Efficient and effective robust airfoil design optimization is proposed by integrating stochastic expansions and utility theory. In this work, the stochastic expansions are generated efficiently using non-intrusive polynomial chaos (NIPC) expansions. The robust design problem is formulated using utility functions which transfer a targeted response using a prescribed mathematical function to represent the designers' risk preferences. The proposed approach is demonstrated using examples of lift-constrained airfoil drag minimization in transonic viscous flow using the Mach number as an uncertain variable in the range of 0.70 to 0.75. The results are compared with the common problem formulation for robust design of the minimizing the sum of the mean and standard deviation of performance metric, as well as with single- and multi-point deterministic optimization. The approach is demonstrated on two numerical test cases, one at relatively low lift coefficient of 0.5, and the other one at a high lift of 0.824. The constraints differ between the cases as well. In both cases, the proposed approach with utility function formulation achieves the most insensitive responses compared with the standard robust problem formulation and the single- and multi-point deterministic problem formulations.