两阶段随机串联系统中的1.73最优阶梯库存(R,nQ)策略
A 5/3 Guarantee for Echelon Stock (R,nQ) Policies in Two-Stage Stochastic Serial Systems
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中文总结 AI 辅助
针对两阶段串联库存系统，提出阶梯库存(R,nQ)策略，证明其成本至多为最优的1.73倍，且数值实验显示比率接近1。
中文摘要 AI 辅助
我们考虑一个经典的两阶段连续审查串联库存系统，其中需求为速率为lambda的单位泊松过程，缺货成本率为p，提前期为L_1和L_2，阶梯持有成本率为h_1和h_2，固定装运成本为K_1和K_2。经典的阶梯库存(R,nQ)策略提供了一个简单的操作规则：阶段1订购固定数量的Q_1批，阶段2一次订购n>=1个这样的批次，并且阶段1的请求需等待直到有完整的批次可用。现有的均匀性保证限制了提前期或诱导批量大小，而已知的全实例保证依赖于原始参数，因此尚不清楚这个受限的整数比率类别是否能在整个原始参数空间上提供任何均匀性保证。我们开发了一种保持设置的成本分配下界，并证明在经典类别内的下确界成本最多为速率平衡的可容许比较类别中最优成本的1.73倍，对于所有lambda、p、L_1、L_2、h_1、h_2、K_1、K_2>=0均成立。该结果适用于精确整数批量大小，并且在两种设置约定下均成立：一种对每次正向发货收费，另一种对每个完整批次收费。在边界情况下，当lambda=0、p=0或h_2=0时，该保证为1，作为下确界的等式，并且当K_2=0时改进为5/3。在数值上，跨越三个广泛的参数网格，找到的最佳(R,nQ)策略成本最多为评估下界的1.098倍，中位数比率为1.007；一个故意对抗性的压力测试在保守的批次成本约定下报告了1.6004。在这一点上，在装运成本核算下的有限搜索发现，允许阶段2装运不完整的Q_1批次没有好处，这与评估下界的松散性一致。因此，简单的(R,nQ)策略将直接固定的批次实施与整个非负原始参数空间上的均匀性保证相结合。
英文摘要
Fixed shipment costs encourage large orders, while inventory and customer backlog costs favor frequent replenishment. In a serial system, the stages must also coordinate their lot sizes and material availability. We study the classical echelon stock $(R,nQ)$ policy: the downstream stage requests a fixed lot, the upstream stage orders an integer multiple of that lot, and transfers contain complete lots. For unit Poisson demand, deterministic lead times, setup and holding costs at both stages, and full backlogging, we prove that the optimized cost in this class is at most $5/3\approx1.667$ of the optimal admissible system cost, uniformly over all nonnegative cost rates, demand rates, and lead times. The guarantee holds even when the selected policy pays a setup charge for every lot while the optimal benchmark pays only once per aggregate shipment. To obtain the guarantee, we use a common inventory calculation that allows independently chosen replenishment frequencies while respecting upstream material availability. We compare its optimized value, an upper bound on the best integer-ratio policy cost, with a novel lower bound on the optimal system cost that combines allocated single-stage costs with the minimum inventory cost when setup costs are zero. We also show that no uniform factor below $(1+\sqrt{3})/2\approx1.366$ is possible for the classical class. Existing numerical experiments place best-found classical policies within $22.7\%$ of an evaluated lower bound across three broad grids. The results quantify both the reliability and the limits of a simple operating rule and explain how shipment consolidation, replenishment timing, and flexibility affect its value.
发表机构
- Rotman School of Management, University of Toronto（多伦多大学罗特曼管理学院）
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