In many decentralized supply chains, upstream firms do not observe market demand directly and instead infer downstream conditions from the order stream. A retailer's replenishment policy therefore plays a dual role: it governs inventory replenishment and shapes the information available for upstream forecasting. This creates a fundamental trade-off. Smoother orders improve upstream predictability, but delaying the response to demand can increase downstream inventory costs. We study how a retailer should optimally smooth demand in a two-tier supply chain with one retailer and one manufacturer when the manufacturer forecasts future orders from the retailer's order history. We propose Binomial Smoothing, a class of replenishment policies that implements delayed demand response by spreading each unit of demand over a finite horizon using binomial weights. The class is interpretable, easy to calibrate, and analytically tractable. Under weakly stationary Gaussian demand satisfying mild regularity conditions, we show that, for any fixed smoothing horizon, the Binomial policy minimizes the manufacturer's forecast error among all policies with the same degree of smoothing. It remains invertible, so the manufacturer can recover demand history from observed orders. More generally, Binomial Smoothing achieves a constant-factor approximation guarantee relative to an optimal policy. Our results yield a broader insight: replenishment policies should be designed not merely to reduce order variance, as in the traditional bullwhip measure, but to reduce the unpredictable component of orders. Carefully designed smoothing can improve supply-chain performance and partially substitute for information sharing, providing a concrete mechanism for coordination without collaboration.

翻译：在许多分散式供应链中，上游企业无法直接观测市场需求，只能通过订单流推断下游状况。因此，零售商的补货策略扮演双重角色：既管理库存补货，又塑造上游预测可用的信息。这产生了根本性权衡：更平滑的订单可提升上游可预测性，但延迟对需求的响应会增加下游库存成本。我们研究当制造商根据零售商的订单历史预测未来订单时，单一零售商与制造商组成的二级供应链中零售商应如何最优平滑需求。我们提出二项式平滑（Binomial Smoothing）——一类使用二项式权重将每单位需求分散到有限时域内的补货策略，该类策略具有可解释性、易校准且解析可处理。在满足温和正则条件的弱平稳高斯需求假设下，我们证明：对于任意固定平滑时域，在所有具有相同平滑程度的策略中，二项式策略可使制造商的预测误差最小化。该策略保持可逆性，因此制造商可从观测到的订单中恢复需求历史。更一般地，二项式平滑相对于最优策略实现了常数因子近似保证。我们的结果带来更广泛启示：补货策略的设计目标不应仅仅是减少订单方差（如传统牛鞭效应度量），而应减少订单中不可预测的成分。精心设计的平滑可以改善供应链绩效并部分替代信息共享，为无需协作即可实现协调提供了具体机制。