AI 中文总结
本文针对分段凹生产成本的批量订货问题，提出基于Monge矩阵搜索的精确算法，在固定断点和边界水平下实现更优复杂度，并通过大量基准验证其有效性和效率。
AI 中文摘要
我们研究了在 $T$ 个时期内具有分段凹生产与持有—缺货成本的单物品批量订货问题。生产具有 $m$ 个共同的正有限断点，库存成本和域具有 $K$ 个共同的有限边界水平。对于固定的 $m,K$，我们给出了一个精确确定性算法，使用 $\text{O}_{m,K}(T^{m+2}\text{α}(T+2))$ 次算术运算，以及一个拉斯维加斯算法，使用 $\text{O}_{m,K}(T^{m+2})$ 次期望运算，其中 α 是反阿克曼函数。这些算法保留了非线性凹成本段，并涵盖了缺货、公共网格库存限制和明确的期末库存策略。在库存边界水平上的结构分解产生了仅断点的前缀和后缀表。通过指定生产期间的批量转换产生了隐式矩阵，这些矩阵在每个生产段上是 Monge 双阶梯矩阵。已建立的矩阵搜索、合并状态顺序和向后评估提供了这些界限。一个更简单的矩形/SMAWK 变体需要 $\text{O}_{m,K}(T^{m+2}\text{log}(T+2))$ 次运算。在 300 个无缺货基准案例上，其目标与先前动态规划和库存状态预言机完全一致。几何平均配对时间比率对于无容量限制案例为 \text{GeoUncap}，对于有容量限制案例为 \text{GeoCap}，比率大于 1 时有利于矩阵搜索。一个 640 案例的基础模型验证套件和一个单独的 624 案例的广义模型套件通过。时间研究没有对广义模型或更强的阶梯例程进行基准测试。所有界限都假设精确算术、常数时间成本查询和显式闭域凹性条件。
英文摘要
We study single-item lot sizing with piecewise-concave production and holding--backlog costs over $T$ periods. Production has $m$ common positive finite breakpoints, and stock costs and domains have $K$ common finite boundary levels. For fixed $m,K$, we give an exact deterministic algorithm using $\OmK(T^{m+2}α(T+2))$ arithmetic operations and a Las Vegas algorithm using $\OmK(T^{m+2})$ expected operations, where $α$ is the inverse Ackermann function. The algorithms retain nonlinear concave cost pieces and cover backlogging, common-grid stock limits, and explicit terminal-stock policies. A structural decomposition at stock boundary levels yields breakpoint-only prefix and suffix tables. Batching transitions by a designated production period produces implicit matrices that are Monge double staircases on each production piece. Established matrix searches, merged state orders, and backward evaluation provide the bounds. A simpler rectangle/SMAWK variant requires $\OmK(T^{m+2}\log(T+2))$ operations. On 300 no-backlogging benchmark cases, its objectives agree exactly with the predecessor dynamic program and an inventory-state oracle. Geometric-mean paired time ratios are \GeoUncap\ for uncapacitated cases and \GeoCap\ for capacitated cases, with ratios above one favoring matrix searching. A 640-case base-model validation suite and a separate 624-case generalized-model suite pass. The timing study does not benchmark the generalized models or the stronger staircase routines. All bounds assume exact arithmetic, constant-time cost queries, and explicit closed-domain concavity conditions.
