This paper addresses a new orthogonal packing and time scheduling problem—3D space-time optimization problem (3D-STO). Given a rectangular sheet and a set of rectangular items, each item needs to be continuously processed with a time length on the sheet, the 3D-STO consists in arranging each item’s loading time, location, and orientation for a certain period such that the total utilization time of the sheet, i.e., the makespan of the schedule, is minimized. The 3D-STO involves a series of 2D rectangular packing problem (2D-RPP) for the whole scheduling period. If the processing time of each item is simply regarded as a space dimension for the packing, then the 3D-STO can be reduced to an NP-hard packing problem—3D strip packing problem (3D-SPP). The 3D-STO differs from the 3D-SPP in that the position and orientation of items can be changed over the processing time (the third dimension) such that the 3D-STO has a much larger search space compared with the 3D-SPP. Hereby, the optimal solution of the 3D-STO is better than or equal to the optimal solution of the 3D-SPP, and a possibly better solution could be found for the 3D-STO model. As a new model, there is no algorithm or benchmark in the literature. Moreover, the 2D-RPP and 3D-SPP algorithms are not suitable for solving the 3D-STO which considers the packing and scheduling simultaneously. Hereby, we propose a caving-degree-based scheduling algorithm (CDS) for the 3D-STO. This is the first proposed 3D-STO algorithm. We also formalize the problem as a mixed integer programming model and solve it by ILOG CPLEX. For evaluation, we provide a synthesized method to generate a total of 195 various benchmark instances with guillotine and nonguillotine cut constraints. The comparative results show that CDS is more effective than the CPLEX solver for the 3D-STO. Also, when comparing CDS for the 3D-STO and the adapted CDS for the 3D-SPP, we see that the new 3D-STO model can enhance the flexibility of the item arrangement and make maximum utilization of the sheet and time.

中文翻译：

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正交包装和调度问题：模型、启发式和基准

本文解决了一个新的正交打包和时间调度问题——3D 时空优化问题 (3D-STO)。给定一个矩形片材和一组矩形物品，每个物品都需要在片材上以一定的时间长度连续处理，3D-STO 包括在一定时期内安排每个物品的加载时间、位置和方向，使得总板材的使用时间，即进度表的完工时间，被最小化。3D-STO 涉及整个调度周期的一系列 2D 矩形包装问题 (2D-RPP)。如果将每个项目的处理时间简单地视为包装的空间维度，那么 3D-STO 可以简化为 NP-hard 包装问题——3D 条状包装问题（3D-SPP）。3D-STO 与 3D-SPP 的不同之处在于，物品的位置和方向可以随着处理时间（第三维）而改变，因此与 3D-SPP 相比，3D-STO 具有更大的搜索空间。因此，3D-STO 的最优解优于或等于 3D-SPP 的最优解，并且可以为 3D-STO 模型找到可能更好的解。作为一种新模型，文献中没有算法或基准。此外，2D-RPP和3D-SPP算法不适用于同时考虑打包和调度的3D-STO问题。因此，我们为 3D-STO 提出了一种基于塌陷度的调度算法（CDS）。这是第一个提出的 3D-STO 算法。我们还将问题形式化为混合整数规划模型，并通过 ILOG CPLEX 解决它。对于评估，我们提供了一种合成方法来生成总共 195 个具有断头台和非断头台切割约束的各种基准实例。比较结果表明，对于 3D-STO，CDS 比 CPLEX 求解器更有效。此外，当比较 3D-STO 的 CDS 和 3D-SPP 的适应 CDS 时，我们看到新的 3D-STO 模型可以增强项目安排的灵活性，并最大限度地利用纸张和时间。