Fuzzy Optimization and Decision Making

, Volume 1, Issue 2, pp 143–159 | Cite as

Fuzzy Markov Chains and Decision-Making

  • Konstantin E. Avrachenkov
  • Elie Sanchez

Abstract

Decision-making in an environment of uncertainty and imprecision for real-world problems is a complex task. In this paper it is introduced general finite state fuzzy Markov chains that have a finite convergence to a stationary (may be periodic) solution. The Cesaro average and the α-potential for fuzzy Markov chains are defined, then it is shown that the relationship between them corresponds to the Blackwell formula in the classical theory of Markov decision processes. Furthermore, it is pointed out that recurrency does not necessarily imply ergodicity. However, if a fuzzy Markov chain is ergodic, then the rows of its ergodic projection equal the greatest eigen fuzzy set of the transition matrix. Then, the fuzzy Markov chain is shown to be a robust system with respect to small perturbations of the transition matrix, which is not the case for the classical probabilistic Markov chains. Fuzzy Markov decision processes are finally introduced and discussed.

fuzzy Markov chains fuzzy Markov decision processes eigen fuzzy sets ergodicity robustness 

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Copyright information

© Kluwer Academic Publishers 2002

Authors and Affiliations

  • Konstantin E. Avrachenkov
    • 1
  • Elie Sanchez
    • 2
  1. 1.INRIA Sophia AntipolisFrance
  2. 2.Laboratoire d'Informatique MédicaleFaculté de MédecineMarseille Cedex 5France

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