Abstract
We consider a cumulative scheduling problem where a task duration and resource consumption are not fixed. The consumption profile of the task, which can vary continuously over time, is a decision variable of the problem to be determined and a task is completed as soon as the integration over its time window of a non-decreasing and continuous processing rate function of the consumption profile has reached a predefined amount of energy. The goal is to find a feasible schedule, which is an NP-hard problem. For the case where functions are concave and piecewise linear, we present two propagation algorithms. The first one is the adaptation to concave functions of the variant of the energetic reasoning previously established for linear functions. Furthermore, a full characterization of the relevant intervals for time-window adjustments is provided. The second algorithm combines a flow-based checker with time-bound adjustments derived from the time-table disjunctive reasoning for the cumulative constraint. Complementarity of the algorithms is assessed via their integration in a hybrid branch-and-bound and computational experiments on small-size instances.
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Notes
Although all results presented in this paper can be adapted when \(b_{i}^{min}=0\), for ease of notation, we assume \(b_{i}^{min} \neq 0\).
Indeed, suppose that f i is constant in some pieces the first one being p ∈{1,…, P i }, which would cause problem to compute \(f_{i}^{-1}\). In that case, f i must be constant on all pieces {p,…, P i } due to concavity. It follows that \(b_{i}^{max}\) can be reduced to the start of piece p.
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Acknowledgments
The authors thank José Verschae for enlightening discussions. This study was partially supported by project “Energy-Efficient and Robust approaches for the Scheduling of Production, Services and Urban Transport”, ECOS/CONICYT, N◦ C13E04.
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This article belongs to the Topical Collection: Integration of Artificial Intelligence and Operations Research Techniques in Constraint Programming
Guest Editors: Michele Lombardi and Domenico Salvagnin
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Nattaf, M., Artigues, C. & Lopez, P. Cumulative scheduling with variable task profiles and concave piecewise linear processing rate functions. Constraints 22, 530–547 (2017). https://doi.org/10.1007/s10601-017-9271-4
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DOI: https://doi.org/10.1007/s10601-017-9271-4