A New Consistency Index for Comparison Matrices in the ANP

  • Daji Ergu
  • Gang Kou
  • Yi Peng
  • Yong Shi
Conference paper
Part of the Lecture Notes in Economics and Mathematical Systems book series (LNE, volume 648)


The inconsistency test is more difficult in the analytical network process (ANP) than in the analytical hierarchy process (AHP) because there are more comparison matrices in the ANP. The consistency ratio (CR) introduced by Saaty for the AHP is used for inconsistency test in the ANP. When the number of comparison matrices increases, the inconsistency test of comparison matrices in the ANP becomes complicated. In an attempt to simplify the inconsistency test, this paper proposes a maximum eigenvalue threshold as the consistency index for ANP, which is mathematically equivalent to the CR method. In addition, a block diagonal matrix is introduced for the comparison matrices in the ANP to conduct inconsistency tests simultaneously for all comparison matrices. An illustrative example is used to show the effectiveness and the simplicity of the proposed maximum eigenvalue threshold method.


Block diagonal matrix Inconsistency test Maximum eigenvalue Maximum eigenvalue threshold The analytical network process (ANP) 



This research has been supported by grants from the National Natural Science Foundation of China (#70901011, #70901015, and # 70921061).


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Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  1. 1.School of Management and EconomicsUniversity of Electronic Science and Technology of ChinaChengduChina
  2. 2.Southwest University for NationalitiesChengduChina
  3. 3.College of Information Science & TechnologyUniversity of Nebraska at OmahaOmahaUSA
  4. 4.Research Center on Fictitious Economy and Data SciencesChinese Academy of SciencesBeijingChina

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