Parallel-batching scheduling problem with family jobs for minimizing makespan

In this paper we consider the problem of scheduling family jobs on a parallel-batching machine under on-line setting, Our objective is to minimize the maximum completion time of the jobs (makespan). A batch processing machine can handle up to B jobs simultaneously. The jobs that are processed together form 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. The jobs from different families are incompatible and thus cannot be put in the same batch. We construct our schedule irrevocably as time proceeds and do not know of the existence of any job until its arrival. We deal with the special variant of the problem: the bounded model in which the capacity of the machine is limited, the jobs only have two distinct arrival times and the jobs come from m families. We provide an on-line algorithm with a worst case ratio of 2 - α/2, where α = (√5-1)/2.