A type-2 fuzzy set(T2FS) is essentially a set generated by expanding the membership degree of a fuzzy set into a type-1 fuzzy set, and it is an effective tool to deal with decision analysis problems in complex and uncertain environment. First, this paper systematically reviews the proposed new mathematical representation definition of type-2 fuzzy sets(T2FSs), and further demonstrates its geometric interpretations under different universe conditions. Second, aiming at the significant advantages of type-2 fuzzy preference relations in dealing with multi-criteria decision-making problems in complex decision-making scenarios, basic research is conducted on type-2 fuzzy preference relations and their applications in multi-criteria decision-making. More specifically, the definition and decision interpretations of type-2 fuzzy preference relations are given based on the new mathematical representation method, and the additive as well as multiplicative consistency conditions of type-2 fuzzy preference relations are defined simultaneously. Then, according to the theory on fuzzy preference relations, the related methods and properties of type-2 fuzzy preference relations are constructed and analyzed to illustrate the scientificity and rationality of type-2 fuzzy preference relations and the proposed new mathematical representation definition of T2FSs. Finally, the effectiveness and feasibility of the multi-criteria decision-making method based on type-2 fuzzy preference relations under the new definition are verified by a case study on tourism product selection and a comparative analysis.