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 ≥ α pj, 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 ≤ α pj, where 0 < α < 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...