Based on the synergistic effect of the shape memory alloy （SMA） and inerter， a novel shape memory alloy-inerter （SMAI） cooperative system has been proposed for higher effectiveness， better robustness and wider engineering application. The closed-form expressions for calculating the displacement variances under random Gaussian white noise base excitations are there? upon derived from the dynamics equations of the structure-SMAI system established by resorting to the equivalent linearization method， and then the optimization criterion is determined. By means of iterative optimization process， the optimization analysis for the structure-SMAI system is performed in frequency domain， and then the design parameters of the SMAI for engineering applica? tion are suggested. Results demonstrate that the SMAI has a better effectiveness of alleviating vibration than the SMA， and a nota? bly small damping demand than the tuned inerter damper （TID） after mass ratio of the inerter exceeding 0.3. To further verifying the above results， a time domain analysis of the structure-SMAI system is carried out using simulated white noise and fluctuating wind speed time histories. In addition， it is found that the out force of the inerter in the SMAI is markedly decreased， thus condu? cive to reducing the damage for the structure from the inerter device. Therefore， the SMAI system has the prospect for engineering application.