Whereas the robust design concept has been well established in the probability theory, it has not been developed in the <b>possibility</b> <b>theory</b>. For problems where accurate statistical information for input data is not available, a possibility-based (or fuzzy set) robust design concept is proposed in this paper by investigating the similarity between the membership function of the fuzzy variable and the cumulative distribution function of the random variable. Based on the probability-possibility consistency principle, a random variable that corresponds to the fuzzy variable is introduced in this paper in order to define the robust design concept for the fuzzy variable. For the system with input fuzzy variables, the robustness measure of the output performance is computed using the performance measure integration (PMI) method, while the integration points are obtained from the inverse possibility analysis by using the maximal possibility search method with interpolation (MPS). For the system with mixed random and fuzzy input variables, the robustness measure of the output performance is computed using PMI method, with the integration points obtained from the inverse mixed analysis by using the maximal failure search method (MFS). A new mixed (random and fuzzy) variable robust design optimization (MVRDO) method is proposed and several numerical examples are used to verify the robust design concept in the <b>possibility</b> <b>theory</b> and the MVRDO formulation.