We deal with a retailer’s multi-product inventory system where customers randomly seek acceptable substitutes if their initial requests are not satisfied. Unsuccessful quests for substitutes would result in lost sales. Motivated by a consulting project as well as other real practices, we also choose to deal with further challenges posed by initially unknown base demand distributions and substitution probabilities; moreover, we aim to develop an offline learning method that (i) takes advantage of given data derived from bygone decisions rather than of any learning-while-doing opportunity, (ii) makes inferences about base-demand distributions and pairwise substitution probabilities without knowing whether there have been demand arrivals after inventory depletions, and (iii) is unhindered by the lack of knowledge about the assortments faced by customers and their purchase-or-no-purchase decisions at their individual arrivals. To address these challenges, we propose an innovative approach based on the Kaplan-Meier estimator that circumvents unrealistic data requirements. Our substitution probability estimates employ carefully designed weighting schemes to facilitate rigorous theoretical analysis through tools such as the Cauchy-Schwartz inequality. Both one-time substitution scenarios and more complex Markov-chain substitution patterns would be accommodated. Using large deviation tools, we establish provably optimal convergence rates of our estimates on top of consistency. The precision in parameter estimates would translate 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.
Zheng et al. (Wed,) studied this question.