Reduction methods of type-2 uncertain variables and their applications to solid transportation problem

Uncertainty theory is a branch of mathematics for dealing with realistic uncertainties arising out of complexity, changeability, and non-decidability of practical environments. An uncertain variable is defined as a function from the uncertainty space to the set of real numbers and is characterized by an uncertainty distribution. This paper proposes the definition of type-2 uncertain variables within the framework of uncertainty theory through introduction of generalized uncertain measures and focuses on more complex twofold uncertainties. Some uncertainty reduction methods associated with type-2 uncertain variables are also proposed for convenience of applicability, including reduction of optimistic value, pessimistic value and expected value. Moreover, four classes of type-2 uncertain variables are reduced to type-1 uncertain variables with specific uncertainty distributions. Type-2 uncertain optimization methods are applied to solving the fixed charge solid transportation problem with the type-2 uncertain parameters, where the solution methods are also provided for the proposed models. Finally, numerical experiments are implemented to demonstrate application and sensitivity analysis of the proposed approaches.