for Operations Research;Hungarian Operational Research Society;Czech Society for Operations Research;Österr. Gesellschaft für Operations Research (ÖGOR);Slovenian Society Informatika - Section for Operational Research;Croatian Operational Research Society, vol. 19(2), pages 157-175, June.
    10. Yoram Wind & Thomas L. Saaty, 1980. "Marketing Applications of the Analytic Hierarchy Process," Management Science, INFORMS, vol. 26(7), pages 641-658, July.
    11. Giancarllo Ribeiro Vasconcelos & Caroline Maria de Miranda Mota, 2019. "Exploring Multicriteria Elicitation Model Based on Pairwise Comparisons: Building an Interactive Preference Adjustment Algorithm," Mathematical Problems in Engineering, Hindawi, vol. 2019, pages 1-14, June.
    12. Abel, Edward & Mikhailov, Ludmil & Keane, John, 2018. "Inconsistency reduction in decision making via multi-objective optimisation," European Journal of Operational Research, Elsevier, vol. 267(1), pages 212-226.
    13. Rezaei, Jafar, 2016. "Best-worst multi-criteria decision-making method: Some properties and a linear model," Omega, Elsevier, vol. 64(C), pages 126-130.
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    Keywords

    gradient method; inconsistency indicator; pairwise comparisons; priority vector;
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