This paper unifies the classical intuitionistic fuzzy additive and multiplicative operations and proposes a generalized intuitionistic fuzzy additive operation and a generalized interval-valued intuitionistic fuzzy additive operation. In particular, it was proved that the classical additive and multiplicative operations on intuitionistic fuzzy sets (IFSs) are two special cases of the newly proposed generalized intuitionistic fuzzy additive operation, while the classical additive and multiplicative operations on interval-valued IFSs (IVIFSs) are two special cases of the generalized interval-valued intuitionistic fuzzy additive one. This paper has three innovation points. First, it introduces two kinds of 'intuitionistic preference factors,' which are the key parameters of the generalized intuitionistic fuzzy additive and the generalized interval-valued intuitionistic additive operations. Second, it proposes a generalized intuitionistic fuzzy additive aggregating operator based on an intuitionistic preference factor and a generalized interval-valued intuitionistic fuzzy additive aggregating operator based on the mean value theorem for integrals. Third, two novel multiple attribute decision-making approaches under the intuitionistic fuzzy environment are proposed. In addition, two examples are given to verify the validity of the proposed generalized fuzzy additive operators and decision-making approaches.