Stochastic decision-making in a fuzzy environment

In this paper the authors consider multistage decision processes in Bellman and Zadeh’s paper ‘Decision-making in a fuzzy environment’ from a mathematical view-point. They propose another recursive equation which solves both stochastic and deterministic multistage decision processes in a fuzzy environment in their sense. The present result for deterministic processes coincides with their result. However, the result for stochastic processes is different from theirs. The present ‘stochastic’ recursive equation is derived through invariant imbedding. On the other hand, their ‘stochastic’ recursive equation is a direct analogy for their deterministic one. As an example, the authors illustrate their numerical data, which verify the equality between simultaneous and sequential optimizations.