An exact extended formulation for the unrelated parallel machine total weighted completion time problem

• Published in: Journal of scheduling Vol. 20; no. 4; pp. 373 - 389
• Main Authors: Bülbül, Kerem, Sen, Halil
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.08.2017; Springer Nature B.V
• Subjects: Artificial Intelligence; Benders decomposition; Business and Management; Calculus of Variations and Optimal Control; Optimization; Completion time; Integer programming; Iterative algorithms; Iterative methods; Job shops; Linear programming; Mixed integer; Operations Research/Decision Theory; Optimization; Preempting; Production scheduling; Scheduling; Supply Chain Management; Transportation problem; Benders decomposition; Cut strengthening; Unrelated parallel machines; Exact method; Weighted completion time; Preemptive relaxation
• DOI: 10.1007/s10951-016-0485-x
• ISSN: 1094-6136

The plethora of research on NP -hard parallel machine scheduling problems is focused on heuristics due to the theoretically and practically challenging nature of these problems. Only a handful of exact approaches are available in the literature, and most of these suffer from scalability issues. Moreover, the majority of the papers on the subject are restricted to the identical parallel machine scheduling environment. In this context, the main contribution of this work is to recognize and prove that a particular preemptive relaxation for the problem of minimizing the total weighted completion time (TWCT) on a set of unrelated parallel machines naturally admits a non-preemptive optimal solution and gives rise to an exact mixed integer linear programming formulation of the problem. Furthermore, we exploit the structural properties of TWCT and attain a very fast and scalable exact Benders decomposition-based algorithm for solving this formulation. Computationally, our approach holds great promise and may even be embedded into iterative algorithms for more complex shop scheduling problems as instances with up to 1000 jobs and 8 machines are solved to optimality within a few seconds.