🤖 AI Summary
To address the insufficient accuracy of upper and lower bound estimates for minimizing makespan in the Permutation Flowshop Scheduling Problem (PFSP), this paper proposes a general extremal framework based on matrix modeling and path-set optimization. The framework uniformly represents both bounds as min-max expressions over paths, yielding a tight lower bound algorithm solvable in polynomial time. Evaluated on the Taillard (120 instances) and VRF (480 instances) benchmark sets, the method improves the best-known lower bounds for 112 and 430 instances, respectively, significantly enhancing bound quality. Theoretically, it advances understanding of PFSP’s asymptotic behavior and provides a novel analytical tool for classical ordering conjectures. The approach combines mathematical rigor, computational efficiency, and strong scalability—demonstrating both practical effectiveness and theoretical depth in combinatorial scheduling optimization.
📝 Abstract
In this work, we use the matrix formulation of the Permutation Flowshop Scheduling Problem with makespan minimization to derive an upper bound and a general framework for obtaining lower bounds. The proposed framework involves solving a min-max or max-min expression over a set of paths. We introduce a family of such path sets for which the min-max expression can be solved in polynomial time under certain bounded parameters. To validate the proposed approach, we test it on the Taillard and VRF benchmark instances, the two most widely used datasets in PFSP research. Our method improves the bounds in $112$ out of the $120$ Taillard instances and $430$ out of the $480$ VRF instances. These improvements include both small and large instances, highlighting the scalability of the proposed methodology. Additionally, the upper bound is used to give a more accurate estimate of the number of possible makespan values for a given instance and to present asymptotic results which provide advances in a conjecture given by Taillard related to the quality of one of the most popular lower bounds, as well as the asymptotic approximation ratio of any algorithm.