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The Weighted Least Square Method Applied to the Binary and Ternary AHP

  • Kazutomo Nishizawa
  • Iwaro Takahashi
Part of the Smart Innovation, Systems and Technologies book series (SIST, volume 4)

Abstract

The weighted least square method is applied to the binary and ternary AHP (Analytic Hierarchy Process) based on the Bradley-Terry model. We transform this problem to have constant variance of errors (this is the basic property for the least square method to give the best estimators). The transformed problem has the form just to be solved by the geometric programming. Moreover the multi-stage Bradley-Terry model is proposed to meet the case where paired comparisons have various weights, which can also be solved by the geometric programming. Various examples solved by other methods are solved by our proposed method. The improved results are recognized.

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References

  1. 1.
    Bradley, R.A., Terry, M.E.: Rank analysis of incomplete block designs –The method of paired comparisons. Biometrika 39, 234–335 (1952)MathSciNetGoogle Scholar
  2. 2.
    Ecker, J.G.: Geometric programming: methods, computations and applications. SIAM Review 22, 338–362 (1980)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Furuya, S.: Gyouretsu to gyouretsushiki. Baihukan, Tokyo (1975) (in Japanese)Google Scholar
  4. 4.
    Nishizawa, K., Takahashi, I.: Estimation methods by stochastic model in binary and ternary AHP. Journal of the Operations Research Society of Japan 50, 101–122 (2007)MathSciNetGoogle Scholar
  5. 5.
    Nishizawa, K., Takahashi, I.: Weighted and logarithmic least square methods for mutual evaluation network system including AHP and ANP. Journal of the Operations Research Society of Japan 52, 221–244 (2009)zbMATHMathSciNetGoogle Scholar
  6. 6.
    Peterson, E.L.: Geometric programming. SIAM Review 18, 1–51 (1976)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Rao, C.R.: Linear statistical inference and its applications. Wiley, New York (1980)Google Scholar
  8. 8.
    Takahashi, I.: AHP applied to binary and ternary comparisons. Journal of the Operations Research Society of Japan 33, 199–206 (1990)zbMATHMathSciNetGoogle Scholar
  9. 9.
    Takahashi, I., Kobayashi, R., Koyanagi, Y.: Tokei kaiseki. Baifukan, Tokyo (1992) (in Japanese)Google Scholar
  10. 10.
    Takahashi, I.: Variable separation principle for mathematical programming. Journal of the Operations Research Society of Japan 6, 82–105 (1964)MathSciNetGoogle Scholar
  11. 11.
    Takahashi, I.: A note on the conjugate gradient method. Information Processing in Japan 5, 45–49 (1965)MathSciNetGoogle Scholar
  12. 12.
    Takeuchi, K.: Kakuritsubunpu to tokeikaiseki Nihon kikakukyoukai, Tokyo (1975) (in Japanese)Google Scholar

Copyright information

© Springer Berlin Heidelberg 2010

Authors and Affiliations

  • Kazutomo Nishizawa
    • 1
  • Iwaro Takahashi
    • 2
  1. 1.Nihon UniversityChibaJapan
  2. 2.University of Tsukuba, Tennodai, TsukubaIbarakiJapan

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