Robust design (RD) has been recognized as one of the most useful approaches to improve the quality of products/processes. Response surface methodology (RSM), a significant tool to perform an RD procedure, is often used to estimate the fitted response functions for the process mean and variance by assuming that experimental errors are normally distributed. When this assumption is violated in many real world industrial situations , other alternative estimation methods (i.e., maximum likelihood estimation and weighted least squares methods) can be considered. The primary objective of this paper is to propose a Weibull regression model as an alternative RD modeling and estimation for the failure-time of a system that is followed by a Weibull distribution. In addition, a new inverse problem-based estimation method in order to improve estimation efficiency for the failure-time data is then proposed. Finally, a simulation study is conducted for verification purposes.