Online unbounded batch scheduling on parallel machines with delivery times

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+ alpha(m), where alpha(m) > 0 is determined by alpha(2)(m) + m alpha(m) = 1. Then we present an online algorithm with a competitive ratio of 1 + 2/ [root m] for the general case.