Learning from Historical Transactions: Robust Supplier Pricing and Stocking with Sparse Data

Abstract

We study a wholesale pricing and capacity problem in which an upstream supplier observes only a small number of historical wholesale prices, the retail prices subsequently chosen by a better-informed retailer, and the corresponding purchase probabilities. These observations are insufficient to recover the full end-market demand curve. We introduce a robust, nonparametric model that combines this limited quantile information with a spectrum of mild shape restrictions spanning regularity and monotone hazard rate. Conventional methods treat each record only as a price--quantile observation and discard how the retail price was generated. We instead view the observed retail price as a downstream decision informed by the retailer's private market knowledge and therefore as an additional source of information about demand. Using inverse optimization, we translate this decision-implied information into exact or error-tolerant restrictions and refine the ambiguity set relative to the price--quantile benchmark. A unified transformation then yields an exact characterization of the jointly feasible least-favorable retailer response and a tractable method for choosing the supplier's wholesale price and capacity. The refined ambiguity set is nested within the benchmark set and therefore delivers a worst-case profit guarantee that is never lower. Numerical experiments across 6 demand environments show material gains with only a few historical transactions; with three observations under regularity, the profit-to-oracle ratio improves by 9.1--11.8 percentage points.