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A direct consistency test and improvement method for the analytic hierarchy process

Author

Listed:
  • Kang Xu

    (Sichuan University
    Hubei University of Automotive Technology)

  • Jiuping Xu

    (Sichuan University)

Abstract

The consistency test is a vital component of pairwise comparison matrices if meaningful results are to be guaranteed, and it has been studied extensively since the analytic hierarchy process was developed by Saaty. However, when using the existing methods, it is imperative to carry out matrix operations, which are usually not intuitional. In this paper, a direct method, from the perspective of 3 tuples, is proposed to test and improve the consistency of a pairwise comparison matrix, which is far more intuitional and easier to understand. In the proposed 3 tuples iterative method, only simple mathematical calculations are needed, without the need for matrix operations. Furthermore, the calculation of Saaty’s consistency ratio is only conducted to verify the proposed method. Some related theorems and propositions are proved mathematically or verified by random simulations. The proposed method is applied to some published examples to verify its effectiveness and practicability. Finally, we provide an improved iterative method, which is based on the 3 tuples iterative method. The improved iterative method and the 3 tuples iterative method are closely linked and each has its own different perspective. Moreover, they are applied on different occasions and reinforce one another.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:fuzodm:v:19:y:2020:i:3:d:10.1007_s10700-020-09323-y
    DOI: 10.1007/s10700-020-09323-y
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    References listed on IDEAS

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    1. Thomas L. Saaty, 1986. "Axiomatic Foundation of the Analytic Hierarchy Process," Management Science, INFORMS, vol. 32(7), pages 841-855, July.
    2. Zhu, Bin & Xu, Zeshui & Zhang, Ren & Hong, Mei, 2016. "Hesitant analytic hierarchy process," European Journal of Operational Research, Elsevier, vol. 250(2), pages 602-614.
    3. 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.
    4. Wu, Zhibin & Huang, Shuai & Xu, Jiuping, 2019. "Multi-stage optimization models for individual consistency and group consensus with preference relations," European Journal of Operational Research, Elsevier, vol. 275(1), pages 182-194.
    5. Saaty, Thomas L., 1990. "How to make a decision: The analytic hierarchy process," European Journal of Operational Research, Elsevier, vol. 48(1), pages 9-26, September.
    6. Saaty, Thomas L., 1978. "Modeling unstructured decision problems — the theory of analytical hierarchies," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 20(3), pages 147-158.
    7. Raman Kumar Goyal & Sakshi Kaushal, 2018. "Deriving crisp and consistent priorities for fuzzy AHP-based multicriteria systems using non-linear constrained optimization," Fuzzy Optimization and Decision Making, Springer, vol. 17(2), pages 195-209, June.
    8. Saaty, Thomas L., 2003. "Decision-making with the AHP: Why is the principal eigenvector necessary," European Journal of Operational Research, Elsevier, vol. 145(1), pages 85-91, February.
    9. Saaty, Thomas L., 1979. "Applications of analytical hierarchies," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 21(1), pages 1-20.
    10. Yoram Wind & Thomas L. Saaty, 1980. "Marketing Applications of the Analytic Hierarchy Process," Management Science, INFORMS, vol. 26(7), pages 641-658, July.
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    Cited by:

    1. Jean-Pierre Magnot & Jiří Mazurek & Viera Cernanova, 2021. "A gradient method for inconsistency reduction of pairwise comparisons matrices," Working Papers hal-03313878, HAL.
    2. Ágoston, Kolos Csaba & Csató, László, 2022. "Inconsistency thresholds for incomplete pairwise comparison matrices," Omega, Elsevier, vol. 108(C).

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