In this paper, the bullwhip effect is analytically measured for a two-stage supply chain, including a manufacturer, and a retailer who serves a market demand. The demand is assumed to be a combination of a first-order autoregressive process, a retail price (set by the retailer), and an asymmetric customer reference price (representing the customer valuation). Furthermore, a series of existence conditions of the bullwhip effect with positively, negatively, and no correlated demands is identified by examining the compound effect of the autocorrelation, retail and reference prices, as well as the demand-price interactions. In addition, it is also found that the bullwhip effect in the case of demand with retail and reference prices is either stronger or weaker than that of demand without these prices in several scenarios. The finding indicates that an under- or over-estimating issue may arise if the retail and reference prices are not properly considered. Moreover, the behaviors of the bullwhip effect with respect to different model parameters are both analytically and numerically explored. The results offer some valuable insights, which lead to useful managerial implications.

在本文中，牛鞭效应是针对一个两阶段的供应链进行分析测量的，包括一个制造商和一个服务于市场需求的零售商。假设需求是一阶自回归过程、零售价格（由零售商设定）和不对称客户参考价格（代表客户估值）的组合。此外，通过考察自相关、零售价和参考价的复合效应，以及需求-价格的相互作用，确定了一系列需求正相关、负相关和不相关的牛鞭效应存在条件。此外，还发现在几种情况下，在有零售和参考价格的需求情况下，牛鞭效应强于或弱于没有这些价格的需求。调查结果表明，如果零售和参考价格没有得到适当考虑，可能会出现低估或高估的问题。此外，对不同模型参数的牛鞭效应行为进行了分析和数值研究。结果提供了一些有价值的见解，从而产生了有用的管理启示。