Improvement of the Weights Due to Inconsistent Pairwise Comparisons in the AHP

  • Kazutomo NishizawaEmail author
Conference paper
Part of the Smart Innovation, Systems and Technologies book series (SIST, volume 57)


One of the most important problems in the Analytic Hierarchy Process (AHP) is consistency of pairwise comparisons by the decision maker. This study focuses on the comparison methods to be used when the weights of the alternatives and criteria in AHP are inconsistent. In general, the weights in AHP use the principal eigenvector of the pairwise comparison matrix. However, for example, due to the decision maker’s misunderstandings, inconsistencies in pairwise comparisons sometimes arise. The consistency of the pairwise comparison matrix is usually determined using Consistency Index (CI) values. In the traditional AHP, when judged inconsistent, repeating the pairwise comparison is usually recommended. However, if the repeated comparison is arbitrarily performed, the results will not be optimal. In fact, to obtain the overall evaluation of alternatives, we often use inconsistent weights, even given the inconsistencies in the latter. Another method for judging the consistency of the pairwise comparison is to use a directed graph. Cycles in a directed graph represent comparison inconsistencies. Therefore in this paper, based on the principal eigenvalue and cycles in the directed graph of the pairwise comparison matrix, a method of correcting the principal eigenvector taking into consideration consistency is proposed.


AHP Pairwise comparison Consistency Index Directed graph Cycles 


  1. 1.
    Belton, V., Gear, T.: On a short-coming of saaty’s method of analytic hierarchies. Omega 11, 228–230 (1983)CrossRefGoogle Scholar
  2. 2.
    Belton, V., Gear, T.: The legitimacy of rank reversal—a comment. Omega 13, 143–145 (1985)CrossRefGoogle Scholar
  3. 3.
    Kinoshita, E., Nakanishi, M.: Proposal of new AHP model in light of dominant relationship among alternatives. J. Oper. Res. Soc. Jpn. 42, 180–197 (1999)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Kinoshita, E., Sugiura, S.: A comparison study of dominant AHP and similar dominant models. J. Res. Inst. Meijo Univ. 7, 115–116 (2008)Google Scholar
  5. 5.
    Nishizawa, K.: A Consistency Improving Method in Binary AHP. J. Oper. Res. Soc. Jpn. 38, 21–33 (1995)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Nishizawa, K.: Normalization method based on dummy alternative with perfect evaluation score in AHP and ANP. Intell. Decis. Technol. 1(SIST 15), 253–262 (2012)Google Scholar
  7. 7.
    Nishizawa, K.: Improving of the weight normalization method on alternatives in AHP and ANP. Smart Digital Futures 2014, pp. 155–163. IOS Press (2014)Google Scholar
  8. 8.
    Nishizawa, K.: The Improvement of Pairwise Comparison Method of the Alternatives in the AHP. Intell. Decis. Technol. 1(SIST 39), 483–491 (2015)Google Scholar
  9. 9.
    Saaty, T.L.: The Analytic Hierarchy Process. McGraw-Hill, New York (1980)zbMATHGoogle Scholar
  10. 10.
    Saaty, T.L.: The Analytic Network Process. RWS Publications, Pittsburgh (1996)Google Scholar
  11. 11.
    Schoner, B., Wedley, W.C., Choo, E.U.: A unified approach to AHP with linking pins. Eur. J. Oper. Res. 13, 384–392 (1993)CrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Nihon UniversityNarashino, ChibaJapan

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