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