Discrete-Time MDP Modeling for Multi-Item Capacitated Lot Sizing with Stochastic Demand Timing
Discrete-Time MDP Modeling for Multi-Item Capacitated Lot Sizing with Stochastic Demand Timing
针对需求时间随机的多物品容量约束批量问题的离散时间 MDP 建模
Abstract: This paper studies a finite-horizon multi-item capacitated lot-sizing problem in which demand quantities are deterministic, while demand-arrival periods are stochastic. Each demand occurs once within a known time window and must be satisfied no later than its deadline. The proposed model makes production and allocation decisions at the demand level, allowing it to represent capacity competition, demand-specific backlog, and allocation-dependent inventory dynamics.
摘要： 本文研究了一个有限时域的多物品容量约束批量问题，其中需求量是确定性的，而需求到达周期是随机的。每项需求在已知的时窗内发生一次，且必须在截止日期前得到满足。所提出的模型在需求层面做出生产和分配决策，从而能够表征产能竞争、特定需求积压以及依赖于分配的库存动态。
The stochastic problem is formulated as a discrete-time Markov decision process (DTMDP), including the state space, feasible actions, transition kernel, and one-period cost function. To isolate the computational effect of stochastic timing, each stochastic instance is first compared with a deterministic counterpart in which each arrival distribution is replaced by its most likely arrival period. This comparison shows that stochastic timing substantially increases the number of states, the number of transitions, solution time, and memory pressure.
该随机问题被建模为离散时间马尔可夫决策过程（DTMDP），包括状态空间、可行动作、转移核和单周期成本函数。为了隔离随机时间对计算的影响，研究首先将每个随机实例与确定性对应实例进行比较，后者将每个到达分布替换为其最可能的到达周期。比较结果表明，随机时间显著增加了状态数量、转移数量、求解时间和内存压力。
A genetic algorithm (GA) is then proposed for the stochastic-timing problem. The GA searches over feasible state-feedback policies and evaluates each policy exactly under the DTMDP transition model. Computational experiments on 330 benchmark instances show that the GA remains close to the exact stochastic solution whenever the latter is available, with an average optimality gap of about $3.44%$.
随后，本文针对随机时间问题提出了一种遗传算法（GA）。该遗传算法在可行状态反馈策略中进行搜索，并在 DTMDP 转移模型下对每种策略进行精确评估。在 330 个基准实例上的计算实验表明，在能够获得精确随机解的情况下，遗传算法的结果与之非常接近，平均最优性差距约为 $3.44%$。
On the difficult benchmark instances, comprising 90 test cases, the GA remains below the $5%$ optimality-gap threshold and achieves an average optimization speedup of $6.89 \pm 1.41$ at the $95%$ confidence level. For instances that cannot be solved exactly on the available hardware, an empirical Bellman-time regression is used to estimate the missing exact resolution time and extrapolate the expected GA speedup.
在包含 90 个测试用例的困难基准实例上，遗传算法的最优性差距保持在 $5%$ 以下，并在 $95%$ 置信水平下实现了 $6.89 \pm 1.41$ 的平均优化加速比。对于现有硬件无法精确求解的实例，研究使用经验贝尔曼时间回归（empirical Bellman-time regression）来估计缺失的精确求解时间，并推断预期的遗传算法加速比。