Optimal Semi-online Algorithm for Scheduling on a Batch Processing Machine

We consider two semi-online scheduling problems on a single batch (processing) machine with jobs’ nondecreasing processing times and jobs’ nonincreasing processing times, respectively. Our objective is to minimize the makespan. A batch processing machine can handle up to B jobs simultaneously. We study an unbounded model where B = ∞. The jobs that are processed together construct a batch, and all jobs in a batch start and complete at the same time. The processing time of a batch is given by the longest processing time of any job in the batch. Jobs arrive over time. Let pj denote the processing time of job Jj . Given job Jj and its following job Jj+1, we assume that pj+1 ≥ αpj , where α ≥ 1 is a constant number, for the first problem with jobs’ nondecreasing processing times. For the second problem, we assume that pj+1 ≤ αpj , where 0 < α < 1 is a constant number. We propose an optimal algorithm for both problems with a competitive ratio √ α2+4−α 2 + 1 for the first problem and √ 4α+1+1 2 for the second problem.