On-line scheduling on a batch processing machine with unbounded batch size to minimize the makespan

We present on-line algorithms to minimize the makespan on a single batch processing machine. We consider a parallel batching machine that can process up to b jobs simultaneously. Jobs in the same batch complete at the same time. Such a model of a batch processing machine has been motivated by burn-in ovens in final testing stage of semiconductor manufacturing. We deal with the on-line scheduling problem when jobs arrive over time. We consider a set of independent jobs. Their number is not known in advance. Each job is available at its release date and its processing requirement is not known in advance. This general problem with infinite machine capacity is noted 1|p - batch, rj, b = [infinity]|Cmax. Deterministic algorithms that do not insert idle-times in the schedule cannot be better than 2-competitive and a simple rule based on LPT achieved this bound [Z. Liu, W. Yu, Scheduling one batch processor subject to job release dates, Discrete Applied Mathematics 105 (2000) 129-136]. If we are allowed to postpone start of jobs, the performance guarantee can be improved to 1.618. We provide a simpler proof of this best known lower bound for bounded and unbounded batch sizes. We then present deterministic algorithms that are best possible for the problem with unbounded batch size (i.e., b = [infinity]) and agreeable processing times (i.e., there cannot exist an on-line algorithm with a better performance guarantee). We then propose another algorithm that leads to a best possible algorithm for the general problem with unbounded batch size. This algorithm improves the best known on-line algorithm (i.e. [G. Zhang, X. Cai, C.K. Wong, On-line algorithms for minimizing makespan on batch processing machines, Naval Research Logistics 48 (2001) 241-258]) in the sense that it produces a shortest makespan while ensuring the same worst-case performance guarantee.