With respect to the multi-attribute group decision-making problems in which the attribute values take the form of the interval grey linguistic variables, the multi-attribute group decision making method based on the interval grey uncertain linguistic variable generalized hybrid averaging (IGULGHA) operator is proposed. Firstly, the operation rules, the properties, and the comparing method of the interval grey uncertain linguistic variables are introduced. Then some aggregation operators are defined, such as interval grey uncertain linguistic variable generalized ordered weighted aggregation (IGULGOWA) operator, and interval grey uncertain linguistic variable generalized hybrid aggregation (IGULGHA) operator; some properties of these operators, such as commutativity, idempotency, monotonicity and bounded are proved, then the decision making methods based on these operators are proposed to solve the group decision making problems. Finally, a numerical example is used to show the proposed method, and the influence of the different position weight vector and the different parameter on decision making is analyzed. The result shows the approach is simple and effective.