Online batching scheduling with delivery times related to processing times

We study a single batch machine on-line scheduling problem with delivery times, which is related to processing times. Once the processing of a job is completed it is delivered to the destination, the objective is to minimize the time by which all jobs have been delivered. For each job Jj, its arrival time, processing time and delivery time are denoted by rj, pj and qj, respectively. Two variants are considered: 1) pj≥ βqj, we provide an optimal on-line algorithm with competitive ratio (√5+1)/2; 2) qj= βqj, we provide an on-line algorithm with competitive ratio(√β²+2β+5−β+1)/2, and give a lower bound (√β+5)/(β+1)+1)/2.