A Flow-Based Formulation for Parallel Machine Scheduling Using Decision Diagrams

We present a new flow-based formulation for identical parallel machine scheduling with a regular objective function and without idle time. The formulation is constructed with the help of a decision diagram that represents all job sequences that respect specific ordering rules. These rules rely on a partition of the planning horizon into, generally nonuniform, periods and do not exclude all optimal solutions, but they constrain solutions to adhere to a canonical form. The new formulation has numerous variables and constraints, and hence we apply a Dantzig-Wolfe decomposition to compute the linear programming relaxation in reasonable time; the resulting lower bound is stronger than the bound from the classical time-indexed formulation. We develop a branch-and-price framework that solves three instances from the literature for the first time. We compare the new formulation with the time-indexed and arc time–indexed formulation by means of a series of computational experiments. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms—Discrete. Funding: This work was partially funded by the European Union’s Horizon 2020 research and innovation program under [Marie Skłodowska-Curie Grant 754462]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2022.0301 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2022.0301 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .