Arc-flow approach for single batch-processing machine scheduling

TL;DR

This paper introduces an arc-flow based optimization model for single batch-processing machine scheduling with non-identical jobs, effectively solving large instances to optimality by representing the problem as flows in graphs.

Contribution

The paper presents a novel arc-flow formulation that efficiently handles large-scale scheduling problems with diverse job sizes and processing times, outperforming existing models.

Findings

Superior performance on benchmark instances

Optimal solutions for instances with up to 100 million jobs

Effective for large-scale problems with multiple identical jobs

Abstract

We address the problem of scheduling jobs with non-identical sizes and distinct processing times on a single batch processing machine, aiming at minimizing the makespan. The extensive literature on this NP-hard problem mostly focuses on heuristics. Using an arc flow-based optimization approach, we construct an ingenious formulation that represents it as a problem of determining flows in graphs. The size of the formulation increases with the number of distinct sizes and processing times among the jobs, but it does not increase with the number of jobs, which makes it very effective to solve large instances to optimality, especially when multiple jobs have equal size and processing time. We compare our model to other models from the literature showing its clear superiority on benchmark instances, and proving optimality of random instances with up to 100 million jobs.