To overcome the drawback to the overlarge mass of Tuned Mass Damper （TMD）， Tuned Inerter Damper （TID） has been obtained by replacing mass element of TMD with inerter. A base-isolated structure， simplified as a 2-degree-of-freedom （2DOF） system， with supplemental TID was studied by stochastic response analysis with an optimal method based on performance requirement， which considered the target displacement of isolation layer as a constraint， and acceleration of superstructure as an optimal object. The optimal designs of TID with two control criterions and the comparison with the same structure with supplemental Linear Viscous Damper （LVD）， which is a unitary damper， were conducted. The analysis results indicated that the displacement of isolation layer and the acceleration of superstructure could be reduced simultaneously by the attachment of TID， which was significantly more advantageous than LVD.