By focusing on various influences arose from environmental change, sustainability has become a major conception among many fields, including utility allocation. On the other hand, game-theoretical methods have always been adopted to analyze the reasonability of utility allocation rules. In many real-world situations, however, participants and its energetic levels (decisions) should be essential factors simultaneously. By focusing on both the participants and its energetic levels (decisions), we introduce the restrained core to investigate utility allocation under fuzzy transferable-utility (TU) models. In order to analyze the reasonability for the restrained core, two axiomatic results are further provided by applying several types of reductions. Since the restrained core infringes a specific converse steadiness property, a converse steady enlargement of the restrained core is also introduced to investigate how extensive the violation of this specific converse steadiness property is. This converse steady enlargement is smallest converse steady measuration that contains the restrained core.
