We consider batch scheduling on m machines to minimize the makespan. Each job has a given set of machines to be assigned. Each machine can process several jobs simultaneously as a batch, and the machines may have different batch capacities. We study two models: (i) scheduling on equal-speed batch machines under a nested processing set restriction, where the machines have the same processing speed, and (ii) scheduling on uniform batch machines under a tree-hierarchical processing set restriction, where the machines have different processing speeds. For both models we design polynomial-time approximation algorithms to solve them. The algorithms have a worst-case ratio of 2 for non-identical batch capacities and a worst-case ratio of \(2-1/m\) for identical batch capacities.

我们考虑在m台机器上进行批处理调度以最小化制造时间。每个作业都有一组给定的机器要分配。每台机器可以同时处理多个作业作为一个批次，并且这些机器可能具有不同的批次容量。我们研究了两个模型：（i）在嵌套处理集限制下的等速批处理机器上的调度，其中机器具有相同的处理速度，以及（ii）在树层次处理集限制下的均匀批处理机器上的调度，其中机器具有不同的处理速度。对于这两个模型，我们设计了多项式时间逼近算法来解决它们。对于不同批次容量，算法的最坏情况比率为 2，对于相同批次容量，最坏情况比率为\(2-1/m\)。