Abstract: Our work focuses on various complexity measures for two-handed assembly sequences. For many products, there exist an exponentially large set of valid sequences, and a natural goal is to use automated systems to select wisely from the choices. Although there has been a great deal of algorithmic success for finding feasible assembly sequences, there has been very little success towards optimizing the costs of sequences. We attempt to explain this lack of progress, by proving the inherent difficulty in finding optimal, or even near-optimal, assembly sequences. To begin, we define, "virtual assembly sequencing," a graph-theoretic problem that is a generalization of assembly sequencing, focusing on the combinatorial aspect of the family of feasible assembly sequences while temporarily separating out the specific geometric assumptions inherent to assembly sequencing. We formally prove the hardness of finding even near-optimal sequences for most cost measures in our generalized framework. As a special case, we prove equally strong inapproximability results for the problem of scheduling with AND/OR precedence constraints. Finally, we re-introduce the geometry, and continue by realizing several of these hardness results in rather simple geometric settings. We are able to show strong inapproximability results, for example using an assembly consisting solely of unit disks in the plane.
Keywords: assembly sequencing, approximability, cost measures, AND/OR scheduling, non-directional blocking graphs