Optimizing the design of electric machines is a vital step in ensuring the economical use of active materials. The three-dimensional flux paths in axial flux PM (AFPM) machines necessitate the use of computationally expensive 3D electromagnetic analysis. Furthermore, a large number of design evaluations is required to find the optimum, causing the total computation time to be excessively long. In view of this, a two-level surrogate assisted algorithm capable of handling such expensive optimization problems is introduced, which substantially reduces the number of FEA evaluations. The proposed algorithm is employed to optimally design an AFPM machine within a specified envelope, identifying the limits of cost and efficiency. Optimized designs with different rotor polarities are systematically compared in order to form the basis for a set of generalized design rules.