Improved algorithms for single machine serial-batch scheduling to minimize makespan and maximum cost

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.