This paper addresses the single-item capacitated lot sizing problem with a 1-breakpoint all-units quantity discount in a monotonic setting where the purchase prices are non-increasing over the planning horizon. For this case, we establish several novel properties of the optimal solution and develop a hybrid dynamic programming approach that maintains a compact representation of the solution space by storing only essential information about the states and using linear equations for intermediate values. Our algorithm runs in \(O(n\log n)\) time, where \(n\) denotes the number of periods. Our result is an improvement over the previous state-of-the-art algorithm, which has an \(O(n^2)\) time complexity.

翻译：本文研究了在采购价格随规划周期非递增的单调环境下，具有单断点全单位数量折扣的单产品有容量批量规划问题。针对此情形，我们建立了最优解的几个新颖性质，并开发了一种混合动态规划方法。该方法通过仅存储状态的关键信息并利用线性方程计算中间值，保持了解空间的紧凑表示。我们的算法运行时间为 \(O(n\log n)\)，其中 \(n\) 表示周期数。我们的结果改进了先前时间复杂度为 \(O(n^2)\) 的最先进算法。