a small increase of the overall makespan. We do this by splitting bags into two groups $G_1$ and $G_2$ with $G_1$ only holding a constant number of bags. We then show that to find a feasible distribution of all large jobs, that it is sufficient to schedule all large jobs from $G_1$ without conflicts. Even when multiple jobs of the same bag in $G_2$ are on one machine, we can repair the schedule without increasing the makespan. We use this insight to distribute large jobs to machines with an MILP using a constant number of integral variables.

For the second group $G_2$ especially the placement of the small jobs can cause problems, since we can have many bags with a large number of small jobs. Additionally these jobs might conflict with large jobs already in place. To solve this problem we introduce a new instance transformation that we apply before constructing and solving our MILP. This transformation splits large and small jobs of bags in $G_2$ in separate bags, while only increasing the height of the schedule marginally. To schedule bags only containing small jobs we can apply some easy heuristics such as largest processing time first (LPT) [@graham] with 