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.

中文翻译：

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具有可变任务配置文件和凹分段线性处理速率函数的累积调度

我们考虑任务时间和资源消耗不固定的累积调度问题。任务的消耗曲线可以随时间连续变化，它是要确定的问题的决策变量，并且只要在其时间窗内对任务的非递减和连续处理速率函数进行积分，就可以完成任务。消耗曲线已达到预定义的能量量。目的是找到一个可行的时间表，这是一个NP难题。对于凹函数和分段线性函数，我们提出了两种传播算法。第一个是先前为线性函数建立的能量推理变体对凹函数的适应。此外，提供了用于时间窗调整的相关间隔的完整特征。第二种算法将基于流的检查器与从针对累积约束的时间表析取推理得出的时限调整相结合。通过在小型实例的混合分支定界和计算实验中集成算法，可以评估算法的互补性。