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By decision-making in a fuzzy environment is meant a decision process in which the goals and/or the constraints, but not necessarily the system under control, are fuzzy in nature. This means that the goals and/or the constraints Constitute classes Of alternatives whose boundaries are not sharply defined.An example of a fuzzy constraint is: "The cost of A should not be substantially higher than a," where a is a specified constant. Similarly, an example of a fuzzy goal is: "x should be in the vicinity of x 'I where x is a constant. The underlined words are the sources of fuzziness in these examples. 0, 0Fuzzy goals and fuzzy constraints can be defined precisely as fuzzy sets in the space of alternatives. A fuzzy decision, then, may be viewed as an intersection of the given goals and constraints. A maximizing decision is defined as the set of points in the space of alternatives at which the membership function of a fuzzy decision attains its maximum value.The use of these concepts is illustrated by examples involving multistage decision processes in which the system under -iiicontrol is either deterministic or stochastic. By using dynamic programming, the determination of a maximizing decision is reduced to the solution of a system of functional equations. A reverse-flow technique is described for the solution of a functional equation arising in connection with a decision process in which the termination time is defined implicitly by the condition that the process stops when the system under control enters a specified set of states in its state space.

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