This paper studies a new mixed batch scheduling problem that arises in vacuum heat treatment. A mixed batch machine can process at most a given number of jobs simultaneously. The processing time of a batch is the weighted sum of the maximum processing time and the total processing time of jobs in the batch. The objective is to minimize the makespan. We first prove that the problem on a single machine can be solved in polynomial time, while the problem on multiple identical machines is NP -hard. Then, we develop a pseudopolynomial time exact algorithm when the number of machines is fixed. Further, we analyze the worst-case performance ratio of a full batch longest processing time algorithm and design Algorithm LPT-Greedy with improved worst-case performance.

中文翻译：

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相同机器上的混合批次调度

本文研究了真空热处理中出现的一个新的混合批次调度问题。一台混合批处理机最多可以同时处理给定数量的作业。一个批次的处理时间是该批次中作业的最大处理时间与总处理时间的加权和。目标是最小化完工时间。我们首先证明单台机器上的问题可以在多项式时间内解决，而多台相同机器上的问题是NP-hard的。然后，当机器数量固定时，我们开发了一个伪多项式时间精确算法。此外，我们分析了全批次最长处理时间算法的最坏情况性能比，并设计了具有改进的最坏情况性能的算法 LPT-Greedy。