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Some Remarks on the Mean-Based Prioritization Methods in AHP

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Computational Collective Intelligence (ICCCI 2016)

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 9875))

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Abstract

EVM (eigenvector method) and GMM (geometric mean method) are probably the two most popular priority deriving techniques for AHP (Analytic Hierarchy Process). Although much has already been discussed about these methods, one frequently repeated question is: what do they have in common? In this paper we show that both these methods can be constructed based on the same principle that the priority of one alternative should correspond to the weighted mean of priorities of other alternatives. We also show how the accepted principle can be used to construct priority deriving methods for the generalized (non-reciprocal) pairwise comparisons matrices.

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Notes

  1. 1.

    The vector associated with the principal eigenvalue (spectral radius) of A.

  2. 2.

    Note that accepting \(w(c_{n})=1\), due to (7), we immediately get a solution of (11) in the form \([a_{1n},a_{2n},\ldots ,a_{n-1,n},1]\).

  3. 3.

    It is enough to notice that:

    $$ \frac{w_{r}}{w_{k}}=\frac{\left( \prod _{j=1}^{n}a_{rj}\right) ^{\frac{1}{n}}}{\left( \prod _{j=1}^{n}a_{kj}\right) ^{\frac{1}{n}}}=\left( \prod _{j=1}^{n}\frac{a_{rj}}{a_{kj}}\right) ^{\frac{1}{n}}=\left( \prod _{j=1}^{n}a_{rj}a_{jr}a_{rk}\right) ^{\frac{1}{n}}=\left( \prod _{j=1}^{n}a_{rk}\right) ^{\frac{1}{n}}=a_{rk} $$

    for every \(k,r=1,\ldots ,n\).

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Authors and Affiliations

  1. Faculty of Electrical Engineering, Automatics, Computer Science and Biomedical Engineering, AGH University of Science and Technology, Al. Mickiewicza 30, 30-059, Cracow, Poland

    Konrad Kułakowski

  2. Faculty of Applied Mathematics, Al. Mickiewicza 30, 30-059, Cracow, Poland

    Anna Kȩdzior

Authors
  1. Konrad Kułakowski
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  2. Anna Kȩdzior
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Correspondence to Konrad Kułakowski .

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Editors and Affiliations

  1. Wrocław University of Technology, Wrocław, Poland

    Ngoc-Thanh Nguyen

  2. Aristotle University of Thessaloniki, Thessaloniki, Greece

    Lazaros Iliadis

  3. Department of Forestry and Management, Democritus University of Thrace, Orestiada, Thrace, Greece

    Yannis Manolopoulos

  4. Wrocław University of Technology, Wrocław, Poland

    Bogdan Trawiński

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Kułakowski, K., Kȩdzior, A. (2016). Some Remarks on the Mean-Based Prioritization Methods in AHP. In: Nguyen, NT., Iliadis, L., Manolopoulos, Y., Trawiński, B. (eds) Computational Collective Intelligence. ICCCI 2016. Lecture Notes in Computer Science(), vol 9875. Springer, Cham. https://doi.org/10.1007/978-3-319-45243-2_40

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  • DOI: https://doi.org/10.1007/978-3-319-45243-2_40

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