Estimation and Optimization for a Multi-product Inventory System with Stockout-based Substitution

We deal with a retailer's multi-product inventory system where customers randomly seek acceptable substitutes if their initial requests are not satisfied. If the substitute is also unavailable, the sales request is lost. Besides the already complex multi-product inventory management problem involving substitutions, we face further challenges in the absence of knowledge regarding base demand distributions and substitution probabilities. In their stead, all we can rely on are historical sales data that are compromised by censoring and substitution effects. We develop a method that carefully screens the data and subject the useful portion to a Kaplan-Meier type of estimation. Notably, a very effective estimation of the substitution probabilities is also developed. Using large-deviation tools, we establish guaranteed convergence rates of our estimates on top of consistency. The precision in parameter estimates also translates into accuracy in replenishment decisions. For inventory management, we take advantage of a submodularity property to obtain an exact algorithm for the two-product case and a good heuristic for the general multi-product problem. Computational studies based on simulated and actual data confirm the merits of our approach