Further Discussions on Induced Bias Matrix Model for the Pair-Wise Comparison Matrix

Abstract

The inconsistency issue of pairwise comparison matrices has been an important subject in the study of the analytical network process. Most inconsistent elements can efficiently be identified by inducing a bias matrix only based on the original matrix. This paper further discusses the induced bias matrix and integrates all related theorems and corollaries into the induced bias matrix model. The theorem of inconsistency identification is proved mathematically using the maximum eigenvalue method and the contradiction method. In addition, a fast inconsistency identification method for one pair of inconsistent elements is proposed and proved mathematically. Two examples are used to illustrate the proposed fast identification method. The results show that the proposed new method is easier and faster than the existing method for the special case with only one pair of inconsistent elements in the original comparison matrix.

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References

  1. 1.

    Sun, L., Greenberg, B.: Multicriteria group decision making: optimal priority synthesis from pairwise comparisons. J. Optim. Theory Appl. 130(2), 317–339 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  2. 2.

    Lipovetsky, S.: Global priority estimation in multiperson decision making. J. Optim. Theory Appl. 140(1), 77–91 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  3. 3.

    Saaty, T.L.: Axiomatic foundation of the analytic hierarchy process. Manag. Sci. 32(7), 841–855 (1986)

    Article  MATH  MathSciNet  Google Scholar 

  4. 4.

    Blankmeyer, E.: Approaches to consistency adjustment. J. Optim. Theory Appl. 54(3), 479–488 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  5. 5.

    Saaty, R.W.: The analytic hierarchy process—what it is and how it is used. Math. Model. 9(3–5), 161–176 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  6. 6.

    Harker, P., Vargas, L.: The theory of ratio scale estimation: Saaty’s analytic hierarchy process. Manag. Sci. 33(11), 1383–1403 (1987)

    Article  MathSciNet  Google Scholar 

  7. 7.

    Saaty, T.L.: How to make a decision: the analytic hierarchy process. Eur. J. Oper. Res. 48(1), 9–26 (1990)

    Article  MATH  Google Scholar 

  8. 8.

    Xu, Z., Wei, C.: A consistency improving method in the analytic hierarchy process. Eur. J. Oper. Res. 116, 443–449 (1999)

    Article  MATH  Google Scholar 

  9. 9.

    Li, H., Ma, L.: Detecting and adjusting ordinal and cardinal inconsistencies through a graphical and optimal approach in AHP models. Comput. Oper. Res. 34(3), 780–798 (2007)

    Article  MATH  Google Scholar 

  10. 10.

    Cao, D., Leung, L.C., Law, J.S.: Modifying inconsistent comparison matrix in analytic hierarchy process: a heuristic approach. Decis. Support Syst. 44, 944–953 (2008)

    Article  Google Scholar 

  11. 11.

    Iida, Y.: Ordinality consistency test about items and notation of a pairwise comparison matrix in AHP. In: Proceedings of the International Symposium on the Analytic Hierarchy Process (2009). http://www.isahp.org/2009Proceedings/Final_Papers/32_Iida_Youichi_ConsistencyTest_in_Japan_REV_FIN.pdf

    Google Scholar 

  12. 12.

    Koczkodaj, W.W., Szarek, S.J.: On distance-based inconsistency reduction algorithms for pairwise comparisons. Log. J. IGPL 18(6), 859–869 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  13. 13.

    Peng, Y., Kou, G., Wang, G., Wu, W., Shi, Y.: Ensemble of software defect predictors: an AHP-based evaluation method. Int. J. Inf. Technol. Decis. Mak. 10(1), 187–206 (2011)

    Article  Google Scholar 

  14. 14.

    Kou, G., Lu, Y., Peng, Y., Shi, Y.: Evaluation of classification algorithms using MCDM and rank correlation. Int. J. Inf. Technol. Decis. Mak. 11(1), 197–225 (2012). doi:10.1142/S0219622012500095

    Article  Google Scholar 

  15. 15.

    Ergu, D., Kou, G., Peng, Y., Shi, Y.: A simple method to improve the consistency ratio of the pair-wise comparison matrix in ANP. Eur. J. Oper. Res. 213(1), 246–259 (2011). doi:10.1016/j.ejor.2011.03.014

    Article  MATH  MathSciNet  Google Scholar 

  16. 16.

    Ergu, D., Kou, G.: Questionnaire design improvement and missing item scores estimation for rapid and efficient decision making. Ann. Oper. Res. (2011). doi:10.1007/s10479-011-0922-3

    MATH  Google Scholar 

  17. 17.

    Ergu, D., Kou, G., Shi, Y., Shi, Y.: Analytic network process in risk assessment and decision analysis. Comput. Oper. Res. (2011). doi:10.1016/j.cor.2011.03.005

    MATH  Google Scholar 

  18. 18.

    Ergu, D., Kou, G., Peng, Y., Shi, Y., Shi, Y.: The analytic hierarchy process: task scheduling and resource allocation in cloud computing environment. J. Supercomput. (2011). doi:10.1007/s11227-011-0625-1

    MATH  Google Scholar 

  19. 19.

    Ergu, D., Kou, G., Peng, Y., Shi, Y., Shi, Y.: BIMM: a bias induced matrix model for incomplete reciprocal pairwise comparison matrix. J. Multi-Criteria Decis. Anal. 18(1), 101–113 (2011). doi:10.1002/mcda.472

    Article  MathSciNet  Google Scholar 

  20. 20.

    Saaty, T.L.: The Analytical Hierarchy Process. McGraw-Hill, New York (1980)

    Google Scholar 

  21. 21.

    Bozóki, S., Fülöp, J., Poesz, A.: On pairwise comparison matrices that can be made consistent by the modification of a few elements. Cent. Eur. J. Oper. Res. 19, 157–175 (2011)

    Article  MATH  Google Scholar 

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Acknowledgements

This research has been partially supported by grants from the National Natural Science Foundation of China (#70901015 and #71222108), the Fundamental Research Funds for the Central Universities and Program for New Century Excellent Talents in University (NCET-10-0293).

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Correspondence to Gang Kou.

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Communicated by Po-Lung Yu.

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Ergu, D., Kou, G., Fülöp, J. et al. Further Discussions on Induced Bias Matrix Model for the Pair-Wise Comparison Matrix. J Optim Theory Appl 161, 980–993 (2014). https://doi.org/10.1007/s10957-012-0223-2

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Keywords

  • Analytic network process (ANP)
  • The induced bias matrix model (IBMM)
  • Inconsistency identification
  • Reciprocal pairwise comparison matrix (RPCM)