Squirrel cage induction motors design requires making numerous trade-offs, especially between its audible electromagnetic noise level, its efficiency and its material cost. However, adding the vibro-acoustic and thermal models to the usual electrical model of the motor drastically increases the simulation time. A finite element approach is then inconceivable, especially if the model has to be coupled to an evolutionary optimization algorithm. A fast simulation tool of the variable-speed induction machine, based on electrical, mechanical, acoustic and thermal analytical models, has therefore been elaborated. It has been validated at different stages with both tests and finite element method (FEM) simulations. This model is then coupled to a mixed-variable constrained Non-dominating Sorting Genetic Algorithm (NSGA-II) with a stochastic repair algorithm. Some global optimizations with respect to several objectives (noise level, efficiency and material cost), including thermal constraints, are finally presented, and several convenient visualizations of multi-dimensional solutions are presented.