Copula-based Estimation Method of the Expected Value Function for Robust Design Optimization

The expected value function (EVF), which defined as function of mean value with respect to the design parameters, can simplify the solving process of the RDO with nested-double loop. The traditional method for estimating the expected value function requires a nested procedure and the computational cost depends on the product of the optimal steps and the numbers of the random variable samples. In this paper, a novel method based on copula is proposed to estimate the EVF. In the proposed method, the EVF is derived as the copula function related to the design parameters and the model output, which can be estimated with a single loop samples and the computational cost of the copula-based method does not depend on the total number of design parameters or the number of the optimal steps. Additionally, a vine-copula is introduced to the EVF so that the estimation for the EVF by two-dimensional kernel density estimation (KDE) method can be extended to the multi-dimensional design parameters problems. The present example with two cases demonstrate the effectiveness of the proposed copula-based method.