# Energetic reasoning for energy-constrained scheduling with a continuous resource

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## Abstract

This paper addresses a scheduling problem with continuous resources and energy constraints. Given a set of non-preemptive activities, each activity requires a continuously divisible resource whose instantaneous usage is limited in maximum and minimum, its processing satisfying a time window and a total energy (time \(\times \) resource-usage) requirement. The goal consists of getting a schedule satisfying all the constraints. The problem, which we refer to as the Energy-Constrained Scheduling Problem with Continuous Resources (CECSP), is a generalization of the well-known cumulative scheduling problem for which the “energetic reasoning” or “left-shift/right-shift” necessary feasibility condition yielded a popular polynomially computable satisfiability test. The paper presents a generalization of the energetic reasoning for the CECSP, defining precisely the activity minimum consumptions and exhibiting a polynomial number of relevant time intervals on which it is sufficient to apply the satisfiability tests. A strongly polynomial energetic reasoning satisfiability test can be derived for the considered generalization, which also yields a short proof for the complexity of the original algorithm. Some limits of the approach, as well as an approximation framework for more general resource consumption functions, are also addressed.

## Keywords

Scheduling with continuous resources Energy requirement Energetic reasoning satisfiability test Complexity## Notes

### Acknowledgments

This research was part of OCrE project supported by CNRS/INSIS (Centre National de la Recherche Scientifique/Institut des Sciences de l’Ingénierie et des Systèmes). The authors are indebted to David Rivreau for enlightening discussions. A preliminary version of this work was presented as a “Late breaking abstract” at CPAIOR 2011, Berlin, and we thank an anonymous referee for pointing out to us the interest of studying the fixed start time case. Finally, the authors warmly thank Margaux Nattaf for her help on improving the revised version of the paper.

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