On Maximum Guaranteed Payoff in a Fuzzy Matrix Decision-Making Problem with a Fuzzy Set of States

The current study delves into a fuzzy matrix decision-making problem involving fuzzy sets of states. It establishes that a maximum guaranteed payoff constitutes a type-2 fuzzy set defined on the real line. Additionally, it provides the associated type-2 membership function. Moreover, the paper illustrates that the maximum guaranteed payoff type-2 fuzzy set of the decision-making problem can be broken down, based on the secondary membership grades, into a finite collection of fuzzy numbers. Each of these fuzzy numbers represents the maximum guaranteed payoff of the corresponding decision-making problem with a crisp set of states. This set corresponds to a specific cut of the original fuzzy set of states. Some properties of the maximum guaranteed payoff type-2 fuzzy set are investigated, and illustrative examples are provided. Since the problem formulation is symmetrical with respect to alternatives and states of nature, the results obtained can be used in the case of a fuzzy set of alternatives.