Markov-type fuzzy decision processes with a discounted reward on a closed interval

We formulate a new multi-stage decision process with Markov-type fuzzy transition, which is termed Markov-type fuzzy decision process. In the general framework of the decision process, both state and action are assumed to be fuzzy. The transition of states is defined using the fuzzy relation with Markov property and the discounted total reward is described as a fuzzy number on a closed bounded interval. To discuss the optimization problem, a partial order of convex fuzzy numbers is introduced. In this paper the discounted total reward associated with an admissible stationary policy is characterized by a unique fixed point of the contractive mapping. Moreover, the optimality equation for the fuzzy decision model is derived under some continuity conditions. Also, an illustrated example is given to explain the theoretical results and the computation in the paper.