Bounds for the Permutation Flowshop Scheduling Problem: New Framework and Theoretical Insights

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.