Real-world multi-objective engineering design optimization problems often have parameters with uncontrollable variations. The aim of solving such problems is to obtain solutions that in terms of objectives and feasibility are as good as possible and at the same time are least sensitive to the parameter variations. Such solutions are said to be robust optimum solutions. In order to investigate the trade-off between the performance and robustness of optimum solutions, we present a new Robust Multi-Objective Genetic Algorithm (RMOGA) that optimizes two objectives: a fitness value and a robustness index. The fitness value serves as a measure of performance of design solutions with respect to multiple objectives and feasibility of the original optimization problem. The robustness index, which is based on a non-gradient based parameter sensitivity estimation approach, is a measure that quantitatively evaluates the robustness of design solutions. RMOGA does not require a presumed probability distribution of uncontrollable parameters and also does not utilize the gradient information of these parameters. Three distance metrics are used to obtain the robustness index and robust solutions. To illustrate its application, RMOGA is applied to two well-studied engineering design problems from the literature.