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Incomplete pairwise comparison and consistency optimization


  • Fedrizzi, Michele
  • Giove, Silvio


This paper proposes a new method for calculating the missing elements of an incomplete matrix of pairwise comparison values for a decision problem. The matrix is completed by minimizing a measure of global inconsistency, thus obtaining a matrix which is optimal from the point of view of consistency with respect to the available judgements. The optimal values are obtained by solving a linear system and unicity of the solution is proved under general assumptions. Some other methods proposed in the literature are discussed and a numerical example is presented.
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Suggested Citation

  • Fedrizzi, Michele & Giove, Silvio, 2007. "Incomplete pairwise comparison and consistency optimization," European Journal of Operational Research, Elsevier, vol. 183(1), pages 303-313, November.
  • Handle: RePEc:eee:ejores:v:183:y:2007:i:1:p:303-313

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    References listed on IDEAS

    1. Herrera-Viedma, E. & Herrera, F. & Chiclana, F. & Luque, M., 2004. "Some issues on consistency of fuzzy preference relations," European Journal of Operational Research, Elsevier, vol. 154(1), pages 98-109, April.
    2. Yoram Wind & Thomas L. Saaty, 1980. "Marketing Applications of the Analytic Hierarchy Process," Management Science, INFORMS, vol. 26(7), pages 641-658, July.
    3. Xu, Zeshui & Da, Qingli, 2005. "A least deviation method to obtain a priority vector of a fuzzy preference relation," European Journal of Operational Research, Elsevier, vol. 164(1), pages 206-216, July.
    4. Fishburn, P.C., 1984. "SSB Utility theory: an economic perspective," Mathematical Social Sciences, Elsevier, vol. 8(1), pages 63-94, August.
    5. Herrera, F. & Herrera-Viedma, E. & Chiclana, F., 2001. "Multiperson decision-making based on multiplicative preference relations," European Journal of Operational Research, Elsevier, vol. 129(2), pages 372-385, March.
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    Cited by:

    1. repec:spr:grdene:v:24:y:2015:i:2:d:10.1007_s10726-014-9386-6 is not listed on IDEAS
    2. Csató, László, 2013. "Rangsorolás páros összehasonlításokkal. Kiegészítések a felvételizői preferencia-sorrendek módszertanához
      [Paired comparisons ranking. A supplement to the methodology of application-based preferenc
      ," Közgazdasági Szemle (Economic Review - monthly of the Hungarian Academy of Sciences), Közgazdasági Szemle Alapítvány (Economic Review Foundation), vol. 0(12), pages 1333-1353.
    3. Lundy, Michele & Siraj, Sajid & Greco, Salvatore, 2017. "The mathematical equivalence of the “spanning tree” and row geometric mean preference vectors and its implications for preference analysis," European Journal of Operational Research, Elsevier, vol. 257(1), pages 197-208.
    4. Liu, Fang & Zhang, Wei-Guo & Wang, Zhong-Xing, 2012. "A goal programming model for incomplete interval multiplicative preference relations and its application in group decision-making," European Journal of Operational Research, Elsevier, vol. 218(3), pages 747-754.
    5. repec:spr:grdene:v:27:y:2018:i:2:d:10.1007_s10726-018-9554-1 is not listed on IDEAS
    6. repec:eee:ejores:v:265:y:2018:i:1:p:239-247 is not listed on IDEAS
    7. Bozóki, Sándor & Csató, László & Temesi, József, 2016. "An application of incomplete pairwise comparison matrices for ranking top tennis players," European Journal of Operational Research, Elsevier, vol. 248(1), pages 211-218.
    8. László Csató, 2013. "Ranking by pairwise comparisons for Swiss-system tournaments," 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. 21(4), pages 783-803, December.
    9. Wang, Zhou-Jing & Li, Kevin W., 2015. "A multi-step goal programming approach for group decision making with incomplete interval additive reciprocal comparison matrices," European Journal of Operational Research, Elsevier, vol. 242(3), pages 890-900.
    10. repec:spr:fuzodm:v:17:y:2018:i:2:d:10.1007_s10700-017-9267-y is not listed on IDEAS
    11. Török, Ádám & Telcs, András & Kosztyán, Zsolt Tibor, 2013. "Hallgatói preferencia-sorrendek készítése az egyetemi jelentkezések alapján
      [Preparing student preference rankings based on applications for university]
      ," Közgazdasági Szemle (Economic Review - monthly of the Hungarian Academy of Sciences), Közgazdasági Szemle Alapítvány (Economic Review Foundation), vol. 0(3), pages 290-317.
    12. Entani, Tomoe & Sugihara, Kazutomi, 2012. "Uncertainty index based interval assignment by Interval AHP," European Journal of Operational Research, Elsevier, vol. 219(2), pages 379-385.
    13. Siraj, Sajid & Mikhailov, Ludmil & Keane, John A., 2015. "Contribution of individual judgments toward inconsistency in pairwise comparisons," European Journal of Operational Research, Elsevier, vol. 242(2), pages 557-567.
    14. Silvia Bortot & Ricardo Alberto Marques Pereira, 2011. "Inconsistency and non-additive Choquet integration in the Analytic Hierarchy Process," DISA Working Papers 2011/06, Department of Computer and Management Sciences, University of Trento, Italy, revised 29 Jul 2011.

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    JEL classification:

    • C15 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Statistical Simulation Methods: General


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