Tropical implementation of the Analytical Hierarchy Process decision method

Nikolai Krivulin, Sergey Sergeev

Research output: Contribution to journalArticlepeer-review

3 Citations (Scopus)
124 Downloads (Pure)


We apply methods and techniques of tropical optimization to develop a new theoretical and computational framework for the implementation of the Analytic Hierarchy Process in multi-criteria problems of rating alternatives from pairwise comparison data. In this framework, we first consider the minimax Chebyshev approximation of pairwise comparison matrices by consistent matrices in the logarithmic scale. Recasting this approximation problem as a problem of tropical pseudo-quadratic programming, we then write out a closed-form solution to it. This solution might be either a unique score vector (up to a positive factor) or a set of different score vectors. To handle the problem when the solution is not unique, we develop tropical optimization techniques of maximizing and minimizing the Hilbert seminorm to find those vectors from the solution set that are the most and least differentiating between the alternatives with the highest and lowest scores, and thus are well representative of the entire solution set.
Original languageEnglish
Pages (from-to)31-51
Number of pages21
JournalFuzzy Sets and Systems
Early online date29 Oct 2018
Publication statusPublished - 15 Dec 2019


  • Tropical optimization
  • Max-algebra
  • Pairwise comparison
  • Log-Chebyshev approximation
  • Analytic Hierarchy Process
  • Hilbert distance

ASJC Scopus subject areas

  • General Mathematics


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