This paper studies the bicriteria problem of scheduling $n$ jobs on a serial-batch machine to minimize makespan and maximum cost simultaneously. A serial-batch machine can process up to $b$ jobs as a batch, where $b$ is known as the batch capacity. When a new batch starts, a constant setup time is required for the machine. Within each batch, the jobs are processed sequentially, and thus the processing time of a batch equals the sum of the processing times of its jobs. All the jobs in a batch have the same completion time, namely, the completion time of the batch. The main result is an $O(n^3)$-time algorithm which can generate all Pareto optimal points for the bounded model ($b<n$) without precedence relation. The algorithm can be modified to solve the unbounded model ($b\ge n$) with strict precedence relation in $O(n^3)$ time as well. The results improve the previously best known running time of $O(n^4)$ for both the bounded and unbounded models.

翻译：本文研究在串行批处理机上调度$n$个工件以同时最小化完工时间和最大成本的双目标问题。串行批处理机最多可处理$b$个工件作为一个批次，其中$b$称为批容量。当新批次开始时，机器需要固定的准备时间。在每个批次内，工件按顺序处理，因此批次的处理时间等于其包含工件处理时间之和。同一批次中的所有工件具有相同的完成时间，即该批次的完成时间。主要成果是提出一种$O(n^3)$时间算法，可为无优先约束的有界模型（$b<n$）生成所有帕累托最优点。该算法经修改后同样能以$O(n^3)$时间复杂度求解具有严格优先约束的无界模型（$b\ge n$）。这些结果将先前有界与无界模型的最佳已知时间复杂度从$O(n^4)$进行了改进。