A critical step when designing a successful product family is to determine a cost-saving platform configuration along with an optimally distinct set of product variants that target different market segments. Numerous optimization-based approaches have been proposed to help resolve the tradeoff between platform commonality and the ability to achieve distinct performance targets for each variant. However, the high dimensionality of an "all-in-one" algorithm for optimizing the joint problem of 1) platform variable selection, 2) platform design and 3) variant design makes most of these approaches impractical when a large number of products is considered. Many existing approaches have restricted the scope of the problem by fixing platform configuration a priori, limiting platform configuration to an all-or-none component sharing strategy, or by solving subsets of the joint problem in stages, sacrificing optimality. In this study, we propose a single-stage optimization approach for solving the joint product family problem with generalized commonality using an efficient decomposition solution strategy involving multi-objective genetic algorithms (MOGAs). The proposed approach overcomes prior limitations by introducing a generalized two-dimensional commonality chromosome and decomposing the joint formulation into a two-level GA, where the upper-level determines the optimal platform configuration while each lower-level designs one of the individual variants in the family. Moreover, all sub-problems run in parallel, and the upper-level GA coordinates consistency among the lower-levels using the MPI (Message Passing Interface) library. The proposed approach is demonstrated by optimizing a family of three general aviation aircraft, and results outperform those from a non-decomposed GA. Results also show that the commonality-performance Pareto front contains solutions with generalized commonality, suggesting the need to avoid all-or-none component sharing restrictions in order to avoid sub-optimality. Future work in scaling the decomposed CA to larger product families is also discussed.