Quality characteristics (QCs) are important product performance variables that determine customer satisfaction. Their expected values are optimized and their standard deviations are minimized during robust design (RD). Most of RD methodologies consider only a single QC, but a product is often judged by multiple QCs. It is a challenging task to handle dependent and oftentimes conflicting QCs. This work proposes a new robustness modeling measure that uses the maximum quality loss among multiple QCs for problems where the quality loss is the same no matter which QCs or how many QCs are defective. This treatment makes it easy to model RD with multivariate QCs as a single objective optimization problem and also account for the dependence between QCs. The new method is then applied to problems where bivariate QCs are involved. A numerical method for RD with bivariate QCs is developed based on the first order second moment (FOSM) method. The method is applied to the mechanism synthesis of a four-bar linkage and a piston engine design problem.