Scheduling equal-length jobs with arbitrary sizes on uniform parallel batch machines

We consider the problem of scheduling jobs with equal lengths and arbitrary sizes on uniform parallel batch machines with different capacities. Each machine can only process the jobs whose sizes are not larger than its capacity. Several jobs can be processed as a batch simultaneously on a machine, as long as their total size does not exceed the machine's capacity. The objective is to minimize makespan. Under a divisibility constraint, we obtain two efficient exact algorithms. For the general problem, we obtain an efficient 2-approximation algorithm. Previous work has shown that the problem cannot be approximated to within an approximation ratio better than 2, unless P = NP, even when all machines have identical speeds and capacities.