•Propose a novel reliability analysis approach to handle both the input variations and surrogate model uncertainty.•Propose a smooth sensitivity analysis approach to facilitate the reliability-based design optimization (RBDO) process.•Propose a surrogate-based optimization framework to enable the quantification of surrogate model uncertainty in reliability assessment. Surrogate models have been widely employed as approximations of expensive physics-based simulations to alleviate the computational burden in reliability-based design optimization. Ignoring the surrogate model uncertainty due to the lack of training samples will lead to untrustworthy designs in product development. This paper addresses the surrogate model uncertainty in reliability analysis using the equivalent reliability index (ERI) and further develops a new smooth sensitivity analysis approach to facilitate the surrogate model-based product design process. By using the Gaussian process (GP) modeling technique, a Gaussian mixture model (GMM) is constructed for reliability analysis using Monte Carlo simulations. To propagate both input variations and surrogate model uncertainty, the probability of failure is approximated by calculating the equivalent reliability index using the first and second statistical moments of the GMM. The sensitivity of ERI with respect to design variables is analytically derived based on the GP predictions. Three case studies are used to demonstrate the effectiveness and robustness of the proposed approach.