Abstract
We consider a multi-activity shift scheduling problem where the objective is to construct anonymous multi-activity shifts that respect union rules, satisfy the demand and minimize workforce costs. An implicit approach using adapted forward and backward constraints is proposed that integrates both the shift construction and the activity assignment problems. Our computational study shows that using the branch-and-bound procedure of CPLEX 12.6 on the proposed implicit model yields optimal solutions in relatively short times for environments including up to 2970 millions of explicit shifts. Our implicit model is compared to the grammar-based implicit model proposed by Côté et al. (Manag Sci 57(1):151–163, 2011b) on a large set of instances. The results prove that both implicit models have their strengths and weaknesses and are more or less efficient depending on the scheduling environment.
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Acknowledgements
This work was supported by the Fonds de Recherche Nature et Technologie du Québec (FRNTQ) through its new researchers start-up grant. This support is gratefully acknowledged. The authors thank Louis Martin Rousseau, Marie-Claude Côté and Maria Isabel Restrepo for providing their benchmark and their source code, thus allowing a comparison with their grammar-based approach.
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Appendices
Appendix A: The grammar used to adapt the Côté et al. (2011) model to instances in the Dahmen set
All the grammars used are presented in accordance with the notations adopted by Côté et al. (2011). For clarity, they are not stated in Chomsky normal form. For each production P,‘\( \rightarrow _{S,[min,max]} \)’ restricts the subsequences generated to have a position belonging to the set of periods S and a span between min and max periods. When S or [min, max] are not specified for a production, there are no restrictions on its position or span, respectively. The grammar used for the Dahmen set with no restrictions on the number of activities, denoted \( \mathcal {G} \), is defined as follows:
When restrictions on the number of activities are considered (that is, \(NA_{am} , NA_{pm} = 2\)), some of the productions above defining \( \mathcal {G} \) are changed as follows:
Appendix B: Additional results for Section 6
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Dahmen, S., Rekik, M. & Soumis, F. An implicit model for multi-activity shift scheduling problems. J Sched 21, 285–304 (2018). https://doi.org/10.1007/s10951-017-0544-y
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DOI: https://doi.org/10.1007/s10951-017-0544-y