# A cutting plane approach for the multi-machine precedence-constrained scheduling problem

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## Abstract

A cutting-plane approach is developed for the problem of optimally scheduling jobs with arbitrary precedence constraints on unrelated parallel machines to minimize weighted completion time. While the single machine version of this problem has attracted much research efforts, enabling solving problems with up to 100 jobs, not much has been done on the multiple machines case. A novel mixed-integer programming model is presented for the problem with multiple machines. For this model, many classes of valid inequalities that cut off fractional linear programming solutions are developed. This leads to an increase of the linear programming lower bound from 89.3 to 94.6% of the corresponding optimal solution, and a substantial reduction in the computational time of an optimal branch-and-bound algorithm for this problem. This enables us to report optimal solutions for problem instances with up to 25 jobs and 5 machines, which is more than twice the size of problems for which optimal solutions have been reported in the literature thus far. For a special case of the problem—that of minimizing makespan—application of our model helps solve 18 of 27 previously unsolved problem instances to optimality.

## Keywords

Unrelated machine scheduling Precedence-constrained scheduling Optimization Integer programming Valid inequalities## Notes

### Acknowledgements

The authors would like to thank the authors of Coll et al. (2006) for sharing problem instances reported in their work. All of the problem instances on which solutions have been reported in our paper have been made available as an online supplement. We also thank the editor and the two anonymous referees whose comments helped improve the contribution of the paper.

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