Nowadays, energy-efficient scheduling has assumed a key role in ensuring the sustainability of manufacturing processes. In this context, we focus on the bi-objective problem of scheduling a set of jobs on identical parallel machines to simultaneously minimize the maximum completion time and the total energy consumption over a time horizon partitioned into a set of discrete slots. The energy costs are determined by a time-of-use pricing scheme, which plays a crucial role in regulating energy demand and flattening its peaks. First, we uncover a symmetry-breaking property that characterizes the structure of the solution space of the problem. As a consequence, we provide a novel, compact mixed-integer linear programming formulation at the core of an efficient exact solution algorithm. A thorough experimental campaign shows that the use of the novel mathematical programming formulation enables the solution of larger-scale instances and entails a reduction in the computational times as compared to the formulation already available in the literature. Furthermore, we propose a new heuristic that improves the state-of-the-art in terms of required computational effort and quality of solutions. Such a heuristic outperforms the existing heuristics for the problem and is also capable of speeding up the exact solution algorithm when used for its initialization. Finally, we introduce a novel dynamic programming algorithm that is able to compute the optimal timing of the jobs scheduled on each machine to further improve the performance of the new heuristic.

如今，节能调度在确保制造过程的可持续性方面发挥着关键作用。在这种情况下，我们关注在相同的并行机器上调度一组作业的双目标问题，以同时最小化划分为一组离散槽的时间范围内的最大完成时间和总能耗。能源成本由分时定价方案决定，该方案在调节能源需求和压平能源峰值方面发挥着至关重要的作用。首先，我们发现了表征问题解空间结构的对称破缺性质。因此，我们提供了一种新颖、紧凑的混合整数线性规划公式，其核心是高效的精确求解算法。彻底的实验活动表明，与文献中已有的公式相比，使用新颖的数学规划公式可以解决更大规模的实例，并且需要减少计算时间。此外，我们提出了一种新的启发式方法，可以在所需的计算工作量和解决方案的质量方面提高最先进的水平。这种启发式方法优于该问题的现有启发式方法，并且在用于其初始化时还能够加速精确求解算法。最后，我们引入了一种新颖的动态编程算法，该算法能够计算每台机器上调度的作业的最佳时间，以进一步提高新启发式的性能。