The bump-type gas foil bearing (GFB) is widely used in various high-speed oil-free turbomachinery due to its extra features such as high-speed capability without any external oil lubricant. Under the high-speed condition, the performance characteristics of GFB are majorly dependent on its design parameter. Thus, this paper briefly analyzes and optimizes the design parameters of bump-type GFB to improve its performance characteristic. The numerical simulation of bump-type GFB is performed in ANSYS software. The effect of various design parameters such as foil thickness, bump half-length, bump pitch, and bump angle on the output responses are analyzed using a response surface methodology based CCD design matrix. Here, the output responses are structural stiffness and the equivalent stress in foil bearing. To optimize it, the multi-objective GRA technique is used. These results show that the foil thickness is the most influencing and the bump angle is the least influencing design parameter. The optimal value of foil thickness, bump angle, bump half-length and bump pitch are 0.14mm, 63.75 deg., 1.55mm and 4.6mm respectively. At these optimal design parameters, the deviation between the predicted regression model and the numerical results are within 3.5%. Moreover, with these optimal design parameters, the structural stiffness of foil bearings increases by 68.4% and stress distribution reduces by 44.22% compared to the general configuration of foil bearings.