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

  • Daji Ergu
  • Gang KouEmail author
  • János Fülöp
  • Yong Shi


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.


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



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).


  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) CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Lipovetsky, S.: Global priority estimation in multiperson decision making. J. Optim. Theory Appl. 140(1), 77–91 (2009) CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Saaty, T.L.: Axiomatic foundation of the analytic hierarchy process. Manag. Sci. 32(7), 841–855 (1986) CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Blankmeyer, E.: Approaches to consistency adjustment. J. Optim. Theory Appl. 54(3), 479–488 (1987) CrossRefzbMATHMathSciNetGoogle 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) CrossRefzbMATHMathSciNetGoogle 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) CrossRefMathSciNetGoogle Scholar
  7. 7.
    Saaty, T.L.: How to make a decision: the analytic hierarchy process. Eur. J. Oper. Res. 48(1), 9–26 (1990) CrossRefzbMATHGoogle Scholar
  8. 8.
    Xu, Z., Wei, C.: A consistency improving method in the analytic hierarchy process. Eur. J. Oper. Res. 116, 443–449 (1999) CrossRefzbMATHGoogle 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) CrossRefzbMATHGoogle 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) CrossRefGoogle 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). 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) CrossRefzbMATHMathSciNetGoogle 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) CrossRefGoogle 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 CrossRefGoogle 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 CrossRefzbMATHMathSciNetGoogle 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 zbMATHGoogle 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 zbMATHGoogle 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 zbMATHGoogle 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 CrossRefMathSciNetGoogle 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) CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  • Daji Ergu
    • 1
    • 2
  • Gang Kou
    • 1
    Email author
  • János Fülöp
    • 3
  • Yong Shi
    • 4
    • 5
  1. 1.School of Management and EconomicsUniversity of Electronic Science and Technology of ChinaChengduChina
  2. 2.Southwest University for NationalitiesChengduChina
  3. 3.Research Group of Operations Research and Decision Systems, Computer and Automation Research InstituteHungarian Academy of SciencesBudapestHungary
  4. 4.College of Information Science & TechnologyUniversity of Nebraska at OmahaOmahaUSA
  5. 5.Research Center on Fictitious Economy and Data SciencesChinese Academy of SciencesBeijingChina

Personalised recommendations