Cumulative scheduling with variable task profiles and concave piecewise linear processing rate functions
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.
KeywordsContinuous scheduling Continuous resources Concave piecewise linear functions Energy constraints Energetic reasoning
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.
- 4.Derrien, A., & Petit, T. (2014). A new characterization of relevant intervals for energetic reasoning. In International conference on principles and practice of constraint programming, CP 2014 vol. 8656 of lecture notes in computer science (289–297). Springer International Publishing.Google Scholar
- 5.Erschler, J., & Lopez, P. (1990). Energy-based approach for task scheduling under time and resources constraints. In 2nd International workshop on project management and scheduling (pp 115–121). Compiègne.Google Scholar
- 6.Gay, S., Hartert, R., & Schaus, P. (2015). Time-table disjunctive reasoning for the cumulative constraint. In International conference on AI and OR techniques in constraint programming for combinatorial optimization problems, CPAIOR 2015, vol. 9075 of lecture notes in computer science (pp. 157–172). Springer International Publishing.Google Scholar
- 8.Hung, M.N., Le Van, C., & Michel, P. (2005). Non-convex aggregative technology and optimal economic growth. Cahiers de la Maison des Sciences Économiques 2005.95 - ISSN : 1624-0340.Google Scholar
- 11.Lewis, J. Algebra symposium: optimizing fuel consumption. http://homepages.math.uic.edu/~jlewis/math165/asavgcost.pdf.
- 12.Nattaf, M., Artigues, C., Lopez, P., & Rivreau, D. (2015). Energetic reasoning and mixed-integer linear programming for scheduling with a continuous resource and linear efficiency functions. OR Spectrum, 1–34.Google Scholar
- 14.Nattaf, M., Artigues, C., & Lopez, P. (2015). Flow and energy based satisfiability tests for the continuous energy-constrained scheduling problem with concave piecewise linear functions. In CP Doctoral Program 2015 (pp. 70–81). Cork.Google Scholar
- 15.Vilím, P. (2011). Timetable edge finding filtering algorithm for discrete cumulative resources, Integration of AI and OR techniques in constraint programming for combinatorial optimization problems. CPAIOR 2011 vol 6697 of lecture notes in computer science. Berlin: Springer.Google Scholar
- 16.Wolf, A., & Schrader, G. (2006). O(n n) overload checking for the cumulative constraint and its application. In Umeda, M., Wolf, A., Bartenstein, O., Geske, U., Seipel, D., & Takata, O. (Eds.), Declarative programming for knowledge management. INAP 2005 lecture notes in computer science, Vol. 4369. Berlin: Springer.Google Scholar