We evaluate the performance of a Flexible Manufacturing System (FMS), that manufactures multiple part types loaded on general purpose pallets. We model the FMS as an open queueing network with restricted capacity. The performance of the queueing network is approximated by applying a generalization of the so-called aggregation method (or flow-equivalent server method) for queueing networks. The resulting R-dimensional Markov process (where R denotes the number of part types) can be analyzed by the matrix-geometric approach. Unfortunately, the proposed method becomes numerically intractable for large values of R. We therefore also present a type aggregation method that reduces solving the R-dimensional problem to solving R two-dimensional problems. Furthermore, we present an alternative way to generalize the aggregation method. In this generalization the multiple part type network is modeled as a network with a single chain in which parts may change class. Numerical results are given to test and compare the accuracy of the three approximation methods.