发表机构
Mines Saint-Etienne（圣艾蒂安国立高等矿业学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出网格理论，通过加性结构和网格包络，证明多种动态批量问题在多项式时间内可解，核心贡献在于识别加性结构而非资源值数量为多项式性的关键。
AI 中文摘要
为什么某些动态批量问题是多项式的？我们通过引入网格理论来回答这个问题，这是一个基于累计生产和生产界限的加性结构的结构性框架。对于具有生产和生产下界和上界的一般单物品动态批量模型，存在一个最优极端解，其中在每个再生区间内，除最多一个生产量外，其余生产量均位于边界值上。这产生了加性网格，主网格定理确立了最优累计生产轨迹可以被限制在这些离散集合上。尽管产生的网格可能呈指数级大，我们引入了加性维数的概念来刻画那些边界和允许低维表示的产量界限轮廓。我们证明了有界加性维数产生一个多项式可构造的网格包络和一个基于网格的多项式时间动态规划算法。该框架扩展到可分离凹成本，并确立了多个族的多项式可解性，包括恒定容量、最小订购量、固定数量的容量级别、固定次数的多项式容量、周期性容量和分段多项式容量。特别是，即使不同容量值的数量随规划期增长，多项式性也可能成立。因此，网格理论将加性结构，而非不同资源值的数量，确定为多项式可解的充分机制。
英文摘要
Why are some dynamic lot-sizing problems polynomial? We address this question by introducing Grid Theory, a structural framework based on cumulative production and the additive structure of production bounds. For a general single-item dynamic lot-sizing model with lower and upper production bounds, there exists an optimal extreme solution in which, within each regeneration interval, all but at most one production quantity lie on a boundary value. This induces additive grids, and the Main Grid Theorem establishes that an optimal cumulative production trajectory can be restricted to these discrete sets. Although the resulting grids may be exponentially large, we introduce the notion of additive dimension to capture production-bound profiles whose boundary sums admit a low-dimensional representation. We show that bounded additive dimension yields a polynomially constructible grid envelope and a polynomial time grid-based dynamic programming algorithm. The framework extends to separable concave costs and establishes polynomial solvability of several families, including constant capacities, minimum order quantities, a fixed number of capacity levels, fixed-degree polynomial capacities, periodic capacities, and piecewise polynomial capacities. In particular, polynomiality may hold even when the number of distinct capacity values grows with the planning horizon. Grid Theory thus identifies additive structure, rather than the number of distinct resource values, as a sufficient mechanism for polynomial solvability.
