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

Batch processing machine scheduling in uncertain environment attracts more and more attention in the last decade. This paper deals with semi-online scheduling on two parallel batch processing machines with non-decreasing processing time of job. Jobs arrive over time in the online paradigm, and the processing time of any batch is equal to the length of the last arrival job in the batch. We study the unbounded model where each processing batch may contain an unlimited number of jobs, and the objective is to minimize the makespan. Given any job J(j) together with its following job J(j+1), it is assumed that their processing times satisfy p(j+1) >= alpha p(j) where alpha >= 1 is a constant. That is, jobs arrive in a non-decreasing order of processing times. We mainly propose an optimal phi-competitive online algorithm where phi >= 1 is a solution of equation phi(3) + (alpha - 1)phi(2) + (alpha(2) - alpha - 1)phi - alpha(2) = 0.