We consider the problem of scheduling jobs with sizes, release times and delivery times on identical parallel batch machines. All machines have the same capacity. Each machine can simultaneously process several jobs as a batch as long as the total size of these jobs does not exceed the capacity. The release time, processing time and delivery time of a batch are defined to be the latest release time, longest processing time and largest delivery time of all the jobs in the batch, respectively. The objective is to minimize the maximum delivery completion time, i.e., the time by which all jobs are delivered. For this strongly NP-hard problem, there does not exist any polynomial time algorithm with approximation ratio better than 2 unless P=NP, even if all jobs have equal processing times and all release times and delivery times are zero. We present an algorithm with approximation ratio arbitrarily close to 2.

我们考虑在相同的并行批处理计算机上按大小，发布时间和交付时间计划作业的问题。所有机器具有相同的容量。每台计算机可以同时批量处理多个作业，只要这些作业的总大小不超过容量即可。批处理的发布时间，处理时间和交付时间分别定义为批处理中所有作业的最新发布时间，最长处理时间和最大交付时间。目的是使最大交付完成时间（即所有作业的交付时间）最小化。对于这个强烈的NP-hard问题，除非所有作业具有相同的处理时间且所有释放时间和交付时间均为零，否则除非存在P = NP，否则不存在任何近似比大于2的多项式时间算法。