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

Upstream suppliers often set wholesale prices and stocking levels without direct access to detailed end-market demand information, which is instead held by downstream retailers. We study how a supplier can learn from a small number of historical transactions consisting of wholesale prices, retailer-chosen retail prices, and the resulting demand responses. We develop a robust, nonparametric framework that combines inverse optimization with distributionally robust optimization. Historical retailer decisions are used to restrict the set of demand distributions consistent with the observed transactions, after which the supplier optimizes against the least favorable demand response within this ambiguity set. We further accommodate imperfect retailer decisions through two error models (condition error and price error) and derive tractable worst-case response computations based on linear programs and one-dimensional price searches. Numerical experiments across six demand environments show that incorporating historical decision information can generate substantial profit improvements with only a few transactions and that these gains persist under moderate decision errors.