Intuitionistic fuzzy sets are the most widely used fuzzy set extension in the literature. It is a fuzzy set extension in which decision makers specify the degree of membership of the elements in the set as well as their non-membership to the set. Thus, the indecision of the decision makers about the membership of the elements to the set also emerges spontaneously. It is known that discrete intuitionistic fuzzy sets or linear continuous intuitionistic fuzzy sets are used in the literature. In this study, it is aimed to develop non-linear continuous intuitionistic fuzzy sets and to use them in multi-criteria decision making models. CINFUSs consisting of membership and non-membership degrees represented by second-order non-linear functions, have been used to develop the analytical hierarchy process (AHP) and Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) methodology in a fuzzy environment. This study is a milestone in showing how higher order non-linear functions can be used in intuitionistic fuzzy sets. The CINFUS-AHP&TOPSIS methodology has been applied to the solution of the multi-criteria research proposals evaluation for grand funding problem and has been tested for validity and robustness with sensitivity analysis and comparative analysis.