The vector bin packing problem (VBP) is a generalization of bin packing with multiple constraints. In this problem we are required to pack items, represented by p-dimensional vectors, into as few bins as possible. The multiple-choice vector bin packing (MVBP) is a variant of the VBP in which bins have several types and items have several incarnations. We present an exact method, based on an arc-flow formulation with graph compression, for solving MVBP by simply representing all the patterns in a very compact graph. As a proof of concept we report computational results on a variable-sized bin packing data set.