Modeling and optimization for multiple correlated responses with distribution variability

In production design processes, multiple correlated responses with different distributions are often encountered. The existing literature usually assumes that they follow normal distributions for computational convenience, and then analyzes these responses using traditional parametric methods. A few research papers assume that they follow the same type of distribution, such as the t-distribution, and then use a multivariate joint distribution to deal with the correlation. However, these methods give a poor approximation to the actual problem and may lead to the recommended settings that yield substandard products. In this article, we propose a new method for the robust parameter design that can solve the above problems. Specifically, a semiparametric model is used to estimate the margins, and then a joint distribution function is constructed using a multivariate copula function. Finally, the probability that the responses meet the specifications simultaneously is used to obtain the optimal settings. The advantages of the proposed method lie in the consideration of multiple correlation patterns among responses, the absence of restrictions on the response distributions, and the use of nonparametric smoothing to reduce the risk of model misspecification. The results of the case study and the simulation study validate the effectiveness of the proposed method.