The confidence of reliability indicates that reliability has randomness induced by any epistemic uncertainties, and these uncertainties can be reduced and manipulated by additional knowledge. In this paper, the uncertainty of input statistical models is mainly treated in the context of confidence-based design optimization (CBDO). Thus, the objective of this paper is to determine the optimal number of data for reliability-based design optimization (RBDO) under input model uncertainty. The uncertainty of input statistical models due to insufficient data is frequent in practical applications since collecting and testing samples of random variables requires engineering efforts. There are two ways to increase the confidence of reliability to be satisfied, which are shifting design vector and supplementing input data. The purpose of this research is to find balanced optimum accounting for a trade-off between two operations since both operations lead to the growth of overall cost. Therefore, it is necessary to optimally distribute the resources to two costs which are denoted as the operating cost of design vector and the development cost of acquiring new data. In this study, two types of costs are integrated as a bi-objective function, satisfying the probabilistic constraint for the confidence of reliability. The number of data is regarded as design variable to be optimized, and stochastic sensitivity analysis of reliability with respect to the number of data is developed. The proposed bi-objective CBDO can determine the optimal number of input data based on the current dataset. Then, the designers decide the additional number of tests for collecting input data according to the optimum of bi-objective CBDO to minimize the overall cost.