Fuzzy Markovian decision process

Fuzzy Semi-Markov Model for Web Testing

“…Using similar reasoning as in Bhattacharyya [3], the transition fuzzy probabilitiespF i F j and the fuzzy probabilities of the holding timescF i F j (m) for the fuzzy states can be interpreted in matrix notation by means of the corresponding matrices of the initial process. We denote: i = 1, 2, .…”

“…In the second case the actual states can be exactly measured and are observable but the number of states is too large and thus the decisions cannot be practically associated with the exact states of the system. In these situations the decisions are associated with fuzzy states which can be defined as fuzzy sets on the original non fuzzy state space of the system (Bhattacharyya [3]). Hence in this paper, both the fuzzy states and the fuzzy transitions are described as a fuzzy probabilities in a closed bounded interval and throughout the paper, we use the fuzzy probabilities as triangular fuzzy number.…”

“…, N) equals tōP (Fr) = n i=1 μi(Fr)p(i), ∀Fr ∈ SF(2)wherep(i) is the fuzzy probabilitry that the process is in state i initially. Now as described above and taking into account Bhattacharyya[3], the fuzzy transition probabilities of the process can be defined as:pF i F j =P {the process moves to fuzzy state Fj at its next transition / the process entered fuzzy state Fi at its previous transiting } =P {Xn = Fj/Xp = Fi}. Then from (2) and the latter definition we havē pF i F j =P {Xn = Fj, XP = Fi} P {Xp = Fi} = n s=1 n r=1P {Xp = r, Xn = s}μr(Fi)μs(Fj)…”

Markov Chain Lumpability on Fuzzy Partitions

“…to the fuzzy partition 2/ and Xn is called a Markov chain (9) X with fuzzy states [4]. This allows X-f to be analyzed by Markov chain methods.…”

“…randomness and vagueness. The model was proposed by Bhattacharyya [4], with the aim to study a Markov decision process with fuzzy states. However, the author passed over the important question, when the aggregated process is Markovian.…”