We study a single-machine lot-sizing problem, where n types of products need to be scheduled on the machine. Each product is associated with a constant demand rate, maximum production rate and inventory costs per time unit. Every time when the machine switches production between products, sequencing costs are incurred. These sequencing costs depend both on the product the machine just produced and on the product the machine is about to produce. The goal is to find a cyclic schedule minimizing total average costs, subject to the condition that all demands are satisfied. We establish the complexity of the problem, and we prove a number of structural properties largely characterizing optimal solutions. Moreover, we present two algorithms approximating the optimal schedules by augmenting the problem input. Due to the high-multiplicity setting, even trivial cases of the corresponding conventional counterparts become highly non-trivial with respect to the output sizes and computational complexity, even without sequencing costs. In particular, the length of an optimal solution can be exponential in the input size of the problem. Nevertheless, our approximation algorithms produce schedules of a polynomial length and with a good quality compared to the optimal schedules of exponential length.

中文翻译：

---

与测序成本有关的循环批量大小问题

我们研究了一个单机批量大小问题，其中需要在机器上安排 n 种产品。每个产品都与恒定的需求率、最大生产率和单位时间的库存成本相关联。每次机器在产品之间切换生产时，都会产生排序成本。这些排序成本取决于机器刚刚生产的产品和机器即将生产的产品。目标是在满足所有需求的条件下找到最小化总平均成本的循环计划。我们确定了问题的复杂性，并证明了一些主要表征最佳解决方案的结构特性。此外，我们提出了两种通过增加问题输入来逼近最佳调度的算法。由于高多重设置，即使没有排序成本，即使是相应的传统对应物的微不足道的情况，在输出大小和计算复杂性方面也变得非常重要。特别是，最优解的长度可以是问题输入大小的指数。尽管如此，我们的近似算法产生了多项式长度的调度，并且与指数长度的最佳调度相比具有良好的质量。