Optimal Semi-Online Algorithm for Scheduling on Two Parallel Batch Processing Machines

This paper deals with semi-online scheduling on two parallel batch processing machines with jobs’ nondecreasing processing times. Our objective is to minimize the makespan. A batch processing machine can handle up to B jobs simultaneously. We study unbounded model which means that B is sufficiently large. 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. We propose an optimal algorithm with a competitive ratio φ such that φ + (α− 1)φ + (α − α− 1)φ− α = 0 (φ ≥ 1).