Optimal Inventory Allocation for Indifferent Goods Under Dynamic Substitution

A New Structural Insight into Stockout-Based Substitution Stockout-based substitution creates complex stochastic dynamics in inventory systems even in highly symmetric settings. In this paper, Zhou, Wang, and Zhang examine a canonical setting in which a firm allocates a fixed inventory across multiple perfectly substitutable product types (e.g., colors or designs) and customers purchase uniformly from available options. Despite the model’s symmetry, the induced stochastic dynamics are subtle and resist classical convexity or induction-based arguments. The authors introduce a continuous-time embedding of the discrete sales process, converting inventory depletion times into independent Erlang random variables. This probabilistic reformulation enables an exact characterization of marginal values and yields a clean structural result: the optimal allocation is balanced. Beyond solving the symmetric case, the method extends naturally to finite-horizon MNL models, offering new analytical insight into fluid relaxations and dynamic substitution.