This paper addresses the problem of scheduling chain-like structures of tasks on a single multiprocessor resource. In fact, sub-tasks of unit-time length and predefined size are aggregated to composite tasks that have to be scheduled without preemption, but subject to flexibility concerning resource allocation. This setting most closely resembles the problem of malleable task scheduling, with sub-tasks being the smallest atomic unit of allocation. The specific type of malleability is realized using precedence constraints with minimum and maximum time lags. A bin packing model is established for this scheduling problem and a corresponding, dedicated branch-and-bound algorithm is devised, alongside problem-specific bound tightening, symmetry breaking and dominance concepts. The efficacy of the solution approach is demonstrated based on extensive computational experiments, including randomized instances, adapted benchmark instances from the literature, and a small real-world data set. In comparison to mixed-integer and constraint programming formulations, the new method is able to achieve a considerably higher percentage of optimal solutions at computation times that are up to orders of magnitude smaller.

本文解决了在单个多处理器资源上调度任务链状结构的问题。事实上，单位时间长度和预定义大小的子任务被聚合成复合任务，这些任务必须在没有抢占的情况下进行调度，但在资源分配方面具有灵活性。这个设置最接近于可延展任务调度的问题，子任务是最小的原子分配单元。使用具有最小和最大时滞的优先约束来实现特定类型的延展性。为这个调度问题建立了一个装箱模型，并设计了一个相应的专用分支定界算法，以及特定于问题的边界紧缩、对称破坏和优势概念。基于广泛的计算实验证明了解决方法的有效性，包括随机实例、来自文献的改编基准实例和一个小的真实世界数据集。与混合整数和约束规划公式相比，新方法能够在更小几个数量级的计算时间内实现更高百分比的最优解。