The single-machine parallel-batching scheduling problem with family jobs to minimize makespan

In this paper we consider the single-machine parallel-batching scheduling problem with family jobs under on-line setting in the sense that we construct our schedule irrevocably as time proceeds and do not know of the existence of any job until its arrival. Our objective is to minimize the maximum completion time of the jobs (makespan). We deal with the special variant of the problem: the unbounded model in which the machine can handle infinite number of jobs simultaneously,the jobs only have two distinct arrival times and come from m families. we provide an on-line algorithm with a worst case ratio of 2 - α/2, where α = √(5 - 1)/2.