Online unbounded batch scheduling on parallel machines with delivery times

Abstract

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 $q_i\ge q_j$ . For the restrict case, we provide a best possible online algorithm with competitive ratio $1+\alpha _m$, where $\alpha _m>0$ is determined by $\alpha _m^2+m\alpha _m=1$. Then we present an online algorithm with a competitive ratio of $1+2/\lfloor \sqrt{m}\rfloor$ for the general case.