| This dissertation introduces the concepts of the tolerable solution set, united solution set, and controllable solution set of interval-valued fuzzy relational equations. Given a continuous t-norm, it is proved that each of the three types of the solution sets of interval-valued fuzzy relational equations with a max-t-norm composition, if nonempty, is composed of one maximum solution and a finite number of minimal solutions. Necessary and sufficient conditions for the existence of solutions are given. Computational procedures based on the constructive proofs are proposed to generate the complete solution sets. Examples are given to illustrate the procedures. Similarly, it is also proved that each type of solution set of interval-valued fuzzy relational equations with a min-s-norm composition, if nonempty, is composed of one minimum solution and a finite number of maximal solutions.; For interval-valued games, a new method for ranking interval numbers is introduced. Interval-valued cooperative games are defined based on this method. Three axioms as desired properties of an interval-valued cooperative game were proposed. It is proved that a unique payoff function, which is similar to the Shapley value function, exists and satisfies the proposed axioms. Furthermore, this payoff function can be applied to non-superadditive games. |
