In grey correlation degree decision model (GRCM), it is assumed that all the attributes are mutually independent. However, in real decision making problems, the interaction often exists between attributes which leads GRCM to lose effectiveness. For this problem, the fuzzy integral theory is introduced and grey fuzzy integral correlation degree decision model (GRFICM) is established. To solve the model, the Mobius transformation coefficients based on weights and interaction degrees are defined to calculate 2-order additive fuzzy measures. In Mobius transformation coefficients, the weights are determined by the rank correlation analysis method and Mahalanobis-Taguchi Gram-Schmidt jointly, and the interaction relations and interaction degrees are judged by experts. An evaluation of the financial situation of low-rent housing safeguard family is provided as a practical case in order to validate GRCM and GRFICM by comparing. The validation results show that GRFICM makes the decision results more scientific and reasonable, and is more worth of spreading.