We study a two-stage newsvendor model where the initially uncertain mean demand is revealed midstream, allowing for a second, costlier order to be placed. The two-stage process is relevant to many retailers who have access to two supply options: a long-lead, low-cost option where orders need to be placed under much demand uncertainty, and a short-lead, high-cost option after a signal is revealed that updates the mean demand. We introduce a forecast evolution model that describes how the initial forecast for mean demand varies in response to the signal, which generalizes the popular additive and multiplicative martingale model of forecast evolution (MMFE). We show that the optimal first-stage order solves a simple ordinary differential equation (ODE), whereas the second-stage order is analytically available. We characterize asymptotics for the first-stage order and expected profit as the forecasted mean demand grows, and propose a simple, asymptotically optimal heuristic. We apply our study to data from a national retail chain, where we calibrate our model by maximum likelihood estimation (MLE). The comparison of several heuristic policies, as well as two benchmarks, shows the efficacy of our method. When the signal is more uncertain (e.g., for product lines with impulse- and trend-driven purchases), our heuristic with a simple adjusted critical fractile outperforms benchmarks; otherwise, the classic newsvendor solution performs well and is asymptotically optimal. We also extend the approach to distribution-free settings and capacitated systems. This paper was accepted by Jeannette Song, operations management. Funding: This work was supported by the Guangdong Key Lab of Mathematical Foundations for Artificial Intelligence Grant 72192805, the National Natural Science Foundation of China Grant 72401245, and Imperial Business School (School Research Fund). Supplemental Material: The online appendix and data files are available at https://doi.org/10.1287/mnsc.2024.04692 .
Federgruen et al. (Thu,) studied this question.