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 p j denote the processing time of job J j . Given job J j and its following job J j + 1 , we assume that p j + 1 *** ***p j , where *** *** 1 is a constant number, for the first problem with jobs' nondecreasing processing times. For the second problem, we assume that p j + 1 ≤ ***p j , where 0 < *** < 1 is a constant number. We propose an optimal algorithm for both problems with a competitive ratio $\frac{\sqrt{\alpha^2+4}-\alpha}{2}+1$ for the first problem and $\frac{\sqrt{4\alpha+1}+1}{2}$ for the second problem.