Minimizing Total Weighted Completion Time on Uniform Machines with Unbounded Batch∗

In this paper, we consider the scheduling problem of minimizing total weighted completion time on uniform machines with unbounded batch. Each of the machine Ml (l = 1, ⋅ ⋅ ⋅ ,m) has a speed sl and can process up to B(≥ n) jobs simultaneously as a batch. The processing time of a batch denoted by P(B) is given by the processing time of the longest job in it, and the running time of the batch on machine Ml is P(B) sl . We present some useful properties of the optimal schedule, and then design an O(nm+2) time backward dynamic programming algorithm for the problem.