A robust optimization procedure based on a multi-objective genetic algorithm (MOGA) is used to generate airfoil profiles for transonic inviscid flows of dense gases, subject to uncertainties in the upstream thermodynamic conditions. The effect of the random variations on system response is evaluated using a non-intrusive polynomial chaos (PC) based method known as the probabilistic collocation method (PCM). After initial PCM simulations which showed that the dense gas system was highly sensitive to input parameter variation, a multi-objective genetic algorithm coupled to the PCM produced a Pareto front of optimized individual geometries which exhibited improvements in mean performance and/or stability over the baseline NACA0012 airfoil. This type of analysis is essential in improving the feasibility of organic Rankine cycle (ORC) turbines, which are typically designed to recover energy from variable sources such as waste heat from industrial processes.