The paper addresses the robustness of multi-objective optimization problems with fuzzy data, expressed via triangular fuzzy numbers. To this end, we introduced a new robustness approach able to deal with fuzziness in the multi-objective context. The proposed approach is composed of two main contributions: First, new concepts of β-robustness are proposed to analyze fuzziness propagation to the multiple objectives. Second, an extension of our previously proposed evolutionary algorithms is suggested for integrating robustness. These proposals are illustrated on a multi-objective vehicle routing problem with fuzzy customer demands. The experimental results on different instances show the efficiency of the proposed approach.