The harmonic consistency index for the analytic hierarchy process

William E. Stein, Philip Mizzi

Research output: Contribution to journalArticlepeer-review

102 Scopus citations


A new consistency measure, the harmonic consistency index, is obtained for any positive reciprocal matrix in the analytic hierarchy process. We show how this index varies with changes in any matrix element. A tight upper bound is provided for this new consistency measure when the entries of matrix are at most 9, as is often recommended. Using simulation, the harmonic consistency index is shown to give numerical values similar to the standard consistency index but it is easier to compute and interpret. In addition, new properties of the column sums of reciprocal matrices are obtained.

Original languageEnglish (US)
Pages (from-to)488-497
Number of pages10
JournalEuropean Journal of Operational Research
Issue number1
StatePublished - Feb 16 2007


  • AHP
  • Consistency index
  • Multiple criteria analysis
  • Reciprocal matrices

ASJC Scopus subject areas

  • General Computer Science
  • Modeling and Simulation
  • Management Science and Operations Research
  • Information Systems and Management


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