This paper investigates the rotor design optimization of synchronous reluctance machines (SynRMs) by using electromagnetic and structural finite-element simulations. Three conflicting objectives, i.e., average torque, torque ripple, and sound pressure level, were considered using a surrogate-based multi-objective approach. While the stator is fixed, the rotor flux barriers of a 33-slot 8-pole SynRM were geometrically varied to extract optimal design regions. These regions or constraints help decrease the computational time during the sampling procedure of a multiple-barrier design. Different numbers of flux barriers were studied and related to each other. Adding the sound pressure level was observed to affect previous results by spreading the Pareto front solutions across the design space.