This paper addresses a single machine scheduling problem with non-preemptive jobs to minimize the total electricity cost. Two latest trends in the area of the energy-aware scheduling are considered, namely the variable energy pricing and the power-saving states of a machine. Scheduling of the jobs and the machine states are considered jointly to achieve the highest possible savings. Although this problem has been previously addressed in the literature, the reported results of the state-of-the-art method show that the optimal solutions can be found only for instances with up to 35 jobs and 209 intervals within 3 hours of computation. We propose an elegant pre-processing technique called SPACES for computing the optimal switching of the machine states with respect to the energy costs. The optimal switchings are associated with the shortest paths in an interval-state graph that describes all possible transitions between the machine states in time. This idea allows us to implement efficient integer linear programming and constraint programming models of the problem while preserving the optimality. The efficiency of the models lies in the simplification of the optimal switching representation. The results of the experiments show that our approach outperforms the existing state-of-the-art exact method. On a set of benchmark instances with varying sizes and different state transition graphs, the proposed approach finds the optimal solutions even for the large instances with up to 190 jobs and 1277 intervals within an hour of computation.

本文解决了具有非抢占性工作的单机调度问题，以最大程度地降低总电力成本。考虑了节能调度领域中的两个最新趋势，即可变能源定价和机器的节能状态。作业和机器状态的调度共同考虑，以实现最高的节省。尽管此问题先前已在文献中得到解决，但最新方法的报告结果表明，仅在计算的3小时内，最多具有35个作业和209个间隔的实例才能找到最佳解决方案。我们提出了一种优雅的预处理技术，称为SPACES，用于计算与能耗相关的机器状态的最佳切换。最佳切换与间隔状态图中的最短路径相关联，该状态图中描述了机器状态之间在时间上所有可能的转换。这个想法使我们能够在保持最优性的同时实现问题的有效整数线性规划和约束规划模型。模型的效率在于最佳开关表示的简化。实验结果表明，我们的方法优于现有的最新精确方法。在具有不同大小和不同状态转换图的一组基准实例上，即使在一个小时的计算中，对于具有190个作业和1277个间隔的大型实例，所提出的方法也可以找到最佳解决方案。