Minimizing total completion time on a batching machine with job processing time compatibilities

The problem of scheduling n jobs on an unbounded batching machine to minimize the total completion time is studied. The machine can process any number of jobs simultaneously in a batch, subject to an additional constraint that, in the same batch, the job processing times are compatible. There are given normal job processing times. An actual job processing time can exceed its normal value up to a certain percent. This percent is the same for all jobs. Thus, there are processing time intervals for the jobs. The job processing times are compatible if the corresponding processing time intervals intersect. The processing time of a batch is given by the longest processing time of the tasks in the batch and it correspnds to the left endpoint of the intersection of the job processing time intervals in this batch. For the total completion time a dynamic programming algorithm is provided.