Online unbounded batch scheduling on parallel machines with delivery times - Liu, Peihai; Lu, Xiwen (Journal of combinatorial optimization, 2015)
We consider the online unbounded batch scheduling problems on m identical machines subject to release dates and delivery times. Jobs arrive over time and the characteristics of jobs are unknown until their arrival times. Jobs can be processed in a common batch and the batch capacity is unbounded. Once the processing of a job is completed it is independently delivered to the destination. The objective is to minimize the time by which all jobs have been delivered. For each job J j , its processing time and delivery time are denoted by p j and q j , respectively. We first consider a restricted model: the jobs have agreeable processing and delivery times, i.e., for any two jobs J i and J j p i > p j implies q i ≥ q j . For the restrict case, we provide a best possible online algorithm with competitive ratio 1 + α m , where α m > 0 is determined by α m 2 + m α m = 1 . Then we present an online algorithm with a competitive ratio of 1 + 2 / ⌊ m ⌋ for the general case.