带随机需求时间的多品种容量受限批量问题的离散时间马尔可夫决策过程建模

中文摘要 AI 辅助

本文研究有限时间域下的多品种容量受限批量问题，其中需求数量为确定性值，而需求到达时段为随机值。每个需求在已知的时间窗口内发生一次，且必须在其截止日期前得到满足。所提模型在需求层面制定生产与分配决策，使其能够表征产能竞争、需求特定的缺货以及与分配相关的库存动态。该随机问题被建模为离散时间马尔可夫决策过程（DTMDP），包含状态空间、可行动作、转移核以及单周期成本函数。为分离随机时间的计算影响，将每个随机实例与其确定性对应实例（其中每个到达分布被替换为其最可能的到达时段）进行比较。结果显示，随机时间会大幅增加状态数量、转移数量、求解时间以及内存压力。随后针对随机时间问题提出一种遗传算法（GA），该算法在可行的状态反馈策略空间中搜索，并基于DTMDP转移模型精确评估每个策略。在330个基准实例上开展的计算实验表明，当存在精确随机解时，GA的性能与该解接近，平均最优性间隙约为3.44%；在包含90个测试用例的困难基准实例上，GA的最优性间隙始终低于5%的阈值，且在95%置信水平下实现了6.89±1.41的平均优化加速比。对于在可用硬件上无法精确求解的实例，采用经验贝尔曼时间回归方法估计缺失的精确求解时间，并外推GA的预期加速比。

英文摘要

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. 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. 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 $3.44\%$. 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.

发表机构

• Université Paris-Saclay（巴黎萨克雷大学）
• Univ Evry（埃夫里大学）

机构由 AI 辅助整理，请以论文原文为准。