Summary: We consider TU-games with a vector-valued characteristic function defined on a set of permissible coalitions. Within those games we shall propose a generalized core. The idea behind this new core comes from the superadditive duality theory developed for integer programming. We show that this generalized core is non-empty, whereas the classical core may be empty. Furthermore, the weighted payment schemes in the generalized core are stable, satisfy group rationality and have the property that no coalition can obtain an improvement by itself. Moreover, we extend into the multiple criteria case the well-known theorem for single criterion TU-games stating that a TU-game is balanced if and only if its (classical) core is non-empty.